106 THE TIDAL PROBLEM. 



IX. APPLICATION TO BINARY STAR SYSTEMS. 



If the earth and moon were derived by the fission of a parent mass, 

 the process has presumably been exemplified elsewhere. We shall conclude 

 that the earth has had an exceptional origin only as a last resort. It has 

 been many times suggested that the binary stars may have originated by 

 the breaking up of larger masses, and See especially has urged this view 

 and applied Darwin's formulas in an attempt to explain the dimensions 

 and eccentricities of their orbits.^ 



We shall apply the methods developed here to the problem. In order 

 to simpUfy it as much as possible we shall suppose first that the parent 

 mass divided into two similar and equal masses. We know that this rela- 

 tion of the masses of the two members of binary stars is, in a number of 

 cases, nearly fulfilled, and nothing is known to make this assumption seem 

 improbable. We shall assume that immediately after the fission the system 

 moved as a rigid mass. Then the equations for the moment of momentum 

 and energy are F i 2m 



differing from those in section V only in that m, has been replaced by 2m,. 

 The condition for a maximum or minimum of E is [cf. eq. (28)] 



PJ-MP + 2wi=0 (80) 



As has been shown, when M is sufficiently large this equation has two 

 real roots. The smallest M for which there are real roots is that for which 

 the real roots are equal. The condition for equal roots of (80) is 



M=|PJ =1^6^ (81) 



Consider the system moving as a rigid body with the two stars in contact . 



Then their orbital moment of momentum will be — times their rotational 

 moment of momentum, or ^ 



pi=^,{&ndP=D) (82) 



Consequently the whole moment of momentum of the system is, calling 

 this special value of the common period Pg, 



M = (l+Ci)Po' (83) 



In order that this may be at least as great as the value defined in (81) 

 we must have _ , ,o.s 



= 1 



Ci>i 



The value of c^ depends only upon the law of variation of density of the 

 bodies. When they are homogeneous Ci = 0.4. When the density varies 



* Inaugural Dissertation, Berlin, 1892. 



