project | Iuventus+, IP2012 036572 |
funding | Polish Ministry of Science and Higher Education, MNiSW |
PI | Radosław Smolec |
dates | 06.2013 - 11.2015, finished |
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Fig. 1. Possible dynamical scenarios in the HR diagram for type-II Cepheid models of different mass, metallicity and convective parameters. Grey-shaded areas correspond to domains of period-doubling. Filled diamonds mark models in which period-4 pulsation was detected. In models marked with open circles modulation was found. Open squares in the bottom right panel are double-mode F+4O models. Black dashed lines plotted in each panel are lines of constant fundamental mode period, from top to bottom: 2, 4, 8, 12 and 16 days. Loci of various half-integer resonances are marked with coloured dotted lines as indicated in the bottom part of the figure.
We have conducted a large survey of hydrodynamic type-II Cepheid models with two different masses (0.6M_{S} and 0.8M_{S}) and three metallicities, [Fe/H]=-1.0, -1.5 and -2.0. We also used two sets of convective parameters. Our models display a wealth of dynamical behaviours, in particular:
No large-amplitude chaotic behaviour was detected in our models. We note that our survey is limited in luminosity due to dynamical instability that prevents computation of the most luminous models. Read more in MNRAS paper (PDF, arXiv:1512.01550).
Fig. 2. Top left panel: radius variation for the F+4O double-mode model; Bottom left panel: frequency spectrum after prewhitening with the fundamental mode and its harmonics (dashed blue lines); Right panels: amplitude of the fundamental mode (top) and of the fourth overtone (bottom) during the initial phase of model integration.
In our survey of type-II Cepheid models we have detected a very interesting form of pulsation: double-mode pulsation in the fundamental mode and in the fourth overtone (open squares in Fig. 1). Fourth overtone is a special mode: it is trapped in the outer model layers, between the surface and the partial ionization regions (such modes are sometimes called strange, see e.g. Buchler et al. 1997, ADS). Although its linear growth rate is negative, the mode is only weakly damped as compared to the neighbouring overtones. Amplitude of the fourth overtone is very small; the beating pattern is barely visible in the radius variation curve (top left panel in Fig. 2), but the presence of the fourth overtone is clearly revealed in the frequency spectrum (bottom left panel in Fig. 2). This form of pulsation was detected only in the more massive models adjacent to the blue edge of the instability strip (Fig. 1). Fundamental mode periods are 5.6-6.8 days, period ratios, P_{4O}/P_{F}, are around 0.3. The fourth overtone is weakly damped which is well visible in the bottom right panel of Fig. 2: its amplitude drops in the initial phase of model integration and starts to grow only when the amplitude of the fundamental mode (top right panel) becomes significant. Nevertheless, the origin of the beat pulsation is most likely non-resonant. Further analysis is needed and planned. Read more in MNRAS paper (PDF, arXiv:1512.01550).
In a sequence of hydrodynamic BL Herculis type models (L=const, M=const, T_{eff} varied) we have found a wealth of dynamic behaviours characteristic for deterministic chaos. Bifurcation diagram for the model sequence is presented above. It shows the possible values of maximum radii, R_{max}, as a function of model's effective temperature. Many of the phenomena are detected for the first time in the context of pulsating stars. In our models we observe different routes to chaos, crisis bifurcation, stable periodic domains within chaotic bands, type-I and type-III intermittency.
Hydrodynamic BL Her models are similar to other well known physical sytems showing the deterministic chaos. We find close anlogies between our models and even the simplest example of chaos: iteration of the logistic map, i.e. x_{i+1}=kx_{i}(1-x_{i}). Animations below show: (left) the first return map for logistic equation (a plot of x_{i+1} vs. x_{i}, as a function of k), and (right) first return map for maximum radii in the BL Her models models (a plot of R_{max}^{i+1} vs. R_{max}^{i} as a function of T_{eff}). Read more in MNRAS paper (PDF, arXiv:1403.4937).