Presentation on theme: "Mass Estimators in Astrophysics"— Presentation transcript:
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 M½ mv2 = GmM/RM = ½ v2 R /GThis 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 TheoremConsider 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 . riI = d2I/dt2 = Σ mi (ri . ri + ri . ri )Ni=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 elimiateri.miri = - Σ (ri - rj )and note thatΣ miri . ri = 2T (twice the Kinetic Energy)....Gmimj|ri –rj| 3j = 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|3i j=ireversing labelsGmi mjj i=j|rj - ri|3Gmi mjadding|ri - rj|3i j=iGmi mj|ri - rj|
8 ∴ I = 2T + UIf we have a relaxed (or statistically steady) system which is not changing shape or size, d2I/dt2 = I = 02T + U = 0; U = -2T; E = T+U = ½ Uconversely, for a slowly changing or periodic system <T> + <U> = 0....Virial Equilibrium
9 Virial Mass EstimatorWe 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 riand velocity viNN1|ri – rj|i=1i=1 j<i
10 projected radial averaged over solid angle Ω The observables are (1) the l.o.s. time averagevelocity:< v2R,i> Ω = 1/3 vi2projected radial averaged over solid angle Ωi.e. we only see the radial component of motion &vi ~ √3 vrDitto 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 = < >Ωand< >Ω = = = π/2111|ri – rj||ri – rj|sin θij∫0π dθ∫(sinθ)-1 dΩ1sin θij∫0πsinθ dθdΩ
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 haveMVT = Nthis is the Virial Theorem Mass EstimatorΣ vi2 = Velocity dispersion[ Σ (1/Rij)]-1 = Harmonic RadiusΣ vi23πΣ (1/Rij)2 Gi<ji<j
13 This is a good estimator but it is unstable if there exist objects in the system with very small projected separations:x xx x x xxx x x x xx x x xx x xx xall 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 whereRi,c = Projected distance from the centervi = l.o.s. difference from the centerfp = Projection factor which depends on(includes) orbital eccentricitiesfpGN
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 OrbitsRichstone and Tremaine plotted the effect ofeccentricity vs radius on the velocity dispersion profile: