Spiral Structure and the. Stability of Stellar Disks y. Rutgers University, Department of Physics and Astronomy, PO Box 849, Piscataway, NJ 08855, USA

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1 1 Spiral Structure and the Stability of Stellar Disks y J A Sellwood Rutgers University, Department of Physics and Astronomy, PO Box 849, Piscataway, NJ 08855, USA Rutgers Astrophysics Preprint no 182 Abstract The generation of transient spiral waves by swing-amplication in self-gravitating disks is now regarded as the underlying reason for the predominance of spiral patterns in galaxies. The physics of this process is well understood and should be readily applicable to gaseous accretion disks. There is less agreement on a mechanism for the maintenance of long-lived spirals, or the continuous regeneration of fresh spirals, in galaxies, but most current ideas in this area do not seem likely to carry over to gaseous disks. 1. Introduction In some respects, disk galaxies can themselves be regarded as gigantic accretion disks in which most of the eective viscosity comes from the gravitational stresses associated with spiral waves. Thus my main focus here will be on the controversial topic of the origin of spirals in galaxies. In the limited space available, I am able to do no more than summarize my own view of the current state of the theory. 2. Gravity Torques It is easy to state what is required to maximize the gravitational stresses that transport the angular momentum. A formula for the couple exerted on the inner disk by the outer disk seems to have been rst written down by Lynden-Bell & Kalnajs (1972): C z = 1 4G Z S f R f ds: y To appear in: Basic Physics of Accretion Disks S. Kato et al. (eds.)

2 2 Sellwood Here, the integration surface is a right cylinder parallel to the spin axis and f R & f are the radial and azimuthal parts of the perturbation forces that arise from the spiral density variations. Since the couple depends on their product, its magnitude rises directly as the square of the amplitude of a perturbation of constant shape. For a xed amplitude, the couple also peaks for open spiral waves with pitch angles around 45, dropping to zero both for rings and for bars. Strong, open spiral patterns transport the most angular momentum, therefore. Devising a theory for the generation of such disturbances seen in galaxies has proved to be surprisingly dicult, however. 3. Swing Amplication It is now clear that the so-called \swing amplication" process is the fundamental reason for the existence of spiral patterns in shearing disks. It was rst recognized by Goldreich & Lynden-Bell (1965) and by Julian & Toomre (1966), and has been reviewed by Toomre (1981). Any disturbance in the disk induces a large-scale spiral response which looks very promising for a short period. The typical crest-to-crest wavelength at which amplication peaks is G crit = 42 ; 2 where is the disk surface density and is the Lindblad epicycle frequency. Because the spirals are strongest at some nite inclination angle, the preferred azimuthal wavelength, 2R=m, will be greater than crit. Since the inclination angle for peak amplication increases with the rate of shear, the ratio 2R=m to crit is 1-2 for best amplication in a at rotation curve, while it is 2-4 in a Keplerian disk. Thus for open patterns to produce a large torque, we require a high surface density disk, since crit is larger and the shear is reduced. Unfortunately, a swing-amplied spiral winds up and fades rapidly as the remorseless shear continues, and patterns can therefore be long-lived only if the seed disturbance is constantly regenerated or if there is some sort of feed back from trailing to leading waves. 4. Stellar vs Gaseous Disks The process of swing amplication works equally well in both stellar and gaseous disks, but other ideas for the maintenance of strong, open spiral waves in galaxies are specic to particle disks. In particular, the wave-mechanics of feed-back loops or other processes needed to maintain spiral activity over long periods dier in fundamental respects between gaseous and stellar disks. First, stellar disks cannot support compressive (sound) disturbances and non-axisymmetric waves are therefore conned to the region between the

3 Stellar Disk Instabilities 3 Lindblad resonances, straddling co-rotation. Second, strong wave-particle interactions occur at Lindblad resonances in stellar disks, where incident, small-amplitude waves are absorbed. Third, the strong, open spirals that exert considerable gravity torques, must extract and deposit angular momentum from the resonant particles at the ends of the spiral arms (Lynden-Bell & Kalnajs 1972). We therefore expect strong spirals in galaxies to be short-lived since narrow resonances will saturate quickly. In fact, resonant scattering by large-amplitude waves may play a pivotal role in the continuous regeneration of fresh spirals in particle disks (see x6.2). None of these processes occurs in a gaseous disk, where no Lindblad resonances occur. 5. Global Modes Ideally, one might hope that spiral instabilities would emerge as solutions of the global eigen-mode problem for small perturbations to some equilibrium model. The basic set of equations to be solved were written down long ago (Kalnajs 1971) but solutions have been obtained in just a few highly idealized cases. Much more has been learned from local approximations and N-body simulations, and relating these less elegant approaches can be still more informative. A whole zoo of global instabilities has been found so far. Jeans-type instabilities, in which gravity is the destabilizing agent, include axisymmetric modes (Toomre 1964), bar modes (Hohl 1971, Kalnajs 1972), edge modes (Toomre 1989), lop-sided modes (Zang & Hohl 1978, Sellwood 1985), etc. Bending modes (Toomre 1966, Fridman & Polyachenko 1984, Merritt & Sellwood 1994), on the other hand, are driven by anisotropic velocity dispersion and gravity exerts a stabilizing inuence. No galaxy model exhibits all these instabilities simultaneously, of course, and we now have some understanding of the regions of parameter space where each can be expected (Sellwood 1994). Most of these modes seem irrelevant to accretion disks. Lop-sided modes, for example, which might have the most interest for accretion disk dynamics, come in two kinds in stellar disks. The simplest are found in rather bizarre disks with roughly equal counter-rotating streams of stars (Merritt & Stiavelli 1990; Sellwood & Merritt 1994), an impossible situation in gaseous disks. The second are the m = 1 analogue of the well-known bar instability (Zang 1976, Sellwood 1985) which requires an extremely massive disk. The so-called slingamplied modes described by Shu et al. (1990) are not seen in stellar systems because their feed-back is via sound waves reecting o the edge. None of these instabilities leads to mild spiral density waves. Most initially smooth models possess either a violently disruptive instability, such as a bar, or nothing at all. A rare example of a completely stable model is the Mestel disk used by Toomre (1981) to illustrate swing-amplication; the last few

4 4 Sellwood frames of his most revealing Figure 7 illustrates what was already known from Lynden-Bell & Kalnajs (1972) and Mark (1974) that that even a vigorous transient disturbance is damped as the wave \rolls onto the beach" at the Lindblad resonance. In such a situation, the wave action is absorbed at the resonance through wave-particle interaction, in a manner very analogous to Landau damping. 6. Theories of Spiral Waves Since Lindblad resonances were recognized to damp waves, C C Lin and his collaborators have developed a theory of spiral wave generation in smooth disks in which the inner Lindblad resonance is shielded by a \Q-barrier". Bertin et al. (1989) have found slowly growing, tightly wrapped, mildly unstable spiral modes in cool disks with a dynamically hot inner region. They argue that such modes probably lead to low-amplitude, quasi-stationary spiral patterns. Even if these solutions behave as they argue, and the same models do not permit more vigorous evolution, such mild, tightly wrapped waves will transport so little angular momentum (Bertin 1983) that they are of little interest in the context of accretion disks. It has gradually dawned on us, however, that it may be incorrect to assume that disk galaxies are smooth, well mixed systems, and this idea has led to two further theories for the origin of spiral patterns. Further ideas related to bar forcing and tidally induced patterns are clearly important for some galaxies, but are not of much interest in the context of accretion disks Swing-amplied noise Toomre (1990) and Toomre & Kalnajs (1991) emphasize that shot noise associated with a random distribution of particles will be swing amplied. As the inner particles drift by the outer ones, one observes an ever changing \kaleidoscope" of spiral arm fragments. The level of shot noise from stars alone is too low to make strong spirals by this mechanism, but Toomre envisages that uctuations from star clusters and giant molecular clouds would be large enough excite what is observed. This essentially local theory does not attempt to oer an explanation for a \grand-design" pattern, although the most strongly amplied features will be quite large-scale, since crit, which determines the crest-to-crest spacing, is typically comparable to the local radius in many galaxies Groove modes Lovelace & Hohlfeld (1978) and Sellwood & Kahn (1991) showed that a local deciency in the angular momentum density of stars in a disk drives an instability. The latter authors named it the groove instability, since a

5 Stellar Disk Instabilities 5 deciency of particles with a particular angular momentum would, in the absence of random motion, yield a disk with a groove, but the instability persists even when random motions are large-enough to wash out a noticeable density feature. The basic instability originates through a process analogous to Landau excitation in a plasma with a double-peaked distribution function and produces a co-rotating wave-like distortion to the edges of the groove. This would be a mild disturbance, were it not for the enthusiastic support response of the surrounding disk, which since it is again due to the swingamplier, gives greatest encouragement to waves on scales close to crit. The instability saturates when the groove is closed, leaving the disk with an azimuthal density variation at co-rotation which takes some time to disperse; the spiral response of the surrounding disk decays as the forcing perturbation disperses, and the angular momentum it transfers is deposited at the Lindblad resonances. The absorption of angular momentum and resonant scattering is readily observed in the simulations. It creates a deciency of particles in the disk with the resonant angular momentum, which can lead to a new instability with co-rotation of the new wave at the radius of the Lindblad resonance of the old. Other processes, such as wave reection of the disturbed region, may also contribute to new instabilities. Thus a self-perpetuating cycle of instabilities is set up in the simulations, which is ultimately limited by the increasing random motion in the disk. Because the disk amplies the disturbance, each new groove is deeper than the last, and the limiting amplitudes of successive instabilities increase. Moreover, particle noise can provide the initial seed. We therefore conclude that any system of particles, no matter how large, can always produce large amplitude spirals even when the equivalent smooth disk is stable, which is consistent with my experimental results with up to two million particles. Interesting as I nd all this, I once again have to conclude that a theory of spiral wave generation based on resonant scattering can be of little relevance to accretion disks. 7. Conclusions A strongly shearing, massive disk is all that is required for the swing ampli- er to produce large-scale, open, transient spiral patterns. Their amplitude is determined by the magnitude of the input signal, however, and will be very feeble in an initially quiescent disk. If a mechanism is available to produce seed density variations, or if some form of feed-back can occur, then the swing amplier will give us the large-amplitude spirals we desire. But the mechanisms by which this could happen in gaseous accretion disks are unlikely to resemble any of the current ideas for generating spiral waves in galaxies. More is known about the global modes of stellar disks than of gaseous disks, yet it also seems unlikely that this body of theory will prove to be

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