The temperature of earth is maintained by a dynamic equilibrium between sun and Earth. Sun & Earth can be assumed to be black bodies:

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If the distance between earth and sun would increases by a factor of 2, the equilibrium temperature of earth would decreases by factor of $\sqrt{2}$

If the radius of earth doubles without any change in sun. It’s equilibrium temperature would increase by factor of $\sqrt{2}$

If the power output of sun would double by changing the temperature, equilibrium temperature of earth also doubles

If the radius of sun doubles without changing it's power, its surface temperature would decrease by factor of $\sqrt{2}$

answer is B, D.

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Detailed Solution

$\begin{array}{l} P o w e r o u t p u t o f S u n = δ (1) 4 π R_{s}^{2} T_{s}^{4} \\ \begin{array}{l} \Rightarrow T_{S} \propto \frac{1}{\sqrt{R_{S}}} \\ \frac{T_{1}}{T_{2}} = \sqrt{\frac{R_{2}}{R_{1}}} \end{array} \\ \Rightarrow \frac{T_{1}}{T_{2}} = \sqrt{\frac{2 R}{R}} \\ \Rightarrow T_{2} = \frac{T_{1}}{\sqrt{2}} \end{array}$

$\begin{array}{l} P o w e r r e c e i v e d b y E a r t h = \frac{δ (1) 4 π R_{s}^{2} T_{s}^{4}}{4 π d^{2}} π R_{E}^{2} = δ (1) (4 π R_{E}^{2}) T_{E}^{4} \\ \Rightarrow T_{E} \propto \frac{1}{\sqrt{d}} \\ \frac{T_{1}}{T_{2}} = \sqrt{\frac{d_{2}}{d_{1}}} \\ \Rightarrow \frac{T_{1}}{T_{2}} = \sqrt{\frac{2 d}{d}} \\ \Rightarrow T_{2} = \frac{T_{1}}{\sqrt{2}} \end{array}$

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