High-degree Modes Research Articles

Modelling continuum intensity perturbations caused by solar acoustic oscillations

Context. Helioseismology is the study of the Sun’s interior using observations of oscillations at the surface. It suffers from systematic errors, for instance a center-to-limb error in travel-time measurements. Understanding these errors requires an adequate understanding of the nontrivial relationship between wave displacement and helioseismic observables (intensity or velocity). Aims. The wave displacement causes perturbations in the atmospheric thermodynamical quantities which, in turn, perturb the opacity, the optical depth, the source function, and the local ray geometry, thus affecting the emergent intensity. We aim to establish the most complete relationship achieved to date between the wave displacement and the emergent intensity perturbation by solving the radiative transfer problem in the perturbed atmosphere. Methods. We derived an expression for the emergent intensity perturbation caused by acoustic oscillations at any point on the solar disk by applying a first-order perturbation theory. As input perturbations, we considerd adiabatic modes of oscillation of different degrees in a spherically-symmetric solar model. The background and the perturbed intensities are computed by solving the radiative transfer equation considering the main sources of opacity in the continuum (absorption and scattering). Results. We find that for all modes, the perturbations to the thermodynamical quantities are not sufficient to model the intensity perturbations: the geometrical effects due to the wave displacement must always be taken into account as they lead to a difference in amplitude and a phase shift between temperature perturbations at the surface and emergent intensity perturbations. The closer to the limb, the greater the differences. For modes with eigenfrequencies around 3 mHz, we found that the radial and horizontal components of the wave displacement are important, in particular, for high-degree modes. Conclusions. This work presents improvements for the computation of the intensity perturbations, in particular, for high-degree modes. Here, we explain the differences in intensity computations seen in earlier works. The phase shifts and amplitude differences between the temperature and intensity perturbations increase toward the limb. This should prove helpful when interpreting some of the systematic centre-to-limb effects observed in local helioseismology. The computations are fast (3 s for 2000 positions and one frequency for one core) and can be parallelised. This work can be extended to models of the line-of-sight velocity observable.

Core magnetic field imprint in the non-radial oscillations of red giant stars

ABSTRACT Magnetic fields in red giant stars remain a poorly understood topic, particularly in what concerns their intensity in regions far below the surface. In this work, we propose that gravity-dominated mixed modes of high absolute radial order and low angular degree can be used to probe the magnetic field in their radiative cores. Using two poloidal, axisymmetric configurations for the field in the core and the classical perturbative approach, we derive an analytical expression for the magnetic frequency splitting of these oscillation modes. Considering three distinct red giant models, with masses of 1.3, 1.6, and 2.0 M⊙, we find that a field strength of 105 G is necessary in the core of these stars to induce a frequency splitting of the order of a μHz in dipole and quadrupole oscillation modes. Moreover, taking into account observational limits, we estimate that magnetic fields in the cores of red giants that do not present observable magnetic splittings cannot exceed 104 G. Given the general absence of observable splittings in the oscillation spectra of these stars, and assuming that present mode suppression mechanisms are not biased towards certain azimuthal orders and retain all peaks in each multiplet, our results lead us to conclude that internal fields with the considered configurations and strengths above 104 G are not prevalent in red giants.

High-degree gravity modes in the single sdB star HD 4539

ABSTRACT HD 4539 (alias PG 0044 + 097 or EPIC 220641886) is a bright (V = 10.2) long-period V1093 Her-type subdwarf B (sdB) pulsating star that was observed by the Kepler spacecraft in its secondary (K2) mission. We use the K2 light curve (78.7 d) to extract 169 pulsation frequencies, 124 with a robust detection. Most of these frequencies are found in the low-frequency region typical of gravity (g-)modes, but some higher frequencies corresponding to pressure (p-)modes are also detected. Therefore HD 4539 is a hybrid pulsator and both the deep and surface layers of the star can potentially be probed through asteroseismology. The lack of any frequency splitting in its amplitude spectrum suggests that HD 4539 has a rotation period longer than the K2 run and/or that it is seen pole-on. From asymptotic period spacing we see many high-degree modes, up to l = 12, in the spectrum of HD 4539, with amplitudes as low as a few ppm. A large fraction of these modes can be identified and for ∼29 per cent of them we obtain a unique and robust identification corresponding to l ≤ 8. Our study includes also a new determination of the atmospheric parameters of the star. From low-resolution spectroscopy we obtain Teff = 22 800 ± 160 K, log g = 5.20 ± 0.02, and log(N(He)/N(H)) = −2.34 ± 0.05. By fitting the SED we obtain Teff = 23 470$^{+650}_{-210}$ K, R⋆ = 0.26 ± 0.01 R⊙, and M⋆ = 0.40 ± 0.08 M⊙. Moreover, from 11 high-resolution spectra we see the radial velocity variations caused by the stellar pulsations, with amplitudes of ≈150 m s−1 for the main modes, and we can exclude the presence of a companion with a minimum mass higher than a few Jupiter masses for orbital periods below ∼300 d.

More about solar g modes

Context. The detection of asymptotic solar g-mode parameters was the main goal of the GOLF instrument onboard the SOHO space observatory. This detection has recently been reported and has identified a rapid mean rotation of the solar core, with a one-week period, nearly four times faster than all the rest of the solar body, from the surface to the bottom of the radiative zone. Aim. We present here the detection of more g modes of higher degree, and a more precise estimation of all their parameters, which will have to be exploited as additional constraints in modeling the solar core. Methods. Having identified the period equidistance and the splitting of a large number of asymptotic g modes of degrees 1 and 2, we test a model of frequencies of these modes by a cross-correlation with the power spectrum from which they have been detected. It shows a high correlation peak at lag zero, showing that the model is hidden but present in the real spectrum. The model parameters can then be adjusted to optimize the position (at exactly zero lag) and the height of this correlation peak. The same method is then extended to the search for modes of degrees 3 and 4, which were not detected in the previous analysis.Results. g-mode parameters are optimally measured in similar-frequency bandwidths, ranging from 7 to 8 μHz at one end and all close to 30 μHz at the other end, for the degrees 1 to 4. They include the four asymptotic period equidistances, the slight departure from equidistance of the detected periods for l = 1 and l = 2, the measured amplitudes, functions of the degree and the tesseral order, and the splittings that will possibly constrain the estimated sharpness of the transition between the one-week mean rotation of the core and the almost four-week rotation of the radiative envelope. The g-mode periods themselves are crucial inputs in the solar core structure helioseismic investigation.

Observations of the Singlets of Higher-Degree Modes Based on the OSE

In this study, we select 18 SG (superconducting gravimeter) records from 15 GGP stations and 99 vertical, 69 horizontal components of IRIS broad-band seismograms during 2004 Sumatra Earthquake to detect the splitting of higher-degree Earth’s free oscillations modes (0S4, 0S7~0S10, 2S4, 1S5, 2S5, 1S6) and 12 inner-core sensitive modes (25S2, 27S2, 6S3, 9S3, 13S3, 15S3, 11S4, 18S4, 8S5, 11S5, 23S5, 16S6) by using OSE (optimal sequence estimation) method which only considers self-coupling. Results indicate that OSE can completely isolate singlets of high-degree modes in time-domain, effectively resolve the coupled multiplets independently, and reduce the possibility of mode mixing and end effect, showing that OSE could improve some signals’ signal-to-noise ratio. Comparing the results of SG records with seismic data sets suggests that the number of SG records is inadequate to detect all singlets of higher modes. Hence we mainly select plentiful seismograms of IRIS to observe the multiplets of higher modes. We estimate frequencies of the singlets using AR method and evaluate the measurement error using bootstrap method. Besides, we compared the observations with the predictions of PREM-tidal model. This study demonstrates that OSE is effective in isolating singlets of Earth’s free oscillations with higher modes. The experimental results may provide constraints to the construction of 3D Earth model.

A NEW CHALLENGE TO SOLAR DYNAMO MODELS FROM HELIOSEISMIC OBSERVATIONS: THE LATITUDINAL DEPENDENCE OF THE PROGRESSION OF THE SOLAR CYCLE

ABSTRACT The onset of the solar cycle at mid-latitudes, the slowdown in the drift of sunspots toward the equator, the tail-like attachment, and the overlap of successive cycles at the time of minimum activity are delicate issues in models of the αΩ dynamo wave and the flux transport dynamo. Very different parameter values produce similar results, making it difficult to understand the origin of the properties of these solar cycles. We use helioseismic data from the Global Oscillation Network Group to investigate the progression of the solar cycle as observed in intermediate-degree global p-mode frequency shifts at different latitudes and subsurface layers, from the beginning of solar cycle 23 up to the maximum of the current solar cycle. We also analyze those for high-degree modes in each hemisphere obtained through the ring-diagram technique of local helioseismology. The analysis highlights differences in the progression of the cycle below 15° compared to higher latitudes. While the cycle starts at mid-latitudes and then migrates equatorward/poleward, the sunspot eruptions of the old cycle are still ongoing below 15° latitude. This prolonged activity causes a delay in the onset of the cycle and an overlap of successive cycles, whose extent differs in the two hemispheres. Then the activity level rises faster, reaching a maximum characterized by a single-peak structure as opposed to the double peak at higher latitudes. Afterwards the descending phase shows up with a slower decay rate. The latitudinal properties of the progression of the solar cycle highlighted in this study provide useful constraints for discerning among the multitude of solar dynamo models.

A METHOD FOR THE ESTIMATION OF $p$-MODE PARAMETERS FROM AVERAGED SOLAR OSCILLATION POWER SPECTRA

A new fitting methodology is presented which is equally well suited for the estimation of low-, medium-, and high-degree mode parameters from $m$-averaged solar oscillation power spectra of widely differing spectral resolution. This method, which we call the "Windowed, MuLTiple-Peak, averaged spectrum", or WMLTP Method, constructs a theoretical profile by convolving the weighted sum of the profiles of the modes appearing in the fitting box with the power spectrum of the window function of the observing run using weights from a leakage matrix that takes into account both observational and physical effects, such as the distortion of modes by solar latitudinal differential rotation. We demonstrate that the WMLTP Method makes substantial improvements in the inferences of the properties of the solar oscillations in comparison with a previous method that employed a single profile to represent each spectral peak. We also present an inversion for the internal solar structure which is based upon 6,366 modes that we have computed using the WMLTP method on the 66-day long 2010 SOHO/MDI Dynamics Run. To improve both the numerical stability and reliability of the inversion we developed a new procedure for the identification and correction of outliers in a frequency data set. We present evidence for a pronounced departure of the sound speed in the outer half of the solar convection zone and in the subsurface shear layer from the radial sound speed profile contained in Model~S of Christensen-Dalsgaard and his collaborators that existed in the rising phase of Solar Cycle~24 during mid-2010.

ACCURATE CHARACTERIZATION OF HIGH-DEGREE MODES USING MDI OBSERVATIONS

We present the first accurate characterization of high-degree modes, derived using the best Michelson Doppler Imager (MDI) full-disk full-resolution data set available. A 90 day long time series of full-disk 2 arcsec pixel–1 resolution Dopplergrams was acquired in 2001, thanks to the high rate telemetry provided by the Deep Space Network. These Dopplergrams were spatially decomposed using our best estimate of the image scale and the known components of MDI's image distortion. A multi-taper power spectrum estimator was used to generate power spectra for all degrees and all azimuthal orders, up to l = 1000. We used a large number of tapers to reduce the realization noise, since at high degrees the individual modes blend into ridges and thus there is no reason to preserve a high spectral resolution. These power spectra were fitted for all degrees and all azimuthal orders, between l = 100 and l = 1000, and for all the orders with substantial amplitude. This fitting generated in excess of 5.2 × 106 individual estimates of ridge frequencies, line widths, amplitudes, and asymmetries (singlets), corresponding to some 5700 multiplets (l, n). Fitting at high degrees generates ridge characteristics, characteristics that do not correspond to the underlying mode characteristics. We used a sophisticated forward modeling to recover the best possible estimate of the underlying mode characteristics (mode frequencies, as well as line widths, amplitudes, and asymmetries). We describe in detail this modeling and its validation. The modeling has been extensively reviewed and refined, by including an iterative process to improve its input parameters to better match the observations. Also, the contribution of the leakage matrix on the accuracy of the procedure has been carefully assessed. We present the derived set of corrected mode characteristics, which includes not only frequencies, but line widths, asymmetries, and amplitudes. We present and discuss their uncertainties and the precision of the ridge-to-mode correction schemes, through a detailed assessment of the sensitivity of the model to its input set. The precision of the ridge-to-mode correction is indicative of any possible residual systematic biases in the inferred mode characteristics. In our conclusions, we address how to further improve these estimates, and the implications for other data sets, like GONG+ and HMI.

The Sun-as-a-star observations: GOLF & VIRGO on SoHO, and BiSON network

Sun-as-a-star observations are very important for the study of the conditions within the Sun and in particular for the deep interior where higher degree modes do not penetrate. They are also of significance in this era of dramatic advances in stellar asteroseismology as they are comparable to those measured in other stars by asteroseismic missions such as CoRoT, Kepler, and MOST. More than 17 years of continuous measurements of SoHO and more than 30 years of BiSON observations provide very long data sets of uninterrupted helioseismic observations. In this work, we discuss the present status of all these facilities that continue to provide state-of-the-art measurements and invaluable data to improve our knowledge of the deepest layers of the Sun and its structural changes during the activity cycle.

Multi-wavelength analysis of active regions using ring-diagram technique

With the availability of high-cadence and high-resolution Doppler and intensity images from the Solar Dynamics Observatory's Helioseismic Magnetic Imager (HMI) and Atmospheric Imager Assembly (AIA), we analyze the characteristics of high-degree solar acoustic modes in active regions. We apply the ring-diagram technique to Doppler, continuum intensity measurements from HMI, and intensity images from AIA 1600 Å and 1700 Å passband to examine the high-degree mode parameters and sub-surface horizontal flows.

Accurate characterization of high-degree modes using MDI data

We present the first accurate characterization of high-degree modes (i.e., ℓ up to 1000), using the best MDI full-disk full-resolution data set available (90-day long time series, acquired in 2001). The Dopplergrams were spatially decomposed using our best estimate of the image scale and the known components of MDI's image distortion. Multi-tapered power spectra were fitted for all degrees and all azimuthal orders, between ℓ = 100 and ℓ = 1000, and for all orders with substantial amplitude. Fitting at high degrees generates ridge characteristics, characteristics that do not correspond to the underlying mode characteristics. We used a sophisticated forward modeling to recover the best possible estimate of the underlying mode characteristics. We have derived a final set of corrected mode characteristics (frequencies, line widths, asymmetries and amplitudes) and their uncertainties.

Acoustic Mode Frequencies of the Sun During the Minimum Phase Between Solar Cycles 23 and 24

We investigate the spatial and temporal variations of the high-degree mode frequencies calculated over localized regions of the Sun during the extended minimum phase between solar cycles 23 and 24. The frequency shifts measured relative to the spatial average over the solar disk indicate that the correlation between the frequency shift and magnetic field strength during the low-activity phase is weak. The disk-averaged frequency shifts computed relative to a minimal activity period also reveal a moderate correlation with different activity indices, with a maximum linear correlation of about 72%. From the investigation of the frequency shifts at different latitudinal bands, we do not find a consensus period for the onset of solar cycle 24. The frequency shifts corresponding to most of the latitudes in the northern hemisphere and 30 degree south of the equator indicate the minimum epoch to be February 2008, which is earlier than inferred from solar activity indices.

Misleading variations in estimated rotational frequency splittings of solar p modes: consequences for helioseismology and asteroseismology

The aim of this paper is to investigate whether there are any 11-yr or quasi-biennial solar cycle-related variations in solar rotational splitting frequencies of low-degree solar p modes. Although no 11-yr signals were observed, variations on a shorter timescale (~2yrs) were apparent. We show that the variations arose from complications/artifacts associated with the realization noise in the data and the process by which the data were analyzed. More specifically, the realization noise was observed to have a larger effect on the rotational splittings than accounted for by the formal uncertainties. When used to infer the rotation profile of the Sun these variations are not important. The outer regions of the solar interior can be constrained using higher-degree modes. While the variations in the low-l splittings do make large differences to the inferred rotation rate of the core, the core rotation rate is so poorly constrained, even by low-l modes, that the different inferred rotation profiles still agree within their respective 1sigma uncertainties. By contrast, in asteroseismology, only low-l modes are visible and so higher-l modes cannot be used to constrain the rotation profile of stars. Furthermore, we usually only have one data set from which to measure the observed low-l splitting. In such circumstances the inferred internal rotation rate of a main sequence star could differ significantly from estimates of the surface rotation rate, hence leading to spurious conclusions. Therefore, extreme care must be taken when using only the splittings of low-l modes to draw conclusions about the average internal rotation rate of a star.

Kepler observations of the variability in B-type stars

The analysis of the light curves of 48 B-type stars observed by Kepler is presented. Among these are 15 pulsating stars, all of which show low frequencies characteristic of SPB stars. Seven of these stars also show a few weak, isolated high frequencies and they could be considered as SPB/beta Cep hybrids. In all cases the frequency spectra are quite different from what is seen from ground-based observations. We suggest that this is because most of the low frequencies are modes of high degree which are predicted to be unstable in models of mid-B stars. We find that there are non-pulsating stars within the beta Cep and SPB instability strips. Apart from the pulsating stars, we can identify stars with frequency groupings similar to what is seen in Be stars but which are not Be stars. The origin of the groupings is not clear, but may be related to rotation. We find periodic variations in other stars which we attribute to proximity effects in binary systems or possibly rotational modulation. We find no evidence for pulsating stars between the cool edge of the SPB and the hot edge of the delta Sct instability strips. None of the stars show the broad features which can be attributed to stochastically-excited modes as recently proposed. Among our sample of B stars are two chemically peculiar stars, one of which is a HgMn star showing rotational modulation in the light curve.

PULSATIONAL ANALYSIS OF V 588 MON AND V 589 MON OBSERVED WITH THEMOSTANDCoRoTSATELLITES

The two pulsating pre-main sequence (PMS) stars V 588 Mon and V 589 Mon were observed by CoRoT for 23.4 days in March 2008 during the Short Run SRa01 and in 2004 and 2006 by MOST for a total of ~70 days. We present their photometric variability up to 1000 $\mu$ Hz and down to residual amplitude noise levels of 23 and 10 ppm of the CoRoT data for V 588 Mon and V 589 Mon, respectively. The CoRoT imagette data as well as the two MOST data sets allowed for detailed frequency analyses using Period04 and SigSpec. We confirm all previously identified frequencies, improve the known pulsation spectra to a total of 21 frequencies for V 588 Mon and 37 for V 589 Mon and compare them to our PMS model predictions. No model oscillation spectrum with l = 0, 1, 2, and 3 p-modes matches all the observed frequencies. When rotation is included we find that the rotationally split modes of the slower rotating star, V 589 Mon, are addressable via perturbative methods while for the more rapidly rotating star, V 588 Mon, they are not and, consequently, will require more sophisticated modeling. The high precision of the CoRoT data allowed us to investigate the large density of frequencies found in the region from 0 to 300 $\mu$Hz. The presence of granulation appears to be a more attractive explanation than the excitation of high-degree modes. Granulation was modeled with a superposition of white noise, a sum of Lorentzian-like functions and a Gaussian. Our analysis clearly illustrates the need for a more sophisticated granulation model.

Variation of high-degree mode frequencies during the declining phase of solar cycle 23

We investigate the spatial and temporal variation of the high-degree mode frequencies during the declining phase of the solar cycle 23 and the extended minimum between the cycle 23 and 24. We find that the frequency shifts of high-degree modes obtained through the ring-diagram analysis in different phases of the solar cycle are not equally correlated with the local magnetic activity index.

Adiabatic high degree modes of a rotating star

Aims. The influence of the rotation of the Sun on non-radial p-modes with high wave numbers l is studied. To investigate and understand the basic properties of these modes, it is sufficient to consider only the outer layers of the Sun, which can be approximated by a plane layer with constant gravity. Methods. We use a model with a smooth transition between a polytropic convection zone and an isothermal atmosphere. The rotation is simulated by a constant horizontal wind. For this model, using the column mass instead of the geometrical height, the adiabatic wave equation of the pressure perturbation can be reduced to Whittaker’s differential equation. From boundary conditions we obtain the dispersion relation. The geometrical height is a simple elementary function of the column mass. Results. The dispersion relation F(ω,k) = 0 is a higher order algebraic equation in both frequency and horizontal wave number, which must be solved numerically. We analyze the behavior of the dispersion curves of modes with an adiabatic exponent γ = 5/ 3f or layers with polytropic indices n = 3a ndn = 3/2. The f-mode is considered separately. For the understanding of the results we also consider modes of a homogeneous gas. We compare the k − ω diagram of our idealized model with the k − ω diagram of a real solar model.

HELIOSEISMIC DETECTION OF DEEP MERIDIONAL FLOW

Steady meridional flow makes no first-order perturbation to the frequencies of helioseismic normal modes. It does, however, Doppler shift the local wavenumber, thereby distorting the eigenfunctions. For high-degree modes, whose peaks in a power spectrum are blended into continuous ridges, the effect of the distortion is to shift the locations of those ridges. From this blended superposition of modes, one can isolate oppositely directed wave components with the same local horizontal wavenumber and measure a frequency difference which can be safely used to infer the subsurface background flow. But such a procedure fails for the components of the more-deeply-penetrating low-degree modes that are not blended into ridges. Instead, one must analyze the spatial distortions explicitly. With a simple toy model, we illustrate one method by which that might be accomplished by measuring the spatial variation of the oscillation phase. We estimate that by this procedure it might be possible to infer meridional flow deep in the solar convection zone.

HD 50844: a new look atδScuti stars from CoRoT space photometry

Context. Aims. This work presents the results obtained by CoRoT on HD 50844, the only δ Sct star observed in the CoRoT initial run (57.6 d). The aim of these CoRoT observations was to investigate and characterize for the first time the pulsational behaviour of a δ Sct star, when observed at a level of precision and with a much better duty cycle than from the ground.Methods. The 140 016 datapoints were analysed using independent approaches (SigSpec software and different iterative sine-wave fittings) and several checks performed (splitting of the timeseries in different subsets, investigation of the residual light curves and spectra). A level of 10-5 mag was reached in the amplitude spectra of the CoRoT timeseries. The space monitoring was complemented by ground-based high-resolution spectroscopy, which allowed the mode identification of 30 terms. Results. The frequency analysis of the CoRoT timeseries revealed hundreds of terms in the frequency range 0–30 d-1 . All the cross-checks confirmed this new result. The initial guess that δ Sct stars have a very rich frequency content is confirmed. The spectroscopic mode identification gives theoretical support since very high-degree modes (up to ) are identified. We also prove that cancellation effects are not sufficient in removing the flux variations associated to these modes at the noise level of the CoRoT measurements. The ground-based observations indicate that HD 50844 is an evolved star that is slightly underabundant in heavy elements, located on the Terminal Age Main Sequence. Probably due to this unfavourable evolutionary status, no clear regular distribution is observed in the frequency set. The predominant term ( d-1 ) has been identified as the fundamental radial mode combining ground-based photometric and spectroscopic data. Conclusions.

Time-series spectroscopy and photometry of the pulsating subdwarf B star PG 1219+534 (KY UMa)

Aims. We present observations and analysis of time-series spectroscopy and photometry of the pulsating subdwarf B star PG 1219+534 (KY UMa). Subdwarf B stars are blue horizontal branch stars that have shed most of their hydrogen envelopes. Pulsating subdwarf B stars allow a probe into this interesting phase of evolution. Methods. Low-resolution spectra were obtained at the Nordic Optical Telescope and Kitt Peak National Observatory, and photometric observations were obtained at MDM and Baker observatories in 2006. We extracted radial velocity and equivalent width variations from several Balmer and He I lines in individual spectra. The pulsation frequencies were separated via phase binning to detect lineprofile variations in Balmer and helium lines, which were subsequently matched to atmospheric models to infer effective temperature and gravity changes throughout the pulsation cycle. Results. From the photometry we recovered the four previously observed frequencies and detected a new fifth frequency. From the spectra we directly measured radial velocity and equivalent width variations for the four main frequencies and from atmospheric models. We successfully inferred temperature and gravity changes for these four frequencies. We compared amplitude ratios and phase differences of these quantities and searched for outliers that could be identified as high-degree modes. These are the first such measurements for a “normal” amplitude pulsating subdwarf B star, indicating that spectroscopic studies can benefit the majority of pulsating subdwarf B stars.