Introduction

There is now considerable observational evidence for the presence of large amounts of dark matter in the Universe. On the scales of individual galaxies, the mass-to-light ratio of spirals and ellipticals is in the range 10-50 (M/L) , well in excess of the mass-to-light ratio for normal stellar populations (M/L) (e.g., Binney & Tremaine 1987). On larger scales associated with groups and clusters of galaxies, the mass-to-light ratios are even higher, typically (M/L) . Clearly, ordinary ``luminous" matter such as stars and stellar remnants cannot account for such extreme values. On the largest scales, an even higher total mass-to-light ratio for the Universe of (/100 km/s/Mpc) is required for closure density in a critical, universe. Determining the amount of dark matter present in the Universe is thus a key ingredient in understanding cosmological evolution.

In principle, cosmological models can be constrained by the structure of individual galaxies, if we can infer the masses and extents of their dark matter halos. Gunn & Gott (1972) recognized that spherical infall of dissipationless matter onto a density perturbation in an Einstein-de Sitter universe would relax into an object with a density profile , similar to the inferred mass distribution around galaxies (see also Bertschinger 1985). Furthermore, if , mass continues to accrete continuously so that halo masses grow with time as , implying that dark halos should extend to great distances. In the case of the Milky Way, a profile extrapolated from the local rotation curve reaches the background density at a radius Mpc. In contrast, in subcritical universes collapsing objects stop accreting at a redshift, , and therefore have a steeper density profile than they would if (e.g., Hoffman & Shaham 1985; Zurek, Quinn, & Salmon 1988). Measurements of the properties of dark halos in cosmological N-body simulations support the predictions of the spherical accretion calculations (e.g. Zurek, Quinn, & Salmon 1988). For example, the cold dark matter (CDM) model predicts that dark halos associated with galaxies like the Milky Way should extend well beyond 200 kpc at the present time, with approximately isothermal density profiles and masses well in excess of M (e.g., Navarro, Frenk, & White 1995).

The strongest observational constraints on the amount of dark matter around galaxies comes from the kinematics of spirals. The rotation curves of spiral galaxies, as inferred by optical spectra and HI linewidths, are roughly flat out to 10 disk scale lengths, or 30 -- 40 kpc, (e.g. Rubin et al. 1980, 1982, 1985; Kent 1987), contrary to the assumption that light traces mass. The usual interpretation of this finding is that galaxies are surrounded by unseen dark halos. At large radii, the dark halo dominates the mass distribution and so a flat rotation curve implies a density profile and, hence, . The decomposition of rotation curves suggests that the mass ratio of halo to disk plus bulge matter is approximately 10:1 out to the edges of HI gas disks (e.g., Kent 1987). A further outwards extrapolation of the profile implies that halos could be much more massive, but their extents and total masses cannot be determined from rotation curves alone.

The hot, X-ray coronae around some elliptical galaxies also seems to require massive dark halos to support them (e.g., Forman, Jones, & Tucker 1985). If the gas is in hydrostatic equilibrium, the gravitational potential can be determined by analyzing the X-ray emission profile. Typically, a mass model with fits the data well, consistent with the implications of spiral rotation curves, but again in most cases the total masses and extents of the halos cannot be determined unambiguously.

In an effort to probe the nature of halos on scales larger than those covered by rotation curves, Zaritsky & White (1994) have examined the distribution and kinematics of satellites around external galaxies. Since the objects in their sample were chosen to have similar luminosities and Hubble types, Zaritsky & White were able to perform an ensemble average over all of their galaxies to estimate halo masses. Within the uncertainties of projection effects, they conclude that halo masses are M for galaxies with rotation velocities similar to the Milky Way, km s .

Brainerd, Blandford & Smail (1995) have introduced a new technique to study halos which is based on weak gravitational lensing of faint background galaxies by brighter foreground galaxies. Deep CCD exposures show that fainter galaxies have a tendency to be tangentially aligned around the brighter (presumably closer) galaxies in projection. By modeling the galaxy redshift distribution, Brainerd et al. show that this result can be explained by weak gravitational lensing if galaxy halos extend to at least 100 kpc and have characteristic masses at least M . Their detection of the effect is marginal at present, but in principle, this method can be used to determine the masses and extents of halos without any dynamical modeling.

Our own Galaxy provides a unique opportunity to study dark halos. The rotation curve of the Milky Way is nearly flat out to a distance of at least 20 kpc (e.g. Fich & Tremaine 1991) making it similar to external galaxies. Beyond that, the mass distribution has been estimated using various tracers of the gravitational potential which are inaccessible in external galaxies. High velocity stars in the solar neighborhood set a lower limit on the local escape velocity. The analysis by Leonard & Tremaine (1990) gives a lower limit to the mass of the Milky Way of M . Further out, the kinematics of distant globular clusters and dwarf galaxies trace the mass profile in the outer galaxy from 50 to 200 kpc (Hartwick & Sargent 1978; Lynden-Bell, Cannon & Godwin 1983; Little & Tremaine 1987). The most recent estimates based on this approach (e.g., Zaritsky et al. 1989; Kochanek 1995) suggest that the Milky Way's mass is M within a radius of 100 kpc. Unfortunately, this method is particularly sensitive to whether or not the dwarf Leo I is included in the sample. Leo I lies at a large distance from the Galactic center (r=220 kpc) and has a high radial velocity (v = 177 km s); removing it from the analysis lowers the estimated mass within 100 kpc by 50% in Kochanek's (1995) analysis.

By modeling the orbital dynamics of the Large and Small Magellanic Clouds, and by requiring that the Magellanic Stream originated from tidal stripping, Lin et al. (1995) estimate the mass of the Milky Way. At the distance of the LMC (55 kpc), their estimated mass of M is consistent with the mass interior to 50 kpc inferred from satellite kinematics.

Finally, the mass of the Local Group can be estimated from the timing argument (Kahn & Woltjer 1959), which is founded on the assumption that M31 and the Milky Way are bound and are currently on the return portion of a nearly radial orbit which originated at the Big Bang. From the current separation and relative radial velocity of the two galaxies, the derived total mass is between and M for Universe ages between 10 and 20 Gyr (with the higher mass being associated with the lower age). Andromeda has a larger circular velocity and disk scale length than the Milky Way, suggesting that it is times the mass of the Galaxy (e.g., Raychaudury & Lynden-Bell 1989). This sets the mass of the Milky Way at M depending on the age of the Universe, consistent with the mass derived from satellite kinematics (Zaritsky et al. 1989; Kochanek 1995). These total masses imply a halo:disk+bulge mass ratio of 15-30:1 using the estimated disk+bulge mass of M (e.g., Bahcall & Soneira 1980).

In this paper, we seek to probe the structure of dark halos in the outer regions of galaxies in an independent manner, by examining the structure of the often long tidal tails associated with merging galaxies. Among the well-known examples of ongoing mergers, several have prominent tidal tails (Arp 1966) -- e.g. NGC 4038/39 (``The Antennae''), NGC 7252, Arp 193, and Arp 243, to name a few. In projection, the tails in these objects extend out to 10 -- 20 disk scale lengths from the merging pair (Schweizer 1982; Schombert, Wallin & Struck-Marcell 1990; Hibbard 1994). The tails probably extend to even greater physical distances in three dimensions, perhaps 20 -- 40 scale lengths. The Superantennae (IRAS 19254-7245) is an extreme case in which the tails span kpc from tip to tip (Mirabel, Lutz, & Maza 1991; Colina, Lipari, & Maccheto 1991).

The origin of the thin tails extending from interacting galaxies is well understood. Toomre & Toomre (1972) and Wright (1972) demonstrated that thin tails could be produced by tidal forces during a close encounter of two disk galaxies, finally laying to rest lingering doubts about the gravitational origin of these features. However, these calculations modeled the potential of the galaxies as point masses, and did not explore variations in tidal tail morphology due to the presence of dark halos. Faber and Gallagher (1979) later speculated that the lengths of these tidal features might be used to constrain the mass distributions of the outer regions of galaxies. In principle, tidal tails trace the orbit of particles ejected from the disk at some high velocity acquired in the interaction and so trace the potential of the galaxies at large radii. In practice, the strength of the perturbation depends on many factors including the perigalactic separation and velocity, disk orientation, and galaxy mass ratio. An extensive survey of possible galaxy collisions is therefore required to determine this effects.

Simulations employing self-consistent galaxies have put such arguments on a more quantitative footing and moved towards attaining this goal. White (1982) and Negroponte & White (1983) showed that tidal tails are readily produced in self-consistent mergers. Because of computational limitations, however, they were unable to explore parameter space fully, and noted that galaxies with very massive halos might have difficulty producing tails, for the following reasons. When galaxies with massive halos collide, their encounter velocities will be high since the pair falls into a a deep potential well during the encounter. A higher velocity interaction can detune the resonance between the orbital angular frequency and the internal angular frequency of the disk stars, which is needed to produce tidal tails. Moreover, the material forming the tails would have a deeper (and perhaps steeper) potential well to climb while being ejected from each galaxy. In principle, both effects can reduce the masses and lengths of the tails.

Barnes (1988) investigated this issue by simulating collisions of pairs of identical galaxies, each having halo:disk+bulge mass ratios in the sequence 0:1, 4:1, 8:1. Although there seemed to be evidence that encounters of galaxies with more extended halos produce less massive and shorter tails, Barnes concluded that tails are relatively easy to make. However, according to observational and theoretical prejudice, appropriate halo:disk+bulge mass ratios for galaxies may well be considerably larger than those employed by Barnes.

In this paper, we revisit the issue of the sensitivity of tidal tail formation in mergers to the masses and extents of dark halos, but consider larger halos than used in earlier studies. In § 2, we describe galaxy models having halo:disk+bulge mass ratios of 4:1, 8:1, 16:1, and 30:1, but similar rotation curves within 5 disk scale lengths, with the more massive halos extending to larger radii. In § 3, we describe zero energy, prograde encounters of galaxies in which the impact parameter is varied. Disks suffer the strongest tidal response during prograde collisions, so this geometry should be optimal for producing tidal tails. Our simulations demonstrate that long tidal tails are difficult to produce in mergers between galaxies with the most massive halos even under these optimal conditions. We elaborate this point further in § 4, by investigating variations in orbital energy, disk inclinations, and the mass ratio of the progenitors, using NGC 4038/39 and the Local Group as test cases. Finally, we consider the implications of our results for the structure of galaxies and cosmology in general in § 5.