Why is it strange that outer stars in a galaxy are travelling with the same speed as inner while the total mass is also increasing?

Question

Astronomers say that it's strange that the outer stars of a galaxy are travelling with the same speed as the inner stars. Therefore they introduced dark matter to solve this strangeness. The theory that the inner stars 'have to' go faster is that because they are closer to the center so the gravity is more, like the earth is closer to the sun and faster than Neptune.

But what I wonder is, whether the fact that just the outer stars are attracted to more mass, because there is more mass of stars that are within the orbits of the outer stars, explains the equality? So in this view it is not so strange that they have the same speed because they are influenced by more gravity. Or is it that this extra influence is considered to be too little to explain the equality?

Answer 1

This is accounted for in real models of the rotation curves of galaxies. The rotation curve of a galaxy is more complicated than the motion of planets in the solar system, for the reasons that you describe.

A trivial model would consider the galaxy as spherically symmetric and would use the shell theorem to estimate the centripetal acceleration of a star at radius $r$. Thus $$ m r \omega^2 = G\frac{m M(r)}{r^2},$$ where $m$ is the mass of the star, $\omega$ is the angular velocity and $M(r)$ would be the mass of all gravitating matter at radii $<r$. Note that this only applies to a spherically symmetric distribution of mass.

To make further progress demands that you know the density distribution of the gravitating matter. Let's just assume that density $\rho$ is constant for the moment. Then $$ r^3 \omega^2 = G \int \rho\ 4\pi r^2\ dr = \frac{4\pi}{3} G \rho r^3$$ $$ \omega = \sqrt{\frac{4\pi G \rho}{3}}$$

Thus the angular velocity would be constant with radius and the rotation speed, $v = \omega r$ would increase with radius. This I think is the situation that your question supposes and so yes, if there was a constant (or perhaps slowly declining) density of material in the Galaxy then this would produce a rotation curve that increased (or was flat).

The trouble is that the density of material in our Galaxy inferred from the matter that we can see is not constant with radius. It declines rapidly and exponentially, such that there is very little visible matter beyond a radius of about 15 kpc. If we take a situation where we go to radii beyond the gravitating matter, then a similar treatment to the above suggests that $$ m r \omega^{2} = G\frac{mM}{r^2},$$ where $M$ is now the total mass of the (visible) Galaxy. In this case $$ \omega = \sqrt{\frac{GM}{r^{3}}}$$ and the rotation velocity $v = \omega r$ should decline as $1/\sqrt{r}$ (like it does in the solar system).

It is the fact that the rotation velocity of stars and gas at large radii ($>15$ kpc) continues to be flat or even increase that leads to the conclusion that the mass that we can see is not all that there is. i.e That we need "dark matter" to explain the high rotation speeds at radii where there is very little visible matter.

Realistic models of the galaxy do not make the assumption that the visible matter is spherically symmetric (it isn't). But the conclusions I qualitatively set out above hold in the same way.

Answer 2

When you say same speed, I assume you mean same angular velocity (which means more speed on the outside). Back earlier this century, I had a gravitational rope theory about the spokes of a galaxy, which I discussed with an astronomer Lisa Moore (who was doing lectures on cruise, I was doing), where the arms of a galaxy were acting like ropes attached to a spindle held together by gravity and turning at the same angular velocity. She told me that stars were passing through the arms. On further thinking, I realized that was not a problem as there was no solid connection through the arms (just gravity), so the stars could orbit at their own speed (faster near the middle & slower near the outside) while the arms orbited at the same angular velocity as the center. If dark matter was the reason, I would expect the rotational speeds to be only approximately matched as there is nothing binding the center to the outside. When a galaxy like ours turns once every 250,000,000 years I don't know hth they know it is all rotating at the same speed when it has only done one revolution since before the age of the dinosaurs. Their speed detection methods must be pretty sensitive.