The surface temperature of the sun is ${\mathrm{T}}_0$ and it is at average distance d from a planet. The radius of the sun is R. The temperature at which planet radiates the energy is

r = Radius of planet

$T_1 = Temperature of planet at which energy is radiated Power radiated by Sun, P = 4\pi R^2\sigma {T_0}^4$

Energy received by planet = $\frac{P}{4{\mathrm{\pi d}}^2}\times {\mathrm{\pi r}}^2 = \frac{{\mathrm{R}}^2{{\mathrm{T}}_0}^4{\mathrm{\pi r}}^2\mathrm{\sigma }}{{\mathrm{d}}^2}$

Energy radiated by planet = $4{\mathrm{\pi }}^2{{\mathrm{\sigma T}}_1}^4$

For thermal equilibrium,

$\frac{R^2{T_0}^4\mathrm{\pi } {\mathrm{r}}^2\mathrm{\sigma }}{d^2} = 4{\mathrm{\pi r}}^2{{\mathrm{\sigma T}}_1}^4$

${\mathrm{R}}^2{{\mathrm{T}}_0}^4 = 4{\mathrm{d}}^2{{\mathrm{T}}_1}^4$

${\mathrm{T}}_1 = {\mathrm{T}}_{\mathrm{o}}\sqrt{\frac{\mathrm{R}}{2\mathrm{d}}}$