Rotational Curves

In my first article, I discussed about how we calculate velocity of a galaxy or a star from its spectrum. A quick reminder, we use Doppler Effect to calculate its shift and then we use shift to calculate velocity using equations derived from relativity.

The rotation curve (or velocity curve) of a galaxy is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's center. The velocity of these stars or gas is due to the gravitation field acting towards them. As they are orbiting around the galaxy they are also acted upon by Centripetal force. The centripetal force is equal to gravitational force acting on them.

Using Newton’s and Kepler’s law, we can also derive the relation between the velocity , mass and distance of a star from the center of galaxy.

Rotational curve for solar system

Here, the rotation curve looks like this with the assumption that all the mass is in the center, Sun in this case.

But while calculating rotation curves for galaxies it is hard to assume that all the mass is in the center. As even though black holes are present in the center of the galaxy, they do not account for all of the mass. The stars orbiting account for more mass and thus calculation gravitational force becomes more complicated.

For this Isaac Newton worked out a simplification, in which

For example,

If we want to calculate gravitational force on a gas cloud R distance away from the center of galaxy, we can take all the mass present in the sphere of volume 4/3πr^3 in account and ignore the rest.

Using this simplification we can draw out the rotational curve for a galaxy. As we move further from center of galaxy, the volume of sphere in account will increase and the mass present inside it will also increase. Thus the velocity of the given star will increase.

At a point we will reach the boundary of the galaxy, the mass in account will be the greatest at this point and thus the velocity will reach its highest peak.

If we keep moving further, the distance from center of galaxy will keep increasing but there will no significant increase in mass and thus the velocity will keep reducing.

The rotation curve will be something like this.

I hope you have understood what rotation curves are. The rotation curves I showed in the article are theoretical and the observed ones are different from these. We will be discussing about those in my article about “evidences for dark matter”.