Celestial mechanics

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4. Restricted Three-body-problem

• [Lagrange 1772] analytically

solvable: heavy M 1 and M 2

with M 2 < 38,5‰ (M 1+M 2)

and a light m « (M 1+M 2)

• Observing in a frame rotating

with the baseline M 1-M 2

M 1 , M 2 and m remain on a triangle

with equal edges.

• More than mathematical curiosity:

Trojans of Jupiter (N>2000),

Mars + Eureka + 3T, Neptun + 5 T

Saturn‘s natural laboratory:

Thetys (1060 km) + Telesto, Calypso (26 km),

Dione (1118 km) + Helene (32 km),

Janus + Epimetheus (horseshoe orbit)

Application for space-probes: (eternal sun or shadow)

SOHO (L 1 ), WMAP (L 2 ), HERSCHEL+PLANCK (L 2 ), JWST (L 2 ), DARWIN (L 2 ?)

Trojan Exoplanets

L 4

L 3 L 1 L 2

L 5

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3. Orbit types • Energy of planet in two body system • 1 GMm E = 2 µv² − r , µ = Mm M + m m: Planet, M: Star • Semimayoraxis a = - ½GMm/E • Type of orbit E > 0 => hyperbolical E = 0 => parabolical E < 0 => elliptical E = − 1 2 GMm r => circular orbit if v ⊥ r µv² GMm GM = ⇒ v = r r² r • Correction for m→ M in COM System (m « M) GM M v p = , vS r m + M = Gm r m m + M6

4. Restricted Three-body-problem • [Lagrange 1772] analytically solvable: heavy M 1 and M 2 with M 2 < 38,5‰ (M 1+M 2) and a light m « (M 1+M 2) • Observing in a frame rotating with the baseline M 1-M 2 M 1 , M 2 and m remain on a triangle with equal edges. • More than mathematical curiosity: Trojans of Jupiter (N>2000), Mars + Eureka + 3T, Neptun + 5 T Saturn‘s natural laboratory: Thetys (1060 km) + Telesto, Calypso (26 km), Dione (1118 km) + Helene (32 km), Janus + Epimetheus (horseshoe orbit) Application for space-probes: (eternal sun or shadow) SOHO (L 1 ), WMAP (L 2 ), HERSCHEL+PLANCK (L 2 ), JWST (L 2 ), DARWIN (L 2 ?) Trojan Exoplanets L 4 L 3 L 1 L 2 L 5 7

Page 1 and 2: Celestial mechanics Contribution to
Page 3 and 4: 1. Theory of Celestial mechanics Ke
Page 5: 2. Numerical Solution • Simple Eu
Page 9 and 10: Orbits in rotating frame • Yellow

Celestial mechanics

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