105G . THE ASTRONOMICAL JOURNAL VOLUME 112, NUMBER 1 JULY 1996 112. THE CENTERS OF EARLY-TYPE GALAXIES WITH HST. III. NON-PARAMETRIC RECOVERY OF STELLAR LUMINOSITY DISTRIBUTIONS Karl Gebhardt and Douglas Richstone 1996AJ Department of Astronomy, Dennison Building, University of Michigan, Ann Arbor, Michigan 48109 Electronic mail: [email protected], [email protected] Edward A. Ajhar and Tod R. Lauer Kitt Peak National Observatory, National Optical Astronomy Observatories, P. O. Box 26732, Tucson, Arizona 85726 Yong-Ik Byun and John Kormendy Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, Hawaii 96822 Alan Dressler The Observatories of the Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena, California 91101 S. M. Faber and Carl Grillmair UCO/Lick Observatories, Board of Studies in Astronomy and Astrophysics, University of California, Santa Cruz, California 95064 Scott Tremaine Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, M5S 3H8, Canada Received 1995 November 3; revised 1996 March 8 ABSTRACT We have non-parametrically determined the luminosity density profiles and their logarithmic slopes for 42 early-type galaxies observed with HST. Assuming that the isodensity contours are spheroidal, then the luminosity density is uniquely determined from the surface brightness data through the Abel equation. For nearly all the galaxies in our sample, the logarithmic slope of the luminosity density (S = d log v/dlog r) measured at 0.1" (the innermost reliable measurement with the uncorrected HST) is significantly different from zero; i.e., most elliptical galaxies have cusps. There are only two galaxies for which an analytic core (S—>0) cannot be excluded. The distribution of logarithmic slopes at 0.1" appears to be bimodal, confirming the conclusion of Lauer et al [AJ, 110, 2622 (1995)] that early-type galaxies can be divided into two types based on their surface brightness profiles; i.e., those with cuspy cores and those whose steep power-law profiles continue essentially unchanged in to the resolution limit. The peaks in the slope distribution occur 3i S= —0.8 and -1.9. More than half of the galaxies have slopes steeper than -1.0. Taken together with the recent theoretical work of Merritt and Fridman, these results suggest that many (and maybe most) elliptical galaxies are either nearly axisymmetric or spherical near the center, or slowly evolve due to the influence of stochastic orbits. © 1996 American Astronomical Society. 1. INTRODUCTION pending on energy) of the orbits are irregular. This is in sharp contrast with separable triaxial potentials, in which all The mass density distribution for galaxies has important of the orbits are regular (reviewed in de Zeeuw & Franx consequences for the phase space structure of the galaxy 1991). In all of these potentials, the density is well behaved (Gerhard 1987; Hasan & Norman 1990; Lees & Schwarzs- (Taylor series expandable) near the center and, therefore, child 1992; Merritt & Fridman 1995, 1996). In particular, they can be described as having “analytic” cores. Merritt & Fridman (1996) studied the influence of a central Merritt and Fridman’s work strongly suggests that in gen- density cusp on the orbital structure by following orbits in a eral, if the density distribution has a nonzero logarithmic potential with a cusp and building self-consistent models slope near the center, the orbits that pass near the center (i.e., from the orbit library. They have shown that in a triaxial box orbits) will be chaotic. Therefore, if the density distribu- -2 potential with a density varying as p r , which they term tion is triaxial and has d log p/d log r 1/2, or steeper, a “strong cusp,” a fraction 80% of the orbits are irregular, then stochasticity is likely to play a critical role in the struc- that is, they conserve energy but have no other isolating in- ture and evolution of the central regions of the galaxy (Mer- tegrals. This behavior is caused by scattering off the steep ritt & Fridman 1995). The role of a central cusp is less im- inner potential spike. They also showed that if p r-1, portant in the axisymmetric case since orbits do not pass near which they term a “weak cusp,” a fraction 60%-80% (de- the center because they conserve one component of the an- 105 Astron. J. 112 (1), July 1996 0004-6256/96/112(l)/105/9/$6.00 © 1996 Am. Astron. Soc. 105 © American Astronomical Society • Provided by the NASA Astrophysics Data System 105G . 106 GEBHARDT ETAL: EARLY-TYPE GALAXIES. III. 106 112. guiar momentum. Merritt & Fridman’s work emphasizes the a discontinuous mapping from the surface brightness to the important role of high resolution surface brightness distribu- luminosity density near y=0. If the model has y exactly tions in understanding the dynamics of elliptical galaxies and = 0 and a = 2, then the deprojected model will have an ana- spiral bulges. lytic core. For any value of y greater than zero, even by an 1996AJ Lauer et al (1995, Paper I) have presented surface bright- arbitrarily small amount, deprojecting either Eqs. 3 or 4 will ness profiles for early-type galaxies observed with the uncor- lead to an asymptotic value of d log v{r)ld log r at least as rected HST. These can be deprojected to obtain the luminos- steep as —1. However, the radii at which this discontinuity ity density which we assume also represents the mass density occurs are smaller than the HST resolution for most of the profile. The existence of a central massive dark object galaxies in our sample, and this effect will not be as signifi- (MDO) will only increase the central mass concentration, so cant. This point is also illustrated in Figure 8 in Paper I. the luminosity density probably provides a lower limit for This argument suggests that estimates of the density and the mass concentration. Deprojection to obtain the mass den- its derivative at small radii may be suspect in galaxies with sity is formally simple in the case of spherical (or spheroidal) relatively flat inner surface brightness profiles. In view of the symmetry. If the galaxy has a luminosity density v(r), then importance of the density slope in determining dynamics and the surface brightness as a function of projected radius is evolution of the center of the galaxy, and because of the 3 potential sensitivity of some of our earlier conclusions to this v(r)rdr problem, we have recomputed the density and its derivative 1{R) (1) =/,R Vr2-/?2' at 0.1" using non-parametric methods in all of the galaxies reported in Paper I and some additional galaxies (a similar This equation is an Abel integral equation with solution analysis was done on a smaller sample of six galaxies by given by Merritt & Fridman 1995). Various methods to handle Eq. 2 1 C™dl dR have been discussed in the statistics literature and many of 2 2 (2) these are now being introduced to astronomy (Merritt & wir dR ^ -r ' Tremblay 1994, Gebhardt & Fischer 1995). In the present Cast in this way, the solution looks deceptively simple. In case, two obvious choices are a kernel estimator or a smooth- practice any noise in the data is amplified by the construction ing spline fit to the data. We choose the latter method and of v(r), and further amplified by a second differentiation to describe it in some detail below. construct S=d log v(r)/d\og r. In Paper I, we followed the We also rediscuss the question of whether there appear to time honored procedure of fitting a parametric model to the be two classes of objects in this sample, based on the loga- data before deprojecting. The model had the form rithmic slope of the central density. The existence of two a classes continues to appear probable. 2^~ Ib /(Ä) ~ (r/r^)^ 1 + ' (3) 2. DATA This parametric model was fitted to the observed data by a maximum-likelihood technique (Byun et al. 1996, Paper II). The data comprise the surface brightness profiles for the The parametric model can then be inverted numerically us- 45 galaxies in Paper I and 21 galaxies taken from the HST ing Eq. 2 to estimate the stellar density run near the center. archive (presented in Byun et al 1996). All of the measure- While this approach is reasonable and should yield good ments are from the uncorrected HST. The 21 additional gal- estimates of v if the parametric model fits the data well, there axies, originally observed as part of various GO and GTO is some cause for concern in its application. programs, were obtained from the HST archive when they It is easiest to understand this concern by contemplating became publicly available. All images were deconvolved us- Fig. 3 and Eq. 28 of Tremaine et al (1994) (see also Dehnen ing 80 iterations of the the Lucy-Richardson algorithm (Lucy 1993). That paper examined a variety of properties of 1974; Richardson 1972) and the same, high signal-to-noise “77-models,” with density profiles of the form composite PSFs used in Paper I. As in Paper I, we used the _V 1 PSF closest in time to the observation date for each galaxy. Pr,(r) 1 Measurement of surface brightness profiles was described in 477- r’-’a+r) ^' (4) detail in Paper I. The central density slope is not reported for The figure and Eq. 28 show that for 2<rj<?> the central those galaxies which have an AGN, inner dust obscuration, surface brightness is finite (and therefore or central flat-fielding defects. This was 35% of the total d log/(/?)/*/log Æ=0), despite the fact that the central den- initial sample, and we therefore have 42 remaining galaxies sity is infinite and the central value of d log v{r)ld log r var- which are used in the following analysis.