The Leonard–Merritt mass estimator is a formula for estimating the mass of a spherical

stellar system A star system or stellar system is a small number of stars that orbit each other, bound by gravitational attraction. A large group of stars bound by gravitation is generally called a ''star cluster'' or ''galaxy'', although, broadly speaking, ...

using the apparent (angular) positions and

proper motions Proper motion is the astrometric measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distan ...

of its component

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s. The distance to the stellar system must also be known. Like the

virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...

, 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

\langle M(r)\rangle =  
\langle R\left(2V_R^2 + V_T^2\right)\rangle.

The angle brackets denote averages over the ensemble of observed stars.

M(r)

is the mass contained within a distance

r

from the center of the stellar system;

R

is the projected distance of a star from the apparent center;

V_R

and

V_T

are the components of a star's velocity parallel to, and perpendicular to, the apparent radius vector; and

G

is the

gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...

. Like all estimators based on moments of the

Jeans equations The Jeans equations are a set of partial differential equations that describe the motion of a collection of stars in a gravitational field. The Jeans equations relate the second-order velocity moments to the density and potential of a stellar sy ...

, 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. Case (2) applies to a stellar system composed entirely of luminous stars (i.e. no

dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not ab ...

black holes A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

). In a cluster with constant mass-to-light ratio and total mass

M_T

, the Leonard–Merritt estimator becomes:

M_T =  
\langle R \left(2V_R^2 + V_T^2\right)\rangle.

On the other hand, if all the mass is located in a central point of mass

M_0

, then:

M_0 =  
\langle R\left(2V_R^2 + V_T^2\right)\rangle.

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 The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye ...

galaxy A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...

References

{{DEFAULTSORT:Leonard-Merritt Mass Estimator Astrometry Celestial mechanics Stellar astronomy Supermassive black holes

See also

References