This is the energy density.
At this stage, we are in a position to use each of the above moments to further manipulate the equation 0f transfer. We start by multiplying the original equation of transfer: (cos q) dI/ dt = I (q) – j/ s through by 1/4p ò 4p I (q) dw to get:
From here a number of specific assumptions are made in order to not have to evaluate the integral. The main one is the Eddington approximation which will apply to the quantities J, H and K. We will also discriminate the radiation intensity I into two components: I1 (in the forward direction) and I2 (in the backward direction). We can then write as follows:
Further, K = J/3 or (½I1)/3 so K = Ht + const. This follows, since we had: dK/ dC = H or dK = H dt and we know t= 0, hence Ht + const. on integration. From this it follows that:
Problems for the budding astrophysicist:
1) Find the mean intensity if I (q=p/4) is taken over all space.
2) Estimate the net flux, H, passing through the Sun’s surface.
3) 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.
b) The change in (a) over each absorption and re-emission over vertical optical depth.
c) The total absorption and total emission and the relationship between then over all space.
(All content extracted from my book: 'Astronomy & Astrophysics: Notes, Problems and Solutions', Chapters 15, 17)