Chandrasekhar's variational principle

In astrophysics, Chandrasekhar's variational principle provides the stability criterion for a static barotropic star, subjected to radial perturbation, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar.

Statement

A baratropic star with ${\frac {d\rho }{dr}}<0$ and $\rho (R)=0$ is stable if the quantity

${\mathcal {E}}(\rho ')=\int _{V}\left|{\frac {d\Phi }{d\rho }}\right|_{0}\rho '^{2}d\mathbf {x} -G\int _{V}\int _{V}{\frac {\rho '(\mathbf {x} )\rho '(\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}d\mathbf {x} d\mathbf {x'} \quad {\text{where}}\quad \Phi =-G\int _{V}{\frac {\rho (\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}d\mathbf {x} ,$

is non-negative for all real functions $\rho '(\mathbf {x} )$ that conserve the total mass of the star $\int _{V}\rho 'd\mathbf {x} =0$.

where

• $\mathbf {x}$ is the coordinate system fixed to the center of the star
• $R$ is the radius of the star
• $V$ is the volume of the star
• $\rho (\mathbf {x} )$ is the unperturbed density
• $\rho '(\mathbf {x} )$ is the small perturbed density such that in the perturbed state, the total density is $\rho +\rho '$
• $\Phi$ is the self-gravitating potential from Newton's law of gravity
• $G$ is the Gravitational constant