3. THE MILKY WAY HALO

The question of the mass of the Milky Way, which is dominated by the dark halo, was recently reviewed in detail by Fich and Tremaine (1991). I therefore give only a brief description of the techniques that have been used to tackle this question and a summary of recent results. The nature of the DM in the halo of the Galaxy is addressed in Section 10.

The globular cluster system and satellite galaxies of the Milky Way have both been used extensively to estimate the mass and extent of the dark halo. There are two ways that such information may be obtained. The first uses the tidal radii of such objects to probe the Galactic gravitational field. The tidal radius r_t of a satellite object with a perigalactic distance R_p, is related to the mass of the Galaxy through the Roche criterion:

(3.1)

where m is the mass of the globular cluster or satellite galaxy and M(R_p) is the Galactic mass interior to R_p. A more precise expression contains factors of order unity that depend on orbital parameters of the objects being used and the assumed density profile of the dark halo.

Innanen, Harris and Webbink (1983) carried out such an analysis using the globular cluster system of the Milky Way and found that the halo extended to at least 44 kpc with a mass within this radius of 8.9 ± 2.6 x 10¹¹ M. They further claimed that the density profile of the halo was given by (R) R^{-1.73 ± 0.18}. However, such analyses have certain limitations which Innanen et al. (1983) recognized. For instance, the results are dependent on a knowledge of the mass-to-light ratio of the globular clusters. A bigger problem is obtaining reliable values of the tidal radius. Usually this is obtained by a considerable extrapolation of the observed light profile of the cluster. Consequently, errors are potentially large.

It is perhaps because of these problems that more recent attention has focussed on using the dynamics of such objects to estimate the mass of the Milky Way. This involves measuring the radial velocities and distances of globular clusters and/or satellite galaxies. Since these objects also have non-radial velocity components, a statistical form of the virial theorem must be used. For N objects with Galactocentric distances r_i and radial velocities v_i, the estimated mass of the Milky Way is

(3.2)

where A is a constant of order unity that depends on the orbital distribution of the objects (cf. Binney and Tremaine 1987). Early work, summarized by Trimble (1987) and Fich and Tremaine (1991) consistently led to values around 10¹² M and 100 kpc for the mass and radius of the Galactic halo, respectively. However, Little and Tremaine (1987) developed a sophisticated statistical method to attack the problem which also allowed them to assign uncertainties to the mass determinations (see also Arnold 1992). Their result of a halo mass less than 5 x 10¹¹ M at the 95% confidence level was significantly lower than earlier estimates, and implied a halo radius less than about 46 kpc.

Zaritsky et al. (1989) obtained new velocity data for some of the Milky Way satellites, including a result for Leo I which differed substantially from previous values. Incorporating this into the data set and using the method of Little and Tremaine (1987), they obtained a Milky Way halo mass between 8.1 x 10¹¹ M and 2.1 x 10¹² M, a good deal higher than the Little and Tremaine (1987) result. There is no mystery here since the difference is attributable to the revised velocity of Leo I. Further support for the higher mass value was provided by Salucci and Frenk (1989) who considered the effect of the disk on the Milky Way rotation curve and concluded that the halo extended well beyond the 46 kpc derived by Little and Tremaine (1987). Peterson and Latham (1989) obtained a minimum halo mass of 5 x 10¹¹ M assuming that the globular cluster Palomar 15 was bound to the Milky Way, although this result depends somewhat on assumptions about the orbit of the cluster.

Kulessa and Lynden-Bell (1992) have obtained similar results using a maximum likelihood technique. They find a mass of 1.3 x 10¹² M for the Milky Way extending to a radius of 230 kpc. Their favored solution gives a density fall-off for the halo of the form (R) R^-2.4, where R is the distance from the Galactic center. As in the Zaritsky et al. (1989) study, the mass estimate of Kulessa and Lynden-Bell (1992) falls considerably if Leo I is excluded from the data set.

The sensitivity of this measurement technique to the inclusion of Leo I is a little worrying. Indeed, Zaritsky et al. (1989) suggest that timing arguments are a more reliable method of calculating the mass of the Milky Way. This technique was first used to estimate the mass of the Local Group of galaxies (Kahn and Woltjer 1959), and is based on the observation that the Milky Way and M31 are approaching one another. This is interpreted as being due to the gravitational attraction of the two galaxies overcoming the Hubble expansion and pulling them together. The current relative velocity is a function of the time that this attraction has been operating and the masses of the two galaxies. The mass of the individual objects is then obtained by assuming a mass ratio for the two galaxies.

Zaritsky et al. (1989) used such a timing argument for the Milky Way and M31, but also carried out a similar calculation for the Milky Way and Leo I. Combining their results, they obtained a Milky Way mass of (13 ± 2) x 10¹¹ M and a halo radius between about 120 and 210 kpc. The derived radius is dependent on the density profile of the dark halo. The mass is somewhat sensitive to the cosmological model through its dependence on the age of the Universe. Reducing the age of the Universe means that the galaxies have been attracting each other for a shorter time, so more mass must be assigned to them to account for the observed relative velocity. Lake (1992) has emphasized that if the Universe has the critical density required for closure and the cosmological constant is zero, then measurements of the Hubble constant imply an age of the Universe significantly less than that assumed by Zaritsky et al. (1989). This increases the derived mass of the Milky Way.

Other recent work on the mass of the Milky Way includes Merrifield's (1992) study of the rotation curve of the Galaxy from the thickness of the HI layer. This allows the construction of mass models similar to those used in other spiral galaxies (see Section 6 below). This technique gives results consistent with other dynamical estimates of the Milky Way rotation curve, but is only applicable out to about 20 kpc and requires an accurate knowledge of R₀ and ₀ (the distance of the Sun from the Galactic center and the circular velocity at that radius, respectively) if it is to provide an independent measurement of the quantities of interest.

There now seems to be a consensus that the mass of the Milky Way halo is around 10¹² M, with a radial extent of around 100 to 200 kpc. This corresponds to a dark-to-luminous mass ratio around 10. Attempts to detect DM in the Milky Way halo, as well as theoretical constraints on its nature, are discussed in Section 10.