Английская Википедия:Chandrasekhar's variational principle

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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 <math>\frac{d\rho}{dr}<0</math> and <math>\rho(R)=0</math> is stable if the quantity

<math>\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},</math>

is non-negative for all real functions <math>\rho'(\mathbf{x})</math> that conserve the total mass of the star <math>\int_V \rho' d\mathbf{x} = 0</math>.

where

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