Astronomy: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 [math]\displaystyle{ \frac{d\rho}{dr}\lt 0 }[/math] and [math]\displaystyle{ \rho(R)=0 }[/math] is stable if the quantity

[math]\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}, }[/math]

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

where

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