Model the Earth and the Moon as uniform spheres

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · The equivalence principle and gravity as geometry

The Geometry of Gravity, Ch The equivalence principle under test

Problem

Model the Earth and the Moon as uniform spheres, so that the fractional gravitational binding energy of each is

E_{\rm grav}/mc^2 \approx -\tfrac{3}{5}GM/(R_\star c^2)

. Use

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

c = 3.00\times10^8\ \mathrm{m\,s^{-1}}

M_\oplus = 5.972\times10^{24}

kg,

R_\oplus = 6.371\times10^6

M_{\rm Moon} = 7.342\times10^{22}

kg,

R_{\rm Moon} = 1.737\times10^6

m, and take the Sun's acceleration at 1 AU as

5.93\times10^{-3}\ \mathrm{m\,s^{-2}}

and the Moon's sidereal orbital period as

27.32

days. ((a)) Estimate the fractional binding energy of the Earth and of the Moon. ((b)) Take

\eta_{\rm N} = 1

and estimate the resulting differential solar acceleration between the Earth and the Moon,

\Delta a_\odot = \eta_{\rm N}\times\left|(E_{\rm grav}/mc^2)_\oplus - (E_{\rm grav}/mc^2)_{\rm Moon}\right|\times a_{\rm sun}

. ((c)) Estimate the resulting static displacement of the lunar orbit,

\Delta a_\odot/n_{\rm Moon}^2

with

n_{\rm Moon} = 2\pi/T_{\rm orb}

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