To tackle this problem, we need to apply the principles of black body radiation and the concept of thermal equilibrium. The sun, acting as a black body, emits radiation that the Earth absorbs. Since the Earth is also treated as a black body, it will radiate energy back into space. The key here is to find the relationship between the temperatures of the sun and the Earth based on their respective radii and distances.

Understanding Black Body Radiation

A black body is an idealized physical object that absorbs all incoming radiation and re-emits energy based on its temperature. The Stefan-Boltzmann Law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature:

P = σT^4, where σ is the Stefan-Boltzmann constant.

Radiation from the Sun

The total power emitted by the sun can be calculated using its surface temperature (Ts) and radius (Rs). The surface area of the sun is given by:

A_sun = 4πRs²

Thus, the total power (P_sun) radiated by the sun is:

P_sun = A_sun * σTs^4 = 4πRs²σTs^4

Radiation Absorbed by the Earth

The Earth, located at a distance R from the surface of the sun, receives this radiation. The intensity of radiation reaching the Earth can be approximated as parallel rays due to the large distance. The power received by the Earth (P_earth) can be calculated by considering the area over which this power is distributed:

P_earth = P_sun / (4πR²)

Substituting the expression for P_sun, we have:

P_earth = (4πRs²σTs^4) / (4πR²) = (Rs²σTs^4) / R²

Equilibrium Temperature of the Earth

At thermal equilibrium, the power absorbed by the Earth equals the power it radiates. The power radiated by the Earth (P_earth_radiated) is given by:

P_earth_radiated = 4πR_earth²σT^4

Setting the power absorbed equal to the power radiated:

(Rs²σTs^4) / R² = 4πR_earth²σT^4

We can simplify this equation by canceling out σ and rearranging:

Rs²Ts^4 / R² = 4R_earth²T^4

From this, we can isolate T:

T^4 = (Rs²Ts^4) / (4R²R_earth²)

Taking the fourth root gives us:

T = Ts * (Rs / (2R))

Final Result

Thus, the temperature of the Earth (T) can be expressed as:

T = Ts * (Rs / (2R))

This matches option A: T = Ts Rs / 2R.

Summary

In summary, by applying the principles of black body radiation and thermal equilibrium, we derived the relationship between the temperatures of the sun and the Earth. The Earth absorbs radiation from the sun and, at equilibrium, radiates energy at a temperature that depends on the sun's temperature and the geometric factors of their respective sizes and distances.