An exoplanet survey reports the following for a newly characterised rocky planet:…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · 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

An exoplanet survey reports the following for a newly characterised rocky planet: equatorial radius

a = 6.00\times10^{6}\,\mathrm{m}

GM = 3.00\times10^{14}\,\mathrm{m^3\,s^{-2}}

, rotation period

T = 18.0\,\mathrm{hours}

, dynamical form factor

J_2 = 4.50\times10^{-3}

(from the satellite orbit) and dynamical ellipticity

(C-A)/C = 1/71.0

(from a measured precession rate). A separate radar-altimetry survey gives the planet's flattening directly as

f = 1/98.7

. ((a)) Compute the rotational parameter

m = \omega^2a^3/GM

. ((b)) Use Clairaut's theorem,

f = \tfrac32J_2 + \tfrac12m

, to predict the flattening, and check it against the surveyed value. ((c)) Use MacCullagh's formula together with

J_2

and

(C-A)/C

to find

C/Ma^2

. ((d)) Use the Darwin-Radau relation with the surveyed

f

and

m

to find the value of

C/Ma^2

a hydrostatic planet with this shape and spin would need. Compare it with part (c) and say whether this planet can be close to hydrostatic equilibrium.

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