See — Temperature of the Sun and Ages of Stars and Nebulae. 9 



the planet becomes potential energy. Thus the velocity at 

 any instant is equivalent to that which would be produced 

 by letting the planet fall to its position from rest at a dis- 

 tance 2a'. 



Substituting this value of v', we have 



M t =\mV (l + »') (p—s) < 15 > 



In astronomical units k 2 expresses the mass of the sun and 

 Wm' the mass of the planet. Using M for k 2 in this formula 

 we may write 



B. = M* 0, - l~ a ) + »" - £p) (16) 



Now suppose the planet at perihelion to touch the surface 

 of the sun ; then r = R, and E k will become a maximum. 

 The second term of ( 16) is very small on account of the factor 

 m' 2 ; and therefore may be disregarded. In the remaining 



term the part depending on — is small compared to that de- 



a 

 1 



pending on „, and thus we have approximately 



Mm' 

 ~R 



E k = ^ (17) 



Comparing this expression with (4) we see that 



E k : r = 5m' : ZM. (18) 



But E k is the vis viva of a single planet only, and hence we 

 shall have 



» =* -F-l - * = n K 1 



i= 1 * = 1 



and thus for the solar system 



i = S 



yi. = -Lr (19) 



^ kl 447.6 v } 



i — 1 



