Al-Bīrunī And The Planet Mercury

In The Name of God The Merciful The Compassionate
Al-Bīrunī
And
The Planet Mercury
An introduction to Al-Bīrunī’s corrections on planet Mercury’s motion in space
Mohammad Hossein Jamshidi
Nima Ronaghi
Sharif University of Technology Astronomy Group
Guide Professor: Jafar Chavoshi
University of Paris VII (Denis Diderot) History and Epistemology of science
Sharif University of Technology Philosophy of Science Group
1
Copyright 2012 SUT Astronomy Group, Physics Department, Sharif Univ. of Technology, Tehran, Iran
Email: [email protected]
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, electronic, mechanical, photocopying,
recording, scanning or otherwise, except under the terms of the Copyright. Requests to the
Publisher should be addressed to the Astronomy Group, Physics Department, Sharif Univ. of
Technology, Tehran, Iran, or emailed to [email protected]
2
Dedicated to
Holiness Abd Al-Azim Holy Shrine
3
0. Abstract:
From years ago, the strange and maybe unpredictable motions of heavens have
surprised people. The curiosity of man made him try hard to figure out what is
happening overhead, which was an appropriate motive for modeling those complex
motions.
The science of the ancient civilizations was weak compared to modern science
because it contained no real understanding of dynamics. In retrospect, we see that
two stumbling blocks were responsible for this shortcoming. One lay in the
concepts of "mass" and "force," as being determined through departures from
uniform motion. The other lay in the mode of description of the planetary motions.
Here we will first have a review of planetary models from the ancient Greeks up
to now, and then we shall discuss about Islamic astronomers - especially Al-Bīrunī
because of his high accuracy in measuring the angles and positions - planetary
models and what they have done to derive these models. For a more precise
inspection, we will compare our simulation program data with those of Al- Bīrunī
for Mercury.
4
Contents
0. Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. A Brief History of Planetary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
2.1 Geocentric models era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Heliocentric models era . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Modern model for planetary motion . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Where were Al-Bīrunī and other Islamic scientists? . . . . . . . . . . .17
3. Al-Bīrunī’s Correction on Mercury’s Motion in Space. . . . . . . . . . . . . .19
4. The Simulation Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
4.1 Modern planetary model simulation . . . . . . . . . . . . . . . . . . . . . . . . .21
4.2 Al-Bīrunī’s planetary model simulation. . . . . . . . . . . . . . . . . . . . . . 22
5. Epistemological Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6. Appendix: Deriving Kepler laws using Newtonian mechanics. . . . . . . 28
7. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5
1. Introduction
The science of the ancient civilizations was weak compared to modern
science because it contained no real understanding of dynamics. In retrospect,
we see that two stumbling blocks were responsible for this shortcoming. One
lay in the concepts of "mass" and "force," as being determined through
departures from uniform motion. The other lay in the mode of description of the
planetary motions. Let it be understood at the outset that it makes no difference,
from the point of view of describing planetary motions, whether we take the
Earth or the Sun as the center of solar system. Since the issue is one of relative
motion only, there are infinitely many exactly equivalent descriptions referred
to different centers, as we shall see. Here we will first have a review of
planetary models from the ancient Greeks up to now, and then we shall discuss
about Islamic astronomers - especially Al-Bīrunī because of his high accuracy
in measuring the angles and positions - planetary models and what they exactly
done to derive these models. For a more precise inspection, we will compare
our simulation program data with those of Al- Bīrunī for Mercury.
6
2. A Brief History of Planetary Models
From years ago, the strange and maybe unpredictable motions of heavens
have surprised people. The curiosity of man made him try hard to figure out
what is happening overhead, which was an appropriate motive for modeling
those complex motions.
Let us start from the ancient Greeks models of planetary motions to see
what happened in the minds of their developers and how they developed.
2.1
Geocentric models era
There was little chance of ancient astronomers’ arriving at a tolerable picture
of the planetary motions until it had been realized that the Earth is essentially
spherical in shape. Two observations, which led to this conclusion, are worth
mentioning. When one journeys south, new stars not previously visible appear
above the southern horizon, and stars dip lower toward the northern horizon.
The Greeks knew long before Hipparchus that the constellation of the Plow
comes down to the northern horizon when seen from Egypt but not when seen
from Greece. Assuming the stars to be distant objects, this shows that the Earth
is not flat.
More directly, when the Earth comes between the Sun and the Moon, the
shadow that the Earth throws onto the Moon is seen to be circular in shape. By
judging the apparent radius of the shadow, it is possible to determine the ratio
of the distance to the Moon to the radius of the Earth, and Hipparchus
succeeded in obtaining a surprisingly good estimate this way (the ratio is about
60). The spherical shape of the Earth was understood and accepted by the
Greeks from the time of Plato, so there was no impediment to later astronomers
on this score.
The diurnal rotation of the heavens is the most obvious motion to be fitted
into our picture. It makes little difference from a purely kinematical point of
view whether we consider the Earth to be fixed, with the heavens rotating, or
vice versa, although dynamically it would be awkward (but not impossible) to
think in terms of a rotation of the heavens. Indeed, kinematically it is somewhat
simpler, at any rate to begin with, to think in terms of the heavens as rotating.
We have the situation shown in Fig. 2.1, where we think of the stars being
projected onto a distant sphere with the center taken at the Earth, the celestial
sphere. The celestial sphere is considered to rotate about an axis directed
toward the pole P. The circle in which the plane through the center or the Earth
taken perpendicular to the polar axis PP1 cuts the celestial sphere is called the
7
celestial equator. The angle, which, a line drawn from our particular position on
the surface of the Earth to the center makes with this plane, is called our
latitude.
FIGURE 2.1 The determination of latitude and longitude on the celestial sphere. The
latitude of the point X is the angle which the circular arc XB subtends at the Earth E; the
longitude is the angle which AB subtends at E, A being a fixed reference point.
We can obviously set up a system of latitude and longitude on the celestial
sphere as well as on the surface of the Earth. This is what astronomers do in
order to describe the positions of stars and other objects.
Longitudes are reckoned with respect to a standard position, just as
longitudes on the earth are reckoned with respect to a standard place
(Greenwich). Notice that the rotation of the heavens does not affect latitudes
and longitudes on the celestial sphere because the standard position is taken to
partake in the diurnal motion.
For the Sun, Moon and planets we determine latitudes and longitudes in the
same way, but in these cases the values change with time. If, at any particular
moment, a planet, or the Sun or Moon, happens to be very close to a particular
star, we assign essentially the same latitude and longitude to it that we do to the
star, but because the planet or the Moon or Sun, as the case may be, is found to
move with respect to the stars we have to be constantly changing the assigned
latitudes and longitudes. The determination of how the latitudes and longitudes
change with time is precisely the problem before us.
Of all such cases, it is easiest to observe the motion of the Moon with respect
to the stars. This shows up after a few nights of casual observation. If followed
for a month or two, the moon is found to move along a path on the celestial
sphere of the kind shown in Fig. 2.2. Careful observation over many years
shows, however, that the Moon does not quite retrace its path from one month
to the next. Starting with a particular path in a particular month, it takes 18.61
8
years before the Moon returns to that same path, and in this time the angle
which the plane of the path makes with the plane of the celestial equator goes
through all values from about 19° to 29° , and back again to the starting value.
Stone Age man probably already knew this complex behavior. It plays a critical
role in the occurrence of eclipses and for this reason has been important to
astronomers throughout the ages. It must have seemed just as important to
explain this curious behavior of the Moon as it seemed to explain the motions
of the planets. Yet whereas the planetary problem lay reasonably within the
grasp of the Moon was not accessible to them, or to Copernicus or to Kepler.
The broad features of the lunar motion had to await Newton’s theory, while the
fine details were not subject to analysis until the nineteenth century – indeed,
there are problems still extant.
FIGURE 2.2 In addition to sharing the diurnal rotation of the celestial sphere, the Moon
follows a path among the stars which it completes in a lunar month.
The early astronomers could not know which problems they could hope to
solve. Perforce they had to take a shot at everything. Moreover, because
insoluble problems were mixed up with soluble ones their task in dealing with
the soluble ones was made all the harder. This must always be remembered in
attempting to understand the difficulties that beset Ptolemy and Copernicus.
Both expended much effort in attempting to understand the Moon, whereas they
would probably have gone further with less effort if they had ignored this
problem. Moon, whereas they would probably have gone further with less effort
if they had ignored this problem.
The Sun is harder to observe than the Moon, because the glare of the sun
prevents the background of stars from being seen in full sunlight. Nevertheless,
by observing stars close to the Sun immediately before sunrise and immediately
after sunset it is possible to make a determination of the path of the Sun on the
celestial sphere. It is found that the latitude and longitude of the Sun also
change from day to day, and that the Sun returns to its original position in a
9
year. The path followed by the Sun does not change significantly from year to
year, so fortunately we do not have to face the complexities, which bedevil the
motion of the Moon. The path of the Sun projected on the celestial sphere is
shown in Fig 2.3. The plane of path makes an angle of about 23.5° with the
plane of the celestial equator. It is this tilt, which explains the seasons of the
year. For the Northern Hemisphere of the Earth there is midwinter when the
Sun is at B, spring at γ, summer at A, and autumn at γˊ . The points γ and γˊ
are the nodes of the Sun’s orbit, γ being called the first point of Aries.
Longitudes of stars on the celestial sphere, as well as longitudes of planets and
of the Sun and Moon, are reckoned with respect to γ.
FIGURE 2.3 The Sun follows an annual path among the stars, the plane of the path making
an angle of about 23.5° with the plane of the equator.
It makes no difference, kinematically, whether we think of the Sun’s
observed path among the stars as being due to a motion of the earth around the
Sun or of the Sun around the Earth. For the moment, then, let us follow the old
belief that the Sun moves around the Earth. The path drawn in Figure 6 is a
circle, and indeed must not be a circle because, if it were, summer and winter
would be equal, whereas the seasons are found to be unequal. It was already
known to the Greeks that spring-to-summer-to-autumn differs from autumn-towinter-to-spring by three days. Indeed, this fact was probably known long
before the Greeks. It was explained by Hipparchus in the manner of Fig. 2.4(a),
where the point X moves uniformly in a circle around the Earth in a year. The
Sun S is fastened to X by a short stick (not necessarily a real stick) which
maintains a constant direction throughout the annual motion of X. Ugly and
implausible or not, the Hipparchus theory grapples with the facts whereas the
circular picture of Aristarchus fails to do so. It will be clear then why it was so
hard to arrive at the basic elements of a heliocentric theory. The two most
striking bodies in the sky, the Sun and Moon, cause difficulties at the outset,
11
even before we come to the planets. To make progress we have to be
courageous enough (or foolhardy enough) to ignore the complexities of the
Moon’s orbit and also the known inequality of seasons. As an aside, before we
proceed, it may be of interest to compare the Hipparchus theory of Fig. 2.4(a)
with Fig. 2.5(b), which gives the modern view of the inequality of the seasons.
(a)
(b)
FIGURE 2.4 (a) Hipparchus’ theory of the seasons. As X moves around the circle,
the segment drawn from X to the Sun S makes a constant angle with a fixed
direction. (b) P is perihelion point, A is aphelion, and γ is the first point of Aries.
You may wonder how a knowledge of the planetary motions can possibly
make things any easier. Actually it does, as we shall see in the remainder of this
section. First, there is the welcome simplification that the orbits of the planets
are all more or less in the same plane as the path of the Sun.
The first step in breaking into the problem is to consider the observations of
Venus. Venus sometimes runs ahead of the Sun in its path and sometimes it
lags behind, in a kind of oscillatory motion, which we might try to represent in
the manner of Fig. 2.5, with Venus swinging in harmonic motion along a line
through the Sun. In the part of the cycle from R to L, Venus would move more
rapidly forward than the Sun, from L to R it would move less rapidly. At R the
planet would be behind the Sun, at L it would be ahead. This picture therefore
corresponds qualitatively with the observational situation. It fails, however,
when we take account of quantitative aspects of the data. On the model of Fig.2.5, we would expect variations in the brightness of Venus in the half-cycle
from L to R to be exactly matched by similar variations in the opposite haltcycle from R to L. This is incorrect. The two parts, L to R and R to L, are not
11
the same, suggesting that the distance from the Earth to Venus is not the same.
Nor, indeed, is the time of swing L to R equal to the time from R to L, as it
would be for simple harmonic motion.
FIGURE 2.5 The apparent motion of Venus, as seen from the Earth, consists of an
oscillation backward from L to R, then forward from R to L.
A more sophisticated picture of the motion of Venus is shown in Fig. 2.6. We
now have a situation in which the sun S moves uniformly in a circle around the
Earth (ignoring the inequality of the seasons) and in which Venus V moves
uniformly around the Sun in another circle. At point L of this second circle
Venus is moving directly toward the earth, at R Venus is moving away from the
Earth. The situation projected on the sky is like Fig. 2.5, but now the distance of
Venus from the Earth is different over the sections L to R and R to l, and the
times are also different. In addition, we have avoided the embarrassment that
Fig. 2.5 would require Venus to pass through the Sun.
FIGURE 2.6 This model of the motion of Venus provided a first breakthrough in the
understanding of the planetary motions. When projected onto the celestial sphere it gives the
apparent oscillation of Fig. 2.5.
12
The relative-size relation of the two circles of Fig. 2.6 is readily determined
by measuring the angle between the direction of the Sun and the direction of
Venus at the moment when Venus is most ahead of the Sun. A similar angle
when Venus is most behind the Sun can also be measured, and the sum of these
two angles is equal to the angle LER of Fig. 2.6. With this known, the ratio of
the radius SV of the moving circle to the radius ES can be determined. It turns
out to be about 0.723. The time for the motion L to R added to the time back
again from R to L gives the synodic period of Venus in its orbit, 1.599 years.
We now have a model with predictive capacity. Although designed to fit a
set of known data, it can be used to explain further new sets of data. This is a
crucial requirement of a useful theory. Theories which only explain already
known facts usually end up in a blind alley. The eccentricity of the orbit of
Venus happens to be very small, and the eccentricity or the Earth’s orbit is
reasonably small. Therefore, the actual orbits are nearly circle, which means
that Fig. 2.6 is a good approximation to the real situation. This regularity was
the first to be discovered among the planets. It was known to Heraclides and
was therefore available to Hipparchus and Ptolemy as well as to Aristarchus.
From a code-cracking point of view, the case of Venus is a breakthrough.
The situation for Mercury is similar to that for Venus, but because of the
large eccentricity of the orbit of Mercury the approximation given by a simple
model like Fig. 2.6 is not nearly so good. Fortunately, the shortcomings of a
simple model were not so obvious in really ancient times, because reasonably
accurate observations were not easy to make on account of the closeness of
Mercury to the Sun. However, the discrepancies were well known to Ptolemy,
who had considerable difficulty in improving on Fig. 2.6.
The outer planets behave in a way that is both curiously similar to end
curiously different from the case of Venus. Imagine the Sun removed from Fig.
2.6. We should not then be able to compare observationally the direction E to V
with the direction of Sun in the same way as before. We should be aware,
however, that the arcs L to R and R to L were very different, because over the
arc L to R the planet would appear to go backward in the sky, while over R to L
it would appear to go forward. This alternation of backward and forward
motions, with the forward motions winning on balance, is just what is observed
for the outer planets. We look therefore to a model of the kind drawn in Fig.
2.7, but with C now only a geometrical point, since the outer planets are not
observed to oscillate about the position of the Sun. As in the case of Venus, we
seek to find the angle LER by comparing the lengths of the backward and
13
forward motions. The matter is not so straightforward as before, however,
because C is not a visible point. Nevertheless, from a thorough analysis of the
data the angle LER can be determined. This again fixes the ratio of the radius
CP of the small circle to the radius EC of the large circle. And the periods of
time in which C moves around the large circle and P about the small circle can
be obtained from a suitably complete set of data. Once again we arrive at a
model with predictive capacity. The results turn out to be reasonably good for
Jupiter and Saturn, but much less good for Mars-partly because the eccentricity
of the actual orbit of Mars is about twice as great as the eccentricities for Jupiter
and Saturn, and partly because the Earth and Mars approach each other much
more closely.
FIGURE 2.7 This model applies to the motions of planets more distant from the Sun than
the Earth. It is like Fig. 2.6 for Venus, except that the Sun is no longer at the center of the
small circle.
2.2
Heliocentric models era
So far, the route toward understanding the motions of the planets is clearly
marked-with the solar system being the way it is, there is no other way of
making a sensible start on the problem. But now we have alternative routes
ahead. Which we elect to follow will depend on our temperament. If we are
overwhelmingly concerned with the predictive quality of our model (which
would be the point of view of most modern theoretical physicists), we shall
seek to modify Fig. 2.7 in order to secure a better correspondence between
theory and observation. This is just what Ptolemy sought to do. But if we are
concerned not so much with quantitative accuracy as with the qualitative
structure of the problem, we shall be unable to overlook two further strange
aspects of the situation.
14
Its turns out that P in Fig. 2.7 is required to move around the small circle in a
year, the same period as S moves around E in Fig. 2.6. But why should the Sun
be involved with Fig. 2.7 at all? Here we are concerned with the relation of the
planet p, whether Mars, Jupiter, or Saturn, to the Earth. Indeed, in Fig. 2.7 we
were required to remove the Sun from the center of the small circle. Still more
curious perhaps, if we ask where the Sun should be placed with respect to Fig.
2.7, it turns out as in Fig. 2.8: from observation it is found that a line drawn
from the Earth in the direction of the Sun is always parallel to the joining C to
P.
FIGURE 2.8 The Sun is observed to lie along a line through E drawn parallel to CP. The
point K is chosen so that EK=CP.
Clearly, this must mean something, but we cannot decide what from
observation alone. Indeed, if observation had resolved this issue, there would
have been no controversy in astronomical history between geocentric and
heliocentric theories. To distill meaning out of these curious facts we must take
a theoretical step. Somewhere along the line EK of Fig. 2.8 lies the Sun, but
observation does not tell us where (at any rate observations available to the
Greeks or to Copernicus). The inspired step taken by Aristarchus, and later
independently by Copernicus, was to place the Sun at K, where K is chosen
such that the length E to K is equal to the length C to P. It is worth redwing
Fig. 2.8 with the Sun S placed firmly in this position , as in Fig. 2.9(a) is
exactly the same as that described in Fig. 2.9(b) , where the planet P now goes
in a larger circle about S.
It may take the reader a moment or two to become satisfied that Figures
2.9(a) and 2.9(b) do indeed give the same motion of P. The equivalence of
these two pictures was already known to Apollonius, who lived in the third
century B.C., long before Ptolemy (150 A.D.).
15
(a)
(b)
FIGURE 2.9 (a) This is the same diagram as Fig. 2.8, but with the Sun placed by
hypothesis at the point K. (b) The planet P now moves in a circle about S, while S moves in a
circle about E. The motion of P is exactly the same as in Fig. 2.8.
We have a construction of the form of Fig. 2.6 for an inner planet and one of
the forms Fig. 2.9(b) for an outer planet. If they are drawn together, as in Fig.
2.10(a) we can at last take advantage of the fact that the motion of s around E
is kinematically equivalent to the motion of E around S. Thus, Fig. 2.10(a) is
equivalent to Fig. 2.10(b). The respective circles have the same periods of
motion in corresponding circles the relative positions of E, S, V, P must be the
same in the two pictures of all times. Here P can represent either Venus or
Mercury. This is the heliocentric theory of Aristarchus, the theory which
Copernicus was so faithful to.
(a)
(b)
FIGURE 2.2 (a) This combines Figures 2.6 and 2.9(b), with V representing an inner planet
and P representing. (b) The motions in this heliocentric picture are exactly the same as in Fig.
2.10(a).
16
2.3
Modern model for planetary motion
Our modern knowledge from the solar system is so close to Kepler's
model, so we may consider Kepler's model as the modern model for planetary
motions.
The Copernican theory fitted the planets of small eccentricity – Venus, Earth,
Jupiter, and Saturn – very well, to within the errors of observation in some
cases, so there had to be something "Right" about it. The theory thus provided
a base from which to proceed. Progress was certain as soon as a man, or men,
of adequate stature came along. As we know in retrospect, favorable
circumstances came with the observations of Tycho Brahe, and with
appearance of Kepler. Kepler fitted the task of advancing the theory, not only
in stature, but also in the demonic energy with which he tackled the problem of
Mars. The sheer volume of calculation, which he carried through, fits him to be
called the iron man of science.
In the beginning, Kepler contented himself with setting right one or two
minor blemishes in the Copernican theory. Finding this did help the problem of
Mars, he then began a long series of trial calculations, which in effect were a
search, along Copernican lines, for the missing second-order terms in the
planetary eccentricities. Kepler achieved improvements, but not complete
success, and always at the expense of increasing complexity.
Kepler and his successors might well have gone on in this style for
generations without arriving at a satisfactory final solution, for a reason we
now understand clearly. There is no simple mathematical expression for the
way in which the direction of a planet changes with time. Even today, we must
express the longitude as an infinite series of terms when we use time as a free
variable.
After dealing with problem by some complex numerical methods, Kepler
released his triple laws known as Kepler Laws of Planetary Motion; here we
are going to mention them1. (See Fig. 2.2.)
 Kepler’s first law of planetary motion:
The orbit of a planet is an ellipse, one focus of which is in the Sun.
 Kepler’s second law of planetary motion:
1
All Kepler’s laws are proven in the appendix A.
17
The radius vector of a planet sweeps equal areas in equal amounts of time.
 Kepler’s third law of planetary motion:
The ratio of the cubes of the semimajor axes of the orbits of two planets is
Equal to the ratio of the squares of their orbital periods.
(a)
(b)
FIGURE 2.2 (a) r is planet’s position vector in polar coordinates, which changes with time.
(b) The areas of the shaded sectors of the ellipse are equal. According to Kepler’s second law,
it takes equal times to travel distances AB, CD and EF.
2.4
Where were Al-Bīrunī and other Islamic scientists?
“Islamic science at its very outset appears to be purely receptive and derived
from earlier models. But already at a very early stage, there is observable a
tendency to attain as comprehensive as possible a survey of the whole of Greek
scientific activity and reasoning. As concerns the motives underlying this
phenomenon, it may be true that practical aims prevail, and this has been
emphasized repeatedly. But, to those who prefer reading manuscripts to rereading what others have said about the matter, it becomes evident that the true
stimulus or incentive, already in early Abbasid times, perhaps even in the late
Ummayad period, has been a genuine desire for factual knowledge and
understanding. In fact, this desire, and the circumstances that Islamic scientists
never got tired of quenching their thirst for knowledge, is an essential
characteristic of medieval Islam.1”
As Willy Hartner and Matthias Schramm mentioned in their excellent book,
the most famous Islamic scientists - as well as Al-Bīrunī - lived between Greek
1
Al-Bīrunī and the theory of solar apogee, Willy Hartner and Matthias Schramm.
18
and renaissance scientists. However, the most interesting thing here is that in
some cases they had even more science than the renaissance scientists did. As
an example, we will see what Al-Bīrunī done to model the planet Mercury’s
motion, and then we will compare it with Kepler’s planetary model.
19
3. Al-Bīrunī’s Correction on Mercury’s Motion In Space
In this section, we will describe what Al-Bīrunī exactly wrote in his book
Qanun Al-Masudi about Mercury’s motion.
In the first chapter of third volume of Qanun Al-Masudi, Al-Bīrunī had
written:
“The motion of these five stars1 in the sky is a superposition of two motions:
first, displacement between East and West in conjunction with a raise in
Zodiac. Second, a displacement between North and South. … From these five,
the stars Venus and Mercury do not get high distances from the Sun and they
are so hot. … ”
Obviously, Al-Bīrunī – as well as Ptolemy – knew that inner planets rotate
around the Sun. And even he knew that they are so hot. However, what made
him to do these corrections just on Mercury’s motion?
The answer is simple: because of small eccentricity, we can approximately
assume Venus orbit as a circle. But against, Mercury has the largest eccentricity
of the solar system and its complex motion is easier to discover. But the real
problem with observing the Mercury is its closeness to Sun, which makes it so
hard to record its positions.
Al-Bīrunī done some exact observations on Mercury and by analyzing the
data, he understood that the Mercury’s motion is not as what Ptolemy said.
More after he released a correction of high accuracy on Mercury’s motion in
space.
He has written in his book Qanun Al-Masudi:
“The center of large circle (Carrier Heaven2) is not fixed and moves on a small
circle.”
This sentence carries a really deep and fundamental fact with itself, which AlBīrunī certainly knew. We will discuss more about this fact at the end.
1
2
Those are Mercury, Venus, Mars, Jupiter and Saturn.
Falak Al-Hamel
21
FIGURE 3.1 This is exactly what Al-Bīrunī drew. D is the center of the large circle,
the small circle DHT is the one that D rotates on.
Let us describe Al-Bīrunī’s method:
First, we draw the Carrier Heaven with center on D, and then we draw the diameter
ADTJ. After that, we divide DN into three equal pieces using points K and T. Then
we draw a circle with center on K and radius DK, this circle is the carrier of the
Carrier Heaven’s center.
Now assume that initially, when the center of the Carrier Heaven is on D, the
center of the Rotator Heaven1 is on A. As D moves to H, the Carrier Heaven will
look like MB (See Fig. 3.1). With a deeper look in Al-Bīrunī’s text, we get that the
point B rotates in the count clockwise direction and the point H rotates in the
clockwise direction.
1
Falak Al-Tadvir
21
4. The Simulation Program
This section is divided into two parts first is a simulation program for modern
planetary model, and the second is a simulation program for what Al-Bīrunī
did.
4.1
Modern planetary model simulation
This program is written by 3Ds MAX compiler and simulates what exactly
happening in our solar system. This program even takes minor gravitational
interactions between the planets into account.
If you enter the codes bellow in maxscript part of your Autodesk 3Ds MAX and
put the frames number f, 1000000, and turn on the Autokey you will see the
simulation for modern planetary model.
m=#()
for i=1 to 7 do append m (sphere name: (”m”+i as string) radius:10.0 segs:10.0)
M=#(seeslmeyratepplp)
dv=#()
for i=1 to 7 do append Dv [0,0,0]
V=#(sspimissp)
dt=0.01 ; G=6.67*10^-11
for i=1 to 7 do Dv[i]=0
while(SliderTime <1000000) do
(
for i=1 to 7 do Dv[i]=[0,0,0]
for i=1 to 7 do
(
for j=1 to 7 do
if(i !=j)do
dv[i]=Dv[i] +(G*M[j]/(distance m[i].pos m[j].pos)^2)*dt*normalize(m[j].posm[i].pos)
22
)
for i=1 to 7 do
(
V[i]=V[i] + dv[i]
m[i].pos = m[i].pos +dt*V[i]
)
SliderTime = SliderTime +1
)
In addition, if you would like to watch this high accurate model from POV 1 of a
geocentric observer you can add these codes:
P=#()
for i=1 to 6 do append P (point name: (“p”+i as string))
for i= 1 to 6 do P[i].pos =m[i].pos-m[7].pos
Note that the seventh mass is the planet Earth.
4.2
Al-Bīrunī’s planetary model simulation
Here you see the Autodesk 3Ds MAX compiler codes for Al-Bīrunī’s planetary
model simulation, if you enter the codes bellow in maxscript part of your Autodesk
3Ds MAX and put the frames number f, 10000, you will see the simulation for AlBīrunī’s planetary model.
sphere name:"markaz" radius:1.0 segs:30 wirecolor:(color 28 89 177)
sphere name:"sun" radius:2.0 segs:30 wirecolor:(color 226 204 57)
sphere name:"otared" radius:1.0 segs:30 wirecolor:(color 176 26 26)
FTradius = 57.9
FHradius = 149.397
KD = 0.602501
1
Point Of View
23
$markaz.pos.controller = position_script()
$markaz.pos.controller.script ="ct = 100*degTorad currentTime \n $markaz.pos =
KD*[cos(ct),sin(ct),0]"
$sun.pos.controller = position_script()
$sun.pos.controller.script ="ct = 100*degTorad currentTime \n $sun.pos =
FHradius*[cos(ct),-sin(ct),0]+$markaz.pos"
$otared.pos.controller=position_script()
$otared.pos.controller.script = "ct = 4.15009*100*degTorad currentTime \n
$otared.pos= FTradius*[-sin(ct),cos(ct),0]+$sun.pos"
circle name:"DM" radius:KD steps:15 wirecolor:(color 86 173 205)
circle name:"falak_hamel" radius:FHradius steps:15 wirecolor:(color 225 198 87)
circle name:"falak_tadwir" radius:FTradius steps:15 wirecolor:(color 135 6 6)
$falak_hamel.pos = $markaz.pos
$falak_tadwir.pos=$sun.pos
$falak_hamel.parent = $markaz
$falak_tadwir.parent = $sun
point name:"ovalF" pos:[(KD+FHradius)*(1-((KDFHradius)/(KD+FHradius))^2)^0.5,0,0] cross:true box:true
If you track the Sun’s motion around the center of Carrier Heaven, you will see an
ellipse shaped orbit, which is so surprising. In the next section, we will get a
surprising conclusion.
24
5. Epistemological Conclusion
We saw what Al-Bīrunī done to model the planet Mercury’s motion in the
sky. Moreover, using the simulation program we understood that Al-Bīrunī
knew that Sun orbits the Earth in an oval shaped orbit. But watching is not
enough for proving this fact; here, we will prove this using mathematical
methods.
As you know in a uniform circular motion like Fig. 5.1, we have these
equations:
Where r is the length of position vector of the object.
Therefore, the position in Cartesian coordinates is
̂
̂
Y
𝑟
𝜃
o
X
FIGURE 5.1 The object is having a uniform circular motion. is its position vector.
25
Now let us find the position vector of Mercury in Al-Bīrunī’s model:
We draw Fig. 3.1 again, but with some differences (see Fig. 5.2), in this figure
and are position vectors of H and B respectively. By writing their equations, we
will have the Sun’s orbit around the Earth.
̂
̂
̂
̂
The negative mark for y component of
of B.
is because of count clock wise rotating
Thus, the Sun’s position vector comes out to be
⃗
⃗
̂
̂
In addition, we have:
∫ ̇
Where
is the initial angle.
So
̇
̇
But as Al-Bīrunī has written in his book ̇
̇ ,we consider
to be zero.
̇
̇
⃗
If
(
)̂
̂
we have:
⃗
(
)
̂
̂
(5.10)
26
So according to the ellipse equation in Cartesian coordinates we get
X=(
)
Y=
Now if
the curve is an ellipse, and we see:
Thus, Al-Bīrunī said that the Sun’s orbit around the Earth is an ellipse of
course indirectly.
Nevertheless, if
we must rotate the axes by
to get the correct
equation.
Because of relative vectors in mechanics, the Sun’s orbit around the Earth is
mirror of the Earth’s orbit around the Sun. (See Fig. 5.1)
So Al-Bīrunī was right about the Sun’s motion around Earth, Which was a
suitable model for Mercury’s motion.
FIGURE 5.1 The relative vectors in Sun- Earth system.
27
B
𝑟
Y’
Y
D
𝜃
𝑟
𝜃
X’
H
K
FIGURE 5.2 Here the points names are the same as Fig. 3.1.
X
28
Appendix
Deriving Kepler laws using Newtonian mechanics
A.1 Equation of motion
Let the masses of the two bodies be m1 and m2 and the radius vectors in some
fixed inertial coordinate frame r1 and r2 (Fig A.1.). The position of the planet
relative to the Sun is denoted by r = r2−r1. According to Newton’s law of
gravitation, the planet feels a gravitational pull proportional to the masses m1 and
m2 and inversely proportional to the square of the distance r. Since the force is
directed towards the Sun, it can be expressed as
Where G is the gravitational constant.
Newton’s second law tells us that the acceleration ̈ of the planet is
proportional to the applied force:
̈
Combining (A.1) and (A.2), we get the equation of motion of the planet
̈
Since the Sun feels the same gravitational pull, but in the opposite direction, we
can immediately write the equation of motion of the Sun:
̈
FIGURE A.1 The radius vectors of the Sun and a planet in an arbitrary inertial frame are r1
and r2, and r = r2 –r1 is the position of the planet relative to the Sun.
29
We are mainly interested in the relative motion of the planet with respect to the
Sun. To find the equation of the relative orbit, we cancel the masses appearing on
both sides of (A.3) and (A.4), and subtract (A.4) from
(A.3) to get
̈
(A.5)
,
Where we have denoted
.
(A.6)
The solution of (A.5) now gives the relative orbit of the planet. The equation
involves the radius vector and its second time derivative. In principle, the solution
should yield the radius vector as a function of time, r = r (t). Unfortunately, things
are not this simple in practice; in fact, there is no way to express the radius vector
as a function of time in a closed form (i.e. as a finite expression involving familiar
elementary functions). Although there are several ways to solve the equation of
motion, we must resort to mathematical manipulation in one form or another to
figure out the essential properties of the orbit. Next, we shall study one possible
method.
A.2 Solution of the Equation of Motion
The equation of motion (A.5) is a second-order (i.e. contains second derivatives)
vector valued differential equation. Therefore, we need six integration constants or
integrals for the complete solution. The solution is an infinite family of orbits with
different sizes, shapes and orientations. A particular solution (e.g. the orbit of
Jupiter) is selected by fixing the values of the six integrals. The fate of a planet is
unambiguously determined by its position and velocity at any given moment; thus
we could take the position and velocity vectors at some moment as our integrals.
Although they do not tell us anything about the geometry of the orbit, they can be
used as initial values when integrating the orbit numerically with a computer.
Another set of integrals, the orbital elements, contains geometric quantities
describing the orbit in a very clear and concrete way. We shall return to these later.
A third possible set involves certain physical quantities, which we shall derive
next. We begin by showing that the angular momentum remains constant. The
angular momentum of the planet in the heliocentric frame is
L=
r × ̇.
(A.7)
Celestial mechanicians usually prefer to use the angular momentum divided by the
planet’s mass
k= r × ̇ .
(A.8)
31
Let us find the time derivative of this:
̇=r× ̈ + ̇ × ̇ .
The latter term vanishes as a vector product of two parallel vectors. The former
term contains ̇ , which is given by the equation of motion:
̇ =r×(
r/ )
/ ) r ×r = 0.
Thus k is a constant vector independent of time (as is L, of course). Since the
angular momentum vector is always perpendicular to the motion (this follows from
(A.8)), the motion is at all times restricted to the invariable plane perpendicular to
k (Fig. A.2). to find another constant vector, we compute the vector product
k
× ̈:
⁄
̇ )×(
k× ̈ =(
̇
=
̇
The time derivative of the distance r is equal to the projection of ̇ in the direction
of r (Fig. A.3); thus, using the properties of the scalar product, we get ̇
̇⁄ ,
which gives
̇
̇
Hence,
̇⁄
̇⁄
The vector product can also be expressed as
̈
̇
⁄
31
FIGURE A.2. The angular momentum vector k is perpendicular to the radius and
velocity vectors of the planet. Since k is a constant vector, the motion of the planet
is restricted to the plane perpendicular to k
FIGURE A.3. The radial velocity ̇ is the projection of the velocity vector ̇ in the
direction of the radius vector r since k is a constant vector. Combining this with the
previous equation, we have
⁄
̇
and
̇
⁄
(A.10)
Since k is perpendicular to the orbital plane, k× ̇ must lie in that plane. Thus, e is
a linear combination of two vectors in the orbital plane; so e itself must be in the
orbital plane (Fig. A.4). Later we shall see that it points to the direction where the
planet is closest to the Sun in its orbit. This point is called the perihelion. One
more constant is found by computing ̇ · ̈ :
̇ ̈
⁄
̇
̇⁄
̇/
⁄
32
Since we also have
̇ ̈
( ̇ ̇)
we get
(
̇ ̇
)
or
⁄
(A.11)
Here v is the speed of the planet relative to the Sun. The constant h is called the
energy integral; the total energy of the planet is
h. We must not forget that
energy and angular momentum depend on the coordinate frame used. Here we
have used a heliocentric frame, which in fact is in accelerated motion.
FIGURE A.4. The orbit of an object in the gravitational field of another object is a conic
section: ellipse, parabola or hyperbola. Vector e points to the direction of the pericentre,
where the orbiting object is closest to central body. If the central body is the Sun, this
direction is called the perihelion; if some other star, periastron; if the Earth, perigee, etc.
The true anomaly f is measured from the pericentre
33
So far, we have found two constant vectors and one constant scalar. It looks as
though we already have seven integrals, i. e. one too many. But not all of these
constants are independent; specifically, the following two relations hold:
where e and k are the lengths of e and k. The first equation is obvious from the
definitions of e and k. To prove (A.13), we square both sides of (A.10) to get
̇
̇
̇
.
Since k is perpendicular to ̇ , the length of k× ̇ is |k|| ̇ | = kv and (k× ̇ ) · (k× ̇ ) =
. Thus, we have
̇
The last term contains a scalar triple product, where we can exchange the dot
and cross to get k· ̇ ×r. Next we reverse the order of the two last factors. Because
the vector product is anti-commutative, we have to change the sign of the product:
(
)
This completes the proof of (A.13).
The relations (A.12) and (A.13) reduce the number of independent integrals by
two, so we still need one more. The constants we have describe the size, shape and
orientation of the orbit completely, but we do not yet know where the planet is! To
fix its position in the orbit, we have to determine where the planet is at some given
instant of time t = t0, or alternatively, at what time it is in some given direction. We
use the latter method by specifying the time of perihelion passage, the time of
perihelion τ.
A.3 Equation of the orbit and Kepler’s First Law
In order to find the geometric shape of the orbit, we now derive the equation of
the orbit. Since e is a constant vector lying in the orbital plane, we choose it as the
reference direction. We denote the angle between the radius vector r and e by f .
34
The angle f is called the true anomaly. (There is nothing false or anomalous in this
and other anomalies we shall meet later. Angles measured from the perihelion
point are called anomalies to distinguish them from longitudes measured from
some other reference point, usually the vernal equinox.) Using the properties of the
scalar product we get
r · e = re cos f .
But the product r · e can also be evaluated using the definition of e:
̇
⁄
̇
Equating the two expressions of r · e we get
⁄
This is the general equation of a conic section in polar coordinates (Fig. A.4).
the magnitude of e gives the eccentricity of the conic section:
Circle,
Ellipse,
Parabola,
Hyperbola.
Inspecting (A.14), we find that r attains its minimum when f = 0, i.e. in the
direction of the vector e. Thus, e indeed points to the direction of the perihelion.
Starting with Newton’s laws, we have thus managed to prove Kepler’s first law:
The orbit of a planet is an ellipse, one focus of which is in the Sun. Without any
extra effort, we have shown that also other conic sections, the parabola and
hyperbola, are possible orbits.
A.4 Kepler’s Second and Third Law
The radius vector of a planet in polar coordinates is simply
̂
35
Where ̂ is a unit vector parallel with r (Fig. A.5). If the planet moves with
angular velocity f, the direction of this unit vector also changes at the same rate:
̂̇
̇ ̂
Where ̂ is a unit vector perpendicular to ̂ . The velocity of the planet is found
by taking the time derivative of (A.19):
̇
̂̇
̇ ̂
̇
̇ ̂
FIGURE A.5. Unit vectors ̂ and ̂ of the polar coordinate frame.The directions of
these change while the planet moves along its orbit
The angular momentum k can now be evaluated using (A.19) and (A.21):
̇ ̂
̇
where ̂ is a unit vector perpendicular to the orbital plane. The magnitude of k is
̇
The surface velocity of a planet means the area swept by the radius vector per unit
of time. This is obviously the time derivative of some area, so let us call it ̇ . In
terms of the distance r and true anomaly f , the surface velocity is
̇
̇
By comparing this with the length of k (A.23), we find that
36
̇
Since k is constant, so is the surface velocity.
Since the Sun–planet distance varies, the orbital velocity must also vary (Fig. A.6).
From Kepler’s second law it follows that a planet must move fastest when it is
closest to the Sun (near perihelion). Motion is slowest when the planet is farthest
from the Sun at aphelion. We can write (A.25) in the form
and integrate over one complete period:
∫
∫
where P is the orbital period. Since the area of the ellipse is
√
FIGURE A.6. The areas of the shaded sectors of the ellipse are equal. According to
Kepler’s second law, it takes equal times to travel distances AB, CD and EF
Where a and b are the semimajor and semiminor axis and e the eccentricity, we get
√
To find the length of k, we substitute the energy integral h as a function of
semimajor axis (A.16) into (A.13) to get
37
√
When this is substituted into (A.29), we have
This is the exact form of Kepler’s third law as derived from Newton’s laws.
In this form, the law is not exactly valid, even for planets of the solar system,
since their own masses influence their periods. The errors due to ignoring this
effect are very small, however.
Kepler’s third law becomes remarkably simple if we express distances in
astronomical units (AU), times in sidereal years (the abbreviation is unfortunately
a, not to be confused with the semimajor axis, denoted by a somewhat similar
symbol a) and masses in solar masses (M_). Then G =
and
The masses of objects orbiting around the Sun can safely be ignored (except for
the largest planets), and we have the original law
= . This is very useful for
determining distances of various objects whose periods have been observed. For
absolute distances we have to measure at least one distance in meters to find the
length of one AU. Earlier, triangulation was used to measure the parallax of the
Sun or a minor planet, such as Eros, that comes very close to the Earth. Nowadays,
radio telescopes are used as radar to very accurately measure, for example, the
distance to Venus. Since changes in the value of one AU also change all other
distances, the International Astronomical Union decided in 1968 to adopt the value
1 AU = 1.496000×
m. The semimajor axis of Earth’s orbit is then slightly
over one AU.
But constants tend to change. And so, after 1984, the astronomical unit has a
new value,
1AU= 1.49597870×
m.
Another important application of Kepler’s third law is the determination of
masses. By observing the period of a natural or artificial satellite, the mass of the
central body can be obtained immediately. The same method is used to determine
masses of binary stars.
Although the values of the AU and year are accurately known in SI-units, the
gravitational constant is known only approximately. Astronomical observations
give the product G
, but there is no way to distinguish between the
38
contributions of the gravitational constant and those of the masses. The
gravitational constant must be measured in the laboratory; this is very difficult
because of the weakness of gravitation. Therefore, if a precision higher than 2–3
significant digits is required, the SI-units cannot be used. Instead, we have to use
the solar mass as a unit of mass (or, for example, the Earth’s mass after G m has
been determined from observations of satellite orbits).
39
6. References
F. Hoyle: Nicolaus Copernicus, New York 1973
H. Kartunenn: Fundamental Astronomy 5th ed., Springer 2007
W. Hartner: Al-Bīrunī and the theory of solar apogee, 1960
Al-Bīrunī: Qanun Al-Masudi, Beirut