1.5. Spiral stellar wakes

The main inference to be drawn from these analyses is un-
doubtedly that even a seemingly stable, differentially rotat-
ing star disk ought to respond with a remarkable intensity,
and in a distinctly spiral manner, to quite typical forms of
non-axisymmetric forcing. [...] These intense trailing star
responses obviously demand a physical explanation.
Julian & Toomre 1966, pp.829, 827

The response to a traveling point mass is a nice physical idea
although again it can't be very large scale. However I am sure
one can see results of it in galaxies and it could be important
as an observational tool to tell the conditions in a galaxy from
the shapes and angles of `the tails of condensations'.
Lynden-Bell 1964d

However nice the above results from the first part of the JT paper may have appeared, one cannot help noting that in the large they just confirmed the basic GLB picture of the strong transient amplification. The `English signal' that had come to Julian and Toomre in June 1964 must have made them feel that they would not get out of the shadow cast by the GLB-planted spreading tree without an advance in strict description of local swing-amplification in its self-consistence and closure. And, it must be said, that signal did not take them unawares: they had already set themselves the task of answering the question: "How would a thin, differentially rotating, self-gravitating disk of stars respond to the presence of a single, particle-like concentration of interstellar material orbiting steadily within its plane?" (JT, p.810)

"We have thus far mainly talked about putting these disturbances together to obtain among others the density patterns of the steady response of stars to a point mass representing a similarly orbiting gas concentration, for instance. ²³ This task will only be messy, not difficult, [...] but we can already foresee that the response density will be in the form of an elongated hump, inclined roughly at your angle to the radius" (Toomre 1964b).

Volterra-equation methods allowed Julian and Toomre to calculate those responses, essentially by superposing lots of individually shearing waves to obtain a steady pattern of positive and negative disturbed densities in the vicinity of the imposed mass point. And precisely because many of those waves had been strongly swing-amplified, this sum of Fourier harmonics resulted in an awesome trailing stellar wake extending to both sides from the point perturber (Fig.2). As might be expected, the isodensity line inclination was sensitive to the shear rate and much less so to the stability parameter Q; the latter, in its turn, strongly influenced the wake's amplitude, especially at Q 1. JT preferred Q = 1.4, however, as corresponding to our solar vicinity, and for this case they computed disk-thickness corrections. At the assumed thickness 2h 0.1 _T 1 kpc, those reduced the perturbed gravity by no more than 20-30% but did not hurt the general characteristic picture of a steady trailing spiral-shaped wake that impressed one with its severe length scale and amplitude (Fig.3).

Figure 2. A stationary density response of the JT model on the action of a local mass source. Q = 1.4, V(r) = const. (The figure is reproduced from Julian & Toomre 1966)

Figure 3. A polar-coordinate view of the response in Fig.2 corrected for the disk thickness 2h 0.1 T. The galactocentric distance R0 equals T. (The figure is reproduced from Julian & Toomre 1966)

To clarify the dynamical substance, the authors separately considered what happens to a `cold' test star, say, on a larger circular galactic orbit than the imposed point mass, as the differential rotation carries it past this force center, collective forces being ignored (Fig.4). As long as the time interval during which the star is close to the center is only a fraction of an epicyclic period, as in the shearing conditions near the Sun, the radial force component resembles an impulse occurring at the instant of closest passage. It sets the star in epicyclic motion by giving it a radially inward disturbance velocity at the abreast position. Because of this the relative speed of passage reaches a minimum approximately one-quarter epicyclic period later, or some 45° or so downstream of the perturbing mass point. This angle, which is considerably larger in the collective case - toward 70°, as in Figs 3, 4 - shows the direction in which the passing stars are grouped most closely, forming a characteristic phase concentration called a wake.

Figure 4. Trajectory of a test star moving past a local mass source. Q = 1.4 and V(r) = const as in Fig.2, but the collective effects due to mutual attractions of the background stars are turned off. (The figure is reproduced from Julian & Toomre 1966)

What can one say about the JT work in summary? It stands on its own as a complete solution to a well-defined problem, an accurate and ample model, a neat and strict theory (though tedious to compute). Being self-contained, it has no need for subsidiary assumptions, hypotheses, speculations and evaluations. It must have been evident to many thinkers that a steady compact source might create nothing but a steady (what else, if any?) hump of trailing (what other in the face of shear?) orientation. Why had this idea not been worked out earlier? Because fresh physical intuition, mathematical excellence and advanced computing were needed, and all at once. But, all the same, the paper itself impeded general insight into its findings. Written with the feeling of intellectual and aesthetic pleasure of having solved a difficult but important problem, the article contains some unnecessary confusing details, and in other places - through scrupulous and otherwise brilliant style and wording - is too condensed to be accessible without a lot of work by the reader. So it was rather too terse and mathematical in the general climate of tastes and attitudes with which traditional astronomers had encountered the early steps of new, modern galaxy dynamics. It has been no wonder that even several serious dynamicists, let alone ordinary astronomers, have never bothered to read this important paper carefully. ²⁴

²³ "It must have occurred to Bill and myself that the forcing by a point mass was a basic question to be answered, since it amounts essentially to a Green's function approach to this subject." (Toomre) Back.

²⁴ The JT paper had fewer than 20 citations (according to ADS) in the first 10 years after its publication. Back.