J. C. Suárez1, - H. Bruntt2,3 - D. Buzasi2
1 - Instituto de Astrofísica de Andalucía (CSIC), Granada, Spain
2 - Department of Physics, US Air Force Academy, 2354 Fairchild Dr.,
Ste. 2A31, USAF Academy, CO, USA
3 - Copenhagen University, Astronomical Observatory,
Juliane Maries Vej 30, 2100 Copenhagen Ø, Denmark
Received 22 November 2004 / Accepted 15 March 2005
Abstract
We present an asteroseismic study of the fast rotating star
HD 187642 (Altair), recently discovered
to be a Scuti pulsator.
We have computed models taking into account rotation
for increasing rotational velocities. We investigate the relation
between the fundamental
radial mode and the first overtone in the framework
of Petersen diagrams. The effects of rotation on such diagrams, which
become important at rotational velocities above
,
as well as the domain of validity of our seismic tools are discussed.
We also investigate the radial and non-radial modes
in order to constrain models fitting the five most dominant observed
oscillation modes.
Key words: stars: variables: Sct - stars: rotation - stars: oscillations - stars: fundamental parameters -
stars: interiors - stars: individual: Altair
The bright star Altair (
Aql) was recently observed by
Buzasi et al. (2005) (hereafter Paper I) with the star tracker on the Wide-Field Infrared
Explorer (WIRE) satellite. The overall observations span from 18 October until
12 November 1999, with exposures of 0.5 s taken during around 40% of the
spacecraft orbital period (96 min). The analysis of the observations made in
Paper I reveals Altair
to be a low-amplitude variable star (
),
pulsating with at least 7 oscillation modes. These results suggest that
many other non-variable stars may indeed turn out to be variable when
investigated with accurate space observations.
Since Altair lies toward the low-mass end
of the instability strip and no abundance anomalies or Pop II characteristics are shown, the
authors identified it as a Scuti star.
The
Scuti stars are representative of intermediate mass stars with spectral types from A
to F. They are located on and just off the main sequence, in the faint part of
the Cepheid instability strip (luminosity classes V and IV).
Hydrodynamical processes occurring
in stellar interiors remain poorly understood.
Scuti stars seem particularly suitable
for the study of such physical process, e.g. (a) convective overshoot from the core, which
causes extension of
the mixed region beyond the edge of the core as defined by the Schwarzschild
criterion, affecting evolution; and (b) the balance between meridional circulation
and rotationally induced turbulence generates chemical mixing and
angular momentum redistribution (Zahn 1992).
From the observational side, great efforts have been made within last decades
in developing the seismology of Scuti stars within coordinated networks, e.g.:
STEPHI (Michel et al. 2000) or DSN (Breger 2000; Handler 2000).
However, several aspects of
the pulsating behaviour of these stars are not completely understood
(see Templeton et al. 1997). For more details, an interesting review of unsolved
problems in stellar pulsation physics is given in Cox (2002). Due to the
complexity of the oscillation spectra of
Scuti stars, the identification
of detected modes is often difficult and require additional information
(see for instance Breger et al. 1999; Viskum et al. 1998). A unique mode identification is
often impossible and this hampers the seismology studies for
these stars. Additional uncertainties arise from the effect of rapid rotation,
both directly, on the hydrostatic balance in the star and, perhaps more
importantly, through mixing caused by circulation or instabilities induced by
rotation.
Intermediate mass stars, like A type stars ( Scuti stars,
Dor, etc.) are known to be
rapid rotators. Stars with
are no longer
spherically symmetric but are oblate spheroids due to the centrifugal force.
Rotation modifies the structure of the star and thereby the propagation
cavity of the modes. The characteristic pattern of symmetric multiplets split by
rotation is thus broken.
In the framework of a linear perturbation analysis, the second order effects
induce strong asymmetries in the splitting of multiplets (Dziembowski & Goode 1992; Saio 1981)
and shifts which cannot be neglected even for radial modes Soufi et al. (1995).
The star studied here is a very rapidly rotating A type star. Therefore
rotation must be taken
into account, not only when computing equilibrium models but also in the
computation of the oscillation frequencies. The forthcoming space mission
COROT (Baglin et al. 2002), represents a very good opportunity for
investigating such stars since such high frequency resolution data will allow us to
test theoretically predicted effects of rotation.
The paper is structured as follows: in Sect. 2, fundamental
parameters necessary for the modelling of Altair are given. Equilibrium models
as well as our adiabatic oscillation code are described in Sect. 3.
Section 4 presents a discussion of the two seismic approaches followed:
1) considering radial modes and analysing the Petersen diagrams; and 2)
considering radial and non radial modes. We also discuss a possible modal
identification.
Finally, major problems encountered for modelling Altair and conclusions are
presented in Sect. 5.
Several values of the effective temperature and surface gravity of Altair
( Aql) can be found in the literature. Recently, (Erspamer & North 2003)
proposed
K and
dex,
derived from photometric measurements (Geneva system).
From Hipparcos measurements (a parallax of
mas) combined with
the observed V magnitude and the bolometric correction given by
Flower (1996), we obtain a bolometric magnitude of
dex.
Using previous values we thus report a luminosity of
.
Rapid rotation modifies the location of stars in the HR diagram.
Michel et al. (1999) proposed a method to determine the effect of rotation
on photometric parameters. In the framework of Scuti stars, this method was then
further developed by Pérez Hernández et al. (1999), showing errors around 200-300 K and 0.1-0.2 mag in the
effective
temperature and absolute magnitude determination respectively (see Suárez et al. 2002, for
recent results on
Scuti stars in open clusters).
The error box shown in
Fig. 1 has taken such systematic errors into account and it
will be the reference for our modelling.
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Figure 1:
HR diagram containing the whole sample of
models considered in this work (filled circles) as well as the
observational photometric error box. Dotted and dot-dashed lines
correspond to non-rotating evolutionary tracks of 1.70 and
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From photometric measurements, an estimate of the mass (
)
and
the
radius (
)
of the star is given by Zakhozhaj (1979); Zakhozhaj & Shaparenko (1996).
However, Altair is found to rotate quite rapidly. Taking advantage of the fact that Altair
is a nearby star, Richichi & Percheron (2002) measured its radius providing
a diameter of 3.12 mas, which corresponds to a radius of
.
Moreover,
van Belle et al. (2001) showed Altair
as oblate. Their interferometric observations report an equatorial diameter of
3.46 mas (corresponding to a
radius of
)
and a polar diameter of 3.037 mas, for an axial ratio
.
The same authors derived a projected rotational velocity of
.
Furthermore, Royer et al. (2002) reported
values which
range which range from
to
.
In addition to this, recent spectroscopically determined constraints on Altair's inclination
angle i have been established by Reiners & Royer (2004). They provide a range of ibetween 45
and 68
,
yielding therefore a range of possible equatorial
velocities of Altair between 305 and
.
As shown in next sections, such
velocities, representing 70-90% of the break up velocity, place significant limits on our ability to model the star.
The equation of state CEFF (Christensen-Dalsgaard & Daeppen 1992) is used, in which, the Coulombian
correction to the classical EFF (Eggleton et al. 1973) has been included.
The pp chain as well as the CNO cycle nuclear reactions are considered,
in which standard species from 1H to 17O are included. The species D, 7Li and 7Be have been set at equilibrium.
For evolutionary stages considered in this work, a weak electronic screening
in these reactions can be assumed (Clayton 1968).
Opacity tables are taken from the OPAL package (Iglesias & Rogers 1996), complemented at
low temperatures ( K) by the tables provided by
(Alexander & Ferguson 1994). For the atmosphere reconstruction, the
Eddington
law (grey approximation) is considered.
A solar metallicity Z=0.02 is used.
Convective transport is described by the classical Mixing Length theory,
with efficiency and core overshooting parameters set to
and
,
respectively. The latter parameter is prescribed by Schaller et al. (1992) for
intermediate mass stars.
corresponds to the local pressure
scale-height, while
and
represent respectively the mixing length
and the inertial penetration distance of convective elements.
Rotation effects on equilibrium models (pseudo rotating models) has been
considered by modifying the equations (Kippenhahn & Weigert 1990) to include the
spherically symmetric
contribution of the centrifugal acceleration by means of an effective gravity
,
where g corresponds to the local gravity, and
represents the
radial component of the centrifugal acceleration.
During evolution, models are assumed to rotate as a rigid body, and
their total angular momentum is conserved.
In order to cover the range of
given in Sect. 2,
a set of rotational velocities of 50, 100, 150, 200 and
has been
considered. The location in the HR diagram of models considered in this work
is given in Fig. 1.
A wide range of masses and rotational velocities is delimited by the
photometric error box. To illustrate this, a few representative evolutionary tracks are also
displayed. Characteristics of computed equilibrium models (filled circles) are
given in Tables 2-4, for models of
,
and
respectively.
Theoretical oscillation spectra are computed from the equilibrium models described in the previous section. For this purpose the oscillation code Filou (Tran Minh & Léon 1995; Suárez 2002) is used. This code, based on a perturbative analysis, provides adiabatic oscillations corrected for the effects of rotation up to second order (centrifugal and Coriolis forces).
Furthermore, for moderate-high rotational velocities, the effects of near degeneracy are expected
to be significant (Soufi et al. 1998). Two or more modes, close in frequency, are rendered degenerate
by rotation under certain conditions, corresponding to selection rules. In particular these rules select modes
with
the same azimuthal order m and degrees differing by 2 (Soufi et al. 1998). If we consider two generic modes
and
under the aforementioned conditions, near degeneracy
occurs for
,
where
and
represent the eigenfrequency associated to modes
and
respectively, and
represents the stellar rotational
frequency (see Goupil et al. 2000, for more details).
High-amplitude Scuti stars (HADS) display V amplitudes in excess of 0.3 mag and generally oscillate
in radial modes. In contrast, lower-amplitude members of the class present complex spectra,
typically showing non-radial modes.
The amplitude of the observed main frequency is below 0.5 ppt which is several times smaller
than typically detected in non-HADS Scuti stars from ground-based observations.
As shown in Paper I, using the classical period-luminosity relation with the fundamental parameters
given in Sect. 2, and assuming the value of
given by Breger (1979) for
Scuti stars, the frequency of the fundamental radial mode is predicted to
be
,
suggestively close to the observed
.
For the
remaining modes (Table 1) there is no observational evidence to identify them as
radial. Nevertheless, the second dominant mode, f2, will be considered here as the first
overtone.
The present study is divided into two parts. In Sect. 4.1, we consider only the two dominant modes (i.e. f1 and f2, cf. Table 1). The observations are then compared with models through period-ratio vs. period diagrams, from now on called Petersen diagrams (see e.g. Petersen & Christensen-Dalsgaard 1999,1996). Then, in Sect. 4.2 we use the 5 most dominant observed modes to find the best fit to the theoretical models.
Table 1:
Altair observed frequencies (from Paper I). From left to
right, columns provide frequencies in cd-1, in
,
amplitudes in ppm, and finally,
f1/fj=2,...,7 frequency ratios.
Table 2:
Table containing the main characteristics of
computed
models. From left to right,
represents the rotational velocity
in
;
the effective temperature (on a logarithmic scale);
the
luminosity in solar luminosities (on a logarithmic scale);
the stellar radius
in solar radii; g the surface gravity in cgs (on a logarithmic scale);
the age in Myr;
the frequency of
the fundamental radial mode (in
)
and finally,
represents the
ratio between periods of the first overtone and the fundamental radial mode.
Variables indexed with a d represent the same quantities obtained when including
near-degeneracy effects.
Table 3:
Idem Table 2 for
models.
Table 4:
Idem Table 2 for
models.
The well known Petersen diagrams show the variation of
ratios with
,
where
represents the period of the fundamental radial mode, and
the period of the first overtone. They are quite useful
for constraining the mass and metallicity of models given the observed ratio between the
fundamental radial mode and the first overtone. However these diagrams do not consider the effect
of rotation on oscillations
(see Petersen & Christensen-Dalsgaard 1999,1996, for more details).
Problems arise when rapid rotation is taken into account: it
introduces variations in the period ratios even for radial modes. This, combined with the
sensitivity of Petersen diagrams to the metallicity, renders the present
work somewhat limited. A detailed investigation of such combined effects is currently a work
in progress (Suárez & Garrido 2005).
In order to cover the range of effective temperature (from
to
), luminosity and rotational velocity, three sets of 20 models
are computed, where rotational velocity varies from
to
.
Characteristics of the three sets are given in Tables 2-4, corresponding
to sets of
,
and
models respectively.
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Figure 2:
Variation of the first overtone to fundamental radial period ratio
as a function of the rotational velocity. For each mass considered, four curves are
displayed, representing the results obtained modifying the effective
temperature of models from
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Figure 3:
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The impact of these results on Petersen diagrams are shown in Fig. 3.
In this diagram the variation of
is displayed versus
for the three masses studied. As in Fig. 2,
note the trend toward the standard value of
as the rotational velocity decreases.
Considering the
dependence in isolation, a
canonical value
can
be established for a given mass. For the range of masses studied
here, the analysis of three panels of Fig. 3 shows a very
low dependence on mass.
Up to this point, the effect of near degeneracy has
not been taken into account. Near degeneracy
occurs systematically for close modes (in frequency) following certain selection
rules (see Sect. 3.2). It increases the asymmetry of multiplets
and thereby the behaviour of modes. The higher the value of the rotational
velocity, the higher the importance of near degeneracy.
In the present case, for the range of rotational velocities considered, near
degeneracy cannot be neglected. As can be seen in Fig. 4,
not all models present the same behaviour with .
For rotational velocities
up to
,
a double behaviour is shown, one for lower
effective temperatures (
), and one for higher effective
temperatures (
), both remaining linear. This shows the dependence
of
with the evolutionary stage of the star.
However, for higher rotational velocities, particularly on
the right side of the vertical limit line, everything becomes confusing.
At these rotational velocities, the global effect of rotation up to second
order (which includes asymmetries and near degeneracy) on multiplets complicates
the use of
predictions on a Petersen diagram.
In particular, radial modes are affected by rotation through the distortion
of the star (and thereby its propagation cavity), and through near degeneracy
effects, coupling them with
modes (Soufi et al. 1998).
In addition, other factors such as the
evolutionary stage and the metallicity must also be taken into account, which
makes the global dependence of Petersen diagrams on rotation rather
complex.
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Figure 4:
Same as Fig. 2 but considering near
degeneracy.
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Figure 5:
Variation of fundamental mode frequencies
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In Fig. 5 (left and right panels), the combined effect of
rotation and evolution on the fundamental radial mode is shown
for representative models with a rotational velocity of
.
In particular, for the
model, the stellar radius is
approximately
larger
than those of 1.70 and
,
which difference is of the order
of
.
As a result, a clear difference of behaviour
between those models is observed. Such a
difference depends not only on the
mass, but also on the evolutionary stage, the inital rotational
velocity (at ZAMS), metallicity, etc. In Suárez et al. (2005),
for a certain small range of masses,
a similar non-linear behaviour is found for predicted unstable mode ranges.
The results of this work may provide a clue to understand this
unsolved question.
The reader should notice that
at these velocities, when frequencies are corrected for near degeneracy effects, their
variations with
(and thereby with evolution) are more rapid (right
panel) than was the case for uncorrected ones.
Moreover, a shift to higher
frequencies is observed when correcting for near degeneracy, which favors the
selection of lower mass objects.
For low temperatures (i.e. for more evolved models),
stabilization is found for low
frequencies, which
can be explained by the joint action of both near degeneracy and
evolution effects.
Table 5:
Table containing a proposed mode identification for the
observed frequencies given in Table 1, using the oscillations
computed from different models. Values in parentheses correspond to the mode
identification numbers (). The first five models corresponds to those
chosen by their proximity to the fundamental radial mode. The last five,
corresponds to those chosen by minimum
value.
represent
the theoretical frequencies closest to the observed ones.
In the present work, following the prescription of Goupil et al. (2000),
modes are near degenerate when their
proximity in frequency is less or equal to the rotational frequency
of the stellar model (
). Considering
all possibilities, 5 models identify the observed f1 as the
fundamental radial mode: m18, m19, m20 (
), m39 and
m40 (
). For last three models (20, 39 and 40), this happens
without considering near degeneracy effects. For m18 and m19, the
theoretical fundamental mode approaches f1 when considering near
degeneracy.
In Table 5, radial and non radial identifications (discussed
in the next section) are presented
for selected models. The first set of five models corresponds to those selected
by their identification of f1 as the radial fundamental mode. As can be
seen, no identification is possible when trying to fit the whole set of observed
frequencies. However f3 is identified as the third overtone by the
rapid rotating model (
)
m19.
Uncertainties in the observed mass and metallicity are also an important
source of error in determining the correct equilibrium model. Thus, the fact
of obtaining fundamental modes with frequencies lower than
for most
of the models could be explained by an erroneous
position of the photometric box on the HR diagram. In fact, the lower the
mass of model used (always within the errors), the higher the
value.
However, the location in the HR diagram of models with masses lower
than
(with the same metallicity) is not representative of
the Altair observations.
At this stage, there are two crucial aspects to fix. On one hand, it is necessary
to determine the physical conditions which enforce degeneracy between mode
frequencies. A physically selective near-degeneracy could
explain the behaviour of
for very high rotating stars.
On the other
hand, for high rotational velocities, third order effects of rotation are
presumably important.
As neither observational nor theoretical evidence supporting a radial identification of the
observed modes fj exists, in this part of the work, we carry out an analysis
generalised to non-radial modes.
As we did for radial oscillations (see Sect. 3.2), here we compute non-radial adiabatic oscillations for the models of
Tables 2-4.
Assuming the 7 observed modes of Altair pulsate with
,
a possible mode identification is proposed in Table 5.
In order to avoid confusion, only the first 5 observed modes (Table 1) with larger
amplitudes, will be considered.
This constitutes a rough identification based only on the proximity
between observed and theoretical mode frequencies for each model. That is, no additional
information about the
and m values is used.
In order to obtain an estimate of the quality of fits (identification of the whole set
of observed frequencies), the mean square error function is used. The lower its value, the
better the fit for the free parameters considered.
For each model
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Figure 6:
HR diagram showing selected models given in Table 5.
Labels correspond to model numbers.
Square symbols represent models selected by the proximity to the fundamental
radial mode; filled squares are used for those taking into account near degeneracy
effects. Filled circles represent models selected by the minimum ![]() ![]() ![]() ![]() ![]() |
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On the other hand, since no specific clues for the degree
and the azimuthal order mare given, other characteristics of the observed spectrum must be investigated. In particular,
it is quite reasonable to consider some of the observed frequencies as belonging to one
or more rotational multiplets. Although no complete multiplets are found, several sets
of two multiplet members are obtained. Specifically, the observed frequencies f4 and
f5 are identified as members of
multiplets by models
m13, m18, m20,
m39 and m10, and as members of
triplets by models m10
and m19.
Figure 6 shows the location on
a HR diagram of selected models given in Table 5. Notice that around 80% of them
lie in effective temperature and luminosity ranges of
and
respectively. In the area delimited by these models (shaded surface),
models with similar characteristics are found. As can be seen, the shaded area is located in
the central part of the error box. Except models m39, m40 and m58, all
models are situated toward colder and more luminous locations in the box. This is in agreement
with the expected effect of rotation on fundamental parameters (see Sect. 2).
On the other hand, positions of models selected by the proximity to the fundamental
radial mode (squares) can be connected by a straight line. This is an iso-
line
, with
,
and
which basically explains the similar frequencies of their fundamental
radial mode. Considering near-degeneracy, the identification of the fundamental radial mode
and the lowest
value, our best model is m19, which is at the lower
limit of the range of Altair's observed
.
Nevertheless, in order to further constrain our models representative of Altair, it would be
necessary
to obtain additional information on the mode degree and/or azimuthal order of observations.
In this context, spectroscopic analysis may provide information about ,
m and the angle
of inclination of the star, as nonradial pulsations generate Doppler shifts and line profile
variations (Aerts & Eyer 2000). Furthermore, multicolor photometry may also provide information
about
(Garrido et al. 1990).
In the present paper a theoretical analysis of frequencies of HR 6534 (Altair) was presented, where rapid rotation has been properly taken into account in the modelling. The analysis was separated in two parts: 1) considering the observed modes f1 and f2 corresponding to the fundamental radial mode and the first overtone and models were analysed through the Petersen diagrams; and 2) a preliminary modal identification was proposed by considering radial and non-radial oscillations.
Firstly, in the context of radial modes, we studied the isolated effect of rotation on
Petersen diagrams. For the different rotational velocities
considered, the shape of
leads to a limit of
validity of the perturbation theory (up to second order) used at around
.
This limit is mainly given by the behaviour of such period ratios when near degeneracy
is considered, which visibly complicates the interpretation of Petersen diagrams.
Nevertheless, in this procedure, for rotational velocities up to
it is
found that
is lower than 0.77 for lower effective temperatures,
and reciprocally higher than 0.77 for higher effective temperatures (inside
the photometric error box).
The analysis of radial modes also reveals that only a few models identify the
observed f1 as the
fundamental radial mode. This reduces the sample of models to those with
masses around
or lower, on the main sequence and implies models with
different metallicity in order to agree with photometric error box.
These partial results (for a given region in the HR diagram,
for a given range of masses, evolutionary stages and a given metallicity)
constitute a promising tool for seismic investigation, not only of
Scuti stars, but
in general, of multi-periodic rotating stars. A detailed
investigation on the
effect of both rotation and metallicity on Petersen diagrams will be proposed in
a coming paper (Suárez & Garrido 2005).
Secondly, in the context of radial and non radial modes and assuming observed modes pulsating
with
,
a set of 14 models was selected in which five of them
identify the fundamental radial order and at least six others identify two of the observed
frequencies (f4 and f5) as members of
and
multiplets. A range of masses of
[1.70,1.76]
principally for a wide range of evolutionary stages on the main sequence
(ages from 225 to 775 Myr) was obtained. Considering information of both radial
(through Petersen diagrams) and non-radial modes, a set of representative models with rotational
velocities larger than
was obtained.
Further constraints on the models are thus necessary. Such constraints can be obtained by employing
additional information on the mode degree
and/or the azimuthal order m of the observed
modes, which may be inferred from spectroscopy and multicolor photometry
(Garrido et al. 1990). Improvements on adiabatic oscillation computations (including third
order computations) will constitute a coherent and very powerful tool to obtain seismic data
from future space missions like COROT, EDDINGTON and MOST.
Acknowledgements
This work was partially financed by the Spanish Plan Nacional del Espacio, under project ESP2004-03855-C03-01, and by the Spanish Plan Nacional de Astronomía y Astrofísica, under proyect AYA2003-04651. We also thank the anonymous referee for useful comments and corrections which helped us to improve this manuscript.