On the maximum entropy and the problem of moments. An application to stellar kinematics

The maximum entropy approach is used to solve the classical moment problem of stellar kinematics. If an extended set of moments is available, the current method provides a linear estimation algorithm, which is given by a Gramian system of equations, that leads to a fast and suitable estimation of the velocity distribution. In particular, it can be used as an alternative approach for modelling multimodal distributions that can not be described through gaussian mixtures. Two samples drawn from the Hipparcos Catalog are studied. The first one, which is composed of a non-gaussian mixture of early-type and young disk stars, is used to test the ability of the procedure in reconstructing a truncated distribution associated with a bounded stellar sample. By fitting up to tenth moments, the maximum entropy approach gives a realistic portrait of its actual asymmetries. The second one is a nearly complete and kinematically representative sample of the solar neighbourhood, containing thin and thick disk populations. The local velocity distribution can be described by fitting moments up to sixth-order as a product of two exponential functions in the form f = φ1(Q)φ2(h), where Q is a quadratic positive definite form, which gives the background ellipsoidal shape of the distribution, and the function φ2(h), which can be written in terms of the angular momentum integral h, is a perturbation factor, which is quadratic in the V velocity alone, and gives account of the skewness and of the slight shift of the velocity ellipsoids in terms of the rotation velocity.