Initially, we focus on exactly prograde mergers of equal mass disk
galaxies from zero energy orbits. This choice of encounter parameters
is motivated by our desire to match systems such as the Antennae or
NGC 7252, which possess long tidal tails. As shown by Toomre &
Toomre (1972), these features are most easily generated when
comparable mass disk galaxies collide on a prograde orbit. The orbital
energy is also chosen with tail-making in mind: galaxies on high speed
unbound orbits will pass by one another too quickly to form long
tails, while moderately bound orbits have encounter speeds only
marginally slower than zero-energy orbits. Accordingly, we choose a
zero energy orbit for our fiducial set of calculations, and consider
bound orbits separately in § 4.1. The galaxies are placed on their
orbits with an initial separation R chosen such that the dark halos
are just touching; their relative velocity is then given by 
.
Given the orbital energy and disk inclinations, the remaining parameter
to be fixed is the impact parameter, or pericentric distance of the
initial orbit. The most appropriate choice here is less clear. While
collisions with small impact parameters yield a stronger tidal
perturbation, they are also faster, which may inhibit the
tail-building process. Conversely, slower, more distant encounters
have more time to raise tidal tails, but their tidal impulse will be
weaker. Rather than fixing the impact parameter, we explore a range of
possibilities with b=0.6, 1.2, 2.4, and 4.8 
. For a Keplerian
orbit, b is simply the perigalactic separation, 
; however,
galaxies are not point masses, and their extended mass distributions
cause the orbits to diverge from a Keplerian trajectory.
The exact value of 
and the relative velocity at periapse,
, will depend on the impact parameter b and the mass
distribution for the chosen model. (We found that the orbits of the
galaxies until perigalacticon are well-traced by an orbit in the effective
potential 
where 
and
refer to mass and potential of each galaxy. While ad hoc,
this potential predicts 
and 
within 10% for the simulations
in this study.) The range of 
covered
by our calculations is shown in Figure 2, where it can be seen
that 
and b are most discrepant for galaxies
with the most massive, extended halos. In the discussion that follows
we will refer to the models by their impact parameter b rather
than the varying perigalactic separation.
Figure 2: Velocity, 
versus separation, 
at perigalacticon for zero
energy orbits of prograde, equal mass galaxy mergers for the four models.
The points refer to the trajectories of the 16 simulations described in the
text. Lines of constant angular velocity, 
,
at various radii in the disk are also plotted. The intersection with
curves of 
vs. 
show where the orbital angular frequency is
resonant with various disk spin frequencies for different galaxy trajectories
at perigalacticon.
At first glance, it might appear as though the differences
between b and 
may make interpretation of the models
difficult, since for a fixed b we sample different values of 
in each of the models. In fact, since 
varies as well, this
works to our advantage. For each value of b the orbital angular
velocity at periapse 
is roughly constant
in each of the models, as shown by the diagonal lines in Figure 2.
The resonance between this orbital angular velocity and the rotational
angular velocities 
in the disk is an important factor
driving the formation of tidal tails during an interaction. Since the
inner rotation curves of the galaxies are fixed, the different model
disks all have identical 
, and collisions at
fixed b all sample similar values of 
,
regardless of varying 
for the different models.
With four galaxy models and four impact parameters, our fiducial calculations involve a total of 16 different merger simulations. While idealized, these encounters provide a best-case situation for generating long tidal tails; if galaxies with massive halos do not develop extended tails under these conditions, it will be very difficult for them to form long tails under any conditions.