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2. Decomposition of spectra with variable intensities

  To generalize the method described in Paper I, let us suppose the observed spectrum I(x,t) (the variable tex2html_wrap_inline1137 is proportional to logarithm of wavelength expressed in an arbitrary unit tex2html_wrap_inline1139; cf. Simkin 1974) to be composed of spectra tex2html_wrap_inline1141 of n stars which have no intrinsic variability apart from an overall change of their intensities proportional to functions tex2html_wrap_inline1145 which may be caused e.g., by ellipticity or eclipses of binary components or by air-mass or humidity for the telluric spectrum.

If the instantaneous radial velocity (of the star j at the time t) tex2html_wrap_inline1151, the composite spectrum is given by the convolution in x
The Fourier transform (tex2html_wrap_inline1155) of this equation reads

If we have k spectra (k>n) observed at times tex2html_wrap_inline1161 corresponding to various values of tex2html_wrap_inline1163, we can - in principle - fit them by searching for appropriate values of tex2html_wrap_inline1165, tex2html_wrap_inline1167 and tex2html_wrap_inline1169. The velocities tex2html_wrap_inline1171 can be treated either as independent values, or to be given functions of time and certain parameters p, e.g. the orbital elements of the spectroscopic binary. Using the standard method of least squares we arrive at the condition


Here tex2html_wrap_inline1175 is the weight of each Fourier mode y in the exposure l. In practice, we suppose
The weight tex2html_wrap_inline1181 of each exposure can be chosen e.g. in dependence on the number of photon counts. The function w(y) can be introduced as a frequency filter, e.g. to cut off the low frequency modes influenced by the rectification of the spectra.

Figure 1: Continuum and line intensities of uneclipsed components (left) and in the primary eclipse (right)

Because S is bilinear in tex2html_wrap_inline1187, the conditions obtained for them from Eq. (3 (click here)) varying with respect to tex2html_wrap_inline1189 (i.e. the variation of complex conjugate of tex2html_wrap_inline1191) are linear equations
(m=1,...,n), which are moreover independent for each y. Just this independence of Fourier components, which is the consequence of the form of Eq. (2 (click here)) local in the variable y (unlike the form of Eq. (1 (click here)) integral in the variable x), makes the disentangling of the observed spectra easier in Fourier transform than in the wavelength space. Solving this system of equations for each y and substituting tex2html_wrap_inline1203 into Eq. (4 (click here)), S can be optimized only with respect to other parameters, which are much less numerous. It is obvious that Eq. (6 (click here)) is singular for y=0. This corresponds to the fact that the contributions of individual stars to the constant term of I(x) cannot be directly distinguished. The continua of stars are almost constant and in practice unaffected by the Doppler shift. Consequently, a large error of tex2html_wrap_inline1211 can be expected for small values of y. It can thus be convenient to cut-off this range of y choosing here w(y)=0. An indirect method of distinguishing the contributions to the continuum is described in Sect. 3 (click here).

S is bilinear also in the coefficients tex2html_wrap_inline1221. Hence, varying with respect to tex2html_wrap_inline1223, we get the linear set of equations
for these coefficients. Because these coefficients are generally still quite numerous (but less than the Fourier modes of the component spectra), it is advantageous to solve for them directly from these equations before optimizing S with respect to either tex2html_wrap_inline1227 or p, in which it is non-linear.

It is important to keep in mind that the solutions of orbital elements (or individual independent radial velocities), the decomposition of the spectrum and the solution of component intensities are inter-related and their self-consistent solution should be found. This can be achieved iteratively starting from some initial estimate of orbital parameters and line intensities. An inner loop solving successively Eq. (6 (click here)) for tex2html_wrap_inline1231 and then Eq. (7 (click here)) for tex2html_wrap_inline1233 is used in KOREL, while the orbital parameters are solved in an outer loop by simplex method. Practical experience shows, that slower, but more stable convergence of some of coefficients tex2html_wrap_inline1235 can be advantageous to get a better initial estimate. A successive increasing of the number k of exposures can also help the convergence.

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