On the separation of component spectra in binary and higher-multiplicity stellar systems: bias progression and spurious patterns

Context. Powerful methods are available to reconstruct the spectra of stars in orbit around each other, using a time-series of observed, composite spectra. They act either on the Fourier components of the observed spectra, or directly on these spectra in a velocity grid. Aims. We discuss under which conditions spurious patterns can appear in reconstructed spectra, either as a consequence of ( quasi)degeneracy of the equations or as a consequence of bias in the observed spectra. Also we discuss the equivalence of Fourier and direct methods in practice. Methods. We show under which conditions the equations degenerate, and how to evaluate this. We pay special attention to spectra of binary stars and triple systems analysed in Fourier space. We apply the theory to real data sets and to artificial data sets with several types of data-reduction bias constructed to illustrate degeneracies and the transfer of bias to the reconstructed spectra. Results. Quasi-degeneracy of the equations depends on the lack of significant time-variability in the relative light contribution of the stars, on the length of the spectral interval in units of the involved Doppler shifts, on the presence of very faint stellar components, and on the distribution of the observations over the orbital phases. Eclipse spectra, possibly used with different weight in low-and high-frequency Fourier modes, remove quasi-degeneracies. But when the normalisation of the observed spectra is biased in a systematic way with relation to the orbital phase, then the bias amplifies strongly in the reconstructed spectra, particularly for the faintest components. Wavelength-locked bias is transferred more strongly to the spectrum of the star with the lowest velocity amplitude. Unrecognized variations in line strength lead to bias that is larger in spectral regions with high line-density. Most importantly, the bias in all reconstructed spectra is coupled in a unique, predictable way. This coupling allows us to design robust procedures for the removal of bias from the reconstructed spectra. Conclusions. When the spectral features of all stars visible in the observed spectra undergo a significant time-variable dilution effect, then the reconstructed spectra are well-defined. Otherwise, spurious patterns, mostly of low frequency, may be superimposed on the reconstructed spectra. The analysis presented in this paper allows observers to optimise their observing strategy, gives insight in the origin of spurious patterns and indicates ways either to suppress such patterns or to remove them a posteriori from the reconstructed spectra. In this way, a broader range of astrophysical analyses can be applied to the component spectra.