If the temperature of the sun is increased from T to 2 T and...

If the temperature of the sun is increased from $T$ to $2 T$ and its radius from $R$ to $2 R$, then the ratio of the radiant energy received on earth to what it was previously will be
(A) 4
(B) 16
(C) 32
(D) 64

Key Concepts

Combined Effect on Radiant Energy Received

The received radiant energy depends on the star's total luminosity and the distance from the star. Since the distance is constant, only changes in luminosity affect the energy received on Earth. Doubling the sun's temperature increases the energy output by a factor of 16 (due to the T^4 relationship), and doubling the radius increases it by a factor of 4. Multiplying these factors shows that the total increase in radiant energy received is 64 times.

Geometric Scaling (Surface Area Increase)

The surface area of a sphere scales with the square of its radius. When the sun’s radius is doubled, the surface area (and hence the contribution to the overall luminosity) increases by a factor of four. This geometric scaling is a key factor in determining how changes in size affect the radiant energy.

Luminosity

Luminosity is the total energy emitted by a star per unit time. For a spherical star, it depends on both the temperature and the surface area (which is proportional to the square of the radius). An increase in both temperature and radius will significantly boost the luminosity, thereby increasing the energy received on Earth.

Stefan-Boltzmann Law

This law states that the power radiated per unit area of a blackbody is proportional to the fourth power of its temperature. It is crucial because it relates an object's temperature to its total emitted energy, which is essential in understanding how a change in the sun’s temperature affects its luminosity.

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