= Steady radiogenic heating of a conducting sphere
{title2=$T(r)=T_S+q_v(R^2-r^2)/(6\kappa)$}
For constant volumetric <heat generation rate> $q_v$ and <thermal conductivity> $\kappa$, spherical steady <thermal conduction> obeys $\kappa r^{-2}(r^2T')'+q_v=0$. Regularity at the centre and a specified surface <temperature> give the displayed parabolic profile. For a black surface radiating into negligible background radiation, global <energy conservation> gives $\sigma_{\rm SB}T_S^4=q_vR/3$. The central excess scales as $R^2$, while the surface <temperature> scales only as $R^{1/4}$.
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