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3. Transient Events I: Some Wave Morphologies and Their Causes(Yxx)

This and the following chapter are devoted to the consideration of short term responses in galaxy collisions, i.e. those that occur shortly after closest passage. The examples below will be presented more or less in order of increasing disruptiveness. We begin with waves induced in disks, then take up large-scale mass transfer events, and finally the disruption of small companions. As discussed above, the often bizarre morphologies produced shortly after closest approach had a substantial impact on the early morphological groupings in catalogs; most of the nicknames below derive from those systems. Less obvious is the role of these transient features in the larger issues of collisionally enhanced (or suppressed) star formation, the merger process, and fueling active galactic nuclei (AGNs). In fact, their role is probably significant in all these cases, but they also provide unique signposts to help identify the nature of the interaction and the structure of the precursors. For example, the waves discussed below can provide a kind of seismological probe of the collisional galaxy.

In this chapter we consider in turn the types of waves induced in galaxy disks as a function of the spin-angular momentum coupling (whether +, - or 0). There is a great variety of disk waveforms, but the general structure of these waves depends more strongly on the angular momentum coupling than any other variable. We consider these waves "transients" because, generally, wave propagation times in disks are shorter than the companion return or merger timescales. Moreover, as we will discuss in later chapters, the processes of disk heating and phase mixing generally guarantee that waves will damp or disperse within a few propagation times.

There are several reasons for limiting our discussion to waves in disks. First of all, the constituents of bulge and halos have much larger random velocities than those found in disks, so unless the disturbance is large, waves rapidly diffuse. Here "large" disturbance means one that generates velocity perturbations comparable to the thermal velocities in these dynamically hot components. Such large disturbances are probably at least partially disruptive, and so, belong with the cases considered in the following sections. Secondly, halo oscillations have relatively long characteristic timescales.

The discussion in this chapter will also concentrate on two-dimensional waves in thin flat disks. Disk warping is undoubtedly an important effect of most collisions. We will also focus on stellar waves in this section, or on wave behavior that is common to both gas clouds and stars, but important differences will be noted in context.

3.1. Ring Galaxies (YDe0)

P. N. Appleton and the author have recently completed an extensive review of this subject (1996, henceforth AS96), so I will limit this to a brief summary. (References are minimal in this section, but the reader can find many sources of more detailed information in that review.) Collisional ring galaxies are rare. They are the product of a nearly head-on collision between a D-type primary and a substantial companion, i.e. one with mass in the range 10-100% of the primary. A companion with much less mass would not have much of an effect on the primary, while one more massive than the "primary" is possible, but evidently unusual. The basic theory was worked out by Lynds and Toomre (1976, also see Theys and Spiegel 1976, 1977, and Toomre 1978). As the companion approaches and passes through the primary disk, stars and gas clouds assumed to be in circular orbits before the collision, are drawn inward by the extra gravity. As the companion moves away, the unbalanced centripetal force drives an outward rebound. The response is faster in the inner disk and slower in the outer disk, so stars still moving inward meet rebounders moving out, producing a compression wave which propagates outward. If the impact parameter is small this wave is a circular ring.

The Cartwheel galaxy, mentioned above, was probably the first ring galaxy discovered, and still is regarded as a prototype. This despite the fact that its progenitor was an usually late-type galaxy. The outer disk shows no evidence of old stars, though there is plenty of gas. It is also unusual in having two prominent rings, and the so-called spokes - spiral segments between the two rings (see Figure 4). However, there are a couple dozen collisional ring galaxies that have been studied in some detail, with many more candidates awaiting further study. Their progenitors span the whole range of Hubble disk types, e.g., from the "Sacred Mushroom" system, AM 1724-622, studied by Wallin and Struck-Marcell (1994, see Fig. 1) with a very early-type progenitor (e.g. an S0 galaxy) to the Cartwheel.


Figure 4. Hubble Space Telescope image of the "Cartwheel", a prototypical collisional ring galaxy (courtesy P. N. Appleton and NASA).

Three fundamental facts make collisional ring galaxies a very important class, despite their rarity. The first is the symmetry of the collision that produces them. Because this symmetry is needed to produce a circular or nearly circular ring wave, once a collisional (e.g. expanding) ring is identified we immediately know a great deal about the collision. More precisely, there is a growing literature of comparisons between the collisional model and observation, and the general conclusion is that the collision theory is doing very well in accounting for observational features. (Though we note that there are other mechanisms for producing rings in galaxy disks, and we must have sufficient data to distinguish rings produced by these mechanisms before making detailed comparisons to collisional models.) On the other hand, nature has not missed a chance for an ironical twist. In a number of ring systems, the companion galaxy has not been identified. This is often because there are several possible suspects, which is not surprising since galaxies are commonly found in small groups.

The second fact is that the ring compression wave drives strongly enhanced star formation. Theys and Spiegel (1976) discovered that the rings in their modest sample frequently had blue colors indicative of massive young stars. Jeske (1986) and Appleton and Struck-Marcell (1987a) found that ring galaxy systems were relatively strong far-infrared emitters on the basis of IRAS observations. Now, observations of a number of systems in a variety of wavebands confirm the enhanced star formation in almost all cases, except those where the precursor was evidently an early-type, gas-poor disk (see Appleton and Marston 1997, and Appleton 1998 for an update of AS96). Insofar as density wave-driven star formation is understood (see e.g., the reviews of Elmegreen 1992, 1994b) this is not a surprise, strong compressions are supposed to trigger star formation. However, the details of this process are not well understood, and ring waves provide a relatively clean way to study them. In this case the Cartwheel is a prime example. The evidence suggests (e.g. Higdon 1993, 1995) that the ring wave is driving some of the first star formation to occur in the outer disk of the Cartwheel. Moreover, in the Cartwheel, Arp 10 and other systems, the intensity of the current star formation varies around the ring, which models suggest is the result of a variation in wave-strength following a slightly off-center collision. Thus, the ring waves provide a nearly direct confirmation that waves can induce vigorous star formation (i.e. a nonlinear response), and even within a single wave there are indications of a variation of response as a function of wave amplitude. Of course, this is also true in the much more common spiral density waves, but the generally more complicated spatial-temporal variations of those waves, and the effects of various resonances, make it useful to have a very different case like the ring waves to compare to. The range of companion masses implies a corresponding range in the strength of ring waves in different systems, potentially providing a great deal of information about wave-driven star formation from comparisons between systems.

The third fundamental fact is that if the collision is impulsive, and the companion relatively small, then the structure of the ring waves is primarily a function of the distribution of matter in precursor. Thus, ring seismology is possible. The amplitude, width, spacing between successive rings, and azimuthal variations in the case of off-center collisions, can be used to deduce the distribution of dark matter in the precursor. For example, widely spaced rings are a good indication of the presence of a massive halo (Struck-Marcell and Lotan 1990). In most ring galaxies we only see one ring, so the rings are definitely widely spaced, and most of the precursors probably had substantial halos. Models (Struck-Marcell and Higdon 1993) suggest that the Cartwheel is dominated by a large halo. If the collision is not impulsive (e.g., the relative velocity is low), ring seismology should still be possible, but the time-dependent perturbation will have to be modeled.

Let us return to the second point, star formation in rings, for a moment. At high resolution, such as that obtained by the Hubble Space Telescope observations shown in Figure 4, we have detailed information about where in the ring star clusters are formed (Appleton, priv. comm.). Even at lower resolutions, information can be obtained on the relative positions of young stars, old stars, and the gas clouds (Marston and Appleton 1995, Appleton and Marston 1997). For example, these authors find evidence that in large rings the ionized gas is concentrated on the outer edge of the old star wave. This data can provide powerful constraints on theories of the star formation process, and the gas/star wave dynamics.

One important complication, however, is that young star activity, i.e., winds, radiation and supernova explosions, may provide nonlinear feedbacks to the gas dynamics. For example, pushing some gas to the front of the wave, or out of the disk. Figure 4 provides some direct visual evidence for such effects, i.e. the interstellar gas represented by emission and reflection nebulae seems very frothy (to borrow the term of Hunter and Gallagher 1990). The filaments, arcs and shells are all likely consequences of the activity of the young star clusters (e.g., Heiles, Reach, and Koo 1996, and references therein), and in aggregate give a clear impression of turbulence. There are a myriad questions waiting to be addressed: How is this turbulence different from that in the interstellar gas of undisturbed galaxies (model examples of which are given in Passot, Vazquez-Semadeni, and Pouquet 1995)?, How well-developed is it, and over what range of scales does it extend before it is damped or frozen out in the rarefaction region behind the compression wave? How does the turbulence effect the star formation process which generated it? How does it effect the thermal phase balance in the gas? These are very difficult questions, relevant to many types of collisional galaxy, which have hardly begun to be explored. The relative simplicity of ring waves makes them an attractive locale for addressing them.

3.2 Symmetric Caustic Waves

At this point, we will retreat from the complexities of turbulent gas dynamics, and review the simple theory of symmetric stellar waves excited in a planar disk by a collision. Special attention was given to this topic in the review of AS96, so I will omit many details. However, since the key concepts can be generalized and carried over to many other cases, a self-contained overview is needed here. There are three key elements to this theory: 1) an impulsive disturbance (e.g. Alladin and Narasimhan, 1982), 2) followed by epicyclic kinematic motions (Lynds and Toomre 1976), and 3) the development of nonlinear, caustic waveforms (Struck-Marcell and Lotan 1990). The first item actually has two parts: that the disturbance occurs very rapidly compared to other relevant timescales, and that the disturbance can be decoupled from the subsequent evolution. Simple (e.g. analytic) models are based on the idealization of an instantaneous disturbance, but they remain interesting even if this is only approximately true. If the disturbance is persistent, it can't be treated as part of the initial conditions of the dynamical equations, and in general, no conceptually simple model can be constructed. (However, this case can be treated with the perturbation theory described in Chapter 5.)

The first condition above is necessary, but not sufficient, for the second condition. The validity of a kinematic approximation to the motions of stars, gas clouds, and dark matter particles, depends not only on the prompt disappearance of the disturber, but on the constancy of the gravitational potential they move in. In principle, this potential is also perturbed by the collision. However, if it is dominated by a dynamically hot component, like the dark matter halo, and the perturber is not too massive, the halo disturbance may be small compared to that experienced by dynamically cold disk particles. Henceforth, I will refer to assumptions 1) and 2) together as the KIA (kinematic impulse approximation), and 1) as the IA.

If these approximations are valid, then we only need a description of the (kinematic) particle orbits to complete the theory. Depending on the form of the potential, the orbit equations will generally involve elliptical integrals (e.g., Grossman 1996). However, the ancient greeks developed a planetary orbital model that provides a very convenient conceptual and analytic tool here too. This is the famous epicyclic model, in which, the particle is assumed to orbit on a (small) circle, whose center orbits the potential center on a larger orbit. The epicyclic model was first extensively applied to galactic dynamics by Lindblad (1959 and references therein). If we assume circular orbits in the target disk before the collision, and that the impulsive disturbance in the symmetric collision is small (perturbative limit), then the effect of the collision on the orbit will be a sinusoidal oscillation about the initial, "guiding center" radius. That is, an epicycle. This is only an approximation when the disturbance is of finite amplitude, but the comparison of analytic and numerical models suggests that it can be a good one for transient waves.

The epicyclic orbit equations for a star are,

Equation 2 (2)

Equation 3 (3)

where r(q,t) is the instantaneous particle radius, q is the precollision orbital radius, and A(q) is the amplitude of the epicyclic oscillations. In the IA it is assumed that the collision is so rapid that the particles do not move during it, but the force and acceleration induce a velocity change. Thus, the initial radial velocity amplitude is the velocity impulse, and the amplitude A is found by setting t = 0 in equation (3), A = -Delta vr / qkappa. The epicyclic frequency kappa depends on the structure of the gravitational potential (Binney and Tremaine 1987, section 3.2.3), and generally goes in the sense of longer periods at larger radii. This is the origin of the ring compression, which, as noted above, results when outwardly rebounding particles meet infalling particles from larger radii (as a result of the longer epicyclic periods of the latter).

The radial motions of a collection of such particles, in a representative gravitational potential, are shown in Figure 5. In the particular case shown in Figure 5 it is assumed that the companion to primary galaxy mass ratio is 0.25 and that both galaxies have massive dark halos. Specifically, the potential assumed for the primary galaxy gives a rotation curve of the form v = v(gammaa) (r/gammaa)1/n, where gammaa is a constant scale-length. A large value of n (n = 20) is used to make v nearly constant. It is further assumed that the amplitude of the collisional disturbance is constant with radius. Dimensionless units are used, where the scale-length of the graviational potential gammaa , and the product GM(g) have been set to unity.


Figure 5. Radius versus time for representative stars in a kinematic model for a collisional ring galaxy as described in the text.

As stellar orbits at different radii get more out of phase, the orbit crowding phenomenon becomes orbit crossing, and thus, the second and third rings are broader than the first. (Ultimately, the rings overlap and become effectively smoothed out by this "phase mixing".) This is shown in Figure 6, which is a r - vr phase diagram (after figures in Struck-Marcell 1990a, b) with a phase mixed center, an orbit crowding outer ring, and an isolated orbit-crossing ring between. The orbit crossing rings are bounded by sharp edges. They are in fact caustics, formal singularities in the stellar density.


Figure 6. Phase diagram of radial velocity versus radius (r - vr) from the kinematic calculation of Figure 5, at dimensionless time t = 20. The loops are the result of orbit crossing in the inner ring, while the positive velocity wave between radii of r = 3.0 - 4.0 shows the orbit-crowding outer ring.

The conservation of mass in a thin cylindrical annulus implies that the density is given by,

Equation 4 (4)

where rhoo(q) is the initial, unperturbed density profile. Equation (4) applies to regions with a single star stream. In orbit crossing zones (e.g., the inner rings in Figs. 5, 6), the right hand side must be replaced with a sum over terms for each star stream. Singularities occur wherever ðr / ðq = 0, i.e. where the particles some initial radial range Deltaq are squeezed into zero volume, Deltar = 0. Then, formally, rho -> infty, though infinite densities will not occur in real galaxies with finite numbers of stars. Substituting the orbit equation (2) into the caustic condition, allows one to derive an equation for the location of the caustic edges. This, together with equations (2)-(4), provides the basis for a complete analytic model of stellar waves in ring galaxies (see AS96 for details). We will see in subsequent sections how the three elements of this theory can be generalized to less symmetric cases.

Generally, we do not expect the same wave structure in the collisional (see section 1.2) gas component. The stellar orbit crossing zone and its caustic edges should be replaced by a dissipative shock wave. The gas will be heated in this shock, but cooling times are short, and so as a first approximation the shock can be assumed to be isothermal. In the first ring this shock may be relatively weak, because the epicyclic motions in adjacent radial zones are still nearly in phase. The shock in the second ring wave is likely to be much stronger (Appleton and Struck-Marcell 1987b, Struck-Marcell and Appleton 1987). Even so, the thermal physics and the observables are likely to be dominated by the compression induced star formation behind the shock.

Sometimes, however, the gas may behave more like the collisionless stars, as discovered by Gerber, Lamb, and Balsara (1992) in their simulational study of Arp 147. This is the result of a very interesting effect, when the disk is warped by the collision, so gas clouds at different radii are able to execute their radial epicyclic oscillations in different vertical planes, thus becoming collisionless. The gas behavior in any particular situation depends on whether the ratio of the radius of curvature of the warp to the local epicyclic excursion is greater or less than unity.

3.3 Ring Relatives: Bananas, Swallows and Others.

When collisions become less than perfectly cylindrically symmetric, that is, as the impact parameter increases, the diversity of waveforms increases rapidly. In this section I will illustrate this with a few examples, and note how the theory described in the previous section is generalized.

The consequences of a small increase in the impact parameter (relative to the scale length of the gravitational potential) are not terribly dramatic. The result is an asymmetric or partial ring, which looks like a crescent or banana (see Appleton and Struck-Marcell 1987b, Chatterjee 1986). Theoretically, these crescent waves are nearly as simple as the symmetric rings, at least for points at radii greater than the impact radius. There, the impulse is still primarily radial, but generally with an amplitude that depends on distance from the impact point. However, the radius of the compression wave depends primarily on the epicyclic frequencies (in the perturbation limit), so it is little different from the symmetric case. Thus, in this approximation, the wave is still nearly circular, but with an amplitude that varies with azimuth around the ring. (See AS96 for a more complete description.) For the stellar component this means that the caustic wave may not extend to all azimuths, i.e. there may be orbit crossing on the "strong" side, but only orbit crowding on the weak side. As a result the two circular caustic edges of the symmetric wave are replaced by the crescent. Similarly we expect variable compression and shock strengths around the ring.

As the ring propagates outward, the ratio of the impact radius to the ring radius decreases, so the perturbation is more symmetric, and the crescent ends join to form a (weaker) ring. This shown in Figure 7 (from Appleton and Struck-Marcell 1987b). Other numerical models of asymmetric rings can be found in Appleton and James (1990), Gerber (1993), and Struck-Marcell and Higdon (1993).


Figure 7. Contour maps of the gas density for a hydrodynamical simulation of an off-center galaxy collision (Appleton and Struck-Marcell 1987b). Solid contours indicate densities above the initial unperturbed value and dotted contours show lower densities.

The overwhelming majority of real ring galaxies are asymmetric in appearance. This includes both the Cartwheel and the very similar VII Zw 466 ring. Appearances can be a bit deceiving here, since optical/infrared observations usually reflect the number of massive young stars and clusters. The local star formation rate (SFR), and perhaps the stellar mass function, are most likely nonlinear amplifiers of the wave compression. Examples of galaxies with apparently strong crescents include the "Sacred Mushroom" AM1724-622 (see Fig. 1), and most of the objects on page 6.1 of the Arp-Madore photographic atlas.

The crescent is an evolving or "metamorphosing" caustic structure, and thus, we can learn more about it from singularity or catastrophe theory (see e.g. Poston and Stewart 1978, Arnold 1986). While singularity theory is not widely used in astronomy, it has found a couple of niches. One of these is the Zeldovich "pancake" approximation for galaxy and large-scale structure formation (Arnold, Shandarin and Zeldovich 1982), where as a result of gravitational collapse, collisionless dark matter particles form a full range of three-dimensional caustics or singularities. An early, less well known application, is found in the work of Hunter (1973) on spiral density waves. Hunter found that wave characteristics converged to singularities.

Caustics occur in (models of) a number of different types of collisional galaxies. AS96 reviewed the application of singularity theory to collisional galaxies, and suggested that the theory is more generally relevant because,

In two and three spatial dimensions it offers a complete classification of the generic, nonlinear waveforms and their possible evolutions... These include cusps, swallowtails, and pockets or purses (see Arnold 1986), and overlapping combinations.

... (Moreover,) it significantly extends our conceptual model. It takes us from models for individual stellar orbits to the structure of the nonlinear density waves,... It makes us aware of "elementary" waveforms that are more complicated than rings or spirals, and yet not intractably complex... For example, beginning with a model for the orbits, like the kinematic impulse approximation, we can derive analytic expressions for the location of the caustics, which provide a skeletal outline of the waves for any particular set of structural and collision parameters. (AS96)

The procedure for finding ring edge caustics described in the previous section can be generalized with the goal of mapping the edges or "skeletal outline" of the more general waveform. However, even limiting consideration to waves in a thin, unwarped disk, three effects complicate this formalism: 1) the azimuthal dependence of the perturbation amplitude A, 2) the fact there is now an azimuthal component to the velocity impulse (or a torque), and 3) even in the limit that the gravitational potential is dominated by a rigid halo or bulge, there is an impulse on the potential center. Items 2) and 3) may have a small magnitude, but still have important effects. Because of effect 3) halo particles at radii smaller than the impact radius will receive a net impulse toward the intruder, while particles in a spherical shell at large radii will not. Thus, the halo is apparently broken into two kinematic components, though even this dichotomy is too simplified to realistically represent the time-dependent potential. It seems that perturbation theory is rather shakey unless the typical velocity impulse is smaller than the mean halo particle velocities, i.e. the halo is too hot to be significantly perturbed.

We will adopt this hot halo assumption for the moment (but see the analysis of Gerber and Lamb 1994 which does not). Then following AS96 we can write the approximate kinematic equations as,

Equation 5 (5)

These epicyclic orbit equations are very similar to equations (2) and (3). The chief difference is that, as a result of the azimuthal velocity impulse, the epicycle is centered on a new guiding center of radius q'. The initial particle radius q is generally different from q', so this introduces an initial phase psi. The epicyclic frequency is kappa' = kappa(q'), and the mean angular velocity is omegacir(q'). The amplitude A is a function of radius and azimuth. Using the initial conditions (in the IA), and the force balance equation for the guiding center, the perturbed (primed) quantities can be eliminated in favor of pre-collision values and the velocity impulses (see AS96). Then, in this two-dimensional case, the infinite density caustic condition is given by setting the Jacobean determinant equal to zero,

Equation 6 (6)

AS96 describe a limiting case of this equation that has "crescent" solutions, and probably contains other waveforms as well.

Struck-Marcell (1990a), Donner, Engstrom, and Sundelius (1991), and Gerber and Lamb (1994) all used numerical models to study the creation of waves in disks of particles following kinematic orbits like those of equations (5). Donner et al. also numerically solved the caustic determinant equation (6) in specific cases to compare to numerical models. Donner et al. (in planar collisions), and Gerber and Lamb (in collisions nearly perpendicular to the target disk), both found good agreement between the kinematic models and self-consistent N-body simulations at early times. Thus, providing evidence that the effects of the time varying gravitational potential take some time to accumulate.

Struck-Marcell (1990a) and Donner et al. both found evidence for the development of higher order caustics like the cusp, swallowtail and pocket/purse (see Arnold 1986) in their calculations. Perusal of the photographic atlases shows many sytems that appear to match the nonlinear waveforms, but this is very circumstantial evidence. M. Kaufman and collaborators (including the author) are investigating two galaxies that have the appearance of swallowtails (NGC 3145 and NGC 5676) with radio (21 cm) and optical observations, but the results are not yet complete. We are specifically searching for the high velocity dispersions or gaseous shock waves that would characterize the stellar orbit crossing region.

Wallin and Struck-Marcell (1994) compared caustic models to the broad crescent wave in the early-type ring galaxy AM1724-622 (the "Sacred Mushroom"). Interestingly, the crescent was best matched by models with a declining rotation curve in the precursor. Though somewhat unusual, this might be the result of the gravitational potential in the disk being dominated by the strong bulge component. It was hoped that it would be possible to determine if the broad ring edges were in fact sharp caustics, or if not, obtain some measure of the effect of diffusive smoothing processes. Ultimately, such processes, together with phase mixing (see previous section) and the overlap of multiple caustics, will erase all traces of caustic waves in the disk. However, the published simulations indicate that these processes do not strongly effect the first couple of waves. Because of the limited resolution of the data, and the fact that dozens of foreground stars cover this galaxy, which lies near the galactic plane, this study was not definitive. Hubble Space Telescope observations might be able to surmount the difficulties.

The caustics theory for nonlinear waves in collisional disks is still relatively young and untested. Many questions have not yet been fully addressed, including how far can the analytic models be pushed before they become too inaccurate or too algebraically complex to be useful?

3.4. From Rings to Spirals

Toomre (1978) discovered that, as the impact point in a vertical collision is moved out from the disk center to its edge, the ring wave metamorphoses into a spiral. Toomre used a sequence of restricted three-body models (reproduced in Fig. 8) to demonstrate this. Several remarkable and instructive points can be taken from this figure. The first, noted by Toomre, is the disks are not destroyed in any of these cases, though they do suffer significant time-dependent warping or flapping. This result, along with most of the other qualitative features of Figure 8 are confirmed in an analogous series of fully self-consistent star-plus-gas simulations run by Gerber (1993). There are differences, but they are modest given the different simulation techniques and the different structural characteristics in the initial galaxies. (I.e., a rigid point-mass potential in Toomre's versus an initial flat rotation curve structure in Gerber's.)


Figure 8. Toomre's ring-to-spiral transition is illustrated by a sequence numerical model evolutions with progressively decreasing companion impact radii. Each row shows a different model. See text and Toomre 1978 for details.

Secondly, these simulations clearly show caustic edges, and in some cases they appear quite complex. We would hope that these waveforms could be explained by the KIA-caustics theory, but to date no detailed analysis has been done. However, some qualitative aspects can be explained by this theory with little effort. To begin with the line connecting the target disk center and the impact point is a key division. Ahead of this line one component of the velocity impulse will be directed against the particles' rotation velocity, so these particles will be slowed and fall inward. Behind this line particles will have their tangential velocity increased, so they will fly outwards. The infalling particles will compress, forming a region of enhanced density. In fact, since the backward impulse increases to some maximum as azimuth increases from the division line, we can expect an orbit-crossing zone in this compressed region if the impulse is great enough. Thus, in general, a "lips" (crescent) caustic will form and shear into the leading edge of a spiral wave. This feature can be seen in the more off-center of Gerber's model collisions, and in his model Ring4, the wave is strong and caustic-like edges are especially apparent in the dynamically cooler gas particles (see Fig. 9).


Figure 9. A self-consistent, N-body plus Smoothed Particle Hydrodynamics simulation from R. A. Gerber's thesis, showing an incipient spiral in a collision like those that produce ring galaxies. In this case the trajectory of the companion galaxy was nearly perpendicular to the primary disk, and the point of closest approach was at the edge of that disk (see Gerber 1993 for details).

We can sketch a quick derivation of this wave from the KIA caustics theory (neglecting the perturbation of the potential center of the disturbed galaxy). The azimuthal impulse can be written,

Equation 7 (7)

where theta is the azimuth of a disk particle relative to the center-to-impact point line, and r* is the impact radius, and R is the distance between disk particle and the impact point (see SM90, AS96). R is related to r*, and the particle's unperturbed radius q by the law of cosines,

Equation 8 (8)

At each radius, and on each side of the center-to-impact line, there is some azimuth where |Deltavtheta| is a maximum. Physically, particles located at these extremal azimuths feel the strongest pull backward (or forward) in their orbits, and so, would seem to be likely participants in the formation of an orbit crossing zone. More formally, the term ðtheta / ðtheta0 in the caustic condition (6) is proportional to ðDeltavtheta / ðtheta0. Therefore, if the cross terms in equation (6) are small, the zeros of these derivatives yield caustics.

For simplicity, assume that the potential of the perturber is that of a simple point mass. Then the equation ðtheta / ðtheta0 = 0 reduces to a simple quadratic for cos(theta),

Equation 9 (9)

which depends only on the parameter r*/q. Deltavtheta itself depends on the mass of the perturber and other variables. In the small radius limit, q -> 0, the solution is theta = ±90°. When r*/q is about unity, cos(theta) approx sqrt3 - 1, or theta 45°. The full solution curve to equation (9) spirals out from small radii to a nearly linear form at larger radii. Rotational shear will turn this linear feature into a spiral. Qualitatively, this result seems much like the behavior of the numerical models.

For impacts within the disk, this spiral kinematics is superimposed on the radial kinematics discussed above. Figure 8 shows that as the impact point moves out of the disk, the radial ringing diminishes in importance, and tidal stretching becomes more important.

Note that this is a very nonstandard description of a spiral wave in a galactic disk. The caustic spiral is a nonlinear transient, so there is no obvious connection to the classical quasi-stationary (linear) density wave theory (see e.g. Binney and Tremaine 1987; ch. 6, Palmer, 1994, ch. 12). Swing amplification is another process that plays a very important role in interaction driven waves (see Toomre 1981 and below), but in this case there has not been time for significant amplification. This wave is simply the result of correlated initial conditions, and a special aspect of the direct collision is that it is truly impulsive. This example also provides an interesting illustration of the transition from the study of ring waves to more standard tidally induced waves.

3.5. Tidal Spirals and Oculars (YDx+)

The class YDx+ includes encounters in which the companion galaxy flies by in the plane of the primary disk, and in the same orbital sense as the disk rotation, and with a point of closest approach generally located outside the disk (i.e., prograde collisions). It seems odd at first that these collisions can result in more damage than nearly head-on vertical collisions, but the tidal perturbation is sustained for a longer time in this case. The result is the formation of the great bridges, tails and strong spirals hinted at in Holmberg's work, emphasized by Zwicky, and shown convincingly to be tidal remnants in the work of the Toomres and others in the early 70s (see Chapter 1). In this section we will consider the strong spirals, and leave bridges and tails to later sections.

The great "Whirlpool" galaxy M51 is the prototype of these collisional spirals (see Fig. 10 and section 9 of the Arp-Madore atlas). It is also a bridge/tail galaxy, with a connected companion that has attracted much attention in the interacting galaxy literature. At the same time, it has also been a prototype in the spiral density wave literature. The possible connection between the inner and outer phenomena was discussed by the Toomres (1972, also Toomre 1974), but the idea encountered difficulties (Toomre 1978). While there was some evidence for tidal influence on the inner spirals in the 21 cm observations, the restricted three-body simulations didn't produce any such waves. However, by 1980 Toomre had discovered the missing piece of the puzzle in "swing amplification".


Figure 10. Optical image of the "Whirlpool galaxy" M51, whose beautiful spiral arms are likely a result of the ongoing collision, see text. (Digital Sky Survey image courtesy of AURA/STScI.)

Swing amplification, as described by Toomre (1981), depends on the near commensurability of the shearing timescale and the epicyclic (compression) timescale, which is common in galaxy disks. This commensurability works to keep stars in the overdense region for relatively long times, which gives self-gravity the time needed to greatly amplify the density contrast of a spiral wave. Toomre used self-consistent N-body simulations to demonstrate the operation of the process, and simple, "shearing sheet", kinematic models to illustrate the role of the three main component processes: epicyclic "shaking", wave shear and self-gravity. The original paper gives a very clear presentation and other pedagogical summaries can be found in Athanassoula (1984) and Binney and Tremaine (1987, ch. 6.3). The discussion here will be very brief, and more details on all the topics covered can be found in these sources. For a recent technical review see Tagger, Sygnet, and Pellat (1993).

The amount of amplification is surprising - factors of between a few and a hundred resulting from external perturbations of a few percent or less. Toomre points out that the basic process was understood in work on spiral waves carried out more than a decade earlier, but its amplification ability was not. However, the nonlinear wave is generally short-lived, building up to maximum amplitude within about one disk rotation time, and then winding up and decaying on a comparable timescale. These points are illustrated in Figure 11 from Toomre's paper. If the structure of the galaxy disk is such that it possesses an inner Lindblad resonance, then the wave is "absorbed" at the radius of the resonance. Lindblad resonances occur where the epicyclic frequency is commensurate with the wave pattern frequency in the local rotating frame.


Figure 11. The development of a "swing-amplified" trailing spiral wave from an initially leading wave from Toomre (1981). Contours represent fixed fractional excess surface density, and the time between snapshots is one half of the rotation period at the corotation point.

If there is no inner Lindblad resonance, the trailing wave can propagate to the disk center, and reemerge as a leading wave. This leading wave is sheared around into a trailing wave, and in the process, another round of swing amplification occurs. The amount of amplification depends on two parameters. The first is the Toomre Q parameter, which is a measure of how close to gravitational instability the unperturbed disk is. The second parameter Toomre called X = lambda / lambdacrit, where lambda is the (unwrapped) spiral pattern wavelength, and lambdacrit is critical wavelength for gravitational instability. Note that Q, X and lambdacrit are all defined as local quantities, though they typically do not vary drastically across a galaxy disk. X and lambda depend on the number of spiral arms in the global pattern, and since the amplification is very sensitive to the value of X, significant amplification can only occur when there are no more than a few arms.

If the wave can traverse the amplification feedback loop, numerical models show that the end result is a global bar instability. This is an extremely important consequence of a collisional disturbance. However, it is not a transient event, so we defer further discussion to later chapters.

Another feature of the models in Toomre's paper was that the spiral wave started from the center and moved out to include a large part of the disk. This phenomenon offers hope that there would be time for large-scale tidal features to develop, as in M51, before the prominent spirals disappear. In sum, these discoveries bode well for the idea that strong inner spiral waves can be driven by tidal encounters. Unfortunately, the M51 story has not yet ended happily ever after.

Extensive 21 cm. observations (Appleton, Foster and Davies 1986, Rots et al. 1990) revealed unexpected HI features, including a long southern tail coming off the (outer) western arm, and gas clumps north of the companion. These discoveries coincided with the implementation of a new generation of N-body and gas simulation codes (see e.g. Sellwood 1987, and Barnes and Hernquist 1992a), so it is not surprising that a number of new modeling efforts were initiated. These include Hernquist (1990), Howard and Byrd (1990) and Sundelius (1990), and more recently, Toomre (1994) and Byrd and Salo (1995). The models of Barnes, Hernquist and Toomre are based on a passage of a companion on a high eccentricity orbit, while Byrd and collaborators favor two disk passages to account some of the morphological details of the system. Barnes and Hernquist (1992) conclude - "At present, however, none of the calculations offer a really convincing reconstruction of M51's spiral structure ... " (see also the discussion of Barnes 1998).

To compound the modeler's difficulties, new high resolution, multiwaveband observations are being acquired at a steady rate, and they reveal not only more detail, but new phenomena. These include the distribution of ionized, atomic and molecular gas, and star formation across the disk, and especially in the spiral waves (see reviews of Rand and Tilanus 1990, and Casoli 1991, as well as the other observational reports in Combes and Casoli 1991). Casoli (1991) discusses how the sequence of dense cloud buildup, star formation, and subsequent cloud disruption is displayed as expected across one spiral arm, but does not follow this sequence elsewhere. Another example is the recent discovery from infrared imaging that the spirals go deep into the central regions, and wind through three full revolutions (Zaritsky, Rix, and Rieke 1993). The infrared observations also revealed a small bar in the inner regions. These phenomena, together with the large-scale structures, will undoubtedly continue to challenge modelers for a long time yet.

We should not, however, let the details of the M51 system distract us from the general result - that even moderate collisional disturbances can stimulate the formation of strong spiral waves via nonlinear amplification processes. With perfect hindsight, we can see that this is just what was required for M51 types, not only to explain the waves, but also their presence in a disk that does not appear highly disturbed (except in the outer regions). Stronger disturbances bring more wholesale distortions. This point is well illustrated by the ocular galaxies, first studied by Elmegreen et al. (1991).

Elmegreen et al. define the ocular as "a bright oval approximately one-half the size of the galaxy centered on the nucleus with a right angled vertex at each end of the major axis, and spiral arms extend smoothly from each of the flatter sides of the oval..." At first glance the ocular seems to be a relatively pure result of tidal forces, with the oval resulting from tidal compression, while the spiral arms result from tidal stretching plus shear (see Fig. 12 and the images in Elmegreen et al. 1995). The fact that the ocular persists for only about one rotation reinforces this impression. However, the ocular is not simply the result of a homologous compression. Unlike the initial unperturbed disk the ocular has sharp, caustic edges ( in the models).


Figure 12. The collisional system NGC 2207 / IC 2163 illustrates the ocular waveform. Specifically, the disk of the smaller galaxy (IC 2163) has the characteristic eyelid shape and the double-branched spiral arm. The other spiral arm has been hidden or disrupted by the larger galaxy. (Digital Sky Survey image courtesy of AURA/STScI.)

The ocular is formed in prograde collisions where the perturbation is relatively strong, so the azimuthal impulse is substantial. Moreover, it is clear that the sharp edge on the companion side forms first, and it appears in the quadrant ahead of the line connecting the companion to the primary center. Thus, it appears that stars are pulled ahead in the near side quadrant behind the line of centers, and they swing out to apoapse in the leading quadrant, where they are also pulled backwards by the companion. A mirror image process occurs in the other half of the disk, but as a result of the interaction between the stars and the swinging potential center of the primary. It would be interesting to see the trajectories of stars that make up the ocular, but these have not yet been presented in the literature. It would also be interesting to see if the ocular form could be captured in a KIA model. This would probably require an instantaneous approximation to the swing imparted by the primary center as well as the companion.

Elmegreen et al. also found that the formation of an ocular depends on a tidal strength parameter,

Equation 10 (10)

where DeltaT is the time it takes the companion to travel through one radian relative to the primary, and T = (Rgal3 / GMgal) 1/2. (Compare to equation (1) derived from the IA.) Their two-dimensional simulations showed that oculars only formed when the value of this parameter is greater than 0.019. For lower values, substantial spiral waves are still produced, but there is not enough transverse compression to produce the eye shape. They also found that when S > 0.038, the ocular galaxy evolves into a barred galaxy, and beyond the YD+ stage considered here.

3.6. Fan Galaxies and One Arms (YDx-)

Retrograde collisions have never inspired the same interest as prograde collisions. They do not form beautiful two-armed spirals. Toomre and Toomre (1972) included a retrograde encounter among their four numerical examples, but concluded that the effects were "remarkably mild". Eneev et al (1973) presented a simulation that gives a very different impression, probably because the perturbation was stronger. However, they did not discuss the morphology of this model. In summarizing the retrograde encounters in their atlas of N-body simulations of galaxy interactions, Howard et al. (1993) state that they "produce only broad fanlike global patterns, but rich small-scale internal structure."

This is not to say that retrogrades were entirely overlooked in the colliding galaxy renaissance of the 70s. Kalnajs (1975) and Athanassoula (1978) studied the idea that a companion in a retrograde orbit could stimulate the formation of a leading spiral wave (i.e. one whose outer end points in the direction of disk rotation). Kalnajs (1975) presented evidence that M31, the Andromeda galaxy, possessed such a one-arm. This work was followed up by the study of Thomasson et al. (1989), which included analytic work, numerical studies, and a comparison to observation. These works suggested that the one-armed spiral wave is the result of the 1:1 orbital resonance, at which orbits close after one radial oscillation. This resonance plays a role similar to that of the 2:1 Lindblad resonance (two radial oscillations to closure) in the case of the two-armed trailing pattern generated in prograde collisions.

On the face of it the preceeding sentences seem strange. If the radial epicyclic oscillation period depends primarily on the intrinsic mass distribution in the disk galaxy, then how can the number of radial bounces depend on the orientation of the perturber's orbit. This would be no problem if the two types of wave appeared in different parts of the disk, but the simulations show that they both can involve a considerable fraction of the disk. The answer to the paradox is that the orbit closure statements refer to a reference frame rotating with the wave pattern. In general, the stellar orbits are precessing ellipses that don't close in an inertial frame, but as Figure 3 of Thomasson et al. shows closure is nearly achieved in the wave frame (also Figure 16 of Athanassoula 1984).

It is worth a little further digression on this point, which is a matter of fundamental kinematics akin to others considered in this chapter. Consider, for example, a flat rotation curve galaxy, in which the rotation velocity v = constant, the rotation frequency Omega = v/r, and the epicyclic frequency kappa = sqrt2 Omega. This means that in an inertial frame a star goes around a bit less than 3/4 of a circle (255°) in one radial period. Let us suppose, that the wave is defined by stars at a given epicyclic phase, e.g., at minimum radius (point of greatest radial compression). In the case of the leading one-arm this means that the wave pattern merely has to travel (counter to the rotation) through an angle of about 105° to meet the star again at minimum radius, and thus, close the orbit in the wave frame. The wave must be leading because stars at larger radii take longer to traverse their 255° of azimuth, by which time the inner wave has traveled more than 105° (i.e., farther "backwards").

In the two-armed case, the star begins at minimum radius in one arm, and meets the second arm at the next minimum if that arm has also advanced (in the same direction) by about 75°. Because this wave is moving in the rotation direction, it must be trailing to maintain coherence at all radii. We have discussed above how the tidal perturbation in a prograde collision induces the two-armed wave. The one-arm mode is probably excited in retrograde collisions simply because it is the lowest order leading mode.

Thomasson et al. find that the one-arm persists for several disk rotation times, i.e. about as long as the typical collision-induced two-arm pattern. The question then arises - why are there so few examples to be found in the observations. The authors consider a variety of possible answers. They favor a somewhat indirect explanation. A large halo-to-disk mass ratio makes a galaxy stable against Swing Amplification of the m=2 mode. On the contrary, they suggest that this mass ratio may commonly be low, enough to give the m=2 mode a competitive advantage. It appears that the halo they are referring to is that contained within the radius of the stellar disk. This "halo" should also include a bulge if present.

Thomasson et al. also find that the retrograde disturbance has a steeper dependence on separation (1/r4) than the usual tidal force, and that it takes a substantial disturbance to produce the leading arm. Retrograde collisions with small disturbances produce m=2 trailing arms or combined m=1 leading and m=2 trailing patterns. "Rich internal structure" indeed.

Another possible example of this richness is the galaxy NGC 4622, modeled by Byrd, Freeman and Howard (1993). This galaxy possesses an inner leading arm, a ring, and the two outer trailing arms. Byrd et al. found that such features could be produced following a small impact parameter collision with a low mass companion, orbiting in either direct or retrograde senses. However, the retrograde collisions produced the better match to the observed morphology.

A final note on leading one-armed waves - Lotan-Luban (1990) carried out a series of restricted three-body simulations which showed that head-on, low impact parameter collisions (like those that produce ring galaxies) can produce a long-lived one-arm spiral. Generally, this spiral becomes prominant after several ring waves have propagated through the disk, and the ringing has pretty well phase-mixed away. By varying the potential structure she confirmed Thomasson et al.'s result that a substantial halo component is needed to produce the one-arm. She also varied the companion to target mass ratio and found that intermediate mass companions produced the strongest wave. This is not too surprising - high-mass companions caused much disruption in the test particle disk, and low-mass companions didn't produce a sufficiently strong perturbation. Thirdly, she carried out a series of simulations with varying impact angle from head-on (small impact parameter) to in-plane retrograde (with impact parameters greater than the disk radius). Of these, the head-on small impact parameter collisions produce the strongest one-arm waves. This helps account for the fact that some of her simulations seem to make stronger and longer lived waves than those of Thomasson et al. The retrograde planar waves are similar in both works.

Unfortunately, Lotan-Luban's work has not been redone with self-consistent N-body simulations. This is especially important for testing the longevity of these waves.

3.7. Gas vs. Stars in Waves.

Most of the discussion above, and in the literature, on collisional wave morphologies concerns stellar waves. Low amplitude, transient waves mainly depend on kinematics, so to first order there is no significant difference in the behavior of the interstellar gas and stars. However, in nonlinear waves we expect caustic waveforms to develop in the stars and dissipative shocks to develop in the gas. Dissipation could lead to some separation of the two components, unless the gas disk is highly warped or distorted. However, observations have not yet provided any compelling examples of separation in waves, perhaps because strong perturbations lead to highly distorted disks. Extreme separations may occur on large scales in mergers, and large quantities of gas can be funneled to the center (e.g., Negroponte and White 1983, Noguchi 1987, 1990, Barnes and Hernquist 1991, 1992a), at the same time that gas and stars can be thrown out to great distances to form separate shells and ripples (e.g. Hibbard 1995, Hibbard & van Gorkom 1996 and references therein). These topics are discussed below.

A large-scale, but less violent, example is provided by the galaxy NGC4747 (Arp 159) studied by Wevers et al. (1984), whose HI disk seems to have been twisted relative to its stellar disk by 11° in projection! Exactly how this occured remains a mystery, but given the distortions of the outer HI disk of its companion NGC4725 it may well be that a direct collision with a modest impact parameter was involved. In that case, direct cloud collisions might have contributed to the separation, as well as dissipative accretion of gas from the partner, described in the next section. There we will also meet some milder examples of gas-star separation in waves and tidal structures.

In nonmerging collisional galaxies important differences between stellar and gas dynamics result from heating and cooling effects. Young stars winds, UV photoheating, and supernova blasts can heat and push the gas, and disrupt cold clouds. At the least this can boost gas to greater heights above the disk, i.e. making a thick gas disk. This is evident in recent simulations of ring galaxies that include heating and cooling (Struck 1997), though it is a transient effect in that application. The disk gas cools and settles on a timescale comparable to the wave passage time. A more spectacular heating phenomenon is that of superwinds generated by nuclear starbursts (Heckman et al. 1993, Lehnert & Heckman 1996). Collisional galaxies may frequently experience a nuclear starburst phase, driven by gas inflow resulting from dissipation in waves and induced bars (see below).

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