Leonard-Merritt mass estimator

The Leonard–Merritt mass estimator is a formula for estimating the mass of a spherical stellar system using the apparent (angular) positions and proper motions of its component stars. The distance to the stellar system must also be known.

Like the virial theorem, the Leonard–Merritt estimator yields correct results regardless of the degree of velocity anisotropy. Its statistical properties are superior to those of the virial theorem. However, it requires that two components of the velocity be known for every star, rather than just one for the virial theorem.

The estimator has the general form

${\displaystyle \langle M(r)\rangle ={16 \over 3\pi G}\langle R\left(2V_{R}^{2}+V_{T}^{2}\right)\rangle .}$

The angle brackets denote averages over the ensemble of observed stars. ${\displaystyle M(r)}$ is the mass contained within a distance ${\displaystyle r}$ from the center of the stellar system; ${\displaystyle R}$ is the projected distance of a star from the apparent center; ${\displaystyle V_{R}}$ and ${\displaystyle V_{T}}$ are the components of a star's velocity parallel to, and perpendicular to, the apparent radius vector; and ${\displaystyle G}$ is the gravitational constant.

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