380-384] Rate of Escape 319 



orbit. The molecules which leave the earth's atmosphere are therefore 

 those which cross the sphere r = 2a with a velocity equal to or greater 

 than \/ga. 



At a point on the sphere r = 2, of which the latitude is 6, the gas has a 

 mass velocity due to the earth's rotation, equal to 2a<w cos 0. This, however, 

 as we have seen, is small compared with *Jga, so that the effect of the 

 rotation of the earth in determining the dissipation of its atmosphere is 

 slight. We shall neglect it altogether, as our calculations are necessarily of 

 the roughest kind. 



383. There has, however, been an arbitrary element introduced in 

 choosing the radius of the sphere to be 2a. Let us consider the more 



general sphere of unknown radius R drawn in the earth's atmosphere. The 



go? 

 gravitational potential at the surface of this sphere is ^ , so that the critical 



, .. . /2gra 2 . x 



velocity is c = A/ -%- , let us say c . 



Maxwell's Law must of course hold throughout the outer atmosphere, 

 which is in conductive equilibrium, and we shall suppose the sphere of 

 radius R to lie within this part of the atmosphere. If at any point of this 

 sphere we take axes in such a direction that the axis of z coincides with the 

 outward normal, we find that the number of molecules of any type and of 

 class A (i.e. having velocity components with a given range dudvdw) which 

 cross an element dS of the sphere in time dt is 



(759), 



7T 



where v is the molecular density of molecules of.jbhis type at the sphere 

 r = R t and is therefore, by equation (758), given by 



a (R - a) 



-Zhmg - 5 /t7#A\ 



V = V 6 R v ............................ (760). 



Integrating expression (759), we find for the total number of molecules 

 crossing the sphere r = R in the outward direction in time dt 



dte- hm ^ + 2+w ^wdudvdw (761), 



7T 3 JjJ 



where the integration extends over all values of u, v, w for which w is 

 positive and ^- . 7 



M 2 + V 2 + W 2 > C 2 . 



384. It will facilitate integration to divide the integral 



fff 



1=11 Ie- hm(u * +v2+w ^wdudvdw 



into two parts, so that 



