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(A) In thermal equilibrium, the energy radiated by the planet must equal the energy absorbed by the planet from the star.According to the Stefan-Boltz

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$\sqrt{R^{*} / d}$

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(A) In thermal equilibrium, the energy radiated by the planet must equal the energy absorbed by the planet from the star.According to the Stefan-Boltzmann law, the power radiated by the planet of radius $R_{p}$ at temperature $T_{p} = f T^{*}$ is $P_{out} = \sigma (4 \pi R_{p}^{2}) (f T^{*})^{4}$.The power received by the planet from the star is the intensity of radiation at distance $d$ multiplied by the cross-sectional area of the planet: $P_{in} = \left( \frac{\sigma (4 \pi R^{*2}) T^{*4}}{4 \pi d^{2}} \right) (\pi R_{p}^{2}) = \sigma \pi R_{p}^{2} T^{*4} \left( \frac{R^{*}}{d} \right)^{2}$.Equating $P_{in} = P_{out}$:$4 \pi \sigma R_{p}^{2} f^{4} T^{*4} = \pi \sigma R_{p}^{2} T^{*4} \left( \frac{R^{*}}{d} \right)^{2}$.Simplifying the equation:$4 f^{4} = \left( \frac{R^{*}}{d} \right)^{2}$.Taking the fourth root on both sides:$f \propto \sqrt{\frac{R^{*}}{d}}$.

$A$ planet of radius $R_{p}$ is revolving around a star of radius $R^{}$ which is at temperature $T^{}$. The distance between the star and the planet is $d$. If the planet's temperature is $f T^{*}$, then $f$ is proportional to

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$A$ planet of radius $R_{p}$ is revolving around a star of radius $R^{*}$ which is at temperature $T^{*}$. The distance between the star and the planet is $d$. If the planet's temperature is $f T^{*}$, then $f$ is proportional to