APPLICATION TO BINARY STAR SYSTEMS. 



109 



of shrinking, which is measured by k. Therefore the final density may 

 be determined, by the use of (94), in terms of the initial density and the 

 final period. 



We shall assume that P=100 years, or approximately the period of a 

 Centauri and $ Scorpii. There are many binaries known with much longer 

 periods than this, as well as many with shorter periods. Since they have 

 presumably all originated in a similar manner, any correct theory must 

 explain the long periods as well as the shorter ones. We shall assume that 

 the bodies have always remained homogeneous, whence Ci = 0.4, which 

 has been shown to be most favorable to the theory of a large increase of 

 period through tidal friction. From the formula for the period in the 

 two-body problem we find, using the volume times the density for the 

 masses, and assuming that the bodies were originally in contact, that P^ 

 varies inversely as the square root of Oq, or 



n = 



constant 



V^o 



(95) 



The constant of this equation is determined by the fact that the density 

 of the sun is 1.41, while in the case of two such stars as the sun we have 

 found Pq to be equal to 0.2324. Therefore the constant is 0.2760. 



From the fact that the moment of momentum of a body simply shrink- 

 ing must remain constant, we find that 



a, = ar^J K 



(96) 



where a/"' is the initial radius and aj the final. Since the density varies 

 inversely as the cube of the radius we have 



— — ^0 = ~f = ^"1^1 density 



(97) 



By formulas (94), (95), and (97), with P = 100 years, Ci = 0.4, the fol- 

 lowing table has been computed: 



