# Lagrangian point or dark matter?

We know that spiral galaxies spin in a way such that we have to assume that dark matter is responsible for the extra mass required to do so.

My question is, can Lagrangian points (L1 and L2) be used to describe a galaxy's rotation instead?

Can we explain that objects far away from the center of the galaxy have higher velocity because they are at the L2 Lagrangian point of a Lagrangian system which consists of a) the galaxy's super massive black hole at its center, b) a part of its spiral arm c) the far away object in question?

(I'm a computer engineer interested in physics. Please excuse my ignorance)

• I think the problem would be that the L-points are for "point mas" objects that exert Newtonian gravity, not for extended objects such as the spiral arms. Note also that the SMBH gravitational force gets fairly small compared to the other sources of gravitational force. – Kyle Kanos May 31 '14 at 2:08
• If we are able to substitute a point mass for mass of the spiral arm in concern and substitute a point mass for effective mass of all objects near the galactic center, we can think of it as a point mass system, can't we? – Pravin Sonawane May 31 '14 at 3:52
• No, I don't think so. The spiral arms are huge, I don't think we could just ignore that. – Kyle Kanos May 31 '14 at 10:49

## 2 Answers

As noted by Kyle Kanos: the derivation of the Lagrange point heavily relies on the assumption that the object can be seen as a point-like mass.

The Lagrangian points are the constant-pattern solutions of the restricted three-body problem: (Wikipedia) This assumption breaks down, when you consider an extended mass distribution like a galaxy.

The objections posted by the answerers to the original question really just point to the need for a solution to an unrestricted many-body problem to actually answer the question fairly. Since the asker is a computer engineer I would suggest he code a numerical simulation. My intuition is that, although the bodies involved are numerous and extended, that the gravitational fields will interact in interesting ways. I suspect the rough magnitudes involved are accounted for in existing models, but it would be interesting to see if the interacting fields have a significant effect on galactic motion.

• Note that there are about $10^{11}$ stars in the Milky Way, which gives roughly $10^{121}$ pairwise interactions — even running for a million gigahertz-years wouldn't make a dent in computing the motion for the first iteration. Real computations generally treat the interstellar medium, stars and all, as a fluid, and evolve according to gravity and hydrodynamics. – rob Jun 12 '14 at 23:11