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:

Mass Ratio of Components1:3:16
Total Mass of Componentstex2html_wrap_inline116
Inverse Scale Length of Exponential Disktex2html_wrap_inline118
Circular orbital period at radius tex2html_wrap_inline120tex2html_wrap_inline122
All these quantities were measured in arbitray 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 tex2html_wrap_inline130.
The galaxies were started on elliptical orbits with an eccentricty of e=0.5; their pericentric separation was tex2html_wrap_inline132 and the time to pericentre was tex2html_wrap_inline134. Both galaxies were inclined at tex2html_wrap_inline80 to the orbital plane with a pericentric arguement of tex2html_wrap_inline138.
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 place where they will eventually be seen in projection crossing each other. The configuration used is shown in figure 4.

 

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Figure: Figure 1 of Encounters of Disk/Halo Galaxies.

 

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Figure: Figure 2 of Encounters of Disk/Halo Galaxies.

Whereas Toomre and Toomre's test particle model galaxies followed quasi-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. 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:

Reviewing this particular simulation, Barnes considered both the positive and negative aspects of the result; the model encounter successfully reproduced: However, the model result was not without its shortcomings: 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 assymetric tails - and so it would be neither possible to infer precise properties of the progenitors nor to establish the uniqueness of a solution.

Chris Power