On multifractality and synchronization of nonlinear stellar pulsators

Simple models of nonlinear stellar pulsation, whose temporal behavior may reproduce some of the observed features of different classes of variable stars, have been studied. The temporal behavior of dynamical variables of these models exhibits a cascade of period doubling chaos, depending on the specific values of the various control parameters. A multifractal detrended fluctuation analysis (MFDFA) method is further used to identify the scaling behavior of such synthetic time series. The MFDFA of the considered time series, for various models of nonlinear stellar pulsation, shows that the observed multifractal nature is due to long-range correlations. The pulsating star with increased nonadiabaticity and the star with increased convective luminosity, as represented by the simulated data, is shown to possess a strange attractor with noninteger correlation dimension that lies between 2–3. Also the problem of synchronization in coupled nonlinear pulsation models has been investigated using permutation entropy—a complexity measure of the system. The computed order parameter, Γ, representing the correlation of computed permutation entropy for different segments of the simulated time series of displacement of two nonidentical oscillators, has been further used to find the critical coupling parameter for general synchronization of the oscillators.     

Combination frequencies in the Fourier spectra of white dwarfs

Combination frequencies are observed in the Fourier spectra of pulsating DA and DB white dwarfs, along with frequencies that are associated with stellar gravity modes. They appear at the sum and difference frequencies of the stellar modes. Brickhill proposed that the combination frequencies result from mixing of the eigenmode signals by a depth-varying surface convection zone when undergoing pulsation. The depth changes cause time-dependent thermal impedance.
Following Brickhill's proposal, we developed analytical expressions for the amplitudes and phases of these combination frequencies. The parameters that appear in these expressions are the depth of the stellar convection zone when at rest, the sensitivity of this depth towards changes in the stellar effective temperature, the inclination angle of the stellar pulsation axis with respect to the line of sight, and lastly the spherical degrees of the eigenmodes involved in the mixing. Adopting credible values for these parameters, we apply our expressions to DA and DB variable white dwarfs. We find reasonable agreement between theory and observation, although some discrepancies remain unexplained. It is possible to identify the spherical degrees of the pulsation modes using the combination frequencies.     

UBV observations of post-AGB supergiants and peculiarities of their photometric variability

The systematic UBV observations of six variable post-AGB supergiants in 1991–1999 are presented. Their variability is analyzed. The coolest stars V1027 Cyg and V354 Lac exhibit bimodal pulsations with variable amplitudes. Apart from pulsations, the hotter stars V887 Her and IRAS 19386+0155 show light variations associated with a stellar wind. A variable stellar wind appears to be mainly responsible for the photometric variations in the still hotter stars SAO 163075 and IRAS 20572+4919. Distinct trends in the yearly mean brightness have been found in three of the six supergiants studied, with the trend amplitude being independent of the spectral range. They are interpreted as the result of dust envelopes composed of large grains with R=A _V /E(B?V)≥7 becoming optically thin.     

Photometric, spectroscopic and polarimetric variability of the weak-emission T Tauri star HD 288313

Photometric observations over three seasons show HD 288313 to be a light variable with a 2.2636-d period. The observed V amplitudes lie in the range of 0.06–0.15 mag. The star showed appreciable changes in the brightness at maximum and minimum of the light curve from season to season. The (   b − y   ) colour did not show any significant variation during the photometric cycle. The light variation appears to be caused by the rotational modulation of stellar flux by cool starspots distributed asymmetrically across the stellar longitudes. The Hα line strength in HD 288313 varied drastically from completely filled-in emission to almost full absorption, that is typical of a normal star of similar spectral type. The Hα equivalent width is found to show a clear rotational modulation only occasionally. Most of the time, chromospheric active regions are distributed well across the stellar longitudes, thereby suppressing obvious rotational modulations. Broad-band linear polarization measurements show HD 288313 to be a short period, low-amplitude polarization variable. The polarization variation is, apparently, rotationally modulated. Dust grain scattering in a non-spherical circumstellar envelope of a star with inhomogeneities in the surface brightness distribution seems to be the mechanism operating in producing the observed polarization.     

Tomography of stellar non‐radial pulsations

The stellar surface imaging technique is used for studying stellar non‐radial pulsations on the basis of inversions of time series of variable line profiles without making assumptions on the specific shape of the pulsations. The inversion results in an image of the stellar surface in which sectoral and tesseral modes can be distinguished in many cases and the pulsational degree and the azimuthal order can be determined. The capability of the technique is studied with simulated data. Then, the surface imaging technique is applied to high‐resolution spectra of the rapidly rotating Beta Cep‐type star ω¹ Sco, which shows strong line‐profile variations. Stellar surface imaging is concluded to be a useful technique for pulsation‐mode identification. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the effect of turbulent anisotropy on pulsation stability of stars

Within the framework of a non-local time-dependent stellar convection theory, we study in detail the effect of turbulent anisotropy on stellar pulsation stability. The results show that anisotropy has no substantial influence on pulsation stability of g modes and low-order(radial order n_r 5) p modes.The effect of turbulent anisotropy increases as the radial order increases. When turbulent anisotropy is neglected, most high-order(n r 5) p modes of all low-temperature stars become unstable. Fortunately,within a wide range of the anisotropic parameter c_3, stellar pulsation stability is not sensitive to the specific value of c_3. Therefore it is safe to say that calibration errors of the convective parameter c_3 do not cause any uncertainty in the calculation of stellar pulsation stability.     

Introduction

A historical sketch of the nonlinear theory of variable stars is outlined briefly. The main break-through came from the hydrodynamic study of stellar pulsation. From a theoretical point of view, coupling needs to be discussed more carefully. The impact of new opacities on the astrophysical problems is also discussed.     

A merger tree with microsolar mass resolution: application to γ-ray emission from subhalo population

Observations of the southern Cepheid ℓ Car to yield the mean angular diameter and angular pulsation amplitude have been made with the Sydney University Stellar Interferometer at a wavelength of 696 nm. The resulting mean limb-darkened angular diameter is 2.990 ± 0.017 mas (i.e. ± 0.6 per cent) with a maximum-to-minimum amplitude of 0.560 ± 0.018 mas corresponding to 18.7 ± 0.6 per cent in the mean stellar diameter. Careful attention has been paid to uncertainties, including those in measurements, in the adopted calibrator angular diameters, in the projected values of visibility squared at zero baseline, and to systematic effects. No evidence was found for a circumstellar envelope at 696 nm. The interferometric results have been combined with radial displacements of the stellar atmosphere derived from selected radial velocity data taken from the literature to determine the distance and mean diameter of ℓ Car. The distance is determined to be 525 ± 26 pc and the mean radius  169 ± 8 R_⊙  . Comparison with published values for the distance and mean radius shows excellent agreement, particularly when a common scaling factor from observed radial velocity to pulsation velocity of the stellar atmosphere (the p -factor) is used.     

Oscillations de l'atmosphère de l'étoile pulsante

Résumé On étudie les oscillations non-adiabatiques d'une atmosphère stellaire sous l'approximation du premier ordre, en déduisant les solutions générales des équations différentielles. On confirme les conditions aux limites de la pulsation d'étoile. Pour les étoiles variables des supergéantes rouges l'onde progressive est possible dans l'atmosphère, avec l'oscillation stationnaire.

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Non-adiabatic linear oscillations of the stellar atmosphere are studied by deriving the general solution of the equation of the problem. Boundary conditions which we obtained at the stellar surface are confirmed for the pulsation of the cepheid variables. It is shown that the progressive wave coupling with the stellar pulsation is possible in the atmospheres of red supergiants.

     

The effect of artificial viscosity on a hydrodynamic model of Cepheids

The effect of artificial viscosity on a hydrodynamic simulation of stellar radial pulsation is examined for the purpose of studying the dependence of the modal coupling on the sharpness of the shock front. The model used in our study is a 2.5-d first-overtone pulsator in the Cepheid instability strip. By increasing the parameters in the von Neumann–Richtmyer formula of artificial viscosity, we obtained a low growth rate together with a small pulsation amplitude. The time-scale related to pulsation mode-switching is also increased.     

Theoretical rates of pulsation period change in the Galactic Cepheids

Theoretical estimates of the rates of radial pulsation period change in Galactic Cepheids with initial masses 5.5 M _⊙ ≤ M _ZAMS ≤ 13 M _⊙, chemical composition X = 0.7, Z = 0.02 and periods 1.5 day ≤ Π ≤ 100 day are obtained from consistent stellar evolution and nonlinear stellar pulsation computations. Pulsational instability was investigated for three crossings of the instability strip by the evolutionary track in the HR diagram. The first crossing occurs at the post-main sequence helium core gravitational contraction stage which proceeds in the Kelvin-Helmholtz timescale whereas the second and the third crossings take place at the evolutionary stage of thermonuclear core helium burning. During each crossing of the instability strip the period of radial pulsations is a quadratic function of the stellar evolution time. Theoretical rates of the pulsation period change agree with observations but the scatter of observational estimates of \(\dot \Pi\) noticeably exceeds the width of the band \(\left( {\delta \log \left| {\dot \Pi } \right| \leqslant 0.6} \right)\) confining evolutionary tracks in the period-period change rate diagram. One of the causes of the large scatter with very high values of \(\dot \Pi\) in Cepheids with increasing periods might be the stars that cross the instability strip for the first time. Their fraction ranges from 2% for M _ZAMS = 5.5 M _⊙ to 9% for M _ZAMS = 13 M _⊙ and variables α UMi and IX Cas seem to belong to such objects.     

PULSE: a finite element code for solving adiabatic nonradial pulsation equations

We briefly present the nonradial adiabatic pulsation code PULSE first developped for white dwarf asteroseismology and now used to compute adiabatic oscillation properties for various types of stellar objects. Numerical tests show that the code is able to provide the accuracy (for a given stellar model) required to deal with the precision in frequency expected from the COROT long runs. While the ultimate objective is to compare the output of various pulsation codes (see these proceedings), we already emphasize problems that need to be addressed concerning, in particular, the mesh resolution of the input stellar models and its impact on the accuracy at which frequencies can be computed.     

O-bearing Molecules in Carbon-rich Proto-Planetary Objects

We use high-precision multiband photometric data of the first-overtone RR Lyrae star U Comae to investigate the predictive capability of full-amplitude, nonlinear, convective hydrodynamical models. The main outcome of this investigation is that theoretical predictions properly account for the luminosity variations along a full pulsation cycle. Moreover, we find that this approach, because of the strong dependence of this observable and of the pulsation period on stellar parameters, supplies tight constraints on stellar mass, effective temperature, and distance modulus. Pulsational estimates of these parameters appear in good agreement with empirical ones. Finally, a well-defined bump just before the luminosity maximum gave the unique opportunity to calibrate the turbulent convection model adopted for handling the coupling between pulsation and convection.     

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.     

Evolutionary Status and Fundamental Parameters of the Cepheid V1033 Cyg

Astronomy Letters - On the basis of consistent stellar evolution and nonlinear stellar pulsation calculations, the Cepheid V1033 Cyg is shown to be the post-main sequence star at the first crossing...     

The coupling coefficients of radial pulsation in third order

The third order theory of coupling is discussed regarding the radial pulsation of stellar models.     

X-ray variability of the pulsar Vela X-1 as observed with ART-P/Granat

Observations of the X-ray pulsar Vela X-1 with the ART-P telescope onboard the Granat Observatory are presented. Variability on a time scale of several thousand seconds was detected; intensity variations are shown to be accompanied by changes in the source’s spectrum. The hardness was also found to be highly variable on a scale of one pulsation period. The source’s spectrum exhibits an absorption feature at energy ～7 keV, which is apparently attributable to cyclotron scattering/absorption in the neutron-star magnetic field. Weak persistent emission was detected during an X-ray eclipse, which probably resulted from the scattering of pulsar emission in the stellar wind from an optical star.     

Vibrational stability of stars in thermal imbalance: A solution in terms of asymptotic expansions

The solution of the partial differential equation describing the ‘non-isentropic’ oscillations of a star in thermal imbalance has been obtained in terms of asymptotic expansions up to the first order in the parameterII/t _s, whereII is the adiabatic pulsation period for the fundamental mode andt _s , a secular time scale of the order of the Kelvin-Helmholtz time. Use has been made of the zeroth order ‘isentopic’ solution derived in I. The solution obtained allows one to derive unambiguously a general integral expression for the coefficient of vibrational stability for arbitrary stellar models in thermal imbalance. The physical interpretation of this stability coefficient is discussed and its generality and its simplicity are stressed. Application to some simple analytic stellar models in homologous and nonhomologous contraction enables one to recover, in a more straightforward manner, results obtained by Coxet al. (1973). Aizenman and Cox (1974) and Davey (1974). Finally, we emphasize that the inclusion of the effects of thermal imbalance in the stability calculations of realistic evolutionary sequences of stellar models, not considered up to now by the other authors, is quite easy and straightforward with the simple formula derived here.     

Coupled-oscillators

Modal coupling oscillation models for the stellar radial pulsation and coupled-oscillators are reviewed. Coupled-oscillators with the second-order and third-order terms seemed to behave non-systematically. Using the equation by Schwarzschild and Savedoff (1949) with the dissipation term of van del Pol's type which is third-order, we demonstrate the effect of each term. The effects can be understood by the terms of the nonlinear dynamics, which is recently developing, that is. phase-locking, quasi-periodicity, period doubling, and chaos. As the problem of stellar pulsation, especially of double-mode cepheids on the period-ratio, we examine the dependence on the stellar structure from which the coupling constants in the second-order terms are derived. Eigen functions for adiabatic pulsations had been used for the calculation of the constants. It is noted that only two set of the constants are available, that is, for the polytrope model withn = 3 and a cepheid model without convection. Some examples of nonlinear dynamical effects will be shown.It is shown that if the constants were suitable values, the period-ratio of double-mode cepheids is probably realized. The possibility is briefly suggested.     

A linear and nonlinear study of Mira

Both linear and nonlinear calculations of the 331 day, long period variable star Mira have been undertaken to see what radial pulsation mode is naturally selected. Models are similar to those considered in the linear nonadiabatic stellar pulsation study of Ostlie and Cox (1986). Models are considered with masses near one solar mass, luminosities between 4000 and 5000 solar luminosities, and effective temperatures of approximately 3000 K. These models have fundamental mode periods that closely match the pulsation period of Mira. The equation of state for the stellar material is given by the Stellingwerf (1975ab) procedure, and the opacity is obtained from a fit by Cahn that matches the low temperature molecular absorption data for the Population I Ross-Aller 1 mixture calculated from the Los Alamos Astrophysical Opacity Library. For the linear study, the Cox, Brownlee, and Eilers (1966) approximation is used for the linear theory variation of the convection luminosity. For the nonlinear work, the method described by Ostlie (1990) and Cox (1990) is followed. Results showing internal details of the radial fundamental and first overtone modes behavior in linear theory are presented. Preliminary radial fundamental mode nonlinear calculations are discussed. The very tentative conclusion is that neither the fundamental or first overtone mode is excluded from being the actual observed one.