Orbit-superposition models of discrete, incomplete stellar kinematics: application to the Galactic centre

We present a method for fitting orbit-superposition models to the kinematics of discrete stellar systems when the available stellar sample has been filtered by a known selection function. The fitting method can be applied to any model in which the distribution function is represented as a linear superposition of basis elements with unknown weights. As an example, we apply it to Fritz et al.'s kinematics of the innermost regions of the Milky Way's nuclear stellar cluster. Assuming spherical symmetry, our models fit a black hole of mass $M_\bullet=(3.76\pm0.22)\times10^6\,M_\odot$, surrounded by an extended mass $M_\star=(6.57\pm0.54)\times10^6\,M_\odot$ within 4 pc. Within 1 pc the best-fitting mass models have an approximate power-law density cusp $\rho\propto r^{-\gamma}$ with $\gamma=1.3\pm0.3$. We carry out an extensive investigation of how our modelling assumptions might bias these estimates: $M_\bullet$ is the most robust parameter and $\gamma$ the least. Internally the best-fitting models have broadly isotropic orbit distributions, apart from a bias towards circular orbits between 0.1 and 0.3 parsec.Comment: 17 pages, MNRAS accepte

Kinematical signatures of hidden stellar discs

The deprojection of the surface brightness distribution of an axisymmetric galaxy does not have a unique solution unless the galaxy is viewed precisely edge-on. I present an algorithm that finds the full range of smooth axisymmetric density distributions consistent with a given surface brightness distribution and inclination angle, and use it to investigate the effects of this non-uniqueness on the line-of-sight velocity profiles (VPs) of two-integral models of both real and toy disky galaxies viewed at a range of inclination angles. Photometrically invisible face-on disks leave very clear signatures in the minor-axis VPs of the models (Gauss--Hermite coefficients h_4>0.1), provided the disk-to-bulge ratio is greater than about 3%. I discuss the implications of these hitherto neglected disks for dynamical modelling.Comment: 7 pages, 7 figures, uses mn.te

Constraining black hole masses from stellar kinematics by summing over all possible distribution functions

When faced with the task of constraining a galaxy's potential given limited stellar kinematical information, what is the best way of treating the galaxy's unknown distribution function (DF)? Using the example of estimating black hole (BH) masses, I argue that the correct approach is to consider all possible DFs for each trial potential, marginalizing the DF using an infinitely divisible prior. Alternative approaches, such as the widely used maximum penalized likelihood method, neglect the huge degeneracies inherent in the problem and simply identify a single, special DF for each trial potential. Using simulated observations of toy galaxies with realistic amounts of noise, I find that this marginalization procedure yields significantly tighter constraints on BH masses than the conventional maximum-likelihood method, although it does pose a computational challenge which might be solved with the development of a suitable algorithm for massively parallel machines. I show that in practice the conventional maximum-likelihood method yields reliable BH masses with well-defined minima in their chi^2 distributions, contrary to claims made by Valluri, Merritt & Emsellem.Comment: 11 pages, 7 figures, 1 tabl

Rates of tidal disruption of stars by massive central black holes

There is strong evidence for some kind of massive dark object in the centres of many galaxy bulges. The detection of flares from tidally disrupted stars could confirm that these objects are black holes (BHs). Here we present calculations of the stellar disruption rates in detailed dynamical models of real galaxies, taking into account the refilling of the loss cone of stars on disruptable orbits by two-body relaxation and tidal forces in non-spherical galaxies. The highest disruption rates (one star per 10^4 yr) occur in faint (L>10^10 L_sun) galaxies, which have steep central density cusps. More luminous galaxies are less dense and have much longer relaxation times and more massive BHs. Dwarf stars in such galaxies are swallowed whole by the BH and hence do not emit flares; giant stars could produce flares as often as every 10^5 yr, although the rate depends sensitively on the shape of the stellar distribution function. We discuss the possibility of detecting disruption flares in current supernova searches. The total mass of stars consumed over the lifetime of the galaxy is of order 10^6 M_sun, independent of galaxy luminosity; thus disrupted stars may contribute significantly to the present BH mass in galaxies fainter than about 10^9 L_sun.Comment: 14 pages, 5 figures, 1 landscape table. Submitted to MNRA

Recognizing the fingerprints of the Galactic bar: a quantitative approach to comparing model (l,v) distributions to observation

We present a new method for fitting simple hydrodynamical models to the (l,v) distribution of atomic and molecular gas observed in the Milky Way. The method works by matching features found in models and observations. It is based on the assumption that the large-scale features seen in (l,v) plots, such as ridgelines and the terminal velocity curve, are influenced primarily by the underlying large-scale Galactic potential and are only weakly dependent on local ISM heating and cooling processes. In our scheme one first identifies by hand the features in the observations: this only has to be done once. We describe a procedure for automatically extracting similar features from simple hydrodynamical models and quantifying the "distance" between each model's features and the observations. Application to models of the Galactic Bar region (|l|<30deg) shows that our feature-fitting method performs better than \chi^2 or envelope distances at identifying the correct underlying galaxy model.Comment: Accepted for publication in MNRA

The secular evolution of discrete quasi-Keplerian systems. I. Kinetic theory of stellar clusters near black holes

We derive the kinetic equation that describes the secular evolution of a large set of particles orbiting a dominant massive object, such as stars bound to a supermassive black hole or a proto-planetary debris disc encircling a star. Because the particles move in a quasi-Keplerian potential, their orbits can be approximated by ellipses whose orientations remain fixed over many dynamical times. The kinetic equation is obtained by simply averaging the BBGKY equations over the fast angle that describes motion along these ellipses. This so-called Balescu-Lenard equation describes self-consistently the long-term evolution of the distribution of quasi-Keplerian orbits around the central object: it models the diffusion and drift of their actions, induced through their mutual resonant interaction. Hence, it is the master equation that describes the secular effects of resonant relaxation. We show how it captures the phenonema of mass segregation and of the relativistic Schwarzschild barrier recently discovered in $N$-body simulations.Comment: 24 pages, 3 figure

Bayes versus the virial theorem: inferring the potential of a galaxy from a kinematical snapshot

We present a new framework for estimating a galaxy's gravitational potential, Phi(x), from its stellar kinematics by adopting a fully non-parametric model for the galaxy's unknown action-space distribution function, f(J). Having an expression for the joint likelihood of Phi and f, the likelihood of Phi is calculated by using a Dirichlet process mixture to represent the prior on f and marginalising. We demonstrate that modelling machinery constructed using this framework is successful at recovering the potentials of some simple systems from perfect discrete kinematical data, a situation handled effortlessly by traditional moment-based methods, such as the virial theorem, but in which other, more modern, methods are less than satisfactory. We show how to generalise the machinery to account for realistic observational errors and selection functions. A practical implementation is likely to raise some interesting algorithmic and computational challenges.Comment: 19 pages, 7 figures. Updated to match accepted versio

Stellar dynamics in the periodic cube

We use the problem of dynamical friction within the periodic cube to illustrate the application of perturbation theory in stellar dynamics, testing its predictions against measurements from $N$-body simulation. Our development is based on the explicitly time-dependent Volterra integral equation for the cube's linear response, which avoids the subtleties encountered in analyses based on complex frequency. We obtain an expression for the self-consistent response of the cube to steady stirring by an external perturber. From this we show how to obtain the familiar Chandrasekhar dynamical friction formula and construct an elementary derivation of the Lenard--Balescu equation for the secular quasilinear evolution of an isolated cube composed of $N$ equal-mass stars. We present an alternative expression for the (real-frequency) van Kampen modes of the cube and show explicitly how to decompose any linear perturbation of the cube into a superposition of such modes.Comment: 13 pages, submitted to MNRA

Gas flow in barred potentials

We use a Cartesian grid to simulate the flow of gas in a barred Galactic potential and investigate the effects of varying the sound speed in the gas and the resolution of the grid. For all sound speeds and resolutions, streamlines closely follow closed orbits at large and small radii. At intermediate radii shocks arise and the streamlines shift between two families of closed orbits. The point at which the shocks appear and the streamlines shift between orbit families depends strongly on sound speed and resolution. For sufficiently large values of these two parameters, the transfer happens at the cusped orbit as hypothesised by Binney et al. over two decades ago. For sufficiently high resolutions the flow downstream of the shocks becomes unsteady. If this unsteadiness is physical, as appears to be the case, it provides a promising explanation for the asymmetry in the observed distribution of CO.Comment: Accepted for publication in MNRA

Gas flow in barred potentials II. Bar Driven Spiral Arms

Spiral arms that emerge from the ends of a galactic bar are important in interpreting observations of our and external galaxies. It is therefore important to understand the physical mechanism that causes them. We find that these spiral arms can be understood as kinematic density waves generated by librations around underlying ballistic closed orbits. This is even true in the case of a strong bar, provided the librations are around the appropriate closed orbits and not around the circular orbits that form the basis of the epicycle approximation. An important consequence is that it is a potential's orbital structure that determines whether a bar should be classified as weak or strong, and not crude estimates of the potential's deviation from axisymmetry.Comment: Accepted for publication in MNRA