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Physics 1025: Lecture 13 Thumbnail History of Astronomy, Bode’s Law, Kepler’s Laws
Announcements
One additional item re Planets table of last lecture:
The relative solar distances of the planets can be given by Bode’s law published in 1766 (actually
discovered by Titus of Wittenburg in 1766 before the discovery of Uranus, Neptune and Pluto). Start
with 0, then form a series by putting 3 as the next number, then 6, successively doubling each time.
Then add 4 to each number and divide by 10: you then have the planetary solar distances (in AU) except
for the gap at 2.8 AU. In 1910 Ceres was discovered at 2.8 A.U. and since then, of course, thousands of
asteroids with this mean solar distance have been found.
0
4
0.4
Hg
0.39
3
4
0.7
V
0.72
6
4
1.0
E
1.0
12
4
1.6
M
1.52
24
4
2.8
48
4
5.2
J
5.2
96
4
10.0
S
9.54
192
4
19.6
U
19.2
384
4
38.8
N
30.6
768
4
77.2
P
39.9
In the table above the last row lists the actual solar distances of the planets in AU. As noted before Pluto
does not fit into the above, but it was once thought Pluto was an escaped moon of Neptune, and its
escape must have also disrupted Neptune’s orbit (accounting also for the discrepancy in Neptune’s
distance). Bode’s law is only an empirical relation and there has been no formal derivation or
justification for the remarkable fit to planetary distances. We will now briefly give a history of early
models of the solar system and the beginnings of modern astronomy, ending with Kepler’s laws and
Newton’s laws including the Law of Universal Gravitation.
Going back in history, we have Aristarchus (310-230 B.C.) who first deduced the solar distance. More
importantly, he first propounded the heliocentric (sun-centered) model for the solar system, and was so
referenced by Copernicus. Things then regressed and the next important model was by Ptolemy in the
second century AD ; Ptolemy proposed the geocentric (earth-centered)model and worked out an
elaborate system of epicycles, circles rolling on circles to account for retrograde motion. It should be
emphasized that with proper refinement the Ptolemaic model can be arbitrary accurate, but it involves
unnecessary hypotheses like the series of epicycles. During the Middle Ages scientific thought was kept
flowing in the Middle East by the Muslims who compiled many star catalogues. Roger Bacon urged a
calendar reform in 1200 AD and even proposed the idea of a telescope but Ptolemaic theory still
dominated.
About 500 years ago there was dissatisfaction with Ptolemy and of course it was Copernicus (14731543) who proposed the heliocentric model, using offset circles to account for the non-uniform
planetary motions. His model more simply accounted for retrograde motion (see last lecture). Originally
accepted by the church, Copernicus’ work was later denounced. Giordano Bruno defended Copernicus’
work in 1600 and for this was excommunicated and burned at the stake.
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The first modern observer was Tycho Brahe, a Danish nobleman who built the first modern observatory
on the island of Hven after seeing a new star or supernova visible in the daytime in 1572. He spent his
life making thousands of accurate visual observations of stars and planets; he took these tables with him
to Prague where they formed the basis of Kepler’s laws. Tycho knew about parallax but since he didn’t
observe it, his model keeps earth at rest.
Tycho’s assistant in Prague was Kepler who worked with the observations to model planet orbits and
came up with 3 empirical laws:
1609 1. The orbits of the planets are ellipses with the sun at one focus. (concerns shape of orbits)
1609 2. The radius vector to a planet sweeps out equal areas in equal times. (concerns speeds of
planets – this is basically conservation of angular momentum)
1618 3. The squares of the periods of any 2 planets are in the ratio of their solar distances cubed: p2 ~ a3
with the proportionality constant being the mass of the body the planets are orbiting (a later
generalization)
Example: PE= 1 year, aE = 1 AU for earth. Hence for Jupiter, aJ = 5.2 AU so PJ = (aJ3)1/2 = 12.6 years.
Namely for the planets orbiting the sun, P2 = ka3 where k = 1. For other orbiting objects such as
moons the same equation holds but k is proportional to the mass of the body that the moons orbit.
Kepler’s laws have been completely verified and form the basis of orbital motion today (except in
regions of huge masses like black holes or inside atoms, where relativity and quantum mechanics are
needed, respectively). However, again they are completely empirical. Like Tycho, Kepler earned much of
his income from horoscopes and almanacs.
During this time Galileo was laying the foundation of mechanics in Italy, e.g. using a pendulum clock to
time motion down inclined planes (which “slow down” gravity to a measurable quantity). He showed
that the velocity with which an object falls is independent of its weight (contrary to Aristotle) and that
v=gt where g is the acceleration of gravity, or that component acting along the plane. In general he
determined the laws of falling bodies which were the forerunners of Newton’s laws. And, of course, he
made the astronomical discoveries for which he is so famous: the craters on the moon, sunspots, moons
of Jupiter, and phases of Venus.