Probing stellar interiors of intermediate
mass rotating stars
J.C. Suárez 1,2, M-J. Goupil 2
Insituto de Astrofísica de Andalucía, Granada, Spain
Obervatoire de Paris-Meudon, (Lesia), France
3Vars Observatory, Polony
4 Observatoire de la Côte d’Azur, Nice, France
1
2
Abstract
In this work we present preliminary numerical results of adiabatic frequencies in an intermediate mass star model, using a perturbation method up
to second order, taking into account a radial differential rotation. We show, as we predicted, that g and mixed modes are the most affected by
differential rotation. We found differences up to 5 µHz in first and second order corrections to the frequency. This can be a very important step
to try to understand the rotation profile in intermediate mass stars.
Up to date, we ignore how the angular momentum is
transported in intermediate mass stars. The structure of
this kind of stars, with their convective core, induces
instinctively to suppose a differential rotation profile. To
obtain this kind of information we need accurate
observations and numerical simulations of the oscillation
obtained from the different assumptions of the transport
of the angular momentum. The first point can be achieved
with the future space missions COROT and EDDINGTON,
and for the second point, we present here a code which
calculates
the
adiabatic
eigenfrequencies
and
eigenfunctions up to the second order and which takes into
account differential rotation.
Pseudo-rotating models: The way of taking into account
the rotation in the equilibrium models is by using an
effective gravity modified by the effects of the
centrifugal force. We use the evolution code CESAM
(Morel et al. 1995) which introduces this effect in one
dimension as:
2
g eff = g − rΩ 2
3
In figure 2 we represent the shape of the radial
term of the centrifugal acceleration.
Figure 2
When rotation is considered, the global expansion
occurred in the evolution in the Main Sequence is greater
because of the effect of the centrifugal fore. In figure 3,
the curves represent the rotation profile of models with
uniform rotation (dashed line) and differential rotation
(continuous line). This last one is build under the
assumption of a local conservation of the angular
momentum. During the evolution the convective core
shrinks and the envelope expands, thus to satisfy the
conservation of the angular momentum in a local shell, the
rotation frequency must change.
Figure 3
Figure 1
To study the effects of differential rotation on adiabatic frequencies we use equilibrium models in similar physical conditions: For an intermediate
masse-type star of 1.8M๏, we calculate 3 middle-age models (Xc~ 0.3), using three different kind of rotation: Differential rotation, uniform rotation
and no rotation. In figure 1, the continuous line corresponds to the evolution track for a model in differential rotation, where it is obtained by
considering a local conservation of the angular momentum. The dashed line corresponds to the track of a uniform rotating model, and finally the
dotted-dashed line corresponds to the non rotating model.
In figure 3 the shape of the different rotation profiles are represented. The models were constructed as to obtain the same rotation frequency at
the atmosphere. This is necessary to eliminate the effect of the radius when comparing frequencies from the different models.
To obtain the adiabatic eigenfrequencies we use a perturbative
method. Following Soufi et al (1998) we adapt the analytic
formulation to a numerical one up to second order. We obtain the
eigenfrequencies modifyied by rotation as their perturbative
corrections up to the second order:
ω = ω 0 + ω1 + ω 2
Where ω0 is the non perturbed term, ω1 is the first order
correction corresponding to:
ω1 = m Ω (C l − 1 − J l )
The term Jl takes into account the differential rotation. It is
nul when uniform rotation or no rotation are considered.
Finally ω2 is the second order correction which the analytic
expression is:
ω2 =
Ω
2
ω 02
(D0 + m 2D2 )
D0 and D2 are r-dependent. The second term is also m2
dependent which brake the symmetry of multiplets.
Figure 5
Figure 6
For the models presented above, we calculated the eigenfrequencies up to second order. In
figure 5 and 6, we represent the second order correction term vs the order n, in terms of
the adimensional frequencies (upper graphs) and in terms of micro-Hertzs (lower graphs).
In figure 5 is represented the region of g and mixed modes (lower frequencies), and in
figure 6 is focused in the higher frequencies area (p modes region). For this last region,
the difference between uniform rotation (crosses) and differential rotation (rombes) are
around 1-2 µHz. But the biggest differences come from the region of g and mixed modes
with ∆ω~4-5 µHz !!. This confirms the supposition that g and mixed modes can be affected
by a differential rotation profile in the interior region.
This is the first step to try to understand the behavior of modes in presence of
differential rotation. In the future, inversion techniques and construct with observations
from the space will be decisive for the comprehension of the transport of the angular
momentum of intermediate mass stars.