A small, spherical planetesimal of radius generates radiogenic heat at a uniform rate…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Earth, Climate & Energy Physics · The solid Earth: structure, fields and thermal history

The Solid Earth, Ch Dating, heat flow and the thermal history of the Earth

Problem

A small, spherical planetesimal of radius

R

generates radiogenic heat at a uniform rate per unit volume

A_{\rm rad}

(treated as constant, ignoring its slow decay), and conducts that heat with uniform thermal conductivity

k

. Its surface is held at a fixed temperature

T_s

by contact with the surrounding nebula or space, and it has reached a steady state. ((a)) Starting from the heat-conduction equation with a uniform internal source, written in spherically symmetric form, derive the steady-state temperature

T(r)

inside the body, stating why one integration constant must vanish. ((b)) Give the temperature excess at the centre,

T(0) - T_s

, in terms of

A_{\rm rad}

R

and

k

. ((c)) Evaluate this excess for

R = 50\,\mathrm{km}

A_{\rm rad} = 4\times10^{-8}\,\mathrm{W\,m^{-3}}

and

k = 3.0\,\mathrm{W\,m^{-1}\,K^{-1}}

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