2.2. Antispiral theorem

After my paper on the stability of collisionless gravitating
spheres was published (Antonov 1960), I approached the
density-wave theory but did not believe it, mainly because
of the antispiral theorem, anyway known to physicists.
Antonov 2003

In hindsight, I think Lin's judgment was accurate consider-
ing how quick people were to attack his point of view with
proofs of `antispiral theorems' and the like shortly after the
publication of Lin & Shu 1964.
Shu 2001

At the Noordwijk Symposium Prendergast explained to astronomers the general meaning of the theorem. If in linear theory there were to exist a nondissipative global mode of trailing planform that was content to rotate indefinitely without growing or decaying, then a similar mirror-image leading mode must exist as well. "This symmetry property of the equations means only one thing: the system is too simple. Whenever you see a symmetry property, all you have to do is mess up the system a little bit and give up the symmetry. There are a large number of things that will remove the symmetry, [...] there is non-conservation of everything" (Prendergast 1967, p.308-309). ³⁴

The antispiral theorem took on particular sounding after Lynden-Bell and Ostriker (1967) set it out as an application of general principles they worked out for differentially rotating bodies. Lynden-Bell, to whom we owe the idea of this explicit consideration, no doubt knew that it "had many let-outs" hence he "did not think it as restrictive of spiral theories as some others took it to be". For one thing, the theorem could be strictly applied to exponential modes only, and Lynden-Bell hoped that "double modes that might grow as t exp(i t) might well be the ones needed to transfer angular momentum outward" through corotation (Lynden-Bell). ³⁵ Moreover, it did not oblige one at all to mix leading and trailing waves in equal proportion obtaining a cartwheel-type mode, that was no necessity imposed by the equal-frequency condition. Shu recalls that Lin from the beginning "felt sure that one should not do the naïve thing of superimposing equal trailing and leading parts" and that "he probably wanted to discover the reason why before publishing anything", but the Toomre 1964a paper "triggered him into premature action" (Shu). One is to wonder what annoyance for Lin and Shu became Lynden-Bell and Ostriker's antispiral address that appeared just when they thought they got the true mixing mechanism as due to disk overstability. It was imagined to cause slow growth of one of the components, the trailing one in Lin-Shu's `nonlinear' assumption, and then to break the full symmetry in the basically neutral-mode problem by ensuring different radial behavior for the components and, correspondingly and automatically, their unequal mixing.

In his 1968 thesis and, more pointedly, in his papers to follow (Shu 1970a, b) Shu demonstrated one more `let-out' in the antispiral theorem. "The general formulation for the normal modes, he noticed, [...] shows that a certain degree of spiral structure must be present in every mode of oscillation which contain stars in resonance" (S68, p.7). Stars, unlike gas, can resonate with the oscillating gravity field without any continual shattering due to collisions. Mathematically, this is answered by the integrand poles, and even at real frequencies those compel one to make integrations along contours going off the real axis, which provides the solutions with an imaginary part and ensures their general spiral form. The resonant technique of clearing the antispiral hurdle was to Lin and Shu one of the highest points to back up the QSSS as a neutral density wave (Shu 1970a, b; Lin & Shu 1971). <sup>36</sup> It seems curious, however, that they did not refer to any leading component either in 1966 on their short-wavelength spiral proposal for our Galaxy (Lin & Shu 1967), or in 1971 when Shu et al (1971) announced for M51 and M81, apart from their dominant short trailing waves, unmistakable traces of an extra `mode', yet not mirror-reflected - short and leading - but long and again trailing.

³⁴ The feel of symmetry breaking "non-conservation of everything" then prompted Prendergast that there ought to be some way to determine that "the natural way to get the arms is trailing" and that "presumably that would be a direction that would be given [...] by an increase of entropy" (Prendergast 1967, p.309). Back.

³⁵ "I always held the view that angular-momentum transfer is the driving force behind spiral structure. [...] In part the anti-spiral theorem was there because it seemed to point out that what Lin said was much less than the whole story." (Lynden-Bell) Back.

³⁶ Kalnajs already in 1963 had an idea of such a resonance `resolution' of the antispiral theorem in the neutral-wave setting (Kalnajs 1963; see Paper I, Sect. 2.4). Back.