Lecture #4 - History of Astronomy - Ptolemy to Kepler

Ptolemy (125 A.D.)
Ptolemy’s Model
• Designed a complete geometrical
model of the universe that accurately
predicted planetary motions with errors
within 50
• Most of the geometric devices and
basic foundations of his model did not
originate with him but were based on
the models of the early Greeks such
as Aristotle & Hipparchus
• Wrote the Almagest (Greatest)
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– natural motions - no forces
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– included the original works & models of Ptolemy
– included a compilation of past works of Greeks,
especially Hipparchus
– 13 volumes
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Ptolemy’s Equant
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Earth was spherical & at center of
cosmos - GEOCENTRIC
Cosmos is finite in size
Earth has no motions
Sun, Moon Planets exhibit uniform,
circular motions
Used devices of eccentrics, epicycles
& deferents to explain the observed
non-uniform motions of the Sun and
planets along the ecliptic & retrograde
motion.
Introduced equant to explain the
variations in retrograde motions
Ptolemy’s Model
point inside a circle, not at the center, from which motion along the
circumference of the circle would appear to be uniform
opposite the circle’s center from the eccentric (the Earth)
nonphysical geometrical device that broke fundamental
assumption of uniform circular motion
Celestial motions no longer had to be
uniform around the centers of circles
Copernicus’ Model
Copernicus (1473-1543)
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Developed a Heliocentric (Sun centered) model of
the cosmos
Why? Ptolemy’s geocentric model lasted for
centuries mainly because it accurately predicted
celestial motions so there was little reason to discard
it
Copernicus studied the works of Aristotle,
Pythagoras & Plato
an offshoot of Plato’s philosophy asserted that Sun
was godhead of all knowledge
Copernicus objected to equant based on aesthetics equant not faithful to ideal of uniform motion - makes
models to complex
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Copernicus worked on his new
Heliocentric model for 20 years
– Sun was placed at center of
cosmos
– Earth no longer static, but revolved
around Sun once a year & rotated
on axis once a day
His work was published in De
revolutionibus in the year of his
death
De revolutionibus took after the
Almagest in outline and basic intention
- to explain planetary motions
Even though it took 20 years to
develop this model did predict celestial
motions any better than Ptolemy’s
geocentric model
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Heliocentric Model of Copernicus
Heliocentric Model of Copernicus
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Cosmos finite in size
Assumed no forces for heavenly
motions
– Physics of Aristotle
Assumed uniform, circular motions
– done for aesthetics - followed
Aristotle
All heavenly spheres revolve around
the sun & the sun is at the center of the
cosmos
– chosen based on aesthetics and
simplicity
The distance from the earth to the
sphere of stars is much greater than the
distance from the earth to the sun
– accounts for lack of observed
stellar parallax
Retrograde Motion Explained
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The daily motion of the heavens relative to the
horizon results from the earth’s motion on its
axis
– aesthetic appeal since only 1 sphere is rotating
not many
– however, he did not account for the objection
that if the earth rotated, objects should be flung
from the surface
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The apparent motion of the sun relative to the
stars results from the annual revolution of the
earth around the sun
The planets’ retrograde motion occur from the
motion of the earth relative to the other planets
– retrograde explained as a natural result of the
planet’s revolutions about sun - what we
observe is an illusion
Retrograde Motion Explained
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Copernican Model
• Copernicus eliminated epicycles to
explain retrograde motion
• Eliminated equant - kept uniform circular
motion
• Needed to account for variations in
planetary motion so he was forced to
add many smaller circles
• Violated Aristotelian physics & did not
offer new physical ideas to support his
model
• Didn’t predict motions any better than
Ptolemy’s model
When the earth passes any of the outer planets or when the
earth is passed by the inner ones, retrograde motion occurs
Passing is key to retrograde motion in the heliocentric model
Ptolemy
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Tycho Brahe (1546-1601)
Copernicus
• Danish born member of nobility
– Stolen by his uncle and adopted as his own son
– Trained as a lawyer but secretly studied
astronomy after being impressed that mortals
could predict astronomical events
– Very arrogant and hard to get along with and
played the role of a courtly prince with a gold
nose
• Supported by the King Frederick II
of Denmark, he built and
observatory on the island of Hven
with one ton of gold
– His adopted father saved Frederick from
drowning but then died of pneumonia
Tycho’s Observations
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Tycho’s Supernova
He made a long series of very precise observations
of the motions of the planets
– Very accurate since they were made with a very
large quadrant circle (<0.5 arc minute)
– Made repeated observations and took the
average to reduce the errors
– Made observations over many years 1576-1591
– Realized the motions of planets did not match
the predictions of Ptolemy
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Tycho’s Cosmology
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Refuted Copernican system
Placed the earth at the center
– Due to his inability to measure parallax
– believed in Aristotlean physics
Realized that the Ptolemaic system failed
Compromise
– Stationary earth at center
– Sun moves around the earth
– All other celestial bodies move around
the sun
– His worked no better than Ptolemy He needed better mathematics
The supernova of 1572
– Noticed a brilliant star had
appeared in Cassiopeia
– Observed that it faded slowly
over the following 2 years
– Demonstrated that the universe
was changing and imperfect
– Showed that it had the same
position no matter where on earth
it was observed - thus it was at a
great distance
Kepler (1571-1630)
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Was born a commoner in Germany
– His father was a mercenary
– His mother was raised by an aunt who was
burnt as a witch
– Went to college at Tubingen were he was
recognized as a mathematical genius and went
to teach math at a Protestant School in Graz
– While in Graz he felt that the 5 perfect Platonic
solids could explain the motion of the 6 known
planets
– Attempted to get observations of the planets but was incompetent
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Kepler Solves The Mystery of Planetary
Motions
Kepler and Brahe Meet
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Brahe had to leave Hven for Prague because Fredrick II drank
himself to death
Kepler was driven out because the Catholics took over Graz
and drove out the Protestants
Tycho knew he needed a mathematician to help him model his
observations
• It took Kepler 8 years to
solve the motion of Mars
– He tried various combinations of
circular motions
– He finally abandoned circular motion
and tried ovals and other geometric
shapes
– They meet Feb 4, 1600 and fought continually
– Tycho feared that Kepler would eclipse him so he would only show
him data on Mars (his most challenging object)
– Tycho died October 21, 1601 from over partying
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• He found that an ellipse
could produce the motions of
the planets
Kepler got Tycho’s data from the family follow a death bed
request by Tycho “Let me not seem to have died in vain”
Properties of Ellipses
Kepler’s 1st Law of Ellipses 1609
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Ellipses belong to the family of
conic sections (the intersection of a
plane and a cone)
– The shape is defined by the
separation between the two foci
of the ellipse
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• When the foci coincide the
shape is a circle
• As the foci separate from each
other the ellipse gets flatter
• The amount by which an ellipse
differs from a circle is defined
as an ellipse’s eccentricity
Planets move in elliptical orbits with
the sun at one focus
The other focus is empty & located
in space
⇒ Distance between the planet
and the sun changes as the
planet moves along its elliptical
orbit
⇒Planet move along orbits which
have a special and regular
geometrical shape
⇒Gets rid of idea of uniform
circular motions
– Each point on an ellipse has the
property that the sum of its
distances to the two foci is the
same
– The line through the foci to both
sides of the ellipse is called the
major axis
Kepler’s 1st Law
Kepler’s 2nd Law of Equal Areas 1609
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Kepler noted that Mars slowed down when it
approached the sun
He found that a line drawn between the sun
and a planet sweeps out an equal area in an
equal period of time anytime during its orbit
– Planetary motions are non-uniform but vary
in a regular way
– Planets move more slowly when they are
far from the sun and fastest when they are
near the sun
– The increase in speed as a planet moves
toward the sun makes it appear that the
sun is “pulling” on the planet
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Kepler’s 2nd Law
Kepler’s 3nd Harmonic Law 1618
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The Harmonic Law - p2 = k a3
– p is the period of the planets orbit
– a is the planets average distance to the Sun
– k is a constant which is the same for all planets
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Kepler’s Laws
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Kepler’s scheme correctly described
planetary motions when compared to the
accurate observations of Tycho
Together laws 2 and 3 imply that there
exists a force, between the sun and
planets, that weakens with increasing
distance
– Invested Sun with physical properties
necessary for the uniform description of
planetary motions
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Kepler recognized there must be a force at
work but could not figure out what it was.
He speculated it may be magnetism
The more distant a planet from the Sun
– The slower it moves in its orbit
– The longer it takes to complete and orbit
Since a single constant works for all planets
suggests a single physical cause for their
orbits
Astronomy Enters a New Era
• Astronomical objects are viewed as actual
bodies not distant ethereal or mythical
entities
• Realization that the laws of nature are the
same for both the earthly and heavenly
domains
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