Dark Matter — Question 5

Question 5

A point-lens Einstein ring measures a projected gravitational mass

Use $\theta_E^2=(4GM/c^2)D_{ls}/(D_lD_s)$. Let $\theta_E=1.0$ arcsecond, $1$ arcsecond=$4.848\times10^{-6}$ rad and $D_lD_s/D_{ls}=2.0$ Gpc. Use G=$6.67\times10^{-11}$ SI, c=$3.00\times10^8$ m/s, 1 Gpc=$3.086\times10^{25}$ m and $M_\odot=1.99\times10^{30}$ kg.

1. Solve for the lens mass and evaluate it.

2. If the ring radius is measured 10 percent larger with the same distances, find the inferred mass change.

Question 5 - Solution

A point-lens Einstein ring measures a projected gravitational mass

(a) Method and calculation

$$M=\frac{c^2}{4G}\theta_E^2\frac{D_lD_s}{D_{ls}}.$$$$\boxed{M\simeq4.893\times10^{41}\ {\mathrm{kg}}\simeq2.459\times10^{11}M_\odot}.$$

(b) Interpretation and further calculation

$$M\propto\theta_E^2\Rightarrow M^\prime/M=(1.10)^2.$$$$\boxed{\Delta M/M=.21=21\%}.$$

Physical check

The distances are angular diameter distances. An extended lens requires a projected mass model; lensing measures gravitation rather than emitted light.