Steady radiogenic heating of a conducting sphere ($T(r)=T_S+q_v(R^2-r^2)/(6\kappa )$)

For constant volumetric heat generation rate $q_v$ and thermal conductivity $\kappa$, spherical steady thermal conduction obeys $\kappa r^{-2}(r^2{T}^{\prime }{)}^{\prime }+q_v=0$. 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 ${\sigma }_{\mathrm{SB}}{T}_S^4=q_{v}R/3$. The central excess scales as $R^2$, while the surface temperature scales only as $R^{1/4}$.