Two small satellites coast freely (engines off) at a distance from the centre of the Earth

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Newtonian gravitation as a field theory

Relativity and Gravitation, Ch Tidal forces and the Newtonian tidal field

Problem

Two small satellites coast freely (engines off) at a distance

r = 7.00\times10^{6}\ \mathrm{m}

from the centre of the Earth, whose mass is

M = 5.972\times10^{24}\ \mathrm{kg}

. The Earth is treated as a point mass, the mutual gravity of the two satellites is ignored, and their separation is far smaller than

r

, so first-order (tidal) results apply. Choose axes at the position of the first satellite with

x

along the outward radius and

y

z

perpendicular to it. The second satellite lies a distance

\xi_y = 30.0\ \mathrm{m}

from the first along the

y

axis. ((a)) Write the values of

\mathcal{E}_{xx}

\mathcal{E}_{yy}

and

\mathcal{E}_{zz}

for these axes. ((b)) Find the magnitude of the relative acceleration of the two satellites, and say whether they approach or separate.

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