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Chandrasekhar potential energy tensor - Wikipedia
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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.
The Chandrasekhar potential energy tensor is defined as
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W i j = − 1 2 ∫ V ρ Φ i j d x = ∫ V ρ x i ∂ Φ ∂ x j d x {\displaystyle 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
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Φ i j ( x ) = G ∫ V ρ ( x ′ ) ( x i − x i ′ ) ( x j − x j ′ ) | x − x ′ | 3 d x ′ , ⇒ Φ i i = Φ = G ∫ V ρ ( x ′ ) | x − x ′ | d x ′ {\displaystyle \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
It is evident that W i j {\displaystyle W_{ij}} is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor W i j {\displaystyle W_{ij}} is nothing but the potential energy W {\displaystyle W} .
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W = W i i = − 1 2 ∫ V ρ Φ d x = ∫ V ρ x i ∂ Φ ∂ x i d x {\displaystyle 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[edit]
Consider a matter of volume V {\displaystyle V} with density ρ ( x ) {\displaystyle \rho (\mathbf {x} )} . Thus
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W i j = − 1 2 ∫ V ρ Φ i j d x = − 1 2 G ∫ V ∫ V ρ ( x ) ρ ( x ′ ) ( x i − x i ′ ) ( x j − x j ′ ) | x − x ′ | 3 d x ′ d x = − G ∫ V ∫ V ρ ( x ) ρ ( x ′ ) x i ( x j − x j ′ ) | x − x ′ | 3 d x d x ′ = G ∫ V d x ρ ( x ) x i ∂ ∂ x j ∫ V d x ′ ρ ( x ′ ) | x − x ′ | = ∫ V ρ x i ∂ Φ ∂ x j d x {\displaystyle {\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[edit]
The scalar potential is defined as
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χ ( x ) = − G ∫ V ρ ( x ′ ) | x − x ′ | d x ′ {\displaystyle \chi (\mathbf {x} )=-G\int _{V}\rho (\mathbf {x'} )|\mathbf {x} -\mathbf {x'} |d\mathbf {x'} }
then Chandrasekhar[5] proves that
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W i j = δ i j W + ∂ 2 χ ∂ x i ∂ x j {\displaystyle W_{ij}=\delta _{ij}W+{\frac {\partial ^{2}\chi }{\partial x_{i}\partial x_{j}}}}
Setting i = j {\displaystyle i=j} we get ∇ 2 χ = − 2 W {\displaystyle \nabla ^{2}\chi =-2W} , taking Laplacian again, we get ∇ 4 χ = 8 π G ρ {\displaystyle \nabla ^{4}\chi =8\pi G\rho } .
- ^ 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.
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