We construct dynamical models for a sample of 36 nearby galaxies with Hubble Space Telescope (HST ) 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 massto-light ratio !, and a central massive dark object (MDO) of arbitrary mass They provide accept-M a . able Ðts to 32 of the galaxies for some value of and ! ; the four galaxies that cannot be Ðtted have M a kinematically decoupled cores. The mass-to-light ratios inferred for the 32 well-Ðtted galaxies are consistent with the fundamental-plane correlation ! P L0.2, where L is galaxy luminosity. In all but six galaxies the models require at the 95% conÐdence level an MDO of mass M a D 0.006M bulge 4 0.006!L . Five of the six galaxies consistent with are also consistent with this correlation. The other (NGC M a \ 0 7332) has a much stronger upper limit onWe predict the second-moment proÐles that should be M a . 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 of mean [2.28 and standard deviation D0.51. There is also marginal (M a /M bulge ) evidence that is distributed di †erently for "" core ÏÏ and "" power law ÏÏ galaxies, with core galaxies M a having a somewhat steeper dependence on M bulge .

Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object M BH and the velocity dispersion of the host galaxy, of the form logðM BH =M Þ ¼ þ logð = 0 Þ; however, published estimates of the slope span a wide range (3.75-5.3). Merritt & Ferrarese have argued that low slopes (d4) 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 and several others 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 ¼ 4:02 AE 0:32, ¼ 8:13 AE 0:06 for 0 ¼ 200 km s À1 . The M BH -relation has an intrinsic dispersion in log M BH that is no larger than 0.25-0.3 dex and may be smaller if observational errors have been underestimated. In an appendix, we present a simple kinematic model for the velocity-dispersion profile of the Galactic bulge.

We describe a correlation between the mass of a galaxy's central black hole and the luminosity-weighted M bh line-of-sight velocity dispersion within the half-light radius. The result is based on a sample of 26 galaxies, j e including 13 galaxies with new determinations of black hole masses from errors. The -relation is of interest not only for its strong predictive power but also because it implies that M j bh e central black hole mass is constrained by and closely related to properties of the host galaxy's bulge.

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We derive improved versions of the relations between supermassive black hole mass (M BH ) and host-galaxy bulge velocity dispersion (σ) and luminosity (L) (the M-σ and M-L relations), based on 49 M BH measurements and 19 upper limits. Particular attention is paid to recovery of the intrinsic scatter (ε 0 ) in both relations. We find log(M BH /M ) = α + β log(σ/200 km s −1 ) with (α, β, ε 0 ) = (8.12 ± 0.08, 4.24 ± 0.41, 0.44 ± 0.06) for all galaxies and (α, β, ε 0 ) = (8.23 ± 0.08, 3.96 ± 0.42, 0.31 ± 0.06) for ellipticals. The results for ellipticals are consistent with previous studies, but the intrinsic scatter recovered for spirals is significantly larger. The scatter inferred reinforces the need for its consideration when calculating local black hole mass function based on the M-σ relation, and further implies that there may be substantial selection bias in studies of the evolution of the M-σ relation. We estimate the M-L relationship as log(M BH /M ) = α + β log(L V /10 11 L ,V ) of (α, β, ε 0 ) = (8.95 ± 0.11, 1.11 ± 0.18, 0.38 ± 0.09); using only early-type galaxies. These results appear to be insensitive to a wide range of assumptions about the measurement errors and the distribution of intrinsic scatter. We show that culling the sample according to the resolution of the black hole's sphere of influence biases the relations to larger mean masses, larger slopes, and incorrect intrinsic residuals.

Many aspects of the large-scale structure of the universe can be described successfully using cosmological models in which 27 ± 1% of the critical mass-energy density consists of cold dark matter (CDM). However, few-if any-of the predictions of CDM models have been successful on scales of ∼ 10 kpc or less. This lack of success is usually explained by the difficulty of modeling baryonic physics (star formation, supernova and black-hole feedback, etc.). An intriguing alternative to CDM is that the dark matter is an extremely light (m ∼ 10 −22 eV) boson having a de Broglie wavelength λ ∼ 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 or sub-halos smaller than about 10 7 (m/10 −22 eV) −3/2 M do not form, and the abundance of halos smaller than a few times 10 10 (m/10 −22 eV) −4/3 M is substantially smaller in FDM than in CDM; (ii) FDM halos are comprised of a central core that is a stationary, minimum-energy solution of the Schrödinger-Poisson equation, sometimes called a "soliton", surrounded by an envelope that resembles a CDM halo. The soliton can produce a distinct signature in the rotation curves of FDM-dominated systems. (iii) The transition between soliton and envelope is determined by a relaxation process analogous to two-body relaxation in gravitating N-body systems, which proceeds as if the halo were composed of particles with mass ∼ ρλ 3 where ρ is the halo density. (iv) Relaxation may have substantial effects on the stellar disk and bulge in the inner parts of disk galaxies, but has negligible effect on disk thickening or globular cluster disruption near the solar radius. (v) Relaxation can produce FDM disks but an FDM disk in the solar neighborhood must have a half-thickness of at least ∼ 300(m/10 −22 eV) −2/3 pc and a mid-plane density less than 0.2(m/10 −22 eV) 2/3 times the baryonic disk density. (vi) Solitonic FDM sub-halos evaporate by tunneling through the tidal radius and this limits the minimum subhalo mass inside ∼ 30 kpc of the Milky Way to a few times 10 8 (m/10 −22 eV) −3/2 M . (vii) If the dark matter in the Fornax dwarf galaxy is composed of CDM, most of the globular clusters observed in that galaxy should have long ago spiraled to its center, and this problem is resolved if the dark matter is FDM. (viii) FDM delays galaxy formation relative to CDM but its galaxy-formation history is consistent with current observations of high-redshift galaxies and the late reionization observed by Planck. If the dark matter is composed of FDM, most observations favor a particle mass 10 −22 eV and the most significant observational consequences occur if the mass is in the range 1-10×10 −22 eV. There is tension with observations of the Lyman-α forest, which favor m 10-20 × 10 −22 eV and we discuss whether more sophisticated models of reionization may resolve this tension. *

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. We integrate the relevant equations of motion to predict the distribution of orbital elements produced by this process. Binary stars with orbital periods of 0.1 to 10 days, with a median of ∼ 2 d, are produced from binaries with much longer periods (10 d to ∼ 10 5 d), consistent with observations indicating that most or all short-period binaries have distant companions (tertiaries). We also make two new testable predictions: (1) For periods between 3 and 10 d, the distribution of the mutual inclination between the inner binary and the tertiary orbit should peak strongly near 40 • and 140 • . (2) Extrasolar planets whose host stars have a distant binary companion may also undergo this process, in which case the orbit of the resulting hot Jupiter will typically be misaligned with the equator of its host star.