﻿the Insolation of an Atmosphere. 873 



their temperature would continue to increase until the extra 

 emission due to increased temperature balanced the absorption 

 and a new steady state set in — a state of radiative equilibrium. 

 The direct absorption of solar radiation is small and, though 

 important, does not affect the argument. (It is of interest to 

 mention that exactly the same course of argument shows that 

 even in the absence of convection a strictly isothermal atmo- 

 sphere is impossible ; tor the outer portions would not be able 

 to absorb as much as the} r emitted, and so would cool, causing 

 convection.) 



Gold embodied these ideas in analysis, in order to determine 

 the temperature and the height of the tropopause, and he 

 showed that the theory generally was adequate to account 

 for the observed values. His procedure, however, was in 

 part empirical. In the light of Schwarzschild's # theory of 

 radiative equilibrium in a stellar atmosphere, an immediate 

 rough evaluation of the boundary temperature is possible ; 

 if T is this temperature, then T 4 =^T 1 4 , where T x is the 

 effective temperature of the system (earth plus atmosphere) 

 as determined by the amount of energy radiated away into 

 space. This energy is equal to the mean value of the absorption 

 of solar radiation, assuming that the earth is on the average 

 neither losing nor accumulating energy. The value of T 1? 

 deduced by Abbot | from the solar constant and the earth's 

 albedo, is about 254°, giving T = 214°. The observed mean 

 value of the temperature of the stratosphere over the British 

 Isles is about 21U°. Schwarzschild's formula, T 4 = ^T 1 4 , was 

 indeed obtained independently by Humphreys J in this con- 

 nexion, and applied to the stratosphere. Gold, however, did 

 not proceed in this way. Accepting the observed division of 

 the atmosphere into two shells — an inner one in convective 

 equilibrium with a known temperature gradient, and an outer 

 one 'at a uniform temperature, — he determined the height at 

 which the convective gradient should terminate, in order that 

 the atmosphere above this height should, as a whole, gain as 

 much heat by absorption as it lost by radiation ; the temper- 

 ature of the convective region at this height then gave the 

 temperature of the isothermal re ion. It appeared that a 

 satisfactory balance was obtained if the point of division was 

 taken at a height given by p = \p x , where p is the pressure 

 at any height, p x the ground -pressure. It appeared further 

 that there was very nearly a balance of radiation in the upper 



* Gott. Nach. 1906, p. 41. 



t Annals Astropbys. Obs. Smithson. Inst. ii. p. 174 (1908). ' The 

 Sun' ( Appleton, New York, 1912), p. 323. 

 \ Astrophys. Journ. vol. xxix. p. 26 (1909). 



Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov. 1922. 3 L 



