1.1: The Basic Equations of Structure

Before turning to the derivation of the virial theorem, it is appropriate to review the origin of the fundamental structural Equations of stellar astrophysics. This not only provides insight into the basic conservation laws implicitly assumed in the description of physical systems, but by their generality and completeness graphically illustrates the complexity of the complete description that we seek to circumvent. Since lengthy and excellent texts already exist on this subject, our review will of necessity be a sketch. Any description of a physical system begins either implicitly or explicitly from certain general conservation principles. Such a system is considered to be a collection of articles, each endowed with a spatial location and momentum which move under the influence of known forces. If one regards the characteristics of spatial position and momentum as being highly independent, then one can construct a multi-dimensional space through which the particles will trace out unique paths describing their history.

This is essentially a statement of determinism, and in classical terms is formulated in a six-dimensional space called phase-space consisting of three spatial dimensions and three linearly independent momentum dimensions. If one considers an infinitesimal volume of this space, he may formulate a very general conservation law which simply says that the divergence of the flow of particles in that volume is equal to the number created or destroyed within that volume.

The mathematical formulation of this concept is usually called the Boltzmann transport Equation and takes this form:

\[\frac{\partial \psi}{\partial \mathrm{t}}+\sum_{\mathrm{i}=1}^3 \dot{\mathrm{x}}_{\mathrm{i}} \frac{\partial \psi}{\partial \mathrm{x}_{\mathrm{i}}}+\sum_{\mathrm{i}=1}^3 \dot{\mathrm{p}}_{\mathrm{i}} \frac{\partial \psi}{\partial \mathrm{p}_{\mathrm{i}}}=\mathrm{S},\nonumber\]

or in vector notation

\[\frac{\partial \psi}{\partial \mathrm{t}}+\mathbf{v} \cdot \nabla \psi+\mathbf{f} \cdot \nabla_{\mathrm{p}} \psi=\mathrm{S},\label{1.1.1}\]

where ψ is the density of points in phase space, f is the vector sum of the forces acting on the particles and S is the 'creation rate' of particles within the volume. The homogeneous form of this Equation is often called the Louisville Theorem and would be discussed in detail in any good book on Classical Mechanics.

A determination of ψ as a function of the coordinates and time constitutes a complete description of the system. However, rarely is an attempt made to solve Equation \ref{1.1.1} but rather simplifications are made from which come the basic Equations of stellar structure. This is generally done by taking 'moments' of the Equations with respect to the various coordinates. For example, noting that the integral of ψ over all velocity space yields the matter density ρ and that no particles can exist with unbounded momentum, averaging Equation \ref{1.1.1} over all velocity space yields

\[\frac{\partial \rho}{\partial \mathrm{t}}+\nabla \cdot (\mathbf{u} \rho)=\overline{\mathrm{S}},\label{1.1.2}\]

where u is the average stream velocity of the particles and is defined by

\[\mathbf{u}=\frac{1}{\rho} \int \psi \mathbf{v} d \mathbf{v}.\label{1.1.3}\]

For systems where mass is neither created nor destroyed \(\overline{\mathrm{S}}=0\),

and Equation \ref{1.1.2} is just a statement of the conservation of mass. If one multiplies Equation \ref{1.1.1} by the particle velocities and averages again over all velocity space he will obtain after a great deal of algebra the Euler-Lagrange Equations of hydrodynamic flow

\[\frac{\partial \mathbf{u}}{\partial \mathrm{t}}+(\mathbf{u} \cdot \nabla) \mathbf{u}=-\nabla \Psi-\frac{1}{\rho} \nabla \cdot \mathfrak{P}-\frac{1}{\rho} \int \mathbf{S}(\mathbf{v}-\mathbf{u}) \mathrm{dv}.\label{1.1.4}\]

Here the forces f have been assumed to be derivable from a potential Ψ. The symbol \(\mathfrak{P}\) is known as the pressure tensor and has the form

\[\mathfrak{P}=\int \psi(\mathbf{v}-\mathbf{u})(\mathbf{v}-\mathbf{u}) \mathrm{dv}.\label{1.1.5}\]

These rather formidable Equations simplify considerably in the case where many collisions randomize the particle motion with respect to the mean stream velocity u .Under these conditions the last term on the right of Equation \ref{1.1.4} vanishes and the pressure tensor becomes diagonal with each element equal. Its divergence then becomes the gradient of the familiar scalar known as the gas pressure P. If we further consider only systems exhibiting no stream motion we arrive at the familiar Equation of hydrostatic equilibrium

\[\nabla \mathrm{P}=-\rho \nabla \Psi .\label{1.1.6}\]

Multiplying Equation \ref{1.1.1} by v and averaging over v, has essentially turned the Boltzmann transport Equation into an Equation expressing the conservation of momentum. Equation \ref{1.1.6} along with Poisson's Equation for the sources of the potential

\[\nabla^2 \Psi=-4 \pi \mathrm{G} \rho,\label{1.1.7}\]

constitute a complete statement of the conservation of momentum.

Multiplying Equation \ref{1.1.1} by \(\mathbf{v} \cdot \mathbf{v} \text { or } \mathbf{v}^2\) and averaging over all velocity space will produce an Equation which represents the conservation of energy, which when combined with the ideal gas law is

\[\rho \frac{\mathrm{dE}}{\mathrm{dt}}+\rho \nabla \cdot \mathbf{v}=\rho \varepsilon+\chi-\nabla \cdot \mathbf{F},\label{1.1.8}\]

where F is the radiant flux, ε the total rate of energy generation and χ is the energy generated by viscous motions. If one has a machine wherein no mass motions exist and all energy flows by radiation, we have a statement of radiative equilibrium;

\[\nabla \cdot \mathbf{F}=\rho \varepsilon.\label{1.1.9}\]

For static configurations exhibiting spherical symmetry these conservative laws take their most familiar form:

\[\begin{aligned}
\text{Conservation of mass}\quad& \frac{\mathrm{dm}(\mathrm{r})}{\mathrm{dr}}=4 \pi \mathrm{r}^2 \rho . \\[4pt]
\text{Conservation of momentum}\quad& \frac{\mathrm{dP}(\mathrm{r})}{\mathrm{dr}}=-\frac{\mathrm{Gm}(\mathrm{r}) \rho}{\mathrm{r}^2} . \\[4pt]
\text{Conservation of energy}\quad& \frac{\mathrm{dL}(\mathrm{r})}{\mathrm{dr}}=4 \pi \mathrm{r}^2 \rho \varepsilon, \quad \mathrm{L}(\mathrm{r})=4 \pi \mathrm{r}^2 \mathrm{F} .
\end{aligned}\label{1.1.10}\]