We present a family of robust tracer mass estimators to compute the enclosed mass of galaxy haloes from samples of discrete positional and kinematical data of tracers, such as halo stars, globular clusters and dwarf satellites. The data may be projected positions, distances, line‐of‐sight velocities or proper motions. The estimators all assume that the tracer population has a scale‐free density and moves in a scale‐free potential in the region of interest. The circumstances under which the boundary terms can be discarded and the estimator converges are derived. Forms of the estimator tailored for the Milky Way Galaxy and for M31 are given. Monte Carlo simulations are used to quantify the uncertainty as a function of sample size. For the Milky Way Galaxy, the satellite sample consists of 26 galaxies with line‐of‐sight velocities. We find that the mass of the Milky Way within 300 kpc is M300= 0.9 ± 0.3 × 1012 M⊙ assuming velocity isotropy. However, the mass estimate is sensitive to the assumed anisotropy and could plausibly lie between 0.7 × 1012 and 3.4 × 1012 M⊙, if anisotropies implied by simulations or by the observations are used. Incorporating the proper motions of six Milky Way satellites into the data set, we find M300= 1.4 ± 0.3 × 1012 M⊙. The range here if plausible anisotropies are used is still broader, from 1.2 × 1012 to 2.7 × 1012 M⊙. Note that our error bars only incorporate the statistical uncertainty. There are much greater uncertainties induced by velocity anisotropy and by selection of satellite members. For M31, there are 23 satellite galaxies with measured line‐of‐sight velocities, but only M33 and IC 10 have proper motions. We use the line‐of‐sight velocities and distances of the satellite galaxies to estimate the mass of M31 within 300 kpc as M300= 1.4 ± 0.4 × 1012 M⊙ assuming isotropy. There is only a modest dependence on anisotropy, with the mass varying between 1.3 × 1012 and 1.6 × 1012 M⊙. Incorporating the proper motion data set does not change the results significantly. Given the uncertainties, we conclude that the satellite data by themselves yield no reliable insights into which of the two galaxies is actually the more massive. Leo I has long been known to dominate mass estimates for the Milky Way due to its substantial distance and line‐of‐sight velocity. We find that And XII and And XIV similarly dominate the estimated mass of M31. As such, we repeat the calculations without these galaxies, in case they are not bound – although on the balance of the evidence, we favour their inclusion in mass calculations.

We use distant blue horizontal branch stars with Galactocentric distances 16 < r < 48 kpc as kinematic tracers of the Milky Way dark halo. We model the tracer density as an oblate, power law embedded within a spherical power-law potential. Using a distribution function method, we estimate the overall power-law potential and the velocity anisotropy of the halo tracers. We measure the slope of the potential to be γ ∼ 0.4, and the overall mass within 50 kpc is ∼4 × 10 11 M . The tracer velocity anisotropy is radially biased with β ∼ 0.5, which is in good agreement with local solar neighbourhood studies. Our results provide an accurate outer circular velocity profile for the Milky Way and suggest a relatively high-concentration dark matter halo (c vir ∼ 20).