Model a small icy moon as a uniform sphere of radius and total mass

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator

Problem

Model a small icy moon as a uniform sphere of radius R=1.5×106 mR=1.5\times10^{6}\,\mathrm m and total mass M=2.0×1022 kgM=2.0\times10^{22}\,\mathrm{kg}, with specific heat capacity c=1.0×103 J kg−1 K−1c=1.0\times10^{3}\,\mathrm{J\,kg^{-1}\,K^{-1}} and uniform volumetric radiogenic heating f=5.0×10−8 W m−3f=5.0\times10^{-8}\,\mathrm{W\,m^{-3}} throughout its interior. A spacecraft measures the surface heat flux and finds it uniform over the whole surface, q=0.020 W m−2q=0.020\,\mathrm{W\,m^{-2}}, directed outward. ((a)) Steady conduction obeys ∇2u=−f/k\nabla^2u=-f/k with outward heat-flux vector q=−k∇u\mathbf q=-k\nabla u. Using Green's first identity with ϕ=1\phi=1, ψ=u\psi=u (as in any steady conduction problem), state the compatibility condition the total measured outward flux over a closed surface must satisfy, in terms of the total power generated inside, for the interior to be in a steady thermal state. ((b)) Decide whether the moon is in a steady thermal state on this model, and if not, find the rate of change of its mean interior temperature, treating the moon as a uniform lumped mass.

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