Non-radial Modes Research Articles

Considering the incidence rate of RR Lyrae stars with non-radial modes

Over recent years, additional low-amplitude non-radial modes have been detected in many first-overtone RR Lyrae stars. These non-radial modes form a characteristic period ratio with the dominant first overtone of around 0.61. The incidence rate of this phenomenon varies from population to population. It is also strongly dependent on the quality of the analyzed data. Current models aimed at explaining these additional signals involve non-radial modes of degrees of 8 and 9. Using synthetic horizontal branch populations, we investigate the incidence rate of first-overtone RR Lyrae stars with non-radial modes, depending on the population properties, namely, ages and metallicities. We compare our results with the observed results for globular clusters and the vast collection of field first-overtone RR Lyrae stars to test the model predictions. We used synthetic horizontal branches combined with pulsation models to predict how the incidence rate would depend on the age and metallicity of the population. To test whether the results based on synthetic horizontal branches are realistic, we compared them to incidence rates observed by TESS in first-overtone field RR Lyrae stars, using photometric metallicity values from a newly established calibration for TESS. The analysis of synthetic horizontal branches indicates that the incidence rate decreases with decreasing metallicity. We inferred the photometric metallicity for RR Lyrae stars observed by TESS and showed that the theoretical predictions are in agreement with the observations. Using the same method, we also concluded that the metallicity distribution of RR Lyrae stars showing an additional mode with a period-ratio around $0.68$ appears to be different from that of 1) all first-overtone stars and 2) those showing additional non-radial modes.

The VELOCE modulation zoo I. Spectroscopic detection of non-radial modes in the first-overtone Cepheids BG Crucis, QZ Normae, V0391 Normae, and V0411 Lacertae

The photometric observations from the recent decade revolutionized our view on classical pulsators. Low-amplitude signals have been detected photometrically in addition to the dominant high-amplitude radial mode pulsations in many RR Lyrae stars and classical Cepheids. First-overtone pulsators with an additional low-amplitude signal at a period ratio of around 0.61 with the main mode, the so-called 0.61 stars, form the most populous group among these stars. The nature of this signal has been attributed to non-radial pulsations. Another mysterious group are stars in which the additional signal forms a period ratio of around 0.68. These are the 0.68 stars. The origin of the signal remains unknown. Here, we search for similar phenomena in spectroscopic observations of first-overtone classical Cepheids collected as part of the project. We performed a frequency analysis of several parameters derived from cross-correlation functions (CCFs), including radial velocity, the full width at half maximum, the bisector inverse span, and the CCF depth (contrast). Using standard pre-whitening, we searched for additional low-amplitude signals. We identified the location of these stars in various sequences of the Petersen diagram. We detect additional signals in four first-overtone classical Cepheids: BG Cru, QZ Nor, V0391 Nor, and V0411 Lac. We classified BG Cru, QZ Nor, and V0391 Nor as 0.61 stars based on their period ratios. V0411 Lac, however, exhibits a ratio of 0.68 between the two modes, and the additional signal has a longer period. This type of multi-periodicity remains unexplained. CCFs yield the first spectroscopic detections of non-radial pulsation modes in classical Cepheids. This opens an asteroseismic window for pursuing a more detailed understanding of these important stars. While the 0.61 signal of BG Cru, QZ Nor, and V0391 Nor is understood to originate from non-radial modes of moderate degrees, the 0.68 signal of V0411 Lac still lacks a physical explanation.

Frequency and photometric analysis of the δ Scuti stars SDSS J231559.32 + 595654.1 and ASAS J080529-1357.5

We carried out multi-bandpass BVRI CCD photometric observations for the two δ Scuti stars SDSS J231559.32 + 595654.1 (SDSS23 + 59) and ASAS J080529-1357.5 (ASAS08-13) at Kottamia Astronomical Observatory (KAO). The star SDSS23 + 59 has δ Scuti-type with a low amplitude pulsating variable, and the star ASAS08-13 is classified as a high amplitude δ Scuti-type. The time-series light curve data for each star was analyzed using the discrete Fourier transform (DFT), which revealed a dominant fundamental mode (f0) of oscillation and associated harmonics. The photometric analysis of the two stars suggests SDSS23 + 59 is a star that can pulsate in a short period mode (P = 0.111314d ± 0.000012) with low amplitude (0.101–0.09), and the star ASAS08-13 can pulsate in a short period mode (P = 0.10052247 ± 0.00000281) with high amplitude (0.186–0.168 mag). The nonradial modes for SDSS23 + 59 and radial modes for ASAS08-13 are detected. We calculated new times of maximum light for both stars and combined them with those from other sources (ZTF and AAVSO) to update their corresponding linear ephemerides. The O − C diagram gives us a primary indication that the period of the star is increasing at the rate of (1/P) dP/dt = 3.410(0.006)×10 -10 yr−1 and 2.7321(0.0092)x10-6 yr−1 for SDSS23 + 59 and ASAS08-13 respectively. The physical parameters are calculated, and the positions of the stars on the P-Teff, M−L, M−R, and Teff-log g are investigated.

Curve of Growth Analysis of SZ Lyn

We present one high-resolution and a time series of 561 low-resolution follow-up spectroscopic observations of SZ Lyn. It is a high-amplitude δ Scuti-type pulsating star in a binary system. The photometric observations reveal the existence of radial and non-radial oscillation modes in SZ Lyn. In spectroscopy, the variation of equivalent width of the line profiles reflects the temperature variations. The equivalent widths of the Balmer lines, Hα, Hβ, and Hγ were measured over the pulsation cycle of SZ Lyn using time sequence spectra. Hence, the temperature profile of SZ Lyn was derived using the curve of growth analysis. Furthermore, the stellar parameters were determined through the best fit analysis of observed and synthetic high-resolution spectral lines. The best fit determines a model of Teff = 6750 K, log g = 3.5 dex, and vrot = 10 km s−1 for solar abundance.

Similar additional frequency patterns on fundamental- and overtone-mode RR Lyrae stars showing f68 frequencies

Context. Until now, it has been accepted that the additional frequencies in the fundamental-(RRab) and overtone-mode pulsating (RRc and RRd) RR Lyrae stars are of a different nature. RRab stars show frequencies associated with periodic doubling, as well as frequencies at the first and second radial overtones, and linear combinations of these. RRc stars show frequencies with specific ratios (f1/fx ∼ 0.61 or ∼0.63), which are explained by non-radial modes, and frequencies with a ratio of fx/f1 ∼ 0.68, for which there is currently no accepted explanation. Aims. To search for similarities in spectral content, we compared the recently published Fourier spectra of TESS and K2 RRc stars with the spectra of Kepler RRab stars that do not show the Blazhko effect but contain additional frequencies. Methods. We analysed the time series data using standard Fourier methods, and also investigated the possibility of excitation of the second radial overtone mode in RRab stars using numerical hydrodynamical codes. Results. We show that the additional frequencies appear in non-Blazhko RRab stars at the position of the second radial overtone mode, and the pattern they create is very similar to that caused by the additional frequencies with the period ratio of ∼0.68 in RRc stars. The formerly raised hypothesis that the additional frequencies of these RRab stars are due to a second radial overtone is unlikely.

TESS Observations of Seven Newly Identified High-amplitude δ Scuti Stars

We report seven newly identified high-amplitude δ Scuti (HADS) stars. Among them, two stars (TIC 30977864 and TIC 387379145) exhibit pure radial pulsation without the excitation of non-radial modes. TIC 30977864 is classified as a double-mode HADS star, while the other four stars potentially show triple-mode HADS behavior. TIC 112682462 and TIC 255603395 closely resemble RR Lyrae stars based on their light curves, position in the period–luminosity diagram, and the period ratio diagram. However, without spectral observations, it is challenging to ascertain whether these two stars are HADS or RR Lyrae stars. TIC 281695001 exhibits a fundamental frequency amplitude smaller than that of the first overtone, suggesting the presence of intriguing phenomena that necessitate further analysis. We analyzed the relationship between metallicity and period using data provided by Netzel & Smolec. The metallicities of the 176 stars display a broad distribution ranging from −2.0 dex to 0.5 dex, with periods spanning 0.05 to 0.20 days. This random distribution of metallicities may contribute to the dispersion observed in the P1/P0 ratio. To derive more accurate conclusions, future spectroscopic observations of a larger sample of HADS stars are crucial. These observations will provide precise rotational velocities and more accurate determinations of metallicities.

Asteroseismology of double-mode radial δ Scuti stars: AE Ursae Majoris and RV Arietis

ABSTRACT We construct complex seismic models of two high-amplitude δ Sct stars, AE UMa and RV Ari, each pulsating in two radial modes: fundamental and first overtone. Besides the frequencies of two radial modes, the models also reproduce the amplitude of bolometric flux variations (the non-adiabatic parameter f) for the dominant mode. Applying the Monte-Carlo-based Bayesian analysis, we derive strong constraints on the parameters of the model, as well as on the free parameters of the theory. A vast majority of seismic models of the two stars are just at the beginning of hydrogen-shell burning and a small fraction are at the very end of an overall contraction. The stars have a similar age of about 1.6 Gyr for the hydrogen-shell burning phase. Both stars have unusual low overshooting from the convective core, about 0.02 and 0.004 of the pressure scaleheight for AE UMa and RV Ari, respectively. This result presumably indicates that overshooting should vary with time and scale with a decreasing convective core. The efficiency of convection in the envelope of both stars is low and is described by the mixing length parameter αMLT of about 0.3–0.6. The third frequency of RV Ari, confirmed by us in the Transiting Exoplanet Survey Satellite (TESS) photometry, can only be associated with mixed non-radial modes ℓ = 1, g4–g8 or ℓ = 2, g10–g12. We include the dipole mode into our Bayesian modelling and demonstrate its huge asteroseismic potential.

Detailed asteroseismic modelling of RR Lyrae stars with non-radial modes

ABSTRACT Photometric observations from the last decade have revealed additional low-amplitude periodicities in many classical pulsators that are likely due to pulsations in non-radial modes. One group of multimode RR Lyrae stars, the so-called 0.61 stars, is particularly interesting. In these stars, the radial first overtone is accompanied by additional signals with period ratios around 0.61. The most promising explanation for these signals is pulsation in non-radial modes of degrees 8 and 9. If the theory behind the additional signals in the 0.61 stars is substantiated, it would allow us to use non-radial modes to study classical pulsators. We aim to perform asteroseismic modelling of selected 0.61 stars with independently determined physical parameters to test whether this assumption behind the modelling leads to correct results. Namely, we test whether the additional signals are indeed due to non-radial modes of the proposed moderate degrees. We selected a number of RR Lyrae stars that are also 0.61 stars and have good observational constraints on their other physical parameters. We assume that the nature of those modes is correctly explained with non-radial modes of degrees 8 or 9. Using this assumption and observational constraints on physical parameters, we performed asteroseismic modelling to test whether the observed periods and period ratios can be reproduced. For the majority of selected targets, we obtained a good match between observed and calculated periods and period ratios. For a few targets however, the results obtained are ambiguous and not straightforward to interpret.

Unveiling the Evolutionary State of Three B Supergiant Stars: PU Gem, ϵ CMa, and η CMa

We aim to combine asteroseismology, spectroscopy, and evolutionary models to establish a comprehensive picture of the evolution of Galactic blue supergiant stars (BSG). To start such an investigation, we selected three BSG candidates for our analysis: HD 42087 (PU Gem), HD 52089 (ϵ CMa), and HD 58350 (η CMa). These stars show pulsations and were suspected to be in an evolutionary stage either preceding or succeding the red supergiant (RSG) stage. For our analysis, we utilized the 2-min cadence TESS data to study the photometric variability, and we obtained new spectroscopic observations at the CASLEO observatory. We used non-LTE radiative transfer models calculated with CMFGEN to derive their stellar and wind parameters. For the fitting procedure, we included CMFGEN models in the iterative spectral analysis pipeline XTgrid to determine their CNO abundances. The spectral modeling was limited to changing only the effective temperature, surface gravity, CNO abundances, and mass-loss rates. Finally, we compared the derived metal abundances with prediction from Geneva stellar evolution models. The frequency spectra of all three stars show stochastic oscillations and indications of one nonradial strange mode, fr= 0.09321 d−1 in HD 42087 and a rotational splitting centred in f2= 0.36366 d−1 in HD 52089. We conclude that the rather short sectoral observing windows of TESS prevent establishing a reliable mode identification of low frequencies connected to mass-loss variabilities. The spectral analysis confirmed gradual changes in the mass-loss rates, and the derived CNO abundances comply with the values reported in the literature. We were able to achieve a quantitative match with stellar evolution models for the stellar masses and luminosities. However, the spectroscopic surface abundances turned out to be inconsistent with the theoretical predictions. The stars show N enrichment, typical for CNO cycle processed material, but the abundance ratios did not reflect the associated levels of C and O depletion. We found HD 42087 to be the most consistent with a pre-RSG evolutionary stage, HD 58350 is most likely in a post-RSG evolution and HD 52089 shows stellar parameters compatible with a star at the TAMS.

Three approaches for the classification of protoneutron star oscillation modes

ABSTRACT The future detection of gravitational waves (GWs) from a Galactic core-collapse supernova will provide information on the physics inside protoneutron stars (PNS). In this work, we apply three different classification methods for the PNS non-radial oscillation modes: Cowling classification, Generalized Cowling Nomenclature (GCN), and a classification based on modal properties (CBMP). Using PNS models from 3D simulations of core-collapse supernovae, we find that in the early stages of the PNS evolution, typically 0.4 s after the bounce, the Cowling classification is inconsistent, but the GCN and the CBMP provide complementary information that helps to understand the evolution of the modes. In the GCN, we note several avoided crossings as the mode frequencies evolve at early times, while the CBMP tracks the modes across the avoided crossings. We verify that the strongest emission of GWs by the PNS corresponds to the f mode in the GCN, indicating that the mode trapping region alternates between the core and the envelope at each avoided crossing. At later times, approximately 0.4 s after the bounce, the three classification methods present a similar description of the mode spectrum. We use our results to test universal relations for the PNS modes according to their classification and find that the behaviour of the universal relations for f and p modes is remarkably simple in the CBMP.

Evolutionary and Observational Properties of Red Giant Acoustic Glitch Signatures

While solar-like oscillations in red giants have been observed at massive scales by the Kepler mission, few features of these oscillation mode frequencies, other than their global properties, have been exploited for stellar characterization. The signatures of acoustic glitches in mode frequencies have been used for studying main-sequence stars, but the validity of applying such techniques to evolved red giants, particularly pertaining to the inclusion of nonradial modes, has been less well examined. Making use of new theoretical developments, we characterize glitches using the π modes associated with red giant stellar models, and use our procedure to examine for the first time how the properties of the He ii acoustic glitch—specifically its amplitude and associated acoustic depth—vary over the course of evolution up the red giant branch, and with respect to other fundamental stellar properties. We find that the acoustic depths of these glitches, in conjunction with other spectroscopic information, discriminate between red giants in the first-ascent and core-helium-burning phases. We critically reexamine previous attempts to constrain acoustic glitches from nonradial (in particular dipole) modes in red giants. Finally, we apply our fitting procedure to Kepler data, to evaluate its robustness to noise and other observational systematics.

Astrophysical properties of 15062 Gaia DR3 gravity-mode pulsators

Context. Gravito-inertial asteroseismology came into existence thanks to high-precision CoRoT and Kepler space photometric light curves. So far, it has given rise to the internal rotation frequency of a few hundred intermediate-mass stars, yet only several tens of these have been weighed, sized, and age-dated with high precision using asteroseismic modelling. Aims. We aim to increase the sample of optimal targets for future gravito-inertial asteroseismology by assessing the properties of 15062 newly found Gaia DR3 gravity-mode pulsators. We also wish to investigate whether or not there is a connection between their fundamental parameters, the dominant mode, and their spectral line broadening as measured by Gaia. Methods. After reclassifying about 22% of the F-type gravity-mode pulsators as B-type according to their effective temperature, we constructed histograms of the fundamental parameters and mode properties of the 15062 new Gaia DR3 pulsators. We compared these histograms with those of 63 Kepler bona fide class members. We fit errors-in-variables regression models to couple the effective temperature, luminosity, gravity, and oscillation properties to the two Gaia DR3 parameters capturing spectral line broadening for a fraction of the pulsators. Results. We find that the selected 15062 gravity-mode pulsators have properties that are fully in line with those of their well-known Kepler analogues, revealing that Gaia has a role to play in asteroseismology. The dominant ɡ-mode frequency is a significant predictor of the spectral line broadening for the class members for which this quantity has been measured. We show that the Gaia vbroad parameter captures the joint effect of time-independent intrinsic and rotational line broadening and time-dependent tangential pulsational broadening. Conclusions. While the Gaia mission was not designed to detect non-radial oscillation modes, its multitude of data and homogeneous data treatment allow us to identify a vast number of new gravity-mode pulsators that have fundamental parameters and dominant mode properties in agreement with those of such Kepler bona fide pulsators. This large new sample of Gaia DR3 pulsators can be followed up with dedicated high-precision photometric or high-resolution spectroscopic instruments to embark on asteroseismic modelling.

Time series analysis of bright TESS RRc stars: additional modes, phase variations, and more

ABSTRACT Using two years of data from the TESS space telescope, we have investigated the time series of 633 overtone pulsating field RR Lyrae (RRc) stars. The majority of stars (82.8 per cent) contain additional frequencies beyond the main pulsation. In addition to the frequencies previously explained by the ℓ = 8 and 9 non-radial modes, we have identified a group of stars where the additional frequencies may belong to the ℓ = 10 non-radial modes. We found that stars with no additional frequencies are more common among stars with shorter periods, while stars with longer periods almost always show additional frequencies. The incidence rate and this period distribution both agree well with the predictions of recent theoretical models. The amplitude and phase of additional frequencies are varying in time. The frequencies of different non-radial modes appearing in a given star seem to vary on different time-scales. We have determined a 10.4 per cent incidence rate for the Blazhko effect. For several stars we have detected continuous annual-scale phase change without significant amplitude variation. This type of variation offers a plausible explanation for the ‘phase jump’ phenomenon reported in many RRc stars. The main pulsation frequency could show quasi-periodic phase and amplitude fluctuations. This fluctuation is clearly related to additional frequencies present in the star: stars with two non-radial modes show the strongest fluctuations, while stars with no such modes show no fluctuations at all. The summation of the phase fluctuation over time may explain the O−C variations that have long been known for many non-Blazhko RRc stars.

Non-radial oscillation modes in hybrid stars: consequences of a mixed phase

We study the possibility of the existence of a deconfined quark matter in the core of neutron star (NS)s and its relation to non-radial oscillation modes in NSs and hybrid star (HS)s. We use relativistic mean field (RMF) models to describe the nuclear matter at low densities and zero temperature. The Nambu-Jona-Lasinio (NJL) model is used to describe the quark matter at high densities and zero temperature. A Gibbs construct is used to describe the hadron-quark phase transition (HQPT) at large densities. Within the model, as the density increases, a mixed phase (MP) appears at density about 2.5 times the nuclear matter saturation density (ρ 0) and ends at density about 5ρ 0 beyond which the pure quark matter phase appears. It turns out that a stable HS of maximum mass, M = 2.27 M ⊙ with radius R = 14 km (for NL3 parameterisation of nuclear RMF model), can exist with the quark matter in the core in a MP only. HQPT in the core of maximum mass HS occurs at radial distance, rc = 0.27R where the equilibrium speed of sound shows a discontinuity. Existence of quark matter in the core enhances the non-radial oscillation frequencies in HSs compared to NSs of the same mass. This enhancement is significantly large for the g modes. Such an enhancement of the g modes is also seen for a density dependent Bayesian (DDB) parmeterisation of the nucleonic EOS. The non-radial oscillation frequencies depend on the vector coupling in the NJL model. The values of g and f mode frequencies decrease with increase the vector coupling in quark matter.

Asteroseismology: Looking for Axions in the Red Supergiant Star Alpha Ori

In this work, for the first time, we use seismic data as well as surface abundances to model the supergiant α-Ori, with the goal of setting an upper bound on the axion–photon coupling constant g a γ . We find that, in general, stellar models with g a γ ∈ [0.002; 2.0] × 10−10 GeV−1 agree with the observational data, but beyond that upper limit, we do not find stellar models that are compatible with the observational constraints and the current literature. From g a γ = 3.5 × 10−10 GeV−1 on, the algorithm did not find any fitting models. Even so, all the axionic models considered present distinct internal profiles from the reference case, without axions. Moreover, as the axion energy losses become more significant, the behavior of the stellar models becomes more diversified, even with very similar input parameters. Nonetheless, the consecutive increments of g a γ still show systematic tendencies, resulting from the axion energy losses. Moreover, we establish three important conclusions: (1) the increased luminosity and higher neutrino production are measurable effects, possibly associated with axion energy losses; (2) stellar models with axion energy loss show a quite distinct internal structure; and (3) the importance of future asteroseismic missions in observing low-degree nonradial modes in massive stars is emphasized—as internal gravity waves probe the near-core regions, where axion effects are most intense. Thus, more seismic data will allow us to constrain g a γ better and to prove or dismiss the existence of axion energy loss inside massive stars.

First overtone Cepheids of the OGLE Magellanic Cloud Collection – beyond radial modes

ABSTRACT We have analysed Optical Gravitational Lensing Experiment photometry for first overtone classical Cepheids in the Large and Small Magellanic Clouds in search for additional periodicities beyond radial modes. We have used standard consecutive pre-whitening technique in some cases followed by time-dependent pre-whitening. We report new candidates for double-mode radial pulsations. However, majority of signals we have detected cannot be interpreted in terms of radial modes. We report 516 double-periodic stars with period ratios, Px/P1O, in the range 0.60–0.65. We study the properties of this class and implications for model explaining these periodicities. We also report 28 stars in which additional variability is of longer period, below radial fundamental mode, with median period ratio, P1O/Px, of 0.684. This class is an analogue of a class known in RR Lyrae stars. Hundreds of other signals were detected that cannot be attributed to radial modes or the above-mentioned classes. Statistical properties of these signals are analysed. We suggest that majority of these signals correspond to non-radial modes. In particular, a significant fraction can be attributed to non-radial modes of moderate degrees, tightly connected to a class with period ratios in between 0.60 and 0.65. In tens of stars, close to radial mode frequency, relatively large-amplitude and coherent signals are observed, which may represent yet another class. In 27 stars periodic modulation of pulsation was detected. Differences in additional frequency content between the two Clouds are discussed.

A Rapidly Evolving High-amplitude δ Scuti Star Crossing the Hertzsprung Gap

In this work, we report the discovery of the rapidly evolving high-amplitude δ Scuti star KIC 6382916 (J19480292+4146558), which is crossing the Hertzsprung gap. According to the analysis of the archival data, we find three independent pulsation modes, whose amplitudes and frequencies vary distinctly in 4 yr. The linear period variation rates of the first two modes are about 3–4 times larger than the best seismic model constructed by the standard evolution theory, while that of the third one is about 8 times larger than the first two modes. What is more interesting is that almost all the combinations of the third mode have frequency peaks 0.0815 c day−1 away from them in the frequency domain. A framework is proposed to interpret the markedly large frequency and amplitude variation rates of the third mode, in which we employ a new pulsation mode (resonating integration mode) generated by the resonance between a radial p-mode and a nonradial mixed mode. Moreover, global analysis of the interactions between the three independent pulsation modes and their harmonics/combinations is performed based on the interaction diagrams of their amplitudes and phases, which would be a useful tool for future asteroseismology research.

Asteroseismology of RR Lyrae stars with non-radial modes

ABSTRACT The additional signals observed in the frequency spectra of the first-overtone RR Lyrae stars, which form a period ratio around 0.61 with the period of the first overtone, are a common phenomenon for RRc and RRd stars, as well as for first-overtone classical Cepheids. The recently proposed model explains these signals as harmonics of non-radial modes of degrees 8 or 9 in the case of RR Lyrae stars and 7, 8, or 9 in the case of classical Cepheids. We selected at least triple-mode RR Lyrae stars pulsating in radial and non-radial modes for asteroseismic modelling. We assume the identification of the non-radial modes as predicted by the model. We calculated a dense grid of models for RR Lyrae stars using envelope pulsation code. By matching first-overtone period and period ratios, we obtained physical parameters for the selected sample of triple-mode stars. It is the very first attempt of modelling RR Lyrae stars with non-radial modes. We compared our results with predictions of stellar evolution theory, which resulted in a mass discrepancy more noticeable for long-period stars: Pulsation masses seem higher than evolutionary masses. We compared metallicity estimates for RRc stars from a modelled sample with metallicities determined spectroscopically for a sample of RRc stars in the solar neighbourhood: Both distributions are consistent.

The Solar Oscillation Equation and Some of its Particular Solutions

Background: The phenomenon that sun oscillates in characteristic eigen frequencies witnessed on the solar surface has fascinated many solar physicists to go for exact measurements of the internal properties of our nearest star that we are most familiar with. The present work bears the objective to understand the dynamic and nonlinear profile of solar oscillation on the basis of some particular solutions of the fundamental equation of solar oscillation. After the detection of global solar oscillation during 1970’s different physical aspects of solar oscillation started to draw the attention of several astrophysicists both from the perspectives of theoretical investigation as well as observational evidences. Methods: The functional forms of the wave representation term appeared in the wave-equation of our present system has been qualitatively studied by performing linear stability analysis, where the ‘Jacobian’ provides the basis of understanding of the nature of equilibrium points. Analytical means or approaches have been employed to determine the solutions of the said wave equation without using any kind of computational method or simulation at different regions of the number line. Findings: The corresponding equilibrium points of the wave-equations followed from the solar oscillation equation at various chosen functional forms of the complicated wave representation term have been nicely calculated. Sincere analytical attempts have also been taken for the first time to derive the solutions of the solar oscillation equation segmented into several mathematically chosen scenarios by us. Novelty: The analytic solutions of the solar oscillation equation obtained for different particular functional structures explain the existence of nonlinear acoustic modes of frequency in the observed solar oscillations of long period. This result will surely motivate any further quest of chaotic oscillations in both radial and non-radial modes. Keywords: Solar oscillation; Equilibrium point; Jacobian; Bessel function; Eigenvalues

Oscillating magnetized hybrid stars under the magnifying glass of multimessenger observations

ABSTRACTWe model neutron stars as magnetized hybrid stars with an abrupt hadron–quark phase transition in their cores, taking into account current constraints from nuclear experiments and multimessenger observations. We include magnetic field effects considering the Landau level quantization of charged particles and the anomalous magnetic moment of neutral particles. We construct the magnetized hybrid equation of state, and we compute the particle population, the matter magnetization and the transverse and parallel pressure components. We integrate the stable stellar models, considering the dynamical stability for rapid or slow hadron–quark phase conversion. Finally, we calculate the frequencies and damping times of the fundamental and g non-radial oscillation modes. The latter, a key mode to learn about phase transitions in compact objects, is only obtained for stars with slow conversions. For low magnetic fields, we find that one of the objects of the GW170817 binary system might be a hybrid star belonging to the slow extended stability branch. For magnetars, we find that a stronger magnetic field always softens the hadronic equation of state. Besides, only for some parameter combinations a stronger magnetic field implies a higher hybrid star maximum mass. Contrary to previous results, the incorporation of anomalous magnetic moment does not affect the studied astrophysical quantities. We discuss possible imprints of the microphysics of the equation of state that could be tested observationally in the future, and that might help infer the nature of dense matter and hybrid stars.