Solutions To Practical Astrophysics Source Function Problems

 The Problem:

Consider the solar half-sphere and the energy going into it each second.  We know on average photons are absorbed after traveling a distance with optical depth t =1 in the propagation direction. Averaged over all directions this corresponds to a vertical optical depth of t = 2/3. Based on this find:

a) The energy going into the half sphere each second.

Solution: 

This is just:   s(T_eff)⁴   =  p F = D 2p S =D 2p I

b) The change in (a) over each absorption and re-emission over vertical optical depth.

Solution:

Over a vertical optical depth one has t = 2/3, then:

D 2p S/D t   =   p F/ (2/3)  = 3p F/ 2   

or:

DS/D t   =   3p F/ 2 (2p) = 3F/4

c) The total absorption and total emission and the relationship between them, defined over all space.

Solution:

Over all space, the total emission = 4p S and the total absorption = 4p J, and by radiative equilibrium:   4p S = 4p J  so that S = J.