An overview of stellar pulsation theory

In this paper I will give overview of the stellar pulsation theory. Starting with basic equations I will discuss modal properties of oscillations and excitation mechanisms. I also mention briefly the effects of rotation.     

Bifurcation in hydrodynamic models of stellar pulsation

Phenomena of bifurcation in hydrodynamic stellar models of radial pulsation are reviewed. By changing control parameters of models, we can see qualitatively different pulsation behaviors in hydrodynamic models with transitions due to various types of bifurcation.In weakly dissipative models (classical Cepheids), the bifurcation is induced by modal resonances. Two types of the modal resonances found in models are discussed: The higherharmonic resonances of the second overtone mode in the fundamental mode pulsator and of the fourth overtone mode in the first overtone pulsator are relevant to observations. The subharmonic resonance between the fundamental and first overtone modes is confirmed in classical Cepheid models.In strongly dissipative models (less-massive supergiant stars), the bifurcation of nonlinear pulsation is induced by the hydrodynamics of ionization zones as well as modal resonances. The sequence of the bifurcation sometimes leads to chaotic behaviors in nonlinear pulsation. The transition routes from regular to the chaotic pulsations found in models are discussed with respect to the theory of chaos in simple dynamical systems: The cascade of period-doubling bifurcation is confirmed to cause chaotic pulsation in W Virginis models. For models of higher luminosity, the tangent bifurcation is found to lead intermittent chaos.Finally, hydrodynamic models for chaotic pulsation with small amplitudes observed in the post-AGB stars are briefly discussed.     

The importance of radiative transfer in stellar pulsation models

With the advent of new astrophysical opacities it seems appropriate to discuss the need for a full radiative transfer (RT) theory instead of the usual equilibrium diffusion theory used in most nonlinear pulsation codes. Early studies on the importance of RT in the calculation of light curves for Cepheid models showed little effect over diffusion theory. The new opacities though may help to explain the bump mass discrepancy problem. For RR Lyrae models the use of RT theory causes some effects both in the color differences (U-B) as well as in the light curves. New opacities help to explain the period ratios for double mode RR Lyrae and beat Cepheids. A new area of research is in the modeling of stars with high luminosity to mass ratios that show tendencies for doubling and transitions to chaos, such as W Virginis and RV Tauri stars. For these stars it has been shown that RT is necessary in calculating their light curves and that the understanding of the shock dynamics depends on the transfer of lines in the pulsating RT dependent atmospheres (Fokin 1991).     

The coupling coefficients of pulsation for radiative stellar models

The second order theory of coupling is discussed regarding the radial pulsation of stellar models which are constructed ignoring convection. The formula including the nonadiabatic effect is presented. Numerical values given for model classical cepheids are considerably greater than the adiabatic values.     

Intermittent chaos in a subharmonic bifurcation sequence of stellar pulsation models

One of the model sequences which is found to show the period-doubling cascade by Kovacs and Buchler (1988) is re-examined by the hydrodynamic pulsation code with dynamic rezoning of the hydrogen ionization zones (TGRID). It is found that the sequence shows subharmonic bifurcations and chaotic behaviours as the effective temperature, a control parameter in this sequence, is varied, as found by Kovacs and Buchler. The subharmonic cascade, however,is terminated at period 4, and the type I intermittency takes place against the cascade. We conclude that the chaotic behaviour is caused by the intermittency in our model sequence rather than the infinite period-doubling cascade. Finally, we briefly discuss effects of the outburst of highly irregular oscillation due to the intermittency on the tendency in the train of light curves.     

Three-mode Cepheid pulsation

We simulated two models of classical Cepheids with the same effective temperature and luminosity but different masses. Neither model has significant resonance among linear nonadiabatic periods of the fundamental, the first overtone, and the second overtone radial modes. The higher mass model found to approach to the second overtone-only state. The lower mass model shows a complicated three-mode interaction and none of the amplitudes of three modes is negligible. The relation of the higher mass model to the shortest period Cepheid HR 7308, and that of the lower mass model to the double-mode Cepheids are discussed. It appears that three-mode nonresonant coupling should be considered to explain the behaviours of both models.     

Three-mode Cepheid pulsation

Amplitude equations, that are used in the investigation of stellar pulsations, for the three-mode nonresonant coupling case are inferred from that of the two-mode nonresonant coupling case. Characteristics of the fixed points of the amplitude equations are studied. Chaotic behaviour is expected when the amplitude equations consist of more than two modes.     

Trends of stellar entropy along stellar evolution

This paper is devoted to discussing the difference in the thermodynamic entropy budget per baryon in each type of stellar object found in the Universe. We track and discuss the actual decrease of the stored baryonic thermodynamic entropy from the most primitive molecular cloud up to the final fate of matter in black holes, passing through evolved states of matter as found in white dwarfs and neutron stars.We then discuss the case of actual stars with different masses throughout their evolution, clarifying the role of the virial equilibrium condition for the decrease in entropy and related issues. Finally, we discuss the role of gravity in driving the composition and the structural changes of stars with different Main Sequence masses during their evolution up to the final product. Particularly, we discuss the entropy of a black hole in this context arguing that the dramatic increase in its entropy, differently from the other cases, is due to the gravitational field itself.     

The eight-schwabe-cycle pulsation

The shape of the Sun’s secular activity cycle is found to be a saw-tooth curve. The additional Schwabe cycle 4′ (1793–1799) suggested by Usoskin, Mursula, and Kovaltsov (2001a) is taken into account in the telescopic sunspot record (1610–2001). Instead of a symmetrical Gleissberg cycle, a saw-tooth of exactly eight Schwabe sunspot maxima (‘Pulsation’) is found. On average, the last sunspot maximum of an eight-Schwabe-cycle saw-tooth pulsation has been about three times as high as its first maximum. The Maunder Minimum remains an exception to this pattern. The Pulsation is defined as a secular-scale envelope of Schwabe-cycle maxima, whereas the Gleissberg cycle is a result of long-term smoothing of the sunspot series.     

The eight-schwabe-cycle pulsation

The shape of the Sun’s secular activity cycle is found to be a saw-tooth curve. The additional Schwabe cycle 4′ (1793–1799) suggested by Usoskin, Mursula, and Kovaltsov (2001a) is taken into account in the telescopic sunspot record (1610–2001). Instead of a symmetrical Gleissberg cycle, a saw-tooth of exactly eight Schwabe sunspot maxima (‘Pulsation’) is found. On average, the last sunspot maximum of an eight-Schwabe-cycle saw-tooth pulsation has been about three times as high as its first maximum. The Maunder Minimum remains an exception to this pattern. The Pulsation is defined as a secular-scale envelope of Schwabe-cycle maxima, whereas the Gleissberg cycle is a result of long-term smoothing of the sunspot series.     

Patterns of aperiodic pulsation

Techniques for deriving amplitude equations for stellar pulsation are outlined. For the simplest such equations with multiple instabilities, the derivation of a map for the patterns of pulsation phases is described. This map gives the time between two successive maxima of pulsation in terms of the time between the previous pair, under suitable conditions. The phase differences can be regular, chaotic or hyperchaotic.     

MODEST-1: Integrating stellar evolution and stellar dynamics

We summarize the main results from MODEST-1, the first workshop on MOdeling DEnse STellar systems. Our goal is to go beyond traditional population synthesis models, by introducing dynamical interactions between single stars, binaries, and multiple systems. The challenge is to define and develop a software framework to enable us to combine in one simulation existing computer codes in stellar evolution, stellar dynamics, and stellar hydrodynamics. With this objective, the workshop brought together experts in these three fields, as well as other interested astrophysicists and computer scientists. We report here our main conclusions, questions and suggestions for further steps toward integrating stellar evolution and stellar (hydro)dynamics.     

Analytic stellar structure

This paper continues to elaborate on analytic methods to construct models for the internal structure of solar-type stars. Since a detailed stellar model is desired, a nonlinear analytic density distribution in terms of a two-parameter family of models has been assumed. Hydrostatic equilibrium and energy conservation determine the conditions in the gravitationally stabilized stellar fusion reactor. The results show once more that methods of differential and integral calculus provide a laboratory for the application of special functions of mathematical physics in stellar structure.Paper presented at the Second International Conference on Industrial and Applied Mathematics, July 8–12 1991, Washington D.C., U.S.A.     

Multimode stellar pulsations

We apply the two time method to weakly anharmonic (nonlinear) stellar pulsations. The problem contains two small parameters: -the ratio of the dynamical and the thermal timescale and -the measure of the anharmonicity in the oscillator equation. The 0(⁰) system is solved up to 0() by the use of an asymptotic perturbation method. This solution is then used within the framework of the two time method to obtain the slow evolution of the amplitudes and entropy. Both monomode and double-mode pulsations are treated in a non-resonance case.Work supported by the National Science Foundation (AST79-20024).     

Local stellar stability

The traditional linearized stellar-structure equations, which provide information on the global stability of a stellar equilibrium state, are shown to engender a class of continuous modes, if the eigenfunctions are sought in a generalized function space. To endow the intuitive concept of Local Stellar Statility with a precise mathematical meaning, we endeavour to link this notion to the family of continuous spectra. We indicate a simple numerical method to compute the eigenfunctions associated with these modes, and we illustrate our approach by briefly analysing the continuous spectra connected with the radial and nonradial dynamical stability problem.     

Binary stellar winds

Stellar winds from a binary star pair will interact with each other along a contact discontinuity. We discuss qualitatively the geometry of the flow and field resulting from this interaction in the simplest case where the stars and winds are identical. We consider the shape of the critical surface (defined as the surface where the flow speed is equal to the sound speed) as a function of stellar separation and the role of shock waves in the flow field. The effect of stellar spin and magnetic sectors on the field configuration is given. The relative roles of mass loss and magnetic torque in the evolution of orbital parameters are discussed.     

Theory of solar radio pulsation

We show that the observed modulation of some coronal microwave, X-ray and Type III emission into pulses of 10 sec intervals is a consequence of the stimulation of electron cyclotron waves propagated in the whistler mode in dipole-like bipolar regions of dimension 0.2 R . Assuming that a power law spectrum of 10 keV electrons with a slope similar to solar flare protons can be trapped in a bipolar region, we show that whistlers can be generated by pitch angle instability. The resultant 10 sec bounce motion of whistler wave trains leads to enhanced, modulated emission in microwave and X-ray frequencies by pitch angle scattering of MeV electrons, and to modulated Type III emission by scattering with coherent plasma waves. A direct prediction of the theory is the existence of sympathetic pulsations at two sources a fraction of a solar radius apart. A second test of the theory is that modulated Type III emission should show strong polarization.This work was conducted under U.S. Air Force Space and Missile Systems Organization (SAMSO) Contract No. F04701-69-C-0066.     

The pulsation of Delta-Scuti stars

Evolution, linear pulsation, and nonlinear pulsation codes were combined to produce nonlinear models of Delta-Scuti stars in an instability region extending over 3.8T _e<3.95 and 0.6L/L <2.0. The linear analysis upheld the consensus that they are normal Population I stars of about 2M , in stages of evolution corresponding to central hydrogen burning and shell hydrogen burning. The growth rates were very slow; driving was due to an opacity mechanism in the second helium ionization region; periods and period ratios of the lowest modes of the models were in the same range as those observed. A wide range of nonlinear models was investigated. When eigenfunctions from the linear analysis were used as initial velocity profiles, it was found that the dominant peak in the periodogram of the light curve corresponded to the mode initiated. For a small subset of models, limiting amplitudes were identified, and were found to be in close agreement with observed light amplitudes.