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Astron. Astrophys. 354, 1115-1122 (2000)

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Liouville's equation in post Newtonian approximation

II. The post Newtonian modes

Y. Sobouti 1,2,3 and V. Rezania 1

1 Institute for Advanced Studies in Basic Sciences, P.O. Box 45195-159, Zanjan 45195, Iran
2 Physics Department, Shiraz University, Shiraz 71454, Iran
3 Center for Theoretical Physics and Mathematics, AEOI, P.O. Box 11345-8486,Tehran, Iran (sobouti@iasbs.ac.ir; rezania@iasbs.ac.ir)

Received 23 April 1998 / Accepted 3 December 1999


We use the post-Newtonian (pn) order of Liouville's equation to study the normal modes of oscillation of a spherically symmetric relativistic system. Perturbations that are neutral in Newtonian approximation develop into a new sequence of normal modes. In the first pn order; a) their frequency is an order q smaller than the classical frequencies, where q is a pn expansion parameter; b) they are not damped, for there is no gravitational wave radiation in this order; c) they are not coupled with the classical modes in q order; d) because of the spherical symmetry of the underlying equilibrium configuration, they are designated by a pair of angular momentum eigennumbers, ([FORMULA]), of a pair of phase space angular momentum operators ([FORMULA]). The eigenfrequencies are, however, m-independent. Hydrodynamics of these modes is also investigated; a) they generate oscillating macroscopic toroidal motions that are neutral in the classical case; and b) they give rise to an oscillatory [FORMULA] component of the metric tensor that otherwise is zero in the unperturbed system. The conventional classical modes, which in their hydrodynamic behaviour emerge as p and g modes are, of course, perturbed to order q. These, however, have not been of concern in this paper.

Key words: methods: numerical – stars: general – stars: oscillations

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© European Southern Observatory (ESO) 2000

Online publication: February 25, 2000