###### By Joshua Winn, Princeton University

## When it comes to elliptical galaxies, we can estimate its total mass in two ways. One is by measuring the velocity dispersion of the stars and using the virial theorem. The other is by measuring the total luminosity of the galaxy, and calculating what total mass of stars is needed to produce that much light. But what about its rotation velocity, how do we calculate that? Is it different for disk galaxies? Read on more to find out.

### Calculating the Rotation Curve

We perform Doppler spectroscopy of the starlight at different distances from the center of the galaxy. That way, we determine the rotation velocity as a function of radius. That function, *V* of *r*, is called the galaxy’s rotation curve.

What would we expect the rotation curve to look like? Let’s start simple: we’ll assume, contrary to fact, that the galaxy is a point mass *M*. Then, the situation is just like a planet going in a circle around a star: we set the centripetal acceleration equal to the gravitational acceleration, and solve for *V*, giving root *GM* over *r*. So, we’d expect the rotation velocity to decline with increasing radius.

A real galaxy, though, does not have central dominant point mass. Remember, the black hole is tiny compared to the combined mass of all the stars. So, we need to calculate the gravitational force from the whole distribution of mass within the galaxy. And here, things get tricky. Newton taught us that if we’re at a distance *r *within a spherical mass distribution, we’re allowed to ignore all the exterior mass, and pretend the interior mass is lumped together as a single point at the center. That’s a relatively easy problem to solve, and Newton proved we’ll get the same answer as we would by solving the harder problem of a spatially extended mass distribution.

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### Newton’s Theorem and Disk Galaxies

But disk galaxies aren’t spheres, they’re disks, and Newton’s theorem doesn’t apply to flat disks. For a disk, the formula for the rotation curve is much messier. We would need to do integrals involving modified Bessel functions of the second kind, and we’re not going to do that. We’ll do what physicists tend to do; we’ll assume the galaxy is a sphere anyway and we’ll trust that the answer will resemble the right answer even if it differs in detail.

We’ll describe the mass distribution with a function, *M-sub-r*, that tells us how much total mass is enclosed within a sphere of radius *r*. That same kind of ‘enclosed mass function’ was useful when we were modeling the interiors of stars, using the equation of hydrostatic balance. As *r* increases, *M-sub-r* rises, too, since we’re enclosing more and more material, until we get far enough away that we’re enclosing all the mass. Then, in the limit of large *r*, *M-sub-r* would level off to a constant, the total mass of the galaxy. To find the velocity of an orbiting star, we appeal to Newton’s theorem. We simply take our previous result, and replace the constant *M*, with the function, *M-sub-r*.

### The Dark Matter Halo

In the 1970s, measurements of galaxy rotation curves got more accurate and extended to larger *r*. Nearly everyone expected that *M-sub-r* would level off, once *r* was larger than about 10 kiloparsecs, outside the visible disk of stars. That far away, *M-sub-r* should stop increasing, and the rotation velocity should start declining as *one over root r*. But that’s not what was observed. Instead, *V* was found to keep rising! In many galaxies, it levels off to a constant value, but it doesn’t go down, even well outside the disk!

What does this mean? If *V* is observed to be a constant, then we can solve for *M-sub-r*, and we find that it grows in proportion to *r*. Way out there, where there are hardly any stars, as we increase *r* we still enclose more and more mass. It’s the dark matter, again.

More sophisticated analyses, using the best available data, show that the dark matter does, in fact, form a nearly spherical mass distribution, and extends out to hundreds of kiloparsecs. We are led to the stunning conclusion that a spiral galaxy is just a little bit of flotsam spinning around at the center of a much larger and more massive entity, which has been called the dark matter halo.

### Dark Matter

So, what is dark matter? Unfortunately, this is one of the most important unanswered questions in astrophysics. Astronomers and physicists have tried for decades to detect dark matter in some other way besides its gravitational influence and they failed. Theorists have tried to dream up new forms of matter that could avoid detection in all these ways. All this work has narrowed down the possibilities for dark matter.

The idea currently in favor—basically the only one that hasn’t been ruled out—is that dark matter is composed of one or more hitherto unknown fundamental particles. Particles that feel and exert gravity, but that otherwise interact very weakly, if at all, with normal matter. No electromagnetism, no nuclear forces.

### Common Questions about the Rotation Velocity of Elliptical and Disk Galaxies

**Q: How do we determine the rotation velocity as a function of radius?**

We perform Doppler spectroscopy of the starlight at different distances from the center of the galaxy. That way, we determine the rotation velocity as a function of radius.

**Q: What does more sophisticated analyses reveal about the mass distribution of dark matter?**

More sophisticated analyses, using the best available data, show that the dark matter does, in fact, form a nearly spherical mass distribution, and extends out to hundreds of kiloparsecs.

**Q: What is dark matter?**

The idea currently in favor—basically the only one that hasn’t been ruled out—is that dark matter is composed of one or more hitherto unknown fundamental particles. Particles that feel and exert gravity, but that otherwise interact very weakly, if at all, with normal matter.