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 d ρ d r < 0 {\displaystyle {\frac {d\rho }{dr}}<0} and ρ ( R ) = 0 {\displaystyle \rho (R)=0} is stable if the quantity
E ( ρ ′ ) = ∫ V | d Φ d ρ | 0 ρ ′ 2 d x − G ∫ V ∫ V ρ ′ ( x ) ρ ′ ( x ′ ) | x − x ′ | d x d x ′ where Φ = − G ∫ V ρ ( x ′ ) | x − x ′ | d x , {\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 ρ ′ ( x ) {\displaystyle \rho '(\mathbf {x} )} that conserve the total mass of the star ∫ V ρ ′ d x = 0 {\displaystyle \int _{V}\rho 'd\mathbf {x} =0} . where
x {\displaystyle \mathbf {x} } is the coordinate system fixed to the center of the star
R {\displaystyle R} is the radius of the star
V {\displaystyle V} is the volume of the star
ρ ( x ) {\displaystyle \rho (\mathbf {x} )} is the unperturbed density
ρ ′ ( x ) {\displaystyle \rho '(\mathbf {x} )} is the small perturbed density such that in the perturbed state, the total density is ρ + ρ ′ {\displaystyle \rho +\rho '}
Φ {\displaystyle \Phi } is the self-gravitating potential from Newton's law of gravity
G {\displaystyle G} is the Gravitational constant
References