Toomre and Toomre, 1972

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:

• the self attraction of the outer part of a disk is, by definition, small. Hence, by neglecting self gravity the main qualitative conclusions are unlikely to be affected.
• by relying on test particles as tracers it was hoped to make clear the kinematic nature of the observed filamentary structures.

TT modeled 4 interacting systems - Arp 295, M51, NGC 4676 and NGC 4038/9 - but it was their reconstuction of the ``Antennae'' which gave them ``the most pleasure''. This stemmed from prior concerns regarding whether ``it was possible to obtain seemingly crossed tails from tidal interactions''.
In section IV of TT, Survey of Tails, it is noted that ``proper tail making in the sense of escape to infinity of particles from the antitidal side of a victim disk requires the perturbing mass to be at least comparable to the perturbed''. Hence, if we observe an interacting system with two prominent tidal tails emanating from both disks of both galaxies then we can conclude that the masses involved were comparable and that the separation at periapse was small. With this assumption in mind, a mass ratio of exactly unity was assumed in the survey of 3 dimensional tails. This survey was of particular importance for analysis of NGC 4038/9, and in particular figure 18 of this section, shown below, plays a central theme.

 
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:

1. the real tails are unequal in length.
2. the actual NGC 4038 tail is more curved than in the model.
3. the rotation of the real hulks seem to be such that their adjacent ends approach and recede alike whereas the opposite holds true for the models.

In an attempt to remedy these shortcomings, TT made three suggestions:

• unequal tail length (objection 1) can be achieved by introducing slight inequalities in either the masses, inclinations or the initial outer radii.
• the second objection would seem to suggest that a second encounter will occur.
• the third objection, by far the most serious, requires a substantial change of at least the argument from as well as a further reduction in the longitude of viewing.

Despite the shortcomings, the Toomres captured the essence of the Antennae with a simplified 3-body model, and as they indicated, if further progress was to be made then a full N-Body self consistent code was necessary.