We describe a one-parameter family of models of stable spherical stellar
systems in which the phase-space distribution function depends only on energy.
The models have similar density profiles in their outer parts (rho propto
r-4 and central power-law density cusps, rho propto r3-eta,
0< eta le 3. The family contains the Jaffe (1983) and Hernquist (1990)
models as special cases. We evaluate the surface brightness profile, the
line-of-sight velocity dispersion profile, and the distribution function, and
discuss analogs of King's core-fitting formula for determining mass-to-light
ratio. We also generalize the models to a two-parameter family, in which the
galaxy contains a central black hole; the second parameter is the mass of the
black hole. Our models can be used to estimate the detectability of central
black holes and the velocity-dispersion profiles of galaxies that contain
central cusps, with or without a central black hole.
We have obtained V-band images of
45 nearby elliptical galaxies and bulges using
the original Planetary Camera of the Hubble Space Telescope.
The majority of the sample is at distances of 10 to 20 Mpc.
This represents a substantial increase in the number
of galaxies that have been studied at a resolution of a few parsecs.
At this resolution, many galaxies reveal previously
unknown central disks, dust clouds, and nuclear components.
We find that galaxies have two types of brightness profiles.
The first type consists of galaxies that have cores.
These galaxies have brightness profiles that ``break''
from steep outer power-laws to shallow inner cusps.
The Core class includes many galaxies that had cores apparently resolved
from the ground.
The second type consists of galaxies that have profiles that
continue into the resolution limit as steep power-laws,
showing no evidence of cores of any sort.
We thus find that all galaxies studied so far have singular brightness profiles
in the sense that I(r)\sim r^-gamma}as r->0.1''
with 0< gamma< 0.3 at the few parsec scale for galaxies with cores,
and gamma~1 for Power-law galaxies.
No galaxies in our sample have a central region
that is constant in surface brightness.
This implies that the stellar density in these systems
is still increasing steeply at the HST resolution limit.
Many galaxies reach stellar mass-densities of
~5X104MSun pc-3 at the resolution limit,
appearing similar in form to M32 at radii of a few parsecs.
The Core and Power-law profile classes correspond
to the Jaffe et al. (1994) Type I and II profiles;
however, we disagree with their suggestion that the presence
of a central stellar disk is closely related to, or
even determines, profile type.
Power-law galaxies are seen at all ellipticities, and the
majority of them show no evidence for central disks.
The discovery by Kormendy & Richstone of an M• ≃ 109 M&sun;
massive dark object (MDO) in NGC 3115 is confirmed with higher
resolution spectroscopy from the Canada-France-Hawaii Telescope (CFHT)
and the Hubble Space Telescope (HST). Measurements with the CFHT and
Subarcsecond Imaging Spectrograph improve the resolution from sigma * =
0."44 to sigma * = 0."244 ( sigma * = Gaussian dispersion radius of the
point-spread function). The apparent central velocity dispersion rises
from sigma = 295 +/- 9 km s-1 to sigma = 343 +/- 19 km s-1. The Faint
Object Spectrograph and COSTAR-corrected HST provide a further
improvement in resolution using a 0."21 aperture. Then, the measured
sigma = 443 +/- 18 km s-1 is remarkably high, and the wings of the
velocity profiles extend beyond 1200 km s-1 from the line centers.
Similarly, the apparent rotation curve rises much more rapidly than is
observed from the ground. Published dynamical models fit the new
observations reasonably well when "observed" at the improved spatial
resolution; V and sigma are at the high end of the predicted range near
the center. Therefore, M• > 109 M&sun;. The spatial resolution has
now improved by a factor of ~5 since the discovery observations, and the
case for a central MDO has strengthened correspondingly. With HST and
the Second Wide Field and Planetary Camera, NGC 3115 also shows a bright
nucleus. This is very prominent and distinct from the bulge when the
superposed nuclear disk is subtracted. After bulge subtraction, the
nucleus has sigma = 600 +/- 37 km s-1, the largest central dispersion
seen in any galaxy. If the nucleus contained only old stars and not an
MDO, its escape velocity would be ~352 km s-1, much smaller than the
observed velocities of the stars. This is independent proof that an MDO
is present. The new observations put more stringent constraints on the
radius inside which the dark mass lies and strengthen the case that it
is a 2 x 109 M&sun; black hole.
We present a set of structural parameters for the central parts of 57
early-type galaxies observed with the Planetary Camera of the Hubble Space
Telescope. These parameters are based on a new empirical law that successfully
characterizes the centers of early type galaxies. This empirical law assumes
that the surface brightness profile is a combination of two power laws with
different slopes gamma and beta for the inner and outer regions. Conventional
structural parameters such as core radius and central surface brightness are
replaced by break radius r_b, where the transition between power-law slopes
takes place, and surface brightness mu_b at that radius. An additional
parameter alpha describes the sharpness of the break. The structural parameters
are derived using a chi-squared minimization process applied to the mean
surface brightness profiles. The resulting model profiles generally give very
good agreement to the observed profiles out to the radius of 10 arcseconds
imaged by the Planetary Camera. Exceptions include galaxies which depart from
pure power-laws at large radius, those with strong nuclear components, and
galaxies partly obscured by dust. The uncertainties in the derived parameters
are estimated using Monte-Carlo simulations which test the stability of
solutions in the face of photon noise and the effects of the deconvolution
process. The covariance of the structural parameters is examined by computing
contours of constant chi squared in multi-dimensional parameter space.
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 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
cannot be excluded. The distribution of logarithmic slopes at 0.1" appears to
be bimodal, confirming the conclusion of Lauer et al. (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 at -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 & 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.
Hubble Space Telescope WFPC2 images show that the low-luminosity
elliptical galaxy companion to M87, NGC 4486B, has a double nucleus that
resembles the one discovered in M31. The NGC 4486B nucleus comprises two
peaks separated by ~0."15 or 12 pc. Neither peak is coincident with the
galaxy photocenter, which falls between them. The nuclear morphology is
independent of color; thus, the double structure is not likely to arise
from dust absorption. It is also unlikely that the peaks are a binary
stellar system (such as an ongoing merger of the nucleus of a less
luminous system with the nucleus of NGC 4486B), since the decay
timescale is short (<108 yr) and the present environment of NGC 4486B
should inhibit mergers. We suggest that the nuclear morphology of NGC
4486B may be explained by the eccentric-disk model of Tremaine, which
was originally advanced to account for the central structure of M31.
This model requires that NGC 4486B contains a central massive dark
object, which is suggested by the spectroscopic observations of Kormendy
et al. The eccentric disk might be related to the symmetric disk seen at
larger radii.
The discovery by Kormendy of a M• ~= 109 Msolar massive dark object
(MDO) in NGC 4594 is confirmed with higher resolution spectroscopy from
the Canada-France-Hawaii Telescope (CFHT) and the Hubble Space Telescope
(HST). CFHT measurements with the Subarcsecond Imaging Spectrograph
improve the resolution from sigma * = 0."40 to 0."27 Gaussian dispersion
radius of the point-spread function (PSF). The apparent central velocity
dispersion rises from sigma = 250 +/- 7 km s-1 to sigma = 286 +/- 7 km
s-1. As observed with the COSTAR-corrected HST, the Faint Object
Spectrograph, and a 0."21 aperture, sigma = 321 +/- 7 km s-1 is still
higher, and the central rotation curve is very steep. The
highest-M• published dynamical model fits the new observations
reasonably well when "observed" at HST resolution. The spatial
resolution has now improved by a factor of ~5 since the discovery
measurements, and the case for a black hole (BH) has strengthened
correspondingly. We confirm that NGC 4594 has a Seyfert spectrum; H
alpha is ~5200 km s-1 wide at zero intensity. However, gas velocities
are lower than the circular velocities implied by the stars, so they
cannot be used to test the BH case in NGC 4594. The gas may be in a
ring, or it may be associated with patchy dust. HST images with the Wide
Field and Planetary Camera 2 show dust at some aperture positions. NGC
4594 appears to have a bright point nucleus. However, the central
absorption-line strengths are low, consistent with dilution by enough
nonthermal light to explain the "nucleus." There is no evidence for a
distinct nuclear star cluster. NGC 4594 is similar to M87, which also
has a nonthermal nuclear source, and not to M31 and NGC 3115, which have
quiescent BHs and nuclear star clusters.
The stellar kinematics of the dwarf elliptical galaxy NGC 4486B have been
measured in seeing sigma_* =0.22 arcsec with the Canada-France-Hawaii
Telescope. Lauer et al. 1996, ApJ, 471, L79 have shown that NGC 4486B is
similar to M31 in having a double nucleus. We show that it also resembles M31
in its kinematics. The velocity dispersion gradient is very steep: sigma
increases from 116 +- 6 km/s at r = 2" - 6" to 281 +- 11 km/s at the center.
This is much higher than expected for an elliptical galaxy of absolute
magnitude M_B = -16.8: NGC 4486B is far above the scatter in the Faber-Jackson
correlation between sigma and bulge luminosity. Therefore the King core
mass-to-light ratio, M/L_V = 20, is unusually high compared with normal values
for old stellar populations. We construct dynamical models with isotropic
velocity dispersions and show that they reproduce black hole (BH) masses
derived by more detailed methods. We also fit axisymmetric, three-integral
models. Isotropic models imply that NGC 4486B contains a central dark object,
probably a BH, of mass M_BH = 6(+3 -2) x 10^8 M_sun. However, anisotropic
models fit the data without a BH if the ratio of radial to azimuthal
dispersions is ~ 2 at 1". Therefore this is a less strong BH detection than the
ones in M31, M32, and NGC 3115. A 6 x 10^8 M_sun BH is 9% of the mass M_bulge
in stars; even if M_BH is smaller than the isotropic value, M_BH/M_bulge is
likely to be unusually large. Double nuclei are a puzzle because the dynamical
friction timescales for self-gravitating star clusters in orbit around each
other are short. Since both M31 and NGC 4486B contain central dark objects, our
results support models in which the survival of double nuclei is connected with
the presence of a BH (e. g., Tremaine 1995, AJ, 110, 628).
We analyze Hubble Space Telescope surface-brightness profiles of 61
elliptical galaxies and spiral bulges (hereafter "hot" galaxies). The
profiles are parameterized by break radius rb
and break surface brightness
Ib. These are combined with central velocity dispersions, total
luminosities, rotation velocities, and isophote shapes to explore correlations
among central and global properties. Luminous hot galaxies (MV < -22)
have cuspy cores with steep outer power-law profiles that
break at r \approx rb to shallow inner profiles I \propto r^-gamma
with gamma \le 0.3.
Break radii and core luminosities for these objects are approximately
proportional to effective radii and total
luminosities. Scaling relations are presented for
several core parameters as a function of total luminosity.
Cores follow a fundamental plane that parallels the global fundamental
plane for hot galaxies but is 30% thicker. Some of this extra thickness may
be due to the effect of massive black holes (BHs) on central velocity
dispersions. Faint hot galaxies (MV > -20.5) show steep, largely
featureless power-law profiles that lack cores. Measured values of
rb and Ib for these galaxies are limits only.
At a limiting radius of 10 pc, the centers of power-law
galaxies are up to 1000 times denser in mass and luminosity than the
cores of large galaxies. At intermediate
magnitudes -22 < MV < -20.5,
core and power-law galaxies coexist, and
there is a range in rb
at a given luminosity of at least two orders of
magnitude. Here, central properties correlate strongly with global rotation andshape: core galaxies tend to be boxy and slowly rotating, whereas power-law
galaxies tend to be disky and rapidly rotating.
A search for inner disks was conducted to test a
claim in the literature, based on a smaller sample, that power laws originate
from edge-on stellar disks. We find only limited evidence for such disks and
believe that the difference between core and power-law profiles
reflects a
real difference in the spatial distribution of the luminous spheroidal
component of the galaxy. The dense power-law centers of disky, rotating
galaxies are consistent with their formation in gas-rich mergers. The
parallel proposition, that cores are the by-products of gas-free stellar mergers, is
less compelling for at least two reasons: (1) dissipationless hierarchical
clustering does not appear to produce core profiles like those seen; (2) core
galaxies accrete small, dense, gas-free galaxies at a rate sufficient to fill intheir low-density cores if the satellites survived and sank to the center (whether
the satellites survive is still an open question). An alternative
model for core formation involves
the orbital decay of massive black holes (BHs) that are
accreted in mergers: the decaying BHs may heat and
eject stars from the center, eroding a power law if any exists and
scouring out a core. An average BH mass per spheroid of 0.002 times the
stellar mass yields cores in fair agreement with
observed cores and is consistent with the energetics of AGNs and
the kinematic detection of BHs in nearby galaxies. An unresolved issue is why
power-law galaxies also do not have cores if this process operates in all hot
galaxies.
We construct dynamical models for a sample of 36 nearby galaxies with
Hubble Space Telescope photometry and ground-based kinematics. The
models assume that each galaxy is axisymmetric, with a two-integral
distribution function, arbitrary inclination angle, a
position-independent stellar mass-to-light ratio Upsilon, and a
central massive dark object (MDO) of arbitrary mass MBH. They
provide acceptable fits to 32 of the galaxies for some value of
MBH and Upsilon; the four galaxies that cannot be fit have
kinematically decoupled cores. The mass-to-light ratios inferred for
the 32 well-fit galaxies are consistent with the fundamental plane
correlation Upsilon propto L0.2, where L is galaxy luminosity.
In all but six galaxies the models require at the 95% confidence
level an MDO of mass MBH sim 0.006 M
bulge equiv 0.006 Upsilon L. Five of the six galaxies consistent
with MBH=0 are also consistent with this correlation. The
other (NGC 7332) has a much stronger upper limit on MBH. We
predict the second-moment profiles that should be observed at HST
resolution for the 32 galaxies that our models describe well.
We consider various parameterizations for the probability distribution
describing the correlation of the masses of these MDOs with other
galaxy properties. One of the best models can be summarized thus: a
fraction f ~=0.97 of early-type galaxies have MDOs, whose masses
are well described by a Gaussian distribution in log(MBH/M
bulge) of mean -2.28 and standard deviation
~0.51. There is also marginal evidence that MBH is
distributed differently for ``core'' and ``power-law'' galaxies, with
core galaxies having a somewhat steeper dependence on Mbulge.
Black holes, an extreme consequence of the mathematics of General
Relativity, have long been suspected of being the prime movers of
quasars, which emit more energy than any other objects in the
Universe. Recent evidence indicates that supermassive black holes,
which are probably quasar remnants, reside at the centers of most
galaxies. As our knowledge of the demographics of these relics of a
violent earlier Universe improve, we see tantalizing clues that they
participated intimately in the formation of galaxies and have strongly
influenced their present-day structure.
We fit axisymmetric 3-integral dynamical models to NGC3379 using the
line-of-sight velocity distribution obtained from HST/FOS spectra of the galaxy
center and ground-based long-slit spectroscopy along four position angles, with
the light distribution constrained by WFPC2 and ground-based images. We have
fitted models with inclinations from 29 (intrinsic galaxy type E5) to 90
degrees (intrinsic E1) and black hole masses from 0 to 1e9 M_solar. The
best-fit black hole masses range from 6e7 to 2e8 M_solar, depending on
inclination. The velocity ellipsoid of the best model is not consistent with
either isotropy or a two-integral distribution function. Along the major axis,
the velocity ellipsoid becomes tangential at the innermost bin, radial in the
mid-range radii, and tangential again at the outermost bins. For the acceptable
models, the radial to tangential dispersion in the mid-range radii ranges from
1.1 < sigma_r / sigma_t < 1.7. Compared with these 3-integral models,
2-integral isotropic models overestimate the black hole mass since they cannot
provide adequate radial motion. However, the models presented in this paper
still contain restrictive assumptions-namely assumptions of constant M/L and
spheroidal symmetry-requiring yet more models to study black hole properties in
complete generality.
We describe a correlation between the mass M_BH of a galaxy's central black
hole and the luminosity-weighted line-of-sight velocity dispersion sigma_e
within the half-light radius. The result is based on a sample of 26 galaxies,
including 13 galaxies with new determinations of black hole masses from Hubble
Space Telescope measurements of stellar kinematics. The best-fit correlation is
M_BH = 1.2 (+-0.2) x 10^8 M_sun (sigma_e/200 km/s)^(3.75 (+-0.3))over almost
three orders of magnitude in M_BH; the scatter in M_BH at fixed sigma_e is only
0.30 dex and most of this is due to observational errors. The M_BH-sigma_e
relation is of interest not only for its strong predictive power but also
because it implies that central black hole mass is constrained by and closely
related to properties of the host galaxy's bulge.
Black hole (BH) masses that have been measured by reverberation
mapping in active galaxies fall significantly below the correlation
between bulge luminosity and BH mass determined from spatially
resolved kinematics of nearby normal galaxies. This discrepency has
created concern that one or both techniques suffer from systematic
errors. We show that BH masses from reverberation mapping are
consistent with the recently discovered relationship between BH mass
and galaxy velocity dispersion. Therefore systematic errors in
estimating bulge luminosities, not problems with either kind of mass
measurement, are the probable source of the above disagreement. This
result underscores the utility of the BH mass -- velocity dispersion
relationship. Reverberation mapping can now be applied with increased
confidence to galaxies whose active nuclei are too bright or whose
distances are too large for BH searches based on spatially resolved
kinematics.
Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object M
and the velocity dispersion sigma of the host galaxy, of the form log(M/M_sun) = a +
b*log(sigma/sigma_0); however, published estimates of the slope b span a wide range (3.75 to 5.3). Merritt
& Ferrarese have argued that low slopes (<4) arise because of neglect of random measurement errors in
the dispersions and an incorrect choice for the dispersion of the Milky Way Galaxy. We show that these
explanations account for at most a small part of the slope range. Instead, the range of slopes arises mostly
because of systematic differences in the velocity dispersions used by different groups for the same galaxies.
The origin of these differences remains unclear, but we suggest that one significant component of the
difference results from Ferrarese & Merritt's extrapolation of central velocity dispersions to r_e/8 (r_e is
the effective radius) using an empirical formula. Another component may arise from
dispersion-dependent systematic errors in the measurements. A new determination of the slope using 31
galaxies yields b=4.02 +/- 0.32, a=8.13 +/- 0.06, for sigma_0=200 km/s. The M-sigma relation has an
intrinsic dispersion in log M that is no larger than 0.3 dex. In an Appendix, we present a simple model for
the velocity-dispersion profile of the Galactic bulge.
We used Hubble Space Telescope WFPC2 images to identify six
early-type galaxies with surface-brightness profiles that decrease
inward over a limited range of radii near their centers.
The inferred luminosity density profiles of these galaxies have local minima
interior to their core break radii. NGC 3706 harbors a high surface
brightness ring of starlight with radius ~20 pc.
Its central structure may be related to that in the double-nucleus
galaxies M31 and NGC 4486B.
NGC 4406 and NGC 6876 have nearly flat cores that on close
inspection are centrally depressed. Colors for both
galaxies imply that this is not due to dust absorption.
The surface brightness distributions of both galaxies are consistent with
stellar tori that are more diffuse than the sharply defined system in NGC 3706.
The remaining three galaxies are the brightest cluster galaxies in A260, A347,
and A3574. Color information is not available for these objects,
but they strongly resemble NGC 4406 and NGC 6876 in their cores.
The thin ring in NGC 3706 may have formed dissipatively.
The five other galaxies resemble the endpoints of
some simulations of the merging of two gas-free stellar systems,
each harboring a massive nuclear black hole.
In one version of this scenario, diffuse stellar tori are produced
when stars initially
bound to one black hole are tidally stripped away by the second black hole.
Alternatively, some inward-decreasing surface-brightness profiles may
reflect the ejection of stars from a core during
the hardening of the binary black hole created during the merger.
We present axisymmetric, orbit superposition models for 12 galaxies using data
taken with the Hubble Space Telescope (HST) and ground-based
observatories. In each galaxy, we detect a central black hole (BH) and measure
its mass to accuracies ranging from 10% to 70%. We demonstrate that in most
cases the BH detection requires both the< i>HST and ground-based
data. Using the ground-based data alone does provide an unbiased measure of the
BH mass (provided that they are fitted with fully general models), but at a
greatly reduced significance. The
most significant correlation with host galaxy properties is the relation between
the BH mass and the velocity dispersion of the host galaxy; we find no other
equally strong correlation and no second parameter that improves the quality of
the mass-dispersion relation. We are also able to measure the stellar orbital
properties from these general models. The most massive galaxies are strongly
biased to tangential orbits near the BH, consistent with binary BH models, while
lower mass galaxies have a range of anisotropies, consistent with an adiabatic
growth of the BH.
We present stellar kinematics for a sample of 10 early-type galaxies observed
using the STIS aboard the Hubble Space Telescope, and the Modular Spectrograph
on the MDM Observatory 2.4-m telescope. The spectra are used to derive
line-of-sight velocity distributions (LOSVDs) of the stars using a Maximum
Penalized Likelihood method. We use Gauss-Hermite polynomials to parameterize
the LOSVDs and find predominantly negative h4 values (boxy distributions) in
the central regions of our galaxies. One galaxy, NGC 4697, has significantly
positive central h4 (high tail weight). The majority of galaxies have a central
velocity dispersion excess in the STIS kinematics over ground-based velocity
dispersions. The galaxies with the strongest rotational support, as quantified
with v_MAX/sigma_STIS, have the smallest dispersion excess at STIS resolution.
The best-fitting, general, axisymmetric dynamical models (described in a
companion paper) require black holes in all cases, with masses ranging from
10^6.5 to 10^9.3 Msun. We replot these updated masses on the BH/sigma relation,
and show that the fit to only these 10 galaxies has a slope consistent with the
fits to larger samples. The greatest outlier is NGC 2778, a dwarf elliptical
with relatively poorly constrained black hole mass. The two best candidates for
pseudobulges, NGC 3384 and 7457, do not deviate significantly from the
established relation between black hole and sigma. Neither do the three
galaxies which show the most evidence of a recent merger, NGC 3608, 4473, and
4697.
We present observations of 77 early-type galaxies imaged with the PC1 CCD
of HST+WFPC2. ``Nuker law'' parametric fits to the surface brightness
profiles are used to classify the central structure into ``core'' or
``power-law'' forms. Core galaxies are typically rounder than power-law
galaxies. Nearly all power-laws with central ellipticity $\epsilon\geq0.3$
have stellar disks, implying that disks are present in most power-laws with
$\epsilon<0.3,$ but are not visible due to unfavorable geometry. A few
low-luminosity flattened core galaxies also have disks; these may be transition
forms from power-laws to more luminous core galaxies, which lack disks. Several
core galaxies have strong isophote twists interior to their break radii,
although power-laws have interior twists of similar physical
significance when the photometric perturbations implied by the twists are
evaluated. Central color gradients are typically
consistent with the envelope gradients; core galaxies have somewhat weaker
color gradients than power-laws. Nuclei are found in 29% of the cores
and 60% of the power-laws. Nuclei are typically bluer than
the surrounding galaxy. While some nuclei are associated with AGN, just as many
are not; conversely, not all galaxies known to have low-level AGN exhibit
detectable nuclei in the broad-band filters. NGC 4073 and 4382,
are found to have central minima in their intrinsic starlight distributions;
NGC 4382 resembles the double nucleus of M31. In general, the peak
brightness location is coincident with the photocenter
of the core to a typical physical scale $<1$ pc. Five
galaxies, however, have centers significantly displaced from their surrounding
cores; these may be unresolved asymmetric double nuclei. Lastly, as noted by
previous authors, central dust is visible in about half of the galaxies. The
presence and strength of dust correlates with nuclear emission, thus
dust may outline gas that is falling into the central black hole. The
prevalence of dust and its morphology suggest that dust clouds form,
settle to the center, and disappear repeatedly on ~10^8 yr timescales. We
discuss the hypothesis that cores are created by the decay of a massive black
hole binary formed in a merger. Apart from their brightness profiles,
there are no strong differences between cores and power-laws that
demand this scenario; however, the rounder shapes of cores, their lack of disks,
and their reduced color gradients may be consistent with it.
Black hole masses predicted from the $M_\bullet-\sigma$ relationship conflict with those predicted from the $M_\bullet-L$ relationship for the most luminous galaxies, such as brightest cluster galaxies (BCGs). This is because stellar
velocity dispersion, $\sigma,$ increases only weakly with luminosity for BCGs and other giant ellipticals. The $M_\bullet-L$ relationship predicts that the most luminous BCGs may harbor black holes with $M_\bullet$ approaching
$10^{10}M_\odot,$ while the $M_\bullet-\sigma$ relationship essentially always predicts $M_\bullet<3\times10^9M_\odot.$ Lacking direct determination of $M_\bullet$ in a sample of the most luminous galaxies, we advance arguments that
the $M_\bullet-L$ relationship is a plausible or even preferred description for BCGs and other galaxies of similar luminosity.
Under the hypothesis that cores in central stellar density are formed by binary black holes, the inner-core cusp radius, $r_\gamma,$ may be an independent witness of $M_\bullet.$ Using central structural parameters derived from a large sample of early-type galaxies observed by {\it HST}, we argue that $L$ is superior to $\sigma$ as an indicator of $r_\gamma$ in luminous galaxies.
Further, the observed $r_\gamma-M_\bullet$ relationship for 11 core galaxies with measured $M_\bullet$ appears to be consistent with the $M_\bullet-L$ relationship for BCGs.
BCGs have large cores appropriate for their large luminosities that may be difficult to generate with the more modest black hole masses inferred from the $M_\bullet-\sigma$ relationship.
$M_\bullet\sim M$ may be expected to hold for BCGs, if they were formed in dissipationless mergers, which should preserve ratio of black hole to stellar mass, $M.$ This picture appears to be consistent with the slow increase in $\sigma$ with $L$ and the more rapid increase in effective radii, $R_e,$ with $L$ seen in BCGs as compared to less luminous galaxies.
If BCGs have large BHs commensurate with their high luminosities, then the local black hole mass function for $M_\bullet>3\times10^9M_\odot$ may be nearly an order of magnitude richer than what would be inferred from the $M_\bullet-\sigma$ relationship. The volume density of the most luminous QSOs at earlier epochs may favor the predictions from the $M_\bullet-L$ relationship.
We combine several HST investigations on the central structure of early-type galaxies to generate a large sample of surface photometry. The studies selected were those that used the "Nuker law" to characterize the inner light distributions of the galaxies. The sample comprises WFPC1 and WFPC2 V-band observations published earlier by our group, R-band WFPC2 photometry of Rest et al., NICMOS H-band photometry by Ravindranath et al. and Quillen et al., and the brightest cluster galaxy WFPC2 I-band photometry of Laine et al.
The distribution of the logarithmic slopes of the central brightness profiles strongly affirms that the central structure of elliptical galaxies with MV < -19 is bimodal, based on both parametric and nonparametric analysis. At the HST resolution limit, most galaxies are either power-law systems, which have steep cusps in surface brightness, or core systems, which have shallow cusps interior to a steeper envelope brightness distribution.
A rapid transition between the two forms occurs over the luminosity range -22 < MV < -20, with cores dominating at the highest luminosities and power laws at the lowest. There are a few "intermediate" systems that have both cusp slopes and total luminosities that fall within the core/power-law transition, but they are rare and do not fill in the overall bimodal distribution.
The largest galaxies, and in particular central galaxies in clusters, offer
unique insight into understanding the mechanism for the growth of nuclear black
holes. We present Hubble Space Telescope kinematics for NGC 1399, the central
galaxy in Fornax. We find the best-fit model contains a black hole of
(5.1±0.7)× 108MSun,
(at a distance of 21.1 Mpc), a factor of over 2 below the correlation of black
hole mass and velocity dispersion. We also find a dramatic signature for
central tangential anisotropy. The velocity profiles on adjacent sides 0.5'' away
from the nucleus show strong bimodality, and the central spectrum shows a
large drop in the dispersion. Both of these observations point to an orbital
distribution that is tangentially biased. The best-fit orbital model suggests
a ratio of the tangential to radial internal velocity dispersions of 3. This
ratio is the largest seen in any galaxy to date and will provide an important
measure for the mode by which the central black hole has grown.
We determine the mass of the black hole at the center of the spiral
galaxy NGC 4258 by constructing axisymmetric dynamical models of the
galaxy. These models are constrained by high spatial resolution
imaging and long-slit spectroscopy of the nuclear region obtained
with the Hubble Space Telescope, complemented by ground-based
observations extending to larger radii. Our best mass estimate is
M&bull = (3.3±0.2) × 107MSun for a distance of 7.28 Mpc
(statistical errors only). This is within 15% of
(3.82±0.01)× 107MSun,
the mass determined from the kinematics of water
masers (rescaled to the same distance) assuming they are in Keplerian
rotation in a warped disk. Unfortunately, the construction of
accurate dynamical models of NGC 4258 is compromised by an
unresolved active nucleus and color gradients, the latter caused by
variations in the stellar population and/or obscuring
dust. Depending on how these effects are treated, as well as on
assumptions about the ellipticity and inclination of the galaxy, we
obtain black hole masses ranging from 2.4 × 107MSun to
3.6×107MSun. This spread is mainly due to uncertainties in
the stellar mass profile inside the central 2'' (~70 pc).
Obscuration of high-velocity stars by circumnuclear dust (possibly
associated with the masing disk) could lead to an underestimate of
the black hole mass which is hard to correct. These problems are not
present in the ~30 other black hole mass determinations from
stellar dynamics that have been published by us and other groups;
thus, the relatively close agreement between the stellar dynamical
mass and the maser mass in NGC 4258 enhances our confidence in the
black hole masses determined in other galaxies from stellar dynamics
using similar methods and data of comparable quality.
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