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.^[1][2][3]

Consider a fluid mass ${\displaystyle M}$ of volume ${\displaystyle V}$ with density ${\displaystyle \rho (\mathbf {x} ,t)}$ and an isotropic pressure ${\displaystyle p(\mathbf {x} ,t)}$ with vanishing pressure at the bounding surfaces. Here, ${\displaystyle \mathbf {x} }$ 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

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

the pressure moments are

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

the kinetic energy moments are

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

and the Chandrasekhar potential energy tensor moments are

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

where ${\displaystyle G}$ is the gravitational constant.

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

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

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

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

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

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

In steady state, the equation becomes

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

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

In steady state, the equation becomes

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

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

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

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

In steady state, the second order virial equation becomes

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

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

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

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

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

In steady state, the third order virial equation becomes

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

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

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

With ${\displaystyle x_{3}}$ being the axis of rotation, the steady state fourth order virial equation is also derived by Chandrasekhar in 1968.^[4] The equation reads as

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

Consider the Navier-Stokes equations instead of Euler equations,

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

and we define the shear-energy tensor as

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

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

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

This can be easily extended to rotating frame of references.

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