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. Mathematical description. Consider a fluid mass formula_1 of volume formula_2 with density formula_3 and an isotropic pressure formula_4 with vanishing pressure at the bounding surfaces. Here, formula_5 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 Chandrasekhar assumed that the fluid mass is subjected to pressure force and its own gravitational force, then the Euler equations is 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 formula_16 is given by where formula_18 is the Levi-Civita symbol, formula_19 is the centrifugal acceleration and formula_20 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 formula_22 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 formula_22 direction, the equation becomes Steady state fourth order virial equation. With formula_22 being the axis of rotation, the steady state fourth order virial equation is also derived by Chandrasekhar in 1968. 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., formula_32, where formula_33 is the outward unit normal, the second order virial equation then be This can be easily extended to rotating frame of references.