120-123] The Tidal Problem 121 



Equation (359) may now be put in the form 



When IJL is given as a function of the time, this becomes_an equation 

 of motion for e, and so enables us to follow the. changes in the shape of 

 the spheroid. The equation cannot be completely solved, but the general 

 character of the motion is easily traced out. 



122. To represent roughly the approach and passage of a second star, 

 we may suppose //, to start from a zero value and increase slowly at first 

 but afterwards more rapidly. The initial conditions will be e = and e = 0. 

 So long as e is so small that its square may be neglected, equation (362) 

 reduces to 



ee = 37rp(l-e*)(^ - F (e)} . 



\7Tp ^ J 



Provided that /JL changes slowly, the solution will remain very close to 

 that of 



so long as this equation has a solution i.e. so long as //, is less than "12 0777). 

 But as soon as //, exceeds this critical value, f^/Trp F (e) can no longer 

 vanish or remain small, so that no matter how slowly //, increases, e becomes 

 finite and e 2 necessarily becomes appreciable. The eccentricity now increases 

 rapidly, its changes being given by equation (362). This determines the 

 dynamical motion which occurs when statical configurations of equilibrium 

 are no longer possible, and we see that it consists of a passage along the 

 unstable series of spheroids, the rate of motion being determined by equa- 

 tion (362). 



When fj, increases more rapidly, there will be no sharp change in the 

 character of the motion on passing the critical value //, = '125^; the statical 

 and dynamical parts of the motion merge imperceptibly into one another. 



In either case equation (362) shews that the motion may or may not pass 

 the value e 1. In the latter case e increases until a " turning point " is 

 reached, defined by e = 0, after which it decreases, ultimately coming back 

 to rest at the value e when dissipative forces are present. At the turning 

 point e is negative, so that equation (362) shews that //. must be less than 

 7rpF(e) and therefore a fortiori less than the maximum value of TrpF (e) 

 which is '125^/0. Thus e goes on increasing not merely while p is increasing 

 but also through the whole period in which /u > 'I257rp. 



123. A case in which the motion can be fully determined, and is more- 

 over of great importance, is that in which ^ increases and decreases with 

 great rapidity, so that the primary is " impulsively " set into motion before 



