Our goal is to use tidal tails to constrain the amount of dark matter surrounding galaxies at radii larger than those which can be probed by optical or HI rotation curves. Accordingly, we wish to compare mergers of galaxies whose rotation curves are similar in their inner regions, where the mass distribution is relatively well-constrained, but which differ in the total masses and extents of their dark matter halos. Previous techniques for constructing model galaxies (e.g., Barnes 1988; Hernquist 1993a) are not well-suited for this task.
Instead, we employ the methods for constructing equilibrium model galaxies
developed recently by Kuijken & Dubinski
(1995). In particular, they present a set of galaxy
models with different halo masses and extents, all of which reproduce
the observed rotation curve of the Milky Way out to 5 disk scale
lengths (5
). In dimensionless units, the models have 
,
rotation velocity 
, a disk mass, 
,
and a bulge mass 
. Scaling to values appropriate for the
Milky Way, these values correspond to 
kpc, 
km s
, 
, and 
. The halos vary in mass, yielding halo:disk+bulge mass ratios
between 4:1 and 30:1, and vary in radial extent from 
22 to 73 disk scale lengths (see Table 1). The rotation curves of
these models, shown in Figure 1,
are nearly identical within 5
,
differing significantly only at large radii.
Figure 1: Rotation curves for the model galaxies. a) Inner rotation curves,
b) Outer rotation curves
The bulge and halo components of these galaxies are derived from lowered
Evans models for spheroidal systems, with distribution functions that
depend on the exact integrals of motion: the energy, E, and the z
component of angular momentum, 
. The distribution function for
the disks depends on E, 
, and a third ``integral,'' 
, the
vertical energy, which is approximately conserved in cool stellar
disks (see, e.g., Binney & Tremaine 1987). The radial velocity
dispersion profiles correspond initially to a disk with Toomre (1963)
at 
. Evolved in isolation, the models experience no
major transitions at startup, and they are stable against bar
formation over the timescales of interest.
In our calculations, each galaxy is represented by 48,000 particles:
16,000 in the disk, 8,000 in the bulge, and 24,000 in the dark halo.
Because we use a fixed number of halo particles, but vary the halo
masses, individual halo particles have different masses from one run
to another. As a result, the amount of disk heating due to two-body
relaxation is more pronounced in the models with more massive halos.
This effect is compounded by the fact that the more massive galaxies
have a longer pre-collision evolution, as they are started further
apart due to their more extended halos. Models of isolated galaxies
allow us to quantify this disk heating: at the time of collision (when
the tails are formed), Models A and B have 
, while
Models C and D have 
and 5.0, respectively. The
relatively warm disks at the time of encounter caused concern about
the validity of our results, particularly for the models with the most
massive halos. To address this problem, we repeated two of the
experiments with 5 times as many halo particles to reduce the growth
rate of Q and
examine the sensitivity of our results to the disk heating. While the
tidal features that develop in the large N, low Q models are
crisper than those in their high Q counterparts, the morphologies and
lengths of the tails are quite similar in both cases (see §§ 3 and
4.3 below), implying that our conclusions are insensitive to this
aspect of the dynamics.
