The observational material used for the development and the tests of our reconstruction method, and also for its first application, was obtained at the Observatoire de Haute Provence (OHP) in France. The spectra were taken at the 1.52-m telescope by using the Aurélie spectrometer (Gillet et al. 1994) equipped with its mono-dimensional CCD detector (2048-pixel array).
In the tests hereafter, the basic idea is to construct a composite spectrum, made up of two known originals, and then to check on how well the spectral restoration procedure described in Sect.2 is able to reproduce them separately. A successful performance will then provide support for the application of our method to the real binary star IZPer, finally revealing a faint secondary spectrum.
In order to evaluate the stability of the reconstruction method, with respect to noise fluctuations on the S-spectrum, and to possible errors on the value of the shift-parameter k, we prepared some analytically-generated line profiles and derived the S-features. Then we reconstructed the lines by applying our procedure, and compared the results with the originals.
We used both gaussian and lorentzian functions to create realistic line-profiles, with superimposed poissonian noise produced by a random number generator (Fig. 3 (click here)). So a large collection of S-profiles was obtained, for k values ranging from 0.1 up to 2 times the line FWHM. Then we reconstructed the input lines, by checking different values of k that ranged from 0.5 to 1.5 times its original value. From the above tests we derived the following conclusions: i) the noise is amplified by each iteration; ii) an error on k leads to an error in the intensity of the restored line (if the input value of k is overestimated the restored lines are deeper than real, and vice-versa).
However, from a practical point of view, these effects are tolerable
when we do not need too many (
)
iterations for the reconstruction, when the relative error on k
is less than 
, and when the noise is small, so that
on the secondary spectrum. Thus, on the observed
composite spectra, we should have 
where generally the fractional luminosity of the secondary is
.
For testing the method under typical observational conditions and with real spectra, we used simulated observations. By using a modelling code (Bradstreet 1993), we constructed some fictitious binary systems, each consisting of a close pair of well-known bright stars. Then we simulated the systemic spectrum, by combining and adapting the two known spectra, to reproduce the relative intensities, radial velocities at different phases, rotational line-broadening, noise, etc. Finally, we applied the separation procedure, and checked how accurately the extracted spectra matched the input originals, for a variety of conceivable conditions.
Figure 3:
Tests with artificial profiles: left, gaussian blend;
right, lorentzian line. Above: original features (solid),
and S-spectrum (dashed). Below: step-by-step
reconstruction
The spectra used in these simulations (Bédalo 1995)
were taken from the Trieste-Aurélie-Archive (TAA).
This local facility (Ferluga & Mangiacapra 1994) contains a collection of
high-resolution optical spectra - mainly from standard, peculiar and binary
stars - taken at the OHP with Aurélie by observers from Trieste
(about 1000 stored spectra). The TAA provides free on-line data
retrieval
,
and it will allow on-line access to our S-profile conversion
algorithm (Appendix A).
An example of simulation is given by the hypothetical binary EtaVega-2.
This object is composed of the pair A=
Aur and
B=Vega, closely rotating with a supposed period of 2 days.
Assuming (for sp.types B3
) the masses
and 
(cf. Schaifers & Voigt 1981), our model has a separation
between the two stars (while their radii are
and 
).
Fig. 4 (click here) shows the simulated composite spectrum of EtaVega-2, namely
, generated at the first quadrature. By applying the
reconstruction procedure, we extracted a secondary spectrum
. This should be compared with the original spectrum 
of our secondary star (Vega, broadened by
kms
for synchronous rotation).
Figure 4:
EtaVega-2, a simulated binary spectrum (
).
The extracted secondary 
matches the spectrum 
of Vega
The earliest idea of our separation technique, to be applied to
real astronomical objects, was conceived for the eclipsing binary
IZPer, when we first detected that it was double-lined, and
we struggled to isolate the faint secondary spectrum.
A pioneering attempt at line reconstruction was made for H
of IZPer B (Ferluga et al. 1991).
The actual observations of IZPer were performed with
Aurélie, in the framework of a survey program,
in search of double-lined eclipsing binaries
(Ferluga et al. 1993). The spectra of the survey, taken at high
S/N in the range 
,
are now available in the TAA.
We applied the separation procedure to a pair of mean
quadrature spectra, using coadded exposures taken
during various orbital cycles.
The k-parameter for the reconstruction is
provided by the S-profile of MgII 
,
where the dual images of the secondary line are split well apart.
Figure 5:
IZPer at first quadrature (
). The secondary spectrum
and the primary spectrum 
are finally separated
Fig. 5 (click here) displays the resulting spectra of the components.
Note the wide H
wings in the secondary spectrum
, practically unpredictable by simple visual
inspection of the observed systemic spectrum 
. The appearing
of such feature only in the extracted spectrum
is surprising, and one may wonder whether it is
definitely real and not an artifact.
This is easy to prove. First, the existence of a secondary
H
is revealed by a slight mirror-like asymmetry of the systemic
profile at quadratures 
and 
(detectable by careful
overplotting). Second, our reconstruction
is confirmed by simulations, see EtaVega-2 (Fig. 4 (click here)) where
the extracted 
perfectly matches the test-profile 
embedded in the composite spectrum 
.
For IZPerB, the resulting depth of H
and H
is about 
of the systemic continuum. There are also
features of the secondary spectrum reaching only 
of the
primary continuum, as the SiII(3) doublet
4128,4131 seen just above the noise-level.
Error propagation and causes of scatter, in our procedure and in its implementations (Sect.2.5), were tested experimentally (Sects. 3.1, 3.2) under realistic conditions, with the aim of practical application. a) Spikes and defects in the data strongly disturb the extraction, thus preliminary cosmetics is necessary. b) Normalization discrepancies of the input spectra may cause the algorithm to diverge: this is avoided by rectifying the continuum of the S-spectrum. c) Each iteration slightly amplifies the noise, while the two-sided procedure (rightward+leftward) minimizes this effect.
In most cases, the reconstruction is obtained in few iterations
(two for IZPer), simply with the advantage of containing the noise.
A favourable situation occurs when many spectra are available at various
orbital phases: this means more input pairs to be processed,
then more versions of 
and 
to be averaged.
Future work will first be devoted to IZPer itself, and to the application of the separation method to some other eclipsing binaries from the Aurélie survey which show possible secondary lines. Later, application will be extended to other, also non-eclipsing, binary systems.
Finally, we may say that the information provided in this paper is intended to enable anyone who is interested to separate personally his own binary spectra.
Acknowledgements
The authors are indebted to D. Mangiacapra for collaboration in the observations.
Here we shall analyse how information of the secondary component is preserved in the difference-spectrum, and how it can be extracted. In principle, this case is not very different from resolving a blend of two nearby lines, the only peculiarity being that here one component of the blend is in emission, having also the same strength as the other in absorption. So one possible way of studying the difference-spectrum is to consider the S-features as being particular blends, treatable by special deblending methods currently available for spectroscopic data-analysis.
The application of a conventional best-fit, however, could not be the optimum choice when the special goal is just to determine k. Our idea is that, in a merged S-profile, the distortion of the lobes should betray the distance between the two embedded images of the line (while this distortion may be smoothed by fitting). We propose an original method which provides k and the parent-line parameters by directly measuring the shape of the S-profile, for gaussian lines (while the lorentzian case is similar).
Let us represent an absorption line of an intensity-normalized
spectrum with the gaussian:
where 
D, and W
(central wavelength, central depth, and equivalent width)
are the standard parameters of the line.
Then the S-profile is given by:
where k is the shift-parameter, as defined above.
The basic parameters of the S-profile, which can
be directly measured from the difference spectrum,
are the following: 
, and 
(peak wavelength, peak intensity, and equivalent width of the positive lobe),
as represented in Fig. 3.4 (click here). Alternatively to 
, it may be
convenient to measure the quantity 
, that
is the separation between the two S-profile peaks.
Figure A.1
The peak wavelength 
can be derived, in terms of the
original line parameters, from Eq.(A1) with the
condition d
, leading to:
The other quantity 
can be also obtained from Eq.(A1) by substitution:
and the same can be done for
, which
becomes:
Relations (A2), (A3) and (A4) form a system of implicit equations,
where the unknowns are k, D, and W;
while s, 
and 
(parameters of the S-profile),
together with 
(central wavelength of the S-profile),
are measurable quantities.
The solution of this system of equations can be achieved by the
following half-analytical, half-numerical technique. First, from Eq.(A2)
we obtain the term:
Then we substitute this expression in Eq.(A3), writing
the central depth as a function of k:
By substituting the above expressions of 
and D(k) in Eq.(A4),
we finally obtain:
This is an equation in the single unknown k; but it still has
an implicit form, and is rather complicated. Since it seems impossible
to derive the solution analytically, we find the value of k in a
numerical form. In fact, there are actually many different routines
designed to solve cases such as Eq.(A7).
So, we finally use the resulting value of k to calculate
D and W from (A4) and (A5). In conclusion, a practical
algorithm can be established (Bravar & Ferluga 1995), simply making the
conversion 
; this will
be available within the Trieste-Aurélie-Archive
on line (via WWW).
The above conversion is reliable, as far as the S-profile is not
remarkably altered by the noise. Only the S-features
generated by isolated lines can be processed; if more
of them are available, a mean value of 
(same for all lines) can be derived, thus improving the accuracy.
