The Mass of the Coma Cluster

Application of Different Mass Estimators

by Drew BrisbinCarl Ferkinhoff

Astro 7620 - Spring 2010

Estimating the Mass of ClusterImportant for determining the amount of dark matter versus normal matter in the universe.Knowing the mass of a cluster is import for comparing to formation models and understanding the growth of structure in the universe.

How does one determine the mass of a cluster? Heisler, Tremaine, and Bahcall(1985) describe four methods.

1. Virial Method2. Projected Mass3. Median Mass4. Average Mass

What is the mass of the Coma Cluster?How well do they work?

1. Reliability? 2. Affects of contamination?3. Dependence on sample size?

Virial Method

Falls out of the Virial TheoremGalaxies far from the rest recieve extra weighting

fPM characterizes distribution of orbits(eg: radial? isotropic? etc..) 1.5 corrects for observables being relative to centroid(instead of center of mass) Less sensitive to one galaxy projected close to another

Projected Mass Estimator

Median Mass Estimator

Picks the one galaxy pair that best probes the total mass of the cluster.Least suceptable to outliers(or contamination from non-cluster member galaxies)

Average Mass Estimator

Similar to the Median estimator, but now the average is taken.Similar to Virial and Projected in sensitivity to interlopers

fAv = 2.8

Coma Cluster

APOD 2006 March 21

12h 59m 48s RA +27d 58m 48s DECz = 0.0232 +/- 2x10-4

(cz ~ 6900 km/s)line of sight velocity dispersion ~ 900 km/s

From Danese, Zotti, and di Tullio (1980)

Diameter = 319 arcminutes

Sample for determining the mass:

All galaxies within 190 arcmin4660 < Vhelio < 9660

From Kubo et al. 2007 1.88x1015 h-1Msolar

Distribution of heliocentric velocity and distance to cluster center for entire sample.

INTERLOPERS!!!

Mass of the Coma Cluster

MVT=2.149 x 1015 Msun (+/-)0.018 x 1015 Msun

MPM=2.590 x 1015 Msun (+/-)0.047 x 1015 Msun

MMe=2.006 x 1015 Msun (+/-)0.078 x 1015 Msun

MAv =2.1937 x 1015 Msun(+/-)0.041 x 1015 Msun

Comparison of Estimators(1)

How reliable are the estimators?How do they stand up to undersampling?To examine we chose a random sample of N galaxies from our data and estimated massesN=10,20,30,50,100,300,600 We did 100 trials of this and considered estimate fluctuations

Comparison of Estimators(2)

N

StandardDev.

Mass Indicator Standard Deviation over 100 Trials

Comparison of Estimators(3)

N

Median

Mass Indicator Medians over 100 Trials

Comparison of Estimators(3)

Sensitivity to outliersFound mass estimates on full samples

M'VT=2.88 x 1015 Msun=1.34 MVT

M'PM=3.58 x 1015 Msun=1.38 MPM

M'Me=2.53 x 1015 Msun=1.26 MMe

M'Av =3.08 x 1015 Msun=1.40 MAv

ConclusionsUsing four different methods, the Coma Cluster has a mass of:

MVT=2.1488 x 1015 Msun

MPM=2.5902 x 1015 Msun

MMe=2.0061 x 1015 Msun

MAv =2.1937 x 1015 Msun

All four methods give a different mass.All four methods behave similarly to sample size and outliers.

Reliability depends heavily on sample size.For a large enough sample, N>100, the amount of disagreement is significantly larger than the statistical uncertainties.