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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 accepted

Gaussian processes are the ideal tool for modelling the Galactic ISM, combining statistical flexibility with a good match to the underlying physics. In an earlier paper we outlined how they can be employed to construct three-dimensional maps of dust extinction from stellar surveys. Gaussian processes scale poorly to large datasets though, which put the analysis of realistic catalogues out of reach. Here we show how a novel combination of the Expectation Propagation method and certain sparse matrix approximations can be used to accelerate the dust mapping problem. We demonstrate, using simulated Gaia data, that the resultant algorithm is fast, accurate and precise. Critically, it can be scaled up to map the Gaia catalogue., Comment: 15 pages, 6 figures, submitted for publication in MNRAS

It has been known for more than thirty years that the distribution of molecular gas in the innermost 300 parsecs of the Milky Way, the Central Molecular Zone, is strongly asymmetric. Indeed, approximately three quarters of molecular emission comes from positive longitudes, and only one quarter from negative longitudes. However, despite much theoretical effort, the origin of this asymmetry has remained a mystery. Here we show that the asymmetry can be neatly explained by unsteady flow of gas in a barred potential. We use high-resolution 3D hydrodynamical simulations coupled to a state-of-the-art chemical network. Despite the initial conditions and the bar potential being point-symmetric with respect to the Galactic Centre, asymmetries develop spontaneously due to the combination of a hydrodynamical instability known as the "wiggle instability" and the thermal instability. The observed asymmetry must be transient: observations made tens of megayears in the past or in the future would often show an asymmetry in the opposite sense. Fluctuations of amplitude comparable to the observed asymmetry occur for a large fraction of the time in our simulations, and suggest that the present is not an exceptional moment in the life of our Galaxy., Comment: Accepted for publication in MNRAS. Videos of the simulations are available at http://www.ita.uni-heidelberg.de/~mattia/download.html

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., Accepted for publication in MNRAS

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, Accepted for publication in MNRAS

We present a scheme for using stellar catalogues to map the three-dimensional distributions of extinction and dust within our Galaxy. Extinction is modelled as a Gaussian random field, whose covariance function is set by a simple physical model of the ISM that assumes a Kolmogorov-like power spectrum of turbulent fluctuations. As extinction is modelled as a random field, the spatial resolution of the resulting maps is set naturally by the data available; there is no need to impose any spatial binning. We verify the validity of our scheme by testing it on simulated extinction fields and show that its precision is significantly improved over previous dust-mapping efforts. The approach we describe here can make use of any photometric, spectroscopic or astrometric data; it is not limited to any particular survey. Consequently, it can be applied to a wide range of data from both existing and future surveys., Comment: 16 pages, 12 figures. Submitted for publication in MNRAS. Text revised

We use hydrodynamical simulations to construct a new coherent picture for the gas flow in the Central Molecular Zone (CMZ), the region of our Galaxy within $R\leq 500\, \mathrm{pc}$. We relate connected structures observed in $(l,b,v)$ data cubes of molecular tracers to nuclear spiral arms. These arise naturally in hydrodynamical simulations of barred galaxies, and are similar to those that can be seen in external galaxies such as NGC4303 or NGC1097. We discuss a face-on view of the CMZ including the position of several prominent molecular clouds, such as Sgr B2, the $20\,{\rm km\, s^{-1}}$ and $50\,{\rm km\, s^{-1}}$ clouds, the polar arc, Bania Clump 2 and Sgr C. Our model is also consistent with the larger scale gas flow, up to $R\simeq 3\,\rm kpc$, thus providing a consistent picture of the entire Galactic bar region., Comment: Accepted for publication in MNRAS, 12 pages, 12 figures

(abridged) Photometry and long-slit spectroscopy are presented for a sample of 6 galaxies with a low surface brightness stellar disc and a bulge. The stellar and ionised-gas kinematics were measured along the major and minor axis in half of the sample galaxies, whereas the other half was observed only along two diagonal axes. Spectra along two diagonal axes were obtained also for one of the objects with major and minor axis spectra. The kinematic measurements extend in the disc region out to a surface-brightness level mu_R~24mag/arcsec^2 reaching in all cases the flat part of the rotation curve. The stellar kinematics turns out to be more regular and symmetric than the ionised-gas kinematics, which often shows the presence of non-circular, off-plane, and non-ordered motions. This raises the question about the reliability of the use of the ionised gas as the tracer of the circular velocity in the modeling of the mass distribution, in particular in the central regions of low surface brightness galaxies., Accepted for pubblication on MNRAS (11 figures and 15 tables)

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 figures

We present observations of 77 early-type galaxies imaged with the PC1 CCD of the Hubble Space Telescope (HST) WFPC2. "Nuker-law" parametric fits to the surface brightness profiles are used to classify the central structure into "core" or "power-law" forms. Core galaxies are typically rounder than power-law galaxies. Nearly all power-law galaxies with central ellipticities ε ≥ 0.3 have stellar disks, implying that disks are present in power-law galaxies with ε < 0.3 but are not visible because of unfavorable geometry. A few low-luminosity flattened core galaxies also have disks; these may be transition forms from power-law galaxies to more luminous core galaxies, which lack disks. Several core galaxies have strong isophote twists interior to their break radii, although power-law galaxies have interior twists of similar physical significance when the photometric perturbations implied by the twists are evaluated. Central color gradients are typically consistent with the envelope gradients; core galaxies have somewhat weaker color gradients than power-law galaxies. Nuclei are found in 29% of the core galaxies and 60% of the power-law galaxies. Nuclei are typically bluer than the surrounding galaxy. While some nuclei are associated with active galactic nuclei (AGNs), just as many are not; conversely, not all galaxies known to have a low-level AGN exhibit detectable nuclei in the broadband filters. NGC 4073 and 4382 are found to have central minima in their intrinsic starlight distributions; NGC 4382 resembles the double nucleus of M31. In general, the peak brightness location is coincident with the photocenter of the core to a typical physical scale of < 1 pc. Five galaxies, however, have centers significantly displaced from their surrounding cores; these may be unresolved asymmetric double nuclei. Finally, as noted by previous authors, central dust is visible in about half of the galaxies. The presence and strength of dust correlates with nuclear emission; thus, dust may outline gas that is falling into the central black hole. The prevalence of dust and its morphology suggest that dust clouds form, settle to the center, and disappear repeatedly on ∼ 108 yr timescales. We discuss the hypothesis that cores are created by the decay of a massive black hole binary formed in a merger. Apart from their brightness profiles, there are no strong differences between core galaxies and power-law galaxies that demand this scenario; however, the rounder shapes of core, their lack of disks, and their reduced color gradients may be consistent with it. © 2005. The American Astronomical Society. All rights reserved.

We run hydrodynamical simulations of a 2D isothermal non self-gravitating inviscid gas flowing in a rigidly rotating externally imposed potential formed by only two components: a monopole and a quadrupole. We explore systematically the effects of varying the quadrupole while keeping fixed the monopole and discuss the consequences for the interpretation of longitude-velocity diagrams in the Milky Way. We find that the gas flow can constrain the quadrupole of the potential and the characteristics of the bar that generates it. The exponential scale length of the bar must be at least $1.5\rm\, kpc$. The strength of the bar is also constrained. Our global interpretation favours a pattern speed of $\Omega=40\,\rm km s^{-1} {kpc}^{-1}$. We find that for most observational features, there exist a value of the parameters that matches each individual feature well, but is difficult to reproduce all the important features at once. Due to the intractably high number of parameters involved in the general problem, quantitative fitting methods that can run automatic searches in parameter space are necessary., Comment: accepted for publication in MNRAS. This version corrects a typo in the normalization factor that is present in the published version

Observations of nearby galaxies reveal a strong correlation between the mass of the central dark object M and the velocity dispersion sigma of the host galaxy, of the form log(M/M_sun) = a + b*log(sigma/sigma_0); however, published estimates of the slope b span a wide range (3.75 to 5.3). Merritt & Ferrarese have argued that low slopes (, 37 pages, 9 figures

Black hole (BH) masses that have been measured by reverberation mapping in active galaxies fall significantly below the correlation between bulge luminosity and BH mass determined from spatially resolved kinematics of nearby normal galaxies. This discrepancy has created concern that one or both techniques suffer from systematic errors. We show that BH masses from reverberation mapping are consistent with the recently discovered relationship between BH mass and galaxy velocity dispersion. Therefore the bulge luminosities are the probable source of the disagreement, not problems with either method of mass measurement. This result underscores the utility of the BH mass -- velocity dispersion relationship. Reverberation mapping can now be applied with increased confidence to galaxies whose active nuclei are too bright or whose distances are too large for BH searches based on spatially resolved kinematics., Comment: 4 pages, 1 figure, ApJ Letters accepted, minor revisions

We describe a correlation between the mass M_BH of a galaxy's central black hole and the luminosity-weighted line-of-sight velocity dispersion sigma_e within the half-light radius. The result is based on a sample of 26 galaxies, including 13 galaxies with new determinations of black hole masses from Hubble Space Telescope measurements of stellar kinematics. The best-fit correlation is M_BH = 1.2 (+-0.2) x 10^8 M_sun (sigma_e/200 km/s)^(3.75 (+-0.3))over almost three orders of magnitude in M_BH; the scatter in M_BH at fixed sigma_e is only 0.30 dex and most of this is due to observational errors. The M_BH-sigma_e relation is of interest not only for its strong predictive power but also because it implies that central black hole mass is constrained by and closely related to properties of the host galaxy's bulge., Comment: 4 pages, accepted ApJ Letters, figure added and minor corrections

We fit axisymmetric three-integral dynamical models to NGC 3379 using the line-of-sight velocity distribution obtained from Hubble Space Telescope FOS spectra of the galaxy center and ground-based long-slit spectroscopy along four position angles, with the light distribution constrained by WFPC2 and ground-based images. We have fitted models with inclinations from 29° (intrinsic galaxy type E5) to 90° (intrinsic E1) and black hole masses from 0 to 109 M⊙. The best-fit black hole masses range from 6 × 107 to 2 × 108 M⊙, depending on inclination. The preferred inclination is 90° (edge-on); however, the constraints on allowed inclination are not very strong, owing to our assumption of constant M/LV. The velocity ellipsoid of the best model is not consistent with either isotropy or a two-integral distribution function. Along the major axis, the velocity ellipsoid becomes tangential at the innermost bin, radial in the midrange radii, and tangential again at the outermost bins. The rotation rises quickly at small radii owing to the presence of the black hole. For the acceptable models, the radial-to-tangential [(σ2θ + σ2φ)/2] dispersion in the midrange radii ranges over 1.1 < σr/σt < 1.7, with the smaller black holes requiring larger radial anisotropy. Compared with these three-integral models, two-integral isotropic models overestimate the black hole mass since they cannot provide adequate radial motion. However, the models presented in this paper still contain restrictive assumptions - namely, assumptions of constant M/LV and spheroidal symmetry - requiring yet more models to study black hole properties in complete generality.

We present a method for obtaining the likelihood function of distance and extinction to a star given its photometry. The other properties of the star (its mass, age, metallicity and so on) are marginalised assuming a simple Galaxy model. We demonstrate that the resulting marginalised likelihood function can be described faithfully and compactly using a Gaussian mixture model. For dust mapping applications we strongly advocate using monochromatic over bandpass extinctions, and provide tables for converting from the former to the latter for different stellar types., Comment: 14 pages, 12 figures. Accepted for publication in MNRAS. Source code is available at https://github.com/stuartsale

We present a general scheme for constructing Monte Carlo realizations of equilibrium, collisionless galaxy models with known distribution function (DF) f_0. Our method uses importance sampling to find the sampling DF f_s that minimizes the mean-square formal errors in a given set of projections of the DF f_0. The result is a multi-mass N-body realization of the galaxy model in which ``interesting'' regions of phase-space are densely populated by lots of low-mass particles, increasing the effective N there, and less interesting regions by fewer, higher-mass particles. As a simple application, we consider the case of minimizing the shot noise in estimates of the acceleration field for an N-body model of a spherical Hernquist model. Models constructed using our scheme easily yield a factor ~100 reduction in the variance in the central acceleration field when compared to a traditional equal-mass model with the same number of particles. When evolving both models with a real N-body code, the diffusion coefficients in our model are reduced by a similar factor. Therefore, for certain types of problems, our scheme is a practical method for reducing the two-body relaxation effects, thereby bringing the N-body simulations closer to the collisionless ideal., Comment: 8 pages, 8 figures. Added discussion on Nbody codes updated to match published version

This paper presents a fast, economical particle-multiple-mesh N-body code optimized for large-N modelling of collisionless dynamical processes, such as black-hole wandering or bar-halo interactions, occurring within isolated galaxies. The code has been specially designed to conserve linear momentum. Despite this, it also has variable softening and an efficient block-timestep scheme: the force between any pair of particles is calculated using the finest mesh that encloses them both (respecting Newton's third law) and is updated only on the longest timestep of the two (which conserves momentum). For realistic galaxy models with N > 10^6, it is faster than the fastest comparable momentum-conserving tree code by a factor ranging from ~2 (using single timesteps) to ~10 (multiple timesteps in a concentrated galaxy)., 10 pages, 7 figures. Simplified integrator updated to match published version

The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is investigated using the inhomogeneous Balescu-Lenard equation expressed in terms of angle-action variables. The matrix method is implemented numerically in order to model the induced gravitational polarization. Special care is taken to account for the amplification of potential fluctuations of mutually resonant orbits and the unwinding of the induced swing amplified transients. Quantitative comparisons with ${N-}$body simulations yield consistent scalings with the number of particles and with the self-gravity of the disc: the fewer particles and the colder the disc, the faster the secular evolution. Secular evolution is driven by resonances, but does not depend on the initial phases of the disc. For a Mestel disc with ${Q \sim 1.5}$, the polarization cloud around each star boosts up its secular effect by a factor of the order of a thousand or more, promoting accordingly the dynamical relevance of self-induced collisional secular evolution. The position and shape of the induced resonant ridge are found to be in very good agreement with the prediction of the Balescu-Lenard equation, which scales with the square of the susceptibility of the disc. In astrophysics, the inhomogeneous Balescu-Lenard equation may describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic centre. It could be used as a valuable check of the accuracy of ${N-}$body integrators over secular timescales., Comment: 21 pages, 19 figures

This is the second in a series of papers dedicated to unveil the mass structure and orbital content of a sample of flattened early-type galaxies in the Coma cluster. The ability of our orbit libraries to reconstruct internal stellar motions and the mass composition of a typical elliptical in the sample is investigated by means of Monte-Carlo simulations of isotropic rotator models. The simulations allow a determination of the optimal amount of regularization needed in the orbit superpositions. It is shown that under realistic observational conditions and with the appropriate regularization internal velocity moments can be reconstructed to an accuracy of about 15 per cent; the same accuracy can be achieved for the circular velocity and dark matter fraction. In contrast, the flattening of the halo remains unconstrained. Regularized orbit superpositions are applied to a first galaxy in our sample, NGC 4807, for which stellar kinematical observations extend to 3 Reff. The galaxy seems dark matter dominated outside 2 Reff. Logarithmic dark matter potentials are consistent with the data, as well as NFW-profiles, mimicking logarithmic potentials over the observationally sampled radial range. In both cases, the derived stellar mass-to-light ratio agrees well with independently obtained mass-to-light ratios from stellar population analysis. Kinematically, NGC 4807 is characterized by mild radial anisotropy outside r>0.5 Reff, becoming isotropic towards the center. Our orbit models hint at either a distinct stellar component or weak triaxiality in the outer parts of the galaxy., 20 pages, 25 figures, accepted for publication in MNRAS

We present near infra-red (NIR) adaptive optics assisted spectroscopic observations of the CO ($\Delta\mu=2$) absorption bands towards the centre of the giant elliptical galaxy NGC 1399. The observations were made with NAOS-CONICA (ESO VLT) and have a FWHM resolution of 0\farcs15 (14pc). Kinematic analysis of the observations reveals a decoupled core and strongly non-Gaussian line-of-sight velocity profiles (VPs) in the central 0.2 arcsec (19pc). NIR imaging also indicates an asymmetric elongation of the central isophotes in the same region. We use spherical orbit-superposition models to interpret the kinematics, using a set of orthogonal ``eigenVPs'' that allow us to fit models directly to spectra. The models require a central black hole of mass $1.2^{+0.5}_{-0.6}\times10^9M_\odot$, with a strongly tangentially biased orbit distribution in the inner 40pc., Comment: 19 pages, 9 figures. Accepted for publication in MNRAS. High resolution version available at http://www-astro.physics.ox.ac.uk/~rcwh/ngc1399.ps.gz

We analyze the nuclear stellar dynamics of the SB0 galaxy NGC 1023, utilizing observational data both from the Space Telescope Imaging Spectrograph aboard the Hubble Space Telescope and from the ground. The stellar kinematics measured from these long-slit spectra show rapid rotation (V = 70 km/s at a distance of 0.1 arcsec = 4.9 pc from the nucleus) and increasing velocity dispersion toward the nucleus (where sigma = 295 +/- 30 km/s). We model the observed stellar kinematics assuming an axisymmetric mass distribution with both two and three integrals of motion. Both modeling techniques point to the presence of a central dark compact mass (which presumably is a supermassive black hole) with confidence > 99%. The isotropic two-integral models yield a best-fitting black hole mass of (6.0 +/- 1.4) x 10^7 M_sun and mass-to-light ratio (M/L_V) of 5.38 +/- 0.08, and the goodness-of-fit (chi^2) is insensitive to reasonable values for the galaxy's inclination. The three-integral models, which non-parametrically fit the observed line-of-sight velocity distribution as a function of position in the galaxy, suggest a black hole mass of (3.9 +/- 0.4) x 10^7 M_sun and M/L_V of 5.56 +/- 0.02 (internal errors), and the edge-on models are vastly superior fits over models at other inclinations. The internal dynamics in NGC 1023 as suggested by our best-fit three-integral model shows that the velocity distribution function at the nucleus is tangentially anisotropic, suggesting the presence of a nuclear stellar disk. The nuclear line of sight velocity distribution has enhanced wings at velocities >= 600 km/s from systemic, suggesting that perhaps we have detected a group of stars very close to the central dark mass., 21 pages, 12 figures, accepted in the Astrophysical Journal

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., 7 pages, 7 figures, uses mn.tex

We present our mass estimate of the central black hole in the isolated spiral galaxy NGC 4414. Using natural guide star adaptive optics assisted observations with the Gemini Near-Infrared Integral Field Spectrometer (NIFS) and the natural seeing Gemini Multi-Object Spectrographs-North (GMOS), we derived two-dimensional stellar kinematic maps of NGC 4414 covering the central 1.5 arcsec and 10 arcsec, respectively, at a NIFS spatial resolution of 0.13 arcsec. The kinematic maps reveal a regular rotation pattern and a central velocity dispersion dip down to around 105 km/s. We constructed dynamical methods using two different methods: Jeans anisotropic dynamical modeling and axisymmetric Schwarzschild modeling. Both modeling methods give consistent results, but we cannot constrain the lower mass limit and only measure an upper limit for the black hole mass of Mbh= 1.56 x 10^6 Msun(at 3 sigma level) which is at least 1 sigma below the recent Mbh-sigma_e relations. Further tests with dark matter, mass-to-light ratio variation and different light models confirm that our results are not dominated by uncertainties. The derived upper limit mass is not only below the Mbh-sigma_e relation, but is also five times lower than the lower limit black hole mass anticipated from the resolution limit of the sphere of influence. This proves that via high quality integral field data we are now able to push black hole measurements down to at least five times less than the resolution limit., Comment: 20 pages, 25 figures, accepted in A&A