GALAXIES, ELLIPTICAL, DYNAMICS

David Merritt

Elliptical galaxies are the second largest class of galaxies by number, comprising roughly 30% of all observed galaxies, and a larger fraction of galaxies in dense environments such as galaxy clusters. Compared with spiral galaxies, elliptical galaxies are very regular and smooth in appearance, and contain very little gas, dust, or young stars.

Until 1975, most astronomers viewed elliptical galaxies as oblate spheroids flattened by rotation. In this model, elliptical galaxies were thought to be similar to spiral galaxies in their dynamics and formation history, the principal difference being the efficiency of star formation during collapse of the protogalactic cloud (low efficiency in spirals, high in ellipticals). However, that year saw the publication of the first accurate measurements of the rotation velocities of elliptical galaxies, derived from absorption-line spectra of the stars. These studies, and a large body of observational work since that time, have indicated that the majority of bright elliptical galaxies rotate much too slowly for their shapes to be determined by rotation. It is now believed that the motions of the stars in most elliptical galaxies are essentially random; the shapes of these galaxies are determined by the large-scale anisotropy of the stellar motions, that is, the degree to which random velocities are different in different directions. Since rotation is not very important in these galaxies, there is no compelling reason to assume that they are oblate, and it is now generally believed that elliptical galaxies are triaxial, that is, that their figures are ellipsoids (possibly slowly rotating) with three unequal axes. Since 1975, theoretical and observational work on the dynamics of elliptical galaxies has focused on the following problems: determining their three-dimensional shapes, understanding the character of stellar orbits in triaxial potential wells, constructing self-consistent triaxial models, searching for correlations between the kinematical and morphological parameters of elliptical galaxies, deriving their masses and mass distributions, and resolving the structure of their cores.

MORPHOLOGY

To first order, the appearance of an elliptical galaxy on the sky can be described in terms of its surface brightness profile and its apparent shape. Most elliptical galaxies are well described by an empirical surface brightness law first proposed by Gerard de Vaucouleurs:

(R) = _e exp{-7.67[(R / R_e)^1/4 - 1]} ,

where R_e is the effective radius and corresponds to the radius that encloses half of the total integrated luminosity of the galaxy, and _e is the surface brightness at R_e. Deviations from this law are often correlated with a galaxy's environment. For instance, dwarf companions to larger galaxies often show surface brightness profiles that fall off more rapidly than de Vaucouleurs's law, an effect that may be attributable to tidal truncation of the envelopes of these galaxies. The brightest elliptical galaxies, called cD galaxies, have more extensive envelopes than predicted by de Vaucouleurs's law. These galaxies are always located at the centers of galaxy clusters, and their envelopes are thought to consist of tidal debris from other cluster galaxies.

The isophotal contours of most elliptical galaxies are usually elliptical to a high degree of accuracy. However, the elongation and orientation of these isophotal ellipses often varies with position. In contrast to surface brightness, ellipticity shows no characteristic dependence on radius: Flattening sometimes increases, sometimes decreases, and sometimes remains roughly constant with radius. The same is true with regard to the orientation of the isophotal major axis as a function of position. The origin and significance of these isophotal twists is not well understood. One possibility is that they result simply from the galaxy's triaxial form, because a set of nested, triaxial ellipsoids with differing ellipticities will show twisted isophotes if viewed from a direction that does not lie along one of the symmetry axes. Alternatively, the twists may be intrinsic, resulting from tidal interactions with neighboring galaxies.

STELLAR DYNAMICS OF ELLIPTICAL GALAXIES

Galaxies are "collisionless" systems: Each star moves along its orbit under the influence of the smooth gravitational potential of the whole galaxy, and hardly ever comes close enough to another star for its motion to be significantly perturbed by the encounter. The basic time scale that governs the dynamical evolution of a collisionless system is just the time for a typical star to cross it in its orbit, called the "dynamical" or "crossing" time; it is given roughly by

Tdyn	(GMgal / R3gal)-1/2
	5 × 107 yr (Mgal / 1011M)-1/2 (Rgal / 10 kpc)3/2

where M_gal and R_gal are the galaxy's mass and radius, M is the mass of the Sun, and G is the gravitational constant. A typical large elliptical galaxy has a dynamical time near the center of about 10⁸ yr. Because this value is much shorter than the age of the universe (~ 10¹⁰ yr), elliptical galaxies are thought to be well "relaxed"; that is, the spatial distribution of stars in these galaxies should long since have reached a smooth unchanging state, and the potential well through which each star moves should be nearly fixed in time.

Computer modeling has shown that most of the stellar orbits in nonrotating triaxial potentials fill volumes with one of two characteristic shapes. "Box" orbits densely fill regions similar to rectangular parallelepipeds; the basic character of the motion is up and down along the long axis of the galaxy. "Tube" orbits fill roughly doughnut-shaped regions; these orbits circulate around the short or the long axis of the galaxy and avoid the center. Tube orbits are the only orbits present in axisymmetric (oblate or prolate) potentials; it is the box orbits that are uniquely associated with triaxial potentials, and that permit self-consistent triaxial galaxies to exist. Box orbits have the important additional property that a star on such an orbit eventually passes arbitrarily close to the center of the galaxy. This could be important if - as recent observations suggest-some elliptical galaxies contain massive objects (such as black holes) in their cores, since the massive objects will perturb the orbits and induce slow changes in the galaxy's shape.

In addition to the box and tube orbits, which are called "regular," some fraction of the orbits in most triaxial potentials are found to be "irregular," or "stochastic." Stochastic orbits have no well-defined shape; instead they traverse first one, then another, volume, with the transition occurring nearly randomly. It is at present uncertain whether the slow diffusion of these stochastic orbits implies a slow evolution of the structure of elliptical galaxies, on a time scale longer than the dynamical time but shorter than the age of the universe.

Important as orbital studies are for understanding the internal structure of elliptical galaxies, the interpretation of observational data generally involves considerably less-detailed models than the sort described above. This is because the dynamical information available observationally - the rotation velocity and the velocity dispersion of the stars, projected along lines of sight through the galaxy - is not nearly sufficient to constrain a unique orbital model, even if the three-dimensional shape of the galaxy were known. Fortunately, many interesting questions about the dynamics of elliptical galaxies can be answered, at least in part, with the types of observational data currently available. One such question, mentioned above, concerns the degree to which elliptical galaxies are supported by rotation as opposed to velocity anisotropies. Simple calculations based on an isotropic model, in which the random component of the stellar motion is the same in all directions at a given point, indicate that the rotation velocity of an elliptical galaxy with an axis ratio of 1:2 should be roughly comparable to its velocity dispersion along the line of sight. In the early 1970s, advances in photon-detection systems and digital data processing permitted the first accurate determinations of these quantities in a number of elliptical galaxies. The results showed clearly that the majority of bright ellipticals were rotating too slowly, by a factor of about 2, for their flattenings to be explained by rotation; thus the velocity distributions in these galaxies had to be strongly anisotropic. In general, the degree of rotational support is observed to increase as galaxy luminosity decreases, with the least-luminous elliptical galaxies having rotation velocities consistent with that expected for isotropic oblate rotators. More recent work has uncovered significant rotation around the apparent long axis of several elliptical galaxies, a result that is easiest to understand if these galaxies are strongly triaxial or prolate.

Much observational work has concentrated on measuring the dependence of stellar velocity dispersion on radius in a large sample of galaxies. These data can be used in one of two ways. If one assumes that the variation of mass density with radius is known for a galaxy - for instance, by equating the mass density to some fixed multiple (the mass-to-light ratio) of the luminosity density, which is easily measured - then one can use the velocity dispersion profile to understand how the basic character of the stellar motion varies with radius. For instance, if the stellar orbits are predominantly radial (i.e., boxes), then the component of their motion along the line of sight falls off more rapidly with radius than if the orbits are mostly circular (i.e., tubes). Alternatively, if one makes an assumption about the orbital character - for instance, that the stellar motions are approximately isotropic - then the variation of velocity dispersion with radius constrains the distribution of mass in the galaxy, since the typical orbital velocity at every radius depends on the amount of matter producing the gravitational acceleration. Unfortunately, it is impossible to determine both the orbital character and the mass distribution given only the observed velocity dispersions, and this fact has greatly hampered the interpretation of dynamical data. Fortunately, other techniques can sometimes be used to place independent constraints on the mass-to-light ratio; these include measurement of the rotation velocity of the gaseous disks that appear in a small fraction of elliptical galaxies, and observations at x-ray wavelengths of the hot gas that is sometimes present in galactic potential wells. At present, however, there is no elliptical galaxy for which the mass distribution or the character of the stellar orbits has been well determined. It is not yet certain, for instance, whether elliptical galaxies are typically surrounded by the massive dark matter halos that are known to be prevalent around spiral galaxies, although the x-ray data strongly suggest the existence of such halos around a few elliptical galaxies.

CORES AND NUCLEI

Observations of the very centers of elliptical galaxies are hampered by distortions induced by motions in the earth's atmosphere, which limit angular resolution to about one second of arc, regardless of the size of the telescope. It was not until about 1985 that careful observations verified the existence of cores in most elliptical galaxies, that is, regions near the center where the surface brightness levels off to a nearly constant value. (Note that de Vaucouleurs's law predicts an ever-rising central brightness.) Typical core radii, that is, radii at which the surface brightness equals half its central value, range from about 1 kpc for the brightest elliptical galaxies to less than 100 Pc for the fainter ones. Many elliptical galaxies are also observed to have unresolved, pointlike nuclei, with much smaller radii and much higher surface brightnesses. The first elliptical galaxy for which useful dynamical information about its core was obtained was M87, the brightest member of the Virgo galaxy cluster. Because M87 is relatively nearby and very large, its core is easily resolved. Early observations of the core of M87 revealed stellar random velocities that rose steeply toward the center, an effect that was initially attributed to the gravitational influence of a massive central object, perhaps a black hole. It was later pointed out that other models were equally consistent with the data, including models in which the stellar velocities are very anisotropic, or the core is elongated along the line of sight. However, recent studies of several other nearby elliptical galaxies make a much stronger case for massive central objects. For instance, stars within a few parsecs of the center of M32, a dwarf elliptical galaxy in the Local Group, are observed to rotate about the center with a velocity of about 100 km s^-1. The mass required to produce such large rotational velocities can be derived from the classical equation relating centripetal acceleration to force:

where V is the velocity of a star in orbit around a point mass M. The implied mass for the central object in M32 is of order 10⁷ solar masses; because of the high rotation, this estimate is not strongly dependent on the unknown anisotropies. A number of other nearby galaxies (including our own) appear to have dark massive objects in their nuclei, with masses ranging from 10⁶ to 10⁹ solar masses. The nature and origin of the central mass concentrations in elliptical galaxies is still obscure. One possibility is that many elliptical galaxies were once quasars, and that the massive objects are the black holes that once powered their phenomenal radio and optical emission. Alternatively, the nuclei of these galaxies may contain dense clusters of stellar remnants such as neutron stars.

On a somewhat larger scale, the presence of significant rotation in the cores of elliptical galaxies is now believed to be a common phenomenon. Observations of nearby elliptical galaxies show that the cores of 1/4 to 1/2 of these galaxies exhibit strong rotation, often around a different axis than that of the rest of the galaxy. These rapidly rotating cores are thought to be the remains of dwarf galaxies that have spiraled to the center of the larger galaxies, remaining substantially intact during the descent; their large rotation velocities are all that remains of the original orbital velocity of the dwarf.

PARAMETER CORRELATIONS

In spite of their complexity, elliptical galaxies appear to obey certain laws relating their kinematical and morphological properties. The existence of these laws is important, because they suggest that the process of galaxy formation - which is still very poorly understood - somehow imposes constraints on the present-day form of galaxies, preferring certain final configurations over others. One such correlation, mentioned above, is between the luminosity and rotation velocity of elliptical galaxies: Luminous elliptical galaxies exhibit less rotation, measured in terms of the amount required for rotational support, than faint elliptical galaxies. The dependence is not very tight, however. A much tighter correlation is seen between the total luminosity L and the central velocity dispersion of elliptical galaxies:

L ⁴

called the Faber-Jackson relation (after Sandra M. Faber and Robert E. Jackson). This relation is important because it allows one to estimate the intrinsic luminosity of a galaxy from its observed velocity dispersion, and hence to calculate the galaxy's approximate distance. The Faber-Jackson relation, combined with Fish's law (after Robert Fish), which states that the average surface brightness of elliptical galaxies is roughly constant, implies that all elliptical galaxies have roughly the same mass-to-light ratio. Additional correlations are observed between the parameters that characterize the cores of elliptical galaxies. For instance, the core radius r_c depends on total luminosity through the approximate relation

r_c L^5/4

These relations suggest that elliptical galaxies, in spite of their individual complexity, constitute a basically one-parameter family, the single parameter being the total luminosity or the total mass.

Additional Reading

Binney, J. and Tremaine, S.(1987). Galactic Dynamics. Princeton University Press, Princeton.

de Zeeuw, T., ed.(1987). Structure and Dynamics of Elliptical Galaxies. proceedings of IAO Symposium 127. Reidel, Dordrecht.

Martinet, L. and Mayor, M., eds.(1982). Morphology and Dynamics of Galaxies. Geneva Observatory, Geneva.

Merritt, D., ed.(1989). Dynamics of Dense Stellar Systems. Cambridge University Press, Cambridge.

Mihalas, D. and Binney, J.(1981). Galactic Astronomy. W.H. Freeman, San Francisco.

Adapted from The Astronomy and Astophysics Encyclopedia, ed. Stephen P. Maran