Our separation method is based on the analysis of one pair of spectra of the system, obtained at two different orbital configurations. Both spectra must be taken out of eclipse or conjunction.
The method is applicable in most general cases (only secondary spectra with emission lines are excluded). We shall see that the extraction procedure requires, to converge, a secondary spectrum containing at least one window of flat continuum, in the observed spectral range.
Observations at opposite or near-opposite phases are preferable, and at
quadratures (opposite elongations) are optimal. So the displacement
of secondary lines is almost symmetrical in the two phases, and maximum at
quadratures. Conventionally, for the two observed phases 
and 
, we use the index 1 to label the phase with
the most blue-shifted primary (most red-shifted secondary).
There are no special requirements for the adopted wavelength range
, and for the spectral resolution,
except that all instrumental settings must be the same for both spectra.
Also the standard intensity-wavelength calibration, and the (optional)
continuum-normalization as well, should be initially performed in an
identical way.
The procedure is applicable, in principle, even if the input spectra
have no recognizable continuum. Practically, in place of the
continuum level, each input spectrum 
may be normalized with respect to its average intensity
level 
, coinciding with the continuum only if
the integrated contribution of the lines is negligible. However,
for simplicity's sake, we shall still use the word continuum
instead of the expression average intensity, when describing
normalization and renormalization procedures hereafter
(e.g. in Sect.2.4).
The first step of the procedure is to compensate for the radial
velocity of the primary star in our intensity-normalized spectra.
So we get two
velocity-corrected spectra 
and 
, where
each primary line has its laboratory wavelength
.
The composite spectra 
and 
are then subtracted
point-by-point one from the other: in this way the primary lines are
eliminated (assuming the star is not intrinsically variable), and what we
obtain is the difference spectrum
This special spectrum, which we shall also call S-spectrum, contains - in principle - no trace of primary lines (if the primary velocity has been correctly compensated for); while it shows symmetrical excursions around the zero level (that is now the continuum). Each secondary line is represented by an undulated feature, hereafter named S-profile, which consists of the line itself, flanked by (and possibly merged with) its capsized and shifted replication. Fig. 1 (click here) shows how the S-spectrum is obtained by starting from two spectra observed at near-opposite phases.
Figure 1: Generation of the S-spectrum (schematic).
Above: the input spectra 
and 
,
containing primary (dashed) and secondary (solid) features.
Middle: the shifted spectra 
and 
, both with the
primary lines at rest-wavelength. Below: the
difference spectrum 
-
. Note that
s
k for merging profiles (right, dotted)
In practical applications, the secondary star is generally
much fainter than the primary (up to only 1%); so we
shall realistically describe the secondary lines as an
additional effect on the input spectra (though the treatment
is independent of the secondary brightness). Then,
if 
is the spectrum of the primary star at
rest (normalized to the continuum of the system), the
contribution of the secondary star to the two input spectra can
be represented with 
and 
,
defined by:
Note that, while 
is a normal spectrum (continuum=1),
the functions 
and 
have the form of superimposed perturbations
(continuum=0), to be renormalized later (Sect.2.4).
With the above definitions, the S-spectrum in Eq.(1) is reduced to:
where only the secondary signal is left (while the primary 
is cleared off).
Since the twin images 
and -
of the secondary lines
(displaced and mirror-like) are entangled in the S-spectrum above,
they should be first separated, re-oriented and shifted,
and then recombined. We shall use an iterative method, restoring
the line step by step, with an original shift-check-sum technique.
We must state beforehand that, if the two images of a line are well separated within the S-profile, without overlapping, clearly each one is representative of the original feature. So, in this case, there is absolutely no problem.
Problems arise, instead, as soon as we want to recover the shape of a line when its original images partially overlap and neutralize each other, so that a merging S-profile is formed. It happens when the width of a line is large with respect to the velocity separation of input phases. In this case, profile restoration is needed.
Our reconstruction procedure starts from the pair of
composite spectra 
and 
,
which through Eq.(1) provide the difference-spectrum
to be processed. Our purpose is then to
derive, in Eq.(2), the form of its constituent functions
and 
. They are
symmetrical, because 
is a shifted replication of 
.
The Doppler shift, corresponding to the difference in radial velocity
of the secondary component in the two spectra, will be hereafter indicated
by 
. So, it is identically:
with k>0 because of our initial choice of indexes.
Let us now assume for simplicity's sake a secondary spectrum having
a single spectral line, and being equal to the continuum outside
the interval containing that line. Supposing this interval is
for 
;
then, according to Eq.(3), it will be
for 
.
It follows that a particular interval
[
exists,
in which 
displays the line-profile,
while we still have 
. Therefore, in
this interval, from Eq.(2) we can derive
. So, in the interval
[
,
both 
and 
are determined.
Then, by knowing 
in the above interval, we are able to
compute, from Eq.(3), the values of 
in the new interval
.
Now, in this interval, from Eq.(2) we can also
derive 
. Thus, in the new interval
[
,
both functions 
and 
can be obtained.
This leads through Eq.(3) to the values of 
in the next
interval [
, and so on.
In such a way, by iteratively applying Eqs.(3) and (2),
we are able to reconstruct the whole function 
,
representing the contribution of the secondary spectrum.
Figure 2:
Reconstruction of a line-profile, drawn schematically
as a triangle. Above: the original line images -
and 
, displaced by the shift k, which are merged into the
S-spectrum. Below: from the S-spectrum (thick line),
the profile restoration is achieved step by step, here in 2 iterations
(progressively thinner lines):
= S-profile;
(1a) = max
;
(1b) = 
;
(2a) = max
;
(2b) = 
= original line-profile 
.
Note. Index k means that the spectrum is red-shifted by k
The reconstruction can be carried out in a practical way, by repeatedly comparing the S-profile with a shifted and reversed image of itself, which is implemented step by step until the complete line profile is formed. The steps are shown in Fig. 2 (click here), considering a schematic triangle-shaped line. Let us remark how easy the procedure is, involving only elementary operations.
The requested number n of steps is proportional to the width
of the feature to be reconstructed. Since each iteration restores only a
portion of the profile, and the length of this portion is k, then:
[
, 
] being the total width
of the feature. Normally, the integer n is generally a
small number (a few units).
This method can be applied also in the general case when the function
represents a full set of spectral lines, provided that it
still contains one window on the flat continuum, wider than k. This
interval 
, in which 
, is the required
basis for running our reconstruction procedure; otherwise,
the iterations will not converge.
In practice, working with a real spectrum, we can consider that the profile of an isolated line reaches the continuum, outside a suitable interval around its peak. When dealing with an observational S-spectrum, the extremes of our interval can be found, for example, where the far `tails' of S-profile merge into the noise fluctuations of the continuum itself.
Owing to the way it has been defined, the restored function
describes only the shape of the
secondary lines, referring to a `differential' zero-level continuum.
Now, a proper renormalization procedure should be established.
The primary spectrum can be isolated, by simply
removing the secondary lines from the normalized systemic spectrum 
:
Precisely, 
is the spectrum of the primary star A at the phase
, normalized with respect to the total systemic intensity
(that is the 
level).
In order to be consistent with this normalization, the secondary spectrum
needs only to be raised back to its original unity-level continuum
(formerly the 
level):
So, 
is the spectrum of the star B at the same phase,
normalized to the systemic intensity as well.
Our restoration procedure, if concluded at this point, may already be useful for most practical applications. Let us note, however, that it does not match the standard definition of normalization, which is generally referred to the star's own intensity.
To proceed further, one should know independently the fractional
luminosities 
and 
of the two stars
(relative height of their continua
).
Assuming for simplicity's sake a small-enough spectral range 
where the luminosities of the two stars may be considered
constant, the two original intensity-normalized spectra are then:
Finally, 
and 
should be
correctly shifted in 
, to compensate for the orbital motion.
Such fully-restored spectra may be obtained, in principle,
for eclipsing binaries, where 
and 
can be
accurately measured by occultation spectrophotometry.
The algorithms introduced in this section have been verified by using tests and simulations, applied to artificial line-profiles and modelled binary spectra, which will be shown in Sects.3.1 and 3.2. Finally, an application to a true binary star will be presented in Sect.3.3.
Since the iterative procedure takes n steps, any spurious difference between the input data will be replicated n times (and possibly amplified) onto the processed spectra. For this reason, preliminary cosmetics is necessary, to correct CCD defects and/or to remove cosmic-ray spikes. Moreover, the two observational spectra must be carefully equalized before processing, in order to compensate for possible normalization discrepancies.
It is also convenient to contain the propagation of noise, by applying the minimum possible number of iterations, on those parts of the spectrum which do not need complex reconstruction. Generally, sharp metallic lines are already restored at the first step, while iterations may only reconstruct the profiles of broader lines (e.g. Balmer series).
Our method, as defined above, restores the function 
by
proceeding in the sense of increasing wavelength (that is rightward
in Fig. 2 (click here)). The quality of the extraction can be improved by applying the
symmetric procedure, that is by starting from the reversed difference-spectrum
, and using the opposite shift -k; so we can
reconstruct the function 
(performing a leftward
restoration). Thus, we can derive a new version of the primary spectrum
; and finally by
averaging (after shift compensation) the two versions of 
,
we may get a primary spectrum with reduced noise and smaller distortions.
Similarly, we can obtain another version for the secondary spectrum
, allowing us to compute
an improved mean-version also for 
.
Coaddition of multiple exposures, taken at a given phase, may enhance the S/N ratio. In the same way, by employing different pairs of input phases, more S-spectra can be obtained, providing independent reconstructions: their average will further improve the S/N value.
On the other side, problems may arise when attempting the restoration of cool stellar spectra crowded with overlapping lines, and having a poorly-defined continuum. Similarly, edge distortions may arise in the presence of truncated line-profiles at the extremes of the spectrum.
In synthesis, although the method may work also under unfavourable conditions, its full power is displayed only when applied to high-quality observational spectra. Such data are actually provided by state-of-the-art astronomical instrumentation, particularly concerning highly stable, sensitive and flawless receptors. Quality requirements will be even more compelling for the profile-analysis algorithm introduced below.
Our separation method uses the secondary shift k as an input
parameter (this also happens, incidentally, for other Doppler-based methods
mentioned in Sect.2.1). If some secondary lines are already evident on
the input spectra 
and 
, the case is trivial since k can
be directly measured. On the other side, difficulties may arise when the
secondary signal is revealed only after the compensation of primary
lines in the S-spectrum.
As shown in Fig. 1 (click here), only for non-merged S-profiles can one take the peak-to-peak distance s as representative of k. Generally, for all merging situations, the assumption k=s is misleading. As a result, the reconstructed lines risk becoming stronger than real (tests in Sect.3.1).
Given the astronomical importance of some situations of merged profiles, we think it is worthwhile studying a way of determining k independently, before performing the spectral reconstruction. Such a derivation of k may be required, for example, in the following cases: i) when only intermediate phases are available, and the relative Doppler shift may be not large enough to separate, in the S-spectrum, the dual images of metallic lines (gaussian approx.); ii) when the secondary star is so faint that its metallic lines are not detectable, and in the S-spectrum only overlapping HI lines are seen (lorentzian approx.).
The shape of the S-profile is studied analytically in the appendixA, where
a special algorithm is derived. It performs the conversion 
by assuming a gaussian (or lorentzian) approximation for the secondary lines.
As a straightforward result, the determination of k may allow the
computation of the masses of the two components A and B of the system.
For example, let us suppose that our S-spectrum is obtained by difference
from two opposite quadrature phases, correctly shifted to compensate
for the motion of primary lines; then k corresponds to twice
the secondary radial-velocity, plus twice the primary radial-velocity
(just compensated for). Precisely, for circular orbits, we have:
, where
and 
are the maximum radial-velocities (barycentric)
of the two stars. As 
is well measured from primary lines,
the knowledge of k will then provide a preliminary determination of
. This will give us the spectroscopic mass-ratio
which, combined with
other known parameters of the system (e.g., see Batten et al. 1989),
may finally provide the masses and other absolute elements of the binary star.