90 CHAPTER 15. AXISYMMETRIC & TRIAXIAL MODELS the resulting model, or work backwards from a model towards a distribution function. From fèe;j z è to ç

Transcription

1 Chapter 15 Axisymmetric & Triaxial Models Dynamical models are useful in èaè constraining the true shapes of elliptical galaxies and èbè searching for dark matter andèor nuclear black holes. Two-integral models of axisymmetric elliptical galaxies oæer a logical starting point, but often fail to æt the observed kinematics. Schwarzschild's method permits the construction of both axisymmetric and triaxial models which include a third integral. Minor orbit families and chaotic orbits may limit the range of shapes in elliptical galaxies; systems with steep power-law cusps probably evolve toward more axisymmetric shapes over many orbital times Two-Integral Models The simplest distribution functions which can describe an axisymmetric galaxy depend on just two integrals, the energy E and z-component of angular momentum J z : f = fèe;j z è: è15.1è The assumption that f depends only on E and J z is a logical starting point in building models of non-spherical galaxies. One immediate consequence is that the distribution function depends on the R and z components of the velocity only through the combination v 2 + R v2 z; thus in two-integral models the R and z velocity dispersions must be equal: v 2 R = v2 z : è15.2è This equality doesn't hold in the disk of the Milky Way; the radial dispersion is about twice that in the vertical direction èmb81è, so our galaxy can't be described by a two-integral model. But for other galaxies the situation is not so clear, and two-integral models may suæce Distribution functions As always, a dynamically complete description of a galaxy is provided by its distribution function. One can either start with a distribution function and work out the structure of 89

2 90 CHAPTER 15. AXISYMMETRIC & TRIAXIAL MODELS the resulting model, or work backwards from a model towards a distribution function. From fèe;j z è to çèr; zè: Much as in the spherically symmetric case described in Chapter 12, one may adopt a plausible guess for fèe;j z è, derive the density as a function of position and gravitational potential, and solve Poisson's equation for the gravitational potential. Perhaps the most interesting example of this approach is an elegant series of scalefree models with r,2 density proæles ètoomre 1982è; however, these models are somewhat implausible astronomically since the density vanishes along the minor axis. From çèr; zè to fèe;j z è: Conversely, one may try to ænd a distribution function which generates a given çèr; zè. Computing f given ç and æ involves inverting the relation Z çèr; zè = 2ç 0 Z Jm de dj z fèe;j z è; è15.3è R æ where J m = R p 2E, 2æ is the maximum angular momentum allowed at energy E. This equation may beinverted by integral transform techniques or by matrix methods èbt87, Ch èaè, Merritt 1999è, but the result is underconstrained because a star contributes equally to the total density regardless of its sense of motion about the z-axis. Formally, if fèe;j z è yields the desired çèr; zè, then so does fèe;j z è+f o èe;j z è, where f o èe;j z è =,f o èe;,j z èisany odd function of J z. The odd part of the distribution function may be constrained by kinematic data since it determines the net streaming motion in the ç direction. But even if kinematic data is available, this approach doesn't provide a straightforward way to model galaxies from observational data. The reason is that the transformation from the density èand streaming velocityè to the distribution function is unstable; small errors in the input data produce huge errors in the results èe.g. Dejonghe 1986, BT87è. Anumber of two-integral distribution functions are known for analytic density distributions èdejonghe 1986; Hunter & Qian 1993è. An `unbelievably simple' analytic distribution function exists for the mass distribution which generates the axisymmetric logarithmic potential èevans 1993è. This potential, introduced to describe the halos of galaxies èbinney 1981, BT87, Chapter 2.2.2è, has the form,jm æèr; zè = 1 2 v2 0lnèR 2 c + R 2 + z 2 =q 2 è ; è15.4è where v 0 is the velocity scale, R c is the core scale radius, and q is the æattening of the potential èthe mass distribution is even æatterè. The corresponding distribution function has the form fèe;j z è=èaj 2 z + Bè expè4e=v0è+cexpè2e=v 2 0è; 2 è15.5è where A, B, and C are constants. Evans also divides this distribution function up into `luminous' and `dark' components to obtain models of luminous galaxies embedded in massive dark halos; his results illustrate a number of important points, including the non-gaussian line proæles which result when the luminous distribution function is anisotropic. Recent developments have removed some of the mathematical obstacles to the construction of two-integral models matching observed galaxies èhunter & Qian 1993, Kuijken 1995, Dehnen 1995, Merritt 1996è. In particular, it's no longer necessary to represent the stellar density as an analytic function which can be continued onto the complex plane, as demanded by integral transform methods. Progress has also been made in regularizing the transformation algorithm to prevent amplifying the noise present in observational data.

3 15.1. TWO-INTEGRAL MODELS Jeans-equation models Given the diæculties in constructing distribution functions for real galaxies, it's worth considering the simpler problem of modeling observed systems using the Jeans equations. If we assume that the underlying distribution function depends only on E and J z we can simplify the Jeans equations; the radial and vertical dispersions must be everywhere equal and v R v èçv2 è+ ç R R èv2 R,v2è=,ç@æ ; èçv2 è=,ç@æ : è15.6è è15.7è At each R one can calculate the mean squared velocity in the R direction by integrating è15.7è downward from z = 1; once v 2 R is known, è15.6è gives the mean squared velocity in the ç direction. The Jeans equations don't specify how the azimuthal motion is divided into streaming and random components. One popular choice for the streaming velocity is q v ç = kèv 2 ç,v2 R è; where k is a free parameter èsatoh 1980è. The dispersion in the ç direction is è15.8è ç 2 ç = v 2 ç, v ç 2 : è15.9è Note that if k = 1 the velocity dispersion is isotropic and the excess azimuthal motion is entirely due to rotation, while for k é 1 the azimuthal dispersion exceeds the radial dispersion Application to elliptical galaxies Jeans equations models have been constructed of a number of elliptical galaxies èbinney, Davies, & Illingworth 1990; van der Marel, Binney, & Davies 1990; van der Marel 1991; van der Marel et al. 1994b; Cretton & van den Bosch 1999è. The basic procedure is to: 1. Observe the surface brightness æèx 0 ;y 0 è. 2. Deproject to get the stellar density çèr; zè, assuming an inclination angle. 3. Compute the potential æèr; zè, assuming a constant mass-to-light ratio; terms representing unseem mass may be included. 4. Solve the Jeans equations for the mean squared velocities. 5. Divide the azimuthal motion into streaming and random parts using è15.8è. 6. Project the velocities back on to the plane of the sky to get the line-of-sight velocity and dispersion v los èx 0 ;y 0 è and ç los èx 0 ;y 0 è. 7. Compare the predicted and observed kinematics, adjust parameters to improve the match, and go back to step è2.

4 92 CHAPTER 15. AXISYMMETRIC & TRIAXIAL MODELS The inclination angle, mass-to-light ratio, unseen mass distribution, and rotation factor k are the unknown parameters to be determined by trial and error. Some conclusions following from this exercise are that: æ Isotropic oblate rotators èk = 1è generally don't æt the data, even though some of the galaxies modeled lie close to the expected relation between v 0 =ç 0 and æ. æ Some galaxies èe.g. NGC 1052, M32è are well-æt by two-integral models. æ In other cases, the predicted major-axis velocity dispersions are often larger than those observed. If no parameter set yields a model matching the observations then presumably the underlying assumptions of the models are at fault. Within the context of two-integral models one may try to improve the æt by adding dark mass or by letting the rotation parameter k depend on radius. However, the overly-large major-axis dispersions noted above indicate a fundamental problem with two-integral models èmerriæeld 1991è. In such models, the vertical and radial dispersions are always equal. Thus the æattening of such a model is determined by the energy invested in azimuthal motion, so that v 2 ç é v2 = R v2 z;atpoints along the major axis, motion in the ç direction projects along the line of sight, increasing the predicted velocity dispersion. To account for the observed dispersions, we require v 2 R é v2 z. This is only possible if the distribution function depends on a third integral Three-Integral Models Axisymmetric potentials can and often do support orbits which have three integrals of motion: E, J z, and a third integral I 3. No general expression for I 3 is known, although it may be approximated by the total angular momentum jjj in nearly spherical systems, and by the energy of vertical motion in highly æattened systems. The assumption that f = fèe;j z ;I 3 è è15.10è considerably broadens the range of possible axisymmetric models. Three-integral models are also used to describe triaxial systems. Interest in such models was sparked by observational evidence that luminous elliptical galaxies are slow rotators. In some triaxial potentials the equations of motion separate and all three integrals can be explicitly calculated; this greatly simpliæes the model-building process. But these separable potentials are atypical í for example, they support only regular orbits. The status of triaxial models with non-separable potentials is a bit problematic; there's ample observational and numerical evidence that near-equilibrium systems exist, but the range of shapes and density proæles consistent with true equilibrium may be limited by the eæects of chaotic orbits Schwarzschild's Method Schwarzschild è1979è invented apowerful method for constructing both axisymmetric and triaxial models of equilibrium galaxies without explicit knowledge of the integrals of motion. This works as follows:

5 15.2. THREE-INTEGRAL MODELS Specify the mass model çèrè and ænd the corresponding potential. 2. Construct a grid of K cells in position space. 3. Chose initial conditions for a set of N orbits, and for each one, èaè integrate the equations of motion for many orbital periods, and èbè keep track ofhowmuch time the orbit spends in each cell, which is a measure of how much mass the orbit contributes to that cell. 4. Determine non-negative weights for each orbit such that the summed mass in each cell is equal to the mass implied by the original çèrè. The resulting set of orbital weights represents the same information as does the distribution function. Step è4 is the most subtle. Formally, let Mècè be the integral of çèrè over cell c, and let P i ècè be the mass contributed to cell c by orbit i. The task is then to ænd N non-negative quantities Q i such that Mècè = NX i=1 Q i P i ècè è15.11è simultaneously for all cells. It's generally necessary to take N ç K so as to obtain a reasonably rich set of `basis functions'. Solutions have been found using a number ofnu- merical techniques, including linear programming èschwarzschild 1979è, non-negative least squares èpfenniger 1984è, Lucy's method ènewton & Binney 1984è, and maximum entropy èrichstone & Tremaine 1988è. In general, è15.11è has many solutions, reæecting the fact that many diæerent distribution functions are consistent with a given mass model. Some methods allow one to specify additional constraints so as to select solutions with special properties èmaximum rotation, radial anisotropy, etc.è Axisymmetric models Given the extra freedom a third integral oæers, it's no great surprise that axisymmetric models often manage to match observational data sets. More impressive are recent models, computed using a modiæcation of Schwarzschild's è1979è method, which match the bulk kinematics and line-of-sight velocity distributions of real galaxies ècretton et al. 1999è. Such models have been used to look for evidence of dark matter èrix et al. 1997è and to infer the presence of nuclear black holes èvan der Marel et al. 1998; Cretton & van den Bosch 1999è Triaxial models The general problem of modeling triaxial galaxies is illustrated with a couple of special cases. In separable models the allowed orbits are fairly simple, and the job of populating orbits so as to produce the density distribution is well-understood. In scale-free models the allowed orbits are more complex, and it's not clear if such models can be in true equilibrium.

6 94 CHAPTER 15. AXISYMMETRIC & TRIAXIAL MODELS Separable potentials In three dimensions, a separable potential permits four distinct orbit families: æ box orbits, æ short-axis tube orbits, æ inner long-axis tube orbits, and æ outer long-axis tube orbits. The time-averaged angular momentum of a star on a box orbit vanishes; such orbits therefore do not contribute to the net rotation of the system. Short-axis and long-axis tube orbits, on the other hand, preserve a deænite sense of rotation about their respective axes; consequently, their time-averaged angular momenta do not vanish. The total angular momentum vector of a non-rotating triaxial galaxy may thus lie anywhere in the plane containing the short and long axes èlevison 1987è. Using Schwarzschild's method, it is possible to numerically determine orbit populations corresponding to separable triaxial models èstatler 1987è. A somewhat more restricted set of models can be constructed exactly; these models make use of all available box orbits, but only those tube orbits with zero radial thickness èhunter & de Zeeuw 1992è. Apart from the choice of streaming motion, thin tube models are unique. One use of such models is to illustrate the eæects of streaming motion by giving all tube orbits the same sense of rotation; the predicted velocity æelds display a wide range of possibilities èarnold, de Zeeuw, & Hunter 1994è. Non-zero streaming on the projected minor axis is a generic feature of such models; anumber of real galaxies exhibit such motions and thus must be triaxial èfranx, Illingworth, & Heckman 1989aè. Scale-Free Potentials In scale-free models with suæciently steep cusps, box orbits tend to be replaced by minor orbital families known as boxlets ègerhard & Binney 1985, Miralda-Escude & Schwarzschild 1989è. Each boxlet family is associated with a closed and stable orbit arising from a resonance between the motions in the x and y directions. The appearance of boxlets instead of boxes poses a problem for model building because boxlets are `centrophobic' èmeaning that they avoid the centerè and do not provide the elongated density distributions of the box orbits they replace. As a result, the very existence of scale-free triaxial systems is open to doubt ède Zeeuw 1995è; it may turn out that scale-free systems must be axisymmetric. Moreover, some scale-free potentials have chaotic orbits; these may have no integrals of motion apart from the energy E. In principle, such an orbit could wander everywhere within some chaotic region on a phase-space surface of constant E. The actual structure of this surface may bevery complex; regions of stable and unstable motion are often nested in fractal-like patterns. The scale-free elliptic disk is a relatively simple two-dimensional analog of a scale-free triaxial system. Because the model is scale-free, the radial dimension can be folded out

7 15.3. ROTATION, CHAOS, & SECULAR EVOLUTION 95 when applying Schwarzschild's method; thus the calculations are fast èkuijken 1993è. The result is that self-consistent models can be built using available orbits, but when the angular resolution of the calculations is increased the range of possible axial ratios b=a diminishes. Similar results hold for scale-free models in three dimensions. Models have been constructed for triaxial logarithmic potentials with a range of axial ratios b=a and c=a èschwarzschild 1993è. Tubes and regular boxlets provide suæcient variety to produce models with c=a é 0:5, but not æatter. However, over intervals of 50 dynamical times, chaotic orbits behave like `fuzzy regular orbits', and by including them it becomes possible to build nearequilibrium models as æat as c=a =0:3. But these models are not true equilibria; over long times they may become rounder and less triaxial Rotation, Chaos, & Secular Evolution Figure rotation adds a new level of complexity to the orbit structure of triaxial systems. A few models have been constructed using Schwarzschild's method, but little is known about the existence and stability of such systems. N-body experiments indicate that at least some such systems are viable models of elliptical galaxies. Rotation tends to steer orbits away from the center and so may lessen the eæects of central density cusps. Realistic potentials are likely to have some irregular or chaotic orbits, and there is no reason to think that such orbits are systematically avoided by processes of galaxy formation. Over 10 2 or more dynamical times, such orbits tend to produce nearly round density distributions èmerritt & Valluri 1996è. Consequently, itislikely that secular evolution over timescales of 10 2 dynamical times may bechanging the structures of elliptical galaxies èbinney 1982, Gerhard 1986è. The outer regions are not likely to be aæected since dynamical times are long at large radii, but signiæcant changes are expected in the central cusps where dynamical times are as short as 10 6 years ède Zeeuw 1995è.

8 96 CHAPTER 15. AXISYMMETRIC & TRIAXIAL MODELS