A two-layer model planet has a uniform-density core and mantle bounded by coaxial…

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 two-layer model planet has a uniform-density core and mantle bounded by coaxial oblate ellipsoids sharing the same small flattening

f

: a core of density

\rho_1

and mean radius

b

, and a mantle of density

\rho_2

occupying the shell from

b

to the outer mean radius

a

. For a uniform oblate ellipsoid of mass

M_{\rm ell}

, equatorial radius

R

and small flattening

f

, rotational dynamics gives

C_{\rm ell} = \tfrac25M_{\rm ell}R^2

and

A_{\rm ell} = \tfrac15M_{\rm ell}(R^2+c^2)

with

c = R(1-f)

, so that

C_{\rm ell}-A_{\rm ell} \approx \tfrac25M_{\rm ell}R^2f

to first order in

f

. ((a)) By treating the mantle as the difference of two full ellipsoids of density

\rho_2

(one of mean radius

a

, one of mean radius

b

), derive an expression for

J_2 = (C-A)/Ma^2

of the two-layer planet in terms of

\rho_1

\rho_2

a

b

and

f

, where

M

is the total mass. ((b)) Evaluate

J_2

for

b = 3480\,\mathrm{km}

a = 6371\,\mathrm{km}

\rho_1 = 10\,900\,\mathrm{kg\,m^{-3}}

\rho_2 = 4500\,\mathrm{kg\,m^{-3}}

and

f = 1/300

, and compare with the measured Earth value

J_2 = 1.0826\times10^{-3}

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