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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.

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?

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Thermal equilibrium

Thermal equilibrium relies strongly on the idea of equipartition of (kinetic) energy. 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) and these results): $$\left(\frac{M_2}{M_1}\right)\left(\frac{m_2}{m_1}\right)^{3/2}<0.16$$ 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 (see Meylan (2000)), i.e. $$2\langle T\rangle+\langle V\rangle=0$$ where $T$ and $V$ are kinetic and potential energies.

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  • $\begingroup$ And note that virial equilibrium does not require thermal equilibrium because the kinetic energy can be distributed among the stars in any way. $\endgroup$
    – ProfRob
    Oct 15, 2016 at 17:26
  • $\begingroup$ 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? $\endgroup$
    – quantumflash
    Oct 15, 2016 at 19:41
  • $\begingroup$ @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. $\endgroup$
    – ProfRob
    Oct 15, 2016 at 20:38
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    $\begingroup$ @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. $\endgroup$
    – ProfRob
    Oct 16, 2016 at 8:18
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    $\begingroup$ @quantumflash. N-body simulations: Spera et al. (2016) arxiv.org/abs/1604.03943 Trenti & van der Marel (2013) 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. $\endgroup$
    – ProfRob
    Oct 16, 2016 at 8:22

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