In contrast to our earlier studies (DMH, MDH) in which we used only a limited number of galaxy models, the expanded suite of models presented here provides a more complete map of the parameter space of rotation curves, covering the extremes of observational and theoretical interpretations of the dark halo mass distribution. Our models are best viewed as constraints on the depth and gradient of the potential well - in other words, on the shape of the galactic rotation curve. We find that models with rotation curves which are truly flat inhibit the formation of tidal tails. To create long tails, rotation curves must decline, either through a falling halo rotation curve, or a marked disk-halo transition. How then do these constraints compare to other observational constraints on the shape of spiral galaxy rotation curves?
Optical and H1 studies, which at first pointed toward universally flat rotation curves to the limit of detectability (e.g., Rubin et al. 1982; Kent 1987), now reveal a variety of rotation curve shapes. Rotation curves which are truly flat and halo-dominated - such as those from isothermal halo models - are extremely difficult to reconcile with observed tidal tails unless they have a small truncation radius (i.e. just outside the edge of the optical/HI disk). Models with flat rotation curves sit in the bottom right of Region III; these models have such steep potential gradients in their outer regions that only very short and low-mass tails form. The classic merging pairs of the Toomre Sequence are inconsistent with the flat rotation curves of Region III models.
As rotation curve ``libraries'' (e.g., Persic & Salucci 1995; Mathewson & Ford 1996) have expanded, however, the notion of universally flat rotation curves has been superseded. Instead, galaxies appear to possess a variety of rotation curve shapes, and it has been suggested that rotation curve shape is correlated with luminosity and/or Hubble type of the galaxy (e.g., Persic, Salucci, & Stel 1996; McGaugh & de Blok 1998). Persic et al. advocate a ``universal rotation curve'' in which bright spirals have slightly falling rotation curves, while low luminosity disks possess rotation curves which continue to rise to the last observed data point. This classification based on luminosity runs into difficulties when low surface brightness (LSB) galaxies are considered; even luminous LSBs possess slowly rising rotation curves, suggesting that it is surface density, not luminosity, which determines rotation curve shape (McGaugh & de Blok 1998). The decline observed in some high surface brightness disk galaxies may in fact be associated with the presence of a massive disk, rather than a halo with a mass distribution declining faster than r-2. As yet, no clear cut cases exist for declining halo rotation curves in luminous disk galaxies.
Galaxies with falling rotation curves have no problem making long tails as shown by the examples in Regions I and II of Figure 5. The bottom of Region I represents models with compact halo-dominated rotation curves that cut off at moderate radii (large vs and small rs) while Region II represents models with disk-dominated rotation curves with less massive but more extended (small vs and large rs) dark halos. Both maximal disk models and 60% maximal disk models such as advocated by Bottema (1997) are represented in Region II; LSB galaxies with their extremely flat rotation curves are not. Generally, the appearance of the tails in interacting systems such as the Antennae and the Mice could be attributed to galaxies with either disk-dominated or halo-dominated rotation curves. One possible exception is Arp 295 - a widely separated pair (> 100 kpc) with long tails and a connecting bridge (Hibbard 1995). Only disk-dominated models in Region II exhibit this behavior, suggesting that the galaxies in Arp 295 have disk-dominated rotation curves -- the galaxies in Region I simply merge too rapidly to allow a phase with a long connecting bridge. The structure of dark matter halos probably varies from galaxy to galaxy so mergers with long tidal tails may simply be a product of systems with different halo properties but falling rotation curves. Although galaxies with low-mass, compact halos (small rs and small vs) represented in Region I can also make long tails they can be ruled out directly since their rotation curves simply fall too steeply.
One further check for consistency is provided by examining late stages in
the merging process. Figure 5 provides snapshots shortly
after an interaction as seen in systems like the Antennae or the Mice.
NGC 7252 provides a later view when the two galaxies have coalesced with a
single, concentration of light yet memory of their interaction is still
visible in protruding tails. We examined a late stage analogous to NGC 7252
by repeating the calculations using a smaller pericentric separation,
rp=2.0, to decrease the merging time to less than 5 Gyr for all the models.
NGC 7252 has been caught very close to the exact time of coalescence and so
we have run all the models to a time when the galaxy centers are
separated only by < 0.2 scale length (< 1 kpc). The merging times range
from 0.3 to 5.4 Gyr after the time of closest approach (see Table 3)
reflecting the large variation in the strength of dynamical friction during
the interactions. The morphology and kinematics of the remnants and any
protruding tails are shown in Figures 7 and
6. The galaxies in Region I still show the tails
from the primary encounter and some forming secondary tails. These tails
are very long and expanding outward similar to
those in NGC 7252. In Region II, the
primary tails have long since escaped and the remnant is surrounded by
diffuse debris originating from the tidal bridge formed between the galaxies.
There are also diffuse tails formed in the second encounter which could
potentially be identified with those of NGC 7252. Finally, in Region III
the formation of tails in the second encounter is suppressed as strongly as
in the first encounter. The tails which do form are short-lived and
completely infalling when they reach their maximum extent of 10 scale-lengths,
in contrast to those of NGC 7252 which are expanding outwards (Hibbard et al.
1994). At face value, the galaxies in Regions I and II could plausibly be
identified with NGC 7252 at the time of merging while those in Region III are
unlikely candidates. However, the merger remnants of disk-dominated models of
Region II would be surrounded by an extended cloud of diffuse stellar and
gaseous debris. The extended gas in NGC 7252 appears to be largely confined
to the tails (Hibbard et al. 1994) so perhaps the preferred mass model is that
of the compact halo models.
Other less direct observational methods exist for probing the outer rotation curves of spiral galaxies. Zaritsky et al. (1993, 1997ab) have studied the velocity dispersions of satellite galaxies around luminous spirals, and find no evidence for a drop in the velocity dispersion out to a distance of 400 kpc. If rotation curves are truly flat to such large distances (see also Barcons, Lanzetta, & Webb 1995), it is impossible to reconcile our models with observed merging galaxies. However, the data of Zaritsky et al. do permit modestly falling rotation curves, for which our models do produce passable tidal tails (e.g., the rs=4.8, vs=0.71 model). ``Better'' tails (longer, more well defined) need more rapidly declining rotation curves than the Zaritsky et al. data allow.
Leonard & Tremaine's (1990) analysis of high velocity stars implies an escape
velocity at the solar radius of 440-660 km/s or 2 to 3 vc, straddling the
critical value,
where tails are difficult to make. The
lower bound is about ve=440 km/s based on the fastest star in their
analysis. If indeed, ve > 550 km/s in the solar neighborhood, the model
study implies that the Milky Way will not throw off extended tidal tails in
any future interaction with M31.
In summary, there are two requirements for galaxies to make realistic tidal
tails in interactions. First, the ratio of escape to circular velocity at
R=2Rd should be
.
Secondly, rotation curves should be
falling rather than flat or rising at the disk edge. Both of these
requirements can be met with disk-dominated models with large rs and
small vs and a restricted set of halo-dominated models of moderate mass
with small rs and large vs. Some interacting pairs appear to fit
disk-dominated models better (such as Arp 295) while others prefer
compact-halo-dominated models (NGC 7252).
