The geometry of phase mixing (original) (raw)

Abstract

Partially phase-mixed structures in galaxies occupy a complex surface of dimension D in six-dimensional phase space. The appearance of such structures to observers is determined by their projection into a space the dimensionality K of which is determined by the number of observables (e.g. sky position, distance, radial velocity, etc.). We discuss the expected dimensionality of phase-space structures and suggest that the most prominent features in surveys with KD will be stable singularities (catastrophes). The simplest of these are the shells seen in the outer parts of elliptical galaxies.

Introduction

The evolution of the phase-space density of stars in galaxies is determined by the collisionless Boltzmann equation, which states that phase-space flow is incompressible. Thus a cloud of stars in phase space becomes more and more distorted as it evolves; the local or fine-grained density around any comoving point in phase space remains the same, but the coarse-grained density evolves towards a stationary state. This process, known as phase mixing, is the principal mechanism by which stellar systems reach coarse-grained equilibrium. The standard analogy is stirring a glass containing 20 per cent rum and 80 per cent Coke; eventually every finite volume element in the glass contains 20 per cent rum and 80 per cent Coke, even though infinitesimal volume elements are either 100 per cent rum or 100 per cent Coke (e.g. Arnold & Avez 1968).

Galaxies are not thoroughly phase-mixed, because they are at most a few dynamical-times old in their outer parts, and because mergers and star formation continuously add new stars. There is a variety of direct observational evidence for incomplete phase mixing: (i) sharp-edged features (‘shells’) in the outer parts of at least 10 per cent of elliptical galaxies are believed to arise from the recent tidal disruption of small galaxies (e.g. Hernquist & Quinn 1988); (ii) proper-motion and radial-velocity surveys of the local halo reveal clumping in phase space (Majewski, Hawley & Munn 1996); (iii) the tidally disrupted Sagittarius dwarf galaxy (Ibata et al. 1997) provides a concrete example of a cloud in the early stages of phase mixing; (iv) moving groups in the solar neighbourhood may result from dissolved star clusters and associations (Eggen 1965; Dehnen 1998).

Incomplete phase mixing plays a growing role in the interpretation of observations, for several reasons: radial-velocity and proper-motion surveys are rapidly advancing in size and quality; large telescopes with improved image quality permit us to examine the surface-brightness structure of elliptical galaxies with high spatial resolution and signal-to-noise ratio; the Hipparcos mission has dramatically improved the precision of measurements of the phase-space distribution of stars in the solar neighbourhood, and future space-based astrometric missions such as SIM and GAIA will provide proper motions and parallaxes over much of the Galaxy; and surveys such as 2MASS and the Sloan Digital Sky Survey provide stellar data bases of unprecedented size and uniformity.

Virtually all theoretical descriptions of phase mixing have focused on mixing in a two-dimensional phase space, which is easy to visualize (Fig. 1). This paper discusses the geometrical features of partially phase-mixed systems (we shall call these ‘phase structures’) with higher dimensionality, and the projection of this geometry into the observables that are measured in a survey. We shall argue that this geometry can be organized by two integers: the dimension of the phase structure and the dimension of the survey.

Figure 1.

Phase mixing in a two-dimensional phase space. The figure shows the evolution of 5000 points following the equation of motion _ẋ_=-y r, _ẏ_=x r where _r_2_x_2+_y_2. The flow is area-preserving as it arises from the Hamiltonian H(x,y)=r, where x is the momentum conjugate to y.

The Dimension of Phase Structures

A cloud of stars orbiting in an integrable potential (the ‘host galaxy’) can be viewed as a surface or manifold of dimension DN embedded in _N_-dimensional phase space (usually _N_=6). For example, a cold disc has _D_=2 (the disc occupies a two-dimensional surface in configuration space, and the velocity at each position is unique), while a hot galaxy has _D_=6. In a cosmological context, collisionless cold dark matter has _D_=3 (three spatial dimensions but zero random velocity), but once it collapses and phase mixes to form a hot bound structure then _D_=6.

As real stellar systems always have non-zero thickness and velocity dispersion, we are referring to an ‘effective’ or ‘coarse-grained’ dimension; loosely speaking, a structure in a phase space of N dimensions has effective dimension D if its extent in D independent directions is much larger than its extent in the other _N_-D directions. Thus, for example, in Fig. 1 the effective dimension is _D_=2 at _t_=0, _D_=1 at _t_=50 as the phase structure resembles a filament at this stage, and _D_=2 again at sufficiently large times, when phase mixing is complete.

Because the flow of the cloud through phase space is incompressible, the dimension of the manifold does not change in the initial stages of phase mixing, although the shape of the manifold becomes more and more complicated. However, the effective dimension can change over longer times through several distinct processes. The most obvious is that the phase-mixing scale grows smaller and smaller with time, and eventually becomes smaller than the coarse-graining scale. At this point the phase structure has the same (coarse-grained) dimension as the phase space, unless all of the stars in the cloud have the same value of one or more isolating integrals of the motion. Gravitational scattering by small-scale irregularities (e.g. massive objects) can also lead to diffusion of the phase structure and an increase in effective dimension. Finally, if the phase structure occupies a region of phase space where orbits are chaotic, the structure will expand to fill the chaotic region, which may also lead to an increase in effective dimension.

As we see in Fig. 1 at _t_=50, phase mixing can also decrease the number of effective dimensions. For example, consider the cloud resulting from the disruption of a small hot galaxy. The motion of the stars in the cloud can be described by a Hamiltonian H(J), where (w, J) are action-angle variables in the host galaxy potential. The equations of motion are

(1)

The trajectory of the centroid of the cloud is (J0,wc(t))=(J0,w0+ω0_t_), where ω0=ω(**J0) and (J0,w**0) is approximately the location of the centre of the satellite at the disruption time _t_=0.

Similarly, the trajectory of a star in the cloud is (J(t),w(t))=(J0+Δ J,w0+Δw+ω(J0+ΔJ)t), where ∣Δ**J∣≪J0 and ∣Δw**∣≪1. This can be simplified to

(2)

where

(3)

is the Hessian of the Hamiltonian. At large times the extent of the cloud is dominated by the terms ∝t in equation (2), so we may write approximately

(4)

As H is symmetric, it is diagonalizable, that is, there exists an orthogonal matrix A such that

(5)

where D ij(λ)=λ i δ ij, λ={λ i} are the eigenvalues of H, and AtA−1. We now make a canonical transformation to new action-angle variables (J′,w′) using the mixed generating function S(J′,w)=J′A w; thus

6

Equation (4) now simplifies to

7

This result shows that the shape of the expanding cloud is determined by the eigenvalues λ of the Hessian H, which are invariant under the orthogonal transformation A (although not under arbitrary canonical transformations that preserve the action-angle structure). If two of the three eigenvalues are zero (say _λ_2_λ_3=0) then the cloud expands only along the _w_1′ direction in phase space, yielding a one-dimensional ‘tidal streamer’ that grows linearly with time. If one of the eigenvalues is zero (say _λ_3=0) then the cloud expands in two dimensions in phase space. However, if ∣_λ_1Δ_J_1′∣≫∣_λ_2Δ_J_2′∣ or vice versa, then the expansion will still be effectively one-dimensional. Thus the dimension D has been reduced from 6 to 1 or 2, because the cloud expands much faster in some angles than in others, and not at all in action space.

We illustrate these remarks with some examples. The Hamiltonian for the triaxial harmonic oscillator may be written , where ω k is the frequency along axis k. The Hessian for this Hamiltonian is zero, so the disrupted system does not expand at all: the frequencies are independent of the actions and phase mixing does not occur. The Hamiltonian for the Kepler potential is where _J_1=(GMa)1/2 and a is the semimajor axis. The eigenvalues of the Hessian are (−3/_a_2,0,0), so the expanded cloud is one-dimensional, even if its original state was six-dimensional. Simulations of disruptions of small galaxies in a spherical logarithmic potential also yield one-dimensional clouds for at least the first few orbits (Johnston, Hernquist & Bolte 1996), but adding a flattened potential owing to a disc leads to differential precession that eventually smears the cloud into a three-dimensional structure (Helmi & White 1999).

Catastrophes

We now survey the phase space by measuring K of the N phase-space coordinates of each star (we call this _K_-dimensional space the ‘observable space’1), and ask how to detect the _D_-dimensional phase structure. For example, an image or set of sky positions has _K_=2; a data cube (positions and radial velocities) has _K_=3; a proper-motion survey has _K_=4, etc. A bolometric laboratory detector of cold dark matter measures only energy (_K_=1) and moreover is restricted to a single point in configuration space (Sikivie & Ipser 1992).

If D<K the phase structure will appear as a distinct entity in the observable space, which can be detected by standard techniques: plotting the data on paper (_K_=2), visualization software (_K_=3), cluster-finding algorithms, etc. More specialized techniques can be used if the phase structure has known properties: for example, the disruption of distant satellite galaxies in a spherical potential leads to phase structures that are great circles on the sky (Lynden-Bell & Lynden-Bell 1995; Johnston et al. 1996).

In this discussion we concentrate on the more challenging case when D K, so that the phase structure covers the observable space. In this case, the most prominent features in the survey will be the singularities of the phase structure, that is, the locations where the projection of the phase structure into the observable space leads to a singularity. For example, if in Fig. 1 the observations yield only the horizontal coordinate, then near-singularities arise in the plot at _t_=50 when the curve representing the phase structure is vertical.

We can generalize this simple concept to higher dimension using catastrophe theory (Poston & Stewart 1978; Berry & Upstill 1980; Gilmore 1981; Arnold 1986). We begin by setting up a coordinate system (_u_1,…,u D) on the phase structure. We complete the coordinate system of the phase space by (u D+1,…,u N), which are chosen so that u D+1=⋯=u _N_=0 on the phase structure. Thus the phase structure is specified by

(8)

The advantage of specifying the phase structure in terms of the generating function φ is that φ is a single-valued function of the phase-space coordinates.

Next let (_x_1,…,x D) be the coordinates of the observable space, and complete the coordinate system of the phase space by (_y_1,…,y _N_-D) (we call this (_N_-D)-dimensional space the ‘hidden space’2). For given values of the observable coordinates, the location or locations of the phase structure are specified implicitly by the equations

(9)

Singularities in the observable space arise when there are displacements in the hidden space that leave (9) unaltered, that is, for which

(10)

These equations have a non-trivial solution if the determinant of the (_N_-D)×(_N_-D) Hessian matrix vanishes, that is, if

(11)

We can restrict our attention to structurally stable singularities (roughly speaking, ones for which small changes in φ lead to small changes in the locus of the singularity); the reason is that unstable singularities represent a subset of measure zero and hence are atypical, at least for N7 (Zeeman 1977).

Catastrophes are structurally stable singularities of gradient maps. They are classified by their codimension, which is the dimension of the observable space minus the dimension of the singularity. Thus codimension 1 corresponds to a singular point in a survey with 1 observable, a singular line in a survey with 2 observables, etc. The corank of the catastrophe is the minimum value of N_-D, the dimensionality of the hidden space, which is required for the singularity to occur. A third classification is by the degree k of the singularity: the mean density in a neighbourhood of the observable space of radius e around the singularity is ∝_ε_−_k.

Fold

The fold is the only catastrophe with codimension 1. As the corank of the fold is also 1, the simplest example occurs in a phase space with two dimensions, one of which is observable. For example, the generating function

(12)

implies that the phase structure is the parabolic curve x+_y_2=0, and this manifold has a fold catastrophe at _x_=0. If the linear density on the curve is uniform, the observable density is

(13)

and zero otherwise. The square-root divergence in density on one side of the singularity is characteristic of a fold, and implies that the degree .

Examples of fold catastrophes occur in the projection of a spherical shell on to a plane, rainbows, sunlight sparkling on the sea, twinkling of starlight, gravitational lensing, etc.

The simplest examples of fold catastrophes in phase mixing are shell structures in elliptical galaxies, which are believed to arise from disrupted companion galaxies. Shells are one-dimensional structures that appear in two-dimensional observable space (the two coordinates on the sky plane) and hence if they are singularities they must have codimension 1 and hence must be folds. Hernquist & Quinn (1988) carried out simulations of the evolution of disrupted companions in spherical host galaxy potentials, and several of their conclusions can be interpreted in terms of the results we have derived so far. They find that shell formation requires the accretion of either a dynamically cold or a spatially compact companion; these are precisely the conditions required so that the cloud dimension D2 (Section 2), which in turn is required so that singularities are present in a two-dimensional observable space. They distinguish ‘spatial folding’ (projection from three-dimensional configuration space to two-dimensional space) from ‘phase-wrapping’ (projection from phase space into three-dimensional space) but this distinction is not fundamental: both are projections from phase space into observable space. They recognize that spatial folding is an example of a fold catastrophe. Finally, they point out that some shells arise from step functions in surface density rather than square-root singularities; the former are not singularities and hence are not described by catastrophe theory.

Cusp

The cusp is the only catastrophe with codimension 2, and has corank 1. The cusp is a singular point in a two-dimensional observable space, a line in three-dimensional observable space, etc. For example, consider a phase space with _N_=3 dimensions, two of which are observable. The generating function

(14)

implies that the phase structure is given by the surface _y_3+x_2_y+_x_1=0, which is singular (in the sense of equation 11) along the lines

15

Each of these lines is a fold catastrophe, and their junction is a cusp catastrophe.

If the surface density of stars is uniform on the phase structure, the surface density in the observable space (_x_1,_x_2) is given parametrically by

16

where _x_1=-_y_3-x_2_y. At a distance Δ_x_2 from the fold (∣Δ_x_2∣≪∣_x_2∣∼∣_x_1∣2/3≪1), the density Σ∝∣_x_1∣−2/3 for Δ_x_2>0 and Σ∝∣_x_1∣−1/3(-Δ_x_2)−1/2 for Δ_x_2<0 (the characteristic fold behaviour). The degree of the cusp singularity is .

To illustrate the appearance of a cusp catastrophe, the first three panels of Fig. 2 show a survey that samples the surface density (16) with 300, 1000 and 3000 data points in the interval _x_1,_x_2∈[- 1,1]. The fourth panel shows a higher-resolution survey with 3000 points in _x_1,_x_2∈[- 0.1,0.1]. Any cusp in a two-dimensional observable space can be formed from this special case by smooth distortion (i.e. diffeomorphism) of the observable space, and still consists of two fold catastrophes meeting with a common tangent, as in equation (15).

Figure 2.

The cusp catastrophe defined by the generating function (14), assuming a uniform density of stars on the manifold defined by ∂φ/∂_y_=0. The first three panels show samples of 300, 1000 and 3000 points over the square [−1,1]2; the fourth panel shows a sample of 3000 points over [−0.1,0.1]2. The cusp point at the origin unfolds into two fold catastrophes.

Examples of cusp catastrophes include the caustic curve seen on the surface of coffee in a cup, and the critical point of the van der Waals equation of state in the pressure-temperature plane.

One might speculate that some of the dwarf spheroidal satellite galaxies are cusp catastrophes rather than bound equilibrium stellar systems, but their isopleths are approximately elliptical (Irwin & Hatzidimitriou 1995) and do not resemble the density contours that are expected near a cusp (Fig. 3).

Figure 3.

The cusp catastrophe defined by the generating function (14), convolved with a two-dimensional Gaussian with standard deviation _σ_=0.03 (top) and 0.1 (bottom). The contour levels are 0.1,0.15,0.2,…,0.95,1 in units where the maximum surface density is 1.

Catastrophes with codimension 3

There are three catastrophes with codimension 3, the swallowtail, elliptic umbilic, and hyperbolic umbilic. The swallowtail has corank 1, and thus first appears in a phase space with N4 dimensions; the umbilics have corank 2 and thus require N5. All of these singularities are points in a three-dimensional observable space, lines in four-dimensional observable space, etc. Just as a cusp occurs at the junction of two fold curves, these occur at the junction of cusp curves, which are connected by fold surfaces.

Swallowtail

Consider a phase space with _N_=4 dimensions, three of which are observable. The generating function

(17)

implies that the phase structure is given by _y_4+_x_3_y_2+x_2_y+_x_1=0, which is singular (in the sense of equation 11) along the surfaces defined in terms of the parameters (y,_x_3) by

(18)

Each of these surfaces is a fold catastrophe; the folds meet at cusp lines defined by

(19)

The fold surfaces intersect one another along the parabola . The degree of the swallowtail is .

Fig. 4 shows the fold surfaces and cusp lines for the swallowtail catastrophe.

Figure 4.

The swallowtail catastrophe defined by the generating function (17). The top panel shows the fold surfaces and the bottom panel shows the cusp lines.

Elliptic umbilic

Consider a phase space with _N_=5 dimensions, three of which are observable. The generating function

(20)

implies that the phase structure is given by the surface

(21)

which is singular along the surfaces defined parametrically by

(22)

Each of these surfaces is a fold catastrophe; the folds meet at three parabolic cusp lines defined by and confined to planes at angles of 120° that intersect along the _x_3-axis (see Fig. 5). The degree is _k_=1.

Figure 5.

The elliptic umbilic (top) and hyperbolic umbilic (bottom) catastrophes.

Hyperbolic umbilic

The generating function

(23)

implies that the phase structure is given parametrically by

(24)

where e _i_=±1 and _y_1,_y_20. Each of these surfaces is a fold catastrophe; the folds intersect along the coordinate axes _x_1>0, _x_2_x_3=0 and _x_1=0, _x_2>0, _x_3=0 (Fig. 5). The cusp line is defined parametrically by

(25)

The degree is _k_=1.

These exhaust the catastrophes with codimension 3. Catastrophes with higher codimension are only relevant when the observable space has dimension D4. Catastrophes of higher codimension become increasingly complicated, and are less important since we have argued in Section 2 that typical phase structures will often have dimension 3 or less.

Summary

It is important to understand the geometry of phase structures before searching for them. We have focused on two specific aspects of this geometry: the expected dimension of the phase structure in (usually six-dimensional) phase space, and the classification of singularities when a _D_-dimensional phase structure is projected into a _D_-dimensional survey.

Phase mixing of disrupted small, hot galaxies generally leads to three-dimensional phase structures but the dimensionality can be smaller if the potential of the host galaxy has special symmetries (e.g. spherical or Keplerian), or if the eigenvalues of the Hessian (equation 3) are very different in magnitude. Phase mixing of large, cold galaxies leads to two-dimensional phase structures.

The structurally stable singularities in _D_-dimensional surveys are folds (D1), cusps (D2), swallowtails, elliptic umbilics and hyperbolic umbilics (D3).

Even though phase mixing in a fixed potential is one of the simplest processes in galaxy dynamics, there remain many unresolved theoretical issues. How fast are phase structures disrupted by small-scale gravitational irregularities, dynamical chaos, or differential precession? Do phase structures provide a significant source of relaxation in galaxies (Tremaine & Ostriker 1999)? What are the properties of phase structures in galaxies that we expect from standard models of structure and galaxy formation (Sikivie & Ipser 1992; Sikivie 1999)? What statistical measures can we use to identify and characterize the projections of phase structures into surveys?

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©1999 RAS