Chandrasekhar virial equations

In astrophysics, the Chandrasekhar virial equations are a hierarchy of moment equations of the Euler equations, developed by the Indian American astrophysicist Subrahmanyan Chandrasekhar, and the physicist Enrico Fermi and Norman R. Lebovitz.

Mathematical description

Consider a fluid mass $M$ of volume $V$ with density $\rho (𝐱,t)$ and an isotropic pressure $p(𝐱,t)$ with vanishing pressure at the bounding surfaces. Here, $𝐱$ refers to a frame of reference attached to the center of mass. Before describing the virial equations, let's define some moments.

The density moments are defined as

$M=\underset{V}{\int }\rho d𝐱,I_i=\underset{V}{\int }\rho x_{i}d𝐱,I_{ij}=\underset{V}{\int }\rho x_i{x}_{j}d𝐱,I_{ijk}=\underset{V}{\int }\rho x_i{x}_j{x}_{k}d𝐱,I_{ijk\ell }=\underset{V}{\int }\rho x_i{x}_j{x}_k{x}_{\ell }d𝐱,\text{etc.}$

the pressure moments are

$\mathrm{\Pi }=\underset{V}{\int }pd𝐱,{\mathrm{\Pi }}_i=\underset{V}{\int }p{x}_{i}d𝐱,{\mathrm{\Pi }}_{ij}=\underset{V}{\int }p{x}_i{x}_{j}d𝐱,{\mathrm{\Pi }}_{ijk}=\underset{V}{\int }p{x}_i{x}_j{x}_{k}d𝐱\text{etc.}$

the kinetic energy moments are

$T_{ij}=\frac{1}{2}\underset{V}{\int }\rho u_i{u}_{j}d𝐱,T_{ij;k}=\frac{1}{2}\underset{V}{\int }\rho u_i{u}_j{x}_{k}d𝐱,T_{ij;k\ell }=\frac{1}{2}\underset{V}{\int }\rho u_i{u}_j{x}_k{x}_{\ell }d𝐱,\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}$

and the Chandrasekhar potential energy tensor moments are

$W_{ij}=-\frac{1}{2}\underset{V}{\int }\rho {\mathrm{\Phi }}_{ij}d𝐱,W_{ij;k}=-\frac{1}{2}\underset{V}{\int }\rho {\mathrm{\Phi }}_{ij}{x}_{k}d𝐱,W_{ij;k\ell }=-\frac{1}{2}\underset{V}{\int }\rho {\mathrm{\Phi }}_{ij}{x}_k{x}_{\ell }d𝐱,\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\text{where}{\mathrm{\Phi }}_{ij}=G\underset{V}{\int }\rho ({𝐱}^{\prime })\frac{(x_i-{x_i}^{\prime })(x_j-{x_j}^{\prime })}{|𝐱-{𝐱}^{\prime }{|}^3}d{𝐱}^{\prime }$

where $G$ is the gravitational constant.

All the tensors are symmetric by definition. The moment of inertia $I$, kinetic energy $T$ and the potential energy $W$ are just traces of the following tensors

$I=I_{ii}=\underset{V}{\int }\rho |𝐱{|}^{2}d𝐱,T=T_{ii}=\frac{1}{2}\underset{V}{\int }\rho |𝐮{|}^{2}d𝐱,W=W_{ii}=-\frac{1}{2}\underset{V}{\int }\rho \mathrm{\Phi }d𝐱\text{where}\mathrm{\Phi }={\mathrm{\Phi }}_{ii}=\underset{V}{\int }\frac{\rho ({𝐱}^{\prime })}{|𝐱-{𝐱}^{\prime }|}d{𝐱}^{\prime }$

Chandrasekhar assumed that the fluid mass is subjected to pressure force and its own gravitational force, then the Euler equations is

$\rho \frac{d{u}_i}{dt}=-\frac{\partial p}{\partial x_i}+\rho \frac{\partial \mathrm{\Phi }}{\partial x_i},\text{where}\frac{d}{dt}=\frac{\partial }{\partial t}+u_j\frac{\partial }{\partial x_j}$

First order virial equation

$\frac{d^2{I}_i}{d{t}^2}=0$

Second order virial equation

$\frac{1}{2}\frac{d^2{I}_{ij}}{d{t}^2}=2T_{ij}+W_{ij}+{\delta }_{ij}\mathrm{\Pi }$

In steady state, the equation becomes

$2T_{ij}+W_{ij}=-{\delta }_{ij}\mathrm{\Pi }$

Third order virial equation

$\frac{1}{6}\frac{d^2{I}_{ijk}}{d{t}^2}=2(T_{ij;k}+T_{jk;i}+T_{ki;j})+W_{ij;k}+W_{jk;i}+W_{ki;j}+{\delta }_{ij}{\mathrm{\Pi }}_k+{\delta }_{jk}{\mathrm{\Pi }}_i+{\delta }_{ki}{\mathrm{\Pi }}_j$

In steady state, the equation becomes

$2(T_{ij;k}+T_{ik;j})+W_{ij;k}+W_{ik;j}=-{\delta }_{ij}{\mathrm{\Pi }}_K-{\delta }_{ik}{\mathrm{\Pi }}_j$

Virial equations in rotating frame of reference

The Euler equations in a rotating frame of reference, rotating with an angular velocity $\mathrm{\Omega }$ is given by

$\rho \frac{d{u}_i}{dt}=-\frac{\partial p}{\partial x_i}+\rho \frac{\partial \mathrm{\Phi }}{\partial x_i}+\frac{1}{2}\rho \frac{\partial }{\partial x_i}|\mathrm{\Omega }\times 𝐱{|}^2+2\rho {\epsilon }_{i\ell m}{u}_{\ell }{\mathrm{\Omega }}_m$

where ${\epsilon }_{i\ell m}$ is the Levi-Civita symbol, $\frac{1}{2}|\mathrm{\Omega }\times 𝐱{|}^2$ is the centrifugal acceleration and $2𝐮\times \mathrm{\Omega }$ is the Coriolis acceleration.

Steady state second order virial equation

In steady state, the second order virial equation becomes

$2T_{ij}+W_{ij}+{\mathrm{\Omega }}^2{I}_{ij}-{\mathrm{\Omega }}_i{\mathrm{\Omega }}_k{I}_{kj}+2{ϵ}_{i\ell m}{\mathrm{\Omega }}_m\underset{V}{\int }\rho u_{\ell }{x}_{j}d𝐱=-{\delta }_{ij}\mathrm{\Pi }$

If the axis of rotation is chosen in $x_3$ direction, the equation becomes

$W_{ij}+{\mathrm{\Omega }}^2(I_{ij}-{\delta }_{i3}{I}_{3j})=-{\delta }_{ij}\mathrm{\Pi }$

and Chandrasekhar shows that in this case, the tensors can take only the following form

$W_{ij}=\left(\begin{array}{ccc}{W}_{11}& W_{12}& 0\\ W_{21}& W_{22}& 0\\ 0& 0& W_{33}\end{array}\right),I_{ij}=\left(\begin{array}{ccc}{I}_{11}& I_{12}& 0\\ I_{21}& I_{22}& 0\\ 0& 0& I_{33}\end{array}\right)$

Steady state third order virial equation

In steady state, the third order virial equation becomes

$2(T_{ij;k}+T_{ik;j})+W_{ij;k}+W_{ik;j}+{\mathrm{\Omega }}^2{I}_{ijk}-{\mathrm{\Omega }}_i{\mathrm{\Omega }}_{\ell }{I}_{\ell jk}+2{\epsilon }_{i\ell m}{\mathrm{\Omega }}_m\underset{V}{\int }\rho u_{\ell }{x}_j{x}_{k}d𝐱=-{\delta }_{ij}{\mathrm{\Pi }}_k-{\delta }_{ik}{\mathrm{\Pi }}_j$

If the axis of rotation is chosen in $x_3$ direction, the equation becomes

$W_{ij;k}+W_{ik;j}+{\mathrm{\Omega }}^2(I_{ijk}-{\delta }_{i3}{I}_{3jk})=-({\delta }_{ij}{\mathrm{\Pi }}_k+{\delta }_{ik}{\mathrm{\Pi }}_j)$

Steady state fourth order virial equation

With $x_3$ being the axis of rotation, the steady state fourth order virial equation is also derived by Chandrasekhar in 1968. The equation reads as

$\frac{1}{3}(2W_{ij;kl}+2W_{ik;lj}+2W_{il;jk}+W_{ij;k;l}+W_{ik;l;j}+W_{il;j;k})+{\mathrm{\Omega }}^2(I_{ijkl}-{\delta }_{i3}{I}_{3jkl})=-({\delta }_{ij}{\mathrm{\Pi }}_{kl}+{\delta }_{ik}{\mathrm{\Pi }}_{lj}+{\delta }_{il}{\mathrm{\Pi }}_{jk})$

Virial equations with viscous stresses

Consider the Navier-Stokes equations instead of Euler equations,

$\rho \frac{d{u}_i}{dt}=-\frac{\partial p}{\partial x_i}+\rho \frac{\partial \mathrm{\Phi }}{\partial x_i}+\frac{\partial {\tau }_{ik}}{\partial x_k},\text{where}{\tau }_{ik}=\rho \nu (\frac{\partial u_i}{\partial x_k}+\frac{\partial u_k}{\partial x_i}-\frac{2}{3}\frac{\partial u_l}{\partial x_l}{\delta }_{ik})$

and we define the shear-energy tensor as

$S_{ij}=\underset{V}{\int }{\tau }_{ij}d𝐱.$

With the condition that the normal component of the total stress on the free surface must vanish, i.e., $(-p{\delta }_{ik}+{\tau }_{ik})n_k=0$, where $𝐧$ is the outward unit normal, the second order virial equation then be

$\frac{1}{2}\frac{d^2{I}_{ij}}{d{t}^2}=2T_{ij}+W_{ij}+{\delta }_{ij}\mathrm{\Pi }-S_{ij}.$

This can be easily extended to rotating frame of references.

See also

• Virial theorem
• Dirichlet's ellipsoidal problem
• Chandrasekhar tensor