Is it true dark matter was discovered whilst looking at the rotational speed of galaxies?

Question

Just a little something whilst a dark matter pioneer moves onto a better place.

This commentator writes:

My understanding is that dark matter was "discovered" when trying to find out why stars in the outer edge of a galaxy rotates with the same speed as the inner. A ten-fold increase in number of galaxies doesn't seem to affect this.

When reading the history - this simple explanation doesn't quite come across. Perhaps the statement above is oversimplified. For example:

• mass in the galactic plane must be greater than what was observed (doesn't quite talk about comparative rotational velocity)
• estimated its mass based on the motions of galaxies near its edge (doesn't quite talk about comparative rotational velocity)
• some unseen matter provided the mass and associated gravitation attraction to hold the cluster together (doesn't quite talk about comparative rotational velocity)
• measurements of galaxy rotation curves (what is a galaxy rotational curve? Shouldn't it be a straight line if they're rotating at the same angular velocity?)
• gravitational lensing of background objects by galaxy clusters (not talking about comparative rotational velocity?)

My question is: Is it true dark matter was discovered whilst looking at the rotational speed of galaxies?

Answer 1

What you're possibly missing here, judging from your questions, is the link between the mass distribution of a galaxy and the rotation velocities.

The simplest way to obtain a prediction for a tangential velocity component is to look at the Kepler problem, i.e. a central mass, dominating another smaller mass, orbiting it.
Newton's force-law $\vec F = m \dot{\vec{v}}$ relates acceleration and acting force, which is given by gravity.
All acting forces then sum up to change the velocity of a test-particle $\dot{\vec{v}} = \sum_i \frac{\vec F_i}{m}$.

Then we get a number of cases, depending on how many summands there are, and how they are distributed spatially:

• Kepler problem: Only one central $F_i$ exists, which dominates the system's behaviour. For a circular orbit one gets the well-known tangential velocity $v_t = \sqrt{\frac{GM}{r}}$, with $M$ being the mass of the central object and $r$ the distance from it.
• A Galaxy: In a galaxy the visible mass distribution is very different from a stellar system. It is spaced out more equidistantly and thus would give rise to a peak in $v_t$ and then a decrease again, after one gets far away from most of the mass. This is however not observed, and calculating the measured velocities of visible mass backward to the true mass distribution leads to the postulation of dark matter.

A related problem is the virialisation of galaxy clusters. When a bunch of mass particles interact with each other gravitationally for long enough, in this case galaxies, their averaged kinetic energy $<T> = <\frac{1}{2}mv^2>$ becomes half the available gravitational potential energy $<V>$. Thus, under the assumption of virial equilibrium one can measure the total mass in a system by measuring the velocity dispersion. Because galaxy clusters are very old objects, it is usually assumed they display this equilibrium. Departures from this are still ongoing research and for reasons I don't remember the milky way is usually not assumed to be virialised.
Zwicky 1933 realised that the Coma cluster then needs to contain as much as 400 times the observed mass, to explain it's observed velocity dispersion.