- 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 sphericalstellar system using the apparent (angular) positions andproper motions of its componentstars .The distance to the stellar system must also be known.Like thevirial 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 asupermassive black hole .Case (2) applies to a stellar system composed entirely of luminous stars (i.e. nodark matter orblack 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 theMilky 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
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