COMMISSION 27 OF THE I. A. U. INFORMATION BULLETIN ON VARIABLE STARS NUMBER 359 Konkoly Observatory Budapest 1969 June 30 ANALYSIS OF GRADIENT DIAGRAM FOR CEPHEIDS As was found by E. S. Kheylo [1] all cepheids occupy a band on the plane (DeltaU, DeltaV), where DeltaU = dU/dB, DeltaV = dV/dB, the band being elongated along the axis DeltaU. We shall call this plane the gradient diagram. Inside the band all cepheids separate according to their types. This paper is an attempt to explain such a location of cepheids on the gradient diagram. Let us consider the light variations of cepheids to be connected with changes of dimension and of the black body temperature. In this case we can write for the luminosity in the Q-region I_Q ~ R^2 I_Q(T), Q = U,B,V (1) where R denotes the radius of star, I_Q (T) the radiation intensity in the isophotal wave length of Q for temperature T. Isophotal wave lengths were adopted to be 3680A for U 4450A for B and 5460A for V [2]. The Wien formula gives satisfactory accuracy for I_Q(T). For radius changes from R_1 to R_2 and temperature changes from T_1 to T_2, equation (1) gives for the gradients DeltaU = a-C_U d/a-C_B d, DeltaV = a-C_V d/a-C_B d (2) where a = 2 ln R_1/R_2 DeltaR/R2 when changes of radius are small, d = (T_2-T_1)/T_1T_2 = DeltaT/T_1T_2 c_U = 3.91 10^4, c_B = 3.23 10^4, c_V = 2.64 10^4. Educations (2) indicate, that DeltaU and DeltaV are linearly connected. This line passes through the point (1,1) and the point DeltaU = 1.21, DV = 0,815, (3) which corresponds to the changes of the black body temperature only. In the Figure one can see that the cepheids are located below this line. Thus it is impossible to explane light changes of cepheids by the changes of dimension and of the black body temperature. The changes of R or T in the U-region may be assumed to differ from those in the B- and V-regions. In this case d_U is not equal with d_B,V or a_U is not equal with a_B,V. [FIGURE 1] Lines of equal m and the band of cepheids Two hypotheses were considered d_U is not equal with d_B = d_V, a_U = a_B = a_V (4) and d_U = d_B = d_V = d, a_U is not equal with a_B = a_V (5) Better agreement seems to be achieved for the location of cepheids in the gradient diagram in the case (5). Then DeltaU = DeltaU m-1/Delta_V-1 (1-Delta_V), (6) where m = a_U/C_U d is a free parameter. As follows from the Figure, these lines occupy the cepheid band for m=0.23-0.58. Then DeltaU and DeltaV for a cepheid permit to obtain a_U/d = C_U 1-DeltaU/D_U 1-D_V/1-DeltaV = C_U m (7) and a_U/a = D_U 1-DeltaV/D_V-DeltaV - DeltaU 1-D_V/D_V-DeltaV (8) Table 1 Star Type DeltaU DeltaV m a_U/d 10^-4 a_U/a DeltaCep Cdelta 1.29 0.64 0.451 1.77 1.13 Eta Aql " 1.36 .65 .343 1.14 1.02 RS Cas " 1.24 .64 .465 1.82 1.18 XY Cas " 1.27 .75 .223 0.875 1.03 IX Cas " 1.27 .60 .513 2.01 1.16 ST Tau CW 1.19 .67 .448 1.76 1.24 AP Her CW 1.15 .71 .394 1.54 1.31 CC Lyr CW 1.04 .75 .363 1.42 1.69 RR Lyr RR 0.91 .73 .485 1.90 1.30 T Sex RR 0.87 .80 .335 1.32 5.37 EH Lib RR 0.97 .77 .355 1.39 2.20 Table 2. Type DeltaU DeltaV m a_U/d 10^-4 a_U/a RR 0.96 0.77 0.362 1.42 2.24 CW 1.13 .71 .405 1.59 1.35 C 1.31 .66 .428 1.68 1.13 Tables 1 and 2 give the result of calculations according to expressions (7) and (8) for cepheids of different types. We came to the following conclusions: 1. The location of cepheids on the gradient diagram can be explained assuming that amplitude of radius changes in the U-region differs from those in the B- and V-regions. 2. It is possible to calculate the ratio of these amplitudes using the results of photoelectric observations eq. (8). This ratio is the greatest for RR Lyr type and the smallest for classical cepheids. 3. Photoelectric observations permit to obtain the relation between changes of radius and temperature eq. (7). Kiev, June 19, 1969 I. G. KOLESNIK The Main Astronomical Observatory of the Ukrainian Academy of Sciences References [1]. Kheylo, E. S., Inf. Bull. Var. Stars. No 356, 1969 [2]. Landolt-Boernstein. Numerical Data and Functional Relationships in Science and Technology. Group VI. Vol. 1. Springer-Verlag. 1965. p. 346.