314 Planetary Atmospheres [en. xvn 



The pressure in the atmosphere is given by 



and therefore does not vanish anywhere, and becomes infinite at infinity. 

 It is therefore clear that no steady state is possible for the atmosphere of a 

 rotating planet, for there is no physical agency capable of supplying the 

 requisite pressure at the boundary. This simply means that the atmosphere 

 is continually flowing away to infinity, and so rendering a steady state an 

 impossibility. 



376. Let us, however, examine the atmospheric distribution expressed 

 by equation (753), without at present considering how this distribution is to 

 be maintained. The surfaces of equal density for any constituent gas are 

 given by the equation 



ra a 2 (# 2 + ;y 2 ) 2^ a = constant. 



If g is the value of gravity at the planet's surface, the value of gravity at 

 a distance r from the planet's centre is ^ , so that % = m a ga 2 fr, and the 

 equation of the surfaces of equal density may be taken to be 



w 2 O 2 + y 2 ) + ^ = constant ..................... (754). 



The form of these surfaces was first studied by Edward Roche*; a dis- 

 cussion of his results is given in a paper by Prof. G. H. Bryan f. Sufficient 

 information for our present purpose will be obtained by examining the 

 distribution of density in the equatorial plane of the planet. Replacing 

 & + y* by r 2 , we have in the equatorial plane 



r , , 200,*"} 

 Va = Ae~ hm ^ f -J ........................ (755). 



Differentiating, we find that dv a /dr vanishes when 



(756). 



There is therefore a single minimum of density, and . the position of this 

 minimum is the same for each constituent atmosphere, being in fact the 

 series of points at which the apparent eentrifugal force exactly balances the 

 gravitational attraction of the planet. As we pass outwards the density 

 decreases until this minimum is reached, and afterwards increases con- 

 tinually. 



In the case of our earth, -- , the ratio of the apparent centrifugal force 



t/ 



to gravity at the equator, is ^^, so that the minimum of density occurs at a 

 distance from the earth's centre given by 



r = \X~293a = 6'642a. 



* Memoires Acad. Sci. Montpellier. 



t "The Kinetic Theory of Planetary Atmospheres," Phil. Trans, cxcvi. A, p. 12. 



