Near-Gaussian distributions for modelling discrete stellar velocity data with heteroskedastic uncertainties

ABSTRACT The velocity distributions of stellar tracers in general exhibit weak non-Gaussianity encoding information on the orbital composition of a galaxy and the underlying potential. The standard solution for measuring non-Gaussianity involves constructing a series expansion (e.g. the Gauss–Hermite series) that can produce regions of negative probability density. This is a significant issue for the modelling of discrete data with heteroskedastic uncertainties. Here, we introduce a method to construct positive-definite probability distributions by the convolution of a given kernel with a Gaussian distribution. Further convolutions by observational uncertainties are trivial. The statistics (moments and cumulants) of the resulting distributions are governed by the kernel distribution. Two kernels (uniform and Laplace) offer simple drop-in replacements for a Gauss–Hermite series for negative and positive excess kurtosis distributions with the option of skewness. We demonstrate the power of our method by an application to real and mock line-of-sight velocity data sets on dwarf spheroidal galaxies, where kurtosis is indicative of orbital anisotropy and hence a route to breaking the mass–anisotropy degeneracy for the identification of cusped versus cored dark matter profiles. Data on the Fornax dwarf spheroidal galaxy indicate positive excess kurtosis and hence favour a cored dark matter profile. Although designed for discrete data, the analytic Fourier transforms of the new models also make them appropriate for spectral fitting, which could improve the fits of high-quality data by avoiding unphysical negative wings in the line-of-sight velocity distribution.

Modified Hermite Operational Matrix Method for Nonlinear Lane-Emden Problem

Nonlinear Lane –Emden equations are significant in astrophysics and mathematical physics. The aim of this paper is submitted a new approximate method for finding solution to nonlinear Lane-Emden type equations appearing in astrophysics based on modified Hermite integration operational matrix. The suggest technique is based on taking the truncated modified Hermite series of the solution function in the nonlinear Lane-Emden equation and then changed into a matrix equation with the given conditions. Nonlinear system of algebraic equation using collection points is obtained. The present method is applied on some relevant physical problems as nonlinear Lane-Emden type equations.

Numerical solutions of a class of nonlinear ordinary differential equations in Hermite series

The purpose of this paper is to present a Hermite polynomial approach for solving a high-order ODE with non-linear terms under mixed conditions. The method we used is a matrix method based on collocation points together with truncated Hermite series and reduces the solution of equation to solution of a matrix equation which corresponds to a system of non-linear algebraic equations with unknown Hermite coefficients. In addition, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing result in literature.

Radial velocity variability in post-AGB stars: V448 Lac

To investigate the binary hypothesis in the formation of planetary nebulae, we have been doing long-term photometry and radial velocity (RV) monitoring of bright post-AGB stars which possess bipolar or ellipsoidal nebulae but no indication of a disk in their spectral energy distribution, indicative of a binary companion. RV’s are determined by cross correlating high-resolution spectra with a line mask. Stellar variability and companions both deform the cross correlation function (CCF) and induce periodic variations in the RV. To uniformly quantify the asymmetry of the CCF from a Gaussian, we propose to fit the CCF profile with a Gauss-Hermite series and determine all CCF parameters (RV, skewness, FWHM, and depth) in one single fit. We analyze the correlation and time series of these CCF parameters for V448 Lac and conclude that its RV variability is most likely due to stellar pulsation and not to an orbiting body.