318 Planetary Atmospheres [CH. xvn 



high as C., the atmosphere may be regarded as consisting entirely of 

 hydrogen. If we take 



i/o = 4 x 10 14 , 



C = 18-4 x 10 4 , 

 we have, at height z = a, 



v = 



= 340, roughly ; 

 so that, taking a = 2 x 10~ 8 , the mean free path is 



= 1'6 x 10 12 cms., roughly, 



so that even in this least favourable case the free path is many times greater 

 than a. 



We have assumed a temperature of C., although the actual tempera- 

 ture of the outer atmosphere is probably much lower than this. A lower 

 temperature means a greater value for h, and hence a smaller value for v 

 and a greater mean free path. The result is therefore a fortiori true at 

 temperatures lower than that which we have assumed. 



Thus even on the hypothesis of isothermal equilibrium throughout, the 

 calculated free path at a height a above the earth's surface is many thousands 

 of times as great as a : d fortiori the result is true when the density is 

 diminished by the fact of part of the atmosphere being in adiabatic instead 

 of isothermal equilibrium. 



381. It follows that outside a sphere of radius equal to twice the earth's 

 radius, we have an atmosphere in which practically no collisions occur. The 

 molecules are simply describing orbits about the earth uninfluenced by one 

 another, the velocities and directions of these orbits being determined by 

 their last collision, which may be supposed to have taken place inside the 

 sphere r = 2a. Of these orbits, some will be elliptic, some parabolic, and 

 some hyperbolic. As has already been remarked, those molecules which 

 describe parabolic or hyperbolic orbits must be regarded as permanently lost 

 to the earth's atmosphere. 



DISSIPATION OF PLANETARY ATMOSPHERES. 



The Earth. 



382. The number of molecules which cross the sphere r = 2a and 

 describe parabolic or hyperbolic orbits is identical with the number which 

 cross with a velocity equal to or greater than that necessary to carry them 

 to infinity against the earth's attraction. At the surface of a sphere of 

 radius 2a the gravitational potential is \ga, so that a molecule with a 

 velocity c, such that %c 2 = ^ga, will just pass to infinity, describing a parabolic 



