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 $\rho (\mathbf {x} ,t)$ and an isotropic pressure $p(\mathbf {x} ,t)$ with vanishing pressure at the bounding surfaces. Here, $\mathbf {x}$ 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

$M=\int _{V}\rho \,d\mathbf {x} ,\quad I_{i}=\int _{V}\rho x_{i}\,d\mathbf {x} ,\quad I_{ij}=\int _{V}\rho x_{i}x_{j}\,d\mathbf {x} ,\quad I_{ijk}=\int _{V}\rho x_{i}x_{j}x_{k}\,d\mathbf {x} ,\quad I_{ijk\ell }=\int _{V}\rho x_{i}x_{j}x_{k}x_{\ell }\,d\mathbf {x} ,\quad {\text{etc.}}$

the pressure moments are

$\Pi =\int _{V}p\,d\mathbf {x} ,\quad \Pi _{i}=\int _{V}px_{i}\,d\mathbf {x} ,\quad \Pi _{ij}=\int _{V}px_{i}x_{j}\,d\mathbf {x} ,\quad \Pi _{ijk}=\int _{V}px_{i}x_{j}x_{k}d\mathbf {x} \quad {\text{etc.}}$

the kinetic energy moments are

$T_{ij}={\frac {1}{2}}\int _{V}\rho u_{i}u_{j}\,d\mathbf {x} ,\quad T_{ij;k}={\frac {1}{2}}\int _{V}\rho u_{i}u_{j}x_{k}\,d\mathbf {x} ,\quad T_{ij;k\ell }={\frac {1}{2}}\int _{V}\rho u_{i}u_{j}x_{k}x_{\ell }\,d\mathbf {x} ,\quad \mathrm {etc.}$

and the Chandrasekhar potential energy tensor moments are

$W_{ij}=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}\,d\mathbf {x} ,\quad W_{ij;k}=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}x_{k}\,d\mathbf {x} ,\quad W_{ij;k\ell }=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}x_{k}x_{\ell }d\mathbf {x} ,\quad \mathrm {etc.} \quad {\text{where}}\quad \Phi _{ij}=G\int _{V}\rho (\mathbf {x'} ){\frac {(x_{i}-x_{i}')(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}\,d\mathbf {x'}$

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

$I=I_{ii}=\int _{V}\rho |\mathbf {x} |^{2}\,d\mathbf {x} ,\quad T=T_{ii}={\frac {1}{2}}\int _{V}\rho |\mathbf {u} |^{2}\,d\mathbf {x} ,\quad W=W_{ii}=-{\frac {1}{2}}\int _{V}\rho \Phi \,d\mathbf {x} \quad {\text{where}}\quad \Phi =\Phi _{ii}=\int _{V}{\frac {\rho (\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}\,d\mathbf {x'}$

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

$\rho {\frac {du_{i}}{dt}}=-{\frac {\partial p}{\partial x_{i}}}+\rho {\frac {\partial \Phi }{\partial x_{i}}},\quad {\text{where}}\quad {\frac {d}{dt}}={\frac {\partial }{\partial t}}+u_{j}{\frac {\partial }{\partial x_{j}}}$

First order virial equation

${\frac {d^{2}I_{i}}{dt^{2}}}=0$

Second order virial equation

${\frac {1}{2}}{\frac {d^{2}I_{ij}}{dt^{2}}}=2T_{ij}+W_{ij}+\delta _{ij}\Pi$

In steady state, the equation becomes

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

Third order virial equation

${\frac {1}{6}}{\frac {d^{2}I_{ijk}}{dt^{2}}}=2(T_{ij;k}+T_{jk;i}+T_{ki;j})+W_{ij;k}+W_{jk;i}+W_{ki;j}+\delta _{ij}\Pi _{k}+\delta _{jk}\Pi _{i}+\delta _{ki}\Pi _{j}$

In steady state, the equation becomes

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

Virial equations in rotating frame of reference

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

$\rho {\frac {du_{i}}{dt}}=-{\frac {\partial p}{\partial x_{i}}}+\rho {\frac {\partial \Phi }{\partial x_{i}}}+{\frac {1}{2}}\rho {\frac {\partial }{\partial x_{i}}}|\mathbf {\Omega } \times \mathbf {x} |^{2}+2\rho \varepsilon _{i\ell m}u_{\ell }\Omega _{m}$

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

Steady state second order virial equation

In steady state, the second order virial equation becomes

$2T_{ij}+W_{ij}+\Omega ^{2}I_{ij}-\Omega _{i}\Omega _{k}I_{kj}+2\epsilon _{i\ell m}\Omega _{m}\int _{V}\rho u_{\ell }x_{j}\,d\mathbf {x} =-\delta _{ij}\Pi$

If the axis of rotation is chosen in $x_{3}$ direction, the equation becomes

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

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

$W_{ij}={\begin{pmatrix}W_{11}&W_{12}&0\\W_{21}&W_{22}&0\\0&0&W_{33}\end{pmatrix}},\quad I_{ij}={\begin{pmatrix}I_{11}&I_{12}&0\\I_{21}&I_{22}&0\\0&0&I_{33}\end{pmatrix}}$

Steady state third order virial equation

In steady state, the third order virial equation becomes

$2(T_{ij;k}+T_{ik;j})+W_{ij;k}+W_{ik;j}+\Omega ^{2}I_{ijk}-\Omega _{i}\Omega _{\ell }I_{\ell jk}+2\varepsilon _{i\ell m}\Omega _{m}\int _{V}\rho u_{\ell }x_{j}x_{k}\,d\mathbf {x} =-\delta _{ij}\Pi _{k}-\delta _{ik}\Pi _{j}$

If the axis of rotation is chosen in $x_{3}$ direction, the equation becomes

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

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

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

Virial equations with viscous stresses

Consider the Navier-Stokes equations instead of Euler equations,

$\rho {\frac {du_{i}}{dt}}=-{\frac {\partial p}{\partial x_{i}}}+\rho {\frac {\partial \Phi }{\partial x_{i}}}+{\frac {\partial \tau _{ik}}{\partial x_{k}}},\quad {\text{where}}\quad \tau _{ik}=\rho \nu \left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}-{\frac {2}{3}}{\frac {\partial u_{l}}{\partial x_{l}}}\delta _{ik}\right)$

and we define the shear-energy tensor as

$S_{ij}=\int _{V}\tau _{ij}d\mathbf {x} .$

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 $\mathbf {n}$ is the outward unit normal, the second order virial equation then be

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

This can be easily extended to rotating frame of references.

See also

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