294, 295] The Tidal Theory 279 



295. The average time between encounters at a distance equal to or less 

 than R is ; as in formula (545), 



seconds ........................... (593). 



In our present universe, v, the stellar density, is about 5 x 10" 56 , while v 

 averages 40 kms. a second. The average interval between encounters at a 

 distance less than T27 x 10 15 cms. is found to be about 10 18 seconds or 

 3 x 10 10 years. This period is much longer than any reasonable estimate of 

 the age of the universe. Moreover, of the encounters in question, only a few, 

 namely those having small relative velocities, are likely to effect a tidal 

 break-up. Thus tidal break-up is an excessively rare event, and only a small 

 fraction of stars can ever experience it at all. - 



This estimate of frequency of encounters which effect a tidal break-up 

 has of course depended on the assumed density of the broken up star, which 

 we took to be 5'5 x 10~ 12 , corresponding to a solar radius equal to the radius 

 of Neptune's orbit. Greater densities would make tidal break-up still more 

 improbable, the time interval varying as p* for transitory encounters and as 

 p* for slow encounters. No reasonable density could make tidal break-up 

 probable within astronomical time. 



Thus if we suppose the constitution of our stellar universe to have been 

 always as it now is, a tidal break-up would be an abnormal event: the 

 a priori odds against our sun having broken up tidally would be so great 

 that we might feel inclined to discard the tidal theory on the grounds of its 

 inherent improbability. 



We have, however, already had occasion to contemplate an earlier epoch 

 in the evolution of our stellar universe, in which the stars were much closer 

 than now, their relative velocities probably much smaller than now, and their 

 densities very low. Making the appropriate alterations in the numerical data, 

 the mean interval between tidal encounters is greatly reduced. Suppose that 

 in this earlier stage the mass-density in space was of the order of 10~ 18 grammes 

 per cubic cm. Taking the mass of the average star to be 17 times that of the 

 sun, the value of v, the number of stars per cubic cm., would be 3 x 10~ 52 , or 

 10,000 times that previously assumed. The time-scale (593), which varies as 

 \jv, is reduced from 3 x 10 10 years to three million years by this change. The 

 time-scale ought no longer to be compared with the whole supposed age of 

 the universe, but rather with the duration of the epoch in which the stars 

 were closely crowded together. Tidal break-up, even now, can hardly be con- 

 sidered a likely event, but it is considerably more probable than our former 

 calculations would have shewn it to be, and the improbability of close en- 

 counters among the stars no longer provides adequate grounds for rejecting 

 the tidal theory. 



