Chaos in the Stars

Irregular Variable stars

 

R Sct

Lightcurve of the variable star R Scuti
(data from American Association of Variable Star Observers)

1. Time-series analysis of observational lightcurve of the star R Scuti

The nonlinear time-series analysis alows us to infer the properties of the physical dynamics of the star from the mere knowledge of the observed lightcurve.
Using a 'flow reconstruction' technique [1] we have constructed a map that generates the observed lightcurve of R Scuti [2, 3].
The remarkable properties of this map are its low dimension of 4. This means that a simple set of four order first order differential equations can generate the complex lightcurve of R Sct.
The pulsations of this star are quite violent, with the luminosity varying up to factors of 40, with shockwaves and ionization fronts criss-crossing the stellar envelope. The uncovered low dimensional behavior is thus perhaps surprising.

As a byproduct of the time-series analysis we find that the fractal dimension of the attractor is approximately 3.1 - 3.2 .

From the map we also infer the interesting physical result that the complicated lightcurve is the result of the nonlinear, resonant interaction between two modes of pulsation, the first one linearly unstable and the second one, linearly stable, with almost twice the frequency of the first. An analysis of additional stars gives strong evidence that this resonant interacion mechanism is generic, at least for a subgroup or irregularly pulsating stars.

  • Buchler, J. R., Serre, T., Kollath, Z. & Mattei, J. 1995,  A Chaotic Pulsating Star -- The Case of R~Scuti, Physical Review Letters 74, 842.
  • Buchler, J. R., Kollath, Z. & Cadmus, R. 2001, Chaos in the Music of the Spheres, Proceedings of CHAOS 2001, July 22 - 26, Potsdam, Germany, (in press); astro-ph/0106329
  • Buchler, J.R., 1996, Search for Low-Dimensional Chaos in Observational Data, International School of Physics "Enrico Fermi", Course CXXXIII, "Past and Present Variability of the Solar-Terrestrial System: Measurement, Data Analysis and Theoretical Models", Eds. G. Cini Castagnoli & A. Provenzale, 275-288, Società Italiana de Fisica, Bologna, Italy. chao-dyn/9707012
  •  Serre, T., Kollath, Z. & Buchler, J. R., 1996,  Search For Low--Dimensional Nonlinear Behavior in Irregular Variable Stars -- The Global Flow Reconstruction Method, Astronomy & Astrophysics 311, 833.
  •  Buchler, J. R., Kollath, Z., Serre, T. & Mattei, J. 1996,  A Nonlinear Analysis of the Variable Star R Scuti, Astrophysical Journal, 462, 489. astro-ph/9707116
  • Buchler, J.R. 1993, A Dynamical Systems Approach to Nonlinear Stellar Pulsations, in Nonlinear Phenomena in Stellar Variability, Eds. M. Takeuti & J.R. Buchler, Dordrecht: Kluwer Publishers, reprinted from 1993, Ap&SS, 210, 1-31. paper
  • In French: Buchler, J. R. & Kollath, Z. 2001, Du chaos dans la musique des étoiles , (in french), Comptes Rendus du Colloque sur le "Chaos temporel et chaos spatio-temporel", 24-25 sep. 2001, Le Havre, France (in press);    nlin.CD/0109028     postscript file, with higher resolution figures

  • Buchler, J. R., Kolláth, Zoltan and Cadmus, R., 2004, Evidence for Low-Dimensional Chaos in Semiregular Variable Stars, Astrophysical Journal, Vol. 613, pp. 532 - 547.
    http://xxx.lanl.gov/abs/astro-ph/0406109    ApJ    .ps.gz

    2. Numerical Hydrodynamic simulations

    State of the art hydrodynamical simulations of the nonlinear behavior of similar types of stars (W Virginis type stars) that were performed a decade ago [5, 6] had predicted the chaotic nature of these pulsations.
    Indeed, sequences of models show a period doubling cascade as a control parameter (the effective temperature of the equilibrium stellar model) is varied. One-dimensional first return maps of the pulsation maxima for an irregularly pulsating model clearly display a Feigenbaum quadratic maximum.
    The "flow reconstruction" technique indicates that the pulsations take place in a 3D phase-space [7]. The fractal dimension of the attactor is approximately 2.05-2.10. The attractor bears great similarity to the well known Roessler system.

    R Sct Template for the attractor.

    A topological analysis based on the way the orbits of the attractor are organized spatially [8] shows that the attractor is actually only superficially similar to the Roessler system.
     

  • Buchler, J.R. & Kovacs, G. 1987,   Period-Doubling Bifurcations and Chaos in W Virginis Models,   Astrophysical Journal Letters 320, L57-62
  • Kovacs G. & Buchler J.R. 1988,   Regular and Irregular Pulsations in Population II Cepheids , Astrophysical Journal 334, 971.
  • Serre, T., Kollath, Z.  & Buchler, J. R., 1996,   Search For Low--Dimensional Nonlinear Behavior in Irregular Variable Stars -- The Analysis of W Vir Model Pulsations, Astronomy & Astrophysics,  311, 845.
  • Letellier, C., Gouesbet, G., Soufi, F., Buchler, J.R. & Kollath, Z. 1996,  Chaos In Variable Stars: Topological Analysis of W Vir Model Pulsations, Chaos, 6, 466

    Work supported by National Science Foundation. Updated November 2005