6.4. What Are the Spokes?

One topic of especial interest in the Cartwheel is the spokes which so perplexed Zwicky in 1941. Spokes in ring galaxies are rare. Indeed, the Cartwheel spokes are unique among the known collisional ring galaxies. AM0644-741 and II Hz 4 have spiral-like arms interior to the ring, but there are few of them and they are less flocculent in appearance. It is possible that these latter features are the direct result of a low order spiral mode driven by a slightly asymmetric collision. Similarly, the Cartwheel spokes might be the result of a higher order mode in the collisional perturbation, but their morphology makes it seem less likely. We favor the alternative hypothesis that they are the result of internal gravitational instabilities following the impact, and if so, they may provide sensitive constraints on that process. Several of the simulation studies referenced above provide input on this question, including especially Hernquist and Weil (1993), Struck-Marcell and Higdon (1993), and Gerber (1993).

Let us consider the wave-triggered instability hypothesis in a little more detail. Before the collision, assume that the surface density in the disk is below, but near the threshold value for axisymmetric gravitational instability (e.g. Binney and Tremaine 1987, Section 5.3). The disk already knows the approximate instability scale, because the wavelength of the fastest growing mode is determined only by parameters intrinsic to the target galaxy. Finite amplitude stochastic fluctuations can generate flocculent spirals on roughly this scale. After the collision, wave compression increases the surface density, pushing it over the critical value for instability. Unstable growth occurs in the rings, with the flocculent spirals or giant clouds serving as seeds. The simulations of Struck-Marcell and Higdon show that there need not be a one-to-one correspondence between seeds and spokes. Some seeds may merge in the ring, but the number of seeds and spokes (and star-forming knots in the ring) should be roughly the same since they are formed by the same instability. In the rarefaction behind the compression wave high density regions are stretched into spokes. Clearly these high density regions must not become so tightly bound in the wave compression that the shear and expansion are unable to stretch them apart. At the same time we assume that the wave compression is sufficient to lead to gravitational collapse and star formation on smaller scales within these proto-spokes. Some flocculent seeds may consist of such loose agglomerations that they do not grow significantly in the compression and are subsequently disrupted in the rarefaction.

This hypothesis has many interesting consequences. Firstly, no spokes are predicted if the perturbation is small and the pre-collision disk is well below the gravitational instability threshold, e.g., if the gas surface density is very low. However, the mass of the companion galaxy in most ring systems is usually fairly substantial compared with the target galaxy, if the optical or IR luminosity of the companion is any guide. Thus, even if the precollisional gas was quite stable, we might expect a substantial perturbation in most targets, which would be sufficient to push the gas density over the instability threshold. In this case a second factor enters, namely, the instability growth time relative to the compression time. The former is a local quantity, essentially the local free-fall time. The latter is one half of the local epicyclic period, which depends on the global gravitational potential, and so the compression time depends on global structure. Therefore, spoke formation depends on both local and global parameters.

One possible explanation for the apparent rarity of spokes is that most spokes may generally be hard to recognize. A reasonable estimate for the scale of spokes is given by _max, the maximum unstable wavelength. According to the linear perturbation theory of the axisymmetric gravitational instability (see Binney and Tremaine (1987), Section 6.2),

(6.3)

where is the gas surface density. The number of spokes can be estimated as the ring circumference divided by this scale,

(6.4)

If this number is large, we might expect many small spoke segments, which would be poorly resolved on most ground-based CCD images presently available. Moreover, since each such "spokelet" only contains a small fraction of the gas in the disk, the enhancement of star formation within the spokelet may be modest. The brightness contrast between spokelet and inter-spoke regions should be small. Thus, relatively bright spokes, stretching between two rings, or between an outer ring and the nuclear regions, may only form when N_spoke 2. (Note: this criterion is closely related to the dependence of the swing amplification on the parameter X = / _crit, see Toomre 1981). Within the next few years it should be possible to obtain observations of sufficiently high resolution to test these ideas.

In the meantime, we can return to the published simulations, and ask what information they provide about spokes. At first sight, the fully self-gravitational N-body/SPH studies seem to give contradictory results. In the simulations of Hernquist and Weil (1993) strong spokes form, while spokes do not form in the simulations of Gerber (1993), and Gerber, Lamb and Balsara (1994). Spokes, if present, are not strong in the N-body/gas particle simulations of Horellou and Combes (1994). Spokes also formed in the models of Struck-Marcell and Higdon (1993), in which self-gravity was computed only on small scales.

Another mysterious fact is that in all these simulations the precollision target disks were evidently set up near the instability threshold. This statement can be quantified through the use of Toomre's Q parameter for axisymmetric stability (see e.g., Kennicutt 1989). Gerber's simulations were initialized such that Q = 1.5 for the stars throughout the disk (where Q is defined as 0.3_r / (G), and _r is the radial velocity dispersion). Hernquist and Weil (1993) quote a value of Q = 1.3 (apparently also for the stars) in their model. In Struck-Marcell and Higdon (1993), the value of Q is similarly about 2.0-3.0 throughout the disk. Thus, it would appear that these simulations, starting from similar initial conditions, produce different results.

In fact, there are significant differences in the halo and disk density distributions between the various models. Gerber's target rotation curve increases roughly linearly out to about 1.5 disk scale lengths, and then is quite flat (though it does turn over). Hernquist and Weil describe the rotation curve of their target disk as nearly flat out to a cutoff radius, r_c. The rotation curve used by Struck-Marcell and Higdon was of the form v (r / )^1/5, which is slowly rising at radii greater than the scale radius . These differences and those in the initial conditions account for different radial distributions of _max and N_spoke.

In Hernquist and Weil (1993), the value spoke ranges from a few in the inner regions to greater than 10 in the outer disk. According to our calculation, in Gerber's disk N_spoke 11 at r = 1.0 (in computational units, or 0.8 disk scale lengths), and rises steadily at larger radii. Moreover, in Gerber's simulations the typical value of _max is only of the order of a couple of mesh lengths used in the PM calculation. These results are in accord with the discussion above, i.e., spokes are seen in the models of Hernquist and Weil because of the low value of N_spoke. Only small spokelets are expected in Gerber's simulations, and these may be smoothed since their size is comparable to the finite difference algorithm scale. In the models of Struck-Marcell and Higdon (1993), self-gravity is only computed over a small range of wavelengths, but the target disk was initialized such that _max falls within this range. Models with N_spoke of order of a few produced spokes much like those of Hernquist and Weil, though weaker. Models with larger values of N_spoke produced numerous spokelets (see Figure 8 of Struck-Marcell and Higdon 1993). Note also that heating and cooling effects have not been included in these "spoke simulations" (see Section 6.5).

In the figures of Hernquist and Weil, the spokes appear in the gas before the collision, and are then amplified by the ring passage. Their early appearance may be the result of a slightly lower effective Q than the other models, and greater intrinsic amplification because the most unstable mode is of relatively low order.

The story of the formation of the spokes is not yet a closed book. We have recently produced realistic looking spokes behind the second ring in a simulation which only produced rather weak small-scale spokelets (large N_spoke) behind the first ring (Struck, in preparation). This Cartwheel model is based on the premise that the far companion (Galaxy G3 in Higdon's nomenclature; see Section 3.3.1) was the intruder, and that the outer ring is the second, not the first ring. The computational details are as in Struck-Marcell and Higdon (1993), except that the intruder model also contains a gas disk, and heating and cooling processes were included in the model (Section 6.5). The companion mass was also lower (about 15% of the Cartwheel). Such models open up the possibility that the disk need not be on the edge of instability for spoke formation initially, but that two cycles of ring compressional amplification could do the job. This would help to explain the rarity of spokes, since only well evolved ring galaxies would develop them.

We remind the reader that, thus far, no gas has been found unambiguously associated with the spokes as might be expected from the models above and they still remain enigmatic. The discovery of dust lanes crossing the inner ring in the vicinity of the spokes (based on the HST observations of Figure 1) does, however, suggest that the detection of molecules in the spokes may simply be a matter of time and the need for greater sensitivity than is currently possible (see Section 3).