We first briefly summarize the traditional expression for the surface velocity field of non-radial pulsations in the limit of no rotation. We will refer to this as the zero-rotation model. Subsequently, we give a more general description which accounts for the effects of the Coriolis force on the displacement field of the pulsation. This description, which we refer to as the slow-rotation model, is equivalent to those of Saio (1981), Martens & Smeyers (1982) and Aerts & Waelkens (1993). Hereafter we will refer to these three papers jointly as SMA.
The equations that govern linear, isentropic pulsations of a rotating
star are the equations of motion, continuity, conservation of entropy,
and Poisson's equation. In our treatment the distortion of
the spherical equilibrium surface by centrifugal forces is neglected,
as well as the effects of centrifugal forces on the pulsation (which
are proportional to 
, with 
the angular
rotation frequency of the star): only the effects of the Coriolis
force (
) are taken into account. Therefore, our
description is only valid for slowly rotating stars. We will quantify
this restriction in Sect. 5.1 (click here). Following SMA, we assume
that the evolution of the star is a succession of quasi-static states
of hydrostatic and thermal equilibrium, and that the temporal part of
the pulsation can be written as e
, where 
is the angular frequency of pulsation in the corotating frame and t
is time. (Hereafter, we speak of frequencies when we mean
angular frequencies.) We write the perturbed and linearized equations
as
in which 
is the
Lagrangian displacement vector, 
, P and 
are the mass
density, the pressure, and the gravitational potential,
, 
, and 
are their Eulerian
perturbations, 
, and 
is the first
generalized isentropic coefficient. This set of six equations
determines the state of a rotating, pulsating star.
Since our aim is to calculate line-profile variations, we need to
describe the pulsational velocity field at the surface
of the star. Neglect of the term proportional to 
in
Eq. (1 (click here)) leads to the well-known solution of the
perturbation problem for which the angular dependence of the
displacement field of a normal mode is specified by one spherical
harmonic 
and its derivatives.
The spherical harmonic describes the shape of the perturbation as a
function of co-latitude 
measured from the polar axis, and
azimuth 
. We write the spherical harmonics as
and the Associated Legendre polynomial 
as
where 
and 
denote the degree and
azimuthal order of the mode respectively. The Lagrangian
displacement vector for spheroidal modes of a non-rotating star in
spherical coordinates is then given by
with 
and 
the vertical and horizontal displacement
amplitudes. The superscripts 
refer to quantities in the
non-rotating case. 
is a normalization constant
Note that our definition of 
differs from those of SMA
and Unno et al. (1989), but that this difference is not important for our
computations of the line-profile variability since we use the maximum
surface velocity to parameterize the pulsation amplitude (see
Sect. 5.4 (click here)).
The useful parameter k is defined as the
ratio of the horizontal to the vertical amplitude. In the Cowling
approximation (
), k can be evaluated at the stellar
surface by
with G, M and R the gravitational constant, the stellar mass and
radius, respectively. We discuss the validity of the Cowling
approximation for the determination of k in Sect. 2.5 (click here).
The surface velocity field in the
corotating frame is then completely specified by 
, m,
, 
, and 
. We refer to this
description as the zero-rotation model. Note that this does not mean
that the star does not rotate, but rather that we are using the zero-rotation
approximations to describe its pulsational behavior.
We now consider the effects of the Coriolis force on the properties of
the pulsation of a rotating star. Following SMA we expand all unknown
quantities in a zero-rotation part and a correction term due to the
Coriolis force. The correction terms are proportional to the parameter
(which is assumed to be smaller than unity, see
below). The vertical and horizontal Lagrangian displacement
amplitudes a and b become
and similarly, the pulsation frequency is written as (in analogy with
Ledoux 1951)
which describes the frequency splitting caused by the rotation of the
star. The constant 
(Ledoux 1951; Hansen et al.\
1978) depends on the internal structure of the star, and on the degree
and radial order n of the mode, and hence 
contains asteroseismological information.
Since the effects of the centrifugal force 
are neglected, the expansions above are, in
general, only applicable to cases where
. Substituting expansions similar
to those in Eqs. (10 (click here)) and (11 (click here)) for all
quantities in the set of Eqs. (1 (click here)-4 (click here)), and
neglecting all terms of the second and higher orders in
, leads to a new system of equations which needs
to be solved to determine the unknown first-order quantities.
As shown by SMA, the eigenfunctions of the star can now be written as
a superposition of the zero-rotation eigenfunctions (Eq. 7 (click here))
and one spheroidal and two toroidal terms. The three latter terms are
due to the Coriolis force. We express the Lagrangian displacement field
at the surface of a rotating star by
in which the zeroth and first order spheroidal terms are combined in
the first term. We note that, instead of 
, AW
incorrectly use 
and 
to normalize the
toroidal terms in their Eq. (19), and that their implementation of
the velocity field contains a small error (Aerts & Waelkens 1995).
With our definition of 
, the amplitudes 
, 
, and 
in
Eq. (12 (click here)) are given by
Note that for sectoral (
) and radial
(
) modes the term proportional to 
in
Eq. (12 (click here)) equals zero.
For a given stellar model, the amplitude of the first-order spheroidal
correction 
in Eq. (13 (click here)) can be
determined from the zeroth-order solution of the pulsation mode. In
our general treatment we avoid the introduction of an additional
parameter 
, making use of an approximation that is allowed in
a first-order approach: instead of 
, we substitute 
into Eqs. (14 (click here)) and (15 (click here)), which
introduces an error of the second order in 
in
Eq. (12 (click here)). This substitution allows us to study the effect of
the toroidal terms, which arise as a consequence of rotation. The
effect of rotation is then determined only by the rotation parameter
, which becomes the sixth parameter of the
surface velocity field in addition to the five mentioned above:
, m, 
, 
, 
, and
. In the following, we refer to this description
as the slow-rotation model.
In the slow-rotation model, k is modified by rotation and may differ
significantly from Eq. (9 (click here)). AW give an expression for
k which is correct up to the first order in 
,
but their result contains the unknown parameters 
and
and therefore cannot be used in applications. We derive
a new expression to approximate k, up to the first order in
, from the boundary condition
at the surface of the star. Our derivation of k is similar to the
one in the zero-rotation model. Under the assumptions of spherical
symmetry and hydrostatic equilibrium, the condition (16 (click here))
leads to
From the 
-component of the equation of motion of order zero in
(Eq. 1 (click here)), we find that
Similarly, an expression for 
is given by
(see e.g. Eq. (13) in AW). Note that we used the Cowling approximation
(
) in both Eq. (18 (click here)) and (19 (click here)).
We use Eqs. (10 (click here)-11 (click here), 17 (click here)-19 (click here)),
and neglect terms of order 
to obtain
This leads to the following expression for k, correct to the first
order in the parameter 
Relating k to the zero-rotation frequency 
rather than
to the pulsation frequency 
leads to the desired expression,
correct up to the first order in 
in which we used Eqs. (9 (click here)) and (11 (click here)).
Equation (22 (click here)) shows that the structure constant 
enters the description of the surface-velocity field, when first-order
terms due to the rotation are included. In the case of p-modes with
high radial order, 
tends to zero, while for g-modes with
high radial order it tends to 
(Unno et al. 1989).
For high-order p-modes, the term with 
in
Eq. (22 (click here)) vanishes (
). On the other
hand, as we will discuss in Sect. 5.6 (click here), for large
-values (i.e. high-order g-modes) a small change in k
does not change the characteristics of the line-profile variations.
Hence, for these two limiting cases of high order p- and g-modes,
the line profiles do not depend on the actual value for 
.
For cases with intermediate 
-values, 
co-determines the line profiles via the first-order correction of k.
In the slow-rotation model, we assume 
for
simplicity, which is in between the two limiting cases for p- and
g-modes of high radial order. This approximation might not be
accurate for fundamental modes. For numerical calculations of
we refer to Hansen et al. (1978) and to Carroll &
Hansen (1982).
By means of Eq. (22 (click here)), we draw the following conclusions
about the correction of 
by the rotation of the star. In the
case of high-order g-modes (large 
), the first-order
correction term for k is smaller than the zeroth-order component if
< 0.5. This means that for these modes
horizontal motions dominate like in the non-rotating case. For
p-modes (i.e. low 
), however, the first-order correction
term of k may become larger than the zeroth-order term, reflecting
the fact that the rotation can reduce radial motion in favor of
horizontal motion. For certain prograde modes (m < 0), this could
theoretically give rise to negative k-values. However, as will be
discussed in Sect. 5.1 (click here), for the small 
-values
associated with p-modes, the relative rotation rate
is also small. Therefore, the small
-values associated with p-modes will never be corrected by
the rotation in such a way that a large negative k-value is found.
We note that the first-order correction of k is
most important for modes of low degree 
.
Here we address the question: how large is the error in k due to the
Cowling approximation? For this purpose, we have used the computer
code by De Boeck (De Boeck, in preparation) to determine the ratio
of the amplitudes b and a by means of an approach that does not
make use of the Cowling approximation. The code performs a numerical
integration of a system of differential equations describing the
pulsations up to first order in 
in the case of a
polytropic model with index 3. The numerical values for k obtained
in this way were then compared with the ones obtained from
Eq. (22 (click here)), for both p- and g-modes (
and
). For each mode we have considered
, with 
, and
. In all the considered
cases, the two k-values differ less than 5%.
To derive the k-value for a real star, we need to know its
mass, radius, and pulsation frequency in the corotating frame. Since
in practice the uncertainty in 
is likely to be larger than
5%, we conclude that the error in k due to the Cowling
approximation is much smaller than that due to the inaccuracy of the
stellar parameters. The expression for k is therefore accurate
enough for our purposes.