A drag-free satellite in a circular orbit above the Earth's surface carries

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Atomic, Molecular & Optical Physics · The equivalence principle and gravity as geometry

Relativity and Gravitation, Ch Experimental foundations: free fall, Lorentz invariance and clock tests

Problem

A drag-free satellite in a circular orbit

1200\ \mathrm{km}

above the Earth's surface carries, as test bodies, two concentric hollow cylinders of different composition. Take the Earth's mass as

M = 5.972\times 10^{24}\ \mathrm{kg}

and its radius as

R = 6.371\times 10^{6}\ \mathrm{m}

, with

G = 6.674\times 10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}

and

c = 2.998\times 10^{8}\ \mathrm{m\,s^{-1}}

. ((a)) Suppose the two materials have Eötvös parameter

\eta = 5.0\times 10^{-15}

in the Earth's field. Estimate how far apart the cylinders would drift in

8.0\times 10^{6}\ \mathrm{s}

of free fall if no force were applied to keep them concentric, treating their differential acceleration as constant in magnitude and direction over that time. (In the real instrument an electrostatic force keeps them concentric and it is that force which is measured.) ((b)) The satellite data are referred to a clock at the ground station. Estimate the fractional difference in rate between that clock and an identical clock

25\ \mathrm{m}

higher up on the station's mast, taking

g = 9.81\ \mathrm{m\,s^{-2}}

and working to first order in

gh/c^{2}

. ((c)) For comparison, estimate the separation the same

\eta

would produce in a

10\ \mathrm{m}

drop in a laboratory tower.

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