Nonlinear Pulsation Research Articles

Convection and pulsation in red giant stars

AbstractConvection strongly influences the periods, stability and amplitudes of pulsation in red giant stars. For example, changing the efficiency of convection (the mixing length parameter) changes the radius and the effective temperature of a red giant, which in turn changes the pulsation period at a given luminosity. Since essentially all the energy flux outside the nuclear-burning core is carried by convection, it is the variation of convective energy transport throughout the pulsation cycle that determines pulsation stability. In both linear and nonlinear pulsation models, the turbulent viscosity that results from the interaction of pulsation with turbulent convection provides a strong damping effect on pulsation and, in nonlinear models, it determines the limiting amplitude. In this paper, these and other topics are discussed.

Bump Cepheids in the Magellanic Clouds: Metallicities, the Distances to the LMC and SMC, and the Pulsation‐Evolution Mass Discrepancy

We utilize nonlinear pulsation models to reproduce the observed light and color curves for two samples of bump Cepheid variables, 19 from the Large Magellanic Cloud and 9 from the Small Magellanic Cloud. This analysis determines the fundamental parameters mass, luminosity, effective temperature, metallicity, distance and reddening for the sample of stars. The use of light curve shape alone to determine metallicity is a new modelling technique introduced here. The metallicity, distance and reddening distributions for the two samples are in agreement with those of similar stellar populations in the literature. The distance modulus of the Large Magellanic Cloud is determined to be 18.54$\pm$0.018 and the distance modulus of the Small Magellanic Cloud is determined to be 18.93$\pm$0.024. The mean Cepheid metallicities are $Z = 0.0091\pm0.0007$ and $0.0050\pm0.0005$ for the LMC and SMC, respectively. The masses derived from pulsation analysis are significantly less than those predicted by stellar evolutionary models with no or mild convective core overshoot. We show that this discrepancy can not be accounted for by uncertainties in our input opacities or in mass-loss physics. We interpret the observed mass discrepancy in terms of enhanced internal mixing in the vicinity of the convective core during the main-sequence lifetime and find that the overshoot parameter $\Lambda_{c}$ rises from 0.688$\pm$0.009H_p at the mean LMC metallicity to 0.746$\pm$0.009H_p in the SMC.

Modeling of acoustically enhanced bubble growth in supersaturated biological media

Recent measurements and predictions of dissolved nitrogen concentration in tissues of dolphins and whales following repeated dives have indicated that the gas concentration may reach 300% of saturation [Houser et al., ‘‘Can diving-induced tissue nitrogen supersaturation increase the chance of acoustically driven bubble growth in marine mammals?,’’ J. Theor. Biol. 213, 183–195 (2001)]. This has generated renewed interest in whether underwater sound may increase the likelihood of decompression sickness in these cetaceans. One potential mechanism is bubble growth due to rectified diffusion. Eller and Flynn [‘‘Rectified diffusion during nonlinear pulsations of cavitation bubbles,’’ J. Acoust. Soc. Am. 37, 493–503 (1965)] modeled rectified diffusion by solving a Rayleigh-Plesset equation for the acoustically driven bubble dynamics, coupled to a quasianalytical diffusion model that accounts for increase of the equilibrium bubble radius. Crum and Mao [‘‘Acoustically enhanced bubble growth at low frequencies and its implications for human diver and marine mammal,’’ J. Acoust. Soc. Am. 99, 2898–2907 (1996)] used similar models to consider dissolved gas saturations up to 223% and demonstrated significantly enhanced bubble growth at sound-pressure levels above 210 dB (re: 1 μPa). We are revisiting this problem by extending these models and considering wider ranges of parameters. A principal difference is that the gas diffusion is modeled by solving the full partial differential equation numerically. Not only does this approach circumvent analytical approximations, it permits taking into account variations in gas concentration associated with dive-profile-dependent hydrostatic pressure variations. [Work supported by ONR.]

Non-linear axisymmetric pulsations of rotating relativistic stars in the conformal flatness approximation

We study non-linear axisymmetric pulsations of rotating relativistic stars using a general relativistic hydrodynamics code under the assumption of a conformally flat three-metric. We compare the results of our simulations, in which the space-time dynamics is coupled to the evolution of the fluid, to previous results performed in the Cowling approximation in which the space-time dynamics was neglected. We show that the conformal flatness condition has only a small effect on the dynamics of pulsating relativistic stars and the obtained pulsation frequencies are very close to those expected in full general relativity. The pulsations are studied along various sequences of both uniformly and differentially rotating relativistic polytropes with index N = 1. For small pulsation amplitudes we identify several modes, including the lowest-order l = 0, 2 and 4 axisymmetric modes, as well as several axisymmetric inertial modes. Differential rotation significantly shifts mode frequencies to smaller values, increasing the likelihood of detection by current gravitational wave interferometric detectors. We observe an extended avoided crossing between the l = 0 and l = 4 first overtones (previously known to exist from perturbative studies), which is important for correctly identifying mode frequencies in the case of detection. For uniformly rotating stars near the mass-shedding limit, we confirm the existence of the mass-shedding-induced damping of pulsations and argue that it is still relevant for secularly unstable modes, even though the effect is not as strong as was previously found in the Cowling approximation. We also investigate non-linear harmonics of the linear modes and note that rotation changes the pulsation frequencies in a way that would allow for various parametric instabilities between two or three modes to take place. Although this scenario has been explored before for slowly rotating collapse, it could become very interesting in the case of rapidly rotating collapse, where the quasi-radial mode could be in resonance with inertial modes. We assess the detectability of each obtained mode by current gravitational wave detectors and outline how the empirical relations that have been constructed for gravitational wave asteroseismology could be extended to include the effects of rotation.

Propagation of acoustic wave in viscoelastic medium permeated with air bubbles

Based on the modification of the radial pulsation equation of an individual bubble, an effective medium method (EMM) is presented for studying propagation of linear and nonlinear longitudinal acoustic waves in viscoelastic medium permeated with air bubbles. A classical theory developed previously by Gaunaurd (Gaunaurd GC and Uberall H, J. Acoust. Soc. Am., 1978; 63: 1699–1711) is employed to verify the EMM under linear approximation by comparing the dynamic (i.e. frequency-dependent) effective parameters and an excellent agreement is obtained. The propagation of longitudinal waves is hereby studied in detail. The results illustrate that the nonlinear pulsation of bubbles serves as the source of second harmonic wave and the sound energy has the tendency to be transferred to second harmonic wave. Therefore the sound attenuation and acoustic nonlinearity of the viscoelastic matrix are remarkably enhanced due to the system's resonance induced by the existence of bubbles.

Cepheid Pulsation Models at Varying Metallicity and ΔY/ΔZ

In this paper we present an extended set of nonlinear convective pulsation models at varying metallicity and ΔY/ΔZ ratio. The predicted instability strip and bolometric light curves are discussed by comparing the new models with our previous ones. In particular, the dependence on both metal and helium abundances is investigated. By transforming the bolometric light curves into the observational bands, we are able to derive both period-color-luminosity and Wesenheit relations for each selected chemical composition. Synthetic period-luminosity relations are obtained by populating the instability strip according to specific assumptions on the number of pulsators and the mass distribution. These theoretical results are compared with recent accurate data by Sandage et al. and Kervella et al. in order to test the predictive capabilities of the models. We confirm our previous results that the theoretical metallicity correction to the Key Project Cepheid distance scale depends on both the period range and ΔY/ΔZ ratio, becoming important for periods longer than 20 days and ΔY/ΔZ > 1.5.

Long period variables in 47 Tuc: direct evidence for lost mass

We have identified 22 new variable red giants in 47 Tuc and determined periods for another 8 previously known variables. All red giants redder than = 1.8 are variable at the limits of our detection threshold, which corresponds to ≈ 0.1 mag. This colour limit corresponds to a luminosity log L/ and it is considerably below the tip of the RGB at log L/. Linear non-adiabatic models without mass loss on the giant branch can not reproduce the observed PL laws for the low amplitude pulsators. Models that have undergone mass loss do reproduce the observed PL relations and they show that mass loss of the order of 0.3 occurs along the RGB and AGB. The linear pulsation periods do not agree well with the observed periods of the large amplitude Mira variables, which pulsate in the fundamental mode. The solution to this problem appears to be that the nonlinear pulsation periods in these low mass stars are considerably shorter than the linear pulsation periods due to a rearrangement of stellar structure caused by the pulsation. Both observations and theory show that stars evolve up the RGB and first part of the AGB pulsating in low order overtone modes, then switch to fundamental mode at high luminosities.

Non-linear pulsation models of red giants

There are currently unresolved differences between fundamental mode red giant pulsation models and the large amplitude pulsation behaviour of real stars. In particular, current models tend to be driven too hard and they consequently have excessive amplitudes. This may be a result of the convection energy transport approximations used. We have developed a new pulsation code which includes a time-dependent convection theory, turbulent pressure and turbulent viscosity. Exploratory calculations performed using this code show that the turbulent pressure and turbulent energy transport seem to have little effect on pulsation properties such as velocity, radius and light variation. All calculations also reveal the presence of sharp luminosity spikes that are not observed in real stars. The calculations indicate that the inclusion of a turbulent viscosity in the models may be what is required to bring the model pulsation amplitudes down into agreement with observations. Calculations for a high mass and luminosity model show a double-peaked light curve comparable to that observed for high-luminosity Large Magellanic Cloud Miras.

The simulation of non-linear stellar pulsations

Details of a method for the numerical simulation of stellar pulsations and instabilities in the non-linear regime are presented. Important properties of the method are intrinsic mass and energy conservation. These are indispensable if the connection between stellar pulsations and mass loss is to be investigated. A prescription for the construction of initial models is given, which allows for the study of the linear regime of exponential growth of instabilities and a comparison with independent linear stability analyses. The dependence on numerical parameters of the simulations is investigated in detail. As a result, the acoustic luminosity generated by strange mode instabilities in a massive star is found to depend sensitively on the numerical viscosity necessary to handle shock waves. The values derived have to be regarded as lower limits and reach the order of magnitude of the kinetic luminosities observed in the winds of corresponding objects. This result is taken as further indication for a connection between strange mode instabilities and stellar mass loss.

A "Pulsational" Distance Determination for the Large Magellanic Cloud

We present results from the theoretical modelling of the observed B,V light curves of 14 RR Lyrae stars in the Large Magellanic Cloud. The sample includes 7 fundamental and 7 first overtone pulsators covering the metallicity range from -2.12 to -0.79 dex in [Fe/H], with average value of -1.54 dex. Masses, intrinsic luminosities, effective temperatures, and reddenings were derived by fitting high accuracy multiband light curves available for these RR Lyrae stars to nonlinear convective pulsation models. Individual distance moduli were determined for each variable, they lead to an average distance modulus for the Large Magellanic Cloud of (m-M)o= 18.54 +/-0.02 (sigm= 0.09 mag, standard deviation of the average), in good agreement with the long astronomical distance scale.

Automated nonlinear stellar pulsation calculations: Applications to RR Lyrae stars

We describe a methodology that allows us to follow the pulsational behavior of an RR Lyrae model consistently and automatically along its evolutionary track throughout the whole instability strip. It is based on the powerful amplitude equation formalism, and resorts to a judicious combination of numerical hydrodynamical simulations, the analytical signal time-series analysis, and amplitude equations. A large-scale survey of the nonlinear pulsations in RR Lyr instability strip is then presented, and the mode selection mechanism is delineated throughout the relevant regions of parameter space. We obtain and examine two regions with hysteresis, where the pulsational state depends on the direction of the evolutionary tracks, namely a region with either fundamental (RRab) or first overtone (RRc) pulsations and a region with either fundamental (RRab) or double-mode (RRd) pulsations. The regions where stable double-mode (DM, or RRd) pulsations occur are very narrow and hard to find in astrophysical parameter (L, M, , X, Z) space with hydrodynamic simulations, but our systematic and efficient methodology allows us to investigate them with unprecedented detail. It is shown that by simultaneously considering the effects of mode selection and of horizontal branch evolution we can naturally solve one of the extant puzzles involving the topologies of the theoretical and observed instability strips, namely the slope of the fundamental blue edge. The importance of the interplay between mode selection and stellar evolutionary effects is also demonstrated for the properties of double-mode RR Lyr. Finally, the Petersen diagram of double-mode RR Lyr models is discussed for the first time.

Improvement of the CORS method for Cepheids radii determination based on Strömgren photometry

In this paper we present a modified version of the CORS method based on a new calibration of the Surface Brightness function in the Stromgren photometric system. The method has been tested by means of synthetic light and radial velocity curves derived from nonlinear pulsation models. Detailed simulations have been performed to take into account the quality of real observed curves as well as possible shifts between photometric and radial velocity data. The method has been then applied to a sample of Galactic Cepheids with Stromgren photometry and radial velocity data to derive the radii and a new PR relation. As a result we find log R = (1.19 +- 0.09) + (0.74 +- 0.11) log P (r.m.s=0.07). The comparison between our result and previous estimates in the literature is satisfactory. Better results are expected from the adoption of improved model atmosphere grids.

Non-linear pulsations in differentially rotating neutron stars: mass-shedding-induced damping and splitting of the fundamental mode

We study small-amplitude, non-linear pulsations of uniformly and differentially rotating neutron stars employing a two-dimensional evolution code for general-relativistic hydrodynamics. Using Fourier transforms at several points inside the star, both the eigenfrequencies and two-dimensional eigenfunctions of pulsations are extracted. The centrifugal forces and the degree of differential rotation have significant effects on the mode eigenfunction. We find that near the mass-shedding limit, the pulsations are damped due to shocks forming at the surface of the star. This new damping mechanism may set a small saturation amplitude for modes that are unstable to the emission of gravitational waves. After correcting for the assumption of the Cowling approximation (used in our numerical code), we construct empirical relations that predict the range of gravitational-wave frequencies from quasi-periodic post-bounce oscillations in the core collapse of massive stars. We also find that the fundamental quasi-radial mode is split, at least in the Cowling approximation and mainly in differentially rotating stars, into two different sequences.

Updated pulsation models for anomalous Cepheids

A theoretical investigation of the pulsation behavior of so-named “anomalous” Cepheids is presented. The study is based on nonlinear convective pulsation models with $Z=0.0001$ and 0.0004, mass in the range 1.3-2.2 $M_{\odot}$ and various luminosity levels. Based on these computations, we derive periods, bolometric light curves and the edges of the instability strip, showing that a variation of the metal abundance from $Z=0.0001$ to 0.0004 has quite small effects on these quantities. Then, using bolometric corrections and color-temperature transformations, we are able to provide the predicted relations connecting pulsational properties (periods, amplitudes) with magnitudes and colors in the various photometric bands. The theoretical pulsational scenario is compared to observed anomalous Cepheids in dwarf spheroidal galaxies and, in particular, the predicted mass-dependent Period-Magnitude-Amplitude and Period-Magnitude-Color relations are used to estimate individual mass values, as well as to discriminate between fundamental (F) and first-overtone (FO) pulsators.

Determining The Metallicity of Cepheid Stars in The SMC, LMC and The Galaxy

AbstractThe Cepheid Period-Luminosity relation is unquestionably one of the most powerful tools at our disposal for determining the extragalactic distance scale. While significant progress has been made in the past few years towards its understanding and characterisation, both on the observational (e.g. the HST Key Project) and theoretical (e.g. non-linear pulsation models, non-LTE atmospheres etc.) sides, the debate on the influence that chemical composition may have on the Period-Luminosity relation is still unsettled. Current estimates lead to differences in the distance as large as 15%, effectively limiting the accuracy of Cepheids as distance indicators. To further tackle this problem, we have obtained high resolution spectra of a large sample of Cepheids in our Galaxy and the Magellanic Clouds. The superb quality of the data allow us to probe the detailed effects of chemical composition (alpha, iron-group, and heavy elements) over more than a factor of ten in metallicity. Here, we present the first preliminary results of the analysis of iron abundances in a sub-sample of Cepheids.

The Changing Period of R Hya

We discuss the period evolution of the Mira variable R Hya. We show, using observations dating back to 1662, that it has evolved from a stable 500-day period to a stable 380-day period. The onset of the period evolution coincided with a sharp decline in mass-loss rate. We suggest that mass-loss fluctuations over a period of a few hundred years may be common in Miras, and relate to a non-linear pulsation instability.

Theoretical Models for Classical Cepheids. VIII. Effects of Helium and Heavy‐Element Abundance on the Cepheid Distance Scale

Previous nonlinear fundamental pulsation models for classical Cepheids with metal content Z ? 0.02 are implemented with new computations at supersolar metallicity (Z = 0.03, 0.04) and selected choices of the helium-to-metal enrichment ratio ?Y/?Z. On this basis, we show that the location into the H-R diagram of the Cepheid instability strip is dependent on both metal and helium abundance, moving toward higher effective temperatures with decreasing metal content (at fixed Y) or with increasing helium content (at fixed Z). The contributions of helium and metals to the predicted period-luminosity and period-luminosity-color relations are discussed as well as the implications on the Cepheid distance scale. We suggest that the adoption of empirical V and I period-luminosity relations, as inferred by Cepheids at the LMC, to get distance moduli with an uncertainty of ?0.10 mag is fully justified for variables in the short-period range (P ? 10 days), at least with Z ? 0.04 and ?Y/?Z in the range of 2-4. Conversely, at longer periods (P > 10 days) a correction to LMC-based distance moduli may be needed, whose sign and amount depend on the helium and metal content of the Cepheids. Specifically, from fundamental pulsators with Z > 0.008 we derive that the correction (in magnitude) may be approximated as c = -6.03 + 17.80Y - 2.80 log Z + 8.19Y log Z, with a total intrinsic uncertainty of ?0.05 mag, whereas c = -0.23(?0.03) log(Z/0.008) if Z < 0.008. Based on these new results, we show that the empirical metallicity correction suggested by Cepheid observations in two fields of the galaxy M101 may be accounted for, provided that the adopted helium-to-metal enrichment ratio is reasonably high (?Y/?Z ~ 3.5).

The energy efficiency of formation of photons, radicals and ions during single-bubble cavitation.

It is extremely difficult to perform a quantitative analysis of the chemistry associated with multibubble cavitation: unknown parameters include the number of active bubbles, the acoustic pressure acting on each bubble and the bubble size distribution. Single-bubble sonoluminescence (characterized by the emission of picosecond flashes of light) results from nonlinear pulsations of an isolated vapour-gas bubble in an acoustic field. Although the latter offers a much simpler environment in which to study the chemical activity of cavitation, quantitative measurements have been hindered by the tiny amount of reacting gas within a single bubble (typically <10(-13) mol). Here we demonstrate the existence of chemical reactions within a single cavitating bubble, and quantify the sources of energy dissipation during bubble collapse. We measure the yields of nitrite ions, hydroxyl radicals and photons. The energy efficiency of hydroxyl radical formation is comparable to that in multibubble cavitation, but the energy efficiency of light emission is much higher. The observed rate of nitrite formation is in good agreement with the calculated diffusion rate of nitrogen into the bubble. We note that the temperatures attained in single-bubble cavitation in liquids with significant vapour pressures will be substantially limited by the endothermic chemical reactions of the polyatomic species inside the collapsing bubble.

Large Magellanic Cloud Bump Cepheids: Probing the Stellar Mass‐Luminosity Relation

We present the results of non-linear pulsation modelling of 20 bump Cepheids in the LMC. By obtaining a optimal fit to the observed V,R MACHO lightcurves we have placed tight constraints on stellar parameters of M, L, Teff and well as quantities of distance modulus and reddening. We describe the mass-luminosity relation for core-He burning for intermediate mass stars. The mass-luminosity relation depends critically on the level of mixing within the stellar interior over the course of the main-sequence lifetime. Our sample is significantly more luminous than predicted by classical stellar evolutionary models that do not incorporate extension to the convective core. Under the paradigm of convective core overshoot our data implies $\Lambda_c$ of 0.65$\pm0.03l/H_p$. We derive a LMC distance modulus of 18.55$\pm$0.02.

Non-linear radial pulsation models for the early-type helium stars V652 Her and BX Cir

We report new non-linear pulsation models of the helium stars V652 Her and BX Cir. Linear theory has previously shown their pulsations to be due to iron-group bump instability. Recent high-resolution spectroscopic observations have provided high-precision measurements of their radial velocity curves and of their radii. Their masses remain less well determined. A hydrodynamic code including recent OPAL opacity data has been used to construct the models. These are compared with the observational data. In particular, we attempt to reproduce accurately the observed radial velocity and luminosity curves. The results impose additional constraints on those stellar dimensions, including mass, which remain poorly determined by observation. Final results show a model for V652 Her which reproduce the observed velocity and luminosity curves with 0.7 and 23 400 . For BX Cir, the mass must lie between 0.50 and 0.38 if the temperature is in the range 22 400–24 000 . However, the luminosity of the models is smaller than that measured directly by a factor of two.