2.1. Local Dark Matter
The dynamical density of matter in the Solar vicinity can be estimated using vertical oscillations of stars around the galactic plane. The orbital motions of stars around the galactic center play a much smaller role in determining the local density. Ernst Öpik (1915) found that the summed contribution of all known stellar populations (and interstellar gas) is sufficient to explain the vertical oscillations of stars - in other words, there is no need to assume the existence of a dark population. A similar analysis was made by Kapteyn (1922), who first used the term "Dark Matter" to denote invisible matter which existence is suggested by its gravity only. Both Öpik and Kapteyn found that the amount of invisible matter in the Solar neighbourhood is small.
Another conclusion was obtained by Jan Oort (1932). His analysis indicated that the total density, found from dynamical data, exceeds the density of visible stellar populations by a factor of up to 2. This limit is often called the Oort limit. This result means that the amount of invisible matter in the Solar vicinity should be approximately equal to the amount of visible matter.
The local density of matter has been redetermined by various authors many times. Kuzmin (1952b, 1955) and his students Heino Eelsalu (1959) and Mihkel Jõeveer (1972, 1974) confirmed the earlier result by Öpik. A number of other astronomers, including more recently Oort (1960), Bahcall & Soneira (1980), Bahcall (1984), found results in agreement with Oorts original result. This discussion was open until recently; we will describe the present conclusions below.
For long time no distinction between local and global dark matter was made. The realisation, that these two types of dark matter have very different properties and nature came from the detailed study of galactic models, as we shall discuss below.
2.2. Global Dark Matter - clusters, groups and galaxies
A different mass discrepancy was found by Fritz Zwicky (1933). He measured redshifts of galaxies in the Coma cluster and found that the velocities of individual galaxies with respect to the cluster mean velocity are much larger than those expected from the estimated total mass of the cluster, calculated from masses of individual galaxies. The only way to hold the cluster from rapid expansion is to assume that the cluster contains huge quantities of some invisible dark matter. According to his estimate the amount of dark matter in this cluster exceeds the total mass of cluster galaxies at least tenfolds, probably even more. As characteristic in scientific revolutions, early indications of problems in current paradigms are ignored by the community, this happened also with the Zwicky's discovery.
The next step in the study of masses of systems of galaxies was made by Kahn & Woltjer (1959). They paid attention to the fact that most galaxies have positive redshifts as a result of the expansion of the Universe; only the Andromeda galaxy (M31) has a negative redshift of about 120 km/s, directed toward our Galaxy. This fact can be explained, if both galaxies, M31 and our Galaxy, form a physical system. A negative radial velocity indicates that these galaxies have already passed the apogalacticon of their relative orbit and are presently approaching each other. From the approaching velocity, the mutual distance, and the time since passing the perigalacticon (taken equal to the present age of the Universe), the authors calculated the total mass of the double system. They found that Mtot 1.8 × 1012 M. The conventional masses of the Galaxy and M31 are of the order of 2 × 1011 M. In other words, the authors found evidence for the presence of additional mass in the Local Group of galaxies. The authors suggested that the extra mass is probably in the form of hot gas of temperature about 5 × 105 K. Using more modern data Einasto & Lynden-Bell (1982) made a new estimate of the total mass of the Local Group, using the same approach, and found the total mass of 4.5 ± 0.5 × 1012 M. This estimate is in good agreement with new determinations of the sum of masses of M31 and the Galaxy including their dark halos (see below).
A certain discrepancy was also detected between masses of individual galaxies and masses of pairs and groups of galaxies. The conventional approach for the mass determination of pairs and groups of galaxies is statistical. The method is based on the virial theorem and is almost identical to the procedure used to calculate masses of clusters of galaxies. Instead of a single pair or group often a synthetic group is used consisting of a number of individual pairs or groups. These determinations yield for the mass-to-light ratio (in blue light) the values M / LB = 1 ... 20 for spiral galaxy dominated pairs, and M / LB = 5 ... 90 for elliptical galaxy dominated pairs (for a review see Faber & Gallagher 1979). These ratios are larger than found from local mass indicators of galaxies (velocity dispersions at the center and rotation curves of spiral galaxies). However, it was not clear how serious is the discrepancy between the masses found using global or local mass indicators.
2.3. Rotation curves of galaxies
Another problem with the distribution of mass and mass-to-light ratio was detected in spiral galaxies. Babcock (1939) obtained spectra of the Andromeda galaxy M31, and found that in the outer regions the galaxy is rotating with an unexpectedly high velocity, far above the expected Keplerian velocity. He interpreted this result either as a high mass-to-light ratio in the periphery or as a strong dust absorption. Oort (1940) studied the rotation and surface brightness of the edge-on S0 galaxy NGC 3115, and found in the outer regions a mass-to-light ratio ~ 250. Subsequently, Rubin & Ford (1970) and Roberts & Rots (1973) extended the rotation curve of M31 up to a distance ~ 30 kpc, using optical and radio data, respectively. The rotation speed rises slowly with increasing distance from the center of the galaxy and remains almost constant over radial distances of 16-30 kpc, see Fig. 1.
Figure 1. The rotation curve of M31 by Roberts & Whitehurst (1975). The filled triangles show the optical data from Rubin & Ford (1970), the filled circles show the 21-cm measurements made with the 300-ft radio telescope (reproduced by permission of the AAS and the author).
The rotation data allow to determine the distribution of mass, and the photometric data - the distribution of light. Comparing both distributions one can calculate the local value of the mass-to-light ratio. In the periphery of M31 and other galaxies studied the local value of M / L, calculated from the rotation and photometric data, increases very rapidly outwards, if the mass distribution is calculated directly from the rotation velocity. In the periphery old metal-poor halo-type stellar populations dominate. These metal-poor populations have a low M / L 1 (this value can be checked directly in globular clusters which contain similar old metal-poor stars as the halo). In the peripheral region the luminosity of a galaxy drops rather rapidly, thus the expected circular velocity should decrease according to the Keplerian law. In contrast, in the periphery the rotation speeds of galaxies are almost constant, which leads to very high local values of M / L > 200 near the last points with a measured rotational velocity.
Two possibilities were suggested to solve this controversy. One possibility is to identify the observed rotation velocity with the circular velocity, but this leads to the presence in galaxies of an extended population with a very high M / L. The other possibility is to assume that in the periphery of galaxies there exist non-circular motions which distort the rotation velocity.
To make a choice between the two possibilities for solving the mass discrepancy in galaxies more detailed models of galaxies were needed. In particular, it was necessary to take into account the presence in galaxies stellar populations with different physical properties (age, metal content, colour, M / L value).
2.4. Mass paradox in galaxies from Galactic models
Classical models of elliptical galaxies were found from luminosity profiles and calibrated using either central velocity dispersions, or motions of companion galaxies. The luminosity profiles of disks were often approximated by an exponential law, and bulge and halo dominated ellipticals by the de Vaucouleurs (1953b) law.
Models of spiral galaxies were constructed using rotation velocities. As a rule, the rotation velocity was approximated by some simple formula, such as the Bottlinger (1933) law, or a polynomial. The other possibility was to approximate the spatial density (calculated from the rotation data) by a sum of ellipsoids of constant density (the [Schmidt 1956] model). In the first case there exists a danger that, if the velocity law is not chosen well, then the density in the periphery of the galaxy may have unrealistic values (negative density or too high density, leading to an infinite total mass). If the model is built by superposition of ellipsoids of constant density, then the density is not a smooth function of the distance from the center of the galaxy. To avoid these difficulties Kuzmin (1952a, 1956) developed models with a continuous change of the spatial density, and applied the new technique to M31 and our Galaxy. His method allows to apply this approach also for galaxies consisting of several populations.
A natural generalisation of classical galactic models is the use of all available observational data for spiral and elliptical galaxies, both photometric data on the distribution of colour and light, and kinematical data on the rotation and/or velocity dispersion. Further, it is natural to apply identical methods for modeling of galaxies of different morphological type (including our own Galaxy), and to describe explicitly all major stellar populations, such as the bulge, the disk, the halo, as well as the flat population in spiral galaxies, consisting of young stars and interstellar gas (Einasto 1965).
Multi-component models for spiral and elliptical galaxies using photometric data were constructed by Freeman (1970). To combine photometric and kinematic data, mass-to-light ratios of galactic populations are needed. Luminosities and colours of galaxies in various photometric systems result from the physical evolution of stellar populations that can be modeled. The chemical evolution of galaxies was investigated by Tinsley (1968) and Cameron & Truran (1971). Combined population and physical evolution models were calculated for a representative sample of galaxies by Einasto (1972, 1974). The last calculations showed that it was impossible to reproduce the rotation data by known stellar populations only. The only way to eliminate this conflict was to assume the presence of an unknown population - corona - with a very high value of the mass-to-light ratio, and a large radius and mass. Thus, the detailed modeling confirmed earlier results obtained by simpler models. But here we have one serious difficulty - no known stellar population has so large a M / L value.
Additional arguments for the presence of a spherical massive population in spiral galaxies came from the stability criteria against bar formation, suggested by Ostriker & Peebles (1973). Their numerical calculations demonstrated that initially very flat systems become rapidly thicker (during one revolution of the system) and evolve to a bar-like body. In real spiral galaxies a thin population exists, and it has no bar-like form. In their concluding remarks the authors write: "Presumably even Sc and other relatively 'pure' spirals must have some means of remaining stable, and the possibility exists that those systems also have very large, low-luminosity halos. The picture developed here agrees very well with the fact, noted by several authors (see, for example, Rogstad & Shostak 1972), that the mass-to-light ratio increases rapidly with distance from the center in these systems; the increase may be due to the growing dominance of the high mass-to-light halo over the low mass-to-light ratio disk. It also suggests that the total mass of such systems has been severely underestimated. In particular, the finding of Roberts & Rots (1973) that the rotation curves of several nearby spirals become flat at large distances from the nucleus may indicate the presence of very extended halos having masses that diverge rapidly [M(r) prop to r] with distance."