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, the Physicist Enrico Fermi and Norman R. Lebovitz.[1][2][3]
Mathematical description
Consider an ideal fluid mass of volume with density and an isotropic pressure with vanishing pressure at the bounding surface, where refers the frame of reference attached fixed at 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 from their definition. It should be noted that the moment of inertia , kinetic energy and the potential energy are just traces of the following tensors
Chandrasekhar assumed the that 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
See also
References
- ↑ 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.
- ↑ 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.
- ↑ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.
- ↑ 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