A question on Physics. Dark matter, galaxies, and rotational velocity

I am currently reading a book called Wrinkles in Time and while not finished yet, I still have a few questions I would like an answer to (if indeed one exists).

I understand that when trying to measure the speed of rotation of stars around the centre of our galaxy according to Newton’s Laws, the closer to the cetnre the faster around the core the stars should be orbiting (much like Mercury whips around the sun 10x faster than Pluto). When measurements were taken, this was disproven, as the stars on the outer arms of the spiral moved much faster than Newtonian laws allowed. This is currently explained by postulating a huge “ring” of dark matter surrounding the galaxy which affects the motion of these outer rim stars.

The question I have is as follows:

Is it possible that the reason the stars on the outer spiral arms rotate as quickly as they do relative to the inner stars is due to having a greater gravitational relationship with the stars sharing the same sprial arm so the inner stars collectively drag the outer stars along, much like spokes on a ferris wheel moving the outside as simialr to the inside? ( I know this analogy is not quite the same as for this to occur the outer rim would be rotating a lot fast than the inner. However, the fact that it is a spiral can suggest more of a gravitational association which drags them (albeit slower than a ferris wheel analogy) and that in turn is what creats the sprial itself (the latent drag of gravity)).

Would this theory work in Physics and not violate any laws currently? It just seems to me that gravitational effects of stars along the same band would affect eachother moreso than other “outside” sources.

Am I out to lunch? Has this already been considered and dismissed? Am I a genius in waiting? Or is this a totally terrible question?

It seems to me that the gravity of stars that are near you, and closer to the center of the galaxy, would have no net force pulling you forward or backward. Gravity’s not a frictional force (mathematicians would say that it’s conservative), so any tug the sun feels pulling it forward would be balanced by that from other stars a little behind, pulling it backwards.

It isn’t so much a matter of being pulled forward or backward as bing pulled inward.

And there’s a difference between the solar system and galaxies. As you move away from the Sun in the solar system the amount of mass that is contributing to orbital motion remains fairly constant. That is to say, the Sun is the only appreciable mass in the neighborhood.

In galactic motion the amount of mass contributing to orbital motion increases with distance. All them stars’ influence tends to add up.

Can you be a little more detailed?

OK, when Mercury is deciding how fast to orbit, it asks “How much mass is inside of my orbit?”. The answer is “The mass of the Sun”. When Pluto asks that same question, the answer is also “The mass of the Sun”. Actually, Pluto’s answer is the mass of the Sun plus the masses of the other eight planets, the asteroids, and any other junk that happens to be closer to the center than Pluto, but the Sun weighs much more than all of that junk combined, so adding it all in makes hardly any difference.

This is not the case with galaxies. While a galaxy does have a central black hole with a pretty big mass, it doesn’t dominate the total mass of the galaxy. So when a star from the outer edges of the galaxy asks how much mass it’s orbiting around, the answer is much higher than it is for a star near the center.

But here’s the catch. We can see the stars in galaxies, and see where they are. If all of the mass of a galaxy were in stuff like stars that we could see, we would still be able to predict how fast a star should be orbiting. But those aren’t the speeds we actually see, which means that there must be a lot of mass we can’t see. There are a lot of ways you could arrange that mass to get the required effect: Most models put the dark matter in a spherically-symmetric distribution, but you could probably use some sort of ring-shaped distribution as well.

Okay, this I don’t really get. If you were to have a spherically symmetrical uniform thickness shell of homogenous matter surrounding the galaxy (that didn’t collapse in on itself), I was under the impression that this would have zero gravitational effect on the galaxy (proven, one presumes, by integrating the gravitational force on a selected point). Even if you were close to one side of said shell, the pull from all of the mass on the other side of you, even though it is in general much further away, should completely cancel out the small portion close to you.

Of course, I haven’t actually done the full integration. Not sure where I’d begin. I remember taking a stab at it assuming being on the same plane as a ring (the problem in two dimensions). I don’t remember if I was successful.

Anyway, I’ve always had trouble seeing how dark matter could cause this effect without being non-uniformly distributed or being of different densities in different regions, in which case how can we make any suppositions about the impact of the gravitational influence over time at all?


I guess I’m as curious as Canuck.

You’re right that the gravitational force within a spherical shell of matter, due to that shell, is zero. But the dark-matter hypothesis is not that the dark matter surrounds the galaxy in a shell, but that it is distributed throughout (and possibly beyond) the galaxy in some distribution different than the distribution of normal matter. The dark matter within a particle’s orbit does affect its orbital period.

The puzzle is that astronomers have two somewhat-accurate ways of measuring how much matter is in a region, which disagree. Counting stars and using stellar models of mass-luminosity relations, and estimating the density of dust lanes and nebulas, gives one estimate of the “normal” matter distribution of a galaxy. But measuring the orbital velocity distribution of a galaxy gives an estimate of the distribution of the “gravitating” matter. These two distributions are different, so either Newtonian gravitation fails on galactic scales, some of the various mass-luminosity (etc.) methods are wrong, or there is extra matter that we can’t see. Each of these possibilities is being explored.

It appears that dark matter distribution is nonuniform on an intergalactic scale. Here. More detail and better pics here. The distribution of dark matter in and around single galaxies is also thought to be variable. The Milky Way appears to be surrounded by a spherically symmetric cloud of dark matter. Other types of galaxy: dwarf, irregular, or spherical, seem to have their dark matter distributed differently.

One of the more disturbing things about dark matter is just how much of it there needs to be in order to make the galactic orbital velocity curves fit the laws of gravity. I ran across a site claiming that the required dark matter density at the sun’s distance from the center of the galaxy is ~4 times the density of normal matter. Does that imply that there’s a couple of suns worth of mass hiding between us and proxima centauri ?