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.^[1]^[2]^[3] 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_{ij}=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}d\mathbf {x} =\int _{V}\rho x_{i}{\frac {\partial \Phi }{\partial x_{j}}}d\mathbf {x}

where

\Phi _{ij}(\mathbf {x} )=G\int _{V}\rho (\mathbf {x'} ){\frac {(x_{i}-x_{i}')(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}d\mathbf {x'} ,\quad \Rightarrow \quad \Phi _{ii}=\Phi =G\int _{V}{\frac {\rho (\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}d\mathbf {x'}

where

$G$ is the Gravitational constant
$\Phi (\mathbf {x} )$ is the self-gravitating potential from Newton's law of gravity
$\Phi_{ij}$ is the generalized version of $\Phi$
$\rho ({\mathbf {x}})$ is the matter density distribution
$V$ is the volume of the body

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

W=W_{ii}=-{\frac {1}{2}}\int _{V}\rho \Phi d\mathbf {x} =\int _{V}\rho x_{i}{\frac {\partial \Phi }{\partial x_{i}}}d\mathbf {x}

Hence Chandrasekhar tensor can be viewed as the generalization of potential energy.^[4]

Chandrasekhar's Proof

Consider a matter of volume $V$ with density $\rho ({\mathbf {x}})$ . Thus

{\begin{aligned}W_{ij}&=-{\frac {1}{2}}\int _{V}\rho \Phi _{ij}d\mathbf {x} \\&=-{\frac {1}{2}}G\int _{V}\int _{V}\rho (\mathbf {x} )\rho (\mathbf {x'} ){\frac {(x_{i}-x_{i}')(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}d\mathbf {x'} d\mathbf {x} \\&=-G\int _{V}\int _{V}\rho (\mathbf {x} )\rho (\mathbf {x'} ){\frac {x_{i}(x_{j}-x_{j}')}{|\mathbf {x} -\mathbf {x'} |^{3}}}d\mathbf {x} d\mathbf {x'} \\&=G\int _{V}d\mathbf {x} \rho (\mathbf {x} )x_{i}{\frac {\partial }{\partial x_{j}}}\int _{V}d\mathbf {x'} {\frac {\rho (\mathbf {x'} )}{|\mathbf {x} -\mathbf {x'} |}}\\&=\int _{V}\rho x_{i}{\frac {\partial \Phi }{\partial x_{j}}}d\mathbf {x} \end{aligned}}

Chandrasekhar tensor in terms of scalar potential

The scalar potential is defined as

\chi (\mathbf {x} )=-G\int _{V}\rho (\mathbf {x'} )|\mathbf {x} -\mathbf {x'} |d\mathbf {x'}

then Chandrasekhar^[5] proves that

W_{ij}=\delta _{ij}W+{\frac {\partial ^{2}\chi }{\partial x_{i}\partial x_{j}}}

Setting $i=j$ we get $\nabla ^{2}\chi =-2W$ , taking Laplacian again, we get $\nabla ^{4}\chi =8\pi G\rho$ .

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. Galactic Dynamics: (Second Edition). Princeton University Press. pp. 59–60. ISBN 1400828724.
↑ Chandrasekhar, Subrahmanyan. Ellipsoidal figures of equilibrium. Vol. 9. New Haven: Yale University Press, 1969.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Chandrasekhar potential energy tensor

Definition

Chandrasekhar's Proof

Chandrasekhar tensor in terms of scalar potential

See also

References