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



content ourselves with observing that in the exact solution 

 the differential coefficient B'(t) has a singularity at t = 0, 

 becoming infinite * like log t. This is easily proved. 



§ 5. Extension to non-grey absorption. — Let us now suppose 

 that the material has a different coefficient of absorption for 

 the incident radiation, say equal to n times that for its own 

 low temperature radiation ; ?iwill usually be a small fraction. 

 The inward solar intensity at t is now $e~ nT . Hence in the 

 flux equation, (19 J, e~ T must be replaced by e~ nT , and in 

 the equation of radiative equilibrium, (18), $e~ T must be 

 replaced by n$e~ nr . Proceeding as before, we find that 



B(t) =gi±i?[l_(l r in>^,- . . . (2$') 



B^S^t*^, B = iSQ+in), 



Ti : T : 4 : T 4 : = \ + n^ : 1 : i(l+i*). . (29') 



As n->0, T*,-**), To 4 -^^ 4 , and the temperature dis- 

 tribution tends to 



B(T) = B(*+t), 



The limiting case is, in fact, the Schwarzschild case for 

 a constant net flux tt¥. Notice that T£= 2T 4 /??. 



§ 6. Extension to oblique incident radiation. — Next suppose 

 that the external radiation is incident at an angle a with the 

 normal. If we preserve the same intrinsic intensity, the 

 amount incident per unit area is now S cos a and the amount 

 crossing unit area at depth r is S cos «r BrseCft . We can 

 obtain the solution by putting S cos u for S and n sec ol 

 for n in the foregoing formulse. We find 



B(r) = S^^ 2 [cos a -(cosa-in>-^--], (28") 



B M = Scosa(cosa + Jw)/;i, B = -|S(cos a + \n), 

 T,* 4 : Tj 4 : T 4 = cos ex. [\ + n' 1 cos a) : cos a : -J (cos ex, + \ii) 



= i + n-icosa : 1 : £(1 + ±rc sec a). . (29") 

 Notice that T w 4 =2 cos «T 4 /n. 



* Cf. Monthly Notices, lxxxi. p. 367 (1921). 



