In Galactic Bridges and Tails, Toomre and Toomre (hereafter, TT) considered encounters between pairs of galaxies on parabolic orbits; each galaxy was idealized as just a disk of noninteracting test particles initially in orbit about a central mass points. As a result, all explicit self gravity of the disk was neglected; this was justified on two grounds:
Figure 1: Figure 18 on page 623 of Galactic Bridges and
Tails. This is part of the tail survey; in this case, the ellipticity
was for equal mass passages of fixed inclination
but with different orbital arguments of
and . The viewing time is t=6.086 in
their units, corresponding to of orbit travel since
pericenter.
The viewing angles are normal and edge on to the orbit plane; in this
case, the edge on views are from directions which exactly superpose
the centers of victim and satellite.
Figure 2: Figure 18 on page 624 of Galactic Bridges and
Tails.
TT observed that if every bare companion carried an tail of
its own, chosen from amongst the four possibilities after a
visual rotation about the axis normal to the orbital plane, that no
tail crossings were to be found in any view that was roughly
perpendicular to the orbits. In addition to this, they were able to
exclude all other views from near the great circle normal to the line
connecting the masses; this was because of the precondition that no
part of one tail could be much closer to the other than the distance
between the hulks themselves.
Views from along the connecting line, however, produce a pair of
crossed tails which TT compare to a ``pair of symmetric gull
wings''. In this case, they utilised the tail in figure
18 together with its identical twin attached to the perturbing
mass. Similar results are possible for both the and
cases, and almost the same for either case partnered with
ribbon. The result is illustrated in figure 23 of
TT, as shown.
Figure 3: Figure 23 of Galactic Bridges and Tails. In this
instance we see two identical disks of radius 0.75 which
have undergone an encounter with orbit angles
and argument at pericenter
that appeared the same to both. The subscripts refer to the galaxies
NGC 4038 and NGC 4039 respectively. The upper view is exactly normal to
the orbit plane, while the lower view is edge on to the orbit plane
with a viewing direction from the line connecting the two
pericenters. The viewing time is at t=15 in their units, just slightly
past apocenter.
TT chose the precise orbital parameters used in figure 23 to produce a
result in accord with their 1971 NGC 4038/9 movie.
The forces from each massive body were softened gradually at close
range by varying the potential at small radius as:
where as opposed to the standard .
This gravity softening was essential in the mimicking of the mass
distribution; if neglected, too many of the near side particles would
have been extracted from both of the disks. In addition, this gravity
softening facilitated further thinning of the tails.
Whilst the models successfully reproduced the hallmark crossed tails
of the ``Antennae'', TT noted several deficiencies: