A satellite of mass orbits a planet of mass (so and )

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

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

Problem

A satellite of mass

m = 1000\,\mathrm{kg}

orbits a planet of mass

M = 6.0\times10^{24}\,\mathrm{kg}

(so

\mu \approx m

and

k = G_{\rm N}M\mu

). At pericentre its distance is

r_{\rm p} = 7.0\times10^6\,\mathrm m

and its speed is

v_{\rm p} = 8.0\times10^3\,\mathrm{m\,s^{-1}}

. Take

G_{\rm N} = 6.674\times10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}

. ((a)) Find the orbital angular momentum

\ell

, the total energy

E

(using the pericentre data), and hence the semi-major axis

a

of the orbit. ((b)) The planet's oblateness contributes a small extra term

\epsilon H_1 = -\gamma_3/r^3

to the Hamiltonian, with

\gamma_3 = 1.30\times10^{28}\,\mathrm{J\,m^3}

. Using

\langle r^{-3}\rangle = (\mu k)^3/N^3\ell^3

with

N = \sqrt{\mu k^2/(-2E)}

, find the apsidal precession

\Delta\varpi

per orbit. ((c)) A distant perturbing moon has mean motion corresponding to a period

\tau_{\rm pert} = 2.15\times10^4\,\mathrm s

. Compute the satellite's orbital period

\tau_{\rm orb}

from

a

, and hence find, as a fraction of

\nu_{\rm orb}

, the value of

\nu_{\rm orb} - 3\nu_{\rm pert}

. State what this fraction tells you about the reliability of a first-order perturbation series built from this pair of frequencies.

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