4 2 4 D O C . 5 6 G L O B U L A R S TA R C L U S T E R S
Published in Festschrift der Kaiser-Wilhelm-Gesellschaft zur Förderung der Wissenschaften zu ihrem
zehnjährigen Jubiläum dargebracht von ihren Instituten. Berlin: Springer, 1921, pp. 50–52. Also
available are a manuscript of an earlier version of the last paragraphs with calculations on the verso
(GyB Autogr. I/1675), and several pages with calculations of the size and potential energy of a glob-
ular cluster among Einstein’s lecture notes for an introductory course on mechanics at the University
of Zurich in the winter semester 1909–1910 (Vol. 3, Doc. 1). [3 004, 3 005]. The earlier version of the
last paragraphs is presented in note 9 below, the calculations are reproduced in facsimile in
A special session in honor of the tenth anniversary of the foundation of the Kaiser Wilhelm Soci-
ety was held on 18 March 1921 (see Henning and Kazemi 1988, p. 34).
As Einstein had argued in Einstein 1917b (Vol. 6, Doc. 43), the Machian idea that inertia is
reducible to a relation to all the masses in the universe requires that the universe be finite, i.e., roughly
spherical. To allow for a stable universe, he had introduced the cosmological term which, interpreted
within the Newtonian theory of gravitation, is equivalent to a universal constant negative mass density
(see Einstein 1918d [Doc. 3], p. 166).
He outlined how to test this consequence of his theory in Einstein 1921c (Doc. 52), p. 13, almost
simultaneously with the present document: if we knew the statistical distribution of stars within a stel-
lar system such as our galaxy and their masses, we could calculate with Newton’s theory their gravi-
tational field and their average velocities necessary to prevent the galaxy from collapsing. If the
observed velocities prove to be different from the calculated ones, this difference would indicate a
nonzero cosmological constant and give a means for estimating its value, and from that the spatial size
of the universe (according to the relation given in Einstein 1917b [Vol. 6, Doc. 43], p. 152). Since the
additional mass term given by the cosmological constant is very small (half the average mass density
of the universe), this method can be applied only to a large system of low density.
The present document appears to be the result of an attempt at such a calculation, using globular
clusters instead of our galaxy.
It is highly probable that Einstein received the information on the data and simplifying assump-
tions used in this document from Erwin Freundlich (1885–1964), the astronomer working on the
empirical confirmation of general relativity for Einstein’s Kaiser Wilhelm Institute of Physics at the
Potsdam Astrophysical Observatory (see Vol. 8, Introduction, p. xxxviii); they also discussed the
problems in person (see the last sentence of Erwin Freundlich to Einstein, 6 December 1919).
In his proposal of research conceived in 1917, Freundlich envisaged the observation of gravita-
tional light deflection and redshift. He concluded the proposal with a short note that astronomy may
contribute to the test of the new physical theories, e.g., by investigating globular star clusters (see Er-
win Freundlich to Einstein, 17 June 1917 [Vol. 8, Doc. 353]).
In the next four years, Freundlich, simultaneously with his attempt to find observational evidence
for the redshift, performed calculations on the mass and density of stars in the globular cluster M13
from their distribution by spectral type, taking into consideration the deviation of the shape of the
cluster from the spherical (see Erwin Freundlich to Einstein, 6 December 1919; Freundlich’s report
for 1919, January 1920 [GyBP, I. Abt., 34, 2, Mappe Freundlich]; and Erwin Freundlich to Einstein,
24 February 1920). The results were published as Freundlich and Heiskanen 1923.
Einstein not only followed Freundlich’s progress, but also played an active role in the calculations
(see Moritz Schlick to Einstein, 19 December 1919; Freundlich’s report for 1919; “Bericht über die
Versammlung der Astronomischen Gesellschaft zu Potsdam 1921 August 24–27,” Vierteljahrsschrift
der Astronomischen Gesellschaft 56 : 141–164, p. 157; “25. Generalversammlung der Astro-
nomischen Gesellschaft in Potsdam,” Astronomische Nachrichten 214 : 98–104, p. 102; and
Freundlich and Heiskanen 1923).
Toward the end of 1920, Freundlich sent detailed instructions to Einstein on how to calculate the
volume density of globular clusters from the surface density of their projection in the vision radius,
using Hugo von Zeipel’s method (Zeipel 1908) and Schuster’s law (see note 7), and concluded with
the remark that Einstein had asked for them (Erwin Freundlich to Einstein, 14 December 1920). Ap-
parently this was the point when Einstein started his calculations on the diameter of a cluster using
Newton’s theory of gravitation and the virial theorem. Fragments of his handwritten calculations are
reproduced in Appendix A.