A probe orbits a moon of gravitational parameter at circular radius

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Simulating classical dynamical systems

Computational Physics, Ch An orbit that must not drift

Problem

A probe orbits a moon of gravitational parameter

GM = 4.90\times10^{12}\,\mathrm{m^3\,s^{-2}}

at circular radius

r = 1.0\times10^{6}\,\mathrm{m}

, all given in SI units. Following the strategy used for the Earth's orbit about the Sun, choose a distance unit

D = 1.0\times10^{6}\,\mathrm{m}

(so

r = 1.0\,D

) and a time unit

T_u

chosen so that

GM = 4\pi^2\,D^3\,T_u^{-2}

exactly, as was done with the astronomical unit and the year. (a) Find

T_u

in seconds, to three significant figures. (b) Write the four-component system for

(x,y,v_x,v_y)

in the units

D

and

T_u

, and give the initial condition, in these units, for a circular orbit at

r=1.0\,D

that closes after exactly one

T_u

. (c) Convert the initial tangential speed found in part 2 back into SI units (m s

^{-1}