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 edit

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 edit

Consider a matter of volume   with density  . Thus

 

Chandrasekhar tensor in terms of scalar potential edit

The scalar potential is defined as

 

then Chandrasekhar[5] proves that

 

Setting   we get  , taking Laplacian again, we get  .

See also edit

References edit

  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.