# Scott Tremaine - Institute for Advanced Study

## Contact Details

NameScott Tremaine |
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AffiliationInstitute for Advanced Study |
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CityMinneapolis |
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CountryUnited States |
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## Pubs By Year |
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## Pub CategoriesAstrophysics of Galaxies (20) Earth and Planetary Astrophysics (13) Astrophysics (13) Cosmology and Nongalactic Astrophysics (11) High Energy Astrophysical Phenomena (3) High Energy Physics - Phenomenology (3) Physics - Statistical Mechanics (2) High Energy Physics - Theory (2) Instrumentation and Methods for Astrophysics (2) High Energy Physics - Experiment (1) Solar and Stellar Astrophysics (1) Physics - Space Physics (1) Physics - Plasma Physics (1) Physics - Classical Physics (1) Physics - Physics Education (1) Mathematics - Numerical Analysis (1) |

## Publications Authored By Scott Tremaine

Collisional fragmentation is shown to not be a barrier to rocky planet formation at small distances from the host star. Simple analytic arguments demonstrate that rocky planet formation via collisions of homogeneous gravity-dominated bodies is possible down to distances of order the Roche radius ($r_\mathrm{Roche}$). Extensive N-body simulations that include plausible models for fragmentation and merging of gravity-dominated bodies confirm this conclusion and demonstrate that rocky planet formation is possible down to ${\sim}$1. Read More

We examine dense self-gravitating stellar systems dominated by a central potential, such as nuclear star clusters hosting a central supermassive black hole. Different dynamical properties of these systems evolve on vastly different timescales. In particular, the orbital-plane orientations are typically driven into internal thermodynamic equilibrium by vector resonant relaxation before the orbital eccentricities or semimajor axes relax. Read More

**Affiliations:**

^{1}MPIA,

^{2}IAS,

^{3}MPIA

**Category:**Astrophysics of Galaxies

We show that the stellar surface-brightness profiles in disc galaxies---observed to be approximately exponential---can be explained if radial migration efficiently scrambles the individual stars' angular momenta while conserving the circularity of the orbits and the total mass and angular momentum. In this case the disc's distribution of specific angular momenta $j$ should be near a maximum-entropy state and therefore approximately exponential, $dN\propto\exp(-j/\langle j\rangle)dj$. This distribution translates to a surface-density profile that is generally not an exponential function of radius: $\Sigma(R)\propto\exp[-R/R_e(R)]/(RR_e(R))(1+d\log v_c(R)/d\log R)$, for a rotation curve $v_c(R)$ and $R_e(R)\equiv\langle j\rangle/v_c(R)$. Read More

An intriguing alternative to cold dark matter (CDM) is that the dark matter is a light ( $m \sim 10^{-22}$ eV) boson having a de Broglie wavelength $\lambda \sim 1$ kpc, often called fuzzy dark matter (FDM). We describe the arguments from particle physics that motivate FDM, review previous work on its astrophysical signatures, and analyze several unexplored aspects of its behavior. In particular, (i) FDM halos smaller than about $10^7 (m/10^{-22} {\rm eV})^{-3/2} M_\odot$ do not form. Read More

We show that a black-hole binary with an external companion can undergo Lidov-Kozai cycles that cause a close pericenter passage, leading to a rapid merger due to gravitational-wave emission. This scenario occurs most often for systems in which the companion has mass comparable to the reduced mass of the binary and the companion orbit has semi-major axis within a factor of $\sim 10$ of the binary semi-major axis. Using a simple population-synthesis model and 3-body simulations, we estimate the rate of mergers in triple black hole systems in the field to be about six per Gpc$^3$ per year in the absence of natal kicks during black hole formation. Read More

We run simulations to determine the expected distribution of orbital elements of nearly isotropic comets (NICs) in the outer solar system, assuming that these comets originate in the Oort Cloud at thousands of AU and are perturbed into the planetary region by the Galactic tide. We show that the Large Synoptic Survey Telescope (LSST) should detect and characterize the orbits of hundreds to thousands of NICs with perihelion distance outside 5 AU. Observing NICs in the outer solar system is our only way of directly detecting comets from the inner Oort Cloud, as these comets are dynamically excluded from the inner solar system by the giant planets. Read More

Most warm Jupiters (gas-giant planets with $0.1~{\rm AU}\lesssim a \lesssim1$ AU) have pericenter distances that are too large for significant orbital migration by tidal friction. We study the possibility that the warm Jupiters are undergoing secular eccentricity oscillations excited by an outer companion (a planet or star) in an eccentric and/or mutually inclined orbit. Read More

We have examined the effect of slow growth of a central black hole on spherical galaxies that obey S\'ersic or $R^{1/m}$ surface-brightness profiles. During such growth the actions of each stellar orbit are conserved, which allows us to compute the final distribution function if we assume that the initial distribution function is isotropic. We find that black-hole growth leads to a central cusp or ``excess light', in which the surface brightness varies with radius as $R^{-1. Read More

The final "giant-impact" phase of terrestrial planet formation is believed to begin with a large number of planetary "embryos" on nearly circular, coplanar orbits. Mutual gravitational interactions gradually excite their eccentricities until their orbits cross and they collide and merge; through this process the number of surviving bodies declines until the system contains a small number of planets on well-separated, stable orbits. In this paper we explore a simple statistical model for the orbit distribution of planets formed by this process, based on the sheared-sheet approximation and the ansatz that the planets explore uniformly all of the stable region of phase space. Read More

**Affiliations:**

^{1}Institute for Advanced Study

**Category:**Astrophysics of Galaxies

This article reviews Michel Henon's contributions to a diverse set of problems in astrophysical dynamics, including violent relaxation, Saturn's rings, roundoff error in orbit integrations, and planet formation. Read More

Numerous telescopes and techniques have been used to find and study extrasolar planets, but none has been more successful than NASA's Kepler Space Telescope. Kepler has discovered the majority of known exoplanets, the smallest planets to orbit normal stars, and the worlds most likely to be similar to our home planet. Most importantly, Kepler has provided our first look at typical characteristics of planets and planetary systems for planets with sizes as small as and orbits as large as those of the Earth. Read More

Quasars emit more energy than any other objects in the universe, yet are not much bigger than the solar system. We are almost certain that quasars are powered by giant black holes of up to $10^{10}$ times the mass of the Sun, and that black holes of between $10^6$ and $10^{10}$ solar masses---dead quasars---are present at the centers of most galaxies. Our own galaxy contains a black hole of $4. Read More

**Affiliations:**

^{1}Institute for Advanced Study, Princeton,

^{2}Carnegie Observatories,

^{3}University of California, Los Angeles,

^{4}Harvard-Smithsonian Center for Astrophysics

The broad emission lines commonly seen in quasar spectra have velocity widths of a few per cent of the speed of light, so special- and general-relativistic effects have a significant influence on the line profile. We have determined the redshift of the broad H-beta line in the quasar rest frame (determined from the core component of the [OIII] line) for over 20,000 quasars from the Sloan Digital Sky Survey DR7 quasar catalog. The mean redshift as a function of line width is approximately consistent with the relativistic redshift that is expected if the line originates in a randomly oriented Keplerian disk that is obscured when the inclination of the disk to the line of sight exceeds ~30-45 degrees, consistent with simple AGN unification schemes. Read More

Stars bound to a supermassive black hole interact gravitationally. Persistent torques acting between stellar orbits lead to the rapid resonant relaxation of the orbital orientation vectors ("vector" resonant relaxation) and slower relaxation of the eccentricities ("scalar" resonant relaxation), both at rates much faster than two-body or non-resonant relaxation. We describe a new parallel symplectic integrator, N-ring, which follows the dynamical evolution of a cluster of N stars through vector resonant relaxation, by averaging the pairwise interactions over the orbital period and periapsis-precession timescale. Read More

We describe the dynamics and thermodynamics of collisionless particle disks orbiting a massive central body, in the case where the disk mass is small compared to the central mass, the self-gravity of the disk dominates the non-Keplerian force, and the spread in semi-major axes is small. We show that with plausible approximations such disks have logarithmic two-body interactions and a compact phase space, and therefore exhibit thermodynamics that are simpler than most other gravitating systems, which require a confining box and artificial softening of the potential at small scales to be thermodynamically well-behaved. We solve for the microcanonical axisymmetric thermal equilibria and demonstrate the existence of a symmetry-breaking bifurcation into lopsided equilibria. Read More

Many exoplanets in close-in orbits are observed to have relatively high eccentricities and large stellar obliquities. We explore the possibility that these result from planet-planet scattering by studying the dynamical outcomes from a large number of orbit integrations in systems with two and three gas-giant planets in close-in orbits (0.05 AU < a < 0. Read More

**Affiliations:**

^{1}UCLA,

^{2}Carnegie,

^{3}ANU,

^{4}Harvard,

^{5}IAS

**Category:**Cosmology and Nongalactic Astrophysics

A small fraction of quasars have long been known to show bulk velocity offsets in the broad Balmer lines with respect to the systemic redshift of the host galaxy. Models to explain these offsets usually invoke broad-line region gas kinematics/asymmetry around single black holes (BHs), orbital motion of massive (~sub-pc) binary black holes (BBHs), or recoil BHs, but single-epoch spectra are unable to distinguish between these scenarios. The line-of-sight (LOS) radial velocity (RV) shifts from long-term spectroscopic monitoring can be used to test the BBH hypothesis. Read More

We determine the mass of the nuclear black hole ($M$) in NGC 3706, an early type galaxy with a central surface brightness minimum arising from an apparent stellar ring, which is misaligned with respect to the galaxy's major axis at larger radii. We fit new HST/STIS and archival data with axisymmetric orbit models to determine $M$, mass-to-light ratio ($\Upsilon_V$), and dark matter halo profile. The best-fit model parameters with 1$\sigma$ uncertainties are $M = (6. Read More

Satellites in low Earth orbits must accurately conserve their orbital eccentricity, since a decrease in perigee of only 5-10% would cause them to crash. However, these satellites are subject to gravitational perturbations from the Earth's multipole moments, the Moon, and the Sun that are not spherically symmetric and hence do not conserve angular momentum, especially over the tens of thousands of orbits made by a typical satellite. Why then do satellites not crash? We describe a vector-based analysis of the long-term behavior of satellite orbits and apply this to several toy systems containing a single non-Keplerian perturbing potential. Read More

Accretion discs are present around both stellar-mass black holes in X-ray binaries and supermassive black holes in active galactic nuclei. A wide variety of circumstantial evidence implies that many of these discs are warped. The standard Bardeen--Petterson model attributes the shape of the warp to the competition between Lense--Thirring torque from the central black hole and viscous angular-momentum transport within the disc. Read More

We revisit the hypothesis that dense galactic nuclei are formed from inspiraling globular clusters. Recent advances in understanding of the continuous formation of globular clusters over cosmic time and the concurrent evolution of the galaxy stellar distribution allow us to construct a simple model that matches the observed spatial and mass distributions of clusters in the Galaxy and the giant elliptical galaxy M87. In order to compare with observations, we model the effects of dynamical friction and dynamical evolution, including stellar mass loss, tidal stripping of stars, and tidal disruption of clusters by the growing galactic nucleus. Read More

**Affiliations:**

^{1}Carnegie,

^{2}UCLA,

^{3}CfA,

^{4}IAS

**Category:**Cosmology and Nongalactic Astrophysics

We perform a systematic search for sub-parsec binary supermassive black holes (BHs) in normal broad line quasars at z<0.8, using multi-epoch SDSS spectroscopy of the broad Hbeta line. Our working model is that: only one of the two BHs in the binary is active, and dynamically dominates its own broad line region (BLR); the inactive companion BH is orbiting at a distance of a few R_BLR, where R_BLR~0. Read More

The multiple-planet systems discovered by the Kepler mission exhibit the following feature: planet pairs near first-order mean-motion resonances prefer orbits just outside the nominal resonance, while avoiding those just inside the resonance. We explore an extremely simple dynamical model for planet formation, in which planets grow in mass at a prescribed rate without orbital migration or dissipation. We develop an analytic version of this model for two-planet systems in two limiting cases: the planet mass grows quickly or slowly relative to the characteristic resonant libration time. Read More

We simulate the evolution of one-dimensional gravitating collisionless systems from non- equilibrium initial conditions, similar to the conditions that lead to the formation of dark- matter halos in three dimensions. As in the case of 3D halo formation we find that initially cold, nearly homogeneous particle distributions collapse to approach a final equilibrium state with a universal density profile. At small radii, this attractor exhibits a power-law behavior in density, {\rho}(x) \propto |x|^(-{\gamma}_crit), {\gamma}_crit \simeq 0. Read More

**Affiliations:**

^{1}IAS,

^{2}IAS

An analysis of the kinematics of 412 stars at 1-4 kpc from the Galactic mid-plane by Moni Bidin et al. (2012) has claimed to derive a local density of dark matter that is an order of magnitude below standard expectations. We show that this result is incorrect and that it arises from the assumption that the mean azimuthal velocity of the stellar tracers is independent of Galactocentric radius at all heights. Read More

We study the linear perturbations of collisionless near-Keplerian discs. Such systems are models for debris discs around stars and the stellar discs surrounding supermassive black holes at the centres of galaxies. Using a finite-element method, we solve the linearized collisionless Boltzmann equation and Poisson's equation for a wide range of disc masses and rms orbital eccentricities to obtain the eigenfrequencies and shapes of normal modes. Read More

An important class of formation theories for hot Jupiters involves the excitation of extreme orbital eccentricity (e=0.99 or even larger) followed by tidal dissipation at periastron passage that eventually circularizes the planetary orbit at a period less than 10 days. In a steady state, this mechanism requires the existence of a significant population of super-eccentric (e>0. Read More

We describe statistical methods for measuring the exoplanet multiplicity function - the fraction of host stars containing a given number of planets - from transit and radial-velocity surveys. The analysis is based on the approximation of separability - that the distribution of planetary parameters in an n-planet system is the product of identical 1-planet distributions. We review the evidence that separability is a valid approximation for exoplanets. Read More

We examine the possibility that the observed relation between black-hole mass and host-galaxy stellar velocity dispersion (the M-sigma relation) is biased by an observational selection effect, the difficulty of detecting a black hole whose sphere of influence is smaller than the telescope resolution. In particular, we critically investigate recent claims that the M-sigma relation only represents the upper limit to a broad distribution of black-hole masses in galaxies of a given velocity dispersion. We find that this hypothesis can be rejected at a high confidence level, at least for the early-type galaxies with relatively high velocity dispersions (median 268 km/s) that comprise most of our sample. Read More

The shearing sheet is a model dynamical system that is used to study the small-scale dynamics of astrophysical disks. Numerical simulations of particle trajectories in the shearing sheet usually employ the leapfrog integrator, but this integrator performs poorly because of velocity-dependent (Coriolis) forces. We describe two new integrators for this purpose; both are symplectic, time-reversible and second-order accurate, and can easily be generalized to higher orders. Read More

We present the stellar kinematics in the central 2" of the luminous elliptical galaxy M87 (NGC 4486), using laser adaptive optics to feed the Gemini telescope integral-field spectrograph, NIFS. The velocity dispersion rises to 480 km/s at 0.2". Read More

We describe a new finite element method (FEM) to construct continuous equilibrium distribution functions of stellar systems. The method is a generalization of Schwarzschild's orbit superposition method from the space of discrete functions to continuous ones. In contrast to Schwarzschild's method, FEM produces a continuous distribution function (DF) and satisfies the intra element continuity and Jeans equations. Read More

Observations of the spatial distribution and kinematics of young stars in the Galactic centre can be interpreted as showing that the stars occupy one, or possibly two, discs of radii ~0.05-0.5 pc. Read More

Elliptical, lenticular, and early-type spiral galaxies show a remarkably tight power-law correlation between the mass M_BH of their central supermassive black hole (SMBH) and the number N_GC of globular clusters: M_BH=m*N_GC^(1.08+/-0.04) with m=1. Read More

We investigate the survival of planetesimal discs over Gyr timescales, using a unified approach that is applicable to all Keplerian discs of solid bodies -- dust grains, asteroids, planets, etc. Planetesimal discs can be characterized locally by four parameters: surface density, semi-major axis, planetesimal size and planetesimal radial velocity dispersion. Any planetesimal disc must have survived all dynamical processes, including gravitational instability, dynamical chaos, gravitational scattering, physical collisions, and radiation forces, that would lead to significant evolution over its lifetime. Read More

We study the orbital evolution of wide binary stars in the solar neighborhood due to gravitational perturbations from passing stars. We include the effects of the Galactic tidal field and continue to follow the stars after they become unbound. For a wide variety of initial semi-major axes and formation times, we find that the number density (stars per unit logarithmic interval in projected separation) exhibits a minimum at a few times the Jacobi radius r_J, which equals 1. Read More

**Authors:**Kayhan Gultekin

^{1}, Douglas O. Richstone

^{2}, Karl Gebhardt

^{3}, Tod R. Lauer

^{4}, Scott Tremaine

^{5}, M. C. Aller

^{6}, Ralf Bender

^{7}, Alan Dressler

^{8}, S. M. Faber

^{9}, Alexei V. Filippenko

^{10}, Richard Green

^{11}, Luis C. Ho

^{12}, John Kormendy

^{13}, John Magorrian

^{14}, Jason Pinkney

^{15}, Christos Siopis

^{16}

**Affiliations:**

^{1}Dept. of Astronomy, University of Michigan,

^{2}Dept. of Astronomy, University of Michigan,

^{3}Department of Astronomy, University of Texas,

^{4}National Optical Astronomy Observatory,

^{5}School of Natural Sciences, Institute for Advanced Study,

^{6}Department of Physics, Institute of Astronomy, ETH Zurich,

^{7}Universitaets-Sternwarte der Ludwig-Maximilians-Universität,

^{8}Observatories of the Carnegie Institution of Washington,

^{9}University of California Observatories/Lick Observatory, Board of Studies in Astronomy and Astrophysics, University of California, Santa Cruz,

^{10}Department of Astronomy, University of California, Berkeley,

^{11}LBT Observatory, University of Arizona,

^{12}Observatories of the Carnegie Institution of Washington,

^{13}Department of Astronomy, University of Texas,

^{14}Department of Physics, University of Durham,

^{15}Department of Physics and Astronomy, Ohio Northern University,

^{16}Institut d'Astronomie et d'Astrophysique, Université Libre de Bruxelles

We derive improved versions of the relations between supermassive black hole mass (M_BH) and host-galaxy bulge velocity dispersion (sigma) and luminosity (L) (the M-sigma and M-L relations), based on 49 M_BH measurements and 19 upper limits. Particular attention is paid to recovery of the intrinsic scatter (epsilon_0) in both relations. We find log(M_BH / M_sun) = alpha + beta * log(sigma / 200 km/s) with (alpha, beta, epsilon_0) = (8. Read More

The orbital dynamics of most planetary satellites is governed by the quadrupole moment from the equatorial bulge of the host planet and the tidal field from the Sun. On the Laplace surface, the long-term orbital evolution driven by the combined effects of these forces is zero, so that orbits have a fixed orientation and shape. The "classical" Laplace surface is defined for circular orbits, and coincides with the planet's equator at small planetocentric distances and with its orbital plane at large distances. Read More

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 {\em Hubble Space Telescope}, complemented by ground-based observations extending to larger radii. Our best mass estimate is $\MBH = (3. Read More

**Affiliations:**

^{1}Princeton,

^{2}IAS

**Category:**Astrophysics

We conduct a systematic survey of the regions in which distant satellites can orbit stably around the four giant planets in the solar system, using orbital integrations of up to $10^9$ yr. In contrast to previous investigations, we use a grid of initial conditions on a surface of section to explore phase space uniformly inside and outside the planet's Hill sphere (radius $r_{\rm H}$; satellites outside the Hill sphere sometimes are also known as quasi-satellites). Our confirmations and extensions of old results and new findings include the following: (i) many prograde and retrograde satellites can survive out to radii $\sim 0. Read More

Semi-analytic treatments of the evolution of orbits of weakly interacting massive particles (WIMPs) in the solar system suggest that the WIMPs bound to the solar system may enhance the direct detection rate relative to that of the unbound population by up to a factor of order unity, and boost the flux of neutrinos from WIMP annihilation in the Earth by up to two orders of magnitude. To test these important but uncertain results, we perform a suite of numerical orbit integrations to explore the properties of the bound WIMP population as a function of the WIMP mass and the scattering cross section with baryonic matter. For regions of WIMP parameter space presently allowed by experiments, we find that (i) the bound WIMP population enhances the direct detection rate by at most ~1% relative to the rate from unbound halo WIMPs; (ii) it is unlikely that planned km^3-scale neutrino telescopes will detect neutrinos from WIMP annihilation in the Earth; (iii) the event rate from neutrinos produced by WIMP annihilation in the Sun may be much smaller than implied by the usual calculations, which assume that WIMPs scattered onto bound orbits are rapidly thermalized in the Sun. Read More

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 NGC1399, the central galaxy in Fornax. We find the best-fit model contains a black hole of 5. Read More

At least two arguments suggest that the orbits of a large fraction of binary stars and extrasolar planets shrank by 1-2 orders of magnitude after formation: (i) the physical radius of a star shrinks by a large factor from birth to the main sequence, yet many main-sequence stars have companions orbiting only a few stellar radii away, and (ii) in current theories of planet formation, the region within ~0.1 AU of a protostar is too hot and rarefied for a Jupiter-mass planet to form, yet many "hot Jupiters" are observed at such distances. We investigate orbital shrinkage by the combined effects of secular perturbations from a distant companion star (Kozai oscillations) and tidal friction. Read More

Programs to observe evolution in the Mbh-sigma or Mbh-L relations typically compare black-hole masses, Mbh, in high-redshift galaxies selected by nuclear activity to Mbh in local galaxies selected by luminosity L, or stellar velocity dispersion sigma. Because AGN luminosity is likely to depend on Mbh, selection effects are different for high-redshift and local samples, potentially producing a false signal of evolution. This bias arises because cosmic scatter in the Mbh-sigma and Mbh-L relations means that the mean log(L) or log(sigma) among galaxies that host a black hole of given Mbh, may be substantially different than the log(L) or log(sigma) obtained from inverting the Mbh-L or Mbh-sigma relations for the same nominal Mbh. Read More

We combine the results from several HST investigations of the central structure of early-type galaxies to generate a large sample of parameterized surface photometry. The studies included 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. Read More

Black hole masses predicted from the Mbh-sigma relationship conflict with those predicted from the Mbh-L relationship for the most luminous galaxies, such as brightest cluster galaxies (BCGs). This is because stellar velocity dispersion, sigma, increases only weakly with L for BCGs and other giant ellipticals. The Mbh-L relationship predicts that the most luminous BCGs may have Mbh approaching 10^{10}M_sol, while the M-sigma relationship always predicts Mbh<3X10^9M_sol. Read More

We describe a maximum-likelihood method for determining the mass distribution in spherical stellar systems from the radial velocities of a population of discrete test particles. The method assumes a parametric form for the mass distribution and a non-parametric two-integral distribution function. We apply the method to a sample of 161 globular clusters in M87. Read More

**Affiliations:**

^{1}Princeton University,

^{2}Princeton University

**Category:**Astrophysics

Many astronomers have speculated that the solar system contains undiscovered massive planets or a distant stellar companion. The acceleration of the solar system barycenter can constrain the mass and position of the putative companion. In this paper we use the most recent timing data on accurate astronomical clocks (millisecond pulsars, pulsars in binary systems and pulsating white dwarfs) to constrain this acceleration. Read More

We examine the stability of a low-mass stellar system surrounding a massive central object. Examples of such systems include the centers of galaxies or star clusters containing a massive black hole, and the Oort comet cloud. If the self-gravity of the stellar system is the dominant non-Keplerian force, such systems may be subject to slowly growing (secular) lopsided instabilities. Read More

**Authors:**Tod R. Lauer

^{1}, S. M. Faber, Karl Gebhardt, Douglas Richstone, Scott Tremaine, Edward A. Ajhar, M. C. Aller, Ralf Bender, Alan Dressler, Alexei V. Filippenko, Richard Green, Carl J. Grillmair, Luis C. Ho, John Kormendy, John Magorrian, Jason Pinkney, Christos Siopis

**Affiliations:**

^{1}NOAO

**Category:**Astrophysics

We analyze HST+WFPC2 images of 77 early-type galaxies. Brightness profiles are classed into "core" or "power-law" forms. Cores are typically rounder than power-law galaxies. Read More