A Modern Almagest - Richard Fitzpatrick

A Modern Almagest - Richard Fitzpatrick

A Modern Almagest
An Updated Version of Ptolemy’s Model of the Solar System
Richard Fitzpatrick
Professor of Physics
The University of Texas at Austin
Contents
1 Introduction
1.1
Euclid’s Elements and Ptolemy’s Almagest
1.2
Ptolemy’s Model of the Solar System . . .
1.3
Copernicus’s Model of the Solar System . .
1.4
Kepler’s Model of the Solar System . . . .
1.5
Purpose of Treatise . . . . . . . . . . . . .
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2 Spherical Astronomy
2.1
Celestial Sphere . . . . . . . . . . . . . . . .
2.2
Celestial Motions . . . . . . . . . . . . . . .
2.3
Celestial Coordinates . . . . . . . . . . . . .
2.4
Ecliptic Circle . . . . . . . . . . . . . . . . .
2.5
Ecliptic Coordinates . . . . . . . . . . . . . .
2.6
Signs of the Zodiac . . . . . . . . . . . . . .
2.7
Ecliptic Declinations and Right Ascenesions.
2.8
Local Horizon and Meridian . . . . . . . . .
2.9
Horizontal Coordinates . . . . . . . . . . . .
2.10 Meridian Transits . . . . . . . . . . . . . . .
2.11 Principal Terrestrial Latitude Circles . . . . .
2.12 Equinoxes and Solstices . . . . . . . . . . . .
2.13 Terrestrial Climes . . . . . . . . . . . . . . .
2.14 Ecliptic Ascensions . . . . . . . . . . . . . .
2.15 Azimuth of Ecliptic Ascension Point . . . . .
2.16 Ecliptic Altitude and Orientation . . . . . . .
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5
5
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10
11
12
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15
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20
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23
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25
26
27
29
30
3 Dates
63
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2
Determination of Julian Day Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Geometric Planetary Orbit Models
67
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2
MODERN ALMAGEST
4.2
4.3
4.4
4.5
Model of Kepler . . .
Model of Hipparchus
Model of Ptolemy . .
Model of Copernicus
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67
69
71
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5 The Sun
5.1
Determination of Ecliptic Longitude . . . . .
5.2
Example Longitude Calculations . . . . . . .
5.3
Determination of Equinox and Solstice Dates
5.4
Equation of Time . . . . . . . . . . . . . . .
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75
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87
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90
91
6 The Moon
6.1
Determination of Ecliptic Longitude
6.2
Example Longitude Calculations . .
6.3
Determination of Ecliptic Latitude .
6.4
Lunar Parallax . . . . . . . . . . . .
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7 Lunar-Solar Syzygies and Eclipses
7.1
Introduction . . . . . . . . . . . . . . . .
7.2
Determination of Lunar-Solar Elongation
7.3
Example Syzygy Calculations . . . . . . .
7.4
Solar and Lunar Eclipses . . . . . . . . .
7.5
Example Eclipse Calculations . . . . . . .
7.6
Eclipse Statistics . . . . . . . . . . . . . .
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101
101
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107
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8 The Superior Planets
8.1
Determination of Ecliptic Longitude . . . . . . . . . . . . . .
8.2
Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3
Determination of Conjunction, Opposition, and Station Dates
8.4
Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5
Saturn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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115
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119
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125
9 The Inferior Planets
9.1
Determination of Ecliptic Longitude . . . . . . . . . . . . . .
9.2
Venus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3
Determination of Conjunction and Greatest Elongation Dates
9.4
Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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141
141
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145
147
10 Planetary Latitudes
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
10.2 Determination of Ecliptic Latitude of Superior Planet
10.3 Mars . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Jupiter . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Saturn . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Determination of Ecliptic Latitude of Inferior Planet .
10.7 Venus . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8 Mercury . . . . . . . . . . . . . . . . . . . . . . . . .
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157
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159
160
161
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164
166
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CONTENTS
3
11 Glossary
179
12 Index of Symbols
185
13 Bibliography
187
4
MODERN ALMAGEST
Introduction
5
1 Introduction
1.1 Euclid’s Elements and Ptolemy’s Almagest
The modern world inherited two major scientific treaties from the civilization of Ancient Greece.
The first of these, the Elements of Euclid, is a large compendium of mathematical theorems concerning geometry, proportion, and number theory. These theorems were not necessarily discovered
by Euclid himself—being largely the work of earlier mathematicians, such as Eudoxos of Cnidus,
and Theaetetus of Athens—but were arranged by him in a logical manner, so as to demonstrate
that they can all ultimately be derived from five simple axioms. The Elements is rightly regarded
as the first, largely succesful, attempt to construct an axiomatic system in mathematics, and is still
held in high esteem within the scientific community.
The second treatise, the Almagest1 of Claudius Ptolemy, is an attempt to find a simple geometric
explanation for the apparent motions of the sun, the moon, and the five visible planets in the earth’s
sky. On the basis of his own naked-eye observations, combined with those of earlier astronomers
such as Hipparchus of Nicaea, Ptolemy proposed a model of the solar system in which the earth
is stationary. According to this model, the sun moves in a circular orbit, (nearly) centered on the
earth, which maintains a fixed inclination of about 23◦ to the terrestrial equator. Furthermore, the
planets move on the rims of small circles called epicycles, whose centers revolve around the earth
on large eccentric circles called deferents—see Fig. 8.2. The planetary deferents and epicycles also
maintain fixed inclinations,2 which are all fairly close to 23◦ , to the terrestrial equator.
The scientific reputation of the Almagest has not fared as well as that of Euclid’s Elements.
Nowadays, it is a commonly held belief, even amongst scientists, that Ptolemy’s mistaken adherence to the tenets of Aristotelian philosophy—in particular, the immovability of the earth, and the
necessity for heavenly bodies to move uniformly in circles—led him to construct an overcomplicated, unwieldy, and faintly ridiculous model of planetary motion. As is well-known, this model
was superseded in 1543 CE by the heliocentric model of Nicolaus Copernicus, in which the planets
revolve about the sun in circular orbits.3 The Copernican model was, in turn, superseded in the
early 1600’s CE by the, ultimately correct, model of Johannes Kepler, in which the planets revolve
about the sun in eccentric elliptical orbits.
The aim of this treatise is to re-examine the scientific merits of Ptolemy’s Almagest.
1.2 Ptolemy’s Model of the Solar System
Claudius Ptolemy lived and worked in the city of Alexandria, capital of the Roman province of
Egypt, during the reigns of the later Flavian and the Antonine emperors. Ptolemy was heir—via
the writings of Euclid, and later mathematicians such as Apollonius of Perga, and Archimedes of
Syracuse—to the considerable mathematical knowledge of geometry and arithmetic acquired by
1
The true title of this work is Syntaxis Mathematica, which means something like “Mathematical Treatise”. The name
Almagest is probably an Arabic corruption of the work’s later Greek nickname, H Megiste (Syntaxis), meaning “The
Greatest (Treatise)”.
2
Actually, Ptolemy erroneously allowed the inclinations of the deferents and epicycles to vary slightly.
3
In fact, the planets revolve on small circular epicycles, whose centers revolve around the sun on eccentric circular
deferents.
6
MODERN ALMAGEST
the civilization of Ancient Greece. Ptolemy also inherited an extensive Ancient Greek tradition of
observational and theoretical astronomy. The most important astronomer prior to Ptolemy was
undoubtedly Hipparchus of Nicaea (second century BCE), who developed the theory of solar motion used by Ptolemy, discovered the precession of the equinoxes, and collected an extensive set
of astronomical observations—some of which he made himself, and some of which dated back to
Babylonian times—which were available to Ptolemy (probably via the famous Library of Alexandria). Other astronomers who made significant contributions prior to Ptolemy include Meton of
Athens (5th century BCE), Eudoxos of Cnidus (5th/4th century BCE), Callipus of Cyzicus (4th century BCE), Aristarchus of Samos (4th/3rd century BCE), Eratosthenes of Cyrene (3rd/2nd century
BCE), and Menelaus of Alexandria (1st century CE).
Ptolemy’s aim in the Almagest is to construct a kinematic model of the solar system, as seen
from the earth. In other words, the Almagest outlines a relatively simple geometric model which
describes the apparent motions of the sun, moon, and planets, relative to the earth, but does not
attempt to explain why these motions occur (in this respect, the models of Copernicus and Kepler
are similar). As such, the fact that the model described in the Almagest is geocentric in nature is
a non-issue, since the earth is stationary in its own frame of reference. This is not to say that the
heliocentric hypothesis is without advantages. As we shall see, the assumption of heliocentricity
allowed Copernicus to determine, for the first time, the ratios of the mean radii of the various
planets in the solar system.
We now know, from the work of Kepler, that planetary orbits are actually ellipses which are
confocal with the sun. Such orbits possess two main properties. First, they are eccentric: i.e., the
sun is displaced from the geometric center of the orbit. Second, they are elliptical: i.e., the orbit is
elongated along a particular axis. Now, Keplerian orbits are characterized by a quantity, e, called
the eccentricity, which measures their deviation from circularity. It is easily demonstrated that the
eccentricity of a Keplerian orbit scales as e, whereas the corresponding degree of elongation scales
as e2. Since the orbits of the visible planets in the solar system all possess relatively small values
of e (i.e., e ≤ 0.21), it follows that, to an excellent approximation, these orbits can be represented
as eccentric circles: i.e., circles which are not quite concentric with the sun. In other words, we
can neglect the ellipticities of planetary orbits compared to their eccentricities. This is exactly what
Ptolemy does in the Almagest. It follows that Ptolemy’s assumption that heavenly bodies move in
circles is actually one of the main strengths of his model, rather than being the main weakness, as
is commonly supposed.
Kepler’s second law of planetary motion states that the radius vector connecting a planet to the
sun sweeps out equal areas in equal time intervals. In the approximation in which planetary orbits
are represented as eccentric circles, this law implies that a typical planet revolves around the sun at
a non-uniform rate. However, it is easily demonstrated that the non-uniform rotation of the radius
vector connecting the planet to the sun implies a uniform rotation of the radius vector connecting
the planet to the so-called equant: i.e., the point directly opposite the sun relative to the geometric
center of the orbit—see Fig. 1.1. Ptolemy discovered the equant scheme empirically, and used it
to control the non-uniform rotation of the planets in his model. In fact, this discovery is one of
Ptolemy’s main claims to fame.
It follows, from the above discussion, that the geocentric model of Ptolemy is equivalent to a
heliocentric model in which the various planetary orbits are represented as eccentric circles, and in
which the radius vector connecting a given planet to its corresponding equant revolves at a uniform
rate. In fact, Ptolemy’s model of planetary motion can be thought of as a version of Kepler’s
Introduction
7
model which is accurate to first-order in the planetary eccentricities—see Cha. 4. According to the
Ptolemaic scheme, from the point of view of the earth, the orbit of the sun is described by a single
circular motion, whereas that of a planet is described by a combination of two circular motions. In
reality, the single circular motion of the sun represents the (approximately) circular motion of the
earth around the sun, whereas the two circular motions of a typical planet represent a combination
of the planet’s (approximately) circular motion around the sun, and the earth’s motion around the
sun. Incidentally, the popular myth that Ptolemy’s scheme requires an absurdly large number of
circles in order to fit the observational data to any degree of accuracy has no basis in fact. Actually,
Ptolemy’s model of the sun and the planets, which fits the data very well, only contains 12 circles
(i.e., 6 deferents and 6 epicycles).
Ptolemy is often accused of slavish adherence to the tenants of Aristotelian philosophy, to the
overall detriment of his model. However, despite Ptolemy’s conventional geocentrism, his model of
the solar system deviates from orthodox Aristotelism in a number of crucially important respects.
First of all, Aristotle argued—from a purely philosophical standpoint—that heavenly bodies should
move in single uniform circles. However, in the Ptolemaic system, the motion of the planets is
a combination of two circular motions. Moreover, at least one of these motions is non-uniform.
Secondly, Aristotle also argued—again from purely philosophical grounds—that the earth is located
at the exact center of the universe, about which all heavenly bodies orbit in concentric circles.
However, in the Ptolemaic system, the earth is slightly displaced from the center of the universe.
Indeed, there is no unique center of the universe, since the circular orbit of the sun and the circular
planetary deferents all have slightly different geometric centers, none of which coincide with the
earth. As described in the Almagest, the non-orthodox (from the point of view of Aristolelian
philosophy) aspects of Ptolemy’s model were ultimately dictated by observations. This suggests
that, although Ptolemy’s world-view was based on Aristolelian philosophy, he did not hesitate to
deviate from this standpoint when required to by observational data.
From our heliocentric point of view, it is easily appreciated that the epicycles of the superior
planets (i.e., the planets further from the sun than the earth) in Ptolemy’s model actually represent
the earth’s orbit around the sun, whereas the deferents represent the planets’ orbits around the
sun—see Fig. 8.1. It follows that the epicycles of the superior planets should all be the same size
(i.e., the size of the earth’s orbit), and that the radius vectors connecting the centers of the epicycles
to the planets should always all point in the same direction as the vector connecting the earth to the
sun.
We can also appreciate that the deferents of the inferior planets (i.e., the planets closer to the
sun than the earth) in Ptolemy’s model actually represent the earth’s orbit around the sun, whereas
the epicycles represent the planets’ orbits around the sun—see Fig. 9.1. It follows that the deferents
of the inferior planets should all be the same size (i.e., the size of the earth’s orbit), and that the
centers of the epicycles (relative to the earth) should all correspond to the position of the sun
(relative to the earth).
The geocentric model of the solar system outlined above represents a perfected version of
Ptolemy’s model, constructed with a knowledge of the true motions of the planets around the
sun. Not surprisingly, the model actually described in the Almagest deviates somewhat from this
ideal form. In the following, we shall refer to these deviations as “errors”, but this should not be
understood in a perjorative sense.
Ptolemy’s first error lies in his model of the sun’s apparent motion around the earth, which
he inherited from Hipparchus. Figure 1.1 compares what Ptolemy actually did, in this respect,
8
MODERN ALMAGEST
Π
Π
S
S
G
G
C
Q
e
e
A
2e
C
A
Figure 1.1: Hipparchus’ (and Ptolemy’s) model of the sun’s apparent orbit about the earth (right)
compared to the optimal model (left). The radius vectors in both models rotate uniformly. Here, S
is the sun, G the earth, C the geometric center of the orbit, Q the equant, Π the perigee, and A the
apogee. The radius of the orbit is normalized to unity.
compared to what he should have done in order to be completely consistent with the rest of his
model. Let us normalize the mean radius of the sun’s apparent orbit to unity, for the sake of
clarity. Ptolemy should have adopted the model shown on the left in Fig. 1.1, in which the earth
is displaced from the center of the sun’s orbit a distance e = 0.0167 (the eccentricity of the earth’s
orbit around the sun) towards the perigee (the point of the sun’s closest approach to the earth),
and the equant is displaced the same distance in the opposite direction. The instantaneous angular
position of the sun is then obtained by allowing the radius vector connecting the equant to the
sun to rotate uniformly at the sun’s mean orbital angular velocity. Of course, this implies that
the sun rotates non-uniformly about the earth. Ptolemy actually adopted the Hipparchian model
shown on the right in Fig. 1.1. In this model, the earth is displaced a distance 2e from the center
of the sun’s orbit in the direction of the perigee, and the sun rotates at a uniform rate (i.e., the
radius vector CS rotates uniformly). It turns out that, to first-order in e, these two models are
equivalent in terms of their ability to predict the angular position of the sun relative to the earth—
see Cha. 4. Nevertheless, the Hippachian model is incorrect, since it predicts too large (by a factor
of 2) a variation in the radial distance of the sun from the earth (and, hence, the angular size of the
sun) during the course of a year (see Cha. 4). Ptolemy probabaly adopted the Hipparchian model
because his Aristotelian leanings prejudiced him in favor of uniform circular motion whenever this
was consistent with observations. (It should be noted that Ptolemy was not interested in explaining
the relatively small variations in the angular size of the sun during the year—presumably, because
this effect was difficult for him to accurately measure.)
Ptolemy’s next error was to neglect the non-uniform rotation of the superior planets on their
epicycles. This is equivalent to neglecting the orbital eccentricity of the earth (recall that the
epicycles of the superior planets actually represent the earth’s orbit) compared to those of the
Introduction
9
superior planets. It turns out that this is a fairly good approximation, since the superior planets
all have significantly greater orbital eccentricities than the earth. Nevertheless, neglecting the nonuniform rotation of the superior planets on their epicycles has the unfortunate effect of obscuring
the tight coupling between the apparent motions of these planets, and that of the sun. The radius
vectors connecting the epicycle centers of the superior planets to the planets themselves should
always all point exactly in the same direction as that of the sun relative to the earth. When the
aforementioned non-uniform rotation is neglected, the radius vectors instead point in the direction
of the mean sun relative to the earth. The mean sun is a fictitious body which has the same apparent
orbit around the earth as the real sun, but which circles the earth at a uniform rate. The mean sun
only coincides with the real sun twice a year.
Ptolemy’s third error is associated with his treatment of the inferior planets. As we have seen,
in going from the superior to the inferior planets, deferents and epicycles effectively swap roles.
For instance, it is the deferents of the inferior planets, rather than the epicycles, which represent the
earth’s orbit. Hence, for the sake of consistency with his treatment of the superior planets, Ptolemy
should have neglected the non-uniform rotation of the epicycle centers around the deferents of
the inferior planets, and retained the non-uniform rotation of the planets themselves around the
epicycle centers. Instead, he did exactly the opposite. This is equivalent to neglecting the inferior
planets’ orbital eccentricities relative to that of the earth. It follows that this approximation only
works when an inferior planet has a significantly smaller orbital eccentricity than that of the earth.
It turns out that this is indeed the case for Venus, which has the smallest eccentricity of any planet
in the solar system. Thus, Ptolemy was able to successfully account for the apparent motion of
Venus. Mercury, on the other hand, has a much larger orbital eccentricity than the earth. Moreover,
it is particularly difficult to obtain good naked-eye positional data for Mercury, since this planet
always appears very close to the sun in the sky. Consequently, Ptolemy’s Mercury data was highly
inaccurate. Not surprisingly, then, Ptolemy was not able to account for the apparent motion of
Mercury using his standard deferent-epicycle approach. Instead, in order to fit the data, he was
forced to introduce an additional, and quite spurious, epicycle into his model of Mercury’s orbit.
Ptolemy’s fourth, and possibly largest, error is associated with his treatment of the moon. It
should be noted that the moon’s motion around the earth is extremely complicated in nature,
and was not fully understood until the early 20th century CE. Ptolemy constructed an ingenious
geometric model of the moon’s orbit which was capable of predicting the lunar ecliptic longitude
to reasonable accuracy. Unfortunately, this model necessitates a monthly variation in the earthmoon distance by a factor of about two, which implies a similarly large variation in the moon’s
angular diameter. However, the observed variation in the moon’s diameter is much smaller than
this. Hence, Ptolemy’s model is not even approximately correct.
Ptolemy’s fifth error is associated with his treatment of planetary ecliptic latitudes. Given that
the deferents and epicycles of the superior planets represent the orbits of the planets themselves
around the sun, and the sun’s apparent orbit around the earth, respectively, it follows that one
should take the slight inclination of planetary orbits to the ecliptic plane (i.e., the plane of the
sun’s apparent orbit) into account by tilting the deferents of superior planets, whilst keeping their
epicycles parallel to the ecliptic. Similarly, given that the epicycles and deferents of inferior planets
represent the orbits of the planets themselves around the sun, and the sun’s apparent orbit around
the earth, respectively, one should tilt the epicycles of inferior planets, whilst keeping their deferents
parallel to the ecliptic. Finally, since the inclination of planetary orbits are all essentially constant in
time, the inclinations of the epicycles and deferents should also be constant. Unfortunately, when
10
MODERN ALMAGEST
Ptolemy constructed his theory of planetary latitudes he tilted the both deferents and epicyles of all
the planets. Even worse, he allowed the inclinations of the epicycles to the ecliptic plane to vary in
time. The net result is a theory which is far more complicated than is necessary.
The final failing in Ptolemy’s model of the solar system lies in its scale invariance. Using angular
position data alone, Ptolemy was able to determine the ratio of the epicycle radius to that of the
deferent for each planet, but was not able to determine the relative sizes of the deferents of different
planets. In order to break this scale invariance it is necessary to make an additional assumption—
i.e., that the earth orbits the sun. This brings us to Copernicus.
1.3 Copernicus’s Model of the Solar System
The Polish astronomer Nicolaus Copernicus (1473–1543 CE) studied the Almagest assiduously, but
eventually became dissatisfied with Ptolomy’s approach. The main reason for this dissatisfaction
was not the geocentric nature of Ptolomy’s model, but rather the fact that it mandates that heavenly
bodies execute non-uniform circular motion. Copernicus, like Aristotle, was convinced that the
supposed perfection of the heavens requires such bodies to execute uniform circular motion only.
Copernicus was thus spurred to construct his own model of the solar system, which was described
in the book De Revolutionibus Orbium Coelestium (On the Revolutions of the Heavenly Spheres),
published in the year of his death.
The most well-known aspect of Copernicus’s model is the fact that it is heliocentric. As has
already been mentioned, when describing the motion of the sun, moon, and planets relative to the
earth, it makes little practical difference whether one adopts a geocentric or a heliocentric model
of the solar system. Having said this, the heliocentric approach does have one large advantage.
If we accept that the sun, and not the earth, is stationary, then it immediately follows that the
epicycles of the superior planets, and the deferents of the inferior planets, represent the earth’s
orbit around the sun. Hence, all of these circles must be the same size. This realization allows
us to break the scale invariance which is one of the main failings of Ptolemy’s model. Thus, the
ratio of the deferent radius to that of the epicycle for a superior planet, which is easily inferred
from observations, actually corresponds to the ratio of planet’s orbital radius to that of the earth.
Likewise, the ratio of the epicycle radius to that of the deferent for an inferior planet, which is again
easily determined observationally, also corresponds to the ratio of the planet’s orbital radius to that
of the earth. Using this type of reasoning, Copernicus was able to construct the first accurate scale
model of the solar system, and to firmly establish the order in which the planets orbit the sun. In
some sense, this was his main achievement.
Copernicus’s insistence that heavenly bodies should only move in uniform circles lead him to
reject Ptolemy’s equant scheme, and to replace it with the scheme illustrated in Fig. 1.2. According
to Copernicus, a heliocentric planetary orbit is a combination of two circular motions. The first is
motion of the planet around a small circular epicycle, and the second is the motion of the center
of the epicycle around the sun on a circular deferent. Both motions are uniform, and in the same
direction. However, the former motion is twice as fast as the latter. In addition, the sun is displaced
from the center of the deferent in the direction of the perihelion, the displacement being proportional to the orbital eccentricity. Furthermore, the sun’s displacement is three times greater than
the radius of the epicycle. Finally, the radius of the deferent is equal to the major radius of the
planetary orbit. It turns out that Copernicus’ scheme is a marginally less accurate approximation
than Ptolemy’s to a low eccentricity Keplerian orbit (see Cha. 4).
Introduction
11
Π
P
X
(1/2)e
S
(3/2)e
C
A
Figure 1.2: Copernicus’ model of a heliocentric planetary orbit. Here, S is the sun, P the planet, C the
geometric center of the deferent, X the center of the epicycle, Π the perihelion, and A the aphelion. The
radius vectors CX and XP both rotate uniformly in the same direction, but XP rotates twice as fast as
CX. The major radius of the orbit is normalized to unity.
Copernicus modeled the orbit of the earth around the sun using an Hippachian scheme (see
Fig. 1.1) in which the earth moves uniformly around an eccentric circle. Unfortunately, such a
scheme exaggerates the variation in the radial distance between the earth and the sun during the
course of a year by a factor of 2, and so introduces significant errors into the calculation of the
parallax of the planets due to the motion of the earth. On the other hand, Copernicus’ model of the
moon’s orbit around the earth is a considerable improvement on Ptolemy’s, since it does not grossly
exaggerate the monthly variation in the earth-moon distance. Like Ptolemy, Copernicus introduced
an additional spurious epicycle into his model of Mercury’s orbit, and erroneously allowed the
inclination of his planetary orbits to vary slightly in time.
In summary, Copernicus’s model of the solar system contains approximately the same number of
epicycles as Ptolemy’s, the only difference being that Copernicus’ epicycles are much smaller than
Ptolemy’s. Indeed, the model of Copernicus is about as complicated, and not appreciably more
accurate, than that described in the Almagest. In this respect, Copernicus cannot be said to have
demonstrated the correctness of his heliocentric approach on the basis of observational data.
1.4 Kepler’s Model of the Solar System
Johannes Kepler (1571–1630 CE) was fortunate enough to inherit an extensive set of naked-eye
solar, lunar, and planetary angular position data from the Danish astronomer Tycho Brahe (1546–
1601 CE). This data extended over many decades, and was of unprecedented accuracy.
Although Kepler adopted the heliocentric approach of Copernicus, what he effectively first did
was to perfect Ptolemy’s model of the solar system (or, rather, its heliocentric equivalent). Thus,
Kepler replaced Ptolemy’s erroneous equantless model of the sun’s apparent orbit around the earth
12
MODERN ALMAGEST
with a corrected version containing an equant—in the process, halving the eccentricity of the orbit
(see Fig. 1.1). Kepler also introduced equants into the epicycles of the superior and inferior planets. Once he had perfected Ptolemy’s model, the heliocentric nature of the solar system became
manifestly apparent to Kepler. For instance, he found that the epicycles of the superior planets,
the sun’s apparent orbit around the earth, and the deferents of the inferior planets all have exactly
the same eccentricity. The obvious implication is that these circles all correspond to some common
motion within the solar system—in fact, the motion of the earth around the sun.
Once Kepler had corrected the Almagest model, he compared its predictions with his observational data. In particular, Kepler investigated the apparent motion of Mars in the night sky. Kepler
found that his model performed extremely well, but that there remained small differences between
its predictions and the observational data. The maximum discrepancy was about 8 ′ : i.e., about
1/4 the apparent size of the sun. By the standards of naked-eye astronomy, this was a very small
discrepancy indeed. Nevertheless, given the incredible accuracy of Tycho Brahe’s observations, it
was still significant. Thus, Kepler embarked on an epic new series of calculations which eventually
lead him to the conclusion that the planetary orbits are actually eccentric ellipses, rather than eccentric circles. Kepler published the results of his research in Astronomia Nova (New Astronomy)
in 1609 CE. It is interesting to note that had Tycho’s data been a little less accurate, or had the
orbit of Mars been a little less eccentric, Kepler might well have settled for a model which was
kinematically equivalent to a perfected version of the model described in the Almagest. We can
also appreciate that, given the far less accurate observational data available to Ptolemy, there was
no way in which he could have discerned the very small difference between elliptical planetary
orbits and the eccentric circular orbits employed in the Almagest.
1.5 Purpose of Treatise
As we have seen, misconceptions abound regarding the details of Ptolemy’s model of the solar
system, as well as its scientific merit. Part of the reason for this is that the Almagest is an extremely
difficult book for a modern reader to comprehend. For instance, virtually all of its theoretical
results are justified via lengthy and opaque geometric proofs. Moreover, the plane and spherical
trigonometry employed by Ptolemy is of a rather primitive nature, and, consequently, somewhat
unwieldy. Dates are also a major stumbling block, since three different systems are used in the
Almagest, all of which are archaic, and essentially meaningless to the modern reader. Another
difficulty is the unfamiliar, and far from optimal, Ancient Greek method of representing numbers
and fractions. Finally, the terminology employed in the Almagest is, in many instances, significantly
different to that used in modern astronomy textbooks.
The aim of this treatise is to reconstruct Ptolemy’s model of the solar system employing modern
mathematical methods, standard dates, and conventional astronomical terminology. It is hoped
that the resulting model will enable the reader to comprehend the full extent of Ptolemy’s scientific achievement. In fact, the model described in this work is a somewhat improved version of
Ptolemy’s, in that all of the previously mentioned deficiencies have been corrected. Furthermore,
Ptolemy’s equant scheme has been replaced by a Keplerian scheme, expanded to second-order in
the planetary eccentricities. It should be noted, however, that these two schemes are essentially
indistinguishable for small eccentricity orbits. Certain aspects of the Almagest have not been reproduced. For instance, it was not thought necessary to instruct the reader on how to construct
trigonometric tables, or primitive astronomical instruments. Furthermore, no attempted has been
Introduction
13
made to derive any of the model parameters directly from observational data, since the orbital
elements and physical properties of the sun, moon, and planets are, by now, extremely well established. Any detailed discussion of the fixed stars has also been omitted, because stellar positions
are also very well established, and the apparent motion of the stars in the sky is comparatively
straightforward compared to those of solar system objects. What remains is a mathematical model
of the solar system which is surprisingly accurate (the maximum errors in the ecliptic longitudes
of the sun, moon, Mercury, Venus, Mars, Jupiter, and Saturn during the years 1995–2006 CE are
0.7 ′ , 14 ′ , 28 ′ , 10 ′ , 14 ′ , 4 ′ , and 1 ′ , respectively), yet sufficiently simple that all of the necessary
calculations can be performed by hand, with the aid of tables. The form of the calculations, as well
as the layout of the tables, is, for the most part, fairly similar to those found in the Almagest. Many
examples of the use of the tables are provided.
14
MODERN ALMAGEST
Spherical Astronomy
15
2 Spherical Astronomy
2.1 Celestial Sphere
It is often helpful to imagine that celestial objects are attached to a vast sphere centered on the
earth. This fictitious construction is known as the celestial sphere. The earth’s dimensions are
assumed to be infinitesimally small compared to those of the sphere (since the distance of a typical
celestial object from the earth is very much larger than the earth’s radius). It follows that only half
of the sphere is visible from any particular observation site on the earth’s surface. Furthermore, the
angular position of a given celestial object (relative to some fixed celestial reference) is the same at
all such sites. In other words, there is negligible parallax associated with viewing the same celestial
object from different observation sites on the surface of the earth.1
2.2 Celestial Motions
Celestial objects exhibit two different types of motion. The first motion is such that the whole
celestial sphere, and all of the celestial objects attached to it, rotates uniformly from east to west
once every 24 (sidereal) hours, about a fixed axis passing through the earth’s north and south
poles. This type of motion is called diurnal motion, and is a consequence of the earth’s daily
rotation. Diurnal motion preserves the relative angular positions of all celestial objects. However,
certain celestial objects, such as the sun, the moon, and the planets, possess a second motion,
superimposed on the first, which causes their angular positions to slowly change relative to one
another, and to the fixed stars. This intrinsic motion of objects in the solar system is due to a
combination of the earth’s orbital motion about the sun, and the orbital motions of the moon and
the planets about the earth and the sun, respectively.
2.3 Celestial Coordinates
Consider Fig. 2.1. The celestial sphere rotates about the celestial axis, PP ′ , which is the imagined
extension of the earth’s axis of rotation. This axis intersects the celestial sphere at the north celestial
pole, P, and the south celestial pole, P ′ . It follows that the two celestial poles are unaffected by
diurnal motion, and remain fixed in the sky.
The celestial equator, VUV ′ U ′ , is the intersection of the earth’s equatorial plane with the celestial sphere, and is therefore perpendicular to the celestial axis. The so-called vernal equinox,
V, is a particular point on the celestial equator that is used as the origin of celestial longitude.
Furthermore, the autumnal equinox, V ′ , is a point which lies directly opposite the vernal equinox
on the celestial equator. Let the line UU ′ lie in the plane of the celestial equator such that it is
perpendicular to VV ′ , as shown in the figure.
It is helpful to define three, right-handed, mutually perpendicular, unit vectors: v, u, and p.
Here, v is directed from the earth to the vernal equinox, u from the earth to point U, and p from
the earth to the north celestial pole—see Fig. 2.1.
1
The one exception to this rule is the moon, which is sufficiently close to the earth that its parallax is significant—see
Sect. 6.4.
16
MODERN ALMAGEST
P
V
p
′
U
u
G
v
U′
V
P′
Figure 2.1: The celestial sphere. G, P, P ′ , V, and V ′ represent the earth, north celestial pole, south
celestial pole, vernal equinox, and autumnal equinox, respectively. VUV ′ U ′ is the celestial equator,
and PP ′ the celestial axis.
P
R
r
V
′
G
δ
α
R′
V
P′
Figure 2.2: Celestial coordinates. R is a celestial object, and R ′ its projection onto the plane of the
celestial equator, VR ′ V ′ .
Spherical Astronomy
17
Consider a general celestial object, R—see Fig. 2.2. The location of R on the celestial sphere
is conveniently specified by two angular coordinates, δ and α. Let GR ′ be the projection of GR
onto the equatorial plane. The coordinate δ, which is known as declination, is the angle subtended
between GR ′ and GR. Objects north of the celestial equator have positive declinations, and vice
versa. It follows that objects on the celestial equator have declinations of 0◦ , whereas the north and
south celestial poles have declinations of +90◦ and −90◦ , respectively. The coordinate α, which is
known as right ascension, is the angle subtended between GV and GR ′ . Right ascension increases
from west to east (i.e., in the opposite direction to the celestial sphere’s diurnal rotation). Thus,
the vernal and autumnal equinoxes have right ascensions of 0◦ and 180◦ , respectively. Note that α
lies in the range 0◦ to 360◦ . Right ascension is sometimes measured in hours, instead of degrees,
with one hour corresponding to 15◦ (since it takes 24 hours for the celestial sphere to complete one
diurnal rotation). In this scheme, the vernal and autumnal equinoxes have right ascensions of 0
hrs. and 12 hrs., respectively. Moreover, α lies in the range 0 to 24 hrs. (Incidentally, in this treatise,
α is measured relative to the mean equinox at date, unless otherwise specified.) Finally, let r be a
unit vector which is directed from the earth to R—see Fig. 2.2. It is easily demonstrated that
r = cos δ cos α v + cos δ sin α u + sin δ p,
(2.1)
sin δ = r · p,
r·u
tan α =
.
r·v
(2.2)
and
(2.3)
2.4 Ecliptic Circle
During the course of a year, the sun’s intrinsic motion causes it to trace out a fixed circle which
bisects the celestial sphere. This circle is known as the ecliptic. The sun travels around the ecliptic
from west to east (i.e., in the opposite direction to the celestial sphere’s diurnal rotation). Moreover,
the ecliptic circle is inclined at a fixed angle of ǫ = 23◦ 26 ′ to the celestial equator. This angle
actually represents the fixed inclination of the earth’s axis of rotation to the normal to its orbital
plane.2
The vernal equinox, V, is defined as the point at which the ecliptic crosses the celestial equator
from south to north (in the direction of the sun’s ecliptic motion)—see Fig. 2.3. Likewise, the
autumnal equinox, V ′ , is the point at which the ecliptic crosses the celestial equator from north to
south. In addition, the summer solstice, S, is the point on the ecliptic which is furthest north of the
celestial equator, whereas the winter solstice, S ′ , is the point which is furthest south. It follows that
the lines VV ′ and SS ′ are perpendicular. Let QQ ′ be the normal to the plane of the ecliptic which
passes through the earth, as shown in Fig. 2.3. Here, Q is termed the northern ecliptic pole, and Q ′
the southern ecliptic pole. It is easily demonstrated that
2
s = cos ǫ u + sin ǫ p,
(2.4)
q = − sin ǫ u + cos ǫ p,
(2.5)
In fact, ǫ is very slowly decreasing in time. The value of ǫ used in the Almagest is 23◦ 51 ′ . However, the true value
of ǫ in Ptolemy’s day was 23◦ 41 ′ .
18
MODERN ALMAGEST
Q
P
S
ǫ
′
V q
s
U′
U
v
V
S′
P
′
Q′
Figure 2.3: The ecliptic circle. P, P ′ , Q, Q ′ , V, V ′ , S, and S ′ denote the north celestial pole, south
celestial pole, north ecliptic pole, south ecliptic pole, vernal equinox, autumnal equinox, summer solstice, and winter solstice, respectively. VUV ′ U ′ is the celestial equator, VSV ′ S ′ the ecliptic, and PP ′ the
celestial axis.
where s is a unit vector which is directed from the earth to the summer solstice, and q a unit vector
which is directed from the earth to the north ecliptic pole—see Fig. 2.3. We can also write
u = cos ǫ s − sin ǫ q,
(2.6)
p = sin ǫ s + cos ǫ q.
(2.7)
Thus, v, s, and q constitute another right-handed, mutually perpendicular, set of unit vectors.
2.5 Ecliptic Coordinates
It is convenient to specify the positions of the sun, moon, and planets in the sky using a pair of
angular coordinates, β and λ, which are measured with respect to the ecliptic, rather than the
celestial equator. Let R denote a celestial object, and GR ′ the projection of the line GR onto the
plane of the ecliptic, VR ′ V ′ —see Fig. 2.4. The coordinate β, which is known as ecliptic latitude,
is the angle subtended between GR ′ and GR. Objects north of the ecliptic plane have positive
ecliptic latitudes, and vice versa. The coordinate λ, which is known as ecliptic longitude, is the
angle subtended between GV and GR ′ . Ecliptic longitude increases from west to east (i.e., in the
same direction that the sun travels around the ecliptic). (Again, in this treatise, λ is measured
relative to the mean equinox at date, unless specified otherwise.) Note that the basis vectors in the
ecliptic coordinate system are v, s, and q, whereas the corresponding basis vectors in the celestial
coordinate system are v, u, and p—see Figs. 2.1 and 2.3. By analogy with Eqs. (2.1)–(2.3), we can
write
r = cos β cos λ v + cos β sin λ s + sin β q,
(2.8)
Spherical Astronomy
19
Q
R
r
V
′
G
β
λ
R′
V
Q′
Figure 2.4: Ecliptic coordinates. G is the earth, R a celestial object, and R ′ its projection onto the
ecliptic plane, VR ′ V ′ .
sin β = r · q,
r·s
,
tan λ =
r·v
(2.9)
(2.10)
where r is a unit vector which is directed from G to R. Hence, it follows from Eqs. (2.1), (2.4), and
(2.5) that
sin β = cos ǫ sin δ − sin ǫ cos δ sin α,
(2.11)
cos ǫ cos δ sin α + sin ǫ sin δ
.
(2.12)
cos δ cos α
These expressions specify the transformation from celestial to ecliptic coordinates. The inverse
transformation follows from Eqs. (2.2), (2.3), and (2.6)–(2.8):
tan λ =
sin δ = cos ǫ sin β + sin ǫ cos β sin λ,
(2.13)
cos ǫ cos β sin λ − sin ǫ sin β
.
cos β cos λ
(2.14)
tan α =
Figures 2.13 and 2.14 show all stars of visible magnitude less than +6 lying within 15◦ of
the ecliptic. Table 2.1 gives the ecliptic longitudes, ecliptic latitudes, and visible magnitudes of a
selection of these stars which lie within 10◦ of the ecliptic. The figures and table can be used to
convert ecliptic longitude and latitude into approximate position in the sky against the backdrop of
the fixed stars.
2.6 Signs of the Zodiac
The signs of the zodiac are a well-known set of names given to 30◦ long segments of the ecliptic
circle. Thus, the sign of Aries extends over the range of ecliptic longitudes 0◦ –30◦ , the sign of
20
MODERN ALMAGEST
Taurus over the range 30◦ –60◦ , and so on. Note that, as a consequence of the precession of the
equinoxes, the signs of the zodiac no longer coincide with the constellations of the same name (see
Figs. 2.13 and 2.14). The 12 zodiacal signs are listed in the table below. It can be seen from the
table that ecliptic longitude 72◦ corresponds to the twelfth degree of Gemini, and ecliptic longitude
242◦ to the second degree of Sagittarius, etc.
Sign
Aries
Taurus
Gemini
Cancer
Abbr.
Longitude
Sign
AR
TA
GE
CN
0◦ –30◦
30◦ –60◦
60◦ –90◦
90◦ –120◦
Leo
Virgo
Libra
Scorpio
Abbr.
Longitude
Sign
LE
VI
LI
SC
120◦ –150◦
150◦ –180◦
180◦ –210◦
210◦ –240◦
Sagittarius
Capricorn
Aquarius
Pisces
Abbr.
Longitude
SG
CP
AQ
PI
240◦ –270◦
270◦ –300◦
300◦ –330◦
330◦ –360◦
2.7 Ecliptic Declinations and Right Ascenesions.
According to Eqs. (2.13) and (2.14), the celestial coordinates of a point on the ecliptic circle (i.e.,
β = 0) which has ecliptic longitude λ are specified by
sin δ = sin ǫ sin λ,
(2.15)
tan α = cos ǫ tan λ.
(2.16)
The above formulae have been used to construct Tables 2.2 and 2.3, which list the declinations and
right ascensions of a set of equally spaced points on the ecliptic circle.
2.8 Local Horizon and Meridian
Consider a general observation site X on the surface of the earth. (Note that, in the following, it
is tacitly assumed that the site lies the earth’s northern hemisphere. However, the analysis also
applies to sites situated in the the southern hemisphere.) The local zenith Z is the point on the
celestial sphere which is directly overhead at X, whereas the nadir Z ′ is the point which is directly
underfoot—see Fig. 2.5. The horizon is the tangent plane to the earth at X, and divides the celestial
sphere into two halves. The upper half, containing the zenith, is visible from site X, whereas the
lower half is invisible.
Figure 2.6 shows the visible half of the celestial sphere at observation site X. Here, NESW is the
local horizon, and N, E, S, and W are the north, east, south, and west compass points, respectively.
The plane NPZS, which passes through the north and south compass points, as well as the zenith,
is known as the local meridian. The meridian is perpendicular to the horizon. The north celestial
pole lies in the meridian plane, and is elevated an angular distance L above the north compass
point—see Figs. 2.5 and 2.6. Here, L is the terrestrial latitude of observation site X. It is helpful
to define three, right-handed, mutually perpendicular, local unit vectors: e, n, and z. Here, e
is directed toward the east compass point, n toward the north compass point, and z toward the
zenith—see Fig. 2.6.
Figure 2.7 shows the meridian plane at X. Let the line MM ′ lie in this plane such that it is
perpendicular to the celestial axis, PP ′ . Moreover, let M lie in the visible hemisphere. It is helpful
Spherical Astronomy
21
P
N
Z
L
X
axis
S
L
equator
Z′
P′
Figure 2.5: A general observation site X, of latitude L, on the surface of the earth. P, P ′ , Z, and Z ′
denote the directions to the north celestial pole, south celestial pole, zenith, and nadir, respectively. The
line NS represents the local horizon.
Z
meridian
P
E
z
N
L
n
e
S
horizon
W
Figure 2.6: The local horizon and meridian. N, S, E, W denote the north. south, east, and west
compass points, Z the zenith, and P the north celestial pole. NESW is the horizon, and NPZS the
meridian.
22
MODERN ALMAGEST
Z
M
P
p
L
n
N
z
m
S
P′
M′
Z′
Figure 2.7: The local meridian.
to define the unit vector m which is directed toward M, as shown in the diagram. It is easily seen
that
n = cos L p − sin L m,
(2.17)
z = sin L p + cos L m.
(2.18)
Figure 2.8 shows the celestial equator viewed from observation site X. Here, α0 is the right
ascension of the celestial objects culminating (i.e., reaching their highest altitude in the sky) on
the meridian at the time of observation. Incidentally, it is easily demonstrated that all objects
culminating on the meridian at any instant in time have the same right ascension. Note that the
angle α0 increases uniformly in time, at the rate of 15◦ a (sidereal) hour, due to the diurnal motion
of the celestial sphere. It can be seen from the diagram that
m = sin α0 u + cos α0 v,
(2.19)
e = cos α0 u − sin α0 v.
(2.20)
Thus, from Eqs. (2.17) and (2.18),
e = − sin α0 v + cos α0 u,
(2.21)
n = − sin L cos α0 v − sin L sin α0 u + cos L p,
(2.22)
z = cos L cos α0 v + cos L sin α0 u + sin L p.
(2.23)
Similarly, from Eqs. (2.6) and (2.7),
e = − sin α0 v + cos ǫ cos α0 s − sin ǫ cos α0 q,
n = − sin L cos α0 v + (cos L sin ǫ − sin L cos ǫ sin α0) s + (cos L cos ǫ
(2.24)
Spherical Astronomy
23
M
V
U
m
u
E
v
α0
e
W
U′
V′
M′
Figure 2.8: The local celestial equator.
+ sin L sin ǫ sin α0) q,
(2.25)
z = cos L cos α0 v + (sin L sin ǫ + cos L cos ǫ sin α0) s + (sin L cos ǫ
− cos L sin ǫ sin α0) q.
(2.26)
2.9 Horizontal Coordinates
It is convenient to specify the positions of celestial objects in the sky, when viewed from a particular
observation site, X, on the earth’s surface, using a pair of angular coordinates, a and A, which are
measured with respect to the local horizon. Let R denote a celestial object, and XR ′ the projection
of the line XR onto the horizontal plane, NESW—see Fig. 2.9. The coordinate a, which is known
as altitude, is the angle subtended between XR ′ and XR. Objects above the horizon have positive
altitudes, whereas objects below the horizon have negative altitudes. The zenith has altitude 90◦ ,
and the horizon altitude 0◦ . The coordinate A, which is known as azimuth, is the angle subtended
between XN and XR ′ . Azimuth increases from the north towards the east. Thus, the north, east,
south, and west compass points have azimuths of 0◦ , 90◦ , 180◦ , and 270◦ , respectively. Note that
the basis vectors in the horizontal coordinate system are e, n, and z, whereas the corresponding
basis vectors in the celestial coordinate system are v, u, and p—see Figs. 2.1 and 2.6. By analogy
with Eqs. (2.1)—(2.3), we can write
r = cos a sin A e + cos a cos A n + sin a z,
sin a = r · z,
r·e
tan A =
,
r·n
(2.27)
(2.28)
(2.29)
24
MODERN ALMAGEST
Z
R
r
E
R′
a
A
N
X
S
W
Figure 2.9: Horizontal coordinates. R is a celestial object, and R ′ its projection onto the horizontal
plane, NESW.
where r is a unit vector directed from X to R. Hence, it follows from Eqs. (2.1), and (2.22)–(2.23),
that
sin a = sin L sin δ + cos L cos δ cos(α − α0),
(2.30)
cos δ sin(α − α0)
.
cos L sin δ − sin L cos δ cos(α − α0)
(2.31)
tan A =
These expressions allow us to calculate the altitude and azimuth of a celestial object of declination δ
and right ascension α which is viewed from an observation site on the earth’s surface of terrestrial
latitude L at an instant in time when celestial objects of right ascension α0 are culminating at
the meridian. According to Eqs. (2.8), and (2.25)–(2.26), the altitude and azimuth of a similarly
viewed point on the ecliptic (i.e., β = 0) of ecliptic longitude λ are given by
sin a = cos L cos λ cos α0 + sin L sin ǫ sin λ + cos L cos ǫ sin λ sin α0,
cos ǫ sin λ cos α0 − cos λ sin α0
.
tan A =
cos L sin ǫ sin λ − sin L cos λ cos α0 − sin L cos ǫ sin λ sin α0
(2.32)
(2.33)
2.10 Meridian Transits
Consider a celestial object, of declination δ and right ascension α, which is viewed from an observation site on the earth’s surface of terrestrial latitude L. According to Eq. (2.30), the object
culminates, or attains its highest altitude in the sky, when α0 = α. This event is known as an upper
transit. Furthermore, the object attains its lowest altitude in the sky when α0 = 180◦ + α. This
event is known as a lower transit. Both upper and lower transits take place as the object in question
passes through the meridian plane.
Spherical Astronomy
25
According to Eq. (2.30), the altitude of a celestial object at its upper transit satisfies sin a+ =
cos(L − δ), implying that
a+ = 90◦ − |L − δ|.
(2.34)
Likewise, the altitude at its lower transit satisfies sin a− = − cos(L + δ), giving
a− = |L + δ| − 90◦ .
(2.35)
The previous two expressions allow us to group celestial objects into three classes. Objects with
declinations satisfying |L + δ| > 90◦ never set: i.e., their lower transits lie above the horizon. Objects
with declinations satisfying |L − δ| > 90◦ never rise: i.e., their upper transits lie below the horizon.
Finally, objects with declinations which satisfy neither of the two previous inequalities both rise and
set during the course of a day. It follows that all celestial objects appear to rise and set when viewed
from an observation site on the terrestrial equator (i.e., L = 0◦ ) . On the other hand, when viewed
from an observation site at the north pole (i.e., L = 90◦ ), objects north of the celestial equator
never set, whilst objects south of the celestial equator never rise, and vice versa for objects viewed
from the south pole. All three classes of celestial object are present when the sky is viewed from an
observation site on the earth’s surface of intermediate latitude.
2.11 Principal Terrestrial Latitude Circles
According to Eq. (2.15), the sun’s declination varies between −ǫ and +ǫ during the course of a
year. It follows from Eq. (2.34) that it is only possible for the sun to have an upper transit at
the zenith in a region of the earth whose latitude lies between −ǫ and ǫ. The circles of latitude
bounding this region are known as the tropics. Thus, the tropic of Capricorn—so-called because the
sun is at the winter solstice, and, therefore, at the first point of Capricorn (i.e., the zeroth degree
of Capricorn), when it culminates at the zenith at this latitude—lies at L = −23◦ 26 ′ . Moreover,
the tropic of Cancer—so-called because the sun is at the summer solstice, and, therefore, at the first
point of Cancer, when it culminates at the zenith at this latitude—lies at L = +23◦ 26 ′ .
Equations (2.34) and (2.35) imply that the sun does not rise for part of the year, and does not
set for part of the year, in two regions of the earth whose terrestrial latitudes satisfy |L| > 90◦ − ǫ.
These two regions are bounded by the poles and two circles of latitude known as the arctic circles.
The south arctic circle lies at L = −66◦ 34 ′ . Likewise, the north arctic circle lies at L = +66◦ 34 ′ .
The equator, the two tropics, and the two arctic circles constitute the five principal latitude
circles of the earth, and are shown in Fig. 2.10.
2.12 Equinoxes and Solstices
The ecliptic longitude of the sun when it reaches the vernal equinox is λ = 0◦ . It follows, from
Eq. (2.32), that the altitude of the sun on the day of the equinox is given by sin a = cos L cos α0.
Thus, the sun rises when α0 = −90◦ , culminates at an altitude of 90◦ − |L| when α0 = 0◦ , and sets
when α0 = 90◦ . We conclude that the length of the equinoctial day is 180 time-degrees, which is
equivalent to 12 hours (since 15◦ of right ascension cross the meridian in one hour). Thus, day and
night are equally long on the day of the vernal equinox. It is easily demonstrated that the same is
true on the day of the autumnal equinox.
The ecliptic longitude of the sun when it reaches the summer solstice is λ = 90◦ . It follows that
the altitude of the sun on the day of the solstice is given by sin a = sin L sin ǫ + cos L cos ǫ sin α0.
26
MODERN ALMAGEST
N. Arctic Circle
Tropic of Cancer
Equator
Tropic of Capricorn
S. Arctic Circle
Figure 2.10: The principal latitude circles of the earth.
Thus, the sun rises when α0 = − sin−1(tan L tan ǫ), culminates at an altitude of 90◦ − |L − ǫ| when
α0 = 90◦ , and sets when α0 = 180◦ + sin−1(tan L tan ǫ). We conclude that the length of the longest
day of the year in the earth’s northern hemisphere (which, of course, occurs when the sun reaches
the summer solstice) is 180 + 2 sin−1(tan L tan ǫ) time-degrees. Likewise, the length of the shortest
night (which also occurs at the summer solstice) is 180 − 2 sin−1(tan L tan ǫ) time-degrees. These
formulae are only valid for northern latitudes below the arctic circle. At higher latitudes, the sun
never sets for part of the year, and the longest “day” is consequently longer than 24 hours. It
is easily demonstrated that the shortest day in the earth’s northern hemisphere, which takes place
when the sun reaches the winter solstice, is equal to the shortest night, and the longest night (which
also occurs at the winter solstice) to the longest day. Moreover, the sun culminates at an altitude of
90◦ − |L + ǫ| on day of the winter solstice. Again, at latitudes above the arctic circle, the sun never
rises for part of the year, and the longest “night” is consequently longer than 24 hours.
Consider an observation site on the earth’s surface of latitude L which lies above the northern
arctic circle. The declination of the sun on the first day after the spring equinox on which it fails to
set is δ = 90◦ − L. According to Eq. (2.15), its ecliptic longitude on this day is sin−1(cos L/ sin ǫ).
Likewise, the declination of the sun on the day when it starts to set again is δ = 90◦ − L, and
its ecliptic longitude is 180◦ − sin−1(cos L/ sin ǫ). Assuming that the sun travels around the ecliptic
circle at a uniform rate (which is approximately true), the fraction of a year that the sun stays above
the horizon in summer is 0.5 − sin−1(cos L/ sin ǫ)/180◦ . It is easily demonstrated that the fraction
of a year that the sun stays below the horizon in winter is also 0.5 − sin−1(cos L/ sin ǫ)/180◦ .
2.13 Terrestrial Climes
Table 2.4 specifies the length of the longest day, as well as the altitude of the sun when it culminates
at the meridian on the days of the equinoxes and solstices, calculated for a set of observation sites
in the northern hemisphere with equally spaced terrestrial latitudes. This table was constructed
Spherical Astronomy
27
using the formulae in the previous section. The table can be adapted to observation sites in the
earth’s southern hemisphere via the following simple transformation: L → −L, Summer ↔ Winter,
N ↔ S. For instance, at a latitude of −10◦ , the longest day, which corresponds to the winter solstice,
is of length 12h35m. Moreover, on this day, the sun’s upper transit is south of the zenith, at an
altitude of +76◦ 34 ′ .
2.14 Ecliptic Ascensions
Consider the rising, or ascension, of celestial objects at the eastern horizon, as viewed from a
particular observation site on the earth’s surface. If the observation site lies on the terrestrial
equator then all celestial objects appear to ascend at right angles to the horizon. This process is
known as right ascension. On the other hand, if the observation site does not lie at the equator
then celestial objects appear to ascend at an oblique angle to the horizon. This process is known as
oblique ascension. For the case of right ascension, it is easily demonstrated that all celestial objects
with the same celestial longitude ascend simultaneously. Indeed, celestial longitude is generally
known as “right ascension” because, in the case of right ascension, the celestial longitude of an
object (in hours) is simply the time elapsed between the ascension of the vernal equinox, and the
ascension of the object in question.
Let us now consider the ascension of points on the ecliptic. Applying Eq. (2.30) to a point on
the celestial equator (i.e., δ = 0) of right ascension α, we obtain
sin a = cos L cos(α − α0) = cos L sin(α0 − α + 90◦ ).
(2.36)
It follows that we can write
α0 = α − 90◦ ,
(2.37)
where α is the right ascension of the point on the celestial equator which ascends at the eastern
horizon (i.e., a = 0 and da/dt > 0) at the same time that celestial objects of right ascension α0 are
culminating at the meridian. Substituting this result into Eq. (2.32), we get
sin a = cos L cos λ sin α + sin L sin ǫ sin λ − cos L cos ǫ sin λ cos α,
(2.38)
which implies that if a = 0 and da/dt > 0 then
tan λ =
cos L sin α
.
cos L cos ǫ cos α − sin ǫ sin L
(2.39)
This expression specifies the ecliptic longitude, λ, of the point on the ecliptic circle which ascends
simultaneously with a point on the celestial equator of right ascension α. Note, incidentally, that
points on the celestial equator ascend at a uniform rate of 15◦ an hour at all viewing sites on the
earth’s surface (except the poles, where the celestial equator does not ascend at all). The same is
not true of points on the ecliptic. Expression (2.39) can be inverted to give
−1
α = tan
(tan λ cos ǫ) − sin
−1
"
#
sin λ sin ǫ tan L
.
(1 − sin2 λ sin2 ǫ)1/2
(2.40)
The solution of Eq. (2.40) for observation sites lying above the arctic circle is complicated by the
fact that, at such sites, a section of the ecliptic never sets, or descends, and a section never ascends.
28
MODERN ALMAGEST
It is easily demonstrated that the section which never descends lies between ecliptic longitudes λc
and 180◦ − λc, whereas the section which never ascends lies between longitudes 180◦ + λc and
360◦ − λc. Here, λc = sin−1(cos L/ sin ǫ). Points on the ecliptic of longitude λc, 180◦ − λc, 180◦ + λc,
and 360◦ −λc ascend simultaneously with points on the celestial equator of right ascension 360◦ −αc,
αc, 360◦ − αc, and αc, respectively. Here, αc = cos−1(1/ tan L tan ǫ).
Tables 2.5–2.17 list the ascensions of a series of equally spaced points on the ecliptic circle, as
viewed from a set of observation sites in the earth’s northern hemisphere with different terrestrial
latitudes. The tables were calculated with the aid of formula (2.40). Let us now illustrate the use
of these tables.
Consider a day on which the sun is at ecliptic longitude 14LE00 (i.e., 14◦ 00 ′ into the sign of
Leo). What is the length of the day (i.e., the period between sunrise and sunset) at an observation
site on the earth’s surface of latitude +30◦ ? Consulting Table 2.8, we find that the sun ascends
simultaneously with a point on the celestial equator of right ascension 126◦ 32 ′ . Now, the ecliptic is a
great circle on the celestial sphere. Hence, exactly half of the ecliptic is visible from any observation
site on the earth’s surface. This implies that when a given point on the ecliptic circle is ascending,
the point directly opposite it on the circle is descending, and vice versa. Let us term the directly
opposite point the complimentary point. By definition, the difference in ecliptic longitude between
a given point on the ecliptic circle and its complementary point is 180◦ . Thus, the complimentary
point to 14LE00 is 14AQ00. It follows that 14AQ00 ascends at the same time that 14LE00 descends.
In other words, the sun sets when 14AQ00 ascends. Consulting Table 2.8, we find that the sun sets
at the same time that a point on the celestial equator of right ascension 326◦ 23 ′ rises. Thus, in the
time interval between the rising and setting of the sun a 326◦ 23 ′ − 126◦ 32 ′ = 199◦ 51 ′ section of the
celestial equator ascends at the eastern horizon. However, points on the celestial equator ascend at
the uniform rate of 15◦ an hour. Thus, the length, in hours, of the period between the rising and
setting of the sun is 199◦ 51 ′ /15◦ = 13h14m. In other words, the length of the day in question is
13h14m.
The above calculation is slightly inaccurate for a number of reasons. Firstly, it neglects the fact
that the sun is continuously moving on the ecliptic circle at the rate of about 1◦ a day. Secondly,
it neglects the fact that the celestial equator ascends at the rate of 15◦ per sidereal, rather than
solar, hour. A sidereal hour is 1/24th of a sidereal day, which is the time between successive upper
transits of a fixed celestial object, such as a star. On the other hand, a solar hour is 1/24th of a solar
day, which is the mean time between successive upper transits of the sun. A sidereal day is shorter
than a solar day by 4 minutes. Fortunately, it turns out that these first two inaccuracies largely
cancel one another out. Another source of inaccuracy is the fact that, due to refraction of light by
the atmosphere, the sun is actually 1◦ below the horizon when it appears to rise or set. The final
source of inaccuracy is the fact that the sun has a finite angular extent (of about half a degree),
and that, strictly speaking, dawn and dusk commence when the sun’s upper limb rises and sets,
respectively. Of course, our calculation only deals with the rising and setting of the center of the
sun. All in all, the above mentioned inaccuracies can make the true length of a day differ from that
calculated from the ascension tables by up to 15 minutes.
Tables 2.5–2.17 also effectively list the descents of a series of equally spaced points on the ecliptic
circle, as viewed from a set of observation sites in the earth’s southern hemisphere with different
terrestrial latitudes (which are minus those specified in the various tables). For instance, Table 2.6
gives the right ascensions of points on the celestial equator which set simultaneously with points
on the ecliptic, as seen from an observation site at latitude −10◦ .
Spherical Astronomy
29
Consider a day on which the sun is at ecliptic longitude 08SC00. Let us calculate the length of
the day at an observation site on the earth’s surface of latitude −50◦ ? Consulting Table 2.10, we
find that the sun sets simultaneously with a point on the celestial equator of right ascension 233◦ 09 ′ .
Now, the complementary point on the ecliptic to 08SC00 is 08TA00. Consulting Table 2.10 again,
we find that this point sets simultaneously with a point on the celestial equator of right ascension
18◦ 07 ′ . It follows that the sun rises simultaneously with the latter point. Thus, the time interval
between the rising and setting of the sun is 233◦ 09 ′ − 18◦ 07 ′ = 215◦ 02 ′ time-degrees, or 14h20m.
The ascendent, or horoscope, is defined as the point on the ecliptic which is ascending at the
eastern horizon. Suppose that we wish to find the ascendent 2.6 hours after sunrise, as seen from
an observation site of latitude +55◦ , on a day on which the sun has ecliptic longitude 16SC00.
Of course, knowledge of the ascendent at the time of birth is key to drawing up a natal chart in
astrology. Hence, this type of calculation was of great importance to the ancients. Consulting
Table 2.11, we find that, on the day in question, the sun rises simultaneously with a point on
the celestial equator of right ascension 248◦ 46 ′ . Now, 2.6 hours corresponds to 39◦ 00 ′ . Thus, the
ascendent rises simultaneously with a point on the celestial equator of right ascension 248◦ 46 ′ +
39◦ 00 ′ = 287◦ 46 ′ . Consulting Table 2.11 again, we find that, to the nearest degree, the ascendent
at the time in question has ecliptic longitude 13SG00.
Suppose, next, that we wish to find the right ascension, α, of the point on the celestial equator
which culminates simultaneously with a given point on the ecliptic of ecliptic longitude λ. From
Eq. (2.33), we can see that if A = 180◦ then tan A = 0, and tan λ cos ǫ = sin α, or
α = sin−1(tan λ cos ǫ).
(2.41)
However, this expression is identical to expression (2.40), when the latter is evaluated for the
special case L = 0◦ . It follows that our problem can be solved by consulting Table 2.5, which is
the ascension table for the case of right ascension. For instance, on a day on which the ecliptic
longitude of the sun is 08TA00, we find from Table 2.5 that the right ascension of the point on
the celestial equator which culminates simultaneously with the sun (i.e., which culminates at local
noon) is 35◦ 38 ′ . Moreover, this is the case for observation sites at all terrestrial latitudes. Note that
we have effectively calculated the right ascension of the sun on the day in question.
Suppose, finally, that we wish to find the point on the ecliptic which culminates 7 hours after
local noon on the aforementioned day. Since 7 hours corresponds to 105◦ , the right ascension of
the point on the celestial equator which culminates simultaneously with the point is question is
35◦ 38 ′ + 105◦ 00 ′ = 143◦ 38 ′ . Consulting Table 2.5 again, we find that, to the nearest degree, the
ecliptic longitude of the point in question is 21LE00.
2.15 Azimuth of Ecliptic Ascension Point
Consider the azimuth of the point on the ecliptic circle which is ascending at the eastern horizon.
According to Eq. (2.27), the azimuth of any point on the horizon (i.e., a = 0◦ ) satisfies cos A = r · n.
It follows from Eqs. (2.8) and (2.25) that
cos A = − cos λ sin L sin α + sin λ cos L sin ǫ + sin λ sin L cos ǫ cos α.
(2.42)
Here, we have made use of the fact that the point in question also lies on the ecliptic (i.e., β = 0),
as well as the fact that α0 = α − 90◦ , where α is the right ascension of the simultaneously rising
point on the celestial equator. Here, λ is the ecliptic longitude of the point in question, and L the
30
MODERN ALMAGEST
Z
a
D
β
µ
Y
S
λ
C
B
E
Figure 2.11: Parallactic angle in the case where increasing altitude corresponds to increasing ecliptic
latitude. SCBE is the southern horizon, with S and E the south and east compass points, respectively.
DYB is the ecliptic. ZDS the meridian, and Z the zenith. ZYC is an altitude circle.
terrestrial latitude of the observation site. Now, λ and α satisfy Eq. (2.39), as well as the above
equation. Thus, eliminating α between these two equations, we obtain
sin λ sin ǫ
.
(2.43)
cos L
This expression gives the azimuth, A, of the ascending point of the ecliptic as a function of its
ecliptic longitude, λ, and the latitude, L, of the observation site.
For instance, suppose that we wish to find the azimuth of the point at which the sun rises on the
eastern horizon at an observation site of terrestrial latitude +60◦ , on a day on which the sun’s ecliptic longitude is 08PI00. It follows from Eq. (2.43) that A = cos−1[sin(338◦ ) sin(23◦ 26 ′ )/ cos(60◦ )] =
107◦ 20 ′ . We conclude that the sun rises 17◦ 20 ′ to the south of the east compass point on the day in
question. It is easily demonstrated that the sun sets 17◦ 20 ′ south of the west compass point on the
same day (neglecting the slight change in the sun’s ecliptic latitude during the course of the day.)
Likewise, it can easily be shown that, at an observation site of terrestrial latitude −60◦ , the sun also
rises 17◦ 20 ′ to the south of the east compass point on the day in question, and sets 17◦ 20 ′ to the
south of the west compass point.
cos A =
2.16 Ecliptic Altitude and Orientation
Consider a point on the ecliptic circle of ecliptic longitude λ. We wish to determine the altitude of
this point, as well as the angle subtended there between the ecliptic and the vertical, t hours before
or after it culminates at the meridian, as seen from an observation site on the earth’s surface of
latitude L.
The situation is as shown in Fig. 2.11. Here, Y is the point in question, and ZYC an altitude
circle (i.e., a great circle passing through the zenith) drawn through it. We wish to determine the
Spherical Astronomy
31
Z
a
D
Y
β
µ
S
λ
C
B
E
Figure 2.12: Parallactic angle in the case where increasing altitude corresponds to decreasing ecliptic
latitude. SCBE is the southern horizon, with S and E the south and east compass points, respectively.
DYB is the ecliptic. ZDS the meridian, and Z the zenith. ZYC is an altitude circle.
altitude a ≡ CY of point Y, as well as the angle µ ≡ ZYB. Note that µ is defined such that it
lies between the ecliptic in the direction of increasing ecliptic longitude and the altitude circle in
the direction of increasing altitude. Moreover, µ is acute when increasing altitude, a, corresponds
to increasing ecliptic latitude, β, and obtuse when increasing a corresponds to decreasing β. See
Figs. 2.11 and 2.12. Incidentally, this definition is adopted in order to simplify the calculation of
lunar parallax—see Sect. 6.4. In the following, we shall refer to µ as the parallactic angle. However,
it should be noted that, according to the modern definition, the parallactic angle is 90◦ − µ.
From Eqs. (2.15) and (2.16), the declination and right ascension of point Y are given by
sin δ = sin ǫ sin λ,
(2.44)
tan α = cos ǫ tan λ,
(2.45)
respectively. We can also write α0 = α − t, where α0 is the right ascension of the point on the
ecliptic which is culminating (i.e., point D in the diagram), and t is measured in time-degrees.
Note that if t is positive then it measures time before culmination, whereas if it is negative then
its magnitude measures time after culmination. It follows from Eqs. (2.30) and (2.31) that the
altitude and azimuth of point Y satisfy
sin a = sin L sin δ + cos L cos δ cos t,
cos δ sin t
tan A =
,
cos L sin δ − sin L cos δ cos t
(2.46)
(2.47)
respectively.
From Eq. (2.8), the unit vector
r = cos λ v + sin λ s
(2.48)
32
MODERN ALMAGEST
is directed from the observation site to point Y. Furthermore, the unit vector
∂r
= − sin λ v + cos λ s
∂λ
(2.49)
is tangent to the ecliptic circle, at point Y, in the direction of increasing ecliptic longitude. From
Eq. (2.27), the unit vector
r = cos a sin A e + cos a cos A n + sin a z
(2.50)
is directed from the observation site to point Y. Here, a and A are the altitude and azimuth,
respectively, of this point in the sky. Moreover, the unit vector
∂r
∂a
= − sin a sin A e − sin a cos A n − cos a z
≡ (cos A e − sin A n) × r
(2.51)
is a tangent to the altitude circle passing through point Y in the direction of increasing altitude. It
follows from the definition of parallactic angle, and elementary vector algebra, that
cos µ =
∂r ∂r
·
∂λ ∂a
= − sin λ cos A v × e · r + sin λ sin A v × n · r
+ cos λ cos A s × e · r − cos λ sin A s × n · r.
(2.52)
However, according to Eqs. (2.24), (2.25), and (2.48),
v × n · r = − sin L (cos L cos ǫ + sin L sin ǫ sin α0),
(2.53)
v × e · r = sin λ sin ǫ cos α0,
(2.54)
s × e · r = − cos λ sin ǫ cos α0,
(2.55)
s × n · r = cos λ (cos L cos ǫ + sin L sin ǫ sin α0).
(2.56)
The previous five equations can be combined to give
cos µ = − cos A sin ǫ cos(α − t)
− sin A [cos L cos ǫ + sin L sin ǫ sin(α − t)] .
(2.57)
Now, it follows from Eq. (2.26) that
z · q = sin L cos ǫ − cos L sin ǫ sin(α − t).
(2.58)
This quantity is significant because if z · q > 0 then increasing altitude corresponds to increasing
ecliptic latitude, whereas if z · q < 0 then increasing altitude corresponds to decreasing ecliptic
latitude. Thus, in the former case, µ is the solution of (2.57) which lies in the range 0◦ ≤ µ ≤ 180◦ ,
whereas in the latter case it is the solution which lies in the range 180◦ ≤ µ ≤ 360◦ .
According to Eq. (2.46), the critical value of t at which point Y reaches the horizon is given by
cos th = − tan L tan δ.
(2.59)
Spherical Astronomy
33
Of course, the above equation is only soluble if | tan L tan δ| < 1. However, it is easily demonstrated
that if tan L tan δ < −1 then point Y never sets, whereas if tan L tan δ > 1 then point Y never rises.
Note that the value of µ at t = 0 represents the inclination of the ecliptic to the vertical as point
Y culminates. Furthermore, the values of µ at t = th (corresponding to a = 0◦ ) represent the
inclination of the ecliptic to the vertical as point Y rises and sets.
Tables 2.18–2.26 show the altitudes of twelve equally spaced points on the ecliptic, as well as
the parallactic angle at these points, as functions of time, calculated for a series of observation
sites in the earth’s northern hemisphere with equally spaced terrestrial latitudes. The twelve points
correspond to the start of the twelve zodiacal signs, and are named accordingly. Thus, “Aries”
corresponds to ecliptic longitude 0◦ , “Taurus” to ecliptic longitude 30◦ , etc. For each point, four
columns of data are provided. The first column corresponds to the time (in hours and minutes)
either before or after the culmination of the point, the second column gives the altitude of the point
(which is the same in both cases), the third column gives the parallactic angle, µ, for the case in
which the first column indicates time prior to the culmination of the point, and the fourth column
gives the parallactic angle for the opposite case. Data is only provided for cases in which the various
points on the ecliptic lie on or above the horizon.
Now, it can be seen, from the above analysis, that if L → −L, t → t, λ → λ + 180◦ then δ → −δ,
α → α + 180◦ , A → 180◦ − A, cos µ → cos µ, z · q → −z · q, and so a → a, µ → 360◦ − µ. It
follows that Tables 2.18–2.26 can also be used to calculate altitudes and parallactic angles of points
on the ecliptic, as functions of time, for observation sites in the earth’s southern hemisphere. For
example, suppose that we wish to determine the altitude and parallactic angle of the first point of
Gemini (i.e., λ = 60◦ ), as seen from an observation site of terrestrial latitude −10◦ , 3 hours before
and after it culminates at the meridian. In order to do this, we must examine the Sagittarius (i.e.,
λ = 240◦ ) entry in the L = +10◦ ecliptic altitude table: i.e., Table 2.19 (since λ → λ + 180◦ when
L → −L). The fourth row of this entry tells us that t = 03:00 hrs. before culmination the altitude
and parallactic angle of the first point of Gemini are a = 36◦ 26 ′ and µ = 360◦ − 162◦ 11 ′ = 197◦ 49 ′ ,
respectively (since a → a and µ → 360◦ − µ as L → −L). This row also tells us that t = 03:00 hrs.
after culmination the altitude and parallactic angle of the first point of Gemini are a = 36◦ 26 ′ and
µ = 360◦ − 042◦ 17 ′ = 317◦ 43 ′ , respectively
34
MODERN ALMAGEST
Virgo
Leo
Cancer
10
0
-10
180
170
160
150
140
Geminii
130
120
110
Taurus
100
90
10
0
Aries
10
0
-10
90
80
70
60
50
40
30
20
Figure 2.13: Map showing all stars of visual magnitude less than +6 lying within 15◦ of the ecliptic
plane. (a)
Spherical Astronomy
35
Pisces
Aquarius
Capricorn
10
0
-10
360
350
340
330
320
Saggitarius
310
300
290
Scorpio
280
270
190
180
Libra
10
0
-10
270
260
250
240
230
220
210
200
Figure 2.14: Map showing all stars of visual magnitude less than +6 lying within 15◦ of the ecliptic
plane. (b)
36
MODERN ALMAGEST
Aries
λ
◦
Mag.
Name
λ
+2.8
+3.6
γ PEG
η PSC
◦
04 50
10◦ 08 ′
11◦ 28 ′
22◦ 08 ′
23◦ 51 ′
Taurus
Mag.
Name
+2.6
+2.0
β ARI
α ARI
15 5
19◦ 22 ′
Gemini
β
Mag.
Name
λ
+2.9
+0.9
+1.7
η TAU
α TAU
β TAU
◦
02 34
02◦ 56 ′
03◦ 11 ′
07◦ 48 ′
09◦ 46 ′
11◦ 27 ′
17◦ 58 ′
Cancer
β
Mag.
Name
λ
β
′
◦
09 09
26◦ 49 ′
+12 36
+5◦ 23 ′
λ
β
◦
′
03 58
07◦ 39 ′
λ
◦
′
00 00
09◦ 47 ′
22◦ 34 ′
λ
◦
′
◦
′
′
+8 29
+9◦ 58 ′
◦
′
+4 03
−5◦ 28 ′
+5◦ 23 ′
◦
′
05 18
09◦ 06 ′
09◦ 56 ′
20◦ 14 ′
23◦ 13 ′
− 0 49
− 6◦ 44 ′
+ 2◦ 04 ′
+10◦ 06 ′
+ 6◦ 41 ′
λ
β
β
′
′
◦
′
20 47
29◦ 37 ′
29◦ 50 ′
+9 43
+8◦ 49 ′
+0◦ 28 ′
λ
β
′
06 23
13◦ 25 ′
17◦ 34 ′
27◦ 10 ′
◦
′
+0 09
+9◦ 40 ′
+6◦ 06 ′
+0◦ 42 ′
λ
Scorpio
β
Mag.
Name
◦ ′
◦
◦
′
′
′
+2.8
+2.6
α LIB
β LIB
Sagittarius
β
Mag.
Name
+0 20
+8◦ 30 ′
◦
′
+2.3
+2.9
+2.6
+2.9
+1.0
+2.8
+2.4
δ SCO
π SCO
β SCO
σ SCO
α SCO
τ SCO
η OPH
Capricorn
β
Mag.
Name
−1 59
−5◦ 29 ′
+1◦ 00 ′
−4◦ 02 ′
−4◦ 34 ′
−6◦ 08 ′
+7◦ 12 ′
◦
′
+2.9
+1.9
+3.0
+2.0
+1.1
µ GEM
γ GEM
ǫ GEM
α GEM
β GEM
01 16
04◦ 34 ′
06◦ 19 ′
12◦ 23 ′
13◦ 38 ′
16◦ 15 ′
Mag.
Name
λ
+3.0
+2.6
+1.4
ǫ LEO
γ LEO
α LEO
◦
23 24
23◦ 33 ′
+8 37
−2◦ 36 ′
Mag.
Name
λ
β
ρ LEO
θ LEO
ι LEO
β VIR
◦
+3.9
+3.3
+3.9
+3.6
Name
η VIR
γ VIR
δ VIR
ζ VIR
α VIR
+1 22
+2◦ 46 ′
+8◦ 37 ′
+8◦ 38 ′
−2◦ 03 ′
′
+3.0
+2.7
+2.8
+2.0
+2.6
+2.9
γ SGR
δ SGR
λ SGR
σ SGR
ζ SGR
π SGR
Aquarius
β
Mag.
Name
−7 00
−6◦ 28 ′
−2◦ 08 ′
−3◦ 27 ′
−7◦ 11 ′
+1◦ 26 ′
◦
Virgo
◦
′
+3.9
+3.5
+3.4
+3.4
+1.0
Leo
◦
◦
Libra
Mag.
′
06 43
08◦ 52 ′
11◦ 34 ′
21◦ 27 ′
◦
′
+2.9
+2.9
β AQR
δ CAP
Pisces
Mag.
Name
′
+8 14
−8◦ 12 ′
−0◦ 23 ′
+7◦ 15 ′
+3.8
+3.3
+3.7
+3.7
γ AQR
δ AQR
λ AQR
γ PSC
Table 2.1: Ecliptic longitudes (relative to the mean equinox at the J2000 epoch), ecliptic latitudes, and
visual magnitudes of selected bright stars lying within 10◦ of the ecliptic plane.
Spherical Astronomy
λ
Aries
δ
37
Taurus
δ
Gemini
δ
α
λ
α
λ
00◦
+00◦ 00 ′
000◦ 00 ′
00◦
+11◦ 28 ′
027◦ 55 ′
00◦
+20◦ 09 ′
02◦
+00◦ 48 ′
001◦ 50 ′
02◦
+12◦ 10 ′
029◦ 50 ′
02◦
+20◦ 33 ′
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
+01◦ 35 ′
+02◦ 23 ′
+03◦ 10 ′
+03◦ 58 ′
+04◦ 45 ′
+05◦ 31 ′
+06◦ 18 ′
+07◦ 04 ′
+07◦ 49 ′
+08◦ 34 ′
+09◦ 19 ′
+10◦ 02 ′
+10◦ 46 ′
+11◦ 28 ′
003◦ 40 ′
005◦ 30 ′
007◦ 21 ′
009◦ 11 ′
011◦ 02 ′
012◦ 53 ′
014◦ 44 ′
016◦ 36 ′
018◦ 28 ′
020◦ 20 ′
022◦ 13 ′
024◦ 07 ′
026◦ 00 ′
027◦ 55 ′
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
+12◦ 51 ′
+13◦ 31 ′
+14◦ 10 ′
+14◦ 49 ′
+15◦ 26 ′
+16◦ 02 ′
+16◦ 37 ′
+17◦ 11 ′
+17◦ 44 ′
+18◦ 16 ′
+18◦ 46 ′
+19◦ 15 ′
+19◦ 43 ′
+20◦ 09 ′
031◦ 45 ′
033◦ 41 ′
035◦ 38 ′
037◦ 36 ′
039◦ 34 ′
041◦ 33 ′
043◦ 32 ′
045◦ 32 ′
047◦ 33 ′
049◦ 35 ′
051◦ 38 ′
053◦ 41 ′
055◦ 45 ′
057◦ 49 ′
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
+20◦ 57 ′
+21◦ 18 ′
+21◦ 38 ′
+21◦ 57 ′
+22◦ 13 ′
+22◦ 28 ′
+22◦ 42 ′
+22◦ 54 ′
+23◦ 03 ′
+23◦ 12 ′
+23◦ 18 ′
+23◦ 22 ′
+23◦ 25 ′
+23◦ 26 ′
α
λ
α
λ
λ
Cancer
δ
Leo
δ
Virgo
δ
00◦
+23◦ 26 ′
090◦ 00 ′
00◦
+20◦ 09 ′
122◦ 11 ′
00◦
+11◦ 28 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
+23◦ 25 ′
+23◦ 22 ′
+23◦ 18 ′
+23◦ 12 ′
+23◦ 03 ′
+22◦ 54 ′
+22◦ 42 ′
+22◦ 28 ′
+22◦ 13 ′
+21◦ 57 ′
+21◦ 38 ′
+21◦ 18 ′
+20◦ 57 ′
+20◦ 33 ′
+20◦ 09 ′
092◦ 11 ′
094◦ 21 ′
096◦ 32 ′
098◦ 43 ′
100◦ 53 ′
103◦ 03 ′
105◦ 12 ′
107◦ 21 ′
109◦ 30 ′
111◦ 38 ′
113◦ 46 ′
115◦ 53 ′
117◦ 60 ′
120◦ 06 ′
122◦ 11 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
+19◦ 43 ′
+19◦ 15 ′
+18◦ 46 ′
+18◦ 16 ′
+17◦ 44 ′
+17◦ 11 ′
+16◦ 37 ′
+16◦ 02 ′
+15◦ 26 ′
+14◦ 49 ′
+14◦ 10 ′
+13◦ 31 ′
+12◦ 51 ′
+12◦ 10 ′
+11◦ 28 ′
124◦ 15 ′
126◦ 19 ′
128◦ 22 ′
130◦ 25 ′
132◦ 27 ′
134◦ 28 ′
136◦ 28 ′
138◦ 27 ′
140◦ 26 ′
142◦ 24 ′
144◦ 22 ′
146◦ 19 ′
148◦ 15 ′
150◦ 10 ′
152◦ 05 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
+10◦ 46 ′
+10◦ 02 ′
+09◦ 19 ′
+08◦ 34 ′
+07◦ 49 ′
+07◦ 04 ′
+06◦ 18 ′
+05◦ 31 ′
+04◦ 45 ′
+03◦ 58 ′
+03◦ 10 ′
+02◦ 23 ′
+01◦ 35 ′
+00◦ 48 ′
+00◦ 00 ′
α
057◦ 49 ′
059◦ 54 ′
062◦ 00 ′
064◦ 07 ′
066◦ 14 ′
068◦ 22 ′
070◦ 30 ′
072◦ 39 ′
074◦ 48 ′
076◦ 57 ′
079◦ 07 ′
081◦ 17 ′
083◦ 28 ′
085◦ 39 ′
087◦ 49 ′
090◦ 00 ′
α
152◦ 05 ′
153◦ 60 ′
155◦ 53 ′
157◦ 47 ′
159◦ 40 ′
161◦ 32 ′
163◦ 24 ′
165◦ 16 ′
167◦ 07 ′
168◦ 58 ′
170◦ 49 ′
172◦ 39 ′
174◦ 30 ′
176◦ 20 ′
178◦ 10 ′
180◦ 00 ′
Table 2.2: Declinations and right ascensions of points on the ecliptic circle (a).
38
MODERN ALMAGEST
λ
Libra
δ
Scorpio
δ
α
λ
α
λ
00◦
-00◦ 00 ′
180◦ 00 ′
00◦
-11◦ 28 ′
207◦ 55 ′
02◦
-00◦ 48 ′
181◦ 50 ′
02◦
-12◦ 10 ′
209◦ 50 ′
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
-01◦ 35 ′
-02◦ 23 ′
-03◦ 10 ′
-03◦ 58 ′
-04◦ 45 ′
-05◦ 31 ′
-06◦ 18 ′
-07◦ 04 ′
-07◦ 49 ′
-08◦ 34 ′
-09◦ 19 ′
-10◦ 02 ′
-10◦ 46 ′
-11◦ 28 ′
183◦ 40 ′
185◦ 30 ′
187◦ 21 ′
189◦ 11 ′
191◦ 02 ′
192◦ 53 ′
194◦ 44 ′
196◦ 36 ′
198◦ 28 ′
200◦ 20 ′
202◦ 13 ′
204◦ 07 ′
206◦ 00 ′
207◦ 55 ′
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
-12◦ 51 ′
-13◦ 31 ′
-14◦ 10 ′
-14◦ 49 ′
-15◦ 26 ′
-16◦ 02 ′
-16◦ 37 ′
-17◦ 11 ′
-17◦ 44 ′
-18◦ 16 ′
-18◦ 46 ′
-19◦ 15 ′
-19◦ 43 ′
-20◦ 09 ′
211◦ 45 ′
213◦ 41 ′
215◦ 38 ′
217◦ 36 ′
219◦ 34 ′
221◦ 33 ′
223◦ 32 ′
225◦ 32 ′
227◦ 33 ′
229◦ 35 ′
231◦ 38 ′
233◦ 41 ′
235◦ 45 ′
237◦ 49 ′
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
λ
Capricorn
δ
α
λ
Aquarius
δ
α
λ
Sagittarius
δ
α
-20◦ 09 ′
-20◦ 33 ′
-20◦ 57 ′
-21◦ 18 ′
-21◦ 38 ′
-21◦ 57 ′
-22◦ 13 ′
-22◦ 28 ′
-22◦ 42 ′
-22◦ 54 ′
-23◦ 03 ′
-23◦ 12 ′
-23◦ 18 ′
-23◦ 22 ′
-23◦ 25 ′
-23◦ 26 ′
Pisces
δ
00◦
-23◦ 26 ′
270◦ 00 ′
00◦
-20◦ 09 ′
302◦ 11 ′
00◦
-11◦ 28 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
-23◦ 25 ′
-23◦ 22 ′
-23◦ 18 ′
-23◦ 12 ′
-23◦ 03 ′
-22◦ 54 ′
-22◦ 42 ′
-22◦ 28 ′
-22◦ 13 ′
-21◦ 57 ′
-21◦ 38 ′
-21◦ 18 ′
-20◦ 57 ′
-20◦ 33 ′
-20◦ 09 ′
272◦ 11 ′
274◦ 21 ′
276◦ 32 ′
278◦ 43 ′
280◦ 53 ′
283◦ 03 ′
285◦ 12 ′
287◦ 21 ′
289◦ 30 ′
291◦ 38 ′
293◦ 46 ′
295◦ 53 ′
297◦ 60 ′
300◦ 06 ′
302◦ 11 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
-19◦ 43 ′
-19◦ 15 ′
-18◦ 46 ′
-18◦ 16 ′
-17◦ 44 ′
-17◦ 11 ′
-16◦ 37 ′
-16◦ 02 ′
-15◦ 26 ′
-14◦ 49 ′
-14◦ 10 ′
-13◦ 31 ′
-12◦ 51 ′
-12◦ 10 ′
-11◦ 28 ′
304◦ 15 ′
306◦ 19 ′
308◦ 22 ′
310◦ 25 ′
312◦ 27 ′
314◦ 28 ′
316◦ 28 ′
318◦ 27 ′
320◦ 26 ′
322◦ 24 ′
324◦ 22 ′
326◦ 19 ′
328◦ 15 ′
330◦ 10 ′
332◦ 05 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
-10◦ 46 ′
-10◦ 02 ′
-09◦ 19 ′
-08◦ 34 ′
-07◦ 49 ′
-07◦ 04 ′
-06◦ 18 ′
-05◦ 31 ′
-04◦ 45 ′
-03◦ 58 ′
-03◦ 10 ′
-02◦ 23 ′
-01◦ 35 ′
-00◦ 48 ′
-00◦ 00 ′
237◦ 49 ′
239◦ 54 ′
242◦ 00 ′
244◦ 07 ′
246◦ 14 ′
248◦ 22 ′
250◦ 30 ′
252◦ 39 ′
254◦ 48 ′
256◦ 57 ′
259◦ 07 ′
261◦ 17 ′
263◦ 28 ′
265◦ 39 ′
267◦ 49 ′
270◦ 00 ′
α
332◦ 05 ′
333◦ 60 ′
335◦ 53 ′
337◦ 47 ′
339◦ 40 ′
341◦ 32 ′
343◦ 24 ′
345◦ 16 ′
347◦ 07 ′
348◦ 58 ′
350◦ 49 ′
352◦ 39 ′
354◦ 30 ′
356◦ 20 ′
358◦ 10 ′
360◦ 00 ′
Table 2.3: Declinations and right ascensions of points on the ecliptic circle (b).
Spherical Astronomy
39
L
Longest Day
Summer Solstice Noon
Altitude of Sun
+00◦
+05◦
+10◦
+15◦
+20◦
+25◦
+30◦
+35◦
+40◦
+45◦
+50◦
+55◦
+60◦
+65◦
+70◦
+75◦
+80◦
+85◦
+90◦
12h00m
12h17m
12h35m
12h53m
13h13m
13h33m
13h56m
14h21m
14h51m
15h25m
16h09m
17h06m
18h29m
21h07m
61d06h
100d06h
130d02h
156d22h
182d15h
+66◦ 34 ′ N
+71◦ 34 ′ N
+76◦ 34 ′ N
+81◦ 34 ′ N
+86◦ 34 ′ N
+88◦ 26 ′ N
+83◦ 26 ′ S
+78◦ 26 ′ S
+73◦ 26 ′ S
+68◦ 26 ′ S
+63◦ 26 ′ S
+58◦ 26 ′ S
+53◦ 26 ′ S
+48◦ 26 ′ S
+43◦ 26 ′ S
+38◦ 26 ′ S
+33◦ 26 ′ S
+28◦ 26 ′ S
+23◦ 26 ′ S
Equinoctial Noon
Altitude of Sun
+90◦ 00 ′
+85◦ 00 ′
+80◦ 00 ′
+75◦ 00 ′
+70◦ 00 ′
+65◦ 00 ′
+60◦ 00 ′
+55◦ 00 ′
+50◦ 00 ′
+45◦ 00 ′
+40◦ 00 ′
+35◦ 00 ′
+30◦ 00 ′
+25◦ 00 ′
+20◦ 00 ′
+15◦ 00 ′
+10◦ 00 ′
+05◦ 00 ′
+00◦ 00 ′
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
Winter Solstice Noon
Altitude of Sun
+66◦ 34 ′
+61◦ 34 ′
+56◦ 34 ′
+51◦ 34 ′
+46◦ 34 ′
+41◦ 34 ′
+36◦ 34 ′
+31◦ 34 ′
+26◦ 34 ′
+21◦ 34 ′
+16◦ 34 ′
+11◦ 34 ′
+06◦ 34 ′
+01◦ 34 ′
−03◦ 26 ′
−08◦ 26 ′
−13◦ 26 ′
−18◦ 26 ′
−23◦ 26 ′
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
Table 2.4: Terrestrial climes in the earth’s northern hemisphere. The superscripts h, m, and d stand
for hours, minutes, and days, respectively. The symbols S and N indicated that the upper transit of the
sun occurs to the south and north of the zenith, respectively.
40
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
001◦ 50 ′
003◦ 40 ′
005◦ 30 ′
007◦ 21 ′
009◦ 11 ′
011◦ 02 ′
012◦ 53 ′
014◦ 44 ′
016◦ 36 ′
018◦ 28 ′
020◦ 20 ′
022◦ 13 ′
024◦ 07 ′
026◦ 00 ′
027◦ 55 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
027◦ 55 ′
029◦ 50 ′
031◦ 45 ′
033◦ 41 ′
035◦ 38 ′
037◦ 36 ′
039◦ 34 ′
041◦ 33 ′
043◦ 32 ′
045◦ 32 ′
047◦ 33 ′
049◦ 35 ′
051◦ 38 ′
053◦ 41 ′
055◦ 45 ′
057◦ 49 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
181◦ 50 ′
183◦ 40 ′
185◦ 30 ′
187◦ 21 ′
189◦ 11 ′
191◦ 02 ′
192◦ 53 ′
194◦ 44 ′
196◦ 36 ′
198◦ 28 ′
200◦ 20 ′
202◦ 13 ′
204◦ 07 ′
206◦ 00 ′
207◦ 55 ′
Scorpio
λ
α
◦
00
207◦ 55 ′
◦
02
209◦ 50 ′
◦
04
211◦ 45 ′
◦
06
213◦ 41 ′
◦
08
215◦ 38 ′
◦
10
217◦ 36 ′
◦
12
219◦ 34 ′
◦
14
221◦ 33 ′
◦
16
223◦ 32 ′
◦
18
225◦ 32 ′
◦
20
227◦ 33 ′
◦
22
229◦ 35 ′
◦
24
231◦ 38 ′
◦
26
233◦ 41 ′
◦
28
235◦ 45 ′
◦
30
237◦ 49 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
057◦ 49 ′
059◦ 54 ′
062◦ 00 ′
064◦ 07 ′
066◦ 14 ′
068◦ 22 ′
070◦ 30 ′
072◦ 39 ′
074◦ 48 ′
076◦ 57 ′
079◦ 07 ′
081◦ 17 ′
083◦ 28 ′
085◦ 39 ′
087◦ 49 ′
090◦ 00 ′
Sagittarius
λ
α
◦
00
237◦ 49 ′
◦
02
239◦ 54 ′
◦
04
242◦ 00 ′
◦
06
244◦ 07 ′
◦
08
246◦ 14 ′
◦
10
248◦ 22 ′
◦
12
250◦ 30 ′
◦
14
252◦ 39 ′
◦
16
254◦ 48 ′
◦
18
256◦ 57 ′
◦
20
259◦ 07 ′
◦
22
261◦ 17 ′
◦
24
263◦ 28 ′
◦
26
265◦ 39 ′
◦
28
267◦ 49 ′
◦
30
270◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
090◦ 00 ′
092◦ 11 ′
094◦ 21 ′
096◦ 32 ′
098◦ 43 ′
100◦ 53 ′
103◦ 03 ′
105◦ 12 ′
107◦ 21 ′
109◦ 30 ′
111◦ 38 ′
113◦ 46 ′
115◦ 53 ′
117◦ 60 ′
120◦ 06 ′
122◦ 11 ′
Capricorn
λ
α
◦
00
270◦ 00 ′
◦
02
272◦ 11 ′
◦
04
274◦ 21 ′
◦
06
276◦ 32 ′
◦
08
278◦ 43 ′
◦
10
280◦ 53 ′
◦
12
283◦ 03 ′
◦
14
285◦ 12 ′
◦
16
287◦ 21 ′
◦
18
289◦ 30 ′
◦
20
291◦ 38 ′
◦
22
293◦ 46 ′
◦
24
295◦ 53 ′
◦
26
297◦ 60 ′
◦
28
300◦ 06 ′
◦
30
302◦ 11 ′
Leo
α
122◦ 11 ′
124◦ 15 ′
126◦ 19 ′
128◦ 22 ′
130◦ 25 ′
132◦ 27 ′
134◦ 28 ′
136◦ 28 ′
138◦ 27 ′
140◦ 26 ′
142◦ 24 ′
144◦ 22 ′
146◦ 19 ′
148◦ 15 ′
150◦ 10 ′
152◦ 05 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
152◦ 05 ′
153◦ 60 ′
155◦ 53 ′
157◦ 47 ′
159◦ 40 ′
161◦ 32 ′
163◦ 24 ′
165◦ 16 ′
167◦ 07 ′
168◦ 58 ′
170◦ 49 ′
172◦ 39 ′
174◦ 30 ′
176◦ 20 ′
178◦ 10 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
302◦ 11 ′
◦
02
304◦ 15 ′
◦
04
306◦ 19 ′
◦
06
308◦ 22 ′
◦
08
310◦ 25 ′
◦
10
312◦ 27 ′
◦
12
314◦ 28 ′
◦
14
316◦ 28 ′
◦
16
318◦ 27 ′
◦
18
320◦ 26 ′
◦
20
322◦ 24 ′
◦
22
324◦ 22 ′
◦
24
326◦ 19 ′
◦
26
328◦ 15 ′
◦
28
330◦ 10 ′
◦
30
332◦ 05 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
332◦ 05 ′
333◦ 60 ′
335◦ 53 ′
337◦ 47 ′
339◦ 40 ′
341◦ 32 ′
343◦ 24 ′
345◦ 16 ′
347◦ 07 ′
348◦ 58 ′
350◦ 49 ′
352◦ 39 ′
354◦ 30 ′
356◦ 20 ′
358◦ 10 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.5: Right ascensions of the ecliptic for latitude 0◦ .
Spherical Astronomy
41
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
001◦ 42 ′
003◦ 23 ′
005◦ 05 ′
006◦ 47 ′
008◦ 29 ′
010◦ 12 ′
011◦ 55 ′
013◦ 38 ′
015◦ 21 ′
017◦ 05 ′
018◦ 49 ′
020◦ 34 ′
022◦ 19 ′
024◦ 05 ′
025◦ 52 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
025◦ 52 ′
027◦ 39 ′
029◦ 27 ′
031◦ 16 ′
033◦ 05 ′
034◦ 55 ′
036◦ 46 ′
038◦ 38 ′
040◦ 31 ′
042◦ 25 ′
044◦ 19 ′
046◦ 15 ′
048◦ 11 ′
050◦ 09 ′
052◦ 07 ′
054◦ 07 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
181◦ 59 ′
183◦ 57 ′
185◦ 56 ′
187◦ 54 ′
189◦ 53 ′
191◦ 52 ′
193◦ 52 ′
195◦ 51 ′
197◦ 51 ′
199◦ 51 ′
201◦ 52 ′
203◦ 53 ′
205◦ 54 ′
207◦ 56 ′
209◦ 58 ′
Scorpio
λ
α
◦
00
209◦ 58 ′
◦
02
212◦ 00 ′
◦
04
214◦ 03 ′
◦
06
216◦ 07 ′
◦
08
218◦ 11 ′
◦
10
220◦ 16 ′
◦
12
222◦ 21 ′
◦
14
224◦ 27 ′
◦
16
226◦ 33 ′
◦
18
228◦ 40 ′
◦
20
230◦ 47 ′
◦
22
232◦ 55 ′
◦
24
235◦ 04 ′
◦
26
237◦ 13 ′
◦
28
239◦ 22 ′
◦
30
241◦ 32 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
054◦ 07 ′
056◦ 07 ′
058◦ 08 ′
060◦ 10 ′
062◦ 13 ′
064◦ 17 ′
066◦ 22 ′
068◦ 28 ′
070◦ 34 ′
072◦ 41 ′
074◦ 49 ′
076◦ 58 ′
079◦ 07 ′
081◦ 16 ′
083◦ 26 ′
085◦ 37 ′
Sagittarius
λ
α
◦
00
241◦ 32 ′
◦
02
243◦ 42 ′
◦
04
245◦ 53 ′
◦
06
248◦ 03 ′
◦
08
250◦ 15 ′
◦
10
252◦ 26 ′
◦
12
254◦ 38 ′
◦
14
256◦ 50 ′
◦
16
259◦ 02 ′
◦
18
261◦ 14 ′
◦
20
263◦ 26 ′
◦
22
265◦ 37 ′
◦
24
267◦ 49 ′
◦
26
270◦ 01 ′
◦
28
272◦ 12 ′
◦
30
274◦ 23 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
085◦ 37 ′
087◦ 48 ′
089◦ 59 ′
092◦ 11 ′
094◦ 23 ′
096◦ 34 ′
098◦ 46 ′
100◦ 58 ′
103◦ 10 ′
105◦ 22 ′
107◦ 34 ′
109◦ 45 ′
111◦ 57 ′
114◦ 07 ′
116◦ 18 ′
118◦ 28 ′
Capricorn
λ
α
◦
00
274◦ 23 ′
◦
02
276◦ 34 ′
◦
04
278◦ 44 ′
◦
06
280◦ 53 ′
◦
08
283◦ 02 ′
◦
10
285◦ 11 ′
◦
12
287◦ 19 ′
◦
14
289◦ 26 ′
◦
16
291◦ 32 ′
◦
18
293◦ 38 ′
◦
20
295◦ 43 ′
◦
22
297◦ 47 ′
◦
24
299◦ 50 ′
◦
26
301◦ 52 ′
◦
28
303◦ 53 ′
◦
30
305◦ 53 ′
Leo
α
118◦ 28 ′
120◦ 38 ′
122◦ 47 ′
124◦ 56 ′
127◦ 05 ′
129◦ 13 ′
131◦ 20 ′
133◦ 27 ′
135◦ 33 ′
137◦ 39 ′
139◦ 44 ′
141◦ 49 ′
143◦ 53 ′
145◦ 57 ′
148◦ 00 ′
150◦ 02 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
150◦ 02 ′
152◦ 04 ′
154◦ 06 ′
156◦ 07 ′
158◦ 08 ′
160◦ 09 ′
162◦ 09 ′
164◦ 09 ′
166◦ 08 ′
168◦ 08 ′
170◦ 07 ′
172◦ 06 ′
174◦ 04 ′
176◦ 03 ′
178◦ 01 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
305◦ 53 ′
◦
02
307◦ 53 ′
◦
04
309◦ 51 ′
◦
06
311◦ 49 ′
◦
08
313◦ 45 ′
◦
10
315◦ 41 ′
◦
12
317◦ 35 ′
◦
14
319◦ 29 ′
◦
16
321◦ 22 ′
◦
18
323◦ 14 ′
◦
20
325◦ 05 ′
◦
22
326◦ 55 ′
◦
24
328◦ 44 ′
◦
26
330◦ 33 ′
◦
28
332◦ 21 ′
◦
30
334◦ 08 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
334◦ 08 ′
335◦ 55 ′
337◦ 41 ′
339◦ 26 ′
341◦ 11 ′
342◦ 55 ′
344◦ 39 ′
346◦ 22 ′
348◦ 05 ′
349◦ 48 ′
351◦ 31 ′
353◦ 13 ′
354◦ 55 ′
356◦ 37 ′
358◦ 18 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.6: Oblique ascensions of the ecliptic for latitude +10◦ .
42
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
001◦ 33 ′
003◦ 06 ′
004◦ 38 ′
006◦ 12 ′
007◦ 45 ′
009◦ 18 ′
010◦ 52 ′
012◦ 26 ′
014◦ 01 ′
015◦ 36 ′
017◦ 12 ′
018◦ 48 ′
020◦ 25 ′
022◦ 02 ′
023◦ 41 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
023◦ 41 ′
025◦ 20 ′
026◦ 59 ′
028◦ 40 ′
030◦ 22 ′
032◦ 04 ′
033◦ 48 ′
035◦ 32 ′
037◦ 18 ′
039◦ 04 ′
040◦ 52 ′
042◦ 41 ′
044◦ 31 ′
046◦ 23 ′
048◦ 15 ′
050◦ 09 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
182◦ 07 ′
184◦ 15 ′
186◦ 23 ′
188◦ 30 ′
190◦ 38 ′
192◦ 46 ′
194◦ 54 ′
197◦ 02 ′
199◦ 11 ′
201◦ 20 ′
203◦ 29 ′
205◦ 38 ′
207◦ 48 ′
209◦ 58 ′
212◦ 09 ′
Scorpio
λ
α
◦
00
212◦ 09 ′
◦
02
214◦ 20 ′
◦
04
216◦ 31 ′
◦
06
218◦ 42 ′
◦
08
220◦ 54 ′
◦
10
223◦ 07 ′
◦
12
225◦ 20 ′
◦
14
227◦ 33 ′
◦
16
229◦ 46 ′
◦
18
232◦ 00 ′
◦
20
234◦ 14 ′
◦
22
236◦ 29 ′
◦
24
238◦ 44 ′
◦
26
240◦ 59 ′
◦
28
243◦ 14 ′
◦
30
245◦ 30 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
050◦ 09 ′
052◦ 04 ′
054◦ 00 ′
055◦ 57 ′
057◦ 56 ′
059◦ 56 ′
061◦ 57 ′
063◦ 59 ′
066◦ 02 ′
068◦ 07 ′
070◦ 13 ′
072◦ 19 ′
074◦ 27 ′
076◦ 35 ′
078◦ 45 ′
080◦ 55 ′
Sagittarius
λ
α
◦
00
245◦ 30 ′
◦
02
247◦ 45 ′
◦
04
250◦ 01 ′
◦
06
252◦ 16 ′
◦
08
254◦ 32 ′
◦
10
256◦ 48 ′
◦
12
259◦ 03 ′
◦
14
261◦ 18 ′
◦
16
263◦ 33 ′
◦
18
265◦ 48 ′
◦
20
268◦ 02 ′
◦
22
270◦ 16 ′
◦
24
272◦ 29 ′
◦
26
274◦ 42 ′
◦
28
276◦ 53 ′
◦
30
279◦ 05 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
080◦ 55 ′
083◦ 07 ′
085◦ 18 ′
087◦ 31 ′
089◦ 44 ′
091◦ 58 ′
094◦ 12 ′
096◦ 27 ′
098◦ 42 ′
100◦ 57 ′
103◦ 12 ′
105◦ 28 ′
107◦ 44 ′
109◦ 59 ′
112◦ 15 ′
114◦ 30 ′
Capricorn
λ
α
◦
00
279◦ 05 ′
◦
02
281◦ 15 ′
◦
04
283◦ 25 ′
◦
06
285◦ 33 ′
◦
08
287◦ 41 ′
◦
10
289◦ 47 ′
◦
12
291◦ 53 ′
◦
14
293◦ 58 ′
◦
16
296◦ 01 ′
◦
18
298◦ 03 ′
◦
20
300◦ 04 ′
◦
22
302◦ 04 ′
◦
24
304◦ 03 ′
◦
26
306◦ 00 ′
◦
28
307◦ 56 ′
◦
30
309◦ 51 ′
Leo
α
114◦ 30 ′
116◦ 46 ′
119◦ 01 ′
121◦ 16 ′
123◦ 31 ′
125◦ 46 ′
128◦ 00 ′
130◦ 14 ′
132◦ 27 ′
134◦ 40 ′
136◦ 53 ′
139◦ 06 ′
141◦ 18 ′
143◦ 29 ′
145◦ 40 ′
147◦ 51 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
147◦ 51 ′
150◦ 02 ′
152◦ 12 ′
154◦ 22 ′
156◦ 31 ′
158◦ 40 ′
160◦ 49 ′
162◦ 58 ′
165◦ 06 ′
167◦ 14 ′
169◦ 22 ′
171◦ 30 ′
173◦ 37 ′
175◦ 45 ′
177◦ 53 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
309◦ 51 ′
◦
02
311◦ 45 ′
◦
04
313◦ 37 ′
◦
06
315◦ 29 ′
◦
08
317◦ 19 ′
◦
10
319◦ 08 ′
◦
12
320◦ 56 ′
◦
14
322◦ 42 ′
◦
16
324◦ 28 ′
◦
18
326◦ 12 ′
◦
20
327◦ 56 ′
◦
22
329◦ 38 ′
◦
24
331◦ 20 ′
◦
26
333◦ 01 ′
◦
28
334◦ 40 ′
◦
30
336◦ 19 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
336◦ 19 ′
337◦ 58 ′
339◦ 35 ′
341◦ 12 ′
342◦ 48 ′
344◦ 24 ′
345◦ 59 ′
347◦ 34 ′
349◦ 08 ′
350◦ 42 ′
352◦ 15 ′
353◦ 48 ′
355◦ 22 ′
356◦ 54 ′
358◦ 27 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.7: Oblique ascensions of the ecliptic for latitude +20◦ .
Spherical Astronomy
43
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
001◦ 23 ′
002◦ 45 ′
004◦ 08 ′
005◦ 31 ′
006◦ 54 ′
008◦ 17 ′
009◦ 41 ′
011◦ 05 ′
012◦ 30 ′
013◦ 55 ′
015◦ 21 ′
016◦ 47 ′
018◦ 15 ′
019◦ 42 ′
021◦ 11 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
021◦ 11 ′
022◦ 41 ′
024◦ 11 ′
025◦ 43 ′
027◦ 15 ′
028◦ 49 ′
030◦ 23 ′
031◦ 59 ′
033◦ 37 ′
035◦ 15 ′
036◦ 55 ′
038◦ 36 ′
040◦ 19 ′
042◦ 03 ′
043◦ 48 ′
045◦ 36 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
182◦ 18 ′
184◦ 35 ′
186◦ 53 ′
189◦ 11 ′
191◦ 29 ′
193◦ 47 ′
196◦ 05 ′
198◦ 23 ′
200◦ 42 ′
203◦ 01 ′
205◦ 20 ′
207◦ 39 ′
209◦ 59 ′
212◦ 18 ′
214◦ 38 ′
Scorpio
λ
α
◦
00
214◦ 38 ′
◦
02
216◦ 59 ′
◦
04
219◦ 19 ′
◦
06
221◦ 40 ′
◦
08
224◦ 01 ′
◦
10
226◦ 22 ′
◦
12
228◦ 44 ′
◦
14
231◦ 06 ′
◦
16
233◦ 28 ′
◦
18
235◦ 50 ′
◦
20
238◦ 12 ′
◦
22
240◦ 34 ′
◦
24
242◦ 56 ′
◦
26
245◦ 19 ′
◦
28
247◦ 41 ′
◦
30
250◦ 03 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
045◦ 36 ′
047◦ 24 ′
049◦ 14 ′
051◦ 06 ′
053◦ 00 ′
054◦ 55 ′
056◦ 51 ′
058◦ 50 ′
060◦ 49 ′
062◦ 51 ′
064◦ 54 ′
066◦ 58 ′
069◦ 04 ′
071◦ 12 ′
073◦ 20 ′
075◦ 30 ′
Sagittarius
λ
α
◦
00
250◦ 03 ′
◦
02
252◦ 25 ′
◦
04
254◦ 46 ′
◦
06
257◦ 08 ′
◦
08
259◦ 28 ′
◦
10
261◦ 49 ′
◦
12
264◦ 09 ′
◦
14
266◦ 28 ′
◦
16
268◦ 46 ′
◦
18
271◦ 04 ′
◦
20
273◦ 21 ′
◦
22
275◦ 37 ′
◦
24
277◦ 52 ′
◦
26
280◦ 05 ′
◦
28
282◦ 18 ′
◦
30
284◦ 30 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
075◦ 30 ′
077◦ 42 ′
079◦ 55 ′
082◦ 08 ′
084◦ 23 ′
086◦ 39 ′
088◦ 56 ′
091◦ 14 ′
093◦ 32 ′
095◦ 51 ′
098◦ 11 ′
100◦ 32 ′
102◦ 52 ′
105◦ 14 ′
107◦ 35 ′
109◦ 57 ′
Capricorn
λ
α
◦
00
284◦ 30 ′
◦
02
286◦ 40 ′
◦
04
288◦ 48 ′
◦
06
290◦ 56 ′
◦
08
293◦ 02 ′
◦
10
295◦ 06 ′
◦
12
297◦ 09 ′
◦
14
299◦ 11 ′
◦
16
301◦ 10 ′
◦
18
303◦ 09 ′
◦
20
305◦ 05 ′
◦
22
307◦ 00 ′
◦
24
308◦ 54 ′
◦
26
310◦ 46 ′
◦
28
312◦ 36 ′
◦
30
314◦ 24 ′
Leo
α
109◦ 57 ′
112◦ 19 ′
114◦ 41 ′
117◦ 04 ′
119◦ 26 ′
121◦ 48 ′
124◦ 10 ′
126◦ 32 ′
128◦ 54 ′
131◦ 16 ′
133◦ 38 ′
135◦ 59 ′
138◦ 20 ′
140◦ 41 ′
143◦ 01 ′
145◦ 22 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
145◦ 22 ′
147◦ 42 ′
150◦ 01 ′
152◦ 21 ′
154◦ 40 ′
156◦ 59 ′
159◦ 18 ′
161◦ 37 ′
163◦ 55 ′
166◦ 13 ′
168◦ 31 ′
170◦ 49 ′
173◦ 07 ′
175◦ 25 ′
177◦ 42 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
314◦ 24 ′
◦
02
316◦ 12 ′
◦
04
317◦ 57 ′
◦
06
319◦ 41 ′
◦
08
321◦ 24 ′
◦
10
323◦ 05 ′
◦
12
324◦ 45 ′
◦
14
326◦ 23 ′
◦
16
328◦ 01 ′
◦
18
329◦ 37 ′
◦
20
331◦ 11 ′
◦
22
332◦ 45 ′
◦
24
334◦ 17 ′
◦
26
335◦ 49 ′
◦
28
337◦ 19 ′
◦
30
338◦ 49 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
338◦ 49 ′
340◦ 18 ′
341◦ 45 ′
343◦ 13 ′
344◦ 39 ′
346◦ 05 ′
347◦ 30 ′
348◦ 55 ′
350◦ 19 ′
351◦ 43 ′
353◦ 06 ′
354◦ 29 ′
355◦ 52 ′
357◦ 15 ′
358◦ 37 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.8: Oblique ascensions of the ecliptic for latitude +30◦ .
44
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
001◦ 10 ′
002◦ 20 ′
003◦ 30 ′
004◦ 41 ′
005◦ 52 ′
007◦ 03 ′
008◦ 14 ′
009◦ 26 ′
010◦ 38 ′
011◦ 51 ′
013◦ 05 ′
014◦ 19 ′
015◦ 34 ′
016◦ 50 ′
018◦ 07 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
018◦ 07 ′
019◦ 24 ′
020◦ 43 ′
022◦ 03 ′
023◦ 24 ′
024◦ 46 ′
026◦ 10 ′
027◦ 35 ′
029◦ 02 ′
030◦ 30 ′
031◦ 59 ′
033◦ 31 ′
035◦ 04 ′
036◦ 38 ′
038◦ 15 ′
039◦ 54 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
182◦ 30 ′
185◦ 00 ′
187◦ 31 ′
190◦ 01 ′
192◦ 31 ′
195◦ 02 ′
197◦ 32 ′
200◦ 03 ′
202◦ 34 ′
205◦ 05 ′
207◦ 36 ′
210◦ 08 ′
212◦ 39 ′
215◦ 11 ′
217◦ 43 ′
Scorpio
λ
α
◦
00
217◦ 43 ′
◦
02
220◦ 15 ′
◦
04
222◦ 47 ′
◦
06
225◦ 20 ′
◦
08
227◦ 52 ′
◦
10
230◦ 25 ′
◦
12
232◦ 57 ′
◦
14
235◦ 30 ′
◦
16
238◦ 03 ′
◦
18
240◦ 35 ′
◦
20
243◦ 07 ′
◦
22
245◦ 40 ′
◦
24
248◦ 12 ′
◦
26
250◦ 43 ′
◦
28
253◦ 14 ′
◦
30
255◦ 45 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
039◦ 54 ′
041◦ 34 ′
043◦ 16 ′
045◦ 01 ′
046◦ 48 ′
048◦ 36 ′
050◦ 27 ′
052◦ 20 ′
054◦ 15 ′
056◦ 12 ′
058◦ 12 ′
060◦ 13 ′
062◦ 17 ′
064◦ 23 ′
066◦ 31 ′
068◦ 40 ′
Sagittarius
λ
α
◦
00
255◦ 45 ′
◦
02
258◦ 15 ′
◦
04
260◦ 44 ′
◦
06
263◦ 13 ′
◦
08
265◦ 41 ′
◦
10
268◦ 07 ′
◦
12
270◦ 33 ′
◦
14
272◦ 57 ′
◦
16
275◦ 21 ′
◦
18
277◦ 42 ′
◦
20
280◦ 03 ′
◦
22
282◦ 22 ′
◦
24
284◦ 39 ′
◦
26
286◦ 54 ′
◦
28
289◦ 08 ′
◦
30
291◦ 20 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
068◦ 40 ′
070◦ 52 ′
073◦ 06 ′
075◦ 21 ′
077◦ 38 ′
079◦ 57 ′
082◦ 18 ′
084◦ 39 ′
087◦ 03 ′
089◦ 27 ′
091◦ 53 ′
094◦ 19 ′
096◦ 47 ′
099◦ 16 ′
101◦ 45 ′
104◦ 15 ′
Capricorn
λ
α
◦
00
291◦ 20 ′
◦
02
293◦ 29 ′
◦
04
295◦ 37 ′
◦
06
297◦ 43 ′
◦
08
299◦ 47 ′
◦
10
301◦ 48 ′
◦
12
303◦ 48 ′
◦
14
305◦ 45 ′
◦
16
307◦ 40 ′
◦
18
309◦ 33 ′
◦
20
311◦ 24 ′
◦
22
313◦ 12 ′
◦
24
314◦ 59 ′
◦
26
316◦ 44 ′
◦
28
318◦ 26 ′
◦
30
320◦ 06 ′
Leo
α
104◦ 15 ′
106◦ 46 ′
109◦ 17 ′
111◦ 48 ′
114◦ 20 ′
116◦ 53 ′
119◦ 25 ′
121◦ 57 ′
124◦ 30 ′
127◦ 03 ′
129◦ 35 ′
132◦ 08 ′
134◦ 40 ′
137◦ 13 ′
139◦ 45 ′
142◦ 17 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
142◦ 17 ′
144◦ 49 ′
147◦ 21 ′
149◦ 52 ′
152◦ 24 ′
154◦ 55 ′
157◦ 26 ′
159◦ 57 ′
162◦ 28 ′
164◦ 58 ′
167◦ 29 ′
169◦ 59 ′
172◦ 29 ′
175◦ 00 ′
177◦ 30 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
320◦ 06 ′
◦
02
321◦ 45 ′
◦
04
323◦ 22 ′
◦
06
324◦ 56 ′
◦
08
326◦ 29 ′
◦
10
328◦ 01 ′
◦
12
329◦ 30 ′
◦
14
330◦ 58 ′
◦
16
332◦ 25 ′
◦
18
333◦ 50 ′
◦
20
335◦ 14 ′
◦
22
336◦ 36 ′
◦
24
337◦ 57 ′
◦
26
339◦ 17 ′
◦
28
340◦ 36 ′
◦
30
341◦ 53 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
341◦ 53 ′
343◦ 10 ′
344◦ 26 ′
345◦ 41 ′
346◦ 55 ′
348◦ 09 ′
349◦ 22 ′
350◦ 34 ′
351◦ 46 ′
352◦ 57 ′
354◦ 08 ′
355◦ 19 ′
356◦ 30 ′
357◦ 40 ′
358◦ 50 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.9: Oblique ascensions of the ecliptic for latitude +40◦ .
Spherical Astronomy
45
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
000◦ 53 ′
001◦ 47 ′
002◦ 40 ′
003◦ 34 ′
004◦ 27 ′
005◦ 22 ′
006◦ 16 ′
007◦ 11 ′
008◦ 07 ′
009◦ 03 ′
010◦ 00 ′
010◦ 57 ′
011◦ 56 ′
012◦ 55 ′
013◦ 55 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
013◦ 55 ′
014◦ 56 ′
015◦ 59 ′
017◦ 02 ′
018◦ 07 ′
019◦ 13 ′
020◦ 21 ′
021◦ 31 ′
022◦ 42 ′
023◦ 54 ′
025◦ 09 ′
026◦ 26 ′
027◦ 44 ′
029◦ 05 ′
030◦ 28 ′
031◦ 54 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
182◦ 47 ′
185◦ 34 ′
188◦ 21 ′
191◦ 08 ′
193◦ 55 ′
196◦ 43 ′
199◦ 30 ′
202◦ 18 ′
205◦ 05 ′
207◦ 53 ′
210◦ 41 ′
213◦ 29 ′
216◦ 17 ′
219◦ 06 ′
221◦ 54 ′
Scorpio
λ
α
◦
00
221◦ 54 ′
◦
02
224◦ 43 ′
◦
04
227◦ 32 ′
◦
06
230◦ 20 ′
◦
08
233◦ 09 ′
◦
10
235◦ 58 ′
◦
12
238◦ 46 ′
◦
14
241◦ 34 ′
◦
16
244◦ 23 ′
◦
18
247◦ 10 ′
◦
20
249◦ 58 ′
◦
22
252◦ 45 ′
◦
24
255◦ 31 ′
◦
26
258◦ 16 ′
◦
28
261◦ 01 ′
◦
30
263◦ 45 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
031◦ 54 ′
033◦ 22 ′
034◦ 52 ′
036◦ 25 ′
038◦ 01 ′
039◦ 40 ′
041◦ 22 ′
043◦ 06 ′
044◦ 54 ′
046◦ 45 ′
048◦ 38 ′
050◦ 35 ′
052◦ 35 ′
054◦ 38 ′
056◦ 45 ′
058◦ 54 ′
Sagittarius
λ
α
◦
00
263◦ 45 ′
◦
02
266◦ 27 ′
◦
04
269◦ 09 ′
◦
06
271◦ 48 ′
◦
08
274◦ 27 ′
◦
10
277◦ 03 ′
◦
12
279◦ 38 ′
◦
14
282◦ 11 ′
◦
16
284◦ 42 ′
◦
18
287◦ 10 ′
◦
20
289◦ 36 ′
◦
22
292◦ 00 ′
◦
24
294◦ 20 ′
◦
26
296◦ 39 ′
◦
28
298◦ 54 ′
◦
30
301◦ 06 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
058◦ 54 ′
061◦ 06 ′
063◦ 21 ′
065◦ 40 ′
068◦ 00 ′
070◦ 24 ′
072◦ 50 ′
075◦ 18 ′
077◦ 49 ′
080◦ 22 ′
082◦ 57 ′
085◦ 33 ′
088◦ 12 ′
090◦ 51 ′
093◦ 33 ′
096◦ 15 ′
Capricorn
λ
α
◦
00
301◦ 06 ′
◦
02
303◦ 15 ′
◦
04
305◦ 22 ′
◦
06
307◦ 25 ′
◦
08
309◦ 25 ′
◦
10
311◦ 22 ′
◦
12
313◦ 15 ′
◦
14
315◦ 06 ′
◦
16
316◦ 54 ′
◦
18
318◦ 38 ′
◦
20
320◦ 20 ′
◦
22
321◦ 59 ′
◦
24
323◦ 35 ′
◦
26
325◦ 08 ′
◦
28
326◦ 38 ′
◦
30
328◦ 06 ′
Leo
α
096◦ 15 ′
098◦ 59 ′
101◦ 44 ′
104◦ 29 ′
107◦ 15 ′
110◦ 02 ′
112◦ 50 ′
115◦ 37 ′
118◦ 26 ′
121◦ 14 ′
124◦ 02 ′
126◦ 51 ′
129◦ 40 ′
132◦ 28 ′
135◦ 17 ′
138◦ 06 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
138◦ 06 ′
140◦ 54 ′
143◦ 43 ′
146◦ 31 ′
149◦ 19 ′
152◦ 07 ′
154◦ 55 ′
157◦ 42 ′
160◦ 30 ′
163◦ 17 ′
166◦ 05 ′
168◦ 52 ′
171◦ 39 ′
174◦ 26 ′
177◦ 13 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
328◦ 06 ′
◦
02
329◦ 32 ′
◦
04
330◦ 55 ′
◦
06
332◦ 16 ′
◦
08
333◦ 34 ′
◦
10
334◦ 51 ′
◦
12
336◦ 06 ′
◦
14
337◦ 18 ′
◦
16
338◦ 29 ′
◦
18
339◦ 39 ′
◦
20
340◦ 47 ′
◦
22
341◦ 53 ′
◦
24
342◦ 58 ′
◦
26
344◦ 01 ′
◦
28
345◦ 04 ′
◦
30
346◦ 05 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
346◦ 05 ′
347◦ 05 ′
348◦ 04 ′
349◦ 03 ′
350◦ 00 ′
350◦ 57 ′
351◦ 53 ′
352◦ 49 ′
353◦ 44 ′
354◦ 38 ′
355◦ 33 ′
356◦ 26 ′
357◦ 20 ′
358◦ 13 ′
359◦ 07 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.10: Oblique ascensions of the ecliptic for latitude +50◦ .
46
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
000◦ 42 ′
001◦ 24 ′
002◦ 06 ′
002◦ 48 ′
003◦ 31 ′
004◦ 14 ′
004◦ 57 ′
005◦ 41 ′
006◦ 25 ′
007◦ 10 ′
007◦ 55 ′
008◦ 41 ′
009◦ 28 ′
010◦ 15 ′
011◦ 04 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
011◦ 04 ′
011◦ 54 ′
012◦ 44 ′
013◦ 36 ′
014◦ 30 ′
015◦ 24 ′
016◦ 21 ′
017◦ 18 ′
018◦ 18 ′
019◦ 19 ′
020◦ 23 ′
021◦ 28 ′
022◦ 36 ′
023◦ 46 ′
024◦ 58 ′
026◦ 14 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
182◦ 58 ′
185◦ 57 ′
188◦ 55 ′
191◦ 53 ′
194◦ 52 ′
197◦ 50 ′
200◦ 49 ′
203◦ 48 ′
206◦ 47 ′
209◦ 46 ′
212◦ 46 ′
215◦ 45 ′
218◦ 45 ′
221◦ 45 ′
224◦ 45 ′
Scorpio
λ
α
◦
00
224◦ 45 ′
◦
02
227◦ 46 ′
◦
04
230◦ 46 ′
◦
06
233◦ 46 ′
◦
08
236◦ 46 ′
◦
10
239◦ 47 ′
◦
12
242◦ 47 ′
◦
14
245◦ 47 ′
◦
16
248◦ 46 ′
◦
18
251◦ 46 ′
◦
20
254◦ 44 ′
◦
22
257◦ 42 ′
◦
24
260◦ 39 ′
◦
26
263◦ 36 ′
◦
28
266◦ 31 ′
◦
30
269◦ 25 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
026◦ 14 ′
027◦ 31 ′
028◦ 52 ′
030◦ 16 ′
031◦ 44 ′
033◦ 14 ′
034◦ 48 ′
036◦ 26 ′
038◦ 07 ′
039◦ 52 ′
041◦ 41 ′
043◦ 34 ′
045◦ 31 ′
047◦ 32 ′
049◦ 37 ′
051◦ 45 ′
Sagittarius
λ
α
◦
00
269◦ 25 ′
◦
02
272◦ 17 ′
◦
04
275◦ 08 ′
◦
06
277◦ 57 ′
◦
08
280◦ 44 ′
◦
10
283◦ 29 ′
◦
12
286◦ 12 ′
◦
14
288◦ 52 ′
◦
16
291◦ 29 ′
◦
18
294◦ 03 ′
◦
20
296◦ 33 ′
◦
22
299◦ 01 ′
◦
24
301◦ 25 ′
◦
26
303◦ 45 ′
◦
28
306◦ 02 ′
◦
30
308◦ 15 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
051◦ 45 ′
053◦ 58 ′
056◦ 15 ′
058◦ 35 ′
060◦ 59 ′
063◦ 27 ′
065◦ 57 ′
068◦ 31 ′
071◦ 08 ′
073◦ 48 ′
076◦ 31 ′
079◦ 16 ′
082◦ 03 ′
084◦ 52 ′
087◦ 43 ′
090◦ 35 ′
Capricorn
λ
α
◦
00
308◦ 15 ′
◦
02
310◦ 23 ′
◦
04
312◦ 28 ′
◦
06
314◦ 29 ′
◦
08
316◦ 26 ′
◦
10
318◦ 19 ′
◦
12
320◦ 08 ′
◦
14
321◦ 53 ′
◦
16
323◦ 34 ′
◦
18
325◦ 12 ′
◦
20
326◦ 46 ′
◦
22
328◦ 16 ′
◦
24
329◦ 44 ′
◦
26
331◦ 08 ′
◦
28
332◦ 29 ′
◦
30
333◦ 46 ′
Leo
α
090◦ 35 ′
093◦ 29 ′
096◦ 24 ′
099◦ 21 ′
102◦ 18 ′
105◦ 16 ′
108◦ 14 ′
111◦ 14 ′
114◦ 13 ′
117◦ 13 ′
120◦ 13 ′
123◦ 14 ′
126◦ 14 ′
129◦ 14 ′
132◦ 14 ′
135◦ 15 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
135◦ 15 ′
138◦ 15 ′
141◦ 15 ′
144◦ 15 ′
147◦ 14 ′
150◦ 14 ′
153◦ 13 ′
156◦ 12 ′
159◦ 11 ′
162◦ 10 ′
165◦ 08 ′
168◦ 07 ′
171◦ 05 ′
174◦ 03 ′
177◦ 02 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
333◦ 46 ′
◦
02
335◦ 02 ′
◦
04
336◦ 14 ′
◦
06
337◦ 24 ′
◦
08
338◦ 32 ′
◦
10
339◦ 37 ′
◦
12
340◦ 41 ′
◦
14
341◦ 42 ′
◦
16
342◦ 42 ′
◦
18
343◦ 39 ′
◦
20
344◦ 36 ′
◦
22
345◦ 30 ′
◦
24
346◦ 24 ′
◦
26
347◦ 16 ′
◦
28
348◦ 06 ′
◦
30
348◦ 56 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
348◦ 56 ′
349◦ 45 ′
350◦ 32 ′
351◦ 19 ′
352◦ 05 ′
352◦ 50 ′
353◦ 35 ′
354◦ 19 ′
355◦ 03 ′
355◦ 46 ′
356◦ 29 ′
357◦ 12 ′
357◦ 54 ′
358◦ 36 ′
359◦ 18 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.11: Oblique ascensions of the ecliptic for latitude +55◦ .
Spherical Astronomy
47
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
000◦ 27 ′
000◦ 55 ′
001◦ 23 ′
001◦ 50 ′
002◦ 18 ′
002◦ 46 ′
003◦ 15 ′
003◦ 44 ′
004◦ 13 ′
004◦ 43 ′
005◦ 13 ′
005◦ 44 ′
006◦ 15 ′
006◦ 47 ′
007◦ 20 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
007◦ 20 ′
007◦ 54 ′
008◦ 29 ′
009◦ 05 ′
009◦ 42 ′
010◦ 20 ′
011◦ 00 ′
011◦ 41 ′
012◦ 24 ′
013◦ 08 ′
013◦ 55 ′
014◦ 43 ′
015◦ 34 ′
016◦ 28 ′
017◦ 23 ′
018◦ 22 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
183◦ 13 ′
186◦ 26 ′
189◦ 38 ′
192◦ 51 ′
196◦ 05 ′
199◦ 18 ′
202◦ 31 ′
205◦ 45 ′
208◦ 59 ′
212◦ 13 ′
215◦ 28 ′
218◦ 43 ′
221◦ 58 ′
225◦ 13 ′
228◦ 29 ′
Scorpio
λ
α
◦
00
228◦ 29 ′
◦
02
231◦ 45 ′
◦
04
235◦ 01 ′
◦
06
238◦ 18 ′
◦
08
241◦ 34 ′
◦
10
244◦ 51 ′
◦
12
248◦ 08 ′
◦
14
251◦ 24 ′
◦
16
254◦ 40 ′
◦
18
257◦ 56 ′
◦
20
261◦ 12 ′
◦
22
264◦ 27 ′
◦
24
267◦ 41 ′
◦
26
270◦ 54 ′
◦
28
274◦ 06 ′
◦
30
277◦ 16 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
018◦ 22 ′
019◦ 24 ′
020◦ 29 ′
021◦ 38 ′
022◦ 50 ′
024◦ 07 ′
025◦ 27 ′
026◦ 52 ′
028◦ 22 ′
029◦ 57 ′
031◦ 38 ′
033◦ 23 ′
035◦ 14 ′
037◦ 11 ′
039◦ 13 ′
041◦ 21 ′
Sagittarius
λ
α
◦
00
277◦ 16 ′
◦
02
280◦ 25 ′
◦
04
283◦ 32 ′
◦
06
286◦ 36 ′
◦
08
289◦ 38 ′
◦
10
292◦ 37 ′
◦
12
295◦ 33 ′
◦
14
298◦ 25 ′
◦
16
301◦ 13 ′
◦
18
303◦ 57 ′
◦
20
306◦ 37 ′
◦
22
309◦ 12 ′
◦
24
311◦ 42 ′
◦
26
314◦ 06 ′
◦
28
316◦ 26 ′
◦
30
318◦ 39 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
041◦ 21 ′
043◦ 34 ′
045◦ 54 ′
048◦ 18 ′
050◦ 48 ′
053◦ 23 ′
056◦ 03 ′
058◦ 47 ′
061◦ 35 ′
064◦ 27 ′
067◦ 23 ′
070◦ 22 ′
073◦ 24 ′
076◦ 28 ′
079◦ 35 ′
082◦ 44 ′
Capricorn
λ
α
◦
00
318◦ 39 ′
◦
02
320◦ 47 ′
◦
04
322◦ 49 ′
◦
06
324◦ 46 ′
◦
08
326◦ 37 ′
◦
10
328◦ 22 ′
◦
12
330◦ 03 ′
◦
14
331◦ 38 ′
◦
16
333◦ 08 ′
◦
18
334◦ 33 ′
◦
20
335◦ 53 ′
◦
22
337◦ 10 ′
◦
24
338◦ 22 ′
◦
26
339◦ 31 ′
◦
28
340◦ 36 ′
◦
30
341◦ 38 ′
Leo
α
082◦ 44 ′
085◦ 54 ′
089◦ 06 ′
092◦ 19 ′
095◦ 33 ′
098◦ 48 ′
102◦ 04 ′
105◦ 20 ′
108◦ 36 ′
111◦ 52 ′
115◦ 09 ′
118◦ 26 ′
121◦ 42 ′
124◦ 59 ′
128◦ 15 ′
131◦ 31 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
131◦ 31 ′
134◦ 47 ′
138◦ 02 ′
141◦ 17 ′
144◦ 32 ′
147◦ 47 ′
151◦ 01 ′
154◦ 15 ′
157◦ 29 ′
160◦ 42 ′
163◦ 55 ′
167◦ 09 ′
170◦ 22 ′
173◦ 34 ′
176◦ 47 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
341◦ 38 ′
◦
02
342◦ 37 ′
◦
04
343◦ 32 ′
◦
06
344◦ 26 ′
◦
08
345◦ 17 ′
◦
10
346◦ 05 ′
◦
12
346◦ 52 ′
◦
14
347◦ 36 ′
◦
16
348◦ 19 ′
◦
18
349◦ 00 ′
◦
20
349◦ 40 ′
◦
22
350◦ 18 ′
◦
24
350◦ 55 ′
◦
26
351◦ 31 ′
◦
28
352◦ 06 ′
◦
30
352◦ 40 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
352◦ 40 ′
353◦ 13 ′
353◦ 45 ′
354◦ 16 ′
354◦ 47 ′
355◦ 17 ′
355◦ 47 ′
356◦ 16 ′
356◦ 45 ′
357◦ 14 ′
357◦ 42 ′
358◦ 10 ′
358◦ 37 ′
359◦ 05 ′
359◦ 33 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.12: Oblique ascensions of the ecliptic for latitude +60◦ .
48
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
000◦ 00 ′
000◦ 08 ′
000◦ 16 ′
000◦ 23 ′
000◦ 31 ′
000◦ 39 ′
000◦ 47 ′
000◦ 55 ′
001◦ 04 ′
001◦ 12 ′
001◦ 21 ′
001◦ 29 ′
001◦ 38 ′
001◦ 48 ′
001◦ 57 ′
002◦ 07 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
α
002◦ 07 ′
002◦ 17 ′
002◦ 28 ′
002◦ 39 ′
002◦ 51 ′
003◦ 03 ′
003◦ 16 ′
003◦ 29 ′
003◦ 44 ′
003◦ 59 ′
004◦ 15 ′
004◦ 32 ′
004◦ 51 ′
005◦ 11 ′
005◦ 33 ′
005◦ 56 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
183◦ 32 ′
187◦ 05 ′
190◦ 38 ′
194◦ 10 ′
197◦ 44 ′
201◦ 17 ′
204◦ 51 ′
208◦ 25 ′
212◦ 00 ′
215◦ 35 ′
219◦ 11 ′
222◦ 48 ′
226◦ 25 ′
230◦ 03 ′
233◦ 42 ′
Scorpio
λ
α
◦
00
233◦ 42 ′
◦
02
237◦ 22 ′
◦
04
241◦ 02 ′
◦
06
244◦ 43 ′
◦
08
248◦ 25 ′
◦
10
252◦ 08 ′
◦
12
255◦ 52 ′
◦
14
259◦ 36 ′
◦
16
263◦ 21 ′
◦
18
267◦ 06 ′
◦
20
270◦ 52 ′
◦
22
274◦ 38 ′
◦
24
278◦ 24 ′
◦
26
282◦ 10 ′
◦
28
285◦ 56 ′
◦
30
289◦ 42 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
α
005◦ 56 ′
006◦ 22 ′
006◦ 51 ′
007◦ 22 ′
007◦ 57 ′
008◦ 36 ′
009◦ 19 ′
010◦ 07 ′
011◦ 02 ′
012◦ 04 ′
013◦ 14 ′
014◦ 33 ′
016◦ 02 ′
017◦ 42 ′
019◦ 34 ′
021◦ 39 ′
Sagittarius
λ
α
◦
00
289◦ 42 ′
◦
02
293◦ 27 ′
◦
04
297◦ 10 ′
◦
06
300◦ 52 ′
◦
08
304◦ 31 ′
◦
10
308◦ 08 ′
◦
12
311◦ 41 ′
◦
14
315◦ 10 ′
◦
16
318◦ 34 ′
◦
18
321◦ 51 ′
◦
20
325◦ 01 ′
◦
22
328◦ 02 ′
◦
24
330◦ 54 ′
◦
26
333◦ 35 ′
◦
28
336◦ 04 ′
◦
30
338◦ 21 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
α
021◦ 39 ′
023◦ 56 ′
026◦ 25 ′
029◦ 06 ′
031◦ 58 ′
034◦ 59 ′
038◦ 09 ′
041◦ 26 ′
044◦ 50 ′
048◦ 19 ′
051◦ 52 ′
055◦ 29 ′
059◦ 08 ′
062◦ 50 ′
066◦ 33 ′
070◦ 18 ′
Capricorn
λ
α
◦
00
338◦ 21 ′
◦
02
340◦ 26 ′
◦
04
342◦ 18 ′
◦
06
343◦ 58 ′
◦
08
345◦ 27 ′
◦
10
346◦ 46 ′
◦
12
347◦ 56 ′
◦
14
348◦ 58 ′
◦
16
349◦ 53 ′
◦
18
350◦ 41 ′
◦
20
351◦ 24 ′
◦
22
352◦ 03 ′
◦
24
352◦ 38 ′
◦
26
353◦ 09 ′
◦
28
353◦ 38 ′
◦
30
354◦ 04 ′
Leo
α
070◦ 18 ′
074◦ 04 ′
077◦ 50 ′
081◦ 36 ′
085◦ 22 ′
089◦ 08 ′
092◦ 54 ′
096◦ 39 ′
100◦ 24 ′
104◦ 08 ′
107◦ 52 ′
111◦ 35 ′
115◦ 17 ′
118◦ 58 ′
122◦ 38 ′
126◦ 18 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
126◦ 18 ′
129◦ 57 ′
133◦ 35 ′
137◦ 12 ′
140◦ 49 ′
144◦ 25 ′
148◦ 00 ′
151◦ 35 ′
155◦ 09 ′
158◦ 43 ′
162◦ 16 ′
165◦ 50 ′
169◦ 22 ′
172◦ 55 ′
176◦ 28 ′
180◦ 00 ′
Aquarius
λ
α
◦
00
354◦ 04 ′
◦
02
354◦ 27 ′
◦
04
354◦ 49 ′
◦
06
355◦ 09 ′
◦
08
355◦ 28 ′
◦
10
355◦ 45 ′
◦
12
356◦ 01 ′
◦
14
356◦ 16 ′
◦
16
356◦ 31 ′
◦
18
356◦ 44 ′
◦
20
356◦ 57 ′
◦
22
357◦ 09 ′
◦
24
357◦ 21 ′
◦
26
357◦ 32 ′
◦
28
357◦ 43 ′
◦
30
357◦ 53 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
357◦ 53 ′
358◦ 03 ′
358◦ 12 ′
358◦ 22 ′
358◦ 31 ′
358◦ 39 ′
358◦ 48 ′
358◦ 56 ′
359◦ 05 ′
359◦ 13 ′
359◦ 21 ′
359◦ 29 ′
359◦ 37 ′
359◦ 44 ′
359◦ 52 ′
360◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 2.13: Oblique ascensions of the ecliptic for latitude +65◦ .
Spherical Astronomy
49
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
360◦ 00 ′
359◦ 39 ′
359◦ 18 ′
358◦ 57 ′
358◦ 35 ′
358◦ 14 ′
357◦ 52 ′
357◦ 29 ′
357◦ 06 ′
356◦ 43 ′
356◦ 18 ′
355◦ 53 ′
355◦ 27 ′
355◦ 00 ′
354◦ 32 ′
354◦ 02 ′
Taurus
λ
α
00◦
354◦ 02 ′
02◦
353◦ 30 ′
◦
04
352◦ 57 ′
◦
06
352◦ 21 ′
◦
08
351◦ 42 ′
◦
10
351◦ 00 ′
◦
12
350◦ 14 ′
◦
14
349◦ 23 ′
◦
16
348◦ 26 ′
◦
18
347◦ 20 ′
◦
20
346◦ 04 ′
◦
22
344◦ 32 ′
◦
24
342◦ 37 ′
◦
26
340◦ 03 ′
◦
28
335◦ 55 ′
◦
′
29 19
327◦ 07 ′
Gemini
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Cancer
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
λ
00◦ 41 ′
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Leo
α
032◦ 54 ′
044◦ 26 ′
052◦ 41 ′
059◦ 22 ′
065◦ 22 ′
070◦ 57 ′
076◦ 15 ′
081◦ 21 ′
086◦ 18 ′
091◦ 07 ′
095◦ 49 ′
100◦ 26 ′
104◦ 58 ′
109◦ 27 ′
113◦ 51 ′
118◦ 13 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
118◦ 13 ′
122◦ 31 ′
126◦ 47 ′
131◦ 01 ′
135◦ 13 ′
139◦ 22 ′
143◦ 31 ′
147◦ 37 ′
151◦ 43 ′
155◦ 47 ′
159◦ 51 ′
163◦ 54 ′
167◦ 56 ′
171◦ 57 ′
175◦ 59 ′
180◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
184◦ 01 ′
188◦ 03 ′
192◦ 04 ′
196◦ 06 ′
200◦ 09 ′
204◦ 13 ′
208◦ 17 ′
212◦ 23 ′
216◦ 29 ′
220◦ 38 ′
224◦ 47 ′
228◦ 59 ′
233◦ 13 ′
237◦ 29 ′
241◦ 47 ′
Scorpio
λ
α
◦
00
241◦ 47 ′
◦
02
246◦ 09 ′
◦
04
250◦ 33 ′
◦
06
255◦ 02 ′
◦
08
259◦ 34 ′
◦
10
264◦ 11 ′
◦
12
268◦ 53 ′
◦
14
273◦ 42 ′
◦
16
278◦ 39 ′
◦
18
283◦ 45 ′
◦
20
289◦ 03 ′
◦
22
294◦ 38 ′
◦
24
300◦ 38 ′
◦
26
307◦ 19 ′
◦
28
315◦ 34 ′
◦
′
29 19
327◦ 07 ′
Sagittarius
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Capricorn
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Aquarius
λ
α
◦
′
00 41
032◦ 52 ′
◦
02
024◦ 05 ′
◦
04
019◦ 57 ′
◦
06
017◦ 23 ′
◦
08
015◦ 28 ′
◦
10
013◦ 56 ′
◦
12
012◦ 40 ′
◦
14
011◦ 34 ′
◦
16
010◦ 37 ′
◦
18
009◦ 46 ′
◦
20
009◦ 00 ′
◦
22
008◦ 18 ′
◦
24
007◦ 39 ′
◦
26
007◦ 03 ′
◦
28
006◦ 30 ′
◦
30
005◦ 58 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
005◦ 58 ′
005◦ 28 ′
005◦ 00 ′
004◦ 33 ′
004◦ 07 ′
003◦ 42 ′
003◦ 17 ′
002◦ 54 ′
002◦ 31 ′
002◦ 08 ′
001◦ 46 ′
001◦ 25 ′
001◦ 03 ′
000◦ 42 ′
000◦ 21 ′
000◦ 00 ′
Table 2.14: Oblique ascensions of the ecliptic for latitude +70◦ .
50
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
α
360◦ 00 ′
358◦ 52 ′
357◦ 44 ′
356◦ 35 ′
355◦ 25 ′
354◦ 13 ′
353◦ 00 ′
351◦ 44 ′
350◦ 26 ′
349◦ 05 ′
347◦ 39 ′
346◦ 08 ′
344◦ 30 ′
342◦ 45 ′
340◦ 50 ′
338◦ 42 ′
Taurus
λ
α
00◦
338◦ 42 ′
02◦
336◦ 16 ′
◦
04
333◦ 24 ′
◦
06
329◦ 54 ′
◦
08
325◦ 10 ′
◦
10
316◦ 55 ′
◦
′
11 36
308◦ 12 ′
◦
14
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Gemini
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Cancer
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
19◦ 24 ′
20◦
22◦
24◦
26◦
28◦
30◦
Leo
α
—
—
—
—
—
—
—
—
—
051◦ 50 ′
061◦ 44 ′
073◦ 54 ′
082◦ 31 ′
089◦ 54 ′
096◦ 36 ′
102◦ 52 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
α
102◦ 52 ′
108◦ 49 ′
114◦ 32 ′
120◦ 04 ′
125◦ 27 ′
130◦ 43 ′
135◦ 52 ′
140◦ 57 ′
145◦ 58 ′
150◦ 56 ′
155◦ 50 ′
160◦ 43 ′
165◦ 34 ′
170◦ 23 ′
175◦ 12 ′
180◦ 00 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
α
180◦ 00 ′
184◦ 48 ′
189◦ 37 ′
194◦ 26 ′
199◦ 17 ′
204◦ 10 ′
209◦ 04 ′
214◦ 02 ′
219◦ 03 ′
224◦ 08 ′
229◦ 17 ′
234◦ 33 ′
239◦ 56 ′
245◦ 28 ′
251◦ 11 ′
257◦ 08 ′
Scorpio
λ
α
◦
00
257◦ 08 ′
◦
02
263◦ 24 ′
◦
04
270◦ 06 ′
◦
06
277◦ 29 ′
◦
08
286◦ 06 ′
◦
10
298◦ 16 ′
◦
′
11 36
308◦ 12 ′
◦
14
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Sagittarius
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Capricorn
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Aquarius
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
19◦ 24 ′ 051◦ 50 ′
20◦
043◦ 05 ′
◦
22
034◦ 50 ′
◦
24
030◦ 06 ′
◦
26
026◦ 36 ′
◦
28
023◦ 44 ′
◦
30
021◦ 18 ′
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
α
021◦ 18 ′
019◦ 10 ′
017◦ 15 ′
015◦ 30 ′
013◦ 52 ′
012◦ 21 ′
010◦ 55 ′
009◦ 34 ′
008◦ 16 ′
007◦ 00 ′
005◦ 47 ′
004◦ 35 ′
003◦ 25 ′
002◦ 16 ′
001◦ 08 ′
000◦ 00 ′
Table 2.15: Oblique ascensions of the ecliptic for latitude +75◦ .
Spherical Astronomy
51
Aries
λ
α
00◦
360◦ 00 ′
02◦
357◦ 19 ′
◦
04
354◦ 37 ′
◦
06
351◦ 52 ′
◦
08
349◦ 02 ′
◦
10
346◦ 04 ′
◦
12
342◦ 58 ′
◦
14
339◦ 39 ′
◦
16
336◦ 02 ′
◦
18
331◦ 59 ′
◦
20
327◦ 20 ′
◦
22
321◦ 39 ′
◦
24
313◦ 51 ′
◦
′
25 53
294◦ 01 ′
◦
28
—
30◦
—
Taurus
λ
α
00◦ —
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Gemini
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Cancer
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Leo
λ
α
00◦ —
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Virgo
λ
α
00◦
—
02◦
—
04◦ 07 ′ 066◦ 01 ′
06◦
089◦ 25 ′
◦
08
100◦ 58 ′
◦
10
110◦ 24 ′
◦
12
118◦ 47 ′
◦
14
126◦ 33 ′
◦
16
133◦ 52 ′
◦
18
140◦ 54 ′
◦
20
147◦ 42 ′
◦
22
154◦ 20 ′
◦
24
160◦ 51 ′
◦
26
167◦ 16 ′
◦
28
173◦ 39 ′
◦
30
180◦ 00 ′
Libra
λ
α
◦
00
180◦ 00 ′
◦
02
186◦ 21 ′
◦
04
192◦ 44 ′
◦
06
199◦ 09 ′
◦
08
205◦ 40 ′
◦
10
212◦ 18 ′
◦
12
219◦ 06 ′
◦
14
226◦ 08 ′
◦
16
233◦ 27 ′
◦
18
241◦ 13 ′
◦
20
249◦ 36 ′
◦
22
259◦ 02 ′
◦
24
270◦ 35 ′
◦
′
25 53
294◦ 01 ′
◦
28
—
30◦
—
Scorpio
λ
α
◦
00
—
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Sagittarius
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Capricorn
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Aquarius
λ
α
◦
00
—
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Pisces
λ
α
◦
00
—
02◦
—
04◦ 07 ′ 066◦ 01 ′
06◦
046◦ 09 ′
◦
08
038◦ 21 ′
◦
10
032◦ 40 ′
◦
12
028◦ 01 ′
◦
14
023◦ 58 ′
◦
16
020◦ 21 ′
◦
18
017◦ 02 ′
◦
20
013◦ 56 ′
◦
22
010◦ 58 ′
◦
24
008◦ 08 ′
◦
26
005◦ 23 ′
◦
28
002◦ 41 ′
◦
30
000◦ 00 ′
Table 2.16: Oblique ascensions of the ecliptic for latitude +80◦ .
52
MODERN ALMAGEST
Aries
λ
α
00◦
360◦ 00 ′
02◦
352◦ 42 ′
◦
04
345◦ 11 ′
◦
06
337◦ 07 ′
◦
08
328◦ 02 ′
◦
10
316◦ 53 ′
◦
12
299◦ 32 ′
◦
′
12 40
281◦ 40 ′
◦
16
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Taurus
λ
α
00◦ —
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Gemini
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Cancer
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Leo
λ
α
00◦ —
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Virgo
λ
α
00◦
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
17◦ 20 ′ 078◦ 23 ′
18◦
097◦ 28 ′
◦
20
118◦ 31 ′
◦
22
133◦ 20 ′
◦
24
146◦ 06 ′
◦
26
157◦ 50 ′
◦
28
169◦ 02 ′
◦
30
180◦ 00 ′
Libra
λ
α
◦
00
180◦ 00 ′
◦
02
190◦ 58 ′
◦
04
202◦ 10 ′
◦
06
213◦ 54 ′
◦
08
226◦ 40 ′
◦
10
241◦ 29 ′
◦
12
262◦ 32 ′
◦
′
12 40
281◦ 40 ′
◦
16
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Scorpio
λ
α
◦
00
—
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Sagittarius
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Capricorn
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
16◦
—
18◦
—
20◦
—
22◦
—
24◦
—
26◦
—
28◦
—
30◦
—
Aquarius
λ
α
◦
00
—
02◦ —
04◦ —
06◦ —
08◦ —
10◦ —
12◦ —
14◦ —
16◦ —
18◦ —
20◦ —
22◦ —
24◦ —
26◦ —
28◦ —
30◦ —
Pisces
λ
α
◦
00
—
02◦
—
04◦
—
06◦
—
08◦
—
10◦
—
12◦
—
14◦
—
17◦ 20 ′ 078◦ 23 ′
18◦
060◦ 28 ′
◦
20
043◦ 07 ′
◦
22
031◦ 58 ′
◦
24
022◦ 53 ′
◦
26
014◦ 49 ′
◦
28
007◦ 18 ′
◦
30
000◦ 00 ′
Table 2.17: Oblique ascensions of the ecliptic for latitude +85◦ .
Spherical Astronomy
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
Aries
90◦ 00 ′ 314◦ 53 ′
75◦ 00 ′ 156◦ 34 ′
60◦ 00 ′ 156◦ 34 ′
45◦ 00 ′ 156◦ 34 ′
30◦ 00 ′ 156◦ 34 ′
15◦ 00 ′ 156◦ 34 ′
00◦ 00 ′ 156◦ 34 ′
Taurus
78◦ 32 ′ 249◦ 26 ′
71◦ 12 ′ 196◦ 00 ′
58◦ 04 ′ 178◦ 26 ′
43◦ 52 ′ 170◦ 40 ′
29◦ 20 ′ 165◦ 58 ′
14◦ 42 ′ 162◦ 29 ′
00◦ 00 ′ 159◦ 26 ′
Gemini
69◦ 51 ′ 257◦ 46 ′
65◦ 04 ′ 219◦ 53 ′
54◦ 24 ′ 198◦ 35 ′
41◦ 36 ′ 186◦ 47 ′
28◦ 00 ′ 179◦ 01 ′
14◦ 04 ′ 173◦ 03 ′
00◦ 00 ′ 167◦ 46 ′
Cancer
69◦ 51 ′ 282◦ 14 ′
65◦ 04 ′ 244◦ 21 ′
54◦ 24 ′ 223◦ 03 ′
41◦ 36 ′ 211◦ 14 ′
28◦ 00 ′ 203◦ 28 ′
14◦ 04 ′ 197◦ 30 ′
00◦ 00 ′ 192◦ 14 ′
00◦ 00 ′ 192◦ 14 ′
Leo
69◦ 51 ′ 282◦ 14 ′
65◦ 04 ′ 244◦ 21 ′
54◦ 24 ′ 223◦ 03 ′
41◦ 36 ′ 211◦ 14 ′
28◦ 00 ′ 203◦ 28 ′
14◦ 04 ′ 197◦ 30 ′
00◦ 00 ′ 192◦ 14 ′
Virgo
78◦ 32 ′ 290◦ 34 ′
71◦ 12 ′ 237◦ 09 ′
58◦ 04 ′ 219◦ 35 ′
43◦ 52 ′ 211◦ 49 ′
29◦ 20 ′ 207◦ 07 ′
14◦ 42 ′ 203◦ 37 ′
00◦ 00 ′ 200◦ 34 ′
53
178◦ 15 ′
336◦ 34 ′
336◦ 34 ′
336◦ 34 ′
336◦ 34 ′
336◦ 34 ′
336◦ 34 ′
249◦ 26 ′
302◦ 51 ′
320◦ 25 ′
328◦ 11 ′
332◦ 53 ′
336◦ 23 ′
339◦ 26 ′
257◦ 46 ′
295◦ 39 ′
316◦ 57 ′
328◦ 46 ′
336◦ 32 ′
342◦ 30 ′
347◦ 46 ′
282◦ 14 ′
320◦ 07 ′
341◦ 25 ′
353◦ 13 ′
000◦ 59 ′
006◦ 57 ′
012◦ 14 ′
012◦ 14 ′
282◦ 14 ′
320◦ 07 ′
341◦ 25 ′
353◦ 13 ′
000◦ 59 ′
006◦ 57 ′
012◦ 14 ′
290◦ 34 ′
344◦ 00 ′
001◦ 34 ′
009◦ 20 ′
014◦ 02 ′
017◦ 31 ′
020◦ 34 ′
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
Libra
90◦ 00 ′ 045◦ 07 ′
75◦ 00 ′ 203◦ 26 ′
60◦ 00 ′ 203◦ 26 ′
45◦ 00 ′ 203◦ 26 ′
30◦ 00 ′ 203◦ 26 ′
15◦ 00 ′ 203◦ 26 ′
00◦ 00 ′ 203◦ 26 ′
Scorpio
78◦ 32 ′ 110◦ 34 ′
71◦ 12 ′ 164◦ 00 ′
58◦ 04 ′ 181◦ 34 ′
43◦ 52 ′ 189◦ 20 ′
29◦ 20 ′ 194◦ 02 ′
14◦ 42 ′ 197◦ 31 ′
00◦ 00 ′ 200◦ 34 ′
Sagittarius
69◦ 51 ′ 102◦ 14 ′
65◦ 04 ′ 140◦ 07 ′
54◦ 24 ′ 161◦ 25 ′
41◦ 36 ′ 173◦ 13 ′
28◦ 00 ′ 180◦ 59 ′
14◦ 04 ′ 186◦ 57 ′
00◦ 00 ′ 192◦ 14 ′
Capricorn
66◦ 34 ′ 090◦ 00 ′
62◦ 24 ′ 123◦ 58 ′
52◦ 37 ′ 145◦ 26 ′
40◦ 27 ′ 158◦ 19 ′
27◦ 18 ′ 167◦ 04 ′
13◦ 44 ′ 173◦ 55 ′
00◦ 00 ′ 180◦ 00 ′
Aquarius
69◦ 51 ′ 077◦ 46 ′
65◦ 04 ′ 115◦ 39 ′
54◦ 24 ′ 136◦ 57 ′
41◦ 36 ′ 148◦ 46 ′
28◦ 00 ′ 156◦ 32 ′
14◦ 04 ′ 162◦ 30 ′
00◦ 00 ′ 167◦ 46 ′
Pisces
78◦ 32 ′ 069◦ 26 ′
71◦ 12 ′ 122◦ 51 ′
58◦ 04 ′ 140◦ 25 ′
43◦ 52 ′ 148◦ 11 ′
29◦ 20 ′ 152◦ 53 ′
14◦ 42 ′ 156◦ 23 ′
00◦ 00 ′ 159◦ 26 ′
181◦ 45 ′
023◦ 26 ′
023◦ 26 ′
023◦ 26 ′
023◦ 26 ′
023◦ 26 ′
023◦ 26 ′
110◦ 34 ′
057◦ 09 ′
039◦ 35 ′
031◦ 49 ′
027◦ 07 ′
023◦ 37 ′
020◦ 34 ′
102◦ 14 ′
064◦ 21 ′
043◦ 03 ′
031◦ 14 ′
023◦ 28 ′
017◦ 30 ′
012◦ 14 ′
090◦ 00 ′
056◦ 02 ′
034◦ 34 ′
021◦ 41 ′
012◦ 56 ′
006◦ 05 ′
360◦ 00 ′
077◦ 46 ′
039◦ 53 ′
018◦ 35 ′
006◦ 47 ′
359◦ 01 ′
353◦ 03 ′
347◦ 46 ′
069◦ 26 ′
016◦ 00 ′
358◦ 26 ′
350◦ 40 ′
345◦ 58 ′
342◦ 29 ′
339◦ 26 ′
Table 2.18: Ecliptic altitude and parallactic angle for latitude 0◦ .
54
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:08
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:14
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:17
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:14
00:00
01:00
02:00
03:00
04:00
05:00
MODERN ALMAGEST
Aries
80◦ 00 ′ 066◦ 34 ′
72◦ 02 ′ 122◦ 18 ′
58◦ 32 ′ 137◦ 08 ′
44◦ 08 ′ 142◦ 34 ′
29◦ 30 ′ 145◦ 03 ′
14◦ 46 ′ 146◦ 13 ′
00◦ 00 ′ 146◦ 34 ′
Taurus
88◦ 32 ′ 249◦ 26 ′
75◦ 11 ′ 163◦ 41 ′
60◦ 30 ′ 159◦ 21 ′
45◦ 48 ′ 156◦ 49 ′
31◦ 08 ′ 154◦ 35 ′
16◦ 31 ′ 152◦ 16 ′
01◦ 59 ′ 149◦ 37 ′
00◦ 00 ′ 149◦ 13 ′
Gemini
79◦ 51 ′ 257◦ 46 ′
72◦ 20 ′ 200◦ 37 ′
59◦ 22 ′ 182◦ 38 ′
45◦ 32 ′ 174◦ 04 ′
31◦ 28 ′ 168◦ 13 ′
17◦ 24 ′ 163◦ 15 ′
03◦ 26 ′ 158◦ 22 ′
00◦ 00 ′ 157◦ 07 ′
Cancer
76◦ 34 ′ 270◦ 00 ′
70◦ 22 ′ 220◦ 40 ′
58◦ 23 ′ 200◦ 04 ′
45◦ 04 ′ 189◦ 35 ′
31◦ 23 ′ 182◦ 27 ′
17◦ 38 ′ 176◦ 31 ′
03◦ 58 ′ 170◦ 49 ′
00◦ 00 ′ 169◦ 05 ′
Leo
79◦ 51 ′ 282◦ 14 ′
72◦ 20 ′ 225◦ 05 ′
59◦ 22 ′ 207◦ 06 ′
45◦ 32 ′ 198◦ 31 ′
31◦ 28 ′ 192◦ 40 ′
17◦ 24 ′ 187◦ 42 ′
03◦ 26 ′ 182◦ 50 ′
00◦ 00 ′ 181◦ 34 ′
Virgo
88◦ 32 ′ 290◦ 34 ′
75◦ 11 ′ 204◦ 50 ′
60◦ 30 ′ 200◦ 30 ′
45◦ 48 ′ 197◦ 58 ′
31◦ 08 ′ 195◦ 44 ′
16◦ 31 ′ 193◦ 25 ′
066◦ 34 ′
010◦ 50 ′
356◦ 00 ′
350◦ 34 ′
348◦ 05 ′
346◦ 55 ′
346◦ 34 ′
249◦ 26 ′
335◦ 10 ′
339◦ 30 ′
342◦ 02 ′
344◦ 16 ′
346◦ 35 ′
349◦ 14 ′
349◦ 38 ′
257◦ 46 ′
314◦ 55 ′
332◦ 54 ′
341◦ 29 ′
347◦ 20 ′
352◦ 18 ′
357◦ 10 ′
358◦ 26 ′
270◦ 00 ′
319◦ 20 ′
339◦ 56 ′
350◦ 25 ′
357◦ 33 ′
003◦ 29 ′
009◦ 11 ′
010◦ 55 ′
282◦ 14 ′
339◦ 23 ′
357◦ 22 ′
005◦ 56 ′
011◦ 47 ′
016◦ 45 ′
021◦ 38 ′
022◦ 53 ′
290◦ 34 ′
016◦ 19 ′
020◦ 39 ′
023◦ 11 ′
025◦ 25 ′
027◦ 44 ′
06:00
06:08
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
05:51
00:00
01:00
02:00
03:00
04:00
05:00
05:45
00:00
01:00
02:00
03:00
04:00
05:00
05:42
00:00
01:00
02:00
03:00
04:00
05:00
05:45
00:00
01:00
02:00
03:00
04:00
05:00
05:51
01◦ 59 ′
00◦ 00 ′
190◦ 46 ′
190◦ 22 ′
Libra
80◦ 00 ′ 113◦ 26 ′
72◦ 02 ′ 169◦ 10 ′
58◦ 32 ′ 184◦ 00 ′
44◦ 08 ′ 189◦ 26 ′
29◦ 30 ′ 191◦ 55 ′
14◦ 46 ′ 193◦ 05 ′
00◦ 00 ′ 193◦ 26 ′
Scorpio
68◦ 32 ′ 110◦ 34 ′
63◦ 52 ′ 145◦ 55 ′
53◦ 15 ′ 165◦ 58 ′
40◦ 23 ′ 176◦ 40 ′
26◦ 37 ′ 183◦ 07 ′
12◦ 26 ′ 187◦ 30 ′
00◦ 00 ′ 190◦ 22 ′
Sagittarius
59◦ 51 ′ 102◦ 14 ′
56◦ 26 ′ 129◦ 41 ′
47◦ 48 ′ 149◦ 23 ′
36◦ 26 ′ 162◦ 11 ′
23◦ 44 ′ 170◦ 55 ′
10◦ 20 ′ 177◦ 27 ′
00◦ 00 ′ 181◦ 34 ′
Capricorn
56◦ 34 ′ 090◦ 00 ′
53◦ 29 ′ 115◦ 22 ′
45◦ 31 ′ 134◦ 39 ′
34◦ 44 ′ 147◦ 56 ′
22◦ 30 ′ 157◦ 24 ′
09◦ 29 ′ 164◦ 40 ′
00◦ 00 ′ 169◦ 05 ′
Aquarius
59◦ 51 ′ 077◦ 46 ′
56◦ 26 ′ 105◦ 13 ′
47◦ 48 ′ 124◦ 55 ′
36◦ 26 ′ 137◦ 43 ′
23◦ 44 ′ 146◦ 28 ′
10◦ 20 ′ 153◦ 00 ′
00◦ 00 ′ 157◦ 07 ′
Pisces
68◦ 32 ′ 069◦ 26 ′
63◦ 52 ′ 104◦ 47 ′
53◦ 15 ′ 124◦ 49 ′
40◦ 23 ′ 135◦ 31 ′
26◦ 37 ′ 141◦ 59 ′
12◦ 26 ′ 146◦ 21 ′
00◦ 00 ′ 149◦ 13 ′
030◦ 23 ′
030◦ 47 ′
113◦ 26 ′
057◦ 42 ′
042◦ 52 ′
037◦ 26 ′
034◦ 57 ′
033◦ 47 ′
033◦ 26 ′
110◦ 34 ′
075◦ 13 ′
055◦ 11 ′
044◦ 29 ′
038◦ 01 ′
033◦ 39 ′
030◦ 47 ′
102◦ 14 ′
074◦ 47 ′
055◦ 05 ′
042◦ 17 ′
033◦ 32 ′
027◦ 00 ′
022◦ 53 ′
090◦ 00 ′
064◦ 38 ′
045◦ 21 ′
032◦ 04 ′
022◦ 36 ′
015◦ 20 ′
010◦ 55 ′
077◦ 46 ′
050◦ 19 ′
030◦ 37 ′
017◦ 49 ′
009◦ 05 ′
002◦ 33 ′
358◦ 26 ′
069◦ 26 ′
034◦ 05 ′
014◦ 02 ′
003◦ 20 ′
356◦ 53 ′
352◦ 30 ′
349◦ 38 ′
Table 2.19: Ecliptic altitude and parallactic angle for latitude +10◦ .
Spherical Astronomy
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:16
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:30
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:36
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:30
00:00
01:00
02:00
03:00
04:00
05:00
Aries
70◦ 00 ′ 066◦ 34 ′
65◦ 11 ′ 101◦ 59 ′
54◦ 28 ′ 120◦ 31 ′
41◦ 38 ′ 129◦ 20 ′
28◦ 01 ′ 133◦ 46 ′
14◦ 05 ′ 135◦ 55 ′
00◦ 00 ′ 136◦ 34 ′
Taurus
81◦ 28 ′ 069◦ 26 ′
73◦ 15 ′ 126◦ 58 ′
59◦ 57 ′ 139◦ 10 ′
45◦ 59 ′ 142◦ 26 ′
31◦ 54 ′ 142◦ 53 ′
17◦ 50 ′ 141◦ 53 ′
03◦ 54 ′ 139◦ 48 ′
00◦ 00 ′ 139◦ 00 ′
Gemini
89◦ 51 ′ 257◦ 46 ′
75◦ 55 ′ 165◦ 46 ′
61◦ 52 ′ 162◦ 48 ′
47◦ 52 ′ 159◦ 52 ′
33◦ 59 ′ 156◦ 42 ′
20◦ 15 ′ 153◦ 07 ′
06◦ 46 ′ 148◦ 54 ′
00◦ 00 ′ 146◦ 24 ′
Cancer
86◦ 34 ′ 270◦ 00 ′
75◦ 39 ′ 190◦ 58 ′
61◦ 58 ′ 181◦ 12 ′
48◦ 13 ′ 175◦ 44 ′
34◦ 33 ′ 171◦ 08 ′
21◦ 03 ′ 166◦ 33 ′
07◦ 49 ′ 161◦ 32 ′
00◦ 00 ′ 158◦ 07 ′
Leo
89◦ 51 ′ 282◦ 14 ′
75◦ 55 ′ 190◦ 14 ′
61◦ 52 ′ 187◦ 16 ′
47◦ 52 ′ 184◦ 19 ′
33◦ 59 ′ 181◦ 09 ′
20◦ 15 ′ 177◦ 34 ′
06◦ 46 ′ 173◦ 22 ′
00◦ 00 ′ 170◦ 52 ′
Virgo
81◦ 28 ′ 110◦ 34 ′
73◦ 15 ′ 168◦ 07 ′
59◦ 57 ′ 180◦ 19 ′
45◦ 59 ′ 183◦ 35 ′
31◦ 54 ′ 184◦ 02 ′
17◦ 50 ′ 183◦ 02 ′
55
066◦ 34 ′
031◦ 09 ′
012◦ 37 ′
003◦ 48 ′
359◦ 22 ′
357◦ 13 ′
356◦ 34 ′
069◦ 26 ′
011◦ 53 ′
359◦ 41 ′
356◦ 25 ′
355◦ 58 ′
356◦ 58 ′
359◦ 03 ′
359◦ 51 ′
257◦ 46 ′
349◦ 46 ′
352◦ 44 ′
355◦ 41 ′
358◦ 51 ′
002◦ 26 ′
006◦ 38 ′
009◦ 08 ′
270◦ 00 ′
349◦ 02 ′
358◦ 48 ′
004◦ 16 ′
008◦ 52 ′
013◦ 27 ′
018◦ 28 ′
021◦ 53 ′
282◦ 14 ′
014◦ 14 ′
017◦ 12 ′
020◦ 08 ′
023◦ 18 ′
026◦ 53 ′
031◦ 06 ′
033◦ 36 ′
110◦ 34 ′
053◦ 02 ′
040◦ 50 ′
037◦ 34 ′
037◦ 07 ′
038◦ 07 ′
06:00
06:16
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
05:43
00:00
01:00
02:00
03:00
04:00
05:00
05:29
00:00
01:00
02:00
03:00
04:00
05:00
05:23
00:00
01:00
02:00
03:00
04:00
05:00
05:29
00:00
01:00
02:00
03:00
04:00
05:00
05:43
03◦ 54 ′
00◦ 00 ′
180◦ 57 ′
180◦ 09 ′
Libra
70◦ 00 ′ 113◦ 26 ′
65◦ 11 ′ 148◦ 51 ′
54◦ 28 ′ 167◦ 23 ′
41◦ 38 ′ 176◦ 12 ′
28◦ 01 ′ 180◦ 38 ′
14◦ 05 ′ 182◦ 47 ′
00◦ 00 ′ 183◦ 26 ′
Scorpio
58◦ 32 ′ 110◦ 34 ′
55◦ 14 ′ 135◦ 49 ′
46◦ 51 ′ 153◦ 58 ′
35◦ 41 ′ 165◦ 27 ′
23◦ 06 ′ 172◦ 48 ′
09◦ 48 ′ 177◦ 40 ′
00◦ 00 ′ 180◦ 09 ′
Sagittarius
49◦ 51 ′ 102◦ 14 ′
47◦ 15 ′ 123◦ 13 ′
40◦ 15 ′ 140◦ 14 ′
30◦ 24 ′ 152◦ 37 ′
18◦ 52 ′ 161◦ 33 ′
06◦ 21 ′ 168◦ 11 ′
00◦ 00 ′ 170◦ 52 ′
Capricorn
46◦ 34 ′ 090◦ 00 ′
44◦ 10 ′ 109◦ 49 ′
37◦ 38 ′ 126◦ 24 ′
28◦ 16 ′ 138◦ 59 ′
17◦ 10 ′ 148◦ 24 ′
05◦ 00 ′ 155◦ 40 ′
00◦ 00 ′ 158◦ 07 ′
Aquarius
49◦ 51 ′ 077◦ 46 ′
47◦ 15 ′ 098◦ 46 ′
40◦ 15 ′ 115◦ 46 ′
30◦ 24 ′ 128◦ 10 ′
18◦ 52 ′ 137◦ 05 ′
06◦ 21 ′ 143◦ 44 ′
00◦ 00 ′ 146◦ 24 ′
Pisces
58◦ 32 ′ 069◦ 26 ′
55◦ 14 ′ 094◦ 41 ′
46◦ 51 ′ 112◦ 49 ′
35◦ 41 ′ 124◦ 18 ′
23◦ 06 ′ 131◦ 39 ′
09◦ 48 ′ 136◦ 31 ′
00◦ 00 ′ 139◦ 00 ′
040◦ 12 ′
041◦ 00 ′
113◦ 26 ′
078◦ 01 ′
059◦ 29 ′
050◦ 40 ′
046◦ 14 ′
044◦ 05 ′
043◦ 26 ′
110◦ 34 ′
085◦ 19 ′
067◦ 11 ′
055◦ 42 ′
048◦ 21 ′
043◦ 29 ′
041◦ 00 ′
102◦ 14 ′
081◦ 14 ′
064◦ 14 ′
051◦ 50 ′
042◦ 55 ′
036◦ 16 ′
033◦ 36 ′
090◦ 00 ′
070◦ 11 ′
053◦ 36 ′
041◦ 01 ′
031◦ 36 ′
024◦ 20 ′
021◦ 53 ′
077◦ 46 ′
056◦ 47 ′
039◦ 46 ′
027◦ 23 ′
018◦ 27 ′
011◦ 49 ′
009◦ 08 ′
069◦ 26 ′
044◦ 11 ′
026◦ 02 ′
014◦ 33 ′
007◦ 12 ′
002◦ 20 ′
359◦ 51 ′
Table 2.20: Ecliptic altitude and parallactic angle for latitude +20◦ .
56
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:26
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:48
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:57
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:48
00:00
01:00
02:00
03:00
04:00
05:00
MODERN ALMAGEST
Aries
60◦ 00 ′ 066◦ 34 ′
56◦ 46 ′ 090◦ 43 ′
48◦ 35 ′ 107◦ 28 ′
37◦ 46 ′ 117◦ 20 ′
25◦ 40 ′ 122◦ 53 ′
12◦ 57 ′ 125◦ 42 ′
00◦ 00 ′ 126◦ 34 ′
Taurus
71◦ 28 ′ 069◦ 26 ′
66◦ 49 ′ 104◦ 08 ′
56◦ 33 ′ 121◦ 13 ′
44◦ 24 ′ 128◦ 24 ′
31◦ 35 ′ 131◦ 07 ′
18◦ 36 ′ 131◦ 23 ′
05◦ 42 ′ 129◦ 55 ′
00◦ 00 ′ 128◦ 45 ′
Gemini
80◦ 09 ′ 077◦ 46 ′
73◦ 15 ′ 128◦ 48 ′
61◦ 12 ′ 141◦ 47 ′
48◦ 20 ′ 144◦ 53 ′
35◦ 22 ′ 144◦ 39 ′
22◦ 30 ′ 142◦ 39 ′
09◦ 55 ′ 139◦ 19 ′
00◦ 00 ′ 135◦ 36 ′
Cancer
83◦ 26 ′ 090◦ 00 ′
75◦ 06 ′ 150◦ 38 ′
62◦ 30 ′ 159◦ 40 ′
49◦ 32 ′ 160◦ 38 ′
36◦ 36 ′ 159◦ 05 ′
23◦ 52 ′ 156◦ 10 ′
11◦ 28 ′ 152◦ 05 ′
00◦ 00 ′ 146◦ 59 ′
Leo
80◦ 09 ′ 102◦ 14 ′
73◦ 15 ′ 153◦ 15 ′
61◦ 12 ′ 166◦ 14 ′
48◦ 20 ′ 169◦ 20 ′
35◦ 22 ′ 169◦ 06 ′
22◦ 30 ′ 167◦ 06 ′
09◦ 55 ′ 163◦ 46 ′
00◦ 00 ′ 160◦ 03 ′
Virgo
71◦ 28 ′ 110◦ 34 ′
66◦ 49 ′ 145◦ 17 ′
56◦ 33 ′ 162◦ 22 ′
44◦ 24 ′ 169◦ 33 ′
31◦ 35 ′ 172◦ 16 ′
18◦ 36 ′ 172◦ 32 ′
066◦ 34 ′
042◦ 25 ′
025◦ 40 ′
015◦ 48 ′
010◦ 15 ′
007◦ 26 ′
006◦ 34 ′
069◦ 26 ′
034◦ 43 ′
017◦ 38 ′
010◦ 27 ′
007◦ 44 ′
007◦ 28 ′
008◦ 56 ′
010◦ 06 ′
077◦ 46 ′
026◦ 45 ′
013◦ 46 ′
010◦ 40 ′
010◦ 54 ′
012◦ 54 ′
016◦ 14 ′
019◦ 57 ′
090◦ 00 ′
029◦ 22 ′
020◦ 20 ′
019◦ 22 ′
020◦ 55 ′
023◦ 50 ′
027◦ 55 ′
033◦ 01 ′
102◦ 14 ′
051◦ 12 ′
038◦ 13 ′
035◦ 07 ′
035◦ 21 ′
037◦ 21 ′
040◦ 41 ′
044◦ 24 ′
110◦ 34 ′
075◦ 52 ′
058◦ 47 ′
051◦ 36 ′
048◦ 53 ′
048◦ 37 ′
06:00
06:26
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
05:33
00:00
01:00
02:00
03:00
04:00
05:00
05:11
00:00
01:00
02:00
03:00
04:00
05:00
05:02
00:00
01:00
02:00
03:00
04:00
05:00
05:11
00:00
01:00
02:00
03:00
04:00
05:00
05:33
05◦ 42 ′
00◦ 00 ′
171◦ 04 ′
169◦ 54 ′
Libra
60◦ 00 ′ 113◦ 26 ′
56◦ 46 ′ 137◦ 35 ′
48◦ 35 ′ 154◦ 20 ′
37◦ 46 ′ 164◦ 12 ′
25◦ 40 ′ 169◦ 45 ′
12◦ 57 ′ 172◦ 34 ′
00◦ 00 ′ 173◦ 26 ′
Scorpio
48◦ 32 ′ 110◦ 34 ′
46◦ 05 ′ 129◦ 26 ′
39◦ 28 ′ 144◦ 41 ′
30◦ 03 ′ 155◦ 36 ′
18◦ 58 ′ 163◦ 03 ′
06◦ 54 ′ 168◦ 00 ′
00◦ 00 ′ 169◦ 54 ′
Sagittarius
39◦ 51 ′ 102◦ 14 ′
37◦ 49 ′ 118◦ 43 ′
32◦ 08 ′ 132◦ 59 ′
23◦ 45 ′ 144◦ 13 ′
13◦ 33 ′ 152◦ 43 ′
02◦ 11 ′ 159◦ 04 ′
00◦ 00 ′ 160◦ 03 ′
Capricorn
36◦ 34 ′ 090◦ 00 ′
34◦ 40 ′ 105◦ 49 ′
29◦ 18 ′ 119◦ 46 ′
21◦ 17 ′ 131◦ 05 ′
11◦ 27 ′ 139◦ 56 ′
00◦ 23 ′ 146◦ 47 ′
00◦ 00 ′ 146◦ 59 ′
Aquarius
39◦ 51 ′ 077◦ 46 ′
37◦ 49 ′ 094◦ 15 ′
32◦ 08 ′ 108◦ 32 ′
23◦ 45 ′ 119◦ 46 ′
13◦ 33 ′ 128◦ 16 ′
02◦ 11 ′ 134◦ 37 ′
00◦ 00 ′ 135◦ 36 ′
Pisces
48◦ 32 ′ 069◦ 26 ′
46◦ 05 ′ 088◦ 17 ′
39◦ 28 ′ 103◦ 33 ′
30◦ 03 ′ 114◦ 27 ′
18◦ 58 ′ 121◦ 54 ′
06◦ 54 ′ 126◦ 51 ′
00◦ 00 ′ 128◦ 45 ′
050◦ 05 ′
051◦ 15 ′
113◦ 26 ′
089◦ 17 ′
072◦ 32 ′
062◦ 40 ′
057◦ 07 ′
054◦ 18 ′
053◦ 26 ′
110◦ 34 ′
091◦ 43 ′
076◦ 27 ′
065◦ 33 ′
058◦ 06 ′
053◦ 09 ′
051◦ 15 ′
102◦ 14 ′
085◦ 45 ′
071◦ 28 ′
060◦ 14 ′
051◦ 44 ′
045◦ 23 ′
044◦ 24 ′
090◦ 00 ′
074◦ 11 ′
060◦ 14 ′
048◦ 55 ′
040◦ 04 ′
033◦ 13 ′
033◦ 01 ′
077◦ 46 ′
061◦ 17 ′
047◦ 01 ′
035◦ 47 ′
027◦ 17 ′
020◦ 56 ′
019◦ 57 ′
069◦ 26 ′
050◦ 34 ′
035◦ 19 ′
024◦ 24 ′
016◦ 57 ′
012◦ 00 ′
010◦ 06 ′
Table 2.21: Ecliptic altitude and parallactic angle for latitude +30◦ .
Spherical Astronomy
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:39
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:11
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:25
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:11
00:00
01:00
02:00
Aries
50◦ 00 ′ 066◦ 34 ′
47◦ 44 ′ 083◦ 43 ′
41◦ 34 ′ 097◦ 21 ′
32◦ 48 ′ 106◦ 41 ′
22◦ 31 ′ 112◦ 28 ′
11◦ 26 ′ 115◦ 35 ′
00◦ 00 ′ 116◦ 34 ′
Taurus
61◦ 28 ′ 069◦ 26 ′
58◦ 32 ′ 091◦ 45 ′
51◦ 05 ′ 106◦ 59 ′
41◦ 12 ′ 115◦ 28 ′
30◦ 13 ′ 119◦ 34 ′
18◦ 47 ′ 120◦ 50 ′
07◦ 21 ′ 120◦ 00 ′
00◦ 00 ′ 118◦ 26 ′
Gemini
70◦ 09 ′ 077◦ 46 ′
66◦ 21 ′ 107◦ 24 ′
57◦ 35 ′ 123◦ 23 ′
46◦ 53 ′ 130◦ 11 ′
35◦ 31 ′ 132◦ 22 ′
24◦ 03 ′ 131◦ 54 ′
12◦ 47 ′ 129◦ 33 ′
02◦ 01 ′ 125◦ 32 ′
00◦ 00 ′ 124◦ 34 ′
Cancer
73◦ 26 ′ 090◦ 00 ′
69◦ 09 ′ 123◦ 52 ′
59◦ 48 ′ 139◦ 36 ′
48◦ 49 ′ 145◦ 21 ′
37◦ 23 ′ 146◦ 36 ′
25◦ 57 ′ 145◦ 23 ′
14◦ 49 ′ 142◦ 24 ′
04◦ 14 ′ 137◦ 54 ′
00◦ 00 ′ 135◦ 32 ′
Leo
70◦ 09 ′ 102◦ 14 ′
66◦ 21 ′ 131◦ 51 ′
57◦ 35 ′ 147◦ 50 ′
46◦ 53 ′ 154◦ 39 ′
35◦ 31 ′ 156◦ 49 ′
24◦ 03 ′ 156◦ 21 ′
12◦ 47 ′ 154◦ 00 ′
02◦ 01 ′ 150◦ 00 ′
00◦ 00 ′ 149◦ 01 ′
Virgo
61◦ 28 ′ 110◦ 34 ′
58◦ 32 ′ 132◦ 54 ′
51◦ 05 ′ 148◦ 08 ′
57
066◦ 34 ′
049◦ 25 ′
035◦ 47 ′
026◦ 27 ′
020◦ 40 ′
017◦ 33 ′
016◦ 34 ′
069◦ 26 ′
047◦ 06 ′
031◦ 52 ′
023◦ 23 ′
019◦ 17 ′
018◦ 01 ′
018◦ 51 ′
020◦ 25 ′
077◦ 46 ′
048◦ 09 ′
032◦ 10 ′
025◦ 21 ′
023◦ 11 ′
023◦ 39 ′
026◦ 00 ′
030◦ 00 ′
030◦ 59 ′
090◦ 00 ′
056◦ 08 ′
040◦ 24 ′
034◦ 39 ′
033◦ 24 ′
034◦ 37 ′
037◦ 36 ′
042◦ 06 ′
044◦ 28 ′
102◦ 14 ′
072◦ 36 ′
056◦ 37 ′
049◦ 49 ′
047◦ 38 ′
048◦ 06 ′
050◦ 27 ′
054◦ 28 ′
055◦ 26 ′
110◦ 34 ′
088◦ 15 ′
073◦ 01 ′
03:00
04:00
05:00
06:00
06:39
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
05:20
00:00
01:00
02:00
03:00
04:00
04:48
00:00
01:00
02:00
03:00
04:00
04:34
00:00
01:00
02:00
03:00
04:00
04:48
00:00
01:00
02:00
03:00
04:00
05:00
05:20
41◦ 12 ′
30◦ 13 ′
18◦ 47 ′
07◦ 21 ′
00◦ 00 ′
156◦ 37 ′
160◦ 43 ′
161◦ 59 ′
161◦ 09 ′
159◦ 35 ′
Libra
50◦ 00 ′ 113◦ 26 ′
47◦ 44 ′ 130◦ 35 ′
41◦ 34 ′ 144◦ 13 ′
32◦ 48 ′ 153◦ 33 ′
22◦ 31 ′ 159◦ 20 ′
11◦ 26 ′ 162◦ 27 ′
00◦ 00 ′ 163◦ 26 ′
Scorpio
38◦ 32 ′ 110◦ 34 ′
36◦ 41 ′ 124◦ 53 ′
31◦ 29 ′ 137◦ 16 ′
23◦ 46 ′ 146◦ 52 ′
14◦ 20 ′ 153◦ 47 ′
03◦ 49 ′ 158◦ 26 ′
00◦ 00 ′ 159◦ 35 ′
Sagittarius
29◦ 51 ′ 102◦ 14 ′
28◦ 15 ′ 115◦ 14 ′
23◦ 40 ′ 126◦ 57 ′
16◦ 41 ′ 136◦ 40 ′
07◦ 57 ′ 144◦ 17 ′
00◦ 00 ′ 149◦ 01 ′
Capricorn
26◦ 34 ′ 090◦ 00 ′
25◦ 03 ′ 102◦ 38 ′
20◦ 41 ′ 114◦ 10 ′
13◦ 58 ′ 123◦ 56 ′
05◦ 30 ′ 131◦ 48 ′
00◦ 00 ′ 135◦ 32 ′
Aquarius
29◦ 51 ′ 077◦ 46 ′
28◦ 15 ′ 090◦ 47 ′
23◦ 40 ′ 102◦ 30 ′
16◦ 41 ′ 112◦ 13 ′
07◦ 57 ′ 119◦ 50 ′
00◦ 00 ′ 124◦ 34 ′
Pisces
38◦ 32 ′ 069◦ 26 ′
36◦ 41 ′ 083◦ 44 ′
31◦ 29 ′ 096◦ 07 ′
23◦ 46 ′ 105◦ 43 ′
14◦ 20 ′ 112◦ 38 ′
03◦ 49 ′ 117◦ 18 ′
00◦ 00 ′ 118◦ 26 ′
064◦ 32 ′
060◦ 26 ′
059◦ 10 ′
060◦ 00 ′
061◦ 34 ′
113◦ 26 ′
096◦ 17 ′
082◦ 39 ′
073◦ 19 ′
067◦ 32 ′
064◦ 25 ′
063◦ 26 ′
110◦ 34 ′
096◦ 16 ′
083◦ 53 ′
074◦ 17 ′
067◦ 22 ′
062◦ 42 ′
061◦ 34 ′
102◦ 14 ′
089◦ 13 ′
077◦ 30 ′
067◦ 47 ′
060◦ 10 ′
055◦ 26 ′
090◦ 00 ′
077◦ 22 ′
065◦ 50 ′
056◦ 04 ′
048◦ 12 ′
044◦ 28 ′
077◦ 46 ′
064◦ 46 ′
053◦ 03 ′
043◦ 20 ′
035◦ 43 ′
030◦ 59 ′
069◦ 26 ′
055◦ 07 ′
042◦ 44 ′
033◦ 08 ′
026◦ 13 ′
021◦ 34 ′
020◦ 25 ′
Table 2.22: Ecliptic altitude and parallactic angle for latitude +40◦ .
58
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
06:55
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:43
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
08:04
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:43
00:00
01:00
MODERN ALMAGEST
Aries
40◦ 00 ′ 066◦ 34 ′
38◦ 23 ′ 078◦ 49 ′
33◦ 50 ′ 089◦ 20 ′
27◦ 02 ′ 097◦ 15 ′
18◦ 45 ′ 102◦ 34 ′
09◦ 35 ′ 105◦ 36 ′
00◦ 00 ′ 106◦ 34 ′
Taurus
51◦ 28 ′ 069◦ 26 ′
49◦ 32 ′ 084◦ 17 ′
44◦ 15 ′ 096◦ 05 ′
36◦ 43 ′ 103◦ 58 ′
27◦ 52 ′ 108◦ 27 ′
18◦ 23 ′ 110◦ 17 ′
08◦ 46 ′ 110◦ 00 ′
00◦ 00 ′ 108◦ 01 ′
Gemini
60◦ 09 ′ 077◦ 46 ′
57◦ 51 ′ 096◦ 00 ′
51◦ 51 ′ 109◦ 08 ′
43◦ 40 ′ 116◦ 42 ′
34◦ 26 ′ 120◦ 14 ′
24◦ 50 ′ 120◦ 57 ′
15◦ 18 ′ 119◦ 34 ′
06◦ 11 ′ 116◦ 25 ′
00◦ 00 ′ 113◦ 05 ′
Cancer
63◦ 26 ′ 090◦ 00 ′
60◦ 58 ′ 110◦ 03 ′
54◦ 38 ′ 123◦ 43 ′
46◦ 12 ′ 131◦ 02 ′
36◦ 50 ′ 134◦ 04 ′
27◦ 13 ′ 134◦ 17 ′
17◦ 44 ′ 132◦ 27 ′
08◦ 45 ′ 128◦ 55 ′
00◦ 34 ′ 123◦ 50 ′
00◦ 00 ′ 123◦ 24 ′
Leo
60◦ 09 ′ 102◦ 14 ′
57◦ 51 ′ 120◦ 27 ′
51◦ 51 ′ 133◦ 35 ′
43◦ 40 ′ 141◦ 10 ′
34◦ 26 ′ 144◦ 41 ′
24◦ 50 ′ 145◦ 24 ′
15◦ 18 ′ 144◦ 01 ′
06◦ 11 ′ 140◦ 52 ′
00◦ 00 ′ 137◦ 33 ′
Virgo
51◦ 28 ′ 110◦ 34 ′
49◦ 32 ′ 125◦ 26 ′
066◦ 34 ′
054◦ 19 ′
043◦ 48 ′
035◦ 53 ′
030◦ 34 ′
027◦ 32 ′
026◦ 34 ′
069◦ 26 ′
054◦ 34 ′
042◦ 46 ′
034◦ 53 ′
030◦ 24 ′
028◦ 34 ′
028◦ 51 ′
030◦ 50 ′
077◦ 46 ′
059◦ 33 ′
046◦ 25 ′
038◦ 50 ′
035◦ 19 ′
034◦ 36 ′
035◦ 59 ′
039◦ 08 ′
042◦ 27 ′
090◦ 00 ′
069◦ 57 ′
056◦ 17 ′
048◦ 58 ′
045◦ 56 ′
045◦ 43 ′
047◦ 33 ′
051◦ 05 ′
056◦ 10 ′
056◦ 36 ′
102◦ 14 ′
084◦ 00 ′
070◦ 52 ′
063◦ 18 ′
059◦ 46 ′
059◦ 03 ′
060◦ 26 ′
063◦ 35 ′
066◦ 55 ′
110◦ 34 ′
095◦ 43 ′
02:00
03:00
04:00
05:00
06:00
06:55
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
05:04
00:00
01:00
02:00
03:00
04:00
04:16
00:00
01:00
02:00
03:00
03:55
00:00
01:00
02:00
03:00
04:00
04:16
00:00
01:00
02:00
03:00
04:00
05:00
05:04
44◦ 15 ′
36◦ 43 ′
27◦ 52 ′
18◦ 23 ′
08◦ 46 ′
00◦ 00 ′
137◦ 14 ′
145◦ 07 ′
149◦ 36 ′
151◦ 26 ′
151◦ 09 ′
149◦ 10 ′
Libra
40◦ 00 ′ 113◦ 26 ′
38◦ 23 ′ 125◦ 41 ′
33◦ 50 ′ 136◦ 12 ′
27◦ 02 ′ 144◦ 07 ′
18◦ 45 ′ 149◦ 26 ′
09◦ 35 ′ 152◦ 28 ′
00◦ 00 ′ 153◦ 26 ′
Scorpio
28◦ 32 ′ 110◦ 34 ′
27◦ 08 ′ 121◦ 21 ′
23◦ 09 ′ 131◦ 02 ′
17◦ 03 ′ 138◦ 58 ′
09◦ 22 ′ 144◦ 55 ′
00◦ 37 ′ 148◦ 57 ′
00◦ 00 ′ 149◦ 10 ′
Sagittarius
19◦ 51 ′ 102◦ 14 ′
18◦ 36 ′ 112◦ 20 ′
15◦ 00 ′ 121◦ 40 ′
09◦ 22 ′ 129◦ 39 ′
02◦ 10 ′ 136◦ 05 ′
00◦ 00 ′ 137◦ 33 ′
Capricorn
16◦ 34 ′ 090◦ 00 ′
15◦ 22 ′ 099◦ 56 ′
11◦ 54 ′ 109◦ 10 ′
06◦ 27 ′ 117◦ 13 ′
00◦ 00 ′ 123◦ 24 ′
Aquarius
19◦ 51 ′ 077◦ 46 ′
18◦ 36 ′ 087◦ 53 ′
15◦ 00 ′ 097◦ 12 ′
09◦ 22 ′ 105◦ 12 ′
02◦ 10 ′ 111◦ 38 ′
00◦ 00 ′ 113◦ 05 ′
Pisces
28◦ 32 ′ 069◦ 26 ′
27◦ 08 ′ 080◦ 12 ′
23◦ 09 ′ 089◦ 53 ′
17◦ 03 ′ 097◦ 49 ′
09◦ 22 ′ 103◦ 46 ′
00◦ 37 ′ 107◦ 49 ′
00◦ 00 ′ 108◦ 01 ′
083◦ 55 ′
076◦ 02 ′
071◦ 33 ′
069◦ 43 ′
070◦ 00 ′
071◦ 59 ′
113◦ 26 ′
101◦ 11 ′
090◦ 40 ′
082◦ 45 ′
077◦ 26 ′
074◦ 24 ′
073◦ 26 ′
110◦ 34 ′
099◦ 48 ′
090◦ 07 ′
082◦ 11 ′
076◦ 14 ′
072◦ 11 ′
071◦ 59 ′
102◦ 14 ′
092◦ 07 ′
082◦ 48 ′
074◦ 48 ′
068◦ 22 ′
066◦ 55 ′
090◦ 00 ′
080◦ 04 ′
070◦ 50 ′
062◦ 47 ′
056◦ 36 ′
077◦ 46 ′
067◦ 40 ′
058◦ 20 ′
050◦ 21 ′
043◦ 55 ′
042◦ 27 ′
069◦ 26 ′
058◦ 39 ′
048◦ 58 ′
041◦ 02 ′
035◦ 05 ′
031◦ 03 ′
030◦ 50 ′
Table 2.23: Ecliptic altitude and parallactic angle for latitude +50◦ .
Spherical Astronomy
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:22
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
08:37
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
09:14
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
Aries
30◦ 00 ′ 066◦ 34 ′
28◦ 53 ′ 075◦ 04 ′
25◦ 40 ′ 082◦ 40 ′
20◦ 42 ′ 088◦ 46 ′
14◦ 29 ′ 093◦ 08 ′
07◦ 26 ′ 095◦ 43 ′
00◦ 00 ′ 096◦ 34 ′
Taurus
41◦ 28 ′ 069◦ 26 ′
40◦ 12 ′ 079◦ 11 ′
36◦ 37 ′ 087◦ 35 ′
31◦ 15 ′ 093◦ 51 ′
24◦ 40 ′ 097◦ 53 ′
17◦ 24 ′ 099◦ 50 ′
09◦ 55 ′ 099◦ 56 ′
02◦ 36 ′ 098◦ 20 ′
00◦ 00 ′ 097◦ 20 ′
Gemini
50◦ 09 ′ 077◦ 46 ′
48◦ 44 ′ 089◦ 05 ′
44◦ 49 ′ 098◦ 24 ′
39◦ 04 ′ 104◦ 52 ′
32◦ 12 ′ 108◦ 33 ′
24◦ 49 ′ 109◦ 55 ′
17◦ 21 ′ 109◦ 22 ′
10◦ 11 ′ 107◦ 09 ′
03◦ 39 ′ 103◦ 29 ′
00◦ 00 ′ 100◦ 29 ′
Cancer
53◦ 26 ′ 090◦ 00 ′
51◦ 57 ′ 102◦ 07 ′
47◦ 53 ′ 111◦ 53 ′
41◦ 58 ′ 118◦ 24 ′
35◦ 01 ′ 121◦ 55 ′
27◦ 35 ′ 123◦ 01 ′
20◦ 09 ′ 122◦ 11 ′
13◦ 03 ′ 119◦ 43 ′
06◦ 36 ′ 115◦ 51 ′
01◦ 09 ′ 110◦ 43 ′
00◦ 00 ′ 109◦ 17 ′
Leo
50◦ 09 ′ 102◦ 14 ′
48◦ 44 ′ 113◦ 33 ′
44◦ 49 ′ 122◦ 52 ′
39◦ 04 ′ 129◦ 19 ′
32◦ 12 ′ 133◦ 01 ′
24◦ 49 ′ 134◦ 23 ′
17◦ 21 ′ 133◦ 49 ′
10◦ 11 ′ 131◦ 37 ′
03◦ 39 ′ 127◦ 57 ′
59
066◦ 34 ′
058◦ 04 ′
050◦ 28 ′
044◦ 22 ′
040◦ 00 ′
037◦ 25 ′
036◦ 34 ′
069◦ 26 ′
059◦ 40 ′
051◦ 17 ′
045◦ 00 ′
040◦ 58 ′
039◦ 01 ′
038◦ 55 ′
040◦ 31 ′
041◦ 31 ′
077◦ 46 ′
066◦ 27 ′
057◦ 08 ′
050◦ 41 ′
046◦ 59 ′
045◦ 37 ′
046◦ 11 ′
048◦ 23 ′
052◦ 03 ′
055◦ 04 ′
090◦ 00 ′
077◦ 53 ′
068◦ 07 ′
061◦ 36 ′
058◦ 05 ′
056◦ 59 ′
057◦ 49 ′
060◦ 17 ′
064◦ 09 ′
069◦ 17 ′
070◦ 43 ′
102◦ 14 ′
090◦ 55 ′
081◦ 36 ′
075◦ 08 ′
071◦ 27 ′
070◦ 05 ′
070◦ 38 ′
072◦ 51 ′
076◦ 31 ′
08:37
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
07:22
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
04:37
00:00
01:00
02:00
03:00
03:22
00:00
01:00
02:00
02:45
00:00
01:00
02:00
03:00
03:22
0:00
01:00
02:00
03:00
04:00
04:37
00◦ 00 ′
124◦ 56 ′
Virgo
41◦ 28 ′ 110◦ 34 ′
40◦ 12 ′ 120◦ 20 ′
36◦ 37 ′ 128◦ 43 ′
31◦ 15 ′ 135◦ 00 ′
24◦ 40 ′ 139◦ 02 ′
17◦ 24 ′ 140◦ 59 ′
09◦ 55 ′ 141◦ 05 ′
02◦ 36 ′ 139◦ 29 ′
00◦ 00 ′ 138◦ 29 ′
Libra
30◦ 00 ′ 113◦ 26 ′
28◦ 53 ′ 121◦ 56 ′
25◦ 40 ′ 129◦ 32 ′
20◦ 42 ′ 135◦ 38 ′
14◦ 29 ′ 140◦ 00 ′
07◦ 26 ′ 142◦ 35 ′
00◦ 00 ′ 143◦ 26 ′
Scorpio
18◦ 32 ′ 110◦ 34 ′
17◦ 31 ′ 118◦ 22 ′
14◦ 36 ′ 125◦ 33 ′
10◦ 02 ′ 131◦ 37 ′
04◦ 11 ′ 136◦ 18 ′
00◦ 00 ′ 138◦ 29 ′
Sagittarius
09◦ 51 ′ 102◦ 14 ′
08◦ 56 ′ 109◦ 45 ′
06◦ 13 ′ 116◦ 48 ′
01◦ 56 ′ 122◦ 57 ′
00◦ 00 ′ 124◦ 56 ′
Capricorn
06◦ 34 ′ 090◦ 00 ′
05◦ 40 ′ 097◦ 28 ′
03◦ 02 ′ 104◦ 30 ′
00◦ 00 ′ 109◦ 17 ′
Aquarius
09◦ 51 ′ 077◦ 46 ′
08◦ 56 ′ 085◦ 18 ′
06◦ 13 ′ 092◦ 20 ′
01◦ 56 ′ 098◦ 29 ′
00◦ 00 ′ 100◦ 29 ′
Pisces
18◦ 32 ′ 069◦ 26 ′
17◦ 31 ′ 077◦ 14 ′
14◦ 36 ′ 084◦ 24 ′
10◦ 02 ′ 090◦ 28 ′
04◦ 11 ′ 095◦ 09 ′
00◦ 00 ′ 097◦ 20 ′
079◦ 31 ′
110◦ 34 ′
100◦ 49 ′
092◦ 25 ′
086◦ 09 ′
082◦ 07 ′
080◦ 10 ′
080◦ 04 ′
081◦ 40 ′
082◦ 40 ′
113◦ 26 ′
104◦ 56 ′
097◦ 20 ′
091◦ 14 ′
086◦ 52 ′
084◦ 17 ′
083◦ 26 ′
110◦ 34 ′
102◦ 46 ′
095◦ 36 ′
089◦ 32 ′
084◦ 51 ′
082◦ 40 ′
102◦ 14 ′
094◦ 42 ′
087◦ 40 ′
081◦ 31 ′
079◦ 31 ′
090◦ 00 ′
082◦ 32 ′
075◦ 30 ′
070◦ 43 ′
077◦ 46 ′
070◦ 15 ′
063◦ 12 ′
057◦ 03 ′
055◦ 04 ′
069◦ 26 ′
061◦ 38 ′
054◦ 27 ′
048◦ 23 ′
043◦ 42 ′
041◦ 31 ′
Table 2.24: Ecliptic altitude and parallactic angle for latitude +60◦ .
60
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
08:15
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
MODERN ALMAGEST
Aries
20 00
066◦ 34 ′
◦
′
19 17
071◦ 57 ′
◦
′
17 14
076◦ 53 ′
◦
′
14 00
081◦ 00 ′
◦
′
09 51
084◦ 04 ′
◦
′
05 05
085◦ 56 ′
◦
′
00 00
086◦ 34 ′
Taurus
31◦ 28 ′ 069◦ 26 ′
30◦ 42 ′ 075◦ 20 ′
28◦ 30 ′ 080◦ 39 ′
25◦ 05 ′ 084◦ 55 ′
20◦ 46 ′ 087◦ 54 ′
15◦ 53 ′ 089◦ 31 ′
10◦ 46 ′ 089◦ 48 ′
05◦ 45 ′ 088◦ 49 ′
01◦ 06 ′ 086◦ 40 ′
00◦ 00 ′ 085◦ 55 ′
Gemini
40◦ 09 ′ 077◦ 46 ′
39◦ 20 ′ 084◦ 21 ′
37◦ 00 ′ 090◦ 08 ′
33◦ 25 ′ 094◦ 37 ′
28◦ 58 ′ 097◦ 34 ′
24◦ 00 ′ 098◦ 58 ′
18◦ 53 ′ 098◦ 58 ′
13◦ 55 ′ 097◦ 40 ′
09◦ 23 ′ 095◦ 15 ′
05◦ 33 ′ 091◦ 50 ′
02◦ 37 ′ 087◦ 38 ′
00◦ 46 ′ 082◦ 51 ′
00◦ 09 ′ 077◦ 46 ′
Cancer
43◦ 26 ′ 090◦ 00 ′
42◦ 36 ′ 096◦ 54 ′
40◦ 12 ′ 102◦ 56 ′
36◦ 33 ′ 107◦ 31 ′
32◦ 03 ′ 110◦ 27 ′
27◦ 04 ′ 111◦ 47 ′
21◦ 57 ′ 111◦ 38 ′
17◦ 00 ′ 110◦ 13 ′
12◦ 31 ′ 107◦ 40 ′
08◦ 44 ′ 104◦ 10 ′
05◦ 51 ′ 099◦ 54 ′
04◦ 03 ′ 095◦ 05 ′
◦
′
12:00
◦
′
066 34
061◦ 11 ′
056◦ 15 ′
052◦ 08 ′
049◦ 04 ′
047◦ 12 ′
046◦ 34 ′
069◦ 26 ′
063◦ 31 ′
058◦ 12 ′
053◦ 56 ′
050◦ 58 ′
049◦ 20 ′
049◦ 03 ′
050◦ 02 ′
052◦ 12 ′
052◦ 56 ′
077◦ 46 ′
071◦ 12 ′
065◦ 25 ′
060◦ 56 ′
057◦ 59 ′
056◦ 34 ′
056◦ 35 ′
057◦ 52 ′
060◦ 18 ′
063◦ 43 ′
067◦ 55 ′
072◦ 42 ′
077◦ 46 ′
090◦ 00 ′
083◦ 06 ′
077◦ 04 ′
072◦ 29 ′
069◦ 33 ′
068◦ 13 ′
068◦ 22 ′
069◦ 47 ′
072◦ 20 ′
075◦ 50 ′
080◦ 06 ′
084◦ 55 ′
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
08:15
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
03:44
00:00
01:00
02:00
03:00
03:44
03◦ 26 ′
090◦ 00 ′
Leo
40◦ 09 ′ 102◦ 14 ′
39◦ 20 ′ 108◦ 48 ′
37◦ 00 ′ 114◦ 35 ′
33◦ 25 ′ 119◦ 04 ′
28◦ 58 ′ 122◦ 01 ′
24◦ 00 ′ 123◦ 26 ′
18◦ 53 ′ 123◦ 25 ′
13◦ 55 ′ 122◦ 08 ′
09◦ 23 ′ 119◦ 42 ′
05◦ 33 ′ 116◦ 17 ′
02◦ 37 ′ 112◦ 05 ′
00◦ 46 ′ 107◦ 18 ′
00◦ 09 ′ 102◦ 14 ′
Virgo
31◦ 28 ′ 110◦ 34 ′
30◦ 42 ′ 116◦ 29 ′
28◦ 30 ′ 121◦ 48 ′
25◦ 05 ′ 126◦ 04 ′
20◦ 46 ′ 129◦ 02 ′
15◦ 53 ′ 130◦ 40 ′
10◦ 46 ′ 130◦ 57 ′
05◦ 45 ′ 129◦ 58 ′
01◦ 06 ′ 127◦ 48 ′
00◦ 00 ′ 127◦ 04 ′
Libra
20◦ 00 ′ 113◦ 26 ′
19◦ 17 ′ 118◦ 49 ′
17◦ 14 ′ 123◦ 45 ′
14◦ 00 ′ 127◦ 52 ′
09◦ 51 ′ 130◦ 56 ′
05◦ 05 ′ 132◦ 48 ′
00◦ 00 ′ 133◦ 26 ′
Scorpio
08◦ 32 ′ 110◦ 34 ′
07◦ 52 ′ 115◦ 42 ′
05◦ 56 ′ 120◦ 28 ′
02◦ 53 ′ 124◦ 35 ′
00◦ 00 ′ 127◦ 04 ′
Pisces
08◦ 32 ′ 069◦ 26 ′
07◦ 52 ′ 074◦ 33 ′
05◦ 56 ′ 079◦ 20 ′
02◦ 53 ′ 083◦ 26 ′
00◦ 00 ′ 085◦ 55 ′
090◦ 00 ′
102◦ 14 ′
095◦ 39 ′
089◦ 52 ′
085◦ 23 ′
082◦ 26 ′
081◦ 02 ′
081◦ 02 ′
082◦ 20 ′
084◦ 45 ′
088◦ 10 ′
092◦ 22 ′
097◦ 09 ′
102◦ 14 ′
110◦ 34 ′
104◦ 40 ′
099◦ 21 ′
095◦ 05 ′
092◦ 06 ′
090◦ 29 ′
090◦ 12 ′
091◦ 11 ′
093◦ 20 ′
094◦ 05 ′
113◦ 26 ′
108◦ 03 ′
103◦ 07 ′
099◦ 00 ′
095◦ 56 ′
094◦ 04 ′
093◦ 26 ′
110◦ 34 ′
105◦ 27 ′
100◦ 40 ′
096◦ 34 ′
094◦ 05 ′
069◦ 26 ′
064◦ 18 ′
059◦ 32 ′
055◦ 25 ′
052◦ 56 ′
Table 2.25: Ecliptic altitude and parallactic angle for latitude +70◦ .
Spherical Astronomy
00:00
01:00
02:00
03:00
04:00
05:00
06:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
Aries
10◦ 00 ′ 066◦ 34 ′
09◦ 39 ′ 069◦ 11 ′
08◦ 39 ′ 071◦ 36 ′
07◦ 03 ′ 073◦ 40 ′
04◦ 59 ′ 075◦ 15 ′
02◦ 35 ′ 076◦ 14 ′
00◦ 00 ′ 076◦ 34 ′
Taurus
21◦ 28 ′ 069◦ 26 ′
21◦ 07 ′ 072◦ 11 ′
20◦ 04 ′ 074◦ 44 ′
18◦ 26 ′ 076◦ 52 ′
16◦ 19 ′ 078◦ 26 ′
13◦ 53 ′ 079◦ 23 ′
11◦ 18 ′ 079◦ 38 ′
08◦ 44 ′ 079◦ 12 ′
06◦ 21 ′ 078◦ 08 ′
04◦ 20 ′ 076◦ 30 ′
02◦ 47 ′ 074◦ 25 ′
01◦ 48 ′ 072◦ 00 ′
01◦ 28 ′ 069◦ 26 ′
Gemini
30◦ 09 ′ 077◦ 46 ′
29◦ 47 ′ 080◦ 44 ′
28◦ 43 ′ 083◦ 27 ′
27◦ 02 ′ 085◦ 42 ′
24◦ 53 ′ 087◦ 19 ′
22◦ 25 ′ 088◦ 14 ′
19◦ 50 ′ 088◦ 25 ′
17◦ 17 ′ 087◦ 53 ′
14◦ 56 ′ 086◦ 44 ′
12◦ 56 ′ 085◦ 01 ′
11◦ 25 ′ 082◦ 51 ′
10◦ 28 ′ 080◦ 24 ′
10◦ 09 ′ 077◦ 46 ′
Cancer
33◦ 26 ′ 090◦ 00 ′
33◦ 04 ′ 093◦ 04 ′
31◦ 59 ′ 095◦ 53 ′
30◦ 17 ′ 098◦ 10 ′
28◦ 07 ′ 099◦ 49 ′
25◦ 39 ′ 100◦ 43 ′
23◦ 03 ′ 100◦ 53 ′
20◦ 31 ′ 100◦ 19 ′
61
066◦ 34 ′
063◦ 57 ′
061◦ 32 ′
059◦ 28 ′
057◦ 53 ′
056◦ 54 ′
056◦ 34 ′
069◦ 26 ′
066◦ 40 ′
064◦ 07 ′
061◦ 59 ′
060◦ 25 ′
059◦ 29 ′
059◦ 14 ′
059◦ 39 ′
060◦ 43 ′
062◦ 21 ′
064◦ 26 ′
066◦ 51 ′
069◦ 26 ′
077◦ 46 ′
074◦ 48 ′
072◦ 05 ′
069◦ 51 ′
068◦ 14 ′
067◦ 19 ′
067◦ 08 ′
067◦ 39 ′
068◦ 49 ′
070◦ 32 ′
072◦ 41 ′
075◦ 09 ′
077◦ 46 ′
090◦ 00 ′
086◦ 56 ′
084◦ 07 ′
081◦ 50 ′
080◦ 11 ′
079◦ 17 ′
079◦ 07 ′
079◦ 41 ′
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
00:00
01:00
02:00
03:00
04:00
05:00
06:00
18◦ 11 ′
16◦ 12 ′
14◦ 42 ′
13◦ 45 ′
13◦ 26 ′
099◦ 06 ′
097◦ 21 ′
095◦ 09 ′
092◦ 39 ′
090◦ 00 ′
Leo
30◦ 09 ′ 102◦ 14 ′
29◦ 47 ′ 105◦ 12 ′
28◦ 43 ′ 107◦ 55 ′
27◦ 02 ′ 110◦ 09 ′
24◦ 53 ′ 111◦ 46 ′
22◦ 25 ′ 112◦ 41 ′
19◦ 50 ′ 112◦ 52 ′
17◦ 17 ′ 112◦ 21 ′
14◦ 56 ′ 111◦ 11 ′
12◦ 56 ′ 109◦ 28 ′
11◦ 25 ′ 107◦ 19 ′
10◦ 28 ′ 104◦ 51 ′
10◦ 09 ′ 102◦ 14 ′
Virgo
21◦ 28 ′ 110◦ 34 ′
21◦ 07 ′ 113◦ 20 ′
20◦ 04 ′ 115◦ 53 ′
18◦ 26 ′ 118◦ 01 ′
16◦ 19 ′ 119◦ 35 ′
13◦ 53 ′ 120◦ 31 ′
11◦ 18 ′ 120◦ 46 ′
08◦ 44 ′ 120◦ 21 ′
06◦ 21 ′ 119◦ 17 ′
04◦ 20 ′ 117◦ 39 ′
02◦ 47 ′ 115◦ 34 ′
01◦ 48 ′ 113◦ 09 ′
01◦ 28 ′ 110◦ 34 ′
Libra
10◦ 00 ′ 113◦ 26 ′
09◦ 39 ′ 116◦ 03 ′
08◦ 39 ′ 118◦ 28 ′
07◦ 03 ′ 120◦ 32 ′
04◦ 59 ′ 122◦ 07 ′
02◦ 35 ′ 123◦ 06 ′
00◦ 00 ′ 123◦ 26 ′
080◦ 54 ′
082◦ 39 ′
084◦ 51 ′
087◦ 21 ′
090◦ 00 ′
102◦ 14 ′
099◦ 16 ′
096◦ 33 ′
094◦ 18 ′
092◦ 41 ′
091◦ 46 ′
091◦ 35 ′
092◦ 07 ′
093◦ 16 ′
094◦ 59 ′
097◦ 09 ′
099◦ 36 ′
102◦ 14 ′
110◦ 34 ′
107◦ 49 ′
105◦ 16 ′
103◦ 08 ′
101◦ 34 ′
100◦ 37 ′
100◦ 22 ′
100◦ 48 ′
101◦ 52 ′
103◦ 30 ′
105◦ 35 ′
108◦ 00 ′
110◦ 34 ′
113◦ 26 ′
110◦ 49 ′
108◦ 24 ′
106◦ 20 ′
104◦ 45 ′
103◦ 46 ′
103◦ 26 ′
Table 2.26: Ecliptic altitude and parallactic angle for latitude +80◦ .
62
MODERN ALMAGEST
Dates
63
3 Dates
3.1 Introduction
Following modern astronomical practice, in this treatise we shall specify dates by means of Julian
day numbers. According to this scheme, days are numbered consecutively from January 1, 4713
BCE, which is designated day zero. For instance, October 14, 1066 CE (the date of the battle of
Hastings) is day 2 110 701. Each Julian day commences at 12:00 universal time (UT).
3.2 Determination of Julian Day Numbers
The Julian day number of a given day can be determined from Tables 3.1–3.3. The date must be
expressed in terms of the Gregorian calendar.
The procedure is as follows:
1. Enter the table of century years (Table 3.1) with the century year immediately preceding the
date in question, and take out the tabular value. If the century year is marked with a †, note
this fact for use in step 2.
2. Enter the table of years of the century (Table 3.2) with the last two digits of the year in
question, and take out the tabular value. If the century year used in step 1 was marked with
a †, diminish the tabular value by one day, unless the tabular value is zero. If the year in
question is a leap year, marked with a ∗, note this fact for use in step 3.
3. Enter the table of the days of the year (Table 3.3) with the day in question, and take out the
tabular value. If the year in question is a leap year and the table entry falls after February 28,
add one day to the tabular value. The sum of the values obtained in steps 1, 2, and 3 then
gives the Julian day number of the date in question.
Example 1: June 10, 1992 CE:
1. Century year
2. Year of the century
3. Day of the year
Julian day number
† 1900
∗ 92
June 10
33 603 - 1 =
161+1 =
2 415 020
33 602
162
2 448 784
Observe that in step 2 the tabular value has been diminished by 1 because 1900 is a common year
(marked with a † in Table 3.1). In step 3, the tabular value has been increased by 1 because 1992
is a leap year (marked with a ∗ in Table 3.2), and the date falls after February 28.
Example 2: January 18, 1824 CE:
64
1. Century year
2. Year of the century
3. Day of the year
Julian day number
MODERN ALMAGEST
† 1800
∗ 24
January 18
8 766 - 1 =
18 =
2 378 496
8 765
18
2 387 279
Observe that in step 2 the tabular value has been diminished by 1 because 1800 is a common year
(marked with a † in Table 3.1). In step 3, the tabular value has not been increased by 1, despite the
fact that 1824 is a leap year (marked with an ∗ in Table 3.2), because the date falls before February
28.
We can specify the time of day (in universal time), as well as the date, by means of fractional
Julian day numbers. For instance, t = 2 448 784.0 JD corresponds to 12:00 UT on June 10, 1992
CE, whereas t = 2 448 784.5 JD corresponds to 24:00 UT later the same day.
Dates
65
† 1800
† 1900
2000
2 378 496
2 415 020
2 451 544
Table 3.1: Julian Day Number: Century Years. † Common years. All years are CE. From “The History
and Practice of Ancient Astronomy”, J. Evans (Oxford University Press, Oxford UK, 1998).
§0
1
2
3
∗4
5
6
7
∗8
9
10
11
∗ 12
13
14
15
∗ 16
17
18
19
0
336
731
1 096
∗ 20
1 461
1 827
2 192
2 557
∗ 24
2 922
3 288
3 653
4 018
∗ 28
4 383
4 749
5 114
5 479
∗ 32
5 844
6 210
6 575
6 940
∗ 36
21
22
23
25
26
27
29
30
31
33
34
35
37
38
39
7 305
7 671
8 036
8 401
∗ 40
8 766
9 132
9 497
9 862
∗ 44
10 227
10 593
10 958
11 323
∗ 48
11 688
12 054
12 419
12 784
∗ 52
13 149
13 515
13 880
14 245
∗ 56
41
42
43
45
46
47
49
50
51
53
54
55
57
58
59
14 610
14 976
15 341
15 706
∗ 60
16 071
16 437
16 802
17 167
∗ 64
17 532
17 898
18 263
18 628
∗ 68
18 993
19 359
19 724
20 089
∗ 72
20 454
20 820
21 185
21 550
∗ 76
61
62
63
65
66
67
69
70
71
73
74
75
77
78
79
21 915
22 281
22 646
22 011
∗ 80
23 376
23 742
24 107
24 472
∗ 84
24 837
25 203
25 568
25 933
∗ 88
26 298
26 664
27 029
27 394
∗ 92
27 759
28 125
28 490
28 855
∗ 96
81
82
83
85
86
87
89
90
91
93
94
95
97
98
99
29 220
29 586
29 951
30 316
30 681
31 047
31 412
31 777
32 142
32 508
32 873
33 238
33 603
33 969
34 334
34 699
35 064
35 430
35 795
36 160
Table 3.2: Julian Day Number: Years of the Century. ∗ Leap year. § Leap year unless century is marked
†. In centuries marked †, subtract one day from the tabulated values for the years 1 through 99. From
“The History and Practice of Ancient Astronomy”, J. Evans (Oxford University Press, Oxford UK, 1998).
66
MODERN ALMAGEST
Day
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
1
2
3
4
5
1
2
3
4
5
32
33
34
35
36
60
61
62
63
64
91
92
93
94
95
121
122
123
124
125
152
153
154
155
156
182
183
184
185
186
213
214
215
216
217
244
245
246
247
248
274
275
276
277
278
305
306
307
308
309
335
336
337
338
339
6
7
8
9
10
6
7
8
9
10
37
38
39
40
41
65
66
67
68
69
96
97
98
99
100
126
127
128
129
130
157
158
159
160
161
187
188
189
190
191
218
219
220
221
222
249
250
251
252
253
279
280
281
282
283
310
311
312
313
314
340
341
342
343
344
11
12
13
14
15
11
12
13
14
15
42
43
44
45
46
70
71
72
73
74
101
102
103
104
105
131
132
133
134
135
162
163
164
165
166
192
193
194
195
196
223
224
225
226
227
254
255
256
257
258
284
285
286
285
288
315
316
317
318
319
345
346
347
348
349
16
17
18
19
20
16
17
18
19
20
47
48
49
50
51
75
76
77
78
79
106
107
108
109
110
136
137
138
139
140
167
168
169
170
171
197
198
199
200
201
228
229
230
231
232
259
260
261
262
263
289
290
291
292
293
320
321
322
323
324
350
351
352
353
354
21
22
23
24
25
21
22
23
24
25
52
53
54
55
56
80
81
82
83
84
111
112
113
114
115
141
142
143
144
145
172
173
174
175
176
202
203
204
205
206
233
234
235
236
237
264
265
266
267
268
294
295
296
297
298
325
326
327
328
329
355
356
357
358
359
26
27
28
29
30
31
26
27
28
29
30
31
57
58
59
∗
85
86
87
88
89
90
116
117
118
119
120
146
147
148
149
150
151
177
178
179
180
181
207
208
209
210
211
212
238
239
240
241
242
243
269
270
271
272
273
299
300
301
302
303
304
330
331
332
333
334
360
361
362
363
364
365
Table 3.3: Julian Day Number: Days of the Year. ∗ In leap year, after February 28, add 1 to the
tabulated value. From “The History and Practice of Ancient Astronomy”, J. Evans (Oxford University
Press, Oxford UK, 1998).
Geometric Planetary Orbit Models
67
4 Geometric Planetary Orbit Models
4.1 Introduction
In this section, Kepler’s geometric model of a geocentric planetary orbit is examined in detail, and
then compared to the less accurate geometric models of Hipparchus, Ptolemy, and Copernicus. In
the following, all orbits are viewed from the northern ecliptic pole.
4.2 Model of Kepler
Kepler’s geometric model of a heliocentric planetary orbit is summed up in his three well-known
laws of planetary motion. According to Kepler’s first law, all planetary orbits are ellipses which are
confocal with the sun and lie in a fixed plane. Moreover, according to Kepler’s second law, the radius
vector which connects the sun to a given planet sweeps out equal areas in equal time intervals.
D
Q
B
P
b
r
T
E
A
C
a
S
R
Π
ea
Figure 4.1: A Keplerian orbit.
Consider Figure 4.1. ΠPBA is half of an elliptical planetary orbit. Furthermore, C is the geometric center of the orbit, S the focus at which the sun is located, P the instantaneous position of
the planet, Π the perihelion point (i.e., the planet’s point of closest approach to the sun), and A
the aphelion point (i.e., the point of furthest distance from the sun). The ellipse is symmetric about
ΠA, which is termed the major axis, and about CB, which is termed the minor axis. The length
CA ≡ a is called the orbital major radius. The length CS represents the displacement of the sun
from the geometric center of the orbit, and is generally written e a, where e is termed the orbital
eccentricity, where 0 ≤ e ≤ 1. The length CB ≡ b = a (1 − e2)1/2 is called the orbital minor radius.
The length SP ≡ r represents the radial distance of the planet from the sun. Finally, the angle
RSP ≡ T is the angular bearing of the planet from the sun, relative to the major axis of the orbit,
and is termed the true anomaly.
ΠQDA is half of a circle whose geometric center is C, and whose radius is a. Hence, the circle
passes through the perihelion and aphelion points. R is the point at which the perpendicular from P
68
MODERN ALMAGEST
meets the major axis ΠA. The point where RP produced meets circle ΠQDA is denoted Q. Finally,
the angle SCQ ≡ E is called the elliptic anomaly.
Now, the equation of the ellipse ΠPBA is
x 2 y2
+
= 1,
a2 b2
(4.1)
where x and y are the perpendicular distances from the minor and major axes, respectively. Likewise, the equation of the circle ΠQDA is
x′ 2 y′ 2
+ 2 = 1.
a2
a
(4.2)
y
b
= ,
y′
a
(4.3)
b
RP
= .
RQ
a
(4.4)
Hence, if x = x ′ then
and it follows that
Now, CS = e a. Furthermore, it is easily demonstrated that SR = r cos T , RP = r sin T , CR =
a cos E, and RQ = a sin E. Consequently, Eq. (4.4) yields
r sin T = b sin E = a (1 − e2)1/2 sin E.
(4.5)
Also, since SR = CR − CS, we have
r cos T = a (cos E − e).
(4.6)
Taking the square root of the sum of the squares of the previous two equations, we obtain
r = a (1 − e cos E),
(4.7)
which can be combined with Eq. (4.6) to give
cos E − e
.
1 − e cos E
(4.8)
t − t∗
Area ΠPS
=
,
πab
τ
(4.9)
cos T =
Now, according to Kepler’s second law,
where t is the time at which the planet passes point P, t∗ the time at which it passes the perihelion
point, and τ the orbital period. However,
Area ΠPS = Area SRP + Area ΠRP =
1 2
r cos T sin T + Area ΠRP.
2
(4.10)
But,
Area ΠRP =
b
Area ΠRQ,
a
(4.11)
Geometric Planetary Orbit Models
69
since RP/RQ = b/a for all values of T . In addition,
Area ΠRQ = Area ΠQC − Area RQC =
1
1
E a2 − a2 cos E sin E.
2
2
(4.12)
Hence, we can write
b a2
1
t − t∗
(E − cos E sin E).
π a b = r2 cos T sin T +
τ
2
a 2
(4.13)
According to Eqs. (4.5) and (4.6), r sin T = b sin E, and r cos T = a (cos E − e), so the above
expression reduces to
M = E − e sin E,
(4.14)
where
M=
2π
τ
(t − t∗ )
(4.15)
is an angle which is zero at the perihelion point, increases uniformly in time, and has a repetition
period which matches the period of the planetary orbit. This angle is termed the mean anomaly.
In summary, the radial and angular polar coordinates, r and T , respectively, of a planet in a
Keplerian orbit about the sun are specified as implicit functions of the mean anomaly, which is a
linear function of time, by the following three equations:
M = E − e sin E,
r = a (1 − e cos E),
cos T
=
cos E − e
.
1 − e cos E
(4.16)
(4.17)
(4.18)
It turns out that the earth and the five visible planets all possess low eccentricity orbits characterized
by e ≪ 1. Hence, it is a good approximation to expand the above three equations using e as a small
parameter. To second-order, we get
E = M + e sin M + (1/2) e2 sin 2M,
(4.19)
r = a (1 − e cos T − e2 sin2 T ),
(4.20)
T
= E + e sin E + (1/4) e2 sin 2E.
(4.21)
Finally, these equations can be combined to give r and T as explicit functions of the mean anomaly:
r
a
= 1 − e cos M + e2 sin2 M,
(4.22)
T
= M + 2 e sin M + (5/4) e2 sin 2M.
(4.23)
4.3 Model of Hipparchus
Hipparchus’ geometric model of the apparent orbit of the sun around the earth can also be used
to describe a heliocentric planetary orbit. The model is illustrated in Fig. 4.2. The orbit of the
planet corresponds to the circle ΠPDA (only half of which is shown), where Π is the perihelion
point, P the planet’s instantaneous position, and A the aphelion point. The diameter ΠSCA is
70
MODERN ALMAGEST
the effective major axis of the orbit (to be more exact, it is the line of apsides), where C is the
geometric center of circle ΠPDA, and S the fixed position of the sun. The radius CP of circle
ΠPDA is the effective major radius, a, of the orbit. The distance SC is equal to 2 e a, where e is the
orbit’s effective eccentricity. The angle PCΠ is identified with the mean anomaly, M, and increases
linearly in time. In other words, as seen from C, the planet P moves uniformly around circle ΠPDA
in a counterclockwise direction. Finally, SP is the radial distance, r, of the planet from the sun, and
angle PSΠ is the planet’s true anomaly, T .
D
P
L
T
M
A
S
C
Π
K
Figure 4.2: A Hipparchian orbit.
Let us draw the straight-line KSL parallel to CP, and passing through point S, and then complete
the rectangle PCKL. Simple geometry reveals that CK = PL = 2 e a sin M, KS = 2 e a cos M, and
SL = a − 2 e a cos M. Moreover, SP2 = SL2 + PL2, which implies that
r
= (1 − 4 e cos M + 4 e2)1/2.
a
(4.24)
Now, T = M + q, where q is angle PSL. However,
sin q =
PL
2 e sin M
=
.
SP
(1 − 4 e cos M + 4 e2)1/2
(4.25)
Finally, expanding the previous two equations to second-order in the small parameter e, we
obtain
r
= 1 − 2 e cos M + 2 e2 sin2 M,
(4.26)
a
T
= M + 2 e sin M + 2 e2 sin 2M.
(4.27)
It can be seen, by comparison with Eqs. (4.22) and (4.23), that the relative radial distance, r/a,
in the Hipparchian model deviates from that in the (correct) Keplerian model to first-order in e
(in fact, the variation of r/a is greater by a factor of 2 in the former model), whereas the true
Geometric Planetary Orbit Models
71
anomaly, T , only deviates to second-order in e. We conclude that Hipparchus’ geometric model of
a heliocentric planetary orbit does a reasonably good job at predicting the angular position of the
planet, relative to the sun, but significantly exaggerates (by a factor of 2) the variation in the radial
distance between the two during the course of a complete orbital rotation.
4.4 Model of Ptolemy
Ptolemy’s geometric model of the motion of the center of an epicycle around a deferent can also
be used to describe a heliocentric planetary orbit. The model is illustrated in Fig. 4.3. The orbit of
the planet corresponds to the circle ΠPDA (only half of which is shown), where Π is the perihelion
point, P the planet’s instantaneous position, and A the aphelion point. The diameter ΠSCQA is
the effective major axis of the orbit, where C is the geometric center of circle ΠPDA, S the fixed
position of the sun, and Q the location of the so-called equant. The radius CP of circle ΠPDA is
the effective major radius, a, of the orbit. The distances SC and CQ are both equal to e a, where
e is the orbit’s effective eccentricity. The angle PQΠ is identified with the mean anomaly, M, and
increases linearly in time. In other words, as seen from Q, the planet P moves uniformly around
circle ΠPDA in a counterclockwise direction. Finally, SP is the radial distance, r, of the planet from
the sun, and angle PSΠ is the planet’s true anomaly, T .
D
P
L
A
Q
M
C
T
S
Π
K
Figure 4.3: A Ptolemaic orbit.
Let us draw the straight-line KSL parallel to QP, and passing through point S, and then complete
the rectangle PQKL. Simple geometry reveals that QK = PL = 2 e a sin M, KS = 2 e a cos M, and
SL = ρ − 2 e a cos M, where ρ = QP. The cosine rule applied to triangle CQP yields CP2 =
CQ2 + QP2 − 2 CQ QP cos M, or ρ2 − 2 e a cos M ρ − a2 (1 − e2) = 0, which can be solved to give
ρ/a = e cos M + (1 − e2 sin2 M)1/2. Moreover, SP2 = SL2 + PL2, which implies that
r
= [1 − 2 e cos M (1 − e2 sin2 M)1/2 + e2 + 2 e2 sin2 M]1/2.
a
(4.28)
72
MODERN ALMAGEST
Now, T = M + q, where q is angle PSL. However,
sin q =
PL
2 e sin M
.
=
2
SP
[1 − 2 e cos M (1 − e sin2 M)1/2 + e2 + 2 e2 sin2 M]1/2
(4.29)
Finally, expanding the previous two equations to second-order in the small parameter e, we
obtain
r
a
= 1 − e cos M + (3/2) e2 sin2 M,
(4.30)
T
= M + 2 e sin M + e2 sin 2M.
(4.31)
It can be seen, by comparison with Eqs. (4.22)–(4.23) and (4.26)–(4.27), that Ptolemy’s geometric
model of a heliocentric planetary orbit is significantly more accurate than Hipparchus’ model, since
the relative radial distance, r/a, and the true anomaly, T , in the former model both only deviate
from those in the (correct) Keplerian model to second-order in e.
4.5 Model of Copernicus
Copernicus’ geometric model of a heliocentric planetary orbit is illustrated in Fig. 4.4. The planet
P rotates on a circular epicycle YP whose center X moves around the sun on the eccentric circle
ΠXDA (only half of which is shown). The diameter ΠSCA is the effective major axis of the orbit,
where C is the geometric center of circle ΠXDA, and S the fixed position of the sun. When X is at
Π or A the planet is at its perihelion or aphelion points, respectively. The radius CX of circle ΠXDA
is the effective major radius, a, of the orbit. The distance SC is equal to (3/2) e a, where e is the
orbit’s effective eccentricity. Moreover, the radius XP of the epicycle is equal to (1/2) e a. The angle
XCΠ is identified with the mean anomaly, M, and increases linearly in time. In other words, as seen
from C, the center of the epicycle X moves uniformly around circle ΠXDA in a counterclockwise
direction. The angle PXY, where Y is point at which CX produced meets the epicycle, is equal to
the mean anomaly M. In other words, the planet P moves uniformly around the epicycle YP, in an
counterclockwise direction, at twice the speed that point X moves around circle ΠXDA. Finally, SP
is the radial distance, r, of the planet from the sun, and angle PSΠ is the planet’s true anomaly, T .
Let us draw the straight-line KSL parallel to CX, and passing through point S, and then complete the rectangle XCKL. Simple geometry reveals that CK = XL = (3/2) e a sin M, KS =
(3/2) e a cos M, and SL = a − (3/2) e a cos M. Let PZ be drawn normal to XY, and let it meet
KSL produced at point W. Simple geometry reveals that ZW = XL, ZP = (1/2) e a sin M, and
XZ = LW = (1/2) e a cos M. It follows that WP = ZW + ZP = XL + ZP = 2 e a sin M, and
SW = SL + LW = SL + XZ = a − e a cos M. Moreover, SP2 = SW 2 + WP2, which implies that
r
= (1 − 2 e cos M + e2 + 3 e2 sin2 M)1/2.
a
(4.32)
Now, T = M + q, where q is angle PSW. However,
sin q =
2 e sin M
WP
=
.
SP
(1 − 2 e cos M + e2 + 3 e2 sin2 M)1/2
(4.33)
Geometric Planetary Orbit Models
73
D
P
X
Y
Z
L
M
A
T
S
C
W
Π
K
Figure 4.4: A Copernican orbit.
Finally, expanding the previous two equations to second-order in the small parameter e, we
obtain
r
a
= 1 − e cos M + 2 e2 sin2 M,
(4.34)
T
= M + 2 e sin M + e2 sin 2M.
(4.35)
It can be seen, by comparison with Eqs. (4.22)–(4.23) and (4.30)–(4.31), that, as is the case for
Ptolemy’s model, both the relative radial distance, r/a, and the true anomaly, T , in Copernicus’
geometric model of a heliocentric planetary orbit only deviate from those in the (correct) Keplerian
model to second-order in e. However, the deviation in the Ptolemaic model is slightly smaller than
that in the Copernican model. To be more exact, the maximum deviation in r/a is (1/2) e2 in the
former model, and e2 in the latter. On the other hand, the maximum deviation in T is (1/4) e2 in
both models.
74
MODERN ALMAGEST
The Sun
75
5 The Sun
5.1 Determination of Ecliptic Longitude
Our solar longitude model is sketched in Figure 5.1. From a geocentric point of view, the sun, S,
appears to execute a (counterclockwise) Keplerian orbit of major radius a, and eccentricity e, about
the earth, G. As has already been mentioned, the circle traced out by the sun on the celestial sphere
is known as the ecliptic circle. This circle is inclined at 23◦ 26 ′ to the celestial equator, which is the
projection of the earth’s equator onto the celestial sphere. Suppose that the angle subtended at the
earth between the vernal equinox (i.e., the point at which the ecliptic crosses the celestial equator
from south to north) and the sun’s perigee (i.e., the point of closest approach to the earth) is ̟.
This angle is termed the longitude of the perigee, and is assumed to vary linearly with time: i.e.,
(5.1)
̟ = ̟0 + ̟1 (t − t0).
S
A
G
λ
T
̟
Π
Υ
Figure 5.1: The apparent orbit of the sun about the earth. Here, S, G, Π, A, ̟, T , λ, and Υ represent
the sun, earth, perigee, apogee, longitude of the perigee, true anomaly, ecliptic longitude, and vernal
equinox, respectively. View is from northern ecliptic pole. The sun orbits counterclockwise.
The sun’s ecliptic longitude is defined as the angle subtended at the earth between the vernal
equinox and the sun. Hence, from Fig. 5.1,
λ = ̟ + T,
(5.2)
where T is the true anomaly (see Cha. 4). By analogy, the mean longitude is written
λ̄ = ̟ + M,
(5.3)
76
MODERN ALMAGEST
where M is the mean anomaly (see Cha. 4). It follows from Eq. (4.23) that
λ = λ̄ + q,
(5.4)
q = 2 e sin M + (5/4) e2 sin 2M,
(5.5)
where
is called the equation of center. Note that λ, λ̄, T , and M are usually written as angles in the range
0◦ to 360◦ , whereas q is generally written as an angle in the range −180◦ to +180◦ .
The mean longitude increases uniformly with time (since both ̟ and M increase uniformly
with time) as
λ̄ = λ̄0 + n (t − t0),
(5.6)
where λ̄0 is termed the mean longitude at epoch, n the rate of motion in mean longitude, and t0 the
epoch. We can also write
M = M0 + ñ (t − t0),
(5.7)
where
M0 = λ̄0 − ̟0
(5.8)
ñ = n − ̟1
(5.9)
is called the mean anomaly at epoch, and
the rate of motion in mean anomaly.
Our procedure for determining the ecliptic longitude of the sun is described below. The requisite
orbital elements (i.e., e, n, ñ, λ̄0, and M0) for the J2000 epoch (i.e., 12:00 UT on January 1, 2000
CE, which corresponds to t0 = 2 451 545.0 JD) are listed in Table 5.1. These elements are calculated
on the assumption that the vernal equinox precesses at the uniform rate of −3.8246 × 10−5 ◦ /day.
The ecliptic longitude of the sun is specified by the following formulae:
λ̄ = λ̄0 + n (t − t0),
(5.10)
M = M0 + ñ (t − t0),
(5.11)
q = 2 e sin M + (5/4) e2 sin 2M,
(5.12)
λ = λ̄ + q.
(5.13)
These formulae are capable of matching NASA ephemeris data during the years 1995–2006 CE (see
http://ssd.jpl.nasa.gov/) with a mean error of 0.2 ′ and a maximum error of 0.7 ′ .
The ecliptic longitude of the sun can be calculated with the aid of Tables 5.3 and 5.4. Table 5.3
allows the mean longitude, λ̄, and mean anomaly, M, of the sun to be determined as functions of
time. Table 5.4 specifies the equation of center, q, as a function of the mean anomaly.
The procedure for using the tables is as follows:
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the sun’s ecliptic longitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t−t0,
where t0 = 2 451 545.0 is the epoch.
The Sun
77
2. Enter Table 5.3 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆λ̄ and ∆M. If ∆t is negative then the corresponding values are also negative. The
value of the mean longitude, λ̄, is the sum of all the ∆λ̄ values plus the value of λ̄ at the
epoch. Likewise, the value of the mean anomaly, M, is the sum of all the ∆M values plus the
value of M at the epoch. Add as many multiples of 360◦ to λ̄ and M as is required to make
them both fall in the range 0◦ to 360◦ . Round M to the nearest degree.
3. Enter Table 5.4 with the value of M and take out the corresponding value of the equation of
center, q, and the radial anomaly, ζ. (The latter step is only necessary if the ecliptic longitude
of the sun is to be used to determine that of a planet.) It is necessary to interpolate if M is
odd.
4. The ecliptic longitude, λ, is the sum of the mean longitude, λ̄, and the equation of center,
q. If necessary, convert λ into an angle in the range 0◦ to 360◦ . The decimal fraction can be
converted into arc minutes using Table 5.2. Round to the nearest arc minute.
Two examples of the use of this procedure are given below.
5.2 Example Longitude Calculations
Example 1: May 5, 2005 CE, 00:00 UT:
According to Tables 3.1–3.3, t = 2 453 495.5 JD. Hence, t − t0 = 2 453 495.5 − 2 451 545.0 = 1 950.5
JD. Making use of Table 5.3, we find:
t(JD)
λ̄(◦ )
M(◦ )
+1000
+900
+50
+.5
Epoch
265.647
167.083
49.280
0.493
280.458
762.961
42.961
265.600
167.040
49.280
0.493
357.588
840.001
120.001
Modulus
Rounding the mean anomaly to the nearest degree, we obtain M ≃ 120◦ . It follows from Table 5.4
that
q(120◦ ) = 1.641◦ ,
so
λ = λ̄ + q = 42.961 + 1.641 = 44.602 ≃ 44◦ 36 ′ .
Here, we have converted the decimal fraction into arc minutes using Table 5.2, and then rounded
the final result to the nearest arc minute.
Following the practice of the Ancient Greeks (and modern-day astrologers), we shall express
ecliptic longitudes in terms of the signs of the zodiac, which are listed in Sect. 2.6. The ecliptic
longitude 44◦ 36 ′ is conventionally written 14TA36: i.e., 14◦ 36 ′ into the sign of Taurus. Thus, we
conclude that the position of the sun at 00:00 UT on May 5, 2005 CE was 14TA36.
78
MODERN ALMAGEST
Example 2: December 25, 1800 CE, 00:00 UT:
According to Tables 3.1–3.3, t = 2 378 854.5 JD. Hence, t−t0 = 2 378 854.5−2 451 545.0 = −72 690.5
JD. Making use of Table 5.3, we find:
t(JD)
λ̄(◦ )
M(◦ )
-70,000
-2,000
-600
-90
-.5
Epoch
−235.315
−171.295
−231.388
−88.708
−0.493
280.458
−446.741
273.259
−232.017
−171.200
−231.360
−88.704
−0.493
357.588
−366.186
353.814
Modulus
We conclude that M ≃ 354◦ . From Table 5.4,
q(354◦ ) = −0.204◦ ,
so
λ = λ̄ + q = 273.259 − 0.204 = 273.055 ≃ 273◦ 03 ′ .
Thus, the position of the sun at 00:00 UT on December 25, 1800 CE was 3CP03.
5.3 Determination of Equinox and Solstice Dates
We can also use Tables 5.3 and 5.4 to calculate the dates of the equinoxes and solstices, and, hence,
the lengths of the seasons, in a given year. The vernal equinox (i.e., the point on the sun’s apparent
orbit at which it passes through the celestial equator from south to north) corresponds to λ = 0◦ ,
the summer solstice (i.e., the point at which the sun is furthest north of the celestial equator) to
λ = 90◦ , the autumnal equinox (i.e., the point at which the sun passes through the celestial equator
from north to south) to λ = 180◦ , and the winter solstice (i.e., the point at which the sun is furthest
south of the celestial equator) to λ = 270◦ —see Fig. 5.2. Furthermore, spring is defined as the
period between the spring equinox and the summer solstice, summer as the period between the
summer solstice and the autumnal equinox, autumn as the period between the autumnal equinox
and the winter solstice, and winter as the period between the winter solstice and the following
vernal equinox. Consider the year 2000 CE. For the case of the vernal equinox, we can first estimate
the time at which this event takes place by approximating the solar longitude as the mean solar
longitude: i.e.,
λ ≃ λ̄ = λ̄0 + n (t − t0) = 280.458 + 0.98564735 (t − t0),
We obtain
t ≃ t0 + (360 − 280.458)/0.98564735 ≃ t0 + 81 JD.
Calculating the true solar longitude at this time, using Tables 5.3 and 5.4, we get λ = 2.177◦ . Now,
the actual vernal equinox occurs when λ = 0◦ . Thus, a much better estimate for the date of the
The Sun
79
SS
93.6
92.8
A
C
AE
λ
G
VE
Π
88.9
89.9
WS
Figure 5.2: The sun’s apparent orbit around the earth, G, showing the vernal equinox (VE), summer
solstice (SS), autumnal equinox (AE), and winter solstice (WS). Here, λ, Π, A, and C are the ecliptic
longitude, perigee, apogee, and geometric center of the orbit, respectively. The lengths of the seasons
(in days) are indicated.
vernal equinox is
t = t0 + 81 − 2.177/0.98564735 ≃ t0 + 78.8 JD,
which corresponds to 7:00 UT on March 20. Similar calculations show that the summer solstice
takes place at
t = t0 + 171.6 JD,
corresponding to 2:00 UT on June 21, that the autumnal equinox takes place at
t = t0 + 265.2 JD,
corresponding to 17:00 UT on September 22, and that the winter solstice takes place at
t = t0 + 355.1 JD,
corresponding to 14:00 UT on December 21. Thus, the length of spring is 92.8 days, the length
of summer 93.6 days, and the length of autumn 89.9 days. Finally, the length of winter is the
length of the tropical year (i.e., the time period between successive vernal equinoxes), which is
360/0.98564735 = 325.24 days, minus the sum of the lengths of the other three seasons. This gives
88.9 days.
Figure 5.2 illustrates the relationship between the equinox and solstice points, and the lengths
of the seasons. The earth is displaced from the geometric center of the sun’s apparent orbit in
the direction of the solar perigee, which presently lies between the winter solstice and the vernal
80
MODERN ALMAGEST
equinox. This displacement (which is greatly exaggerated in the figure) has two effects. Firstly, it
causes the arc of the sun’s apparent orbit between the summer solstice and autumnal equinox to be
longer than that between the winter solstice and the vernal equinox. Secondly, it causes the sun to
appear to move faster in winter than in summer, in accordance with Kepler’s second law, since the
sun is closer to the earth in the former season. Both of these effects tend to lengthen summer, and
shorten winter. Hence, summer is presently the longest season, and winter the shortest.
5.4 Equation of Time
At any particular observation site on the earth’s surface, local noon is defined as the instant in
time when the sun culminates at the meridian. However, as a consequence of the inclination of
the ecliptic to the celestial equatior, as well as the uneven motion of the sun around the ecliptic,
the time interval between successive local noons, which is known as a solar day, is not constant,
but varies throughout the year. Hence, if we were to define a second as 1/86, 400 of a solar day
then the length of a second would also vary throughout the year, which is clearly undesirable. In
order to avoid this problem, astronomers have invented a fictitious body called the mean sun. The
mean sun travels around the celestial equator (from west to east) at a constant rate which is such
that it completes one orbit every tropical year. Moreover, the mean sun and the true sun coincide
at the spring equinox. Local mean noon at a particular observation site is defined as the instance
in time when the mean sun culminates at the meridian. Since the orbit of the mean sun is not
inclined to the celestial equator, and the mean sun travels around the celestial equator at a uniform
rate, the time interval between successive mean noons, which is known as a mean solar day, takes
the constant value of 24 hours, or 86,400 seconds, throughout the year. Universal time (UT) is
defined such that 12:00 UT coincides with mean noon every day at an observation site of terrestrial
longitude 0◦ . If we define local time (LT) as LT = UT − φ(◦ )/15◦ hrs., where φ is the terrestrial
longitude of the observation site, then 12:00 LT coincides with mean noon every day at a general
observation site on the earth’s surface.
According to the above definition, the right ascension, ᾱ, of the mean sun satisfies
ᾱ = λ̄,
(5.14)
where λ̄ is the sun’s mean ecliptic longitude. Moreover, it follows from Eqs. (2.16) and (5.4) that
the right ascension of the true sun is given by
tan α = cos ǫ tan(λ̄ + q),
(5.15)
where ǫ is the inclination of the ecliptic to the celestial equator, q(M) the sun’s equation of center,
and M its mean anomaly. Now, neglecting the small time variation of the longitude of the sun’s
perigee [i.e., setting ̟1 = 0 in Eq. (5.1)], we can write [see Eqs. (5.6), (5.7), and (5.9), as well as
Table 5.1]
M = λ̄ + M0 − λ̄0 = λ̄ + 77.213◦ .
(5.16)
It follows that, to first order in the solar eccentricity, e, we have
∆α = ᾱ − α = λ − tan−1(cos ǫ tan λ) − 2 e sin M,
(5.17)
M = λ + 77.213◦ .
(5.18)
where
The Sun
81
Now,
∆t = ∆α(◦ )/15◦
(5.19)
represents the time difference (in hours) between local noon and mean local noon (since right
ascension crosses the meridian at the uniform rate of 15◦ an hour), and is known as the equation of
time. If ∆t is positive then local noon occurs before mean local noon, and vice versa.
The equation of time specifies the difference between time calculated using a sundial or sextant—
which is known as solar time—and time obtained from an accurate clock—which is known as mean
solar time. Table 5.5 shows the equation of time as a function of the sun’s ecliptic longitude. It
can be seen that the difference between solar time and mean solar time can be as much as 16
minutes, and attains its maximum value between the autumnal equinox and the winter solstice,
and its minimum value between the winter solstice and vernal equinox.
82
MODERN ALMAGEST
Object
Mercury
Venus
Sun
Mars
Jupiter
Saturn
a (AU)
0.387098
0.723334
1.000000
1.523706
5.202873
9.536651
e
0.205636
0.006777
0.016711
0.093394
0.048386
0.053862
n (◦ /day)
4.09237703
1.60216872
0.98564735
0.52407118
0.08312507
0.03350830
ñ (◦ /day)
4.09233439
1.60213040
0.98560025
0.52402076
0.08308100
0.03348152
λ̄0 (◦ )
252.087
181.973
280.458
355.460
34.365
50.059
M0 (◦ )
174.693
49.237
357.588
19.388
19.348
317.857
Table 5.1: Keplerian orbital elements for the sun and the five visible planets at the J2000 epoch (i.e.,
12:00 UT, January 1, 2000 CE, which corresponds to t0 = 2 451 545.0 JD). The elements are optimized for use in the time period 1800 CE to 2050 CE. Source: Jet Propulsion Laboratory (NASA),
http://ssd.jpl.nasa.gov/. The motion rates have been converted into tropical motion rates assuming a uniform precession of the equinoxes of 3.8246 × 10−5 ◦ /day.
The Sun
83
00.0 ′
00.2 ′
00.4 ′
00.6 ′
00.8 ′
01.0 ′
01.2 ′
01.4 ′
01.6 ′
01.8 ′
02.0 ′
02.2 ′
02.4 ′
02.6 ′
02.8 ′
03.0 ′
03.2 ′
03.4 ′
03.6 ′
03.8 ′
04.0 ′
04.2 ′
04.4 ′
04.6 ′
04.8 ′
05.0 ′
05.2 ′
05.4 ′
05.6 ′
05.8 ′
06.0 ′
06.2 ′
06.4 ′
06.6 ′
06.8 ′
07.0 ′
07.2 ′
07.4 ′
07.6 ′
07.8 ′
08.0 ′
08.2 ′
08.4 ′
08.6 ′
08.8 ′
09.0 ′
09.2 ′
09.4 ′
09.6 ′
09.8 ′
.000
.003
.007
.010
.013
.017
.020
.023
.027
.030
.033
.037
.040
.043
.047
.050
.053
.057
.060
.063
.067
.070
.073
.077
.080
.083
.087
.090
.093
.097
.100
.103
.107
.110
.113
.117
.120
.123
.127
.130
.133
.137
.140
.143
.147
.150
.153
.157
.160
.163
10.0 ′
10.2 ′
10.4 ′
10.6 ′
10.8 ′
11.0 ′
11.2 ′
11.4 ′
11.6 ′
11.8 ′
12.0 ′
12.2 ′
12.4 ′
12.6 ′
12.8 ′
13.0 ′
13.2 ′
13.4 ′
13.6 ′
13.8 ′
14.0 ′
14.2 ′
14.4 ′
14.6 ′
14.8 ′
15.0 ′
15.2 ′
15.4 ′
15.6 ′
15.8 ′
16.0 ′
16.2 ′
16.4 ′
16.6 ′
16.8 ′
17.0 ′
17.2 ′
17.4 ′
17.6 ′
17.8 ′
18.0 ′
18.2 ′
18.4 ′
18.6 ′
18.8 ′
19.0 ′
19.2 ′
19.4 ′
19.6 ′
19.8 ′
.167
.170
.173
.177
.180
.183
.187
.190
.193
.197
.200
.203
.207
.210
.213
.217
.220
.223
.227
.230
.233
.237
.240
.243
.247
.250
.253
.257
.260
.263
.267
.270
.273
.277
.280
.283
.287
.290
.293
.297
.300
.303
.307
.310
.313
.317
.320
.323
.327
.330
20.0 ′
20.2 ′
20.4 ′
20.6 ′
20.8 ′
21.0 ′
21.2 ′
21.4 ′
21.6 ′
21.8 ′
22.0 ′
22.2 ′
22.4 ′
22.6 ′
22.8 ′
23.0 ′
23.2 ′
23.4 ′
23.6 ′
23.8 ′
24.0 ′
24.2 ′
24.4 ′
24.6 ′
24.8 ′
25.0 ′
25.2 ′
25.4 ′
25.6 ′
25.8 ′
26.0 ′
26.2 ′
26.4 ′
26.6 ′
26.8 ′
27.0 ′
27.2 ′
27.4 ′
27.6 ′
27.8 ′
28.0 ′
28.2 ′
28.4 ′
28.6 ′
28.8 ′
29.0 ′
29.2 ′
29.4 ′
29.6 ′
29.8 ′
.333
.337
.340
.343
.347
.350
.353
.357
.360
.363
.367
.370
.373
.377
.380
.383
.387
.390
.393
.397
.400
.403
.407
.410
.413
.417
.420
.423
.427
.430
.433
.437
.440
.443
.447
.450
.453
.457
.460
.463
.467
.470
.473
.477
.480
.483
.487
.490
.493
.497
30.0 ′
30.2 ′
30.4 ′
30.6 ′
30.8 ′
31.0 ′
31.2 ′
31.4 ′
31.6 ′
31.8 ′
32.0 ′
32.2 ′
32.4 ′
32.6 ′
32.8 ′
33.0 ′
33.2 ′
33.4 ′
33.6 ′
33.8 ′
34.0 ′
34.2 ′
34.4 ′
34.6 ′
34.8 ′
35.0 ′
35.2 ′
35.4 ′
35.6 ′
35.8 ′
36.0 ′
36.2 ′
36.4 ′
36.6 ′
36.8 ′
37.0 ′
37.2 ′
37.4 ′
37.6 ′
37.8 ′
38.0 ′
38.2 ′
38.4 ′
38.6 ′
38.8 ′
39.0 ′
39.2 ′
39.4 ′
39.6 ′
39.8 ′
.500
.503
.507
.510
.513
.517
.520
.523
.527
.530
.533
.537
.540
.543
.547
.550
.553
.557
.560
.563
.567
.570
.573
.577
.580
.583
.587
.590
.593
.597
.600
.603
.607
.610
.613
.617
.620
.623
.627
.630
.633
.637
.640
.643
.647
.650
.653
.657
.660
.663
40.0 ′
40.2 ′
40.4 ′
40.6 ′
40.8 ′
41.0 ′
41.2 ′
41.4 ′
41.6 ′
41.8 ′
42.0 ′
42.2 ′
42.4 ′
42.6 ′
42.8 ′
43.0 ′
43.2 ′
43.4 ′
43.6 ′
43.8 ′
44.0 ′
44.2 ′
44.4 ′
44.6 ′
44.8 ′
45.0 ′
45.2 ′
45.4 ′
45.6 ′
45.8 ′
46.0 ′
46.2 ′
46.4 ′
46.6 ′
46.8 ′
47.0 ′
47.2 ′
47.4 ′
47.6 ′
47.8 ′
48.0 ′
48.2 ′
48.4 ′
48.6 ′
48.8 ′
49.0 ′
49.2 ′
49.4 ′
49.6 ′
49.8 ′
.667
.670
.673
.677
.680
.683
.687
.690
.693
.697
.700
.703
.707
.710
.713
.717
.720
.723
.727
.730
.733
.737
.740
.743
.747
.750
.753
.757
.760
.763
.767
.770
.773
.777
.780
.783
.787
.790
.793
.797
.800
.803
.807
.810
.813
.817
.820
.823
.827
.830
50.0 ′
50.2 ′
50.4 ′
50.6 ′
50.8 ′
51.0 ′
51.2 ′
51.4 ′
51.6 ′
51.8 ′
52.0 ′
52.2 ′
52.4 ′
52.6 ′
52.8 ′
53.0 ′
53.2 ′
53.4 ′
53.6 ′
53.8 ′
54.0 ′
54.2 ′
54.4 ′
54.6 ′
54.8 ′
55.0 ′
55.2 ′
55.4 ′
55.6 ′
55.8 ′
56.0 ′
56.2 ′
56.4 ′
56.6 ′
56.8 ′
57.0 ′
57.2 ′
57.4 ′
57.6 ′
57.8 ′
58.0 ′
58.2 ′
58.4 ′
58.6 ′
58.8 ′
59.0 ′
59.2 ′
59.4 ′
59.6 ′
59.8 ′
Table 5.2: Arc minute to decimal fraction conversion table.
.833
.837
.840
.843
.847
.850
.853
.857
.860
.863
.867
.870
.873
.877
.880
.883
.887
.890
.893
.897
.900
.903
.907
.910
.913
.917
.920
.923
.927
.930
.933
.937
.940
.943
.947
.950
.953
.957
.960
.963
.967
.970
.973
.977
.980
.983
.987
.990
.993
.997
84
MODERN ALMAGEST
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
136.474
272.947
49.421
185.894
322.367
98.841
235.315
11.788
148.262
136.002
272.005
48.007
184.010
320.012
96.015
232.017
8.020
144.022
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
265.647
171.295
76.942
342.589
248.237
153.884
59.531
325.179
230.826
265.600
171.200
76.801
342.401
248.001
153.601
59.202
324.802
230.402
100
200
300
400
500
600
700
800
900
98.565
197.129
295.694
34.259
132.824
231.388
329.953
68.518
167.083
98.560
197.120
295.680
34.240
132.800
231.360
329.920
68.480
167.040
10
20
30
40
50
60
70
80
90
9.856
19.713
29.569
39.426
49.282
59.139
68.995
78.852
88.708
9.856
19.712
29.568
39.424
49.280
59.136
68.992
78.848
88.704
1
2
3
4
5
6
7
8
9
0.986
1.971
2.957
3.943
4.928
5.914
6.900
7.885
8.871
0.986
1.971
2.957
3.942
4.928
5.914
6.899
7.885
8.870
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.099
0.197
0.296
0.394
0.493
0.591
0.690
0.789
0.887
0.099
0.197
0.296
0.394
0.493
0.591
0.690
0.788
0.887
Table 5.3: Mean motion of the sun. Here, ∆t = t − t0, ∆λ̄ = λ̄ − λ̄0, and ∆M = M − M0. At epoch
(t0 = 2 451 545.0 JD), λ̄0 = 280.458◦ , and M0 = 357.588◦ .
The Sun
85
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
0.000
0.068
0.136
0.204
0.272
0.339
0.406
0.473
0.538
0.604
0.668
0.731
0.794
0.855
0.916
0.975
1.033
1.089
1.145
1.198
1.251
1.301
1.350
1.397
1.443
1.487
1.528
1.568
1.606
1.642
1.676
1.707
1.737
1.764
1.789
1.812
1.833
1.851
1.867
1.881
1.893
1.902
1.909
1.913
1.915
1.915
1.671
1.670
1.667
1.662
1.654
1.645
1.633
1.620
1.604
1.587
1.567
1.545
1.522
1.497
1.469
1.440
1.409
1.377
1.342
1.306
1.269
1.229
1.189
1.146
1.103
1.058
1.011
0.964
0.915
0.865
0.815
0.763
0.710
0.656
0.602
0.547
0.491
0.435
0.378
0.321
0.263
0.205
0.147
0.089
0.030
-0.028
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
1.915
1.912
1.907
1.900
1.891
1.879
1.865
1.849
1.830
1.809
1.787
1.762
1.735
1.705
1.674
1.641
1.606
1.569
1.530
1.490
1.447
1.403
1.358
1.310
1.261
1.211
1.160
1.107
1.052
0.997
0.940
0.882
0.824
0.764
0.703
0.642
0.580
0.517
0.454
0.390
0.326
0.261
0.196
0.131
0.065
0.000
-0.028
-0.086
-0.144
-0.202
-0.260
-0.317
-0.374
-0.431
-0.486
-0.542
-0.596
-0.650
-0.703
-0.755
-0.806
-0.856
-0.906
-0.954
-1.001
-1.046
-1.091
-1.134
-1.175
-1.216
-1.254
-1.292
-1.327
-1.362
-1.394
-1.425
-1.454
-1.482
-1.507
-1.531
-1.553
-1.574
-1.592
-1.608
-1.623
-1.636
-1.647
-1.655
-1.662
-1.667
-1.670
-1.671
180
182
184
186
188
190
192
194
196
198
200
202
204
206
208
210
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246
248
250
252
254
256
258
260
262
264
266
268
270
0.000
-0.065
-0.131
-0.196
-0.261
-0.326
-0.390
-0.454
-0.517
-0.580
-0.642
-0.703
-0.764
-0.824
-0.882
-0.940
-0.997
-1.052
-1.107
-1.160
-1.211
-1.261
-1.310
-1.358
-1.403
-1.447
-1.490
-1.530
-1.569
-1.606
-1.641
-1.674
-1.705
-1.735
-1.762
-1.787
-1.809
-1.830
-1.849
-1.865
-1.879
-1.891
-1.900
-1.907
-1.912
-1.915
-1.671
-1.670
-1.667
-1.662
-1.655
-1.647
-1.636
-1.623
-1.608
-1.592
-1.574
-1.553
-1.531
-1.507
-1.482
-1.454
-1.425
-1.394
-1.362
-1.327
-1.292
-1.254
-1.216
-1.175
-1.134
-1.091
-1.046
-1.001
-0.954
-0.906
-0.856
-0.806
-0.755
-0.703
-0.650
-0.596
-0.542
-0.486
-0.431
-0.374
-0.317
-0.260
-0.202
-0.144
-0.086
-0.028
270
272
274
276
278
280
282
284
286
288
290
292
294
296
298
300
302
304
306
308
310
312
314
316
318
320
322
324
326
328
330
332
334
336
338
340
342
344
346
348
350
352
354
356
358
360
-1.915
-1.915
-1.913
-1.909
-1.902
-1.893
-1.881
-1.867
-1.851
-1.833
-1.812
-1.789
-1.764
-1.737
-1.707
-1.676
-1.642
-1.606
-1.568
-1.528
-1.487
-1.443
-1.397
-1.350
-1.301
-1.251
-1.198
-1.145
-1.089
-1.033
-0.975
-0.916
-0.855
-0.794
-0.731
-0.668
-0.604
-0.538
-0.473
-0.406
-0.339
-0.272
-0.204
-0.136
-0.068
-0.000
-0.028
0.030
0.089
0.147
0.205
0.263
0.321
0.378
0.435
0.491
0.547
0.602
0.656
0.710
0.763
0.815
0.865
0.915
0.964
1.011
1.058
1.103
1.146
1.189
1.229
1.269
1.306
1.342
1.377
1.409
1.440
1.469
1.497
1.522
1.545
1.567
1.587
1.604
1.620
1.633
1.645
1.654
1.662
1.667
1.670
1.671
Table 5.4: Anomalies of the sun.
86
MODERN ALMAGEST
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Aries
∆t
-07m 28s
-06m 51s
-06m 15s
-05m 38s
-05m 01s
-04m 24s
-03m 48s
-03m 12s
-02m 36s
-02m 01s
-01m 28s
+00m 55s
+00m 23s
+00m 06s
+00m 34s
+01m 02s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Taurus
∆t
+01m 02s
+01m 27s
+01m 50s
+02m 12s
+02m 31s
+02m 48s
+03m 03s
+03m 16s
+03m 26s
+03m 34s
+03m 40s
+03m 43s
+03m 43s
+03m 41s
+03m 37s
+03m 30s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Gemini
∆t
+03m 30s
+03m 21s
+03m 10s
+02m 56s
+02m 41s
+02m 23s
+02m 04s
+01m 43s
+01m 20s
+00m 57s
+00m 32s
+00m 06s
+00m 20s
+00m 47s
-01m 14s
-01m 42s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Libra
∆t
+07m 28s
+08m 10s
+08m 53s
+09m 34s
+10m 14s
+10m 53s
+11m 30s
+12m 06s
+12m 41s
+13m 13s
+13m 43s
+14m 12s
+14m 38s
+15m 01s
+15m 22s
+15m 40s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Scorpio
∆t
+15m 40s
+15m 55s
+16m 08s
+16m 17s
+16m 23s
+16m 26s
+16m 26s
+16m 23s
+16m 16s
+16m 06s
+15m 52s
+15m 36s
+15m 15s
+14m 52s
+14m 25s
+13m 55s
Sagittarius
λ
∆t
◦
00
+13m 55s
◦
02
+13m 22s
◦
04
+12m 46s
◦
06
+12m 08s
◦
08
+11m 26s
◦
10
+10m 42s
◦
12
+09m 55s
◦
14
+09m 07s
◦
16
+08m 16s
◦
18
+07m 23s
◦
20
+06m 29s
◦
22
+05m 33s
◦
24
+04m 37s
◦
26
+03m 39s
◦
28
+02m 41s
◦
30
+01m 42s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Cancer
∆t
-01m 42s
-02m 09s
-02m 36s
-03m 03s
-03m 29s
-03m 53s
-04m 17s
-04m 39s
-05m 00s
-05m 19s
-05m 35s
-05m 50s
-06m 03s
-06m 14s
-06m 22s
-06m 27s
Capricorn
λ
∆t
◦
00
+01m 42s
◦
02
+00m 43s
◦
04
+00m 15s
◦
06
-01m 13s
◦
08
-02m 11s
◦
10
-03m 07s
◦
12
-04m 03s
◦
14
-04m 57s
◦
16
-05m 50s
◦
18
-06m 41s
◦
20
-07m 30s
◦
22
-08m 16s
◦
24
-09m 01s
◦
26
-09m 42s
◦
28
-10m 22s
◦
30
-10m 58s
Leo
∆t
-06m 27s
-06m 30s
-06m 31s
-06m 29s
-06m 24s
-06m 16s
-06m 06s
-05m 54s
-05m 38s
-05m 20s
-05m 00s
-04m 37s
-04m 12s
-03m 45s
-03m 15s
-02m 44s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Virgo
∆t
-02m 44s
-02m 11s
-01m 36s
+00m 59s
+00m 21s
+00m 18s
+00m 58s
+01m 40s
+02m 22s
+03m 05s
+03m 49s
+04m 32s
+05m 16s
+06m 00s
+06m 44s
+07m 28s
Aquarius
λ
∆t
◦
00
-10m 58s
◦
02
-11m 32s
◦
04
-12m 02s
◦
06
-12m 30s
◦
08
-12m 54s
◦
10
-13m 16s
◦
12
-13m 34s
◦
14
-13m 49s
◦
16
-14m 01s
◦
18
-14m 09s
◦
20
-14m 15s
◦
22
-14m 17s
◦
24
-14m 17s
◦
26
-14m 13s
◦
28
-14m 07s
◦
30
-13m 58s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Pisces
∆t
-13m 58s
-13m 46s
-13m 31s
-13m 14s
-12m 55s
-12m 33s
-12m 10s
-11m 44s
-11m 17s
-10m 48s
-10m 17s
-09m 45s
-09m 12s
-08m 38s
-08m 03s
-07m 28s
λ
00◦
02◦
04◦
06◦
08◦
10◦
12◦
14◦
16◦
18◦
20◦
22◦
24◦
26◦
28◦
30◦
Table 5.5: The equation of time. The superscripts m and s denote minutes and seconds.
The Moon
87
6 The Moon
6.1 Determination of Ecliptic Longitude
The orbit of the moon around the earth is strongly perturbed by the gravitational influence of the
sun. It follows that we cannot derive an accurate lunar longitude model from Keplerian orbit theory
alone. Instead, we shall employ a greatly simplified version of modern lunar theory. According to
such theory, the time variation of the ecliptic longitude of the moon is fairly well represented by
the following formulae (see http://jgiesen.de/moonmotion/index.html, or Astronomical Algorithms, J. Meeus, Willmann-Bell, 1998):
λ̄ = λ̄0 + n (t − t0),
(6.1)
M = M0 + ñ (t − t0),
(6.2)
F̄ = F̄0 + n̆ (t − t0),
(6.3)
D̃ = λ̄ − λS,
(6.4)
q1 = 2 e sin M + 1.430 e2 sin 2M,
(6.5)
q2 = 0.422 e sin(2D̃ − M),
(6.6)
q3 = 0.211 e (sin 2D̃ − 0.066 sin D̃),
(6.7)
q4 = −0.051 e sin MS,
(6.8)
q5 = −0.038 e sin 2F̄,
(6.9)
λ = λ̄ + q1 + q2 + q3 + q4 + q5.
(6.10)
Here, λS and MS are the longitude and mean anomaly of the sun, respectively. Moreover, e, λ, λ̄, F̄,
and qi are the eccentricity, longitude, mean longitude, mean argument of latitude, and ith anomaly
of the moon, respectively. The moon’s first anomaly is due to the eccentricity of its orbit, and is
very similar in form to that obtained from Keplerian orbit theory (see Cha. 4). The moon’s second,
third, and fourth anomalies are knows as evection, variation, and the annual inequality, respectively,
and originate from the perturbing influence of the sun. Finally, the moon’s fifth anomaly is called
the reduction to the ecliptic, and is a consequence of the fact that the moon’s orbit is slightly tilted
with respect to the plane of the ecliptic. Note that Ptolemy’s lunar theory only takes the first two
lunar anomalies into account. The moon’s orbital elements—e, n, ñ, n̆, λ̄0, M0, and F0—for the
J2000 epoch are listed in Table 6.1. Note that the lunar perigee precesses in the direction of the
moon’s orbital motion at the rate of n − ñ = 0.11140 ◦ per day, or 360◦ in 8.85 years. This very large
precession rate (more than 2000 times the corresponding precession rate for the sun’s apparent
orbit) is another consequence of the strong perturbing influence of the sun on the moon’s orbit.
The above formulae are capable of matching NASA ephemeris data during the years 1995–2006 CE
with a mean error of 5 ′ and a maximum error of 14 ′ .
The ecliptic longitude of the moon can be calculated with the aid of Tables 6.2 and 6.3. Table 6.2
allows the lunar mean longitude, λ̄, mean anomaly, M, and mean argument of latitude, F̄, to be
determined as functions of time. Table 6.3 specifies the lunar anomalies, q1–q5, as functions of
their various arguments.
88
MODERN ALMAGEST
The procedure for using the tables is as follows:
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the moon’s ecliptic longitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t =
t − t0, where t0 = 2 451 545.0 is the epoch.
2. Calculate the ecliptic longitude, λS, and the mean anomaly, MS, of the sun using the procedure set out in Sect. 5.1.
3. Enter Table 6.2 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆λ̄, ∆M, and ∆F̄. If ∆t is negative then the values are minus those shown in the
table. The value of the mean longitude, λ̄, is the sum of all the ∆λ̄ values plus the value of λ̄
at the epoch. Likewise, the value of the mean anomaly, M, is the sum of all the ∆M values
plus the value of M at the epoch. Finally, the value of the mean argument of latitude, F̄, is
the sum of all the ∆F̄ values plus the value of F̄ at the epoch. Add as many multiples of 360◦
to λ̄, M, and F̄ as is required to make them all fall in the range 0◦ to 360◦ .
4. Form D̃ = λ̄ − λS.
5. Form the five arguments a1 = M, a2 = 2D̃ − M, a3 = D̃, a4 = MS, a5 = 2F̄. Add as many
multiples of 360◦ to the arguments as is required to make them all fall in the range 0◦ to 360◦ .
Round each argument to the nearest degree.
6. Enter Table 6.3 with the value of each of the five arguments a1–a5 and take out the value of
each of the five corresponding anomalies q1–q5. It is necessary to interpolate if the arguments
are odd.
7. The moon’s ecliptic longitude is given by λ = λ̄ + q1 + q2 + q3 + q4 + q5. If necessary, convert λ
into an angle in the range 0◦ to 360◦ . The decimal fraction can be converted into arc minutes
using Table 5.2. Round to the nearest arc minute.
Two examples of the use of this procedure are given below.
6.2 Example Longitude Calculations
Example 1: May 5, 2005 CE, 00:00 UT:
From Sect. 5.1, t − t0 = 1950.5 JD, λS = 44.602◦ , and MS = 120.001◦ . Making use of Table 6.2, we
find:
t(JD)
λ̄(◦ )
M(◦ )
F̄(◦ )
+1000
+900
+50
+.5
Epoch
216.396
338.757
298.820
6.588
218.322
1078.883
358.883
104.993
238.494
293.250
6.532
134.916
778.185
58.185
269.350
26.415
301.468
6.615
93.284
697.132
337.132
Modulus
The Moon
89
It follows that
D̃ = λ̄ − λS = 358.883 − 44.602 = 314.281◦ .
Thus,
a1 = M ≃ 58◦ ,
a2 = 2D̃ − M = 2 × 314.281 − 58.185 = 570.377 ≃ 210◦ ,
a3 = D̃ ≃ 314◦ ,
a4 = MS ≃ 120◦ ,
a5 = 2F̄ = 2 × 337.132 = 674.264 ≃ 314◦ .
Table 6.3 yields
q1(a1) = 5.555◦ , q2(a2) = −0.663◦ , q3(a3) = −0.631◦ ,
q4(a4) = −0.139◦ , q5(a5) = 0.086◦ .
Hence,
λ = λ̄ + q1 + q2 + q3 + q4 + q5 = 358.883 + 5.555 − 0.663 − 0.631 − 0.139 + 0.086 = 363.091◦ ,
or
λ = 3.091 ≃ 3◦ 05 ′ .
Thus, the ecliptic longitude of the moon at 00:00 UT on May 5, 2005 CE was 3AR05.
Example 2: December 25, 1800 CE, 00:00 UT:
From Sect. 5.1, t − t0 = −72 690.5 JD, λS = 273.055◦ , and MS = 353.814◦ . Making use of Table 6.2, we find:
t(JD)
λ̄(◦ )
M(◦ )
F̄(◦ )
-70,000
-2,000
-600
-90
-.5
Epoch
−27.752
−72.793
−345.838
−105.876
−6.588
218.322
−340.525
19.475
−149.506
−209.986
−278.996
−95.849
−6.532
134.916
−605.953
114.047
−134.519
−178.701
−17.610
−110.642
−6.615
93.284
−354.803
5.197
Modulus
It follows that
D̃ = λ̄ − λS = 19.475 − 273.055 = −253.580◦ .
Thus,
a1 = M ≃ 114◦ ,
a2 = 2D̃ − M = −2 × 253.580 − 114.047 = −621.207 ≃ 99◦ ,
a3 = D̃ ≃ 106◦ ,
a4 = MS ≃ 354◦ ,
a5 = 2F̄ = 2 × 5.197 = 10.394 ≃ 10◦ .
Table 6.3 yields
q1(a1) = 5.562◦ , q2(a2) = 1.311◦ , q3(a3) = −0.394◦ ,
q4(a4) = 0.017◦ , q5(a5) = −0.021◦ .
90
MODERN ALMAGEST
Hence,
λ = λ̄ + q1 + q2 + q3 + q4 + q5 = 19.475 + 5.562 + 1.311 − 0.394 + 0.017 − 0.021 = 25.950◦ ,
or
λ = 25.950 ≃ 25◦ 57 ′ .
Thus, the ecliptic longitude of the moon at 00:00 UT on December 25, 1800 CE was 25AR57.
L
M
N
F
G
Ω
M′
N′
Υ
Figure 6.1: The orbit of the moon about the earth. Here, G, L, N, N ′ , Ω, F, and Υ represent the earth,
moon, ascending node, descending node, longitude of the ascending node, argument of latitude, and
vernal equinox, respectively. View is from northern ecliptic pole. The moon orbits counterclockwise.
6.3 Determination of Ecliptic Latitude
A model of the moon’s ecliptic latitude is needed in order to predict the occurrence of solar and
lunar eclipses. Figure 6.1 shows the moon’s orbit about the earth. The plane of this orbit is fixed,
but slightly tilted with respect to the plane of the ecliptic (i.e., the plane of the sun’s apparent
orbit about the earth). Let the two planes intersect along the line of nodes, NGN ′ . Here, N is
the point at which the orbit crosses the ecliptic plane from south to north (in the direction of the
moon’s orbital motion), and is termed the ascending node. Likewise, N ′ is the point at which the
orbit crosses the ecliptic plane from north to south, and is called the descending node. Incidentally,
the line of nodes must pass through point G, since the earth is common to the ecliptic plane and
the plane of the lunar orbit. The angle, Ω, subtended between the radius vector GΥ, connecting
the earth to the vernal equinox, and the line GN, is known as the longitude of the ascending node.
Note, incidentally, that the ascending node precessess in the opposite direction to the moon’s orbital
motion at the rate n̆ − n = 5.2954 × 10−2 ◦ per day, or 360◦ in 18.6 years. This unusually large
precession rate is another consequence of the sun’s strong perturbing influence on the moon’s orbit.
Let the line MGM ′ lie in the plane of the moon’s orbit such that it is perpendicular to NGN ′ . The
The Moon
91
inclination, i, of the moon’s orbital plane is the angle that GM subtends with its projection onto
the ecliptic plane. Likewise, the moon’s ecliptic longitude, β, is the angle that GL subtends with its
projection onto the ecliptic plane. Simple geometry yields sin β = sin i sin F, where F is the angle
between GN and GL. This angle is termed the argument of latitude. Now, it is easily seen that
F ≃ λ − Ω, where λ is the moon’s ecliptic longitude (i.e., the angle subtended between GΥ and
GL). Here, we are assuming that the orbital inclination i is relatively small. The mean argument of
latitude is defined F̄ = λ̄ − Ω. Hence, our model for the moon’s ecliptic latitude becomes
F = F̄ + q1 + q2 + q3 + q4 + q5,
sin β = sin i sin F.
(6.11)
(6.12)
The value of the lunar orbital inclination, i, for the J2000 epoch is specified in Table 6.1. The above
model is capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean
error of 6 ′ , and a maximum error of 11 ′ .
The ecliptic latitude of the moon can be calculated with the aid of Table 6.4. The procedure for
using this table is as follows:
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the moon’s ecliptic latitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t−t0,
where t0 = 2 451 545.0 is the epoch.
2. Calculate the lunar mean argument of latitude, F̄, and the five lunar anomalies, q1–q5, using
the procedure outlined earlier in this section.
3. Form the argument F = F̄ + q1 + q2 + q3 + q4 + q5. Add as many multiples of 360◦ to F as is
required to make it fall in the range 0◦ to 360◦ . Round F to the nearest degree.
4. Enter Table 6.4 with the value of F and take out the lunar ecliptic latitude, β. It is necessary
to interpolate if F is odd.
For example, we have already seen that at 00:00 UT on May 5, 2005 CE the lunar mean argument of latitude, and the lunar anomalies, were F̄ = 337.132◦ , and q1 = 5.555◦ , q2 = −0.663◦ ,
q3 = −0.631◦ , q4 = −0.139◦ , and q5 = 0.086◦ , respectively. Hence, F = F̄ + q1 + q2 + q3 + q4 + q5 =
337.132 + 5.555 − 0.663 − 0.631 − 0.139 + 0.086 ≃ 341◦ . Thus, according to Table 6.4, the ecliptic
latitude of the moon at 00:00 UT on May 5, 2005 CE was −1.680◦ ≃ −1◦ 41 ′ .
6.4 Lunar Parallax
Now, it turns out that the moon is sufficiently close to the earth that its position in the sky is
significantly modified by parallax. All of our previous analysis applies to a hypothetical observer
situated at the center of the earth. Consider a real observer situated on the earth’s surface. It
can be seen from Fig. 6.2 that the altitude of the moon is a ′ for the real observer, and a for the
hypothetical observer. Simple trigonometry reveals that a ′ = a − δa, which implies that the real
observer sees the moon at a lower altitude than the hypothetical observer. Let R be the radius of
the earth, and r the distance from the center of the earth to the moon. More simple trigonometry
yields
R
sin δa = cos a ′ .
(6.13)
r
92
MODERN ALMAGEST
L
δa
X
a′
r
R
a
C
Figure 6.2: The moon, L, as viewed by a hypothetical observer, C, at the center of the earth, and a real
observer, X, on the surface of the earth.
Let us assume that the moon’s orbit is elliptical to first order in its eccentricity. It follows, from
Cha. 4, that
r ≃ aM (1 − e cos M),
(6.14)
where aM, e, and M are major radius, eccentricity, and mean anomaly of the lunar orbit. Assuming
that δa is small, we obtain
δa ≃ δa0 cos a (1 + e cos M),
(6.15)
where δa0 = R/aM = 0.0166 = 56.98 ′ (since R = 6371 km and aM = 384, 399 km).
According to Eq. (6.15), lunar parallax can be written in the form
δa = δ(a) [1 + ζ(M)],
(6.16)
where a, a − δa, and M are the moon’s geocentric altitude (i.e., the altitude seen from the center
of the earth), true altitude, and mean anomaly, respectively. The functions δ(a) = δa0 cos a and
ζ(M) = e cos M are tabulated in Table 6.5. It can be seen from the table that lunar parallax
increases with decreasing lunar altitude, reaching a maximum value of about 57 ′ when the moon
is close to the horizon. For example, if a = 44◦ 00 ′ and M = 100◦ then Table 6.5 yields δ = 41.050 ′
and ζ = −0.00953. Hence, δa = 41.050 (1 − 0.00953) ≃ 41 ′ , and the true altitude of the moon
becomes 43◦ 19 ′ .
It now remains to investigate how parallax affects the moon’s ecliptic longitude and latitude.
Figure 6.3 shows a detail of Fig. 2.11. Point Y is the moon’s geocentric position on the celestial
sphere. DB is a line passing through this point which is parallel to the local ecliptic circle, whereas
ZC is a small section of an altitude circle passing through Y. The angle subtended between the
ecliptic and the altitude circle is the parallactic angle, µ. Let F be the true position of the moon. It
follows that δa = YF. The changes in the moon’s ecliptic longitude and latitude are δλ = YE and
−δβ = EF, respectively. Here, we are considering the case where increasing altitude corresponds to
The Moon
93
Z
a
D
β
µ
E
Y
λ
B
F
C
Figure 6.3: Parallactic shifts in the moon’s ecliptic longitude and latitude.
increasing ecliptic latitude. Assuming that the arcs δa, δλ, and δβ are all fairly small, the triangle
YEF can be treated as a plane triangle. Hence, we obtain
δλ = −δa cos µ,
(6.17)
δβ = −δa sin µ.
(6.18)
As is easily demonstrated, the above formulae also apply to the case in which increasing altitude
corresponds to decreasing ecliptic latitude.
For example, consider a day on which the geocentric ecliptic longitude and mean anomaly of
the moon are λ = 210◦ (i.e., 00SC00) and M = 90◦ , respectively. Suppose that the moon is viewed
from an observation site located at terrestrial latitude +10◦ . The “Scorpio” entry in Table 2.19 gives
the moon’s geocentric altitude, a, as a function of time, as well as the value of the parallactic angle
µ. Making use of this data, in combination with Table 6.5 and Eqs. (6.17) and (6.18), we can
calculate the parallax-induced changes in the moon’s ecliptic longitude and latitude as it transits
the sky. Data from such a calculation is given in the table below. The first column specifies time
since the moon’s upper transit (thus, t = +1 hrs. means one hour after the upper transit), the
second column gives the moon’s geocentric altitude, the third column the parallactic angle, the
fourth column the decrease in the moon’s real altitude due to parallax, and the fifth and sixth
columns the parallax-induced changes in its ecliptic longitude and latitude, respectively. It can
be seen that parallax causes the moon’s apparent location to shift by almost 2◦ relative to the
fixed stars as it transits the sky. Note that the above calculation is somewhat inaccurate because
it does not take into account the moon’s motion along the ecliptic (which can easily amount to 6◦
during the course of a night). However, the calculation does illustrate how the data contained in
Tables 2.18–2.26, in combination with the data in Table 6.5, permits the parallax-induced shift in
the moon’s ecliptic position to be calculated for a wide range of different lunar phases, observation
sites, and observation times.
94
MODERN ALMAGEST
t (hrs.)
−5.51
−5.00
−4.00
−3.00
−2.00
−1.00
+0.00
+1.00
+2.00
+3.00
+4.00
+5.00
+5.51
a
µ
δa
δλ
δβ
00◦ 00 ′
12◦ 26 ′
26◦ 37 ′
40◦ 23 ′
53◦ 15 ′
63◦ 52 ′
68◦ 32 ′
63◦ 52 ′
53◦ 15 ′
40◦ 23 ′
26◦ 37 ′
12◦ 26 ′
00◦ 00 ′
190◦ 22 ′
187◦ 30 ′
183◦ 07 ′
176◦ 40 ′
165◦ 58 ′
145◦ 55 ′
110◦ 34 ′
075◦ 13 ′
055◦ 11 ′
044◦ 29 ′
038◦ 01 ′
033◦ 39 ′
030◦ 47 ′
57 ′
56 ′
51 ′
43 ′
34 ′
25 ′
21 ′
25 ′
34 ′
43 ′
51 ′
56 ′
57 ′
+56 ′
+55 ′
+51 ′
+43 ′
+33 ′
+21 ′
+07 ′
−06 ′
−20 ′
−31 ′
−40 ′
−46 ′
−49 ′
+10 ′
+07 ′
+03 ′
−03 ′
−08 ′
−14 ′
−20 ′
−24 ′
−28 ′
−30 ′
−32 ′
−31 ′
−29 ′
The Moon
e
0.054881
95
n(◦ /day)
13.17639646
ñ(◦ /day)
13.06499295
n̆(◦ /day)
13.22935027
λ̄0(◦ )
218.322
M0(◦ )
134.916
F0(◦ )
93.284
i(◦ )
5.161
Table 6.1: Orbital elements of the moon for the J2000 epoch (i.e., 12:00 UT, January 1, 2000 CE,
which corresponds to t0 = 2 451 545.0 JD).
96
MODERN ALMAGEST
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
3.965
7.929
11.894
15.858
19.823
23.788
27.752
31.717
35.681
329.930
299.859
269.788
239.718
209.648
179.577
149.506
119.436
89.366
173.503
347.005
160.508
334.011
147.513
321.016
134.519
308.022
121.524
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
216.396
72.793
289.189
145.586
1.982
218.379
74.775
291.172
147.568
104.993
209.986
314.979
59.972
164.965
269.958
14.951
119.944
224.937
269.350
178.701
88.051
357.401
266.751
176.102
85.452
354.802
264.152
100
200
300
400
500
600
700
800
900
237.640
115.279
352.919
230.559
108.198
345.838
223.478
101.117
338.757
226.499
92.999
319.498
185.997
52.496
278.996
145.495
11.994
238.494
242.935
125.870
8.805
251.740
134.675
17.610
260.545
143.480
26.415
10
20
30
40
50
60
70
80
90
131.764
263.528
35.292
167.056
298.820
70.584
202.348
334.112
105.876
130.650
261.300
31.950
162.600
293.250
63.900
194.550
325.199
95.849
132.294
264.587
36.881
169.174
301.468
73.761
206.055
338.348
110.642
1
2
3
4
5
6
7
8
9
13.176
26.353
39.529
52.706
65.882
79.058
92.235
105.411
118.588
13.065
26.130
39.195
52.260
65.325
78.390
91.455
104.520
117.585
13.229
26.459
39.688
52.917
66.147
79.376
92.605
105.835
119.064
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.318
2.635
3.953
5.271
6.588
7.906
9.223
10.541
11.859
1.306
2.613
3.919
5.226
6.532
7.839
9.145
10.452
11.758
1.323
2.646
3.969
5.292
6.615
7.938
9.261
10.583
11.906
Table 6.2: Mean motion of the moon. Here, ∆t = t−t0, ∆λ̄ = λ̄− λ̄0, ∆M = M−M0, and ∆F̄ = F̄− F̄0.
At epoch (t0 = 2 451 545.0 JD), λ̄0 = 218.322◦ , M0 = 134.916◦ , and F̄0 = 93.284◦ .
The Moon
97
Arg. (◦ )
q1 (◦ )
q2 (◦ )
q3 (◦ )
q4 (◦ )
q5 (◦ )
Arg. (◦ )
q1 (◦ )
q2 (◦ )
q3 (◦ )
q4 (◦ )
q5 (◦ )
000/(360)
002/(358)
004/(356)
006/(354)
008/(352)
010/(350)
012/(348)
014/(346)
016/(344)
018/(342)
020/(340)
022/(338)
024/(336)
026/(334)
028/(332)
030/(330)
032/(328)
034/(326)
036/(324)
038/(322)
040/(320)
042/(318)
044/(316)
046/(314)
048/(312)
050/(310)
052/(308)
054/(306)
056/(304)
058/(302)
060/(300)
062/(298)
064/(296)
066/(294)
068/(292)
070/(290)
072/(288)
074/(286)
076/(284)
078/(282)
080/(280)
082/(278)
084/(276)
086/(274)
088/(272)
090/(270)
0.000
0.237
0.473
0.709
0.943
1.176
1.408
1.637
1.864
2.088
2.310
2.527
2.741
2.951
3.157
3.358
3.554
3.746
3.931
4.111
4.285
4.454
4.615
4.770
4.919
5.061
5.195
5.323
5.443
5.555
5.660
5.757
5.847
5.929
6.002
6.068
6.126
6.176
6.218
6.252
6.278
6.296
6.306
6.308
6.302
6.289
0.000
0.046
0.093
0.139
0.185
0.230
0.276
0.321
0.366
0.410
0.454
0.497
0.540
0.582
0.623
0.663
0.703
0.742
0.780
0.817
0.853
0.888
0.922
0.955
0.986
1.017
1.046
1.074
1.100
1.125
1.149
1.172
1.193
1.212
1.230
1.247
1.262
1.276
1.288
1.298
1.307
1.314
1.320
1.324
1.326
1.327
0.000
0.045
0.089
0.133
0.177
0.219
0.261
0.301
0.339
0.376
0.411
0.444
0.475
0.504
0.529
0.553
0.573
0.591
0.605
0.617
0.625
0.630
0.632
0.631
0.627
0.620
0.609
0.595
0.579
0.559
0.536
0.511
0.483
0.453
0.420
0.385
0.348
0.309
0.269
0.227
0.184
0.139
0.094
0.048
0.002
-0.044
-0.000
-0.006
-0.011
-0.017
-0.022
-0.028
-0.033
-0.039
-0.044
-0.050
-0.055
-0.060
-0.065
-0.070
-0.075
-0.080
-0.085
-0.090
-0.094
-0.099
-0.103
-0.107
-0.111
-0.115
-0.119
-0.123
-0.126
-0.130
-0.133
-0.136
-0.139
-0.142
-0.144
-0.147
-0.149
-0.151
-0.153
-0.154
-0.156
-0.157
-0.158
-0.159
-0.159
-0.160
-0.160
-0.160
-0.000
-0.004
-0.008
-0.012
-0.017
-0.021
-0.025
-0.029
-0.033
-0.037
-0.041
-0.045
-0.049
-0.052
-0.056
-0.060
-0.063
-0.067
-0.070
-0.074
-0.077
-0.080
-0.083
-0.086
-0.089
-0.092
-0.094
-0.097
-0.099
-0.101
-0.103
-0.106
-0.107
-0.109
-0.111
-0.112
-0.114
-0.115
-0.116
-0.117
-0.118
-0.118
-0.119
-0.119
-0.119
-0.119
090/(270)
092/(268)
094/(266)
096/(264)
098/(262)
100/(260)
102/(258)
104/(256)
106/(254)
108/(252)
110/(250)
112/(248)
114/(246)
116/(244)
118/(242)
120/(240)
122/(238)
124/(236)
126/(234)
128/(232)
130/(230)
132/(228)
134/(226)
136/(224)
138/(222)
140/(220)
142/(218)
144/(216)
146/(214)
148/(212)
150/(210)
152/(208)
154/(206)
156/(204)
158/(202)
160/(200)
162/(198)
164/(196)
166/(194)
168/(192)
170/(190)
172/(188)
174/(186)
176/(184)
178/(182)
180/(180)
6.289
6.268
6.239
6.203
6.160
6.109
6.051
5.986
5.915
5.836
5.751
5.660
5.562
5.458
5.348
5.233
5.111
4.985
4.853
4.716
4.575
4.428
4.277
4.122
3.963
3.799
3.632
3.462
3.288
3.111
2.931
2.748
2.562
2.375
2.184
1.992
1.798
1.603
1.406
1.207
1.008
0.807
0.606
0.404
0.202
0.000
1.327
1.326
1.324
1.320
1.314
1.307
1.298
1.288
1.276
1.262
1.247
1.230
1.212
1.193
1.172
1.149
1.125
1.100
1.074
1.046
1.017
0.986
0.955
0.922
0.888
0.853
0.817
0.780
0.742
0.703
0.663
0.623
0.582
0.540
0.497
0.454
0.410
0.366
0.321
0.276
0.230
0.185
0.139
0.093
0.046
0.000
-0.044
-0.090
-0.136
-0.182
-0.226
-0.270
-0.313
-0.354
-0.394
-0.432
-0.468
-0.502
-0.533
-0.562
-0.589
-0.613
-0.634
-0.652
-0.667
-0.678
-0.687
-0.693
-0.695
-0.694
-0.689
-0.682
-0.671
-0.657
-0.640
-0.620
-0.597
-0.571
-0.542
-0.511
-0.477
-0.442
-0.404
-0.364
-0.322
-0.279
-0.235
-0.189
-0.143
-0.095
-0.048
-0.000
-0.160
-0.160
-0.160
-0.159
-0.159
-0.158
-0.157
-0.156
-0.154
-0.153
-0.151
-0.149
-0.147
-0.144
-0.142
-0.139
-0.136
-0.133
-0.130
-0.126
-0.123
-0.119
-0.115
-0.111
-0.107
-0.103
-0.099
-0.094
-0.090
-0.085
-0.080
-0.075
-0.070
-0.065
-0.060
-0.055
-0.050
-0.044
-0.039
-0.033
-0.028
-0.022
-0.017
-0.011
-0.006
-0.000
-0.119
-0.119
-0.119
-0.119
-0.118
-0.118
-0.117
-0.116
-0.115
-0.114
-0.112
-0.111
-0.109
-0.107
-0.106
-0.103
-0.101
-0.099
-0.097
-0.094
-0.092
-0.089
-0.086
-0.083
-0.080
-0.077
-0.074
-0.070
-0.067
-0.063
-0.060
-0.056
-0.052
-0.049
-0.045
-0.041
-0.037
-0.033
-0.029
-0.025
-0.021
-0.017
-0.012
-0.008
-0.004
-0.000
Table 6.3: Anomalies of the moon. The common argument corresponds to M, 2D̃ − M, D̃, MS, and
2F̄ for the case of q1, q2, q3, q4, and q5, respectively. If the argument is in parenthesies then the
anomalies are minus the values shown in the table.
98
MODERN ALMAGEST
F(◦ )
000/180
002/178
004/176
006/174
008/172
010/170
012/168
014/166
016/164
018/162
020/160
022/158
024/156
026/154
028/152
030/150
032/148
034/146
036/144
038/142
040/140
042/138
044/136
046/134
048/132
050/130
052/128
054/126
056/124
058/122
060/120
062/118
064/116
066/114
068/112
070/110
072/108
074/106
076/104
078/102
080/100
082/098
084/096
086/094
088/092
090/090
β(◦ )
0.000
0.180
0.360
0.539
0.718
0.896
1.073
1.248
1.422
1.595
1.765
1.933
2.099
2.263
2.423
2.581
2.735
2.887
3.034
3.178
3.319
3.455
3.587
3.714
3.837
3.956
4.070
4.178
4.282
4.380
4.473
4.561
4.643
4.719
4.790
4.855
4.913
4.966
5.013
5.054
5.088
5.117
5.139
5.154
5.164
5.167
F(◦ )
(180)/(360)
(182)/(358)
(184)/(356)
(186)/(354)
(188)/(352)
(190)/(350)
(192)/(348)
(194)/(346)
(196)/(344)
(198)/(342)
(200)/(340)
(202)/(338)
(204)/(336)
(206)/(334)
(208)/(332)
(210)/(330)
(212)/(328)
(214)/(326)
(216)/(324)
(218)/(322)
(220)/(320)
(222)/(318)
(224)/(316)
(226)/(314)
(228)/(312)
(230)/(310)
(232)/(308)
(234)/(306)
(236)/(304)
(238)/(302)
(240)/(300)
(242)/(298)
(244)/(296)
(246)/(294)
(248)/(292)
(250)/(290)
(252)/(288)
(254)/(286)
(256)/(284)
(258)/(282)
(260)/(280)
(262)/(278)
(264)/(276)
(266)/(274)
(268)/(272)
(270)/(270)
Table 6.4: Ecliptic latitude of the moon. The latitude is minus the value shown in the table if the
argument is in parenthesies.
The Moon
99
Arg. (◦ )
δ( ′ )
100 ζ
Arg. (◦ )
000/360
002/358
004/356
006/354
008/352
010/350
012/348
014/346
016/344
018/342
020/340
022/338
024/336
026/334
028/332
030/330
032/328
034/326
036/324
038/322
040/320
042/318
044/316
046/314
048/312
050/310
052/308
054/306
056/304
058/302
060/300
062/298
064/296
066/294
068/292
070/290
072/288
074/286
076/284
078/282
080/280
082/278
084/276
086/274
088/272
090/270
57.067
57.032
56.928
56.754
56.511
56.200
55.820
55.371
54.856
54.274
53.625
52.911
52.133
51.291
50.387
49.421
48.395
47.310
46.168
44.969
43.716
42.409
41.050
39.642
38.185
36.682
35.134
33.543
31.911
30.241
28.533
26.791
25.016
23.211
21.378
19.518
17.635
15.730
13.806
11.865
9.910
7.942
5.965
3.981
1.992
0.000
5.488
5.485
5.475
5.458
5.435
5.405
5.368
5.325
5.276
5.219
5.157
5.088
5.014
4.933
4.846
4.753
4.654
4.550
4.440
4.325
4.204
4.078
3.948
3.812
3.672
3.528
3.379
3.226
3.069
2.908
2.744
2.577
2.406
2.232
2.056
1.877
1.696
1.513
1.328
1.141
0.953
0.764
0.574
0.383
0.192
0.000
(180)/(180)
(178)/(182)
(176)/(184)
(174)/(186)
(172)/(188)
(170)/(190)
(168)/(192)
(166)/(194)
(164)/(196)
(162)/(198)
(160)/(200)
(158)/(202)
(156)/(204)
(154)/(206)
(152)/(208)
(150)/(210)
(148)/(212)
(146)/(214)
(144)/(216)
(142)/(218)
(140)/(220)
(138)/(222)
(136)/(224)
(134)/(226)
(132)/(228)
(130)/(230)
(128)/(232)
(126)/(234)
(124)/(236)
(122)/(238)
(120)/(240)
(118)/(242)
(116)/(244)
(114)/(246)
(112)/(248)
(110)/(250)
(108)/(252)
(106)/(254)
(104)/(256)
(102)/(258)
(100)/(260)
(098)/(262)
(096)/(264)
(094)/(266)
(092)/(268)
(090)/(270)
Table 6.5: Parallax of the moon. The arguments of δ and ζ are a and M, respectively. δ and ζ take
minus the values shown in the table if their arguments are in parenthesies.
100
MODERN ALMAGEST
Lunar-Solar Syzygies and Eclipses
101
7 Lunar-Solar Syzygies and Eclipses
7.1 Introduction
Let λS and λM represent the ecliptic longitudes of the sun and the moon, respectively. The lunarsolar elongation is defined
D = λM − λS.
(7.1)
Since the moon is only visible because of light reflected from the sun, there is a fairly obvious
relationship between lunar-solar elongation and lunar phase—see Fig. 7.1. For instance, a new
moon corresponds to D = 0◦ , a quarter moon to D = 90◦ or 270◦ , and a full moon to D = 180◦ .
New moons and full moons are collectively known as lunar-solar syzygies.
Quarter Moon
Light from Sun
Full Moon
D
Earth
New Moon
Figure 7.1: The phases of the moon.
7.2 Determination of Lunar-Solar Elongation
We can determine the lunar-solar elongation by combining the solar and lunar models described in
the previous two chapters. Our elongation model is as follows:
(7.2)
D̄ = λ̄M − λ̄S,
2
q1 = 2 eM sin MM + 1.430 e sin 2MM,
(7.3)
q2 = 0.422 eM sin(2D̄ − MM),
(7.4)
q3 = 0.211 eM (sin 2D̄ − 0.066 sin D̄),
(7.5)
q4 = −(0.051 eM + 2 eS) sin MS − (5/4) eS2 sin 2MS,
(7.6)
102
MODERN ALMAGEST
q5 = −0.038 eM sin 2F̄M,
(7.7)
D = D̄ + q1 + q2 + q3 + q4 + q5.
(7.8)
Here, eS, MS, and λ̄S are the eccentricity, mean anomaly, and mean longitude of the sun’s apparent
orbit about the earth, respectively. Moreover, eM, MM, λ̄M, and F̄M are the eccentricity, mean
anomaly, mean longitude, and mean argument of latitude of the moon’s orbit, respectively.
The lunar-solar elongation can be calculated with the aid of Tables 7.1 and 7.2. Table 7.1 allows
the mean lunar-solar elongation, D̄, the mean lunar argument of latitude, F̄M, the mean anomaly
of the sun, MS, and the mean anomaly of the moon, MM, to be determined as functions of time.
Table 7.2 specifies the anomalies q1–q5 as functions of their various arguments.
The procedure for using the tables is as follows:
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the lunar-solar elongation is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t − t0,
where t0 = 2 451 545.0 is the epoch.
2. Enter Table 7.1 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆D̄, ∆F̄M, ∆MS, and ∆MM. If ∆t is negative then the values are minus those
shown in the table. The value of the mean lunar-solar elongation, D̄, is the sum of all the ∆D̄
values plus the value of D̄ at the epoch. Likewise, the value of the mean lunar argument of
latitude, F̄M, is the sum of all the ∆F̄M values plus the value of F̄M at the epoch. Moreover,
the value of the solar mean anomaly, MS, is the sum of all the ∆MS values plus the value of
MS at the epoch. Finally, the value of the lunar mean anomaly, MM, is the sum of all the
∆MM values plus the value of MM at the epoch. Add as many multiples of 360◦ to D̄, F̄M,
MS, and MM as is required to make them all fall in the range 0◦ to 360◦ .
3. Form the five arguments a1 = MM, a2 = 2D̄ − MM, a3 = D̄, a4 = MS, a5 = 2F̄M. Add as
many multiples of 360◦ to the arguments as is required to make them all fall in the range 0◦
to 360◦ . Round each argument to the nearest degree.
4. Enter Table 7.2 with the value of each of the five arguments a1–a5 and take out the value of
each of the five corresponding anomalies q1–q5. It is necessary to interpolate if the arguments
are odd.
5. The lunar-solar elongation is given by D = D̄ + q1 + q2 + q3 + q4 + q5. If necessary, convert D
into an angle in the range 0◦ to 360◦ . The decimal fraction can be converted into arc minutes
using Table 5.2.
In order to facilitate the calculation of syzygies, the above model has been used to contruct
Table 7.3, which lists the dates and fractional Julian day numbers of the first new moons of the
years 1900–2099 CE. Two examples of syzygy calculations are given below.
7.3 Example Syzygy Calculations
Example 1: Sixth new moon in 2004 CE:
From Table 7.3, the date of first new moon in 2004 CE is 2453026.4 JD. Now, the lunar-solar
Lunar-Solar Syzygies and Eclipses
103
elongation increases at the mean rate nM − nS = 13.17639646 − 0.98564735 = 12.1907491◦ per day,
or 360◦ in 29.53 days—the latter time period is known as a synodic month. Hence, a rough estimate
for the date of the sixth new moon in 2004 CE is five synodic months after that of the first: i.e.,
2453026.4 + 5 × 29.53 ≃ 2453174.1 JD. It follows that ∆t = 2453174.1 − 2451545.0 = 1629.1 JD. Let
us calculate the lunar-solar elongation at this date. From Table 7.1:
t(JD)
D̄(◦ )
F̄M(◦ )
MS(◦ )
MM(◦ )
+1000
+600
+20
+9
+.1
Epoch
310.749
114.449
243.815
109.717
1.219
297.864
1077.813
357.813
269.350
17.610
264.587
119.064
1.323
93.284
765.218
45.218
265.600
231.360
19.712
8.870
0.099
357.588
883.229
163.229
104.993
278.996
261.300
117.585
1.306
134.916
899.096
179.096
Modulus
Thus,
a1 = MM ≃ 179◦ ,
a2 = 2D̄ − MM = 2 × 357.813 − 179.082 ≃ 177◦ ,
a3 = D̄ ≃ 358◦ ,
a4 = MS ≃ 163◦ ,
a5 = 2F̄M = 2 × 45.218 ≃ 90◦ .
Table 7.2 yields
q1(a1) = 0.101◦ , q2(a2) = 0.070◦ , q3(a3) = −0.045◦ ,
q4(a4) = −0.596◦ , q5(a5) = −0.119◦ .
Hence,
D = D̄ + q1 + q2 + q3 + q4 + q5 = 357.813 + 0.101 + 0.070 − 0.045 − 0.596 − 0.119 ≃ 357.22◦ .
Now, the actual new moon takes place when D = 360.00◦ . Thus, a far better estimate for the date
of the sixth new moon in 2004 CE is 2453174.10 + (360.00 − 357.22)/12.1907491 = 2453174.33 JD.
This corresponds to 20:00 UT on June 17th.
Example 2: Third full moon in 1982 CE:
From Table 7.3, the fractional Julian day number of first new moon in 1982 CE is 2444994.7
JD, which corresponds to January 25th. Since there is more than half a synodic month between
this event and the start of year, we conclude that the first full moon in 1982 CE took place before
January 25th. Hence, a rough estimate for the date of the third full moon in 1982 CE is one and a
half synodic months after that of the first new moon: i.e., 2444994.7 + 1.5 × 29.53 ≃ 2445039.0 JD. It
follows that ∆t = 2445039.0 − 2451545.0 = −6506.0 JD. Let us calculate the lunar-solar elongation
at this date. From Table 7.1:
104
MODERN ALMAGEST
t(JD)
D̄(◦ )
F̄M(◦ )
MS(◦ )
MM(◦ )
-6000
-500
-6
Epoch
−64.495
−335.375
−73.144
297.864
−175.150
184.131
−176.102
−134.675
−79.376
93.284
−296.869
63.062
−153.601
−132.800
−5.914
357.588
65.273
65.273
−269.958
−52.496
−78.390
134.916
−265.928
94.072
Modulus
Thus,
a1 = MM ≃ 94◦ ,
a2 = 2D̄ − MM = 2 × 184.850 − 94.072 ≃ 276◦ ,
a3 = D̄ ≃ 185◦ ,
a4 = MS ≃ 65◦ ,
a5 = 2F̄M = 2 × 63.062 ≃ 126◦ .
Table 7.2 yields
q1(a1) = 6.239◦ , q2(a2) = −1.320◦ , q3(a3) = 0.119◦ ,
q4(a4) = −1.896◦ , q5(a5) = −0.097◦ .
Hence,
D = D̄ + q1 + q2 + q3 + q4 + q5 = 184.850 + 6.239 − 1.320 + 0.119 − 1.896 − 0.097 ≃ 187.895◦ .
Now, the actual full moon takes place when D = 180.00◦ . Thus, a far better estimate for the date of
the third full moon in 1982 CE is 2445039.0 + (180.00 − 187.90)/12.1907491 = 2445038.35 JD. This
corresponds to 20:00 UT on March 9th.
7.4 Solar and Lunar Eclipses
A solar eclipse—or, more accurately, a lunar-solar occultation—occurs when the moon blocks the
light of the sun. Clearly, this is only possible at a new moon—see Fig. 7.1. On the other hand,
a lunar eclipse occurs when the moon falls into the shadow of the earth. Of course, this is only
possible at a full moon. It follows that eclipses can only take place at lunar-solar syzygies.
In order to determine whether a particular lunar-solar syzygy conincides with an eclipse, we
first need to calculate the angular radii of the sun, the moon, and the earth’s shadow in the sky.
Using the small angle approximation, the angular radius of the sun is given by ρS = RS/rS, where
RS is the solar radius, and rS the earth-sun distance. However, rS ≃ aS (1 − eS cos MS), where
aS, eS, and MS are the major radius, eccentricity, and mean anomaly of the sun’s apparent orbit
around the earth, respectively (see Cha. 4). Hence,
ρS ≃ ρS0 (1 + eS cos MS),
(7.9)
where ρS0 = RS/aS = 6.960 × 105 km/1.496 × 108 km ≃ 15.99 ′ . Likewise, the angular radius of the
moon is
ρM ≃ ρM0 (1 + eM cos MM),
(7.10)
where ρM0 = RM/aM = 1743 km/384, 399 km ≃ 15.59 ′ . Here, RM, aM, eM, and MM are the
radius of the moon, and the major radius, eccentricity, and mean anomaly of the moon’s orbit,
Lunar-Solar Syzygies and Eclipses
105
ρS
RS
x
RE
rS
RU
rM
Sun
Earth
ρU
Moon
Figure 7.2: The earth’s umbra.
respectively. As was shown in the previous chapter, lunar parallax causes the angular position of
the moon in the sky to shift by up to
δM =
RE
= δM0 (1 + eM cos MM),
rM
(7.11)
where δM0 = RE/aM = 6371 km/384, 399 km = 56.98 ′ . Here, RE is the radius of the earth. Finally,
simple trigonometry reveals that the angular size of the earth’s shadow (i.e., umbra) at the radius
of the moon’s orbit is
ρU = δM − ρS.
(7.12)
This can be seen from Fig. 7.2. The radius of the umbra at the position of the moon is RU = RE −x =
RE − rM ρS. Hence, the angular radius of the umbra is ρU = RU/rM = δM − ρS. Incidentally, the
identification of two of the angles in the figure with ρS = RS/rS follows because RS ≫ RE.
A solar eclipse does not take place every new moon, nor a lunar eclipse every full moon, because
of the inclination of the moon’s orbit to the ecliptic plane, which causes the moon to pass either
above or below the sun, or the earth’s shadow, respectively, in the majority of cases. It follows that
the critical parameter which determines the occurrence of eclipses is the ecliptic latitude of the
moon at syzygy, βsyz. Of course, once the date and time of a syzygy has been established, βsyz can
be calculated from Table 6.4. However, the lunar argument of latitude, F, must first be determined
using
(7.13)
F = F̄M + q1 + q2 + q3 + q4′ + q5,
where F̄M comes from Table 7.1, q1, q2, q3, and q5 are obtained from Table 7.2, and q4′ is the q4
from Table 6.3. For instance, we have seen that for the third new moon of 1982 CE, F̄M = 63.131,
MS ≃ 65◦ , q1 = 6.239◦ , q2 = −1.320◦ , q3 = 0.119◦ , and q5 = −0.097◦ . According to Table 6.3,
q4′ (MS) = −0.145◦ . Hence, F = F̄M + q1 + q2 + q3 + q4′ + q5 = 63.139 + 6.239 − 1.320 + 0.119 −
0.145 − 0.097 = 67.926 ≃ 68◦ . It follows from Table 6.4 that βsyz = 4.790◦ ≃ 4◦ 47 ′ .
The criterion for a lunar eclipse is particularly simple, since it is not complicated by lunar
parallax. A total lunar eclipse, in which the moon is completely immersed in the earth’s shadow,
106
MODERN ALMAGEST
Moon
Moon
ρM
ρM
ρU
ρU
βsyz
βsyz
Ecliptic
Earth’s umbra
Earth’s umbra
Figure 7.3: The limiting cases for a total lunar eclipse (left) and a partial lunar eclipse (right).
must take place at a full moon if |βsyz| < ρU − ρM (see Fig. 7.3), or equivalently
|βsyz| < δM − ρM − ρS,
(7.14)
and either a total or a partial lunar eclipse, in which the moon is only partially immersed in the
earth’s shadow, must take place if |βsyz| < ρU + ρM (see Fig. 7.3), or equivalently
|βsyz| < δM + ρM − ρS.
(7.15)
Note that lunar eclipses are simultaneously visible at all observation sites on the earth for which
the moon is above the horizon, since the earth’s shadow is larger than the moon, and the relative
position of the moon and the earth’s shadow is not affected by parallax (since both the moon and
the shadow are the same distance from the earth).
The criterion for a solar eclipse is modified by lunar parallax, which causes the angular position
of the moon relative to the sun to shift by up to δM from its geocentric position. The amount of the
shift depends on the observation site. However, a site can always be found at which the shift takes
its maximum value in any particular direction. Note that the sun has negligible parallax, since it
is much further from the earth than the moon. Taking parallactic shifts into account, a total solar
eclipse, in which the sun is totally obscured by the moon, must take place if ρM > ρS and
|βsyz| < δM + ρM − ρS,
(7.16)
an annular solar eclipse, in which all of the sun apart from a thin outer ring is obscured by the
moon, must take place if ρS > ρM and
|βsyz| < δM + ρS − ρM,
(7.17)
and either a total, an annular, or a partial solar eclipse, in which the sun is only partially obscured
by the moon, must take place if
|βsyz| < δM + ρM + ρS.
(7.18)
Lunar-Solar Syzygies and Eclipses
107
As a consequence of lunar parallax, and the fact that the angular sizes of the sun and moon in
the sky are very similar, solar eclipses are only visible in very localized regions of the earth. Note,
finally, that the above criteria represent necessary, but not sufficient, conditions for the occurrence
of the various eclipses with which they are associated. This is the case because the point of closest
approach of the moon and the earth’s shadow, in the case of a lunar eclipse, and the moon and sun,
in the case of a solar eclipse, does not necessarily occur exactly at the syzygy, due to the inclination
of the moon’s orbit to the ecliptic. However, since the said inclination is fairly gentle, the above
criteria turn out to be very accurate predictors of eclipses.
The criterion for a total lunar eclipse can be written |βsyz| < βMt, where
βMt = 25.41 ′ + δβ1(MM) − δβ2(MM) − δβ3(MS).
(7.19)
Here, the functions δβ1 = δM0 eM cos MM, δβ2 = ρM0 eM cos MM, and δβ3 = ρS0 eS cos MS are
tabulated in Table 7.4. The criterion for any type of lunar eclipse becomes |βsyz| < βM, where
βM = 56.59 ′ + δβ1(MM) + δβ2(MM) − δβ3(MS).
(7.20)
The criterion for a total solar eclipse can be written |βsyz| < βSt and βSt > βSa, where
βSt = 56.59 ′ + δβ1(MM) + δβ2(MM) − δβ3(MS),
(7.21)
βSa = 57.39 ′ + δβ1(MM) − δβ2(MM) + δβ3(MS),
(7.22)
and
The criterion for an annular solar eclipse is |βsyz| < βSa and βSa > βSt. Finally, the criterion for
any type of solar eclipse is |βsyz| < βS, where
βS = 88.57 ′ + δβ1(MM) + δβ2(MM) + δβ3(MS).
(7.23)
7.5 Example Eclipse Calculations
Let us use our model to examine the lunar-solar syzygies of the year 1992 CE, in order to see
whether any of them were associated with solar or lunar eclipses. The table below shows the dates
and times of the new moons of 1992 CE, calculated using the method described at the beginning
of this section. Also shown is the magnitude of the moon’s ecliptic latitude at each syzygy, |βsyz|,
calculated from Eqs. (6.12) and (7.13), as well as the critical values of this parameter for a general,
total, and annular solar eclipse. The latter are calculated from Eqs. (7.21)–(7.23). It can be seen
that the criterion for a total solar eclipse (i.e., |βsyz| < βSt and βSt > βSa) is satisfied for the syzygy
marked with a T, the criterion for an annular solar eclipse (i.e., |βsyz| < βSa and βSa > βSt) for the
syzygy marked with an A, and the criterion for a partial solar eclipse (i.e., βSt, βSa < |βsyz| < βS)
for the syzygy marked with a P. It is easily verified that a total solar eclipse, an annular solar eclipse,
and a partial solar eclipse did indeed take place in 1992 CE at the dates and times indicated.
108
Date
04/01/1992
03/02/1992
04/03/1992
03/04/1992
02/05/1992
01/06/1992
30/06/1992
29/07/1992
28/08/1992
26/09/1992
26/10/1992
24/11/1992
24/12/1992
MODERN ALMAGEST
Time (UT)
23:00
16:00
09:00
00:00
14:00
01:00
12:00
21:00
05:00
14:00
00:00
11:00
01:00
βS( ′ )
85.0
84.9
85.7
87.1
88.7
90.3
91.5
92.2
92.3
91.8
90.7
89.2
87.4
βSt( ′ )
52.5
52.5
53.4
55.1
57.0
58.8
60.1
60.7
60.7
59.9
58.5
56.8
54.9
βSa( ′ )
55.5
55.4
55.8
56.5
57.4
58.3
59.0
59.4
59.5
59.2
58.7
57.8
56.9
|βsyz|( ′ )
22.8
177.9
282.7
307.5
245.8
115.6
46.3
195.3
290.2
304.8
234.8
99.3
64.3
A
T
P
The table below shows the dates and times of the full moons of 1992 CE. Also shown is the
magnitude of the moon’s ecliptic latitude at each syzygy, as well as the critical values of this parameter for a general and a total lunar eclipse. The latter are calculated from Eqs. (7.20) and
(7.19), respectively. It can be seen that the criterion for a total lunar eclipse (i.e., |βsyz| < βMt)
is satisfied for the syzygy marked with a T, whereas the criterion for a partial lunar eclipse (i.e.,
βMt < |βsyz| < βM) is satisfied for the syzygy marked with a P. It is easily verified that a total
lunar eclipse, and a partial lunar eclipse did indeed take place in 1992 CE at the dates and times
indicated.
Date
19/01/1992
18/02/1992
18/03/1992
17/04/1992
16/05/1992
15/06/1992
14/07/1992
13/08/1992
12/09/1992
11/10/1992
10/11/1992
10/12/1992
08/01/1993
Time (UT)
20:00
05:00
14:00
01:00
13:00
03:00
20:00
13:00
05:00
21:00
12:00
01:00
12:00
βM( ′ )
60.3
60.1
59.3
57.9
56.3
54.7
53.4
52.8
53.1
54.3
55.8
57.5
59.0
βMt( ′ )
27.4
27.3
26.9
26.2
25.3
24.4
23.7
23.3
23.5
24.1
24.9
25.8
26.7
|βsyz|( ′ )
104.2
237.8
305.6
288.5
190.8
39.4
123.4
251.6
308.6
278.2
169.6
13.8
145.4
P
T
7.6 Eclipse Statistics
Consider a very large collection of lunar-solar syzygies. For such a collection, we expect the lunar
argument of latitude, F, the lunar mean anomaly, MM, and the solar mean anomaly, MS, to be
statistically independent of one another, and randomly distributed in the range 0◦ to 360◦ . Using
this insight, we can easily calculate the probability that a new moon is coincident with a solar
Lunar-Solar Syzygies and Eclipses
109
eclipse, or a full moon with a lunar eclipse, using Eq. (6.12) and the criteria (7.19)–(7.23). For a
new moon we find:
Probability of total solar eclipse:
Probability of annular solar eclipse:
Probability of partial solar eclipse:
Probability of any solar eclipse:
4.2%
7.7%
6.6%
18.5%
For a full moon we get:
Probability of total lunar eclipse:
Probability of partial lunar eclipse:
Probability of any lunar eclipse:
5.2%
6.5%
11.7%
Thus, we can see that, over a long period of time, the ratio of the number of total/annular solar
eclipses to the number of partial solar eclipses is about 9/5, whereas the ratio of the number of
partial lunar eclipses to the number of total lunar eclipses is approximately 5/4. Furthermore, the
ratio of the number of solar eclipses to the number of lunar eclipses is about 11/7. Since there
are 12.37 synodic months in a year, the mean number of solar eclipses per year is approximately
12.37 × 0.185 ≃ 2.3, whereas the mean number of lunar eclipses per year is about 12.37 × 0.117 ≃
1.4. Clearly, solar eclipses are more common that lunar eclipses. On the other hand, at a given
observation site on the earth, lunar eclipses are much more common than solar eclipses, since the
former are visible all over the earth, whereas the latter are only visible in a very localized region.
110
MODERN ALMAGEST
∆t(JD)
∆D̄(◦ )
∆F̄M(◦ )
∆MS(◦ )
∆MM(◦ )
∆t(JD)
∆D̄(◦ )
∆F̄M(◦ )
∆MS(◦ )
∆MM(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
227.491
94.982
322.473
189.964
57.455
284.947
152.438
19.929
247.420
173.503
347.005
160.508
334.011
147.513
321.016
134.519
308.022
121.524
136.002
272.005
48.007
184.010
320.012
96.015
232.017
8.020
144.022
329.930
299.859
269.788
239.718
209.648
179.577
149.506
119.436
89.366
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
310.749
261.498
212.247
162.996
113.746
64.495
15.244
325.993
276.742
269.350
178.701
88.051
357.401
266.751
176.102
85.452
354.802
264.152
265.600
171.200
76.801
342.401
248.001
153.601
59.202
324.802
230.402
104.993
209.986
314.979
59.972
164.965
269.958
14.951
119.944
224.937
100
200
300
400
500
600
700
800
900
139.075
278.150
57.225
196.300
335.375
114.449
253.524
32.599
171.674
242.935
125.870
8.805
251.740
134.675
17.610
260.545
143.480
26.415
98.560
197.120
295.680
34.240
132.800
231.360
329.920
68.480
167.040
226.499
92.999
319.498
185.997
52.496
278.996
145.495
11.994
238.494
10
20
30
40
50
60
70
80
90
121.907
243.815
5.722
127.630
249.537
11.445
133.352
255.260
17.167
132.294
264.587
36.881
169.174
301.468
73.761
206.055
338.348
110.642
9.856
19.712
29.568
39.424
49.280
59.136
68.992
78.848
88.704
130.650
261.300
31.950
162.600
293.250
63.900
194.550
325.199
95.849
1
2
3
4
5
6
7
8
9
12.191
24.381
36.572
48.763
60.954
73.144
85.335
97.526
109.717
13.229
26.459
39.688
52.917
66.147
79.376
92.605
105.835
119.064
0.986
1.971
2.957
3.942
4.928
5.914
6.899
7.885
8.870
13.065
26.130
39.195
52.260
65.325
78.390
91.455
104.520
117.585
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.219
2.438
3.657
4.876
6.095
7.314
8.534
9.753
10.972
1.323
2.646
3.969
5.292
6.615
7.938
9.261
10.583
11.906
0.099
0.197
0.296
0.394
0.493
0.591
0.690
0.788
0.887
1.306
2.613
3.919
5.226
6.532
7.839
9.145
10.452
11.758
Table 7.1: Mean motion of the lunar-solar elongation. Here, ∆t = t − t0, ∆D̄ = D̄ − D̄0, ∆F̄M =
F̄M − F̄M0, ∆MS = MS − MS0, and ∆MM = MM − MM0. At epoch (t0 = 2 451 545.0 JD),
D̄0 = 297.864◦ , F̄M0 = 93.284◦ , MS0 = 357.588◦ , and MM0 = 134.916◦ .
Lunar-Solar Syzygies and Eclipses
111
Arg. (◦ )
q1 (◦ )
q2 (◦ )
q3 (◦ )
q4 (◦ )
q5 (◦ )
Arg. (◦ )
q1 (◦ )
q2 (◦ )
q3 (◦ )
q4 (◦ )
q5 (◦ )
000/(360)
002/(358)
004/(356)
006/(354)
008/(352)
010/(350)
012/(348)
014/(346)
016/(344)
018/(342)
020/(340)
022/(338)
024/(336)
026/(334)
028/(332)
030/(330)
032/(328)
034/(326)
036/(324)
038/(322)
040/(320)
042/(318)
044/(316)
046/(314)
048/(312)
050/(310)
052/(308)
054/(306)
056/(304)
058/(302)
060/(300)
062/(298)
064/(296)
066/(294)
068/(292)
070/(290)
072/(288)
074/(286)
076/(284)
078/(282)
080/(280)
082/(278)
084/(276)
086/(274)
088/(272)
090/(270)
0.000
0.237
0.473
0.709
0.943
1.176
1.408
1.637
1.864
2.088
2.310
2.527
2.741
2.951
3.157
3.358
3.554
3.746
3.931
4.111
4.285
4.454
4.615
4.770
4.919
5.061
5.195
5.323
5.443
5.555
5.660
5.757
5.847
5.929
6.002
6.068
6.126
6.176
6.218
6.252
6.278
6.296
6.306
6.308
6.302
6.289
0.000
0.046
0.093
0.139
0.185
0.230
0.276
0.321
0.366
0.410
0.454
0.497
0.540
0.582
0.623
0.663
0.703
0.742
0.780
0.817
0.853
0.888
0.922
0.955
0.986
1.017
1.046
1.074
1.100
1.125
1.149
1.172
1.193
1.212
1.230
1.247
1.262
1.276
1.288
1.298
1.307
1.314
1.320
1.324
1.326
1.327
0.000
0.045
0.089
0.133
0.177
0.219
0.261
0.301
0.340
0.376
0.411
0.444
0.475
0.504
0.529
0.553
0.573
0.591
0.605
0.617
0.625
0.631
0.633
0.632
0.627
0.620
0.609
0.596
0.579
0.559
0.537
0.511
0.483
0.453
0.420
0.385
0.348
0.309
0.269
0.227
0.184
0.140
0.094
0.049
0.003
-0.044
-0.000
-0.074
-0.148
-0.221
-0.294
-0.367
-0.440
-0.511
-0.583
-0.653
-0.723
-0.791
-0.859
-0.926
-0.991
-1.055
-1.118
-1.179
-1.239
-1.297
-1.354
-1.409
-1.462
-1.513
-1.562
-1.609
-1.655
-1.698
-1.739
-1.778
-1.815
-1.849
-1.881
-1.911
-1.938
-1.963
-1.985
-2.006
-2.023
-2.038
-2.051
-2.061
-2.068
-2.073
-2.075
-2.075
-0.000
-0.004
-0.008
-0.012
-0.017
-0.021
-0.025
-0.029
-0.033
-0.037
-0.041
-0.045
-0.049
-0.052
-0.056
-0.060
-0.063
-0.067
-0.070
-0.074
-0.077
-0.080
-0.083
-0.086
-0.089
-0.092
-0.094
-0.097
-0.099
-0.101
-0.103
-0.106
-0.107
-0.109
-0.111
-0.112
-0.114
-0.115
-0.116
-0.117
-0.118
-0.118
-0.119
-0.119
-0.119
-0.119
090/(270)
092/(268)
094/(266)
096/(264)
098/(262)
100/(260)
102/(258)
104/(256)
106/(254)
108/(252)
110/(250)
112/(248)
114/(246)
116/(244)
118/(242)
120/(240)
122/(238)
124/(236)
126/(234)
128/(232)
130/(230)
132/(228)
134/(226)
136/(224)
138/(222)
140/(220)
142/(218)
144/(216)
146/(214)
148/(212)
150/(210)
152/(208)
154/(206)
156/(204)
158/(202)
160/(200)
162/(198)
164/(196)
166/(194)
168/(192)
170/(190)
172/(188)
174/(186)
176/(184)
178/(182)
180/(180)
6.289
6.268
6.239
6.203
6.160
6.109
6.051
5.986
5.915
5.836
5.751
5.660
5.562
5.458
5.348
5.233
5.111
4.985
4.853
4.716
4.575
4.428
4.277
4.122
3.963
3.799
3.632
3.462
3.288
3.111
2.931
2.748
2.562
2.375
2.184
1.992
1.798
1.603
1.406
1.207
1.008
0.807
0.606
0.404
0.202
0.000
1.327
1.326
1.324
1.320
1.314
1.307
1.298
1.288
1.276
1.262
1.247
1.230
1.212
1.193
1.172
1.149
1.125
1.100
1.074
1.046
1.017
0.986
0.955
0.922
0.888
0.853
0.817
0.780
0.742
0.703
0.663
0.623
0.582
0.540
0.497
0.454
0.410
0.366
0.321
0.276
0.230
0.185
0.139
0.093
0.046
0.000
-0.044
-0.090
-0.136
-0.181
-0.226
-0.270
-0.313
-0.354
-0.394
-0.432
-0.468
-0.501
-0.533
-0.562
-0.589
-0.613
-0.633
-0.651
-0.666
-0.678
-0.687
-0.692
-0.695
-0.693
-0.689
-0.682
-0.671
-0.657
-0.640
-0.620
-0.596
-0.571
-0.542
-0.511
-0.477
-0.441
-0.404
-0.364
-0.322
-0.279
-0.235
-0.189
-0.143
-0.095
-0.048
-0.000
-2.075
-2.073
-2.067
-2.060
-2.050
-2.037
-2.022
-2.004
-1.984
-1.962
-1.937
-1.910
-1.881
-1.850
-1.816
-1.780
-1.742
-1.702
-1.660
-1.616
-1.570
-1.522
-1.473
-1.422
-1.369
-1.314
-1.258
-1.201
-1.142
-1.082
-1.020
-0.958
-0.894
-0.829
-0.764
-0.697
-0.630
-0.561
-0.493
-0.423
-0.354
-0.283
-0.213
-0.142
-0.071
-0.000
-0.119
-0.119
-0.119
-0.119
-0.118
-0.118
-0.117
-0.116
-0.115
-0.114
-0.112
-0.111
-0.109
-0.107
-0.106
-0.103
-0.101
-0.099
-0.097
-0.094
-0.092
-0.089
-0.086
-0.083
-0.080
-0.077
-0.074
-0.070
-0.067
-0.063
-0.060
-0.056
-0.052
-0.049
-0.045
-0.041
-0.037
-0.033
-0.029
-0.025
-0.021
-0.017
-0.012
-0.008
-0.004
-0.000
Table 7.2: Anomalies of the lunar-solar elongation. The common argument corresponds to MM,
2D̄ − MM, D̄, MS, and 2F̄M for the case of q1, q2, q3, q4, and q5, respectively. If the argument is in
parenthesies then the anomalies are minus the values shown in the table.
112
01/1/1900
20/1/1901
09/1/1902
28/1/1903
17/1/1904
05/1/1905
24/1/1906
14/1/1907
03/1/1908
22/1/1909
11/1/1910
30/1/1911
19/1/1912
07/1/1913
26/1/1914
15/1/1915
05/1/1916
23/1/1917
12/1/1918
02/1/1919
21/1/1920
09/1/1921
28/1/1922
17/1/1923
06/1/1924
24/1/1925
14/1/1926
03/1/1927
22/1/1928
11/1/1929
29/1/1930
18/1/1931
07/1/1932
25/1/1933
15/1/1934
05/1/1935
24/1/1936
12/1/1937
01/1/1938
20/1/1939
09/1/1940
27/1/1941
16/1/1942
06/1/1943
25/1/1944
14/1/1945
03/1/1946
22/1/1947
11/1/1948
29/1/1949
MODERN ALMAGEST
2415021.07
2415405.10
2415759.37
2416143.18
2416497.16
2416851.26
2417235.21
2417589.74
2417944.40
2418328.50
2418682.98
2419066.89
2419420.95
2419774.94
2420158.78
2420513.10
2420867.69
2421251.81
2421606.44
2421960.84
2422344.71
2422698.72
2423082.50
2423436.61
2423791.03
2424175.10
2424529.77
2424884.35
2425268.33
2425622.50
2426006.29
2426360.28
2426714.48
2427098.46
2427453.06
2427807.72
2428191.80
2428546.18
2428900.28
2429284.06
2429638.09
2430021.96
2430376.39
2430731.02
2431115.14
2431469.71
2431824.00
2432207.84
2432561.83
2432945.62
18/1/1950
07/1/1951
26/1/1952
15/1/1953
05/1/1954
24/1/1955
13/1/1956
01/1/1957
19/1/1958
09/1/1959
28/1/1960
16/1/1961
06/1/1962
25/1/1963
14/1/1964
02/1/1965
21/1/1966
10/1/1967
29/1/1968
18/1/1969
07/1/1970
26/1/1971
16/1/1972
04/1/1973
23/1/1974
12/1/1975
01/1/1976
19/1/1977
09/1/1978
28/1/1979
17/1/1980
06/1/1981
25/1/1982
14/1/1983
03/1/1984
21/1/1985
10/1/1986
29/1/1987
19/1/1988
07/1/1989
26/1/1990
15/1/1991
04/1/1992
22/1/1993
11/1/1994
01/1/1995
20/1/1996
09/1/1997
28/1/1998
17/1/1999
2433299.83
2433654.33
2434038.42
2434393.09
2434747.59
2435131.53
2435485.62
2435839.60
2436223.43
2436577.73
2436961.75
2437316.39
2437671.03
2438055.06
2438409.35
2438763.37
2439147.16
2439501.26
2439885.18
2440239.70
2440594.36
2440978.45
2441332.94
2441687.14
2442070.95
2442424.94
2442779.11
2443163.08
2443517.66
2443901.76
2444256.39
2444610.80
2444994.69
2445348.71
2445702.73
2446086.61
2446441.01
2446825.06
2447179.72
2447534.31
2447918.30
2448272.48
2448626.47
2449010.28
2449364.46
2449718.95
2450103.02
2450457.68
2450841.75
2451196.14
06/1/2000
24/1/2001
13/1/2002
02/1/2003
21/1/2004
10/1/2005
29/1/2006
19/1/2007
08/1/2008
26/1/2009
15/1/2010
04/1/2011
23/1/2012
11/1/2013
01/1/2014
20/1/2015
10/1/2016
27/1/2017
17/1/2018
06/1/2019
24/1/2020
13/1/2021
02/1/2022
21/1/2023
11/1/2024
29/1/2025
18/1/2026
07/1/2027
26/1/2028
14/1/2029
04/1/2030
23/1/2031
12/1/2032
01/1/2033
20/1/2034
09/1/2035
28/1/2036
16/1/2037
05/1/2038
24/1/2039
14/1/2040
02/1/2041
21/1/2042
11/1/2043
30/1/2044
18/1/2045
07/1/2046
26/1/2047
15/1/2048
04/1/2049
2451550.25
2451934.05
2452288.07
2452642.34
2453026.37
2453381.00
2453765.09
2454119.66
2454473.97
2454857.81
2455211.80
2455565.88
2455949.82
2456304.31
2456658.97
2457043.05
2457397.56
2457781.49
2458135.58
2458489.57
2458873.41
2459227.70
2459582.26
2459966.36
2460321.00
2460705.02
2461059.31
2461413.34
2461797.14
2462151.23
2462505.61
2462889.67
2463244.33
2463598.93
2463982.91
2464337.11
2464720.92
2465074.91
2465429.07
2465813.06
2466167.63
2466522.30
2466906.36
2467260.77
2467644.65
2467998.68
2468352.69
2468736.58
2469090.97
2469445.59
23/1/2050
12/1/2051
02/1/2052
19/1/2053
08/1/2054
27/1/2055
16/1/2056
05/1/2057
24/1/2058
14/1/2059
03/1/2060
21/1/2061
10/1/2062
29/1/2063
18/1/2064
06/1/2065
25/1/2066
15/1/2067
05/1/2068
23/1/2069
12/1/2070
01/1/2071
20/1/2072
08/1/2073
27/1/2074
16/1/2075
06/1/2076
24/1/2077
14/1/2078
03/1/2079
22/1/2080
10/1/2081
28/1/2082
18/1/2083
07/1/2084
25/1/2085
15/1/2086
04/1/2087
23/1/2088
11/1/2089
30/1/2090
19/1/2091
09/1/2092
27/1/2093
16/1/2094
06/1/2095
25/1/2096
13/1/2097
02/1/2098
21/1/2099
2469829.70
2470184.29
2470538.61
2470922.45
2471276.44
2471660.25
2472014.42
2472368.90
2472753.00
2473107.66
2473462.19
2473846.12
2474200.23
2474584.01
2474938.03
2475292.30
2475676.33
2476030.96
2476385.61
2476769.64
2477123.96
2477478.00
2477861.78
2478215.85
2478599.77
2478954.26
2479308.92
2479693.03
2480047.54
2480401.77
2480785.57
2481139.55
2481523.38
2481877.65
2482232.21
2482616.33
2482970.97
2483325.41
2483709.30
2484063.34
2484447.11
2484801.19
2485155.56
2485539.63
2485894.29
2486248.90
2486632.90
2486987.11
2487341.11
2487724.89
Table 7.3: Dates and fractional Julian day numbers of the first new moons of the years 1900–2099
CE.
Lunar-Solar Syzygies and Eclipses
113
Arg. (◦ )
δβ1 ( ′ )
δβ2 ( ′ )
δβ3 ( ′ )
Arg. (◦ )
000/360
002/358
004/356
006/354
008/352
010/350
012/348
014/346
016/344
018/342
020/340
022/338
024/336
026/334
028/332
030/330
032/328
034/326
036/324
038/322
040/320
042/318
044/316
046/314
048/312
050/310
052/308
054/306
056/304
058/302
060/300
062/298
064/296
066/294
068/292
070/290
072/288
074/286
076/284
078/282
080/280
082/278
084/276
086/274
088/272
090/270
3.128
3.126
3.120
3.111
3.097
3.080
3.059
3.035
3.007
2.975
2.939
2.900
2.857
2.811
2.762
2.709
2.652
2.593
2.530
2.465
2.396
2.324
2.250
2.173
2.093
2.010
1.926
1.838
1.749
1.657
1.564
1.468
1.371
1.272
1.172
1.070
0.967
0.862
0.757
0.650
0.543
0.435
0.327
0.218
0.109
0.000
0.856
0.855
0.854
0.851
0.847
0.843
0.837
0.830
0.822
0.814
0.804
0.793
0.782
0.769
0.755
0.741
0.726
0.709
0.692
0.674
0.655
0.636
0.615
0.594
0.573
0.550
0.527
0.503
0.478
0.453
0.428
0.402
0.375
0.348
0.321
0.293
0.264
0.236
0.207
0.178
0.149
0.119
0.089
0.060
0.030
0.000
0.267
0.267
0.267
0.266
0.265
0.263
0.261
0.259
0.257
0.254
0.251
0.248
0.244
0.240
0.236
0.231
0.227
0.222
0.216
0.211
0.205
0.199
0.192
0.186
0.179
0.172
0.165
0.157
0.149
0.142
0.134
0.125
0.117
0.109
0.100
0.091
0.083
0.074
0.065
0.056
0.046
0.037
0.028
0.019
0.009
0.000
(180)/(180)
(178)/(182)
(176)/(184)
(174)/(186)
(172)/(188)
(170)/(190)
(168)/(192)
(166)/(194)
(164)/(196)
(162)/(198)
(160)/(200)
(158)/(202)
(156)/(204)
(154)/(206)
(152)/(208)
(150)/(210)
(148)/(212)
(146)/(214)
(144)/(216)
(142)/(218)
(140)/(220)
(138)/(222)
(136)/(224)
(134)/(226)
(132)/(228)
(130)/(230)
(128)/(232)
(126)/(234)
(124)/(236)
(122)/(238)
(120)/(240)
(118)/(242)
(116)/(244)
(114)/(246)
(112)/(248)
(110)/(250)
(108)/(252)
(106)/(254)
(104)/(256)
(102)/(258)
(100)/(260)
(098)/(262)
(096)/(264)
(094)/(266)
(092)/(268)
(090)/(270)
Table 7.4: Lunar-solar eclipse functions. The arguments of δβ1, δβ2, and δβ3 are MM, MM, and MS,
respectively. δβ1, δβ2, and δβ3 take minus the values shown in the table if their arguments are in
parenthesies.
114
MODERN ALMAGEST
The Superior Planets
115
8 The Superior Planets
8.1 Determination of Ecliptic Longitude
Figure 8.1 compares and contrasts heliocentric and geocentric models of the motion of a superior
planet (i.e., a planet which is further from the sun than the earth), P, as seen from the earth, G.
The sun is at S. In the heliocentric model, we can write the earth-planet displacement vector, P, as
the sum of the earth-sun displacement vector, S, and the sun-planet displacement vector, P ′ . The
geocentric model, which is entirely equivalent to the heliocentric model as far as the relative motion
of the planet with respect to the earth is concerned, and is much more convenient, relies on the
simple vector identity
P = S + P ′ ≡ P ′ + S.
(8.1)
In other words, we can get from the earth to the planet by one of two different routes. The first
route corresponds to the heliocentric model, and the second to the geocentric model. In the latter
model, P ′ gives the displacement of the so-called guide-point, G ′ , from the earth. Since P ′ is also
the displacement of the planet, P, from the sun, S, it is clear that G ′ executes a Keplerian orbit
about the earth whose elements are the same as those of the orbit of the planet about the sun. The
ellipse traced out by G ′ is termed the deferent. The vector S gives the displacement of the planet
from the guide-point. However, S is also the displacement of the sun from the earth. Hence, it is
clear that the planet, P, executes a Keplerian orbit about the guide-point, G ′ , whose elements are
the same as the sun’s apparent orbit about the earth. The ellipse traced out by P about G ′ is termed
the epicycle.
deferent
planetary orbit
S
G
S
P′
Earth’s orbit
Heliocentric
P
P′
G
epicycle
P
G′
P
P
S
Geocentric
Figure 8.1: Heliocentric and geocentric models of the motion of a superior planet. Here, S is the sun,
G the earth, and P the planet. View is from the northern ecliptic pole.
Figure 8.2 illustrates in more detail how the deferent-epicycle model is used to determine the
ecliptic longitude of a superior planet. The planet P orbits (counterclockwise) on a small Keplerian
116
MODERN ALMAGEST
P
′
a
A′
C′
T ′ Π′
̟′
G′
Υ
a
G
A
C
ea
T
Π
̟
Υ
Figure 8.2: Planetary longitude model. View is from northern ecliptic pole.
orbit Π ′ PA ′ about guide-point G ′ , which, in turn, orbits the earth, G, (counterclockwise) on a
large Keplerian orbit ΠG ′ A. As has already been mentioned, the small orbit is termed the epicycle,
and the large orbit the deferent. Both orbits are assumed to lie in the plane of the ecliptic. This
approximation does not introduce a large error into our calculations because the orbital inclinations
of the visible planets to the ecliptic plane are all fairly small. Let C, A, Π, a, e, ̟, and T denote the
geometric center, apocenter (i.e., the point of furthest distance from the central object), pericenter
(i.e., the point of closest approach to the central object), major radius, eccentricity, longitude of the
pericenter, and true anomaly of the deferent, respectively. Let C ′ , A ′ , Π ′ , a ′ , e ′ , ̟ ′ , and T ′ denote
the corresponding quantities for the epicycle.
P
r′ sin µ
r′
µ
θ
G′
G
r
B
r′ cos µ
Figure 8.3: The triangle GBP.
Let the line GG ′ be produced, and let the perpendicular PB be dropped to it from P, as shown
The Superior Planets
117
in Fig. 8.3. The angle µ ≡ PG ′ B is termed the epicyclic anomaly (see Fig. 8.4), and takes the form
µ = T ′ + ̟ ′ − T − ̟ = λ̄ ′ + q ′ − λ̄ − q,
(8.2)
where λ̄ and q are the mean longitude and equation of center for the deferent, whereas λ̄ ′ and q ′
are the corresponding quantities for the epicycle—see Cha. 5. The epicyclic anomaly is generally
written in the range 0◦ to 360◦ . The angle θ ≡ PGG ′ is termed the equation of the epicycle, and is
usually written in the range −180◦ to +180◦ . It is clear from the figure that
tan θ =
sin µ
,
r/r ′ + cos µ
(8.3)
where r ≡ GG ′ and r ′ ≡ G ′ P are the radial polar coordinates for the deferent and epicycle,
respectively. Moreover, according to Equation (4.22), r/r ′ = (a/a ′ ) z, where
1−ζ
,
1 − ζ′
z=
(8.4)
and
ζ = e cos M − e 2 sin2 M,
(8.5)
ζ ′ = e ′ cos M ′ − e ′ 2 sin2 M ′
(8.6)
are termed radial anomalies. Finally, the ecliptic longitude of the planet is given by (see Fig. 8.4)
(8.7)
λ = λ̄ + q + θ.
Now,
θ(µ, z) ≡ tan−1
sin µ
′
(a/a ) z + cos µ
(8.8)
is a function of two variables, µ and z. It is impractical to tabulate such a function directly. Fortunately, whilst θ(µ, z) has a strong dependence on µ, it only has a fairly weak dependence on z. In
fact, it is easily seen that z varies between zmin = z̄ − δz and zmax = z̄ + δz, where
1 + e e′
,
1 − e′ 2
e + e′
.
δz =
1 − e′ 2
z̄ =
(8.9)
(8.10)
Let us define
z̄ − z
.
(8.11)
δz
This variable takes the value −1 when z = zmax , the value 0 when z = z̄, and the value +1 when
z = zmin . Thus, using quadratic interpolation, we can write
ξ=
θ(µ, z) ≃ Θ−(ξ) δθ−(µ) + θ̄(µ) + Θ+(ξ) δθ+(µ),
(8.12)
where
θ̄(µ) = θ(µ, z̄),
(8.13)
δθ−(µ) = θ(µ, z̄) − θ(µ, zmax ),
(8.14)
δθ+(µ) = θ(µ, zmin ) − θ(µ, z̄),
(8.15)
118
MODERN ALMAGEST
and
Θ−(ξ) = −(1/2) ξ (ξ − 1),
(8.16)
Θ+(ξ) = +(1/2) ξ (ξ + 1).
(8.17)
This scheme allows us to avoid having to tabulate a two-dimensional function, whilst ensuring that
the exact value of θ(µ, z) is obtained when z = z̄, zmin , or zmax . The above interpolation scheme is
very similar to that adopted by Ptolemy in the Almagest.
Our procedure for determining the ecliptic longitude of a superior planet is described below. It
is assumed that the ecliptic longitude, λS, and the radial anomaly, ζS, of the sun have already been
calculated. The latter quantity is tabulated as a function of the solar mean anomaly in Table 5.4. In
the following, a, e, n, ñ, λ̄0, and M0 represent elements of the orbit of the planet in question about
the sun, and eS represents the eccentricity of the sun’s apparent orbit about the earth. (In general,
the subscript S denotes the sun.) In particular, a is the major radius of the planetary orbit in units in
which the major radius of the sun’s apparent orbit about the earth is unity. The requisite elements
for all of the superior planets at the J2000 epoch (t0 = 2 451 545.0 JD) are listed in Table 5.1. The
ecliptic longitude of a superior planet is specified by the following formulae:
λ̄ = λ̄0 + n (t − t0),
(8.18)
M = M0 + ñ (t − t0),
(8.19)
q = 2 e sin M + (5/4) e2 sin 2M,
(8.20)
ζ = e cos M − e2 sin2 M,
(8.21)
(8.22)
µ = λS − λ̄ − q,
θ̄ = θ(µ, z̄) ≡ tan−1
sin µ
,
a z̄ + cos µ
(8.23)
δθ− = θ(µ, z̄) − θ(µ, zmax ),
(8.24)
δθ+ = θ(µ, zmin ) − θ(µ, z̄),
(8.25)
1−ζ
,
1 − ζS
z̄ − z
ξ =
,
δz
θ = Θ−(ξ) δθ− + θ̄ + Θ+(ξ) δθ+,
(8.28)
λ = λ̄ + q + θ.
(8.29)
z =
(8.26)
(8.27)
Here, z̄ = (1 + e eS)/(1 − eS2), δz = (e + eS)/(1 − eS2), zmin = z̄ − δz, and zmax = z̄ + δz. The constants
z̄, δz, zmin , and zmax for each of the superior planets are listed in Table 8.1. Finally, the functions
Θ± are tabulated in Table 8.2.
For the case of Mars, the above formulae are capable of matching NASA ephemeris data during
the years 1995–2006 CE with a mean error of 3 ′ and a maximum error of 14 ′ . For the case of
Jupiter, the mean error is 1.6 ′ and the maximum error 4 ′ . Finally, for the case of Saturn, the mean
error is 0.5 ′ and the maximum error 1 ′ .
The Superior Planets
119
8.2 Mars
The ecliptic longitude of Mars can be determined with the aid of Tables 8.3–8.5. Table 8.3 allows
the mean longitude, λ̄, and the mean anomaly, M, of Mars to be calculated as functions of time.
Next, Table 8.4 permits the equation of center, q, and the radial anomaly, ζ, to be determined
as functions of the mean anomaly. Finally, Table 8.5 allows the quantities δθ−, θ̄, and δθ+ to be
calculated as functions of the epicyclic anomaly, µ.
The procedure for using the tables is as follows:
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the ecliptic longitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t − t0,
where t0 = 2 451 545.0 is the epoch.
2. Calculate the ecliptic longitude, λS, and radial anomaly, ζS, of the sun using the procedure
set out in Sect. 5.1.
3. Enter Table 8.3 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆λ̄ and ∆M. If ∆t is negative then the corresponding values are also negative. The
value of the mean longitude, λ̄, is the sum of all the ∆λ̄ values plus the value of λ̄ at the
epoch. Likewise, the value of the mean anomaly, M, is the sum of all the ∆M values plus the
value of M at the epoch. Add as many multiples of 360◦ to λ̄ and M as is required to make
them both fall in the range 0◦ to 360◦ . Round M to the nearest degree.
4. Enter Table 8.4 with the value of M and take out the corresponding value of the equation of
center, q, and the radial anomaly, ζ. It is necessary to interpolate if M is odd.
5. Form the epicyclic anomaly, µ = λS − λ̄ − q. Add as many multiples of 360◦ to µ as is required
to make it fall in the range 0◦ to 360◦ . Round µ to the nearest degree.
6. Enter Table 8.5 with the value of µ and take out the corresponding values of δθ−, θ̄, and δθ+.
If µ > 180◦ then it is necessary to make use of the identities δθ± (360◦ − µ) = −δθ± (µ) and
θ̄(360◦ − µ) = −θ̄(µ).
7. Form z = (1 − ζ)/(1 − ζS).
8. Obtain the values of z̄ and δz from Table 8.1. Form ξ = (z̄ − z)/δz.
9. Enter Table 8.2 with the value of ξ and take out the corresponding values of Θ− and Θ+. If
ξ < 0 then it is necessary to use the identities Θ+(ξ) = −Θ−(−ξ) and Θ−(ξ) = −Θ+(−ξ).
10. Form the equation of the epicycle, θ = Θ− δθ− + θ̄ + Θ+ δθ+.
11. The ecliptic longitude, λ, is the sum of the mean longitude, λ̄, the equation of center, q, and
the equation of the epicycle, θ. If necessary convert λ into an angle in the range 0◦ to 360◦ .
The decimal fraction can be converted into arc minutes using Table 5.2. Round to the nearest
arc minute. The final result can be written in terms of the signs of the zodiac using the table
in Sect. 2.6.
Two examples of this procedure are given below.
120
MODERN ALMAGEST
Example 1: May 5, 2005 CE, 00:00 UT:
From Sect. 5.1, t − t0 = 1 950.5 JD, λS = 44.602◦ , MS ≃ 120◦ . Hence, it follows from Table 5.4 that
ζS(MS) = −8.56 × 10−3. Making use of Table 8.3, we find:
t(JD)
λ̄(◦ )
M(◦ )
+1000
+900
+50
+.5
Epoch
164.071
111.664
26.204
0.262
355.460
657.661
297.661
164.021
111.619
26.201
0.262
19.388
321.491
321.491
Modulus
Given that M ≃ 321◦ , Table 8.4 yields
q(321◦ ) = −7.345◦ ,
ζ(321◦ ) = 6.912 × 10−2.
Thus,
µ = λS − λ̄ − q = 44.602 − 297.661 + 7.345 = 114.286 ≃ 114◦ ,
where we have rounded the epicylic anomaly to the nearest degree. It follows from Table 8.5 that
δθ−(114◦ ) = 3.853◦ ,
θ̄(114◦ ) = 39.209◦ ,
δθ+(114◦ ) = 4.612◦ .
Now,
z = (1 − ζ)/(1 − ζS) = (1 − 6.912 × 10−2)/(1 + 8.56 × 10−3) = 0.9230.
However, from Table 8.1, z̄ = 1.00184 and δz = 0.11014, so
ξ = (z̄ − z)/δz = (1.00184 − 0.9230)/0.11014 ≃ 0.72.
According to Table 8.2,
Θ−(0.72) = 0.101,
Θ+(0.72) = 0.619,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = 0.101 × 3.853 + 39.209 + 0.619 × 4.612 = 42.453◦ .
Finally,
λ = λ̄ + q + θ = 297.661 − 7.345 + 42.453 = 332.769 ≃ 332◦ 46 ′ .
Thus, the ecliptic longitude of Mars at 00:00 UT on May 5, 2005 CE was 2PI46.
Example 2: December 25, 1800 CE, 00:00 UT:
From Sect. 5.1, t − t0 = −72, 690.5 JD, λS = 273.055◦ , MS ≃ 354◦ . Hence, it follows from Table 5.4 that ζS(MS) = 1.662 × 10−2. Making use of Table 8.3, we find:
The Superior Planets
121
t(JD)
λ̄(◦ )
M(◦ )
-70,000
-2,000
-600
-90
-.5
Epoch
−324.983
−328.142
−314.443
−47.166
−0.262
355.460
−659.536
60.464
−321.453
−328.042
−314.412
−47.162
−0.262
19.388
−991.943
88.057
Modulus
Given that M ≃ 88◦ , Table 8.4 yields
q(88◦ ) = 10.739◦ ,
ζ(88◦ ) = −5.45 × 10−3,
so
µ = λS − λ̄ − q = 273.055 − 60.464 − 10.739 = 201.852 ≃ 202◦ .
It follows from Table 8.5 that
δθ−(202◦ ) = −5.980◦ ,
θ̄(202◦ ) = −32.007◦ ,
δθ+(202◦ ) = −8.955◦ .
Now,
z = (1 − ζ)/(1 − ζS) = (1 + 5.45 × 10−3)/(1 − 1.662 × 10−2) = 1.02244,
so
ξ = (z̄ − z)/δz = (1.00184 − 1.02244)/0.11014 ≃ −0.19.
According to Table 8.2,
Θ−(−0.19) = −0.113,
Θ+(−0.19) = −0.077,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = −0.113 × 5.980 − 32.007 − 0.077 × 8.955 = −30.642◦ .
Finally,
λ = λ̄ + q + θ = 60.464 + 10.739 − 30.642 = 40.561 ≃ 40◦ 34 ′ .
Thus, the ecliptic longitude of Mars at 00:00 UT on December 25, 1800 CE was 10TA34.
8.3 Determination of Conjunction, Opposition, and Station Dates
Figure 8.4 shows the geocentric orbit of a superior planet. Recall that the vector G ′ P is always
parallel to the vector connecting the earth to the sun. It follows that a so-called conjunction, at which
the sun lies directly between the planet and the earth, occurs whenever the epicyclic anomaly, µ,
takes the value 0◦ . At a conjunction, the planet is furthest from the earth, and has the same ecliptic
longitude as the sun, and is, therefore, invisible. Conversely, a so-called opposition, at which the
earth lies directly between the planet and the sun, occurs whenever µ = 180◦ . At an opposition,
the planet is closest to the earth, and also directly opposite the sun in the sky, and, therefore, at its
brightest. Now, a superior planet rotates around the epicycle at a faster angular velocity than its
122
MODERN ALMAGEST
P
G′
θ
λ̄ + q
G
µ
Υ
Figure 8.4: The geocentric orbit of a superior planet. Here, G, G ′ , P, µ, θ, λ̄, q, and Υ represent
the earth, guide-point, planet, epicyclic anomaly, equation of the epicycle, mean longitude, equation
of center, and spring equinox, respectively. View is from northern ecliptic pole. Both G ′ and P orbit
counterclockwise.
guide-point rotates around the deferent. Moreover, both the planet and guide-point rotate in the
same direction. It follows that the planet is traveling backward in the sky (relative to the direction
of its mean motion) at opposition. This phenomenon is called retrograde motion. The period of
retrograde motion begins and ends at stations—so-called because when the planet reaches them it
appears to stand still in the sky for a few days whilst it reverses direction.
Tables 8.3–8.5 can be used to determine the dates of the conjunctions, oppositions, and stations
of Mars. Consider the first conjunction after the epoch (January 1, 2000 CE). We can estimate
the time at which this event occurs by approximating the epicyclic anomaly as the so-called mean
epicyclic anomaly:
µ ≃ µ̄ = λ̄S − λ̄ = λ̄0S − λ̄0 + (nS − n) (t − t0) = 284.998 + 0.46157617 (t − t0).
We obtain
t ≃ t0 + (360 − 284.998)/0.46157617 ≃ t0 + 162 JD.
A calculation of the epicyclic anomaly at this time, using Tables 8.3–8.5, yields µ = −9.583◦ . Now,
the actual conjunction occurs when µ = 0◦ . Hence, our next estimate is
t ≃ t0 + 162 + 9.583/0.46157617 ≃ t0 + 183 JD.
A calculation of the epicyclic anomaly at this time gives 0.294◦ . Thus, our final estimate is
t = t0 + 183 − 0.294/0.461557617 = t0 + 182.4 JD,
which corresponds to July 1, 2000 CE.
The Superior Planets
123
Consider the first opposition of Mars after the epoch. Our first estimate of the time at which
this event takes place is
t ≃ t0 + (540 − 284.998)/0.46157617 ≃ t0 + 552 JD.
A calculation of the epicyclic anomaly at this time yields µ = 188.649◦ . Now, the actual opposition
occurs when µ = 180◦ . Hence, our second estimate is
t ≃ t0 + 552 − 8.649/0.46157617 ≃ t0 + 533 JD.
A calculation of the epicyclic anomaly at this time gives 181.455◦ . Thus, our third estimate is
t ≃ t0 + 533 − 1.455/0.46157617 ≃ t0 + 530 JD.
A calculation of the epicyclic anomaly at this time yields 180.244◦ . Hence, our final estimate is
t = t0 + 530 − 0.244/0.46157617 = t0 + 529.5 JD,
which corresponds to June 13, 2001 CE. Incidentally, it is clear from the above analysis that the
mean time period between successive conjunctions, or oppositions, of Mars is 360/0.46157617 =
779.9 JD, which is equivalent to 2.14 years.
Let us now consider the stations of Mars. We can approximate the ecliptic longitude of a
superior planet as
λ ≃ λ̄ + θ̄,
(8.30)
where
θ̄ = tan−1
sin µ̄
,
ā + cos µ̄
(8.31)
and ā = a z̄. Note that dλ̄/dt = n and dµ̄/dt = nS − n. It follows that
dλ
≃n+
dt
ā cos µ̄ + 1
(nS − n).
1 + 2 ā cos µ̄ + ā2
(8.32)
Now, a station corresponds to dλ/dt = 0 (i.e., a local maximum or minimum of λ), which gives
cos µ̄ ≃ −
(ā2 + nS/n)
.
ā (1 + nS/n)
(8.33)
For the case of Mars, we find that µ̄ = 163.3◦ or 196.7◦ . The first solution corresponds to the socalled retrograde station, at which the planet switches from direct to retrograde motion. The second
solution corresponds to the so-called direct station, at which the planet switches from retrograde to
direct motion. The mean time interval between a retrograde station and the following opposition,
or between an opposition and the following direct station, is (180 − 163.3)/0.46157617 ≃ 36 JD.
Unfortunately, the only option for accurately determining the dates at which the stations occur is
to calculate the ecliptic longitude of Mars over a range of days centered 36 days before and after
its opposition.
Table 8.6 shows the conjunctions, oppositions, and stations of Mars for the years 2000–2020
CE, calculated using the techniques described above.
124
MODERN ALMAGEST
8.4 Jupiter
The ecliptic longitude of Jupiter can be determined with the aid of Tables 8.7–8.9. Table 8.7 allows
the mean longitude, λ̄, and the mean anomaly, M, of Jupiter to be calculated as functions of time.
Next, Table 8.8 permits the equation of center, q, and the radial anomaly, ζ, to be determined
as functions of the mean anomaly. Finally, Table 8.9 allows the quantities δθ−, θ̄, and δθ+ to be
calculated as functions of the epicyclic anomaly, µ. The procedure for using the tables is analogous
to the previously described procedure for using the Mars tables. One example of this procedure is
given below.
Example: May 5, 2005 CE, 00:00 UT:
From before, t − t0 = 1 950.5 JD, λS = 44.602◦ , MS ≃ 120◦ , and ζS = −8.56 × 10−3. Making
use of Table 8.7, we find:
t(JD)
λ̄(◦ )
M(◦ )
+1000
+900
+50
+.5
Epoch
83.125
74.813
4.156
0.042
34.365
196.501
196.501
83.081
74.773
4.154
0.042
19.348
181.398
181.398
Modulus
Given that M ≃ 181◦ , Table 8.8 yields
q(181◦ ) = −0.091◦ ,
ζ(181◦ ) = −4.838 × 10−2.
Thus,
µ = λS − λ̄ − q = 44.602 − 196.501 + 0.091 = −151.808 ≃ 208◦ ,
where we have rounded the epicylic anomaly to the nearest degree. It follows from Table 8.9 that
δθ−(208◦ ) = −0.447◦ ,
θ̄(208◦ ) = −6.194◦ ,
δθ+(208◦ ) = −0.522◦ .
Now,
z = (1 − ζ)/(1 − ζS) = (1 + 4.838 × 10−2)/(1 + 8.56 × 10−3) = 1.0395.
However, from Table 8.1, z̄ = 1.00109 and δz = 0.06512, so
ξ = (z̄ − z)/δz = (1.00109 − 1.0395)/0.06512 ≃ −0.59.
According to Table 8.2,
Θ−(−0.59) = −0.469,
Θ+(−0.59) = −0.121,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = 0.469 × 0.447 − 6.194 + 0.121 × 0.522 = −5.921◦ .
The Superior Planets
125
Finally,
λ = λ̄ + q + θ = 196.501 − 0.091 − 5.921 = 190.489 ≃ 190◦ 29 ′ .
Thus, the ecliptic longitude of Jupiter at 00:00 UT on May 5, 2005 CE was 10LI29.
The conjunctions, oppositions, and stations of Jupiter can be investigated using analogous methods to those employed earlier to examine the conjunctions, oppositions, and stations of Mars. We
find that the mean time period between successive oppositions or conjunctions of Jupiter is 1.09
yr. Furthermore, on average, the retrograde and direct stations of Jupiter occur when the epicyclic
anomaly takes the values µ = 125.6◦ and 234.4◦ , respectively. Finally, the mean time period between a retrograde station and the following opposition, or between the opposition and the following direct station, is 60 JD. The conjunctions, oppositions, and stations of Jupiter during the years
2000–2010 CE are shown in Table 8.10.
8.5 Saturn
The ecliptic longitude of Saturn can be determined with the aid of Tables 8.11–8.13. Table 8.11
allows the mean longitude, λ̄, and the mean anomaly, M, of Saturn to be calculated as functions of
time. Next, Table 8.12 permits the equation of center, q, and the radial anomaly, ζ, to be determined
as functions of the mean anomaly. Finally, Table 8.13 allows the quantities δθ−, θ̄, and δθ+ to be
calculated as functions of the epicyclic anomaly, µ. The procedure for using the tables is analogous
to the previously described procedure for using the Mars tables. One example of this procedure is
given below.
Example: May 5, 2005 CE, 00:00 UT:
From before, t − t0 = 1 950.5 JD, λS = 44.602◦ , MS ≃ 120◦ , and ζS = −8.56 × 10−3. Making
use of Table 8.11, we find:
t(JD)
λ̄(◦ )
M(◦ )
+1000
+900
+50
+.5
Epoch
33.508
30.157
1.675
0.017
50.059
115.416
115.416
33.482
30.133
1.674
0.017
317.857
383.163
23.163
Modulus
Given that M ≃ 23◦ , Table 8.12 yields
q(23◦ ) = 2.561◦ ,
ζ(23◦ ) = 4.913 × 10−2.
Thus,
µ = λS − λ̄ − q = 44.602 − 115.416 − 2.561 = −73.375 ≃ 287◦ ,
where we have rounded the epicylic anomaly to the nearest degree. It follows from Table 8.13 that
δθ−(287◦ ) = −0.353◦ ,
θ̄(287◦ ) = −5.551◦ ,
δθ+(287◦ ) = −0.405◦ .
126
MODERN ALMAGEST
Now,
z = (1 − ζ)/(1 − ζS) = (1 − 4.913 × 10−2)/(1 + 8.56 × 10−3) = 0.9428.
However, from Table 8.1, z̄ = 1.00118 and δz = 0.07059, so
ξ = (z̄ − z)/δz = (1.00118 − 0.9428)/0.07059 ≃ 0.83.
According to Table 8.2,
Θ−(0.83) = 0.071,
Θ+(0.83) = 0.759,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = −0.071 × 0.353 − 5.551 − 0.759 × 0.405 = −5.883◦ .
Finally,
λ = λ̄ + q + θ = 115.416 + 2.561 − 5.883 = 112.094 ≃ 112◦ 06 ′ .
Thus, the ecliptic longitude of Saturn at 00:00 UT on May 5, 2005 CE was 22CN06.
The conjunctions, oppositions, and stations of Saturn can be investigated using analogous methods to those employed earlier to examine the conjunctions, oppositions, and stations of Mars. We
find that the mean time period between successive oppositions or conjunctions of Saturn is 1.035
yr. Furthermore, on average, the retrograde and direct stations of Saturn occur when the epicyclic
anomaly takes the values µ = 114.5◦ and 245.5◦ , respectively. Finally, the mean time period between a retrograde station and the following opposition, or between the opposition and the following direct station, is 69 JD. The conjunctions, oppositions, and stations of Saturn during the years
2000–2010 CE are shown in Table 8.14.
The Superior Planets
127
Planet
Mercury
Venus
Mars
Jupiter
Saturn
z̄
δz
zmin
zmax
1.04774
1.00016
1.00184
1.00109
1.00118
0.23216
0.02349
0.11014
0.06512
0.07059
0.81558
0.97667
0.89170
0.93597
0.93059
1.27990
1.02365
1.11198
1.06602
1.07177
Table 8.1: Constants associated with the epicycles of the inferior and superior planets.
ξ
Θ−
Θ+
ξ
Θ−
Θ+
ξ
Θ−
Θ+
ξ
Θ−
Θ+
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.22
0.23
0.24
0.25
0.000
0.005
0.010
0.015
0.019
0.024
0.028
0.033
0.037
0.041
0.045
0.049
0.053
0.057
0.060
0.064
0.067
0.071
0.074
0.077
0.080
0.083
0.086
0.089
0.091
0.094
0.000
0.005
0.010
0.015
0.021
0.026
0.032
0.037
0.043
0.049
0.055
0.061
0.067
0.073
0.080
0.086
0.093
0.099
0.106
0.113
0.120
0.127
0.134
0.141
0.149
0.156
0.25
0.26
0.27
0.28
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.40
0.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
0.50
0.094
0.096
0.099
0.101
0.103
0.105
0.107
0.109
0.111
0.112
0.114
0.115
0.117
0.118
0.119
0.120
0.121
0.122
0.123
0.123
0.124
0.124
0.125
0.125
0.125
0.125
0.156
0.164
0.171
0.179
0.187
0.195
0.203
0.211
0.219
0.228
0.236
0.245
0.253
0.262
0.271
0.280
0.289
0.298
0.307
0.317
0.326
0.336
0.345
0.355
0.365
0.375
0.50
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0.68
0.69
0.70
0.71
0.72
0.73
0.74
0.75
0.125
0.125
0.125
0.125
0.124
0.124
0.123
0.123
0.122
0.121
0.120
0.119
0.118
0.117
0.115
0.114
0.112
0.111
0.109
0.107
0.105
0.103
0.101
0.099
0.096
0.094
0.375
0.385
0.395
0.405
0.416
0.426
0.437
0.447
0.458
0.469
0.480
0.491
0.502
0.513
0.525
0.536
0.548
0.559
0.571
0.583
0.595
0.607
0.619
0.631
0.644
0.656
0.75
0.76
0.77
0.78
0.79
0.80
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.90
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1.00
0.094
0.091
0.089
0.086
0.083
0.080
0.077
0.074
0.071
0.067
0.064
0.060
0.057
0.053
0.049
0.045
0.041
0.037
0.033
0.028
0.024
0.019
0.015
0.010
0.005
0.000
0.656
0.669
0.681
0.694
0.707
0.720
0.733
0.746
0.759
0.773
0.786
0.800
0.813
0.827
0.841
0.855
0.869
0.883
0.897
0.912
0.926
0.941
0.955
0.970
0.985
1.000
Table 8.2: Epicyclic interpolation coefficients. Note that Θ± (ξ) = −Θ∓ (−ξ).
128
MODERN ALMAGEST
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
200.712
41.424
242.135
82.847
283.559
124.271
324.983
165.694
6.406
200.208
40.415
240.623
80.830
281.038
121.246
321.453
161.661
1.868
200.409
40.819
241.228
81.638
282.047
122.456
322.866
163.275
3.685
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
164.071
328.142
132.214
296.285
100.356
264.427
68.498
232.569
36.641
164.021
328.042
132.062
296.083
100.104
264.125
68.145
232.166
36.187
164.041
328.082
132.123
296.164
100.205
264.246
68.287
232.328
36.368
100
200
300
400
500
600
700
800
900
52.407
104.814
157.221
209.628
262.036
314.443
6.850
59.257
111.664
52.402
104.804
157.206
209.608
262.010
314.412
6.815
59.217
111.619
52.404
104.808
157.212
209.616
262.020
314.425
6.829
59.233
111.637
10
20
30
40
50
60
70
80
90
5.241
10.481
15.722
20.963
26.204
31.444
36.685
41.926
47.166
5.240
10.480
15.721
20.961
26.201
31.441
36.681
41.922
47.162
5.240
10.481
15.721
20.962
26.202
31.442
36.683
41.923
47.164
1
2
3
4
5
6
7
8
9
0.524
1.048
1.572
2.096
2.620
3.144
3.668
4.193
4.717
0.524
1.048
1.572
2.096
2.620
3.144
3.668
4.192
4.716
0.524
1.048
1.572
2.096
2.620
3.144
3.668
4.192
4.716
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.052
0.105
0.157
0.210
0.262
0.314
0.367
0.419
0.472
0.052
0.105
0.157
0.210
0.262
0.314
0.367
0.419
0.472
0.052
0.105
0.157
0.210
0.262
0.314
0.367
0.419
0.472
Table 8.3: Mean motion of Mars. Here, ∆t = t − t0, ∆λ̄ = λ̄ − λ̄0, ∆M = M − M0, and ∆F̄ = F̄ − F̄0.
At epoch (t0 = 2 451 545.0 JD), λ̄0 = 355.460◦ , M0 = 19.388◦ , and F̄0 = 305.796◦ .
The Superior Planets
129
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
0.000
0.417
0.833
1.249
1.662
2.072
2.479
2.882
3.281
3.674
4.062
4.443
4.817
5.184
5.542
5.892
6.233
6.564
6.885
7.195
7.494
7.782
8.059
8.323
8.575
8.814
9.040
9.252
9.452
9.637
9.809
9.967
10.111
10.241
10.357
10.458
10.546
10.619
10.678
10.722
10.753
10.770
10.773
10.763
10.739
10.702
9.339
9.333
9.312
9.279
9.232
9.171
9.098
9.011
8.911
8.799
8.674
8.537
8.388
8.227
8.054
7.870
7.675
7.470
7.254
7.029
6.794
6.550
6.297
6.036
5.768
5.491
5.208
4.919
4.623
4.322
4.016
3.705
3.389
3.071
2.749
2.424
2.097
1.768
1.438
1.107
0.776
0.444
0.114
-0.217
-0.545
-0.872
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
10.702
10.652
10.589
10.514
10.426
10.326
10.214
10.091
9.957
9.811
9.655
9.489
9.313
9.127
8.932
8.727
8.514
8.293
8.064
7.827
7.583
7.332
7.074
6.810
6.540
6.264
5.983
5.696
5.405
5.110
4.810
4.506
4.199
3.889
3.575
3.259
2.940
2.619
2.296
1.971
1.645
1.317
0.989
0.660
0.330
0.000
-0.872
-1.197
-1.519
-1.839
-2.155
-2.468
-2.776
-3.081
-3.380
-3.675
-3.964
-4.248
-4.527
-4.799
-5.065
-5.324
-5.576
-5.822
-6.060
-6.292
-6.515
-6.731
-6.939
-7.139
-7.331
-7.515
-7.690
-7.857
-8.015
-8.165
-8.306
-8.438
-8.562
-8.676
-8.782
-8.878
-8.966
-9.044
-9.113
-9.173
-9.224
-9.265
-9.298
-9.321
-9.335
-9.339
180
182
184
186
188
190
192
194
196
198
200
202
204
206
208
210
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246
248
250
252
254
256
258
260
262
264
266
268
270
0.000
-0.330
-0.660
-0.989
-1.317
-1.645
-1.971
-2.296
-2.619
-2.940
-3.259
-3.575
-3.889
-4.199
-4.506
-4.810
-5.110
-5.405
-5.696
-5.983
-6.264
-6.540
-6.810
-7.074
-7.332
-7.583
-7.827
-8.064
-8.293
-8.514
-8.727
-8.932
-9.127
-9.313
-9.489
-9.655
-9.811
-9.957
-10.091
-10.214
-10.326
-10.426
-10.514
-10.589
-10.652
-10.702
-9.339
-9.335
-9.321
-9.298
-9.265
-9.224
-9.173
-9.113
-9.044
-8.966
-8.878
-8.782
-8.676
-8.562
-8.438
-8.306
-8.165
-8.015
-7.857
-7.690
-7.515
-7.331
-7.139
-6.939
-6.731
-6.515
-6.292
-6.060
-5.822
-5.576
-5.324
-5.065
-4.799
-4.527
-4.248
-3.964
-3.675
-3.380
-3.081
-2.776
-2.468
-2.155
-1.839
-1.519
-1.197
-0.872
270
272
274
276
278
280
282
284
286
288
290
292
294
296
298
300
302
304
306
308
310
312
314
316
318
320
322
324
326
328
330
332
334
336
338
340
342
344
346
348
350
352
354
356
358
360
-10.702
-10.739
-10.763
-10.773
-10.770
-10.753
-10.722
-10.678
-10.619
-10.546
-10.458
-10.357
-10.241
-10.111
-9.967
-9.809
-9.637
-9.452
-9.252
-9.040
-8.814
-8.575
-8.323
-8.059
-7.782
-7.494
-7.195
-6.885
-6.564
-6.233
-5.892
-5.542
-5.184
-4.817
-4.443
-4.062
-3.674
-3.281
-2.882
-2.479
-2.072
-1.662
-1.249
-0.833
-0.417
-0.000
-0.872
-0.545
-0.217
0.114
0.444
0.776
1.107
1.438
1.768
2.097
2.424
2.749
3.071
3.389
3.705
4.016
4.322
4.623
4.919
5.208
5.491
5.768
6.036
6.297
6.550
6.794
7.029
7.254
7.470
7.675
7.870
8.054
8.227
8.388
8.537
8.674
8.799
8.911
9.011
9.098
9.171
9.232
9.279
9.312
9.333
9.339
Table 8.4: Deferential anomalies of Mars.
130
µ
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
MODERN ALMAGEST
δθ−
θ̄
0.000
0.000
0.396
0.792
1.187
1.583
1.979
2.374
2.770
3.165
3.560
3.955
4.350
4.745
5.140
5.534
5.928
6.322
6.716
7.110
7.503
7.896
8.288
8.680
9.072
9.464
9.855
10.246
10.636
11.026
11.415
11.804
12.192
12.580
12.968
13.354
13.741
14.126
14.511
14.896
15.279
15.662
16.045
16.426
16.807
17.187
17.566
0.025
0.049
0.074
0.099
0.123
0.148
0.173
0.197
0.222
0.247
0.272
0.297
0.322
0.347
0.372
0.397
0.422
0.447
0.472
0.497
0.523
0.548
0.573
0.599
0.625
0.650
0.676
0.702
0.728
0.754
0.780
0.806
0.833
0.859
0.886
0.913
0.939
0.966
0.994
1.021
1.048
1.076
1.103
1.131
1.159
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
0.000
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
1.159
17.566
17.945
18.322
18.699
19.075
19.450
19.824
20.196
20.568
20.939
21.309
21.677
22.045
22.411
22.776
23.139
23.502
23.863
24.222
24.581
24.938
25.293
25.647
25.999
26.349
26.698
27.045
27.390
27.734
28.075
28.415
28.753
29.088
29.421
29.752
30.081
30.408
30.732
31.054
31.373
31.689
32.003
32.314
32.622
32.927
33.228
1.329
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
2.679
33.228
33.527
33.822
34.114
34.403
34.688
34.969
35.246
35.519
35.788
36.053
36.313
36.568
36.819
37.065
37.306
37.541
37.771
37.996
38.214
38.426
38.632
38.831
39.023
39.209
39.386
39.556
39.718
39.872
40.017
40.153
40.279
40.396
40.502
40.598
40.683
40.756
40.816
40.864
40.899
40.920
40.926
40.918
40.893
40.851
40.793
3.125
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
5.180
40.793
40.716
40.619
40.503
40.366
40.206
40.024
39.817
39.584
39.325
39.038
38.721
38.373
37.992
37.577
37.126
36.638
36.110
35.541
34.929
34.271
33.567
32.813
32.007
31.149
30.235
29.265
28.236
27.146
25.996
24.783
23.506
22.167
20.764
19.299
17.774
16.189
14.549
12.857
11.116
9.333
7.513
5.662
3.788
1.898
0.000
6.547
0.028
0.056
0.084
0.113
0.141
0.169
0.197
0.226
0.254
0.282
0.311
0.339
0.368
0.396
0.425
0.453
0.482
0.511
0.540
0.568
0.597
0.626
0.656
0.685
0.714
0.744
0.773
0.803
0.833
0.863
0.893
0.923
0.953
0.984
1.014
1.045
1.076
1.107
1.138
1.169
1.201
1.233
1.265
1.297
1.329
1.187
1.216
1.244
1.273
1.302
1.331
1.360
1.390
1.419
1.449
1.479
1.510
1.540
1.571
1.602
1.633
1.665
1.696
1.728
1.761
1.793
1.826
1.859
1.893
1.927
1.961
1.995
2.030
2.065
2.100
2.136
2.172
2.209
2.246
2.283
2.321
2.359
2.397
2.436
2.475
2.515
2.555
2.596
2.637
2.679
1.362
1.394
1.427
1.461
1.494
1.528
1.562
1.596
1.630
1.665
1.700
1.735
1.771
1.807
1.843
1.879
1.916
1.953
1.991
2.029
2.067
2.106
2.145
2.184
2.224
2.264
2.305
2.346
2.387
2.429
2.472
2.515
2.558
2.602
2.647
2.692
2.737
2.784
2.830
2.878
2.926
2.975
3.024
3.074
3.125
2.721
2.764
2.807
2.851
2.895
2.940
2.985
3.031
3.078
3.125
3.173
3.221
3.270
3.320
3.370
3.421
3.472
3.525
3.578
3.631
3.686
3.741
3.796
3.853
3.910
3.968
4.026
4.086
4.146
4.206
4.268
4.330
4.393
4.456
4.520
4.584
4.649
4.715
4.780
4.847
4.913
4.980
5.047
5.113
5.180
3.176
3.228
3.281
3.335
3.390
3.445
3.501
3.558
3.616
3.675
3.735
3.796
3.857
3.920
3.984
4.049
4.115
4.182
4.251
4.321
4.391
4.464
4.537
4.612
4.688
4.765
4.844
4.925
5.007
5.091
5.176
5.262
5.351
5.441
5.533
5.626
5.721
5.818
5.917
6.018
6.120
6.224
6.330
6.438
6.547
5.246
5.312
5.378
5.442
5.506
5.568
5.628
5.687
5.744
5.797
5.848
5.895
5.938
5.976
6.009
6.036
6.056
6.069
6.072
6.066
6.050
6.022
5.980
5.925
5.854
5.766
5.660
5.535
5.389
5.221
5.030
4.815
4.576
4.311
4.021
3.707
3.368
3.005
2.621
2.217
1.796
1.360
0.913
0.458
0.000
Table 8.5: Epicyclic anomalies of Mars. All quantities are in degrees. Note that θ̄(360◦ − µ) = −θ̄(µ),
and δθ± (360◦ − µ) = −δθ± (µ).
6.658
6.771
6.885
7.001
7.118
7.235
7.354
7.474
7.594
7.714
7.833
7.952
8.069
8.184
8.297
8.405
8.509
8.607
8.698
8.780
8.852
8.911
8.955
8.982
8.988
8.972
8.929
8.855
8.747
8.601
8.411
8.174
7.886
7.541
7.138
6.673
6.145
5.553
4.899
4.186
3.420
2.608
1.760
0.886
0.000
The Superior Planets
131
Event
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Date
01/07/2000
12/05/2001
13/06/2001
19/07/2001
10/08/2002
29/07/2003
28/08/2003
27/09/2003
15/09/2004
02/10/2005
07/11/2005
09/12/2005
23/10/2006
15/11/2007
24/12/2007
31/01/2008
05/12/2008
20/12/2009
29/01/2010
10/03/2010
04/02/2011
24/01/2012
03/03/2012
14/04/2012
17/04/2013
01/03/2014
08/04/2014
20/05/2014
14/06/2015
17/04/2016
22/05/2016
29/06/2016
27/07/2017
27/06/2018
27/07/2018
27/08/2018
02/09/2019
10/09/2020
13/10/2020
13/11/2020
λ
10CN13
29SG00
22SG44
15SG02
18LE05
10PI20
05PI03
29AQ55
23VI06
23TA31
15TA06
08TA24
29LI44
12CN36
02CN45
24GE15
14SG08
19LE35
09LE45
00LE20
15AQ42
23VI01
13VI42
03VI51
28AR06
27LI31
19LI00
09LI04
23GE28
08SG45
01SG43
22SC55
04LE11
09AQ35
04AQ22
28CP43
09VI43
28AR08
20AR59
15AR05
Table 8.6: The conjunctions, oppositions, and stations of Mars during the years 2000–2020 CE. (R)
indicates a retrograde station, and (D) a direct station.
132
MODERN ALMAGEST
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
111.251
222.501
333.752
85.003
196.253
307.504
58.755
170.006
281.256
110.810
221.620
332.430
83.240
194.050
304.860
55.670
166.480
277.290
110.812
221.624
332.437
83.249
194.061
304.873
55.685
166.498
277.310
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
83.125
166.250
249.375
332.500
55.625
138.750
221.875
305.001
28.126
83.081
166.162
249.243
332.324
55.405
138.486
221.567
304.648
27.729
83.081
166.162
249.244
332.325
55.406
138.487
221.569
304.650
27.731
100
200
300
400
500
600
700
800
900
8.313
16.625
24.938
33.250
41.563
49.875
58.188
66.500
74.813
8.308
16.616
24.924
33.232
41.541
49.849
58.157
66.465
74.773
8.308
16.616
24.924
33.232
41.541
49.849
58.157
66.465
74.773
10
20
30
40
50
60
70
80
90
0.831
1.663
2.494
3.325
4.156
4.988
5.819
6.650
7.481
0.831
1.662
2.492
3.323
4.154
4.985
5.816
6.646
7.477
0.831
1.662
2.492
3.323
4.154
4.985
5.816
6.646
7.477
1
2
3
4
5
6
7
8
9
0.083
0.166
0.249
0.333
0.416
0.499
0.582
0.665
0.748
0.083
0.166
0.249
0.332
0.415
0.498
0.582
0.665
0.748
0.083
0.166
0.249
0.332
0.415
0.498
0.582
0.665
0.748
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.008
0.017
0.025
0.033
0.042
0.050
0.058
0.067
0.075
0.008
0.017
0.025
0.033
0.042
0.050
0.058
0.066
0.075
0.008
0.017
0.025
0.033
0.042
0.050
0.058
0.066
0.075
Table 8.7: Mean motion of Jupiter. Here, ∆t = t − t0, ∆λ̄ = λ̄ − λ̄0, ∆M = M − M0, and ∆F̄ = F̄ − F̄0.
At epoch (t0 = 2 451 545.0 JD), λ̄0 = 34.365◦ , M0 = 19.348◦ , and F̄0 = 293.660◦ .
The Superior Planets
133
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
0.000
0.205
0.410
0.614
0.818
1.020
1.221
1.420
1.617
1.812
2.004
2.194
2.380
2.563
2.742
2.918
3.089
3.256
3.419
3.576
3.729
3.877
4.019
4.156
4.287
4.413
4.532
4.645
4.752
4.853
4.947
5.035
5.116
5.190
5.257
5.318
5.372
5.419
5.459
5.492
5.518
5.537
5.549
5.554
5.553
5.545
4.839
4.835
4.826
4.810
4.787
4.758
4.723
4.681
4.633
4.579
4.519
4.453
4.382
4.304
4.221
4.132
4.038
3.938
3.834
3.724
3.610
3.491
3.368
3.240
3.108
2.973
2.834
2.691
2.545
2.396
2.244
2.089
1.932
1.773
1.611
1.448
1.283
1.117
0.950
0.782
0.613
0.444
0.274
0.105
-0.065
-0.234
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
5.545
5.530
5.508
5.479
5.444
5.403
5.355
5.301
5.241
5.175
5.102
5.024
4.941
4.851
4.757
4.657
4.551
4.441
4.326
4.207
4.082
3.954
3.821
3.684
3.543
3.399
3.251
3.100
2.945
2.787
2.627
2.464
2.298
2.131
1.961
1.789
1.615
1.439
1.263
1.085
0.905
0.725
0.545
0.363
0.182
0.000
-0.234
-0.403
-0.571
-0.737
-0.903
-1.067
-1.230
-1.391
-1.550
-1.707
-1.862
-2.014
-2.163
-2.310
-2.454
-2.595
-2.732
-2.867
-2.997
-3.124
-3.248
-3.367
-3.482
-3.594
-3.701
-3.803
-3.902
-3.995
-4.085
-4.169
-4.249
-4.324
-4.394
-4.459
-4.519
-4.574
-4.624
-4.669
-4.709
-4.743
-4.772
-4.796
-4.815
-4.828
-4.836
-4.839
180
182
184
186
188
190
192
194
196
198
200
202
204
206
208
210
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246
248
250
252
254
256
258
260
262
264
266
268
270
0.000
-0.182
-0.363
-0.545
-0.725
-0.905
-1.085
-1.263
-1.439
-1.615
-1.789
-1.961
-2.131
-2.298
-2.464
-2.627
-2.787
-2.945
-3.100
-3.251
-3.399
-3.543
-3.684
-3.821
-3.954
-4.082
-4.207
-4.326
-4.441
-4.551
-4.657
-4.757
-4.851
-4.941
-5.024
-5.102
-5.175
-5.241
-5.301
-5.355
-5.403
-5.444
-5.479
-5.508
-5.530
-5.545
-4.839
-4.836
-4.828
-4.815
-4.796
-4.772
-4.743
-4.709
-4.669
-4.624
-4.574
-4.519
-4.459
-4.394
-4.324
-4.249
-4.169
-4.085
-3.995
-3.902
-3.803
-3.701
-3.594
-3.482
-3.367
-3.248
-3.124
-2.997
-2.867
-2.732
-2.595
-2.454
-2.310
-2.163
-2.014
-1.862
-1.707
-1.550
-1.391
-1.230
-1.067
-0.903
-0.737
-0.571
-0.403
-0.234
270
272
274
276
278
280
282
284
286
288
290
292
294
296
298
300
302
304
306
308
310
312
314
316
318
320
322
324
326
328
330
332
334
336
338
340
342
344
346
348
350
352
354
356
358
360
-5.545
-5.553
-5.554
-5.549
-5.537
-5.518
-5.492
-5.459
-5.419
-5.372
-5.318
-5.257
-5.190
-5.116
-5.035
-4.947
-4.853
-4.752
-4.645
-4.532
-4.413
-4.287
-4.156
-4.019
-3.877
-3.729
-3.576
-3.419
-3.256
-3.089
-2.918
-2.742
-2.563
-2.380
-2.194
-2.004
-1.812
-1.617
-1.420
-1.221
-1.020
-0.818
-0.614
-0.410
-0.205
-0.000
-0.234
-0.065
0.105
0.274
0.444
0.613
0.782
0.950
1.117
1.283
1.448
1.611
1.773
1.932
2.089
2.244
2.396
2.545
2.691
2.834
2.973
3.108
3.240
3.368
3.491
3.610
3.724
3.834
3.938
4.038
4.132
4.221
4.304
4.382
4.453
4.519
4.579
4.633
4.681
4.723
4.758
4.787
4.810
4.826
4.835
4.839
Table 8.8: Deferential anomalies of Jupiter.
134
MODERN ALMAGEST
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
0.000
0.000
0.161
0.322
0.483
0.644
0.805
0.965
1.126
1.286
1.446
1.606
1.766
1.925
2.084
2.242
2.400
2.558
2.715
2.872
3.028
3.184
3.339
3.494
3.648
3.801
3.954
4.106
4.257
4.407
4.557
4.705
4.853
5.000
5.146
5.292
5.436
5.579
5.721
5.862
6.002
6.141
6.278
6.415
6.550
6.684
6.816
0.000
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
0.366
6.816
6.948
7.077
7.206
7.333
7.459
7.583
7.705
7.826
7.946
8.063
8.180
8.294
8.407
8.517
8.626
8.734
8.839
8.942
9.044
9.143
9.240
9.336
9.429
9.520
9.609
9.696
9.780
9.862
9.942
10.019
10.094
10.167
10.237
10.304
10.369
10.431
10.491
10.548
10.602
10.654
10.702
10.748
10.791
10.831
10.868
0.410
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
0.649
10.868
10.902
10.933
10.961
10.986
11.008
11.026
11.041
11.053
11.062
11.067
11.069
11.068
11.063
11.054
11.042
11.027
11.008
10.985
10.959
10.929
10.895
10.858
10.817
10.772
10.723
10.671
10.614
10.554
10.490
10.422
10.350
10.274
10.194
10.110
10.023
9.931
9.835
9.736
9.632
9.524
9.413
9.297
9.178
9.055
8.927
0.736
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.616
8.927
8.796
8.661
8.522
8.380
8.233
8.083
7.929
7.771
7.610
7.445
7.276
7.104
6.929
6.750
6.568
6.383
6.194
6.003
5.808
5.610
5.410
5.206
5.000
4.792
4.581
4.367
4.151
3.933
3.713
3.491
3.267
3.041
2.814
2.585
2.354
2.123
1.890
1.656
1.421
1.185
0.949
0.712
0.475
0.238
0.000
0.713
0.008
0.017
0.025
0.033
0.042
0.050
0.058
0.067
0.075
0.083
0.092
0.100
0.108
0.116
0.125
0.133
0.141
0.149
0.158
0.166
0.174
0.182
0.191
0.199
0.207
0.215
0.223
0.231
0.239
0.248
0.256
0.264
0.272
0.280
0.288
0.296
0.304
0.311
0.319
0.327
0.335
0.343
0.351
0.358
0.366
0.009
0.019
0.028
0.037
0.046
0.056
0.065
0.074
0.084
0.093
0.102
0.111
0.121
0.130
0.139
0.148
0.158
0.167
0.176
0.185
0.194
0.204
0.213
0.222
0.231
0.240
0.249
0.258
0.268
0.277
0.286
0.295
0.304
0.313
0.322
0.331
0.339
0.348
0.357
0.366
0.375
0.384
0.392
0.401
0.410
0.374
0.381
0.389
0.396
0.404
0.411
0.419
0.426
0.434
0.441
0.448
0.455
0.462
0.470
0.477
0.484
0.490
0.497
0.504
0.511
0.517
0.524
0.531
0.537
0.543
0.550
0.556
0.562
0.568
0.574
0.580
0.585
0.591
0.596
0.602
0.607
0.612
0.617
0.622
0.627
0.632
0.636
0.641
0.645
0.649
0.418
0.427
0.436
0.444
0.453
0.461
0.470
0.478
0.486
0.495
0.503
0.511
0.519
0.527
0.535
0.543
0.551
0.559
0.567
0.574
0.582
0.590
0.597
0.605
0.612
0.619
0.626
0.633
0.640
0.647
0.654
0.661
0.667
0.674
0.680
0.686
0.692
0.698
0.704
0.710
0.715
0.721
0.726
0.731
0.736
0.653
0.657
0.661
0.664
0.668
0.671
0.674
0.677
0.680
0.682
0.684
0.686
0.688
0.690
0.692
0.693
0.694
0.695
0.695
0.696
0.696
0.696
0.695
0.695
0.694
0.693
0.691
0.690
0.688
0.686
0.683
0.680
0.677
0.674
0.670
0.666
0.662
0.657
0.652
0.647
0.641
0.636
0.629
0.623
0.616
0.741
0.746
0.750
0.755
0.759
0.763
0.766
0.770
0.773
0.777
0.780
0.782
0.785
0.787
0.789
0.791
0.793
0.794
0.795
0.796
0.796
0.797
0.797
0.796
0.796
0.795
0.794
0.792
0.790
0.788
0.786
0.783
0.780
0.776
0.772
0.768
0.764
0.759
0.753
0.748
0.742
0.735
0.728
0.721
0.713
0.609
0.601
0.593
0.585
0.576
0.567
0.558
0.548
0.538
0.528
0.517
0.506
0.495
0.484
0.472
0.459
0.447
0.434
0.421
0.407
0.393
0.379
0.365
0.350
0.336
0.320
0.305
0.289
0.274
0.258
0.241
0.225
0.208
0.192
0.175
0.158
0.140
0.123
0.106
0.088
0.071
0.053
0.035
0.018
0.000
Table 8.9: Epicyclic anomalies of Jupiter. All quantities are in degrees. Note that θ̄(360◦ − µ) = −θ̄(µ),
and δθ± (360◦ − µ) = −δθ± (µ).
0.705
0.697
0.688
0.679
0.669
0.659
0.649
0.638
0.627
0.615
0.603
0.590
0.577
0.564
0.550
0.536
0.522
0.507
0.492
0.476
0.460
0.444
0.427
0.410
0.393
0.375
0.357
0.339
0.321
0.302
0.283
0.264
0.244
0.225
0.205
0.185
0.165
0.145
0.124
0.104
0.083
0.062
0.042
0.021
0.000
The Superior Planets
135
Event
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Date
08/05/2000
29/09/2000
28/11/2000
25/01/2001
14/06/2001
02/11/2001
01/01/2002
01/03/2002
20/07/2002
04/12/2002
02/02/2003
04/04/2003
22/08/2003
04/01/2004
04/03/2004
05/05/2004
22/09/2004
02/02/2005
03/04/2005
05/06/2005
22/10/2005
04/03/2006
04/05/2006
06/07/2006
22/11/2006
06/04/2007
06/06/2007
07/08/2007
23/12/2007
09/05/2008
09/07/2008
08/09/2008
24/01/2009
15/06/2009
14/08/2009
13/10/2009
28/02/2010
23/07/2010
21/09/2010
18/11/2010
λ
17TA53
11GE13
06GE08
01GE10
23GE30
15CN41
10CN37
05CN37
27CN11
18LE06
13LE06
08LE03
28LE55
18VI54
13VI58
08VI55
29VI21
18LI53
14LI00
08LI58
29LI16
18SC54
14SC03
09SC02
29SC34
19SG49
14SG57
09SG58
01CP03
22CP23
17CP30
12CP33
04AQ23
27AQ01
22AQ04
17AQ10
09PI43
03AR20
28PI19
23PI26
Table 8.10: The conjunctions, oppositions, and stations of Jupiter during the years 2000–2010 CE.
(R) indicates a retrograde station, and (D) a direct station.
136
MODERN ALMAGEST
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
335.083
310.166
285.249
260.332
235.415
210.498
185.581
160.664
135.747
334.815
309.630
284.446
259.261
234.076
208.891
183.706
158.522
133.337
334.779
309.559
284.338
259.118
233.897
208.677
183.456
158.236
133.015
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
33.508
67.017
100.525
134.033
167.541
201.050
234.558
268.066
301.575
33.482
66.963
100.445
133.926
167.408
200.889
234.371
267.852
301.334
33.478
66.956
100.434
133.912
167.390
200.868
234.346
267.824
301.302
100
200
300
400
500
600
700
800
900
3.351
6.702
10.052
13.403
16.754
20.105
23.456
26.807
30.157
3.348
6.696
10.044
13.393
16.741
20.089
23.437
26.785
30.133
3.348
6.696
10.043
13.391
16.739
20.087
23.435
26.782
30.130
10
20
30
40
50
60
70
80
90
0.335
0.670
1.005
1.340
1.675
2.010
2.346
2.681
3.016
0.335
0.670
1.004
1.339
1.674
2.009
2.344
2.679
3.013
0.335
0.670
1.004
1.339
1.674
2.009
2.343
2.678
3.013
1
2
3
4
5
6
7
8
9
0.034
0.067
0.101
0.134
0.168
0.201
0.235
0.268
0.302
0.033
0.067
0.100
0.134
0.167
0.201
0.234
0.268
0.301
0.033
0.067
0.100
0.134
0.167
0.201
0.234
0.268
0.301
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.003
0.007
0.010
0.013
0.017
0.020
0.023
0.027
0.030
0.003
0.007
0.010
0.013
0.017
0.020
0.023
0.027
0.030
0.003
0.007
0.010
0.013
0.017
0.020
0.023
0.027
0.030
Table 8.11: Mean motion of Saturn. Here, ∆t = t− t0, ∆λ̄ = λ̄− λ̄0, ∆M = M − M0, and ∆F̄ = F̄− F̄0.
At epoch (t0 = 2 451 545.0 JD), λ̄0 = 50.059◦ , M0 = 317.857◦ , and F̄0 = 296.482◦ .
The Superior Planets
137
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
0.000
0.230
0.459
0.688
0.916
1.143
1.368
1.591
1.811
2.029
2.245
2.456
2.665
2.869
3.070
3.266
3.457
3.644
3.825
4.002
4.172
4.337
4.495
4.648
4.793
4.933
5.065
5.191
5.310
5.421
5.525
5.622
5.711
5.793
5.867
5.933
5.992
6.043
6.086
6.122
6.149
6.169
6.182
6.186
6.183
6.172
5.386
5.383
5.372
5.354
5.328
5.296
5.256
5.209
5.156
5.095
5.027
4.953
4.873
4.785
4.692
4.592
4.486
4.375
4.257
4.134
4.006
3.873
3.735
3.591
3.444
3.292
3.136
2.976
2.813
2.646
2.476
2.302
2.127
1.949
1.768
1.586
1.402
1.217
1.030
0.842
0.654
0.465
0.276
0.087
-0.102
-0.290
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
6.172
6.154
6.128
6.095
6.055
6.007
5.953
5.891
5.823
5.748
5.666
5.578
5.484
5.384
5.277
5.165
5.048
4.924
4.796
4.662
4.524
4.380
4.232
4.080
3.923
3.763
3.598
3.430
3.259
3.084
2.906
2.725
2.542
2.356
2.168
1.977
1.785
1.591
1.396
1.199
1.001
0.802
0.602
0.402
0.201
0.000
-0.290
-0.478
-0.664
-0.850
-1.034
-1.217
-1.397
-1.576
-1.753
-1.927
-2.098
-2.267
-2.433
-2.596
-2.755
-2.911
-3.063
-3.211
-3.356
-3.496
-3.632
-3.764
-3.892
-4.015
-4.133
-4.246
-4.354
-4.458
-4.556
-4.649
-4.737
-4.820
-4.897
-4.969
-5.035
-5.095
-5.150
-5.200
-5.243
-5.281
-5.313
-5.339
-5.360
-5.374
-5.383
-5.386
180
182
184
186
188
190
192
194
196
198
200
202
204
206
208
210
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246
248
250
252
254
256
258
260
262
264
266
268
270
0.000
-0.201
-0.402
-0.602
-0.802
-1.001
-1.199
-1.396
-1.591
-1.785
-1.977
-2.168
-2.356
-2.542
-2.725
-2.906
-3.084
-3.259
-3.430
-3.598
-3.763
-3.923
-4.080
-4.232
-4.380
-4.524
-4.662
-4.796
-4.924
-5.048
-5.165
-5.277
-5.384
-5.484
-5.578
-5.666
-5.748
-5.823
-5.891
-5.953
-6.007
-6.055
-6.095
-6.128
-6.154
-6.172
-5.386
-5.383
-5.374
-5.360
-5.339
-5.313
-5.281
-5.243
-5.200
-5.150
-5.095
-5.035
-4.969
-4.897
-4.820
-4.737
-4.649
-4.556
-4.458
-4.354
-4.246
-4.133
-4.015
-3.892
-3.764
-3.632
-3.496
-3.356
-3.211
-3.063
-2.911
-2.755
-2.596
-2.433
-2.267
-2.098
-1.927
-1.753
-1.576
-1.397
-1.217
-1.034
-0.850
-0.664
-0.478
-0.290
270
272
274
276
278
280
282
284
286
288
290
292
294
296
298
300
302
304
306
308
310
312
314
316
318
320
322
324
326
328
330
332
334
336
338
340
342
344
346
348
350
352
354
356
358
360
-6.172
-6.183
-6.186
-6.182
-6.169
-6.149
-6.122
-6.086
-6.043
-5.992
-5.933
-5.867
-5.793
-5.711
-5.622
-5.525
-5.421
-5.310
-5.191
-5.065
-4.933
-4.793
-4.648
-4.495
-4.337
-4.172
-4.002
-3.825
-3.644
-3.457
-3.266
-3.070
-2.869
-2.665
-2.456
-2.245
-2.029
-1.811
-1.591
-1.368
-1.143
-0.916
-0.688
-0.459
-0.230
-0.000
-0.290
-0.102
0.087
0.276
0.465
0.654
0.842
1.030
1.217
1.402
1.586
1.768
1.949
2.127
2.302
2.476
2.646
2.813
2.976
3.136
3.292
3.444
3.591
3.735
3.873
4.006
4.134
4.257
4.375
4.486
4.592
4.692
4.785
4.873
4.953
5.027
5.095
5.156
5.209
5.256
5.296
5.328
5.354
5.372
5.383
5.386
Table 8.12: Deferential anomalies of Saturn.
138
µ
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
MODERN ALMAGEST
δθ−
θ̄
0.000
0.000
0.095
0.190
0.284
0.379
0.474
0.568
0.662
0.757
0.851
0.945
1.038
1.132
1.225
1.318
1.410
1.502
1.594
1.686
1.777
1.868
1.958
2.048
2.138
2.227
2.315
2.403
2.490
2.577
2.663
2.749
2.834
2.918
3.002
3.085
3.167
3.248
3.329
3.409
3.488
3.566
3.644
3.720
3.796
3.871
3.944
0.006
0.011
0.017
0.023
0.028
0.034
0.040
0.045
0.051
0.057
0.062
0.068
0.074
0.079
0.085
0.090
0.096
0.102
0.107
0.113
0.118
0.124
0.129
0.134
0.140
0.145
0.151
0.156
0.161
0.167
0.172
0.177
0.182
0.188
0.193
0.198
0.203
0.208
0.213
0.218
0.223
0.228
0.233
0.238
0.242
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
0.000
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
0.242
3.944
4.017
4.089
4.160
4.230
4.299
4.367
4.433
4.499
4.564
4.627
4.689
4.750
4.810
4.869
4.926
4.982
5.037
5.091
5.143
5.194
5.243
5.292
5.338
5.384
5.428
5.470
5.511
5.551
5.589
5.625
5.660
5.694
5.726
5.756
5.784
5.811
5.837
5.861
5.883
5.903
5.922
5.939
5.954
5.967
5.979
0.276
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
0.391
5.979
5.989
5.997
6.004
6.008
6.011
6.012
6.011
6.008
6.004
5.997
5.989
5.979
5.966
5.952
5.937
5.919
5.899
5.877
5.854
5.828
5.801
5.772
5.740
5.707
5.672
5.635
5.596
5.555
5.512
5.467
5.421
5.372
5.322
5.270
5.215
5.159
5.101
5.042
4.980
4.917
4.851
4.784
4.716
4.645
4.573
0.450
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.322
4.573
4.499
4.423
4.346
4.267
4.186
4.104
4.020
3.935
3.848
3.760
3.670
3.578
3.486
3.392
3.296
3.199
3.101
3.002
2.901
2.800
2.697
2.593
2.488
2.382
2.275
2.167
2.059
1.949
1.839
1.727
1.616
1.503
1.390
1.276
1.162
1.047
0.932
0.816
0.700
0.584
0.467
0.351
0.234
0.117
0.000
0.375
0.006
0.013
0.019
0.026
0.032
0.039
0.045
0.052
0.058
0.064
0.071
0.077
0.084
0.090
0.096
0.103
0.109
0.115
0.122
0.128
0.134
0.141
0.147
0.153
0.159
0.165
0.171
0.178
0.184
0.190
0.196
0.202
0.208
0.214
0.219
0.225
0.231
0.237
0.243
0.248
0.254
0.260
0.265
0.271
0.276
0.247
0.252
0.256
0.261
0.265
0.270
0.274
0.279
0.283
0.287
0.292
0.296
0.300
0.304
0.308
0.312
0.316
0.320
0.323
0.327
0.330
0.334
0.337
0.341
0.344
0.347
0.350
0.353
0.356
0.359
0.362
0.365
0.367
0.370
0.372
0.375
0.377
0.379
0.381
0.383
0.385
0.387
0.388
0.390
0.391
0.282
0.287
0.292
0.298
0.303
0.308
0.313
0.318
0.323
0.328
0.333
0.338
0.343
0.347
0.352
0.356
0.361
0.365
0.370
0.374
0.378
0.382
0.386
0.390
0.394
0.398
0.401
0.405
0.408
0.412
0.415
0.418
0.421
0.424
0.427
0.430
0.433
0.435
0.438
0.440
0.442
0.444
0.446
0.448
0.450
0.393
0.394
0.395
0.396
0.397
0.397
0.398
0.399
0.399
0.399
0.399
0.399
0.399
0.399
0.399
0.398
0.397
0.397
0.396
0.395
0.394
0.392
0.391
0.389
0.387
0.386
0.384
0.381
0.379
0.377
0.374
0.371
0.368
0.365
0.362
0.359
0.355
0.352
0.348
0.344
0.340
0.336
0.331
0.327
0.322
0.452
0.453
0.454
0.456
0.457
0.458
0.459
0.459
0.460
0.460
0.461
0.461
0.461
0.460
0.460
0.460
0.459
0.458
0.457
0.456
0.455
0.454
0.452
0.450
0.448
0.446
0.444
0.442
0.439
0.437
0.434
0.431
0.427
0.424
0.420
0.417
0.413
0.409
0.404
0.400
0.395
0.390
0.386
0.380
0.375
0.318
0.313
0.308
0.302
0.297
0.292
0.286
0.280
0.274
0.269
0.262
0.256
0.250
0.243
0.237
0.230
0.223
0.216
0.209
0.202
0.195
0.187
0.180
0.172
0.165
0.157
0.149
0.142
0.134
0.126
0.118
0.109
0.101
0.093
0.085
0.076
0.068
0.060
0.051
0.043
0.034
0.026
0.017
0.009
0.000
0.370
0.364
0.358
0.352
0.346
0.340
0.333
0.327
0.320
0.313
0.306
0.299
0.292
0.284
0.276
0.269
0.261
0.253
0.244
0.236
0.228
0.219
0.210
0.202
0.193
0.184
0.175
0.166
0.156
0.147
0.138
0.128
0.118
0.109
0.099
0.089
0.080
0.070
0.060
0.050
0.040
0.030
0.020
0.010
0.000
Table 8.13: Epicyclic anomalies of Saturn. All quantities are in degrees. Note that θ̄(360◦ − µ) =
−θ̄(µ), and δθ± (360◦ − µ) = −δθ± (µ).
The Superior Planets
139
Event
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Station (R)
Opposition
Station (D)
Conjunction
Date
10/05/2000
12/09/2000
19/11/2000
24/01/2001
25/05/2001
26/09/2001
03/12/2001
08/02/2002
09/06/2002
11/10/2002
17/12/2002
22/02/2003
24/06/2003
25/10/2003
31/12/2003
07/03/2004
08/07/2004
08/11/2004
13/01/2005
22/03/2005
23/07/2005
22/11/2005
27/01/2006
05/04/2006
07/08/2006
06/12/2006
10/02/2007
19/04/2007
22/08/2007
19/12/2007
24/02/2008
03/05/2008
04/09/2008
31/12/2008
08/03/2009
17/05/2009
17/09/2009
13/01/2010
22/03/2010
30/05/2010
01/10/2010
λ
20TA26
00GE59
27TA29
24TA03
04GE22
14GE59
11GE29
08GE02
18GE28
29GE06
25GE36
22GE08
02CN39
13CN15
09CN46
06CN17
16CN50
27CN21
23CN52
20CN23
00LE56
11LE19
07LE51
04LE22
14LE51
25LE04
21LE38
18LE09
28LE32
08VI34
05VI10
01VI41
11VI56
21VI46
18VI23
14VI56
25VI01
04LI40
01LI18
27VI51
07LI46
Table 8.14: The conjunctions, oppositions, and stations of Saturn during the years 2000–2010 CE. (R)
indicates a retrograde station, and (D) a direct station.
140
MODERN ALMAGEST
The Inferior Planets
141
9 The Inferior Planets
9.1 Determination of Ecliptic Longitude
Figure 9.1 compares and contrasts heliocentric and geocentric models of the motion of an inferior
planet (i.e., a planet which is closer to the sun than the earth), P, as seen from the earth, G. The
sun is at S. As before, in the heliocentric model the earth-planet displacement vector, P, is the sum
of the earth-sun displacement vector, S, and the sun-planet displacement vector, P ′ . On the other
hand, in the geocentric model S gives the displacement of the guide-point, G ′ , from the earth. Since
S is also the displacement of the sun, S, from the earth, G, it is clear that G ′ executes a Keplerian
orbit about the earth whose elements are the same as those of the apparent orbit of the sun about
the earth. This implies that the sun is coincident with G ′ . The ellipse traced out by G ′ is termed
the deferent. The vector P ′ gives the displacement of the planet, P, from the guide-point, G ′ . Since
P ′ is also the displacement of the planet, P, from the sun, S, it is clear that P executes a Keplerian
orbit about the guide-point whose elements are the same as those of the orbit of the planet about
the sun. The ellipse traced out by P about G ′ is termed the epicycle.
Earth’s orbit P
P
deferent
G
P′
P
S
′
P
P
S
S
planetary orbit
epicycle
Heliocentric
G
G′
Geocentric
Figure 9.1: Heliocentric and geocentric models of the motion of an inferior planet. Here, S is the sun,
G the earth, and P the planet. View is from the northern ecliptic pole.
As we have seen, the deferent of a superior planet has the same elements as the planet’s orbit
about the sun, whereas the epicycle has the same elements as the sun’s apparent orbit about the
earth. On the other hand, the deferent of an inferior planet has the same elements as the sun’s
apparent orbit about the earth, whereas the epicycle has the same elements as the planet’s orbit
about the sun. It follows that we can formulate a procedure for determining the ecliptic longitude
of an inferior planet by simply taking the procedure used in the previous section for determining
the ecliptic longitude of a superior planet and exchanging the roles of the sun and the planet.
Our procedure is described below. As before, it is assumed that the ecliptic longitude, λS, and
the radial anomaly, ζS, of the sun have already been calculated. In the following, a, e, n, ñ, λ̄0,
142
MODERN ALMAGEST
and M0 represent elements of the orbit of the planet in question about the sun, whereas eS is the
eccentricity of the sun’s apparent orbit about the earth. Again, a is the major radius of the planetary
orbit in units in which the major radius of the sun’s apparent orbit about the earth is unity. The
requisite elements for all of the inferior planets at the J2000 epoch (t0 = 2 451 545.0 JD) are listed
in Table 5.1. The ecliptic longitude of an inferior planet is specified by the following formulae:
λ̄ = λ̄0 + n (t − t0),
(9.1)
M = M0 + ñ (t − t0),
(9.2)
q = 2 e sin M + (5/4) e 2 sin 2M,
(9.3)
ζ = e cos M − e 2 sin2 M,
(9.4)
(9.5)
µ = λ̄ + q − λS,
θ̄ = θ(µ, z̄) ≡ tan−1
sin µ
,
a−1 z̄ + cos µ
(9.6)
δθ− = θ(µ, z̄) − θ(µ, zmax ),
(9.7)
δθ+ = θ(µ, zmin ) − θ(µ, z̄),
(9.8)
1 − ζS
,
1−ζ
z̄ − z
ξ =
,
δz
θ = Θ−(ξ) δθ− + θ̄ + Θ+(ξ) δθ+,
z =
λ = λS + θ.
(9.9)
(9.10)
(9.11)
(9.12)
Here, z̄ = (1 + e eS)/(1 − e 2), δz = (e + eS)/(1 − e 2), zmin = z̄ − δz, and zmax = z̄ + δz. The constants
z̄, δz, zmin , and zmax for each of the inferior planets are listed in Table. 2.5. Finally, the functions
Θ± are tabulated in Table 2.17.
For the case of Venus, the above formulae are capable of matching NASA ephemeris data during
the years 1995–2006 CE with a mean error of 2 ′ and a maximum error of 10 ′ . For the case of
Mercury, given its relatively large eccentricity of 0.205636, it is necessary to modify the formulae
slightly by expressing q and ζ to third-order in the eccentricity:
q = [2 e − (1/4) e3] sin M + (5/4) e2 sin 2M + (13/12) e3 sin 3M,
(9.13)
ζ = −(1/2) e2 + [e − (3/8) e3] cos M + (1/2) e2 cos 2M + (3/8) e3 cos 3M.
(9.14)
With this modification, the mean error is 6 ′ and the maximum error 28 ′ .
9.2 Venus
The ecliptic longitude of Venus can be determined with the aid of Tables 9.1–9.3. Table 9.1 allows
the mean longitude, λ̄, and the mean anomaly, M, of Venus to be calculated as functions of time.
Next, Table 9.2 permits the equation of center, q, and the radial anomaly, ζ, to be determined
as functions of the mean anomaly. Finally, Table 9.3 allows the quantities δθ−, θ̄, and δθ+ to be
calculated as functions of the epicyclic anomaly, µ.
The procedure for using the tables is as follows:
The Inferior Planets
143
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the ecliptic longitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t − t0,
where t0 = 2 451 545.0 is the epoch.
2. Calculate the ecliptic longitude, λS, and radial anomaly, ζS, of the sun using the procedure
set out in Sect. 5.1.
3. Enter Table 9.1 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆λ̄ and ∆M. If ∆t is negative then the corresponding values are also negative. The
value of the mean longitude, λ̄, is the sum of all the ∆λ̄ values plus value of λ̄ at the epoch.
Likewise, the value of the mean anomaly, M, is the sum of all the ∆M values plus the value
of M at the epoch. Add as many multiples of 360◦ to λ̄ and M as is required to make them
both fall in the range 0◦ to 360◦ . Round M to the nearest degree.
4. Enter Table 9.2 with the value of M and take out the corresponding value of the equation of
center, q, and the radial anomaly, ζ. It is necessary to interpolate if M is odd.
5. Form the epicyclic anomaly, µ = λ̄ + q − λS. Add as many multiples of 360◦ to µ as is required
to make it fall in the range 0◦ to 360◦ . Round µ to the nearest degree.
6. Enter Table 9.3 with the value of µ and take out the corresponding values of δθ−, θ̄, and δθ+.
If µ > 180◦ then it is necessary to make use of the identities δθ± (360◦ − µ) = −δθ± (µ) and
θ̄(360◦ − µ) = −θ̄(µ).
7. Form z = (1 − ζS)/(1 − ζ).
8. Obtain the values of z̄ and δz from Table 2.5. Form ξ = (z̄ − z)/δz.
9. Enter Table 2.17 with the value of ξ and take out the corresponding values of Θ− and Θ+. If
ξ < 0 then it is necessary to use the identities Θ+(ξ) = −Θ−(−ξ) and Θ−(ξ) = −Θ+(−ξ).
10. Form the equation of the epicycle, θ = Θ− δθ− + θ̄ + Θ+ δθ+.
11. The ecliptic longitude, λ, is the sum of the ecliptic longitude of the sun, λS, and the equation
of the epicycle, θ. If necessary convert λ into an angle in the range 0◦ to 360◦ . The decimal
fraction can be converted into arc minutes using Table 5.2. Round to the nearest arc minute.
The final result can be written in terms of the signs of the zodiac using the table in Sect. 2.6.
Two examples of this procedure are given below.
Example 1: May 5, 2005 CE, 00:00 UT:
From Cha. 8, t − t0 = 1 950.5 JD, λS = 44.602◦ , and ζS = −8.56 × 10−3. Making use of Table 9.1, we find:
144
MODERN ALMAGEST
t(JD)
λ̄(◦ )
M(◦ )
+1000
+900
+50
+.5
Epoch
162.169
1.952
80.108
0.801
181.973
427.003
67.003
162.130
1.917
80.107
0.801
49.237
294.192
294.192
Modulus
Given that M ≃ 294◦ , Table 9.2 yields
q(294◦ ) = −0.712◦ ,
ζ(294◦ ) = 2.72 × 10−3,
so
µ = λ̄ + q − λ̄S = 67.003 − 0.712 − 44.602 = 21.689 ≃ 22◦ .
It follows from Table 9.3 that
δθ−(22◦ ) = 0.126◦ ,
θ̄(22◦ ) = 9.212◦ ,
δθ+(22◦ ) = 0.129◦ .
Now,
z = (1 − ζS)/(1 − ζ) = (1 + 8.56 × 10−3)/(1 − 2.72 × 10−3) = 1.01131.
However, from Table 2.5, z̄ = 1.00016 and δz = 0.02349, so
ξ = (z̄ − z)/δz = (1.00016 − 1.01131)/0.02349 ≃ −0.48.
According to Table 2.17,
Θ−(−0.48) = −0.355,
Θ+(−0.48) = −0.125,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = −0.355 × 0.126 + 9.212 − 0.125 × 0.129 = 9.151◦ .
Finally,
λ = λ̄S + θ = 44.602 + 9.151 = 53.753 ≃ 53◦ 45 ′ .
Thus, the ecliptic longitude of Venus at 00:00 UT on May 5, 2005 CE was 23TA45.
Example 2: December 25, 1800 CE, 00:00 UT:
From Cha. 8, t − t0 = −72 690.5 JD, λS = 273.055◦ , and ζS = 1.662 × 10−2. Making use of Table 9.1, we find:
The Inferior Planets
145
t(JD)
λ̄(◦ )
M(◦ )
-70,000
-2,000
-600
-90
-.5
Epoch
−191.810
−324.337
−241.301
−144.195
−0.801
181.973
−720.471
359.529
−189.128
−324.261
−241.278
−144.192
−0.801
49.237
−850.423
229.577
Modulus
Given that M ≃ 230◦ , Table 9.2 yields
q(230◦ ) = −0.592◦ ,
ζ(230◦ ) = −4.38 × 10−3,
so
µ = λ̄ + q − λ̄S = 359.529 − 0.592 − 273.055 = 85.882 ≃ 86◦ .
It follows from Table 9.3 that
δθ−(86◦ ) = 0.589◦ ,
θ̄(86◦ ) = 34.482◦ ,
δθ+(86◦ ) = 0.607◦ .
Now,
z = (1 − ζS)/(1 − ζ) = (1 − 1.662 × 10−2)/(1 + 4.38 × 10−3) = 0.97909,
so
ξ = (z̄ − z)/δz = (1.00016 − 0.97909)/0.02349 ≃ 0.90.
According to Table 2.17,
Θ−(0.90) = 0.045,
Θ+(0.90) = 0.855,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = 0.045 × 0.589 + 34.482 + 0.855 × 0.607 = 35.027◦ .
Finally,
λ = λ̄S + θ = 273.055 + 35.027 = 308.082 ≃ 308◦ 5 ′ .
Thus, the ecliptic longitude of Venus at 00:00 UT on December 25, 1800 CE was 8AQ5.
9.3 Determination of Conjunction and Greatest Elongation Dates
The geocentric orbit of an inferior planet is similar to that of the superior planet shown in Fig. 8.4,
except for the fact that the sun is coincident with guide-point G ′ in the former case. It follows that
it is impossible for an inferior planet to have an opposition with the sun (i.e, for the earth to lie
directly between the planet and the sun). However, inferior planets do have two different kinds of
conjunctions with the sun. A superior conjuction takes place when the sun lies directly between the
planet and the earth. Conversely, an inferior conjunction takes place when the planet lies directly
between the sun and the earth. It is clear from Fig. 8.4 that a superior conjunction corresponds to
µ = 0◦ , and an inferior conjunction to µ = 180◦ . Now, the equation of the epicycle, θ, measures
the angular separation between the planet and the sun (since the sun lies at the guide-point). It
146
MODERN ALMAGEST
is evident from Figure 8.4 that θ attains a maximum and a minimum value each time the planet
revolves around its epicycle. In other words, there is a limit to how large the angular separation
between an inferior planet and the sun can become. The maximum value is termed the greatest
eastern elongation of the planet, whereas the modulus of the minimum value is termed the greatest
western elongation.
Tables 9.1–9.3 can be used to determine the dates of the conjunctions and greatest elongations
of Venus. Consider the first superior conjunction after the epoch (January 1, 2000 CE). We can
estimate the time at which this event occurs by approximating the epicyclic anomaly as the mean
epicyclic anomaly:
µ ≃ µ̄ = λ̄ − λ̄S = λ̄0 − λ̄0S + (n − nS) (t − t0) = 261.515 + 0.61652137 (t − t0).
Thus,
t ≃ t0 + (360 − 261.515)/0.61652137 ≃ t0 + 160 JD.
A calculation of the epicyclic anomaly at this time, using Tables 9.1–9.3, yields µ = −1.267◦ . Now,
the actual conjunction takes place when µ = 0◦ . Hence, our final estimate is
t = t0 + 160 + 1.267/0.61652137 = t0 + 162.1 JD,
which corresponds to June 11, 2000 CE.
Consider the first inferior conjunction of Venus after the epoch. Our first estimate of the time at
which this event takes place is
t ≃ t0 + (540 − 261.515)/0.61652137 ≃ t0 + 452 JD.
A calculation of the epicyclic anomaly at this time yields µ = 178.900◦ . Now, the actual conjunction
takes place when µ = 180◦ . Hence, our final estimate is
t = t0 + 452 + 1.100/0.61652137 = t0 + 453.8 JD,
which corresponds to March 30, 2001 CE. Incidentally, it is clear from the above analysis that the
mean time period between successive superior, or inferior, conjunctions of Venus is 360/0.61652137 =
583.9 JD, which is equivalent to 1.60 years.
Consider the greatest elongations of Venus. We can approximate the equation of the epicycle as
−1
θ ≃ θ̄ = tan
sin µ̄
,
−1
ā + cos µ̄
(9.15)
where µ̄ is the mean epicyclic anomaly, and ā = a/z̄. It follows that
dθ̄
ā−1 cos µ̄ + 1
=
.
dµ̄
1 + 2 ā−1 cos µ̄ + ā−2
(9.16)
Now, θ̄ attains its maximum or minimum value when dθ̄/dµ̄ = 0: i.e., when
µ̄ = cos−1(−ā).
(9.17)
For the case of Venus, we obtain µ̄ = 136.3◦ or 223.7◦ . The first solution corresponds to the greatest eastern elongation, and the second to the greatest western elongation. Substituting back into
The Inferior Planets
147
Eq. (9.15), we find that θ̄ = ±46.3◦ . Hence, the mean value of the greatest eastern or western
elongation of Venus is 46.3◦ . The mean time period between a greatest eastern elongation and the
following inferior conjunction, or between an inferior conjunction and the following greatest western elongation, is (180 − 136.3)/0.61652137 ≃ 71 JD. Unfortunately, the only option for accurately
determining the dates at which the greatest elongations occur is to calculate the equation of the
epicycle of Venus over a range of days centered 71 days before and after an inferior conjunction.
Table 9.4 shows the conjunctions, and greatest elongations of Venus for the years 2000–2015
CE, calculated using the techniques described above.
9.4 Mercury
The ecliptic longitude of Mercury can be determined with the aid of Tables 9.5–9.7. Table 9.5 allows
the mean longitude, λ̄, and the mean anomaly, M, of Mercury to be calculated as functions of time.
Next, Table 9.6 permits the equation of center, q, and the radial anomaly, ζ, to be determined
as functions of the mean anomaly. Finally, Table 9.7 allows the quantities δθ−, θ̄, and δθ+ to be
calculated as functions of the epicyclic anomaly, µ. The procedure for using the tables is analogous
to the previously described procedure for using the Venus tables. One example of this procedure is
given below.
Example: May 5, 2005 CE, 00:00 UT:
From Cha. 8, t − t0 = 1 950.5 JD, λS = 44.602◦ , and ζS = −8.56 × 10−3. Making use of Table 9.5, we find:
t(JD)
λ̄(◦ )
M(◦ )
+1000
+900
+50
+.5
Epoch
132.377
83.139
204.619
2.046
252.087
647.268
314.268
132.334
83.101
204.617
2.046
174.693
596.791
236.791
Modulus
Given that M ≃ 237◦ , Table 9.6 yields
q(237◦ ) = −16.974◦ ,
ζ(237◦ ) = −1.367 × 10−1,
so
µ = λ̄ + q − λ̄S = 314.268 − 16.974 − 44.602 = 252.692 ≃ 253◦ .
It follows from Table 9.7 that
δθ−(253◦ ) = −4.005◦ ,
θ̄(253◦ ) = −21.609◦ ,
δθ+(253◦ ) = −6.182◦ .
Now,
z = (1 − ζS)/(1 − ζ) = (1 + 8.56 × 10−3)/(1 + 1.367 × 10−1) = 0.8873.
148
MODERN ALMAGEST
However, from Table 2.5, z̄ = 1.04774 and δz = 0.23216, so
ξ = (z̄ − z)/δz = (1.04774 − 0.8873)/0.23216 ≃ 0.69.
According to Table 2.17,
Θ−(0.69) = 0.107,
Θ+(0.69) = 0.583,
so
θ = Θ− δθ− + θ̄ + Θ+ δθ+ = −0.107 × 4.005 − 21.609 − 0.583 × 6.182 = −25.642◦ .
Finally,
λ = λ̄S + θ = 44.602 − 25.642 = 18.960 ≃ 18◦ 58 ′ .
Thus, the ecliptic longitude of Mercury at 00:00 UT on May 5, 2005 CE was 18AR58.
The conjunctions and elongations of Mercury can be investigated using analogous methods to
those employed earlier to examine the conjunctions and elongations of Venus. We find that the
mean time period between successive superior, or inferior, conjunctions of Mercury is 116 days.
On average, the greatest eastern and western elongations of Mercury occur when the epicyclic
anomaly takes the values µ = 111.7◦ and 248.3◦ , respectively. Furthermore, the mean value of the
greatest eastern or western elongation is 21.7◦ . Finally, the mean time period between a greatest
eastern elongation and the following inferior conjunction, or between the inferior conjunction and
the following greatest western elongation, is 22 JD. The conjunctions and elongations of Mercury
during the years 2000–2002 CE are shown in Table 9.8.
The Inferior Planets
149
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
181.687
3.374
185.062
6.749
188.436
10.123
191.810
13.498
195.185
181.304
2.608
183.912
5.216
186.520
7.824
189.128
10.432
191.736
181.381
2.761
184.142
5.523
186.904
8.284
189.665
11.046
192.426
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
162.169
324.337
126.506
288.675
90.844
253.012
55.181
217.350
19.518
162.130
324.261
126.391
288.522
90.652
252.782
54.913
217.043
19.174
162.138
324.276
126.414
288.552
90.690
252.828
54.966
217.105
19.243
100
200
300
400
500
600
700
800
900
160.217
320.434
120.651
280.867
81.084
241.301
41.518
201.735
1.952
160.213
320.426
120.639
280.852
81.065
241.278
41.491
201.704
1.917
160.214
320.428
120.641
280.855
81.069
241.283
41.497
201.710
1.924
10
20
30
40
50
60
70
80
90
16.022
32.043
48.065
64.087
80.108
96.130
112.152
128.173
144.195
16.021
32.043
48.064
64.085
80.107
96.128
112.149
128.170
144.192
16.021
32.043
48.064
64.086
80.107
96.128
112.150
128.171
144.192
1
2
3
4
5
6
7
8
9
1.602
3.204
4.807
6.409
8.011
9.613
11.215
12.817
14.420
1.602
3.204
4.806
6.409
8.011
9.613
11.215
12.817
14.419
1.602
3.204
4.806
6.409
8.011
9.613
11.215
12.817
14.419
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.160
0.320
0.481
0.641
0.801
0.961
1.122
1.282
1.442
0.160
0.320
0.481
0.641
0.801
0.961
1.121
1.282
1.442
0.160
0.320
0.481
0.641
0.801
0.961
1.121
1.282
1.442
Table 9.1: Mean motion of Venus. Here, ∆t = t − t0, ∆λ̄ = λ̄ − λ̄0, ∆M = M − M0, and ∆F̄ = F̄ − F̄0.
At epoch (t0 = 2 451 545.0 JD), λ̄0 = 181.973◦ , M0 = 49.237◦ , and F̄0 = 105.253◦ .
150
MODERN ALMAGEST
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
0.000
0.027
0.055
0.082
0.109
0.136
0.163
0.189
0.216
0.242
0.268
0.293
0.318
0.343
0.367
0.391
0.414
0.437
0.460
0.481
0.502
0.523
0.543
0.562
0.580
0.598
0.615
0.631
0.647
0.662
0.675
0.688
0.701
0.712
0.722
0.732
0.741
0.748
0.755
0.761
0.766
0.770
0.773
0.775
0.776
0.777
0.678
0.677
0.676
0.674
0.671
0.667
0.663
0.657
0.651
0.644
0.636
0.628
0.618
0.608
0.597
0.586
0.573
0.560
0.547
0.532
0.517
0.502
0.485
0.468
0.451
0.433
0.414
0.395
0.376
0.356
0.335
0.315
0.293
0.272
0.250
0.228
0.205
0.183
0.160
0.137
0.113
0.090
0.066
0.043
0.019
-0.005
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
0.777
0.776
0.774
0.772
0.768
0.764
0.758
0.752
0.745
0.737
0.728
0.718
0.707
0.695
0.683
0.670
0.656
0.641
0.625
0.609
0.592
0.574
0.555
0.536
0.516
0.496
0.475
0.453
0.431
0.409
0.385
0.362
0.338
0.313
0.289
0.263
0.238
0.212
0.186
0.160
0.134
0.107
0.080
0.054
0.027
0.000
-0.005
-0.028
-0.052
-0.075
-0.099
-0.122
-0.145
-0.168
-0.191
-0.214
-0.236
-0.258
-0.279
-0.301
-0.322
-0.342
-0.362
-0.382
-0.401
-0.420
-0.438
-0.456
-0.473
-0.490
-0.506
-0.521
-0.536
-0.550
-0.563
-0.576
-0.588
-0.599
-0.610
-0.620
-0.629
-0.637
-0.645
-0.652
-0.658
-0.663
-0.668
-0.671
-0.674
-0.676
-0.677
-0.678
180
182
184
186
188
190
192
194
196
198
200
202
204
206
208
210
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246
248
250
252
254
256
258
260
262
264
266
268
270
0.000
-0.027
-0.054
-0.080
-0.107
-0.134
-0.160
-0.186
-0.212
-0.238
-0.263
-0.289
-0.313
-0.338
-0.362
-0.385
-0.409
-0.431
-0.453
-0.475
-0.496
-0.516
-0.536
-0.555
-0.574
-0.592
-0.609
-0.625
-0.641
-0.656
-0.670
-0.683
-0.695
-0.707
-0.718
-0.728
-0.737
-0.745
-0.752
-0.758
-0.764
-0.768
-0.772
-0.774
-0.776
-0.777
-0.678
-0.677
-0.676
-0.674
-0.671
-0.668
-0.663
-0.658
-0.652
-0.645
-0.637
-0.629
-0.620
-0.610
-0.599
-0.588
-0.576
-0.563
-0.550
-0.536
-0.521
-0.506
-0.490
-0.473
-0.456
-0.438
-0.420
-0.401
-0.382
-0.362
-0.342
-0.322
-0.301
-0.279
-0.258
-0.236
-0.214
-0.191
-0.168
-0.145
-0.122
-0.099
-0.075
-0.052
-0.028
-0.005
270
272
274
276
278
280
282
284
286
288
290
292
294
296
298
300
302
304
306
308
310
312
314
316
318
320
322
324
326
328
330
332
334
336
338
340
342
344
346
348
350
352
354
356
358
360
-0.777
-0.776
-0.775
-0.773
-0.770
-0.766
-0.761
-0.755
-0.748
-0.741
-0.732
-0.722
-0.712
-0.701
-0.688
-0.675
-0.662
-0.647
-0.631
-0.615
-0.598
-0.580
-0.562
-0.543
-0.523
-0.502
-0.481
-0.460
-0.437
-0.414
-0.391
-0.367
-0.343
-0.318
-0.293
-0.268
-0.242
-0.216
-0.189
-0.163
-0.136
-0.109
-0.082
-0.055
-0.027
-0.000
-0.005
0.019
0.043
0.066
0.090
0.113
0.137
0.160
0.183
0.205
0.228
0.250
0.272
0.293
0.315
0.335
0.356
0.376
0.395
0.414
0.433
0.451
0.468
0.485
0.502
0.517
0.532
0.547
0.560
0.573
0.586
0.597
0.608
0.618
0.628
0.636
0.644
0.651
0.657
0.663
0.667
0.671
0.674
0.676
0.677
0.678
Table 9.2: Deferential anomalies of Venus.
The Inferior Planets
µ
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
δθ−
θ̄
0.000
0.000
0.420
0.839
1.259
1.679
2.098
2.518
2.937
3.357
3.776
4.195
4.614
5.033
5.452
5.870
6.289
6.707
7.125
7.543
7.960
8.378
8.795
9.212
9.628
10.045
10.461
10.876
11.292
11.707
12.121
12.536
12.950
13.363
13.776
14.189
14.601
15.013
15.424
15.834
16.245
16.654
17.063
17.472
17.880
18.287
18.694
0.006
0.011
0.017
0.023
0.028
0.034
0.040
0.045
0.051
0.057
0.062
0.068
0.074
0.079
0.085
0.091
0.097
0.102
0.108
0.114
0.120
0.126
0.131
0.137
0.143
0.149
0.155
0.161
0.167
0.173
0.179
0.185
0.191
0.197
0.203
0.209
0.216
0.222
0.228
0.234
0.241
0.247
0.254
0.260
0.267
151
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
0.000
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
0.267
18.694
19.100
19.505
19.910
20.314
20.717
21.119
21.521
21.921
22.321
22.720
23.119
23.516
23.912
24.308
24.702
25.095
25.487
25.879
26.269
26.658
27.045
27.432
27.817
28.201
28.583
28.964
29.344
29.722
30.099
30.474
30.847
31.219
31.589
31.958
32.324
32.689
33.052
33.412
33.771
34.127
34.482
34.834
35.183
35.530
35.875
0.274
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
0.629
35.875
36.217
36.557
36.893
37.227
37.558
37.886
38.210
38.531
38.849
39.164
39.474
39.781
40.084
40.383
40.677
40.968
41.253
41.534
41.810
42.081
42.346
42.606
42.860
43.108
43.349
43.585
43.813
44.034
44.248
44.453
44.651
44.840
45.021
45.191
45.353
45.503
45.644
45.772
45.889
45.994
46.085
46.163
46.226
46.274
46.305
0.649
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
1.344
46.305
46.320
46.317
46.294
46.252
46.188
46.102
45.992
45.857
45.695
45.505
45.284
45.032
44.745
44.422
44.060
43.657
43.210
42.716
42.173
41.577
40.923
40.210
39.433
38.587
37.669
36.675
35.599
34.437
33.186
31.840
30.396
28.850
27.200
25.442
23.577
21.604
19.526
17.347
15.071
12.707
10.266
7.760
5.202
2.610
0.000
1.408
0.006
0.012
0.017
0.023
0.029
0.035
0.041
0.046
0.052
0.058
0.064
0.070
0.076
0.082
0.087
0.093
0.099
0.105
0.111
0.117
0.123
0.129
0.135
0.141
0.147
0.153
0.159
0.165
0.172
0.178
0.184
0.190
0.196
0.203
0.209
0.215
0.222
0.228
0.235
0.241
0.248
0.254
0.261
0.267
0.274
0.273
0.280
0.286
0.293
0.300
0.307
0.313
0.320
0.327
0.334
0.341
0.348
0.355
0.363
0.370
0.377
0.385
0.392
0.400
0.407
0.415
0.423
0.431
0.439
0.447
0.455
0.463
0.471
0.480
0.488
0.497
0.506
0.514
0.523
0.532
0.541
0.551
0.560
0.569
0.579
0.589
0.599
0.609
0.619
0.629
0.281
0.288
0.294
0.301
0.308
0.315
0.322
0.329
0.336
0.344
0.351
0.358
0.366
0.373
0.381
0.388
0.396
0.404
0.411
0.419
0.427
0.435
0.443
0.452
0.460
0.468
0.477
0.485
0.494
0.503
0.512
0.521
0.530
0.539
0.548
0.558
0.567
0.577
0.587
0.597
0.607
0.617
0.628
0.638
0.649
0.640
0.650
0.661
0.672
0.683
0.694
0.706
0.718
0.729
0.742
0.754
0.766
0.779
0.792
0.805
0.818
0.832
0.846
0.860
0.874
0.889
0.904
0.919
0.934
0.950
0.966
0.983
1.000
1.017
1.034
1.052
1.070
1.089
1.108
1.127
1.147
1.167
1.188
1.209
1.230
1.252
1.275
1.297
1.321
1.344
0.660
0.671
0.682
0.693
0.705
0.717
0.729
0.741
0.753
0.766
0.779
0.792
0.805
0.818
0.832
0.846
0.860
0.875
0.890
0.905
0.920
0.936
0.952
0.968
0.985
1.002
1.019
1.037
1.055
1.074
1.093
1.112
1.132
1.152
1.173
1.194
1.216
1.238
1.260
1.284
1.307
1.331
1.356
1.382
1.408
1.369
1.393
1.418
1.444
1.470
1.496
1.523
1.550
1.577
1.605
1.632
1.660
1.688
1.715
1.742
1.769
1.795
1.820
1.844
1.867
1.888
1.906
1.921
1.933
1.942
1.945
1.943
1.934
1.918
1.893
1.859
1.814
1.758
1.688
1.604
1.505
1.391
1.261
1.116
0.956
0.783
0.598
0.404
0.204
0.000
Table 9.3: Epicyclic anomalies of Venus. All quantities are in degrees. Note that θ̄(360◦ − µ) = −θ̄(µ),
and δθ± (360◦ − µ) = −δθ± (µ).
1.434
1.461
1.489
1.517
1.546
1.575
1.605
1.636
1.667
1.698
1.730
1.762
1.794
1.827
1.859
1.892
1.924
1.955
1.986
2.015
2.043
2.069
2.092
2.112
2.128
2.140
2.146
2.145
2.137
2.119
2.091
2.050
1.996
1.926
1.840
1.735
1.612
1.468
1.305
1.123
0.922
0.707
0.478
0.242
0.000
152
MODERN ALMAGEST
Event
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Date
11/06/2000
17/01/2001
30/03/2001
08/06/2001
14/01/2002
22/08/2002
31/10/2002
11/01/2003
18/08/2003
29/03/2004
08/06/2004
17/08/2004
31/03/2005
03/11/2005
13/01/2006
25/03/2006
27/10/2006
09/06/2007
18/08/2007
28/10/2007
09/06/2008
14/01/2009
27/03/2009
05/06/2009
11/01/2010
19/08/2010
29/10/2010
08/01/2011
16/08/2011
27/03/2012
06/06/2012
15/08/2012
28/03/2013
01/11/2013
11/01/2014
23/03/2014
25/10/2014
06/06/2015
15/08/2015
26/10/2015
λ
20GE46
14PI23
09AR36
01TA39
23CP59
15LI08
07SC58
03SG31
25LE20
25TA04
17GE52
09CN29
10AR33
28SG27
23CP36
18AQ07
04SC13
03LE20
24LE45
18VI19
18GE41
12PI03
07AR19
29AR25
21CP24
12LI49
05SC34
01SG06
23LE13
22TA50
15GE43
07CN18
08AR12
26SG02
21CP08
15AQ44
01SC51
01LE09
22LE33
16VI02
Elongation
47.1◦ E
45.8◦ W
46.0◦ E
47.0◦ W
46.0◦ E
45.8◦ W
47.1◦ E
46.5◦ W
45.4◦ E
46.5◦ W
47.1◦ E
45.8◦ W
46.0◦ E
47.0◦ W
46.0◦ E
45.8◦ W
47.1◦ E
46.5◦ W
45.4◦ E
46.4◦ W
Table 9.4: The conjunctions and greatest elongations of Venus during the years 2000–2015 CE.
The Inferior Planets
153
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
∆t(JD)
∆λ̄(◦ )
∆M(◦ )
∆F̄(◦ )
10,000
20,000
30,000
40,000
50,000
60,000
70,000
80,000
90,000
243.770
127.541
11.311
255.081
138.852
22.622
266.392
150.162
33.933
243.344
126.688
10.032
253.376
136.720
20.063
263.407
146.751
30.095
243.422
126.844
10.266
253.688
137.110
20.533
263.955
147.377
30.799
1,000
2,000
3,000
4,000
5,000
6,000
7,000
8,000
9,000
132.377
264.754
37.131
169.508
301.885
74.262
206.639
339.016
111.393
132.334
264.669
37.003
169.338
301.672
74.006
206.341
338.675
111.010
132.342
264.684
37.027
169.369
301.711
74.053
206.395
338.738
111.080
100
200
300
400
500
600
700
800
900
49.238
98.475
147.713
196.951
246.189
295.426
344.664
33.902
83.139
49.233
98.467
147.700
196.934
246.167
295.401
344.634
33.868
83.101
49.234
98.468
147.703
196.937
246.171
295.405
344.640
33.874
83.108
10
20
30
40
50
60
70
80
90
40.924
81.848
122.771
163.695
204.619
245.543
286.466
327.390
8.314
40.923
81.847
122.770
163.693
204.617
245.540
286.463
327.387
8.310
40.923
81.847
122.770
163.694
204.617
245.541
286.464
327.387
8.311
1
2
3
4
5
6
7
8
9
4.092
8.185
12.277
16.370
20.462
24.554
28.647
32.739
36.831
4.092
8.185
12.277
16.369
20.462
24.554
28.646
32.739
36.831
4.092
8.185
12.277
16.369
20.462
24.554
28.646
32.739
36.831
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.409
0.818
1.228
1.637
2.046
2.455
2.865
3.274
3.683
0.409
0.818
1.228
1.637
2.046
2.455
2.865
3.274
3.683
0.409
0.818
1.228
1.637
2.046
2.455
2.865
3.274
3.683
Table 9.5: Mean motion of Mercury. Here, ∆t = t − t0, ∆λ̄ = λ̄ − λ̄0, ∆M = M − M0, and ∆F̄ = F̄ − F̄0.
At epoch (t0 = 2 451 545.0 JD), λ̄0 = 252.087◦ , M0 = 174.693◦ , and F̄0 = 204.436◦ .
154
MODERN ALMAGEST
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
M(◦ )
q(◦ )
100 ζ
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
74
76
78
80
82
84
86
88
90
0.000
1.086
2.169
3.247
4.316
5.376
6.422
7.454
8.467
9.460
10.431
11.377
12.298
13.190
14.052
14.882
15.680
16.443
17.171
17.862
18.517
19.133
19.710
20.249
20.748
21.208
21.629
22.010
22.353
22.656
22.922
23.150
23.342
23.497
23.617
23.703
23.755
23.775
23.764
23.723
23.652
23.553
23.428
23.276
23.100
22.900
20.564
20.544
20.487
20.391
20.257
20.085
19.876
19.631
19.350
19.035
18.685
18.303
17.889
17.445
16.971
16.469
15.941
15.388
14.811
14.212
13.593
12.954
12.299
11.628
10.942
10.245
9.536
8.818
8.091
7.359
6.621
5.880
5.137
4.392
3.648
2.905
2.165
1.429
0.697
-0.030
-0.750
-1.463
-2.168
-2.864
-3.551
-4.229
90
92
94
96
98
100
102
104
106
108
110
112
114
116
118
120
122
124
126
128
130
132
134
136
138
140
142
144
146
148
150
152
154
156
158
160
162
164
166
168
170
172
174
176
178
180
22.900
22.677
22.433
22.168
21.884
21.580
21.259
20.920
20.566
20.195
19.809
19.409
18.996
18.569
18.129
17.677
17.212
16.737
16.250
15.752
15.243
14.724
14.196
13.657
13.109
12.552
11.985
11.410
10.827
10.236
9.637
9.030
8.417
7.796
7.170
6.538
5.900
5.257
4.610
3.959
3.304
2.647
1.987
1.326
0.663
0.000
-4.229
-4.896
-5.552
-6.197
-6.831
-7.452
-8.062
-8.659
-9.243
-9.815
-10.373
-10.918
-11.450
-11.969
-12.473
-12.964
-13.441
-13.904
-14.353
-14.787
-15.207
-15.613
-16.004
-16.380
-16.741
-17.087
-17.418
-17.733
-18.032
-18.316
-18.583
-18.835
-19.070
-19.288
-19.490
-19.675
-19.842
-19.993
-20.126
-20.242
-20.340
-20.420
-20.483
-20.528
-20.555
-20.564
180
182
184
186
188
190
192
194
196
198
200
202
204
206
208
210
212
214
216
218
220
222
224
226
228
230
232
234
236
238
240
242
244
246
248
250
252
254
256
258
260
262
264
266
268
270
0.000
-0.663
-1.326
-1.987
-2.647
-3.304
-3.959
-4.610
-5.257
-5.900
-6.538
-7.170
-7.796
-8.417
-9.030
-9.637
-10.236
-10.827
-11.410
-11.985
-12.552
-13.109
-13.657
-14.196
-14.724
-15.243
-15.752
-16.250
-16.737
-17.212
-17.677
-18.129
-18.569
-18.996
-19.409
-19.809
-20.195
-20.566
-20.920
-21.259
-21.580
-21.884
-22.168
-22.433
-22.677
-22.900
-20.564
-20.555
-20.528
-20.483
-20.420
-20.340
-20.242
-20.126
-19.993
-19.842
-19.675
-19.490
-19.288
-19.070
-18.835
-18.583
-18.316
-18.032
-17.733
-17.418
-17.087
-16.741
-16.380
-16.004
-15.613
-15.207
-14.787
-14.353
-13.904
-13.441
-12.964
-12.473
-11.969
-11.450
-10.918
-10.373
-9.815
-9.243
-8.659
-8.062
-7.452
-6.831
-6.197
-5.552
-4.896
-4.229
270
272
274
276
278
280
282
284
286
288
290
292
294
296
298
300
302
304
306
308
310
312
314
316
318
320
322
324
326
328
330
332
334
336
338
340
342
344
346
348
350
352
354
356
358
360
-22.900
-23.100
-23.276
-23.428
-23.553
-23.652
-23.723
-23.764
-23.775
-23.755
-23.703
-23.617
-23.497
-23.342
-23.150
-22.922
-22.656
-22.353
-22.010
-21.629
-21.208
-20.748
-20.249
-19.710
-19.133
-18.517
-17.862
-17.171
-16.443
-15.680
-14.882
-14.052
-13.190
-12.298
-11.377
-10.431
-9.460
-8.467
-7.454
-6.422
-5.376
-4.316
-3.247
-2.169
-1.086
-0.000
-4.229
-3.551
-2.864
-2.168
-1.463
-0.750
-0.030
0.697
1.429
2.165
2.905
3.648
4.392
5.137
5.880
6.621
7.359
8.091
8.818
9.536
10.245
10.942
11.628
12.299
12.954
13.593
14.212
14.811
15.388
15.941
16.469
16.971
17.445
17.889
18.303
18.685
19.035
19.350
19.631
19.876
20.085
20.257
20.391
20.487
20.544
20.564
Table 9.6: Deferential anomalies of Mercury.
The Inferior Planets
µ
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
δθ−
θ̄
0.000
0.000
0.270
0.540
0.809
1.079
1.348
1.618
1.887
2.156
2.425
2.693
2.961
3.229
3.497
3.764
4.031
4.298
4.564
4.829
5.094
5.359
5.622
5.886
6.148
6.410
6.672
6.932
7.192
7.451
7.709
7.967
8.223
8.479
8.734
8.987
9.240
9.491
9.742
9.991
10.240
10.487
10.732
10.977
11.220
11.462
11.702
0.038
0.075
0.113
0.150
0.188
0.225
0.263
0.301
0.338
0.376
0.414
0.451
0.489
0.527
0.564
0.602
0.640
0.678
0.716
0.753
0.791
0.829
0.867
0.905
0.943
0.981
1.019
1.057
1.095
1.133
1.172
1.210
1.248
1.286
1.325
1.363
1.402
1.440
1.479
1.517
1.556
1.594
1.633
1.672
1.711
155
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
µ
δθ−
θ̄
δθ+
0.000
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
1.711
11.702
11.941
12.179
12.415
12.650
12.882
13.114
13.343
13.571
13.797
14.021
14.244
14.464
14.683
14.899
15.113
15.326
15.536
15.743
15.949
16.152
16.353
16.551
16.747
16.940
17.131
17.319
17.504
17.686
17.865
18.042
18.215
18.385
18.552
18.716
18.876
19.033
19.186
19.336
19.482
19.625
19.763
19.898
20.029
20.155
20.277
2.403
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
3.450
20.277
20.395
20.509
20.618
20.722
20.822
20.917
21.007
21.091
21.171
21.246
21.315
21.378
21.436
21.488
21.534
21.575
21.609
21.636
21.658
21.673
21.681
21.682
21.676
21.663
21.643
21.616
21.580
21.538
21.487
21.428
21.361
21.286
21.202
21.109
21.008
20.898
20.778
20.649
20.511
20.363
20.206
20.038
19.861
19.673
19.475
5.113
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
4.257
19.475
19.267
19.048
18.818
18.578
18.326
18.064
17.791
17.506
17.210
16.903
16.585
16.255
15.914
15.561
15.197
14.822
14.436
14.039
13.630
13.211
12.780
12.339
11.888
11.426
10.955
10.474
9.983
9.483
8.974
8.457
7.932
7.399
6.859
6.312
5.759
5.200
4.636
4.067
3.494
2.917
2.337
1.755
1.171
0.586
0.000
7.325
0.052
0.104
0.156
0.208
0.260
0.313
0.365
0.417
0.469
0.521
0.574
0.626
0.678
0.731
0.783
0.836
0.889
0.941
0.994
1.047
1.100
1.153
1.206
1.259
1.312
1.366
1.419
1.473
1.526
1.580
1.634
1.688
1.742
1.796
1.851
1.905
1.960
2.015
2.070
2.125
2.180
2.236
2.291
2.347
2.403
1.749
1.788
1.827
1.866
1.905
1.944
1.983
2.022
2.061
2.100
2.139
2.178
2.217
2.257
2.296
2.335
2.374
2.413
2.453
2.492
2.531
2.570
2.609
2.649
2.688
2.727
2.766
2.805
2.844
2.882
2.921
2.960
2.999
3.037
3.075
3.114
3.152
3.190
3.227
3.265
3.302
3.340
3.377
3.413
3.450
2.459
2.515
2.572
2.628
2.685
2.742
2.799
2.857
2.914
2.972
3.030
3.088
3.146
3.205
3.263
3.322
3.381
3.441
3.500
3.560
3.620
3.680
3.740
3.801
3.862
3.923
3.984
4.045
4.107
4.169
4.231
4.293
4.355
4.417
4.480
4.543
4.606
4.669
4.732
4.795
4.859
4.922
4.986
5.049
5.113
3.486
3.522
3.557
3.592
3.627
3.662
3.696
3.729
3.762
3.795
3.827
3.858
3.889
3.919
3.949
3.977
4.005
4.032
4.059
4.084
4.109
4.132
4.155
4.176
4.197
4.216
4.233
4.250
4.265
4.278
4.291
4.301
4.310
4.317
4.322
4.326
4.327
4.326
4.323
4.318
4.311
4.301
4.289
4.274
4.257
5.177
5.241
5.305
5.368
5.432
5.496
5.559
5.623
5.686
5.749
5.812
5.874
5.937
5.999
6.060
6.121
6.182
6.242
6.301
6.360
6.418
6.475
6.532
6.587
6.641
6.695
6.747
6.797
6.846
6.894
6.940
6.984
7.026
7.067
7.105
7.140
7.173
7.204
7.231
7.256
7.277
7.295
7.309
7.319
7.325
4.237
4.214
4.188
4.159
4.127
4.092
4.053
4.011
3.966
3.917
3.865
3.809
3.749
3.686
3.619
3.548
3.473
3.394
3.311
3.224
3.133
3.039
2.940
2.838
2.732
2.622
2.508
2.391
2.271
2.147
2.019
1.889
1.756
1.620
1.481
1.340
1.197
1.052
0.905
0.756
0.607
0.456
0.304
0.152
0.000
Table 9.7: Epicyclic anomalies of Mercury. All quantities are in degrees. Note that θ̄(360◦ − µ) =
−θ̄(µ), and δθ± (360◦ − µ) = −δθ± (µ).
7.327
7.324
7.317
7.304
7.286
7.262
7.233
7.197
7.155
7.106
7.050
6.986
6.915
6.836
6.749
6.654
6.549
6.436
6.314
6.182
6.041
5.890
5.729
5.558
5.377
5.186
4.985
4.774
4.554
4.324
4.084
3.835
3.578
3.312
3.038
2.757
2.469
2.174
1.875
1.570
1.261
0.949
0.634
0.317
0.000
156
MODERN ALMAGEST
Event
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Inferior Conjunction
Greatest Elongation
Superior Conjunction
Greatest Elongation
Date
15/01/2000
15/02/2000
01/03/2000
28/03/2000
09/05/2000
09/06/2000
06/07/2000
27/07/2000
22/08/2000
06/10/2000
30/10/2000
15/11/2000
25/12/2000
28/01/2001
13/02/2001
11/03/2001
23/04/2001
22/05/2001
16/06/2001
09/07/2001
05/08/2001
19/09/2001
14/10/2001
29/10/2001
04/12/2001
12/01/2002
27/01/2002
21/02/2002
07/04/2002
04/05/2002
27/05/2002
21/06/2002
21/07/2002
01/09/2002
27/09/2002
13/10/2002
14/11/2002
26/12/2002
λ
25CP08
13PI44
11PI23
10PI35
18TA59
13CN27
14CN39
15CN03
29LE16
09SC17
06SC58
04SC02
04CP18
27AQ07
24AQ23
23AQ05
03TA22
24GE13
25GE26
26GE18
13LE32
22LI50
20LI51
17LI50
12SG45
10AQ33
07AQ42
05AQ55
17AR27
04GE54
05GE48
07GE04
28CN06
06LI10
04LI35
01LI44
21SC39
24CP01
Elongation
18.1◦ E
27.9◦ W
24.2◦ E
19.7◦ W
25.6◦ E
19.3◦ W
18.4◦ E
27.5◦ W
22.6◦ E
21.1◦ W
26.6◦ E
18.5◦ W
18.9◦ E
26.6◦ W
21.0◦ E
22.8◦ W
27.3◦ E
18.0◦ W
19.8◦ E
Table 9.8: The conjunctions and greatest elongations of Mercury during the years 2000–2002 CE.
Planetary Latitudes
157
10 Planetary Latitudes
10.1 Introduction
Up to now, we have neglected the fact that the orbits of the five visible planets about the sun are all
slightly inclined to the plane of the ecliptic. Of course, these inclinations cause the ecliptic latitudes
of the said planets to take small, but non-zero, values. In the following, we shall outline a model
which is capable of predicting these values.
10.2 Determination of Ecliptic Latitude of Superior Planet
Figure 10.1 shows the orbit of a superior planet. As we have already mentioned, the deferent and
epicycle of such a planet have the same elements as the orbit of the planet in question around the
sun, and the apparent orbit of the sun around the earth, respectively. It follows that the deferent
and epicycle of a superior planet are, respectively, inclined and parallel to the ecliptic plane. (Recall
that the ecliptic plane corresponds to the plane of the sun’s apparent orbit about the earth.) Let
the plane of the deferent cut the ecliptic plane along the line NGN ′ . Here, N is the point at which
the deferent passes through the plane of the ecliptic from south to north, in the direction of the
mean planetary motion. This point is called the ascending node. Note that the line NGN ′ must
pass through point G, since the earth is common to the plane of the deferent and the ecliptic plane.
Now, it follows from simple geometry that the elevation of the guide-point G ′ above of the ecliptic
plane satisfies v = r sin i sin F, where r is the length GG ′ , i the fixed inclination of the planetary
orbit (and, hence, of the deferent) to the ecliptic plane, and F the angle NGG ′ . The angle F is
termed the argument of latitude. We can write (see Cha. 8)
F = F̄ + q,
(10.1)
where F̄ is the mean argument of latitude, and q the equation of center of the deferent. Note that F̄
increases uniformly in time: i.e.,
F̄ = F̄0 + n̆ (t − t0).
(10.2)
Now, since the epicycle is parallel to the ecliptic plane, the elevation of the planet above the said
plane is the same as that of the guide-point. Hence, from simple geometry, the ecliptic latitude of
the planet satisfies
v
β = ′′ ,
(10.3)
r
where r ′′ is the length GP, and we have used the small angle approximation. However, it is apparent
from Fig. 8.3 that
r ′′ = (r2 + 2 r r ′ cos µ + r ′ 2)1/2,
(10.4)
where r ′ the length G ′ P, and µ the equation of the epicycle. But, according to the analysis in
Cha. 8, r/r ′ = a z, where a is the planetary major radius in units in which the major radius of the
sun’s apparent orbit about the earth is unity, and z is defined in Eq. (8.4). Thus, we obtain
β = h β0,
(10.5)
158
MODERN ALMAGEST
where
β0(F) = sin i sin F
(10.6)
is termed the deferential latitude, and
i−1/2
h
h(µ, z) = 1 + 2 (a z)−1 cos µ + (a z)−2
(10.7)
the epicyclic latitude correction factor.
P
G′
F
N
G
N′
Figure 10.1: Orbit of a superior planet. Here, G, G ′ , P, N, N ′ , and F represent the earth, guide-point,
planet, ascending node, descending node, and argument of latitude, respectively. View is from northern
ecliptic pole.
In the following, a, e, n, ñ, n̆, λ̄0, M0, F̄0, and i are elements of the orbit of the planet in
question about the sun, and eS, ζS, and λS are elements of the sun’s apparent orbit about the earth.
The requisite elements for all of the superior planets at the J2000 epoch (t0 = 2 451 545.0 JD) are
listed in Tables 5.1 and 10.1. Employing a quadratic interpolation scheme to represent F(µ, z) (see
Cha. 8), our procedure for determining the ecliptic latitude of a superior planet is summed up by
the following formuale:
λ̄ = λ̄0 + n (t − t0),
(10.8)
M = M0 + ñ (t − t0),
(10.9)
(10.10)
F̄ = F̄0 + n̆ (t − t0),
2
q = 2 e sin M + (5/4) e sin 2M,
(10.11)
ζ = e cos M − e2 sin2 M,
(10.12)
F = F̄ + q,
(10.13)
β0 = sin i sin F,
(10.14)
Planetary Latitudes
159
(10.15)
µ = λS − λ̄ − q,
h
i−1/2
h̄ = h(µ, z̄) ≡ 1 + 2 (a z̄)−1 cos µ + (a z̄)−2
δh− = h(µ, z̄) − h(µ, zmax ),
δh+ = h(µ, zmin ) − h(µ, z̄),
1−ζ
,
1 − ζS
z̄ − z
,
ξ =
δz
h = Θ−(ξ) δh− + h̄ + Θ+(ξ) δ h+,
z =
β = h β0.
,
(10.16)
(10.17)
(10.18)
(10.19)
(10.20)
(10.21)
(10.22)
Here, z̄ = (1 + e eS)/(1 − eS2), δz = (e + eS)/(1 − eS2), zmin = z̄ − δz, and zmax = z̄ + δz. The constants
z̄, δz, zmin , and zmax for each of the superior planets are listed in Table 8.1. Finally, the functions
Θ± are tabulated in Table 8.2.
For the case of Mars, the above formulae are capable of matching NASA ephemeris data during
the years 1995–2006 CE with a mean error of 0.3 ′ and a maximum error of 1.5 ′ . For the case of
Jupiter, the mean error is 0.2 ′ and the maximum error 0.5 ′ . Finally, for the case of Saturn, the mean
error is 0.05 ′ and the maximum error 0.08 ′ .
10.3 Mars
The ecliptic latitude of Mars can be determined with the aid of Tables 8.3, 10.2, and 10.3. Table 8.3
allows the mean argument of latitude, F̄, of Mars to be calculated as a function of time. Next,
Table 10.2 permits the deferential latitude, β0, to be determined as a function of the true argument
of latitude, F. Finally, Table 10.3 allows the quantities δh−, h̄, and δh+ to be calculated as functions
of the epicyclic anomaly, µ.
The procedure for using the tables is as follows:
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the ecliptic latitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t − t0, where
t0 = 2 451 545.0 is the epoch.
2. Calculate the planetary equation of center, q, ecliptic anomaly, µ, and interpolation parameters Θ+ and Θ− using the procedure set out in Cha. 8.
3. Enter Table 8.3 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆F̄. If ∆t is negative then the corresponding values are also negative. The value of
the mean argument of latitude, F̄, is the sum of all the ∆F̄ values plus the value of F̄ at the
epoch.
4. Form the true argument of latitude, F = F̄ + q. Add as many multiples of 360◦ to F as is
required to make it fall in the range 0◦ to 360◦ . Round F to the nearest degree.
5. Enter Table 10.2 with the value of F and take out the corresponding value of the deferential
latitude, β0. It is necessary to interpolate if F is odd.
160
MODERN ALMAGEST
6. Enter Table 10.3 with the value of µ and take out the corresponding values of δh−, h̄, and
δh+. If µ > 180◦ then it is necessary to make use of the identities δh± (360◦ − µ) = δh± (µ)
and h̄(360◦ − µ) = h̄(µ).
7. Form the epicyclic latitude correction factor, h = Θ− δh− + h̄ + Θ+ δh+.
8. The ecliptic latitude, β, is the product of the deferential latitude, β0, and the epicyclic latitude
correction factor, h. The decimal fraction can be converted into arc minutes using Table 5.2.
Round to the nearest arc minute.
One example of this procedure is given below.
Example: May 5, 2005 CE, 00:00 UT:
From Cha. 8, t − t0 = 1 950.5 JD, q = −7.345◦ , µ = 114.286◦ , Θ− = 0.101, and Θ+ = 0.619.
Making use of Table 8.3, we find:
t(JD)
F̄(◦ )
+1000
+900
+50
+.5
Epoch
164.041
111.637
26.202
0.262
305.796
607.938
247.938
Modulus
Thus,
F = F̄ + q = 247.938 − 7.345 = 240.593 ≃ 241◦ .
It follows from Table 10.2 that
β0(241◦ ) = −1.615◦ .
Since µ ≃ 114◦ , Table 10.3 yields
δh−(114◦ ) = −0.017,
h̄(114◦ ) = 1.056,
δh+(114◦ ) = −0.027,
so
h = Θ− δh− + h̄ + Θ+ δh+ = −0.101 × 0.017 + 1.056 − 0.619 × 0.027 = 1.038.
Finally,
β = h β0 = −1.038 × 1.615 = −1.676 ≃ −1◦ 41 ′ .
Thus, the ecliptic latitude of Mars at 00:00 UT on May 5, 2005 CE was −1◦ 41 ′ .
10.4 Jupiter
The ecliptic latitude of Jupiter can be determined with the aid of Tables 8.7, 10.4, and 10.5. Table 8.7 allows the mean argument of latitude, F̄, of Jupiter to be calculated as a function of time.
Next, Table 10.4 permits the deferential latitude, β0, to be determined as a function of the true
Planetary Latitudes
161
argument of latitude, F. Finally, Table 10.5 allows the quantities δh−, h̄, and δh+ to be calculated
as functions of the epicyclic anomaly, µ. The procedure for using these tables is analogous to the
previously described procedure for using the Mars tables. One example of this procedure is given
below.
Example: May 5, 2005 CE, 00:00 UT:
From Cha. 8, t − t0 = 1 950.5 JD, q = −0.091◦ , µ = 208.192◦ , Θ− = −0.469, and Θ+ = −0.121.
Making use of Table 8.7, we find:
t(JD)
F̄(◦ )
+1000
+900
+50
+.5
Epoch
83.081
74.773
4.154
0.042
293.660
455.710
95.710
Modulus
Thus,
F = F̄ + q = 95.710 − 0.091 = 95.619 ≃ 96◦ .
It follows from Table 10.4 that
β0(96◦ ) = 1.297◦ .
Since µ ≃ 208◦ , Table 10.5 yields
δh−(208◦ ) = 0.014,
h̄(208◦ ) = 1.197,
δh+(208◦ ) = 0.016,
so
h = Θ− δh− + h̄ + Θ+ δh+ = −0.469 × 0.014 + 1.197 − 0.121 × 0.016 = 1.188.
Finally,
β = h β0 = 1.188 × 1.297 = 1.541 ≃ 1◦ 32 ′ .
Thus, the ecliptic latitude of Jupiter at 00:00 UT on May 5, 2005 CE was 1◦ 32 ′ .
10.5 Saturn
The ecliptic latitude of Saturn can be determined with the aid of Tables 8.11, 10.6, and 10.7.
Table 8.11 allows the mean argument of latitude, F̄, of Saturn to be calculated as a function of
time. Next, Table 10.6 permits the deferential latitude, β0, to be determined as a function of
the true argument of latitude, F. Finally, Table 10.7 allows the quantities δh−, h̄, and δh+ to
be calculated as functions of the epicyclic anomaly, µ. The procedure for using these tables is
analogous to the previously described procedure for using the Mars tables. One example of this
procedure is given below.
Example: May 5, 2005 CE, 00:00 UT:
162
MODERN ALMAGEST
From Cha. 8, t − t0 = 1 950.5 JD, q = 2.561◦ , µ = 286.625◦ , Θ− = 0.071, and Θ+ = 0.759.
Making use of Table 8.11, we find:
t(JD)
F̄(◦ )
+1000
+900
+50
+.5
Epoch
33.478
30.130
1.674
0.017
296.482
361.781
1.781
Modulus
Thus,
F = F̄ + q = 1.781 + 2.561 = 4.342 ≃ 4◦ .
It follows from Table 10.6 that
β0(4◦ ) = 0.173◦ .
Since µ ≃ 287◦ , Table 10.7 yields
δh−(287◦ ) = −0.002,
h̄(287◦ ) = 0.966,
δh+(287◦ ) = −0.003,
so
h = Θ− δh− + h̄ + Θ+ δh+ = −0.071 × 0.002 + 0.966 − 0.759 × 0.003 = 0.964.
Finally,
β = h β0 = 0.964 × 0.173 = 0.167 ≃ 0◦ 10 ′ .
Thus, the ecliptic latitude of Saturn at 00:00 UT on May 5, 2005 CE was 0◦ 10 ′ .
10.6 Determination of Ecliptic Latitude of Inferior Planet
Figure 10.2 shows the orbit of an inferior planet. As we have already mentioned, the epicycle and
deferent of such a planet have the same elements as the orbit of the planet in question around the
sun, and the apparent orbit of the sun around the earth, respectively. It follows that the epicycle and
deferent of an inferior planet are, respectively, inclined and parallel to the ecliptic plane. Let the
plane of the epicycle cut the ecliptic plane along the line NG ′ N ′ . Here, N is the point at which the
epicycle passes through the plane of the ecliptic from south to north, in the direction of the mean
planetary motion. This point is called the ascending node. Note that the line NG ′ N ′ must pass
through the guide-point, G ′ , since the sun (which is coincident with the guide-point) is common to
the plane of the planetary orbit and the ecliptic plane. Now, it follows from simple geometry that
the elevation of the planet P above the guide-point, G ′ , satisfies v = r ′ sin i sin F, where r ′ is the
length G ′ P, i the fixed inclination of the planetary orbit (and, hence, of the epicycle) to the ecliptic
plane, and F the angle NG ′ P. The angle F is termed the argument of latitude. We can write (see
Cha. 9)
F = F̄ + q,
(10.23)
Planetary Latitudes
163
where F̄ is the mean argument of latitude, and q the equation of center of the epicycle. Note that F̄
increases uniformly in time: i.e.,
F̄ = F̄0 + n̆ (t − t0).
(10.24)
Now, since the deferent is parallel to the ecliptic plane, the elevation of the planet above the said
plane is the same as that of the planet above the guide-point. Hence, from simple geometry, the
ecliptic latitude of the planet satisfies
v
β = ′′ ,
(10.25)
r
where r ′′ is the length GP, and we have used the small angle approximation. However, it is apparent
from Fig. 8.3 that
r ′′ = (r2 + 2 r r ′ cos µ + r ′ 2)1/2,
(10.26)
where r the length GG ′ , and µ the equation of the epicycle. But, according to the analysis in Cha. 9,
r ′ /r = a/z, where a is the planetary major radius in units in which the major radius of the sun’s
apparent orbit about the earth is unity, and z is defined in Eq. (9.9). Thus, we obtain
β = h β0,
(10.27)
β0(F) = a sin i sin F
(10.28)
where
is termed the epicyclic latitude, and
i−1/2
h
h(µ, z) = z2 + 2 a z cos µ + a2
(10.29)
the deferential latitude correction factor.
P
N′
F
G
′
N
G
Figure 10.2: Orbit of an inferior planet. Here, G, G ′ , P, N, N ′ , and F represent the earth, guide-point,
planet, ascending node, descending node, and argument of latitude, respectively. View is from northern
ecliptic pole.
164
MODERN ALMAGEST
In the following, a, e, n, ñ, n̆, λ̄0, M0, F̄0, and i are elements of the orbit of the planet in
question about the sun, and eS, ζS, and λS are elements of the sun’s apparent orbit about the earth.
The requisite elements for all of the superior planets at the J2000 epoch (t0 = 2 451 545.0 JD) are
listed in Tables 5.1 and 10.1. Employing a quadratic interpolation scheme to represent F(µ, z) (see
Cha. 8), our procedure for determining the ecliptic latitude of a superior planet is summed up by
the following formuale:
λ̄ = λ̄0 + n (t − t0),
(10.30)
M = M0 + ñ (t − t0),
(10.31)
F̄ = F̄0 + n̆ (t − t0),
(10.32)
q = 2 e sin M + (5/4) e2 sin 2M,
2
(10.33)
2
ζ = e cos M − e sin M,
(10.34)
F = F̄ + q,
(10.35)
β0 = a sin i sin F,
(10.36)
(10.37)
µ = λ̄ + q − λ̄S,
h
i−1/2
h̄ = h(µ, z̄) ≡ z̄2 + 2 a z̄ cos µ + a2
δh− = h(µ, z̄) − h(µ, zmax ),
δh+ = h(µ, zmin ) − h(µ, z̄),
,
(10.38)
(10.39)
(10.40)
1 − ζS
,
1−ζ
z̄ − z
ξ =
,
δz
h = Θ−(ξ) δh− + h̄ + Θ+(ξ) δ h+,
(10.43)
β = h β0.
(10.44)
z =
(10.41)
(10.42)
Here, z̄ = (1 + e eS)/(1 − e2), δz = (e + eS)/(1 − e2), zmin = z̄ − δz, and zmax = z̄ + δz. The constants
z̄, δz, zmin , and zmax for each of the inferior planets are listed in Table 8.1. Finally, the functions Θ±
are tabulated in Table 8.2.
For the case of Venus, the above formulae are capable of matching NASA ephemeris data during
the years 1995–2006 CE with a mean error of 0.7 ′ and a maximum error of 1.8 ′ . For the case of
Mercury, with the augmentations to the theory described in Cha. 9, the mean error is 1.6 ′ and the
maximum error 5 ′ .
10.7 Venus
The ecliptic latitude of Venus can be determined with the aid of Tables 9.1, 10.8, and 10.9. Table 9.1
allows the mean argument of latitude, F̄, of Venus to be calculated as a function of time. Next,
Table 10.8 permits the epicyclic latitude, β0, to be determined as a function of the true argument of
latitude, F. Finally, Table 10.9 allows the quantities δh−, h̄, and δh+ to be calculated as functions
of the epicyclic anomaly, µ.
The procedure for using the tables is as follows:
Planetary Latitudes
165
1. Determine the fractional Julian day number, t, corresponding to the date and time at which
the ecliptic latitude is to be calculated with the aid of Tables 3.1–3.3. Form ∆t = t − t0, where
t0 = 2 451 545.0 is the epoch.
2. Calculate the planetary equation of center, q, ecliptic anomaly, µ, and interpolation parameters Θ+ and Θ− using the procedure set out in Cha. 9.
3. Enter Table 9.1 with the digit for each power of 10 in ∆t and take out the corresponding
values of ∆F̄. If ∆t is negative then the corresponding values are also negative. The value of
the mean argument of latitude, F̄, is the sum of all the ∆F̄ values plus the value of F̄ at the
epoch.
4. Form the true argument of latitude, F = F̄ + q. Add as many multiples of 360◦ to F as is
required to make it fall in the range 0◦ to 360◦ . Round F to the nearest degree.
5. Enter Table 10.8 with the value of F and take out the corresponding value of the epicyclic
latitude, β0. It is necessary to interpolate if F is odd.
6. Enter Table 10.9 with the value of µ and take out the corresponding values of δh−, h̄, and
δh+. If µ > 180◦ then it is necessary to make use of the identities δh± (360◦ − µ) = δh± (µ)
and h̄(360◦ − µ) = h̄(µ).
7. Form the deferential latitude correction factor, h = Θ− δh− + h̄ + Θ+ δh+.
8. The ecliptic latitude, β, is the product of the epicyclic latitude, β0, and the deferential latitude
correction factor, h. The decimal fraction can be converted into arc minutes using Table 5.2.
Round to the nearest arc minute.
One example of this procedure is given below.
Example: May 5, 2005 CE, 00:00 UT:
From Cha. 9, t − t0 = 1 950.5 JD, q = −0.712◦ , µ = 21.689◦ , Θ− = −0.355, and Θ+ = −0.125.
Making use of Table 9.1, we find:
t(JD)
F̄(◦ )
+1000
+900
+50
+.5
Epoch
162.138
1.924
80.107
0.801
105.253
350.223
350.223
Modulus
Thus,
F = F̄ + q = 350.223 − 0.712 = 349.511 ≃ 350◦ .
It follows from Table 10.8 that
β0(350◦ ) = −0.423◦ .
166
MODERN ALMAGEST
Since µ ≃ 22◦ , Table 10.9 yields
δh−(22◦ ) = 0.008,
h̄(22◦ ) = 0.591,
δh+(22◦ ) = 0.008,
so
h = Θ− δh− + h̄ + Θ+ δh+ = −0.355 × 0.008 + 0.591 − 0.125 × 0.008 = 0.587.
Finally,
β = h β0 = −0.587 × 0.423 = −0.248 ≃ −0◦ 15 ′ .
Thus, the ecliptic latitude of Venus at 00:00 UT on May 5, 2005 CE was −0◦ 15 ′ .
10.8 Mercury
The ecliptic latitude of Mercury can be determined with the aid of Tables 9.5, 10.10, and 10.11.
Table 9.5 allows the mean argument of latitude, F̄, of Mercury to be calculated as a function of
time. Next, Table 10.10 permits the epicyclic latitude, β0, to be determined as a function of the
true argument of latitude, F. Finally, Table 10.11 allows the quantities δh−, h̄, and δh+ to be
calculated as functions of the epicyclic anomaly, µ. The procedure for using the tables is analogous
to the previously described procedure for using the Venus tables. One example of this procedure is
given below.
Example: May 5, 2005 CE, 00:00 UT:
From Cha. 9, t − t0 = 1 950.5 JD, q = −16.974◦ , µ = 252.692◦ , Θ− = 0.107, and Θ+ = 0.583.
Making use of Table 9.5, we find:
t(JD)
F̄(◦ )
+1000
+900
+50
+.5
Epoch
132.342
83.108
204.617
2.046
204.436
626.549
266.549
Modulus
Thus,
F = F̄ + q = 266.549 − 16.974 = 249.575 ≃ 250◦ .
It follows from Table 10.10 that
β0(250◦ ) = −2.511◦ .
Since µ ≃ 253◦ , Table 10.11 yields
δh−(253◦ ) = 0.184,
h̄(253◦ ) = 1.037,
δh+(253◦ ) = 0.272,
so
h = Θ− δh− + h̄ + Θ+ δh+ = 0.107 × 0.184 + 1.037 + 0.583 × 0.272 = 1.215.
Planetary Latitudes
167
Finally,
β = h β0 = −1.215 × 2.511 = −3.051 ≃ −3◦ 03 ′ .
Thus, the ecliptic latitude of Mercury at 00:00 UT on May 5, 2005 CE was −3◦ 03 ′ .
168
MODERN ALMAGEST
Object
Mercury
Venus
Mars
Jupiter
Saturn
i(◦ )
6.9190
3.3692
1.8467
1.3044
2.4860
n̆ (◦ /day)
4.09234221
1.60213807
0.52404094
0.08308122
0.03347795
F̄0 (◦ )
204.436
105.253
305.796
293.660
296.482
Table 10.1: Additional Keplerian orbital elements for the five visible planets at the J2000 epoch (i.e.,
12:00 UT, January 1, 2000 CE, which corresponds to t0 = 2 451 545.0 JD). The elements are optimized for use in the time period 1800 CE to 2050 CE. Source: Jet Propulsion Laboratory (NASA),
http://ssd.jpl.nasa.gov/.
Planetary Latitudes
169
F(◦ )
β0 (◦ )
F(◦ )
000/180
002/178
004/176
006/174
008/172
010/170
012/168
014/166
016/164
018/162
020/160
022/158
024/156
026/154
028/152
030/150
032/148
034/146
036/144
038/142
040/140
042/138
044/136
046/134
048/132
050/130
052/128
054/126
056/124
058/122
060/120
062/118
064/116
066/114
068/112
070/110
072/108
074/106
076/104
078/102
080/100
082/098
084/096
086/094
088/092
090/090
0.000
0.064
0.129
0.193
0.257
0.321
0.384
0.447
0.509
0.571
0.631
0.692
0.751
0.809
0.867
0.923
0.978
1.032
1.085
1.137
1.187
1.235
1.283
1.328
1.372
1.414
1.455
1.494
1.531
1.566
1.599
1.630
1.660
1.687
1.712
1.735
1.756
1.775
1.792
1.806
1.818
1.828
1.836
1.842
1.845
1.846
(180)/(360)
(182)/(358)
(184)/(356)
(186)/(354)
(188)/(352)
(190)/(350)
(192)/(348)
(194)/(346)
(196)/(344)
(198)/(342)
(200)/(340)
(202)/(338)
(204)/(336)
(206)/(334)
(208)/(332)
(210)/(330)
(212)/(328)
(214)/(326)
(216)/(324)
(218)/(322)
(220)/(320)
(222)/(318)
(224)/(316)
(226)/(314)
(228)/(312)
(230)/(310)
(232)/(308)
(234)/(306)
(236)/(304)
(238)/(302)
(240)/(300)
(242)/(298)
(244)/(296)
(246)/(294)
(248)/(292)
(250)/(290)
(252)/(288)
(254)/(286)
(256)/(284)
(258)/(282)
(260)/(280)
(262)/(278)
(264)/(276)
(266)/(274)
(268)/(272)
(270)/(270)
Table 10.2: Deferential ecliptic latitude of Mars. The latitude is minus the value shown in the table if
the argument is in parenthesies.
170
MODERN ALMAGEST
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
-0.025
0.604
0.604
0.604
0.604
0.605
0.605
0.605
0.605
0.606
0.606
0.606
0.607
0.607
0.608
0.609
0.609
0.610
0.611
0.611
0.612
0.613
0.614
0.615
0.616
0.617
0.618
0.619
0.621
0.622
0.623
0.625
0.626
0.627
0.629
0.631
0.632
0.634
0.636
0.637
0.639
0.641
0.643
0.645
0.647
0.649
0.652
-0.028
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
-0.025
0.652
0.654
0.656
0.659
0.661
0.664
0.666
0.669
0.672
0.674
0.677
0.680
0.683
0.686
0.689
0.693
0.696
0.699
0.703
0.706
0.710
0.714
0.718
0.722
0.726
0.730
0.734
0.738
0.743
0.747
0.752
0.757
0.762
0.767
0.772
0.777
0.782
0.788
0.793
0.799
0.805
0.811
0.817
0.823
0.830
0.836
-0.029
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
-0.025
0.836
0.843
0.850
0.857
0.865
0.872
0.880
0.888
0.896
0.904
0.912
0.921
0.930
0.939
0.948
0.958
0.968
0.978
0.988
0.999
1.010
1.021
1.032
1.044
1.056
1.069
1.082
1.095
1.108
1.122
1.137
1.151
1.167
1.182
1.198
1.215
1.232
1.249
1.267
1.286
1.305
1.325
1.345
1.366
1.388
1.410
-0.031
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.015
1.410
1.433
1.457
1.482
1.507
1.533
1.560
1.588
1.616
1.646
1.676
1.708
1.740
1.773
1.807
1.843
1.879
1.916
1.955
1.994
2.034
2.075
2.117
2.160
2.203
2.247
2.292
2.337
2.382
2.427
2.472
2.517
2.560
2.603
2.644
2.683
2.721
2.755
2.787
2.816
2.840
2.861
2.878
2.890
2.897
2.899
0.003
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.028
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.029
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.026
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.025
-0.029
-0.029
-0.029
-0.029
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.025
-0.024
-0.024
-0.024
-0.024
-0.024
-0.024
-0.023
-0.023
-0.023
-0.023
-0.022
-0.022
-0.022
-0.021
-0.021
-0.021
-0.020
-0.020
-0.019
-0.019
-0.018
-0.018
-0.017
-0.016
-0.015
-0.015
-0.014
-0.013
-0.012
-0.011
-0.010
-0.008
-0.007
-0.006
-0.004
-0.003
-0.001
0.001
0.003
0.005
0.007
0.010
0.012
0.015
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.031
-0.030
-0.030
-0.030
-0.030
-0.030
-0.030
-0.029
-0.029
-0.029
-0.029
-0.028
-0.028
-0.027
-0.027
-0.027
-0.026
-0.025
-0.025
-0.024
-0.023
-0.023
-0.022
-0.021
-0.020
-0.019
-0.017
-0.016
-0.015
-0.013
-0.011
-0.009
-0.007
-0.005
-0.003
-0.000
0.003
0.018
0.021
0.025
0.028
0.032
0.037
0.041
0.046
0.051
0.057
0.063
0.069
0.076
0.084
0.092
0.100
0.109
0.118
0.128
0.139
0.151
0.163
0.175
0.189
0.202
0.217
0.232
0.248
0.264
0.280
0.297
0.314
0.331
0.348
0.364
0.380
0.395
0.408
0.421
0.432
0.442
0.449
0.455
0.458
0.459
Table 10.3: Epicyclic latitude correction factor for Mars. µ is in degrees. Note that h̄(360◦ −µ) = h̄(µ),
and δh± (360◦ − µ) = δh± (µ).
0.006
0.009
0.013
0.017
0.021
0.026
0.031
0.036
0.043
0.049
0.056
0.064
0.073
0.083
0.093
0.104
0.117
0.130
0.145
0.161
0.178
0.197
0.218
0.240
0.265
0.291
0.319
0.349
0.382
0.416
0.453
0.491
0.530
0.571
0.612
0.654
0.694
0.734
0.770
0.804
0.833
0.857
0.874
0.885
0.889
Planetary Latitudes
171
F(◦ )
β0 (◦ )
F(◦ )
000/180
002/178
004/176
006/174
008/172
010/170
012/168
014/166
016/164
018/162
020/160
022/158
024/156
026/154
028/152
030/150
032/148
034/146
036/144
038/142
040/140
042/138
044/136
046/134
048/132
050/130
052/128
054/126
056/124
058/122
060/120
062/118
064/116
066/114
068/112
070/110
072/108
074/106
076/104
078/102
080/100
082/098
084/096
086/094
088/092
090/090
0.000
0.046
0.091
0.136
0.182
0.226
0.271
0.316
0.360
0.403
0.446
0.489
0.531
0.572
0.612
0.652
0.691
0.729
0.767
0.803
0.838
0.873
0.906
0.938
0.969
0.999
1.028
1.055
1.081
1.106
1.130
1.152
1.172
1.192
1.209
1.226
1.240
1.254
1.266
1.276
1.284
1.292
1.297
1.301
1.303
1.304
(180)/(360)
(182)/(358)
(184)/(356)
(186)/(354)
(188)/(352)
(190)/(350)
(192)/(348)
(194)/(346)
(196)/(344)
(198)/(342)
(200)/(340)
(202)/(338)
(204)/(336)
(206)/(334)
(208)/(332)
(210)/(330)
(212)/(328)
(214)/(326)
(216)/(324)
(218)/(322)
(220)/(320)
(222)/(318)
(224)/(316)
(226)/(314)
(228)/(312)
(230)/(310)
(232)/(308)
(234)/(306)
(236)/(304)
(238)/(302)
(240)/(300)
(242)/(298)
(244)/(296)
(246)/(294)
(248)/(292)
(250)/(290)
(252)/(288)
(254)/(286)
(256)/(284)
(258)/(282)
(260)/(280)
(262)/(278)
(264)/(276)
(266)/(274)
(268)/(272)
(270)/(270)
Table 10.4: Deferential ecliptic latitude of Jupiter. The latitude is minus the value shown in the table
if the argument is in parenthesies.
172
MODERN ALMAGEST
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
-0.008
0.839
0.839
0.839
0.839
0.839
0.839
0.840
0.840
0.840
0.840
0.841
0.841
0.841
0.842
0.842
0.843
0.843
0.844
0.845
0.845
0.846
0.847
0.847
0.848
0.849
0.850
0.851
0.852
0.853
0.854
0.855
0.856
0.857
0.858
0.859
0.860
0.861
0.863
0.864
0.865
0.867
0.868
0.870
0.871
0.873
0.874
-0.009
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
-0.007
0.874
0.876
0.877
0.879
0.881
0.883
0.884
0.886
0.888
0.890
0.892
0.894
0.896
0.898
0.900
0.902
0.904
0.906
0.909
0.911
0.913
0.916
0.918
0.920
0.923
0.925
0.928
0.930
0.933
0.935
0.938
0.941
0.944
0.946
0.949
0.952
0.955
0.958
0.961
0.964
0.967
0.970
0.973
0.976
0.979
0.982
-0.008
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
-0.002
0.982
0.985
0.988
0.992
0.995
0.998
1.002
1.005
1.008
1.012
1.015
1.019
1.022
1.026
1.029
1.033
1.036
1.040
1.044
1.047
1.051
1.055
1.058
1.062
1.066
1.069
1.073
1.077
1.080
1.084
1.088
1.092
1.095
1.099
1.103
1.107
1.110
1.114
1.118
1.121
1.125
1.129
1.132
1.136
1.140
1.143
-0.003
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.009
1.143
1.147
1.150
1.154
1.157
1.160
1.164
1.167
1.170
1.173
1.177
1.180
1.183
1.186
1.189
1.192
1.194
1.197
1.200
1.202
1.205
1.207
1.210
1.212
1.214
1.216
1.218
1.220
1.222
1.224
1.225
1.227
1.228
1.230
1.231
1.232
1.233
1.234
1.235
1.236
1.236
1.237
1.237
1.237
1.238
1.238
0.010
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.009
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.002
-0.002
-0.002
-0.008
-0.008
-0.008
-0.008
-0.008
-0.008
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.007
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.002
-0.002
-0.002
-0.001
-0.001
-0.001
-0.001
-0.001
-0.000
-0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.005
0.005
0.005
0.006
0.006
0.006
0.006
0.007
0.007
0.007
0.008
0.008
0.008
0.008
0.009
0.009
-0.002
-0.002
-0.002
-0.002
-0.001
-0.001
-0.001
-0.001
-0.001
-0.000
-0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.005
0.005
0.005
0.006
0.006
0.006
0.007
0.007
0.007
0.008
0.008
0.008
0.009
0.009
0.009
0.010
0.010
0.010
0.009
0.010
0.010
0.010
0.010
0.011
0.011
0.011
0.012
0.012
0.012
0.012
0.013
0.013
0.013
0.014
0.014
0.014
0.014
0.015
0.015
0.015
0.015
0.015
0.016
0.016
0.016
0.016
0.016
0.016
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.018
0.018
0.018
0.018
0.018
0.018
0.018
Table 10.5: Epicyclic latitude correction factor for Jupiter. µ is in degrees. Note that h̄(360◦ − µ) =
h̄(µ), and δh± (360◦ − µ) = δh± (µ).
0.011
0.011
0.011
0.012
0.012
0.012
0.013
0.013
0.013
0.014
0.014
0.014
0.015
0.015
0.015
0.016
0.016
0.016
0.017
0.017
0.017
0.017
0.018
0.018
0.018
0.018
0.019
0.019
0.019
0.019
0.019
0.020
0.020
0.020
0.020
0.020
0.020
0.020
0.021
0.021
0.021
0.021
0.021
0.021
0.021
Planetary Latitudes
173
F(◦ )
β0 (◦ )
F(◦ )
000/180
002/178
004/176
006/174
008/172
010/170
012/168
014/166
016/164
018/162
020/160
022/158
024/156
026/154
028/152
030/150
032/148
034/146
036/144
038/142
040/140
042/138
044/136
046/134
048/132
050/130
052/128
054/126
056/124
058/122
060/120
062/118
064/116
066/114
068/112
070/110
072/108
074/106
076/104
078/102
080/100
082/098
084/096
086/094
088/092
090/090
0.000
0.087
0.173
0.260
0.346
0.432
0.517
0.601
0.685
0.768
0.850
0.931
1.011
1.089
1.167
1.243
1.317
1.390
1.461
1.530
1.597
1.663
1.726
1.788
1.847
1.904
1.958
2.011
2.060
2.108
2.152
2.194
2.234
2.270
2.304
2.335
2.364
2.389
2.411
2.431
2.447
2.461
2.472
2.479
2.484
2.485
(180)/(360)
(182)/(358)
(184)/(356)
(186)/(354)
(188)/(352)
(190)/(350)
(192)/(348)
(194)/(346)
(196)/(344)
(198)/(342)
(200)/(340)
(202)/(338)
(204)/(336)
(206)/(334)
(208)/(332)
(210)/(330)
(212)/(328)
(214)/(326)
(216)/(324)
(218)/(322)
(220)/(320)
(222)/(318)
(224)/(316)
(226)/(314)
(228)/(312)
(230)/(310)
(232)/(308)
(234)/(306)
(236)/(304)
(238)/(302)
(240)/(300)
(242)/(298)
(244)/(296)
(246)/(294)
(248)/(292)
(250)/(290)
(252)/(288)
(254)/(286)
(256)/(284)
(258)/(282)
(260)/(280)
(262)/(278)
(264)/(276)
(266)/(274)
(268)/(272)
(270)/(270)
Table 10.6: Deferential ecliptic latitude of Saturn. The latitude is minus the value shown in the table
if the argument is in parenthesies.
174
MODERN ALMAGEST
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
-0.006
0.905
0.905
0.905
0.905
0.905
0.905
0.906
0.906
0.906
0.906
0.906
0.907
0.907
0.907
0.908
0.908
0.908
0.909
0.909
0.909
0.910
0.910
0.911
0.911
0.912
0.913
0.913
0.914
0.914
0.915
0.916
0.916
0.917
0.918
0.919
0.920
0.920
0.921
0.922
0.923
0.924
0.925
0.926
0.927
0.928
0.929
-0.006
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
-0.005
0.929
0.930
0.931
0.932
0.933
0.934
0.935
0.937
0.938
0.939
0.940
0.942
0.943
0.944
0.945
0.947
0.948
0.949
0.951
0.952
0.954
0.955
0.957
0.958
0.960
0.961
0.963
0.964
0.966
0.967
0.969
0.971
0.972
0.974
0.975
0.977
0.979
0.981
0.982
0.984
0.986
0.987
0.989
0.991
0.993
0.995
-0.005
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
-0.001
0.995
0.996
0.998
1.000
1.002
1.004
1.006
1.007
1.009
1.011
1.013
1.015
1.017
1.019
1.020
1.022
1.024
1.026
1.028
1.030
1.032
1.034
1.036
1.037
1.039
1.041
1.043
1.045
1.047
1.049
1.050
1.052
1.054
1.056
1.058
1.060
1.061
1.063
1.065
1.067
1.068
1.070
1.072
1.073
1.075
1.077
-0.001
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.005
1.077
1.078
1.080
1.081
1.083
1.084
1.086
1.087
1.089
1.090
1.091
1.093
1.094
1.095
1.097
1.098
1.099
1.100
1.101
1.103
1.104
1.105
1.106
1.107
1.107
1.108
1.109
1.110
1.111
1.111
1.112
1.113
1.113
1.114
1.114
1.115
1.115
1.116
1.116
1.116
1.116
1.117
1.117
1.117
1.117
1.117
0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.006
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.005
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.004
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.003
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.002
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.000
-0.000
-0.000
-0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.004
0.004
0.004
0.005
0.005
0.005
0.005
0.005
0.005
-0.001
-0.001
-0.000
-0.000
-0.000
-0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.002
0.002
0.002
0.002
0.002
0.002
0.003
0.003
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.004
0.004
0.004
0.005
0.005
0.005
0.005
0.005
0.005
0.006
0.006
0.006
0.006
0.005
0.005
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.006
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.009
0.009
0.009
0.009
0.009
Table 10.7: Epicyclic latitude correction factor for Saturn. µ is in degrees. Note that h̄(360◦ − µ) =
h̄(µ), and δh± (360◦ − µ) = δh± (µ).
0.006
0.006
0.006
0.007
0.007
0.007
0.007
0.007
0.007
0.007
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
Planetary Latitudes
175
F(◦ )
β0 (◦ )
F(◦ )
000/180
002/178
004/176
006/174
008/172
010/170
012/168
014/166
016/164
018/162
020/160
022/158
024/156
026/154
028/152
030/150
032/148
034/146
036/144
038/142
040/140
042/138
044/136
046/134
048/132
050/130
052/128
054/126
056/124
058/122
060/120
062/118
064/116
066/114
068/112
070/110
072/108
074/106
076/104
078/102
080/100
082/098
084/096
086/094
088/092
090/090
0.000
0.085
0.170
0.255
0.339
0.423
0.506
0.589
0.671
0.753
0.833
0.912
0.991
1.068
1.143
1.218
1.291
1.362
1.432
1.500
1.566
1.630
1.692
1.752
1.810
1.866
1.919
1.970
2.019
2.066
2.109
2.151
2.189
2.225
2.258
2.289
2.316
2.341
2.363
2.382
2.399
2.412
2.422
2.430
2.434
2.436
(180)/(360)
(182)/(358)
(184)/(356)
(186)/(354)
(188)/(352)
(190)/(350)
(192)/(348)
(194)/(346)
(196)/(344)
(198)/(342)
(200)/(340)
(202)/(338)
(204)/(336)
(206)/(334)
(208)/(332)
(210)/(330)
(212)/(328)
(214)/(326)
(216)/(324)
(218)/(322)
(220)/(320)
(222)/(318)
(224)/(316)
(226)/(314)
(228)/(312)
(230)/(310)
(232)/(308)
(234)/(306)
(236)/(304)
(238)/(302)
(240)/(300)
(242)/(298)
(244)/(296)
(246)/(294)
(248)/(292)
(250)/(290)
(252)/(288)
(254)/(286)
(256)/(284)
(258)/(282)
(260)/(280)
(262)/(278)
(264)/(276)
(266)/(274)
(268)/(272)
(270)/(270)
Table 10.8: Epicyclic ecliptic latitude of Venus. The latitude is minus the value shown in the table if the
argument is in parenthesies.
176
µ
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
MODERN ALMAGEST
δh−
h̄
0.008
0.580
0.580
0.580
0.580
0.580
0.581
0.581
0.581
0.582
0.582
0.582
0.583
0.583
0.584
0.584
0.585
0.586
0.586
0.587
0.588
0.589
0.590
0.591
0.592
0.593
0.594
0.595
0.596
0.597
0.599
0.600
0.601
0.603
0.604
0.606
0.608
0.609
0.611
0.613
0.614
0.616
0.618
0.620
0.622
0.624
0.627
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.009
0.009
0.009
0.009
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
0.008
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
0.009
0.627
0.629
0.631
0.633
0.636
0.638
0.641
0.643
0.646
0.649
0.652
0.655
0.658
0.661
0.664
0.667
0.670
0.674
0.677
0.681
0.684
0.688
0.692
0.696
0.700
0.704
0.708
0.712
0.717
0.721
0.726
0.730
0.735
0.740
0.745
0.751
0.756
0.761
0.767
0.773
0.778
0.784
0.791
0.797
0.803
0.810
0.009
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
0.012
0.810
0.817
0.824
0.831
0.838
0.846
0.854
0.861
0.870
0.878
0.886
0.895
0.904
0.913
0.923
0.933
0.943
0.953
0.964
0.975
0.986
0.997
1.009
1.021
1.034
1.047
1.060
1.074
1.088
1.103
1.118
1.133
1.149
1.166
1.183
1.200
1.219
1.237
1.257
1.277
1.298
1.319
1.342
1.365
1.389
1.413
0.013
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.032
1.413
1.439
1.466
1.493
1.522
1.552
1.583
1.615
1.648
1.683
1.719
1.756
1.795
1.836
1.878
1.922
1.968
2.016
2.065
2.117
2.170
2.226
2.283
2.343
2.405
2.469
2.535
2.603
2.673
2.744
2.816
2.890
2.964
3.037
3.111
3.182
3.252
3.318
3.380
3.436
3.486
3.529
3.564
3.589
3.604
3.609
0.033
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.010
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.013
0.013
0.013
0.013
0.013
0.013
0.013
0.014
0.014
0.014
0.014
0.014
0.015
0.015
0.015
0.015
0.016
0.016
0.016
0.016
0.017
0.017
0.017
0.018
0.018
0.019
0.019
0.019
0.020
0.020
0.021
0.021
0.022
0.022
0.023
0.024
0.024
0.025
0.026
0.026
0.027
0.028
0.029
0.030
0.031
0.032
0.013
0.013
0.013
0.013
0.013
0.014
0.014
0.014
0.014
0.014
0.015
0.015
0.015
0.015
0.016
0.016
0.016
0.017
0.017
0.017
0.017
0.018
0.018
0.019
0.019
0.019
0.020
0.020
0.021
0.021
0.022
0.022
0.023
0.023
0.024
0.025
0.025
0.026
0.027
0.028
0.029
0.030
0.031
0.032
0.033
0.033
0.034
0.036
0.037
0.039
0.040
0.042
0.044
0.046
0.048
0.050
0.053
0.055
0.058
0.061
0.065
0.068
0.072
0.076
0.081
0.086
0.091
0.097
0.103
0.110
0.117
0.125
0.133
0.142
0.151
0.161
0.172
0.183
0.194
0.205
0.217
0.228
0.239
0.249
0.258
0.267
0.273
0.278
0.281
0.282
0.034
0.035
0.037
0.038
0.040
0.041
0.043
0.045
0.047
0.049
0.052
0.054
0.057
0.060
0.063
0.067
0.071
0.075
0.080
0.085
0.090
0.096
0.102
0.109
0.117
0.125
0.134
0.144
0.155
0.166
0.178
0.191
0.204
0.218
0.232
0.247
0.262
0.276
0.290
0.302
0.313
0.322
0.329
0.333
0.334
Table 10.9: Deferential latitude correction factor for Venus. µ is in degrees. Note that h̄(360◦ − µ) =
h̄(µ), and δh± (360◦ − µ) = δh± (µ).
Planetary Latitudes
177
F(◦ )
β0 (◦ )
F(◦ )
000/180
002/178
004/176
006/174
008/172
010/170
012/168
014/166
016/164
018/162
020/160
022/158
024/156
026/154
028/152
030/150
032/148
034/146
036/144
038/142
040/140
042/138
044/136
046/134
048/132
050/130
052/128
054/126
056/124
058/122
060/120
062/118
064/116
066/114
068/112
070/110
072/108
074/106
076/104
078/102
080/100
082/098
084/096
086/094
088/092
090/090
0.000
0.093
0.186
0.279
0.372
0.464
0.556
0.646
0.736
0.826
0.914
1.001
1.087
1.171
1.254
1.336
1.416
1.494
1.570
1.645
1.717
1.788
1.856
1.922
1.986
2.047
2.105
2.162
2.215
2.266
2.314
2.359
2.401
2.441
2.477
2.511
2.541
2.568
2.592
2.613
2.631
2.646
2.657
2.665
2.670
2.672
(180)/(360)
(182)/(358)
(184)/(356)
(186)/(354)
(188)/(352)
(190)/(350)
(192)/(348)
(194)/(346)
(196)/(344)
(198)/(342)
(200)/(340)
(202)/(338)
(204)/(336)
(206)/(334)
(208)/(332)
(210)/(330)
(212)/(328)
(214)/(326)
(216)/(324)
(218)/(322)
(220)/(320)
(222)/(318)
(224)/(316)
(226)/(314)
(228)/(312)
(230)/(310)
(232)/(308)
(234)/(306)
(236)/(304)
(238)/(302)
(240)/(300)
(242)/(298)
(244)/(296)
(246)/(294)
(248)/(292)
(250)/(290)
(252)/(288)
(254)/(286)
(256)/(284)
(258)/(282)
(260)/(280)
(262)/(278)
(264)/(276)
(266)/(274)
(268)/(272)
(270)/(270)
Table 10.10: Epicyclic ecliptic latitude of Mercury. The latitude is minus the value shown in the table
if the argument is in parenthesies.
178
µ
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
MODERN ALMAGEST
δh−
h̄
0.099
0.719
0.719
0.719
0.719
0.719
0.720
0.720
0.720
0.720
0.721
0.721
0.722
0.722
0.723
0.723
0.724
0.725
0.725
0.726
0.727
0.728
0.729
0.730
0.731
0.732
0.733
0.734
0.735
0.737
0.738
0.739
0.741
0.742
0.743
0.745
0.747
0.748
0.750
0.752
0.754
0.755
0.757
0.759
0.761
0.763
0.765
0.099
0.099
0.099
0.099
0.099
0.099
0.099
0.099
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.101
0.101
0.101
0.101
0.101
0.101
0.102
0.102
0.102
0.102
0.103
0.103
0.103
0.104
0.104
0.104
0.105
0.105
0.105
0.106
0.106
0.106
0.107
0.107
0.108
0.108
0.109
0.109
0.110
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
µ
δh−
h̄
δh+
0.137
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
0.110
0.765
0.768
0.770
0.772
0.774
0.777
0.779
0.782
0.784
0.787
0.790
0.793
0.795
0.798
0.801
0.804
0.807
0.811
0.814
0.817
0.820
0.824
0.827
0.831
0.835
0.838
0.842
0.846
0.850
0.854
0.858
0.862
0.866
0.871
0.875
0.880
0.884
0.889
0.894
0.899
0.904
0.909
0.914
0.919
0.924
0.930
0.152
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
0.152
0.930
0.935
0.941
0.946
0.952
0.958
0.964
0.970
0.976
0.983
0.989
0.996
1.002
1.009
1.016
1.023
1.030
1.037
1.044
1.052
1.059
1.067
1.074
1.082
1.090
1.098
1.107
1.115
1.123
1.132
1.141
1.149
1.158
1.167
1.176
1.185
1.195
1.204
1.214
1.223
1.233
1.243
1.253
1.263
1.273
1.283
0.217
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
0.274
1.283
1.293
1.303
1.313
1.324
1.334
1.344
1.355
1.365
1.375
1.386
1.396
1.406
1.417
1.427
1.437
1.447
1.457
1.467
1.476
1.486
1.495
1.504
1.513
1.522
1.530
1.538
1.546
1.554
1.561
1.568
1.575
1.581
1.587
1.592
1.597
1.602
1.606
1.610
1.613
1.615
1.618
1.619
1.621
1.621
1.622
0.451
0.137
0.137
0.137
0.137
0.137
0.137
0.137
0.137
0.137
0.138
0.138
0.138
0.138
0.138
0.138
0.139
0.139
0.139
0.139
0.140
0.140
0.140
0.141
0.141
0.141
0.142
0.142
0.142
0.143
0.143
0.144
0.144
0.145
0.145
0.146
0.146
0.147
0.147
0.148
0.149
0.149
0.150
0.151
0.151
0.152
0.110
0.111
0.111
0.112
0.112
0.113
0.113
0.114
0.115
0.115
0.116
0.117
0.117
0.118
0.119
0.120
0.120
0.121
0.122
0.123
0.124
0.124
0.125
0.126
0.127
0.128
0.129
0.130
0.131
0.132
0.133
0.135
0.136
0.137
0.138
0.139
0.141
0.142
0.143
0.145
0.146
0.147
0.149
0.150
0.152
0.153
0.154
0.154
0.155
0.156
0.157
0.158
0.159
0.160
0.161
0.162
0.163
0.164
0.165
0.166
0.167
0.168
0.169
0.170
0.172
0.173
0.174
0.176
0.177
0.179
0.180
0.182
0.183
0.185
0.186
0.188
0.190
0.192
0.193
0.195
0.197
0.199
0.201
0.203
0.206
0.208
0.210
0.212
0.215
0.217
0.153
0.155
0.157
0.158
0.160
0.162
0.164
0.165
0.167
0.169
0.171
0.173
0.175
0.178
0.180
0.182
0.184
0.187
0.189
0.192
0.194
0.197
0.199
0.202
0.205
0.208
0.210
0.213
0.216
0.219
0.223
0.226
0.229
0.232
0.236
0.239
0.243
0.246
0.250
0.254
0.258
0.262
0.265
0.269
0.274
0.220
0.222
0.225
0.228
0.231
0.234
0.237
0.240
0.243
0.246
0.250
0.253
0.257
0.261
0.264
0.268
0.272
0.276
0.281
0.285
0.290
0.294
0.299
0.304
0.309
0.315
0.320
0.326
0.331
0.337
0.343
0.350
0.356
0.363
0.370
0.377
0.384
0.392
0.400
0.408
0.416
0.424
0.433
0.442
0.451
0.278
0.282
0.286
0.290
0.295
0.299
0.304
0.308
0.313
0.317
0.322
0.326
0.331
0.335
0.340
0.345
0.349
0.354
0.358
0.363
0.367
0.371
0.376
0.380
0.384
0.388
0.392
0.396
0.399
0.403
0.406
0.409
0.412
0.415
0.417
0.420
0.422
0.424
0.425
0.427
0.428
0.429
0.429
0.430
0.430
0.461
0.471
0.481
0.491
0.501
0.512
0.523
0.534
0.546
0.558
0.570
0.582
0.594
0.607
0.620
0.633
0.646
0.659
0.672
0.686
0.699
0.713
0.726
0.739
0.753
0.766
0.779
0.791
0.804
0.816
0.827
0.838
0.849
0.858
0.868
0.876
0.884
0.891
0.897
0.903
0.907
0.911
0.913
0.914
0.915
Table 10.11: Deferential latitude correction factor for Mercury. µ is in degrees. Note that h̄(360◦ −µ) =
h̄(µ), and δh± (360◦ − µ) = δh± (µ).
Glossary
179
11 Glossary
Altitude: The angle subtended at the observer by the radius vector connecting a celestial object to an
observer on the earth’s surface, and the vector’s projection onto the horizontal plane. Object’s
above/below the horizon have positive/negative altitudes.
Altitude Circle: A great circle on the celestial sphere which passes through the local zenith at a given
observation site on the earth’s surface.
Anomaly: Any deviation in an orbit from uniform circular motion which is concentric with the central body.
Anomaly is also used as another word for angle.
Apocenter: Point on a Keplerian orbit which is furthest from the central body. If the central body is the
sun, then the apocenter is generally termed the aphelion. Likewise, if the central body is the earth,
then the apocenter is termed the apogee.
Arctic Circles: Two latitude circles on the earth’s surface which are equidistant from the equator. Above the
arctic circles, the sun never sets for part of the year, and never rises for part of the year.
Argument of Latitude: Angle subtended at the central body by the radius vectors connecting the central
body to the orbiting body, and the central body to the ascending node, in a Keplerian orbit.
Ascendent: Point on ecliptic circle which is ascending at any given time on the eastern horizon.
Ascending Node: Point on a Keplerian orbit at which the orbital plane crosses the ecliptic plane from
south to north in the direction of motion of the orbiting body.
Autumnal Equinox: The point at which the ecliptic circle crosses the celestial equator from north to south
(in the direction of the sun’s apparent motion along the ecliptic).
Azimuth: Angle subtended at the observer by the projection of the vector connecting a celestial object to
an observer on the earth’s surface onto the horizontal plane, and the vector connecting the north
compass point to the observer. Azimuth increases clockwise (i.e., from the north to the east) looking
at the horizontal plane from above.
Celestial Axis: An imaginary extension of the earth’s axis of rotation which pierces the celestial sphere at
the two celestial poles. The sphere’s diurnal motion is about this axis.
Celestial Coordinates: Angular coordinate system whose fundamental plane is the celestial plane, and
whose poles are the celestial poles. The polar and azimuthal angles in this system are called declination and right ascension, respectively.
Celestial Equator: The intersection of the imagined extension of the earth’s equatorial plane with the
celestial sphere.
Celestial Plane: The plane containing the earth’s equator.
Celestial Poles: The two points at which the celestial axis pierces the celestial sphere. The north celestial
pole lies to the north of the celestial plane, whereas the south celestial pole lies to the south. The
celestial poles are the only two points on the celestial sphere whose positions are unaffected by diurnal
motion.
Celestial Sphere: An imaginary sphere of infinite radius which is concentric with the earth. All objects in
the sky are thought of as attached to this sphere.
Compass Points: At a given observation site on the earth’s surface, the north, east, south, and west compass
points lie on the local horizon due north, east, south, and west, respectively, of the observer.
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Conjunction: Two celestial objects are said to be in conjunction when they have the same ecliptic longitude. For an inferior planet in conjunction with the sun, the conjunction is said to be superior if the
planet is further from the earth than the sun, and inferior if the sun is further from the earth than the
planet.
Culmination: A celestial object is said to culminate on a given day when it attains its maximum altitude in
the sky.
Declination: Angle subtended at the earth’s center by the radius vector connecting a celestial object to the
earth’s center, and the vector’s projection onto the celestial plane. Object’s to the north/south of the
celestial equator have positive/negative declinations.
Deferent: Large circle centered on the sun about which the guide point rotates in a geocentric planetary
orbit.
Deferential Latitude: Ecliptic latitude a superior planet has by virtue of the inclination of its deferent.
Deferential Latitude Correction Factor: Correction to the ecliptic latitude of an inferior planet due to
the finite size of its deferent.
Diurnal Motion: Daily rotation of the celestial sphere, and the objects attached to it, from east to west
(looking south in the earth’s northern hemisphere) about the celestial axis.
Eccentricity: Measure of the displacement along the major axis of the central body from the geometric
center in a Keplerian orbit.
Ecliptic Axis: Normal to the ecliptic plane which passes through the center of the earth.
Ecliptic Circle: Apparent path traced out by the sun on the celestial sphere during the course of a year.
Ecliptic Coordinates: Angular coordinate system whose fundamental plane is the ecliptic plane, and whose
poles are the ecliptic poles.
Ecliptic Latitude: Angle subtended at the earth’s center by the radius vector connecting a celestial object to
the earth’s center, and the vector’s projection onto the ecliptic plane. Objects to the north/south of
the ecliptic circle have positive/ecliptic latitudes.
Ecliptic Longitude: Angle subtended at the earth’s center by the projection of the vector connecting a celestial object to the earth’s center onto the ecliptic plane, and the vector connecting the vernal equinox
to the earth’s center. Ecliptic longitude increases counter-clockwise (i.e., from the west to the east)
looking at the ecliptic plane from the north.
Ecliptic Plane: Plane containing the mean orbit of the earth about the sun.
Ecliptic Poles: The two points at which the ecliptic axis pierces the celestial sphere. The north ecliptic
pole lies to the north of the ecliptic plane, whereas the south ecliptic pole lies to the south.
Elongation: Difference in ecliptic longitude between two celestial objects.
Epicycle: Small circle, centered on the guide point, about which a planet rotates in a geocentric planetary
orbit.
Epicyclic Anomaly: Angle subtended between the radius vectors connecting the earth to the guide-point,
and the guide-point to the planet, in a geocentric planetary orbit.
Epicyclic Latitude: Ecliptic latitude an inferior planet has by virtue of the inclination of its epicycle.
Epicyclic Latitude Correction Factor: Correction to the ecliptic latitude of a superior planet due to the
finite size of its epicycle.
Epoch: Standard time at which the orbital elements of an orbiting body in the solar system are specified.
Glossary
181
Equant: Point about which the orbiting body appears to rotate uniformly in a Keplerian orbit of low eccentricity. The equant is diagrammatically opposite the central body with respect to the geometric center
of the orbit.
Equation of Center: Difference between the true anomaly and the mean anomaly in a Keplerian orbit.
Equation of Epicycle: Elongation of a planet from its guide-point in a geocentric planetary orbit.
Equation of Time: Time interval between local noon and mean local noon.
Equinoxes: The two opposite points on the ecliptic circle which the sun reaches on the days of the year
that day and night are equally long.
Evection: An anomaly of the moon’s orbit about the earth which is associated with the perturbing influence
of the sun.
Geocentric Planetary Orbit: An orbit in which a planet rotates about a guide point in a small circle called
an epicycle, and the guide point rotates about the earth in a large circle called a deferent.
Great Circle: Circle on the surface of a sphere produced by the intersection of a plane which bisects the
sphere.
Greatest Elongation: Greatest elongation of an inferior planet from the sun. If the planet is to the east/west
of the sun then the elongation is called the greatest eastern/western elongation.
Guide-Point: Center of an epicycle in a geocentric planetary orbit.
Horizon: Tangent plane to the earth’s surface, at a given observation site, which divides the celestial sphere
into visible and invisible hemispheres.
Horizontal Coordinates: Angular coordinate system whose fundamental plane is the horizontal plane,
and whose poles are the zenith and nadir.
Horizontal Plane: Plane containing the local horizon.
Horoscope: Point on the ecliptic circle which is ascending at a given time on the eastern horizon.
Inclination: Maximum angle subtended between the plane of a Keplerian orbit and the ecliptic plane.
Inclination of Ecliptic: Inclination of the ecliptic plane to the equatorial plane.
Inferior Planet: A planet which is closer to the sun than the earth.
Julian Day Number: Number ascribed to a particular day in a scheme in which days are numbered consecutively from January 1, 4713 BCE, which is designated day zero. Julian days start at 12:00 UT.
Keplerian Orbit: Ellipse which is confocal with the central object. The radius vector connecting the central
and orbiting bodies sweeps out equal areas in equal time intervals.
Local Mean Noon: Instant in time at which the mean sun attains its upper transit.
Local Noon: Instant in time at which the sun attains its upper transit.
Longitude of Ascending Node: Angle subtended at the central body by the radius vectors connecting the
central body to the ascending node, and the central body to the vernal equinox, in a Keplerian orbit.
Longitude of Pericenter: Angle subtended at the central body by the radius vectors connecting the central
body to the pericenter, and the central body to the vernal equinox, in a Keplerian orbit.
Major Axis: Longest diameter which passes through the geometric center of a Keplerian orbit.
Major Radius: Half the length of the major axis of a Keplerian orbit.
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Mean Anomaly: Angle which would be subtended at the central body by the radius vectors connecting the
central body to the orbiting body, and the central body to the pericenter, in a Keplerian orbit, if the
orbiting body were to rotate about the central body with a uniform angular velocity.
Mean Argument of Latitude: Value the argument of latitude would have if the orbiting body in a Keplerian orbit were to rotate about the central body at a fixed angular velocity.
Mean Argument of Latitude at Epoch: Value of the mean argument of latitude of a Keplerian orbit at
the epoch.
Mean (Ecliptic) Longitude: Value the ecliptic longitude would have if the orbiting body in a Keplerian
orbit were to rotate about the central body at a fixed angular velocity.
Mean (Ecliptic) Longitude at Epoch: Value of the mean longitude of a Keplerian orbit at the epoch.
Mean Solar Day: Time interval between successive local mean noons.
Mean Solar Time: Time calculated using the mean sun.
Mean Sun: Fictitious body which travels around the celestial equator (from west to east looking south in
the earth’s northern hemisphere) at a uniform rate, and completes one orbit every tropical year.
Meridian Plane: Plane passing through the zenith and the north and south compass points at a given
observation site on the earth’s surface.
Minor Axis: The minor axis of a Keplerian orbit is the shortest diameter which passes through the geometric center.
Minor Radius: The minor radius of a Keplerian orbit is half the length of the minor axis.
Nadir: Point on the celestial sphere which is directly underfoot at a given observation site on the earth’s
surface.
Opposition: Two celestial objects are said to be in opposition when their ecliptic longitudes differ by 180◦ .
Orbital Elements: Eight quantities which completely specify a Keplerian orbit: i.e., major radius, eccentricity, rate of motion of mean longitude, rate of motion of mean anomaly, mean longitude at
epoch, mean anomaly at epoch, inclination, rate of motion in mean argument of latitude, mean
argument of latitude at epoch.
Parallactic Angle: Angle subtended between the ecliptic circle and an altitude circle.
Parallax: Apparent change in position of a nearby celestial object in the sky when it is viewed at different
points on the earth’s surface.
Pericenter: Point on a Keplerian orbit which is closest to the central body. If the central body is the sun,
then the pericenter is generally termed the perihelion. Likewise, if the central body is the earth, then
the pericenter is termed the perigee.
Precession of Equinoxes: A slow movement of the vernal equinox relative to the fixed stars which causes
the ecliptic longitude of a fixed star to increase steadily at the rate of 50.3 ′′ per year.
Prograde Motion: Motion of a superior planet in the sky in the same direction to that of its mean motion.
Radial Anomaly: Difference between the length of the radius vector connecting the central body to the
orbiting body, in a Keplerian orbit, and the major radius.
Rate of Motion in Mean Anomaly: Time derivative of the mean anomaly of a Keplerian orbit.
Rate of Motion in Mean Argument of Latitude: Time derivative of the mean argument of latitude of a
Keplerian orbit.
Rate of Motion in Mean Longitude: Time derivative of the mean longitude of a Keplerian orbit.
Glossary
183
Retrograde Motion: Motion of a superior planet in the sky in the opposite direction to that of its mean
motion.
Right Ascension: Angle subtended at the earth’s center by the projection of the vector connecting a celestial
body to the earth’s center onto the celestial plane, and the vector connecting the vernal equinox to
the earth’s center. Right ascension increases counter-clockwise (i.e., from the west to the east) looking
at the celestial plane from the north.
Seasons: Spring is the time interval between the vernal equinox and the summer solstice, summer the
interval between the summer solstice and the autumnal equinox, autumn the interval between the
autumnal equinox and the winter solstice, and winter the interval between the winter solstice and
the next spring equinox.
Sidereal Day: Time interval between successive upper transits of a fixed star.
Sidereal Time: Time calculated using the fixed stars.
Solar Day: Time interval between successive local noons.
Solar Time: Time calculated using the sun.
Solstices: The two opposite points on the ecliptic circle which the sun reaches on the longest and shortest
days of the year.
Station: Point in the orbit of a superior planet at which it switches from prograde to retrograde motion,
or vice versa. The former station is called a retrograde station, whereas the latter is called a prograde
station.
Summer Solstice: Most northerly point on the ecliptic circle.
Superior Planet: A planet further from the sun than the earth.
Synodic Month: Mean time interval between successive new moons.
Syzygy: Conjunction or opposition of the sun and the moon.
Transit: On a given day, and at a given observation site on the earth’s surface, a celestial object is said
to transit when it crosses the meridian plane. The object simultaneously attains either its highest
or lowest altitude in the sky. The transit is called an upper/lower transit when the object attains its
highest/lowest altitude.
True Anomaly: Angle subtended at the central body by the radius vectors connecting the central body to
the orbiting body, and the central body to the pericenter, in a Keplerian orbit.
Tropical Year: Time interval between successive vernal equinoxes.
Tropics: Two latitude circles on the earth’s surface which are equidistant from the equator. Between the
tropics the sun culminates both to the north and south of the zenith during the course of a year.
Outside the tropics, the sun culminates either only to the north or only to the south of the zenith.
Universal Time: Time defined such that mean local noon coincides with 12:00 UT every day at an observation site of terrestrial longitude 0◦ .
Vernal Equinox: Point at which the ecliptic circle crosses the celestial equator from south to north (in the
direction of the sun’s apparent motion along the ecliptic).
Winter Solstice: Most southerly point on the ecliptic circle.
Zenith: Point on the celestial sphere which is directly overhead at a given observation site on the earth’s
surface.
Zodiac: The signs of the zodiac are conventional names given to 30◦ segments of the ecliptic circle.
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Index of Symbols
185
12 Index of Symbols
A: azimuth.
MM : mean anomaly of moon.
a: major radius, altitude.
MS : mean anomaly of sun.
aS : major radius of sun.
µ: epicyclic anomaly, parallactic angle.
α: right ascension.
n: rate of motion in mean longitude.
ᾱ: right ascension of mean sun.
ñ: rate of motion in mean anomaly.
b: minor radius.
n̆: rate of motion in mean argument of latitude.
β: ecliptic latitude.
q: equation of center.
D: lunar-solar elongation.
qi : lunar anomalies.
D̄: mean lunar-solar elongation.
RE : radius of earth.
D̃: semi-mean lunar-solar elongation.
RM : radius of moon.
δ: declination.
RS : radius of sun.
δM : parallax of moon.
r: radial distance.
e: eccentricity.
ρS : angular radius of sun.
eM : eccentricity of moon.
ρM : angular radius of moon.
eS : eccentricity of sun.
ρU : angular radius of earth’s umbra.
ǫ: inclination of ecliptic.
t: time.
E: elliptic anomaly.
t0 : epoch.
ζ: radial anomaly.
T : true anomaly.
θ: equation of epicycle.
τ: orbital period.
F: argument of latitude.
̟: longitude of perigee.
F̄: mean argument of latitude.
Ω: longitude of ascending node.
F̄0 : mean argument of latitude at epoch.
F̄M : mean argument of latitude of moon.
λ: ecliptic longitude.
λS : ecliptic longitude of sun.
λ̄: mean longitude.
λ̄0 : mean longitude at epoch.
λ̄M : mean longitude of moon.
λ̄S : mean longitude of sun.
L: terrestrial latitude.
M: mean anomaly.
M0 : mean anomaly at epoch.
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Bibliography
187
13 Bibliography
Primary Sources:
The Almagest, C. Ptolemy, tr. G.J. Toomer (Princeton University Press, 1998).
On the Revolutions: Nicholas Copernicus Complete Works, N. Copernicus, tr. E. Rosen (The Johns
Hopkins University Press, 1992).
Selections From Kepler’s Astronomia Nova, J. Kepler, tr. W.H. Donahue (Green Lion Press, 2005).
Secondary Sources:
The Exact Sciences in Antiquity, O. Neugebauer (Dover, 1969).
A Survey of the Almagest, O. Pedersen (Univ Press of Southern Denmark, 1974).
The History and Practice of Ancient Astronomy, J. Evans (Oxford University Press, 1998).
The Eye of Heaven: Ptolemy, Copernicus, Kepler, O. Gingerich (American Institute of Physics, 1993).
Astronomical Algorithms, J. Meeus (Willmann-Bell, 1998).
The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure
of Dynamical Theories, J. Barbour (Oxford University Press, 2001).