Leonard-Merritt mass estimator

The Leonard-Merritt mass estimator is a formula firstderived by Peter Leonard and David Merritt [Leonard, P. and Merritt, D. (1989). [http://adsabs.harvard.edu/abs/1989ApJ...339..195L| "The mass of the open star cluster M35 as derived from proper motions"] ] for estimating the mass of a spherical stellar systemusing 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 correctresults regardless of the degree of velocity anisotropy,although its statistical properties are generally superior to thoseof the virial theorem.

The estimator has the general form

$$

langle M(r) angle = {16over 3pi G} langle Rleft(2V_R^2 + V_T^2 ight) angle.

The angle brackets denote averages over the ensemble of observed stars.$M(r)$ is the mass contained within a distance $r$ from the centerof the stellar system; $R$ is the projected distanceof a star from the apparent center; $V\_R$ and $V\_T$ arethe components of a star's velocity parallel to, andperpendicular to, the apparent radius vector;and $G$ is the gravitational constant.

Like all estimators based on moments of the Jeans equations(including the virial theorem),the Leonard-Merritt estimator requires that one make an ad hoc assumption about the radial distribution of mass,or equivalently about the relative distribution ofmass and light.As a result, it is most useful when applied tostellar systems that have one of two properties:(1) All or almost all of the mass resides in a central object;(2) the mass is distributed in the same way as the observed stars.Case (1) applies to the nucleus of a galaxy containing a
supermassive black hole.Case (2) applies to a stellar system composed entirely of luminous stars (i.e. no dark matter or black holes).

In a cluster with constant mass-to-light ratio and total mass $M\_T$,the Leonard-Merritt estimator becomes

$$

M_T = {32over 3pi G} langle R left(2V_R^2 + V_T^2 ight) angle.

On the other hand, if all the mass is located in a central point of mass $M\_0$, then

$M\_0\; =\; \{16over\; 3pi\; G\}\; langle\; Rleft(2V\_R^2\; +\; V\_T^2\; ight)\; angle.$

In its second form, the Leonard-Merritt estimator has been successfully used tomeasure the mass of the supermassive black hole at the center of the Milky Way galaxy [Schödel, R., Ott, T., Genzel, R., Eckart, A., Mouawad, N., and Alexander, T. (2003). [http://adsabs.harvard.edu/abs/2003ApJ...596.1015S| "Stellar Dynamics in the Central Arcsecond of Our Galaxy"] ] .

See also

# Supermassive black hole
# Virial theorem
# Proper motion
# Globular cluster

References