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



outer limit of the atmosphere ; if p is the density at height h, 

 k(li) the mass-absorption coefficient, then 



r oo 



■(h) = \ k(h)pdh. 



n. 



Letl(r) be the intensity of radiation at t in a direction 6 

 with the outward vertical, where 0<#<^7r; and let I'(t) 

 be the intensity at i/r with the inward vertical, where 

 <.^r < \<tt. Assume the material is grey (i.e. has an 

 absorption coefficient the same for all the wave-lengths that 

 are important — in this case the wave-lengths that are pre- 

 dominant in the low-temperature radiation considered). 

 Let B(t) be the intensity of black body radiation for the 

 temperature ruling at the point r ; and let 7rF(r) be the net 

 upward flux of energy per unit area across a horizontal plane 

 at t. Then 



C0S *S = I - B ' « 



cos^'=B-I', ..... (2) 



fi* fin 



!F(t) =1 I (t) sin cos OdO-\ I'(t) sin «f cos yjr dy. 



Jo Jo Q v 



. . . (6) 



Consider the expression 



7rF(V)-7rF(y'), 0'>t"). 



Here 7tF(t') is the net amount of radiant energy entering 

 the lower boundary of the layer (V, t"), ttF(t") the net 

 amount leaving the upper boundary. Hence the difference 

 is the excess of absorption over emission for the whole layer 

 (V, t"). Tims F(t) behaves as an integral, whether or no 

 radiative equilibrium holds ; this is interesting, for in certain 

 forms of radiative equilibrium it appears naturally as an 

 integral* of (1) and (2) in the form F = const. 



Let t 2 be the value of r at the earth's surface. Now 

 suppose with Gold that the complete atmosphere t=0 

 to t = t 1 consists of two shells — an outer one at a uniform 

 temperature from t = to t = t 2 (say), and an inner one in 

 convective equilibrium from r = r 2 to t = t x . Then the outer 

 one will be in radiative equilibrium as a whole, provided 



F(t 2 )-F(0) = 0, ....... (4) 



and this is the equation which determines t 2 . 



* Monthly Notices, lxxxi. p. 862 (1921). 



