# Something I'm confused about regarding dark matter...

As I understand it, the outer stars of galaxies revolve around the centers of their galaxies too fast for the the visible matter in the galaxy to account for. Dark matter is necessary to postulate how that can be. But every description of dark matter I’ve read describes it as a large halo of mass surrounding the visible galaxy. Wouldn’t the dark matter then be pulling out on the stars of galactic rims, causing them to escape their galaxies?

The distribution of mass you see for the mix of ordinary and dark matter is simply reversed out from the measured velocities of the visible mass. The distribution is what makes the system work, rather than having any deeper significance.

By “halo” they don’t mean that the dark matter is only outside the galaxy. There is a bunch of dark matter throughout the galaxy, too, it just extends far beyond the radius of the visible matter.

It is a theorem in gravity that for a spherically symmetric configuration of matter, the matter outside a given radius from the centre has no net effect of the gravitational pull at that radius. So for a particular star, orbiting at a particular radius, only the matter (both ordinary and dark) within that radius exerts a net pull (inwards).

When you look at a typical spiral galaxy, most of the light comes from the bulge in the center, because that’s where most of the galaxy’s stars are. So, that should be where most of the gravity is coming from. Therefore, a star closer to the center should feel a stronger pull, and therefore be orbiting faster, than a star further out.

Instead, we don’t see any drop-off in gravitational strength or orbital speed. This means that most of the mass of the galaxy is not concentrated in the center, but uniformly distributed everywhere.

This is what is meant by a “halo” of dark matter. Galaxies are spherical blobs of dark matter, rather than the disks of stars they appear to be.

In other words, if there’s a star halfway out from the center of the galaxy, all the dark matter farther out on that side is pulling the star outward. But all the dark matter on the other side (past the center of the galaxy, or just even with the center, but on either side) is pulling the start toward the center. Turns out, if the dark matter is evenly distributed, the math works out that all the pulling from all dark matter farther out (on all sides) actually cancels out. But the dark matter further in doesn’t cancel, and ends up pulling the start straight toward the center.

leachim and Quercus are correct but I’m not sure they’ve explained it in a way that’s easily understandable by a novice. Let me try.

Imagine a large hollow spherical shell, composed of any arbitrary material (not necessarily dark matter). Now if you go inside this shell, the theorem that leachim referred to says that you will feel no gravitational force from the shell. You will float weightlessly inside, no matter where you are. This is counterintuitive, because you might think that you would feel weightless only at the center, but if you were closer to one side you would be pulled to that side. It turns out that that is not the case. It’s a rather simple calculus exercise to calculate the force anywhere in the shell, and it’s always zero. (As an aside, this is usually overlooked by “hollow Earth” fantasies, which normally assume that there would be a gravitational gradient holding things against the inside of the shell. That’s wrong; things would float around inside.)

Now imagine a galaxy embedded in a spherical halo of dark matter, with the center of the halo coinciding with the center of the galaxy. Take one particular star in the galaxy. We can draw a sphere with its center at the galaxy’s center, and its surface intersecting the star; that is, the radius of the sphere is the distance of the star from the center of the galaxy. Now conceptually let this sphere divide the halo into two parts, the part inside the sphere and the part outside. Ignoring the inside part for a moment, we can see that the outside part is a hollow spherical shell, just as we discussed above. By the theorem alluded to, that part exerts no gravitational force on the star, since the star is (barely) inside the sphere. So we can ignore the outside part. Only the dark matter inside the sphere exerts any gravitational force, and it is directed toward the center of the halo.

–Mark

Got it! Thanks all.

The spherical shell theorem is important, but the halo distribution isn’t a spherical shell. It is a disc, and it varies in density along its radius. What the astronomers do is to work out the mass and velocity distribution of luminous matter, using a range of observations, and then numerically model what additional mass is needed to make the entire system work. Not only does the dark matter have to control the luminous matter, the entire ensemble has to be stable. It is more complex as galaxies have different shapes, what works for a simple spiral won’t work for an elliptical or a spherical galaxy. The answer seems to be that you can’t just add a bit of dark matter in a few places, you end up with what is essentially a much larger galaxy, one that has the luminous bit embedded in it. There is no difference between the luminous stuff and the dark stuff when working this out, it is just that some bits of the galaxy we can see, and some we can’t, so we have to work out the shape from analysing the bit we can see.

So then the complete galaxy, visible and dark matter, is still disc shaped? Is is just that the dakr matter is a fatter, larger disc?

All the models I’ve seen for dark matter halos have them as spherically symmetric. They do vary in density with distance, but not with direction.

Spherical is a good basic approximation, but the halo can be squished quite a bit. The latest computer models suggest that halos can routinely be twice as wide in one direction versus another. Recent models using observations of a Milky Way satellite galaxy suggest a ratio of 0.72 for the Milky Way halo’s narrow-versus-wide thickness, with the wide direction perpendicular to the galactic plane. Note that these computer simulations are non-trivial and do have known deficiencies (such as the core-cusp problem), but they do show that halos needn’t be (and observationally may not be) all that spherical.

You can also have multi-component and/or self-interacting dark matter that leads to more interesting halo shapes, including disks that have an orientation unrelated to the visible matter disk. These models are particularly relevant for direct dark matter detection searches. But in the simplest cases, a round-ish blob is a good mental picture.