Leonard-Merritt mass estimator
Encyclopedia
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 star
s. 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
The angle brackets denote averages over the ensemble of observed stars.
is the mass contained within a distance 
from the center of the stellar system; 
is the projected distance of a star from the apparent center; 
and 
are the components of a star's velocity parallel to, and perpendicular to, the apparent radius vector; and 
is the gravitational constant
.
Like all estimators based on moments of the Jeans equations
, the Leonard–Merritt estimator requires an assumption about the relative distribution of mass and light. As a result, it is most useful when applied to stellar systems that have one of two properties:
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
, the Leonard–Merritt estimator becomes:
On the other hand, if all the mass is located in a central point of mass
, then:
In its second form, the Leonard–Merritt estimator has been successfully used to measure the mass of the supermassive black hole
at the center of the Milky Way
galaxy
.
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s. 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
The angle brackets denote averages over the ensemble of observed stars.
is the mass contained within a distance from the center of the stellar system; is the projected distance of a star from the apparent center; and are the components of a star's velocity parallel to, and perpendicular to, the apparent radius vector; and is the gravitational constantGravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
.
Like all estimators based on moments of the Jeans equations
Jeans equations
The Jeans equations describe the motion of a collection of stars in a gravitational field.If ν = ν is the density of stars in space, as a function of position x = and time t, v = is the velocity, and Φ = Φ is the gravitational potential,the...
, the Leonard–Merritt estimator requires an assumption about the relative distribution of mass and light. As a result, it is most useful when applied to stellar systems that have one of two properties:
- All or almost all of the mass resides in a central object, or,
- 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
Supermassive black hole
A supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...
. Case (2) applies to a stellar system composed entirely of luminous stars (i.e. no dark matter
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
or black holes).
In a cluster with constant mass-to-light ratio and total mass
, the Leonard–Merritt estimator becomes:On the other hand, if all the mass is located in a central point of mass
, then:In its second form, the Leonard–Merritt estimator has been successfully used to measure the mass of the supermassive black hole
Supermassive black hole
A supermassive black hole is the largest type of black hole in a galaxy, in the order of hundreds of thousands to billions of solar masses. Most, and possibly all galaxies, including the Milky Way, are believed to contain supermassive black holes at their centers.Supermassive black holes have...
at the center of the Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...
galaxy
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
.