Line Profiles from Discrete Kinematic Data

We develop a method to extract the shape information of line profiles from discrete kinematic data. The Gauss-Hermite expansion, which is widely used to describe the line of sight velocity distributions extracted from absorption spectra of elliptical galaxies, is not readily applicable to samples of discrete stellar velocity measurements, accompanied by individual measurement errors and probabilities of membership. We introduce two parameter families of probability distributions describing symmetric and asymmetric distortions of the line profiles from Gaussianity. These are used as the basis of a maximum likelihood estimator to quantify the shape of the line profiles. Tests show that the method outperforms a Gauss-Hermite expansion for discrete data, with a lower limit for the relative gain of approx 2 for sample sizes N approx 800. To ensure that our methods can give reliable descriptions of the shape, we develop an efficient test to assess the statistical quality of the obtained fit. As an application, we turn our attention to the discrete velocity datasets of the dwarf spheroidals of the Milky Way. In Sculptor, Carina and Sextans the symmetric deviations are consistent with velocity distributions more peaked than Gaussian. In Fornax, instead, there is an evolution in the symmetric deviations of the line profile from a peakier to more flat-topped distribution on moving outwards. These results suggest a radially biased orbital structure for the outer parts of Sculptor, Carina and Sextans. On the other hand, tangential anisotropy is favoured in Fornax. This is all consistent with a picture in which Fornax may have had a different evolutionary history to Sculptor, Carina and Sextans.