Analytical solutions to the mass-anisotropy degeneracy with higher order Jeans analysis: a general method

The Jeans analysis is often used to infer the total density of a system by relating the velocity moments of an observable tracer population to the underlying gravitational potential. This technique has recently been applied in the search for dark matter (DM) in objects such as dwarf spheroidal galaxies where the presence of DM is inferred via stellar velocities. A precise account of the density is needed to constrain the expected gamma-ray flux from DM self-annihilation and to distinguish between cold and warm DM models. Unfortunately, the traditional method of fitting the second-order Jeans equation to the tracer dispersion suffers from an unbreakable degeneracy of solutions due to the unknown velocity anisotropy of the projected system. To tackle this degeneracy, one can appeal to higher moments of the Jeans equation. By introducing an analogue to the Binney anisotropy parameter at fourth order, beta(') we create a framework that encompasses all solutions to the fourth-order Jeans equations rather than the restricted range imposed by the separable augmented density. The condition beta(') = f(beta) ensures that the degeneracy is lifted and we interpret the separable augmented density system as the order-independent case beta(') = beta. For a generic choice of beta('), we present the line-of-sight projection of the fourth moment and how it could be incorporated into a joint likelihood analysis of the dispersion and kurtosis. The framework is then extended to all orders such that constraints may be placed to ensure a physically positive distribution function. Having presented the mathematical framework, we then use it to make preliminary analyses of simulated dwarf spheroidal data leading to interesting results which strongly motivate further study.