376 
The Galactic System of Stars [ch. xiv 
be small in comparison, and this will result in a flattening perpendicular to 
the direction of motion of the cluster. Thus clusters ought to shew flattening 
at right angles both to the galactic axis and to their own direction of motion. 
Rasmuson’s investigation indicates that most clusters shew precisely the 
required flattening. 
Expansion and Evolution of Clusters. 
349. The calculations already given in Chapter xii enable us to estimate 
the rate at which a cluster expands. For simplicity let us consider a cluster 
bombarded in all directions equally, so that 
where I denotes the total impulsive force at an encounter. 
From formula (285*5) we find that 
7 m 
4J>’ 
where the stars are all supposed to have equal mass m. On summing over 
all encounters which occur in unit time, after the manner adopted in § 287, 
we find 
^ T " 7ry 2 m 2 
— (£2) = 
dt KÇ ' 12 A 
48A V, 
>£)■ 
where 2 p a is the mean distance between stars, and 2is the distance of 
approach for an encounter which deflects a star’s orbit through a right angle. 
On inserting the numerical values (cf. § 287) 
log e (J) = 11-9, A = 0*755 vp, V 0 = 10«, m = 2 x 10 33 , 
this becomes 
J t (P) = 0-33 = 1-4 x 10 20 . 
For a point at which £ = 20 parsecs, the expectation of rate of growth is 
only about a centimetre a second, and a growth of a further 20 parsecs would 
require 3 x 10 12 years. 
While the cluster is growing in size, the velocities of its members 
will be gradually deviating from their parallel paths, so that the cluster is 
losing its identity both by its members being scattered in space and by their 
motions becoming scattered in direction. The formulae of § 287 shew that 
with the numerical values already used (m = in! — 2 x 10 33 , i/ = 4x 10 -67 , 
F=10 kms. a second) a period of 20,000 million years will see about 
one star in 1000 knocked entirely out of the cluster by a violent encounter, 
while the average angle between the directions of motion of the surviving 
members will be about one degree. After 500,000 million years one- 
fortieth of the original members of the cluster will have been lost by violent 
encounters, while the mean angle of the velocities of the remainder is