**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).
^{36} 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.