2.4. Kalnajs' search for spiral modes

One can draw a parallel between the attempts to talk about
galactic evolution at the present time and the attempts to
understand stellar evolution before the sources of energy in
the stars were understood.
G. R. Burbidge 1962, p.291

The study of stellar systems, such as our own galaxy, is not
limited by a lack of understanding of the underlying princi-
ples, but rather by the difficulty of solving the differential
equations which govern the time evolution of the system.
Kalnajs 1962, p.i

Agris Kalnajs began his undergraduate studies in Electrical Engineering at MIT in 1955. As a good student, he participated in a special course which emphasized physics and mathematics, and provided summer employment in the Microwave Research Lab at Raytheon, making measurements for computer modeling of magnetrons. There he learned about such things as electron motions in crossed electric and magnetic field, waves carrying positive and negative energies, modes, coupled modes, parametric amplification. All this proved to be really useful in a quite different field when he arrived in 1959 in the astronomy department at Harvard University and got involved in galaxy dynamics. ⁵⁵

In the fall of 1961 Kalnajs made a research examination on "Stellar kinematics" (Kalnajs 1962). ⁵⁶ The task was to calculate self-consistent radial oscillations in a rotating stellar disk as a tentative explanation for the `local' arms in our Galaxy. Their short spacing L 3 kpc justified the small-scale analysis in the plane of a homogeneous thin sheet. Kalnajs solved the Vlasov and Poisson equations as an initial-value problem and obtained an equation for the radial oscillations and a dispersion relation which was formally correct. ⁵⁷ As he was interested in short waves, he made an asymptotic evaluation of the integral expression, and in the process left out "a factor 2 or something of that order" (Kalnajs). This and the reduced disk response at the short waves ( ~ 1 kpc) made him conclude that , because the self-gravity effects became "too small to be interesting" (Kalnajs): all the solutions oscillated and were traveling waves that, in passing, "tend to gather up the low dispersion objects such as gas" (Kalnajs 1962, p. ii). As a plausible "arm-like density wave" generator, an oval-shaped body at the Galaxy center was mentioned.

The error in this asymptotic evaluation was uncovered in the summer of 1963 when Kalnajs and Toomre finally got together, compared and crosschecked their notes, and detected each other's technical errors. Kalnajs looked anew at his radial-oscillation theory and re-evaluated the dispersion relation, this time into the form in which it entered his thesis (Kalnajs 1965). ⁵⁸ In modern notation - whose convenience and clarity we owe undoubtedly to Lin - and without the uninteresting stellar disk thickness correction going through that original 1961-63 analysis, ⁵⁹ it is

(9)

where

(10)

is Kalnajs' version of a factor to account for the role played by random motions of stars. There is no such play in the limit x = 0, relation (9) then reduces to Toomre's cold-disk result (5) that shows the gravity term proportional to the wavenumber and growing without bound. Now random motions arrest this growth: the total contribution of gravity only reaches a maximum at x₀ 1, still giving rise to instability (² < 0) if large enough, and for x > > 1 it becomes small. In the solar neighborhood that value of x₀ points to a radial wavelength ₀ 6 kpc, the one concluded by Toomre from his neutral stability analysis. Its commensurability with the radial size of the Galactic disk makes the local theory somewhat suspect.

"When I wrote my Research Examination I was under the impression that the spacing between the spiral arms was about 1.5 kpc. After Toomre and I got together, it became clear to me that the 1.5 kpc waves/fluctuations were not the important modes of the Galaxy. [...] Also by the fall of 1963 I had obtained my own copy of Danver's thesis (thanks to my uncle who was at Lund University). Danver had measured the spiral patterns and came up with a typical pitch angle of 16°.6. This implies scales even larger than 6 kpc. [...] By this time Alar had published his disk models, and I could use them to estimate the scales at which these disks were most responsive, and they convinced me that a WKBJ approach [see Sect. 3.1] was too crude [...] and that - unlike plasma - galaxies were too inhomogeneous. [...] So the future was `global modes and integral equations'." (Kalnajs)

Once he realized this fact, Kalnajs lost interest in the local theories, which were good for the stable small-scale solutions, and turned to global modes as the correct approach to the oscillation problem. In the fall of 1963 he presented to his thesis committee at Harvard "An outline of a thesis on the topic `Spiral structure in galaxies'" (Kalnajs 1963), summarizing his ideas for a new theory of steady spiral waves. Because this document has been almost unknown, a long quotation from it appears to be quite appropriate. ⁶⁰

"A feature peculiar to highly flattened stellar systems is the appearance of spiral markings, called arms. These features are most prominently displayed by the gaseous component of the galaxy and the young hot stars which excite the gas. However, the density fluctuations can still be seen in the stellar component, appearing much fainter, but also more regular.

The division of the galaxy into two components, gaseous and stellar, appears natural when one considers the dynamical behavior of these two subsystems. The gaseous component is partly ionized and is therefore subject to magnetic as well as gravitational forces, and has a very uneven distribution in the galactic plane. The stellar system is quite regular, its dynamics being governed by the long-range gravitational forces arising from the galaxy as a whole; the density of stars is sufficiently low that binary encounters between stars may be ignored. The stellar component, which is the more massive, cannot support density fluctuations on a scale much smaller that the mean deviation of the stars from a circular orbit (or the scale of the peculiar motions). The gas, on the other hand, would support smaller-scale fluctuations - at least in the absence of magnetic effects. The fact that observed spiral arms are not much narrower than the smallest scale that the stars will tolerate suggests that stars must participate actively in the spiral patterns.

There is a fundamental difficulty, however, in the assumption that spiral arms are entirely stellar: if an arm can exist and does not grow in time, then its mirror image is also a possible configuration. This follows from the time-reversibility of the equations of motion combined with their invariance under spatial inversion. Thus the leading or trailing character cannot be decided on the basis of a linearized theory if we insist on permanency of the spiral markings. The observations indicate, however, that nature in fact prefers trailing spiral arms. Thus a plausible theory of spiral structure must include both the stars and the gas.

I regard the galaxy as consisting of two components, gas and stars, coupled by gravitational forces. The stars provide the large scale organization and the gas discriminates between leading and trailing arms. ([Footnote in the original text]: The stellar system can be thought of as a resonator, and the gas would then be the driver which excites certain of the normal modes.) If the coupling is not too strong, one may at first consider the two subsystems separately, and afterwards allow for their interaction. Unfortunately, one cannot evaluate the magnitude of the coupling without calculating the normal modes of the two subsystems. For the gaseous component, only the crudest type of analysis is possible at present, since one should include non-linear terms in the equations governing the gas motion in order to be realistic. The stellar component, on the other hand, is sufficiently smooth that a linearized theory should apply, and the problem of determining the normal modes can be formulated, and, with a little effort, solved.

I have chosen as my thesis topic the investigation of the stellar normal modes in the plane of a model galaxy. [...] Some qualitative features of the equations indicate that the type of spiral disturbance with two arms is preferred. This result does not seem to depend critically on the model, which is encouraging. The final proof has to be left to numerical calculations, which are not yet complete." (Kalnajs 1963, p.1-3)

It is seen therefore that Kalnajs was envisaging the disk of stars as a resonator in which global spiral-wave modes are developed. If stationary, the leading and the trailing components are just mirror-imaged, so that, superimposed, they give no spiral pattern. However, due to slow non-reversible processes occurring in real galaxies, the symmetry is violated.

In support of his normal-mode concept, Kalnajs considered large-scale non-axisymmetric disturbances to a hot inhomogeneous flat stellar disk, and derived for them a general integral equation whose complicated frequency dependence implied a discrete wave spectrum. He also pointed out the role of Lindblad's condition (4). When satisfied, large parts of the galactic disk could support coherent oscillations for the m = 2 mode, whereas for larger m's there would be Lindblad resonances within the disk. Stars in these regions feel the perturbing wave potential at their own natural frequency,

(11)

thus undergoing strong orbital displacement and making the m > 2 modes lose integrity ⁶¹ . Hence Kalnajs concluded that his "formulation of the problem" shows a dynamical preference for two-armed spirals and "gives little insight of what to expect in both the shape of the disturbances and their time dependence when m > 2" (Kalnajs 1963, p.13).

A summarizing exposition of the subject Kalnajs gave in his PhD thesis "The Stability of Highly Flattened Galaxies" presented at Harvard in May 1965 (Kalnajs 1965); ⁶² it contained an extended discussion lavish in ideas and technicalities. At the same time, the thesis became in fact Kalnajs' official public debut, so that to it as a reference point should we attach chronology when confronting certain factual points in the spiral history of the 1960s.

⁵⁵ "It was probably David Layzer's course in classical dynamics which steered me towards stellar dynamics. I rather liked David's approach: he strived for elegance. He put a lot of thought in his lectures". (Kalnajs) Back.

⁵⁶ As this was only an unpublished internal document, its outline below is mainly to illustrate how Kalnajs was then progressing. Back.

⁵⁷ Following Landau's method correctly describing small oscillations in homogeneous electrostatic plasma, an arbitrary disturbance is initially imposed on the stellar sheet and its evolution is traced out. With time, the dependence on the initial conditions dies away, and the result is provided by the integrand poles whose expression - the dispersion relation - connects the established wave parameters. Back.

⁵⁸ "Strictly speaking, I was the first to write down the dispersion relation. But that is not the important thing. What is more important is who made the best use of that equation. And here it was Toomre, who used it to discuss the stability of the Galactic disk - a distinctly more fundamental topic than the subject of my Research Examination. [... ] By the time we got together in 1963, that is probably the way we understood our respective contributions". (Kalnajs) Back.

⁵⁹ The thickness corrections were worth considering for wavelengths as short as 1.5 kpc as they reduced the radial force by a factor of 2 or 3, but for 6 kpc the reduction was some 20%-30% at most. Back.

⁶⁰ "I do not recall exactly when I first learned that Lin was also interested in spiral density waves (it was probably a talk he gave at MIT), but at that stage our relations were most cordial and I also felt that my understanding of this topic was more thorough than his. So having produced a written document, I am pretty sure that I would have found it difficult not to boast about my achievements" (Kalnajs). "A written document" there refers to the "Outline" which at least Toomre received from Kalnajs in November 1963. Back.

⁶¹ A combination - m is called the Doppler-shifted wave frequency, one reckoned in a reference system corotating with disk material. The shift is due to the fact that waves are naturally carried along by flows. Back.

⁶² Kalnajs' thesis committee members were Layzer, Lin and Toomre, as officially confirmed from Harvard. Back.