SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 321, 1024-1026 (1997)

Previous Section Next Section Title Page Table of Contents

4. The g and p components of the displacement field

Using a gauged version of Helmholtz theorem (Sobouti 1981), one may decompose a general displacement vector into an irrotational and a "weighted" solenoidal component. Thus

[EQUATION]

where

[EQUATION]

Here [FORMULA] is the unit vector in [FORMULA] direction, and [FORMULA] and [FORMULA] are two scalars. Evidently both components are poloidal and mutually orthogonal, [FORMULA] =0. Next, we define the dimensionless parameter [FORMULA] Schwarzschild's criterion for convective neutrality is [FORMULA]. In such a fluid one readily sees that [FORMULA] and [FORMULA]. That is, the oscillatory motions of the fluid are of purely p -type and are driven mainly by the compression forces [FORMULA]. The g -type motions are neutral, [FORMULA] It can also be shown (Sobouti & Silverman 1978) that to the first order of smallness in [FORMULA], the p oscillations retain their pure p nature of Eq. (8a), while the g motions develop into a sequence of new and long period oscillations of the type of Eq. (8b). The latter are driven by the buoyancy forces, [FORMULA] For larger values of [FORMULA] the two types get more and more coupled. One last remark: If [FORMULA], then [FORMULA] ; the fluid is stable to convective motions and the oscillatory g modes develop. If [FORMULA] then [FORMULA] and convective motions are set up. The imaginary frequency, [FORMULA], then indicates the rate of the exponential growth of the convective motions.

Note added in revision: Toroidal modes of the fluid in the present scheme have the form [FORMULA], [FORMULA] scalar. In the Newtonian models considered in this paper they are neutral and remain neutral upon exposure to gravitational waves. The referee, however, has informed us of a recent work of Kokkotas (in press) where he shows that in relativistic stars they also are excited by gravitational waves.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
helpdesk.link@springer.de