Planetary Science

Text book: Planetary Sciences, Updated 2nd Edition, by I. de Pater and J. Lissauer

. In this problem, you will make some calculations which would be useful for planning spacecraft missions to Jupiter and to other planets. To simplify matters, you may assume that the planets move on circular orbits.

• Calculate the velocity (relative to Earth) at Earth’s orbit of the Hohmann transfer orbit that is tangent to both Earth’s orbit and Jupiter’s orbit.

• Calculate the minimum velocity necessary to launch a spacecraft from the surface of Earth to Jupiter, ignoring Earth’s rotation.

• Calculate the minimum velocity necessary to launch a spacecraft from the Earth’s equator to Jupiter, including Earth’s rotation but ignoring its obliquity.

• Calculate the time required for a spacecraft moving along a Hohmann transfer orbit to travel from Earth to Jupiter.

• Repeat part (b) for spacecraft sent on Hohmann transfer orbits to Venus and to Mercury.

• Textbook 2.29.D. Two moons are spiraling outwards, away from their planet, due to tidal forces. They become locked in a stable orbital resonance which requires them to maintain a constant ratio of orbital periods (see, e.g., Peale 1976). Calculate the energy available for tidal heating in equilibrium (i.e., assume that the moons’ orbital eccentricities do not change). Your answer should depend on the masses of the two moons, mI and mII, and of the primary, mp, on the angular velocities of the moons, nI and nII, and on (the z-components of) the tidal torques exerted by the planet on each of the moons, L̇I(p) and L̇II(p). (Hint: Use conservation of angular momentum and energy:

d/dt (LI + LII) = L̇I(p) + L̇II(p),  (2.65a)

d/dt (EI + EII) = L̇I(p)nI + L̇II(p)nII − H, (2.65b)

where LI and LII are the orbital angular momenta of the moons, EI and EII are the orbital energies, and H is the heating rate. Technically, the rotational angular momenta of the moons should also be included; however, their values are so small that you may ignore them.)