General spherical anisotropic Jeans models of stellar kinematics: including proper motions and radial velocities

Cappellari (2008) presented a flexible and efficient method to model the stellar kinematics of anisotropic axisymmetric and spherical stellar systems. The spherical formalism could be used to model the line-of-sight velocity second moments allowing for essentially arbitrary radial variation in the anisotropy and general luminous and total density profiles. Here we generalize the spherical formalism by providing the expressions for all three components of the projected second moments, including the two proper motion components. A reference implementation is now included in the public JAM package available at http://purl.org/cappellari/softwareComment: 3 pages, 2 figures, LaTeX. Not submitted anywhere but here. Software implementing the update to the JAM method described in this paper is available at http://purl.org/cappellari/softwar

Online Neural Path Guiding with Normalized Anisotropic Spherical Gaussians

The variance reduction speed of physically-based rendering is heavily affected by the adopted importance sampling technique. In this paper we propose a novel online framework to learn the spatial-varying density model with a single small neural network using stochastic ray samples. To achieve this task, we propose a novel closed-form density model called the normalized anisotropic spherical gaussian mixture, that can express complex irradiance fields with a small number of parameters. Our framework learns the distribution in a progressive manner and does not need any warm-up phases. Due to the compact and expressive representation of our density model, our framework can be implemented entirely on the GPU, allowing it produce high quality images with limited computational resources

A nonlinear dynamics approach to Bogoliubov excitations of Bose-Einstein condensates

We assume the macroscopic wave function of a Bose-Einstein condensate as a superposition of Gaussian wave packets, with time-dependent complex width parameters, insert it into the mean-field energy functional corresponding to the Gross-Pitaevskii equation (GPE) and apply the time-dependent variational principle. In this way the GPE is mapped onto a system of coupled equations of motion for the complex width parameters, which can be analyzed using the methods of nonlinear dynamics. We perform a stability analysis of the fixed points of the nonlinear system, and demonstrate that the eigenvalues of the Jacobian reproduce the low-lying quantum mechanical Bogoliubov excitation spectrum of a condensate in an axisymmetric trap.Comment: 7 pages, 3 figures, Proceedings of the "8th International Summer School/Conference Let's Face Chaos Through Nonlinear Dynamics", CAMTP, University of Maribor, Slovenia, 26 June - 10 July 201

Mass estimates from stellar proper motions: The mass of ω\omega Centauri

We lay out and apply methods to use proper motions of individual kinematic tracers for estimating the dynamical mass of star clusters. We first describe a simple projected mass estimator and then develop an approach that evaluates directly the likelihood of the discrete kinematic data given the model predictions. Those predictions may come from any dynamical modelling approach, and we implement an analytic King model, a spherical isotropic Jeans equation model and an axisymmetric, anisotropic Jeans equation model.We apply these approaches to the enigmatic globular cluster omega Centauri, combining the proper motion from van Leeuwen et al (2000) with improved photometric cluster membership probabilities. We show that all mass estimates based on spherical isotropic models yield $(4.55\pm 0.1) \times 10^6 M_{\odot} [D/5.5 \pm 0.2 kpc]^3$, where our modelling allows us to show how the statistical precision of this estimate improves as more proper motion data of lower signal-to-noise are included. MLM predictions, based on an anisotropic axisymmetric Jeans model, indicate for $\omega$ Cen that the inclusion of anisotropies is not important for the mass estimates, but that accounting for the flattening is: flattened models imply $(4.05\pm 0.1) \times 10^6 M_{\odot} [D/5.5 \pm 0.2 kpc]^3$, 10% lower than when restricting the analysis to a spherical model. The best current distance estimates imply an additional uncertainty in the mass estimate of 12%.Comment: Accepted for publication in MNRA

Measuring the inclination and mass-to-light ratio of axisymmetric galaxies via anisotropic Jeans models of stellar kinematics

We present a simple and efficient anisotropic generalization of the semi-isotropic (two-integral) axisymmetric Jeans formalism which is used to model the stellar kinematics of galaxies. The following is assumed: (i) a constant mass-to-light ratio M/L and (ii) a velocity ellipsoid that is aligned with cylindrical coordinates (R,z) and characterized by the classic anisotropy parameter beta_z=1-sigma_z^2/sigma_R^2. Our simple models are fit to SAURON integral-field observations of the stellar kinematics for a set of fast-rotator early-type galaxies. With only two free parameters (beta_z and the inclination) the models generally provide remarkably good descriptions of the shape of the first (V) and second (V_rms=sqrt{V^2+sigma^2}) velocity moments, once a detailed description of the surface brightness is given. This is consistent with previous findings on the simple dynamical structure of these objects. With the observationally-motivated assumption that beta_z>0, the method is able to recover the inclination. The technique can be used to determine the dynamical mass-to-light ratios and angular momenta of early-type fast-rotators and spiral galaxies, especially when the quality of the data does not justify more sophisticated modeling approaches. This formalism allows for the inclusion of dark matter, supermassive black holes, spatially varying anisotropy, and multiple kinematic components.Comment: 16 pages, 7 figures, LaTeX. Published in MNRAS. Software implementing the JAM method described in this paper is available at http://www-astro.physics.ox.ac.uk/~mxc/idl

Relation between the eigenfrequencies of Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian in a time-dependent variational approach

We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates, and the eigenvalues of the Jacobian stability matrix in a variational approach which maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap, and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibits a Rydberg-like structure, which can be explained by means of a quantum defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussians combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction, and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find a good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates, and the eigenfrequencies of the excitation modes are compared.Comment: 15 pages, 12 figure

Real-Time Hand Tracking Using a Sum of Anisotropic Gaussians Model

Real-time marker-less hand tracking is of increasing importance in human-computer interaction. Robust and accurate tracking of arbitrary hand motion is a challenging problem due to the many degrees of freedom, frequent self-occlusions, fast motions, and uniform skin color. In this paper, we propose a new approach that tracks the full skeleton motion of the hand from multiple RGB cameras in real-time. The main contributions include a new generative tracking method which employs an implicit hand shape representation based on Sum of Anisotropic Gaussians (SAG), and a pose fitting energy that is smooth and analytically differentiable making fast gradient based pose optimization possible. This shape representation, together with a full perspective projection model, enables more accurate hand modeling than a related baseline method from literature. Our method achieves better accuracy than previous methods and runs at 25 fps. We show these improvements both qualitatively and quantitatively on publicly available datasets.Comment: 8 pages, Accepted version of paper published at 3DV 201

N-body realizations of cuspy dark matter haloes

We describe an algorithm for generating equilibrium initial conditions for numerical experiments with dark matter haloes. Our haloes are modelled using a general form for the mass density p{r), making it possible to represent most of the popular density profiles in the literature. The finite mass 7-models and the cuspy density profiles found in recent high-resolution cosmological TV-body simulations having a density power-law fall-off at large distances proportional to are included as special cases. The algorithm calculates the phase-space distribution function of each model assuming spherical symmetry and either an isotropic velocity dispersion tensor or an anisotropic velocity dispersion tensor of the type proposed by Osipkov and Merritt. The particle velocities are assigned according to the exact velocity distribution, making this method ideal for experiments requiring a high degree of stability. Numerical tests confirm that the resulting models are highly stable. This approach is motivated by the instabilities that arise when a local Maxwellian velocity distribution is adopted. For example, after approximating the velocity distribution by a Gaussian we show that a Hernquist halo with an initial r(^-1) density cusp immediately develops a constant density core. Moreover, after a single crossing time the orbital anisotropy has evolved over the entire system. Previous studies that use this approximation to construct halo or galaxy models could be compromised by this behaviour. Using the derived distribution functions we show the exact 1-d velocity distributions and we compare them with the Gaussian velocity distributions with the same second moment for different distances from the halo centre. We show that instabilities arise because a Gaussian velocity distribution is a very poor approximation to the true velocity distribution of particles. We also perform a series of numerical simulations evolving several dark matter halo models in isolation, with the intention of checking the stability of the initialization procedure in both configuration and velocity space. A subset of the models are evolved under the assumption that the velocity distribution at any given point is a Gaussian and the time evolution of the density profiles and velocity structure is monitored. Finally, a number of applications are discussed, including issues of relaxation in dark matter haloes as well as mergers of haloes in scattering experiments

A simple method for evaluating low-energy electron-molecule scattering cross sections using discrete basis functions

We present a simple, approximate method for calculating low-energy electron-molecule scattering cross sections using only the results of a basis set diagonalization of the molecular Hamiltonian. The method is based on the approximate conservation of orbital angular momentum in collisions between slow electrons and molecules lacking a permanent dipole moment (low l spoiling). Results are presented for e--H2, and e--N2, in the static-exchange approximation