Constructing distribution functions for spherical galaxies

Given a complete set of photometric and spectroscopic observations of a spherical galaxy, and an adopted form of the potential $$\Phi(r)$$, Binney & Mamon (1982) showed how to calculate stellar density $$\nu(r)$$ and radial velocity dispersion $$\sigma_\text r(r)$$ profiles. However, their work left open the question of whether these profiles can be derived from a non-negative distribution function, $$f$$. We have developed an algorithm for constructing a non-negative distribution function $$f(\epsilon, J)$$ that generates given runs of $$\nu \enspace \text {and}\enspace \sigma_\text r$$, in the adopted Φ. Tests of the algorithm demonstrate its ability to reproduce a distribution function which yields specified $$\nu \enspace \text {and}\enspace \sigma_\text r$$ profiles, and illustrate the behaviour of the algorithm when $$\nu \enspace \text {and}\enspace \sigma_\text r$$ have been derived from a partly negative distribution function. We have applied our algorithm to the $$\nu \enspace \text {and}\enspace \sigma_\text r$$ profiles of M87 obtained by Binney & Mamon. The resulting distribution function yields surface brightness and projected velocity dispersion profiles consistent with the observations of Young et al. (1978) and Sargent et al. (1978) to within the estimated observational errors.