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.
[1][2][3] Consider a fluid mass
with vanishing pressure at the bounding surfaces.
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
All the tensors are symmetric by definition.
The moment of inertia
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 In steady state, the equation becomes In steady state, the equation becomes The Euler equations in a rotating frame of reference, rotating with an angular velocity
is the Levi-Civita symbol,
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 In steady state, the third order virial equation becomes If the axis of rotation is chosen in
direction, the equation becomes 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 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.,
is the outward unit normal, the second order virial equation then be This can be easily extended to rotating frame of references.