VLT multi-object spectroscopy of 33 eclipsing binaries in the Small Magellanic CloudBased on observations made with the FLAMES-GIRAFFE multi-object spectrograph mounted on the Kuyen VLT telescope at ESO-Paranal Observatory (Swiss GTO programme 072.A-0474A; PI: P. North)

# VLT multi-object spectroscopy of 33 eclipsing binaries in the Small Magellanic Cloud1

New distance and depth of the SMC, and a record-breaking apsidal motion
###### Key Words.:
stars: early type – stars: binaries: eclipsing – stars: binaries: spectroscopic – stars: fundamental parameters – galaxies: Magellanic Clouds – distance scale
2

## Abstract

Context:

Aims:Our purpose is to provide reliable stellar parameters for a significant sample of eclipsing binaries, which are representative of a whole dwarf and metal-poor galaxy. We also aim at providing a new estimate of the mean distance to the SMC and of its depth along the line of sight for the observed field of view.

Methods:We use radial velocity curves obtained with the ESO FLAMES facility at the VLT and light curves from the OGLE-II photometric survey. The radial velocities were obtained by least-squares fits of the observed spectra to synthetic ones, excluding the hydrogen Balmer lines.

Results:Our sample contains 23 detached, 9 semi-detached and 1 overcontact systems. Most detached systems have properties consistent with stellar evolution calculations from single-star models at the standard SMC metallicity , though they tend to be slightly overluminous. The few exceptions are probably due to third light contribution or insufficient signal-to-noise ratio. The mass ratios are consistent with a flat distribution, both for detached and semi-detached/contact binaries. A mass-luminosity relation valid from 4 to 18 is derived. The uncertainties are in the 2 to 11 % range for the masses, in the 2 to 5 % range for the radii and in the 1 to 6 % range for the effective temperatures. The average distance modulus is ( kpc). The moduli derived from the and from the data are consistent within  mag. The depth of the SMC is, for our field, of  mag or  kpc under the assumption of a gaussian distribution of stars along the line of sight. Three systems show significant apsidal motion, one of them with an apsidal period of 7.6 years, the shortest known to date for a detached system with main sequence stars.

Conclusions:

## 1 Introduction

Since the late 1990s, the usefulness of extragalactic eclipsing binaries has been emphasized in a number of papers. The reader can notably refer to the excellent reviews from Clausen ((2004)) and Guinan ((2004), (2007)). The two major contributions of eclipsing binaries (hereafter EBs) to astrophysics are to provide (1) fundamental mass and radius measurements for the component stars, allowing to test stellar evolution models, and (2) precise distance moduli (DMi) derived from the luminosities calculated from the combination of the absolute radii with the effective temperatures. Until a purely geometrical distance determination is feasible, Paczyński ((2001)) considers that detached EBs are the most promising distance indicators to the Magellanic Clouds. Besides, Wyithe & Wilson ((2002), hereafter WW02) remarked that semi-detached EBs are even more promising, since their parameters are better constrained.

The renewal of interest in extragalactic EBs, especially EBs in the Magellanic Clouds, has been stimulated by the release of a huge number of light curves as a byproduct of automated microlensing surveys (EROS, MACHO, OGLE) with 1-m class telescopes. As photometry is only half of the story, high resolution spectrographs attached to 4-m class or larger telescopes had to be used to obtain reliable radial velocity (RV) curves. Four B-type EB systems belonging to the Large Magellanic Cloud (LMC) were accurately characterized in a series of papers from Guinan et al. ((1998)), Ribas et al. ((2000), (2002)) and Fitzpatrick et al. ((2002), (2003)). More recently, from high resolution, high spectra obtained with UVES at the ESO VLT, the analysis of eight more LMC systems was presented by González et al. ((2005)). A corner stone in the study of EBs in the SMC was set up with the release of two papers from Harries et al. ((2003), hereinafter HHH03) and Hilditch et al. ((2005), hereinafter HHH05) giving the fundamental parameters of a total of 50 EB systems of spectral types O and B. The spectroscopic data were obtained with the 2dF multi-object spectrograph on the 3.9-m Anglo-Australian Telescope. To our knowledge this is the first use of multi-object spectroscopy in the field of extragalactic EBs. Recently, even the distance to large spiral galaxies were measured on the basis of two EBs in M31 (Ribas et al. (2005), Vilardell et al. (2010)) and one in M33 (Bonanos et al. (2006)).

The huge asymmetry, between the number of EB light curves published so far and the very small number of RV curves, is striking. If one considers the SMC, the new OGLE-II catalog of EBs in the SMC (Wyrzykowski et al. 2004) contains 1350 light curves and currently only 50 of these systems have moderately reliable RV curves. This paper reduces a little this imbalance by releasing the analysis of 28 more EB systems plus revised solutions for 5 systems previously described by HHH03 and HHH05. The RV measurements were derived from muti-object spectroscopic observations made with the VLT FLAMES facility.

Another strong motivation for increasing the number of fully resolved binaries is to settle the problem of the distribution of the mass ratio of detached binaries with early B primaries. Recently, two papers were published supporting two diametrically opposed conclusions: van Rensbergen et al. ((2006)), whose work is based on the 9th Catalogue of Spectroscopic Binaries (Pourbaix et al. (2004)), support the view that the -distribution (where is the mass ratio) follows a Salpeter-like decreasing power law; however, from the examination of the homogeneous sample of the 21 detached systems characterized by HHH03 and HHH05, Pinsonneault & Stanek ((2006)) draw the conclusion that the proportion of close detached systems with mass ratio far outnumbers what can be expected from either a Salpeter or a flat -distribution (the “twins” hypothesis). Finally, let us mention that the -distribution of semi-detached (i.e. evolved) systems is no more settled, the statistics strongly depending on the method used to find the mass ratios, i.e. from the light curve solution or from SB2 spectra (van Rensbergen et al. (2006)).

Although the controversy about the characteristic distance to the SMC seems to be solved in favour of a mid position between the “short” and the “long” scales, distance data and line of sight depth remain vital for comparison with theoretical models concerning the three-dimensional structure and the kinematics of the SMC (Stanimirović et al. (2004)).

Our contribution provides both qualitative and quantitative improvements over previous studies. Thanks to the VLT GIRAFFE facility, spectra were obtained with a resolution three times that in HHH03/05’s study. Another strong point is the treatment of nebular emission. The SMC is known to be rich in H ii regions (Fitzpatrick (1985), Torres & Carranza (1987)). Consequently, strong Balmer lines in emission are very often present in the spectra of the binary systems under study. Therefore, it appeared rapidly that it was essential to find a consistent way to deal with this “third component” polluting most double-lined (SB2) spectra.

We present the observations in Section 2. The reduction of the spectroscopic data and the interpretation of both photometric and spectroscopic data are described in Section 3, where the errors are also discussed in detail. The individual binary systems are described in Section 4, while the sample as a whole is discussed in relation with the SMC properties in Section 5.

## 2 Observations

The targets, astrometry included, were selected from the first OGLE photometric catalog. The GIRAFFE field of view (FoV) constrained to choose systems inside a 25′-diameter circle. Other constraints were mag, at least 15 well-behaved detached light curves (for the SMC field) and finally seven bump cepheids in the FoV (for another program). The positions in the sky of 33 objects studied in this paper are shown in Fig. 1. The epoch, exposure time, air mass, seeing and age of moon for each of the 16 CCD frames are gathered in Table 1.

The relation between our own 1–33 labeling and the OGLE names can be found in Table 3, which lists the basic parameters of the systems. The coordinates are from Wyrzykowski et al. ((2004)). The orbital periods and epochs of the primary minimum are close to those listed by Wyrzykowski et al. ((2004)), but the periods were improved as far as possible using the radial velocity curves determined in this work. That was worth the effort, since spectroscopic observations were performed more than three years after the last photometric ones. Since the times of the photometric minima are quite sharply defined, the uncertainty on the period (mentioned between parentheses in Table 3) is based on the uncertainty of the spectroscopically defined epoch of the primary minimum. The latter is quite precise for circular orbits; for eccentric orbits, it is less accurate, because the Kepler equation had to be solved and the solution is affected by the uncertainty on the eccentricity. We have decided here to adopt the dynamic definition of the primary and secondary components, rather than the photometric one. In other words, the primary component is always the more massive one, and the primary minimum always corresponds to the eclipse of the primary by the secondary component. As a consequence, it may happen that the so-called primary minimum is not the deeper one. Figure 2 shows the histogram of the periods. The strong observational bias in favour of short periods is conspicuous.

For all but two binaries, the light curves come from the new version of the OGLE-II catalog of eclipsing binaries detected in the SMC (Wyrzykowski et al. (2004)). This catalog is based on the Difference Image Analysis (DIA) catalog of variable stars in the SMC (see http://sirius.astrouw.edu.pl/ogle/ogle2/ smc_ecl/index.html). The data were collected from 1997 to 2000. The systems 4 121084 and 5 100731 were selected from the first version of the catalog (standard PSF photometry) but for an unknow reason they do no appear anymore in the new version. Nevertheless, they were retained as there is no objective reason to exclude them.

The DIA photometry is based on -band observations (between 202 and 312 points per curve). and light curves were also used in spite of a much poorer sampling (between 22–28 points/curve and 28–46 points/curve in and respectively). To give an idea of the accuracy of the OGLE photometry, the objects studied in this paper have an average magnitude and scatter in the range to . These values were calculated from the best-fitting synthetic light curves. For the two other bands, we get to for and to for . The quality of an observed light curve can be better expressed by comparing the depth of the primary minimum to the average RMS scatter . These ratios are shown in Table 16. This permits to classify the light curves in five categories: low (), low-to-medium (), medium (), medium-to-high () and high () quality. According to this scheme, most band light curves (58%) belong to the low-to-medium and medium quality categories, one-third (33%) in the medium-to-high and high quality categories, and the remaining 9% in the poor quality category. This classification scheme is not useful for the other bands, the low sampling being the limiting factor.

VLT FLAMES/GIRAFFE spectroscopy was obtained by one of us (FR) during eight consecutive nights from November 16 to 23, 2003. The spectrograph was used in the low resolution (LR2) Medusa mode: resolving power , bandwidth  Å centered on  Å. The most prominent absorption lines in the blue part of early-B stars spectra are: H, He i , H, He i , H, He i , and He i . For late-O stars, He ii and He ii gain in importance. Two fields, one in the SMC and one in the LMC, were alternatively observed at a rate of four exposures per night with an integration time of 2595 s for all but one epoch. Therefore, 16 spectra per target were obtained, with a total of 104 targets in the SMC and 44 in the LMC. The LMC SB2 systems are being analyzed and will be presented in another paper.

Beside the spectra of the objects, 21 sky spectra were obtained for each exposure in the SMC. The parameters related to the spectroscopic observations are gathered in Table 1. The observed signal-to-noise ratios () were determined for each smoothed spectrum (see Section 3.1) in the continuum between 4195 and 4240 Å. For each object, two values are presented in Table 11: the highest and lowest values for an exposure of 2595 s. Not surprisingly, the short exposure of 707 s (due to a technical problem) was useful for the brightest objects only. For a given binary with s, the ratio of the highest to the lowest is 2.

## 3 Data reductions and analysis

### 3.1 Spectroscopic data reduction

The basic reduction and calibration steps including velocity correction to the heliocentric reference frame for the spectra were performed with the GIRAFFE Base Line Data Reduction Software (BLDRS) (see http://girbldrs.sourceforge.net). Sky subtraction, a critical step for faint objects, was done as follows: for each epoch an average sky spectrum was computed from the 21 sky spectra measured over the whole FoV. For a given epoch the sky level was found to vary slightly across the field, but interpolating between spectra was not considered a valuable alternative. Local sky variations with respect to the average spectrum are given in Table 8. The values are read as follows: for example, the sky position labeled S19 is on average (i.e. over all epochs) 20% brighter than the mean (i.e. over all sky positions) sky spectrum. The variations were found to be between about and %. Normalization to the continuum, cosmic-rays removal and Gaussian smoothing ( = 3.3 pix) were performed with standard NOAO/PyRAF tasks.

The first 60 Å of the spectra, i.e for wavelengths between 3940 and 4000 Å, were suppressed. The reason is that below 4000Å the is getting very poor and therefore there is no reliable way to place the continuum. Furthermore, the region around H was found to be strongly contaminated by the interstellar Ca ii H and K absorption lines. The last few  Å (above 4565 Å) were equally found unusable because of a strongly corrupted signal.

### 3.2 Analysis

For historical reasons, the analysis has been made in essentially two steps.

First, RG did a complete analysis of all systems, using the KOREL code (Hadrava (1995), (2004)) to obtain both the radial velocity curves and the disentangled spectra of the individual components. Then, the simultaneous analysis of light and RV curves was made with the 2003 version of the Wilson-Devinney (WD) Binary Star Observables Program (Wilson & Devinney (1971); Wilson (1979), (1990)) via the PHOEBE interface (Prša & Zwitter (2005)). However, simulations performed following the referee’s request regarding this early version of the work, showed that the amplitude of the RV curve and the mass ratio were not recovered with the expected robustness. More details about these simulations are given below (Section 3.5.1). Although, on average, the right values are recovered, one particular solution may be off by as much as five percent (one sigma) or ten percent (two sigma), which was deemed unsatisfactory3.

Thus, a second, almost independent analysis was made by PN, using a least-squares fit of synthetic binary spectra to observed unsmoothed spectra for the RV determinations. The latter technique seemed much more robust, according to the same simulation: the and amplitudes are recovered to better than one percent – at least for the particular binary that was simulated – and the dispersion of the values of the small eccentricity () is no larger than two percent. Small systematic errors may result from temperature or rotation mismatch, but they remain smaller than the uncertainties of the previous analysis.

We took the opportunity of this new analysis to define a more objective determination of the effective temperatures of the components. Instead of a visual fit of a spectrum close to quadrature, we used a least-squares fit of synthetic binary spectra to all observed spectra falling out of eclipses. Then, the error on the effective temperatures could be naturally defined as the RMS scatter of the results. More details are given in the next sections and in the following discussion of individual binaries.

### 3.3 Photometry: quality check

The quality of the band light curves was discussed in Section 2. Despite the high range of RMS scatter, we can expect a very accurate determination of the out-of-eclipse magnitude because of the large number of data points (300). This is not the case with the and bands. A much poorer sampling can lead to erroneous zero-level computation in the light-curve analysis step and result in wrong colour-index determination. Therefore, it is necessary to perform a quality check of the photometric data. This was done in the form of colour-colour diagrams of our sample. Figure 3 presents the three diagrams that can be obtained from the three colour indices , and . These are the values at quadratures, i.e an average value characterizing a “hybrid” star of intermediate properties with respect to the two components of a particular system. Not surprisingly, most binaries are found on a relatively narrow linear strip. For any diagram, the scatter of the objects is low because the reddening line is almost parallel to the sequence. For example, the ratio , determined from Eq. 11 in Section 3.12, is close to 0.69, the slope of the sequence vs. . Nevertheless, four outliers appear, which are marked with open symbols. In principle, there is a possibility to restore a bad colour index, as illustrated by the example of the system 5 261267: Inspecting the three light curves, one can suspect that the cause of the discrepancy lies in a poorly sampled band light curve in the out-of-eclipse domain. The other two light curves ( and ) seem more reliable. Therefore, only the colour-index is reliable for this system. But the two other indices, and , can safely be interpolated from the value under the assumption that the system lies on the linear strip. The method is illustrated by the dashed lines in the diagrams. Of course, this reconstruction of two bad indices from a good one is not possible, unless only one of the three light curves is unreliable (either or ). In this particular example, the situation is not so clear-cut because the reconstructed indices ( and ) would imply too low a value. Therefore, all four outliers will be excluded in the final estimate of the distance modulus.

On the upper right diagram of Fig. 3, red crosses indicate the intrinsic colours computed below (Section 3.11). Their positions, slightly below the regression line of the sequence, is entirely compatible with the reddening arrow and the observed colours, which inspires confidence regarding the colour excesses determined in Section 3.11.

### 3.4 Synthetic spectra

Except for the first two steps of the analysis, i.e. (a) the simultaneous disentangling of the composite spectra and retrieving of the RV curves through the KOREL code (Hadrava (1995), (2004)) and (b) the simultaneous analysis of the light and RV curves through the WD/PHOEBE code, the search for the parameters of a binary relies heavily on synthetic spectra. Indeed, the systemic velocity, the projected rotational velocities, the ratio of radii and the primary temperature are found by comparing the observed spectra (disentangled and composite as well) with a library of synthetic spectra. Actually, two libraries were used. For objects with effective temperature , we used the BSTAR2006 library recently released by Lanz & Hubeny ((2007)). For the few objects above , we took the OSTAR2002 library previously released by the same authors ((2003)). Both libraries are available at the TLUSTY Web site (verb+http://nova.astro.umd.edu/index.html+).

The grids with a metallicity suitable for the SMC, , i.e. one-fifth of the solar metallicity, were chosen. Concerning the BSTAR2006 library, we took the grid with a microturbulent velocity of 2 km s. Both grids of spectra are based on NLTE line-blanketed model atmospheres. The temperature step is 1000 K below and 2500 K for early-O stars. We restrained to surface gravities (0.25 dex step) . The spectra were convolved with the appropriate rotational profiles ( = 30, 40, 50, 75, 100, 150, 200, 250 and 300 km s) and with a Gaussian instrumental profile (resolution of 0.67 Å) by mean of the program ROTIN3 (http://nova.astro.umd.edu/Synspec43/synspec-frames-rotin.html). Beside a grid of continuum normalized spectra, a grid of flux spectra was generated for colour indices calculation through synthetic photometry.

Formally, a normalized synthetic composite spectrum is computed at a given orbital phase from the normalized component spectra , the radial velocities of both components and the light dilution factors :

 Fnclnλ(ϕ)=S∑j=Pℓj,clnλ(ϕ)Fnj,clnλ∗δ(clnλ−Vj(ϕ,\@vecp)) (1)

is the Dirac function and is the set of orbital parameters. The spectra are expressed in the logarithmic wavelength scale. The light dilution factors were calculated from light-curve modeling (Ilijić et al. (2004)):

 ℓP,λ(ϕ)=11+LλϵS(ϕ)ϵP(ϕ)andℓS,λ(ϕ)=1−ℓP,λ(ϕ) (2)

where , the flux of component in a given passband normalized by the value at quadrature, accounts for possible out-of-eclipse variability due to departure from sphericity and reflection effects (Fitzpatrick et al. (2003)); is the ratio of the continuum monochromatic luminosities, i.e. the ratio of the mean surface brightnesses times the ratio of radii,

 ϵj(ϕ)≡fj,λ(ϕ)fqj,λandLλ≡LS,λLP,λ=JS,λJP,λ(RSRP)2 (3)

We checked that can safely be considered constant through the  Å spectral interval and can be safely identified with the luminosity ratio in the Cousins -band of effective wavelength  Å.

As explained above (Section 3.2), the radial velocities were determined using two independent methods. As a first step, a disentangling method was used, which allowed to give a first estimate of the parameters of all systems. As a second and final step, a least-squares method was used. Each step is described in turn below.

#### First step: spectrum disentangling

Simon & Sturm ((1994)) were the first to propose a method allowing the simultaneous recovery of the individual spectra of the components and of the radial velocities. Another method aimed at the same results, but using Fourier transforms to save computing time, was proposed almost simultaneously by Hadrava ((1995)). The advantages of these methods are that they need no hypothesis about the nature of the components of the binary system, except that their individual spectra remain constant with time. Contrary to the correlation techniques, no template is needed. In addition to getting at once the radial velocities and orbital elements, one gets the individual spectra of the components (“disentangling”), with a signal-to-noise ratio which significantly exceeds that of the observed composite spectra. For instance, in the case of a binary system with two components of equal brightness and observed 16 times with , the signal to noise ratio of each disentangled spectrum would be (the factor being due to the fact that there are two components). That is important, because the nature of the components can then be determined safely. In this work we use this advantage to determine the effective temperatures of some components, but for brighter binaries observed at higher resolution and S/N, it would also be possible to determine photospheric abundances. Other details about these techniques and their applications (including abundance determinations) can be found in e.g. Hensberge et al. ((2000)), Pavlovski & Hensberge ((2005)) and Hensberge & Pavlovski ((2007)).

The radial velocities were determined from the lines of He i (4471, 4388, 4144, 4026) only. We preferred to avoid the H Balmer lines (as did Fitzpatrick et al. (2002)) because (1) their large width make them more sensitive to systematics due e.g. to wrong placement of the continuum, and (2) because of moderate to strong nebular emission polluting most systems (only 6/33 systems were found devoid of any emission). Consequently, four regions with a width of 80 Å centered on the four He i lines were cut from each spectrum of the series. Since the KOREL code makes use of the Fourier transform of the spectra, both edges of each spectral region were fixed to 1 by hammering of the signal to 1 with a cosine bell function (Hanning window). KOREL was run with out-of-eclipse spectra only, although the line-strength factors, i.e. the contributions of each component to the system continuum, could be obtained in principle as results of the KOREL analysis. However, most of our spectra were found to have a too low to provide reliable results. Therefore, the selection was performed using the out-of-eclipse phase ranges given by the light curves. The period was taken from Table 3, which are slightly improved values relative to those of Wirzykowski et al. ((2004)), as explained in Section 2. A first estimate of the epoch of periastron passage , of the eccentricity and of the longitude of periastron were determined from the light curves. In the case of eccentric systems, a first solution was found neglecting apsidal motion. The only orbital parameters allowed to converge were the primary semi-amplitude and the mass ratio . For each system, KOREL was run with a grid of values . The solution with the minimum sum of squared residuals as defined by Hadrava ((2004)) was retained as the best solution. For eccentric systems, a second run was performed letting , , and free to converge ( being determined by photometry). It is important to notice that the four spectral regions were analyzed simultaneously, i.e in a single run of KOREL. Each region was weighted according to the of each He i line (). To circumvent the difficulty to measure s inside the lines, they were estimated from the values calculated with the GIRAFFE Exposure Time Calculator of the ESO. The calculated values were then normalized to the measured value between 4195 and 4240 Å. The non-Keplerian correction and Rossiter effect were calculated from the WD/PHOEBE solution.

Beside the simultaneous retrieving of RV curves, orbital parameters and disentangled spectra, the KOREL code is able to disentangle spectra for a given orbital solution (, , and fixed). A final run of KOREL with this mode was then used to disentangle the regions around the Balmer and He ii 4200 and 4542 lines. Indeed, He ii lines and a number of Si iii-iv lines are very useful to set the temperature of hot components.

#### Testing the robustness by simulations

In order to test possible biases on the determination of and by KOREL, we have simulated ten sets of nine out-of-eclipse composite spectra of the system 5 266131. We used the fitted radial velocity curves to shift the synthetic spectra of each component and add them at the observed phases, using the adopted luminosity ratio in the band. We used the synthetic spectra with parameters closest to the observed ones, from the grid of Munari et al. ((2005)). A Gaussian noise was added to the composite synthetic spectra, such that the signal-to-noise ratio varied from 37 to 71 (see Table 11) and assuming that the ratio is inversely proportional to the seeing given in Table 1. The KOREL code was run on each of these ten simulated datasets, and the averages of the resulting and values were computed. We found and instead of the input values and respectively. The simulation then gives instead of the input . Thus, the parameters obtained from the simulated spectra agree with the input value to within , so that no significant systematic error is to be feared. However, the RMS scatter of the and parameters proved disappointingly large, about and respectively. This means that the uncertainty on the RV amplitude reaches about 5%, which translates into 15% for the masses.

#### Second step: least-squares RV determination

On the basis of the first analysis, we selected two synthetic spectra from the OSTAR2002 and BSTAR2006 libraries for the two components of each system, with the parameters closest to the estimations. A chi-square was computed as the quadratic sum of the differences between the observed spectrum and the composite synthetic one, for arbitrary radial velocities. However, we did not use the complete spectra: the hydrogen Balmer lines were suppressed because of their large width and because they are mixed with nebular emission in a number of cases. A SuperMongo (Lupton & Monger (2000)) procedure implementing the amoeba minimization algorithm was used, letting the two radial velocities and the blue intensity ratio free to converge. The radial velocities are essentially constrained by five He i lines (, 4026, 4144, 4388 and 4471). Convergence was generally fast and robust, in the sense that the results did not depend on the initial guess values. Some iterations were necessary, however, to clearly identify the primary and secondary components, so that the right model be attributed to the right component.

A preliminary analysis of the radial velocities was then performed using an interactive code (Lucke & Mayor (1980)), which allowed to assess the quality of the RV curves (especially the RMS scatter of the residuals) and obtain first orbital elements.

#### Simulations

The same ten sets of nine composite synthetic spectra, described above, was used to test the least-squares method of RV determination. The results proved very encouraging, since they follow distributions, the sigma of which amount to only 0.8 and 0.6% of the means, for the amplitudes and respectively. The sigma of the eccentricity distribution is 1.8% of , and the argument of the periastron has .

The effect of a mismatch was tested by using template spectra with effective temperatures smaller by 3300 K, respectively 4080 K for the primary and secondary components, compared to the temperatures used to build the artificial ”observed” spectra (the projected rotational velocities were also smaller by about 40  km s ). The amplitudes changed by , resp. % only for the primary and secondary components. Increasing the temperatures by 2700, resp. 1860 K (and the by 40  km s ) lead to relative differences % and %. Thus, the mismatch that can be expected will not induce systematic errors much larger than about one percent, which is in general smaller than the random error. In view of its robustness, we have adopted the least-squares technique for RV determination, rather than the results of the KOREL code. However, we are aware that the above comparison between the two techniques may not be quite fair, because the KOREL code recovers the individual spectra from the data, while the least-squares fit uses external template spectra. In that sense the advantage of the least-squares fit may prove somewhat artificial.

### 3.6 Apsidal motion

The WD code allows to determine the time derivative of the argument of the periastron. That possibility was used for all eccentric systems but one (4 175333), the latter having the smallest eccentricity of all. The systems for which a significant apsidal motion was found in that way were examined further by subdividing the photometric data into four consecutive time series, and examined using an interactive version of the EBOP16 code (Etzel (1980)). The angle (if precise enough) and the quantities ajusted for each time series were then examined for systematic variation. That allowed to better visualize the effect of apsidal motion on the light curves, and to assess better its significance.

### 3.7 Wilson-Devinney analysis

#### First step

For each system, a preliminary photometric solution had been found (before taking thre radial velocities into account) by the application of the method of multiple subsets (MMS) (Wilson & Biermann (1976)). The groups of subsets used were essentially the same as those advocated by Wyithe & Wilson ((2001), (2002)) (Table 2). That allowed to provide fairly precise values of and that were then introduced in the KOREL analysis. Then, all three light curves and both RV curves provided by KOREL were analyzed simultaneously using the WD code. That does not imply, however, that photometric and spectroscopic data were analyzed in a really simultaneous way, since results from the preliminary light curve analysis were used in the KOREL analysis; it is rather an iterative analysis. The light curve is the most constraining one, thanks to the large number of points, but the and light curves are very important too, since they provide accurate out-of-eclipse and magnitudes. The mass ratio was fixed to the value found by KOREL. The semi-major orbital axis , treated as a free parameter, allows to scale the masses and radii. In a first run, the temperature of the primary was arbitrarily fixed to 26 000 K. Second-order parameters like albedos and gravity darkening exponents were fixed to 1.0. Metallicities were set at . The limb-darkening coefficients were automatically interpolated after each fit from the van Hamme tables (van Hamme (1993)). The code needs an estimation of the standard deviations of the observed curves in order to assign a weight to each curve. These ’s were calculated from the sums of squares of residuals of the individual curves, as advocated by Wilson and van Hamme ((2004)). These values were refined for subsequent runs.

A fine tuning run was performed with the primary temperature found after analyzing the observed spectra. The standard uncertainties on the whole set of parameters were estimated in a final iteration by letting them free to converge.

The standard procedure described above is sufficient for symmetric light curves only. For systems displaying a small depression before the primary minimum, as it is occasionally the case with semi-detached systems, it is necessary to introduce a cool spot on the primary component. Obviously, this step is performed after obtaining the symmetric best-fit solution. The spot is characterized by four parameters, i.e. colatitude, longitude, angular radius and temperature factor. As the observed feature can be described by a large number of combinations of the four parameters related to the spot (high degeneracy), the spot was arbitrarily put on the equator of the primary (i.e. colatitude of ) and the three other parameters were optimized alternately following the MMS. In case of high propensity to diverge, one of the three free parameters was set to an arbitrary value, the MMS being performed on the two remaining parameters.

In this first step, the WD analysis was performed using the photometric convention, according to which the primary star is the one that is eclipsed near phase zero, i.e. the star with the higher mean surface brightness in a given passband (). It followed that in some cases may not necessarily be . In this paper, we have finally adopted the dynamic convention in order to avoid confusion.

For the detached systems, the orbit was considered as circular when the eccentricity given by the WD code was smaller than its estimated error.

#### Second step

Both photometric and RV curves were analyzed simultaneously, fixing the effective temperature of the primary component to the spectroscopic value (see below for the determination of the latter). For semi-detached and contact systems, there is no need to fix any other parameter. For detached systems, however, the ratio of radii is very poorly constrained by photometry alone when the eclipses are partial, which is the case of all detached systems in our sample. Therefore, we adopted the brightness ratio determined by spectroscopy, and fixed the potential of the primary, , to a value such that the brightness ratio in the blue band matched the spectroscopic one within the uncertainties. The potential depends on both radius and mass, but the latter is constrained by the RV curve, so that fixing a potential is equivalent to fixing a radius. In some cases it was not possible to reproduce the spectroscopic brightness ratio without degrading the photometric fit, so we gave priority to the latter.

### 3.8 Systemic velocity and projected rotational velocities

#### First step

The component spectra of the four regions centred on the He i lines were normalized with the help of the KORNOR program (Hadrava (2004)). The systemic velocity was found from the disentangled spectra of the four regions centred on the He i lines. The observed spectra were cross-correlated via the IRAF task against synthetic spectra computed for the estimated , and . The values gathered in Table 11 were obtained as the -weighted averages of the individual velocities calculated for each line.

The projected rotational velocities, , were tentatively measured by calibrating the FWHMs of the He i lines against a grid of (FWHM, ) values obtained from synthetic spectra (Hensberge et al. (2000)). The FWHMs were computed from Gaussian or Voigt profiles fitting via the IRAF task. The values retrieved by this method were often found unsatisfactory when comparing observed and synthetic spectra retrospectively. The problem proved to lie in the high sensitivity of the FWHM measurement to the continuum placement. Therefore, a synchronous rotational velocity was assumed for most circular binaries unless profile fitting proved this hypothesis wrong. Anyway, this assumption is certainly justified for short-period systems, i.e. binaries with (North & Zahn (2003)) where the ratio is the star radius divided by the separation. In the case of eccentric systems, pseudo-synchronization was assumed (Mazeh (2008), Eq. 5.1). For a given star, its pseudo-synchronous rotational velocity is computed from its radius and the pseudo-synchronization frequency of the binary. This equilibrium frequency, close to the orbital periastron frequency, is given in Hut ((1981)).

#### Second step

Contrary to the first step, when the KOREL code was used, we do not need to define the systemic velocity a posteriori here. The least-squares method directly provides “absolute” radial velocities (i.e. not only relative ones), even though mismatch might bias them by a few km s. Thus the systemic velocity naturally flows from the WD analysis, which includes the RV curves.

As in the first step, rotational velocities were derived from the assumption of synchronous (for circular orbits) or pseudo-synchronous (for eccentric orbits) spin motion. No clear departure from this assumption could be seen on the spectra.

### 3.9 Spectroscopic luminosity and ratio of radii

#### First step

As mentioned above, and as emphasized repeatedly by Andersen (e.g. (1980)) and rediscovered by Wyithe & Wilson ((2001), hereafter WW01), the ratio of radii of an EB displaying partial eclipses is poorly constrained by its light curve. The ratio of monochromatic luminosities is equally not well recovered in fitting light curves of simulated EBs (i.e. EBs with previously known parameters). On the contrary, the surface brightness ratio and consequently the derived effective temperature ratio is, in general, reliably recovered. The sum of the radii is also very well constrained. The poor constraining of is very well illustrated by Fig. 3 in González et al. ((2005)).

Since our sample comprises only systems with partial eclipses, , then must be determined in order to find reliable radii and surface gravities. We followed the procedure described in González et al. ((2005)). The ratio of the monochromatic luminosities can be expressed by Eq. 4 (Hilditch (2001)):

 Lλ = (WSλWPλ)obs(WPλ(TPeff,loggP,ZP,YP)WSλ(TSeff,loggS,ZS,YS))true (4) = (RSRP)2(TSeffTPeff)410−0.4(BCPλ(TPeff)−BCSλ(TSeff))

where is the radius, the effective temperature and the bolometric correction in the -band of component . In our case, and the bolometric correction in the -band is given by , where the bolometric correction in the -band is interpolated from values given by Lanz & Hubeny ((2003), (2007)) and the intrinsic colour index is computed from synthetic photometry. is the equivalent width (EW) of line for component . The ‘obs’ index means that the EWs are measured in an observed (out of eclipse) spectrum of the binary. These “apparent” EWs are then normalized by the true values measured in synthetic component spectra. The dependence of the true EWs on the effective temperature, surface gravity, metallicity and helium abundance () is emphasized.

In principle, from the analysis of a set of four lines, knowing the masses, metallicities and helium abundances, it should be possible to derive a purely spectroscopic solution for the two radii and the two effective temperatures. If we further assume that the sum of the radii is known from the light-curves analysis as well as the temperature ratio, the analysis of only two lines is in principle sufficient to determine a mixed photometric-spectroscopic ratio of radii. However, since EWs are sensitive to a possible continuum misplacement, we preferred to fit the observed line profiles with a synthetic composite spectrum or to determine the ratio of the EWs of two different lines in the same component. The latter methods are more reliable than blind application of Eq. 4 to estimate the effective temperatures, at least in case of spectra with moderate .

For a given chemical composition, true undiluted EWs depend on both the effective temperature and the surface gravity of the stars. Moreover, this dependence is not always monotonic even if we restrict to late O to early B stars, the He i lines having a maximum strength at 20 000 K. Consequently, in order to avoid the hassle of working with non-explicit equations, for a given line, Eq. 4 was solved with the photometric temperature ratio and the true EWs values corresponding to the photometric and a first guess of . is then used with and a first guess of to compute the ratio of radii . Combining and , the new and values are obtained straightforwardly. The small error introduced in the chain because of using approximate values for the true EWs could be removed after iterating one more time.

Nevertheless, this method is not very efficient when the observed EWs have large uncertainties as in case of a composite spectrum of low . In this case, a more pragmatic approach consists in optimizing both and in a single step by looking for the best-fitting synthetic composite spectrum for a given pair (, ) and the and constraints.

#### Second step

Here the luminosity ratio in the blue is simply one of the three parameters determined by the non-linear least-squares algorithm amoeba, the other two parameters being the effective temperatures (see more details below, Section 3.10). So the luminosity ratio is determined in a very homogeneous way, and an error estimate naturally arises through the RMS scatter of the resulting values. This does not guarantee, however, that the results are free from any bias. In particular, one may suspect that, in the temperature regime where the strength of all lines (H and He ones) vary in the same way with temperature, some degeneracy may arise between the temperatures and the luminosity ratio. Since that temperature regime spans roughly from to , it means that the luminosity ratio of the majority of systems may be fragile. Nevertheless, a posteriori examination of the resulting HR diagrams does not confirm this fear, even though a few systems fail to match the evolutionary tracks.

### 3.10 Effective temperatures

Once reliable and values have been found, a way for setting the temperature of the primary must be found (the temperature of the secondary is a by-product via the photometric temperature ratio).

For late O and early B stars, the H and H Balmer lines are far better temperature indicators than He i lines (Huang & Gies(2005), (2005)). Therefore, the most direct way to determine the effective temperatures of both components of a given system would consist in calibrating the equivalent widths measured on the normalized disentangled spectra with those obtained with a library of synthetic spectra. Unfortunately, this is not always possible because of the high proportion of systems contaminated by H and H nebular emission lines. Thus, most spectra of individual components are not reliable around the Balmer lines.

#### First step

A safer method consists in comparing an observed composite spectrum close to quadrature with a synthetic composite spectrum computed at the same orbital phase. The spectra retained for the temperature determination are those with or . This method is quite sensitive to the continuum placement. A low and/or strong emission lines can hinder a reliable profile fitting.

Another method is a variant of the traditional spectral type vs. temperature calibration. In the traditional method, the line strengths ratio of two lines are measured and compared to the values obtained from a series of reference spectra whose spectral types are known. The effective temperature is then found via a spectral type - calibration scale. As emphasized by HHH03/05, this technique is efficient for O-B1 stars but far less straightforward for later types. Above this limit, the relative strength of the He ii 4542 and Si iii 4553 lines is a reliable tool, as is the relative strength of the Si iv 4089-4116 and He i 4121 lines. For temperatures below 29-30 000 K, the problem is the lack of exploitable metallic lines. Unfortunately, the faint Si ii lines are totally undetectable. For later B stars, the only detectable metallic line is Mg ii 4481, but this line is often severely buried in the noise for most components.

#### Second step

The method is the same, qualitatively speaking, as that of the first step, consisting in fitting a composite synthetic spectrum to the observed one near quadrature. However, the least-squares fit method allows in principle to use all out-of-eclipse spectra, and provides a much more objective estimate of the temperatures. Since the radial velocities are known, the only parameters which have to be fit are the effective temperatures of both components and the blue luminosity ratio. As mentioned in Section 3.9.2, the fit is quite robust at both ends of the temperature range of our sample: at the cool end, the He i lines increase in strength with temperature, while the H i Balmer lines decrease; in addition, the Mg ii line decreases very fast. At the hot end, both H and He i lines decrease with temperature, but the He ii lines begin to appear. In the intermediate range, all lines vary more or less in parallel, which may lead to degeneracy when the S/N ratio is poor.

For all systems showing a significant nebular emission in the core of the hydrogen Balmer lines, we simply removed a  Å  wavelength interval centered on the emission line, in both observed and synthetic composite spectra. But, contrary to the synthetic spectra used for RV determinations, here we include the H Balmer lines in the fit, except for their very centres.

The RMS scatter of the fitted effective temperatures is typically of the order of  K, and is even smaller than that for one third of the sample. Although formally, the error bar on should be set to that scatter divided by the square root of the number of spectra, we chose to put it equal to the scatter itself. Indeed, visual examination of the observed and model spectra show that the temperature effect is often very subtle, so we feel this choice is more realistic.

The fit proved to depend somewhat on the normalization of the spectra. The latter were first normalized using an automatic procedure with fixed continuum regions. Then, another automatic procedure was used, which corrected the first normalization with the help of a pair of synthetic spectra with preliminary stellar parameters. That normalization resulted in a slightly higher continuum and was found satisfactory in general, except for the bluer end of the spectrum. A final normalization was made by hand, which was adopted in most cases, but not in all because the continuum proved sometimes too high. The temperature determination was run on the automatically normalized spectra as well as on the manually normalized ones. The results were found to depend little on the normalization, which was not unexpected since the last two normalizations did not differ much from one another.

The average temperatures were computed on all out-of-eclipse spectra on the one hand, and on a selection of those for which the RV difference is greater than 300  km s (250 or even 200  km s for longer period systems) on the other hand. The selection often resulted in a smaller scatter of the temperature, though not in every case. The average temperature was weighted with the inverse of the chi-square provided by the amoeba procedure.

### 3.11 Synthetic photometry and reddening

Intrinsic colour indices are needed for two purposes: the computation of the colour excess for a given system and the computation of the band bolometric corrections of the individual components. In the first case, is phase-dependent and characterizes the binary as a whole, while in the second case is the usual (constant) colour characteristic of a given star. Both types of colour indices were computed from synthetic photometry, i.e. from synthetic stellar spectra and the response functions of the filters. The general formula for the phase-dependent of a binary is given by

 (B−V)0(ϕ)=−2.5log∫SBF0λ⊕(ϕ)dλ∫SVdλ∫SVF0λ⊕(ϕ)dλ∫SBdλ+CB−V (5)

where is the response function of the band filter (Bessell (1990)), is the synthetic composite flux spectrum at a given epoch and mag is the zero point (Bessell et al. (1998)). The synthetic composite spectra (i.e. the theoretical unreddened flux received on Earth) were calculated from the synthetic component spectra (i.e. the surface fluxes), the radial velocities and the radii :

 F0λ⊕(ϕ) = 1d2S∑j=PR2jF0j,λ∗δ(clnλ−Vj(ϕ,\@vecp)) (6) = (RPd)2S∑j=PwjF0j,λ∗δ(clnλ−Vj(ϕ,\@vecp))

where and . Substituting in Eq. (5) by the expression given by Eq. (6), one sees that Eq. (5) does not depend on the distance to the system.

A similar procedure was used to compute the colour indices, taking and the appropriate response functions and . This index is needed for the determination of the distance modulus from the band photometric observations.

Alternatively, the intrinsic colour index could be estimated via a colour-temperature relation from the literature. This kind of calibration being established with stars from the solar neighborhood (e.g. Flower (1996)), the coefficients of the fit are in turn representative of stars with a solar metallicity. Consequently, synthetic photometry with spectra was considered more reliable for our objects. Both and were computed for a grid of temperatures and surface gravities. Because the fluxes are not given beyond  Å  in the OSTAR2002 library (which contains fluxes for stars hotter than 30 000 K), we had to complement the flux distribution of stars hotter than  K by the appropriate Kurucz fluxes4 in order to be able to compute the index. We chose the grid with a metallicity of . Because of that inhomogeneity, the indices of binary systems hosting components with  K may be slightly less reliable than those of the other systems (the systems 4 110409, 4 121084, 4 175149, 5 32412, 5 38089 and 5 266015 are in this case). For a given object, the colour indices were linearly interpolated from these grids.

To estimate the uncertainties, however, the following first-order approximations valid for early-B stars of intermediate values were used (see Section 3.13),

 (B−V)0 ≈ 1.768−0.455logTeff   for   15000≤Teff≤32000 (7) ≈ 0.228−0.113logTeff   for   32000≤Teff≤37500 (V−I)0 ≈ 1.958−0.502logTeff   for   15000≤Teff≤32000 ≈ 0.502−0.178logTeff   for   32000≤Teff≤37500

Once the colour excess is determined, one must make an assumption about the value of the extinction parameter . This parameter is assumed equal to the standard value for each system. It is worth mentioning that this parameter suffers from a rather large uncertainty. For the SMC bar, Gordon et al. ((2003)) propose the mean value from a sample of four stars with ranging from to . Nevertheless, because of the small size of Gordon et al.’s sample, the more conservative standard value was retained for this paper. Moreover, the contribution of to the total error budget for the DMi is low. The difference between a DM calculated with Gordon et al. ’s value and the standard value is given by . Thus, with , the DM is slightly increased by 0.036 mag if is adopted.

In order to determine the distance modulus from the band data, the ratio of the absorptions in the infrared and optical bands is calculated from the relationship,

 AλAV=a(x)+b(x)1RVorAλ=EB−V(a(x)RV+b(x)) (8)

with . The coefficients and are computed from the polynomials given in Eq. 3a-b from Cardelli et al. ((1989)). An effective wavelength of 7980Å is taken for the Cousins band filter (Bessell (1990)). Hence, it follows that . That value contrasts with the value of 0.479 given by Cardelli et al. ((1989)) in their Table 3 for the band, because they adopt another effective wavelength.

### 3.12 Distance modulus: V− or I−band approach?

There are two ways of determining the distance modulus of a particular system, depending on whether one takes visual or infrared data:

 5logd−5 = Vq−MqV−AqV (9) = Iq−MqI−AqI ≈ Iq−MqV+(V−I)q0−0.600AqV

The safest choice is to take the expression which leads to the value with the smallest uncertainty. The band magnitude at quadrature is far more reliable than the band value because there is typically 10 more points in the band light curve. Moreover, the contribution of the interstellar absorption is lower at infrared wavelengths. Indeed, it is easy to demonstrate that the variance of the visual absorption is always larger than the sum of the contributions due to the infrared absorption and the intrinsic colour:

 σ2AqV>σ2AqI+σ2(V−I)q0withσ2(V−I)q0≈σ2(B−V)q0 (10)

Consequently, the band calculation must be preferred, at least for system for which there is a reliable way to obtain the index. We have seen that this is the case of systems where both components have K, while for hotter binaries, the synthetic spectra of the OSTAR2002 library had to be complemented by Kurucz models beyond 7500 Å. In spite of this slight inhomogeneity, we have adopted the indices computed in this way, rather than refrain from computing the distance modulus for these hot binaries.

In order to check the purely synthetic intrinsic index, both and band formulations of Eq. 9 can be used to derive the following relation between the two colour excesses:

 EV−IEB−V=0.400RV (11)

A simple modification of this equation gives,

 (V−I)0=(V−I)−0.400RVEB−V (12)

This is a semi-empirical dereddening, as is calculated from a synthetic intrinsic index. The indices so obtained agree within mag for 4 of the 6 hot binaries; the differences reaches  mag for the system 5 32412, but this is less than the estimated error on the distance modulus. For 5 38089, the difference is  mag, but this system has abnormal colours and so will be excluded from the statistics of the distance moduli.

### 3.13 Uncertainties

#### Distance modulus and related parameters

The calculus of the uncertainties in a number of parameters relies on some assumptions and simplifications. These points are discussed in this paragraph. The uncertainty in the distance modulus of a particular binary is determined via the standard rules of the propagation of errors for independent variables,

 σ25logd−5 ≈ σ2MqV+σ2(V−I)q0+ (13) (0.600RV)2(σ2(B−V)q+σ2(B−V)q0)

This expression was easily derived from the infrared formulation of Eq. 9. The uncertainty in the magnitude ( mmag) and the term are negligible and consequently omitted from Eq. 13. It is worth noting that formally the intrinsic colour indices and and the visual absolute magnitude of the system are not independent variables, as they depend ultimately on the effective temperatures and radii of the stars. Nevertheless, for the sake of simplicity these variables were considered as independent in the following development. The uncertainty on can be expressed as a function of the uncertainties on the absolute magnitudes of the components and

 σ2MqV=1(10−0.4MPV+10−0.4MSV)2S∑j=P(10−0.4MjVσMjV)2 (14)

The absolute visual magnitude of a component is found via the absolute bolometric magnitudes and the visual bolometric correction ,

 MV(R,Teff,Z)=Mbol(R,Teff)−BCV(Teff,Z,logg) (15)

There is now an important point emphasized by Clausen ((2000)). and depend both on the effective temperature of the star and thus are not independent variables. Consequently, from the bolometric corrections calculated by Lanz and Hubeny ((2003), (2007)), we find that for late-O and early-B stars the bolometric correction can be given by

 BCV ≈ α−βlogTeffwith α = 21.72±0.28magandβ=5.51±0.07mag (16)

with the associated uncertainty

 σBCjV≈βln10σTjeffTjeff (17)

Combining Eq. 15 and 16 and expressing as a function of effective temperature and radius , the visual absolute magnitude can be written in turn as a function of and (see Clausen (2000) for more details). It follows that the uncertainty in is given by

 σ2MjV≈1(ln10)2⎡⎢⎣(5σRjRj)2+⎛⎜⎝(10−β)σTjeffTjeff⎞⎟⎠2⎤⎥⎦ (18)

Thus the uncertainty in (‘10’ factor) is partially cancelled by the uncertainty in (’’ factor).

The uncertainties in the intrinsic colour indices at quadrature, and , are estimated via the approximation for early-B stars given in Section 3.11. Since the colour indices relate, not to a single star, but to a binary system, one defines an “equivalent” effective temperature of the system (which is the effective temperature of a single star with the same colour index as the system) by inverting Eq. 7. Then, a rough estimate of the error on the intrinsic colour index is obtained by propagating the error on , which is identified with that on , the temperature of the primary. In general, the error on the absolute magnitude of the system will outweigh by far the other terms on the right side of Eq. 13.

Masses and radii depend heavily on the radial velocity semi-amplitudes . The WD code does not provide directly the errors on the masses and radii, but only those on the semi-major axis , on the mass ratio , the inclination , the effective temperature of the secondary and the potentials. It is in principle possible to derive the errors on the masses from those on and using the third Kepler law (which gives the total mass) and appropriate propagation formulae. However, the resulting errors tend to appear underestimated, so we preferred to use the errors on given the bina code that was used for a preliminary interpretation of the RV curves alone. The errors on the masses are then obtained through the formula

 σ2MP,S=(σM1,2sin3isin3i)2+(3cosisin4i)2σ2i (19)

and may be twice larger than those obtained from WD. Still, they have to be considered rather as lower limits to the real uncertainties, because they are based only on the scatter of the radial velocities around the fitted curve. Systematic errors may arise, however, from the choice of the template spectra used to obtain the RV values, and from the unavoidable fact that these synthetic spectra can never perfectly match the real ones.

The relation between the uncertainties in the masses and pertaining variables can be found in Hilditch ((2001)), for example. The absolute radius of component is obtained from

 Rj=arj(Σr,k) (20)

where is the semi-major axis and the relative radius of component . The relative radii are considered as functions of two independent variables, the sum and the ratio of the relative radii. The sum of the radii is obtained reliably from the analysis of the light curves. The ratio of the radii is obtained either from the spectroscopic luminosity ratio, in the case of detached systems (unless the minima are so deep that the photometric estimate proves accurate enough), or from the light curve in the case of semi-detached and contact systems. The individual radii are thus given by

 rP=11+kΣrandrS=Σr−rP (21)

with the associated uncertainties

 σ2rP,S=(αP,S1+k)2σ2Σr+(Σr(1+k)2)2σ2k (22)

where and . is robustly determined from the light curve. For the sake of simplicity, the variance in was taken from the EBOP solution of the curve, by scaling the error on obtained by fixing the ratio of radii.

The spectroscopic ratio of radii is obtained by inverting Eq. 4

 k≡RSRP=L1/2B(TSTP)2⋅