Chandrasekhar virial equations

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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]

Contents

Mathematical description

Consider a fluid mass of volume with density and an isotropic pressure with vanishing pressure at the bounding surfaces. Here, 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

the pressure moments are

the kinetic energy moments are

and the Chandrasekhar potential energy tensor moments are

where is the gravitational constant.

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

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

First order virial equation

Second order virial equation

In steady state, the equation becomes

Third order virial equation

In steady state, the equation becomes

Virial equations in rotating frame of reference

The Euler equations in a rotating frame of reference, rotating with an angular velocity is given by

where is the Levi-Civita symbol, is the centrifugal acceleration and is the Coriolis acceleration.

Steady state second order virial equation

In steady state, the second order virial equation becomes

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

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

Steady state third order virial equation

In steady state, the third order virial equation becomes

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

Steady state fourth order virial equation

With being the axis of rotation, the steady state fourth order virial equation is also derived by Chandrasekhar in 1968. [4] The equation reads as

Virial equations with viscous stresses

Consider the Navier-Stokes equations instead of Euler equations,

and we define the shear-energy tensor as

With the condition that the normal component of the total stress on the free surface must vanish, i.e., , where is the outward unit normal, the second order virial equation then be

This can be easily extended to rotating frame of references.

See also

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References

  1. Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids" (PDF). Ap. J. 136: 1037–1047. Bibcode :  1962ApJ...136.1037C. doi : 10.1086/147456. 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. Bibcode :  1953ApJ...118..116C. doi : 10.1086/145732. 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