5. Equilibria of collisionless stellar systems¶
The concept of equilibrium is of great importance in the observational study of galaxies. The basic reason for this comes from considering Newton’s second law: the gravitational force sets the acceleration of bodies, but does not directly set their positions and velocities. Yet, positions and velocities are all that we can observe for stars and gas in galaxies. Without further assumptions, positions and velocities themselves contain no direct information about the mass distribution in galaxies. That a dynamical system is in equilibrium is one of the most useful of such additional assumptions that one can make and one that allows us to probe the mass and orbital distributions of our own galaxy, the Milky Way, and of external galaxies in detail. We will start our study of dynamical equilibria in this chapter with some general results and their specific application to spherical systems.
Before starting on a mathematical description of dynamical equilibria, it is worth asking why we think this might be a good assumption. If you have ever watched a movie of a simulation of the formation of galaxies in the cosmological setting, you will know that galaxies are not in any equilibrium state when they form and keep being perturbed by infalling gas and galaxies. In section 5.1 below, we will demonstrate that the two-body relaxation time—on which time scale we can expect an equilibrium distribution to form through energy exchanges between pairs of bodies—is \(t_\mathrm{relax} \approx t_\mathrm{dyn}\,N/[8\,\ln N] \gg t_\mathrm{dyn}\) and is many orders of magnitude larger than the current age of the Universe for all galaxies. Thus, we cannot rely on two-body relaxation for galaxies to reach an equilibrium state.
Fortunately, there are non-collisional effects that drive stellar systems to an equilibrium state, in particular violent relaxation and phase mixing, and these effects act on the dynamical time scale rather than the two-body relaxation time scale (we discuss these effects in more detail in Chapter 17.3). Since the dynamical time is typically a few 100 Myr in galaxies, this means that galaxies have typically reached a quasi-equilibrium state at the present time. For any particular galaxy, this quasi-equilibrium can be perturbed, e.g., because of a merger with another galaxy, and the equilibrium assumption should therefore not be made blindly, but overall galaxies are close to being in dynamical equilibrium. An example of violent relaxation and phase-mixing for a one-dimensional, gravitating system is shown in the following animation:
Figure 5.1: Violent relaxation and phase mixing in one dimension.
What is shown is the phase-space distribution of \(N\) particles moving under the influence of their mutual gravity in one dimension. The system starts out in a very inhomogeneous state in phase-space—all particles have almost zero velocity over a wide range of positions—but over the course of just a few orbits, the distribution reaches a quasi-equilibrium.
Because the dynamical time behaves as \(t_\mathrm{dyn} \propto R/v\) and in galaxies \(v \approx\) constant or decreasing at large \(R\), the dynamical time at large distances (e.g., 100 kpc) becomes large and closer to the age of the Universe. The assumption of equilibrium is therefore more suspect in such places. This is also seen in Figure 5.1, where phase-mixing is faster near the center of the mass distribution, because the dynamical time is shorter there.
We start this chapter with a detailed discussion of whether and on what time scales we can approximate a distribution of \(N\) bodies as a smooth distribution rather than a collection of point masses and derive an approximate expression for the relaxation time scale of stellar systems. We then introduce one of the most basic relations for an equilibrium dynamical system: the virial theorem. The virial theorem is a blunt tool that relates the global spatial and kinematic properties to the overall gravitational potential and therefore does not lead to detailed knowledge about a galaxy’s mass distribution. However, it clearly shows the relation between kinematics, spatial scale, and gravitational mass that underlies all of the more sensitive methods that we will discuss later in this chapter and in later chapters. We then continue with a more rigorous discussion of the evolution of collisionless stellar systems and apply it to the equilibrium of spherical systems.
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