# Why don't stars have Keplerian orbits?

In a comment to Rob Jeffries' answer to this question on spherical galaxies, Incnis Mrsi commented

There should exist the entire range of orbits in a spherically symmetric system: near-circular, highly excentric (but not elliptical because galatic orbits are not Keplerian!)

Why shouldn't the orbits of stars be Keplerian? I can think of a few reasons (but I don't know if they're correct):

• The influence of dark matter
• Interactions with gas and dust in the galactic arms

Are either of these explanations correct?

• There's a basic problem with that parenthetical remark. MOND and dark matter are diametrically opposed to one another. MOND proponents say that scientists do know mass (more or less); they say that it's the model of gravitation that is the cause of the galactic rotation problem. Dark matter proponents say that scientists do know how gravity works (more or less); they say that unknown mass (aka "dark matter") is the cause of the galactic rotation problem. – David Hammen Nov 24 '14 at 23:23
• @DavidHammen That's my misunderstanding. I had heard the terms used in conjunction with each other before, but I must have misinterpreted the information given. Thank you. – HDE 226868 Nov 24 '14 at 23:25

Why shouldn't the orbits of stars be Keplerian?

The answer is simple. Keplerian orbits are predicated on a single central point mass. That assumption fails to some extent even in a solar system. It fails massively in a galaxy. A galaxy is not a point mass.

Elliptical orbits are direct consequence of orbiting entirely outside a spherically symmetric mass. Even if you assume that a galaxy has a spherically symmetric mass distribution, the amount of mass at a radial distance less than that of the star would be changing (assuming some eccentricity). Once that happens, the orbit is no longer an ellipse.

• Yes, it's a simple as that. (And the comment on my answer is completely correct). – Rob Jeffries Nov 24 '14 at 22:56

Not Keplerian, because it is not a conic-section. It is not even explained by Newtonian gravity. In contrast, Kepler's laws are explained by newtonian gravity.

The lowest orbital-energy from Keplerian orbit is circular. And the orbits of stars are observed to be approximately circular. Hence: $$\frac{mv^2}{r} = \frac{GMm}{r^2} \quad\Longrightarrow\quad v = \sqrt{\frac{GM}{r}}$$

So, a circular keplerian orbit would imply speed dependent of the distance from the star to the center, proportional to $r^{-1/2}$. However, this is not observed. The observation seems to indicate that exists a certain "independence" of the distance from the center. Therefore, the orbits are not keplerian.

Since gas and dust is observed, then this must not be the problem. To fix this small problem, one possible solution is to postulate $M(r)\propto r$. Since this is not observed, then must be some non-observed kind of matter, some dark matter.

Another solution is to say the force is not $F = \frac{GMm}{r^2}$, and then invent a force that works in this case as well: MOND, which you've pointed out.