Question

3) Consider a spherical planet in our solar system with radius, R, that behaves like a perfect blackbody, absorbing all of the sunlight hitting its surface and radiating light isotropically according to its temperature. At what range of distances from the Sun could this planet support liquid water on its surface?

Hint: Solve for the equilibrium temperature of the planet where the light energy it absorbs equals the energy radiated away, and then find the distances where this temperature is between 273 ≤ T ≤ 373 K. This range of distances is called the “habitable zone”, which can be calculated for any star. Compare your answer for the Sun to the size of Earth’s orbit, 1 AU.

Answer 1

Let d be the distance to the planet.

the power of solar radiation received per square meter of the planet is given by

Where L is the luminosity of the sun

L = 3.86*10^26 W

The total power emitted by the planet is equal to the total power received by it.

where A is the area of the surface where the light falls.

(r is the radius of the planet.

The power emitted by a black body at temperature T is given by

Equating all the equations,

At T = 273 K,

At T = 373 K,

The radius of earth's orbit is 1AU = 1.5*10^11 m

The earth is very close compared to this distance. This is because earth reflects most of the sunlight received by it and is not a perfect black body as in the question.