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.

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

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

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

where

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