292-294] The Tidal Theory 277 



The action between the two masses in a tidal encounter may be of varying 

 degrees of rapidity. These we have classified as slow, intermediate, and 

 transitory. 



In a " slow " encounter, the changes are so leisurely that an equilibrium 

 theory of the tides is supposed to give a good enough approximation. For 

 these we found that break-up would occur if the secondary mass (M') ap- 

 proached to within about 2'2 x (M'fM)* radii of the primary. This limiting 

 distance was approximately the same both for the incompressible mass and 

 for Roche's model, so that it may reasonably be expected to be about the 

 same also for all intermediate types of structure. 



At the other end of the scale come " transitory " encounters. Here the 

 tidal forces are supposed to be impulsive ; their function is to set the parts 

 of the primary into relative motion and the break-up occurs in the subsequent 

 motion of the primary under its own internal forces. Unfortunately it has 

 so far only proved possible to work out the details of this motion for the 

 incompressible model. 



294. We have found ( 130) that, with relative velocities of the order of 

 present stellar velocities (40 kms. a second), all encounters except the very 

 closest ones may be classified as transitory in the very closest ones, the 

 action is still more rapid, but the forces may not be treated as impulsive 

 because the primary has departed substantially from its original spherical 

 shape before the tidal forces disappear. We have found that a transitory 

 encounter will break up an incompressible mass if 



.(591), 



0*675 wp* 



where R is the periastron distance, M' the mass of the tide-raising body, 

 v the relative velocity, p the density of the primary, and 7 the gravitation 

 constant (now restored). 



Our system is unlikely to have been broken up by a very massive star, 

 for these are rare. It is likely to have been broken up by a star of mass 

 rather above the average, for massive stars are more likely to effect a break-up 

 than lighter ones. For definiteness, let us assign to M ' a value equal to 

 twice the sun's mass, or 4 x 10 33 grammes. Equation (591) becomes 



R< 1-24 x lO^xtr*?"* ..................... (592). 



Taking v = 40 kms. a second and p = 5'5 x 10~ 12 , this being the density of 

 our sun expanded to a sphere filling the orbit of Neptune, we find the limit 

 for R to be 4'05 x 10 14 cms., which is slightly less than the radius of Neptune's 

 orbit. 



Thus with a secondary of double the sun's mass, a relative velocity of 

 40 kms. a second would require actual grazing contact before tidal forces 



