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(D) The radiant power (luminosity) of the sun is given by the Stefan-Boltzmann law: $E = \sigma A T^4$.Since the surface area $A = 4\pi R^2$,we have $

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$64$

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(D) The radiant power (luminosity) of the sun is given by the Stefan-Boltzmann law: $E = \sigma A T^4$.Since the surface area $A = 4\pi R^2$,we have $A \propto R^2$.Therefore,the radiant energy $E \propto R^2 T^4$.Let $E_1$ be the initial energy and $E_2$ be the final energy.Given $R_1 = R$,$T_1 = T$ and $R_2 = 2R$,$T_2 = 2T$.The ratio is given by $\frac{E_2}{E_1} = \frac{R_2^2 T_2^4}{R_1^2 T_1^4}$.Substituting the values: $\frac{E_2}{E_1} = \frac{(2R)^2 (2T)^4}{R^2 T^4} = \frac{4R^2 \cdot 16T^4}{R^2 T^4} = 4 \cdot 16 = 64$.

If the temperature of the sun were to increase from $T$ to $2T$ and its radius from $R$ to $2R$,then the ratio of the radiant energy received on earth to what it was previously will be

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