1. Mass Estimators in Astrophysics
One of the most fundamental physical parameters of astrophysical systems is their mass. There are several techniques used to estimate masses (absolute measures are only possible in a few very well defined cases) that rely on the general balance between gravity and motion in systems that are in equilibrium.

2. Mass Estimators: The simplest case = Bound Circular Orbit
Consider a test particle in orbit, mass m, velocity v, radius R, around a body of mass M
½ mv2 = GmM/R 
M = ½ v2 R /G
This formula gets modified for elliptical orbits. What about complex systems of particles?

3. For Multi-Object Systems
There are several estimators we use, the two most popular (and simplest) are the Virial Estimator, which is based on the Virial Theorem, and the Projected Mass Estimator.
For these estimators, all you need are the positions and velocities of the test particles (e.g. the galaxies in the system). The velocities we measure are usually only the

4. Line-of-sight (l.o.s.) Velocities, so some correction must be made to account for velocities (and thus energy) perpendicular to the l.o.s. Ditto, the positions of objects we usually measure are just the positions on the plane of the sky, so some correction needs to be made for physical separations along the l.o.s.
We usually simplify by assuming that our systems are spherical.

5. The Virial Theorem
Consider a moment of inertia for a system of N particles and its derivatives:
I = ½ Σ mi ri . ri (a moment of inertia)
I = dI/dt = Σ mi ri . ri
I = d2I/dt2 = Σ mi (ri . ri + ri . ri ) N
i=1 . . ..

6. Σ miri . ri = 2T (twice the Kinetic Energy) ..
Assume that the N particles have mi and ri and are self gravitating --- their mass forms the overall potential.
We can use the equation of motion to elimiate
ri.
miri = - Σ (ri - rj )
and note that
Σ miri . ri = 2T (twice the Kinetic Energy) .. ..
Gmimj
|ri –rj| 3
j = i
. .

7. = - ½ Σ Σ = U the potential energy
Then we can write (after substitution)
I – 2T = - Σ Σ ri . (ri – rj)
= - Σ Σ rj . (rj – ri)
= - ½ Σ Σ (ri - rj).(ri – rj)
= - ½ Σ Σ = U the potential energy ..
Gmi mj
|ri - rj|3
i j=i
reversing labels
Gmi mj
j i=j
|rj - ri|3
Gmi mj
adding
|ri - rj|3
i j=i
Gmi mj
|ri - rj|

8. ∴ I = 2T + U
If we have a relaxed (or statistically steady) system which is not changing shape or size, d2I/dt2 = I = 0
2T + U = 0; U = -2T; E = T+U = ½ U
conversely, for a slowly changing or periodic system <T> + <U> = 0 .. ..
Virial Equilibrium

9. Virial Mass Estimator
We use the Virial Theorem to estimate masses of astrophysical systems (e.g. Zwicky and Smith and the discovery of Dark Matter)
Go back to:
Σ mi<vi2> = ΣΣ Gmimj < >
where < > denotes the time average, and we have N point masses of mass mi, position ri
and velocity vi N N 1
|ri – rj|
i=1
i=1 j<i

10. projected radial averaged over solid angle Ω
The observables are (1) the l.o.s. time average
velocity:
< v2R,i> Ω = 1/3 vi2
projected radial averaged over solid angle Ω
i.e. we only see the radial component of motion &
vi ~ √3 vr
Ditto for position, we see projected radii,
R = θ d , d = distance, θ = angular separation

11. 1 and < >Ω = = = π/2 ∫0π dθ ∫(sinθ)-1 dΩ ∫0πsinθ dθ
So taking the average projection,
< >Ω = < >Ω
and
< >Ω = = = π/2 1 1 1
|ri – rj|
|ri – rj|
sin θij
∫0π dθ
∫(sinθ)-1 dΩ 1
sin θij
∫0πsinθ dθ dΩ

12. Σ vi2 = Velocity dispersion Σ vi2 3π Σ (1/Rij)
Thus after taking into account all the projection effects, and if we assume masses are the same so that Msys = Σ mi = N mi we have
MVT = N
this is the Virial Theorem Mass Estimator
Σ vi2 = Velocity dispersion
[ Σ (1/Rij)]-1 = Harmonic Radius
Σ vi2 3π
Σ (1/Rij)
2 G
i<j
i<j

13. This is a good estimator but it is unstable if there exist objects in the system with very small projected separations:
x x
x x x xx
x x x x x
x x x x
x x x
x x
all the potential energy is in this pair!

14. Projected Mass Estimator
In the 1980’s, the search for a stable mass estimator led Bahcall & Tremaine and eventually Heisler, Bahcall & tremaine to posit a new estimator with the form
~ [dispersion x size ]

15. Ri,c = Projected distance from the center fp
Derived PM Mass estimator checked against simulations:
MP = Σ vi2 Ri,c where
Ri,c = Projected distance from the center
vi = l.o.s. difference from the center
fp = Projection factor which depends on
(includes) orbital eccentricities fp GN

16. The projection factor depends fairly strongly on the average eccentricities of the orbits of the objects (galaxies, stars, clusters) in the system:
fp = 32/π for primarily Radial Orbits
= 16/π for primarily Isotropic Orbits
= 8/π for primarily Circular Orbits
Richstone and Tremaine plotted the effect of
eccentricity vs radius on the velocity dispersion profile:

17. Expected projected l.o.s. sigmas
Richstone & Tremaine

18. Applications: Coma Cluster σ ~ 1000 km/s MVT = 1.69 x 1015 MSun
MPM = 1.75 x 1015 MSun
M31 Globular Cluster System
σ ~ 155 km/s MPM = 3.1+/-0.5 x 1011 MSun

19. The Coma Cluster

21. Andromeda Galaxy = M31

22. M31 Globular Clusters

24. Velocity Histogram