﻿874 Mr. E. A. Milne on Radiative Equilibrium: 



layer of the convective region extending from p=*ipi to 

 p = ±p 1 , from which Gold deduced that in this layer the 

 convection would be small. 



It is the object of this paper to point out a certain difficulty 

 in Gold's work, and to consider an idealized problem which 

 is suggested by it in the absorption of radiation by an atmo- 

 sphere subject to insolation. 



Since the paper was first written, the author has become 

 aware of a paper by Emden* which anticipates portions of it. 

 Emden criticized Gold's theory on certain points, and investi- 

 gated the general theory of the radiative equilibrium of an 

 atmosphere by a method similar to that of the present paper. 

 Where necessary, the paper has been recast to take account 

 of Emden 's work. 



§2. Criticism of Gold's solution. — One of the most in- 

 teresting points in Gold's discussion is his isolation and 

 explicit formulation of the condition for a convective atmo- 

 sphere. In such an atmosphere, transfers of energy are 

 being effected both by radiation and by convection, and 

 across any plane there will be a net radiative flux and a 

 convective flux. Now convection can only transfer heat 

 upwards, not downwards. But assuming a steady state, the 

 upward convective flux plus the net radiative flux must be 

 equal to the downward solar flux. Hence the net radiative 

 flux (as due to the earth and atmosphere together) mast be 

 less than the downward solar flux. But the downward solar 

 flux at any point cannot exceed its value at the boundary ; 

 and at the boundary the downward solar flux must be equal 

 to the outward flux due to the earth and atmosphere. Hence 

 another form of the condition is : the net out ir ard flux across 

 any plane must be less than its value at the boundary. Again, 

 the upper layers must be gaining more heat by convection 

 from below than they are losing to layers above. Hence, 

 for a steady state, emission of radiation must exceed absorption 

 in the upper layers (for emission must equal absorption plus 

 net gain by convection) . Whenever these inequalities become 

 equalities, radiative equilibrium holds ; if they become re- 

 versed the state cannot be a steady one, for it would involve 

 convection downwards f. 



* " Uber Strahlungsgleichgewicht und atmospharische Strahlung : 

 ein Beitrag zur Theorie der oberen Inversion," Sitz. d. K. Ah ad. Wiss. 

 %u Munchen, 1913, pp. 55-142. 



f Gold's conditions have been applied by the writer to stellar 

 atmospheres in a paper recently communicated to the Royal Society. 



