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3. MODELLING AXISYMMETRIC GALAXIES

Motion in an axisymmetric potential is qualitatively simpler than in a fully triaxial one due to conservation of angular momentum about the symmetry axis. Defining the effective potential

Equation 21   (21)

where (R, z, phi) are cylindrical coordinates and Lz = R2 phidot = constant, the equations of motion are

Equation 22   (22)

and phidot = Lz / R2. These equations describe the two-dimensional motion of a star in the (R, z), or meridional, plane which rotates non-uniformly about the symmetry axis. Motion in axisymmetric potentials is therefore a 2 DOF problem.

Every trajectory in the meridional plane is constrained by energy conservation to lie within the zero-velocity curve, the set of points satisfying E = Phieff(R, z). While the equations of motion (22) can not be solved in closed form for arbitrary Phi(R, z), numerical integrations demonstrate that most orbits do not densely fill the zero-velocity curve but instead remain confined to narrower, typically wedge-shaped regions ([Ollongren 1962]); in three dimensions, the orbits are tubes around the short axis. (1) The restriction of the motion to a subset of the region defined by conservation of E and Lz is indicative of the existence of an additional conserved quantity, or third integral I3, for the majority of orbits. Varying I3 at fixed E and Lz is roughly equivalent to varying the height above and below the equatorial plane of the orbit's intersection with the zero velocity curve. In an oblate potential, extreme values of I3 correspond either to orbits in the equatorial plane, or to "thin tubes," orbits which have zero radial action and which reduce to precessing circles in the limit of a nearly spherical potential. In prolate potentials, two families of thin tube orbits may exist: "outer" thin tubes, similar to the thin tubes in oblate potentials, and "inner" thin tubes, orbits similar to helices that wind around the long axis ([Kuzmin 1973]).

The area enclosed by the zero velocity curve tends to zero as Lz approaches Lc(E), the angular momentum of a circular orbit in the equatorial plane. In this limit, the orbits may be viewed as perturbations of the planar circular orbit, and an additional isolating integral can generally be found ([Verhulst 1979]). As Lz is reduced at fixed E, the amplitudes of allowed motions in R and z increases and resonances between the two degrees of freedom begin to appear. Complete integrability is unlikely in the presence of resonances, and in fact one can find often small regions of stochasticity at sufficiently low Lz in axisymmetric potentials. However the fraction of phase space associated with chaotic motion typically remains small unless Lz is close to zero ([Richstone 1982]; [Lees & Schwarzschild 1992]; [Evans 1994]). The most important resonances at low Lz in oblate potentials are omegaz / omegaR = 1 : 1, which produces the banana orbit in the meridional plane, and omegaz / omegaR = 3 : 4, the fish orbit. The banana orbit bifurcates from the R-axial (i.e. planar) orbit at high E and low Lz, causing the latter to lose its stability; the corresponding three-dimensional orbits are shaped like saucers with central holes. The fish orbit bifurcates from the thin tube orbit typically without affecting its stability. In prolate potentials, the banana orbit does not exist and higher-order bifurcations first occur from the thin, inner tube orbit ([Evans 1994]).

Once the orbital families in an axisymmetric potential have been identified, one can search for a population of orbits that reproduces the kinematical data from some observed galaxy. In practice, this procedure is made difficult by lack of information about the distribution of mass that determines the gravitational potential and about the intrinsic elongation or orientation of the galaxy's figure. Faced with these uncertainties, galaxy modellers have often chosen to tackle simpler problems with well-defined solutions. One such problem is the derivation of the two-integral distribution function f (E, Lz) that self-consistently reproduces a given mass distribution rho(R, z). Closely related is the problem of finding three-integral f's for models based on integrable, or Stäckel, potentials. These approaches make little or no use of kinematical data and hence are of limited applicability to real galaxies. More sophisticated algorithms can construct the family of three-integral f's that reproduce an observed luminosity distribution in any assumed potential Phi(R, z), in addition to satisfying an additional set of constraints imposed by the observed velocities. Most difficult, but potentially most rewarding, are approaches that attempt to simultaneously infer f and Phi in a model-independent way from the data. These different approaches are discussed in turn below.

3.1. Two-Integral Models

One can avoid the complications associated with resonances and stochasticity in axisymmetric potentials by simply postulating that the phase space density is constant on hypersurfaces of constant E and Lz, the two classical integrals of motion. Each such piece of phase space generates a configuration-space density

Equation 23   (23)

where vm = sqrt[vR2 + vz2], the velocity in the meridional plane; deltarho is defined to be nonzero only at points (R, z) reached by an orbit with the specified E and Lz. The total density contributed by all such phase-space pieces is

Equation 24   (24)

where f+ is the part of f even in Lz, f+(E, Lz) = 1/2 [f (E, Lz) + f (E, - Lz)]; the odd part of f affects only the degree of streaming around the symmetry axis. Equation (24) is a linear relation between known functions of two variables, rho(R, z) and Phi(R, z), and an unknown function of two variables, f+(E, Lz); hence one might expect the solution for f+ to be unique. Formal inversions were presented by Lynden-Bell (1962a), Hunter (1975) and Dejonghe (1986) using integral transforms; however these proofs impose fairly stringent conditions on rho. Hunter & Qian (1993) showed that the solution can be formally expressed as a path integral in the complex Phi-plane and calculated a number of explicit solutions. Even if a solution may be shown to exist, finding it is rarely straightforward since one must invert a double integral equation. Analytic solutions can generally only be found for potential-density pairs such that the "augmented density," rho(R, Phi), is expressible in simple form ([Dejonghe 1986]).

An alternative approach is to represent f and rho discretely on two-dimensional grids; the double integration then becomes a matrix operation which can be inverted to give f. Results obtained in this straightforward way tend to be extremely noisy because of the strong ill-conditioning of the inverse operation, however (e.g. [Kuijken 1995], Figure 2). Regularization of the inversion can be achieved via a number of schemes. The functional form of f can be restricted by representation in a basis set that includes only low order, i.e. slowly varying, terms ([Dehnen & Gerhard 1994]; [Magorrian 1995]), or by truncated iteration from some smooth initial guess ([Dehnen 1995]). Neither of these techniques deals in a very flexible way with the ill-conditioning. An alternative approach is suggested by modern techniques for function estimation: one recasts equation (24) as a penalized-likelihood problem, the solution to which is smooth without being otherwise restricted in functional form (Merritt 1996).

Figure 2

Figure 2. Contours of constant chi2 that measure the goodness-of-fit of axisymmetric models to kinematical data for M32 (van der Marel et al. 1998). The abscissa is the mass of a central point representing a nuclear black hole; the ordinate is the mass-to-light ratio of the stars. These two parameters together specify the potential Phi(R, z). Each dot represents a single model constructed by varying the orbital population so as to best reproduce the observations in the specified potential. The plateau of nearly-constant chi2 corresponds to models in which changes in Phi can be compensated for by changes in the orbital population, leaving the goodness-of-fit essentially unchanged.

The two-integral f's corresponding to a large number of axisymmetric potential-density pairs have been found using these techniques; compilations are given by Dejonghe (1986) and by Hunter & Qian (1993). A few of these solutions may be written in closed form ([Lynden-Bell 1962a]; [Lake 1981]; [Batsleer & Dejonghe 1993]; [Evans 1993], 1994) but most can be expressed only as infinite series or as numerical representations on a grid. Since the existence of such solutions is not in question, the most important issue addressed by these studies is the positivity of the derived f's. If f falls below zero for some E and Lz, one may conclude either that no self-consistent distribution function exists for the assumed mass model or (more securely) that any such function must depend on a third integral. For example, Batsleer & Dejonghe (1993) derived analytic expressions for f (E, Lz) corresponding to the Kuzmin-Kutuzov (1962) family of mass models, whose density profile matches that of the isochrone in the spherical limit. They found that f becomes negative when the (central) axis ratio of a prolate model exceeds the modest value of ~ 1.3. A similar result was obtained by Dejonghe (1986) for the prolate branch of Lynden-Bell's (1962a) family of axisymmetric models. By contrast, the two-integral fs corresponding to oblate mass models typically remain non-negative for all values of the flattening.

The failure of two-integral f's to describe prolate models can be understood most simply in terms of the tensor virial theorem. Any f (E, Lz) implies isotropy of motion in the meridional plane, since E is symmetric in vR and vz and Lz depends only on vphi. Now the tensor virial theorem states that the mean square velocity of stars in a steady-state galaxy must be highest in the direction of greatest elongation. In an oblate galaxy, this can be accomplished by making either sigmaR2 or < v2phi > large compared to sigmaz2. But sigmaR = sigmaz in a two-integral model, hence the flattening must come from large phi- velocities, i.e. f must be biased toward orbits with large Lz. Such models may be physically unlikely but will never require negative f's. In a prolate galaxy, however, the same argument implies that the number of stars on nearly-circular orbits must be reduced as the elongation of the model increases. This strategy eventually fails when the population of certain high-Lz orbits falls below zero. The inability of two-integral f's to reproduce the density of even moderately elongated prolate spheroids suggests that barlike or triaxial galaxies are generically dependent on a third integral.

The "isotropy" of two-integral models allows one to infer a great deal about their internal kinematics without even deriving f (E, Lz). The Jeans equations that relate the potential of an axisymmetric galaxy to gradients in the velocity dispersions are

Equation 25   (25)
Equation 26   (26)

with nu the number density of stars and sigma = sigmaR = sigmaz the velocity dispersion in the meridional plane. If nu and Phi are specified, these equations have solutions

Equation 27   (27)
Equation 28   (28)

The uniqueness of the solutions is a consequence of the uniqueness of the even part of f; the only remaining freedom relates to the odd part of f, i.e. the division of $ \overline{v_{\phi}^2}$ into mean motions and dispersions about the mean, $ \overline{v_{\phi}^2}$ = $ \overline{v_{\phi}}^{2}_{}$ + $ \sigma^{2}_{\phi}$. A model with streaming motions adjusted such that sigmaphi = sigmaR = sigmaz everywhere is called an "isotropic oblate rotator" since the model's flattening may be interpreted as being due completely to its rotation. The expressions (27, 28) have been evaluated for a number of axisymmetric potential-density pairs ([Fillmore 1986]; [Dejonghe & de Zeeuw 1988]; [Dehnen & Gerhard 1994]; [Evans & de Zeeuw 1994]); the qualitative nature of the solutions is only weakly dependent on the choices of nu and Phi.

The relative ease with which f (E, Lz) and its moments can be computed given nu and Phi has tempted a number of workers to model real galaxies in this way. The approach was pioneered by Binney, Davies & Illingworth (1990) and has been very widely applied ([van der Marel, Binney & Davies 1990]; [van der Marel 1991]; [Dejonghe 1993]; [van der Marel et al. 1994]; [Kuijken 1995]; [Dehnen 1995]; [Qian et al. 1995]). Typically, a model is fit to the luminosity density and the potential is computed assuming that mass follows light, often with an additional central point mass representing a black hole. The even part of f or its moments are then uniquely determined, as discussed above. The observed velocities are not used at all in the construction of f+ except insofar as they determine the normalization of the potential. Models constructed in this way have been found to reproduce the kinematical data quite well in a few galaxies, notably M32 ([Dehnen 1995]; [Qian et al. 1995]). The main shortcoming of this approach is that it gives no insight into how wide a range of three-integral f's could fit the same data. Furthermore, if the model fails to reproduce the observed velocity dispersions, one does not know whether the two-integral assumption or the assumed form for Phi (or both) are incorrect.

3.2. Models Based on Special Potentials

The motion in certain special potentials is simple enough that the third integral can be written in closed form, allowing one to derive tractable expressions for the (generally non-unique) three-integral distribution functions that reproduce rho(R, z). Such models are mathematically motivated and tend to differ in important ways from real galaxies, but the hope is that they may give insight into more realistic models. Dejonghe & de Zeeuw (1988) pioneered this approach by constructing three-integral f's for the Kuzmin-Kutuzov (1962) family of mass models, which have a potential of Stäckel form and hence a known I3. They wrote f = f1(E, Lz) + f2(E, Lz, I3) and chose a simple parametric form for f2, f2 = |E|l Lzm(Lz + I3)n. The contribution of f2 to the density was then computed and the remaining part of rho was required to come from f1.

Bishop (1987) pointed out that the mass density of any oblate Stäckel model can be reconstructed from the thin short-axis tube orbits alone. The density at any point in Bishop's "shell" models is contributed by a set of thin tubes that differ in only one parameter, their turning point. The distribution of turning points that reproduces the density along every shell in the meridional plane can be found by solving an Abel equation. If all the orbits in such a model are assumed to circulate in the same direction about the symmetry axis, the result is the distribution function with the highest total angular momentum consistent with the assumed distribution of mass. Bishop constructed shell-orbit distribution functions corresponding to a number of oblate Stäckel models. De Zeeuw & Hunter (1990) applied Bishop's algorithm to the Kuzmin-Kutuzov models, and Evans, de Zeeuw & Lynden-Bell (1990) derived shell models based on flattened isochrones. Hunter et al. (1990) derived expressions analogous to Bishop's for the orbital distribution in prolate shell models in which the two families of thin tube orbits permit a range of different solutions for a given mass model.

The ease with which thin-orbit distribution functions can be derived has motivated a number of schemes in which f is assumed to be close to fshell, i.e. in which the orbits have a small but nonzero radial thickness. Robijn & de Zeeuw (1996) wrote f = fshell × g(E, Lz, I3) with g a specified function and described an iterative scheme for finding fshell. They used their algorithm to derive a number of three-integral f's corresponding to the Kuzmin-Kutuzov models. De Zeeuw, Evans & Schwarzschild (1996) noted that, in models where the equipotential surfaces are spheroids with fixed axis ratios (the "power-law" galaxies), one can write an approximate third integral that is nearly conserved for tube orbits with small radial thickness. This "partial integral" reduces to the total angular momentum in the spherical limit; its accuracy in non-spherical models is determined by the degree to which thin tube orbits deviate from precessing circles. Evans, Häfner & de Zeeuw (1997) used the partial integral to construct approximate three-integral distribution functions for axisymmetric power-law galaxies.

The restriction of the potential to Stäckel form implies that the principal axes of the velocity ellipsoid are aligned with the same spheroidal coordinates in which the potential is separable ([Eddington 1915]). This fact allows some progress to made in finding solutions to the Jeans equations. Dejonghe & de Zeeuw (1988) and Evans & Lynden-Bell (1989) showed that specification of a single kinematical function, e.g. the velocity anisotropy, over the complete meridional plane is sufficient to uniquely determine the second velocity moments everywhere in a Stäckel potential. In the limiting case sigmaz = sigmaR, their result reduces to equations (27, 28). Evans (1992) gave a number of numerical solutions to the Jeans equations based on an assumed form for the radial dependence of the anisotropy. Arnold (1995) showed that similar solutions could be found whenever the velocity ellipsoid is aligned with a separable coordinate system, even if the underlying potential is not separable.

3.3. General Axisymmetric Models

In all of the studies outlined above, restrictions were placed on f or Phi for reasons of mathematical convenience alone. One would ultimately like to infer both functions in an unbiased way from observational data, a difficult problem for which no very general solution yet exists. An intermediate approach consists of writing down physically-motivated expressions for Phi(R, z) and nu(R, z), then deriving a numerical representation of f (E, Lz, I3) that reproduces nu as well as any other observational constraints in the assumed potential. For instance, nu might be derived from the observed luminosity density and Phi obtained via Poisson's equation under the assumption that mass follows light. The primary motivation for such an approach is that the relation between f and the data is linear once Phi has been specified, which means that solutions for f can be found using standard techniques like quadratic programming (Dejonghe 1989). Models so constructed are free of the biases that result from placing arbitrary restrictions on f; furthermore, if the expression for Phi is allowed to vary over some set of parameters, one can hope to assign relative likelihoods to different models for the mass distribution.

Most observational constraints take the form of moments of the line-of-sight velocity distribution, and it is appropriate to ask how much freedom is allowed in these moments once Phi and nu have been specified. The Jeans equations for a general axisymmetric galaxy are similar to the ones given above for two-integral models, except that sigmaz and sigmaR are now distinct functions and the velocity ellipsoid can have nonzero $ \overline{v_Rv_z}$, corresponding to a tilt in the meridional plane:

Equation 29   (29)
Equation 30   (30)

Unlike the two-integral case, the solutions to these equations are expected to be highly nonunique since the shape and orientation of the velocity ellipsoid in the meridional plane are free to vary - a consequence of the dependence of f on a third integral. Fillmore (1986) carried out the first thorough investigation of the range of possible solutions; he considered oblate spheroidal galaxies with de Vaucouleurs density profiles, and computed both internal and projected velocity moments for various assumed elongations and orientations of the models. Fillmore forced the velocity ellipsoid to have one of two, fixed orientations: either aligned with the coordinate axes ($ \overline{v_Rv_z}$ = 0), or radially aligned, i.e. oriented such that one axis of the ellipsoid was everywhere directed toward the center. He then computed solutions under various assumptions about the anisotropies. Solutions with large sigmaphi tended to produce large line-of-sight velocity dispersions sigmap along the major axis, and contours of sigmap that were more flattened than the isophotes. Solutions with large sigmaR had more steeply-falling major axis profiles and sigmap contours that were rounder than the isophotes, or even elongated in the z-direction. These differences were strongest in models seen nearly edge-on. Fillmore suggested that the degree of velocity anisotropy could be estimated by comparing the velocity dispersion gradients along the major and minor axes.

Dehnen & Gerhard (1993) carried out an extensive study in which they constructed explicit expressions for f (E, Lz, I3); in this way they were able to avoid finding solutions of the moment equations that corresponded to negative f's. They approximated I3 using the first-order resonant perturbation theory of Gerhard & Saha (1991) described above; their mass model was the same one used in that study, a flattened isochrone. Dehnen & Gerhard made the important point that the mathematically simplest integrals of motion are not necessarily the most useful physically. They defined new integrals Sr and Sm, called "shape invariants," as algebraic functions of E, Lz and I3. The radial shape invariant Sr is an approximate measure of the radial extent of an orbit, while the meridional shape invariant Sm measures the extent of the orbit above and below the equatorial plane. Two-integral distribution functions of the form f = f (E, Sm) are particularly interesting since they assign equal phase space densities to orbits of all radial extents Sr, leading to roughly equal dispersions in the R - and phi - directions. Classical two-integral models, f = f (E, Lz), accentuate the nearly circular orbits to an extent that is probably unphysical. Dehnen & Gerhard also investigated choices for f that produced radially-aligned velocity ellipsoids with anisotropies that varied from pole to equator.

The most general, but least elegant, way to construct f in a specified potential is to superpose individual orbits, integrated numerically. Richstone (1980, 1982, 1984) pioneered this approach by building scale-free oblate models with nu ~ r-2 in a self-consistent, logarithmic potential. Levison & Richstone (1985a, b) generalized the algorithm to models with a logarthmic potential but a more realistic luminosity distribution, nu ~ r-3. Fillmore & Levison (1989) carried out a survey of highly-flattened oblate models with a de Vaucouleurs surface brightness distribution and with two choices for the gravitational potential, self-consistent and logarithmic. They found that the range of orbital shapes was sufficient to produce models in either potential with similar observable properties; for instance, models could be constructed in both potentials with velocity dispersion profiles that increased or decreased along either principal axis over a wide range of radii. Hence they argued that it would be difficult to infer the presence of a dark matter halo based on the observed slope of the velocity dispersion profile alone.

Orbit-based algorithms like Fillmore & Levison's have now been written by a number of groups ([Gebhardt et al. 1998]; [van der Marel et al. 1998]; [Valluri 1998]). In spite of Fillmore & Levison's discouraging conclusions about the degeneracy of solutions, the most common application of these algorithms is to potential estimation, i.e. inferring the form of Phi(R, z) based on observed rotation curves and velocity dispersion profiles. A standard approach is to represent Phi in terms of a small set of parameters; for every choice of parameters, the f is found that best reproduces the kinematical data, and the optimum Phi is defined in terms of the parameters for which the derived f provides the best overall fit. For instance, Phi may be written

Equation 31   (31)

where M/L is the mass-to-light ratio of the stars, PhiL is the "potential" corresponding to the observed luminosity distribution, and Mh is the mass of a central black hole. An example is given in Figure 2 which shows chi2 contours in (Mh, M/L)-space derived from ground-based and HST data for M32 ([van der Marel et al. 1998]). The expected degeneracy appears as a plateau of nearly constant chi2; this plateau reflects the freedom to adjust a three-integral f in response to changes in Phi such that the goodness-of-fit to the data remains precisely unchanged. When the potential is represented by just two parameters, this non-uniqueness appears as a ridge line in parameter space, since the virial theorem implies a unique relation between the two parameters that define the potential ([Merritt 1994]). Imperfections in the data or the modelling algorithm broaden this ridge line into a plateau, often with spurious local minima. The extreme degeneracy of models derived from such data means that is usually impossible to learn much about the potential that could not have been inferred from the virial theorem alone.

3.4. The Axisymmetric Inverse Problem

Modelling of elliptical galaxies has evolved in a very different way from modelling of disk galaxies, where it was recognized early on that most of the information about the mass distribution is contained in the velocities, not in the light. By contrast, most attempts at elliptical galaxy modelling have used the luminosity as a guide to the mass, with the velocities serving only to normalize the mass-to-light ratio. One could imagine doing much better, going from the observed velocities to a map of the gravitational potential. The difficulties in such an "inverse problem" approach are considerable, however. The desired quantity, Phi, appears implicitly as a non-linear argument of f, which itself is unknown and must be determined from the data. There exist few uniqueness proofs that would even justify searching for an optimal solution, much less algorithms capable of finding those solutions.

A notable attempt was made by Merrifield (1991), who asked whether it was possible to infer a dependence of f on a third integral in a model-independent way. Merrifield pointed out that the velocity dispersions along either the major or minor axes of an edge-on, two-integral axisymmetric galaxy could be independently used to evaluate the kinetic energy term in the virial theorem. A discrepancy between the two estimates might be taken as evidence for a dependence of f on a third integral. Merrifield's test may be seen as a consequence of the fact that f (E, Lz) is uniquely determined in an axisymmetric galaxy with known Phi and nu. However, as Merrifield emphasized, a spatially varying M/L could mimic the effects of a dependence of f on a third integral.

An algorithm for simultaneously recovering f (E, Lz) and Phi(R, z) in an edge-on galaxy, without any restrictions on the relative distribution of mass and light, was presented by Merritt (1996). The technique requires complete information about the rotational velocity and line-of-sight velocity dispersion over the image of the galaxy. One can then deproject the data to find unique expressions for sigma(R, z), sigmaphi(R, z) and $ \overline{v_{\phi}}$(R, z). Once these functions are known, the potential follows immediately from either of the Jeans equations (25, 26); f+(E, Lz) is also uniquely determined, as described above. The odd part of f is obtained from the deprojected $ \overline{v_{\phi}}$(R, z). This work highlights the impossibility of ruling out two-integral f's for axisymmetric galaxies based on observed moments of the velocity distribution, since the potential can always be adjusted in such a way as to reproduce the data without forcing f to depend on a third integral.

The algorithm just described may be seen as the generalization to edge-on axisymmetric systems of algorithms that infer f (E) and Phi(r) in spherical galaxies from the velocity dispersion profile (e. g. [Gebhardt & Fischer 1995]). The spherical inverse problem is highly degenerate if f is allowed to depend on L2 as well as E (e.g. [Dejonghe & Merritt 1992]), and one expects a similar degeneracy in the axisymmetric inverse problem if f is allowed to depend on I3. Thus the situation is even more discouraging than envisioned by Fillmore & Levison (1989), who assumed that the data were restricted to the major or minor axes: even knowledge of the velocity moments over the full image of a galaxy is likely to be consistent with a large number of (f,Phi) pairs. Distinguishing between these possible solutions clearly requires additional information, and one possible source is line-of-sight velocity distributions (LOSVD's), which are now routinely measured with high precision ([Capaccioli & Longo 1994]). In the spherical geometry, LOSVD's have been shown to be effective at distinguishing between different f (E, L2) - Phi(R, z) pairs that reproduce the velocity dispersion data equally well ([Merritt & Saha 1993]; [Gerhard 1993]; [Merritt 1993]). A second possible source of information is proper motions, which in the spherical geometry allow one to infer the variation of velocity anisotropy with radius ([Leonard & Merritt 1989]); however most elliptical galaxies are too distant for stellar proper motions to be easily measured. A third candidate is X-ray gas, from which the potential can in principle be mapped using the equation of hydrostatic equilibrium ([Sarazin 1988]).

All of the techniques described above begin from the assumption that the luminosity distribution nu(R, z) is known. Rybicki (1986) pointed out the remarkable fact that nu is uniquely constrained by the observed surface brightness distribution of an axisymmetric galaxy only if the galaxy is seen edge-on, or if some other restrictive condition applies, e.g. if the isodensity contours are assumed to be coaxial ellipsoids with known axis ratios. Gerhard & Binney (1996) constructed axisymmetric density components that are invisible when viewed in projection and showed how the range of possible nu's increases as the inclination varies from edge-on to face-on. Kochanek & Rybicki (1996) developed methods to produce families of density components with arbitrary equatorial density distributions; such components typically look like disks. Romanowsky & Kochanek (1997) explored how uncertainties in deprojected nu's affect computed values of the kinematical quantities in two-integral models with constant mass-to-light ratios. They found that large variations could be produced in the meridional plane velocities but that the projected profiles were generally much less affected.

These studies suggest that the dynamical inverse problem for axisymmetric galaxies is unlikely to have a unique solution except under fairly restrictive conditions. This fact is useful to keep in mind when evaluating axisymmetric modelling studies, in which conclusions about the preferred dynamical state of a galaxy are usually affected to some degree by restrictions placed on the models for reasons of computational convenience only.



1 Because of their boxlike shapes in the meridional plane, such orbits were originally called "boxes" even though their three-dimensional shapes are more similar to doughnuts. Back.

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