Improved observational techniques reveal features which challenge the very definition of elliptical galaxies. Many ellipticals contain disks or rings of stars, dust, or gas; others show signs of recent accretions in the form of shells of luminosity.
High signal-to-noise CCD images of elliptical galaxies reveal that non-elliptical isophotes are common (e.g. Lauer 1985). To detect such distortions, the surface brightness along a trial ellipse is expanded as
-- \ (1) I(theta) = | a_i cos(i theta) + b_i sin(i theta) , / -- i = 0where theta is the azimuthal angle measured with respect to the major axis. Coefficients a_1, b_1 are used to refine the ellipse's center, and a_2, b_2 are used to refine its position angle and ellipticity. When a perfectly elliptical isophote is correctly fit, only a_0 is remains. Non-elliptical isophotes yield nonzero a_i, b_i for i > 2 (lower-order coefficients are used to optimize the fit). If the image is symmetric, the first nonzero coefficients are a_4 and b_4. The presence a disk is suggested if a_4 > 0; for highly inclined disks a_6 is of comparable magnitude (Rix & White 1990).
In a sample of 42 early-type galaxies, ~10% yielded a_4/a_0 > 0.02 and appear to be likely candidates for systems with embedded disks (Lauer 1985). However, the actual percentage of ellipticals with embedded disks could be much larger. Photometric models combining a spheroidal r^1/4 bulge and an exponential disk indicate that embedded disks must be either very substantial or nearly edge-on to be readily detected. Detection statistics are consistent with the hypothesis that all ellipticals with a_4 > 0 contain disks contributing ~20% of the total light (Rix & White 1990).
Photometric modeling may also be used to examine the possibility of recovering disk parameters by decomposing the observed images. The practice of finding the best-fit ellipse and attributing only the residuals to the disk systematically and drastically underestimates the luminosity contributed by the latter (Rix & White 1990).
Surface-brightness data cannot unambiguously diagnose the presence of a disk since some ellipticals may have intrinsically pointy isophotes for unrelated reasons. Line-of-sight velocity profiles could provide a more definitive diagnosis (Rix & White 1990).
To measure velocity profiles, galactic absorption-line spectra are compared with the spectra of template stars. Such comparison may be performed in several different ways (Rix & White 1992):
Fourier transform methods invoke the convolution theorem; the transform of the observed spectrum is (approximately) equal to the transform of the velocity profile times the transform of the template spectrum (Sargent et al. 1977, Franx et al. 1989).
Cross-correlation methods measure the correlation between the observed and template spectra as a function of the relative shift in wavelength (Tonry & Davis 1979, Bender 1990).
Direct methods synthesize the observed spectrum by adding up shifted versions of the template spectrum. This is somewhat expensive computationally but avoids the `belling' needed to produce well-behaved transforms of finite spectral intervals, allows masking of discrepant features to reduce mismatch between template and galactic spectra, and permits a straightforward error analysis (Rix & White 1992).
Non-Gaussian line profiles may be characterized using the Gauss-Hermite series (van der Marel & Franx 1993):
-- alpha(w) \ v - V (2) LP(v) = gamma -------- | h_j H_j(w) , w = ----- , sigma / sigma -- j = 0where gamma is the line strength, sigma is the Gaussian velocity dispersion, V is the system velocity, and
-1/2 2 (3) alpha(x) = (2 pi) exp(- x / 2)is a normalized Gaussian with unit dispersion. The set of functions H_j(x) are Hermite polynomials of degree j; these are orthogonal with respect to the weight function alpha(x)^2:
1/2 / 2 (4) 2 pi | dx H_j(x) H_k(x) alpha(x) = delta_jk /
In the application of Eq. 2, the parameters gamma, sigma, and V are chosen by requiring that h_0 = 1 and h_1 = h_2 = 0; non-Gaussian line profiles yield nonzero h_j for j > 2. In particular, h_3 parametrizes the skewness of the line profile, while h_4 measures whether the profile is more or less peaked than a Gaussian.
Application of these techniques has revealed line profiles characteristic of embedded stellar disks in many ellipticals (Franx & Illingworth 1988, Bender 1990, Rix & White 1992, van der Marel & Franx 1993, van der Marel et al. 1994). Embedded disks yield profiles h_3 values with opposite signs on opposite sides of the galaxy's center, implying the presence of a rapidly rotating component. The characteristics of these embedded disks show considerable variation: NGC 5322 has a relatively `warm' disk, with v/sigma = ~1.4, while the disk in NGC 3610 is quite `cold', with v/sigma = ~4.5 (Rix & White 1992).
Isophote shape is known to be correlated with rotation velocity: Elliptical galaxies with a_4 > 0 tend to be rapid rotators (KD89). Kinematically cold disks may be partly responsible for this correlation, since at some radii such disks may make very substantial contributions to the light and will tend to dominate line-of-sight velocity measurements parametrized by simple Gaussians (Rix & White 1990). For example, purely Gaussian fits to composite line profiles may overestimate rotation curve amplitudes by > 30% (van der Marel & Franx 1993).
A large fraction of elliptical galaxies have dust lanes produced by cold interstellar material distributed in disks or rings (e.g. van Gorkom 1992). Velocity measurements show that in many cases this cold gas counter-rotates or is otherwise kinematically distinct from the underlying stellar component; a likely interpretation is that this material has been accreted since the galaxy formed.
Unlike stars, the gas can only settle down on closed (and stable) orbits. In theory this might allow determination of the fundamental planes of a galaxy's potential, but in practice settling times are long enough that this is questionable (KD89).
The settling process is driven by differential precession of neighboring orbits; if the precession rate varies as a function of radius, gas clouds will collide and dissipate kinetic energy.
Analytic estimates show that settling times at ~1R_e are longer than 10^9 years; thus at any moment only some of the gas can possibly have settled (Steiman-Cameron & Durisen 1988, 1990).
Numerical models confirm the long settling times found analytically (Christodoulou et al. 1992) and illustrate the importance of dissipation (Katz & Rix 1992). In oblate potentials the gas settles into a long-lived warped disk with precession rate constant in radius (Sparke 1986). In prolate potentials, gas disks with constant precession rates are unstable and fall to the center in only few dynamical times.
IC 2006 has a counter-rotating equatorial ring extending to 6.5R_e (19 kpc for H_0 = 50 km/s/Mpc) containing ~5 * 10^8 solar masses of neutral hydrogen (Schweizer et al. 1989). The velocity variation around the ring shows that it is accurately circular, and the gravitational potential must be very nearly round in the ring plane (de Zeeuw 1995). To explain the ring's nearly flat rotation curve, the M/L ratio must increase by a factor of ~3.2 from 0.5R_e to the outer edge of the ring.
NGC 4278 has an extended gas disk with a distinct bend in the zero-velocity contour. Axisymmetric potentials do not reproduce this bend, while a triaxial model with b/a < 0.9 provides a rather good fit (Lees 1991).
NGC 4753 is an S0 galaxy with complex dust lanes resembling a highly twisted disk seen nearly edge-on. The dust distribution is well-fit by a model in which the ring precesses at a rate proportional to r^-1, implying that `most of the galaxy's mass is unseen, is nearly spherically distributed, and has a nearly scale-free spatial distribution' (Steiman-Cameron et al. 1992).
NGC 5128 (Centarus A) is a nearby elliptical galaxy with an active nucleus. The body of the galaxy is crossed by a dust lane due to a warped disk, while the extended envelope surrounding this object contains a number of narrow shells similar to those detected in other elliptical galaxies (Malin et al. 1983). Attempts to model the kinematics of the gas disk in a oblate potential fail, but a prolate model does a good job of reproducing the observed velocities (Quillen et al. 1992). The implied age of the disk is only of order 10^8 years; such an age is probably consistent with a simple merger model needed to account for the shells.
As the example of NGC 5128 indicates, ellipticals may accrete not just gas but also whole galaxies. Tidal forces disrupt the victim, suddenly releasing a spray of stars which orbit in the potential of the larger galaxy. In a simple encounter geometry where a small galaxy falls in along the major axis of the larger one, a set of interleaved shells develop. Each shell consists of stars which have just the right energy to reach apocenter at that radius; because more energetic stars take longer to reach their apocenters, each shell propagates outward from the center of the galaxy and the total number of shells increases with time. Eventually the contrast of the shells becomes so low that they can no longer be discerned in the surface photometry.
The large number of elliptical galaxies with shells or similar features implies that many ellipticals have experienced accretion events. Order-of-magnitude estimates imply that most galaxies with shells acquired these stars within the last few 10^9 years, and that the luminosity accreted in these events may amount to ~10% of the total. Consequently the notion of elliptical galaxies as isolated equilibrium systems must be viewed as only a first approximation to a much more complex (and interesting) situation.
Due date: 2/23/95
14. This homework is a bit more open-ended than previous ones; it is worth 40 points instead of the usual 10. The assignment is to review the web pages on elliptical galaxies, note connections between different lectures, and find places where links could be inserted to make these connections explicit. For example, the Jeans equations might be linked back to the collisionless Boltzmann equation, and forward to various relevant sections on model building.
The choice of how to present these linkages is up to you, but please do not present me with marked-up copies of the pages! You might, for example, make a list of topics and note which ones connect to one another, or you might try to represent the same information graphically.
At a minimum you should try to connect the main topic of each lecture to those of other lectures; such an answer, if reasonably complete, is worth half-credit. However, I expect you to look for more detailed connections and to give specific rather than vague answers; it's not worth much to simply say that everything is connected to everything else!
Last modified: February 16, 1995