Astron. Astrophys. 332, 526-540 (1998)

Appendix A: mathematic justification of algorithm

A result of the spectral observations with the BTA scanner in a high time resolution mode is a file with the coordinates (2x1024 channels) and the time of detection (resolution 32 ms) for every photoelectron, which was recorded during one exposure. Every channel corresponds to the wavelength and can be calibrated with the help of a He-Ne-Ar lamp spectrum. The flux of photoelectrons is described by the function , where is the wavelength of the associated channel and t is the time. Further we assume but not underline a discrete character of .

In a general case one can present this function as the discrete Fourier transform

where ; T is the time of exposure; is the time of integration of one spectrum; is the number of spectra in the exposure; and

The Fourier coefficients are

where

are the spectra integrated during the time ; dt is the time resolution of the photon-counting system and is equal to 32 ms for the BTA scanner; can be presented as

Our algorithm is aimed at the calculation of the Fourier coefficients, which will give information about the power, amplitude and phase of oscillations. The calculation of the Fourier coefficients can be significantly simplified in one special case. We consider the case when the period P, corresponding to the frequency of the Fourier coefficients, is equal exactly to the integer number of the integration time, or

where k is this integer number and it is constant for the algorithm.
The corresponding frequency in (A3) is

where . It is possible to rewrite the exponential member in (A3) as

In our special case we can calculate the Fourier coefficient , which we rewrite as

where and . For the coefficient the calculations are similar. We rewrite the previous equation as

The cosine is a periodical function and it is possible to combine the members in the previous equation in the following way

We introduce the designations

where , and then

or in a compact form

and similar for

The last two formulae permit accurate calculation of the Fourier coefficients, but only for the special case. For this it is necessary to calculate for . This is a well-known procedure of data folding in k bins, where every bin is a spectrum. These formulae (A14,A15) show the relationship between the result of data folding and the Fourier coefficients or power of oscillations at the associated period.

In the real spectral observations the size of the spectrograph slit is comparable with the size of a star on the slit. Small variations in guiding produce broad-band variations in spectra. Our algorithm is aimed at searching for the narrow-band spectral oscillations and the broad-band oscillations must be suppressed. For this every bin must be divided by , where

is a polynomial of 3rd-4th power (it depends on a spectrum) and is called as a continuous spectrum. The calculation of the continuous spectrum can be done using various methods. We choose the wavelength bands in the spectrum without emission or absorption lines . The corresponding values of spectral intensity (as the mean value near , where  Å and a dispersion of 1 Å/channel is assumed) can be determined automatically and the desirable polynomial can be fitted. The normalized bins are

where .
The conversion of the spectra in every bin to relative units suppresses all broad-band spectral variations. One can rewrite the formulae (A14,A15) for calculations of the Fourier coefficients in relative units

The power of spectral oscillations is

In this consideration we have shown how to convert the result of data folding in the power of oscillations in relative units.

The basic idea of our algorithm is freedom in the choice of the spectrum parameter such as the integration time (). It means that it is possible for every period of interest P in the range

where ms and T is the time of exposure, to define and to integrate spectra from the original data (a file with coordinates of photons) with an accuracy limited by ms.
Now we formulate this simple algorithm as:

[1.] The choice of the number of bins (it depends on an astrophysical problem), for example, this number is .

[2.] The choice of the periods of interest in the region (it also depends on an astrophysical problem), for example, .

[3.] For every period of interest BEGIN :

[3.1] Spectrum integrations from the original data with

and data folding in bins. Every bin contains two spectra (object and sky).

[3.2] Subtraction of a sky spectrum in every bin and conversion of every bin to relative units (normalization to continuous spectrum)

where j=1,2,...,k.

[3.4] Calculation of the power (or amplitude, phase) in relative units (A18,A19,A20) for every channel (wavelength) in the spectrum.

[3.5] GO TO 3.1. For all periods of interest.

[4.] As a result we obtain a dependence of the power of oscillations on the wavelength and on the period in the desirable range .

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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