COMMISSION 27 OF THE I. A. U. INFORMATION BULLETIN ON VARIABLE STARS NUMBER 270 Konkoly Observatory Budapest 1968 April 24 ANALYSIS OF CEPHEID LIGHT CURVES The observed light variation of a cepheid variable star in, e.g., V photometric system can be regarded as a sum of two components: Delta V=Delta m_v+Delta m_R (1) where Delta m_v=-2,5 Log F_v1/F_v2 is due to the variation of radiative flux F_v, and Delta m_R=-5 Log R_1/R_2 represents the photometric effect of changes of the radius R. We present here a method to separate these two components. Let us assume that for each star during the whole period the changes of fluxes: F_v in the V system and F_B in the B system, expressed in terms of Delta m_v and Delta m_B are in proportion to each other: Delta m_v=a Delta m_B (2) From this we have also the proportionality of Delta m_v to Delta(B-V) Delta m_v=k Delta(m_B-m_v)=k Delta(B-V) (3) with k=a/(1-a). This relation can be considered as a generalization of Wesselink's method, which is usually used in the form: for two phases with Delta(B-V)=0, there is also Delta m_v=0 and Delta V=Delta m_R. According to the formulae (1) and (3) we assume for every two phases the relation: Delta V=k Delta(B-V)+Delta m_R (4) The photometric observations yield the values (B-V) measured simultaneously with V magnitudes. So the calculation of Delta m_v and Delta m_R requires only a value of k, which can be obtained from the following considerations. In most cases the variations of V magnitudes are mainly resulting from the variations of Delta m_v, with component Delta m_R being less important. So we try to determine the coefficient k, or its mean value over the period, from the condition that for a sufficiently large number of Deltam_B=Delta V-k Delta(B-V) we expect to have Summa(Delta m_R)^2=min. (5) This condition if fulfilled when k=Summa[Delta V. Delta(B-V)]/Summa[Delta(B-V)]^2 (6) We have applied this method to 29 cepheid variables: 19 C Delta and 10 CW stars. The phases were calculated always from minimum of (B-V), which especially for CW stars may differ from the phase of minimum V (maximum brightness). As an initial phase, the phase of mean value of (B-V) on the descending branch was chosen. In most cases it turned out to be about 0,25. The values of V_0 and (B-V)_0, for this phase are used to calculate the differences occurring in the formula (6) Delta V=V-V_0, Delta(B-V)=(B-V)-(B-V)_0. As a rule 10 to 20 values for uniformly spaced phases, read from the V and (B-V) curves, were sufficient to get the coefficient k. Than the values of Delta m_v=k Delta(B-V) and Delta m_R=Delta V-Delta m_v were calculated. The Delta m_R curves have the shapes similar to the Delta R curves obtained by integrating the radial velocities: flat maximum near 0,3 and sharper minimum near 0,8 phase. The amplitude of Delta m_R denoted by A_mR is only partly dependent on k value. From the Delta R curves one can get the amplitudes of Delta R=R_max-R_min. By combining them with A_mR, one can calculate the mean value of radius R: A_mR=5 Log R_max/R_min=5 log (R+0,5 Delta R)/(R--0,5 Delta R) (7) The numerical results are presented in the Table. From these values we can draw the following conclusions: Table No Star log P Type k Am_R Delta R km R km 1 SU Cas 0,290 Cdelta 2,13 0,^m08 7,3*10^5 20*10^6 2 RT Aur 0,571 Cdelta 1,74 0,21 20,8 22 3 T Vul 0,647 Cdelta 1,78 0,16 27,0 37 4 FF Aql 0,673 Cdelta 1,93 0,15 14,7 21 5 delta Cep 0,730 Cdelta 1,62 0,22 40 40 6 U Aql 0,846 Cdelta 1,71 0,25 48 42 7 eta Aql 0,856 Cdelta 1,76 0,30 46 33 8 W Sgr 0,880 Cdelta 1,55 0,15 48 70 9 W Gem 0,898 Cdelta 1,38 0,30 48 35 10 S Sge 0,923 Cdelta 1,53 0,22 52 51 11 zeta Gem 1,006 Cdelta 1,21 0,22 48 48 12 TT Aql 1,138 Cdelta 1,37 0,40 116 63 13 X Cyg 1,216 Cdelta 1,48 0,31 173 121 14 Y Oph 1,233 Cdelta 1,64 0,20 51 55 15 T Mon 1,431 Cdelta 1,29 0,53 263 108 16 SV Vul 1,655 Cdelta 0,77 0,54 430 174 17 W Vir 1,236 CW 2,14 0,92 194 46 18 VZ Aql 0,222 CW 1,64 0,45 19 AU Peg 0,380 CW 1,14 0,27 20 V465 Oph 0,454 CW 2,41 0,47 21 V532 Cyg 0,516 Cdelta 1,97 0,11 22 VY Cyg 0,895 Cdelta 1,86 0,27 23 AP Her 1,018 CW 1,46 0,47 24 V1077 Sgr 1,128 CW 0,98 1,00 25 V410 Sgr 1,139 CW 1,22 0,69 26 SZ Cyg 1,179 Cdelta 1,35 0,32 27 AL Sct 1,192 CW 1,38 0,90 28 CC Lyr 1,380 CW 1,53 0,31 29 TW Cap 1,407 CW 2,29 0,75 Photometric data: No 1-16 Mitchell et al. Bol, Obs. Tonantzintla Tacubaya 3, 153, 1964. No 17-29 Kwee K. Braun L. B.A.N. Sup, Ser. 2, 3, 1967. Radial velocities from different sources. [FIGURE 1] 1. For given P the amplitudes A_mR for Cdelta stars are systematically smaller than those for CW stars. On the diagramm we can fix the position of line A_mR=1/3 Log P+0,10 separating the region of A_mR, for Cdelta from such a region for CW stars. This fact may be used as a purely photometric criterion for classification of Cdelta and CW stars. E.g. CC Lyr classified as a CW star, according to our criterion belongs to the Cdelta stars. 2. The values of R obtained by means of the foregoing procedure are consistent with the values obtained by other authors using Wesselink's method. The results for a greater number of stars and a more detailed discussion will be published in the Acta Astronomica. Wroclaw, April 17, 1968. A. OPOLSKI Astronomical Observatory Wroclaw