Barnes, 1988

In Encounters between Disk/Halo Galaxies, Barnes used prototype bulge/disk/halo model galaxies. These models consisted of a central bulge, a thin rotating disk and an optional spherical extended halo. The exact details of the simulated encounter which would produce the observed features of NGC 4038/9 followed those of Toomre and Toomre; the initial conditions corresponded closely to those used in Galactic Bridges and Tails, although some modifications were necessary to account for the presence of the massive dark halo.
The main characteristics of the model galaxies were as follows:

All these quantities were measured in arbitrary units with G=1. Scaling to our galaxy, these numbers correspond to a length of 40 kpc, a time of 250Myrs and a mass of .
The galaxies were started on elliptical orbits with an eccentricity of e=0.5; their pericentric separation was and the time to pericentre was . Both galaxies were inclined at to the orbital plane with a pericentric argument of .
These particular initial conditions will lead to a slow symmetric prograde encounter with the two disks inclined so as to sling tidal tails high above the orbital plane where they will eventually be seen in projection crossing each other. The configuration used is shown in figure 4.

 
Figure 4: Figure 1 of Encounters of Disk/Halo Galaxies, illustrating the geometry of the encounter.

 
Figure 5: Figure 2 of Encounters of Disk/Halo Galaxies, showing the resulting configuration.

Whereas Toomre and Toomre's test particle model galaxies followed Keplerian trajectories, the orbits of Barnes' self consistent model galaxies quickly decayed. The tidal coupling of orbital to internal motion was so effective that the relative orbit decayed in less than one initial orbital period. This was due to the presence of the massive dark halo which effectively soaked up the energy and angular momentum, as shown previously by van Albada and van Gorkom. Barnes noted that at the time when TT's model most closely matched observations, his models had already merged. However, his models do produce a configuration that resembled observations but at a much earlier time - t=1.8 in his arbitrary time units, corresponding to roughly one quarter of the initial orbital period after pericentre.
Further analysis of the encounter revealed several interesting features:

• the interaction slewed the inner galactic disks by counterclockwise, almost edge on to our line of sight, indicating the presence of powerful torques on the disks during the encounter.
• distinct peaks in the dark matter distribution were observed to coincide with the luminous tails, ``like a dark, distended body surrounding a luminous spine''.
• the visible tails were confined within the halo; this is a consequence of the dark matter, which initially shared phase space with the luminous matter, tagging along to the present stage.
• the more curving left hand tail of the model matches closely the appearance of the south western tail as revealed in both deep optical images (Schweizer, 1978) and high resolution HI mapping (Van der Hulst, 1979).
• Barnes was keen to stress that the asymmetry between the two tails was not due to any macroscopic asymmetry in the initial conditions but was instead due to fluctuations in the galaxies which are swing amplified by the disk and sheared out by tidal forces during the encounter, thus further populating one of the resulting tidal arms.

Reviewing this particular simulation, Barnes considered both the positive and negative aspects of the result; the model encounter successfully reproduced:

• the narrowness of the tidal tails. This demonstrated the requirement for relatively cold disks. Previous attempts to self consistently model encounters of this kind, such as those of Negroponte (1987), produced stubby tails, clearly at odds with observation.
• the proximity of the hulks. This is a distinct improvement on TT's model; although TT foresaw the importance of orbital decay due to tidal friction, they certainly could not have anticipated the rapidity of the decay. By following the bulges (which trace the dynamical centres of the galaxies) Barnes was able to observe that these bulges barely separated after pericentre, plunging back together after one third of an orbiral revolution.

However, the model result was not without its shortcomings:

• The model galaxies, as shown in figure 5, are approaching one another along the line of sight with velocities comparable to the circular velocity of each disk. However, studies in HII (Rubin, Ford and D'Odorico, 1970) and HI (Mahony et al) indicate only a small line of sight difference - . Thus the model fails to match the observed kinematics of the main hulks.
• Conceptually there are problems associated with the very existence of this e=0.5 orbit. The orbital period for such an encounter is longer than the Hubble time! Due consideration would seem to suggest that the galaxies experienced a close parabolic encounter - but such encounters do not result in configurations such as the one observed in the Antennae.

In conclusion, Barnes noted that the model of NGC 4038/9 came subjectively close to matching the appearance of the Antennae; based on this observation, it was felt, with some confidence, that the basic elements of the encounter were correct. He did caution, however, that the quality of the match was somewhat enhanced by the presence of chance fluctuations which could grow to macroscopic proportions at the right time and place to produce asymmetric tails - and so it would be neither possible to infer precise properties of the progenitors nor to establish the uniqueness of a solution.