l Lecture 5

AST 180: Lecture 5

Planetary Motion

Planetary Motions on the Sky

 The apparent path of Mars in 2005-2006 is shown above. Mars path will be retrograde in Fall 2005.
Movie of Martian Motion

 To account for uneven and retrograde motion in the Ptolemaic system, pre-Copernican astronomers invoked additional circles of motion called epicycles.
 Epicycles could add speed to a planets orbit as in the above figure Or they could induce retrograde motion as above.

Movie of retrograde motion in the Ptolemaic System

 Unfortunately, one epicycle was not enough to fit the observations. So astronomers added epicycle upon epicycle (up to 28 in Ptolemy's system) to try to get predicted positions that would agree with the observations. Moreover, some of the epicycles were not centered on previous orbital paths (the "deferent") but were offset. So simplicity was being sacrificed in order to keep "circular" motion alive.

Movie of retrograde motion in the Heliocentric System

 Today we know that retrograde motion is due to the fact that all the planets are rotating around the Sun. Copernicus re-introduced this idea in the 16th century in order to reduce the number of epicycles needed in the calculations. In order to stave off persecution, his posthumously published manuscript contained a disclaimer saying this was just a mathematical convention that had nothing to do with reality. Nevertheless, many subscribed to the view that theories which predict the same result with fewer assumptions are more likely to be true. This is the idea behind Occam's Razor. Tycho Brahe made very accurate measurements of Mars' positions. These showed that even in the Copernican system, epicycles were required! So Brahe believed in a variation of the Ptolemaic system in which the planets went around the Sun, but the Sun went around the Earth. Eventually, Kepler showed that Brahe's data were best explained by planets moving on slightly elliptical orbits with NO epicycles need!

 Galileo made telescope observations that supported the Copernican view - Satellites rotating around Jupiter; Phases of Venus; Topography on the moon, etc. For this he was sent into exile by the Catholic Church. But Isaac Newton finally showed that a simple theory of gravity explains why the planets move on elliptical orbits. This finally convinced western civilization that Galileo was right and the Church was wrong. In order to make his point, Newton had to invent Calculus along the way! His accomplishment was certainly one of the most astounding in physics, prompting the famous couplet by Alexander Pope: "Nature, and Nature's Laws lay hid in Night, God said, "Let Newton Be! and All was Light."

 In the Copernican view, planetary positions as seen from the earth correspond to solar system positions as shown in the figure at left and are referred to by a variety of terms. Opposition, Conjunction, and Elongation refer to how a planet is aligned with the line of site between Earth and Sun. Using these geometries, Copernicus was able to calculate the relative distances of each of the known planets from the Sun. He did remarkably well, as illustrated in the table to the right.
 The period of a planet's orbit is the time it takes for it to go around once. The sidereal period is the time it takes to go around once relative to the stars (sidereal refers to the stars). The synoptic period (synoptic refers to taking the same point of view) is the time it takes for the planet to re-appear at an identical configuration as seen from earth (see figure at right).

Parallax

Tycho Brahe used the parallax effect to show that a newly discovered Supernova was very distant. This same effect led him to believe the Copernican view was incorrect, however! This was because it implied that one should see the nearest stars wobble in position as the Earth moved in its orbit, and Brahe detected no such wobble. Stellar parallax (and the distance to the nearest stars) was not detected until the 19th century. The stars are much further away than anyone imagined in Brahe's day, and this fact makes the parallax effect very small and hard to measure! Currently, the US Naval Observatory Flagstaff Station has the premiere program to measure parallaxes of nearby faint dwarfs.

 Origin of the Parallax Effect Movie of Stellar Parallax

Kepler's Laws

 Johannes Kepler tried to fit Tycho Brahe's measurements of Mars' motion with a Copernican model. Eventually he had the insight to give up on circles and try fitting planetary motion with ellipses.

 Kepler's first two laws are illustrated in the figure to the right and in an animation linked further to the right: 1) The orbit of a planet about the Sun is an ellipse with the Sun at one focus. 2) A line joining a planet and the Sun sweeps out equal areas in equal intervals of time. Movie of Kepler's First Two Laws

 Kepler's Third Law 3) The square of a planet's sidereal periods around the Sun is directly proportional to the cube of the length of its oribt's semi-major axis, or. P2 = a 3 Kepler's Third Law is well illustrated in the table t left.

Galileo's Discoveries

At about the time Kepler was working out his laws of planetary motion, Galileo was turning a telescope to the heavens. His observations of Venus showed that the planet's size varied with phase in a way that could not be explained in the Ptolemaic System.

Galileo's observations of Jupiter's Moons showed four satellites orbiting another planet (i.e., not the Earth!) in a manner that obeyed Kepler's laws. This also supported the Copernican system.