APPLICATION TO BINARY STAR SYSTEMS. 107 



according to the Laplacian law and vanishes at the surface (/i=180°), 

 we find from (21) that Ci = 0.26. The more the matter is condensed toward 

 the centers of the bodies, the smaller in general Cj will be. It is apparent 

 from these figures that c^ can not be much greater than 0.33, and therefore 

 that M can not greatly exceed the value for which there are equal roots. 

 Consequently the two roots of (80) can differ but little, which means that 

 tidal friction is not competent to drive two equal stars originating in this 

 way far from each other. 



Let us suppose c, >0.33 so that E has both a maximum and a minimum. 

 By (82) we have 2Wi = CiPo^' Using the value of M given in (83), equation 

 (80) becomes 



Pi-il+c,)P,iP+c,P,i=0 (85) 



Since one of the roots of this equation is P^^ we may factor it into P^— Po* 

 and 



P-c,P,^Pi-c,P,iPi-c,P,=0 (86) 



whose real root gives the minimum value of E. To solve this cubic in 

 PMet 



(87) 



Then by the theory of cubic equations * 



A=|^n*+^'p-l^ = i^(Ci)Po* (88) 



Let us apply these equations to the case of two stars each equal in mass 

 and dimensions to the sun. Taking the radius of the sun at 433,000 miles, 

 we find from 



that in this case 



Po = 0.2324 



We shall take for Cj the largest possible value, that is 0.4, which belongs 

 to a homogeneous sphere. We find from (88) the corresponding largest 

 value of Pi to be 



Pi = 0.307 



which is'the period for a separationof the centers of the bodies of 1,042,400 

 miles. The original separation was 866,000 miles. That is, under the 

 assumption that each of the two components of a binary system is equal 

 to the sun in mass and volume, and that they remain of constant size and 



• Burnside and Panton, Theory of Equations, p. 108. 



