Chandrasekhar potential energy tensor

In astrophysics, Chandrasekhar potential energy tensor provides the gravitational potential of a body due to its own gravity created by the distribution of matter across the body, named after the Indian American astrophysicist Subrahmanyan Chandrasekhar. The Chandrasekhar tensor is a generalization of potential energy in other words, the trace of the Chandrasekhar tensor provides the potential energy of the body.

Definition

The Chandrasekhar potential energy tensor is defined as

$W_{i j} = - \frac{1}{2} \int_{V} ρ Φ_{i j} d 𝐱 = \int_{V} ρ x_{i} \frac{\partial Φ}{\partial x_{j}} d 𝐱$

where

$Φ_{i j} (𝐱) = G \int_{V} ρ (𝐱^{'}) \frac{(x_{i} - {x_{i}}^{'}) (x_{j} - {x_{j}}^{'})}{| 𝐱 - 𝐱^{'} |^{3}} d 𝐱^{'}, \Rightarrow Φ_{i i} = Φ = G \int_{V} \frac{ρ (𝐱^{'})}{| 𝐱 - 𝐱^{'} |} d 𝐱^{'}$

where

$G$ is the Gravitational constant
$Φ (𝐱)$ is the self-gravitating potential from Newton's law of gravity
$Φ_{i j}$ is the generalized version of $Φ$
$ρ (𝐱)$ is the matter density distribution
$V$ is the volume of the body

It is evident that $W_{i j}$ is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor $W_{i j}$ is nothing but the potential energy $W$ .

$W = W_{i i} = - \frac{1}{2} \int_{V} ρ Φ d 𝐱 = \int_{V} ρ x_{i} \frac{\partial Φ}{\partial x_{i}} d 𝐱$

Hence Chandrasekhar tensor can be viewed as the generalization of potential energy.

Chandrasekhar's Proof

Consider a matter of volume $V$ with density $ρ (𝐱)$ . Thus

$\begin{matrix} W_{i j} & = - \frac{1}{2} \int_{V} ρ Φ_{i j} d 𝐱 \\ = - \frac{1}{2} G \int_{V} \int_{V} ρ (𝐱) ρ (𝐱^{'}) \frac{(x_{i} - {x_{i}}^{'}) (x_{j} - {x_{j}}^{'})}{| 𝐱 - 𝐱^{'} |^{3}} d 𝐱^{'} d 𝐱 \\ = - G \int_{V} \int_{V} ρ (𝐱) ρ (𝐱^{'}) \frac{x_{i} (x_{j} - {x_{j}}^{'})}{| 𝐱 - 𝐱^{'} |^{3}} d 𝐱 d 𝐱^{'} \\ = G \int_{V} d 𝐱 ρ (𝐱) x_{i} \frac{\partial}{\partial x_{j}} \int_{V} d 𝐱^{'} \frac{ρ (𝐱^{'})}{| 𝐱 - 𝐱^{'} |} \\ = \int_{V} ρ x_{i} \frac{\partial Φ}{\partial x_{j}} d 𝐱 \end{matrix}$

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

$χ (𝐱) = - G \int_{V} ρ (𝐱^{'}) | 𝐱 - 𝐱^{'} | d 𝐱^{'}$

then Chandrasekhar proves that

$W_{i j} = δ_{i j} W + \frac{\partial^{2} χ}{\partial x_{i} \partial x_{j}}$

Setting $i = j$ we get $\nabla^{2} χ = - 2 W$ , taking Laplacian again, we get $\nabla^{4} χ = 8 π G ρ$ .

References

↑ Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids" (PDF). Ap. J. 136: 1037–1047. Bibcode:1962ApJ...136.1037C. doi:10.1086/147456. Retrieved March 24, 2012.
↑ Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field" (PDF). Ap. J. 118: 116. Bibcode:1953ApJ...118..116C. doi:10.1086/145732. Retrieved March 24, 2012.
↑ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.
↑ Binney, James; Tremaine, Scott (30 October 2011). Galactic Dynamics (Second ed.). Princeton University Press. pp. 59–60. ISBN 978-1400828722.
↑ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.

[1] Chandrasekhar, S; Lebovitz NR (1962). "The Potentials and the Superpotentials of Homogeneous Ellipsoids" (PDF). Ap. J. 136: 1037–1047. Bibcode:1962ApJ...136.1037C. doi:10.1086/147456. Retrieved March 24, 2012.

[2] Chandrasekhar, S; Fermi E (1953). "Problems of Gravitational Stability in the Presence of a Magnetic Field" (PDF). Ap. J. 118: 116. Bibcode:1953ApJ...118..116C. doi:10.1086/145732. Retrieved March 24, 2012.

[3] Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.

[4] Binney, James; Tremaine, Scott (30 October 2011). Galactic Dynamics (Second ed.). Princeton University Press. pp. 59–60. ISBN 978-1400828722.

[5] Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.

Chandrasekhar potential energy tensor

Definition

Chandrasekhar's Proof

Chandrasekhar tensor in terms of scalar potential

See also

References