1. Battle of the Mass Estimators (Based on Bahcall and Tremaine, 1981) Nick Cowan UW Astronomy January 25, 2005

2. Outline Weighing Galaxies The Virial Theorem The Projected Mass Method Monte Carlo Simulations Application to Clusters of Galaxies

3. Weighing Galaxies Distance to galaxy +Radial velocities of dwarf galaxies +Projected distance to dwarf galaxies =Mass of galaxy …but it ain’t easy! M R1R1 R2R2 v1v1 v2v2

4. The Virial Theorem (The way its always been done) If gravity is the only force acting on a group of particles we have 2 + = 0. For test particles in the vicinity of a point mass, this can be written as GM = /. But we only see the projection onto the celestial sphere: M = (3  /2G) /.

5. The Virial Theorem (continued) But the Virial Theorem Estimator has problems: Biased Inefficient Inconsistent

6. The Virial Theorem is Biased For N=1, it under-estimates the mass for high eccentricity: /M = 0.463 and over-estimates the mass for low eccentricity: /M = 1.388. This bias is most extreme for small N but does not always diminish very rapidly (or at all!) with increasing N

7. The Virial Theorem is Inefficient The variance of 1/R is Var(1/R) = - 2. But = . Therefore the variance of 1/R is formally infinite. Thus the standard deviation of M VT goes as sqrt(lnN)/sqrt(N) rather than 1/sqrt(N), as one would expect.

8. The Virial Theorem is Inconsistent The 1/R term ensures that M VT depends most on nearby test particles. As N goes to infinity, it is possible for a finite set of test particles to be responsible for a finite share of M VT. For a typical sample of test particles, M VT will not be converging on the correct answer, even as N .

9. The Projected Mass Method (we can do better!) Define the projected mass: q = v z 2 R/G. The mean projected mass is related to the actual point mass by = (  M/32)(3-2 ) But we don’t know !

10. The Projected Mass Method: Choosing Linear Orbits = 1 Use M L Isotropic Orbits = 1 / 2 Use M I Circular Orbits = 0

11. The Projected Mass Method (Continued) The Projected Mass Method gives equal weight to all test particles, regardless of their mass or distance from the point mass. Thus, it makes maximal use of all available information.

12. Monte Carlo Simulations Simulate a point mass and its test particles. Output “typical” samples of radial velocities and projected separations. See which mass estimator comes closest to the real answer.

13. Results of MC Simulations

14. More Results of MC Simulations

15. Application to Galaxy Clusters 1.Take spectra for all galaxies in the same field as the galaxy you wish to weigh. 2.Using their redshift, pick out those galaxies which are gravitationally bound. 3.Measure the angular separation of the galaxies to get their projected distance. 4.Use the mass estimator of your choice to determine the galaxy’s mass (at least to within a factor of h -1 ).

16. Application to 3C 273 3C 273 has z = 0.158

17. Application to 3C 273 (continued) Now repeat for M101 and M31…

18. Shortcomings of the PMM Systematic error of order unity due to incomplete knowledge of. Central galaxy must be much more massive than orbiting dwarfs. Radius of central halo must be smaller than distance to dwarfs. Good statistics requires large N, but large N leads to contamination.

19. Recap We would like to estimate the mass of a large galaxy based purely on the redshift and projected distance of nearby dwarf galaxies. Both the Virial Theorem and the Projected Mass Method can do this. In principle, the PMM is better than the VT, but in practice it doesn’t really matter which one you use.