2.2: Higher Order Virial Equations

In the last section it became clear that both the scalar and tensor forms of the virial theorem are obtained by taking spatial moments of the Equations of motion. Chandrasekhar⁹ was apparently the first to note this and to inquire into the utility of taking higher moments of the Equations of motion. There certainly is considerable precedent for this in mathematical physics. As already noted, moments in momentum-space of the Boltzmann transport Equation yield expressions for the conservation of mass, momentum and energy. Spatial moments of the transport Equation of a photon gas can be used to obtain the Equation of radiative transfer. Approximate solutions to the resulting Equations can be found if suitable assumptions such as the existence of an Equation of state are made to "close" the moment Equations. Such is the origin of such diverse expressions as the Eddington approximation in radiative transfer, the diffusion approximation in radiative transfer, the diffusion approximation in gas dynamics and many others. Usually, the higher the order of the moment expressions, the less transparent their physical content. Nevertheless, in the spirit of generality, Chandrasekhar investigated the properties of the first several moment Equations. In a series of papers, Chandrasekhar and Lebovitz^{10, 11} and later Chandrasekhar^{12, 13} developed these expressions as far as the fourth-order moments of the Equations of motion.

Since for no moment expressions other than those of the first moment do any terms ever appear that can be identified with the Virial of Claussius, it is arguable as to whether they should be called virial expressions at all. However, since it is clear that this investigation was inspired by studies of the classical virial theorem, I will briefly review their development. Recall the Euler-Lagrange Equation of hydrodynamic flow developed in Chapter I, Equation \ref{1.1.4}

\[\frac{\partial \mathbf{u}}{\partial \mathrm{t}}+(\mathbf{u} \cdot \nabla) \mathbf{u}=-\nabla \Psi-\frac{1}{\rho} \nabla \cdot \mathfrak{P}-\frac{1}{\rho} \int \mathbf{S}(\mathbf{v}-\mathbf{u}) \mathrm{dv}.\label{2.2.1}\]

Quite simply the nth order "virial Equations" of Chandrasekhar are generated by taking (n-l) outer tensor products of the radius vector r and Equation \ref{2.2.1}. The result is then integrated over all physical space. This leads to a set of tensor Equations containing tensors of rank n. If we assume that particle collisions are isotropic, then the source term of Equation \ref{2.2.1} vanishes and \(\nabla \cdot \mathfrak{P}=\nabla \mathrm{P}\). The symbolic representation for the nth order "virial Equation" can then be written as:

\[\int_\mathrm{v} \mathbf{r}^{(\mathrm{n}-1)} \rho \frac{\mathrm{d} \mathbf{u}}{\mathrm{dt}} \mathrm{dV}+\int_\mathrm{v} \mathbf{r}^{(\mathrm{n}-1)} \rho \nabla \Phi+\int_\mathrm{v} \mathbf{r}^{(\mathrm{n}-1)} \nabla \mathrm{PdV}=0.\label{2.2.2}\]

Recalling our arguments in Chapter I, section 4, about conservation of mass, it follows from Equation \ref{1.4.6} that

\[\frac{\mathrm{d}}{\mathrm{dt}}\left(\int_{\mathrm{v}} \rho \mathrm{QdV}\right)=\int_{\mathrm{v}} \rho \frac{\mathrm{dQ}}{\mathrm{dt}} \mathrm{dV},\label{2.2.3}\]

where Q is any point-defined property of the medium. Thus, the first order "virial Equations", correspond to the integrals of the Equation of motion themselves over volume. So

\[\frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{v}} \rho \mathbf{u} \mathrm{dV}+\int_{\mathrm{v}} \rho \nabla \Phi \mathrm{dV}+\int_{\mathrm{v}} \nabla \mathrm{PdV}=0.\label{2.2.4}\]

Noting that \(\nabla \mathrm{Q}=\nabla \cdot (\mathbb{1} \mathrm{Q} / 3)\), the second and third integrals are zero by the divergence theorem and Equation \ref{2.2.4} becomes

\[\frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{dt}^2}\left(\int_{\mathrm{v}} \rho \mathbf{r}\mathrm{dV}\right)=0,\label{2.2.5}\]

essentially telling us that the center of mass \(\left(\int_\mathrm{v} \rho \mathbf{r}\operatorname{dV}\right)\) is not being accelerated. Setting n = 2 we arrive at the second order "virial Equations" that are the tensor version of Lagrange's identity that we discussed in the previous section.

As in Equation \ref{2.2.2}, we can represent the nth order "virial Equations" as

\[\int_\mathrm{v} \rho \mathbf{r}^{(n-1)} \frac{d \mathbf{u}}{d t} \mathrm{dV}+\int_\mathrm{v} \rho \mathbf{r}^{(n-1)} \nabla \Phi \mathrm{dV}+\int_\mathrm{v} \mathbf{r}^{(n-1)} \nabla P \mathrm{dV}=0,\label{2.2.6}\]

which after use of continuity and the divergence theorem becomes

\[\frac{\mathrm{d}}{\mathrm{dt}} \int_{\mathrm{v}} \rho \mathbf{r}^{(\mathrm{n}-1)} \mathbf{u }\mathrm{dV}-\int_{\mathrm{v}} \rho \frac{\mathrm{d}}{\mathrm{dt}}\left(\mathbf{r}^{(\mathrm{n}-1)}\right) \mathbf{u}\mathrm{dV}+\int_{\mathrm{v}} \rho \mathbf{r}^{(\mathrm{n}-1)} \nabla \Phi \mathrm{dV}+\int_{\mathrm{v}} \mathrm{P} \mathbb1{} \cdot \nabla\left(\mathbf{r}^{(\mathrm{n}-1)}\right) \mathrm{dV}=0 .\label{2.2.7}\]

Since the outer product in general does not commute, the integrals of the second and fourth term become strings of tensors of the form

\[\begin{aligned}
&\rho \frac{\mathrm{d}}{\mathrm{dt}}\left(\mathbf{r}^{(\mathrm{n}-1)}\right) \mathbf{u}=\mathbf{u}\left(\mathbf{r}^{(\mathrm{n}-2)}\right) \mathbf{u}+\mathbf{r} \mathbf{u}\left(\mathbf{r}^{(\mathrm{n}-3)}\right) \mathbf{u}+\cdots+\mathbf{r}^{(\mathrm{n}-2)} \mathbf{u} \mathbf{u},\\[4pt]
&\text{and}\\[4pt]
&\mathrm{P} \nabla\left(\mathbf{r}^{(\mathrm{n}-1)}\right)=\mathrm{P}\left(\mathbb{1} \mathbf{r}^{(\mathrm{n}-2)}+\mathbf{r} \mathbb{1} \mathbf{r}^{(\mathrm{n}-3)}+\cdots+\mathbf{r}^{(\mathrm{n}-2)} \mathbb{1}\right).
\end{aligned}\label{2.2.8}\]

However, the first term can be written as

\[\frac{d}{d t} \int_\mathrm{v} \rho \mathbf{r}^{(n-1)} \mathbf{u} d V=\frac{1}{n!} \frac{d^2}{d t^2} \int_\mathrm{v} \rho \mathbf{r}^{(n)} d V.\label{2.2.9}\]

Thus, each term in Equation \ref{2.2.7} represents one or more tensors of rank n, the first of which is the second time derivative of a generalized moment of inertia tensor and the last three are all 'energy-like' tensors. From Equation \ref{2.2.8}, it is clear that the second integral will generate tensors which are spatial moments of the kinetic energy distributions while the last term will produce moment tensors of the pressure distribution. The third integral is, however, the most difficult to rigorously represent. For n = 2 we know it is just the total potential energy. Chandrasekhar8 shows how these tensors can be built up from the generalized Newtonian tensor potential or alternatively from a series of scalar potentials which obey the Equations

\[\begin{aligned}
\nabla^2 \Psi & =-4 \pi \mathrm{G} \rho \\[4pt]
\nabla^4 \mathfrak{I} & =-8 \pi \mathrm{G} \rho \\[4pt]
\nabla^6 \Re & =-32 \pi \mathrm{G} \rho\\[4pt]
&\cdot\\[4pt]&\cdot\\[4pt]&\cdot
\end{aligned}\label{2.2.10}\]

Thus, we have formulated a representation of what Chandrasekhar calls the "higher order virial Equations". They are, in fact, spatial tensor moments of the Equations of motion. We may expect them to be of importance in the same general way as the virial theorem itself. That is, in stationary systems the left hand side of Equation \ref{2.2.7} vanishes and the result is a system of identities between the various tensor energy moments. Keeping in mind that any continuum function can be represented in terms of a moment expansion, Equation \ref{2.2.7} must thus contain all of the information concerning the structure of the system. These Equations thus represent an alternate form to the solution of the Equation of motions. Like most series expansions, it is devoutly to be wished that they will converge rapidly and the "higher order" tensors can usually be neglected.