Velocity Dispersion Measurements of Galaxies

First direct evidence of dark matter came from Frids Ricky. He did some observations which revealed dark matter for the first time. His observations considered the overall motion within the galaxy cluster.

Extended objects are galaxy clusters and they are considered bound structures. These galaxies are moving with respect to cluster center but do not fly off. We look at the overall motion of the galaxy.

Assumption: Velocities are Represent of Underlying Potential

Every galaxy will have its own proper motion within the cluster and Hubble Flow Component. The smaller galaxies are smaller, most of the pght comes from M31 and MW, there are several dwarf galaxies. For our crude analysis, we can only use M31 and MW and evaluate dynamic mass of the local group.

There is a relative velocity between us and M31. It is crude, but it is true. The story begins long back when M31 and MW were close to each other, because they were members of a cluster they were moving away from each other. After some time they reach the maximum separation, then come closer to each other.

Let us say that the maximum separation that can ever reach is $r_{max}$. Now they have a separation called r. Let M be the combined mass of MW and M31. We do not know when $r_{max}$ is reached.

$$frac{GM}{r_{max}} = :Potential : at :r_{max}$$

When these galaxies were coming close to each other at some instant r, then the energy of the system will be −

$$frac{1}{2}sigma^2 = frac{GM}{r} = frac{GM}{r_{max}}$$

σ is relative velocity of both the galaxies. M is reduced mass only, but the test mass is 1. σ is velocity of any object at distance r from the centre of the cluster. We bepeve that this cluster is in dynamic equation because virial theorem holds. So, galaxies cannot come with different velocity.

How much time would these galaxies take to reach the maximum distance?

To understand this, let us consider the following equation.

$$frac{1}{2}left ( frac{dr}{dt} ight )^2 = frac{GM}{r} - frac{GM}{r_{max}}$$

$$t_{max} = int_{0}^{r_{max}} dt = int_{0}^{r_{max}} frac{dr}{sqrt{2GM}}left ( frac{1}{r} - frac{1}{r_{max}} ight )^2$$

$$t_{max} = frac{pi r_{max}^{frac{3}{2}}}{2sqrt{2GM}}$$

Where, M = dynamical mass of local group. The Total time from the start till the end of colpsion is $2t_{max}$. Therefore,

$$2t_{max} = t_0 + frac{D}{sigma}$$

And $t_0$ is the present age of universe.

If actual $t_{max} < RHS$, then we have a lower pmit for the time. $D/sigma$ is the time when they will colpde again. Here, we have assumed σ to be constant.

$$t_{max} = frac{t_0}{2} + frac{D}{2sigma}$$

$$r_{max} = t_{max} imes sigma = 770K_{pc}$$

Here, σ = relative velocity between MW and M31.

$$M_{dynamic} = 3 imes 10^{12}M_0$$

$$M_{MW}^{lum} = 3 imes 10^{10}M_0$$

$$M_{M31}^{lum} = 3 imes 10^{10}M_0$$

But practically, dynamic mass is found out considering every galaxy within the cluster. The missing mass is the dark matter and Frids Ricky noticed that the galaxies in the coma cluster are moving too fast. He predicted the existence of neutron stars the year after neutron stars were discovered and used Palomar telescope to find the supernova.

Points to Remember

First direct evidence of dark matter came from Frids Ricky.

Extended objects are galaxy clusters and they are considered bound structures.

Dynamic mass is found out considering every galaxy within the cluster.