Question

As a red giant, the Sun’s luminosity will be about 2000 times greater than it is now. Calculate the surface temperature of the Earth under these conditions assuming that the Earth is still at 1 AU and that the photosphere of the sun is well within 1 AU.

the answer should be about 1900 K but im not sure what am i getting wrong in my calculations, please help!

Answer 1

The surface temperature of the earth can be calculated approximately from a radiative balance:

Where is the solar constant, pi R²_E is the cross sectional area of the earth (the area which "intercepts" the heat), alpha is the albedo of the earth (this part accounts for the fraction of the energy that is reflected), 4pi R²_E is the surface area of the earth, sigma is the Stefan-Bolztmann constant and T the equilibrium temperature of the earth. If we make them equal and then solve for T we get:

So we can see that T is proportional to the fourth root of the solar constant S. On the other hand the solar constant is proportional to the luminusity:

So if the luminosity of the sun were to increase by a factor 2000 so would the solar constant, and then the temperature of the earth would increase by a factor equal to the fourth root of 2000 = 6.7 times approx. Right now the mean Temperature calculated with this model is about 20°C = 290 K (S=1360W/m^2) then the temperature in presence of the red giant would be 290 K x 6.7 = 1943K approx