2.2 Physical properties of large-scale structures

2.2.1 Masses and the mass to light ratio of galaxies; dark matter

In cosmology, mass and mass distribution are among the most important physical parameters. The masses of galaxies can be determined directly from their rotation curves and from membership in binary systems and groups and clusters of galaxies, making in each case appropriate assumptions. Gravitional lensing (see Sect. 2.1.1) promises to become an important tool for mass determination in the near future. Masses can be derived indirectly from the mass to light ratio for those galaxies which are too distant for direct measurements from rotation curves.

For clusters of galaxies, masses were first determined from the velocity dispersion of the cluster members under the assumption that the virial theorem can be applied. This method was used by Zwicky (1933) to determine the mass of the Coma cluster and by Smith (1936) to derive the mass of the Virgo cluster.

Methods of mass determination for both galaxies and clusters of galaxies were presented by Zwicky (1937c):

``Present estimates of the masses of nebulae are based on observations of the luminosities and internal rotations of nebulae. It is shown that both these methods are unreliable; that from the observed luminosities and extragalactic systems only lower limits for the values of their masses can be obtained . . ., and that from internal rotations alone no determination of the masses of nebulae is possible . . .
. . . three new methods for the determination of nebular masses are discussed, each of which makes use of a different fundamental principle of physics.
Method (1) is based on the virial theorem of classical mechanics. The application of this theorem to the Coma cluster leads to a minimum value = 4.5 · 10¹⁰ M for the average mass of its member nebulae.
Method (2) calls for the calls for the observation among nebulae of certain gravitational lens effects.
[Method (3)] gives a generalization of the principles of ordinary statistical mechanics to the whole system of nebulae, which suggests a new and powerful method which ultimately should enable us to determine the masses of all types of nebulae. This method is very flexible and is capable of many modes of application. It is proposed, in particular, to investigate the distribution of nebulae in individual great clusters.''

The last method is one of the early instances of using methods of probability theory to describe galaxy clustering (see Sect. 2.3.1).

The luminosity function can be used to derive the total luminous mass of a cluster from the mass/light ratio of its individual members.

The earliest determination of the mass to light ratio was made by Öpik (1922) for our own Galaxy in order to be used in his determination of the distance d to the Andromeda nebula. His value is

Mass = 2.6 Luminosity (solar units)

(the most recently determined value is 2.7).

With the rotation speed at a given angular radius and the apparent magnitude of the nebula within this radius, the assumption that the centripetal acceleration balances the gravitational acceleration and the ratio M / L from our Galaxy, he had all the data needed to determine d.

Rotation values were first reported by Wolf (M81, 1914) and Slipher (NGC 4594, 1914), and later by Pease (1916, 1918).

The same ratio as employed by Öpik was used by Hubble (1926) to determine the mass of the Andromeda galaxy after he had measured its distance.

Most of the early cosmologists (Lemaître 1931b, Zwicky 1933, Hubble 1934) were aware of the possible existence of dark matter. The problem of hidden mass will not be discussed here, though it has become increasingly more debated in cosmology, and in some models is given an important role in the formation of galaxies.