arXiv subjects

J. M. Joel Ong

Publications and source records attributed to J. M. Joel Ong.

At least 19 recordsLinked to original sources

arXiv2026

Unified Kraft Break at ~6500 K: A Newly Identified Single-Star Obliquity Transition Matches the Classical Rotation Break

The stellar obliquity transition, defined by a $\textit{T}_{\rm eff}$ cut separating aligned from misaligned hot Jupiter systems, has long been assumed to coincide with the rotational Kraft break. Yet the commonly quoted obliquity transition (6100 or 6250 K) sits a few hundred kelvin cooler than the rotational break (~6500 K), posing a fundamental inconsistency. We show this offset arises primarily from binaries/multiple-star systems, which drive the cooler stellar obliquity transition ($6105^{+123}_{-133}$ K), although the underlying cause remains ambiguous. After removing binaries and higher-order multiples, the single-star stellar obliquity transition shifts upward to $6447^{+85}_{-119}$ K, in excellent agreement with the single-star rotation break ($6510^{+97}_{-127}$ K). This revision has two immediate consequences for understanding the origin and evolution of spin-orbit misalignment. First, the upward shift reclassifies some hosts previously labeled `hot' into the cooler regime; consequently, there are very few RM measurements of non-hot-Jupiter planets around genuinely hot stars ($T_{\rm eff}\gtrsim6500\,\mathrm{K}$), and previously reported alignment trends for these classes of systems (e.g., warm Jupiters and compact multi-planet systems) lose the power to discriminate the central question: are large misalignments unique to hot-Jupiter-like planets that can be delivered by high-$e$ migration, or are hot stars intrinsically more misaligned across architectures? Second, a single-star stellar obliquity transition near $6500\,\mathrm{K}$, coincident with the rotational break, favors tidal dissipation in outer convective envelopes; as these envelopes thin with increasing $T_{\rm eff}$, inertial-wave damping and magnetic braking weaken in tandem.

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arXiv2026

Extending asteroseismic magnetometry across the diverse landscape of magnetic structures

Magnetic fields have now been asteroseismically measured in the cores of many red giants. However, most interpretations of these measurements assume that the magnetic field is far below the critical field strength known to be exceeded by red giants exhibiting gravity-mode suppression. A recent method based on the traditional approximation of rotation and magnetism accurately predicts mode frequencies under fields up to this critical value by modeling gravity waves as individual magnetogravity ``polarizations'' which propagate through a waveguide-like mode cavity. So far, this formalism has been limited to magnetic fields which are axisymmetric about the rotation axis. In this study, we extend this approach by calculating the polarizations of magnetogravity waves under arbitrarily shaped magnetic fields under potentially rapid rotation. We consider the special cases of a dipolar magnetic field misaligned with the rotation axis as well as a dipole-plus-quadrupole magnetic field with no rotational symmetry. We show that non-axisymmetric field configurations can induce avoided crossings between polarizations, and that waves in such systems can convert between magnetogravity polarizations as they propagate, especially when the magnetic field strength is locally below a stratification-dependent threshold value. This threshold is distinct from the critical field strength for gravity-mode suppression, and is instead similar to the magnetic field strength at which perturbation theory breaks down.

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arXiv2026

On the feasibility of inverting the rotation of the solar core with mixed f/g modes

Context: Thanks to helioseismology, the rotation profile of the Sun has been measured with great precision down to 20% of its total radius. This rotation profile is used as a calibration to infer the rotation of other stars as well as a test of angular momentum transport theory in stellar interiors. However, the deepest 20% of the layers remain out of reach of current observations, preventing astronomers to discriminate between currently competing angular momentum transport mechanisms. Aims: The main obstacle is that no global oscillations modes sensitive to rotation (non-zero degree l) reaching the solar core have been detected yet, as nonradial p modes cannot reach it and g modes are evanescent at the surface and still elude detection. In this work, we propose and examine a new method to constrain the rotation of the core of the Sun, which does not require direct observation of solar g modes. Methods: It is based on a recent prediction that g modes in the radiative interior couple with f modes in the outer parts of the star. These mixed f /g are at the same time sensitive to the rotation of the core and able to reach the surface. These modes can be used together to build average inversion kernels and perform an inversion of the rotation of the solar core. Results: We find that the oscillations' spectrum of the Sun should present 6 mixed f /g modes that can be used to measure the rotation rate of the Sun at r = 0.07 and 0.2R. We estimate that the uncertainty on the measurements should be small enough to distinguish between competing scenarios of angular momentum transport in the Sun.

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arXiv2026

Asteroseismic imprints of strong non-axisymmetric fields in the cores of red giants

To date, magnetic fields have been asteroseismically measured in nearly one hundred red giant cores. However, most analyses assume weak magnetic fields and slow rotation so that perturbation theory can be applied. The "traditional approximation of rotation and magnetism" (TARM) method can predict gravity-mode frequencies under strong magnetic fields and rapid rotation rates. So far, this formalism requires the magnetic field to be symmetric about the rotation axis. We generalize the TARM formalism to apply to arbitrary magnetic field geometries, including cases where the magnetic and rotation axes are misaligned, as well as fields with no symmetry axis at all. The resulting gravity modes exhibit a rich diversity of wave behavior, including oblique pulsation and avoided crossings. We also clarify the domains of validity of perturbation theory and the TARM formalism.

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arXiv2026

Can Asteroseismic Structure Inversions Be Performed in Structure-Dependent Coordinates?

Yes. Unlike other applications of observational asteroseismology, "structure inversions" constrain the physical properties of stellar interiors in a model-independent fashion. However, existing techniques -- which parameterise these quantities as functions of the physical radial or mass coordinate -- break down when applied to stars which differ substantially from the Sun. These difficulties may be overcome by operating in coordinate systems that have long been known to more naturally suit the physical characteristics of the measured normal modes. We derive expressions for transforming inversion kernels in the acoustic and buoyancy radial coordinates, rather than in the physical radius, and make available a numerically performant implementation. These modified inversions directly address several specific known shortcomings of existing inversion procedures. Using the buoyancy radius in gravity-mode and mixed-mode pulsators permits meaningful comparisons of stars and models with differently-sized convective and radiative zones, which defeat standard inversions. Even in pressure-mode oscillators, inversions in the acoustic radial coordinate eliminate the methodological requirement for the mass and radius of the true star being needed to be known in advance.

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arXiv2026

CHARA Interferometry and TESS Asteroseismology of the Core-Helium Burning Red Giant $κ$ Cyg

We present a detailed study of the secondary red clump star, $κ$ Cyg, by combining long-baseline visible interferometry using the PAVO beam combiner at the CHARA Array with high-precision asteroseismology from TESS. This dual approach allowed for a stringent test of stellar evolutionary models in the core helium-burning phase, which remains a regime of significant theoretical uncertainty. Using the PAVO interferometric data and fitting the limb-darkened intensity profile directly, we measured $R = 8.65\pm0.10 \rm R_\odot$. We fitted the spectral energy distribution (SED) using Phoenix model atmospheres and calculated $L = 44.46 \pm 1.09 \rm L_\odot$ and $T_{\rm eff} = 5066^{+47}_{-50} \mathrm{K}$. Using 16 sectors of TESS photometry, we detected clear solar-like oscillations in $κ$ Cyg. Through comparison of oscillation frequencies with MESA grids using either predictive mixing (PM) or exponential overshooting (OS), we found that models reproducing the oscillation frequencies systematically overestimate the stellar radius, with overshooting models performing only marginally better. The same models also under-predict the observed dipole-mode period spacing ($ΔΠ_1$). By inspecting the phase offset ($ε_p$), we conclude that models misrepresent the interior structure of the star. Our results demonstrate that matching envelope-dominated asteroseismic observables alone is insufficient to ensure a correct core or even global structure, and highlight the need for improved treatments of convective boundary mixing in the models of core helium-burning (CHeB) stars.

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arXiv2026

The KPF-SLOPE Survey - Small, Compact Multi-Planet Systems Appear Spin-Orbit Aligned

The angle between stellar spin axes and planetary orbits -- stellar obliquity -- probes the dynamics of planetary migration and evolution. The obliquities of giant planets have been extensively studied because they are the most easily measured. Smaller planets, while more difficult to measure, have the advantage of better reflecting the dynamics of planetary systems because they trigger negligible back-reactions onto the host star. This paper introduces a new observational campaign called the Small, Low-mass Oblique Planets Experiment (SLOPE) survey with the Keck Planet Finder (KPF) spectrograph, and presents four new obliquity measurements. The SLOPE survey focuses on planets smaller than Saturn across a variety of system architectures. The sky-projected obliquities of the four planets measured -- TOI-1386b, TOI-480b, TOI-4596b, and TOI-1823b -- are all consistent with spin-orbit alignment. We validate the planetary nature of TOI-4596b with a significant obliquity detection. Including these measurements, we conducted a statistical analysis of the obliquities of sub-Saturn size planets in different planetary system architectures. Compared to other architectures, those in compact multi-planet systems reside in orbits that appear preferentially aligned with the stellar equator with 6 sigma confidence.

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arXiv2026

Anti-Solar Differential Rotation May Have Revived Magnetic Braking in the Subgiant 31 Aquilae

Recent observations have shown that sufficiently slow rotation disrupts the organization of large-scale magnetic field in older main-sequence stars, leading to weakened magnetic braking (WMB) and a collapse in the efficiency of the global stellar dynamo. Recent simulations predict a shift from solar-like to anti-solar differential rotation (DR) at slower rotation rates, which typically do not occur on the main-sequence due to WMB. However, physical expansion on the subgiant branch can eventually slow the stellar rotation beyond this threshold, yielding a non-cycling large-scale field that revives magnetic braking. We combine asteroseismology from the Transiting Exoplanet Survey Satellite (TESS) with spectropolarimetry from the Large Binocular Telescope (LBT) to test these predictions in the old metal-rich subgiant 31 Aql. The LBT observations reveal a strong large-scale magnetic field in this star, and archival measurements of its chromospheric emission over 50 years confirm that it is non-cycling, as predicted. The star exhibits a variety of rotation periods during different observing seasons, consistent with DR but with no means of distinguishing between solar-like and anti-solar patterns. We incorporate the TESS observations to estimate the current wind braking torque of 31 Aql, demonstrating that it supports revived magnetic braking in this old subgiant. We also use rotational evolution modeling to place a preliminary constraint on the stellar Rossby number for the transition to anti-solar DR. Future refinements in both asteroseismic observations and rotational modeling may yield improvements to this initial analysis.

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arXiv2026

The Effect of Different Methods for Accounting for $α$-enhancement on the Asteroseismic Modeling of Metal-Poor Stars

Constraining stellar models using asteroseismic and spectroscopic observations is a powerful method for precisely determining the fundamental properties of stars in different kinematic components of our galaxy. We use spectroscopy and individual oscillation mode frequencies to perform a homogeneous modeling study of eight evolved metal-poor stars enhanced in $α$-elements. We compare a full treatment of $α$-enhancement against an ad hoc correction to the total metallicity and show that the stellar properties inferred from asteroseismic modeling using both sets of models agree with each other. Additionally, we find that the uncertainties on stellar parameters derived from the both $α$-enhanced modeling methods are comparable. This is in qualitative disagreement with existing works showing red-giant ages constrained by only the global asteroseismic parameters to depend strongly on the opacities and abundances assumed in 1D modeling. We also show that the observed frequency of maximum oscillation power ($ν_{\text{max}}$) is larger than the value predicted from applying the $ν_{\text{max}}$ scaling relation to the masses, radii, and temperatures inferred from the detailed modeling. This discrepancy is pronounced at low metallicities, consistent with recent findings indicating a breakdown of the $ν_{\text{max}}$ scaling relation for metal-poor stars. Understanding the extent to which the $ν_{\text{max}}$ scaling relation fails for low-metallicity solar-like oscillators through detailed modeling will enable more accurate mass and age determinations for hundreds of giant stars in the Galactic Halo for which only global asteroseismic parameters are available.

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arXiv2026

A Test of Substellar Evolutionary Models with High-Precision Ages from Asteroseismology and Gyrochronology for the Benchmark System HR 7672AB

We present high-precision measurements for HR~7672AB, composed of a Sun-like (G0V) star and an L~dwarf companion. Three nights of precise (70 cm/s) radial velocity (RV) asteroseismology with the Keck Planet Finder clearly detect 5-minute oscillations from the primary HR~7672A, and modeling of the frequency spectrum yields an asteroseismic age of $1.87\pm0.65$~Gyr. We also determine a gyrochronological age of $2.58\pm0.47$~Gyr, and we combine these two results for a final age of $2.26\pm0.40$~Gyr. In addition, we obtained new RVs for HR~7672A and new astrometry for the companion HR~7672B. From a joint orbit fit, we measured a dynamical mass of $1.111\pm0.017$~$\text{M}_\odot$ for HR~7672A and $75.39\pm0.67$~$\text{M}_{\text{Jup}}$ for HR~7672B. This places the companion near the stellar/substellar boundary and thus particularly sensitive to differences in model predictions. The joint precision in host star age (18\% uncertainty) and companion mass (0.9\% uncertainty) makes HR~7672AB an exceptional substellar benchmark. Combined with the companion's luminosity, we use these measurements to test predictions from six brown dwarf cooling models. The best agreement occurs with the Chabrier et al. (2023) models, which incorporate a new equation of state, resulting in predictions that agree within $<$0.3$σ$ with all the observations. The other 5 sets of models agree at the 1--3$σ$ level depending on the particular test, and some models struggle to predict a sufficient low luminosity for HR~7672B at any age given its dynamical mass. We also detected a weak seismic signal in near-simultaneous TESS photometry of HR~7672A, with the resulting RV-to-photometry oscillation amplitude ratio consistent with solar values.

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arXiv2025

Impact of near-degeneracy effects on linear rotational inversions for red-giant stars

Accurate estimates of internal red-giant rotation rates are a crucial ingredient for constraining and improving current models of stellar rotation. Asteroseismic rotational inversions are a method to estimate these internal rotation rates. In this work, we focus on the observed differences in the rotationally-induced frequency shifts between prograde and retrograde modes, which were ignored in previous works when estimating internal rotation rates of red giants using inversions. We systematically study the limits of applicability of linear rotational inversions as a function of the evolution on the red-giant branch and the underlying rotation rates. We solve for the oscillation mode frequencies in the presence of rotation in the lowest-order perturbative approach. This enables a description of the differences between prograde and retrograde modes through the coupling of multiple mixed modes. We compute synthetic rotational splittings taking these near-degeneracy effects into account. We use red-giant models with one solar mass, a large frequency separation between 16 and 9 microhertz and core rotation rates between 500 and 1500 nHz covering the regime of observed parameters of Kepler red-giant stars. Finally, we use these synthetic data to quantify the systematic errors of internal rotation rates estimated by means of rotational inversions in the presence of near-degeneracy effects. We show that the systematic errors in the estimated rotation rates introduced by near-degeneracy effects surpass observational uncertainties for more evolved and faster rotating stars. The estimated rotation rates of some of the previously analysed red giants suffer from significant systematic errors that have not been taken into account yet. Notwithstanding, reliable analyses with existing inversion methods are feasible for a number of red giants within the parameter ranges determined here.

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arXiv2025

Precise Asteroseismic Ages for the Helmi Streams

The Helmi streams are remnants of a dwarf galaxy that was accreted by the Milky Way and whose stars now form a distinct kinematic and chemical substructure in the Galactic halo. Precisely age-dating these typically faint stars of extragalactic origin has been notoriously difficult due to the limitations of using only spectroscopic data, interferometry, or coarse asteroseismic measurements. Using observations from NASA's Transiting Exoplanet Survey Satellite, we report the detailed asteroseismic modeling of two of the brightest red giants within the Helmi streams, HD 175305 and HD 128279. By modeling the individual oscillation mode frequencies and the spectroscopic properties of both stars, we determine their fundamental properties including mass, radius, and age ($τ$). We report $τ= 11.16 \pm 0.91$ Gyr for HD 175305 and $τ= 12.52 \pm 1.05$ Gyr for HD 128279, consistent with previously inferred star-formation histories for the Helmi streams and the differential chemical abundances between the two stars. With precise ages for individual stream members, our results reinforce the hypothesis that the Helmi streams' progenitor must have existed at least 12 Gyr ago. Our results also highlight that the ages of metal-poor, $α$-enhanced red giants can be severely underestimated when inferred using global asteroseismic parameters instead of individual mode frequencies.

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arXiv2025

Stellar Obliquity of the Ultra-Short-Period Planet System HD 93963

We report an observation of the Rossiter-McLaughlin (RM) effect of the transiting planet HD 93963 Ac, a mini-Neptune planet orbiting a G0-type star with an orbital period of $P_{\rm{c}} = 3.65\,\mathrm{d}$, accompanied by an inner super-Earth planet with $P_{\rm{b}} = 1.04\,\mathrm{d}$. We observed a full transit of planet c on 2024 May 3rd UT with Keck/KPF. The observed RM effect has an amplitude of $\sim 1\,\mathrm{m\,s}^{-1}$ and implies a sky-projected obliquity of $λ= 14^{+17}_{-19}$ degrees for HD 93963 Ac. Our dynamical analysis suggests that the two inner planets are likely well aligned with the stellar spin, to within a few degrees, thus allowing both to transit. Along with WASP-47, 55 Cnc, and HD 3167, HD 93963 is the fourth planetary system with an ultra-short-period planet and obliquity measurement(s) of any planet(s) in the system. HD 93963, WASP-47, and 55 Cnc favor largely coplanar orbital architectures, whereas HD 3167 has been reported to have a large mutual inclination ($\sim$100$^\circ$) between its transiting planets b and c. In this configuration, the probability that both planets transit is low. Moreover, one planet would quickly evolve to be non-transiting due to nodal precession. Future missions such as ESO/PLATO should detect the resulting transit duration variations. We encourage additional obliquity measurements of the HD 3167 system to better constrain its orbital architecture.

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arXiv2025

It's not just a phase: oblique pulsations in magnetic red giants and other stochastic oscillators

Magnetic fields play a significant role in stellar evolution. In the last few years, asteroseismology has enabled the measurement of strong magnetic fields $10^4$--$10^6\,\mathrm{G}$ in the cores of dozens of red giants, and is the only known way to directly measure internal stellar magnetic fields. However, current data are still interpreted assuming that these fields are too weak or too axisymmetric to affect the orientation of the pulsations (i.e., make the pulsations ``oblique''), rendering stronger field strengths beyond the reach of existing asteroseismic searches. We show that, even when an oblique pulsator is also stochastic (such as in a red giant with a strong non-axisymmetric magnetic field), geometric effects will cause the signal to contain frequency components which remain in perfect relative phase with each other. This perfect phase relationship persists even over timescales in which stochasticity erases absolute phase information. This perfect relative coherence is a distinctive observational signature of oblique pulsation that does not require a model for mode frequencies to search for. However, due to its dependence on phase, this effect will not be evident in the power spectral density alone, and phase information should be retained in order to detect it. Coherence-based searches for oblique pulsations may pave the way to measurements of magnetic fields of currently inaccessible strengths in red giants, as well as some main-sequence and compact pulsators.

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arXiv2025

K-dwarf Radius Inflation and a 10-Gyr Spin-down Clock Unveiled through Asteroseismology of HD 219134 from the Keck Planet Finder

We present the first asteroseismic analysis of the K3\,V planet host HD~219134, based on four consecutive nights of radial velocities collected with the Keck Planet Finder. We applied Gold deconvolution to the power spectrum to disentangle modes from sidelobes in the spectral window, and extracted 25 mode frequencies with spherical degrees $0\leq\ell\leq3$. We derive the fundamental properties using five different evolutionary-modeling pipelines and report a mass of 0.763 $\pm$ 0.020 (stat) $\pm$ 0.007 (sys) M$_\odot$, a radius of 0.748 $\pm$ 0.007 (stat) $\pm$ 0.002 (sys) R$_\odot$, and an age of 10.151 $\pm$ 1.520 (stat) $\pm$ 0.810 (sys) Gyr. Compared to the interferometric radius 0.783 $\pm$ 0.005~R$_\odot$, the asteroseismic radius is 4\% smaller at the 4-$σ$ level -- a discrepancy not easily explained by known interferometric systematics, modeling assumptions on atmospheric boundary conditions and mixing lengths, magnetic fields, or tidal heating. HD~219134 is the first main-sequence star cooler than 5000~K with an asteroseismic age estimate and will serve as a critical calibration point for stellar spin-down relations. We show that existing calibrated prescriptions for angular momentum loss, incorporating weakened magnetic braking with asteroseismically constrained stellar parameters, accurately reproduce the observed rotation period. Additionally, we revised the masses and radii of the super-Earths in the system, which support their having Earth-like compositions. Finally, we confirm that the oscillation amplitude in radial velocity scales as $(L/M)^{1.5}$ in K dwarfs, in contrast to the $(L/M)^{0.7}$ relation observed in G dwarfs. These findings provide significant insights into the structure and angular momentum loss of K-type stars.

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arXiv2025

TOI-6324b: An Earth-Mass Ultra-Short-Period Planet Transiting a Nearby M Dwarf

We report the confirmation of TOI-6324 b, an Earth-sized (1.059 $\pm$ 0.041 R$_\oplus$) ultra-short-period (USP) planet orbiting a nearby ($\sim$20 pc) M dwarf. Using the newly commissioned Keck Planet Finder (KPF) spectrograph, we have measured the mass of TOI-6324 b 1.17 $\pm$ 0.22 M$_\oplus$. Because of its extremely short orbit of just $\sim$6.7 hours, TOI-6324 b is intensely irradiated by its M dwarf host, and is expected to be stripped of any thick, H/He envelope. We were able to constrain its interior composition and found an iron core mass fraction (CMF = 27$\pm$37%) consistent with that of Earth ($\sim$33%) and other confirmed USPs. TOI-6324 b is the closest to Earth-sized USP confirmed to date. TOI-6324 b is a promising target for JWST phase curve and secondary eclipse observations (Emission Spectroscopy Metric = 25) which may reveal its surface mineralogy, day-night temperature contrast, and possible tidal deformation. From 7 sectors of TESS data, we report a tentative detection of the optical phase curve variation with an amplitude of 42$\pm$28 ppm.

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arXiv2025

Resolving an Asteroseismic Catastrophe: Structural Diagnostics from p-mode Phase Functions off the Main Sequence

On the main sequence, the asteroseismic small frequency separation $δν_{02}$ between radial and quadrupole p-modes is customarily interpreted to be a direct diagnostic of internal structure. Such an interpretation is based on a well-known integral estimator relating $δν_{02}$ to a radially-averaged sound-speed gradient. However, this estimator fails, catastrophically, when evaluated on structural models of red giants: their small separations must therefore be interpreted differently. We derive a single expression which both reduces to the classical estimator when applied to main-sequence stellar models, yet reproduces the qualitative features of the small separation for stellar models of very evolved red giants. This expression indicates that the small separations of red giants scale primarily with their global seismic properties as $δν_{02} \propto Δν^2/ν_\mathrm{max}$, rather than being in any way sensitive to their internal structure. Departures from this asymptotic behaviour, during the transition from the main-sequence to red giant regimes, have been recently reported in open-cluster Christensen-Dalsgaard (C-D) diagrams from K2 mission data. Investigating them in detail, we demonstrate that they occur when the convective envelope boundary passes a specific acoustic distance -- roughly a third of a wavelength at $ν_\mathrm{max}$ -- from the centre of the star, at which point radial modes become maximally sensitive to the position of the boundary. The shape of the corresponding features on $ε_p$ and C-D (or $r_{02}$) diagrams may be useful in constraining the nature of convective boundary mixing, in the context of undershooting beneath a convective envelope.

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arXiv2024

Signatures of Core-Envelope Rotational Misalignment in the Mixed-Mode Asteroseismology of Kepler-56

Existing asteroseismic rotational measurements assume that stars rotate around a single axis. However, tidal torques from misaligned companions, or their possible engulfment, may bring the rotational axis of a star's envelope out of alignment with its core, breaking azimuthal symmetry. I derive perturbative expressions for asteroseismic signatures of such hitherto unexamined rotational configurations, under the ``shellular approximation'' of constant rotation rates on radially stratified mass shells. In the aligned case, the distribution of power between multiplet components is determined by the inclination of the rotational axis; radial differential misalignment causes this to vary from multiplet to multiplet. I examine in particular detail the phenomenology of gravitoacoustic mixed modes as seen in evolved sub- and red giants, where near-resonance avoided crossings may break geometrical degeneracies. Upon applying the revised asteroseismic observational methodology that results from this theoretical discussion to revisit Kepler-56 -- a red giant with a misaligned planetary system -- I find that its core and envelope rotate around different rotational axes. While the rotational axis of its core is indeed misaligned from the orbit normal of its transiting planets (consistently with earlier studies), its envelope's rotational axis is close to lying in the sky plane, and may well be aligned with them. More detailed asteroseismic modelling, and spectroscopic follow-up, will be required to fully elucidate the full spin-orbit geometry of the Kepler-56 system, and potentially discriminate between hypotheses for how it formed.

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