Astronomy: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]\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

[1][2][3]

References

  1. 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).
  2. Chandrasekhar, Subrahmanyan. Hydrodynamic and hydromagnetic stability. Courier Corporation, 2013.
  3. Binney, James, and Scott Tremaine. Galactic dynamics. Princeton university press, 2011.