1.5. Circulation theory of quasi-stationary spirals

The suggestion that the patterns are density waves is old
and was first explored by Bertil Lindblad. His emphasis was
mainly on kinematics and less on collective effects on a large
scale, though many of the kinematical effects he discovered
can still be seen in the collective modes.
Kalnajs 1971, p.275

His details were unconvincing, but no one can accuse him of
missing the big picture.
Toomre 1996, p.3

P.O. Lindblad's experiments with flat galaxies were planned to clarify the dispersion-orbit theory. They started with a plane system of several annular formations arranged by N 200 mutually attracting points, and the development of "small deviations in shape and density of a bisymmetrical nature" (Lindblad 1963, p.3), applied to one of the rings, was studied. Two waves propagating along it were shown to rise first, one running slightly faster and the other slower than unperturbed particles, thus invoking a pair of corotation resonances, one on each side from the ring. These induced a leading spiral; soon it rearranged into a trailing one and smeared out almost completely, but some trailing arms then re-appeared, owing evidently to a small oval structure retained at the center. This led P.O. Lindblad to propose that galactic spirals may involve a quasi-periodic phenomenon of trailing-arm formation, breakup and re-formation. ³⁴

B. Lindblad, however, got captivated by another view of these results. He even lost of his earlier dispersion-orbit enthusiasm and turned in 1961-62 to a concept "On the possibility of a quasi-stationary spiral structure in galaxies" (Lindblad 1963) in the presence of differential rotation. ³⁵

"The morphological age of spiral galaxies as estimated [...] from considerations of the evolutionary process connected with star formation from gaseous matter ranges between 10⁹ and 10¹⁰ years. In consequence it is natural to assume that the typical spiral structure is not an ephemeral phenomenon in the systems but has a certain steadiness in time [...and] to investigate how far gravitational forces alone can explain a spiral structure of a fair degree of permanence" (Lindblad 1964, p.103).

To begin with, Lindblad introduced an axisymmetric flat stellar system in differential rotation and, echoing the N -body pictures, imposed on it an initial trailing spiral pattern formed by some extra amount of stars. His calculations of the effect upon a nearby test star from such a spiral arm showed that, as it sheared, the star approached it and fell in, having no other chance to leave it than making slight epicyclic oscillations. Such an assimilation of material in just one galactic turn or so worked well against shearing deformation of spiral arms, through their exchange in angular momentum with stars attracted. As the result, the pattern's angular speed became the same all over, meaning its quasi-stationarity. Now two dynamically different regions arose in the system, an inner region with stars moving faster than the spiral, and an outer one, tuned oppositely; they were divided by a corotation region, where the material orbits at nearly the same rate as the pattern.

For a true stationary pattern not only its permanence in shape was needed, but also a balance of the stars' travel in and out of the arms. The latter was secured in Lindblad's eyes by his circulation theory (Lindblad 1963, 1964) developed in the framework of a trailing two-armed spiral model, each arm making one full convolution (or a bit more), comparably inside and outside corotation (Fig.4). Actually, each arm ended where, according to analytical estimates, its stars were effectively attracted by the next-to-last arm (outside corotation) and fell in it "in a shower of orbits". The assimilated stars kept moving slower than the spiral, thus having an along-arm ascent until a repeated flow down. Inside corotation (the region of much less interest to Lindblad), the circulation was set up as well, but in the opposite direction: stars captured by spiral arms got drawn down along them until sucked upward by the next-to-innermost spiral convolution.

Figure 4. Circulation of material in a galaxy having a quasi-stationary spiral structure. The general rotation is clockwise, points F mark the corotation radius. See the text for more details. (The figure is reproduced from Lindblad 1964)

The original spiral theories by Bertil Lindblad passed into oblivion. Among the causes for the passage were the feeble empirical base of the 1920s-40s, the frightening bulk of mathematics and scant help from the first computers even during the 1950s, a constant flux of changes in Lindblad's latest inferences and the rather opaque prose of his abundant articles, ³⁶ and above all a lack of quantatively checkable predictions. Yes, one can readily agree that

"all problems that in later developments turned out to be important in the theory of spiral structure had, in one way or another, already been touched upon or even studied by Lindblad" (Dekker 1975, p.18)

as well as that

"such complex collective dynamics was perhaps too hard for anyone, no matter how talented, in those mid-20^th-century decades before computers, plasma physics, or any inkling of massive halos" (Toomre 1996, p.3).

but also true is that all of the spiral undertakings by Lindblad, however ingenious and farsighted they may appear to have been in retrospect, got sunk ingloriously in the silence of time.

An interesting question is: why? Why did it come to be that the true master of theory and observation had long been surprisingly close to but never quite at the point of recognition - opened in the 1960s to a pleiad of fresh theorists - that spiral structure is mainly a collective wave phenomenon in shearing galaxies? One can only suppose that Lindblad did not reach, let alone exploit, such wave-mechanical ideas partly because they were not in the air yet, but perhaps mainly because he was impeded by his life-long emphases on the orbits of individual particles. All his efforts on galaxy dynamics were fed by the stellar-epicycle concept, the pearl of his scientific youth. This set the trend for Lindblad's theories, and whenever some such orbital attack fell short of its destination, he did not get on with searching for totally different ways of continuing, but instead renewed his attack time and again under his old epicyclic-orbit colors.

³⁴ "I was delighted to see them [P.O. Lindblad's results] as evidence as to how much one could do already then (!) by way of interesting numerical studies with some hundreds of particles - in that sense his work was very inspiring. Yet [...] it also struck me that his study really dealt with not much more than the transient breakup of inherently unstable configurations of some 4 or 5 artificially introduced rings of material" that imitated "a revolving disk - one which [...] should be fiercely unstable if begun just as cold. [...] But, again, as a sample of what could already be done, P.O. Lindblad's work was indeed like a breath of fresh air". (Toomre) Back.

³⁵ Lebedinski was another one who in his cosmogony of galaxies and stars admitted - still earlier - "the dynamical possibility of the formation of quasi-stable spiral arms rotating with a constant angular velocity for all the spiral" (Lebedinski 1954, p.30). Yet since Jeans' 1920s that idea, as such, did not sound as a novel dynamical motive. It got a really new sounding only when the fact of global galactic shearing was finally conceived. Back.

³⁶ "It has not been possible to do justice to all phases of Lindblad's researches", Chandrasekhar `complained' already in 1942, but nonetheless he gave a "more or less complete bibliography" including 25 Lindblad's writings on the spiral problem (Chandrasekhar 1942). "The flow of his publications can be understood if one realizes that he thought in the form of a paper. When attacking a problem he started writing the paper at once". (P.O. Lindblad) Back.