Chandrasekhar Potential Energy Tensor

In astrophysics, Chandrasekhar potential energy tensor provides the gravitational potential of a body due to its own gravity created by the distribution of matter across the body, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar tensor is a generalization of potential energy in other words, the trace of the Chandrasekhar tensor provides the potential energy of the body.

Definition

The Chandrasekhar potential energy tensor is defined as

:$W_ = -\frac \int_V \rho \Phi_ d\mathbf =\int_V \rho x_i \frac d\mathbf$
where
:$\Phi_(\mathbf) = G \int_V \rho(\mathbf) \frac d\mathbf, \quad \Rightarrow \quad \Phi_ = \Phi = G \int_V \frac d\mathbf$
where
*$G$ is the Gravitational constant
*$\Phi(\mathbf)$ is the self-gravitating potential from Newton's law of gravity
*$\Phi_$ is the generalized version of $\Phi$
*$\rho(\mathbf)$ is the matter density distribution
*$V$ is the volume of the body

It is evident that $W_$ is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor $W_$ is nothing but the potential energy $W$.

:$W= W_ = -\frac \int_V \rho \Phi d\mathbf = \int_V \rho x_i \frac d\mathbf$

Hence Chandrasekhar tensor can be viewed as the generalization of potential energy.

Chandrasekhar's Proof

Consider a matter of volume $V$ with density $\rho(\mathbf)$. Thus

:$\begin W_ &= -\frac \int_V \rho \Phi_ d\mathbf \\ &= - \frac G \int_V \int_V \rho(\mathbf)\rho(\mathbf) \fracd\mathbfd\mathbf \\ &= -G \int_V \int_V \rho(\mathbf)\rho(\mathbf) \fracd\mathbfd\mathbf \\ &= G \int_V d\mathbf\rho(\mathbf)x_i \frac \int_V d\mathbf \frac\\ &= \int_V \rho x_i \frac d\mathbf \end$

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

:$\chi(\mathbf) = -G \int_V \rho(\mathbf) , \mathbf-\mathbf, d\mathbf$

then ChandrasekharChandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969. proves that

:$W_ = \delta_ W + \frac$

Setting $i=j$ we get $\nabla^2\chi = -2W$, taking Laplacian again, we get $\nabla^4\chi = 8\pi G \rho$.

See also

* Virial theorem
* Chandrasekhar virial equations

References

{{Reflist

Stellar dynamics
Astrophysics