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?


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.

  • $\begingroup$ And note that virial equilibrium does not require thermal equilibrium because the kinetic energy can be distributed among the stars in any way. $\endgroup$ – Rob Jeffries Oct 15 '16 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 '16 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. $\endgroup$ – Rob Jeffries Oct 15 '16 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. $\endgroup$ – Rob Jeffries Oct 16 '16 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. $\endgroup$ – Rob Jeffries Oct 16 '16 at 8:22

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