Question

A planet of radius R is revolving a round sun in an orbit of radius r. If the temperature of sun is Ts and its radius is Rs. the equilibrium temp of the plant assuming both sun and planet as black bodies is $T=Ts\sqrt{\frac{Rs}{nr}}$. find n.

Answer 1

Heat radiated per unit time by the planet, $P_p=\sigma e4\pi R^2{T}^4$

Heat radiated per unit time by the sun, $P_s=\sigma e4\pi R_s^2{T}_s^4$

where $e=1$ as both the planet and the sun are black bodies.

Imagine a sphere of radius $r$ around the sun.

Heat radiated by the sun and incident per unit surface area per unit time of the sphere is $\frac{P_s}{4\pi r^2}$

Heat incident on the planet , $P_a=\frac{P_s}{4\pi r^2}\pi R^2$ as cross section area of earth is $\pi R^2$

Heat absorbed per unit time by the planet, $P_a=\sigma 4\pi R_s^2{T}_s^4\frac{\pi R^2}{4\pi r^2}$

Now as the planet is is equilibrium, $P_a=P_p$

$\sigma 4\pi R_s^2{T}_s^4\frac{\pi R^2}{4\pi r^2}=\sigma 4\pi R^2{T}^4$

$⟹T=T_s\sqrt{\frac{R_s}{2r}}$