330-333] 
Dynamical Discussion 
363 
Steady motion of a System of Stars. 
331. With a view to searching for an explanation of this and other observed 
phenomena in the galactic system, we proceed to a dynamical discussion of 
the motions of a system of stars, which move under one another’s gravitational 
attractions. 
Consider first a spherical space in which stars are scattered with fairly 
uniform density, the mean density of matter being p. The gravitational force 
at a distance r from the centre is a force towards the centre, and under 
this force a star will describe an elliptic orbit with a period equal to 
. / — seconds (331T). 
V 7p 
It is noteworthy that this period depends only on the density and not on 
the radius r of the system. 
In the neighbourhood of the sun, we have supposed there to be a star 
density of one star per ten cubic parsecs, and if the average star is supposed 
to be of mass equal to the sun, the average density of matter in space is found 
to be 7 x 10 -24 grammes per cubic cm. The period of an orbit described in a 
field of matter of this density is found, from formula (331T), to be about 150 
million years. 
Thus a star moving in the neighbourhood of the sun, or in regions of 
space of equal star density, will have its path turned through an angle of 
360° in about 150 million years by the gravitational forces arising from the 
stars as a whole. Our process of averaging the density through space has not 
allowed for the special deviations of path produced by near encounters with 
individual stars, but formula (287'4) shews that in 150 million years the 
expectation of the total deflection produced by such near encounters is only 
a few minutes of arc. 
Apart from the precise figures we have used, this makes it clear that the 
motion of a star is determined almost exclusively by the main gravitational 
field of the system to which it belongs, so that, apart from exceptional cases, 
the influence of near stars may be neglected. 
332. This at once explains the continued existence of moving star clusters 
(§ 26). The members of these clusters are so far apart that the encounters 
with near stars must be different for each. These encounters produce only 
very slight effect on the motion, so that, the stars continue to describe approxi 
mately parallel courses under the uniform gravitational field of the stars as a 
whole. Detailed calculations are given in § 349 below. 
333. The problem of tracing the paths of stars now reduces to the problem 
of motion in a steady gravitational field, the field being produced by the stars 
themselves. In brief, the statistical problem of stellar motion is that of the 
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