A spherically symmetric planet of mass and reference radius carries one small extra…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Earth, Climate & Energy Physics · The solid Earth: structure, fields and thermal history

The Solid Earth, Ch The shape and gravity of the Earth: potential, geoid, isostasy and flexure

Problem

A spherically symmetric planet of mass

M

and reference radius

a

carries one small extra point mass

\mu=2.0\times10^{-6}M

at radius

r_0=0.9a

on the equator, at east longitude

\phi_0=30^\circ

. Work to first order in

\mu/M

, so the spherical part contributes nothing beyond degree 0 and the small shift of the centre of mass can be ignored at degree 2. The Schmidt quasi-normalised functions of degree 2 are

P_2^0(\cos\theta)=\tfrac12(3\cos^2\theta-1)

P_2^1(\cos\theta)=\sqrt3\sin\theta\cos\theta

and

P_2^2(\cos\theta)=\tfrac{\sqrt3}{2}\sin^2\theta

. ((a)) Use

C_{nm}=(1/Ma^n)\int r'^nP_n^m(\cos\theta')\cos m\phi'\,\mathrm dm

, and its sine partner for

S_{nm}

, to find

C_{20}

C_{21}

S_{21}

C_{22}

and

S_{22}

. What does the mass do to

J_2

? ((b)) The mass is moved along the equator to

\phi_0=120^\circ

, and then to

\phi_0=210^\circ

. How do

C_{22}

and

S_{22}

change each time, and why?

Browse the problem library