Chandrasekhar's variational principle

Template:Short description 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 $\frac{d\rho }{dr}<0$ and $\rho (R)=0$ is stable if the quantity

$ℰ({\rho }^{\prime })=\underset{V}{\int }{\left|\frac{d\mathrm{\Phi }}{d\rho }\right|}_0\rho {\text{'}}^{2}d𝐱-G\underset{V}{\int }\underset{V}{\int }\frac{{\rho }^{\prime }(𝐱){\rho }^{\prime }({𝐱}^{\prime })}{|𝐱-{𝐱}^{\prime }|}d𝐱d{𝐱}^{\prime }\text{where}\mathrm{\Phi }=-G\underset{V}{\int }\frac{\rho ({𝐱}^{\prime })}{|𝐱-{𝐱}^{\prime }|}d𝐱,$

is non-negative for all real functions ${\rho }^{\prime }(𝐱)$ that conserve the total mass of the star $\underset{V}{\int }{\rho }^{\prime }d𝐱=0$.

where

• $𝐱$ 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
• $\rho (𝐱)$ is the unperturbed density
• ${\rho }^{\prime }(𝐱)$ is the small perturbed density such that in the perturbed state, the total density is $\rho +{\rho }^{\prime }$
• $\mathrm{\Phi }$ is the self-gravitating potential from Newton's law of gravity
• $G$ is the Gravitational constant