1.3. Spiral regeneration, take two
In order to continue the problem you must then do some-
thing nonlinear, or you must simply publish the results. The
authors mentioned did something non-linear.
Prendergast, 1967, p.309
The discovery of strong amplification of shearing formations in self-gravitating systems must have surprised most of the cosmogonists and dynamicists who used to think of galaxies as figures of basically uniform rotation. In essence, with this elementary and natural `microprocess' Goldreich and Lynden-Bell struck upon a powerful engine for generating spiral structures, at least in a transient way. Still what they had dealt with so far was a wave-propagation problem that gave no closed dynamical picture, even in local setting. It left untouched the vital points of fresh-wave sources (one-time, periodic or permanent; external or internal; distributed or compact) and resulting responses. This engaged our authors throughout much of the remainder of their work where they tried to build a home by way of "reasonable speculation which [they] probably felt was justified by [their] solid result" (Goldreich).
Within this speculation, "the predicted return to oscillatory character need not occur". With isothermal gas, it gets "energetically advantageous" and "energetically possible for the nonlinear modes to continue to condense rather than to revert to oscillatory behavior" (GLB, pp.139, 150), because energy released during the gravitational collapse cannot be stored as thermal energy and is radiated away. 14 A thing to stop the growth and revert the system to its initial state (else it is no machine) is to break the energy replenishment of the gas layer. This done, it flattens and gets less stable. Closer to marginal stability, the swing amplifier is turned on, it applies to various existing perturbations, 15 and analysis has it best tuned on those with y 8 h, h being the layer's half-thickness. In the nonlinear stage, genuine trailing arms have been formed and hot stars are born in growing condensations. They stir up the interstellar gas, however, and it swells, recovering stability. The amplifier is turned off, the spiral arms break down, the star formation stops, the hot stars fade away, the gas layer thins, and the cycle repeats - local structures on a scale y 1-2 kpc are periodically regenerated everywhere in the gas layer, the only responsive galactic ingredient.
Now what to do with the disk of stars, another licensed player in galaxy dynamics? Its natural length scale differs from that of the gas layer almost exactly in the ratio of their column (surface) densities, or typically roughly 10:1. At such a hostile difference their actual coupling cannot stop the interstellar gas, well able to cool itself, from tending to have severe gravitational instabilities of its own. Yet GLB reckoned that Jeans instability "occurs for stars in much the same way as it occurs for gas" so that the "spiral arm formation should [...] be regarded [...] as an instability of the whole star-gas mixture". 16 Thus they coopted the star disk into their basic gas-dynamical scheme and got a condominium with an essentially stellar `effective' density and - clearly - T-comparable characteristic scale y 10 kpc, "embarrassingly large for something deduced from a small-scale approximation". "From a local theory we cannot produce any preference for the formation of symmetrical two-arm spirals", GLB recognized, but found it "however [...] likely that the instability leading to them is a somewhat more organized form of the one discussed here" (GLB, p.151).
The GLB paper was closed with a "Note added in proof" whose reproduction here will allow us to turn conveniently to the subject of the remaining sections of this chapter.
"We have heard from Dr Toomre and Mr Julian of further work on zero thickness stellar disks including a discussion of sheared modes. These behave very similarly to their gaseous counterparts discussed here. This work was independent of ours although the same sheared coordinates have been invented by them." (GLB, p.157-158)
14 One knew well the interstellar gas as being heated up by star-formation regions and particularly by supernovae, but also dissipative, tending to self-cooling and forming clumps at least slightly bounded by self-gravity. This invited a no less than two-component gas scheme with molecular clouds as its discrete, dense, cold and inelastic part. Such a mix badly approximates to an isothermal gas sheet, however, and it carries over no better to acoustic waves. "It is not clear at all how one may go about describing the collective behavior of such a medium - Kalnajs reasoned. - Clearly an application of the hydrodynamical equations (including magnetic effects), correct in principle though, leads to a problem of unmanageable proportions" (Kalnajs 1965, p.56). He thus "pretty much avoided gas dynamics." (Kalnajs) Toomre had taken some such action as if continuing his star-disk-stability study, and even submitted a special paper to ApJ (Toomre 1965), but then he dropped it suddenly and was never upset about having retreated (Toomre). In contrast, Goldreich and Lynden-Bell, who claimed priority to gas models, just had to be content with their simplest isothermal treatment, saying that it was "not a bad approximation" overall and, anyway, "not significant for the linear mathematics from which we obtained our main results." (Goldreich) Back.
15 "We felt that there were lots of disturbances in galaxies once one mode had become nonlinear and so there would be no difficulty in having a small component to amplify. We were not concerned with any feedback loop at that time and to this day I am less than sure of its existence in real unbarred galaxies as opposed to theorist's models." (Lynden-Bell) Back.
16 The GLB gas treatment of galactic disks reflected Lynden-Bell's earlier devotion to cosmogony and, in its frames, to Ledoux-oriented analytical tradition of treating flat systems (Ledoux 1951). GLB believed that even the largest-scale galaxy dynamics features the gas as the colder and more pliable dynamical component, and the idea of general `equivalent stability' of gas and star models hit them on the fact that those obey the same Jeans-instability criterion G 2 when infinite and in uniform rotation.
Lynden-Bell to Toomre: "We treated everything as gas not because we think the gas is dominant (except possibly as a triggering mechanism) but because in those cases where the transition from stability to instability can be worked out for both a star distribution function and an equivalent gas system they both become unstable at the same point. At present I only have a proof of this for star clusters whose distribution functions depend on energy only and I am not sure what equations of state the anisotropic pressure of a gas should obey if it is to go unstable in the same way as the stars in the disk of a galaxy. However I think this would probably make our basic philosophy clearer. This work on the equivalence of stability is almost all that is directly relevant that is happening here at present". (Lynden-Bell 1964a) ("I had already found very little difference between stars and gas in the Jeans instability criterion so had little compunction in solving the gas problem with the velocity dispersion of stars replaced by the sound velocity of the gas." (Lynden-Bell))
Toomre to Lynden-Bell: "I understood that your main motive was not so much to gather what a supposedly smooth gas disk would do by itself, as to mimic the likely behavior of a disk of stars. At least in a vague, intuitive sense I agree with you that the pressure should give neutral stability results that should at the worst be of the correct order of magnitude. [...Still] the evident gross unevenness in the way the interstellar matter appears to be distributed in most galaxies would have meant that such initially smooth analyses could not directly be relevant." (Toomre 1964c)
Lynden-Bell to Toomre: "I convinced myself that star and gas systems (apart from static) normally have different critical stability criteria. This floors my earlier hope though I did prove a nice theorem for static systems." (Lynden-Bell 1964c) "When I get a typist to do it I will also send you a lengthened version of the paper I delivered at IAU Symposium No 25 on stability of collisionless systems. This is great fun though not applied to spiral problems." (Lynden-Bell 1964d) Back.