A calculation of structural parameters of spherical stellar systems by means of characteristic functions method. I. Theory

To calculate structural parameters of stellar systems such as an effective radius and central space (or surface) density, the method of characteristic functions is suggested. The characteristic function of the system is a Fourier image of their normalized space density profile f₃(r). In the case of spherical symmetry the probability distribution of r (Q₃(r) = (3/a³)r²f₃(r)) and its orthogonal projections have the same characteristic functions. This fact is used to calculate the effective radii of a few star cluster models (King law, Plummer model and Gausian profile). It is shown, that the characteristic function for King law clusters tends to a finite generalised function if the concentration parameter c is large. The expression for the effective radius (at c ≫ 1) is given. The formula of the effective radius in the Plummer model as well as the relation between the one-dimensional central velocity dispersion and the root mean square velocity are obtained. It is shown, that in the Gaussian model and for King law clusters the effective radius (half-mass visual radius) can differ from the effective (harmonic) radius a few times. This fact should be taken into account in estimating the mass-to-light ratio from the virial mass of such systems using the King radius.