Questions about Subsolar Temperature

[MathIsFun]

1. The problem statement, all variables, and given/known data
I'm having a little trouble understanding planetary subsolar temperatures. The first equation comes from viewing the absorbing and radiating areas of an object as the same and the second equation comes from viewing the absorbing area as B and the radiating area as C.

Homework Equations

T_{ss}=\left(\frac{L\left(1-A\right)}{4\pi d^{2} \sigma}\right)^{\frac{1}{4}} T=\left(\frac{BL\left(1-A\right)}{4\pi Cd^{2} \sigma}\right)^{\frac{1}{4}}

L is the luminosity of the star, A is the bond albedo of the object, and d is the star-object distance. These equations are derived from equating the luminosity of the object with the product of the absorbed flux from the star and the absorption area of the object.

The Attempt at a Solution

I understand that the second equation is a more general form, but how do you know when the special case applied in the first equation holds? I think this would require that the energy that is absorbed be retransmitted through the same area before it can be distributed throughout the object, but how can you tell when this will occur? Does the second equation give the subsolar temperature in a general case, or is subsolar temperature specifically when you take B=C?

Thanks

 

[haruspex]

From reading https://en.m.wikipedia.org/wiki/Effective_temperature, I gather that for a spherical planet, taken as a whole, B is πr². If it is rapidly rotating then we take emission as being from the whole surface, so C=4πr²=4B. For a tidally locked planet the radiation is only from the lit half, so for that half C=2B. For a small subsolar patch B=C.