﻿880 Mr. E. A. Milne on Radiative Equilibrium : 



any plane parallel to the surface there will be a net outward 

 flux of radiation derived from the material just balancing the 

 inward flux of the residual solar radiation. In the far interior 

 the latter will be greatly attenuated, and consequently the 

 outward flux there must be small too. We should expect, 

 therefore, that the temperature gradient in the far interior 

 would be small; and this proves to be the case. In fact, 

 not only is there a definite limiting temperature at the outer 

 boundary, as in the Schwarzschild ease, but there is also a 

 definite limiting temperature in the far interior. This is one 

 of the most interesting characteristics of the model we are 

 discussing. 



Let t be the optical thickness measured from the outer 

 boundary to any point ; I, I' the outward and inward 

 intensities at any point at angles and yjr with the normal ; 

 B(t) the intensity of black radiation for the temperature at 

 the point r ; 7r8 the intensity of the parallel beam of incident 

 solar radiation defined as the energy incident per second per 

 unit area normal to the beam. Here I and I' are to refer 

 only to radiation derived from the material. It must be noted 

 that since we have assumed the solar radiation to constitute 

 a parallel beam, the definition of its intensity is necessarily 

 different from the standard definition for conically spreading 

 pencils *. 



The residual solar intensity at any depth t is 7r$e~r. The 

 equations of transfer are 



cos0^ = I-B, (16) 



cos*g' = B-I'. . ..... (17) 



The amount of energy emitted by an element dv per second 

 is 4:7rkpBdv. That absorbed is 



kpdv[§Id(o + ^I'dco'], 



together with _. , ' 



& 7r$e- T kpdv. 



Hence the equation of radiative equilibrium is 



( 9 *I(t) sin0d0+ ( "*I'(t) sin-f if + iSe-' = 2B(r). (18) 

 Jo Jo 



The flux relation follows from (16), (17), (18), namely, 



I I(t) sin 6 cos 0d0—\ V{t) sin^ cos^d^ = iSe~ r . (1.9) 

 Jo Jo 



* See Planck, Warmestrahlung, p. 15 (3rd edition). 



