The 'Keplerian' revolution in our understanding of RR Lyrae stars

R. Szabó1 , Z. Kolláth1 ,2, L. Molnár1 ,2, E. Plachy1

1 Konkoly Observatory, Research Center for Astronomy and Earth Sciences, Hungarian Academy of Sciences,

H-1 1 21 Budapest, HUNGARY, [email protected] 2University of West Hungary, Faculty of Natural Sciences, Institute of Mathematics and Physics, Szombathely

Fig. 1 . Known RR Lyrae stars in the Kepler field. Red Fig. 2. An example for a non-Blazhko RRab star in the Kepler Fig. 3. Period doubling in the Q5 short-cadence data of the denotes Blazhko-modulated RRab stars, blue dots are non- field: FN Lyr (KIC 69361 1 5). We show part of the Q5 short Blazhko-modulated RR Lyrae, the eponym. Note the large modulated RRab variables, whereas green shows RRc cadence data. Note the constant amplitude. difference between consecutive maxima at the beginning of the pulsators. shown data set.

Period doubling (PD) Radial resonances

RR Lyrae stars are large amplitude, horizontal-branch pulsators. Their variability has We successfully reproduced period doubling in 1 -D full hydrodynamical models, and been well-understood and explained by low-order radial pulsation modes established that the physical origin of PD is a high-order (9:2) resonance between the (fundamental mode: RRab, first overtone: RRc or both: RRd). The RR Lyrae Working fundamental mode and the 9th radial overtone (Szabó et al. 201 0, Kolláth et al. 201 1 ). Group of the Kepler Asteroseismic Science Consortium (KASC) made a careful pre- It is a strange mode in the sense that it is trapped in the outer atmosphere and it can selection of all known RR Lyrae variables in the Kepler field (see Fig. 1 .). The Kepler be even excited, whereas any higher order (n>2) radial overtones are ususally heavily space telescope observed this sample and delivered observations of unprecedented damped. This resonance is at work for a relatively large portion of the mass - quality. - - parameter space.

Blazhko effect has been dicovered more than a century ago and currently lacks a Buchler & Kolláth (201 1 ) used the amplitude equation formalism and demonstrated plausible physical explanation. It is the conspicuous amplitude and phase modulation that if the 9:2 resonance between the fundamental mode and the 9th overtone is of about half of the RRab stars (RRc stars show a somewhat lower percentage.) present, then in a large part of the parameter space regular, irregular, and even chaotic modulations occur naturally. In other words, the 9:2 resonance that causes the Period doubling was noticed in the first long cadence Kepler light curves period doubling is also capable of inducing the Blazhko effect itself. We call this (Kolenberg et al. 201 0, Szabó et al. 201 0) as the alternating maxima in some of theory the radial resonance paradigm. The radial resonance paradigm is currently the the Blazhko RR Lyrae stars. PD (or period-2 state) manifests itself in the frequency best one among the many that tries to explain the Blazhko effect. Additional domain as the presence of half-integer frequencies between the dominant resonances and the presence of nonradial modes are also highly probable, and finding pulsational frequency and its harmonics. The majority of Blazhko-modulated stars them should be of prime priority. show PD, at least temporarily, but it has never been seen in non-modulated RR Lyrae stars (an example is shown in Fig. 2.), even with the precision allowed by Full hydro computations showed that our models often approach other high-order Kepler (Szabó et al. 201 0, Nemec et al. 201 1 ). This fact demonstrates that there is resonant states (1 4:1 9, 20:27, etc.) between the fundamental and the first overtone a strong connection between period doubling and the Blazhko effect. In Fig. 3. we modes, creating a huge variety of complex dynamical behaviors, such as triple-mode see a characteristic and very strong PD phase of RR Lyrae, the prototype of its state or low-dimensional chaos. class.

Triple-mode state Additional frequencies Conclusions

We investigated the hydro models with PD that showed In addition to the traditional frequency solution, extra * We have uncovered an incredible variety of frequencies can be frequently found in Blazhko RR period-6 behavior as well (where the height of maxima dynamical behaviors with Kepler, never before seen in repeats after 6 pulsation periods), indicating that the Lyrae stars. Most of these frequencies tend to show up RR Lyrae stars. model can temporarily be close to the 3:4 resonance around the predicted frequencies of the first and between the fundamental and first overtone. This second overtones (see e.g. Benkõ et al. 201 0). period-6 characteristic was found in the Kepler Nonradial modes can be unstable in RR Lyrae models, * In particular, period doubling is intimately related to observations of RR Lyr, the prototype. Based on the and these modes are most easily excited in the vicinity the enigmatic Blazhko effect. models, we predicted the presence of the first overtone of radial overtones, therefore their origin can be either radial or non-radial modes. with low amplitude. * The origin of the period doubling is a high-order resonance between radial modes. Indeed, Q5-Q6 Kepler data of RR Lyrae, the prototype showed the first overtone frequency with high Chaos significance (Molnár et al. 201 2). Based on this result, * This very same resonance might be the long-sought the presence of the first radial overtone itself is well- physical mechanism of the mysterious Blazhko- A series of period-doubling bifurcations can lead to modulation. established, but additional, non-radial modes cannot be chaos. We were able to show such bifurcation cascades completely ruled out, either. This three-mode state in RR Lyrae models. There are indications that the (fundamental, first and ninth radial overtones) modulation of some of the Kepler RRab stars with * We demonstrated that in RR Lyrae, the prototype, the represents a new pulsational behavior, which differs shorter modulations periods may be the result of low- first radial overtone is also excited, hence a new, triple- from the nature of the well-known double-mode (RRd) dimensional chaos. For the long-period Blazhko stars mode state was found. pulsators, enables an even more diverse bonanza of more observational data would be essential. We complex dynamical states, including chaos and emphasize that the radial resonance model is able to foreshadows the application of nonlinear predict chaotic modulation cycles, hence observations * There are indications that in some cases both the asteroseismology to RR Lyrae stars. seem to confirm the theoretical predictions. pulsation and the modulation can be chaotic.

References Acknowledgements

1 . Benkõ, J. M., Kolenberg, K., Szabó, R.: MNRAS, 409, 1 585, 201 0 This project has been supported by the Lendület-2009 Young Researchers Program of the Hungarian Academy of Sciences and the HUMAN MB08C 81 01 3 grant of the MAG Zrt, the 2. Buchler, J. R., Kolláth, Z.: ApJ, 731 , 24, 201 1 Hungarian OTKA grant K83790 and the KTIA URKUT_1 0-1 -201 1 -001 9 grant. 3. Kolenberg, K., Szabó, R., Kurtz, D.W. et al.: ApJL, 71 3, 1 98, 201 0 4. Kolláth, Z., Molnár, L., Szabó, R.: MNRAS, 41 4, 1 1 1 1 , 201 1 The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-201 3) under grant agreement no. 2691 94 (IRSES/ASK) and in 5. Molnár, L., Kolláth, Z., Szabó, R. et al.: ApJL, 757, 1 3, 201 2 part by the National Science Foundation under Grant No. NSF PHY05-51 1 64. 6. Nemec, J., Smolec, R., Benkõ, J. M. et al.: MNRAS, 41 7. 1 022, 201 1 7. Szabó, R., Kolláth, Z., Molnár, L. et al.: MNRAS, 409, 1 244, 201 0 R. Szabó was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the University of Sydney IRCA grant and the Discretional Funds of the MTA CSFK's Director General.