5.5. The Jeans theorem¶
The Jeans equations are useful in using observational data to constrain the mass distribution of galaxies, but they suffer from the issue that they do not close in general and that solutions obtained under closure assumptions do not necessarily correspond to distribution functions that are everywhere positive: \(\forall \vec{x},\vec{v}: f(\vec{x},\vec{v}) \geq 0\). Therefore, we now move on to discussing full solutions \(f(\vec{x},\vec{v})\) of the equilibrium collisionless Boltzmann Equation (5.33).
The Jeans theorem is an important theorem that significantly simplifies the problem of solving the equilibrium Boltzmann equation. It concerns integrals of motion, which we discussed in Chapter 4.3. From the definition of an integral of motion \(I(\vec{x},\vec{v})\) in Equation (4.44), we have that \begin{equation}\label{eq-const-integral-1} \frac{\mathrm{d} I(\vec{x},\vec{v})}{\mathrm{d} t} = 0\,. \end{equation} If we write the derivative in terms of the partial derivatives with respect to \(\vec{x}\) and \(\vec{v}\), we find that \begin{equation}\label{eq-const-integral} \frac{\mathrm{d} I}{\mathrm{d} t} = \dot{\vec{x}} \,\frac{\partial I}{\partial \vec{x}}+ \dot{\vec{v}}\,\frac{\partial I}{\partial \vec{v}}= 0\,. \end{equation} Comparing this to Equation (5.33), we see that \(I(\vec{x},\vec{v})\) is a solution of the equilibrium collisionless Boltzmann equation. This immediately leads to the Jeans theorem:
Jeans theorem: Any function of the integrals of motion is a solution of the equilibrium collisionless Boltzmann equation. Furthermore, any solution of the equilibrium collisionless Boltzmann equation only depends on the phase-space coordinates \((\vec{x},\vec{v})\) through the integrals of motion.
Proof: Both statements directly follow from Equations (5.52) and (5.53): If \(f \equiv f(I_1,\ldots,I_n)\) is any function of \(n\) integrals of the motion, then \(\frac{\mathrm{d}f}{\mathrm{d} t} = \sum_i \frac{\partial f}{\partial I_i}\,\frac{\mathrm{d} I_i}{\mathrm{d} t} = 0\), by virtue of Equation (5.52). Conversely, if \(f(\vec{x},\vec{v})\) is a solution of the equilibrium collisionless Boltzmann equation, then Equation (5.53) holds for \(f(\vec{x},\vec{v})\) and therefore \(f(\vec{x},\vec{v})\) is an integral of the motion, straightforwardly satisfying the second part of the theorem.
A stronger version of this theorem (which we will not prove) states that the distribution function of a steady-state system in which all but a negligible fraction of orbits are regular can be written as a function of three independent, non-isolating integrals only and these may be chosen to be the three actions for regular orbits discussed in Chapter 3.4.3. This stronger version means that the generally useless non-isolating integrals (see Chapter 4.3) can be ignored when building the distribution function. Instead, when finding solutions of the equilibrium collisionless Boltzmann equations, we should constrain ourselves to looking for solutions that are functions of the isolating integrals of the motion and that are positive for all \((\vec{x},\vec{v})\). For spherical systems, these integrals are \(E\) and \(\vec{L}\). Therefore, we only need to consider functions \(f \equiv f(E,\vec{L})\). If we are furthermore interested in self-consistent solutions where \(f\) sources the potential or in spherically-symmetric tracer distributions embedded in a spherical potential, the distribution function cannot depend on the orientation of the orbital plane (or the resulting [tracer] density distribution would not be spherically symmetric). Therefore, for spherical systems we generally only consider equilibrium distribution functions \(f \equiv f(E,L)\) (or also including the sign of \(L\) when considering rotation).
Before continuing, it is important to note that even though we will consider distribution functions of the form \(f(E,\vec{L})\), the distribution function is always considered to be a probability density on the phase-space coordinates. That is, \(f(\vec{x},\vec{v})\) has units of one over phase-space volume, because its integral over phase-space volume is a unitless number (the fraction of bodies in that phase-space volume). Therefore, even when we use a distribution function \(f \equiv f(E)\), as we do below, the distribution function still gives the probability of finding stars in a small volume around (\(\vec{x},\vec{v}\)): \(p(\vec{x},\vec{v}) = f(E[\vec{x},\vec{v}])\), not the probability of finding stars with energies in a small range around \(E\) (which would have units of one over energy, because its integral over an energy range would be a unitless number). Because canonical transformations preserve the phase-space volume element, it is well defined for any set of canonical coordinates \((\vec{q},\vec{p})\) to consider \(f(\vec{q},\vec{p})\) to represent the probability of finding stars in a small volume around \((\vec{q},\vec{p})\).
Seeking solutions of the collisionless Boltzmann equation through the Jeans theorem comes in general in two flavors: (i) finding solutions that generate a given density profile (and potential) and (ii) finding solutions that are constrained by observational data; in both cases the solution may or may not be self consistent. The first case is a purely theoretical problem—albeit one whose results might be applied to interpreting observational data—and we can search for the simplest distribution function that generates a given density profile. For the second case, we typically want to find the simplest solution that is consistent with the data (applying Occam’s razor) and this solution may be significantly more complicated.
The Jeans theorem makes it clear that there are many, and typically an infinite number, of solutions to the equilibrium collisionless Boltzmann equation. This is in stark contrast with the situation, e.g., in thermodynamics, where there is typically a single equilibrium solution for a general physical context (e.g., the equilibrium velocity distribution in an ideal gas is the Maxwell–Boltzmann distribution). The fundamental reason for this is the collisionless nature of the dynamics within galaxies, which leads to the absence of a collision operator in the Boltzmann equation. With collisions, the right-hand side of Equation (5.29) would be a collision operator \(C[f]\) instead of zero and the presence of this term would significantly restrict possible equilibrium distributions by requiring that \(C[f] = 0\).
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