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^[1]^[2]^[3]

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

References

↑ Chandrasekhar, S. "A general variational principle governing the radial and the non-radial oscillations of gaseous masses." VI. Ellipsoidal Figures of Equilibrium 1.2 (1960).
↑ Chandrasekhar, Subrahmanyan. Hydrodynamic and hydromagnetic stability. Courier Corporation, 2013.
↑ Binney, James, and Scott Tremaine. Galactic dynamics. Princeton university press, 2011.

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