COMMISSION 27 OF THE I. A. U. INFORMATION BULLETIN ON VARIABLE STARS NUMBER 8 Konkoly Observatory Budapest 9 April 1962 A REMARK ON THE MULTIPERIODICITY OF SOME PULSATING STARS In connection with the problem of multiperiodicity of some pulsating stars we propose the interpretation of the observed phenomena by means of modulated oscillations (similar to the modulations introduced in radiotechnique). Let us consider the following simple example: The oscillation f(t) with the frequency {alpha} in the form f(t) = A cos 2*pi*alpha*t is changed by means of modulation with the frequency {beta} to: F(t) = (A+A_1* cos 2*pi*beta*t) cos 2*pi*[alpha*t+B*sin 2*pi*beta*t]. This modification is equivalent to the amplitude modulation from A to A+A_1*cos2*pi*beta*t with the frequency modulation from alpha*t to alpha*t+B*sin 2*pi*beta*t. When looking for periodicity of such modulated oscillation we must represent it by means of the series: F(t) = a_0 cos2*pi*alpha*t+a_1*cos 2*pi*(alpha+beta)*t+ a_2*cos 2*pi*(alpha+2*beta)*t+... b_1*cos 2*pi*(alpha-beta)*t+ b_2*cos 2*pi*(alpha-2*beta)*t+... where a_0 = J_0*(2*pi*B)*A a_1 = J_1*(2*pi*B)*A + 1/2*[J_0*(2*pi*B) + J_2*(2*pi*B)]*A_1 ; a_2 = J_2*(2*pi*B)*A + 1/2*[J_1*(2*pi*B) + J_3*(2*pi*B)]*A_1 ; ..... b_1 =-J_1*(2*pi*B)*A + 1/2 [J_0*(2*pi*B) + J_2*(2*pi*B)]*A_1 b_2 = J_2*(2*pi*B)*A - 1/2 [J_1*(2*pi*B) + J_3 (2*pi*B)]*A_1 and J_i (2*pi*B) denote Bessel functions. It is clear that by means of modulation we get the whole system of frequencies of the type: alpha +- k*beta, k=0,1,2... with different amplitudes. We are of the opinion that many of the observed phenomenon can be regarded as the effect of the high frequency oscillation alpha with a suitable modulation in low frequency beta. The above considerations can be applied to the following stars: 1. The long period variations of RR Lyr according to A.M. Fringant [1] can be regarded as amplitude and frequency modulation: alpha=1/P_0, beta=1/P_1, P_1=72*P_0 2. According to one of our publications [2] the variations of 12(DD) Lac can be described not by 2 or more short periods but by means of one short period P_s with the modulation in a long period P_L. The modulation exhibits the character of amplitude and frequency modulation and the long period P_L, was found in the variations of many physical properties of the star. 3. The variations of nu Eri according to A. van Hoof [3] are represented by means of a great number of short periods. Some of them can be again regarded as the result of the modulation of the fundamental frequency alpha=1/P_0=omega_0=omega_24 alpha-2*beta = omega_02 = 5,6375 beta alpha-beta = omega_13 = 5,7019 0,0644 alpha = omega_0 = 5,7634 0,0615 alpha+beta = omega_35 = 5,8275 0,0641 The mean value of beta=0,0633 corresponds to the long period P_L=15d,790 and the value of 2*beta to the period P_L=7d,895. The last one is equal to the beat period given in paper [3]. Probably also other frequencies can be added to the same scheme: alpha+2*beta = omega_46 = 5,8994 alpha+3*beta = omega'_F = 5,9439 alpha-3*beta = omega_F = 5,5866. We can remark that this method of representation of complicated variations observed in some variable stars (e. g. with the Blashko-effect) requires from the theoretical point of view the explanation of only two frequencies alpha and beta. References: [1] Anne-Marie Fringant - Journal des Observat. 44.p. 187, 1961. [BIBCODE 1961JO.....44..165F ] [2] A. Opolski and T. Ciurla - Acta Astronomica 11 p. 231, 1961. [BIBCODE 1961AcA....11..231O ] [3] A. van Hoof - Z. Astrophys 55 p. 106, 1961. [BIBCODE 1961ZA.....53..106V ] Antoni OPOLSKI and Tadeusz CIURLA Wroclaw Astronomical Observatory Poland