How to break the density-anisotropy degeneracy in spherical stellar systems

We present a new non-parametric Jeans code, GravSphere, that recovers the density $\rho(r)$ and velocity anisotropy $\beta(r)$ of spherical stellar systems, assuming only that they are in a steady-state. Using a large suite of mock data, we confirm that with only line-of-sight velocity data, GravSphere provides a good estimate of the density at the projected stellar half mass radius, $\rho(R_{1/2})$, but is not able to measure $\rho(r)$ or $\beta(r)$, even with 10,000 tracer stars. We then test three popular methods for breaking this $\rho-\beta$ degeneracy: using multiple populations with different $R_{1/2}$; using higher order `Virial Shape Parameters' (VSPs); and including proper motion data. We find that two populations provide an excellent recovery of $\rho(r)$ in-between their respective $R_{1/2}$. However, even with a total of $\sim 7,000$ tracers, we are not able to well-constrain $\beta(r)$ for either population. By contrast, using 1000 tracers with higher order VSPs we are able to measure $\rho(r)$ over the range $0.5 < r/R_{1/2} < 2$ and broadly constrain $\beta(r)$. Including proper motion data for all stars gives an even better performance, with $\rho$ and $\beta$ well-measured over the range $0.25 < r/R_{1/2} < 4$. Finally, we test GravSphere on a triaxial mock galaxy that has axis ratios typical of a merger remnant, $[1:0.8:0.6]$. In this case, GravSphere can become slightly biased. However, we find that when this occurs the data are poorly fit, allowing us to detect when such departures from spherical symmetry become problematic.