Issue |
A&A
Volume 507, Number 1, November III 2009
|
|
---|---|---|
Page(s) | 397 - 404 | |
Section | Stellar atmospheres | |
DOI | https://doi.org/10.1051/0004-6361/200912554 | |
Published online | 08 September 2009 |
A&A 507, 397-404 (2009)
Notes on disentangling of spectra
II. Intrinsic line-profile variability
due to Cepheid pulsations
,
P. Hadrava1 - M. Slechta2 - P. Skoda2
1 - Astronomical Institute,
Academy of Sciences, Bocní II 1401,
141 31 Praha 4, Czech Republic
2 - Astronomical Institute,
Academy of Sciences, Fricova 298,
251 65 Ondrejov, Czech Republic
Received 22 May 2009 / Accepted 2 August 2009
Abstract
Context. The determination of pulsation velocities
from observed
spectra of Cepheids is needed for the Baade-Wesselink calibration of
these primary distance markers.
Aims. The applicability of the Fourier-disentangling
technique
for the determination of pulsation velocities of Cepheids and other
pulsating stars is studied.
Methods. The KOREL-code was modified to enable
fitting of free
parameters of a prescribed line-profile broadening function
corresponding to the radial pulsations of the stellar atmosphere. It
was applied to spectra of Cep
in the H-alpha region observed with the Ondrejov 2-m telescope.
Results. The telluric lines were removed using
template-constrained disentangling, phase-locked variations of
line-strengths were measured and the curves of pulsational velocities
obtained for several spectral lines. It is shown that the amplitude and
phase of the velocities and line-strength variations depend on the
depth of line formation and the excitation potential.
Conclusions. The disentangling of pulsations in the
Cepheid spectra may be used for distance determination.
Key words: line: profiles - techniques: spectroscopic - stars: variables: Cepheids - stars: atmospheres
1 Introduction
The method of Fourier disentangling of spectra was developed by Hadrava (1995) from the method of cross-correlation (cf., e.g., Hill 1993) to decompose, from a series of observed spectra of multiple stars, contributions of the individual components and to simultaneously find orbital parameters, or, more generally, to fit free parameters of the physics governing the Doppler shifts and line-profile variations. One of the advantages of disentangling compared to cross-correlation is that it does not require a template spectrum of a star with similar spectral type or a model atmosphere.
Zucker & Mazeh (2006) introduced their method of Template Independent RAdial-VELocity measurement (TIRAVEL) for single-lined spectroscopic binaries (SB1). Their method uses each exposure from a series of observations as a template for cross-correlation with all other exposures. Zucker and Mazeh mentioned that the KOREL-code (cf., e.g., Hadrava 2004b) for Fourier disentangling is also template-independent. Nevertheless, they advocated use of their TIRAVEL based on the alleged advantage that it assumes the individual radial velocities to be free variables not bounded by an orbital motion. However, KOREL enables the convergence of individual radial velocities of any component also. Moreover, KOREL can do this not only for SB1, which is an extremely simple case, but also for two or more components. The option of free velocities is rarely used, because in practice it is more advantageous to take the orbital motion of multiple stars into account and to solve directly for the orbital parameters. Zucker & Mazeh mentioned the case of a third component as an example of when free radial velocities are needed. However, just in this case the solving for orbital parameters of both the close and the wide orbit (taking into account also the light-time effect) simultaneously with the determination of radial velocities is advantageous, because it checks the consistency of the possibly small perturbation with the source spectra better than a two-step procedure of determination and subsequent solution of the radial-velocity curve.
One reason to solve in some cases for individual radial velocities
independent of
orbital motion is technical, e.g. in the case of an unreliable
wavelength
scale of the observed spectra, as it has been done in the study by Yan
et al.
(2008). However,
a more important reason is seen
in cases when the observed wavelength shifts of spectral lines are not
due to overall orbital motion of the star but due to some other
effects.
One such case is the Doppler shift caused by pulsations of the stars.
The use of Fourier disentangling in the spectroscopic studies of
pulsating stars has been outlined by Hadrava (2004a,b).
Here we shall
demonstrate this method in practice in the case of the star Cep.
The study of either radially or non-radially pulsating stars is
important as a clue to probe the inner structure of stars. In addition,
the period-luminosity (PL) relation of Cepheids and some other radial
pulsators is used as one of the few primary methods for distance
determination. One needs to calibrate this PL-relation, and that
can be done by the Baade-Wesselink method (Baade 1926;
Wesselink 1946).
In this method, the spectroscopically measured pulsation velocity is
integrated over the course of the period to yield the changes of
stellar
radius, which, combined with photometric or interferometric variations,
may
reveal the distance to the star. However, the spectroscopic measurement
is complicated by the line-profile variations caused not only by the
radial
motion of the stellar atmosphere (which must be integrated over the
visible
part of the stellar disc projected locally to the line of sight) but
also due
to changes of other physical conditions (e.g. the temperature and
density)
in line-forming regions of different lines. Consequently, the radial
velocities
measured by different methods reflect only indirectly the
instantaneous pulsation velocity
of the stellar surface. It is thus
used to introduce the so called projection factor
,
which can be estimated either theoretically (e.g. Nardetto
et al. 2004)
or observationally (Nardetto et al. 2008, and
citations therein). An
alternative method directly matching the Doppler shifts and asymmetries
of spectral lines with a proper model of line-profiles has been
introduced
by Gray & Stevenson (2007). Their
approach is, in principle, equivalent
to the above mentioned disentangling of pulsations (Hadrava 2004a,b).
We summarize the method of pulsation disentangling and its
use in Sect. 2
and we test it on our observations of
the Cep
in Sect. 3.
In Sect. 4
we discuss possibilities of further development of the method.
2 Disentangling of radial pulsations
The line-profile variations (LPVs) caused by motion of the stellar atmosphere are often modelled by integrating Doppler shifted spectra of unperturbed (mostly plane-parallel) model atmospheres over the stellar surface. This approximation is used for study of the rotational broadening as well as LPVs due to radial or non-radial pulsations even though it is obvious that the motion may change the structure of the atmosphere and hence also the radiative transfer and the line formation in them. In the case of Cepheids, it is observed that different lines formed in different layers of the atmosphere have slightly different phase-dependence of LPVs (cf. Breitfellner & Gillet 1993; Butler 1993 etc.), which reflects (besides other effects) the changes of velocity gradients within the atmosphere. The basic idea of the Baade-Wesselink method also assumes that the spectroscopic and photometric or interferometric variations are caused by pulsations of the same stellar surface. However, the dilution of the atmosphere and its temperature variations and the presence of stellar winds in these stars implies that the motion of layers with given optical depths may differ from the local velocity of the gas, which then influences the profiles and shifts of the spectral lines.
A safe way (outlined and followed, e.g., by Fokin 1991, 2003 etc.) to treat all these effects properly would be to construct physically self-consistent models of atmospheres of pulsating stars, to calculate the observable quantities (spectra, light-curves, visibility functions) and to match the values of free parameters (including the distance) to the real data. However, the modelling itself is still computationally very demanding, not to speak of the inverse problem of solving for the free parameters. It is thus worth simultaneously following an alternative way of fitting the observations by simplified models, which take into account the most important effects and could be modified if a discrepancy with respect to self-consistent models or with respect to observations were found.
Let us assume now that the observed spectrum I(x,t)
(as a function
of the logarithmic wavelength x and
time t) is given by the integral
over the visible part (i.e. where the directional cosine


the spectrum is given by
where
For purely radial pulsations synchronous on the whole stellar surface, the local radial velocity


where R=R(t) is the instantaneous radius of the star. The Doppler shift which can be measured by the first moment is given by the mean value of xof the broadening and it reads
In agreement with Getting (1934), the projection factor p is thus


i.e. to an absorption occurring in the centre of the stellar disc only. This calculation is valid exactly for radial-velocity measurement by the moment method only. For other methods, like the deepest point of the profile, bisectors, the fit by two semi-Gaussian curves or the cross-correlation method, the result of radial-velocity measurement of the asymmetric lines is less certain. It also depends on the instrumental broadening. Because the standard Fourier disentangling (which assumes the broadening function given by Eq. (7) only) is a handy method of radial-velocity measurement, it is worth investigating its properties (and its p-factor in particular) when applied to Cepheids, or modifying it by taking into account the proper broadening function so that it will avoid the p-factor completely and will directly provide the pulsational velocities.
It is generally understood that limb darkening is crucial for
the asymmetry of the lines and thus also for setting the projection
factor.
However, determination of its proper value is still a problem (cf.
Monta
és Rodriguez & Jeffery
2001;
Marengo et al. 2002). The
limb-darkening corresponding to the radiation in continuum is
sometimes accepted also in the lines, arguing that it ``varies slowly
with the wavelength and can be taken to be constant over the span of
a spectral line'' (Gray 2005, p. 436).
However, the spectral flux in
continuum also varies slowly with the wavelength, but it cannot be
taken to be constant over any line because it is just its fast
variation across the line-profile that is seen as the spectral
lines. A simple
model of radiative transfer in stellar atmospheres reveals that weak
lines are usually dominated by the central part of the visible stellar
disc and hence the differences between the radiation in the lines and
in continuum has the limb darkening close to 1.0 (cf. Hadrava 1997).
This is confirmed by the observed behaviour of the line strengths
during eclipses and it is also consistent with the values
of the
projection factor in Cepheids. However, detailed non-LTE models
of stellar atmospheres show that the limb darkening within strong lines
(e.g. Balmer lines) is more complex (cf. Hadrava & Kubát 2003).
It can be represented as a superposition of a part without limb
darkening,
a part with linear darkening, as well as of higher order terms,
which also may include an emission (especially at the outer edges of
the stellar disc if the spherical instead of the plane-parallel
symmetry
is taken into account). We shall thus investigate separately the
cases of simulated pulsationally broadened lines without any limb
darkening (k=0) and with linearly proportional
broadening (k=1).
We deconvolve these profiles (and in the next section also real
observed spectra) by disentangling with broadening functions for the
same cases (k=0, 1) and by the standard
disentangling (i.e.
).
The generalization of disentangling for line-profile variability has
been described by Hadrava (1997
and 1998, for
the line-strength variability and
the changes in shape of line-profile, respectively; unfortunately, the
formulae
were incorrectly processed in the print of the later paper). The
explicit form
of the pulsational broadening and its Fourier transform was given for
the case of limb-darkening equal to 1 by Hadrava (2004a,b).
The Fourier
transform (from the space of functions of the variable x
to functions
of y) of this profile (Eq. (5) for k=1)
has the form
Here we need also the case with zero limb-darkening (k=0), for which the Fourier transform reads
and also the standard disentangling, for which


We have calculated several sets of simulated data by direct
integration
in the x-space to always convolve one fixed
Lorentzian line-profile
(
)
with
broadening functions either
or
for instantaneous
values of velocity
of a harmonic (i.e. sinusoidal) pulsation
at 20 values of t uniformly
covering the period of the pulsation.
The Lorentzian profile was chosen because it contains both a narrow
core and wide wings. An example of such Lorentzian profiles with
the intrinsic semi-halfwidth
corresponding to 30 km s-1
broadened
by
with a semi-amplitude of the pulsational velocity
equal to 100 km s-1 is
presented in Fig. 1.
(The
step of 100 km s-1 is marked
by the ticks on the x-axis.)
It can be seen that the profiles (cf. the thick lines with offsets
proportional to the phase) have the highest asymmetry at extremes
of the pulsational velocity, the high-velocity wing always being
steeper than the wider low-velocity wings reaching the rest wavelength
of the line. Qualitatively the same feature also results if
the pulsational broadening is applied to non-Lorentzian line
profiles and is in agreement with the line-profile variations
observed in Cepheids (cf., e.g., Breitfellner & Gillet 1993;
Nardetto
et al. 2006; Gray
& Stevenson 2007).
It is used to characterize such
profiles quantitatively by bi-Gaussian fits (cf. Nardetto
et al. 2006),
which, however, are not physically substantiated.
![]() |
Figure 1:
Standard disentangling (with |
Open with DEXTER |
Figure 1
also shows the disentangling of these simulated
profiles by the standard KOREL disentangling (i.e. ).
For spectra containing lines of two or more component stars,
the disentangling simultaneously decomposes them and fits
the orbital parameters, while in the present case it leads
to a simpler problem of fitting all the input spectra by a mean profile
(which is drawn by the lowermost line) scaled in the line-strength and
shifted in x. The best fits to the
individual input profiles are overplotted
in Fig. 1
by the thin lines. The disentangled mean profile is
symmetric here (which need not be exactly the case if the pulsational
velocity includes some higher overtones in addition to the basic
sinusoidal mode, or if the spectra do not cover the period uniformly).
Yet, it reproduces the input profiles relatively well, so that for real
data the difference between the true profiles and the fit by a
simplified
model may be hidden in the noise. The coincidence is even better for
input profiles broadened by
and disentangled by
.
In both cases the coincidence is worse if the amplitude
of the pulsational velocity exceeds more the intrinsic width of the
line.
The disentangled profile is wider than the intrinsic profile
(this is well seen e.g. in the uppermost input line, for which
), whenever the disentangling
is performed using a smaller
broadening (i.e.
with higher k)
than the broadening
used for the simulation of the data. If the input profiles are
disentangled with the same broadening for which they were
simulated, their fit as well as the reconstruction of the
intrinsic profile is perfect within the numerical precision
of the simulation.
The disentangled line-strength factors are higher (up to
nearly +0.08) for phases with low pulsational velocities and
smaller
(nearly -0.08) at extremes of the velocity.
The disentangling of this and other simulated datasets is
performed here with velocities bound to a sinusoidal pulsation,
which was also assumed in the creation of the data. This is done
by formally assuming that the velocity obeys a circular motion.
The ratio of the true amplitude
of the velocity chosen for
the simulation and its disentangled value
gives a mean value
of the projection factor p for a particular
combination of
the broadening functions. The values of pulsational velocities
which give the best fit of the mean disentangled profile with
the individual input profiles are also calculated and they
reveal that the projection factor is slightly phase-dependent.
However, the differences are very small - in the particular
case shown in Fig. 1
the free radial velocity is about
0.62 km s-1 (i.e.
nearly 1%) higher than the harmonic one at their
extremes and about 0.47 km s-1
lower at medium (non-zero) values
of velocity. The profiles reconstructed with both velocities
are drawn in Fig. 1
by the thin lines, but they are not
distinguishable within the precision of the graphics. We thus neglect
these differences which are of the order of the precision of the
computation and we give the mean values of projection factors
only for several combinations of broadening
functions used for the simulation (
,
k=0,1)
and the disentangling (
,
)
and
the velocity-amplitude to line-width ratios
in
Table 1.
Table 1: Projection factors for simulated line-profiles.
It can be seen from these results that for lines broader than their
Doppler shift (
)
the standard disentangling (with
)
yields p-factors agreeing within the
numerical errors (which are of the order 10-3
here) with the
moment method of the radial-velocity measurement, i.e.
or
for lines with unit (
)
or zero
(
)
limb darkening. For lines with intrinsic widths
smaller than the pulsationally induced shifts and asymmetries,
the standard disentangling is more sensitive to the position of the
deeper parts of the profile and the p-factor
decreases slightly closer
to the value 1. The disentangling with the proper
limb-darkening in
the line (i.e.
for
and
for
)
has p=1, it means that
it directly provides the pulsational velocities and it is desirable
to use it for the Baade-Wesselink calibration. The use of improper
broadening functions (i.e.
for
or
vice versa), however, results in an error of about 10% in
radial velocity. We thus need either to find
the proper limb darkening across the line-profiles from detailed
model atmospheres, or to distinguish which model fits the observed
line-profiles better.
3 Disentangling of the observed spectra
To test the disentangling of pulsations on real data we started
spectroscopic observations of Cep using the 700-mm
camera
of the spectrograph in the Coudé focus of the Ondrejov 2-m
telescope equipped with LN2-cooled SITe CCD detector ST-005A
(
15-
m pixels).
Sixty nine medium-resolution
spectra (
)
with a linear dispersion of 17 Å/mm (0.25 Å/pix)
in H
region
(6250-6770 Å) obtained between August 19, 2008 and
April 16, 2009 (mostly by Slechta) are used in this study.
See on-line Tables 3
or 4
for the journal
of observations.
The spectra were reduced in IRAF
using the standard packages ccdproc, doslit
and rv
(for more details of the processing see Skoda & Slechta 2002).
The spectral region around the H-line contains many
atmospheric
water-vapour lines which complicate the measurement of stellar spectra
by standard methods (cf., e.g., Kiss & Vinkó 2000).
However, Fourier disentangling with variable line strengths is not only
suitable
to remove the telluric lines (cf. Hadrava 1997, 2004a,b, 2006a), but
at
the same time it enables one to use them for an additional check or
correction of the wavelength scale similarly to their use in classical
methods (cf., e.g., Butler & Bell 1997). As
the first step
of the disentangling we chose the spectral region 6511-6521 Å
sampled in 1024 bins (i.e. with a step in radial velocity of
0.45 km s-1 per bin)
to find the line-strength coefficients for the telluric lines. The
Doppler
shifts of the telluric lines with respect to the heliocentric
wavelength-scale
were calculated in the Keplerian approximation of the annual motion
which
is provided by the PREKOR-code from the coordinates of the target star
(cf.
Hadrava 2004b)
and the telluric lines were disentangled by the standard
disentangling using the broadening function
.
The stellar lines in this region were disentangled as a superposition
of two
systems of lines using
with
free radial velocities and each one with its own free line-strength
factors. To avoid uncertainties in low Fourier modes, which could cause
anticorrelated distortions of the stellar and telluric component
continua, we used disentangling constrained by a template (cf. Hadrava
2006b)
for the telluric lines. The template had been calculated by
disentangling spectra of the star 68u Her (i.e. spectroscopic
binary HD 156633, also taken with the Ondrejov
2 m-telescope), which gives a telluric spectrum with a
satisfactorily flat
continuum. The differences between the prescribed annual motion of the
telluric lines and the radial velocities of these lines disentangled as
the best fits of individual exposures were then used as corrections of
the wavelength scale in the preparation of spectral regions for
subsequent
disentangling of the stellar lines. Owing to application of a method
of enhanced precision (Hadrava 2009),
these corrections were found with
sub-pixel resolution. These wavelength corrections can be applied
to exposures with sufficiently strong telluric lines only. In our case,
the depth of telluric lines in exposures where they are weakest is
about
one half of their mean depth, so that the correction could be applied
to
all exposures.
To disentangle the pulsational velocities in the observed spectra of
Cep
we chose first several narrow spectral regions, each
one containing a single dominant spectral line listed in Table 2.
These regions were sampled in 256 bins each with a step of
radial velocity
per bin given in the Table (rvpb
in km s-1). In some of the
regions, blends
with some weak lines can be seen, which partly decrease the quality of
the disentangling, however, their influence can be neglected. The
relatively
low spectral resolution of our original data (about
12 km s-1) provides only
a rough sampling of line profiles in individual exposures, which have
half-depth widths comparable to the amplitude of the Doppler shifts,
i.e.
only about 4 to 6 times larger.
The disentangled line profiles are significantly smoother due to
the averaging of a large number of the exposures. Nevertheless, the
limited
quality of the data does not enable us to convincingly decide
which of the models
,
and
best
fits the observed line-profile variations
or even to search for their best linear combination, which could
be expected for more general limb-darkening within the line-profile.
We thus performed the disentangling of all regions for each of these
models separately. In all regions, the integrated (O-C)2
of the
fitted spectra was the largest for the model
,
while the residual noise for the pulsational broadenings
and
was nearly the same (within about 4%) without
any evident regular preference of one model or the other. We thus
illustrate the results on the case
,
which
corresponds to the unit limb darkening and is thus the most
advantageous
from the theoretical point of view (see above).
Table 2: List of disentangled spectral lines.
![]() |
Figure 2:
Phase-dependence of pulsational velocity v in
km s-1 for individual spectral lines
disentangled with |
Open with DEXTER |
Figure 2
shows the pulsational velocity disentangled
using the model
which is given in the on-line
Table 3.
For the purpose of visualizing
the results (i.e. calculation of phase) we use the ephemeris
![]() |
(10) |
Our data do not allow us to check or improve the period, and the scatter in its published values (e.g.


The fit of the observed spectra by the standard
disentangling (
)
is invariant with respect to adding a constant
to all disentangled radial velocities and a simultaneous shift of the
disentangled spectrum in the logarithmic wavelength for the
corresponding
value. If the position of the stellar spectrum disentangled from all
exposures (transformed first into heliocentric wavelength scale) is
fixed
to the laboratory wavelengths of identified stellar lines, the measured
radial velocities correspond to instantaneous heliocentric radial
velocities. Neglecting the systematic errors caused by the line-profile
distortions, the heliocentric radial velocity of the star could be
estimated as the velocity averaged over the pulsational period
![]() |
(11) |
Having the values of velocities measured with some errors in several discrete times only, we fit them using least-squares method in the standard manner by Fourier series (cf. Schaltenbrand & Tammann 1971; or Moffett & Barnes 1985, etc.)
The mean velocity


Unlike the standard disentangling, the disentangling with
broadening
functions
and
is sensitive to
the value of the true radial velocity of the star's centre of mass.
At the phases when the pulsational velocity is equal to zero, the
consequent
line-profile distortion disappears, the instantaneous line-profile
coincides
(within the errors) with the disentangled one and its Doppler shift is
given
by the overall motion of the star only. The Doppler shift of the
disentangled
spectrum should thus correspond to the intrinsic radial velocity of the
star. However, because a systematic shift of all pulsational velocities
can be relatively well compensated by a shift of the disentangled
spectrum,
the convergence of the solution to the true radial velocity is very
slow
and the value of the velocity is poorly defined, in particular for the
rough
sampling of the line profiles in our spectra. (To speed up the
convergence
we introduced an additive term in the velocity, which is possible to
converge
explicitly by the simplex method.) The values for which we
achieve the best fit of the observed spectra differ somewhat from
the values v0 found from the
fit by Eq. (12).
This
discrepancy may be due to the mentioned observational errors, but it
may
also reflect an influence of effects neglected in the simple models
given
by the broadening functions
and
.
For instance the gradients of pulsational velocities or the stellar
wind
overimposed on the pulsations may contribute a distortion resembling
P-Cyg shape to the line profile. More precise measurements as well as
models
of line formation will be needed to solve this question.
![]() |
Figure 3:
Phase-dependence of line-strength factors s for
individual spectral lines disentangled with |
Open with DEXTER |
Similarly, the line-strength factors (see Fig. 3 and
on-line Table 4)
of the lines were expanded into
the Fourier series and the amplitudes s1
of the first component
are also given in Table 2.
The negative values of this amplitude
are assigned to the lines which are approximately in an antiphase
with the other lines. These are the two lines of ionized elements
in our set, for which ,
while for the lines
of neutral atoms we have
.
This amplitude
(anti-)correlates well with the excitation potential
(+ ionization potential
,
cf. Moore et al. 1966) of
the lower level of the line. The largest deviation can be seen for
the Ca I line, which, however, may be blended with the
Fe I (169) line
6344.15 Å with
eV.
These variations of line
strengths measured by the method of relative line photometry (Hadrava
1997)
thus enable one to easily find the changes of
atmosphere temperature in the course of the Cepheid pulsation period
(cf. Krockenberger et al. 1998;
Kovtyukh & Gorlova 2000).
![]() |
Figure 4:
Phase-dependence of pulsational velocity in km s-1
for spectral lines in the H |
Open with DEXTER |
![]() |
Figure 5:
Phase-dependence of line-strength factors s for
spectral lines in the H |
Open with DEXTER |
As a next step we performed disentangling of a wide spectral
region (6506-6618 Å) around the H line. We chose
sampling in 1024 bins with a step of 5 km s-1
per bin. The telluric
lines were disentangled using a template obtained from spectra of
68u Her again. The pulsational velocities and line-strength
factors obtained by disentangling of the whole spectral region into
one stellar component are given in Tables 3 and 4
(in columns labeled H
)
and drawn (by full circles
in green) in Fig. 4
and 5,
respectively. The semiamplitude of the
first Fourier mode of the pulsational velocity is v1=21.40 km s-1;
the line-strength factors have a semiamplitude s1=0.062
and phase
shift (
)
comparable to that of the ionized
atoms in Fig. 3.
The phase dependence of residual spectra
is drawn in Fig. 6.
(The wavelength
is drawn
on a logarithmic scale and labeled in Å.) It can be seen here
that the strength of the H
wide wings is almost in antiphase with
its narrow core as well as with strengths of weaker metallic lines.
The residuals in the wings are thus in absorption and the narrow
lines in emission around phase 0.0 around maximum expansion
(cf.
the bottom and uppermost lines) and it is reversed around phase
0.5 at the infall (the lines around the middle of the figure).
Residuals of some of the narrow lines have a shape of P-Cyg or
inverse P-Cyg profiles in some phases. This indicates that the Doppler
shifts of these lines differ from the other lines.
![]() |
Figure 6:
Residual spectra in the H |
Open with DEXTER |
To decrease the residual spectrum, we must allow each line to vary
its line strengths in agreement with the phase-locked changes of the
atmosphere temperature. We thus also disentangled the Hregion into
two independent components, starting with the two solutions
from Table 2,
which are the opposite extremes in s1,
i.e. the solution for the line Fe I (13)
6358.69 Å and Si II (2)
6347.10 Å. The spectrum splits into two components, as it is
shown by the upper two lines in Fig. 7. The third
very bottom
line in this figure gives the telluric spectrum. It can be seen here
that the H
line
is contained mostly in the second component
obtained from the initial approximation corresponding to the
Si II line,
while the narrow metallic lines are distributed between both
components,
each one to a different proportion. A small contribution to the core
of H
also appears in the first spectrum. This contribution has
an inverse P-Cyg shape.
We let the velocities and line-strengths of both these components in
all spectra converge to the best fit. The (O-C)2
of residual
spectra decreased to 12% in comparison with the previous
one-component
(+ 1 telluric component) solution. The resulting velocities are
labeled as H
or H
in Table 3
and
drawn in Fig. 4
by (blue) open circles or (red) crosses for
the components developed from the solutions for neutral and ionized
atoms,
respectively. The first Fourier modes of the pulsational velocities are
v1=20.18 km s-1
or v1=17.38 km s-1,
resp.
The phase shift of the first mode of velocities converged to
for the first component, unlike the values
found for the second component
of H
region, its one-component solution, as well as for the
solutions of individual lines listed in Table 2 (with the
exception of the value
for the line Fe II).
The line-strength factors (given in Table 4 and
Fig. 5)
have
semiamplitudes s1=0.577
with phase shift (
)
comparable to that of neutral atoms in Fig. 3 for the first
component and s1=-0.152
for the second component. The negative sign
here denotes again the approximate antiphase (
,
cf. Fig. 5).
Let us note that the line C I (22) entirely appears
and the lines Fe II (40) and
Sc II (19) are more pronounced in the second
component also containing the major part of the high excitation H
line (
eV), while the other
low excitation Fe I lines
identified in Fig. 7
and Table 5
(where estimates of
equivalent widths in the first and second component are given) are by
greater or equal part contained in the first ``low excitation''
component
of the spectrum.
![]() |
Figure 7:
Spectrum of |
Open with DEXTER |
Table
3:
List of spectral lines disentangled in the H
region.
These results are consistent with those for individual lines presented above and with the variability of radial velocities and equivalent widths found using classical methods by different authors cited there.
4 Conclusions
Our measurements of pulsational velocities of Cep using
a newly generalized version KOREL09 of the code for Fourier
disentangling and line-strength photometry confirmed that the phase
variations of velocities and strengths of different spectral lines
depend on the depth of their formation in the stellar atmosphere and
on the excitation potential of their lower levels.
This means that any elaboration of the
Baade - Wesselink method to a higher precision must take into account
the structure and dynamics of Cepheid atmospheres, where the
instantaneous
effective stellar radii corresponding to either the photometric or
interferometric version of the method do not precisely follow the
true motion of any particular layer of the atmosphere or velocities
found from Doppler distortions of any particular stellar line. Our
method of disentangling spectra provides a tool for observational
probing of the structure of the pulsating stellar atmospheres.
Originally designed and well proved for disentangling of spectra of multiple stars, our method can also be used for studies of Cepheids in binaries, which can yield more information about the basic physical parameters of the stars and thus also about the PL-relation in different conditions. Further sophistications of the method are possible, e.g. directly fitting the coefficients of the Fourier series given by Eq. (12) instead of independent velocities in individual exposures to distinguish the orbital and pulsational changes of radial velocities. These coefficients could be then used for the distance determination in the way introduced by Krockenberger et al. (1997) or for the classification according to Deb & Singh (2009).
Another possible future improvement is to take into account for
the pulsational broadening functions the results of radiative transfer
and line formation in differentially pulsating atmospheres. This is
particularly challenging for non-radial pulsations, where the
approximation of an unperturbed moving atmosphere is even less
physically
substantiated but even more commonly used to avoid the complexity
of the problem. Note that the output of residual spectra in rest
wavelengths of any component or the centre of mass of a multiple
system was implemented in the KOREL code by its author as a ``first
aid'' for studying pulsational or other perturbations of spectra not
directly included in the model of the broadening functions .
However, such an approach should not be used as a black box without
understanding its underlying assumptions, namely that the free
parameters of the neglected effect (e.g. the pulsations) do not
correlate with the parameters taken into account (e.g. orbital
parameters). If we find a solution with a simplified model, it is
not proof that its assumptions are correct, because neglecting
a problem is not a true solution of the problem. The disentangling
of pulsations in spectra of single or multiple stars thus requires
one to include the pulsational variations in the model by
which we fit the observations. We have demonstrated such an approach
here
in its simplest form. A future sophistication will be needed to account
for more accurate observations.
The authors thank the unknown referee and A. Han for useful comments. This work has been done in the framework of the Center for Theoretical Astrophysics (Ref. LC06014) with a support of grant GACR 202/09/0772.
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Online Material
Table
4:
Pulsational velocities v for individual lines (cf.
Table 2)
and H
region at all exposures.
Table
5:
Line-strength factors for individual lines (cf. Table 2) and H
region at all exposures.
Footnotes
- ... pulsations
- This study uses the spectra from the Ondrejov 2-m telescope.
- ...
- Tables 3 and 4 are only available in electronic form at http://www.aanda.org
- ... IRAF
- IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under cooperative agreement with the National Science Foundation.
All Tables
Table 1: Projection factors for simulated line-profiles.
Table 2: List of disentangled spectral lines.
Table
3: List of spectral lines disentangled in the H
region.
Table
4: Pulsational velocities v for
individual lines (cf. Table 2) and H
region at all exposures.
Table
5: Line-strength factors for individual lines (cf.
Table 2)
and H
region at all exposures.
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