The surface temperature of the sun is $T_{0}$ and it is at average distance $d$ from a planet. The radius of the sun is $R$ . What is the temperature at which planet radiates the energy.

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Solution

Given: The surface temperature of the sun is $T_{0}$ and it is at average distance d from a planet. The radius of the sun is R.
To find the temperature at which planet radiates the energy.
Solution:
Let $P$ be power radiated by the sun
Given, $R$ is radius of the sun, $d$ is average distance between sun and planet and $T_{0}$ is the surface temperature of the sun
if $T$ is the temperature at which planet radiates the energy.
Energy received by the planet = $W A = σ T^{4} (4 π d^{2})$
where $W = σ T^{4}$ is the flux and $A$ is the area
Energy radiated by planet = $(4 π R^{2}) σ T_{0}^{4}$
For thermal equilibrium,
$σ T^{4} (4 π d^{2}) = (4 π R^{2}) σ T_{0}^{4} ⟹ T^{4} = \frac{R^{2} T_{0}^{4}}{d^{2}} ⟹ T = {(\frac{R^{2} T_{0}^{4}}{d^{2}})}^{\frac{1}{4}}$
is the temperature at which planet radiates the energy.

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The surface temperature of the sun is T0 and it is at average distance d from a planet. The radius of the sun is R. What is the temperature at which planet radiates the energy.

The surface temperature of the sun is $T_{0}$ and it is at average distance $d$ from a planet. The radius of the sun is $R$ . What is the temperature at which planet radiates the energy.