Non-radial Oscillation Modes Research Articles

Relativistic Cowling approximation for fluid oscillation modes of color superconducting self-bound stars

AbstractThe investigation of the quasi-normal modes of oscillation of compact stars can reveal much information about their equation of state and internal structure mainly through the analysis of the expected emission of gravitational waves. In this work we study non-radial oscillation modes of strange stars consisting of color superconducting quark matter. We focus on the fundamental and pressure oscillation modes within the frame of the Cowling approximation. We discuss the observable features that may allow a differentiation among hadronic stars, strange stars, and strange stars with color superconductivity.

MODELINGKEPLEROBSERVATIONS OF SOLAR-LIKE OSCILLATIONS IN THE RED GIANT STAR HD 186355

We have analysed oscillations of the red giant star HD 186355 observed by the NASA Kepler satellite. The data consist of the first five quarters of science operations of Kepler, which cover about 13 months. The high-precision time-series data allow us to accurately extract the oscillation frequencies from the power spectrum. We find the frequency of the maximum oscillation power, {\nu}_max, and the mean large frequency separation, {\Delta}{\nu}, are around 106 and 9.4 {\mu}Hz respectively. A regular pattern of radial and non-radial oscillation modes is identified by stacking the power spectra in an echelle diagram. We use the scaling relations of {\Delta}{\nu} and {\nu}_max to estimate the preliminary asteroseismic mass, which is confirmed with the modelling result (M = 1.45 \pm 0.05 M_sun) using the Yale Rotating stellar Evolution Code (YREC7). In addition, we constrain the effective temperature, luminosity and radius from comparisons between observational constraints and models. A number of mixed l = 1 modes are also detected and taken into account in our model comparisons. We find a mean observational period spacing for these mixed modes of about 58 s, suggesting that this red giant branch star is in the shell hydrogen-burning phase.

Asteroseismology of red giants from the first four months ofKeplerdata: Fundamental stellar parameters

Clear power excess in a frequency range typical for solar-type oscillations in red giants has been detected in more than 1000 stars, which have been observed during the first 138 days of the science operation of the NASA Kepler satellite. This sample includes stars in a wide mass and radius range with spectral types G and K, extending in luminosity from the bottom of the giant branch up to high-luminous red giants. The high-precision asteroseismic observations with Kepler provide a perfect source for testing stellar structure and evolutionary models, as well as investigating the stellar population in our Galaxy. We fit a global model to the observed frequency spectra, which allows us to accurately estimate the granulation background signal and the global oscillation parameters, such as the frequency of maximum oscillation power. We find regular patterns of radial and non-radial oscillation modes and use a new technique to automatically identify the mode degree and the characteristic frequency separations between consecutive modes of the same spherical degree. In most cases, we can also measure the small separation. The seismic parameters are used to estimate stellar masses and radii and to place the stars in an H-R diagram by using an extensive grid of stellar models that covers a wide parameter range. Using Bayesian techniques throughout our analysis allows us to determine reliable uncertainties for all parameters. We provide accurate seismic parameters and their uncertainties for a large sample of red giants and determine their asteroseismic fundamental parameters. We investigate the influence of the stars' metallicities on their positions in the H-R diagram. We study the red-giant populations in the red clump and bump and compare them to a synthetic population and find a mass and metallicity gradient in the red clump and clear evidence of a secondary-clump population.

SEISMIC DIAGNOSTICS OF RED GIANTS: FIRST COMPARISON WITH STELLAR MODELS

The clear detection with CoRoT and KEPLER of radial and non-radial solar-like oscillations in many red giants paves the way to seismic inferences on the structure of such stars. We present an overview of the properties of the adiabatic frequencies and frequency separations of radial and non-radial oscillation modes for an extended grid of models. We highlight how their detection allows a deeper insight into the internal structure and evolutionary state of red giants. In particular, we find that the properties of dipole modes constitute a promising seismic diagnostic tool of the evolutionary state of red-giant stars. We compare our theoretical predictions with the first 34 days of KEPLER data and predict the frequency diagram expected for red giants in the COROT exofield in the galactic center direction.

Evidence for a sharp structure variation inside a red-giant star

The availability of precisely determined frequencies of radial and non-radial oscillation modes in red giants is finally paving the way for detailed studies of the internal structure of these stars. We look for the seismic signature of regions of sharp structure variation in the internal structure of the CoRoT target HR7349. We analyse the frequency dependence of the large frequency separation and second frequency differences, as well as the behaviour of the large frequency separation obtained with the envelope auto-correlation function. We find evidence for a periodic component in the oscillation frequencies, i.e. the seismic signature of a sharp structure variation in HR7349. In a comparison with stellar models we interpret this feature as caused by a local depression of the sound speed that occurs in the helium second-ionization region. Using solely seismic constraints this allows us to estimate the mass (M=1.2^{+0.6}_{-0.4} Msun) and radius (R=12.2^{+2.1}_{-1.8} Rsun) of HR7349, which agrees with the location of the star in an HR diagram.

Line-profile variations of stochastically excited oscillations in four evolved stars

Since solar-like oscillations were first detected in red-giant stars, the presence of non-radial oscillation modes has been debated. Spectroscopic line-profile analysis was used in the first attempt to perform mode identification, which revealed that non-radial modes are observable. Despite the fact that the presence of non-radial modes could be confirmed, the degree or azimuthal order could not be uniquely identified. Here we present an improvement to this first spectroscopic line-profile analysis. Aims: We aim to study line-profile variations of stochastically excited solar-like oscillations in four evolved stars to derive the azimuthal order of the observed mode and the surface rotational frequency. Methods: Spectroscopic line-profile analysis is applied to cross-correlation functions, using the Fourier Parameter Fit method on the amplitude and phase distributions across the profiles. Results: For four evolved stars, beta Hydri (G2IV), epsilon Ophiuchi (G9.5III), eta Serpentis (K0III) and delta Eridani (K0IV) the line-profile variations reveal the azimuthal order of the oscillations with an accuracy of ~1. Furthermore, our analysis reveals the projected rotational velocity and the inclination angle. From these parameters we obtain the surface rotational frequency. Conclusions: We conclude that line-profile variations of cross-correlation functions behave differently for different frequencies and that they provide additional information in terms of the surface rotational frequency and azimuthal order.

Effects of rotation on the evolution and asteroseismic properties of red giants

The influence of rotation on the properties of red giants is studied in the context of the asteroseismic modelling of these stars. While red giants exhibit low surface rotational velocities, we find that the rotational history of the star has a large impact on its properties during the red giant phase. In particular, for stars massive enough to ignite He burning in non-degenerate conditions, rotational mixing induces a significant increase of the stellar luminosity and shifts the location of the core helium burning phase to a higher luminosity in the HR diagram. This of course results in a change of the seismic properties of red giants at the same evolutionary state. As a consequence the inclusion of rotation significantly changes the fundamental parameters of a red giant star as determined by performing an asteroseismic calibration. In particular rotation decreases the derived stellar mass and increases the age. Depending on the rotation law assumed in the convective envelope and on the initial velocity of the star, non-negligible values of rotational splitting can be reached, which may complicate the observation and identification of non-radial oscillation modes for red giants exhibiting moderate surface rotational velocities. By comparing the effects of rotation and overshooting, we find that the main-sequence widening and the increase of the H-burning lifetime induced by rotation (Vini=150 km/s) are well reproduced by non-rotating models with an overshooting parameter of 0.1, while the increase of luminosity during the post-main sequence evolution is better reproduced by non-rotating models with overshooting parameters twice as large. This is due to the fact that rotation not only increases the size of the convective core but also changes the chemical composition of the radiative zone.

Non-radial oscillation modes with long lifetimes in giant stars

Towards the end of their lives, stars like the Sun greatly expand to become red giant stars. Such evolved stars could provide stringent tests of stellar theory, as many uncertainties of the internal stellar structure accumulate with age. Important examples are convective overshooting and rotational mixing during the central hydrogen-burning phase, which determine the mass of the helium core, but which are not well understood. In principle, analysis of radial and non-radial stellar oscillations can be used to constrain the mass of the helium core. Although all giants are expected to oscillate, it has hitherto been unclear whether non-radial modes are observable at all in red giants, or whether the oscillation modes have a short or a long mode lifetime, which determines the observational precision of the frequencies. Here we report the presence of radial and non-radial oscillations in more than 300 giant stars. For at least some of the giants, the mode lifetimes are of the order of a month. We observe giant stars with equally spaced frequency peaks in the Fourier spectrum of the time series, as well as giants for which the spectrum seems to be more complex. No satisfactory theoretical explanation currently exists for our observations.

Solar-like Oscillations of the Red Giant ɛ Ophiuchi

Solar-like oscillations have recently been observed in the red giant ɛ Ophiuchi (G9.5III). The large frequency separation is found to be 4.8 μHz, non-radial oscillation mode has been shown to exist. Based on the observed frequency of oscillations and the position of ɛ Ophiuchi on the Hertzsprung-Russell diagram and by adopting a method of combined calculation of stellar evolution and oscillations, some preliminary constraints on the mass, metal abundance, age and radius of this star have been obtained.

Frequency analysis of Cepheids in the Large Magellanic Cloud: new types of classical Cepheid pulsators

We have performed a detailed systematic search for multiperiodicity in the Population I Cepheids of the Large Magellanic Cloud. In this process, we have identified for the first time several new types of Cepheid pulsational behaviour. We have found two triple-mode Cepheids pulsating simultaneously in the first three radial overtones. In 9 per cent of the first overtone (FO) Cepheids, we have detected weak but well-resolved secondary periodicities. They appear either very close to the primary pulsation frequency or at a much higher frequency with a characteristic period ratio of 0.60–0.64. In either case, the secondary periodicities must correspond to non-radial modes of oscillation. This result presents a major challenge to the theory of stellar pulsations, which predicts that such modes should not be excited in Cepheid variables. Non-radial modes have also been found in three of the fundamental first overtone (FU/FO) double-mode Cepheids, but no such oscillations have been detected in single-mode Cepheids pulsating in the FU mode. In 19 per cent of double-mode Cepheids pulsating in the first two radial overtones (FO/SO type), we have detected a Blazhko-type periodic modulation of amplitudes and phases. Both modes are modulated with a common period, which is always longer than 700 d. Variations of the two amplitudes are anticorrelated, and maximum of one amplitude always coincides with minimum of the other. We have compared observations of modulated FO/SO Cepheids with predictions of theoretical models of the Blazhko effect, showing that the currently most popular models cannot account for properties of these stars. We propose that the Blazhko effect in FO/SO Cepheids can be explained by a non-stationary resonant interaction of one of the radial modes with another, perhaps non-radial, mode of oscillations.

Radial and Nonradial Oscillation Modes in Rapidly Rotating Stars

We use two-dimensional stellar models and a two-dimensional finite difference integration of the linearized pulsation equations to calculate nonradial oscillations. This approach allows us to directly calculate the pulsation modes for a distorted rotating star without treating the rotation as a perturbation. We are also able to express the finite difference solution in the horizontal direction as a sum of multiple spherical harmonics for any given mode. Using these methods, we have investigated the effects of increasing rotation and the number of spherical harmonics on the calculated eigenfrequencies and eigenfunctions and compared the results to perturbation theory. We use 10 M☉ models with velocities ranging from 0 to 420 km s−1 (0.89Ωc) and examine low-order p-modes. We find that one spherical harmonic remains reasonable up to a rotation rate around 300 km s−1 (0.69Ωc) for the radial fundamental mode, but can fail at rotation rates as low as 90 km s−1 (0.23Ωc) for the l = 2 p2 mode, based on the eigenfrequencies alone. Depending on the mode in question, a single spherical harmonic may fail at lower rotation rates if the shape of the eigenfunction is taken into consideration. Perturbation theory, in contrast, remains valid up to relatively high rotation rates for most modes. We find the lowest failure surface equatorial velocity is 120 km s−1 (0.30Ωc) for the l = 2 p2 mode, but failure velocities between 240 and 300 km s−1 (0.58Ωc–0.69Ωc) are more typical.

High-resolution spectroscopy of theδScuti star 44 Tauri: photospheric element abundances and mode identification

Aims. We carried out a detailed spectroscopic study of the δ Sct star 44 Tau to investigate its element abundances and stellar parameters as well as to provide an identification of its non-radial oscillation modes. Methods. We acquired high-resolution high-signal-to-noise ratio time-series spectroscopy of 44 Tau from two observing sites in Europe during a time span of 30 days. We determined atmospheric element abundances and stellar properties using spectroscopic methods. Due to the low value of v sini the mode identification was carried out by applying the moment method. Results. A very low value of the projected rotational velocity of v sini = 2 ± 1k m s −1 was determined. We detected spectroscopically 12 independent frequencies between 6.8 and 11.6 d −1 and four combination frequencies by analyzing the line moments as well as the pixel-intensities across the profiles of carefully selected Fe absorption lines. The mode identification applying the moment method yielded six axisymmetric pulsation modes. Two frequencies are low-degree prograde pulsation modes. A rather low value of log g = 3.6 was determined implying an evolved evolutionary status of 44 Tau. The derived atmospheric element abundances are close to solar. Conclusions. Our analysis revealed that 44 Tau is an exceptionally slowly rotating δ Scuti star pulsating with a rich spectrum of axisymmetric pulsation modes. This simplifies the seismic modeling since rotational perturbations of higher order can be neglected.

Gravitational waves from nonlinear couplings of radial and polar nonradial modes in relativistic stars

The postbounce oscillations of newly-born relativistic stars are expected to lead to gravitational-wave emission through the excitation of nonradial oscillation modes. At the same time, the star is oscillating in its radial modes, with a central density variation that can reach several percent. Nonlinear couplings between radial oscillations and polar nonradial modes lead to the appearance of combination frequencies (sums and differences of the linear mode frequencies). We study such combination frequencies using a gauge-invariant perturbative formalism, which includes bilinear coupling terms between different oscillation modes. For typical values of the energy stored in each mode we find that gravitational waves emitted at combination frequencies could become detectable in galactic core-collapse supernovae with advanced interferometric or wideband resonant detectors.

NEUTRON STAR MATTER EQUATION OF STATE AND GRAVITATIONAL WAVE EMISSION

The EOS of strongly interacting matter at densities ten to fifteen orders of magnitude larger than the typical density of terrestrial macroscopic objects determines a number of neutron star properties, including the pattern of gravitational waves emitted following the excitation of nonradial oscillation modes. This paper reviews some of the approaches employed to model neutron star matter, as well as the prospects for obtaining new insights from the experimental study of gravitational waves emitted by neutron stars.

Gravitational signals due to tidal interactions between white dwarfs and black holes

In this paper, we compute the gravitational signal emitted when a white dwarf moves around a black hole on a closed or open orbit using the affine-model approach. We compare the orbital and the tidal contributions to the signal, assuming that the star moves in a safe region where, although very close to the black hole, the strength of the tidal interaction is insufficient to provoke the stellar disruption. We show that for all considered orbits the tidal signal presents sharp peaks corresponding to the excitation of the non-radial oscillation modes of the star, the amplitude of which depends on how deep the star penetrates the black hole tidal radius and on the type of orbit. Further structure is added to the emitted signal by the coupling between the orbital and the tidal motions.

On the excitation of PG 1159-type pulsations

Stability properties are presented of dipole and quadrupole nonradial oscillation modes of model stars that experienced a late helium shell flash on their way to the white-dwarf cooling domain. The computed instability domains are compared with the observed hot variable central stars of planetary nebulae and the GW Vir pulsators.

Disentangling component spectra of , a spectroscopic binary with a pulsating primary

We analyse the complex short-term line-profile variability of the spectroscopic binary Cep star after orbit subtraction, before and after spectral disentangling. We refine the known oscillation frequency of the star: c d-1 and detect . Variability is also found at frequencies near c d-1 and c d-1 or their aliases. These frequencies are not significant if we consider the spectra alone, but they survive our selection after the consideration that they were derived previously from independent ground-based and space photometry by different teams. Moreover, we find dominant variability in the equivalent width with a frequency in the interval c d-1 which we interpret as the rotational frequency frot of the star. The complex window function does not allow us to determine definite values for . The variability with f1 is interpreted as a prograde non-radial oscillation mode with spherical wavenumbers or . The additional frequencies are explained in terms of rotational modulation superposed to the main oscillation. We also point out that we cannot disprove the variability in to originate from co-rotating structures. KOREL disentangling preserves the large-amplitude line-profile variability but its performance for complex low-amplitude variability remains to be studied in detail.

Fourier decomposition and frequency analysis of the pulsating stars with $P < $ 1 d in the OGLE database

We present the results of a systematic search for multiperiodic pulsators among the Galactic Bulge RR Lyrae stars of the OGLE-1 sample. We identify one “canonical” double-mode variable (RRd star) pulsating in two radial modes. In 38 stars we detect secondary periodicities very close to the primary pulsation frequency. This type of multiperiodic variables constitute ~23% of RRab and ~5% of RRc population of the Bulge. With the observed period ratios of the secondary periods must correspond to nonradial modes of oscillation. Their beating with the primary (radial) pulsation leads to a long-term amplitude and phase modulation, known as the Blazhko effect. The Blazhko RRab variables occur more frequently in the Galactic Bulge than in the LMC. The opposite tendency is seen in case of the RRd stars. The differences of incidence rates are most likely caused by different metallicity of the two populations. We discuss pulsation properties of the OGLE-1 Blazhko stars and compare them with predictions of theoretical models. We argue, that the oblique magnetic pulsator model of Shibahashi ([CITE]) cannot account for the observations and should be ruled out.

Non-radial oscillation modes as a probe of density discontinuities in neutron stars

A phase transition occurring in the inner core of a neutron star could be associated with a density discontinuity that would affect the frequency spectrum of the non-radial oscillation modes in two ways. First, it would produce a softening of the equation of state, leading to more compact equilibrium configurations and changing the frequency of the fundamental and pressure modes of the neutron star. Secondly, a new non-zero frequency g mode would appear, associated with each discontinuity. These discontinuity g modes have typical frequencies larger than those of g modes previously studied in the literature (thermal, core g modes or g modes caused by chemical inhomogeneities in the outer layers), and smaller than that of the fundamental mode; therefore they should be distinguishable from the other modes of non-radial oscillation. In this paper we investigate how high-density discontinuities change the frequency spectrum of the non-radial oscillations, within the framework of the general relativistic theory of stellar perturbations. Our purpose is to understand whether a gravitational signal, emitted at the frequencies of the quasi-normal modes, may give some clear information on the equation of state of the neutron star and, in particular, on the parameters that characterize the density discontinuity. We discuss some astrophysical processes that may be associated with the excitation of these modes, and estimate how much gravitational energy should the modes convey to produce a signal detectable by high-frequency gravitational detectors.

Frequency Analysis of δ Scuti and RR Lyrae Stars in the OGLE-1 Database

AbstractWe discuss the results of a systematic search for multi-periodic pulsations in Galactic Bulge δ Scuti and RR Lyrae stars. Six “normal” double-mode variables pulsating in two radial modes have been identified (5 δ Scuti-type and 1 RR Lyrae-type). In 37 RR Lyrae stars secondary periodicities very close to the primary pulsation frequency have been detected. These periodicities correspond to nonradial modes of oscillation. They are found in ∼ 23% of RRab and in ∼ 3% of RRc variables of our sample.