Rotational Splitting Research Articles

Near-degeneracy effects on the frequencies of rotationally-split mixed modes in red giants

Context. The Kepler space mission has made it possible to measure the rotational splittings of mixed modes in red giants, thereby providing an unprecedented opportunity to probe the internal rotation of these stars. Aims. Asymmetries have been detected in the rotational multiplets of several red giants. This is unexpected since all the red giants whose rotation profiles have been measured thus far are found to rotate slowly, and low rotation, in principle, produces symmetrical multiplets. Our aim here is to explain these asymmetries and find a way of exploiting them to probe the internal rotation of red giants. Methods. We show that in the cases where asymmetrical multiplets were detected, near-degeneracy effects are expected to occur, because of the combined effects of rotation and mode mixing. Such effects have not been taken into account so far. By using both perturbative and non-perturbative approaches, we show that near-degeneracy effects produce multiplet asymmetries that are very similar to the observations. We then propose and validate a method based on the perturbative approach to probe the internal rotation of red giants using multiplet asymmetries. Results. We successfully apply our method to the asymmetrical l = 2 multiplets of the Kepler young red giant KIC 7341231 and obtain precise estimates of its mean rotation in the core and the envelope. The observed asymmetries are reproduced with a good statistical agreement, which confirms that near-degeneracy effects are very likely the cause of the detected multiplet asymmetries. Conclusions. We expect near-degeneracy effects to be important for l = 2 mixed modes all along the red giant branch (RGB). For l = 1 modes, these effects can be neglected only at the base of the RGB. They must therefore be taken into account when interpreting rotational splittings and as shown here, they can bring valuable information about the internal rotation of red giants.

Solar cycle variations of rotation and asphericity in the near-surface shear layer

The precise shape of the Sun is sensitive to the influence of gravity, differential rotation, local turbulence and magnetic fields. So its precise measurement is a long-standing astrometric objective. It has been previously shown by different methods that the solar shape exhibits asphericity that evolves with the solar cycle. Thanks to the Michelson Doppler Imager (MDI) on Solar and Heliospheric Observatory (SoHO) and the Helioseismic and Magnetic Imager (HMI) aboard NASA's Solar Dynamics Observatory (SDO), and their capability to observe with an unprecedented accuracy the surface gravity oscillation (f) modes, it is possible to extract information concerning the coefficients of rotational frequency splitting, a1, a3 and a5, that measure the latitudinal differential rotation, together with the a2, a4 and a6 asphericity coefficients. Analysis of these helioseismology data with time for almost two solar cycles, from 1996 to 2017, reveals a close correlation of the a1 and a5 coefficients with the solar activity, whilst a3 exhibits a long-term trend and a weak correlation with the solar activity in the current solar cycle indicating a substantial change of the global solar rotation, potentially associated with a long-term evolution of the solar cycles. Looking in more details, the asphericity coefficients, a2, a4 and a6 are more strongly associated with the solar cycle when applying a time lag of respectively 0.1, 1.6 and −1.6 years. The magnitude of a6-coefficient varies in phase with the sunspot number (SN), but its amplitude is ahead of the SN variation. The latest measurements made in mid 2017 indicate that the magnitude of the a6-coefficient has probably reached its minimum; therefore, the next solar minimum can be expected by the end of 2018 or in the beginning of 2019. The so-called “seismic radius” in the range of f-mode angular degree: ℓ=137−299 exhibits a temporal variability in anti-phase with the solar activity; its relative value decreased by ∼2.3×10−5 in Solar Cycle 23 and ∼1.7×10−5 in Cycle 24. Such results will be useful for better understanding the physical mechanisms which act inside the Sun, and so, better constrain dynamo models for forecasting the solar cycles.

Asymptotic g modes: Evidence for a rapid rotation of the solar core

Context. Over the past 40 years, helioseismology has been enormously successful in the study of the solar interior. A shortcoming has been the lack of a convincing detection of the solar g modes, which are oscillations driven by gravity and are hidden in the deepest part of the solar body – its hydrogen-burning core. The detection of g modes is expected to dramatically improve our ability to model this core, the rotational characteristics of which have, until now, remained unknown. Aims. We present the identification of very low frequency g modes in the asymptotic regime and two important parameters that have long been waited for: the core rotation rate, and the asymptotic equidistant period spacing of these g modes. Methods. The GOLF instrument on board the SOHO space observatory has provided two decades of full-disk helioseismic data. The search for g modes in GOLF measurements has been extremely difficult because of solar and instrumental noise. In the present study, the p modes of the GOLF signal are analyzed differently: we search for possible collective frequency modulations that are produced by periodic changes in the deep solar structure. Such modulations provide access to only very low frequency g modes, thus allowing statistical methods to take advantage of their asymptotic properties. Results. For oscillatory periods in the range between 9 and nearly 48 h, almost 100 g modes of spherical harmonic degree 1 and more than 100 g modes of degree 2 are predicted. They are not observed individually, but when combined, they unambiguously provide their asymptotic period equidistance and rotational splittings, in excellent agreement with the requirements of the asymptotic approximations. When the period equidistance has been measured, all of the individual frequencies of each mode can be determined. Previously, p-mode helioseismology allowed the g-mode period equidistance parameter P0 to be bracketed inside a narrow range, between approximately 34 and 35 min. Here, P0 is measured to be 34 min 01 s, with a 1 s uncertainty. The previously unknown g-mode splittings have now been measured from a non-synodic reference with very high accuracy, and they imply a mean weighted rotation of 1277 ± 10 nHz (9-day period) of their kernels, resulting in a rapid rotation frequency of 1644 ± 23 nHz (period of one week) of the solar core itself, which is a factor 3.8 ± 0.1 faster than the rotation of the radiative envelope. Conclusions. The g modes are known to be the keys to a better understanding of the structure and dynamics of the solar core. Their detection with these precise parameters will certainly stimulate a new era of research in this field.

Limits on radial differential rotation in Sun-like stars from parametric fits to oscillation power spectra

Rotational shear in Sun-like stars is thought to be an important ingredient in models of stellar dynamos. Thanks to helioseismology, rotation in the Sun is characterized well, but the interior rotation profiles of other Sun-like stars are not so well constrained. Until recently, measurements of rotation in Sun-like stars have focused on the mean rotation, but little progress has been made on measuring or even placing limits on differential rotation. Using asteroseismic measurements of rotation we aim to constrain the radial shear in five Sun-like stars observed by the NASA Kepler mission: KIC004914923, KIC005184732, KIC006116048, KIC006933899, and KIC010963065. We used stellar structure models for these five stars from previous works. These models provide the mass density, mode eigenfunctions, and the convection zone depth, which we used to compute the sensitivity kernels for the rotational frequency splitting of the modes. We used these kernels as weights in a parametric model of the stellar rotation profile of each star, where we allowed different rotation rates for the radiative interior and the convective envelope. This parametric model was incorporated into a fit to the oscillation power spectrum of each of the five Kepler stars. This fit included a prior on the rotation of the envelope, estimated from the rotation of surface magnetic activity measured from the photometric variability. The asteroseismic measurements without the application of priors are unable to place meaningful limits on the radial shear. Using a prior on the envelope rotation enables us to constrain the interior rotation rate and thus the radial shear. In the five cases that we studied, the interior rotation rate does not differ from the envelope by more than approximately +/-30%. Uncertainties in the rotational splittings are too large to unambiguously determine the sign of the radial shear.

Two-site Photometry and Spectroscopy of the Rapidly Pulsating sdB Star EC 22221-3152**Based on observations at the SOAR telescope, a collaboration between CPNq-Brazil, NOAO, UNC, and MSU.

We present follow-up photometry and spectroscopy of the rapidly pulsating subdwarf B star EC 22221-3152. Using the SAAO 1.0-m telescope and the 0.4-m Panchromatic Robotic Optical Monitoring and Polarimetry Telescopes (PROMPT) at the Cerro Tololo Inter-American Observatory (CTIO), we monitored the star photometrically for one week. By combining data from both telescopes, we were able to resolve 18 significant periodicities in the light curve. Seven of these signals consist of combination frequencies, a first harmonic, and possible rotational splittings that would be indicative of an 8.2-d rotation period. The remaining 11 periodicities appear to represent independent oscillation modes, including three not originally detected in the discovery work. Time series spectroscopy obtained with the Goodman spectrograph on the 4.1-m SOAR telescope reveals possible velocity variations corresponding to two of the observed pulsation modes. From atmospheric model fits to the spectra, we derive K, log g = 5.86 ± 0.15, and log N(He)/N(H) = −1.4 ± 0.3 and confirm the star’s status as one of the hottest sdBVr stars currently known.

Rotational Splitting and Asteroseismic Modeling of the δ Scuti Star EE Camelopardalis

Abstract According to the rotational splitting law of g modes, the frequency spectra of EE Cam can be disentangled only with oscillation modes of ℓ = 0, 1, and 2. Fifteen sets of rotational splits were found, and they contain five sets of ℓ = 1 multiplets and 10 sets of ℓ = 2 multiplets. The rotational period of EE Cam is deduced to be days. When we do model fittings, we use two nonradial oscillation modes (f 11 and f 32), and the fundamental radial mode f 1. The fitting results show that of the best-fitting model is much smaller than those of other theoretical models. The physical parameters of the best-fitting model are M = 2.04 M ⊙, Z = 0.028, T eff = 6433 K, , R = 4.12 R ⊙, , and . Furthermore, we find that f 11 and f 32 are mixed modes, which mainly characterize the features of the helium core. The fundamental radial mode f 1 mainly restricts the features of the stellar envelope. Finally, the acoustic radius and the period separation Π0 are determined to be 5.80 hr and 463.7 s, respectively, and the size of the helium core of EE Cam is estimated to be M He = 0.181 M ⊙ and R He = 0.0796 R ⊙.

First Results from the Hertzsprung SONG Telescope: Asteroseismology of the G5 Subgiant Star μ Herculis*

Abstract We report the first asteroseismic results obtained with the Hertzsprung Stellar Observations Network Group Telescope from an extensive high-precision radial-velocity observing campaign of the subgiant μ Herculis. The data set was collected during 215 nights in 2014 and 2015. We detected a total of 49 oscillation modes with l values from zero to three, including some l = 1 mixed modes. Based on the rotational splitting observed in l = 1 modes, we determine a rotational period of 52 days and a stellar inclination angle of 63°. The parameters obtained through modeling of the observed oscillation frequencies agree very well with independent observations and imply a stellar mass between 1.11 and 1.15 M ⊙ and an age of Gyr. Furthermore, the high-quality data allowed us to determine the acoustic depths of the He ii ionization layer and the base of the convection zone.

A Deep Test of Radial Differential Rotation in a Helium-atmosphere White Dwarf.I. Discovery of Pulsations in PG 0112+104

Abstract We present the detection of non-radial oscillations in a hot, helium-atmosphere white dwarf using 78.7 days of nearly uninterrupted photometry from the Kepler space telescope. With an effective temperature >30,000 K, PG 0112+104 becomes the hottest helium-atmosphere white dwarf known to pulsate. The rich oscillation spectrum of low-order g-modes includes clear patterns of rotational splittings from consecutive sequences of dipole and quadrupole modes, which can be used to probe the rotation rate with depth in this highly evolved stellar remnant. We also measure a surface rotation rate of 10.17404 hr from an apparent spot modulation in the K2 data. With two independent measures of rotation, PG 0112+104 provides a remarkable test of asteroseismic inference.

EXPLORING THE HELIUM CORE OF THE δ SCUTI STAR COROT 102749568 WITH ASTEROSEISMOLOGY

ABSTRACT Based on regularities in rotational splitting, we seek possible multiplets for the observed frequencies of CoRoT 102749568. There are 21 sets of multiplets identified, including four sets of multiplets with l = 1, nine sets of multiplets with l = 2, and eight sets of multiplets with l = 3. In particular, there are three complete triplets (f 10, f 12, f 14), (f 31, f 34, f 35), and (f 41, f 43, f 44). The rotational period of CoRoT 102749568 is estimated to be days. When doing model fittings, three l = 1 modes (f 12, f 34, and f 43) and the radial first overtone f 13 are used. Our results shows that the three nonradial modes (f 12, f 34, and f 43) are mixed modes, which mainly provide constraints on the helium core. The radial first overtone f 13 mainly provides constraint on the stellar envelope. Hence the size of the helium core of CoRoT 102749568 is determined to be = 0.148 ± 0.003 M ⊙ and = 0.0581 ± 0.0007 R ⊙. The fundamental parameters of CoRoT 102749568 are determined to be M = 1.54 ± 0.03 M ⊙, , , , K, R = 2.916 ± 0.039 R ⊙, and L = 17.12 ± 1.13 L ⊙.

EPIC 211779126: a rare hybrid pulsating subdwarf B star richly pulsating in both pressure and gravity modes

We present our analysis of EPIC 211779126, a pulsating subdwarf B star discovered with the Kepler spacecraft during K2 Campaign 5. We found 154 frequencies in the g -mode region as well as 29 frequencies in the p -mode region. This makes EPIC 211779126 a rare hybrid pulsator with a rich pulsation spectrum in both regions. We successfully identified modal degrees and relative radial orders of most of the g -modes using asymptotic period spacing, and modal degrees of some of the p -modes using rotational splitting. We detected trapped modes, which are a very important feature for constraining theoretical models. Our ground-based spectroscopic observations revealed no companion, therefore EPIC 211779126 is likely a single sdB star. Using p -mode multiplets, we derived a rotation period of approximately 16 days, making EPIC 211779126 the fastest rotating non-binary subdwarf B pulsator observed with Kepler . However, we do not find any resolved multiplets among the high-amplitude g -mode pulsations that correspond to the rotation rate inferred from the p -mode splittings. This may indicate that the star’s core is rotating more slowly than its envelope.

Seismic diagnosis from gravity modes strongly affected by rotation

Most of the information we have about the internal rotation of stars comes from modes that are weakly affected by rotation, for example by using rotational splittings. In contrast, we present here a method, based on the asymptotic theory of Prat et al. (2016), which allows us to analyse the signature of rotation where its effect is the most important, that is in low-frequency gravity modes that are strongly affected by rotation. For such modes, we predict two spectral patterns that could be confronted to observed spectra and those computed using fully two-dimensional oscillation codes.

Rapidly rotating red giants

Stellar oscillations give seismic information on the internal properties of stars. Red giants are targets of interest since they present mixed modes, which behave as pressure modes in the convective envelope and as gravity modes in the radiative core. Mixed modes thus directly probe red giant cores, and allow in particular the study of their mean core rotation. The high-quality data obtained by CoRoT and Kepler satellites represent an unprecedented perspective to obtain thousands of measurements of red giant core rotation, in order to improve our understanding of stellar physics in deep stellar interiors. We developed an automated method to obtain such core rotation measurements and validated it for stars on the red giant branch. In this work, we particularly focus on the specific application of this method to red giants having a rapid core rotation. They show complex spectra where it is tricky to disentangle rotational splittings from mixed-mode period spacings. We demonstrate that the method based on the identification of mode crossings is precise and efficient. The determination of the mean core rotation directly derives from the precise measurement of the asymptotic period spacing {\Delta}{\Pi}1 and of the frequency at which the crossing of the rotational components is observed.

Asteroseismic Properties of Solar-type Stars Observed with the NASA K2 Mission: Results from Campaigns 1–3 and Prospects for Future Observations

We present an asteroseismic analysis of 33 solar-type stars observed in short cadence (SC) during Campaigns (C) 1–3 of the NASA K2 mission. We were able to extract both average seismic parameters and individual mode frequencies for stars with dominant frequencies up to ∼3300 μHz, and we find that data for some targets are good enough to allow for a measurement of the rotational splitting. Modeling of the extracted parameters is performed by using grid-based methods using average parameters and individual frequencies together with spectroscopic parameters. For the target selection in C3, stars were chosen as in C1 and C2 to cover a wide range in parameter space to better understand the performance and noise characteristics. For C3 we still detected oscillations in 73% of the observed stars that we proposed. Future K2 campaigns hold great promise for the study of nearby clusters and the chemical evolution and age–metallicity relation of nearby field stars in the solar neighborhood. We expect oscillations to be detected in ∼388 SC targets if the K2 mission continues until C18, which will greatly complement the ∼500 detections of solar-like oscillations made for SC targets during the nominal Kepler mission. For ∼30–40 of these, including several members of the Hyades open cluster, we furthermore expect that inference from interferometry should be possible.

A new asteroseismic diagnostic for internal rotation in γ Doradus stars

With four years of nearly-continuous photometry from Kepler, we are finally in a good position to apply asteroseismology to $\gamma$ Doradus stars. In particular several analyses have demonstrated the possibility to detect non-uniform period spacings, which have been predicted to be directly related to rotation. In the present work, we define a new seismic diagnostic for rotation in $\gamma$ Doradus stars that are too rapidly rotating to present rotational splittings. Based on the non uniformity of their period spacings, we define the observable $\Sigma$ as the slope of the period spacing when plotted as a function of period. We provide a one-to-one relation between this observable $\Sigma$ and the internal rotation, which applies widely in the instability strip of $\gamma$ Doradus stars. We apply the diagnostic to a handful of stars observed by Kepler. Thanks to g-modes in $\gamma$ Doradus stars, we are now able to determine the internal rotation of stars on the lower main sequence, which is still not possible for Sun-like stars.

Asteroseismology of theδScuti star HD 50844

Aims. We aim to probe the internal structure and investigate more detailed information of the $\delta$ Scuti star HD 50844 with asteroseismology. Methods. We analyse the observed frequencies of the $\delta$ Scuti star HD 50844 obtained by Balona (2014), and search for possible multiplets based on the rotational splitting law of g-mode. We tried to disentangle the frequency spectra of HD 50844 by means of the rotational splitting only. We then compare them with theoretical pulsation modes, which correspond to stellar evolutionary models with various sets of initial metallicity and stellar mass, to find the best-fitting model. Results. There are three multiplets including two complete triplets and one incomplete quintuplet, in which mode identifications for spherical harmonic degree $l$ and azimuthal number $m$ are unique. The corresponding rotational period of HD 50844 is found to be 2.44$^{+0.13}_{-0.08}$ days. The physical parameters of HD 50844 are well limited in a small region by three modes identified as nonradial ones ($f_{11}$, $f_{22}$, and $f_{29}$) and by the fundamental radial mode ($f_{4}$). Our results show that the three nonradial modes ($f_{11}$, $f_{22}$, and $f_{29}$) are all mixed modes, which mainly represent the property of the helium core. The fundamental radial mode ($f_{4}$) mainly represents the property of the stellar envelope. In order to fit these four pulsation modes, both the helium core and the stellar envelope must be matched to the actual structure of HD 50844. Finally, the mass of the helium core of HD 50844 is estimated to be 0.173 $\pm$ 0.004 $M_{\odot}$ for the first time. The physical parameters of HD 50844 are determined to be $M=$ 1.81 $\pm$ 0.01 $M_{\odot}$, $Z=$ 0.008 $\pm$ 0.001. $T_{\rm eff}=$ 7508 $\pm$ 125 K, log$g=$ 3.658 $\pm$ 0.004, $R=$ 3.300 $\pm$ 0.023 $R_{\odot}$, $L=$ 30.98 $\pm$ 2.39 $L_{\odot}$.

A diagnostic for localizing red giant differential rotation

Abstract We present a simple diagnostic that can be used to constrain the location of the differential rotation in red giants with measured mixed mode rotational splittings. Specifically, in red giants with radii ∼4 Rȯ, the splittings of p-dominated modes (sound wave-dominated) relative to those of g-dominated modes (internal gravity wave-dominated) are sensitive to how much of the differential rotation resides in the outer convection zone versus the radiative interior of the red giant. An independently measured surface rotation rate significantly aids breaking degeneracies in interpreting the measured splittings. We apply our results to existing observations of red giants, particularly those of Kepler-56, and find that most of the differential rotation resides in the radiative region rather than in the convection zone. This conclusion is consistent with results in the literature from rotational inversions, but our results are insensitive to some of the uncertainties in the inversion process and can be readily applied to large samples of red giants with even a modest number of measured rotational splittings. We argue that differential rotation in the radiative interior strongly suggests that angular momentum transport in red giants is dominated by local fluid instabilities rather than large-scale magnetic stresses.

Nonradial and radial period changes in theδScuti star 4 CVn

The nature of period and amplitude changes in nonradial pulsators is presently unknown. It is therefore important to examine the correlations between these changes in stars with a large number of simultaneously excited pulsation modes. However, the small amplitudes require extensive high-precision photometry covering many years. We present 702 nights of high-precision photometry of the evolved Delta Scuti variable 4 CVn obtained from 2005 - 2012 with a dedicated telescope. We detected 64 frequencies, of which 38 can be identified as combinations and harmonics. The relative amplitudes of the combination frequencies are similar to those found in 44 Tau and show no evidence for resonant mode coupling. Significant period and amplitude changes are detected for the dominant modes. The known prograde and retrograde modes show period changes with opposite signs, while the radial mode exhibits only small, cyclical period changes. For each mode, the period changes are constant over the eight years and range from (1/P)dP/dt = -16 to +13 parts per million. On the other hand, the amplitude variations show no systematic behavior between different pulsation modes. The behavior of the prograde, axisymmetric, and retrograde modes indicates a constant decrease in the rotational splitting over the eight years.

UNEXPECTED SERIES OF REGULAR FREQUENCY SPACING OF δ SCUTI STARS IN THE NON-ASYMPTOTIC REGIME. I. THE METHODOLOGY

ABSTRACT A sequence search method was developed to search the regular frequency spacing in δ Scuti stars through visual inspection and an algorithmic search. We searched for sequences of quasi-equally spaced frequencies, containing at least four members per sequence, in 90 δ Scuti stars observed by CoRoT. We found an unexpectedly large number of independent series of regular frequency spacing in 77 δ Scuti stars (from one to eight sequences) in the non-asymptotic regime. We introduce the sequence search method presenting the sequences and echelle diagram of CoRoT 102675756 and the structure of the algorithmic search. Four sequences (echelle ridges) were found in the 5–21 d−1 region where the pairs of the sequences are shifted (between 0.5 and 0.59 d−1) by twice the value of the estimated rotational splitting frequency (0.269 d−1). The general conclusions for the whole sample are also presented in this paper. The statistics of the spacings derived by the sequence search method, by FT (Fourier transform of the frequencies), and the statistics of the shifts are also compared. In many stars more than one almost equally valid spacing appeared. The model frequencies of FG Vir and their rotationally split components were used to formulate the possible explanation that one spacing is the large separation while the other is the sum of the large separation and the rotational frequency. In CoRoT 102675756, the two spacings (2.249 and 1.977 d−1) are in better agreement with the sum of a possible 1.710 d−1 large separation and two or one times, respectively, the value of the rotational frequency.

MULTI-PERIOD G-MODE PULSATIONS OF A PRE-HE-WD STAR IN THE ECLIPSING BINARY KIC 9164561

ABSTRACT We report the discovery of a new pulsating pre-He-WD in the EL CVn-type binary KIC 9164561. Light curve modeling and frequency analysis of the binary system were carried out based on short-cadence Kepler photometry. Combined with the radial-velocity solution, revised physical parameters of the binary system were determined. The component KIC 9164561B was confirmed to be a pre-He-WD star with M = 0.213 ± 0.012 M ⊙ , R = 0.283 ± 0.006 R ⊙ , and T eff = 10650 ± 200 K. In addition to the light changes due to an eclipse, pulsational light variations of the pre-He-WD star were detected. The Fourier analysis reveals at least 52 frequencies, with the dominant one at 313.4 μHz. A brief mode identification indicates that the pre-He-WD star pulsates in g-modes with an average period spacing of Δ Π l = 1 = 80.87 s. Moreover, a number of multiplets due to rotational splitting were identified from the frequency spectra. The rotational period of the pulsating pre-He-WD star was found to equal to the orbital period, indicating that KIC 9164561B is in synchronous rotation.

Near-uniform internal rotation of the main-sequence γ Doradus pulsator KIC 7661054

We used Kepler photometry to determine the internal rotation rate of KIC 7661054, a chemically normal γ Dor star on the main sequence at spectral type F2.5 V. The core rotation period of 27.25 ± 0.06 d is obtained from the rotational splittings of a series of dipole g modes. The surface rotation period is calculated from a spectroscopic projected rotation velocity and a stellar radius computed from models. Literature data, obtained without inclusion of macroturbulence as a line-broadening mechanism, imply that the surface rotates much more quickly than the core, while our detailed analysis suggests that the surface may rotate slightly more quickly than the core and that the rotation profile is uniform within the 1-σ uncertainties. We discuss the pitfalls associated with the determination of surface rotation rates of slow rotators from spectroscopy in the absence of asteroseismic constraints. A broad signal is observed at low frequency, which we show cannot be attributed to rotation, contrary to previous suggestions concerning the origin of such signals.