﻿the Insolation of an Atmosphere. 875 



Now Gold applied these conditions in various ways to show 

 under what circumstances a convective atmosphere c;ui or 

 c mnot exist: e.g., he showed that a convective atmosphere 

 cannot extend indefinitely, yet must extend above p = jjj) l . 

 But he did not point out that his final solution was incon- 

 sistent with these conditions. We shall show that although 

 on the assumptions made the layer (£/?i, 0) is neither gaining 

 nor losing heat as a whole, yet its upper portions are emitting 

 more than they are absorbing, and its lower portions absorbing- 

 more than they arc emitting; consequently the upper layers 

 must cool and sink, the lower ones warm and rise, convection 

 will occur, and the state of isothermal equilibrium must be 

 destroyed. Further, although the layer (ijt>i, \p\) satisfies 

 the conditions for convective equilibrium as a whole, emission 

 exceeding absorption, in the upper portions absorption exceeds 

 emission, so that a steady convective state in this region is 

 not possible ; the smallness of the excess of emission ever 

 absorption for the whole layer, attributed by Gold to the 

 slightness of convection required, is merely the result of the 

 excess in the lower portions being balanced by the deficiency 

 in the upper ones. 



Actually we can prove a more precise result than this, 

 under very general conditions. We shall show that the 

 excess of absorption over emission at the base of Gold's 

 isothermal layer, per unit optical mass, is numerically equal 

 to the excess of emission over absorption at the top, whatever 

 the temperature distribution in the convective layer and 

 whatever the law connecting the coefficient of absorption 

 with height. To do this we shall employ the approximate 

 form of the equations of transfer of radiant energy. It may 

 be mentioned here that though Gold uses the ex-dct formulse 

 (involving Ei functions) which take full account of the 

 spherical divergence of the radiation, his results can be 

 obtained more simply to the same degree of precision by 

 using the approximate formulae and by making free use of 

 the optical thickness and the net flux of radiation. The 

 quite small errors of the approximate formulae are swallowed 

 up in the uncertainty of the numerical data that have finally 

 to be employed. The uncertainty arises in the final trans- 

 lation of the optical thicknesses into actual thicknesses ; but, 

 as in other cases of radiative equilibrium, many of the results 

 hold in a form independent of the numerical values of the 

 absorption coefficients. 



Let t be the optical thickness measured inwards from the 



3 L 2 



