So, what is a rotation curve? It is the shape of the graph we get when we look at how fast portions of a collection move when that collection is rotating. Let's consider three examples. First, rotation as a solid body, a disk, or if you are old enough to remember seeing one, a record. The distance along an arc is just the angle moved times the radius so the velocity is just the angular velocity times the radius. This gives as a rotation curve that is a straight line.
Next let's consider a system of planets orbiting around a central star.We start with Newton's equation for gravity, F=GMm/r2 and the equation for centrifugal force, F=mv2/r. Here M is the mass of the star, m is the mass of the planet and r is the distance. To be stable the two forces must be equal so we set these equal and do a bit of algebra and ignore the M and G because those things don't change and we get that v is proportional to 1/√r. That curve looks like this:
A bit of a diversion for some important information is needed before we go further and this is the most technical part so be prepared. There is a beautiful theorem, first proved by Newton, called the Shell Theorem. It states that for a point outside a spherically symmetric shell (like a perfect soap bubble) of matter of uniform density, the gravity is exactly the same as if all of the mass were concentrated at the center. A bit more surprising is that from *anywhere* on the inside the gravitational force is zero. Everything balances and cancels out. The most common proof of this is done using calculus which is why I said that post would *almost* be calculus free. Newton didn't quite use calculus and a fully geometrical proof also exists.
Let's say you are in a near a spherical collection of uniformly distributed stars. From the outside the gravity could be considered to be due to the total mass of the collection right at the center. For a point inside we can ignore everything outside and consider only the material inside and again concentrated at the center. If you'd like a slightly different approach take a look at this video where the gravity at various points *inside* the Earth is considered.
Armed with this let's now consider stars in a spiral galaxy. What should we measure when we look at the speed of stars in a spiral galaxy? The stars near the center are inside of the visible bulk of the galaxy so they are subject only to the gravitational force of the stars closer to the center which are a small fraction of the mass of the galaxy. Those stars should rotate slowly. As we go further out the stars are subject to gravity of more and more stars so their velocity should increase. Once we get past the central bulge of the galaxy the amount of material that is added when we increase distance rapidly decreases so that eventually almost all of the mass in interior to the star's orbit so the curve should resemble the downward curve for the solar system. We expect something like this:
However, when Vera Rubin looked at the Andromeda galaxy she saw something like this (idealized) graph:
As expected the curve rose as more of the dense central region of the galaxy was involved in holding the stars in orbit. But, rather than dropping off as you would think looking at the visible objects it remained at about that level. What does this tell us? First it tells us that the stars aren't a good indicator of the amount of gravitating material there is in the galaxy. To be more specific, since the curve is flat once we get outside of the central bulge of the galaxy we can use the equations above to see how much mass there is as we move away from the center of the galaxy. Remember that the formula for the force of gravity is F=GMm/r2 but now we want to see how the gravitating mass should change with distance so we write F=GM(r)m/r2 and set it equal as the other formula as we did above F=mv2 /r. After a bit if algebra where we get rid of the things that are taken to be constant we get that M(r)∝r. The rotation curve is telling us that as we go well past the most of the visible parts of the galaxy we continue to see the effect of mass that increases with the radius. Since the volume of a sphere increases as the cube of the radius that means the density of the material is dropping of as 1/r2.
Since the dark matter halo that galaxies are embedded in are so much larger than the visible parts we can't use this information to determine the shape and density of these haloes toward their outer edges. It is likely that, if we could see material out much further than we can in real galaxies that the rotation curves would drop indicating that after some distance there was very little additional matter.
This is being investigated in at least two ways. One is large scale numerical simulations of the evolution of the universe taking into account all of the physics we know. These are very difficult and are limited by the amount of computation power and memory available but they are getting consistent result that look a lot like our real universe. This indicates that we aren't likely to be missing anything big. The other is more observations of galaxies with two new telescopes LSST and WFIRST which will look at the effects of dark matter by observing the way the light of very distant object is gravitationally lensed by dark matter surrounding nearer objects.
