﻿the Insolation of an Atmosphere. 895 



would permit the escape of extra radiation which would 

 have no effect in controlling the temperature. But the 

 boundary temperature is not necessarily lower : it may be 

 higher or lower, according to the spectral regions in which 

 the air is not transparent. It has recently been shown by 

 the author * that it" a thin layer of material is exposed 

 on one side to black radiation of effective temperature T\ 

 and is in radiative equilibrium, then it will take up a 

 temperature T which is equal to 2~*T 1 (the Schwarzscnild 

 value) when the material is grey, but which, whatever the 

 optical properties of the material, will satisfy the inequalities 



T 1 >T >iT 1 ; 



and that T will approximate to T 1 if the material is trans- 

 parent save in the extreme ultra-violet, to ^T\ if transparent 

 save in the extreme infra-red. (Here Ci ultra-violet " and 

 14 infra-red " must be interpreted as relative to the value 

 of Xmax. corresponding to T 1 .) If, for simplicity, we 

 neglect the absorption of solar radiation, the theorem can 

 be applied as it stands to the stratosphere. It shows that 

 the temperature will be less than for uniform absorption 

 only if the absorption occurs principally on the long wave- 

 length side of Xmax. — in this case about 10 /j,. But water- 

 vapour is least opaque | to long wave-length radiation in 

 the region 7 /u. to 20 //,, more particularly in the region 

 8 /jl to 12 fi. There is, indeed, important carbon-dioxide 

 absorption J in the region 13 fj, to 16 //,, but this is usually 

 considered to be not large compared with the water-vapour 

 absorption. Without a detailed numerical investigation 

 it would be difficult to estimate the resultant effect ; but 

 if carbon-dioxide absorption is important near the equator, 

 it should be more important in higher latitudes, and this 

 would go against the argument. If a relation T = eT 1 

 holds above the equator in virtue of selective radiative 

 equilibrium, it ought to hold too in higher latitudes ; for it 

 is difficult to see why the atmosphere at 20 km. above the 

 equator should be optically different from that in higher 

 latitudes in the direction of being relatively more absorptive 

 above the equator on the red side of X ma x.- Moreover, 

 if we do modify the selective optical properties of the 

 atmosphere from equator to pole, then Gold's explanation 



* " The Temperature in the Outer Atmosphere of a Star," Monthly 

 Notices R.A.S. lxxii. p. 368 (1922). See also Fabry, Astrophvs. 

 Journ. xlv. p. 269 (1917). 



t Abbot, Annals Astrophys. Obs. Smithson. Inst. ii. p. 167 (1908). 



X See, for example, Humphreys, ' Physics of the Air,' p. 88, 1920. 



