<p><strong>Thermal equilibrium</strong></p>

<p>Thermal equilibrium relies strongly on the idea of <a href="https://en.wikipedia.org/wiki/Equipartition_theorem" rel="nofollow">equipartition of (kinetic) energy</a>. In a stellar system, this means that the total kinetic energy is divided evenly amongst all the stars. This doesn't imply that the velocities are all the same; they can't, because not all the masses are the same.</p>

<p><strong>Dynamical equilibrium</strong></p>

<p>Dynamical equilibrium means that over dynamical timescales, the system is stable - basically, it will not succumb to core collapse due to a gravothermal instability. Note that it may not be possible for a system to reach thermal equilibrium, even if it is in dynamical equilibrium. In a system with two main types of stars, it must satisfy the Spitzer stability condition (see <a href="http://arxiv.org/pdf/astro-ph/0111057v1.pdf" rel="nofollow">Fregeau et al. (2001)</a> and <a href="http://modest15s.net/wp-content/uploads/2015/04/Spera.pdf" rel="nofollow">these results</a>):
$$\left(\frac{M_2}{M_1}\right)\left(\frac{m_2}{m_1}\right)^{3/2}&lt;0.16$$
where $M_1$ and $M_2$ are the total masses of types 1 and 2.</p>

<p><strong>Virial equilibrium</strong></p>

<p>Virial equilibrium comes about when the system satisfies the <a href="https://en.wikipedia.org/wiki/Virial_theorem" rel="nofollow">virial theorem</a> (see <a href="https://cds.cern.ch/record/432240/files/0003390.pdf" rel="nofollow">Meylan (2000)</a>), i.e.
$$2\langle T\rangle+\langle V\rangle=0$$
where $T$ and $V$ are kinetic and potential energies.</p>

CC BY-SA 3.0
2016-10-15T17:26:00.717Z
And note that virial equilibrium does not require thermal equilibrium because the kinetic energy can be distributed among the stars in any way.
ProfRob
CC BY-SA 3.0
2016-10-15T19:41:56.370Z
Thanks both, just to make sure I understand: so the combination of thermal and dynamical equilibrium is "thermodynamical equilibrium" and it involves equipartition of energy among the constituents of the system, as well as long-term dynamical stability. Virial equilibrium seems much easier to work with rather than thermal or dynamical equilibrium because it involves long-term time average and it's not always clear that an observed, e.g., globular cluster is in thermal equilibrium. @RobJeffries & HDE: is gravothermal instability and its implications still studied intensively by astrophysicists?
quantumflash
CC BY-SA 3.0
2016-10-15T20:38:02.120Z
@quantumflash Yes it is. Another instance to consider - the virial theorem fails where objects have internal degrees of freedom. In clusters this means binaries are problematic. It also now appears likely that clusters never reach equipartition.
ProfRob
CC BY-SA 3.0
2016-10-15T23:04:41.647Z
Dear @RobJeffries, can you please offer a nice review or even recent research paper that explores the idea that stellar clusters never reach equipartition? Does this mean using the virial theorem to derive the mass of a globular cluster is not *perfectly* correct?
quantumflash
CC BY-SA 3.0
2016-10-16T08:18:14.983Z
@quantumflash I said they never reach equipartition. They are likely to be in virial equilibrium (approximately). Virial equilibrium does not demand that the KE is distributed among the stars equally.
ProfRob
CC BY-SA 3.0
2016-10-16T08:22:32.413Z
@quantumflash. N-body simulations: Spera et al. (2016) https://arxiv.org/abs/1604.03943 Trenti & van der Marel (2013) https://arxiv.org/abs/1302.2152 .  I am working on a paper that confirms the results of the former paper using radial velocity measurements in a rich open cluster.
ProfRob
CC BY-SA 3.0
2016-10-16T18:39:59.883Z
Very cool @RobJeffries! Can I just ask how one learns enough N-body simulation tools to be able to carry out such theoretical experiments? Is simply reading Binney & Tremaine good enough, and then I should be able to just program simple gravity-based N-body simulations? And would N-body simulations as in those papers run on my own laptop (e.g., using Python), or would they need super-computers?
quantumflash
CC BY-SA 3.0
2016-10-16T18:53:16.517Z
@quantumflash Some codes are public. I can't comment on your speed question; it depends on $N$.  But I would think you might be limited to a few hundred stars. http://www.ast.cam.ac.uk/~sverre/web/pages/nbody.htm
ProfRob
CC BY-SA 3.0
2016-10-17T04:10:45.817Z
@RobJeffries Thanks again! So generally one needs to use these public codes to do their N-body simulations? It is not possible to write up your own N-body solver in, e.g., python?
quantumflash
CC BY-SA 3.0
2016-03-31T18:00:46.687Z
2016-03-31T18:00:46.687Z
HDE 226868
<p>I'm currently going through Binney &amp; Tremaine (2008) on my own to learn about stellar dynamics. I also have been perusing additional online resources such as <a href="http://www.scholarpedia.org/article/N-body_simulations_(gravitational)">this scholarpedia wiki</a>. </p>

<p>Often when distinguishing between collision-less vs collisional stellar systems, the virial theorem is invoked along with the equations for "crossing time" (also known as "dynamical time") and "relaxation time." A large galaxy is said to be collision-less because its relaxation time is many orders of magnitude higher than its age, whereas a dense stellar system (e.g., a globular cluster) is collisional because its relaxation time is less than its age. </p>

<p>But what is the relationship between this so-called "relaxed" state and virial, dynamical, and thermodynamic equilibrium? What do the three different kinds of equilibria intuitively mean? </p>

<p>For example, I have heard that large galaxies are assumed to be in virial equilibrium and then people derive "dynamical masses" (why not "virial masses"?). What would it take and/or mean for a large elliptical galaxy to be not just in virial equilibrium, but also in dynamical or thermodynamic equilibrium?</p>

<galaxy><galactic-dynamics><star-systems><stellar-dynamics>
Stellar systems: what is the difference between virial, dynamical and thermodynamic equilibrium?
quantumflash
# Stellar systems: what is the difference between virial, dynamical and thermodynamic equilibrium?

(Asked by: username_0 on 2016-03-30)

I'm currently going through Binney & Tremaine (2008) on my own to learn about stellar dynamics. I also have been perusing additional online resources such as [this scholarpedia wiki](http://www.scholarpedia.org/article/N-body_simulations_(gravitational)).

Often when distinguishing between collision-less vs collisional stellar systems, the virial theorem is invoked along with the equations for "crossing time" (also known as "dynamical time") and "relaxation time." A large galaxy is said to be collision-less because its relaxation time is many orders of magnitude higher than its age, whereas a dense stellar system (e.g., a globular cluster) is collisional because its relaxation time is less than its age.

But what is the relationship between this so-called "relaxed" state and virial, dynamical, and thermodynamic equilibrium? What do the three different kinds of equilibria intuitively mean?

For example, I have heard that large galaxies are assumed to be in virial equilibrium and then people derive "dynamical masses" (why not "virial masses"?). What would it take and/or mean for a large elliptical galaxy to be not just in virial equilibrium, but also in dynamical or thermodynamic equilibrium?

**Tags:** galaxy, galactic-dynamics, star-systems, stellar-dynamics


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## 1 answers

### Answer by username_1 on 2016-03-31. Score: 3 | Accepted answer

**Thermal equilibrium**

Thermal equilibrium relies strongly on the idea of [equipartition of (kinetic) energy](https://en.wikipedia.org/wiki/Equipartition_theorem). In a stellar system, this means that the total kinetic energy is divided evenly amongst all the stars. This doesn't imply that the velocities are all the same; they can't, because not all the masses are the same.

**Dynamical equilibrium**

Dynamical equilibrium means that over dynamical timescales, the system is stable - basically, it will not succumb to core collapse due to a gravothermal instability. Note that it may not be possible for a system to reach thermal equilibrium, even if it is in dynamical equilibrium. In a system with two main types of stars, it must satisfy the Spitzer stability condition (see [Fregeau et al. (2001)](http://arxiv.org/pdf/astro-ph/0111057v1.pdf) and [these results](http://modest15s.net/wp-content/uploads/2015/04/Spera.pdf)):
$$\left(\frac{M\_2}{M\_1}\right)\left(\frac{m\_2}{m\_1}\right)^{3/2}<0.16$$
where $M\_1$ and $M\_2$ are the total masses of types 1 and 2.

**Virial equilibrium**

Virial equilibrium comes about when the system satisfies the [virial theorem](https://en.wikipedia.org/wiki/Virial_theorem) (see [Meylan (2000)](https://cds.cern.ch/record/432240/files/0003390.pdf)), i.e.
$$2\langle T\rangle+\langle V\rangle=0$$
where $T$ and $V$ are kinetic and potential energies.

> **Comments:**
> **username_2**: And note that virial equilibrium does not require thermal equilibrium because the kinetic energy can be distributed among the stars in any way.  
> **username_0**: Thanks both, just to make sure I understand: so the combination of thermal and dynamical equilibrium is "thermodynamical equilibrium" and it involves equipartition of energy among the constituents of the system, as well as long-term dynamical stability. Virial equilibrium seems much easier to work with rather than thermal or dynamical equilibrium because it involves long-term time average and it's not always clear that an observed, e.g., globular cluster is in thermal equilibrium. @RobJeffries & HDE: is gravothermal instability and its implications still studied intensively by astrophysicists?  
> **username_2**: @username_0 Yes it is. Another instance to consider - the virial theorem fails where objects have internal degrees of freedom. In clusters this means binaries are problematic. It also now appears likely that clusters never reach equipartition.  
> **username_0**: Dear @RobJeffries, can you please offer a nice review or even recent research paper that explores the idea that stellar clusters never reach equipartition? Does this mean using the virial theorem to derive the mass of a globular cluster is not *perfectly* correct?  
> **username_2**: @username_0 I said they never reach equipartition. They are likely to be in virial equilibrium (approximately). Virial equilibrium does not demand that the KE is distributed among the stars equally.  
> **username_2**: @username_0. N-body simulations: Spera et al. (2016) https://arxiv.org/abs/1604.03943 Trenti & van der Marel (2013) https://arxiv.org/abs/1302.2152 .  I am working on a paper that confirms the results of the former paper using radial velocity measurements in a rich open cluster.  
> **username_0**: Very cool @RobJeffries! Can I just ask how one learns enough N-body simulation tools to be able to carry out such theoretical experiments? Is simply reading Binney & Tremaine good enough, and then I should be able to just program simple gravity-based N-body simulations? And would N-body simulations as in those papers run on my own laptop (e.g., using Python), or would they need super-computers?  
> **username_2**: @username_0 Some codes are public. I can't comment on your speed question; it depends on $N$.  But I would think you might be limited to a few hundred stars. http://www.ast.cam.ac.uk/~sverre/web/pages/nbody.htm  
> **username_0**: @RobJeffries Thanks again! So generally one needs to use these public codes to do their N-body simulations? It is not possible to write up your own N-body solver in, e.g., python?  

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