Chandrasekhar virial equations explained

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]}

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 = \int _{V} \rho d x , I _{i} = \int _{V} \rho x _{i} d x , I _{ij} = \int _{V} \rho x _{i} x _{j} d x , I _{ijk} = \int _{V} \rho x _{i} x _{j} x _{k} d x , I _{ijk\ell} = \int _{V} \rho x _{i} x _{j} x _{k} x _{\ell} d x , etc.

the pressure moments are

\Pi = \int _{V} p d x , \Pi _{i} = \int _{V} p x _{i} d x , \Pi _{ij} = \int _{V} p x _{i} x _{j} d x , \Pi _{ijk} = \int _{V} p x _{i} x _{j} x _{kdx} etc.

the kinetic energy moments are

T _{ij} = (1)/(2) \int _{V} \rho u _{i} u _{j} d x , T _{ij;k} = (1)/(2) \int _{V} \rho u _{i} u _{j} x _{k} d x , T _{ij;k\ell} = (1)/(2) \int _{V} \rho u _{i} u _{j} x _{kx} _{\ell} d x , etc.

and the Chandrasekhar potential energy tensor moments are

W _{ij} = - (1)/(2) \int _{V} \rho \Phi _{ij} d x , W _{ij;k} = - (1)/(2) \int _{V} \rho \Phi _{ij} x _{k} d x , W _{ij;k\ell} = - (1)/(2) \int _{V} \rho \Phi _{ij} x _{k} x _{\ell} d x , etc. where \Phi _{ij} = G\int _{V} \rho( x' ) ((x _{i-x} _{i')(x} _{j-x} _{j')})/(| x - x' | ^{3}) d 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 tensorsI = I _{ii} = \int _{V} \rho | x | ^{2} d x , T = T _{ii} = (1)/(2) \int _{V} \rho | u | ^{2} d x , W = W _{ii} = - (1)/(2) \int _{V} \rho \Phi d x where \Phi = \Phi _{ii} = \int _{V} (\rho( x' ))/(| x - x' |) d x'

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

\rho (du _{i})/(dt) = - (\partial p)/(\partial x _{i}) + \rho (\partial \Phi)/(\partial x _{i}) , where (d)/(dt) = (\partial)/(\partial t) + u _{j} (\partial)/(\partial x _{j})

First order virial equation

(d ^{2 I} _{i})/(dt ^{2}) =0

Second order virial equation

(1)/(2) (d ^{2 I} _{ij})/(dt ^{2}) = 2 T _{ij} + W _{ij} + \delta _{ij} \Pi

In steady state, the equation becomes

2 T _{ij} + W _{ij} = - \delta _{ij} \Pi

Third order virial equation

(1)/(6) (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

\Omega is given by\rho (du _{i})/(dt) = - (\partial p)/(\partial x _{i}) + \rho (\partial \Phi)/(\partial x _{i}) + (1)/(2) \rho (\partial)/(\partial x _{i}) | \Omega x x | ^{2} + 2 \rho \varepsilon _{i\ell} u _{\ell} \Omega _{m}

where

\varepsilon _{i\ell} is the Levi-Civita symbol, (1)/(2) | \Omega x x | ^{2} is the centrifugal acceleration and 2 u x \Omega is the Coriolis acceleration.

Steady state second order virial equation

In steady state, the second order virial equation becomes

2 T _{ij} + W _{ij} + \Omega ^{2} I _{ij} - \Omega _{i\Omega} _{kI} _{kj} + 2 \epsilon _{i\ell} \Omega _{m} \int _{V} \rho u _{\ell} x _{j} d x = - \delta _{ij} \Pi

If the axis of rotation is chosen in

x _{3} direction, the equation becomesW _{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}, 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} + 2 \varepsilon _{i\ell} \Omega _{m} \int _{V} \rho u _{\ell} x _{j} x _{k} d x = - \delta _{ij} \Pi _{k} - \delta _{ik} \Pi _{j}

If the axis of rotation is chosen in

x _{3} direction, the equation becomesW _{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.^[4] The equation reads as(1)/(3) ( 2 W _{ij;kl} + 2 W _{ik;lj} + 2 W _{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 (du _{i})/(dt) = - (\partial p)/(\partial x _{i}) + \rho (\partial \Phi)/(\partial x _{i}) + (\partial \tau _{ik})/(\partial x _{k}) , where \tau _{ik} = \rho\nu\left( (\partial u _{i})/(+ \partial x _{k}) (\partial u _{k})/(- \partial x _{i}) (2)/(3) (\partial u _{l})/(\partial x _{l}) \delta _{ik} \right)

and we define the shear-energy tensor as

S _{ij} = \int _{V} \tau _{ij} d 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 n is the outward unit normal, the second order virial equation then be(1)/(2) (d ^{2 I} _{ij})/(dt ^{2}) = 2 T _{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

Notes and References

1. Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids" (PDF). Ap. J. 136: 1037–1047. . . Retrieved March 24, 2012.
2. Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field" (PDF). Ap. J. 118: 116. . . Retrieved March 24, 2012.
3. Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.
4. Chandrasekhar, S. (1968). The virial equations of the fourth order. The Astrophysical Journal, 152, 293. http://repository.ias.ac.in/74364/1/93-p-OCR.pdf