384-388] Rate of Escape 321 



386. This expression is not, however, independent of R, as we might 

 at first have expected it to be. The reason for this is as follows. In the 

 complete atmosphere, supposing it to be constituted according to Maxwell's 

 Law throughout, there will be a number of molecules describing orbits which 

 never pass within a sufficiently small distance from the earth's centre for 

 the chance of collision to be appreciable. Some of these describe hyperbolic 

 or parabolic orbits, passing from infinity past the earth to infinity again 

 without collision. Now if p is the distance of the apse of any orbit from the 

 earth's centre, it is clear that a molecule describing this orbit will be counted 

 in expression (763) as escaping from the earth's atmosphere if R > p, but not 

 if R<p. We should therefore expect expression (763) to increase with R 

 as is in fact seen to be the case. 



It is questionable whether molecules of the kind just described ought to 

 be supposed to exist in the actual atmosphere. The analysis by which the 

 specification of the steady state is arrived at, takes no account of the length 

 of time required for the establishment of this steady state. In the present 

 instance the steady state implies the arrival of molecules which have 

 described hyperbolic and parabolic orbits from infinity. It is therefore 

 obvious that it will require infinite time to establish the steady state. 



On the other hand, molecules which are supposed to describe orbits 

 in the regions in which no collisions occur have no influence on the rest of 

 the atmosphere and may therefore be removed without disturbing the 

 equilibrium of the remainder of the atmosphere. In nature these molecules 

 cannot be supposed to exist. They would be counted in our estimate of the 

 escape of molecules from the atmosphere by taking R great. We therefore 

 obtain the most accurate results by taking R as small as possible, and the 

 error vanishes altogether when we reduce the sphere of radius R to such 

 a size that collisions may be regarded as frequent everywhere inside it. 



387. In the case of a rotating atmosphere, we found that there must be 

 supposed to be a complete atmosphere extending to infinity, lying outside 

 the region in which practically no collisions occur. This atmosphere can 

 be treated in the same way in which individual molecules coming from 

 infinity have been treated. It can be supposed to be removed bodily without 

 disturbing the equilibrium of the remainder of the atmosphere. 



It must, nevertheless, be noticed that the order of magnitude of expression 

 (763) is determined solely by the exponential e - zhm v a , so that the value of 

 Zhmga determines whether the escape of molecules is appreciable or not. 

 This criterion, as we should expect, is independent of R. 



388. The total number of molecules of any specified kind in the planet's 

 outer atmosphere, supposed limited by spheres of radii a and R, is 



f 



J a 



J. 21 



