﻿the Insolation of an Atmosphere. 889 



From the equations 



I + r + i/^-» rseCa = 2B, (26") 



I-I' = Scos«^- nrseea , . . . (27") 

 r = B-^-' irseca (cosa + in). 



Hence 



B, = B(T 1 )+iS(cosa — in)r" r i seca . . . (37) 



If cos a>£w. which will usually be the case in applications, 

 B, is greater than B(r 1 ). Thus the temperature of the 

 surface exceeds that or! the material (say air} in contact 

 with it. Hence convection currents would be set up, and 

 the state o£ radiative equilibrium would be destroyed. 1 his 

 is a simple way of demonstrating the impossibility o£ the 

 existence of a state of radiative equilibrium throughout 

 the entire atmosphere. 



§ 11. The "greenhouse" effect. — Inserting in (27) the 

 value of B(t x ) from (28"), we have 



B s = S cos a[(cos * + ±ri) — (cos a — ^?i)g- ?l7 "i seCa ]/n. (38) 



Now if the black surface were exposed to the direct in- 

 solation 7rS cos a, without the intervention oi an atmosphere, 

 it would take up a temperature T/ given by 



crT/ 4 /V = B s ' = S COS a. 



Hence 



T s * (cosa-KKl-e-T— ) 

 JM-1+ • • • (39) 



Thus, when cosa>^i, the surface is maintained at a 

 temperature higher than it would be in the absence of 

 an atmosphere. The ratio T s 4 /T s ' 4 increases as n decreases, 

 the limit as n— >0 being 1-fTi. The case of diffuse incident 

 radiation is roughly given by putting cosa = ^, and then the 

 condition is n< 1, i. e. that the atmosphere or " protecting 

 layer " must be more transparent to the incident radiation 

 than to the radiation returned. This is the radiation part of 

 the " greenhouse"" or "heat-trap" effect, which is some- 

 times the subject of fallacious statements ; it must of course 

 be distinguished from that part of the effect which is due to 

 the prevention of convection. 



§ 12. Extension to a partially convective atmosphere. — We 

 will now generalize the problem a little further. Suppose 

 that we have a state of affairs in which the material above a 



Phil Mag. Ser. 6. Vol. U. No. 263. Nov. 1922. 3 M 



