One big and as-yet unanswered problem in modern physics is the dark matter problem. The problem is astrophysical: a number of observations, from galactic to universal scales, show a difference between the mass distributions observed directly (in visible stars, galaxies etc) and indirectly (through its gravitational effects). As the name of the problem suggests, it looks as though there is a lot of extra matter that we can't see (because it's dark).

The canonical example of such an observation is also the first one made (by the brilliant but prickly

Zwicky), that of galactic rotation curves. In particular, let us focus on objects (stars, globular clusters) orbiting a galaxy but not really part of it. We are in the limit of weak gravitational fields and small speeds, so Newtonian mechanics is adequate. The gravitational force due to the galaxy drops off with the standard inverse-squared law:

$F = \frac{G M m}{R^2}$

Here,

*G* is Newton's constant;

*M* and

*m* are the masses of the galaxy and the object orbiting it, respectively;

*R* is the distance between them and

*F* the force. Using Newton's second law of motion gives us the acceleration:

$a = \frac{G M}{R^2}$

Lastly, we use the relation between acceleration and velocity for objects moving in a circle:

$a = \frac{V^2}{R} ; \therefore V = \sqrt{\frac{G M}{R}}$

The main point is that we expect the speeds of objects orbiting a galaxy to decrease as they get further away from it. We can extend this for objects within the galaxy itself, but then we need to take the finite size of the galaxy into account. The result is that we expect the orbital speed to increase with distance within the galaxy, then decrease with distance outside it.

What we see looks like this: