Question

Mars has no atmosphere and its surface temperature can be calculated with a simple energy balance model. Its radius is 3400 km and its albedo is 0.25, i.e. 25% of the incoming solar radiation is reflected back into space (without being absorbed by the surface). Assuming that in the distant past the sun had less power than today and the radiant flux reaching Mars was only I_Mars = 494 W/m², what was the surface temperature of Mars? Give your answer in degrees kelvin (K) to three significant figures.

Answer 1

In this problem, distance between Sun and Mars is not given.

Hence , we assume simple energy balance equation.

Step 1: Power absorbed by mars

P_{abs = (1-a) F where a= 0.25 albedo cooefficient , F= heat flux reaching surface}

Step 2: Power radiated by mars  

Using Stefan-Boltzmann Law and Assuming mars as a Perfectly Black Body

P_rad = where   ^{Stefan's Constant of Radiation}

Step 3:ByEnergyBalance Equation

Power absorbed= power radiated

370.5 = 5.67 x 10^-8 x T⁴

T⁴ =65.3439 x 10⁸

T = 2.8431 x 10 ²

T = 284.31 K

Summary : Consider heat flux coming to planet uniformly over entire surface but the part of heat will be absorbed only other than albedo!

Thus calculate power absorbed. Then assuming the planet as a perfectly black body it must radiate the heat at all temperatures. Since temperature of planet

must remain constant, it must radiate the same power with which it absorbs the power. For radiation one has to use Stefan's law. Equating the two we can find the temperature of the planet.

Note: If distance of planet from sun is given then we have to use other formulae for heat absorbed.