K. Uytterhoeven1 - M. Briquet1, - C. Aerts1,2 -
J. H. Telting3 - P. Harmanec4,5 - K. Lefever1 -
J. Cuypers6
1 - Institute of Astronomy, Katholieke Universiteit Leuven,
Celestijnenlaan 200 B, 3001 Leuven, Belgium
2 - Department of
Astrophysics, University of Nijmegen, PO Box 9010, 6500 GL
Nijmegen, The Netherlands
3 - Nordic Optical Telescope,
Apartado de Correos 474, 38700 Santa Cruz de La Palma, Spain
4 -
Astronomical Institute of the Charles University, V
Holesovickách 2, 180 00 Praha 8, Czech Republic
5 -
Astronomical Institute, Academy of Sciences, 251 65 Ondrejov, Czech
Republic
6 - Royal Observatory of Belgium, Ringlaan 3, 1180 Brussel, Belgium
Received 10 June 2004 / Accepted 31 October 2004
Abstract
We analyse the complex short-term Si III line-profile variability of the
spectroscopic binary Cep star
Sco after orbit subtraction, before and
after spectral disentangling. We refine the known oscillation frequency of the
star:
f1=4.99922 c d-1 and detect 2f1. Variability is also found at
frequencies near
c d-1 and
c d-1 or their
aliases. These frequencies are not significant if we consider the spectra alone,
but they survive our selection after the consideration that they were derived
previously from independent ground-based and space photometry by different
teams. Moreover, we find dominant variability in the equivalent width with a
frequency in the interval
[0.22,0.30] c d-1 which we interpret as the
rotational frequency
of the star. The complex window function does
not allow us to determine definite values for
.
The
variability with f1 is interpreted as a prograde non-radial oscillation mode
with spherical wavenumbers
or (1,-1). The additional
frequencies are explained in terms of rotational modulation superposed to the
main oscillation. We also point out that we cannot disprove the variability in
Sco to originate from co-rotating structures. KOREL disentangling preserves
the large-amplitude line-profile variability but its performance for complex
low-amplitude variability remains to be studied in detail.
Key words: binaries: spectroscopic - stars: oscillations - line: profiles - stars:
individual: Sco
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Figure 1:
A randomly chosen set of the normalised disentangled Si III profiles of
![]() |
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Significant progress in the modelling of the internal stellar structure of
stars was achieved recently by Aerts et al. (2003a) for the star
HD 129929 and by Pamyatnykh et al. (2004) for the star HD 29248. Through
seismic modelling of their multiplet non-radial pulsation (NRP) frequencies, it
was shown that the overshooting parameter is less than 0.15 times the pressure
scale height and that non-rigid rotation is present, with the rotation near the
core about 3 times faster than the surface rotation. It is remarkable that
these two stars have such a similar internal structure and a natural question is
if more rapid rotators have the same internal behaviour.
From their list of 26 confirmed line-profile (LP) variable Cep stars, Aerts
& De Cat (2003) found a quite clear separation between slow
(
km s-1) and moderate (
km s-1)
rotators. Most of the moderate rotators are members of
a spectroscopic binary. In this paper, we report on the intrinsic variations
in one of these moderate rotators among the
Cep stars with the hope to
provide sufficient frequency information for future seismic modelling.
In a recent study of the bright B 1.5 III star Sco (HD 160578, HR 6580,
HIP 86670, V = 2
4), Harmanec et al. (2004, hereafter termed Paper I) applied
KOREL
(Hadrava 1995) disentangling to derive the
orbital parameters and to unravel the contributions of both components to the line-profile variability (LPV) observed in the Si III 4552 and 4567 Å lines of a set of 699 spectra. The binary has an eccentric orbit
(e=0.50) with a period of some 196 days. We refer to Paper I for details
and for a complete history of the variability studies of this star, as well as
for the first evaluation of the KOREL disentangling procedure. One of the
important findings of Paper I was that only the primary is a LP
variable.
In the current paper we investigate in detail the LPV of
the primary of
Scorpii, based on 699 profiles obtained with and without KOREL disentangling. Some observed profiles are shown in
Fig. 1. In Paper I we showed that the
dominant intrinsic frequency could be recovered from the disentangled profiles
in the rest frame of the primary. We recall that the synodic frequency
4.99922 c d-1 (called f1 hereafter) was found in the residuals of the radial
velocities after orbit subtraction, while the sidereal frequency 5.00425 c d-1,
known from ground-based photometry (Lomb & Shobbrook 1975) and recovered in the
WIRE photometry (Cuypers et al. 2004), was detected in the KOREL line-strength
variations, whereby the difference between the two frequencies is exactly equal
to the orbital frequency. In this paper we elaborate further on the profiles
with the goal to interpret their variations in terms of NRP
modes, given that the star is a confirmed
Cep variable. Shobbrook &
Lomb (1972) and Lomb & Shobbrook (1975) classified the star as such after their
discovery of the frequency 5.00425 c d-1 in the light changes. These authors also suspected the presence of the frequency 4.86784 c d-1 (hereafter called
f2) as a result of a beat-phenomenon. Uytterhoeven et al. (2001) already
showed the presence of these two frequencies in the LPV of
the subset of 422 spectra gathered during 8 consecutive nights of continuous
monitoring of the star, without having disentangled the lines of the primary in
these spectra. We improve their results here by considering all the 699
available LP data. In particular we aim to search in our spectroscopic data for the
presence of another frequency of 5.2767 c d-1 (hereafter called f4)
suggested by Lomb & Shobbrook (1975) and three additional frequencies recently
found by Cuypers et al. (2004) in white-light photometry gathered
by the WIRE satellite: 5.6950 c d-1 (f3), 2.2498 c d-1 (f5) and
0.2809 c d-1.
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Figure 2: Top ( bottom): line profile measurements taken on HJD 2 450 742.4691 (HJD 2 450 194.9224). The observed spectrum of the primary shifted with +53.39km s-1 (-55.34 km s-1) is given by the full black line and the KOREL disentangled profile in the rest frame of the primary by the dotted line. The gray profile is the observed one from which the disentangled profile of the secondary is subtracted after applying its orbital shift of -57.84 km s-1 (62.70 km s-1) and to which a subsequent orbital shift towards the rest frame of the primary +53.39 km s-1 (-55.34 km s-1) is applied. The vertical lines indicate the integration boundaries we used to calculate the velocity moments from the Si III profiles centered at 4552.654 Å and 4567.872 Å. At the bottom of each figure the residual disentangled and secondary removed spectra with respect to the observed spectrum are also shown, after an arbitrary shift towards flux level 0.88. |
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The basic physical parameters of both components were derived in Paper I from
the orbital solution and from photometric and parallax results available in the
literature. In order to have an independent check of them, and to derive the
flux contribution of the secondary, we observed Sco twice with the
CORALIE échelle spectrograph attached to the 1.2 m Leonard Euler telescope in
La Silla, Chile during one night in March 2004. Both spectra were reduced
following the standard procedure and were co-added to obtain the highest
possible signal-to-noise ratio (
).
To derive good estimates of
and
of both stars
we focused on the regions of the Si II (4128 Å, 5050 Å),
Si III (4560 Å, 4820 Å, 4716 Å, 5739 Å), Si IV
(4116 Å, 4089 Å), He I (4026 Å, 4387 Å, 4471 Å, 4922 Å,
4713 Å, 6678 Å), He II (4686 Å), H
,
H
,
H
,
H
and H
line profiles. These
observed lines were compared with theoretical ones computed for
spherically symmetric non-LTE atmosphere models by means of the newest version
of the FASTWIND code (Santolaya-Rey et al. 1997).
The input parameters to obtain the theoretical profiles are
,
,
the microturbulent
velocity and the particle number ratio n(He)/n(H). The gravity and temperature
are most constrained by respectively the wings of the Balmer lines and the
comparison between the Si II, Si III, Si IV lines. We
restricted to atmospheres with solar helium composition.
There is only one He II line present in the spectrum and it is extremely
weak. It concerns the He II 4686 Å line, which is indeed the first
such line that occurs as the temperature raises above 24 000 K. The other
He II lines (e.g. the ones at 6406 Å, 6527 Å and 6683 Å) are
absent. This implies an upper limit of 25 000 K for the effective temperature
of the primary, which was the estimate derived in Paper I. Further, no
Si IV lines are detected, while the Si III lines are strong. The
Si II doublet at 4128 Å and 4130 Å, while severely blended due to the
rotation, is clearly present. Fitting the lines for a single star with
K gives too strong Si III profiles and too weak
Si II lines. We therefore conclude that the secondary has a
significant contribution to the Si III lines and is mainly responsible
for the Si II lines. The Balmer wings, on the other hand, lead us to
between 3.6 and 4.0 assuming a single star.
We subsequently merged the lines of two contributing stars according to
different flux ratios using the relative orbital velocity of the components and
the
estimates derived in Paper I. The best overall fit to observed
line profiles occurs for the following parameters:
K,
for the primary and
K,
for the
secondary. The uncertainties are 2000 K for the temperature and 0.2 dex for
the gravity, for both stars. These results are entirely compatible with those
of Paper I. They lead to a ratio of the flux of the primary to the one of the
secondary between 1.5 and 2.5.
We refer to Paper I for a detailed description of the 699 profiles used in this
study. The data have a time span of some 3 years, including two intensive monitoring campaigns lasting 3 and 8 nights. By means of the KOREL disentangling procedure the spectrum of the
spectroscopic binary was decomposed in the following parts: the disentangled
profile of the primary, the disentangled profile of the secondary and the
residual spectrum calculated in either the rest frame of the primary or the rest
frame of the secondary. As already pointed out in Paper I, the KOREL code is
based on the assumption that the line profiles do not change shape, a
condition that is evidently violated in the current case of the primary of Sco.
One of the main goals of this paper is to investigate what the implications of
this violation are on the interpretation of the LPV.
Before Paper I, KOREL disentangling was not yet applied as an intermediate step in the interpretation of LPV due to oscillations. Therefore, we carefully checked its performance by employing three different procedures to analyse the intrinsic variability of the primary. In the first procedure we consider the "original profiles'' and simply shift them to the rest frame of the primary, i.e. we ignore the contribution of the secondary to the line. KOREL provides the disentangled spectra with an unknown shift of the continuum (Hadrava 1997). For the second procedure, we normalised (i.e. re-rectified) the disentangled spectrum of the primary produced by KOREL before co-adding the residuals. Hereafter we will refer to these profiles as the "disentangled profiles''. In the third procedure we shift the normalised KOREL disentangled profile of the secondary according to its orbital velocity, we subtract it from the observed profile assuming equal fluxes of both components and we finally move this resulting profile to the rest frame of the primary. These profiles will be referred to as the "secondary subtracted profiles''. We thus incorporate the two extreme cases that the secondary has no and equal contributions to the observed flux while KOREL does not take into account any flux ratio information.
Table 1:
Amplitudes (in km s-1) of the frequencies
c d-1,
f1 = 4.99922 c d-1 and 2f1 for the EW ( top);
amplitudes (in km s-1) of f1, f2, f3 and 2f1 for
( bottom)
of the Si III 4552.654 Å ( left) and Si III 4567.872 Å ( right) line of
Sco,
calculated from the original observed spectra and from the disentangled
spectra. The last column denotes the total variance reduction (f.v.) derived
from the least-squares fits to the data with the considered frequencies.
In Fig. 2 we make a comparison between the profiles resulting from these three different types of numerical operation for opposite elongations. We also give the residual disentangled and secondary removed spectra with respect to the original spectrum. One can see that the KOREL disentangled profile of the primary and the one obtained after subtraction of the secondary's profile differ only slightly in line depth and not in shape of the line features. This implies that KOREL's solution is not too different from the one obtained assuming equal fluxes for the Si III triplet, while we found a flux ratio between 1.5 and 2.5 from the échelle spectrum in the previous section. In the following we report the results of the frequency analyses on the line diagnostics derived from the profiles obtained with the three different procedures.
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Figure 3: Lomb-Scargle periodogram of the RV ( left) and EW ( right) derived from the disentangled Si III 4552.654 Å profiles of the primary. On top of each panel, the window function, shifted according to the highest peak in the periodogram, is shown. |
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We considered the variations of the first three velocity moments (
,
,
;
definition by Aerts et al. 1992), of the equivalent width
(EW) and of the normalised flux at each pixel across the profile (Intensity
Period Search, IPS) for the two bluest Si III lines. In calculating the moments
we took the fixed integration boundaries indicated as vertical lines in
Fig. 2. We searched for frequencies in the interval [0,15] c d-1 with a frequency step of 0.00001 c d-1. As the IPS analysis is very sensitive to
the quality of the data, we selected only the subset of 397 spectra with
signal-to-noise ratio above 500 for that. As the major part of this subset
consists of spectra obtained during the 8 nights of July 1997, we searched for
frequencies with a frequency step of 0.001 c d-1 .
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Figure 4:
The first three normalised moments (without prewhitening in case of
f1; after prewhitening with f1 in case of f2 and f3) of the Si III 4552.654 Å line of ![]() ![]() ![]() ![]() |
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All line diagnostics show one dominant frequency for the three
followed procedures:
c d-1. The difference between this frequency and
f1 = 4.99922 c d-1 equals the
value of the highest peak in the window function, which is located at 0.01292
c d-1. Therefore, these two frequencies are not distinguishable in our data.
The Lomb-Scargle periodogram of the radial velocity (RV) is shown in the left
panel of Fig. 3 while phase diagrams of the moments for f1 can be
found in the left panels of Fig. 4. We note that power also
occurs at the first harmonic 2f1 in the RV (see
Fig. 3).
The EW variations show a dominant frequency at
c d-1 or its alias
c d-1 at a relative amplitude of some 5% for the disentangled profiles.
These two frequency peaks have an amplitude about twice as large as the one for
f1 in the disentangled profiles, as can be seen in Fig. 3. We find
that
f1=4.99922 c d-1 and its first harmonic are only present in the EW
variations of the disentangled spectra and not in the profiles derived
from the other two procedures.
We therefore tentatively conclude that KOREL did a good job in allowing the
detection of realistic periodic EW variations with frequency f1 at
level in the line profiles resulting from co-addition of the
disentangled profile of the primary to the residual profiles after
disentangling. Depending on their effective temperature, EW variations of a few
percent are indeed expected and observed in several
Cep stars due to
their dominant oscillation (De Ridder et al. 2002). It remains to be further
studied how well the code would perform in different circumstances, i.e. different oscillations and amplitudes in stars with different rotational
velocities.
The phase difference between the RV, EW and light
variations is very useful information when it comes to evaluate physical
models. We therefore computed these phase differences from a fit with the
synodic frequency f1 to EW,
and the KOREL line intensity (LI) described
in Paper I. The results are
and
expressed in
radians. The former
phase difference is in remarkable agreement with the theoretical predictions
found for non-adiabatic NRP of low degree and low radial
order by De Ridder et al. (2002) for stellar parameters similar to those of
Sco. That the EW and LI are in phase and not in antiphase is what can be
expected for the Si III triplet subject to NRP for
temperatures near 26 000 K (De Ridder et al. 2002).
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Figure 5: Lomb-Scargle periodograms of the residuals of the EW ( left) and first velocity moment ( right) of the disentangled Si III 4552.654 Å profiles of the primary after prewhitening with f1 = 4.99922 c d-1. The frequencies reported in the literature are indicated by dashed lines: 4.8678 c d-1, 5.2767 c d-1 (Lomb & Shobbrook 1975), 0.2809 c d-1, 2.2498 c d-1 and 5.6950 c d-1 (WIRE, Cuypers et al. 2004). |
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After prewhitening with the dominant frequency f1, the highest peaks in the
frequency spectrum of the different moments appear in the interval 0-2 c d-1. In
particular, a high peak, already clearly visible in the left panel of
Fig. 3, occurs in the residual
data near 0.9 c d-1. This
frequency's amplitude in the residuals of
is about half the one of
f1. Such a low frequency is not expected for p-modes in
Cep stars
and its meaning or implication is not clear at this stage of the analysis. We checked that this frequency is not induced by zeropoint differences between
the different nights. Such a situation is not uncommon in multiperiodic
Cep stars and one tends to continue the analysis with a search for
higher frequencies first (see, e.g., Aerts et al. 2004a for an extended
discussion of such a situation for the
Cep star
Eri) before
returning to the low-frequency domain, which is also what we do here for
Sco,
the more so since we have a complex window function and prewhitening with a
frequency near 0.9 c d-1 might introduce changes in the values of the subsequent
frequencies.
Candidate frequencies in the expected range of frequencies related to p-modes in Cep stars occur only at low amplitudes slightly above the noise level
after prewhitening with f1. We stress that any of the frequencies listed
below would not be accepted if we were to adopt, e.g., the
4
signal-to-noise ratio criterion as introduced by Breger et al. (1993) and often used to determine the significance of a frequency nowadays, to
our spectroscopic data alone. The reason why we continue the frequency search in
quite some detail is twofold:
We also detected a frequency in the interval
[0.227;0.281] c d-1 in all datasets, to which we cannot assign the most probable value due to the poor
and complex window functions. A physical
explanation might be given in terms of the rotational period of the primary.
Indeed, in Paper I we estimated the range of the rotational period to be between
3
56 and 3
68 (corresponding frequencies between 0.272 c d-1 and 0.281 c d-1),
which is consistent with the derived interval.
The variations in the moments of
the disentangled profiles show stronger variability at low frequencies (between
0-2 c d-1) than those of the original and secondary removed profiles. Moreover,
f2 (in case of the 4552.678 Å profile) and f4 do not contribute to the
variability derived from the disentangled spectra, while these frequencies do
appear in the other two sets of profiles. We must therefore conclude that KOREL disentangling affects the low-amplitude LPV of Sco.
The complex frequency spectrum of
Sco implies that this star is not the best target to
derive the consequences of spectral disentangling for the interpretation of
LPV in full detail. In order to synthesize the advantages of
disentangling as an intermediate step in the interpretation of the LPV, simulations of theoretically generated profiles for multiperiodic
oscillations of very different kinds are needed. Such a study is beyond the
scope of our current paper, but our experience obtained from
Sco constitutes a
good starting point for such a detailed simulation study.
In the following we will work with
f1=4.99922 c d-1 and with
.
It is important for the reader to realise that we cannot assign definitive
values to the latter three frequencies because none of the available datasets
allow us to discriminate between the true and alias values. It is therefore
evident that any interpretation of the variability with f1 on the one hand,
and with
on the other hand will be at a very different
level of confidence. This has to be kept in mind in evaluating the results
reported in the following section.
First we assume the observed LPV to be caused entirely by oscillations with frequencies f1, f2 and f3. We used a combination of several mode-identification techniques to identify the associated pulsation modes.
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Figure 6:
IPS diagrams for f1, f2 and f3 and their first harmonic,
calculated from the Si III 4552.654 Å ( left) and 4567.874 Å ( right) line of
the disentangled profiles. The top panel shows the average line profile. Each of
the three sub-panels shows the distribution of the phase (expressed in ![]() |
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![]() |
Figure 7:
Mean profile and phase and amplitude diagrams for f1 centered at
4552.654 Å. The solid black line represents the diagnostics derived from the
disentangled spectra. The other lines are derived from theoretically
calculated profiles, with different ![]() ![]() |
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By means of the phase and amplitude distributions across the LP
(IPS Method, see Telting & Schrijvers
1997), given in Fig. 6, we assigned wavenumbers
to
each of the frequencies
f1, f2, f3.
According to the relationships between
(|m|) and the phase
difference
(
)
derived by Telting & Schrijvers
(1997) we find that all three modes have degrees below 5 (see Table 2).
We subsequently generated theoretical LP time series,
following Telting & Schrijvers (1997) and Schrijvers & Telting
(1999), for the main
frequency f1 by considering the different sets of wavenumbers resulting from
Table 2 and compare them with the observed ones in
Fig. 7. We
find good agreement between the observed and theoretical line diagnostics for
the (1,-1) or (2,-1) solution. Despite our massive exploration of the free
pulsation parameter set, the theoretical models do not explain the amplitude
distribution in all of its details, nor the slightly asymmetric mean
profile. This is not suprising, given that
,
f2 and f3contribute to the LPV albeit at low amplitude while we
averaged them out. Despite this shortcoming, we can easily exclude zonal modes
and the
,
|m|=2 mode, as well as the higher degrees.
For the additional lower-amplitude modes we can only narrow down the intervals
of
and |m| (see Fig. 8): the f2 mode, respectively f3mode, is most likely a tesseral or a sectoral one with
,
respectively
.
More precise identifications are not
possible because the amplitude and phase behaviour is very similar for different
combinations (see also Aerts et al. 2003b for another example of such a
situation in the
Cep star 16 Lac).
Table 2:
Blue-to-red phase differences of f and its first harmonic together
with estimates for
and m for f1, f2 and f3 (Telting & Schrijvers 1997). The phase
differences
are given in
radians. Due to the bad
quality of the IPS diagnostics associated to 2f3 (see
Fig. 6), we cannot provide information on the m-value
of this mode.
In addition, we also used the Moment Method (MM) by
Briquet & Aerts (2003). To identify the mode corresponding to f1,
we removed the contribution of the low-amplitude variability by averaging out
all the observed profiles in phase bins of 0.025 of the oscillation cycle and we
worked with these 40 averaged observed profiles. Their first three velocity
moments are shown in Fig. 9. This figure shows clearly the presence of
the harmonic 2f1, which is confirmed by the Fourier transform on the
moments. The amplitudes of the first moment computed from the averaged spectra
are respectively 2.20 km s-1 and 0.47 km s-1 for f1 and 2f1.
The occurrence of harmonics and/or coupling terms is not exceptional for
Cep stars, see examples in Heynderickx (1992), Mathias et al. (1994),
Aerts et al. (1995), Handler et al. (2004) and Aerts et al. (2004a).
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Figure 8:
Phase distribution across the 4552.654 Å profile for sinusoidal fits
with f2 ( left) and f3 ( right). The solid black line represents the
diagnostics derived from the disentangled spectra. The other lines are derived
from theoretically calculated profiles, with different ![]() ![]() |
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We calculated the discriminant
(see Briquet & Aerts 2003, for the
definition and usage) assuming a stellar mass and radius of
and
leading to K=0.095 for f1. For each tested
,
the oscillation
amplitude is fixed by the condition that the theoretical amplitude of the first
moment has to be equal to the observed one. In the case of
Sco the observed
amplitudes exclude
-values higher than
for the main mode (see
criterion derived in Briquet & Aerts 2003), which is a lower upper limit than
from the IPS analysis.
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Figure 9: The first three velocity moments (dots) computed from 40 averaged spectra in the region near Si III 4552.654 Å over the pulsation cycle of f1 = 4.99922 c d-1. The crossed and dashed lines correspond respectively to theoretical moments computed for the first two best solutions of the upper part of Table 3. The full line corresponds to the best solution for a model with f1 and 2f1 (indicated in bold in Table 4). |
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The MM is implemented for a linear pulsation theory so that it
assumes the first moment to be well described by a single cosine function. In
spite of the presence of 2f1, we performed a mode identification by
considering only f1. This is justified because the amplitude of the harmonic
is about 4 times lower than the one of f1. This type of approach turned out
to be valid for the Cep star BW Vul, for which the relative
contribution of the first harmonic to the radial-velocity variation is also a
factor near 1/4 of the one of the frequency itself (Aerts et al. 1995). The
results of the MM are given in
Table 3. The lower the discriminant
,
the better the
agreement between theoretical and observed moment values. We conclude that
f1 corresponds to a prograde mode with
or (1,-1). This
is in perfect agreement with the results found from the IPS
Method. Figure 9 compares the theoretical moment values with the observed
ones for the two best solutions given in bold in Table 3.
Table 3:
List of the five lowest values of the discriminant
(in
km s-1) for the mode identification of f1 by the MM. i is the inclination angle,
expressed in degrees. The amplitude(s) of the radial part of the pulsation
velocity
(
), the projected rotational velocity
and the
intrinsic LP width
are all expressed in km s-1.
Table 4:
List of the ten lowest values of the discriminant
for the mode identification of f1 and 2f1 by the MM. The symbols are similar as those in Table 3.
Subsequently, we performed a mode identification by also considering 2f1 as
an additional independent mode. The best combinations are listed in Table 4.
We find that
2f1 is also a prograde mode in this model. We note that taking
2f1 into account leads to a smaller discriminant value as expected (more free
parameters) and allows a much better fit to the observed moments, particularly
the second one, as illustrated in Fig. 9. We also point out that the
simpler assumption of having modes with the same
values for f1 and
2f1 does not lead to a very good fit and results in discriminant values that
are higher than taking only f1 into account (two last rows in
Table 4).
Considering a (2,-1) or (1,-1) mode for f1, we checked the parameter
space in
which leads to a discriminant of the order of the two
lowest values given in Table 3. This allows us to restrict the
interval for the equatorial velocity of the star. We found i
[45
; 90
]
and
[94; 100] km s-1. It is
well known that the error of the estimate for the inclination from the
discriminant is unfortunately large; see De Ridder et al. (2005) for a
discussion of the statistical properties and the error propagation of the
discriminant.
We also identified the three modes
f1, f2, f3 simultaneously by using our
multiperiodic version of the MM (Briquet & Aerts 2003) on the 699
moment values of the disentangled spectra. We found that
and
correspond to (1,-1) or (2,-1) with a preference for
(1,-1). This result is not compatible with the IPS diagnostics that favour
and
.
We have to conclude that the IPS
diagnostics and the MM give different results so that the interpretation of f2 and f3 in terms of a stellar pulsation model remains insecure.
The presence of spots, due to either temperature or abundance inhomogeneities,
in combination with rotation, can also be the cause of the complex low-amplitude
LPV superposed on the effects of the main oscillation mode
of Sco. Arguments in favor of such a model are the dominance of
over f1 in the EW (see
Table 2), which is not expected for pure NRP,
and the shape and behaviour of the colour-diagrams of
Sco (Lomb &
Shobbrook 1975, their Fig. 9), which cannot be simply
interpreted as the results of temperature and radius changes of an
oscillation, as noted by the latter authors. Rotational modulation occurs for the prototype of the
Cep stars
(Telting et al. 1997) and also for V2052 Oph, which is a He-strong
Cep star (Neiner et al. 2003). These two stars are confirmed magnetic
variables, while no magnetic field has been detected for
Scorpii.
An approach to model surface spots is based on the idea of an
inhomogeneous distribution of (a) certain element(s) on the stellar surface. The
surface abundance distribution of elements can be obtained from
Doppler Imaging (DI),
which inverts rotationally modulated LPVs into a
two-dimensional abundance distribution. Below we present the modelling of the Si
distribution on the surface of
Scorpii, whereby the surface temperature
and gravity were kept fixed.
The DI technique was applied by using a code called INVERS11, implemented and kindly made available to us by Prof. N. Piskunov. We refer to Piskunov & Rice (1993) and Rice (1996), who describe DI methods in general, for more information. For the spectral synthesis of local line profiles, model atmospheres, assuming solar abundances, were calculated with Kurucz's ATLAS9 programs (Kurucz 1992, 1993). The Vienna Atomic Line Database (VALD, Piskunov et al. 1995; Ryabchikova et al. 1999; Kupka et al. 1999) provided the atomic line data we needed. Spectra were synthesized with the program SYNTH (Piskunov 1992). We started with solar abundances (-4.49 dex for silicon) and adjusted them by comparing observed and calculated spectra.
We used the following input parameters for the construction of
the surface map:
K,
,
km s-1 and
.
Any change of these values
within the uncertainties do not affect the results of the mapping
(see Briquet et al. 2004 for an evaluation of small changes of the input
parameters on the final map).
To remove the contribution of
f1, f2, f3 to the variability of Sco we
averaged out all the observed profiles in phase bins of 0.025 of the rotational
cycle and we worked with these 40 averaged observed profiles. The Si
map derived from the Si III 4552.654 Å line is shown in Fig. 10. We
obtained very similar maps by using the Si III 4567.872 Å line and by using
both lines together. We find that an overabundant spot compared to the rest of
the stellar surface is present at the pole. The surface part between longitude
and
is not visible and consequently we cannot derive
the surface abundance near the other pole. We also see the presence of three
underabundant spots just above the equator, two of them being more pronounced
than the third one. We conclude that an inhomogeneous abundance distribution, after averaging out
the dominant oscillation, is compatible with the observed LPVs of
Sco.
![]() |
Figure 10:
The Si distribution on the surface of ![]() ![]() ![]() ![]() ![]() |
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Next, we consider LPVs resulting from one dominant prograde
NRP mode with frequency f1 in combination with a non-homogeneous
intensity distribution at the stellar surface rotating with frequency
.
In the computations we adopted the value
c
d-1, as suggested in Paper I.
The rotational frequency and radius estimate of Sco imply that the star rotates
at about 20 to 25% of its break-up velocity. Therefore, the oblateness of the
star implies a small gravity gradient, and hence, according to the Von Zeipel
theorem, a flux gradient from the equator to the pole. At 25% of the break-up
velocity, the relative flux difference between pole and equator is about 10%
(see, e.g., Fig. 1 in Aerts et al. 2004b), which is of the same order than the
relative abundance variation between the equatorial and polar regions found from
the Si map. Gravity darkening may therefore explain part of the gradient found
in Fig. 10, but not the 3 spots near the equator. Given that we aim only at a
modelling of the general characteristics of the observed variability without
fine-tuning or optimising the results, and that we do not have a code available
that determines the multiperiodic NRP for an oblate star, we
assume the surface variations to be caused entirely by a spotted inhomogeneous
surface intensity distribution on a spherical star in the calculations
described below.
A physical model consisting of a combination of a large-amplitude monoperiodic
pulsation and a low-amplitude rotational modulation effect seems quite natural
and would not only allow us to give an explanation for the detection of
,
f2 and f3, but also for f4 and f5 found in respectively
ground-based photometry by Lomb & Shobbrook (1975), and the WIRE satellite data
by Cuypers et al. (2004). It is indeed the case that the frequencies f2,
f3, f4 and f5 can be related to f1 by means of the rotational
frequency, as is already remarked by Lomb & Shobbrook (1975). We are in the
unfortunate situation that we cannot provide exact relations between the
frequencies due to our unability to discriminate the frequencies from their
aliases.
We generated LPVs for the 699
observational times for Sco by combined usage of the publicly available code
BRUCE (Townsend 1997) which simulates line profiles for a non-radial pulsating star with a code that calculates LPVs for a spherical star
with a spotted intensity (kindly made available by Dr. L. Balona). The local
intensity within the spots, which, in reality, is determined by the local
effective temperature and abundance, is a free input parameter in units of the
intensity of the surroundings. The parameters for the oscillation mode were
chosen according to our best mode identification for f1 and additional
estimates thereof obtained in previous sections and Paper I.
We considered a simple modulation model with one circular spot in which the
intensity has a higher value than the surroundings and with a radius of
at the pole and three circular spots with a lower intensity with
the same radius at latitude
equally distant by
in
longitude. Such an intensity distribution corresponds to the abundance pattern
derived from the DI.
We computed the first moment variation of the synthetic spectra
and performed a frequency analysis in the same way as done for the
real observations. The whole periodogram of the first moment, before
and after prewhitening with f1, looks
similar to the one of the observed data (left panel Fig. 3
and right panel Fig. 5).
In particular, we find a high amplitude at 3 times the used rotational
frequency, which occurs near 0.9 c d-1, just as for the real data. This may
indicate that this peak is due to
3 spots in a configuration as in the Doppler map shown in Fig. 10. A peak at f2 with
amplitude of the same order as the observed ones is also present. All this is
rather convincing evidence that a rotational
modulation effect occurs in Sco, since this frequency f2 was not put in our
theoretical model. The frequency f3 is hidden in the noise.
Subsequently we calculated the phase distribution across the line profile from the synthetic spectra for a sinusoidal fit with f2. The synthetic phase path approaches the bumpy character of the observed phase distribution with similar quality as the one for the multiperiodic oscillation model shown in Fig. 8.
The character of line diagnostics, in particular of moment variations, due to
strong abundance spots on a star or due to high-amplitude oscillations in a star
are readily distinguishable from each other (see, e.g., Briquet et al. 2004). In the case of Sco, however, our experiments show that it is not so
evident to discriminate between a multiperiodic pulsation model and a model with
monoperiodic pulsation and rotational modulation induced by the occurrence of a
non-homogeneous surface temperature and/or abundance distribution. The firm
proof of rotational modulation would require accurate relations between the
detected frequencies and the rotation frequency of the star, which we cannot
establish due to the limitations of our dataset. In any case, our example of
Sco shows that the presence of frequencies in the range of expected pulsation
frequencies for
Cep pulsators is not a sufficient condition to
conclude the variability to be due to multiperiodic oscillations. In the case
of
Sco , the frequencies f1 and
are sufficient to explain the
occurrence of the frequencies f2, f3 in the line diagnostics at low
amplitude if we allow for an inhomogeneous surface pattern. On the other hand,
it is difficult to exclude firmly the presence of additional pulsation modes
with degree
and with beat frequencies near
for an
oblate star with gravity darkening, as this would lead to a similar velocity and
temperature distribution on the surface of the star as the proposed models with
the temperature or abundance spots. In any case, such a model is also
rotationally modulated and so this seems to be a necessary ingredient given that
is dominant in the EW variations.
Besides the two mentioned physical models to describe the complex LPV and the observed frequencies of Sco in terms of oscillations, we
consider a third possibility which occurred to us from the following
observation.
The LPV of
Sco, characterised by the line moments, can
also be reconciled with a single frequency
= f1/17 = 0.29407
c d-1 and a complicated phase curve. Only a low insignificant peak appears near
in the classic periodogram, but a sine fit of
of
Sco with
the frequency
fixed and its 25 harmonics leads to a variance
reduction of 77%. This is much higher than the result obtained from the fit
with
f1, f2, f3, 2f1 fixed (see Table 1). Note that a good description of
the complicated phase curve requires many more free parameters than the
multiperiodic solution (53 versus 9) but the phase curves (see
Fig. 11) show a good coherence over several years of observations and
much lower scatter than the plots for frequencies f2 and f3 after
subsequent prewhitening shown in Fig. 4.
For comparison, a fit with
c d-1 fixed and 3 of its
most dominant harmonics (which are
,
,
and
), involving also 9 free parameters, leads to a variance
reduction of 65%, which is still higher than the best value of 55% listed in
Table 1. Similar albeit less convincing results are obtained when fixing the
frequencies f1/16 or f1/18 instead of
.
Note also that a
fit with f1, 2f1,
and f2, again including 9 free
parameters, leads to an even better variance reduction of 72%. This latter fit
would point to 3 spots in the stellar atmosphere and is basically identical to
the model discussed in Sect. 5.2.2.
We applied the Bayes Information Criterion (see, e.g., Koen & Laney
2000) and found that adding more than 9 free parameters to the model
would not lead to a significant improvement.
The upper left panel of Figs. 4 and 11 show
remarkable similarity with similar figures obtained for the LP
variable
Per by Harmanec (1999, Figs. 2 and 3). Harmanec (1999)
interpreted the variability of the latter star, as well as the variability of
the rapid rotator
Oph, in terms of circumstellar structures co-rotating
slightly above the photosphere rather than multiperiodic oscillations. There is
a long and stubborn debate about the correct interpretation of the LPV of rapidly-rotating early-type stars (see, e.g., Baade & Balona
1994; Harmanec 1989, 1999, and references therein). This
debate could also be held for a confirmed
Cep star, casting some doubt
on the interpretation of
Sco as a NRP star. Indeed, the frequency
can be reconciled with the rotational frequency of the star and
the LPV can then be qualitatively explained in terms of 17
co-rotating structures, such as envisioned by Harmanec (1989).
Figure 11 shows that the spacing of the structures is quite regular
(corresponding to the dominant frequency f1) but the amplitude
and mean value of individual cycles is different (as the need for many
harmonics confirms) remaining at the same time stable at various phases
of the rotational period over at least a decade. This implies a rather
regular spacing but a slightly different strength of the individual
putative corotating spokes.
However, we refrain from any more detailed discussion at the moment since it
cannot be held on equally quantitative grounds as for the physical models
discussed above. The problem is that there exists, unfortunately, no
quantitative theoretical description of the model of co-rotating circumstellar
structures (see also Harmanec 1999, for details). The first step towards a
quantitative description is presented by Clarke (2003), but this author carried
out only relatively simple simulations and did not provide a prediction of the
amplitudes and phases of the LPV for the circumstellar
spokes in different filters and/or spectroscopic data. This is also the
reason why we are unable to indicate what kind of additional data would help to
test the hypothesis of co-rotating structures further.
![]() |
Figure 11:
A phase plot of the first two moments of the disentangled Si III 4552.654
line profiles folded with
![]() |
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For the moment we conclude that the evidence presented in this subsection is not
strong enough to reject the oscillatory nature of Sco, especially since the
model which combines rotational modulation superimposed on the
Cep
oscillation seems to explain the observed variability in a physically adequate
way, similarly as for other well studied
Cep stars. We point out,
however, that a unique identification of all the frequencies of
Sco requires a
data set with a longer time base and a better defined window function than the
one at hand.
We have considered Sco as a test case for the application of KOREL
disentangling as an intermediate step in the interpretation of short-period
LPV in early-type stars. We investigated its performance by
means of a comparison between three sets of profiles obtained from different
procedures after orbital subtraction. We find that the LPV
due to the high amplitude mode of
Sco is well preserved in all the line
diagnostics derived from the KOREL disentangled spectra. This is by no means
evident as KOREL was certainly not designed for this purpose and in fact, we
violate one of its starting principles: that there be no shape variations in the
profiles. Our experiment has shown that the short-term large-amplitude
variability is well interpreted as a sort of random noise by the code. It is
therefore evident that, in the case of large-amplitude short-term LP
variables in close binaries, KOREL disentangling is an extremely useful tool.
It would be interesting to carry out a similar performance test of KOREL
disentangling on a case study where also the secondary shows obvious
LPVs.
As for the low-amplitude variability, we were unable to evaluate the
capabilities of KOREL in detail. This is probably due to our inability to
derive accurately the low-amplitude frequencies of Sco. Indeed, we found
evidence for three frequencies
besides the dominant
frequency 4.99922 c d-1, but these are not significant in the
spectroscopic data. We accepted them as real because they were derived from two
additional independent photometric datasets (Lomb & Shobbrook 1975; Cuypers et
al. 2004). The derivation and interpretation of these low-amplitude
frequencies is somewhat different for the disentangled versus non-disentangled
spectra but it is not clear whether the results for the disentangled profiles
are more appropriate than for the original ones. For the moment, we therefore
have to refrain from claiming KOREL to handle variability with amplitude below
1 km s-1 in an appropriate way, which is by no means a negative outcome
of our study. A good future testcase to check the validity of KOREL for a
multiperiodic non-radial oscillator is
Cru (Aerts et al. 1998)
although most progress will be achieved by combining the analyses of real
datasets with simulations.
The analysis of the LPV of the pulsating primary Sco, based
on large dataset consisting of 699 spectra, covering a time span of 3 years,
resulted in a new understanding of the intrinsic variability of the star. We
identified the dominant frequency as a prograde NRP mode with
spherical wavenumbers
or (1,-1), whereby the contribution of
at least its first harmonic 2f1 is also needed to explain the observed
variations of the velocity moments. Moreover, we found strong evidence for the
presence of a rotational modulation effect in the Si III line profiles. We
presented a simple temperature or abundance spot model which allowed us to
explain the frequencies f2 and f3 suggested here and also in the
literature before (Lomb & Shobbrook 1975; Cuypers et al. 2004).
Our detailed analysis of Sco stresses that the derivation of
frequencies in the range of the pulsation frequencies expected for
Cep
pulsators resulting from Fourier decomposition does not necessarily prove the
existence of multiperiodic oscillations. A comparison between observations and
a theoretical physical model is needed to check the real cause of the observed
frequencies considering all the LP diagnostics. In the present paper
we performed such a comparison for a multiperiodic oscillation model and for a
modulated monoperiodic oscillation model, keeping in mind that a circumstellar
patch model is not straightforward to prove nor disprove. We conclude that
Sco has one dominant NRP mode
rotationally modulated by an inhomogeneous surface temperature and/or abundance
distribution, as
Cep (Telting
et al. 1997), a slow rotator, and V2052 Oph (Neiner et al. 2003), a moderate
rotator. In all three stars, one dominant frequency occurs in the radial
velocity variations while the rotational frequency is dominant in the EW variations. The latter hence played a crucial role in the interpretation of the
LPV for these three
Cep stars.
As the combination of one NRP mode and inhomogeneities at the
stellar surface of a rotating star seems to explain the very complex
LP character of Sco quite well, the question rises if the same model
might explain similar variations observed in the complex spectral lines of other
moderate rotators among the
Cep stars, e.g.
Per (De Cat
et al. 2000, and references therein),
Vir (e.g. Smith 1985a,b),
Sco (Telting & Schrijvers 1998; Berdyugina et al. 2003),
Cen (Schrijvers & Telting 2002). In many of these cases, the frequency
spectrum is dominated by one frequency while additional frequencies appear near
the noise level as in
Sco. A similar modulated oscillation model as presented
in this paper may be interesting to check as an explanation for their complex
variability. To do so, however, one needs high signal-to-noise
high-spatial and temporal resolution spectroscopy with a time base of years.
Acknowledgements
The authors acknowledge the referee, Dr. G. Handler, for his valuable comments, Dr. P. Hadrava for sharing his code KOREL and for his advice on its use, Prof. N. Piskunov for sharing the codes INVERS11 and SYNTH, Dr. R. Townsend for the code BRUCE and Dr. L. Balona for making some of his codes available. We also thank the Vienna group for putting the Vienna Atomic Line Database at our disposal. This study has benefited greatly from the senior fellowship awarded to P.H. by the Research Council of the University of Leuven which allowed his three-month stay at this university. His research was also supported from the research plans J13/98: 113200004 of Ministry of Education, Youth and Sports and AV 0Z1 003909 and project K2043105 of the Academy of Sciences of the Czech Republic. K.U. is supported by the Fund for Scientific Research - Flanders (FWO) under project G.0178.02 and C.A. and K.L. by the Research Fund K.U. Leuven under grant GOA/2003/04.