2. THE SOLAR NEIGHBORHOOD

The velocities and distributions of stars perpendicular to the Galactic plane reflect the amount of mass in the local disk of the Milky Way. Velocities can be used to obtain the stellar velocity dispersion perpendicular to the plane < v²_z >, whereas distances provide the density distribution perpendicular to the plane, (z). The gravitational potential due to the disk, (z), is related to these quantities through the Jeans equation:

(2.1)

so that measurements of (z) and v_z provide an estimate of the density of the disk in the solar neighborhood.

Oort (1932) pioneered attempts to measure the disk density and found that the local disk mass exceeded the value which could be accounted for in visible material (Oort 1960). The problem was later tackled by Bahcall (1984a, b, c) who used star counts of both F dwarfs and K giants, as well as a range of Galactic mass models, to investigate the presence of DM in the solar neighborhood. He numerically solved the combined Poisson and Boltzmann equations, given by

(2.2)

and

(2.3)

respectively. Here _disk(z) is the total disk density, _halo^eff the density contribution due to the halo, and _i(z) is the density of the ith stellar component with corresponding velocity dispersion < v_z² >_i. Bahcall (1984b) assumed that the disk was made up of a finite number of isothermal components to obtain his results. His findings supported the earlier work of Oort and suggested that the density of unseen material in the Milky Way disk was at least 50% that of the observed material. From rotation curve constraints, the DM was found to be distributed in a disk and to have an exponential scale-height less than 0.7 kpc. Thus at the time of Trimble's (1987) review, the evidence pointed towards a significant component of DM in the Milky Way disk. However, more recent developments have altered this view.

Bienaymé, Robin, and Crézé (1987) adopted a different approach to estimating the mass density in the Galactic plane. They constructed Galactic models, including the central bulge, visible disk, dark halo and disk DM, and incorporated a model of the disk stellar populations. The relative contributions of the various terms were constrained by the requirement that the resulting potential should be consistent with observations of the Milky Way rotation curve. Acceptable models had to provide self-consistent solutions to the Boltzmann and Poisson equations. The stellar density laws were translated into star count predictions.

In contrast to earlier studies, Bienaymé et al. (1987) found little evidence for disk DM. Their technique suggested a dynamical mass density of 0.09 to 0.12 M pc^-3, limiting the local disk DM to a density less than 0.03 M pc^-3, corresponding to a surface density in DM of 24 M pc^-2. Their best fit model had a DM density of just 0.01 M pc^-3, although no DM was also an acceptable solution. Moreover, Bienaymé et al. (1987) found that a moderate flattening of the dark halo would remove the need for even this small amount of disk DM.

In a subsequent paper, Crézé, Robin, and Bienaymé (1989) addressed the question of why their technique gave different results from those of Oort (1960) and Bahcall (1984b). Their study essentially reversed the Bahcall models to calculate predicted star counts, which were then compared to observed star counts. Crézé et al. (1989) concluded that most Bahcall models, both with and without disk DM, were consistent with the observed star counts, and that the extant data simply did not provide strong constraints on the local mass density. In other words, Crézé et al. (1989) believed that the errors associated with earlier mass determinations were much larger than previously claimed. Thus there was no conflict between the low or zero disk DM densities inferred by Bienaymé et al. (1987) and the much higher DM fractions found by Bahcall.

Around the same time, Kuijken and Gilmore (1989a) introduced a new technique to estimate the surface mass density of the local disk. Their method also used a tracer population of stars. However, unlike the methods of Bahcall and Oort, it employed the full observed distribution function of velocities and distances of the stellar tracers, rather than assuming that the stellar population could be represented by a finite number of isothermal distributions. This distribution function, along with the spatial density distribution of the same tracer population, was then used to determine the local surface mass density.

This method was applied by Kuijken and Gilmore (1989b) to a sample of K dwarfs in the direction of the south galactic pole. They obtained the surface mass density of visible material by integrating the stellar population near the Sun through the derived gravitational force perpendicular to the disk, and adding to this the interstellar gas. By removing the contribution of the Galactic dark halo, Kuijken and Gilmore (1989b) found a total dynamical surface mass density of the disk in the solar neighborhood of 46 ± 9 M pc^-2. Their value for the surface mass density in identified material was 48 ± 8 M pc^-2, implying that there was no disk DM in the solar neighborhood. Moreover, Kuijken and Gilmore (1989c) analyzed the earlier F and K star samples used by Bahcall and, like Crézé et al. (1989), concluded that the available data were either internally inconsistent, or that they provided no reliable evidence for disk DM.

Yet another approach was taken by Knapp (1988; also DMW), who used gas rather than stars to trace the Galactic potential. From an analysis of the relation between velocity dispersion and scale-height of molecular hydrogen in the Milky Way, she obtained an estimate of the mid-plane mass density. The inferred disk mass could be accounted for entirely by known luminous material.

The chief protagonists in this debate then embarked on a discussion in the literature which centered on the correct method of analyzing the data. Gould (1990a) applied a maximum likelihood method to the tracer population of Kuijken and Gilmore (1989b) and found that, even if the same physical assumptions were retained, the disk surface mass density was higher than claimed by Kuijken and Gilmore, with a value of 54 ± 8 M pc^-2. On the basis of this finding, Gould (1990a) claimed that the Kuijken and Gilmore (1989b) study did not severely constrain the amount of disk DM, and that it was consistent with the earlier result of Bahcall. Kuijken and Gilmore (1991) rediscussed their method and argued that their technique was more objective than Gould's, still claiming no disk DM within 1.1 kpc of the Galactic plane. In a further study, Kuijken (1991) investigated the presence of DM within 160 pc of the plane, again finding no DM.

The details of this debate are beyond the scope of the present review. In any case, no consensus seems to have been reached on the ``correct'' method of analysis. As Gould (1989) pointed out, different tracers are sensitive to different hypothetical DM components, which complicates direct comparison between results derived from different data sets.

Part of the apparent discrepancy between these results was probably caused by the systematic uncertainties which dominated the Bahcall calculation of the local mass density. This problem was noted by Bahcall in his original analysis and studied further by Gould (1990b). The uncertainties include distance scale errors, non-isothermality of the stellar sample, smoothing of the star counts, spurious stellar density gradients, and other problems arising from the use of data drawn from different samples. In an attempt to remove some of the confusion, Gould (1990b) devised a technique to assign statistical significance to the local disk DM density derived using the Bahcall method. He also presented a strategy for obtaining a suitable sample of stars that could be used with the Bahcall method to measure accurately the DM surface density.

Bahcall, Flynn, and Gould (1992) have adopted Gould's (1990b) statistical test and observing strategy. They have used a survey of K giants carried out by Flynn and Freeman (1993) that was specifically designed to address this question. This sample is expected to be relatively free from systematic effects. The principal result of Bahcall et al. (1992) is that the hypothesis that there is no disk DM is consistent with the data at a level of 14%. They conclude that the odds are 6-to-1 in favor of the existence of disk DM. Assuming that the DM is distributed in the same way as the visible disk material, the best fit model has 53% more DM than visible matter.

Despite the Bahcall et al. (1992) result, the presence of DM in the solar neighborhood seems a less likely proposition than at the time of Trimble's (1987) review. While it can be argued that the Bahcall et al. (1992) work is the most comprehensive investigation of disk DM, their result is somewhat at odds with other independent studies. It is certainly possible that there is some DM in the Milky Way disk, but there appears to be no clear need to invoke such a component. Thus while this issue is not yet closed, the consensus seems to be that the burden of proof has fallen on the advocates of disk DM.