2 3 2 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)
For the calculation of one must know the spatial density for the stars of the
cluster. It is well known that it can be represented in a satisfying manner by the
empirical formula:
. (3)
Here, a is a length proportional to the radius of the cluster, 2a is the radius where
the density has sunk to about 2% of the central density. Furthermore, is the
mean density of the stellar matter within the cluster at a specific point. One does
not commit a substantial error if one calculates as if matter were distributed con-
tinuously with the density . In this manner one gets
(4)
k is the gravitational constant, A is a numerical factor which I find to be about 0.6.
For the radius of the star cluster one gets from (1), taking (2) and (4) into
account,
. (5)
If one puts for the star cluster in Hercules N = 2000, m = 15 masses of the sun
v = 26km/sec, one obtains
According to the apparent brightness of the brightest stars of the cluster, one has
to assume the distance from us such that its radius cannot be less than 100 light
years. Therefore, there must be an error in our assumption.
I had the opportunity to discuss the present difficulty in detail with my col-
leagues at the institute of astrophysics in Potsdam. The result was that with our
present knowledge of masses and distribution of fixed stars, one of my assumptions
is substantially in error. The majority by far of the fixed stars in a star cluster had
to be considerably lower in luminosity than the approximately 2,000 stars that ap-
pear on the photographic plate under short exposure, but without need to assume
their masses as substantially smaller than those of the brightest stars. From pictures
L N
mv-
2
2
---------
⋅ =
Φ ρ
[7]
ρ
3
4π
------
N
a3
---- -
1
r2⎞
a2⎠
----⎟ - +
⎝
⎜
⎛
5
2
- –--
=
ρm
[p. 52]
Φ
ρm
Φ
A----------------.
kN
2
m
2
a
=
[8]
2a 1.2
kNm
v2
-----------
=
[9]
2a 0.65
1018cm
0.65 Lightyears. = ⋅ =
[10]