I Definition of 'dynamical equilibrium'?

[yahastu]

"Assuming that the system is in dynamical equilibrium – that the magnitude of the total potential energy is equal to twice the kinetic energy – a so-called “dynamical mass” can be derived..."
https://astrobites.org/2012/03/16/what-defines-a-galaxy/

I am trying to understand what is meant by dynamical equilibrium in the above quote. It seems that he is talking about the Virial theorem:
https://en.wikipedia.org/wiki/Virial_theorem

In other words, it seems that "a system in dynamical equilibrium" may be defined as "a system where the virial theorem holds."

However, I am curious under what conditions the virial theorem does or does not hold. I think it is one of those things that only becomes approximately true for very large systems, but I'm wondering if it makes any additional assumptions about the dynamics of that system. Does anyone know?

 

[PeterDonis]

t seems that "a system in dynamical equilibrium" may be defined as "a system where the virial theorem holds."

For the particular case you are giving, yes, that appears to be what is meant.

I am curious under what conditions the virial theorem does or does not hold.

In general, it holds for the time averages of the kinetic and potential energies for a closed system.

I'm wondering if it makes any additional assumptions about the dynamics of that system.

Only those that are necessary for the concepts of kinetic and potential energy to be well-defined.

 

[yahastu]

In general, it holds for the time averages of the kinetic and potential energies for a closed system.

I remain confused about the underlying rationale behind this theory because it presumes a fixed ratio between kinetic and potential energy which seems to be at odds with conservation of energy. In the classic example of a ball being dropped, potential energy is converted into kinetic energy so that the total energy remains conserved. According to this theorem, however, there is an approximately fixed ratio between kinetic and potential energy in any closed system, which would imply that when potential energy goes down, kinetic energy would also have to go down to maintain that virial ratio.

If we were talking about a more specific closed system -- for example that of a massive object in orbit -- then it makes sense to me that the total potential and kinetic energy of this body would remain in a fixed ratio due to the stability of it's orbit. However, you're saying that it holds for ANY type of closed system, right? Even things that have nothing to do with planetary orbits? In that case, I am confused.

Second question: when applied to galaxies as described by the author, the author suggests using the velocity dispersion as a proxy for the entire system, rather than needing to directly sum up the kinetic energies of each body individually. However, it seems to me that any attempt based on amortized statistics would suffer from cancellation effects when some objects are moving in opposite directions. For example if two objects are moving in the opposite direction, the true sum of kinetic energy may be high, despite their average velocity being zero. what's the rationale behind cancellation effects not being relevant when looking at galaxies?
https://en.wikipedia.org/wiki/Velocity_dispersion

 

[PeterDonis]

t presumes a fixed ratio between kinetic and potential energy which seems to be at odds with conservation of energy.

Remember that it's a closed system, and that it's only time averages.

which would imply that when potential energy goes down, kinetic energy would also have to go down to maintain that virial ratio

No, the kinetic energy goes up as the potential energy goes down. More precisely, as the potential energy gets more negative, the kinetic energy gets more positive. The theorem says that the (time average of) kinetic energy is minus one-half the (time average of) potential energy.

you're saying that it holds for ANY type of closed system, right?

Any type of classical closed system.

it seems to me that any attempt based on amortized statistics would suffer from cancellation effects when some objects are moving in opposite directions

He's talking about the spread of velocities about the average velocity, not about zero. Also, by "average velocity" he means the average velocity in a particular part of the galaxy; this changes as you go from one side of the galaxy to the other.

 

[yahastu]

No, the kinetic energy goes up as the potential energy goes down. More precisely, as the potential energy gets more negative, the kinetic energy gets more positive. The theorem says that the (time average of) kinetic energy is minus one-half the (time average of) potential energy.

Aha, that minus sign makes more sense!

So basically, it seems that what the virial theorem is saying is that all systems have a tendency toward having double the magnitude of potential energy as kinetic energy. In other words, you could add external energy into a system to increase the kinetic energy, but once you let it alone it would cool down and eventually reach a stable state again where the kinetic energy was half the potential energy.

I presume, then, that the simplest possible visualization of this would be 2 point masses undergoing a single attractive force between them. Regardless of their initial position the two point masses would come together and stabilize at a distance apart such that their kinetic energy was half the potential energy between them, right?

 

[PeterDonis]

So basically, it seems that what the virial theorem is saying is that all systems have a tendency toward having double the magnitude of potential energy as kinetic energy.

All bound, closed systems. And you have the energies backwards: the kinetic energy is half the magnitude of the potential energy. (If the kinetic energy were larger than the potential energy, the system would not be bound.)

you could add external energy into a system to increase the kinetic energy, but once you let it alone it would cool down and eventually reach a stable state again where the kinetic energy was half the potential energy.

Basically, yes, but note that while you are adding energy to the system and then letting it cool down, the system will not be closed. So the virial theorem won't apply during the process, only once it's reached equilibrium again at the end.

Regardless of their initial position the two point masses would come together and stabilize at a distance apart such that their kinetic energy was half the potential energy between them, right?

No. Remember that the theorem is about time averages. For a closed, bound system of two point masses, the virial theorem says that the time average of the kinetic energy is minus one-half the time average of the potential energy. Here "time average" means "average over one orbit", since a two-body closed bound system (at least if we assume Newtonian gravity) will have a closed orbit for each body around the system's center of mass. And this orbit will be entirely determined by the initial conditions; the system won't "equilibrate" (it can't, since by hypothesis there are only two bodies and nowhere else for any energy to go).

 

[yahastu]

Thanks for your clarifications.
I think fundamentally why this holds is because it is the ratio of potential energy to kinetic energy for an object in orbit, and if you put any particles in a closed system they will go into orbits. Makes sense

 

[PeterDonis]

I think fundamentally why this holds is because it is the ratio of potential energy to kinetic energy for an object in orbit,

Yes, for a two-body system in Newtonian gravity, you can solve the orbit exactly and derive the theorem that way.