D O C . 5 6 G L O B U L A R S T A R C L U S T E R S 2 3 1

to us, then one can also determine the distance of the star cluster from us. From

the apparent radius of the star cluster follows—at least within the order of magni-

tude—the true radius of the star cluster, and for the latter one has found values of

100–500 light years in this manner.

It is probably safe to assume that the bright stars of the star cluster are approx-

imately similar to the absolutely bright stars in our neighborhood. For the latter

ones it has been established—using Doppler’s principle—that they move relative

to each other with a mean velocity of about 26

km/sec,1

and we can probably

assume that this is also the order of magnitude of the mean velocity of the bright

stars of the star cluster relative to the center of gravity of the latter, the more so as

it has been shown that the mean velocity of the stars of different spectral types

also agree with each other within the order of magnitude.

We also assume that the distribution of stars in a stellar cluster is stationary in-

sofar as the latter does not substantially change its radius and its star distribution

(when considered from a statistical point of view) over a time during which indi-

vidual stars of the cluster traverse a (curvilinear) path that is large compared to the

radius of the cluster. It can hardly be questioned that this condition is satisfied for

the radially symmetric and statistical distribution shared by many stellar clusters.

Then it is possible to apply the virial theorem by Clausius to the star cluster as a

whole, by treating individual stars as material points. In the case of Newtonian

forces this yields, as H. Poincaré probably was the first to show,

. (1)

L is here the combined kinetic energy of all the stars of the cluster; is the nega-

tive potential energy which is to be attributed to the cluster if the zero point of the

potential energy of the stars is defined such that it vanishes when the distance

between stars approaches infinity.

In order to be able to draw conclusions from equation (1), I make approximate

assumptions about the structure of the cluster. I treat the stars of the cluster that are

imaged on the photographic plate under short exposure as being all of equal mass

m, and let N be the number of these types of stars within the whole cluster. Fur-

thermore, I assume for the time being that the less luminous stars, that is, also the

smaller ones in the cluster, do not substantially contribute to the gravitational field

of the cluster, such that they can be neglected in the calculation of L and . One

then immediately gets, if v is the (quadratic) mean of velocity

1More

precisely: relative to the center of gravity of the system to which they belong.

[4]

[p. 51]

[5]

[6]

L

1

2

-- -

Φ =

Φ

Φ

v

v2

=

⎝

⎠,

⎛ ⎞