Stellar pulsation
Stellar pulsations are caused by expansions and contractions in the outer layers as a star seeks to maintain equilibrium.
These giant stars are observed to undergo pulsations ranging from weak irregularity, when one can still define an average cycling time or period, (as in most RV Tauri and semiregular variables) to the near absence of repetitiveness in the irregular variables.
[2][3] Stellar evolution and pulsation theories suggest that these irregular stars have a much higher luminosity to mass (L/M) ratios.
Many stars are non-radial pulsators, which have smaller fluctuations in brightness than those of regular variables used as standard candles.
[4][5] A prerequisite for irregular variability is that the star be able to change its amplitude on the time scale of a period.
This coupling is measured by the relative linear growth- or decay rate κ (kappa) of the amplitude of a given normal mode in one pulsation cycle (period).
numerical stellar modeling and linear stability analysis show that κ is at most of the order of a couple of percent for the relevant, excited pulsation modes.
On the other hand, the same type of analysis shows that for the high L/M models κ is considerably larger (30% or higher).
These two properties are very general and occur for oscillatory systems in many other fields such as population dynamics, oceanography, plasma physics, etc.
The weak nonlinearity and the long time scale of the amplitude variation allows the temporal description of the pulsating system to be simplified to that of only the pulsation amplitudes, thus eliminating motion on the short time scale of the period.
[9][10][11] For example, in the case of two non-resonant modes, a situation generally encountered in RR Lyrae variables, the temporal evolution of the amplitudes A1 and A2 of the two normal modes 1 and 2 is governed by the following set of ordinary differential equations
No other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients.
In this picture, the boundaries of the instability strip where pulsation sets in during the star's evolution correspond to a Hopf bifurcation.
[17] The existence of a center manifold eliminates the possibility of chaotic (i.e. irregular) pulsations on the time scale of the period.
The light curves of intrinsic variable stars with large amplitudes have been known for centuries to exhibit behavior that goes from extreme regularity, as for the classical Cepheids and the RR Lyrae stars, to extreme irregularity, as for the so-called Irregular variables.
Low-dimensional chaos in stellar pulsations is the current interpretation of this established phenomenon.
The regular behavior of the Cepheids has been successfully modeled with numerical hydrodynamics since the 1960s,[18][19] and from a theoretical point of view it is easily understood as due to the presence of center manifold which arises because of the weakly dissipative nature of the dynamical system.
The boundaries of the instability strip where pulsation sets in during the star's evolution correspond to a Hopf bifurcation.
In contrast, the irregularity of the large amplitude Population II stars is more challenging to explain.
The variation of the pulsation amplitude over one period implies large dissipation, and therefore there exists no center manifold.
It is now established that the mechanism behind the irregular light curves is an underlying low dimensional chaotic dynamics (see also Chaos theory).
The computational fluid dynamics numerical forecasts for the pulsations of sequences of W Virginis stellar models exhibit two approaches to irregular behavior that are a clear signature of low dimensional chaos.
The first indication comes from first return maps in which one plots one maximum radius, or any other suitable variable, versus the next one.
The sequence of models shows a period doubling bifurcation, or cascade, leading to chaos.
[26][27] The following shows a similar visualization of the period doubling cascade to chaos for a sequence of stellar models that differ by their average surface temperature T. The graph shows triplets of values of the stellar radius (Ri, Ri+1, Ri+2) where the indices i, i+1, i+2 indicate successive time intervals.
The presence of low dimensional chaos is also confirmed by another, more sophisticated, analysis of the model pulsations which extracts the lowest unstable periodic orbits and examines their topological organization (twisting).
[28] The method of global flow reconstruction[29] uses a single observed signal {si} to infer properties of the dynamical system that generated it.
Takens' theorem guarantees that under very general circumstances the topological properties of this reconstructed evolution operator are the same as that of the physical system, provided the embedding dimension N is large enough.
The fractal dimension of the dynamics of R Scuti as inferred from the computed Lyapunov exponents lies between 3.1 and 3.2.
[32] This resonance mechanism is not limited to R Scuti, but has been found to hold for several other stars for which the observational data are sufficiently good.
