F-type Stars Research Articles

Spatial distribution of exoplanet candidates based on Kepler and Gaia data

Context. Surveying the spatial distribution of exoplanets in the Galaxy is important for improving our understanding of planet formation and evolution. Aims. We aim to determine the spatial gradients of exoplanet occurrence in the Solar neighbourhood and in the vicinity of open clusters. Methods. We combined Kepler and Gaia DR2 data for this purpose, splitting the volume sampled by the Kepler mission into certain spatial bins. We determined an uncorrected and bias-corrected exoplanet frequency and metallicity for each bin. Results. There is a clear drop in the uncorrected exoplanet frequency with distance for F-type stars (mainly for smaller planets), a decline with increasing distance along the Galactic longitude l = 90°, and a drop with height above the Galactic plane. We find that the metallicity behaviour cannot be the reason for the drop of the exoplanet frequency around F stars with increasing distance. This might have only contributed to the drop in uncorrected exoplanet frequency with the height above the Galactic plane. We argue that the above-mentioned gradients of uncorrected exoplanet frequency are a manifestation of a single bias of undetected smaller planets around fainter stars. When we correct for observational biases, most of these gradients in exoplanet frequency become statistically insignificant. Only a slight decline of the planet occurrence with distance for F stars remains significant at the 3σ level. Apart from that, the spatial distribution of exoplanets in the Kepler field of view is compatible with a homogeneous one. At the same time, we do not find a significant change in the exoplanet frequency with increasing distance from open clusters. In terms of byproducts, we identified six exoplanet host star candidates that are members of open clusters. Four of them are in the NGC 6811 (KIC 9655005, KIC 9533489, Kepler-66, Kepler-67) and two belong to NGC 6866 (KIC 8396288, KIC 8331612). Two out of the six had already been known to be cluster members.

Finding binaries from phase modulation of pulsating stars with Kepler – VI. Orbits for 10 new binaries with mischaracterized primaries

ABSTRACT Measuring phase modulation in pulsating stars has proven to be a highly successful way of finding binary systems. The class of pulsating main-sequence A and F variables, known as δ Scuti stars consists of particularly good targets for this, and the Kepler sample of these has been almost fully exploited. However, some Keplerδ Scuti stars have incorrect temperatures in stellar properties catalogues, and were missed in previous analyses. We used an automated pulsation classification algorithm to find 93 new δ Scuti pulsators among tens of thousands of F-type stars, which we then searched for phase modulation attributable to binarity. We discovered 10 new binary systems and calculated their orbital parameters, which we compared with those of binaries previously discovered in the same way. The results suggest that some of the new companions may be white dwarfs.

Chemical mixing in low mass stars

Context. When modelling stars with masses higher than 1.2 M⊙ with no observed chemical peculiarity, atomic diffusion is often neglected because, on its own, it causes unrealistic surface abundances compared with those observed. The reality is that atomic diffusion is in competition with other transport processes. Rotation is one of the processes able to prevent excessively strong surface abundance variations. Aims. The purpose of this study is to quantify the opposite or conjugated effects of atomic diffusion (including radiative acceleration) and rotationally induced mixing in stellar models of low mass stars, and to assess whether rotational mixing is able to prevent the strong abundance variations induced by atomic diffusion in F-type stars. Our second goal is to estimate the impact of neglecting both rotational mixing and atomic diffusion in stellar parameter inferences for stars with masses higher than 1.3 M⊙. Methods. Using the Asteroseismic Inference on a Massive Scale (AIMS) stellar parameter inference code, we infer the masses and ages of a set of representative artificial stars for which models were computed with the Code d’Evolution Stellaire Adaptatif et Modulaire (CESTAM; the T stands for Transport) evolution code, taking into account rotationally induced mixing and atomic diffusion, including radiative acceleration. The observed constraints are asteroseismic and classical properties. The grid of stellar models used for the optimization search include neither atomic diffusion nor rotationally induced mixing. The differences between real and retrieved parameters then provide an estimate of the errors made when neglecting transport processes in stellar parameter inference. Results. We show that for masses lower than 1.3 M⊙, rotation dominates the transport of chemical elements and strongly reduces the effect of atomic diffusion, with net surface abundance modifications similar to solar values. At higher mass, atomic diffusion and rotation are competing equally. Above 1.44 M⊙, atomic diffusion dominates in stellar models with initial rotation lower than 80 km s−1 producing a chemical peculiarity which is not observed in Kepler Legacy stars. This indicates that a transport process of chemical elements is missing, probably linked to the missing transport process of angular momentum needed to explain rotation profiles in solar-like stars. Importantly, neglecting rotation and atomic diffusion (including radiative acceleration) in the models, when inferring the parameters of F-type stars, may lead to respective errors of ≈5%, ≈2.5%, and ≈25% for stellar masses, radii, and ages. Conclusions. Atomic diffusion (including radiative acceleration) and rotational mixing should be taken into account in stellar models in order to determine accurate stellar parameters. When atomic diffusion and shellular rotation are both included, they enable stellar evolution codes to reproduce the observed metal and helium surface abundances for stars with masses up to 1.4 M⊙ at solar metallicity. However, if rotation is actually uniform for these stars (as observations seem to indicate), then an additional chemical mixing process is needed together with a revised formulation of rotational mixing. For higher masses, an additional mixing process is needed in any case.

The Pre-He White Dwarf in the Post-mass Transfer Binary EL CVn

Abstract EL CVn is the prototype of a class of eclipsing binaries that consist of an A- or F-type main-sequence star and a hot, low-mass, pre-He white dwarf (pre-He WD), the stripped down remains of the former mass donor. Here we present the first direct detection and characterization of the spectrum of the pre-He WD in EL CVn that was made possible through far-UV spectroscopy with the Hubble Space Telescope Cosmic Origins Spectrograph. These spectra straddle the wavelength range where flux dominance shifts from the pre-He WD to the A star. Radial velocities of both components were measured from the far-UV spectra and new optical spectra from the Apache Point Observatory Astrophysical Research Consortium Echelle Spectrograph. We also obtained fast cadence photometry of the eclipses with the pt5m telescope at the Roque de los Muchachos Observatory. A combined analysis of the velocities and light curve yields the component masses and radii. We applied a Doppler tomography algorithm to reconstruct the individual spectra, and we compared these to models to estimate the effective temperatures. The pre-He WD has low mass (0.176 ± 0.004 M ⊙), is small (0.284 ± 0.003 R ⊙), and is relatively hot (11,800 ± 400 K), and these parameters are approximately consistent with predictions for a star stripped through stable mass transfer. The spectral lines of the pre-He WD show that its atmosphere is H-rich, He-depleted, and metal-poor, probably as the result of elemental diffusion that has occurred since mass transfer ceased.

Transmission Spectroscopy of WASP-79b from 0.6 to 5.0 μm

Abstract As part of the Panchromatic Exoplanet Treasury program, we have conducted a spectroscopic study of WASP-79b, an inflated hot Jupiter orbiting an F-type star in Eridanus with a period of 3.66 days. Building on the original WASP and TRAPPIST photometry of Smalley et al., we examine Hubble Space Telescope (HST)/Wide Field Camera 3 (WFC3) (1.125–1.650 μm), Magellan/Low Dispersion Survey Spectrograph (LDSS)-3C (0.6–1 μm) data, and Spitzer data (3.6 and 4.5 μm). Using data from all three instruments, we constrain the water abundance to be −2.20 ≤ log(H2O) ≤ −1.55. We present these results along with the results of an atmospheric retrieval analysis, which favor inclusion of FeH and H− in the atmospheric model. We also provide an updated ephemeris based on the Smalley, HST/WFC3, LDSS-3C, Spitzer, and Transiting Exoplanet Survey Satellite (TESS) transit times. With the detectable water feature and its occupation of the clear/cloudy transition region of the temperature/gravity phase space, WASP-79b is a target of interest for the approved James Webb Space Telescope (JWST) Director’s Discretionary Early Release Science (ERS) program, with ERS observations planned to be the first to execute in Cycle 1. Transiting exoplanets have been approved for 78.1 hr of data collection, and with the delay in the JWST launch, WASP-79b is now a target for the Panchromatic Transmission program. This program will observe WASP-79b for 42 hr in four different instrument modes, providing substantially more data by which to investigate this hot Jupiter.

Determination of the star-spot covering fraction as a function of stellar age from observational data

ABSTRACT The association of star-spots with magnetic fields leads to an expectation that quantities which correlate with magnetic field strength may also correlate with star-spot coverage. Since younger stars spin faster and are more magnetically active, assessing whether star-spot coverage correlates with shorter rotation periods and stellar youth tests these principles. Here, we analyse the star-spot covering fraction versus stellar age for M-, G-, K-, and F-type stars based on previously determined variability and rotation periods of over 30 000 Kepler main-sequence stars. We determine the correlation between age and variability using single and dual power-law best fits. We find that star-spot coverage does indeed decrease with age. Only when the data are binned in an effort to remove the effects of activity cycles of individual stars, do statistically significant power-law fits emerge for each stellar type. Using bin averages, we then find that the star-spot covering fraction scales with the X-ray to bolometric ratio to the power λ with 0.22 ± 0.03 < λ < 0.32 ± 0.09 for G-type stars of rotation period below 15 d and for the full range of F- and M-type stars. For K-type stars, we find two branches of λ separated by variability bins, with the lower branch showing nearly constant star-spot coverage and the upper branch λ ∼ 0.35 ± 0.04. G-type stars with periods longer than 15 d exhibit a transition to steeper power law of λ ∼ 2.4 ± 1.0. The potential connection to previous rotation-age measurements suggesting a magnetic breaking transition at the solar age, corresponding to period of 24.5 is also of interest.

Chromospheric emission of solar-type stars with asteroseismic ages

ABSTRACT Stellar magnetic activity decays over the main-sequence life of cool stars due to the stellar spin-down driven by magnetic braking. The evolution of chromospheric emission is well studied for younger stars, but difficulties in determining the ages of older cool stars on the main sequence have complicated such studies for older stars in the past. Here, we report on chromospheric Ca ii H and K line measurements for 26 main-sequence cool stars with asteroseismic ages older than a gigayear and spectral types F and G. We find that for the G stars and the cooler F-type stars that still have convective envelopes the magnetic activity continues to decrease at stellar ages above 1 Gyr. Our magnetic activity measurements do not show evidence for a stalling of the magnetic braking mechanism, which has been reported for stellar rotation versus age for G- and F-type stars. We also find that the measured $R^{\prime }_{\mathrm{ HK}}$ indicator value for the cool F stars in our sample is lower than predicted by common age–activity relations that are mainly calibrated on data from young stellar clusters. We conclude that, within individual spectral type bins, chromospheric magnetic activity correlates well with stellar age even for old stars.

ODEA: Orbital Dynamics in a complex Evolving Architecture

Context. Mixed-variable symplectic integrators are widely used in orbital dynamics. However, they have been developed for Solar system-type architectures, and can not handle evolving hierarchy, in particular in systems with two or more stellar components. Such configuration may have occurred in the history of HD 106906, a tight pair of F-type stars surrounded by a debris disk and a planetary-mass companion on a wide orbit. Aims. We present the new algorithm ODEA, based on the symplectic algorithm Swift HJS, that can model any system (binary,...) with unstable architecture. We study the peculiar system HD 106906 as a testcase for the code. Methods. We define and compute a criterion based on acceleration ratios to indicate when the initial hierarchy is not relevant anymore. A new hierarchy is then computed. The code is applied to study the two recently evidenced fly-bys that occurred on system HD 106906, to determine if they could account for the wide orbit of the planet. Thousands of simulations have been performed to account for the uncertainty on the perturbers coordinates and velocities. Results. The algorithm is able to handle any change of hierarchy, temporary or not. We used it to fully model HD 106906 encounters. The simulations confirm that the fly-bys could have stabilized the planet orbit, and show that it can account for the planet probable misalignment with respect to the disk plane as well as the disk morphology. However, that requires a small distance at closest approach (≲0.05 pc), and this configuration is not guaranteed. Conclusions. ODEA is a very good choice for the study of non-Solar type architecture. It can now adapt to an evolving hierarchy, and is thus suitable to study capture of planets and dust. Further observations of the perturbers, in particular their radial velocity, are required to conclude on the effects of the fly-by on system HD 106906.

Discovery and Seismic Analysis of an EL CVn-type Binary with Hybrid δ Sct-γ Dor Pulsations

We report the discovery of an EL CVn-type binary consisting of a low-mass, pre-helium white dwarf and the first detection of hybrid δ Sct-γ Dor pulsations in such binary systems. Based on the four years Kepler data, we determined comprehensive photometric solution of the eclipsing binary KIC 8113154. The light-curve modeling reveals that it is a detached system containing a thermally bloated, low-mass, pre-He white dwarf with the mass of 0.26 ± 0.02 M⊙ and the radius of 0.13 ± 0.01 R⊙. After removal of the binary model from the observed Kepler data, multiple frequency analysis is applied to the light residuals. The Fourier spectrum shows low-order p-modes and high-order g-mode pulsations that very likely stem from the F-type primary component star, which could be classified as a new δ Sct-γ Dor hybrid. We detected 111 frequencies with signal-to-noise amplitude ratios larger than 4.0. From these frequencies, we identified 17 high-order quadrupole (l = 2) g modes, including eight zonal (m = 0) and nine prograde (m = 2) ones, on the basis of which we derived the asymptotic period spacing of the g modes and the internal rotation rate of the convective core. This is significantly smaller than the orbital frequency, indicating that the core and envelope of the primary star in KIC 8113154 rotate differentially.

A Mass-accreting Gamma Doradus Pulsator with a Synchronized Core in Kepler Eclipsing Binary KIC 7385478

Abstract The short-period (P ≈ 1.7 days), Algol-type eclipsing binary KIC 7385478 consists of an F-type primary star (M 1 ≈ 1.71M ⊙) and an evolved K-type secondary (M 2 ≈ 0.37M ⊙). We study the variability of the Kepler light curve and attribute many frequency peaks in the Fourier spectrum to the spot modulation. These frequencies are in the form of orbital harmonics and are highly variable in amplitude. They are most likely from the mass-accreting primary star. In addition, we identify a series of prograde dipole g modes from the primary star that show a quasi-linear period spacing pattern and are very stable in amplitude. The period spacing pattern reveals an asymptotic period spacing value in agreement with fundamental parameters of the primary star and also implies that the near-convective-core rotation rate is almost the same as the orbital period. Thus, both the surface and the core of this Gamma Dor pulsator have synchronized with the binary orbit. We find that a lower stellar mass ≈1.50M ⊙ and higher effective temperature are needed in order to be compatible with the asteroseismic constraints from single-star evolutionary models.

TESS Spots a Compact System of Super-Earths around the Naked-eye Star HR 858

Abstract Transiting Exoplanet Survey Satellite (TESS) observations have revealed a compact multiplanet system around the sixth-magnitude star HR 858 (TIC 178155732, TOI 396), located 32 pc away. Three planets, each about twice the size of Earth, transit this slightly evolved, late F-type star, which is also a member of a visual binary. Two of the planets may be in mean motion resonance. We analyze the TESS observations, using novel methods to model and remove instrumental systematic errors, and combine these data with follow-up observations taken from a suite of ground-based telescopes to characterize the planetary system. The HR 858 planets are enticing targets for precise radial velocity observations, secondary eclipse spectroscopy, and measurements of the Rossiter–McLaughlin effect.

The partial ionization zone of heavy elements in F-stars: a study on how it correlates with rotation

ABSTRACT We study the relation between the internal structures of 10 benchmark main-sequence F-stars and their rotational properties. Stellar rotation of main-sequence F-type stars can be characterized by two distinct rotational regimes. Early-type F-stars are usually rapid rotators with periods typically below 10 d, whereas later-type F-stars have longer rotation periods. Specifically, and since the two rotational regimes are tightly connected to the effective temperatures of the stars, we investigate in detail the characteristics of the partial ionization zones in the outer convective envelopes of these stars, which in turn, depend on the internal temperature profiles. Our study shows that the two rotational regimes might be distinguished by the relative locations of the partial ionization region of heavy elements and the base of the convective zone. Since in all these stars is expected a dynamo-driven magnetic field where the shear layer between convective and radiative zones (tachocline) plays an important role, this result suggests that the magnetic field may be related to the combined properties of convection and ionization.

New Substellar Discoveries from Kepler and K2: Is There a Brown Dwarf Desert?

Abstract We present the discoveries of a brown dwarf (BD) and a low-mass star from the Kepler and K2 missions. The newly discovered BD is EPIC 212036875b and the low-mass star is KOI-607b. EPIC 212036875b has a mass of M b = 52.3 ± 1.9 M J, a radius of R b = 0.874 ± 0.017 R J, and orbits its host star in P = 5.169885 ± 0.000027 days. Its host star is a late F-type star with M ⋆ = 1.288 ± 0.065 M ⊙, R ⋆ = 1.498 ± 0.025 R ⊙, and T eff = 6238 ±60 K. KOI-607b has a mass of M b = 95.1 ± 3.4 M J, a radius of R b = 1.089 ± 0.089 R J, and an orbital period of P = 5.89399148 ± 0.00000060 days. The primary star in the KOI-607 system is a G dwarf with M ⋆ =0.993 ± 0.052 M ⊙, R ⋆ = 0.915 ± 0.031 R ⊙, and T eff = 5418 ± 87 K. We also revisit a BD, CWW 89Ab, that was previously published by Nowak et al. (under the designation EPIC 219388192b). CWW 89Ab is one of two known transiting BDs associated with a star cluster, which illustrates the need for more BDs with accurate masses, radii, and reliable age determinations to test theoretical models. We find that the newly discovered BD, EPIC 212036875b, falls in the middle of the so-called BD desert, indicating that EPIC 212036875b is either a particularly rare object, or the BD desert may not be so dry after all.

On the properties of the magnetic Chemically Peculiar B, A, and F-type stars

AbstractWe present a preliminary analysis of the Strömgren uvby photometry of the magnetic CP stars obtained using the Four College Automated Photometric Telescope for its 21.5 years of operation ending in Fall 2012. We summarize the photometry for all the FCAPT mCP stars that have been published to date. We do not find any significant correlation between the amplitudes of variation in the uvby filters and the periods. A small number of stars show anomalous behaviour of the v filter which will be discussed in a future study.

3D Modeling of the Structure and Dynamics of a Main-Sequence F-type Star

AbstractCurrent state-of-the-art computational modeling makes it possible to build realistic models of stellar convection zones and atmospheres that take into account chemical composition, radiative effects, ionization, and turbulence. The standard 1D mixing-length-based evolutionary models are not able to capture many physical processes of the stellar interior dynamics. Mixing-length models provide an initial approximation of stellar structure that can be used to initialize 3D radiative hydrodynamics simulations which include realistic modeling of turbulence, radiation, and other phenomena.In this paper, we present 3D radiative hydrodynamic simulations of an F-type main-sequence star with 1.47 solar mass. The computational domain includes the upper layers of the radiation zone, the entire convection zone, and the photosphere. The effects of stellar rotation is modeled in the f-plane approximation. These simulations provide new insight into the properties of the convective overshoot region, the dynamics of the near-surface, highly turbulent layer, and the structure and dynamics of granulation. They reveal solar-type differential rotation and latitudinal dependence of the tachocline location.

Resolving Power of Asteroseismic Inversion of the Kepler Legacy Sample

AbstractThe Kepler Asteroseismic Legacy Project provided frequencies, separation ratios, error estimates, and covariance matrices for 66 Kepler main sequence targets. Most of the previous analysis of these data was focused on fitting standard stellar models. We present results of direct asteroseismic inversions using the method of optimally localized averages (OLA), which effectively eliminates the surface effects and attempts to resolve the stellar core structure. The inversions are presented for various structure properties, including the density stratification and sound speed. The results show that the mixed modes observed in post-main sequence F-type stars allow us to resolve the stellar core structure and reveal significant deviations from the evolutionary models obtained by the grid-fitting procedure to match the observed oscillation frequencies.

First resolved observations of a highly asymmetric debris disc around HD 160305 with VLT/SPHERE

Context. Direct imaging of debris discs gives important information about their nature, their global morphology, and allows us to identify specific structures possibly in connection with the presence of gravitational perturbers. It is the most straightforward technique to observe planetary systems as a whole. Aims. We present the first resolved images of the debris disc around the young F-type star HD 160305, detected in scattered light using the VLT/SPHERE instrument in the near infrared. Methods. We used a post-processing method based on angular differential imaging and synthetic images of debris discs produced with a disc modelling code (GRaTer) to constrain the main characteristics of the disc around HD 160305. All of the point sources in the field of the IRDIS camera were analysed with an astrometric tool to determine whether they are bound objects or background stars. Results. We detect a very inclined (~82°) ring-like debris disc located at a stellocentric distance of about 86 au (deprojected width ~27 au). The disc displays a brightness asymmetry between the two sides of the major axis, as can be expected from scattering properties of dust grains. We derive an anisotropic scattering factor g > 0.5. A second right-left asymmetry is also observed with respect to the minor axis. We measure a surface brightness ratio of 0.73 ± 0.18 between the bright and the faint sides. Because of the low signal-to-noise ratio (S/N) of the images we cannot easily discriminate between several possible explanations for this left-right asymmetry, such as perturbations by an unseen planet, the aftermath of the breakup of a massive planetesimal, or the pericenter glow effect due to an eccentric ring. Two epochs of observations allow us to reject the companionship hypothesis for the 15 point sources present in the field.

The Flare Catalog and the Flare Activity in the Kepler Mission

Abstract We present a flare catalog of the Kepler mission using the long-cadence data of Data Release 25. This catalog comprises 3420 flare stars and 162,262 flare events. A comparison shows that the flare catalogs of previous studies are seriously polluted by various false-positive signals and artifacts. The incidence of flare stars rises with decreasing temperature, which accords with the theoretical analysis. The flare frequency distributions (FFDs) from F-type stars to M-type stars obey a power-law relation with α ∼ 2, indicating that they have the same mechanism on generating flares. The remarkable incidence and the deviation of FFDs on A-type flare stars imply that they generate flares in a different way. The activity–rotation relation is consistent with previous studies at low temperature bands, whereas it becomes dispersive with increasing temperature. Combined with the Gyrochronology, we find that the mixing of stars of two different dynamos gives rise to the dispersion. We thereby propose a scenario on understanding the activity–rotation relation across the Hertzsprung-Russell diagram. Based on the scenario and the correspondence of dynamo with regard to activity and rotation, we suggest a new expression on the activity–rotation relation, in which the segmentation is on the basis of the dynamo rather than the rotation period. The rotation distribution of flare stars shows that about 70% of flare stars rotate faster than 10 days and the rate approaches 95% at 30 days. Based on the incidence and the rotation distribution of flare stars, we estimate that a superflare with energy ∼1034 erg occurs on the Sun at least once in 5500 yr.

High-resolution spectroscopy of Boyajian’s star during optical dimming events

Boyajian's star is an apparently normal main sequence F-type star with a very unusual light curve. The dipping activity of the star, discovered during the Kepler mission, presents deep, asymmetric, and aperiodic events. Here we present high resolution spectroscopic follow-up during some dimming events recorded post-Kepler observations, from ground-based telescopes. We analise data from the HERMES, HARPS-N and FIES spectrographs to characterise the stellar atmosphere and to put some constraints on the hypotheses that have appeared in the literature concerning the occulting elements. The star's magnetism, if existing, is not extreme. The spots on the surface, if present, would occupy 0.02% of the area, at most. The chromosphere, irrespective of the epoch of observation, is hotter than the values expected from radiative equilibrium, meaning that the star has some degree of activity. We find no clear evidence of the interstellar medium nor exocoments being responsible for the dimmings of the light curve. However, we detect at 1-2 sigma level, a decrease of the radial velocity of the star during the first dip recorded after the \emph{\emph{Kepler}} observations. We claim the presence of an optically thick object with likely inclined and high impact parameter orbits that produces the observed Rossiter-McLaughlin effect.

Asteroseismology of main-sequence F stars with Kepler: overcoming short mode lifetimes

Asteroseismology is a powerful way of determining stellar parameters and properties of stars like the Sun. However, main-sequence F-type stars exhibit short mode lifetimes relative to their oscillation frequency, resulting in overlapping radial and quadrupole modes. The goal of this paper is to use the blended modes for asteroseismology in place of the individual separable modes. We used a peak-bagging method to measure the centroids of radial-quadrupole pairs for 66 stars from the \textit{Kepler} LEGACY sample, as well as {\theta} Cyg, HD 49933, HD 181420, and Procyon. We used the relative quadrupole-mode visibility to estimate a theoretical centroid frequency from a grid of stellar oscillation models. The observed centroids were matched to the modelled centroids with empirical surface correction to calculate stellar parameters. We find that the stellar parameters returned using this approach agree with the results using individual mode frequencies for stars, where those are available. We conclude that the unresolved centroid frequencies can be used to perform asteroseismology with an accuracy similar to that based on individual mode frequencies.