A small moon orbits a planet of mass

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Astrophysics & Cosmology · Celestial mechanics and planetary systems

Action and Symmetry, Ch Adiabatic invariants, canonical perturbation theory and integrable systems

Problem

M

. Because the moon's mass is much less than

M

, its reduced mass is

\mu = 9.0\times10^{22}\,\mathrm{kg}

, essentially the moon's own mass. The orbit has semi-major axis

a = 4.0\times10^{8}\,\mathrm{m}

, eccentricity

e = 0.050

and period

\tau_{\rm orb} = 3.00

days. The planet's oblateness adds to the two-body Hamiltonian a small extra term

\epsilon H_1 = -\gamma_3/r^3

, with

\gamma_3 = 4.0\times10^{52}\,\mathrm{J\,m^3}

, in addition to the Kepler potential

V(r) = -k/r

with

k = G_{\rm N}M\mu

. ((a)) Using Kepler's third law to get

G_{\rm N}M

from

a

and

\tau_{\rm orb}

, find

k

and the orbital angular momentum

\ell

, then use the action-variable method (averaging

-\gamma_3/r^3

over the fast angle) to find the apsidal precession

\Delta\varpi

per orbit, in radians. ((b)) A second, farther moon of the same planet has period

\tau_2 = 6.10

days. Writing

n_1 = 2\pi/\tau_{\rm orb}

and

n_2 = 2\pi/\tau_2

for the two mean motions, evaluate

|n_1 - 2n_2|/n_1

and say, using the resonance-denominator argument, roughly by what factor a first-order perturbation term at this near-commensurability would be magnified compared with a generic (non-resonant) term.

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