COMMISSION 27 OF THE I. A. U. INFORMATION BULLETIN ON VARIABLE STARS Number 2528 Konkoly Observatory Budapest 1 June 1984 HU ISSN 0374-0676 COMMENTS ON THE P-L-C RELATION OF THE CLASSICAL CEPHEIDS The problem of P-L-C relation for classical Cepheids: M_ = a log P + b _0 + c (1) is discussed nowadays for the reason of various numerical values proposed for the colour term coefficient b and because of many critical remarks concerning the methods of calculation of this quantity, see Clube and Dave (1983). Here we present the arguments in favour of a large value of the colour term coefficient for galactic and LMC Cepheids in accordance with Brodie and Madore's (1980) results. 1. The paper by Fernie (1984) contains the list of Cepheid radii obtained by Wesselink method. The _0 for these stars have been obtained from the values taken from a catalogue by Schaltenbrand and Tammann (1971) and reddenings from Dean et al. (1978) and Pel (1978). On the basis of these data we got for 20 most reliable radii and log P<1 the following P-R-C relation: log R = 1.029 log P - 0.572 _0 + 1.259 (2) +-0.113 +-0.181 +-0.077 s.d. = 0.042 The small value of the colour term coefficient -0.572 justified the application in this case of the standard least squares method, which due to the narrow range of _0 and its correlation with log P leads to a systematic lowering of greater values of coefficients. In order to pass from log R to the absolute magnitudes, M_, we use the formula: M_ = -5 log R + Sv, (3) where Sv is the surface brightness: Sv = 42.312 - 10 log Te - B.C. (4) According to van Genderen (1983), for Cepheids B.C. = 0.430 - 0.603 _0 (5) and log Te = 3.870 - 0.175 _0 (6) So we got Sv = 3.182 + 2.353 _0 (7) and the P-L-C relation: M_ = -5.295 log P + 5.213 _0 - 3.115 (8) From this example we see that the small values for b are not acceptable because of significant dependence of Sv on _0. Therefore b should be greater than 2.353 and in this case amounts to 5.213. 2. As the next group of stars we consider the long period Cepheids in LMC. The following numerical data have been taken from the paper by van Genderen (1983): log P, Vjo which we assume to be equal to M_ + Mod, and (B-V)je instead of _0. In this case we use the following procedure, avoiding the least squares method: Eq. (1) means that the Cepheids are placed on the plane in the three dimensional space: log P-M_ - _0. We divide the group of the investigated stars into two halves with shorter and longer periods and calculate for both groups the mean values: log P1, M_1, _01 and log P2, M_2, _02. We assume that the points with the coordinates so obtained are placed also on the plane defined by eq. (1). The projections of these points on the log P-M_ plane have the coordinates log P1, M_1 and log P2, M_2 and they determine the P-L relation as the straight line: M_ = g log P + h (9) The individual deviations of stars from this line: Delta M_ = M_ - g log P - h (10) are due to the arrangement of stars on the P-L-C plane and should not be treated only as errors. On the contrary, their existence is a proof of reality of the plane defined by eq. (1). Similarly we got the P-C relation as a straight line _0 = d log P + e (11) on the log P-_0 plane passing through the points log P1, _01 and log P2, _02. So it is possible to calculate the similar deviations for individual stars: Delta _0 = _0 - d log P - e (12) From the geometry of this problem it follows that the deviations delta M_ and delta _0 and the quantities occurring in eqs. (1), (9), and (11) are related as follows: Delta M_ = b Delta _0, (13) a = g - b d, c = h - be (14) This method applied to 19 long-period Cepheids in LMC according to van Genderen's paper (1983) led to the following results: Vjo = M_ + Mod = -2.516 log P + 16.413, (B-V)je = _0 = 0.472 log P + 0.078 [FIGURE 1] Figure 1 Determination of the coefficient b for the LMC Cepheids The deviations Delta Vjo = Delta M_ and Delta (B-V)je= Delta _0 are plotted in Figure 1. Using again the mean values of these quantities for positive and negative Delta M_ we got from eq. (13) b = 5.82 and from eqs. (14): a = -5.26 and c + Mod = 15.96. The same numerical data, treated by standard least squares method, led to the values: a = -3.474, b = 2.307, c + Mod = 16.040. As it was stated above in this case the least squares method gives significantly lower value for the coefficient b. But, as is shown in Figure 1, b = 2.307 does not suit with the observational points. 3. Finally the M_ and _0 values for 51 galactic Cepheids with _0 < 0.85 have been taken from the author's paper, (Opolski, 1982) and subjected to the same method as LMC stars. The results are as follows: M_ = -2.635 log P - 1.971, o = 0.424 log P + 0.304, a = -5.310, b = 6.31, c = -3.457 In order to get in this problem the results without systematic errors introduced by the least squares method it is enough to change the form of the P-L-C relation to: _0 = x log P + y M_ + z (15) In this case the small numerical values of the coefficients x and y and the greater range of the M_ variability allow to get proper results using the least squares method. In this way for stars in the LMC taken again from van Genderen's paper (1983) we got: (B-V)je = 0.884 log P + 0.173 Vjo - 2.726 +-0.130 +-0.051 +-0.839 s.d. = 0.048 From this we have: Vjo= -5.106 log P + 5.774(B-V)je + 15.739, whereas the application of the same method directly to the eq. (1) gives: Vjo = -3.474 log P + 2.307(B-V)je + 16.040 +-0.357 +-0.686 +-0.257 s.d. = 0.176 But this solution has a systematic error. The differences between the observed and calculated Vjo are correlated with these quantities. The negative differences are predominating in the range of smaller Vjo while the positive ones are connected mostly with larger Vjo. Therefore they do not have the character of accidental errors. Similarly for galactic Cepheids (Opolski, 1982) we got: _0 = 0.982 log P + 0.191 M_ + 0.646 +-0.095 +-0.035 +-0.076 s.d. = 0.054 or M_ = -5.141 log P + 5.238 _0 - 3.384 For this case eq. (1) gives directly: M_ =-3.548 log P + 1.998_0 - 2.540, +-0.206 +-0.367 +-0.131 s.d. = 0.176, with the similar systematic error as in the foregoing example. Also the result obtained by Martin et al. (1979) for 26 stars in the LMC achieved by the maximum likelihood fit: a = -3.80, b = 2.70, c+ Mod = 16.41, is encumbered with the systematic errors. It is worth to notice that nine SMC Cepheids, (van Genderen, 1983), give more consistent results: (B-V)je - 1.349 log P + 0.338 Vjo - 5.789 +-0.151 +-0.063 +-1.032 s.d. = 0.038 or Vjo = -3.988 log P + 2.957(B-V)je + 17.119 The relation obtained directly from eq. (1) for these stars is the following: Vjo = -3.702 log P + 2.454(B-V)je + 17.009 +-0.283 +-0.454 +-0.220 s.d. = 0.101 In this case the smaller value of the coefficient b causes that the lowering from 2.957 to 2.454 is not so significant as in the other examples. A. OPOLSKI and T. CIURLA Wroclaw University Observatory 51-622 Wroclaw, Poland References: Brodie, J.P. and Madore, B.F., 1980, M.N.R.A.S., 191, 841 [BIBCODE 1980MNRAS.191..841B ] Clube, S.V.M. and Dave, J.A., 1983, Astr.Ap., 122, 255 [BIBCODE 1983A&A...122..255C ] Dean, J.F., Warren, P.R. and Cousins, A.W.J., 1978, M.N.R.A.S., 183, 569 [BIBCODE 1978MNRAS.183..569D ] Fernie, J.D., 1984, in press. [BIBCODE 1984ApJ...282..641F ] Martin, W.L., Warren, P.R. and Feast, M.W., 1979, M.N.R.A.S., 188, 139 [BIBCODE 1979MNRAS.188..139M ] Opolski, A., 1982, Comm.Konkoly Obser. Budapest, No. 83, 227 [BIBCODE 1982CoKon..83..227O ] [CoKon No. 83] Pel, J.W., 1978, Astron.Astrophys., 62, 75 [BIBCODE 1978A&A....62...75P ] Schaltenbrand, R. and Tammann, G.A., 1971, Astr.Ap.Sup.Ser., 4, 265 [BIBCODE 1971A&AS....4..265S ] van Genderen, A.M., 1983, Astr.Ap., 124, 223 [BIBCODE 1983A&A...124..223V ]