Consecutive Radial Orders Research Articles

Buoyancy glitches in pulsating stars revisited

Sharp structural variations induce specific signatures on stellar pulsations that can be studied to infer localised information on the stratification of the star. This information is key to improve our understanding of the physical processes that lead to the structural variations and how to model them. Here we revisit and extend the analysis of the signature of different types of buoyancy glitches in gravity-mode and mixed-mode pulsators presented in earlier works, including glitches with step-like, Gaussian-like, and Dirac-δ-like shapes. In particular, we provide analytical expressions for the perturbations to the periods and show that these can be reliably used in place of the expressions provided for the period spacings, with the advantage that the use of the new expressions does not require modes with consecutive radial orders to be observed. Based on a comparison with two limit cases and on simulated data, we further tested the accuracy of the expression for the Gaussian-like glitch signature whose derivation in an earlier work involved a significant approximation. We find that the least reliable glitch parameter inferred from fitting that expression is the amplitude, which can be up to a factor of two larger than the true amplitude, reaching this limit when the glitch is small. We further discuss the impact on the glitch signature of considering a glitch in the inner and outer half of the g-mode cavity, emphasising the break of symmetry that takes place in the case of mixed-mode pulsators.

Probing the temperature gradient in the core boundary layer of stars with gravito-inertial modes

Aims. We investigated the thermal and chemical structure in the near-core region of stars with a convective core by means of gravito-inertial modes. We determined the probing power of different asteroseismic observables and fitting methodologies. We focus on the case of the B-type star KIC 7760680, rotating at a quarter of its critical rotation velocity. Methods. We computed grids of 1D stellar structure and evolution models for two different prescriptions of the temperature gradient and mixing profile in the near-core region. We determined which of these prescriptions is preferred according to the prograde dipole modes detected in 4 yr Kepler photometry of KIC 7760680. We considered different sets of asteroseismic observables and compared the outcomes of the regression problem for a χ2 and a Mahalanobis distance merit function, where the latter takes into account realistic uncertainties for the theoretical predictions and the former does not. Results. Period spacings of modes with consecutive radial order offer a better diagnostic than mode periods or mode frequencies for asteroseismic modelling of stars revealing only high-order gravito-inertial modes. We find KIC 7760680 to reveal a radiative temperature gradient in models with convective boundary mixing, but less complex models without such mixing are statistically preferred for this rotating star, revealing extremely low vertical envelope mixing. Conclusions. Our results strongly suggest the use of measured individual period spacing values for modes of consecutive radial order as an asteroseismic diagnostic for stellar modelling of B-type pulsators with gravito-inertial modes.

Constraining stellar evolution theory with asteroseismology of γ Doradus stars using deep learning

Context. The efficiency of the transport of angular momentum and chemical elements inside intermediate-mass stars lacks proper calibration, thereby introducing uncertainties on a star’s evolutionary pathway. Improvements require better estimation of stellar masses, evolutionary stages, and internal mixing properties. Aims. Our aim was to develop a neural network approach for asteroseismic modelling, and test its capacity to provide stellar masses, ages, and overshooting parameter for a sample of 37 γ Doradus stars; these parameters were previously determined from their effective temperature, surface gravity, near-core rotation frequency, and buoyancy travel time Π0. Here our goal is to perform the parameter estimation from modelling of individual periods measured for dipole modes with consecutive radial order rather than from Π0. We assess whether fitting these individual mode periods increases the capacity of the parameter estimation. Methods. We trained neural networks to predict theoretical pulsation periods of high-order gravity modes (n ∈ [15, 91]), and to predict the luminosity, effective temperature, and surface gravity for a given mass, age, overshooting parameter, diffusive envelope mixing, metallicity, and near-core rotation frequency. We applied our neural networks for Computing Pulsation Periods and Photospheric Observables (C-3PO) to our sample and compute grids of stellar pulsation models for the estimated parameters. Results. We present the near-core rotation rates (from the literature) as a function of the inferred stellar age and critical rotation rate. We assessed the rotation rates of the sample near the start of the main sequence assuming rigid rotation. Furthermore, we measured the extent of the core overshoot region and find no correlation with mass, age, or rotation. Finally, for one star in our sample, KIC 12066947, we find indications of mode coupling in the period spacing pattern which we cannot reproduce with mode trapping. Conclusions. The neural network approach developed in this study allows the derivation of stellar properties dominant for stellar evolution, such as mass, age, and extent of core-boundary mixing. It also opens a path for future estimation of mixing profiles throughout the radiative envelope, with the aim of inferring these profiles for large samples of γ Doradus stars.

Comparing the asteroseismic properties of pulsating pre-extremely low mass white dwarf and δ Scuti stars

Context. Pulsating extremely low-mass pre-white dwarf stars (pre-ELMV), with masses between ~0.15 M⊙ and ~0.30 M⊙, constitute a new class of variable stars showing g- and possibly p-mode pulsations with periods between 320 and 6000 s (frequencies between 14.4 and 270 c/d), driven by the κ mechanism operating in the second He ionization zone. On the other hand, main sequence δ Scuti stars, with masses between 1.2 and 2.5 M⊙, pulsate in low-order g and p modes with periods in the range [700–28 800] s (frequencies in the range [3–123] c/d), driven by the κ mechanism operating in the He II ionization zone and the turbulent pressure acting in the HI ionization layer. Interestingly enough, the instability strips of pre-ELM white dwarf and δ Scuti stars nearly overlap in the Teff vs. log g diagram, leading to a degeneracy when spectroscopy is the only tool to classify the stars and pulsation periods only are considered. Aims. Pre-ELM white dwarf and δ Scuti stars are in very different stages of evolution and therefore their internal structure is very distinct. This is mirrored in their pulsational behavior, thus employing asteroseismology should allow us to distinguish between these groups of stars despite their similar atmospheric parameters. Methods. We have employed adiabatic and non-adiabatic pulsation spectra for models of pre-ELM white dwarfs and δ Scuti stars, and compare their pulsation periods, period spacings, and rates of period change. Results. Unsurprisingly, we found substantial differences in the period spacing of δ Scuti and pre-ELM white dwarf models. Even when the same period range is observed in both classes of pulsating stars, the modes have distinctive signature in the period spacing and period difference values. For instance, the mean period difference of p-modes of consecutive radial orders for δ Scuti model are at least four times longer than the mean period spacing for the pre-ELM white dwarf model in the period range [2000–4600] s (frequency range [18.78–43.6] c/d). In addition, the rate of period change is two orders of magnitudes larger for the pre-ELM white dwarfs compared to δ Scuti stars. In addition, we also report the discovery of a new variable star, SDSSJ075738.94+144827.50, located in the region of the Teff versus log g diagram where these two kind of stars coexist. Conclusions.The characteristic spacing between modes of consecutive radial orders (p as well as g modes) and the large differences found in the rates of period change for δ Scuti and pre-ELM white dwarf stars suggest that asteroseismology can be employed to discriminate between these two groups of variable stars. Furthermore, we found that SDSSJ075738.94+144827.50 exhibits a period difference between p modes characteristic of a δ Sct star, assuming consecutive radial order for the observed periods.

Characterizing the observational properties of δ Sct stars in the era of space photometry from the Kepler mission

The {\delta} Sct stars are a diverse group of intermediate-mass pulsating stars located on and near the main sequence within the classical instability strip in the Hertzsprung-Russell diagram. Many of these stars are hybrid stars pulsating simultaneously with pressure and gravity modes that probe the physics at different depths within a star's interior. Using two large ensembles of {\delta} Sct stars observed by the Kepler Space Telescope, the instrumental biases inherent to Kepler mission data and the statistical properties of these stars are investigated. An important focus of this work is an analysis of the relationships between the pulsational and stellar parameters, and their distribution within the classical instability strip. It is found that a non-negligible fraction of main sequence {\delta} Sct stars exist outside theoretical predictions of the classical instability boundaries, which indicates the necessity of a mass-dependent mixing length parameter to simultaneously explain low- and high-radial order pressure modes in {\delta} Sct stars within the Hertzsprung-Russell diagram. Furthermore, a search for regularities in the amplitude spectra of these stars is also presented, specifically the frequency difference between pressure modes of consecutive radial order. In this work, it is demonstrated that an ensemble-based approach using space photometry from the Kepler mission is not only plausible for {\delta} Sct stars, but that it is a valuable method for identifying the most promising stars for mode identification and asteroseismic modelling. The full scientific potential of studying {\delta} Sct stars is as yet unrealised. The ensembles discussed in this paper represent a high-quality data set for future studies of rotation and angular momentum transport inside A and F stars using asteroseismology.

Mitigating the mass dependence in the Δν scaling relation of red giant stars

The masses and radii of solar-like oscillators can be estimated through the asteroseismic scaling relations. These relations provide a direct link between observables, i.e. effective temperature and characteristics of the oscillation spectra, and stellar properties, i.e. mean density and surface gravity (thus mass and radius). These scaling relations are commonly used to characterize large samples of stars. Usually, the Sun is used as a reference from which the structure is scaled. However, for stars that do not have a similar structure as the Sun, using the Sun as a reference introduces systematic errors as large as 10\% in mass and 5\% in radius. Several alternatives for the reference of the scaling relation involving the large frequency separation (typical frequency difference between modes of the same degree and consecutive radial order) have been suggested in the literature. In a previous paper, we presented a reference function with a dependence on both effective temperature and metallicity. The accuracy of predicted masses and radii improved considerably when using reference values calculated from our reference function. However, the residuals indicated that stars on the red-giant branch possess a mass dependence that was not accounted for. Here, we present a reference function for the scaling relation involving the large frequency separation that includes the mass dependence. This new reference function improves the derived masses and radii significantly by removing the systematic differences and mitigates the trend with $\nu_{\rm max}$ (frequency of maximum oscillation power) that exists when using the solar value as a reference.

Signatures of internal rotation discovered in theKeplerdata of five slowly pulsating B stars

Massive stars are important building blocks of the Universe, and their stellar structure and evolution models are fundamental cornerstones of various fields in modern astrophysics. The precision of these models is limited by our lack of understanding of various internal mixing processes that significantly influence the lifetime of these objects (e.g. core overshoot, chemical mixing, or the internal differential rotation). Our goal is to calibrate models by extending the sample of available seismic studies of slowly pulsating B (SPB) stars, providing input for theoretical modelling efforts that will deliver precise constraints on the parameters describing the internal mixing processes in these objects. We used spectral synthesis and disentangling techniques to derive fundamental parameters and to determine precise orbital parameters from high-resolution spectra. We employed custom masks to construct light curves from the Kepler pixel data and used standard time-series analysis tools to construct a set of significant frequencies for each target. These sets were first filtered from combination frequencies, and then screened for period spacing patterns. We detect gravity mode period series of modes of the same degree l with consecutive radial order n in four new and one revisited SPB star. These predominantly prograde dipole series (covering ~10-40 radial orders) are influenced by rotation and carry signatures of chemical mixing processes. Our spectroscopic analysis, in addition to placing each object inside the SPB instability strip and identifying KIC 4930889 as an SB2 binary, reveals that KIC 11971405 is a fast rotator that shows very weak Be signatures. Together with the observed photometric outbursts this illustrates that this Be star is a fast rotating SPB star. We hypothesise that the outbursts might be connected to its very densely compressed oscillation spectrum.

Evolutionary states of red-giant stars from grid-based modelling

From its surface properties it can be difficult to determine whether a red-giant star is in its helium-core-burning phase or only burning hydrogen in a shell around an inert helium core. Stars in either of these stages can have similar effective temperatures, radii and hence luminosities, i.e. they can be located at the same position in the Hertzsprung-Russell diagram. Asteroseismology -- the study of the internal structure of stars through their global oscillations -- can provide the necessary additional constraints to determine the evolutionary states of red-giant stars. Here, we present a method that uses grid-based modelling based on global asteroseismic properties ($\nu_{\rm max}$, frequency of maximum oscillation power; and $\Delta\nu$, frequency spacing between modes of the same degree and consecutive radial orders) as well as effective temperature and metallicity to determine the evolutionary phases. This method is applicable even to timeseries data of limited length, although with a small fraction of miss-classifications.

Deciphering the period spacing pattern in the oscillation spectrum of the SPB star KIC 7760680

We present the analysis of KIC 7760680, the rotating Slowly Pulsating B-type star identified in the Kepler photometry. The oscillation spectrum of the star exhibits a series of 36 frequencies which are quasi-equally spaced in period. We confirm that this series can be associated with prograde dipole modes of consecutive radial orders. In our studies, the effects of rotation were included in the MESA equilibrium models as well as in the puslational calculations in the framework of the traditional approximation. We find that pulsational models computed with the OPLIB opacities best reproduce the observed frequency range. The modified opacity data with an enhancement of the opacity at $\log T=5.3$, 5.46 and 5.06 were tested as well. Increasing the OPLIB opacities by about 50% at $\log T=5.3$ is sufficient to excite modes in the whole range of 36 frequency peaks of KIC 7760680.

Testing the asymptotic relation for period spacings from mixed modes of red giants observed with theKeplermission

Dipole mixed pulsation modes of consecutive radial order have been detected for thousands of low-mass red-giant stars with the NASA space telescope Kepler. Such modes have the potential to reveal information on the physics of the deep stellar interior. Different methods have been proposed to derive an observed value for the gravity-mode period spacing, the most prominent one relying on a relation derived from asymptotic pulsation theory applied to the gravity-mode character of the mixed modes. Our aim is to compare results based on this asymptotic relation with those derived from an empirical approach for three pulsating red-giant stars. We developed a data-driven method to perform frequency extraction and mode identification. Next, we used the identified dipole mixed modes to determine the gravity-mode period spacing by means of an empirical method and by means of the asymptotic relation. In our methodology, we consider the phase offset, $\epsilon_{\mathrm{g}}$, of the asymptotic relation as a free parameter. Using the frequencies of the identified dipole mixed modes for each star in the sample, we derived a value for the gravity-mode period spacing using the two different methods. These differ by less than 5%. The average precision we achieved for the period spacing derived from the asymptotic relation is better than 1%, while that of our data-driven approach is 3%. Good agreement is found between values for the period spacing derived from the asymptotic relation and from the empirical method. Full abstract in PDF file.

ASTEROSEISMIC FINGERPRINTS OF ROTATION AND MIXING IN THE SLOWLY PULSATING B8 V STAR KIC 7760680

We present the first detection of a rotationally affected series consisting of 36 consecutive high-order sectoral dipole gravity modes in a slowly pulsating B (SPB) star. The results are based on the analysis of four years of virtually uninterrupted photometric data assembled with the Kepler Mission, and high-resolution spectra acquired using the HERMES spectrograph at the 1.2 meter Mercator Telescope. The spectroscopic measurements place KIC7760680 inside the SPB instability strip, near the cool edge. The photometric analysis reveals the longest unambiguous series of gravity modes of the same degree l with consecutive radial order n, that carries clear signatures of chemical mixing and rotation. With such exceptional observational constraints, this star should be considered as the Rosetta Stone of SPBs for future modelling, and bring us a step closer to the much needed seismic calibration of stellar structure models of massive stars.

Asteroseismic surface gravity for evolved stars

Context: Asteroseismic surface gravity values can be of importance in determining spectroscopic stellar parameters. The independent log(g) value from asteroseismology can be used as a fixed value in the spectroscopic analysis to reduce uncertainties due to the fact that log(g) and effective temperature can not be determined independently from spectra. Since 2012, a combined analysis of seismically and spectroscopically derived stellar properties is ongoing for a large survey with SDSS/APOGEE and Kepler. Therefore, knowledge of any potential biases and uncertainties in asteroseismic log(g) values is now becoming important. Aims: The seismic parameter needed to derive log(g) is the frequency of maximum oscillation power (nu_max). Here, we investigate the influence of nu_max derived with different methods on the derived log(g) values. The large frequency separation between modes of the same degree and consecutive radial orders (Dnu) is often used as an additional constraint for the determination of log(g). Additionally, we checked the influence of small corrections applied to Dnu on the derived values of log(g). Methods We use methods extensively described in the literature to determine nu_max and Dnu together with seismic scaling relations and grid-based modeling to derive log(g). Results: We find that different approaches to derive oscillation parameters give results for log(g) with small, but different, biases for red-clump and red-giant-branch stars. These biases are well within the quoted uncertainties of ~0.01 dex (cgs). Corrections suggested in the literature to the Dnu scaling relation have no significant effect on log(g). However somewhat unexpectedly, method specific solar reference values induce biases of the order of the uncertainties, which is not the case when canonical solar reference values are used.

Why the peculiar δ Scuti star HD 187547 is a superstar

The δ Scuti pulsators occupy a region in the Hertzsprung-Russell diagram where several physical processes occur: the subsurface convection layers change from being deep and vigorous to being shallow and ineffective to transport energy. This transition has a large impact not only on pulsational stability but also on stellar evolution, activity, transport of angular momentum, mixing processes, etc.. It is therefore of great interest to understand how exactly the stellar structure changes with increasing temperature and mass. Theoretical models (Houdek et al. 1999; Samadi et al. 2002) predicted that the convection in the outer layers of δ Scuti stars is still efficient enough to excite solar-like oscillations. The Kepler target, HD 187547 (a.k.a. Superstar), was the first δ Scuti star to suggest that solar-like oscillations are indeed present in this type of stars (Antoci et al. 2011). There were several reasons to conclude that HD 187547 is a δ Scuti/solar-like hybrid pulsator. (1) The peaks at high frequencies are modes of pulsations approximately equidistantly spaced, as expected for high radial order pressure modes; these peaks are not combination frequencies as it is sometimes observed in δ Scuti stars. (2) The opacity mechanism cannot excite a continuous frequency range as observed in HD 187547 (Pamyatnykh 2000). (3) The identification as an Am star consistent with the low v sini, makes it very unlikely to be a δ Scuti/roAp hybrid, because strong large-scale magnetic fields, a necessity for roAp pulsators, have never been detected in Am stars (Auriere et al. 2010). (4) Although a large number of Am stars are found in binary systems, we find no evidence in the observed spectra for a companion, i.e. no significant RV shift over 170 days can be detected and the absorption lines can perfectly be reproduced by assuming a slowly-rotating chemically peculiar Am star. This means that the peaks at high frequencies are unlikely to be from a companion, because such a star would be an A or F-type star and would be visible in the spectrum and this is not the case. (5) The large separation Δν, which is the frequency difference between consecutive radial orders of the same degree l, predicts the frequency of maximum power (νmax) to be exactly where the mode with the highest amplitude in the supposed stochastic region is observed. Note that the Δν and νmax scaling relation is valid only for stochastic oscillations (e.g. Huber et al. 2011). (6) The statistical properties (Chang & Gough 1998) of the modes interpreted to be solar-like were very similar to what is expected for stochastic signals in general.

Modelling a high-mass red giant observed by CoRoT

The G6 giant HR\,2582 (HD\,50890) was observed by CoRoT for approximately 55 days. Mode frequencies are extracted from the observed Fourier spectrum of the light curve. Numerical stellar models are then computed to determine the characteristics of the star (mass, age, etc...) from the comparison with observational constraints. We provide evidence for the presence of solar-like oscillations at low frequency, between 10 and 20\,$\mu$Hz, with a regular spacing of $(1.7\pm0.1)\mu$Hz between consecutive radial orders. Only radial modes are clearly visible. From the models compatible with the observational constraints used here, We find that HR\,2582 (HD\,50890) is a massive star with a mass in the range (3--\,5\,$M_{\odot}$), clearly above the red clump. It oscillates with rather low radial order ($n$ = 5\,--\,12) modes. Its evolutionary stage cannot be determined with precision: the star could be on the ascending red giant branch (hydrogen shell burning) with an age of approximately 155 Myr or in a later phase (helium burning). In order to obtain a reasonable helium amount, the metallicity of the star must be quite subsolar. Our best models are obtained with a mixing length significantly smaller than that obtained for the Sun with the same physical description (except overshoot). The amount of core overshoot during the main-sequence phase is found to be mild, of the order of 0.1\,$H_{\rm p}$.

Suppressing g Modes in Shell Hydrogen-Burning δ Scuti Stars

For shell hydrogen-burning δ Scuti stars, many more unstable modes are predicted than are observed. For example, for 4 CVn and δ Scuti itself, only 18 and 6 modes are observed, respectively, whereas several hundred ℓ = 0, 1 and 2 rotationally-split modes are predicted. The predicted modes have a mixed p- and g-mode character, with many g-type nodes present in the H-exhausted core and shell H-burning region, where the Brunt-Väisälä frequency is large.Here we explore whether the predicted frequency spectrum can be made to agree better with the observed spectrum if the g-mode character of the pulsation modes is partially suppressed. Additional motivation for this approach is provided by the observation by Breger et al. (1999) that 4 modes in 4 CVn identified as ℓ = 1 have nearly equal frequency spacings of ∼14 μHz, reminiscent of consecutive radial orders from a pure p-mode spectrum. We consider 2.1- and 2.3-M⊙ evolutionary models that match the observed Teff, L, and log g, plus an identified radial mode frequency of 4 CVn and δ Scuti. For these models, we experimented with setting the Brunt-Väisälä frequency to zero from the model center to the edge of or beyond H-depleted core, so that the g-type portion of the waves becomes evanescent in this region.