Reliability of the measured velocity anisotropy of the Milky Way stellar halo

Abstract: Determining the velocity distribution of halo stars is essential for estimating the mass of the Milky Way and for inferring its formation history. Since the stellar halo is a dynamically hot system, the velocity distribution of halo stars is well described by the 3-dimensional velocity dispersions $(\sigma_r, \sigma_\theta, \sigma_\phi)$, or by the velocity anisotropy parameter $\beta=1-(\sigma_\theta^2+\sigma_\phi^2)/(2\sigma_r^2)$. Direct measurements of $(\sigma_r, \sigma_\theta, \sigma_\phi)$ consistently suggest $\beta =0.5$-$0.7$ for nearby halo stars. In contrast, the value of $\beta$ at large Galactocentric radius $r$ is still controversial, since reliable proper motion data are available for only a handful of stars. In the last decade, several authors have tried to estimate $\beta$ for distant halo stars by fitting the observed line-of-sight velocities at each radius with simple velocity distribution models (local fitting methods). Some results of local fitting methods imply $\beta<0$ at $r \gtrsim 20 \;\rm{kpc}$, which is inconsistent with recent predictions from cosmological simulations. Here we perform mock-catalogue analyses to show that the estimates of $\beta$ based on local fitting methods are reliable only at $r \leq 15 \;\rm{kpc}$ with the current sample size ($\sim10^3$ stars at a given radius). As $r$ increases, the line-of-sight velocity (corrected for the Solar reflex motion) becomes increasingly closer to the Galactocentric radial velocity, so that it becomes increasingly more difficult to estimate tangential velocity dispersion $(\sigma_\theta, \sigma_\phi)$ from line-of-sight velocity distribution. Our results suggest that the forthcoming Gaia data will be crucial for understanding the velocity distribution of halo stars at $r \geq 20\;\rm{kpc}$.