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



approximate solution, however, it is quicker to employ the 

 approximate forms of equations (16) to (19), obtained in 

 the usual way. These are 



1^1 IdV 



2 dr~ ' 2 dr 



= I-B, T^=B-r, . . (24), (25) 



I + r + iS<r r = 2B, •. (26) 



I-V = $e- T (27) 



From the two latter, 



I =B + iS6- 



I' = B-fS< 



T 



But I ; (0) = 0. Consequently B = fS. Inserting this 

 approximate value of I in (24), we find 



dr 4 ^ ' 



whence, using the value of B already found, 



B(T) = fS(l-K r ) (28) 



It follows that there is a limiting temperature in the far 

 interior, given by B^ =|S. If ' now T is the boundary 

 temperature, T x the effective temperature of the whole 

 mass viewed from the outside, T^ the temperature in the 

 far interior, and a Stefan's constant, we have 



o-T 4 = ttB = f 7rS, 

 crT^ = ttBoc — §7rS, 



and thus 



o-IY = ttS, 

 TirV :T 4 = !■: 1 :| (29) 



It is important to notice that T^ is different from T 1? 

 contrary to what might have been anticipated ; also that 

 the relation between T } and T is different from that in the 

 Schwarzschild case, where the' net flux is the same at all 

 depths. Notice also TJ = 2T 4 . 



These values and the general distribution of temperature 

 given by (28) are only approximations. To test them, 

 let us re-employ (28) in (20) to obtain I and so check 

 the radiative equilibrium at the boundary and the net flux 



