﻿the Insolation of an Atmosphere. b'77 



We now approximate. Setting t = 2r, t 1 = 2r 1 , etc., and 

 using t as the variable specifying position, equations (1), 

 (2), (3) can be written approximately * 



§ = I-B, g-B-r, . . .(5), (6) 



F(t)= 1(0 -I'M, (7) 



and the equation for t 2 is 



l(t 2 )-V(t 2 )-l(0) = 0, (8) 



since the incident radiation I'(0) is zero f. Solving (5) 

 and (6) with the assumption that the air near the ground 

 has the same temperature as the ground and that the earth 

 radiates like a black body, we find 



l(t) = e * f B(0«-*«fr-f B(tj)e-K-*>dt, . . (9) 



V{t)= e-tyBfte'dt. (10) 



These can be inserted in (8), and t 2 determined as soon 

 as B(t) is known as a function of t in the convective 

 region. 



Now the excess of absorption over emission in a small 

 element of volume dv is 



//K/r[jlJa>+jIW-47rB] 



= 2tt kp dv^^ n J sin OdO + f^T sin fdyjr- 2Bl 



= 27rkpdv[I(t)+T(t)-2B(t)] 



approximately. Denote the expression in square brackets 

 by E(£). Then for the values of the excess of absorption 

 over emission at the top and bottom of the isothermal region 

 we have respectively 



E(0) = I(0)-2B(0), 



E(* 2 ) = I(* 2 )+I'(< 2 )-2B(t 2 ), . . . (11) 

 or. using (8), 



E(0) = I(i 2 )-I'(« 2 )-2B(0). . . . (12) 



* For details, see e. <j. Monthly Notices, lxxxi. p. 363 (1921). 

 t Ignoring solar radiation. See below. 



