368 
The Galactic System of Stars 
[ch. XIV 
Star-streaming in the Galactic System. 
340. The hypothesis that the observed star-streaming arises from a state 
of steady-motion admits of a quantitative test which is more exacting than 
the qualitative test so far considered. 
As we have already seen (§ 15), Kapteyn has given a first approximation to 
the density of star-distribution in the regions surrounding the sun, according 
to which the surfaces of equal density are similar ellipsoids. It is easy to 
calculate the gravitational potential arising from such a distribution of stars, 
so that V may be regarded as known at every point. Different forms of the 
function f in formula (339T) will of course give different distribution of star- 
density, and on excluding all those which do not give the observed distribution 
of stars, we are left with all the states of steady-motion which are possible for 
the observed distribution of stars in the sky. It is of interest to discuss how 
far the star-streaming of such steady motions agrees with the star-streaming 
which is actually observed. 
341. The first attack on the problem was made by Kapteyn in 1922*. 
It was based on the assumption that each star belonged to one or other of two 
streams of stars which were revolving about the galactic axis, and that, relative 
to the general motion of the streams to which they belonged, the motions of 
the individual stars obeyed a Maxwellian law of distribution, the mean velocity 
being the same throughout the galactic system. Kapteyn was of opinion that 
this last assumption might be “considered doubtful” in its application to the 
stellar system. From a study of radial velocities he took the mean value of 
a single component of velocity to be 10’3 kilometres a second. As we have 
already seen from Poincaré’s theorem, the mean velocity in a system of stars 
moving in steady motion is determined by the mean gravitational potential 
throughout the system. Kapteyn having estimated the star-density through 
out the system, could estimate the gravitational potential as soon as he 
assigned a definite mass to the average star. He found that this gravitational 
potential would give his assumed stellar velocities if each star had an average 
mass equal to about 1*7 times that of the sun. 
This calculation was based on Kapteyn’s estimate that the distribution of 
stars in the central regions of the galactic system was at the rate of 0'0451 per 
cubic parsec. This figure is too small, since more stars than this are already 
known in the neighbourhood of the sun, and the average mass of a star 
must be correspondingly reduced. Kapteyn’s result shewed, in effect, that to 
produce the observed velocities, the average density of matter in the 
neighbourhood of the sun must be equal to (P0451 x 1*7 times the mass of the 
sun per cubic parsec. If we take the density of stars to be one per ten cubic 
parsecs (§ 8), we get the same density of matter by supposing the mass of 
the average star to be 0'77 times the mass of the sun. 
* Astrophys. Journ. lv. (1922), p. 302.