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In spherical stellar systems, a given mass density allows an infinity of distribution functions. This indeterminacy is illustrated with a one-parameter family of anisotropic models. They all satisfy the Plummer law in the mass density, but have different velocity dispersions. Moreover, the stars are not confined to a particular subset of the total accessible phase space. This family is explored analytically in detail (the moments, various energy distributions, the line profiles can all be calculated). Even when both the mass density and the velocity dispersion profiles are required to be the same, a degeneracy in the model space persists, which can be shown with a three-parameter generalization of the above family. The observational consequence of the degeneracy are studied by calculating the observable line profiles obtained with different distribution functions. This provides an estimate of the accuracy needed in the observed shape of the line profiles in order to be of use in the determination of the distribution function.
H. Dejonghe (Thu,) studied this question.