Start with the Solar System. Almost all of the mass is gathered into the Sun alone, and the planets may be neglected.
Running a circular orbit of radius takes a force toward the centre of . What supplies it is gravitation, . Setting them equal and cancelling leaves .
The sits under a root in the denominator, so the farther a planet, the slower it runs. Neptune goes round far more slowly than the Earth. This falling off of as is called Keplerian rotation.
What makes it work is that there is no mass outside the centre. The enclosed mass does not depend on , so is fixed by alone.
Now do the same for a spiral galaxy. Its light is strongly concentrated toward the centre, so it is natural to expect the mass to be as well. If so, the outer stars ought to slow in Keplerian fashion.
The measurement is made with spectra. One side of the disc approaches and the other recedes, so the same emission line is shifted blue on one side and red on the other. The size of that shift is the rotation speed at that radius.
On measuring, however, does not fall. It rises near the centre and then runs almost flat however far out you go. The points in the figure are the measurements; the faint curve is the Keplerian expectation.
Measure out to where the light of the galaxy fades away and it is still flat. Expectation and measurement part company more and more the farther out they go.
How to read the disagreement. In , the is the mass enclosed within radius . In the Solar System it was constant, but a galaxy has stars outside the centre too, so it grows with .
So let us build by counting what shines. The number of stars and the amount of gas both fall away sharply toward the outside. So levels off partway and barely grows beyond that.
Put that levelled-off into the equation and, outside that radius, it is the same as a constant , so falls as . That is the faint curve from before.
The shaded region between the two curves in the figure is what is missing. It widens the farther out you go.
Read the flatness of straight back into the equation. A constant gives , so has to keep growing in proportion to .
Which is to say that mass keeps being added at the same rate out where the light has already faded. Something that is neither star nor gas and gives off no light is spread out in a sphere far beyond the disc. This is called the dark matter halo.
The two lines in the figure are that argument. The mass built up from what shines lies down partway, while the mass demanded by a flat rotation curve carries straight on.
The disagreement is not a quirk of one galaxy. The same thing happens in nearly every spiral galaxy measured. The next article puts the same question again on a larger gathering, the cluster of galaxies.