Dark Matter — Question 4

Question 4

A virial estimate relates cluster mass to velocity dispersion

Use an ideal virial estimator $M=3\sigma_{\mathrm{los}}^2R/G$. Let line-of-sight dispersion be 800 km/s, radius R=1000 kpc, and G=$4.30\times10^{-6}$ kpc (km/s)$^2/M_\odot$.

1. Calculate the cluster dynamical mass.

2. If the visible baryonic mass is $8.0\times10^{13}M_\odot$, find its fraction and the inferred nonluminous mass.

Question 4 - Solution

A virial estimate relates cluster mass to velocity dispersion

(a) Method and calculation

$$\boxed{M=3(800)^2(1000)/(4.30\times10^{-6})=4.47\times10^{14}M_\odot}.$$$$3\sigma_{\mathrm{los}}^2=\langle v^2\rangle\quad\text{for isotropic velocities}.$$

(b) Interpretation and further calculation

$$\boxed{f_b=(8.0\times10^{13})/(4.4651\times10^{14})=.17917}.$$$$\boxed{M_{\mathrm{nonlum}}\simeq3.67\times10^{14}M_\odot}.$$

Physical check

The geometric coefficient is supplied as part of the model. Nonequilibrium mergers, velocity anisotropy and radius definitions can bias the estimate.