COMMISSION 27 OF THE I. A. U.
         INFORMATION BULLETIN ON VARIABLE STARS
                     Number 1347

                                  Konkoly Observatory
                                  Budapest
                                  1977 October 10





 PRACTICAL FORMULAE TO SEARCH PERIODICITIES IN VARIABLE STARS


     The distribution of the dates in the series of observations
of variable stars are often very irregular. The spacing between
consecutive dates may range from hours to weeks or even months.
Such unlikeness forbids the use of classical methods to search
periodicities, or, at least, makes its use very hazardous.
Nevertheless, the use of the fast Fourier transform is almost
generalized among astronomers under the pretext that the results
obtained are fairly good. In fact we obtained some good results
in the study of some BY Dra stars observed at the I.T.A.
Astronomical Observatory before 1974. However this has not been
always true and practical questions arose when comparing some
completely different results obtained for HD 319139 using
the fast Fourier transform or the Lafler-Kinman method. The light
curve of BY Dra stars are characterized by small amplitude and
great noise, and in 1975 we adopted a new technique to deal
with series where the dates are arbitrarily spaced. This technique
is a natural generalization of the fast Fourier transform and
gives more accurate estimates.


     In the figure, we show the periodogram obtained
for 21 V-magnitudes of the star HD 319139 observed by C.A.O. Torres
and I.C. Busko at Cerro Tololo in 1974 (solid line); we also
compare it to the periodogram obtained with the classical
fast Fourier transform (dashed line).
    Recently, in a review paper presented in the Colloquium on
Close Binary Stars held in San Paulo we have shown the way in
which the technique may be generalized to more complex cases
like those were the data consist of N pieces of continuous
records or where the data have different precisions.
    A typical example of problem involving unevenly spaced
data of different precisions is the problem of the search of
periodicities (and beats) in irregular, semi-regular or
long-period variable stars. In this case the measurement I(w)
of the power spectrum at the trial frequency omega is given by
the expressions in the next page.

                          [FIGURE 1]

x_j=2(Pi)(omega)t_j

(a_0)^-2 = Summa w_j

(a_1)^-2 = Summa w_j (cos^2 x_j) - (a_0)^2(Summa w_j cos x_j)^2

(a_2)^-2 = Summa w_j (sin^2 x_j) - (a_0)^2 (Summa w_j sin^2 x_j)^2 
           - (a_1)^2 (Summa w_j cos x_j sin x_j)^2 
           - (a_0)^4 (a_1)^2 (Summa w_j cos x_j)^2 *Summa w_j sin x_j )^2
           + 2(a_0)^2 (a_1)^2 (Summa w_j cos x_j) (Summa w_j sin x_j) (Summa w_j cos x_j sin x_j)

c_1 = a_2 Summa w_j f(t_j) cos x_j

c_2 = a_2 Summa w_j f(t_j) sin x_j - a_1 a_2 c_1 (Summa w_j cos x_j sin x_j - (a_0)^2 Summa w_j cos x_j Summa w_j sin x_j)

I(omega) = (c_1)^2 + (c_2)^2

    In these formulae t_j are the observation dates and w_j
the weights of the observations. The measures f(t_j) are referred
to their mean value, i.e., they are shifted in a way such that
Summa w_j f(t ) = 0. All other symbols are intrinsic quantities.
     The light curves under study may be formed by several waves,
not necessarily harmonic (as in Blazhko effect). In such cases,
before searching for a secondary period, it is necessary to filter
the time-series from the main wave. If omega_1 is the main frequency,
the filtered time-series is given by

               g(t_j) = f(t_j) - d(t_j)

where

               d(t_j) = d0 + d_1 cos x_j + d_2 sin x_j



               (x_j = 2 Pi omega t_j)


and

d_0 = - (a_0)^2 ( d_1 Summa w_j cos x_j + d_2 Summa w_j sin x_j)

d_1 = a_1 c_1 + a_2 (a_1)^2 c_2 ( (a_0)^2 Summa w_j cos x_j Summa w_k sin x_k - Summa w_j cos x_j sin x_j )

d_2 = a_2 c_2

     The secondary frequencies are searched in the periodogram
of the filtered series g(t_j).
     The epoch of the minimum in the main wave is one of the
two roots of tg x_0 = d_2/d_1.





                           S. FERRAZ-MELLO
                      Universidade de Sao Paulo
                     Departamento de Astronomia

                              (Brazil)