Research

Asteroseismology of evolved compact stars

During various stages of their evolution, stars cross at least one phase of pulsational instability that converts them into variable stars. Stellar oscillations travel through the stellar interiors and carry with them information about the internal structure of the star, which can then be studied by observing the changes in the surface brightness of the stars. The interpretation of detected stellar oscillations, known as asteroseismology, provides a unique way to probe and calibrate stellar interiors. By comparing observational data to theoretical stellar model predictions, we can fine-tune the physical descriptions used for these models. Thanks to the advance of high precision and high duty cycle photometric monitoring from space, unprecedented asteroseismic measurements and tools have become available especially for pulsating stars that are in late-stage of stellar evolution.

Compact pulsating stars including hot subdwarf B stars (sdBs) and white dwarfs (WDs) show brightness fluctuations with periods in the range of ∼ 100 − 14 000 s and ∼ 100 − 7 000 s, respectively. The typical amplitudes for both types can amount to up to 0.4 mag. In order to detect and characterize these small fluctuations, uninterrupted high-cadence time-series photometry is the most suited method. Over the last decade, space missions like Kepler/K2 and now Transiting Exoplanet Survey Satellite (TESS) have accumulated a unique data set. Such a data set has greatly improved our knowledge of stellar structure and evolution via the application of asteroseismology. My research focuses on identifying and characterizing variable compact stars using high-precision photometry obtained by space telescopes.

Non-radial oscillations in stars are characterized by three quantized numbers: n(k) for radial overtone, l for surface degree, and m for azimuthal order. Two common methods to identify these are asymptotic period spacings and rotational multiplets. We apply these seismic methods to find patterns in pulsating sdBs and WDs. For WDs, comparing asymptotic period spacing from models with observed spacing yields mass. Pulsation periods also infer mass, temperature, and internal structure. Rotating pulsators show frequency splitting into 2l + 1 components, revealing rotation via rotational multiplets. This assesses rotation in pulsating sdBs and WDs.

The ongoing TESS space mission has so far had great success looking for extrasolar planets as well as studying the pulsations of many different classes of pulsating stars, including pulsating WDs, pre-WDs and sdBs. I have focused on the asteroseismic characterization of pulsating compact stars with Kepler/K2 and TESS since 2017. In here, I highlighted some of our recent findings along with concise introduction of each types. For a comprehensive understanding, I recommend referring to the recent great reviews authored by Córsico et al. 2019 in the case of white dwarfs, Heber 2016 and A. E. Lynas-Gray 2021 in the case of sdBs, which offer detailed insights into these topics.

White dwarfs

White dwarf stars are the compact remnants of stellar evolution for all low- to intermediate-mass stars in our Galaxy, including the Sun. They play a unique and fundamental role for our understanding of the formation and evolution of stars and planetary systems. An important subset of WDs presents photometric variability caused by a range of phenomena, including pulsations, binarity, and more recently transits by both planets and planetary debris.

One of the most important characteristic of WDs is that, in the course of their evolution, all of them seem to cross at least one phase of pulsational instability that converts WDs into pulsating stars. Stellar pulsations propagate throughout the star and carry with them information about the conditions within the deep stellar interior.

Advancements in telescopes and detectors have led to the discovery of pulsations in new types of white dwarfs throughout the cooling stage, spanning different masses. Here, I concentrated on three main instability strips in the Hertzsprung–Russell diagram, including H-rich (DAVs) and H-deficient (DOVs and DBVs) types.

ZZ Ceti stars (DA variables)

The majority (around 80%) of the identified WDs are characterized by Hydrogen(H)-rich (DA) atmospheres (Kepler et al. 2019). The most populous class of pulsating WDs are ZZ Ceti stars (DA variables) with effective temperature ∼10 500 − 13 000 K and 7.5 < logg < 9.2. These stars have been studied extensively thanks to Kepler/K2 and TESS observations and recently, the number of known DAVs has been increased to 500 objects (Romero et al. 2022).

Position of established ZZ Ceti stars on the temperature (Teff) versus surface gravity (logg) plane (Córsico 2022), represented by black circles. The dashed and solid horizontal lines depicted on the diagram represent the evolutionary paths of white dwarfs with varying masses.

Asteroseismological analysis of the polluted ZZ Ceti star G 29 − 38 with TESS

One of the most notable examples of pulsating DA is the polluted ZZ Ceti star G29-38. It is one of the brightest (𝑉 = 13.1) and closest (𝑑 = 17.51 pc) pulsating white dwarfs with a H-rich atmosphere (DAV/ZZ Ceti class). It was observed by the TESS spacecraft in sectors 42 and 56. The atmosphere of G29−38 is polluted by heavy elements that are expected to sink out of visible layers on short timescales. The photometric TESS data set spans ∼ 51 days in total, and from this, we identified around 60 significant pulsation frequencies, which enable us to probe the internal conditions of G29-38 and determine the fundamental parameters. The following figures present the light curves of G29−38 from sector 42 (blue dots) and sector 56 (red dots) and Fourier transforms (FTs) of G29−38 computed from the sector 42 light curves (blue lines) and from the sector 56 light curves (red lines). The FT concentrates on the frequencies detected in the gravity-mode pulsation range. For the FT of sector 56, the amplitudes are inverted to improve clarity and comparison. See Uzundag et al. (2023) for more details.

V777 Her stars (DB variables)

DBV stars are helium-atmosphere WDs pulsating with g-mode periods (120 to 1080 s) and effective temperatures (22 400 to 32 000 K) (Winget & Kepler 2008; Córsico et al. 2019). The DBV class was theoretically predicted (Winget et al. 1982b) before confirmation through observations (Winget et al. 1982a). While the current count of identified DBVs is relatively low (47 objects, as reported in Vanderbosch et al. 2022), these pulsating WDs are of significant interest due to the unresolved origins.

Location of the known DBV WDs on the Teff −logg diagram (Córsico 2022), marked with blue circles. Thin solid curves show the DB WD evolutionary tracks from [6] for different stellar masses. The location of the DBV stars observed with TESS are emphasized with large cyan circles.

Asteroseismology of the DBV star GD 358

GD 358, a prominent instance of this type, was observed by TESS in sector 25. The light curve of GD 358 and Fourier transform of the 120-sec cadence data is shown in the left side. The black dots show the residual flux while the red lines present all prewhitened variations from the light curve. We reduced TESS observations of the DBV star GD 358 and detected 26 periodicities from the TESS light curve of this DBV star using a standard pre-whitening. The oscillation frequencies are associated with nonradial g(gravity)-mode pulsations with periods from ∼422 s to ∼1087 s. Moreover, we detected 8 combination frequencies between ∼543 s and ∼295 s.

We performed a detailed asteroseismological analysis using fully evolutionary DB WD models computed accounting the complete prior evolution of their progenitors and found the fundamental parameters of GD 358. We detected potential frequency multiplets for GD 358, which we use to identify the harmonic degree (l) of the pulsation modes and rotation period. See Córsico, Uzundag et al. 2022 for further details.

GW Vir stars (DO variables)

GW Vir variable stars are the hottest known class of pulsating WDs and pre-WDs, with 75 000 K ≤ Teff ≤ 250 000 K and 5.3 ≤ logg ≤ 8. GW Vir stars include some objects that are still surrounded by a nebula, called the variable planetary nebula nuclei (PNNVs), and some objects that lack a nebula, which are called DOVs. Both groups (DOVs and PNNVs) are frequently referred to as GW Vir variable stars. They can be either the outcome of single star evolution (late thermal pulse scenario, LTP, or very late thermal pulse scenario, VLTP) or binary star evolution (double WD merger). Thus far about 50 PG1159 stars have been identified (Werner et al., 2021). Amongst them, approximately 50% (22 objects; see Uzundag et al. 2021, 2022) have been discovered to be pulsating.

Location of the known GW Vir variable stars in the log(Teff) − logg plane (Córsico 2022) depicted on the right side with red circles. Thin solid curves show the post-born again evolutionary tracks from [7] for different stellar masses. The location of the GW Vir stars observed with TESS are emphasized with large orange circles.

Kepler and TESS Observations of PG 1159-035

PG 1159-035 serves as the prototype for the PG 1159 hot white dwarf (WD) spectroscopic class. The PG1159 spectroscopic class is identified by pronounced hydrogen deficiency and the presence of high-excitation lines such as He II, C IV, O VI, and NV. These stars are among the hottest known pulsating stars. The asteroseismological analysis of these stars on the basis of PG 1159 evolutionary models that take into account the complete evolution of the progenitor stars. The details of the models can be found in Althaus et al. (2005) and Miller Bertolami & Althaus (2006).

During the Kepler satellite K2 mission, PG 1159-035 was observed. We conducted a comprehensive asteroseismic analysis of the acquired data. Our analysis reveals a total of 107 frequencies, with 32 corresponding to l = 1 modes, 27 to l = 2 modes, and 8 to combination frequencies. Notably, the detected combination frequencies and modes with exceptionally high k values represent novel findings. Examining the multiplet structure, we observe an average splitting of 4.0 ± 0.4 μHz for l = 1. This suggests a rotation period of 1.4 ± 0.1 days. For a comprehensive understanding of the topic, I highly recommend reading Oliveira da Rosa et al. (2022), where you will find detailed insights and extensive analysis on this subject.

K2 data FT in the range of asymmetric modes. Vertical blue lines indicate the component dipole modes, the red ones indicate the component quadrupole modes.

Internal chemical profile of the asteroseismic model of PG 1159−035 (M = 0.565 M⊙ , Teff = 129 600 K) in terms of the outer fractional mass. The locations of the O/C and O/C/He chemical interfaces are indicated with gray regions.

Discovery of new pulsating GW Vir pulsating white dwarfs with TESS

In this project, our objective was to search for GW Vir stars, which are hydrogen(H)-deficient pulsating pre-white dwarf stars characterized by atmospheres rich in carbon, oxygen, and helium. We processed and analyzed high-precision TESS photometric light curves of four target stars, deriving their oscillation frequencies. Additionally, we obtained low-resolution spectra for each TESS target and fitted model atmospheres to derive fundamental atmospheric parameters. Employing asteroseismological analysis based on GW Vir white dwarf evolutionary models, which consider complete progenitor star evolution, we searched for patterns of uniform period spacings to constrain stellar mass. By meticulously matching observed frequencies with those computed from models, we characterized four new GW Vir stars using high-quality data from the TESS space mission and follow-up spectroscopy. Detailed results can be found in Uzundag et al. 2021, 2022.

The location of the two new GW Vir stars TIC 0403800675 and TIC 1989122424 (emphasized with a large red dot symbol and error bars) in the Hertzsprung-Russell diagram. Black curves show the post−born again evolutionary tracks from Miller Bertolami & Althaus (2006) for different stellar masses. Both stars share the same effective temperatures (Teff = 110 000 ± 10 000 K) and luminosities (log(L⋆/L⊙) = 1.68+/-0.2).

The internal chemical profile of the asteroseismological model of TIC 333432673 (M = 0.589M⊙, Teff = 117 560 K) in terms of the outer fractional mass. The locations of the O/C and O/C/He chemical interfaces are indicated with gray regions.

Hot subdwarf B stars

Hot subluminous stars of spectral type B (sdB) have been identified as core helium-burning stars. They are located between the Horizontal Branch (HB) and white dwarf cooling track, on a narrow path on the so-called the Extreme Horizontal Branch (EHB). The location of hot subdwarfs on the Hertzsprung-Russell diagram is shown on the right side. heir effective temperatures are spanning from 20 000 to 40 000 K. Their masses are typically around 0.47 solar masses and radii between 0.15 and 0.35 solar radii, making them compact objects. These stars are He-burning cores, which are surrounded by thin hydrogen envelopes (Menvelope < 0.01M⊙). Such small mass of the hydrogen envelope does not allow sdB stars to climb up the Asymptotic Giant Branch (AGB). Therefore, after helium depletion in the core, sdB stars will evolve into subdwarf O (sdO) stars that burn helium in a shell surrounding the C/O core. Soon after, they move directly to the white dwarf cooling tracks.

Pulsating hot subdwarf B stars

Over the last decade the space missions, in particular Kepler, then K2 and now NASA mission of TESS have completely transformed our understanding of hot subdwarf B variable stars. In here, I will briefly overview the current state-of-the-art analysis of hot sdB pulsating stars from the perspective of recent space missions.

Distribution of the pulsating sdB stars in the effective temperature–surface gravity plane. The location of short-period pressure-mode pulsators of the V361 Hya type are indicated in blue, while the long-period gravity-mode variables of the V1093 Her type are shown in red. The hybrid pulsators, showing simultaneously both p-modes and g-modes, are shown in blue and red.

A significant advances in our understanding of sdB stars was initiated by Kilkenny et al. (1997), who discovered rapid pulsations in hot sdBs known as V361 Hya stars (often referred to as short-period sdBV stars). Multiperiodic pulsations with periods ranging from 100 to 800 seconds are detected in the V361 Hya stars. The pulsational modes in this region correspond to low-degree, low-order pressure (p)-modes with photometric amplitudes as small as a few percent of their mean brightness. The p-mode sdB pulsators are found in a temperature range between 28 000 K and 35 000 K and with the surface gravity logg in the interval of 5-6 dex.

The long-period sdB pulsators known as V1093 Her stars were discovered by Green et al. (2003). These stars show brightness variations with periods of up to a few hours and have amplitudes smaller than 0.1 per cent of their mean brightness. The gravity-mode sdB pulsators are somewhat cooler, with temperatures ranging from 22 000 K to 30 000 K and log g from 5 dex to 5.5 dex.

Some sdB pulsators showing both gravity- and pressure-modes have been discovered among the two described families of pulsating sdB stars. These stars are referred to as hybrid sdB pulsating stars with temperatures ranging from 28 000 K to 32 000 K. These objects are especially significant because they allow us to study both the core structure and the outer layers of the sdBVs using asteroseismology.

Lomb–Scargle periodogram (LSP) of new sdB stars showing the gravity-mode region of the frequency spectrum. The panels are sorted from top to bottom with the decreasing brightness of targets. In the insert of each panel we give the TESS input catalog number, effective temperature (in Kelvin) and surface gravity (in cgs units) respectively. Blue dashed lines correspond to 0.1% false-alarm probability confidence level.

LSP of TIC 260795163. The residuals after prewhitening are shown with orange color. The red dots are showing the dipole modes and the blue ones are displaying the quadrupole modes. If there is no unique identification, the period is shown with both colors. The horizontal red line correspond to 0.1% FAP confidence level. The right panel shows the residuals between the observed and the fitted periods. See Uzundag et al. (2021) for more details on mode identification and the mean period spacing computations.

NTT/EFOSC2 spectrum of newly discovered pulsating sdBs with the best matching Tlusty/XTgrid atmospheric models. The spectra are dominated by the H Balmer-series marked at the bottom and few weak He I lines marked at the top of the figure.

Properties of the two sets of stellar evolution models discussed in the text as compared with those observed in pulsating sdB stars. Blue circles indicate the stars were observed by TESS, green circles indicate those from Kepler/K2. Red tracks correspond to models computed with a moderate CBM prescription while cyan tracks correspond to those computed under the extreme assumption of a strict Schwarzschild criterion. Masses of the models are MZAHB = 0.467, 0.4675, 0.468, and 0.47 Msun, and darker colors correspond to more massive models. Thick magenta lines show the locus of the model at the zero-age extreme horizontal branch (ZAEHB). Light-colored bands around the curves in the bottom right panel indicate the typical variance of ∆P around the mean value.

In this work, a detailed asteroseismic and spectroscopic analysis of pulsating sdB stars observed with Kepler/K2 and TESS were produced in order to compare the observations with model predictions based on our stellar evolution computations coupled with adiabatic pulsation computations. The stellar evolution models were computed using the LPCODE stellar evolution code, and computed l=1 g-mode frequencies with the adiabatic non-radial pulsation code LP-PUL. Derived observational mean period spacings were then compared to the mean period spacings from detailed stellar evolution computations coupled with the adiabatic pulsation computations of g-modes. In agreement with the expectations from theoretical arguments and previous asteroseismological works, we find that the mean period spacings obtained for models with small convective cores, as predicted by a pure Schwarzschild criterion, are incompatible with the observations. We find that models with a standard, modest convective boundary mixing at the boundary of the convective core are in better agreement with the observed mean period spacings and are therefore more realistic.

Volume-limited sample of low-mass red giant stars, the progenitors of hot subdwarf stars

Another important aspect of the final evolution of stars is the impact of binarity since a high percentage of stars reside in binary systems. Stars are mostly found in binary and multiple systems: at least half of all solar-like stars have companions, and for the most massive stars, that percentage increases up to 80%. A significant number of these systems will interact during their evolution, changing the structure and evolution of the components and resulting in the formation of exotic objects not predicted by standard stellar evolution models. However, some key details concerning binary interactions are still uncertain.

Thanks to the data from the Gaia mission it is now possible to select complete all-sky samples of hot subdwarf stars and cross-match them with the wealth of spectroscopic data provided in the literature (Geier et al., 2017, 2019). This is the first step towards the compilation of all-sky, volume-complete samples with minimal selection effects, which can be directly compared to quantitative models. Currently, the volume-limited spectroscopic survey of all sdBs within 500 pc has been completed (Dawson et al., in prep.).

The orbital periods of the sdB+MS binaries produced by the stable Roche-lobe-overflow (RLOF) channel range from 500 days to more than 1600 days. Long-term observational campaigns have been conducted in an effort to find these systems over the past few decades and the orbital parameters of 26 systems for stable RLOF sdB binaries with periods between 500 and 1500 days have recently been solved (Vos et al. 2020). These systems are referred to as ”composite” binaries since both stars can be seen in the spectrum, and they account for 30-40% of all sdBs.

In this project, we aim at compiling a homogeneous volume limited sample of pre (low-mass RGB + MS) and post (sdB + MS) RLOF interaction systems that will enable us to constrain the physics of stable mass transfer and will give the observational evidence for correlation between the key parameters governing the formation of hot sdB stars. We are interested in the initial condition of the progenitor population of sdB stars (before they have started binary interactions), we focus on the lower luminosity RGBs as those stars are clearly the H-shell burning around an inert He-core and have not yet undergone binary interaction. We assembled a sample of low-mass red giant candidates using data from the Gaia mission and several ground-based spectroscopic surveys. The candidates were chosen based on the color, absolute magnitude, and proper motion cuts of their Gaia DR2. In order to find RGB + MS binary systems within 200 pc, we focused on targets in the southern hemisphere and performed a spectroscopic survey of red giant stars.

Left: Color-magnitude diagram of all low-mass (0.7−2.3 M) RG candidates within 200 pc (blue dots) from Gaia DR2 for the southern hemisphere stars. As a comparison, the Gaia DR2 color-magnitude diagram of the 100 pc clean sample is shown in gray. The slightly larger gray dots within the selected sample represent the low-mass RG candidates within 500 pc. The red circles show the low-mass RG candidates that were observed with EULER/CORALIE. Right: sky locations (Galactic coordinates, Aitoff projection) of the volume-limited low-mass RG sample within 200 pc with respect to the galactic coordinate system use the same color-coding.

The Gaia colour-magnitude diagram. Full blue circles show composite-spectrum sdB+MS binaries and empty blue circles show the single-lined sdB+MS binaries. sdA+MS binaries are similarly coded in green circles. The systems that will form an sdB+MS or sdA+MS binary are shown with orange squares at the moment when the interaction phase starts. As a comparison, the Gaia colour-magnitude diagram of the Hipparcos sample is shown in grey (Vos et al. 2020).

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