Seismology of stellar envelopes: probing the outer layers of a star through the scattering of acoustic waves

The outer layers of Sun-like stars are regions of rapid spatial variation which modulate the p-mode frequencies by partially reflecting the constituent acoustic waves. With the accuracy that has been achieved by current solar observations, and that is expected from imminent stellar observations, this modulation can be observed from the spectra of the low-degree modes. We present a new and simple theoretical calculation to determine the leading terms in an asymptotic expansion of the outer phase of these modes, which is determined by the structure of the surface layers of the star. Our procedure is to compare the stellar envelope with a plane-parallel polytropic envelope, which we regard as a smooth reference background state. Then we can isolate a seismic signature of the acoustic phase and relate it to the stratification of the outer layers of the convection zone. One can thereby constrain theories of convection that are used to construct the convection zones of the Sun and Sun-like stars. The accuracy of the diagnostic is tested in the solar case by comparing the predicted outer phase with an exact numerical calculation.     

The use of frequency-separation ratios for asteroseismology

The systematic patterns of separations between frequencies of modes of different degree and order are a characteristic of p-mode oscillations of stars. The frequency separations depend on the internal structure of the star and so measuring them in the observed oscillation spectra of variable stars gives valuable diagnostics of the interior of a star. Roxburgh & Vorontsov proposed using the ratio of the so-called small frequency separation to the large frequency separation as a diagnostic of the stellar interior, and demonstrated that this ratio was less sensitive than the individual frequency separations themselves to uncertain details of the near-surface structure. Here we derive kernels relating the frequency separation ratio to structure, and show why the ratio is relatively insensitive to the near-surface structure in terms of the very small amplitude of the kernels in the near-surface layers. We also investigate the behaviour of the separation ratio for stars of different masses and ages, and demonstrate the usefulness of the ratio in the so-called asteroseismic Hertzsprung–Russell diagram.     

Sounding stellar cycles with Kepler – I. Strategy for selecting targets

The long-term monitoring and high photometric precision of the Kepler satellite will provide a unique opportunity to sound the stellar cycles of many solar-type stars using asteroseismology. This can be achieved by studying periodic changes in the amplitudes and frequencies of the oscillation modes observed in these stars. By comparing these measurements with conventional ground-based chromospheric activity indices, we can improve our understanding of the relationship between chromospheric changes and those taking place deep in the interior throughout the stellar activity cycle. In addition, asteroseismic measurements of the convection zone depth and differential rotation may help us determine whether stellar cycles are driven at the top or at the base of the convection zone. In this paper, we analyse the precision that will be possible using Kepler to measure stellar cycles, convection zone depths and differential rotation. Based on this analysis, we describe a strategy for selecting specific targets to be observed by the Kepler Asteroseismic Investigation for the full length of the mission, to optimize their suitability for probing stellar cycles in a wide variety of solar-type stars.     

Anomalous variations in low-degree helioseismic mode frequencies

We compare changes in the frequencies of solar acoustic modes with degree between 0 and 2, as derived from Global Oscillation Network Group (GONG), Birmingham Solar Oscillations Network (BiSON) and Michelson Doppler Imager (MDI) spectra obtained between 1995 and 2003. We find that, after the solar-activity dependence has been removed from the frequencies, there remain variations that appear to be significant, and are often well correlated between the different data sets. We consider possible explanations for these fluctuations, and conclude that they are likely to be related to the stochastic excitation of the modes. The existence of such fluctuations has possible relevance to the analysis of other low-degree acoustic mode spectra such as those from solar-type stars.     

High-frequency modes in solar-like stars

p-mode oscillations in solar-like stars are excited by the outer convection zone in these stars and reflected close to the surface. The p modes are trapped inside an acoustic cavity, but the modes only stay trapped up to a given frequency [known as the acoustic cut-off frequency  (ν_ac)  ] as modes with larger frequencies are generally not reflected at the surface. This means that modes with frequency larger than the acoustic cut-off frequency must be travelling waves. The high-frequency modes may provide information about the physics in the outer layers of the stars and the excitation source and are therefore highly interesting as it is the estimation of these two phenomena that cause some of the largest uncertainties when calculating stellar oscillations.
High-frequency modes have been detected in the Sun, in β Hydri and in α Cen A and α Cen B by smoothing the so-called echelle diagram and the large frequency separation as a function of frequency has been estimated. The large frequency separation has been compared with a simple model of the acoustic cavity which suggests that the reflectivity of the photosphere is larger at high frequency than predicted by standard models of the solar atmosphere and that the depth of the excitation source is larger than what has been estimated by other models and might depend on the order n and degree l of the modes.     

An asteroseismic signature of helium ionization

We investigate the influence of the ionization of helium on the low-degree acoustic oscillation frequencies in model solar-type stars. The signature in the oscillation frequencies characterizing the ionization-induced depression of the first adiabatic exponent γ is a superposition of two decaying periodic functions of frequency ν, with 'frequencies' that are approximately twice the acoustic depths of the centres of the He  i and He  ii ionization regions. That variation is probably best exhibited in the second frequency difference  Δ₂ν _{n ,  l} ≡ν _{n −1,  l} − 2ν _{n ,  l} +ν _{n +1,  l}   . We show how an analytic approximation to the variation of γ leads to a simple representation of this oscillatory contribution to Δ₂ν which can be used to characterize the γ variation, our intention being to use it as a seismic diagnostic of the helium abundance of the star. We emphasize that the objective is to characterize γ, not merely to find a formula for Δ₂ν that reproduces the data.     

Diagnostics of the Internal Structure of Stars using the Differential Response Technique

We address the problem of the diagnosing the deep interior structure of stars using acoustic p-modes, and investigate the diagnostic capabilities of two complementary approaches both based on the differential response technique (Vorontsov, 1998): (a) direct calibration using a grid of evolutionary stellar models, and (b) linear and non-linear (with consecutive linearisations) inversion of low-degree frequencies. We apply this analysis to the frequencies of a model of an old 0.8M star, and to the solar frequencies obtained from BiSON measurements, using a 2-D grid of reference models of different mass and age. We explore the convergence and stability of the asteroseismic inversion, performed with the adaptive regularisation technique of Strakhov and Vorontsov (2001).     

Preliminary results on the determination of the theoretical nonadiabatic observable phase lag (ψT) using VIRGO color photometry

The helioseismic instruments aboard the SOHO satellite make it possible to measure solar oscillations as variations of the irradiance (VIRGO) or as variations of the photospheric velocity (GOLF). Theoretically, phase differences between different photometric bands are expected to be around 0 degrees over the p‐mode frequency range. By using VIRGO (red) and VIRGO (blue) data, we find a mean phase shift of 8.05 ± 1.81°, whereas by using VIRGO (green) and VIRGO (blue) data, we got a mean value of –1.04 ± 0.19°. Hence, when the analysis includes the VIRGO infrared range, the Sun's atmosphere does not follow an exact adiabatic behavior. In this study, we use the phase shifts obtained by VIRGO (green) and VIRGO (blue) to determine the non‐adiabatic parameter phase lag (ψ_T) as a function of frequency. To this aim, we applied the non radial linearized formula put in the complex form by Garrido: we found a mean value of ψ_T = 179.95°. The lowest value being ψ_T = 179.90°, the departure from theoretical predictions is less then a tenth of a degree over the entire p mode frequency range. We can state that the solar atmosphere has a behavior close to the adiabatic case, when the phase shifts and amplitude ratios are computed using VIRGO (green) and VIRGO (blue) data. Nevertheless this small deviation is significant. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropy and Dissipation of Turbulence and Their Effects on Solar Models

1 INTRODUCTIONThe maing-length theory (MLT) is the most commonly used approach to calculate convective energy transport in stars and other astrophysical situations. Based on the original idea ofPrandtl (1952) that turbulent parcels trallsfer heat in a similar way as molecules of gas do inthermal conduction, the MLT assumes that convection cells, drived by buoyancy, move thlougha ~ng length 1 and release the heat they carry when they merge with their environment. Themost widely adopted f…     

Changes in convective properties over the solar cycle: effect on p-mode damping rates

Measurements of both solar irradiance and p-mode oscillation frequencies indicate that the structure of the Sun changes with the solar cycle. Balmforth, Gough & Merryfield investigated the effect of symmetrical thermal disturbances on the solar structure and the resulting pulsation frequency changes. They concluded that thermal perturbations alone cannot account for the variations in both irradiance and p-mode frequencies, and that the presence of a magnetic field affecting acoustical propagation is the most likely explanation of the frequency change, in the manner suggested earlier by Gough & Thompson and by Goldreich et al. Numerical simulations of Boussinesq convection in a magnetic field have shown that at high Rayleigh number the magnetic field can modify the preferred horizontal length scale of the convective flow.
Here, we investigate the effect of changing the horizontal length scale of convective eddies on the linewidths of the acoustic resonant mode peaks observed in helioseismic power spectra. The turbulent fluxes in these model computations are obtained from a time-dependent, non-local generalization of the mixing-length formalism. The modelled variations are compared with p-mode linewidth changes revealed by the analysis of helioseismic data collected by the Birmingham Solar-Oscillations Network (BiSON); these low-degree (low- l ) observations cover the complete falling phase of solar activity cycle 22. The results are also discussed in the light of observations of solar-cycle variations of the horizontal size of granules and with results from 2D simulations by Steffen of convective granules.     

Solar-cycle variation of the sound-speed asphericity from GONG and MDI data 1995–2000

We study the variation of the frequency splitting coefficients describing the solar asphericity in both GONG and MDI data, and use these data to investigate temporal sound-speed variations as a function of both depth and latitude during the period 1995–2000 and a little beyond. The temporal variations in even splitting coefficients are found to be correlated to the corresponding component of magnetic flux at the solar surface. We confirm that the sound-speed variations associated with the surface magnetic field are superficial. Temporally averaged results show a significant excess in sound speed around     and latitude of 60°.     

p‐mode power variation with solar atmosphere as observed in the Na D1 and K spectral lines

In this work we investigate p‐mode power variation with solar atmosphere. To this aim, we use THÉMIS observations of the Na D1 (λ 5896 Å) and K (λ 7699 Å) spectral lines. While the formation heights of the K spectral line are essentially located in the photospheric layer, the formation heights of the Na D1 line span a much wider region: from photosphere up to chromosphere. Hence, we had the opportunity to infer p‐mode power variation up to the chromospheric layer. By analyzing power spectra obtained by temporal series at different points of the Na D1 and K spectral lines, we confirm and quantify the increase in p‐mode power towards higher atmospheric layers. Furthermore, the large span in formation heights of the Na D1 line induces a larger enhancement of p‐mode power with solar atmosphere compared to the K spectral line. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling RR Lyrae pulsation: mission (im)possible?

In order to make an in-depth comparison between theory and observations, we analyse the light and velocity curves of various hydrodynamical models simulating RRab stars. The observations are represented by empirical formulae, derived in this and our earlier papers. It is shown that the overwhelming majority of the models tested do not follow the empirical relations regarding the shape of the light curves and the physical parameters. In almost all cases the luminosities predicted from the model light curves are significantly lower than the corresponding model values. The overall discrepancy of the models is an important indication of the limitation of the applicability of the present theoretical light and velocity curves in the determination of the physical parameters of these stars. In transforming the theoretical data to the observed light curves in V colour and in computing the observed radial velocities, it is shown that both bolometric correction and tracing the line-forming regions have considerable effects on the evaluation of the observed quantities. In an effort to resolve the discrepancy between theory and observations, it is suggested that a proper evaluation of the bolometric correction and radial velocity based on complete dynamical atmosphere models may be a useful step in this direction.     

r modes and mutual friction in rapidly rotating superfluid neutron stars

We develop a new perturbative framework for studying the r modes of rotating superfluid neutron stars. Our analysis accounts for the centrifugal deformation of the star, and considers the two-fluid dynamics at linear order in the perturbed velocities. Our main focus is on a simple model system where the total density profile is that of an   n = 1  polytrope. We derive a partially analytic solution for the superfluid analogue of the classical r mode. This solution is used to analyse the relevance of the vortex-mediated mutual friction damping, confirming that this dissipation mechanism is unlikely to suppress the gravitational-wave-driven instability in rapidly spinning superfluid neutron stars. Our calculation of the superfluid r modes is significantly simpler than previous approaches, because it decouples the r mode from all other inertial modes of the system. This leads to the results being clearer, but it also means that we cannot comment on the relevance of potential avoided crossings (and associated 'resonances') that may occur for particular parameter values. Our analysis of the mutual friction damping differs from previous studies in two important ways. First, we incorporate realistic pairing gaps which means that the regions of superfluidity in the star's core vary with temperature. Secondly, we allow the mutual friction parameters to take the whole range of permissible values rather than focusing on a particular mechanism. Thus, we consider not only the weak drag regime, but also the strong drag regime where the fluid dynamics are significantly different.     

Can mass loss and overshooting prevent the excitation of g-modes in blue supergiants?

Thanks to their past history on the main-sequence phase, supergiant massive stars develop a convective shell around the helium core. This intermediate convective zone (ICZ) plays an essential role in governing which g-modes are excited. Indeed, a strong radiative damping occurs in the high-density radiative core but the ICZ acts as a barrier preventing the propagation of some g-modes into the core. These g-modes can thus be excited in supergiant stars by the κ-mechanism in the superficial layers due to the opacity bump of iron, at  log  T = 5.2  . However, massive stars are submitted to various complex phenomena such as rotation, magnetic fields, semiconvection, mass loss, overshooting. Each of these phenomena exerts a significant effect on the evolution and some of them could prevent the onset of the convective zone. We develop a numerical method which allows us to select the reflected, thus the potentially excited, modes only. We study different cases in order to show that mass loss and overshooting, in a large enough amount, reduce the extent of the ICZ and are unfavourable to the excitation of g-modes.     

Solar internal rotation rate and the latitudinal variation of the tachocline

A new set of accurately measured frequencies of solar oscillations is used to infer the rotation rate inside the Sun, as a function of radial distance as well as latitude. We have adopted a regularized least-squares technique with iterative refinement for both 1.5D inversion, using the splitting coefficients, and 2D inversion using individual m splittings. The inferred rotation rate agrees well with earlier estimates showing a shear layer just below the surface and another one around the base of the convection zone. The tachocline or the transition layer where the rotation rate changes from differential rotation in the convection zone to an almost latitudinally independent rotation rate in the radiative interior is studied in detail. No compelling evidence for any latitudinal variation in the position and width of the tachocline is found, although it appears that the tachocline probably shifts to a slightly larger radial distance at higher latitudes and possibly also becomes thicker. However, these variations are within the estimated errors and more accurate data would be needed to make a definitive statement about latitudinal variations.     

Non-adiabatic linear pulsation models for low-mass helium stars

Non-adiabatic linear pulsation models have been calculated for low-mass stars with effective temperatures between 16 000 and 35 000 K, and with surface gravities in the range 3⊙, X =0.00, Z =0.02. It is shown that the Z -bump instability persists to low masses ( M ∼0.4 M_⊙) but is suppressed either by a reduction in metallicity Z or by a selective enhancement of the carbon abundance. An unexpected result is the discovery that Z -bump instability persists at hydrogen abundances X >0.3, although the position of the red edge is sensitive to X . We have found that non-radial pulsations are also excited in the same instability region as radial pulsations.
The implications of these results for individual low-mass helium stars are discussed. It is concluded that Z -bump driven pulsations (radial and/or non-radial) may be excited in some helium-rich subdwarf B stars, representing a possible major extension to the class of variable stars represented by the prototype V652 Her.