Galactic Rotation Curves - an Introduction

It is well known that in the solar system the outer planets not only take longer to orbit the Sun than the inner planets, they also travel at a slower speed. This is an immediate consequence of the fact that nearly all the mass of the solar system is concentrated in the Sun itself. The gravitational field strength g at a distance R from the Sun is given by Newton's law of gravity:

g = GMO / R2

where MO is the mass of the Sun and G is the Gravitational constant equal to 6.67 x 10-11 N m2 kg-1.

Putting the centripetal acceleration equal to the gravitational field strength we obtain:

v2 = GMO / R

It would be worth your while checking this out using some relevant data. Given that the Earth rotates round the Sun in 365 days at a distance 0f 150 million km, you should be able to verify that the mass of our Sun is 2 x 1030 kg.

It is known that the Sun is rotating round the centre of the Galaxy at a speed of 220 km s-1 at a distance of 8 kPc. (where 1 kPc = 3.1 x 1019 m). If we assume that all the mass of the galaxy is concentrated at its centre, what would that mass be? How many solar masses is that? (You should get an answer of the order of 100 billion solar masses.)

As a first estimate of the mass of our Galaxy, this is not too bad - but it completely ignores the fact that our galaxy is far from being a point mass. As you know, it is more like a spinning disc with a bulge at its centre and spiral arms extending out to at least 10 kPc. That is what our near neighbour, the Andromeda galaxy looks like and we believe that our Galaxy is much the same.

So how can we deduce something about the distribution of matter within the galaxy? The answer is by looking at the way stars (and other things too such as clouds of gas) rotate around the core. When we do this, we find a rather surprising thing. Instead of the rotational velocities falling off as thge square root of the distance (like the planets in the solar system) the rotational speed of all the starts in our galaxy is remarkably constant all the way out from near the centre to way beyond the edge of the visible disc. Here is a graph of some recent data (adapted from Xue et al. 2008).

In order to account for this curve, astronomers have had to accept one or other of two rather unpalatable alternatives. Either there is a lot more matter in the Galaxy that that which we can see or Newton's law of gravity is wrong. Notwithstanding the fact that Newton's laws have been superceded by Einstein's General theory of Relativity, the second alternative is not really an option. But accepting the first option begs three vital questions:

Using this program you can find out the answers to the first two questions yourself. But first we need to discuss how best to construct a model of our galaxy.

It is widely accepted that at the centre of most galaxies there is a concentrated massive object - possibly even a black hole. This is the first thing, therefore, that we need - a central core. As we have seen, the gravitational field strength of such a core falls off rapidly according to an inverse square law

Next it is obvious that a galaxy like ours also has a disc of matter which is probably densest near the core and which decreases as you travel to the edge of the disc. It turns out that, within such a disc, the gravitational field strength increases as you move out to the edge but drops off rapidly outside it.

Neither of these effects have any hope of accounting for a uniform rotational velocity out as far as 60 kPc from the centre. We need a third element - a 'halo' of matter surrounding the whole Galaxy and at least three times as big. It is usual to assume that this halo is spherically symmetric (it makes the calculations nice and easy) but it is quite possible that the halo is ellipsoidal instead.

This is what our galaxy looks like then:

A central core, a disc of stars and a halo of dark matter.

Next page