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The integral Newton's and MacLaurin's theorems in tensor form
Author(s) -
Caimmi R.
Publication year - 2003
Publication title -
astronomische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.394
H-Index - 63
eISSN - 1521-3994
pISSN - 0004-6337
DOI - 10.1002/asna.200310083
Subject(s) - ellipsoid , tensor (intrinsic definition) , physics , mathematical analysis , weyl tensor , generalization , limit (mathematics) , volume integral , classical mechanics , potential energy , stress–energy tensor , gravitation , discontinuity (linguistics) , mathematical physics , mathematics , pure mathematics , exact solutions in general relativity , geometry , integral equation , riemann curvature tensor , curvature , astronomy
The current paper deals with the investigation of the gravitational potential of heterogeneous ellipsoids and its extension to the tensor potential, since little attention has been given to this point in the last century. In this view, both integral Newton's and integral MacLaurin's theorems are formulated in tensor form. The generalization is extended to heterogeneous homeoids and focaloidally striated ellipsoids, respectively. A discontinuity in the tensor potential is found across a homogeneous, infinitely thin focaloid, which vanishes in the spherical limit. The potential‐energy tensors related to focaloidally striated ellipsoids are expressed in integral form, depending on the density profile. All the results are particularized to the spherical limit, for which both Newton's and MacLaurin's theorems hold. With the aim of illustrating the procedure, an explicit calculation of the potential‐energy tensors is outlined in the special case of homogeneous, spherical configurations. Finally, an application is made to the Coma cluster of galaxies.