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The bulk of old stars in the Galactic disk have migrated radially by up to several kpc in their lifetimes, yet the disk has remained relatively cool, i.e., the ratio of radial heating to migration has been small. Here, we demonstrate that this small ratio places very strong constraints on which mechanisms could have been responsible for orbital transport in our Galaxy. For instance, Sellwood & Binney's mechanism of nonlinear horseshoe transport by spirals tends to produce too high a ratio of heating to migration, unless the spirals' amplitudes are heavily suppressed away from their corotation resonances, or their pitch angles are significantly larger than is observed. This problem is only made worse if one includes the effect of the Galactic bar, diffusion due to disk or halo substructure, etc. Resonant (but non-horseshoe) scattering by spirals can drive transport consistent with the data, but even this requires some fine-tuning. In short, reproducing both the observed radial migration and the small ratio of heating to migration is a highly nontrivial requirement, and poses a significant challenge to models of the Milky Way's dynamical history, theories of spiral structure, and the identification of 'Milky Way analogues' in cosmological simulations., Comment: 11 pages, 5 figures. Submitted

The plausibility of an unseen planet in the outer solar system, and the expected orbit and mass of such a planet, have long been a topic of inquiry and debate. We calculate the long-term orbital stability of distant trans-Neptunian objects (TNOs), which allows us to expand the sample of objects that would carry dynamical information about a hypothetical unseen planet in the outer solar system. Using this expanded sample, we find statistically significant clustering at the $\sim 3 \sigma$ level for TNOs with semimajor axes $>170\mbox{\;AU}$, in longitude of perihelion ($\varpi$), but not in inclination ($i$), argument of perihelion ($\omega$) or longitude of node ($\Omega$). Since a natural explanation for clustering in $\varpi$ is an unseen planet, we run 300 $n$-body simulations with the giant planets, a disk of test particles representing Kuiper belt objects, and an additional planet with varied initial conditions for its mass, semimajor axis, eccentricity, and inclination. Based on the distribution of test particles after 1--2 Gyr, we compute relative likelihoods given the actual distribution of $\varpi$ as a function of semimajor axis for distant TNOs on stable orbits using a significantly larger sample ($N = 51$) than previous work ($N = 11$). We find that the unseen planet parameters that best fit the data are a mass of $m_p = 4.4 \pm 1.1 \mathrm{\; M_{\oplus}}$, a semimajor axis of $a_p = 290 \pm 30 \mathrm{\; AU}$, an eccentricity of $e_p = 0.29 \pm 0.13$, and an inclination of $i_p = 6.8 \pm 5.0^{\circ}$ (all error bars are 1$\sigma$). Only $0.06\%$ of the Brown & Batygin (2021) reference population produce probabilities within $1\sigma$ of the maximum within our quadrivariate model, indicating that our work identifies a distinct preferred region of parameter space for an unseen planet in the solar system. If such an unseen planet exists, it is likely to be discovered by LSST., Comment: 15 pages, 9 figures, 2 tables. Submitted to AAS Journals

We present IR photometry and HST imaging and spectroscopy of Sab galaxy NGC 4826. Schwarzschild dynamical modeling is used to measure its central black hole mass $M$. Photometric decomposition is used to enable a comparison of $M$ to published scaling relations between black hole masses and properties of host bulges. This decomposition implies that NGC 4826 contains classical and pseudo bulges of approximately equal mass. The classical bulge has best-fit S\'ersic index $n=3.27$. The pseudobulge is made up of three parts, an inner lens ($n=0.18$ at $r\lesssim4^{\prime\prime}$), an outer lens ($n=0.17$ at $r \lesssim 45^{\prime\prime}$), and a $n=0.58$ component required to match the surface brightness between the lens components. The total $V$-band luminosity of the galaxy is $M_{VT}=-21.07$, the ratio of classical bulge to total light is $B/T\simeq0.12$, and the ratio of pseudobulge to total light is $PB/T\simeq0.13$. The outer disk is exponential ($n=1.07$) and makes up $D/T=0.75$ of the light of the galaxy. Our best-fit Schwarzschild model has a black hole mass with $1\sigma$ uncertainties of $M=8.4^{+1.7}_{-0.6}\times10^6\ M_\odot$ and a stellar $K$-band mass-to-light ratio of $\Upsilon_K=0.46\pm0.03\ M_{\odot}\ \mathrm{L}_{\odot}^{-1}$ at the assumed distance of 7.27 Mpc. Our modeling is marginally consistent with $M=0$ at the $3\sigma$ limit. These best-fit parameters were calculated assuming the black hole is located where the velocity dispersion is largest; this is offset from the maximum surface brightness, probably because of dust absorption. The black hole mass -- one of the smallest measured by modeling stellar dynamics -- satisfies the well known correlations of $M$ with the $K$-band luminosity, stellar mass, and velocity dispersion of the classical bulge only in contrast to total (classical plus pseudo) bulge luminosity., Comment: Accepted by ApJ. 27 or so pages, 18 figures I think. Online data is available at https://doi.org/10.7302/kr8z-fj98

Galactic disks lie at the heart of many of the most pressing astrophysical puzzles. There are sophisticated kinetic theories that describe some aspects of galaxy disk dynamics, but extracting quantitative predictions from those theories has proven very difficult, meaning they have shed little light on observations/simulations of galaxies. Here, we begin to address this issue by developing a tractable theory describing fluctuations and transport in thin galactic disks. Our main conceptual advance is to split potential fluctuations into asymptotic wavelength regimes relative to orbital guiding radius and epicyclic amplitude (similar to plasma gyrokinetics), and then to treat separately the dynamics in each regime. As an illustration, we apply our results to quasilinear theory, calculating the angular-momentum transport due to a transient spiral. At each stage we verify our formulae with numerical examples. Our approach should simplify many important calculations in galactic disk dynamics., Comment: 18 pages, 8 figures. Submitted to ApJ

Galaxies are not in a dynamical steady state. They continually undergo perturbations, e.g., from infalling dwarf galaxies and dark-matter substructure. After a dynamical perturbation, stars phase mix towards a new steady state; in so doing they generally form spiral structures, such as spiral density waves in galaxy disks and the Gaia Snail observed in the vertical phase-space density in the solar neighborhood. Structures in phase-space density can be hard to measure accurately, because spatially varying selection effects imprint their own patterns on the density. However, stellar labels such as metallicity, or other element abundances, or stellar masses and ages, can be measured even in the face of complex or unknown spatial selection functions. We show that if the equilibrium galaxy has phase-space gradients in these labels, any perturbation that could raise a spiral wave in the phase-space density will raise a spiral wave in the distribution of labels as well. We work out the relationship between the spiral patterns in the density and in the labels. As an example, we analyze the Gaia Snail and show that its amplitude and dynamical age as derived from elemental abundances (mainly [Mg/Fe]) follow similar patterns to those derived from the phase-space density. Our best model dates the Snail's perturbation to about 400 Myr ago although we find significant variations with angular momentum in the best-fit age. Conceptually, the ideas presented here are related to Orbital Torus Imaging, chemical tagging, and other methods that use stellar labels to trace dynamics., Comment: Submitted to ApJ, comments are welcome

Using the equations of motion from Hill's problem, with added accelerations for different forms of dynamical friction, we provide the (to-date) broadest scale-free study of friction-driven binary formation in gaseous disks and stellar clusters. We focus mainly on binary formation between stellar-mass black holes in active galactic nuclei (AGNi), considering both gas dynamical friction from AGN disks and stellar dynamical friction from the nuclear star cluster. We first find simple, dimensionless friction coefficients that approximate the effects of standard models for gas and stellar dynamical friction. We perform extensive simulations of Hill's problem under such friction, and we present a picture of binary formation through encounters between single stars on nearby orbits, as a function of friction parameter, eccentricity, and inclination. Notably, we find that the local binary formation rate is a linear function of the friction coefficient so long as the friction is weak. Due to the dimensionless nature of our model problem, our findings are generalizable to binary formation at all scales (e.g., intermediate-mass black holes in a star cluster, planetesimals in a gaseous disk)., Comment: 34 pages, 16 figures. Submitted to ApJ. Comments welcome

Modified Newtonian dynamics (MOND), which postulates a breakdown of Newton's laws of gravity/dynamics below some critical acceleration threshold, can explain many otherwise puzzling observational phenomena on galactic scales. MOND competes with the hypothesis of dark matter, which successfully explains the cosmic microwave background and large-scale structure. Here we provide the first solar-system test of MOND that probes the sub-critical acceleration regime. Using the Bekenstein-Milgrom AQUAL formulation, we simulate the evolution of myriads of test particles (planetesimals or comets) born in the trans-Neptunian region and scattered by the giant planets over the lifetime of the Sun to heliocentric distances of $10^2$-$10^5$ au. We include the effects of the Galactic tidal field and passing stars. While Newtonian simulations reproduce the distribution of binding energies of long-period and Oort-cloud comets detectable from Earth, MOND-based simulations do not. This conclusion is robust to plausible changes in the migration history of the planets, the migration history of the Sun, the MOND transition function, effects of the Sun's birth cluster, and the fading properties of long-period comets. For the most popular version of AQUAL, characterized by a gradual transition between the Newtonian and MOND regimes, our MOND-based simulations also fail to reproduce the orbital distribution of trans-Neptunian objects in the detached disk (perihelion > 38 au). Our results do not rule out some MOND theories more elaborate than AQUAL, in which non-Newtonian effects are screened on small spatial scales, at small masses, or in external gravitational fields comparable in strength to the critical acceleration., Comment: 45 pages, 23 figures

We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior to the pericenter of the parabolic orbit. The map shows excellent agreement with direct numerical integrations and elucidates how the dynamics depends on perturber mass and pericenter distance. We also use the map to explore the onset of chaos, statistical descriptions of chaotic transport, and sticking in mean-motion resonances. We discuss implications of our mapping model for the dynamical evolution of the solar system's scattered disk and other highly eccentric trans-Neptunian objects., Comment: submitted to MNRAS

The Hamiltonian used in classical analyses of von Zeipel-Lidov-Kozai or ZLK oscillations in hierarchical triple systems is based on the quadrupole potential from a distant body on a fixed orbit, averaged over the orbits of both the inner and the outer bodies ("double-averaging"). This approximation can be misleading, because the corresponding Hamiltonian conserves the component of angular momentum of the inner binary normal to the orbit of the outer binary, thereby restricting the volume of phase space that the system can access. This defect is usually remedied by including the effects of the octopole potential, or by allowing the outer orbit to respond to variations in the inner orbit. However, in a wide variety of astrophysical systems nonlinear perturbations are comparable to or greater than these effects. The long-term effects of nonlinear perturbations are described by an additional Hamiltonian, which we call Brown's Hamiltonian. At least three different forms of Brown's Hamiltonian are found in the literature; we show that all three are related by a gauge freedom, although one is much simpler than the others. We argue that investigations of ZLK oscillations in triple systems should include Brown's Hamiltonian., Comment: 11 pages, 3 figures. To be published in Monthly Notices of the Royal Astronomical Society

The distribution of stars in the Milky Way disc shows a spiral structure--the Snail--in the space of velocity and position normal to the Galactic mid-plane. The Snail appears as straight lines in the vertical frequency--vertical phase plane when effects from sample selection are removed. Their slope has the dimension of inverse time, with the simplest interpretation being the inverse age of the Snail. Here, we devise and fit a simple model in which the spiral starts as a lopsided perturbation from steady state, that winds up into the present-day morphology. The winding occurs because the vertical frequency decreases with vertical action. We use data from stars in Gaia EDR3 that have measured radial velocities, pruned by simple distance and photometric selection functions. We divide the data into boxels of dynamical invariants (radial action, angular momentum); our model fits the data well in many of the boxels. The model parameters have physical interpretations: one, $A$, is a perturbation amplitude, and one, $t$, is interpretable in the simplest models as the time since the event that caused the Snail. We find trends relating the strength and age to angular momentum: (i) the amplitude $A$ is small at low angular momentum ($<1\,600\mathrm{\,kpc\,km\,s}^{-1}$ or guiding-centre radius $< 7.3\,$kpc), and over a factor of three larger, with strong variations, in the outer disc; (ii) there is no single well-defined perturbation time, with $t$ varying between 0.2 and 0.6 Gyr. Residuals between the data and the model display systematic trends, implying that the data call for more complex models., Comment: 9 pages, 8 figures. Submitted to MNRAS (constructive comments are welcome)

The Gaia snail is a spiral feature in the distribution of solar-neighbourhood stars in position and velocity normal to the Galactic midplane. The snail probably arises from phase mixing of gravitational disturbances that perturbed the disc in the distant past. The most common hypothesis is that the strongest disturbance resulted from a passage of the Sagittarius dwarf galaxy close to the solar neighbourhood. In this paper we investigate the alternative hypothesis that the snail is created by many small disturbances rather than one large one, that is, by Gaussian noise in the gravitational potential. Probably most of this noise is due to substructures in the dark-matter halo. We show that this hypothesis naturally reproduces most of the properties of the snail. In particular it predicts correctly, with no free parameters, that the apparent age of the snail will be $ \sim 0.5$ Gyr. An important ingredient of this model is that any snail-like feature in the solar neighbourhood, whatever its cause, is erased by scattering from giant molecular clouds or other small-scale structure on a time-scale $\lesssim 1$ Gyr., Comment: 9 pages, 6 figures. Submitted to MNRAS

We derive a semi-analytic criterion for the presence of chaos in compact, eccentric multiplanet systems. Beyond a minimum semimajor-axis separation, below which the dynamics are chaotic at all eccentricities, we show that (i) the onset of chaos is determined by the overlap of two-body mean motion resonances (MMRs), like it is in two-planet systems; (ii) secular evolution causes the MMR widths to expand and contract adiabatically, so that the chaotic boundary is established where MMRs overlap at their greatest width. For closely spaced two-planet systems, a near-symmetry strongly suppresses this secular modulation, explaining why the chaotic boundaries for two-planet systems are qualitatively different from cases with more than two planets. We use these results to derive an improved angular-momentum-deficit (AMD) stability criterion, i.e., the critical system AMD below which stability should be guaranteed. This introduces an additional factor to the expression from Laskar and Petit (2017) that is exponential in the interplanetary separations, which corrects the AMD threshold toward lower eccentricities by a factor of several for tightly packed configurations. We make routines for evaluating the chaotic boundary available to the community through the open-source SPOCK package., Comment: Accepted in AJ. Routines for estimating whether compact planetary configurations are chaotic, together with example notebooks are available as part of the SPOCK package: https://github.com/dtamayo/spock

The equilibrium and fluctuations of an ideal gas in a rigid container are studied by every student of statistical mechanics. Here we study the less well-known case when the box is floating freely; in particular we determine the fluctuations of the box in velocity and position due to interactions with the gas it contains. This system is a toy model for the fluctuations in velocity and position of a black hole surrounded by stars at the center of a galaxy. These fluctuations may be observable in nearby galaxies., Comment: 5 pages, no figures, submitted to American Journal of Physics

In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauss, is to use the secular approximation of smearing out the $N$ stars over their Keplerian orbits, using $K$ nodes along each orbit. In this study we propose three novel improvements to this method. First, we re-formulate the discretisation of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order $\ell_{\mathrm{max}}$, reducing the overall computational costs from $O(N^2K^2)$ to $O(NK\ell_{\mathrm{max}}^2)$. Finally, we show that the averaged dynamical system is equivalent to $2N$ interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics., Comment: 20 pages, 11 figures, submitted to AAS

Fuzzy dark matter (FDM) is composed of ultra-light bosons having a de Broglie wavelength that is comparable to the size of the stellar component of galaxies at typical galactic velocities. FDM behaves like cold dark matter on large scales. However, on the scale of the de Broglie wavelength, an FDM halo exhibits density fluctuations that lead to relaxation, a process similar to the two-body relaxation that occurs in classical gravitational N-body systems and is described by the Fokker-Planck equation. We derive the FDM analog of that kinetic equation, and solve it to find the evolution of the velocity distribution in a spatially homogeneous FDM halo. We also determine the dielectric function and the dispersion relation for linear waves in an FDM halo., Comment: 22 pages, 4 figures, submitted to AAS

We address the claim of Weiss et al. (2018) that the radii of adjacent planets in Kepler multi-planet systems are correlated. We explore two simple toy models---in the first the radii of the planets are chosen at random from a single universal distribution, and in the second we postulate several types of system with distinct radius distributions. We show that an apparent correlation between the radii of adjacent planets similar to the one reported by Weiss et al. (2018) can arise in both models. In addition the second model fits the radius and signal-to-noise distribution of the observed planets. We also comment on the validity of a commonly used correction that is used to estimate intrinsic planet occurrence rates, based on weighting planets by the inverse of their detectability., Comment: 13 pages, 8 figures, 1 appendix v2 several points clarified, accepted to AJ. 14 pages, 8 figures, 1 appendix

Some quadruple star systems in the hierarchical 2+2 configuration exhibit orbit-orbit resonances between the two compact binaries. We show that the most important resonances occur at period ratios of 1:1, 3:2 and 2:1. We describe the conditions required for capture and show that they can be satisfied at the 3:2 and 2:1 resonances in binaries that migrate significantly in semimajor axis after circularization, probably through magnetic braking or gravitational radiation., Comment: 13 pages, 1 figure

Supermassive black holes at the centres of galaxies are often surrounded by dense star clusters. For a wide range of cluster properties and orbital radii the resonant relaxation times in these clusters are much shorter than the Hubble time. Since resonant relaxation conserves semimajor axes, these clusters should be in the maximum-entropy state consistent with the given semimajor axis distribution. We determine these maximum-entropy equilibria in a simplified model in which all of the stars have the same semimajor axes. We find that the cluster exhibits a phase transition from a disordered, spherical, high-temperature equilibrium to an ordered low-temperature equilibrium in which the stellar orbits have a preferred orientation or line of apsides. Here `temperature' is a measure of the non-Keplerian or self-gravitational energy of the cluster; in the spherical state, temperature is a function of the rms eccentricity of the stars. We explore a simple two-parameter model of black-hole star clusters -- the two parameters are semimajor axis and black-hole mass --- and find that clusters are susceptible to the lopsided phase transition over a range of ~100 in semimajor axis, mostly for black-hole masses less than $10^{7.5}$ solar masses., Comment: MNRAS, in press

The supermassive black holes found at the centers of galaxies are often surrounded by dense star clusters. The ages of these clusters are generally longer than the resonant-relaxation time and shorter than the two-body relaxation time over a wide range of radii. We explore the thermodynamic equilibria of such clusters using a simple self-similar model. We find that the cluster exhibits a phase transition between a high-temperature spherical equilibrium and a low-temperature equilibrium in which the stars are on high-eccentricity orbits with nearly the same orientation. In the absence of relativistic precession, the spherical equilibrium is metastable below the critical temperature and the phase transition is first-order. When relativistic effects are important, the spherical equilibrium is linearly unstable below the critical temperature and the phase transition is continuous. A similar phase transition has recently been found in a model cluster composed of stars with a single semimajor axis. The presence of the same phenomenon in two quite different cluster models suggests that lopsided equilibria may form naturally in a wide variety of black-hole star clusters., Comment: Journal reference added

The centers of most galaxies contain massive black holes surrounded by dense star clusters. The structure of these clusters determines the rate and properties of observable transient events, such as flares from tidally disrupted stars and gravitational-wave signals from stars spiraling into the black hole. Most estimates of these rates enforce spherical symmetry on the cluster. Here we show that, in the course of generic evolutionary processes, a star cluster surrounding a black hole can undergo a robust phase transition from a spherical thermal equilibrium to a lopsided equilibrium, in which most stars are on high-eccentricity orbits with aligned orientations. The rate of transient events is expected to be much higher in the ordered phase. Better models of cluster formation and evolution are needed to determine whether clusters should be found in the ordered or disordered phase., Comment: 5 pages, 2 figures

Dark matter may be composed of light bosons, ${m_b \sim 10^{-22}\, \mathrm{eV}}$, with a de Broglie wavelength $\lambda \sim 1 \,\mathrm{kpc}$ in typical galactic potentials. Such `fuzzy' dark matter (FDM) behaves like cold dark matter (CDM) on much larger scales than the de Broglie wavelength, but may resolve some of the challenges faced by CDM in explaining the properties of galaxies on small scales ($\lesssim 10\,\mathrm{kpc}$). Because of its wave nature, FDM exhibits stochastic density fluctuations on the scale of the de Broglie wavelength that never damp. The gravitational field from these fluctuations scatters stars and black holes, causing their orbits to diffuse through phase space. We show that this relaxation process can be analyzed quantitatively with the same tools used to analyze classical two-body relaxation in an $N$-body system, and can be described by treating the FDM fluctuations as quasiparticles, with effective mass $\sim 10^7 M_\odot {(1\,\mathrm{kpc}/r)}^2{(10^{-22}\,\mathrm{eV}/m_b)}^3$ in a galaxy with a constant circular speed of $200\,\mathrm{kms}$. This novel relaxation mechanism may stall the inspiral of supermassive black holes or globular clusters due to dynamical friction at radii of a few hundred pc, and can heat and expand the central regions of galaxies. These processes can be used to constrain the mass of the light bosons that might comprise FDM., Comment: 23 pages, 7 figures, published on APJ

One of the fundamental tasks of dynamical astronomy is to infer the distribution of mass in a stellar system from a snapshot of the positions and velocities of its stars. The usual approach to this task (e.g., Schwarzschild's method) involves fitting parametrized forms of the gravitational potential and the phase-space distribution to the data. We review the practical and conceptual difficulties with this approach and describe a novel statistical method for determining the mass distribution that does not require determining the phase-space distribution of the stars. We show that this new estimator out-performs other distribution-free estimators for the harmonic and Kepler potentials., Comment: 11 pages, 4 figures

It is likely that multiple bodies with masses between those of Mars and Earth ("planetary embryos") formed in the outer planetesimal disk of the solar system. Some of these were likely scattered by the giant planets into orbits with semi-major axes of hundreds of AU. Mutual torques between these embryos may lift the perihelia of some of them beyond the orbit of Neptune, where they are no longer perturbed by the giant planets so their semi-major axes are frozen in place. We conduct N-body simulations of this process, and its effect on smaller planetesimals in the region of the giant planets and the Kuiper belt. We find that (i) there is a significant possibility that one sub-Earth mass embryo, or possibly more, is still present in the outer solar system; (ii) the orbit of the surviving embryo(s) typically has perihelion of 40--70 AU, semi-major axis less than 200 AU, and inclination less than 30 degrees; (iii) it is likely that any surviving embryos could be detected by current or planned optical surveys or have a significant effect on solar-system ephemerides; (iv) whether or not an embryo has survived to the present day, their dynamical influence earlier in the history of the solar system can explain the properties of the detached disk (defined in this paper as containing objects with perihelia > 38 AU and semi-major axes between 80 and 500 AU)., Comment: Accepted to A.J

Hot Jupiters (HJs) are short-period giant planets that are observed around ~ 1% of solar-type field stars. One possible formation scenario for HJs is high-eccentricity (high-e) migration, in which the planet forms at much larger radii, is excited to high eccentricity by some mechanism, and migrates to its current orbit due to tidal dissipation occurring near periapsis. We consider high-e migration in dense stellar systems such as the cores of globular clusters (GCs), in which encounters with passing stars can excite planets to the high eccentricities needed to initiate migration. We study this process via Monte Carlo simulations of encounters with a star+planet system including the effects of tidal dissipation, using an efficient regularized restricted three-body code. HJs are produced in our simulations over a significant range of the stellar number density n_*. Assuming the planet is initially on a low-eccentricity orbit with semimajor axis 1 au, for n_* < 1e3 pc^{-3} the encounter rate is too low to induce orbital migration, whereas for n_* > 1e6 pc^{-3} HJ formation is suppressed because the planet is more likely ejected from its host star, tidally disrupted, or transferred to a perturbing star. The fraction of planets that are converted to HJs peaks at ~ 2% for intermediate number densities of ~ 4e4 pc^{-3}. Warm Jupiters, giant planets with periods between 10 and 100 days, are produced in our simulations with an efficiency of up to ~ 0.5%. Our results suggest that HJs can form through high-e migration induced by stellar encounters in the centers of of dense GCs, but not in their outskirts where the densities are lower., Comment: Accepted for publication in AJ. 19 pages, 15 figures

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 with initial bodies ${\gtrsim}1700$ km 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}$1.1 $r_\mathrm{Roche}$. At smaller distances, tidal effects cause collisions to be too fragmenting to allow mass build-up to a final, dynamically stable planetary system. We argue that even differentiated bodies can accumulate to form planets at distances that are not much larger than $r_\mathrm{Roche}$., Comment: 15 pages, 9 figures, accepted for publication in AJ; revised version after addressing referee comments

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. We show that the statistical mechanics of such systems exhibit a striking resemblance to liquid crystals, with analogous ordered-nematic and disordered-isotropic phases. The ordered phase consists of bodies orbiting in a disk in both directions, with the disk thickness depending on temperature, while the disordered phase corresponds to a nearly isotropic distribution of the orbit normals. We show that below a critical value of the total angular momentum, the system undergoes a first-order phase transition between the ordered and disordered phases. At the critical point the phase transition becomes second-order while for higher angular momenta there is a smooth crossover. We also find metastable equilibria containing two identical disks with mutual inclinations between $90^{\circ}$ and $180^\circ$., Comment: 33 pages, 23 figures; minor changes to match the published version

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 $R_e(R)\equiv\langle j\rangle/v_c(R)$. We show that such a profile matches the observed surface-brightness profiles of disc-dominated galaxies as well as the empirical exponential profile. Disc galaxies that exhibit population gradients cannot have fully reached a maximum-entropy state but appear to be close enough that their surface-brightness profiles are well-fit by this idealized model., Comment: 12 pages, 6 figures, published in MNRAS

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. (ii) FDM halos are comprised of a core that is a stationary, minimum-energy configuration called a "soliton", surrounded by an envelope that resembles a CDM halo. (iii) The transition between soliton and envelope is determined by a relaxation process analogous to two-body relaxation in gravitating systems, which proceeds as if the halo were composed of particles with mass $\sim \rho\lambda^3$ where $\rho$ is the halo density. (iv) Relaxation may have substantial effects on the stellar disk and bulge in the inner parts of disk galaxies. (v) Relaxation can produce FDM disks but an FDM disk in the solar neighborhood must have a half-thickness of at least $300 (m/10^{-22} {\rm eV})^{-2/3}$ pc. (vi) Solitonic FDM sub-halos evaporate by tunneling through the tidal radius and this limits the minimum sub-halo mass inside 30 kpc of the Milky Way to roughly $10^8 (m/10^{-22} {\rm eV})^{-3/2} M_\odot$. (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., Comment: 37 pages, 3 figures. Additional references and discussions on clusters and the soliton-to-host-mass relation. Accepted for publication in PRD

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. This value is within the low end of the 90\% credible interval for the total black-hole black-hole merger rate inferred from the current LIGO results. There are many uncertainties in these calculations, the largest of which is the unknown distribution of natal kicks. Even modest natal kicks of $40\mbox{km s}^{-1}$ will reduce the merger rate by a factor of 40. A few percent of these systems will have eccentricity greater than 0.999 when they first enter the frequency band detectable by aLIGO (above 10 Hz)., Comment: Accepted to ApJ

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. Thus the distribution of orbital elements constrains the spatial distribution of comets in the Oort cloud and the environment in which the solar system formed. Additionally, comet orbits can be characterized more precisely when they are seen far from the Sun as they have not been affected by non-gravitational forces., Comment: Accepted for publication in the Astronomical Journal

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. In this model the warm Jupiters migrate periodically, in the high-eccentricity phase of the oscillation when the pericenter distance is small, but are typically observed at much lower eccentricities. We show that the steady-state eccentricity distribution of the warm Jupiters migrating by this mechanism is approximately flat, which is consistent with the observed distribution if and only if we restrict the sample to warm Jupiters that have outer companions detected by radial-velocity surveys. The eccentricity distribution of warm Jupiters without companions exhibits a peak at low eccentricities ($e\lesssim 0.2$) that must be explained by a different formation mechanism. Based on a population-synthesis study we find that high-eccentricity migration excited by an outer planetary companion (i) can account for $\sim20\%$ of the warm Jupiters and most of the warm Jupiters with $e\gtrsim 0.4$, a fraction that is consistent with the observed population of warm Jupiters with outer companions; (ii) can produce most of the observed population of hot Jupiters, with a semimajor axis distribution that matches the observations, but fails to account adequately for $\sim 60\%$ of hot Jupiters with projected obliquities $\lesssim 20^\circ$. Thus $\sim 20\%$ of the warm Jupiters and $\sim 60\%$ of the hot Jupiters can be produced by high-eccentricity migration. We also provide predictions for the expected mutual inclinations and spin-orbit angles of the planetary systems with hot and warm Jupiters produced by high-eccentricity migration., Comment: 22 pages, 13 figures, accepted for publication in the AJ

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 $R^{-1.3}$ (with a weak dependence on S\'ersic index $m$), the line-of-sight velocity dispersion varies as $R^{-1/2}$, and the velocity anisotropy is $\beta\simeq -0.24$ to $-0.28$ depending on $m$. The excess stellar mass in the cusp scales approximately linearly with the black-hole mass, and is typically 0.5--0.85 times the black-hole mass. This process may strongly influence the structure of nuclear star clusters if they contain black holes., Comment: Submitted, 10 pages,13 figures in ApJ (2015)

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. The model provides analytic predictions for the distribution of eccentricities and semimajor axis differences, correlations between orbital elements of nearby planets, and the complete N-planet distribution function, in terms of a single parameter, the "dynamical temperature", that is determined by the planetary masses. The predicted properties are generally consistent with N-body simulations of the giant-impact phase and with the distribution of semimajor axis differences in the Kepler catalog of extrasolar planets. A similar model may apply to the orbits of giant planets if these orbits are determined mainly by dynamical evolution after the planets have formed and the gas disk has disappeared., Comment: 26 pages, 5 figures. Minor revisions in response to suggestions by the referee. To be published in the Astrophysical Journal

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., Comment: Invited talk at the "hommage a Michel Henon" conference held at the "Institut Henri Poincare" Paris, GRAVASCO trimestre, autumn 2013. Proceeding by Hermann Press, editors: Jean-Michel Alimi, Roya Mohayaee, Jerome Perez. Videos of all talks are available at: https://www.youtube.com/playlist?list=PL9kd4mpdvWcBLrN8u04IZW6-akNgLG7-_

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., Comment: 33 pages, 4 figures, submitted to Nature Insight

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 $4.3\times10^6$ solar masses. The mass of the central black hole appears to be closely related to other properties of its host galaxy, such as the total mass in stars, but the origin of this relation and the role that black holes play in the formation of galaxies are still mysteries., Comment: 14 pages, 2 figures. To appear in "From Atoms to the Stars", a special issue of Daedalus (Fall 2014, vol. 143, no. 4)

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. This result also implies that the net line-of-sight inflow/outflow velocities in the broad-line region are much less than the Keplerian velocity when averaged over a large sample of quasars with a given line width., Comment: 27 pages, 8 figures

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. We use N-ring to follow the evolution of clusters containing over 10^4 stars for tens of relaxation times. Among other results, we find that the evolution is dominated by torques among stars with radially overlapping orbits, and that resonant relaxation can be modelled as a random walk of the orbit normals on the sphere, with angular step size ranging from 0.5-1 radian. The relaxation rate in a cluster with a fixed number of stars is proportional to the RMS mass of the stars. The RMS torque generated by the cluster stars is reduced below the torque between Kepler orbits due to apsidal precession and declines weakly with the eccentricity of the perturbed orbit. However since the angular momentum of an orbit also decreases with eccentricity, the relaxation rate is approximately eccentricity-independent for e<0.7 and grows rapidly with eccentricity for e>0.8. We quantify the relaxation using the autocorrelation function of the spherical multipole moments; this decays exponentially and the e-folding time may be identified with the vector resonant relaxation timescale., Comment: 35 pages, 13 figures, accepted for publication in MNRAS

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. We discuss the relation between thermal and dynamical instability in these systems and draw connections to astrophysical settings, as well as to the wider subject of the statistical mechanics of particles with logarithmic long-range interactions, such as point vortices in two-dimensional fluids., Comment: J. Phys. A (Fast Track Communication), published (minor changes from v1), 32 pages, 4 figures

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 < 0.15 AU). We find that at these orbital separations, unstable systems starting with low eccentricities and mutual inclinations ($e\lesssim0.1$, $i\lesssim0.1$) generally lead to planet-planet collisions in which the collision product is a planet on a low-eccentricity, low-inclination orbit. This result is inconsistent with the observations. We conclude that eccentricity and inclination excitation from planet-planet scattering must precede migration of planets into short-period orbits. This result constrains theories of planet migration: the semi-major axis must shrink by 1-2 orders of magnitude without damping the eccentricity and inclination., Comment: 11 pages, 3 figures, accepted for publication in ApJ

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. We have selected a sample of 399 quasars with offset broad H-beta lines from the SDSS DR7 quasar catalog, and have conducted second-epoch optical spectroscopy for 50 of them. Combined with the existing SDSS spectra, the new observations enable us to constrain the LOS RV shifts of broad H-beta lines with a rest-frame baseline of a few years to nearly a decade. Using cross-correlation analysis, we detect significant (99% confidence) radial accelerations in the broad H-beta lines in 24 of the 50 objects. We suggest that 9 of the 24 detections are sub-pc BBH candidates, which show consistent velocity shifts independently measured from a second broad line (either H-alpha or Mg II) without significant changes in the broad-line profiles. Combining the results on the general quasar population studied in Paper I, we find a tentative anti-correlation between the velocity offset in the first-epoch spectrum and the average acceleration between two epochs, which could be explained by orbital phase modulation when the time separation between two epochs is a non-negligible fraction of the orbital period of the motion causing the line displacement. We discuss the implications of our results for the identification of sub-pc BBH candidates in offset-line quasars and for the constraints on their frequency and orbital parameters. [Abridged], Comment: 23 pages, 18 figures, ApJ in press

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 $M = (6.0^{+0.7}_{-0.9}) \times 10^8\ M_{\scriptscriptstyle \odot}$ and $\Upsilon_V = 6.0 \pm 0.2\ M_{\scriptscriptstyle \odot}\ L_{{\scriptscriptstyle \odot},V}^{-1}$ at an assumed distance of 46 Mpc. The models are inconsistent with no black hole at a significance of $\Delta\chi^2 = 15.4$ and require a dark matter halo to adequately fit the kinematic data, but the fits are consistent with a large range of plausible dark matter halo parameters. The ring is inconsistent with a population of co-rotating stars on circular orbits, which would produce a narrow line-of-sight velocity distribution (LOSVD). Instead, the ring's LOSVD has a small value of $|V|/\sigma$, the ratio of mean velocity to velocity dispersion. Based on the observed low $|V|/\sigma$, our orbit modeling, and a kinematic decomposition of the ring from the bulge, we conclude that the stellar ring contains stars that orbit in both directions. We consider potential origins for this unique feature, including multiple tidal disruptions of stellar clusters, a change in the gravitational potential from triaxial to axisymmetric, resonant capture and inclining of orbits by a binary black hole, and multiple mergers leading to gas being funneled to the center of the galaxy., Comment: Accepted by ApJ, 14 pages, 9 figures, 1 Table

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. If only the quadrupole potential from the Earth's equatorial bulge is present, all near-circular orbits are stable. If only the octupole potential is present, all such orbits are unstable. If only the lunar or solar potential is present, all near-circular orbits with inclinations to the ecliptic exceeding 39 degrees are unstable. We describe the behavior of satellites in the simultaneous presence of all of these perturbations and show that almost all low Earth orbits are stable because of an accidental property of the dominant quadrupole potential. We also relate these results to the phenomenon of Lidov-Kozai oscillations., Comment: 10 pages, 1 figure; submitted to American Journal of Physics

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. We show that this description is incomplete, and that torques from the companion star (for X-ray binaries) or the self-gravity of the disc (for active galactic nuclei) can play a major role in determining the properties of the warped disc. Including these effects leads to a rich set of new phenomena. For example, (i) when a companion star is present and the warp arises from a misalignment between the companion's orbital axis and the black hole's spin axis, there is no steady-state solution of the Pringle--Ogilvie equations for a thin warped disc when the viscosity falls below a critical value; (ii) in AGN accretion discs, the warp can excite short-wavelength bending waves that propagate inward with growing amplitude until they are damped by the disc viscosity. We show that both phenomena can occur for plausible values of the black hole and disc parameters, and briefly discuss their observational implications., Comment: 28 pages, 11 figures

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. We find that inspiraling globular clusters form a dense central structure, with mass and radius comparable to the typical values in observed nuclear star clusters (NSCs) in late-type and low-mass early-type galaxies. The density contrast associated with the NSC is less pronounced in giant elliptical galaxies. Our results indicate that the NSC mass as a fraction of mass of the galaxy stellar spheroid scales as M_NSC/M_* = 0.0025 M_{*,11}^{-0.5}. Thus disrupted globular clusters could contribute most of the mass of NSCs in galaxies with stellar mass below 10^{11} Msun. The inner part of the accumulated cluster may seed the growth of a central black hole via stellar dynamical core collapse, thereby relieving the problem of how to form luminous quasars at high redshift. The seed black hole may reach about 10^5 Msun within 1 Gyr of the beginning of globular cluster formation., Comment: 15 pages, submitted to ApJ, revised version after referee comments

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~0.01-0.1 pc is the BLR size. We search for the expected line-of-sight acceleration of the broad line velocity from binary orbital motion by cross-correlating SDSS spectra from two epochs separated by up to several years in the quasar restframe. Out of ~700 pairs of spectra for which we have good measurements of the velocity shift between two epochs (1-sigma error~40 km/s), we detect 28 systems with significant velocity shifts in broad Hbeta, among which seven are the best candidates for the hypothesized binaries. We use the distribution of the observed accelerations (mostly non-detections) to place constraints on the abundance of such binary systems among the general quasar population. Excess variance in the velocity shift is inferred for observations separated by longer than 0.4 yr (quasar restframe). Attributing all the excess to binary motion would imply that most of the quasars in this sample must be in binaries; that the inactive BH must be on average more massive than the active one; and that the binary separation is at most a few times the size of the BLR. However, if this excess variance is partly or largely due to long-term broad line variability, the requirement of a large population of close binaries is much weakened or even disfavored for massive companions. [Abridged], Comment: replaced with accepted version; ApJ, in press

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. In both cases the distribution of systems in period ratio develops a characteristic asymmetric peak-trough structure around the resonance, qualitatively similar to that observed in the Kepler sample. We verify this result with numerical integrations of the restricted three-body problem. We show that for the 3:2 resonance, where the observed peak-trough structure is strongest, our simple model is consistent with the observations for a range of mean planet masses 20-100M_{\oplus}. This mass range is higher than expected, by at least a factor of three, from the few Kepler planets with measured masses, but part of this discrepancy could be due to oversimplifications in the dynamical model or uncertainties in the planetary mass-radius relation., Comment: 16 pages, 9 figures, submitted to ApJ

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 0.47, slightly but significantly shallower than the value {\gamma} = 1/2 suggested previously. This state develops from the initial conditions through a process of phase mixing and violent relaxation. This process preserves the energy ranks of particles. By warming the initial conditions, we illustrate a cross-over from this power-law final state to a final state containing a homogeneous core. We further show that inhomogeneous but cold power-law initial conditions, with initial exponent {\gamma}_i > {\gamma}_crit, do not evolve toward the attractor but reach a final state that retains their original power-law behavior in the interior of the profile, indicating a bifurcation in the final state as a function of the initial exponent. Our results rely on a high-fidelity event-driven simulation technique., Comment: 14 Pages, 13 Figures. Submitted to MNRAS

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. We substitute the assumption, supported by data, that the circular speed is independent of radius in the mid-plane. We demonstrate that the assumption of constant mean azimuthal velocity is implausible by showing that it requires the circular velocity to drop more steeply than allowed by any plausible mass model, with or without dark matter, at large heights above the mid-plane. Using the approximation that the circular velocity curve is flat in the mid-plane, we find that the data imply a local dark-matter density of 0.008 +/- 0.003 Msun/pc^3 = 0.3 +/- 0.1 GeV/cm3, fully consistent with standard estimates of this quantity. This is the most robust direct measurement of the local dark-matter density to date., Comment: ApJ, in press