The Symmetry of Angular Momentum Conservation

[Alfredo Tifi]

TL;DR Summary

Imagine you start from a quasi-spherical rarefied cloud of "stardust" particles with quasi-inertial random movements. After enough time you'll get a condensed body, maybe a star in the centre of mass and other external bodies. Everything would be rotating as in our Solar system.

I suppose that the principle of conservation of angular momentum holds also for a cloud of particles weekly interacting at low pressure, density and temperature. And it should be still applicable when the particles or the atoms would start condensing and forming fusion products or simply solid matter.

The principle of conservation of angular moment keeps its validity even if the condensing cloud start emitting several kinds of radiant energy or sub-atomic particles.

I also know that the only way to keep the angular moment constant while kinetic energy is dissipated uniformly (due to tidal interactions or any kind of internal friction or decay) is to enlarge the radius of rotation of the particles, or of the condensing bodies, along the rotation plane.

The symmetry changes from spherical to disc or "sombrero" type, with a very precise rotation axis.
So, there are centripetal (due to gravity) and centrifugal (due to energy loss) flows of matter.
The system becomes heterogeneous due to clashing opposite flows and local spiral movement arise out of the centre due to the local concentrations of matter.

This scenery is a fantastic example of self-organisation of matter from a randomly distributed matter, but... which is the source of the initial angular moment? It is a sort of unavoidable non-zero fluctuation of the not-exactly balanced momenta of the many particles which are moving in the cloud? Or is it a portion of a "stash" of angular momentum derived from the big-bang?

 

[anorlunda]

Consider any system of two or more particles having nonzero velocities. Except for the case where they will have a head on collision, the multiparticle system has a nonzero angular momentum around the system's center of gravity. Nothing mysterious.

However, I recommend a very entertaining and educational paper on your topic. It discusses how rotational energy is shed by radiation.

The Potato Radius: a Lower Minimum Size for Dwarf Planets

 

[Alfredo Tifi]

Thank you anurlunda;
I enjoyed the "potatoids" and the protoplasm formation.
I'm very happy to learn so many forms of dissipative mechanisms in the dense protoplanetary disk and angular moment transferring.
All the article confirm that the total angular moment is exactly conserved, even though energy isn't (the system is open).
But I'm not satisfied with your answer about the origin of angular momentum. First of posting my question, I considered the two particles with opposite velocities moving in parallel lines, not head-on. There is a tiny angular momentum there. But, imagine to put a mirror somewhere. The two particles behind the mirror have exactly the opposite angular momentum. Now, in the initial cloud, there are seemingly many couples of particles moving in a mirrorlike way. So a multiparticle spherical cloud of dust should have a very tiny angular momentum. Otherwise, if the angular moment is relevant, it should have been inherited from somewhere, on some time.
Larsen (2010) observation that spin momentum of formed stars is tenfold reduced respect to the clumps in molecular clouds is indicative that all clouds seem to have an initial angular moment of the same order of magnitude (proportional to mass content).

 

[anorlunda]

I considered the two particles with opposite velocities moving in parallel lines, not head-on. There is a tiny angular momentum there. But, imagine to put a mirror somewhere.

It is more of a probabilistic thing. Think of a gas cloud with many particles moving in (seemingly) random directions. We could calculate the net angular momentum from all combinations of particles taken two at a time, and summing those tiny contributions. What is the probability of the entire cloud having net zero angular momentum?

So when we see a nebula in space, it appears to be static with no apparent angular momentum. But if it collapses, it is like a figure skater who pulls in her arms to spin faster. After collapse, any (seemingly negligible) net angular momentum in the nebula becomes very significant for the collapsed object.

 

[Alfredo Tifi]

The probability of nonzero AM should be much less as much bigger was the Angular Momentum. That is we should have a zero-peaked distribution of probability if we could prepare a "canonical ensemble" of equally randomly prepared clouds.
But, I don't think we can talk of probability if momentum is exactly conserved.
A neat momentum should be explained by the previous history of the cloud.
So I definitely go back to the Big Bang.

 

[effed3]

If we have a random cloud of -static- particles (the same will happen with random moving particles), subject only to self-gravity, the collapse will never be perfect or ideal, (all particles direct to the center of gravity CG).

Due to the disomogeneities in the distribution the particles will fall toward the CG but falling in orbit (random elliptic paths), and only a small amount of particles will move directly to the CG to collide-coalesce.
The total net angular momentum will arise from the global mean of this random orbital motion, it's very probable that the orbits will never cancel out each others perfectly, so a net (small) angular momentum will arise and this define an axis of rotation in the collapsing cloud.

Next (but really quite at the same time) others effects come: the dynamic braking, a form of friction due to the mutual gravity of the moving particles in the cloud, will brings orbits smaller, the angular momentum increase (as for dancers), some particles will escape the cloud for gravity assist (slowing the other), more particles will collide-coalesce in the center, etc etc so the whole cloud will collapse and rotate faster.

So rotation arise from a random distribution being not perfectly omogeneous.

And the cloud will be develop in disk form, the orbits will be forced to rotate toward the plane perpendicular to the axis of rotation.. but this is another story (but same physics).

 

[Alfredo Tifi]

If we have a random cloud of -static- particles (the same will happen with random moving particles), subject only to self-gravity, the collapse will never be perfect or ideal, (all particles direct to the center of gravity CG).
...
Due to the disomogeneities in the distribution the particles will fall toward the CG but falling in orbit ...
The total net angular momentum will arise from the global mean of this random orbital motion, it's very probable that the orbits will never cancel out each others perfectly, so a net (small) angular momentum will arise and this define an axis of rotation in the collapsing cloud.
...
So rotation arise from a random distribution being not perfectly omogeneous.

I didn't consider the perspective of starting with a cloud of static particles. This is interesting and better because rotation will ensue from a static random configuration. But it is evident that orbital plus spin angular momenta would give a zero sum. In our Solar System everything is rotating counterclockwise, thus the original cloud had a net original angular momentum.
Anyway, I'm not able to quantify if this is small or big an amount. or how much fluctuations in the distribution of matter are relatable to the narrower or wider distribution in the angular momenta of the several bodies.
And, I don't know anything about the distribution of momenta of the several galaxies.

 

[sophiecentaur]

And, I don't know anything about the distribution of momenta of the several galaxies.

I seem to remember reading on PF, somewhere that the opinion is that the net AM is very small / zero. I'm not sure what the evidence is for that conclusion. But you'd have to ask, if there's a preferred axis then why?