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These merger hopes would all be in vain, of course, if the severe kind of dynamical friction which they seem to require proved simply to be unattainable. Fortunately, as this section will now review, such pessimism seems unwarranted for slow and deeply interpenetrating encounters. It remains quite a different story, however, with making sure that two disk systems in some more grazing passage necessary to produce impressive tails can actually stop and merge rapidly enough - i.e., leave those tails still neatly in view when the bodies sink together. No one seems yet to have made any progress worth citing on this second and more pertinent question of dynamics.

Still, to be grateful for what we have got, it is clear now that simply the splash or "violent relaxation" from mutual tidal forces can indeed halt rapidly any two comparable and already spherical stellar systems which happen to blunder head-on through each other with only about the speed developed in free fall from rest at infinity. Better yet, it appears that already the "impulsive tide" approximation of Alladin (1965; see also Sastry and Alladin 1970, 1977) - which in effect extended Chandrasekhar's friction estimates to the very nonuniform situations arising when whole galaxies interpenetrate at high speed - provides a neat and at least semi-quantitative explanation of even this near-parabolic stickiness.

The examples offered by Alladin and Sastry themselves refer mostly to (spherical) mass distributions akin to the classical n = 4 gaseous polytrope. To appreciate the gist of their reasoning, however, it seems preferable to concentrate instead on the n = 5 polytrope known either as the Schuster or the Plummer model. Its volume density is given by the well-Known formula

Equation 1 (1)

where M is the total mass, a is a scale, length, and r is the spherical radius. The equally simple force law

Equation 2 (2)

of this model greatly reduces the chore of calculating the lateral speed

Equation 3 (3)

that a test particle would develop upon rushing past it at distance D, with an immense and nearly constant speed U. And also for this model, it is pleasant to reckon further that if instead of a single particle the passerby consisted of many different stars from an identical system traveling with a (supposedly) constant speed U along an exactly head-on straight trajectory, then the tidally-induced motions vperp of those stars toward that orbit axis would amount to a kinetic energy

Equation 4 (4)

soaked up suddenly by that intruding galaxy.

Alladin and Sastry stressed very properly that such a gain of internal energy can occur only at the expense of the energy of relative motion of the two galaxies. In essence, they said, any such inward splash converts some of that orbital energy into mere stellar-dynamical heat. The only awesome thing about this reasoning is the magnitude of that expected transfer. To assess it quickly, notice that the potential energy released in bringing two undeformed and yet penetrable Plummer models together from infinity to a perfectly superposed state is |W| = (3pi/16) (GM2/a). Of course this equals not only the negative sum of the potential energies of the two systems reckoned individually while still far apart, but also the "orbital" kinetic energy MUesc2/4 developed by those two when overlapping at the very bottom of a (rigid) free fall from rest at great distance. Now suppose both models indeed to be flabby for the purposes of Equation (4), and adopt as the speed U the full relative escape speed Uesc that we just estimated. It then follows at once that the ratio of the lost to the available kinetic energy is fully

Equation 5 (5)

For other polytropes n = 4, 3, and 2, incidentally, laborious numerical integrations (such as Alladin and Sastry were also forced to perform) yield very similar ratios of 46.3, 45.6, and 45.1 per cent.

These striking conclusions can, of course, be faulted for abusing the impulsive and constant-speed assumptions on which they were based. Strictly speaking, it is correct to treat such estimates as merely asymptotic - that is, to infer only that a head-on intrusion of equal n = 3 polytropes, for example, with relative speeds U >> Uesc, will cost that pair a multiple

Equation 6 (6)

of their new and much larger peak kinetic energy. As such a former skeptic, however, I must say that I now regard Equations (5) and (6) as very adequate even when mistreated. What convinced me was not the occasional mergers found by Hohl, Miller and/or Prendergast in their planar 105 -body experiments; it was more the 3-D studies with 100 rings or 2000 mass points described below.

Figure 2 updates the brief report by Toomre (1974) on a numerical experiment in which Larry Cox and I simulated each of two parabolically approaching Plummer models as a beehive of randomly-moving coaxial rings, all interacting with one another via gravity forces softened modestly at close range. Our aim in using these softened rings instead of conventional point masses was to reduce greatly all inter-particle relaxation effects (such as were blamed, perhaps unduly, by Aarseth and Hills 1972 in their own experiment). We wished to concentrate more on the commotion due to the sudden onset of the collective tidal forces. The old diagram gave results for 12+12 rings. Figure 2, nearly as ancient, now repeats the exercise with 50+50. In units of the Plummer scale length a, it shows the axial coordinate z of each ring as a function of time t. One small discordant note: the densest cores in this diagram seem to separate as far as Deltaz = 8a after the first transit, whereas two rigid n = 5 polytropes like here should not even have reached 4a after a presumed 48 per cent loss of kinetic energy at their instant of overlap. At least half this discrepancy, however, seems due to our reduction of the near-gravity.

Figure 2

Figure 2. Head-on impact and merger of two equal Plummer models that arrived with escape speed. The axial coordinates z(t) of one set of 50 softened rings used to represent one such model are shown dotted, the others as solid curves.

Figure 3 was contributed very kindly by van Albada and van Gorkom (1977), as a cousin of an impressive test case already shown in their paper. As if only for variety, it refers to polytropes of index n = 3 - and these were now assigned, at infinity, a relative motion U = Uesc / 2sqrt2 = exactly the root-mean-square internal speed. Owing to 2000 particles an d more accurate treatment of the gravity, Figure 3 is much to be preferred technically to Figure 2. Its chief immediate value, however, lies doubtless in this explicit demonstration that even a moderately hyperbolic initial motion does not yet spoil the merger. Of course, the total energy just ceases to be negative if we double the speed at infinity from the value in Figure 3. At least such a recipe no longer promises a merger - and further experiments quite agree. In fact, van Albada (1976, private communication) reckons empirically that captures cease already when U(0) gtapprox 1.15 Uesc. By contrast, Equation (6) patterned upon Alladin's work places that crossover at just a shade under 1.16 Uesc Not bad for a simple formula.

Figure 3

Figure 3. Head-on impact and merger of two stellar dynamical n = 3 polytropes upon arrival at center with 1.061 times the escape speed Uesc. This diagram by van Albada and van Gorkom shows projected densities at seven instants.

It will not have escaped the reader that, unfortunately, this little success story has referred only to (already) spherical systems taking part in the most symmetric encounters imaginable. As regards disks and their own interplay, it is trivial, of course, to extend both this thinking and the experiments to exactly axial (= face-to-face) penetrations of two very flattened assemblies of rings. And by constraining them to remain axisymmetric, one can even ignore blithely all serious instability questions of the subject. My own experience in that tractable but unrealistic setting has been that while the immediate energy loss runs only around 20 per cent (instead of the high 40s) for a variety of disk models, soon enough they manage to merge also, and they tend to yield outlines (though hardly the full density profiles) resembling E3/E4 galaxies. But all this, I stress again, seems almost irrelevant. It is surely no substitute for the much more difficult studies of off-center impacts of stable self-gravitating disks or, more likely, disk-halo systems. Quite understandably, such studies have been very slow to emerge.

To conclude, the big worry remains that the strength of braking may drop off too rapidly with increasing impact parameter or miss distance, as one seeks circumstances that will also permit the manufacture of tails of the sort summarized in Table 1. Certainly, the off-axis studies of Alladin and Sastry convey the same warning even for encounters of spheres; paraphrasing them again, it seems that the center of one galaxy needs to impact the other system no farther out than about the 1/2 or 3/4-mass radius, lest the rapidity of their sinking cease to be impressive. Ironically, there is apt to be one logical "out" even if it emerges that disk models cannot decelerate fast enough on their own; In principle at least, one can always embed them, prior to any fateful encounter, within some appreciably larger and more massive systems like the much-discussed extensive halos. Such outer parts would by definition interpenetrate and even splatter nicely as those visible disks only graze one another. But what a strange way that would be to make ellipticals!

This work was supported in part by a generous grant from the National Science Foundation.

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