Sept. 27, 2007 1

Apparent motions: Planets, Kepler’s Laws & Orbits

• Review celestial coordinate system: RA, DEC

• RA,DEC of Sun, Moon, and … planets?

• The great observer, Tycho, and analyst, Kepler

• Sidereal vs. synodic periods of planets: Kepler’s 3rd Law

• Ellipses and Kepler’s 1st and 2nd Laws

• Grand overview: orbits

Sept. 27, 2007 2

Celestial coordinate system: RA, DEC• Stars (and galaxies) “fixed” on celestial sphere, so give

them a longitude (RA) and latitude (DEC) coord.• We need to locate Sun, Moon, stars, galaxies on maps of

the sky (“finding charts” as in EL1…)

• RA = longitude coord., units of time (or degrees) with 0 in constellation Aries

• DEC= latitude coord., with zero on Earth’s equator

Sept. 27, 2007 3

How about RA,DEC for Sun and Moon?

• Both are changing (continuously) since we view them from a rotating, orbiting Earth and Moon is itself orbiting Earth. RA,DEC are geocentric coords.

• Stars have fixed RA,DEC (ignoring precession…) since Earth’s spin axis fixed in 3D space (gyroscope)

• DL1 sundial observation: changing DEC of Sun; DL3 will also show changes in RA of Sun

• What about the planets – e.g. Uranus (EL1)?

Sept. 27, 2007 4

March of the Planets…

• Have you seen (last night) Uranus vs. the stars? It’s moving… Which way? Retro, moving W vs. the stars...

• You will (final week of EL1) “fast-forward” Uranus on computer and see it reverse: go East vs. stars. Why?

• Earth is overtaking Uranus (or Mars; or outer planets) ea. year

Mars doing its Retro loop (Uranus does similarly…)

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How often does this happen?

• Approximately each Earth year, but not exactly… rather, with synodic period, S. Consider Jupiter:

• For Jupiter, Tycho observed S = 1.09y

• Kepler postulated this is explained by Jupiter orbiting Sun (like Earth; Copernican model) with period P (relative to fixed stars) vs. Earth orbiting with period E

• Then Earth vs. Jupiter line up (conjunction) with frequency, F (no. of conjunctions per Earth year) of Fs = Fe – Fp, where Fs = synodic, or conjunction, freq. and Fe and Fp are the true (vs. fixed stars) orbital freq. (=orbits/yr).

• And since Period = 1/Frequency, we have 1/S = 1/E – 1/P, as in your textbook

Sept. 27, 2007 6

So what does this mean?

• Tycho used parallax to deduce approx. distances for say Mars vs. Jupiter: Jupiter is more than twice distant

• The synodic period of superior conjunctions of Jupiter is shorter (399d) than for Mars (780d), so the sidereal period of Jupiter is longer since 1/P = 1/E – 1/S

• So more distant planets have longer periods: Kepler deduced his Third Law: P2 = a3, where a=distance from Sun in AU and P = sidereal period in years

Sept. 27, 2007 7

To complete his 1st and 2nd Laws

• Kepler’s First Law: Planets move on ellipses, not perfect circles (which are ellipses with e=0).

• How did Kepler deduce this? By requiring a good fit to the extensive data accumulated by Tycho: circles did not fit! (particularly for Mercury…)

• And how did planets have to move on ellipses to match the positions vs. time data of Tycho? In accordance with Kepler’s 2nd Law: Fastest when closest to Sun, and sweeping out =area in =time