388-391] Rate of Escape 323 



This may fairly be said to be negligible when x = 30, in which case it has 

 the value 3 x 10~ 14 , and to be appreciable (for our present purpose) when 

 x = 15, in which case it has the value 5 x 10~ 8 . In the former instance, half 

 of the upper atmosphere is lost in about two million years ; in the latter, 

 in about one year. 



We can therefore say that an atmosphere is certainly lost if x < 15, and is 

 certainly retained through millions of years if x > 30. 



NUMERICAL VALUES. 



390. The final part of the problem consists of the calculation of 

 numerical values. We shall find the values of C for which x = 15 and x = 30 



respectively. Since x = -^ , these values are 



= T' To' 



The Earth. 



For the earth, 



^=981, a = 6'4xl0 8 , 



so that the critical values of C are 



3-5 x 10 5 , 2-5 x 10 5 . 



For hydrogen, the value (7 = 2'5xl0 5 corresponds to a temperature of 

 230C. (say 500 absolute), the value (7 = 3'5 x 10 5 to^a temperature of about 

 740C. Since the temperature of the present outer atmosphere of the earth 

 is certainly less than 230C., it is clear that there can be no perceptible 

 escape of gas from our present atmosphere. But if at any time in the history 

 of our earth the temperature of the outer atmosphere has been greater than 

 about 740C., the hydrogen atmosphere will have been lost, and the hydrogen 

 in the present atmosphere must be explained as an addition, probably as the 

 result of chemical action at and near the earth's surface, since that time. 



Sun, Moon, and Planets. 



391. Other atmospheres are best treated by reference to the earth as the 

 standard case. The structure of the planet is involved through the term ga, 



M 



and since g is proportional to where M is the mass, and a the radius of 



M 



the planet, it follows that ga will be proportional to . The following table 



a 



212 



