Conceptual Q: Gravity, Angular Momentum, and Galaxies

[tolove]

In short:
Why are galaxies flat?

In more detail:
A very large mass M is at rest in vacuum, rotating at point O. Its axis of rotation is the xz-plane with angular momentum [itex]\vec{L}[/itex] directed along the positive y-axis. What forces do the following masses experience?

(Units of distance are arbitrary and only given for orientation. Please explain if the actual distance in a given orientation is important.)

1) m₁ is placed at rest with position [itex]\vec{r}[/itex] = [itex]\hat{i}[/itex].
2) m₂ is placed at rest with position [itex]\vec{r}[/itex] = [itex]\hat{j}[/itex].
3) m₃ is placed at rest with position [itex]\vec{r}[/itex] = [itex]\hat{i}[/itex] + [itex]\hat{j}[/itex]
4) m₄ is set into circular orbit in the xz-plane, initial position [itex]\vec{r}[/itex] = [itex]\hat{i}[/itex].
5) m₅ is set into circular orbit in the yz-plane, initial position [itex]\vec{r}[/itex] = [itex]\hat{j}[/itex].
6) m₆ is set into circular orbit, at angle ∅ = 45° with the x-axis, initial position [itex]\vec{r}[/itex] = [itex]\hat{i}[/itex] + [itex]\hat{j}[/itex].

Further, a large mass M undergoes rotation. This implies that particles in M are moving with angular velocity w = rv. However, in the case of black holes, what is the radius r? If black holes are not considered a point, then what keeps them from becoming a point?

Thank you very much for any help in understanding these confusing things!

 

[mfb]

Why are galaxies flat?

Disks are the equilibrium shape for non-zero angular momentum.

What forces do the following masses experience?

They all experience a force which is roughly pointing towards the center of the galaxy.
They can all have stable orbits (stable in terms of the global motion), but if those orbits deviate significantly from the disk, they have to pass through this disk frequently, with the possibility to get scattered there. In those scattering processes, their orbits change to be (on average) closer to the disk.

Rotating black holes have a solution with a rotating "ring". It is a mathematical solution to the equations of General Relativity. It is unclear if this ring really represents the matter distribution, however.

 

[tolove]

They can all have stable orbits (stable in terms of the global motion), but if those orbits deviate significantly from the disk, they have to pass through this disk frequently, with the possibility to get scattered there. In those scattering processes, their orbits change to be (on average) closer to the disk.

Rotating black holes have a solution with a rotating "ring". It is a mathematical solution to the equations of General Relativity. It is unclear if this ring really represents the matter distribution, however.

So, assuming it's only one mass in orbit around the larger mass, then there are no forces acting upon the small mass other than centripetal force/gravity? What if a giant amount of dust was thrown into orbit in a perfect sphere around the large mass? Would this eventually form a disk? Why? Would it be on the axis of rotation of the larger mass?

edit: I'm trying to ask, "What's the significance of the larger mass' axis of rotation? Is the larger mass' axis always connected to the axis of the ring? I can imagine the larger mass' axis of rotation being one way, then the ring of smaller masses form in another way, and over time as the smaller masses plummet inward, they even out through gravitational forces. Is there anything else at play here?

I'm not sure if I grasp your sentence on black holes. Is this what you're saying: Mathematically, there is a radius, but whether or not there actually is distance is unknown?

 

[mfb]

So, assuming it's only one mass in orbit around the larger mass, then there are no forces acting upon the small mass other than centripetal force/gravity?

What else would you expect?

What if a giant amount of dust was thrown into orbit in a perfect sphere around the large mass? Would this eventually form a disk?

If the dust has an angular momentum, parts of it would probably form a disk.

Would it be on the axis of rotation of the larger mass?

The rotation of the central mass does not matter (as long as we can neglect frame-dragging).
Usually, the axis of rotation of the central mass is the same as the axis of rotation of the stuff around it.

I'm not sure if I grasp your sentence on black holes. Is this what you're saying: Mathematically, there is a radius, but whether or not there actually is distance is unknown?

In a black hole, the usual ways to define "distance", "radius" and so on fail, or depend on the (arbitrary) coordinate system. In addition, we cannot observe the interior of a black hole.

 

[tolove]

The rotation of the central mass does not matter (as long as we can neglect frame-dragging).
Usually, the axis of rotation of the central mass is the same as the axis of rotation of the stuff around it.

Thank you for the link on frame dragging, I will read into it this weekend. For now though, since you're saying that frame dragging can be neglected, I'll ask some more questions!

Do I have this right? A ring's formation is purely mechanical, smaller masses bumping into each other over time, eventually creating a plane along an arbitrary axis, thereby making it impossible for anything to orbit outside the plane of reference without physical interference.

What causes this ring to be on the same axis as the larger mass' axis of rotation, though? Is this just a simple physical thing as well? Something like, originally, the ring was slightly off the axis' of the large mass, and as matter fell into the large mass over time, the axis of the central mass was physically "pushed" into line.

And as for why the central mass' axis of rotation started similar to the ring? Just how it was created?

Thank you very much, by the way. Almost every question I've posted here on PF, you've helped me.

 

[mfb]

Do I have this right? A ring's formation is purely mechanical, smaller masses bumping into each other over time, eventually creating a plane along an arbitrary axis, thereby making it impossible for anything to orbit outside the plane of reference without physical interference.

Well, it is not impossible - the galaxy, our solar system, the gas giants... they all have stuff not orbiting in the plane. But most of the material ends there.

What causes this ring to be on the same axis as the larger mass' axis of rotation, though?

The common origin of both - the central mass is formed out of the same clump of material as the disk.

 

[Bandersnatch]

Here, I've found this handy program on Khan Academy:
http://www.khanacademy.org/cs/challenge-modeling-accretion-disks/1180451277

It's great for getting an intuitive feel for how things work in a collapsing cloud of particles.
Play with the variables to model a range of situations.

Let me warn you, though, it can eat up a LOT of resources on your computer.

 

[tolove]

Well, it is not impossible - the galaxy, our solar system, the gas giants... they all have stuff not orbiting in the plane. But most of the material ends there.
The common origin of both - the central mass is formed out of the same clump of material as the disk.

Oh, by impossible, I meant another smaller mass entering orbit after the ring has already formed. That new small mass cannot safely orbit anywhere outside the ring that's already present.

"The common origin of both." Alright. This makes sense. Was just wondering if there was another force involved, but you're saying it's much similar.

And to Bander, thank you! This looks very interesting.

 

[mfb]

Oh, by impossible, I meant another smaller mass entering orbit after the ring has already formed. That new small mass cannot safely orbit anywhere outside the ring that's already present.

It can. This is how the gas giants in our solar system acquired many of their moons, and our galaxy got stellar streams.

 

[rbj]

In short:
Why are galaxies flat?

In more detail:
A very large mass M is at rest in vacuum, rotating at point O. Its axis of rotation is the xz-plane with angular momentum [itex]\vec{L}[/itex] directed along the positive y-axis. What forces do the following masses experience?

think of every swirl of turbulence in the primordial universe as a region of space with matter swirling about an axis of spin. what you identify as the y-axis.

gravity acts in all directions but the (fictional) centrifugal force is only acting in directions perpendicular to the y-axis. so the mass is this swirl of turbulence collapses along the y-axis, but because of the centrifugal force along the xz-plane, it does not collapse as much or as quickly.