The surface temperature of the Sun is $T_{o}$ 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 $T_{o} \sqrt{\frac{R}{a d}}$ where $a$ is constant. Find $a$ (Answer upto two digit after the decimal point)

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Solution

$r =$ Radius of planet
$T_{1} =$ Temperature of planet at which energy is radiated
Power radiated by Sun, $P = 4 π R^{2} σ T_{o}^{4}$
Energy received by planet = $\frac{P}{4 π d^{2}} \times π r^{2} = \frac{R^{2} T_{o}^{4} π r^{2} σ}{d^{2}}$
Energy radiated by planet $= 4 π r^{2} σ T_{1}^{4}$
For thermal equilibrium, $\frac{R^{2} T_{o}^{4} π r^{2} σ}{d^{2}} = 4 π r^{2} σ T_{1}^{4}$
$R^{2} T_{o}^{4} = 4 d^{2} T_{1}^{4}$
$T_{1} = T_{0} \sqrt{\frac{R}{2 d}}$

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The surface temperature of the Sun is $T_{o}$ 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 $T_{o} \sqrt{\frac{R}{a d}}$ where $a$ is constant. Find $a$ (Answer upto two digit after the decimal point)