4. Nonlinear effects
4.1. Existence of a limit cycle
For most of the models we started the simulation with small amplitude, which grew until the pulsation settled into a limit cycle oscillation. Some of the models, however, did not have limit cycles in nonlinear regime. For the most luminous models in Fig. 1 the amplitude at the photosphere became so large that it was not possible to continue the simulation. There are similar cases in the less-massive supergiant stars and most of them show chaotic pulsations as steady states of nonlinear pulsation. Although we cannot confirm a steady state of nonlinear pulsation of the present models, we suspect these stars do not have limit cycles. Thus, we expect that observed counterparts of these stars, namely those with P longer than about 150 d (Madore 1985) are not regular pulsators. It is interesting to compare this result with that obtained in Paper I for galactic Cepheids. The models with studied in Paper I have no limit cycles for d, which indicates a dependence of the maximum P on the metallicity, in the sense that such a P is longer for lower metallicities. The observed stars in Galaxy and Magellanic Clouds confirm qualitatively this theoretical result.
4.2. Nonlinear periods
It is well known that the differences among periods in linear adiabatic, linear nonadiabatic and limit cycle pulsation calculations are quite small in short period Cepheids. For longer period Cepheids, however, the differences become significant. There is a strong coupling between acoustic waves and thermal waves, and this makes the nonadiabatic periods quite longer than adiabatic ones in luminous cepheids (Aikawa 1985). We show in Fig. 3 that the pulsation periods at limit cycles are even longer than the linear nondiabatic ones, and the relative difference between linear nonadiabatic () and nonlinear () periods,
increases remarkably for small changes of the luminosity L. The result is that the slope of the theoretical PL relation for longer P is quite smaller than for shorter P.
Carson & Stothers (1984) tried to explain the observed flattening of the PL relation of Cepheids in Magellanic Clouds for d using nonlinear pulsation models, and ascribed this behavior to the low values. However we note that if it was just a matter of , the period-luminosity-color PLC relation should not show a `flattening'. Laney & Stobie (1986) remark that Magellanic Cloud stars with d are subluminous both in the PL and PLC relations, and by about the same amount, and are not unusually red for their periods. We conclude that the reason for such an observed effect is not just the low , but it is the lengthening of the nonlinear P displayed in Fig. 3. Actually the models can explain quantitatively only a small part of the observed effect.
It is possible to conclude from Fig. 3 that the lengthening is typical of low metallicity galaxies, because, on the one hand, it should not be possible to find very long P Cepheids in metal rich galaxies, and on the other hand for larger Z values the lengthening appears to be generally smaller.
© European Southern Observatory (ESO) 2000
Online publication: December 11, 2000