Up: Relative line photometry

# 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 is proportional to logarithm of wavelength expressed in an arbitrary unit ; cf. Simkin 1974) to be composed of spectra of n stars which have no intrinsic variability apart from an overall change of their intensities proportional to functions 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) , the composite spectrum is given by the convolution in x

The Fourier transform () of this equation reads

If we have k spectra (k>n) observed at times corresponding to various values of , we can - in principle - fit them by searching for appropriate values of , and . The velocities 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

where

Here is the weight of each Fourier mode y in the exposure l. In practice, we suppose

The weight 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 , the conditions obtained for them from Eq. (3 (click here)) varying with respect to (i.e. the variation of complex conjugate of ) 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 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 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 . Hence, varying with respect to , 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 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 and then Eq. (7 (click here)) for 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 can be advantageous to get a better initial estimate. A successive increasing of the number k of exposures can also help the convergence.

Up: Relative line photometry

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