*Astron. Astrophys. 321, 1024-1026 (1997)*
## 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
where
Here is the unit vector in
direction, and and
are two scalars. Evidently both components are
poloidal and mutually orthogonal, =0. Next, we
define the dimensionless parameter
Schwarzschild's criterion for convective neutrality is
. In such a fluid one readily sees that
and . That is, the
oscillatory motions of the fluid are of purely *p* -type and are
driven mainly by the compression forces . The
*g* -type motions are neutral, It can also
be shown (Sobouti & Silverman 1978) that to the first order of
smallness in , 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,
For larger values of the
two types get more and more coupled. One last remark: If
, then ; the fluid is
stable to convective motions and the oscillatory *g* modes
develop. If then and
convective motions are set up. The imaginary frequency,
, 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 ,
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.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998
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