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 $M$ of volume $V$ with density $\rho(\mathbf,t)$ and an isotropic pressure $p(\mathbf,t)$ with vanishing pressure at the bounding surfaces. Here, $\mathbf$ 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

: $M = \int_V \rho \, d\mathbf, \quad I_i = \int_V \rho x_i \, d\mathbf, \quad I_ = \int_V \rho x_i x_j \, d\mathbf, \quad I_ = \int_V \rho x_i x_j x_k \, d\mathbf, \quad I_ = \int_V \rho x_i x_j x_k x_\ell \, d\mathbf, \quad \text$

the pressure moments are

:$\Pi = \int_V p \, d\mathbf, \quad \Pi_i = \int_V p x_i \, d\mathbf, \quad \Pi_ = \int_V p x_i x_j \, d\mathbf, \quad \Pi_ = \int_V p x_i x_j x_kd\mathbf \quad \text$

the kinetic energy moments are

:$T_ = \frac 1 2 \int_V \rho u_i u_j \, d\mathbf, \quad T_ = \frac 1 2 \int_V \rho u_i u_j x_k \, d\mathbf, \quad T_ = \frac 1 2 \int_V \rho u_i u_j x_kx_\ell \, d\mathbf, \quad \mathrm$

and the Chandrasekhar potential energy tensor moments are

:$W_ = - \frac \int_V \rho \Phi_ \, d\mathbf, \quad W_ = - \frac 1 2 \int_V \rho \Phi_ x_k \, d\mathbf, \quad W_ = - \frac 1 2 \int_V \rho \Phi_ x_k x_\ell d\mathbf, \quad \mathrm \quad \text \quad \Phi_ = G\int_V \rho(\mathbf) \frac \, d\mathbf$

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 tensors

:$I = I_ = \int_V \rho , \mathbf, ^2 \, d\mathbf, \quad T = T_ = \frac \int_V \rho , \mathbf, ^2 \, d\mathbf, \quad W = W_ = - \frac\int_V \rho \Phi \, d\mathbf \quad \text \quad \Phi = \Phi_ = \int_V \frac \, d\mathbf$

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

:$\rho \frac = - \frac + \rho \frac, \quad \text \quad \frac = \frac + u_j \frac$

First order virial equation

:$\frac =0$

Second order virial equation

:$\frac\frac = 2T_ + W_ + \delta_ \Pi$

In steady state, the equation becomes

:$2T_ + W_ = -\delta_ \Pi$

Third order virial equation

:$\frac \frac = 2 (T_ + T_ + T_) + W_ + W_ + W_ + \delta_ \Pi_k + \delta_\Pi_i + \delta_\Pi_j$

In steady state, the equation becomes

:$2(T_ + T_) + W_ + W_ = - \delta_\Pi_K -\delta_\Pi_j$

Virial equations in rotating frame of reference

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

:$\rho \frac = - \frac + \rho \frac + \frac \rho \frac , \mathbf\times\mathbf, ^2 + 2 \rho \varepsilon_ u_\ell \Omega_m$

where $\varepsilon_$ is the Levi-Civita symbol, $\frac , \mathbf\times\mathbf, ^2$ is the centrifugal acceleration and $2\mathbf u \times \mathbf\Omega$ is the Coriolis acceleration.

Steady state second order virial equation

In steady state, the second order virial equation becomes

: $2T_ + W_ + \Omega^2 I_ - \Omega_i\Omega_kI_ + 2 \epsilon_ \Omega_m \int_V \rho u_\ell x_j \, d\mathbf x = - \delta_ \Pi$

If the axis of rotation is chosen in $x_3$ direction, the equation becomes

:$W_ + \Omega^2 (I_ - \delta_ I_) = -\delta_ \Pi$

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

:$W_ = \begin W_ & W_ & 0 \\ W_ & W_ & 0 \\ 0 & 0 & W_ \end, \quad I_ = \begin I_ & I_ & 0 \\ I_ & I_ & 0 \\ 0 & 0 & I_ \end$

Steady state third order virial equation

In steady state, the third order virial equation becomes

: $2(T_ + T_) + W_ + W_ + \Omega^2 I_ - \Omega_i\Omega_\ell I_ + 2\varepsilon_ \Omega_m \int_V \rho u_\ell x_j x_k \, d\mathbf x = -\delta_\Pi_k - \delta_\Pi_j$

If the axis of rotation is chosen in $x_3$ direction, the equation becomes

:$W_ + W_ + \Omega^2 (I_ - \delta_ I_) = -(\delta_ \Pi_k + \delta_ \Pi_j)$

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.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 The equation reads as

:$\frac(2 W_ + 2 W_ + 2 W_ + W_ + W_ + W_) + \Omega^2 (I_ -\delta_ I_) = - (\delta_ \Pi_ + \delta_ \Pi_ + \delta_ \Pi_)$

Virial equations with viscous stresses

Consider the Navier-Stokes equations instead of Euler equations,

:$\rho \frac = - \frac + \rho \frac + \frac, \quad \text\quad \tau_ = \rho\nu\left(\frac+\frac-\frac \frac\delta_\right)$

and we define the shear-energy tensor as

:$S_ = \int_V \tau_ d\mathbf.$

With the condition that the normal component of the total stress on the free surface must vanish, i.e., $(-p\delta_+\tau_)n_k=0$, where $\mathbf$ is the outward unit normal, the second order virial equation then be

:$\frac\frac = 2T_ + W_ + \delta_ \Pi - S_.$

This can be easily extended to rotating frame of references.

See also

* Virial theorem
* Dirichlet's ellipsoidal problem
* Chandrasekhar tensor

References

{{Reflist

Stellar dynamics
Fluid dynamics