# 7.5.4 Processing steps

Basically, the SB2 processing is very similar to the SB1 described in the Section 7.4.4. However, additional complications have been added to avoid the ambiguity between the two components. The first step of the analysis of the data is the diagram $\mathrm{RV}_{2}$ versus $\mathrm{RV}_{1}$ and the search of the straight line that should be covered along the orbital motion. The slope of this straight line is a function of the mass ratio (also the semi-amplitude ratio). Therefore, the first step is to derive the mass ratio by fitting the straight line through the data points. The adopted fit is a two-dimensional one where the error are accepted on both axes and where the chi-square is based on the distance computed perpendicularly to the fitted line. If the data points are correctly attributed to components 1 and 2, the slope is well defined. On the opposite, if there exists an ambiguity between components 1 and 2, the $\mathrm{RV}_{2}-\gamma$ versus $\mathrm{RV}_{1}-\gamma$ diagram exhibits two straight lines (mirror of each other with respect to the line with a slope equal to –1). It is then possible to exchange the attribution of the pairs of RVs to the individual components. After a few such inversions, the pair of lines can be transformed into a single line. The chosen one is arbitrary but the choice could also be consistent (e.g. always choosing the component 1 as the most massive one). This procedure is of course possible if the signal is not noisy and if the two masses are markedly different. In the opposite case, it is thus possible to remove the ambiguity by exchanging the attribution to components 1 and 2 and by searching to minimise the chi-square statistic based on the dispersion around the straight line. If the mass ratio is close to one, both diagonal lines tend to merge into a single line and the method is no more efficient. A more sophisticated method is then necessary that performs the various shifts along with the period search technique. An illustration of the application of the adopted procedure can be found in the work reported in https://www.cosmos.esa.int/web/gaia/iow_20210427.

The period search technique adopted for SB2 is very similar to the one described for the SB1 case (see Section 7.4.4). However, the condition that the RVs of component 1 must be in antiphase with the RVs of component 2 is an additional source of non-linearity (in the parameters) of the fit. An easy way to get rid of the problem is to perform the period search on the fake RV: given by $\mathrm{RV}_{\mathrm{f}}\,=\,\mathrm{RV}_{2}\,-\,\mathrm{RV}_{1}$ . The modification is not consequent. A more complex problem is coming from the ambiguity between component 1 and component 2. The fake $\mathrm{RV}_{\mathrm{f}}$ is highly dependent on the component 1 versus component 2 attribution. A straightforward way to get rid of that is to consider the value $|{\mathrm{RV}_{\mathrm{f}}}|$ which is independent on the operation of permuting 1 and 2 (at each transit) but the function presents switchback points at the centre of mass velocity (cusp). This introduces in the mathematical curve a lot of power at high harmonics, which combined to the aliasing phenomenon is not good, neither for the Fourier analysis nor for the corresponding search. A possible alternative is $(\,\mathrm{RV}_{2}\,-\,\mathrm{RV}_{1})^{2}$ whose harmonic content is more limited. Actually, a more sophisticated method was adopted that is fully described in Damerdji et al. (2022).

In principle, the 1 and 2 components are correctly ordered. However, in the case of the Gaia DR3, some double stars could have their components very similar from the physical point of view. In this case, the discrimination component 1 versus component 2 could present weaknesses and the components could be inverted in the entry file. This turns out to be extremely problematic. In the case of two too similar objects, an ambiguity appears between two observations acquired either in opposition of phase or in phase. Consequently, it is necessary to perform the period search with the two ambiguous cases at each transit. Actually the period search should be performed with a very special period search technique. Again, it will be fully described in Damerdji et al. (2022).