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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]
Definition
The Chandrasekhar potential energy tensor is defined as
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
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
G
{\displaystyle G}
Gravitational constant
Φ
(
x
)
{\displaystyle \Phi (\mathbf {x} )}
Newton's law of gravity
Φ
i
j
{\displaystyle \Phi _{ij}}
Φ
{\displaystyle \Phi }
ρ
(
x
)
{\displaystyle \rho (\mathbf {x} )}
density distribution
V
{\displaystyle V}
It is evident that
W
i
j
{\displaystyle W_{ij}}
W
i
j
{\displaystyle W_{ij}}
W
{\displaystyle W}
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]
Consider a matter of volume
V
{\displaystyle V}
ρ
(
x
)
{\displaystyle \rho (\mathbf {x} )}
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
The scalar potential is defined as
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]
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}
∇
2
χ
=
−
2
W
{\displaystyle \nabla ^{2}\chi =-2W}
Laplacian again, we get
∇
4
χ
=
8
π
G
ρ
{\displaystyle \nabla ^{4}\chi =8\pi G\rho }
See also
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.
