Dark Matter & Gravity: Exploring Contradictions & Solutions

[cwdesign]

You've all seen the set-up. Bore a hole thru the center of the earth, and jump in. As you approach the center, the only portion of the Earth's mass that exerts an accelerative gravitational influence/geometry on you is that portion between you and the center, the "external" portions balance/cancel out.

So now enlarge the scale to galaxies. They shouldn't rotate as they do, which was one of the the earliest indicators of the existence of dark matter.

All of the illustrations/maps I've seen (e.g. - Bullet Cluster) of the assumed/calculated locations and distributions of dark matter show a homogeneous "cloud" surrounding the galaxies, or even offset as in the case of the Bullet Cluster.

Granted that the scales are vastly different, the principles remain the same. And those two illustrations are apparently contradictory. The dark matter halo beyond the edge of a galaxy should exert no influence on the bodies closer to the galactic center.

The only solution I see to this conundrum (misunderstanding on my part?) is that the density of dark matter increases as it approaches the galactic center, much as the density of regular matter does - I've never seen anything that addresses this. Does this sound like a reasonable assumption, and can anyone point me to articles that address it?

 

[phinds]

You have a misunderstanding about the apparent distribution of dark matter. It is hardly JUST in a halo around the outer edges of a galaxy, it is spread throughout a galaxy. Because of the way it does not interact other than gravitationally, if an individual particle of dark matter travels from somewhere in the halo through the galactic center and out to the other edge, then because of exactly the gravitational distribution you describe in your falling man scenario, the particle travels quickly through the center and very slowly as it gets back out to the same distance out on the other side and turns around to make the trip again, so there IS more in the halo, but nowhere near all of the DM.

 

[cwdesign]

I see what you're saying, and perhaps that's bad wording on my part. By "halo" I did mean a homogeneous cloud of DM that extends beyond the edges of the galaxy and also pervades it with equal density throughout. That's what the illustrations I've seen seem to indicate. The main question is this: is that "cloud" actually homogeneous, or does it increase in density as it approaches the galactic center?

 

[jfizzix]

The idea that the gravitational force you feel while falling towards the center of the Earth is due to the mass between you and the center of the Earth is correct, but only holds for spherically symmetric distributions of matter. With a galaxy being approximately a disk instead of a sphere, the force of gravity changes differently as you move toward the center.

If you calculate the force of gravity from a spherical hollow shell of mass, the force outside the sphere is more or less what one might expect, but the force everywhere inside the sphere, even near the edges is exactly zero. If you do the same calculations for a ring of mass, the force inside the perimeter of the ring is not zero (except in the exact center due to symmetry),

That being said, if the dark matter cloud is spherically symmetric enough, then there's no problem. The outer edges of the galaxy could orbit with the same angular velocity as the inner stars if most of the mass was dark matter, and that dark matter were approximately uniformly distributed throughout a sphere of one galactic radius. The mass enclosed would go as $r^3$, while the orbital period $T$ goes as $\sqrt{\frac{r^3}{m_{encl}}}$, so that it stays constant with radius (approximately).

 

[Chalnoth]

You have a misunderstanding about the apparent distribution of dark matter. It is hardly JUST in a halo around the outer edges of a galaxy, it is spread throughout a galaxy. Because of the way it does not interact other than gravitationally, if an individual particle of dark matter travels from somewhere in the halo through the galactic center and out to the other edge, then because of exactly the gravitational distribution you describe in your falling man scenario, the particle travels quickly through the center and very slowly as it gets back out to the same distance out on the other side and turns around to make the trip again, so there IS more in the halo, but nowhere near all of the DM.

My understanding is that the density of dark matter increases monotonically as you get closer to the galactic center (well, ignoring the fact that the distribution isn't likely to be perfectly smooth).

There *might* be more total dark matter further out, but only because there's more total volume further out.

 

[cwdesign]

My understanding is that the density of dark matter increases monotonically as you get closer to the galactic center (well, ignoring the fact that the distribution isn't likely to be perfectly smooth).

I thought it must be like that, but had never seen it directly expressed. Thanks to all!

 

[Chalnoth]

I thought it must be like that, but had never seen it directly expressed. Thanks to all!

If you want some details, one approximation for the dark matter profile of a galaxy is the NFW profile:
https://en.wikipedia.org/wiki/Navarro–Frenk–White_profile

$$\rho(r) = {\rho_0 \over {r \over R_s}\left(1 + {r \over R_s}\right)^2}$$

This profile is approximately what you get for a dark matter halo that has had enough time to equilibrate. It's not perfect, but it's good enough to get a rough idea of what numerical simulations say a dark matter halo should look like. For example, the profile will look quite a bit different immediately after a pair of galaxies has merged. But it should approach the above profile over time.

 

[cwdesign]

This profile is approximately what you get for a dark matter halo that has had enough time to equilibrate. It's not perfect, but it's good enough to get a rough idea of what numerical simulations say a dark matter halo should look like. For example, the profile will look quite a bit different immediately after a pair of galaxies has merged. But it should approach the above profile over time.

Awesome - and exactly the kind of info I was looking for - thanks! It had seemed to me that the illustrations I'd seen had completely ignored DM density, showing the halos as essentially isotropic - and that just didn't make any sense to me, even in terms of Newtonian dynamics. Granted that I'd not done the maths - more interested in the "big picture". And I need to delve further into the differences between the spiral galactic form (as jfizzix alluded to above) and the elliptical form. That's on me, to satisfy my own curiosity...

All of that said, all of you have given me food for thought aplenty, and have confirmed that I'm barking up the right tree. And I DO appreciate it!