Stellar Encounters 
319 
286, 287] 
To study the frequency of actual collisions, let us examine how often stars 
of radius equal to the sun’s radius 7 x 10 10 cms. will approach so near as to 
touch or to collide. Putting a = 7 x 10 10 in formula (285-3), and using the 
which collisions occur is 2 - 2 x 10 12 cms., so that the time-interval between 
collisions is 
Thus collisions between stars are so rare that they may be disregarded. 
287. A star’s course may not only be turned by violent encounters of the 
kind we have been considering, but also by a succession of feeble encounters, 
none of which is of much effect by itself but which have a cumulative effect 
equal to one big encounter. 
By differentiation of formula (286T), we find that there are 
The cumulative effect of encounters which produce small deflections 
yjrx, yfr 2 , ... is to produce a deflection of which the expectation T 1 is given by 
Let yfr 2 , yjr 2 ,... be all the deflections of amount between two limits a and ¡3 
which occur within a time t. Then, from formula (287'1), we find that 
Let us take the upper limit of deflection to be /3 = \ir, thus considering 
might at first be thought that to take account of all deflections of amount 
less than r, we ought to take a = 0, but such a procedure would be erroneous 
for the following reason. 
Formula (287'2) is only accurate if the deflections yfr 1} i/r 2 , ... are inde 
pendent, and this requires that they should originate in distinct encounters. 
If \fr is allowed to become very small, the corresponding distance a of closest 
approach, as given by equation (285'4), becomes very large, so that there are 
several stars within a distance a at the same instant, and their effects tend 
values for m, m and V 0 already mentioned, we find that the value of p below 
—4=—- = 6 x 10 17 years. 
7TvV 0 p 2 J 
(m + m'Y V 0 3 sui- 
encounters in unit time which produce a deflection of path between yjr and 
yfr +dyfr. For small deflections, this may be put in the form 
S7rvy 2 m 6 dy\r 
(287*1). 
(m + m'Y V 0 3 
q/' 2 = yfrj 2 + y]r 2 2 + 
(287-2). 
’0 8irvy 2 m 6 d\jr 
(m + in y V 0 3 yfr 
the cumulative effect of deflections less than those considered in § 286. It