258 The Evolution of Binary and Multiple Stars [CH. xi 



increase to more than 1 J times its initial value. From formula (582) it follows 

 that (1 + f) I cannot increase to more than 1*56 times its initial value. For 

 bodies at a considerable distance apart f = ; for two similar ellipsoids in 

 contact = 22, which is the maximum value of f. Thus in the whole course 

 of evolution the value of 1 -f f cannot decrease more than in the ratio 1/22:1. 

 It follows that at the very most I cannot increase in a ratio greater than 

 1-56 x 1-22 or T90. 



These calculations refer to a perfectly homogeneous mass. To study the 

 effect of compressibility let us pass to the extreme case of matter so com- 

 pressible that Roche's model ( 149) may be supposed to give an approximation 

 to the arrangement of density. We may put & 2 = k'* and f = 0. The whole 

 momentum is orbital, and the constancy of M requires that I shall remain 

 constant. Thus we may reasonably suppose that compressibility lessens the 

 possible range of increase in I and that the ratio of 90 per cent, just calculated 

 for an incompressible mass is the maximum possible, always provided the mass 

 ratio does not exceed 2^ : 1, and that the system remains free from external 

 disturbance. 



270. Similar calculations can be made with respect to the period. Calling 

 this P we have 



As evolution progresses ^ (1 + f)*, which is proportional to the orbital 

 momentum, will increase, but not in a ratio greater than 1*25 : 1 if M/M' < 2*5. 

 Similarly 1 + f will decrease but not in a ratio beyond 1'22 : 1. The factor 

 (1 e*Y will decrease to an unknown extent, and may decrease beyond limit. 



Thus there is theoretically no limit to the increase of P, but large increases 

 can only occur through 1 e 2 becoming very small, so that a binary in which 

 P has increased largely must have an almost parabolic orbit. Observation 

 has so far revealed no binary with a nearly parabolic orbit ; the largest observed 

 eccentricities are 0'90 found by Aitken for 7 Virginis and 0'88 found by 

 Campbell for #AHetis; for these 1 e 2 = 0'19 N and 023 respectively. The 

 average values of e for binaries of different types will be seen from Campbell's 

 tables given on pp. 255, 6. Campbell has catalogued e for 75 spectroscopic 

 binaries*; in only one case (jB Arietis just mentioned) is e greater than 0*80 ; 

 similarly out of 50 visual binaries, e has a value greater than 0'80 in only three 

 cases (7 Virginis, e = 0'90 ; 7Androm. BC, e = 0'82 ; 99 Herculis, e = 0'81). 

 Thus we may take e = 0'80 as an upper limit for e for the great majority of 

 binaries ; this makes 1 e 2 = 0'36 and the maximum evolutionary decrease in 

 (I - e 2 ) f may be taken to be one of 1 : 0'216. 



* I have excluded Cepheids in view of the uncertainty as to whether these really are binaries 

 or not. 



