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

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The Chandrasekhar potential energy tensor is defined as

 

where

 

where

  •   is the Gravitational constant
  •   is the self-gravitating potential from Newton's law of gravity
  •   is the generalized version of  
  •   is the matter density distribution
  •   is the volume of the body

It is evident that   is a symmetric tensor from its definition. The trace of the Chandrasekhar tensor   is nothing but the potential energy  .

 

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

Chandrasekhar's Proof

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Consider a matter of volume   with density  . Thus

 

Chandrasekhar tensor in terms of scalar potential

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The scalar potential is defined as

 

then Chandrasekhar[5] proves that

 

Setting   we get  , taking Laplacian again, we get  .

See also

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References

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  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.