An Inverse Look at the Center of M15

The observed radial and transverse velocities of individual stars in M15 are implemented as inputs to a fully non-parametric code (CHASSIS) in order to estimate the equilibrium stellar distribution function and the three-dimensional mass density profile. In particular, the paper explores the possibility of the existence of a central black hole in M15 via several runs that utilize the radial velocity data set which offers kinematic measurements closer to the centre of the cluster than the proper motion data. These runs are distinguished from each other in the choice of the initial seed for the cluster characteristics; however, the profiles identified by the algorithm at the end of each run concur with each other, within error bars, thus confirming the robustness of CHASSIS. The recovered density profiles are noted to exhibit unequivocal flattening, inner to about 0.0525pc. Also, the enclosed mass profile is very close to being a power-law function of radius inside 0.1pc and is not horizontal. Simplistically speaking, these trends negate the possibility of the central mass to be concentrated in a black hole, the lower bound on the radius of the sphere of influence of which would be $\gtrsim$0.041pc, had it existed. However, proper analysis suggests that the mass enclosed within the inner 0.01pc could be in the form of a black hole of mass $\sim{10^3}$M$_{\odot}$, under two different scenarios, which are discussed. The line-of-sight velocity dispersion is visually found to be very similar to the observed dispersion profile. The enclosed mass and velocity dispersion profiles calculated from runs done with the proper motion data are found to be consistent with the profiles obtained with the radial velocity data.Comment: Accepted for publication in AJ; 6 figure

Mass Modelling with Minimum Kinematic Information

Mass modelling of early-type systems is a thorny issue. In this paper, we present a new mass modelling formalism for ellipticals that invokes no observations other than the central velocity dispersion ($\sigma_0$) and photometry. The essence of the method lies in choosing a local mass-to-light ratio ($M/L$) profile for a galaxy, with which the deprojected luminosity density distribution (along the major axis coordinate $x$) is scaled. The resulting discontinuous mass density profile is then smoothed, according to a laid out prescription; the local $M/L$ profile that stems from this smoothed mass density, is found to be significantly different from the raw $M/L$ distribution. A suite of model galaxies (both Sersic and cored in nature) is used for extensive experimentation in order to characterise this raw $M/L$ profile and in each case, the mass density recovered from this mass modelling technique is compared to the known mass distribution. We opt to work with a raw $M/L$ profile that is a simple two-stepped function of $x$, with a low inner and higher outer value of $M/L-\Upsilon_{in}$ and $\Upsilon_{out}$, respectively. The only constraint that we have on this profile is in the centre of the galaxy, via $\sigma_0$. This value of $\sigma_0$ is implemented in the virial theorem to obtain an estimate of the central $M/L$ ratio of the galaxy. The fallibility of the virial mass estimate is taken care of, by allowing for a range in the values of $\Upsilon_{in}$ that can be used for a given galaxy model. Moreover, our experiments indicate that $\Upsilon_{out}$ is uniquely known, for a given $\Upsilon_{in}$; for cored galaxies, this functional form is found uniquely dependant on the core radius. The jump radius of the raw $M/L$ profile is chosen to be thrice the effective radius of the galaxy.Comment: 9 figures, accepted for publication in MNRA

Local Phase Space - Shaped by Chaos?

We attempt to understand the state of the local phase space by comparing simulated 2-D velocity distributions to the distribution that is constructed for the solar neighbourhood, from measurements of stellar radial and transverse velocities. The joint perurbation of the central bar in the Galaxy and the spiral pattern is found to be a must, in order to produce successful models of the local phase space. The existence of chaos is found to be an important ingredient in the formation of the observed phase space structure.Comment: 8 pages, 3 figures, proceedings of the conference: "Chaos in Astronomy", Athens, September 2007, G. Contopoulos and P.A. Patsis (eds), to be published by Springe

Total mass distributions of Sersic galaxies from photometry & cent\ ral velocity dispersion

We develop a novel way of finding total mass density profiles in Sersic ellipticals, to about 3 times the major axis effective radius, using no other information other than what is typically available for distant galaxies, namely the observed surface brightness distribution and the central velocity dispersion $\sigma_0$. The luminosity density profile of the observed galaxy is extracted by deprojecting the measured brightness distribution and scaling it by a fiduciary, step-function shaped, $raw$ mass-to-light ratio profile ($M/L$). The resulting raw, discontinuous, total, 3-D mass density profile is then smoothed according to a proposed smoothing prescription. The parameters of this raw $M/L$ are characterised by implementing the observables in a model-based study. The complete characterisation of the formalism is provided as a function of the measurements of the brightness distribution and $\sigma_0$. The formalism, thus specified, is demonstrated to yield the mass density profiles of a suite of test galaxies and is successfully applied to extract the gravitational mass distribution in NGC 3379 and NGC 4499, out to about 3 effective radii.Comment: accepted for publication in A&

An Inverse Problem Approach to Cluster Dynamics

We propose a new non-parametric algorithm that can be implemented to study and characterize stellar clusters. The scheme attempts to simultaneously recover the stellar distribution function and the cluster potential by using projected radii and velocity information about the cluster members. The pair of these functions that is most consistent with the input data is detected by the Metropolis algorithm. In this work, the cluster characteristics recovered by CHASSIS are calibrated against the N-body realizations of two clusters, namely Hyades and Arches. The cluster mass and line-of-sight projected velocity dispersion profiles are correctly reproduced by the algorithm when the cluster obeys the assumption used in the code, namely isotropy in phase space. The results recovered by the code are shown to be insensitive to the choice of the initial parameters. The results are also not influenced by increasing the number of input data points as long as this number exceeds a minimum value which is moderately low for an input data set that obeys the assumptions of isotropy and sphericity.Comment: 30 pages, 12 figures, accepted for publication in the Astronomical Journa

Minimum Distance Estimation of Milky Way Model Parameters and Related Inference

We propose a method to estimate the location of the Sun in the disk of the Milky Way using a method based on the Hellinger distance and construct confidence sets on our estimate of the unknown location using a bootstrap based method. Assuming the Galactic disk to be two-dimensional, the sought solar location then reduces to the radial distance separating the Sun from the Galactic center and the angular separation of the Galactic center to Sun line, from a pre-fixed line on the disk. On astronomical scales, the unknown solar location is equivalent to the location of us earthlings who observe the velocities of a sample of stars in the neighborhood of the Sun. This unknown location is estimated by undertaking pairwise comparisons of the estimated density of the observed set of velocities of the sampled stars, with densities estimated using synthetic stellar velocity data sets generated at chosen locations in the Milky Way disk according to four base astrophysical models. The "match" between the pair of estimated densities is parameterized by the affinity measure based on the familiar Hellinger distance. We perform a novel cross-validation procedure to establish a desirable "consistency" property of the proposed method.Comment: 25 pages, 10 Figures. This version incorporates the suggestions made by the referees. To appear in SIAM/ASA Journal on Uncertainty Quantificatio

A Non-Parametric Estimate of the Mass of the Central Black Hole in the Galaxy

We estimate the mass of the central black hole in our Galaxy from stellar kinematical data published by Ghez et al. (1998) and Genzel et al. (2000). For this we develop a method, related to Merritt (1993), for non-parametrically reconstructing the mass profile and the stellar distribution function in the central region of the Galaxy from discrete kinematic data, including velocity errors. Models are searched using the Metropolis algorithm. We assume that the mass distribution is spherical and the stellar velocity distribution is isotropic, and devise a test of this assumption. From proper motions we obtain an enclosed mass of $2.0\pm{0.7}\times10^6{\rm M}_{\odot}$ within the inner $0.0044\rm pc$, from radial velocities we obtain a mass of $2.2^{+1.6}_{-1.0}\times10^6{\rm M}_{\odot}$ within 0.046pc and from three-dimensional velocities we obtain $1.8^{+0.4}_{-0.3}\times10^6{\rm M}_{\odot}$ within 0.046pc.Comment: 27 pages, 8 figures, accepted for publication in the July issue of A

DOPING: a New Non-parametric Deprojection Scheme

We present a new non-parametric deprojection algorithm DOPING (Deprojection of Observed Photometry using and INverse Gambit), that is designed to extract the three dimensional luminosity density distribution $\rho$, from the observed surface brightness profile of an astrophysical system such as a galaxy or a galaxy cluster, in a generalised geometry, while taking into account changes in the intrinsic shape of the system. The observable is the 2-D surface brightness distribution of the system. While the deprojection schemes presented hitherto have always worked within the limits of an assumed intrinsic geometry, in DOPING, geometry and inclination can be provided as inputs. The $\rho$ that is most likely to project to the observed brightness data is sought; the maximisation of the likelihood is performed with the Metropolis algorithm. Unless the likelihood function is maximised, $\rho$ is tweaked in shape and amplitude, while maintaining positivity, but otherwise the luminosity distribution is allowed to be completely free-form. Tests and applications of the algorithm are discussed.Comment: 8 pages; to be published in IJMP(D) (Feb, 2008 issue), Vol 17, No. 2, as part of proceedings for the 6th International Workshop on Data Analysis in Astronomy, ``Modelling and Simulations in Science'