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.     

Modeling helioseismic modes in the magnetic atmosphere of the Sun

The pulsation of the solar surface is caused by acoustic waves traveling in the solar interior. Thorough analyses of observational data indicate that these f and p helioseismic oscillation modes are not bounced back completely at the surface but they partially penetrate into the atmosphere. Atmospheric effects and their possible observational application are investigated in one‐dimensional magnetohydrodynamic models. It is found that f and p mode frequencies are shifted of the order of μHz due to the presence of an atmospheric magnetic field. This shift varies with the direction of the wave propagation.Resonant coupling of global helioseismic modes to local Alfvén and slow waves reduce the life time of the global modes. The resulting line width of the frequency line is of the order of nHz, and it also varies with propagation angle. These features enable us to use helioseismic observations in magnetic diagnostics of the lower atmosphere. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The visibility of low‐frequency solar acoustic modes

We make predictions of the detectability of low‐frequency p modes. Estimates of the powers and damping times of these low‐frequency modes are found by extrapolating the observed powers and widths of higher‐frequency modes with large observed signal‐to‐noise ratios. The extrapolations predict that the low‐frequency modes will have small signal‐to‐noise ratios and narrow widths in a frequency‐power spectrum. Monte Carlo simulations were then performed where timeseries containing mode signals and normally distributed Gaussian noise were produced. The mode signals were simulated to have the powers and damping times predicted by the extrapolations. Various statistical tests were then performed on the frequency‐amplitude spectra formed from these timeseries to investigate the fraction of spectra in which the modes could be detected. The results of these simulations were then compared to the number of p‐modes candidates observed in real Sun‐as‐a‐star data at low frequencies. The fraction of simulated spectra in which modes were detected decreases rapidly as the frequency of modes decreases and so the fraction of simulations in which the low‐frequency modes were detected was very small. However, increasing the signal‐to‐noise (S/N) ratio of the low‐frequency modes by a factor of 2 above the extrapolated values led to significantly more detections. Therefore efforts should continue to further improve the quality of solar data that is currently available. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Distribution of magnetic flux on the solar surface and low-degree p-modes

The frequencies of solar p-modes are known to change over the solar cycle. There is also recent evidence that the relation between frequency shift of low-degree modes and magnetic flux or other activity indicators differs between the rising and falling phases of the solar cycle, leading to a hysteresis in such diagrams. We consider the influence of the changing large-scale surface distribution of the magnetic flux on low-degree ( l ≤3) p-mode frequencies. To that end, we use time-dependent models of the magnetic flux distribution and study the ensuing frequency shifts of modes with different order and degree as a function of time. The resulting curves are periodic functions (in simple cases just sine curves) shifted in time by different amounts for the different modes. We show how this may easily lead to hysteresis cycles comparable to those observed. Our models suggest that high-latitude fields are necessary to produce a significant difference in hysteresis between odd- and even-degree modes. Only magnetic field distributions within a small parameter range are consistent with the observations by Jiménez-Reyes et al. Observations of p-mode frequency shifts are therefore capable of providing an additional diagnostic of the magnetic field near the solar poles. The magnetic distribution that is consistent with the p-mode observations also appears reasonable compared with direct measurements of the magnetic field.     

The phase function for stellar acoustic oscillations — IV. Solar-like stars

In recent years there has been some progress towards detecting solar-like oscillations in stars. The goal of this challenging project is to analyse frequency spectra similar to that observed for the Sun in integrated light. In this context it is important to investigate what can be learned about the structure and evolution of the stars from such future observations. Here we concentrate on the structure of the upper layers, as reflected in the phase function. We show that it is possible to obtain this function from low-degree p modes, at least for stars on the main sequence. We analyse its dependence on several uncertainties in the structure of the uppermost layers. We also investigate a filtered phase function, which has properties that depend on the layers around the second helium ionization zone.     

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.     

The influence of turbulence on the solar p-mode frequencies I. Free-mode approximation

The influence of turbulence on the frequencies of free acoustic modes in convection zones is considered. The frequencies are modified via the speed of sound by the turbulence-induced alterations of the effective pressure: (i) by the correlated fluctuations of temperature and density and (ii) the pressure part of the Reynolds stress. The two effects shift the frequency of low l p-modes in opposite directions. In addition, the correlation of the density fluctuations with the random velocity — the eddy-mass flow — is also relevant. It is, in a steady state, balanced by a vertical mean velocity. The balance results in a rather small net effect completely disappearing for highly nonradial oscillations. Both effects of the density fluctuations produce a redshift of the low l p-mode frequencies. The Reynolds stress, however, makes a blueshift of the frequencies relative to that computed for a laminar gas. This effect dominates for subsonic turbulences. The applied second-order correlation-approximation, however, only holds for the lowest frequencies, where the KORONAS (solar minimum) data are indicating a blueshift. Of particular importance for the present concept is the expected cycle-variations of the lineshifts, i.e. the consideration of the magnetic modification of the various contributions. Observations may show whether the suggested modifications of the solar oscillation theory are correct.     

Modelling solar atmospheric gravity oscillation modes

Solar oscillations are investigated in a one‐dimensional hydrodynamic plane‐parallel model with an atmosphere. Besides the acoustic pressure (p) modes, the fundamental (f) and Lamb mode, another set of eigenmodes, a group of atmospheric gravity (g) modes, is found in the low‐frequency region of the spectrum. Their frequencies and spatial behaviour are studied. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Imaging bright-spots in the accretion flow near the black hole horizon of Sgr A*

Solar active regions are distinguished by their strong magnetic fields. Modern local helioseismology seeks to probe them by observing waves which emerge at the solar surface having passed through their interiors. We address the question of how an acoustic wave from below is partially converted to magnetic waves as it passes through a vertical magnetic field layer where the sound and Alfvén speeds coincide (the equipartition level), and find that (i) there is no associated reflection at this depth, either acoustic or magnetic, only transmission and conversion to an ongoing magnetic wave; and (ii) conversion in active regions is likely to be strong, though not total, at frequencies typically used in local helioseismology, with lower frequencies less strongly converted. A simple analytical formula is presented for the acoustic-to-magnetic conversion coefficient.     

Effect of Gravity on the Fast Modes of a Horizontal Coronal Slab

Theoretical studies of the normal modes of a coronal slab often neglect gravity, as in Edwin and Roberts (Solar Phys. 71, 239, 1982). Here we study analytically the effect of gravity acting on a horizontal slab as a first step away from a homogeneous medium. Because of the inclusion of gravity, the symmetry of a homogeneous slab is lost, so the normal modes cannot be classified into kink and sausage modes. The presence of gravity also modifies the oscillatory frequencies of the slab, as well as the lower cutoff frequency, resulting in the possible transition between surface and body modes. For general coronal parameters, the dimensionless gravity term turns out to be small, so these effects are also small. A.J. Díaz’s current affiliation: Universitat de les Illes Balears, Palma, E-07122, Spain.     

The Effect of Abnormal Granulation on Acoustic Wave Travel Times and Mode Frequencies

Observations indicate that in plage areas (i.e. in active regions outside sunspots) acoustic waves travel faster than in the quiet Sun, leading to shortened travel times and higher p-mode frequencies. Coupled with the 11-year variation of solar activity, this may also explain the solar cycle variation of oscillation frequencies. While it is clear that the ultimate cause of any difference between the quiet Sun and plage is the presence of magnetic fields of order 100 G in the latter, the mechanism by which the magnetic field exerts its influence has not yet been conclusively identified. One possible such mechanism is suggested by the observation that granular motions in plage areas tend to be slightly “abnormal”, dampened compared to the quiet Sun. In this paper we consider the effect that abnormal granulation observed in active regions should have on the propagation of acoustic waves. Any such effect is found to be limited to a shallow surface layer where sound waves propagate nearly vertically. The magnetically suppressed turbulence implies higher sound speeds, leading to shorter travel times. This time shift Δ τ is independent of the travel distance, while it shows a characteristic dependence on the assumed plage field strength. As a consequence of the variation of the acoustic cutoff with height, Δ τ is expected to be significantly higher for higher frequency waves within the observed regime of 3 – 5 mHz. The lower group velocity near the upper reflection point further leads to an increased envelope time shift, as compared to the phase shift. p-mode frequencies in plage areas are increased by a corresponding amount, Δ ν/ν=ν Δ τ. These characteristics of the time and frequency shifts are in accordance with observations. The calculated overall amplitudes of the time and frequency shifts are comparable to, but still significantly less than (by a factor of 2 to 5), those suggested by measurements.     

Use of genetic algorithms to determine low-l rotational p-mode splittings at high frequencies

We present a comparative study of genetic and standard fitting routines applied to the task of extracting reliable estimates of the rotational splitting of full-disc, low-angular-degree (low- l ) solar p-mode data at high frequencies. 100 artificial proxies of a 10-yr data base of observations made by the Birmingham Solar-Oscillations Network (BiSON) were used to test the two approaches. All sets were analysed over the frequency range from 3000 to 4000 μHz.
Previous work, based on non-linear 'hill-climbing' fitting techniques, has demonstrated the unfortunate tendency for full-disc estimates of the splitting to overestimate the true, underlying values at high frequencies. Here, we show that the resulting bias is less severe when a genetic-fitting approach is adopted. This is largely the result of the number of erroneous 'null-valued' estimates of the splitting being considerably reduced: these estimates are, in effect, re-introduced into the expected normal distribution of fitted splittings. We also illustrate the diverse control one has when using a genetic algorithm as a fitting routine; this diversity is shown to allow further refinement in the estimate of the rotational splitting.
Finally, we address the issue of the reliability of the formal splitting uncertainties returned by the mode fitting, and find that complications arising from the strong anticorrelation between the splittings and their error estimates are not alleviated by the use of the genetic technique.     

Solar p modes of high degree l: coupling by differential rotation

Accurate measurements of solar p-mode frequencies and frequency splittings at high degree l require an adequate theoretical knowledge of the effects of mode coupling, induced by the variation with latitude of the angular velocity of the solar internal rotation. Earlier results for expansion coefficients of composite solutions (coupling coefficients) are due to Woodard. In this paper, the analysis is extended to allow for the dependence of the differential rotation on depth, and the result is expressed in terms of measurable quantities (the rotational splitting coefficients), which makes it convenient for diagnostic purposes. The analysis is based on the approach of quasi-degenerate perturbation theory, and is extended further to address possible effects of mode coupling in the observational line profiles. It is shown, using approximations applicable at high degree l , that the expected line profiles of composite modes in the observational power spectra are not distorted by mode coupling.     

Detection and temporal coherence of p‐modes below 1.4 mHz

Data collected recently by the helioseismic experiments aboard the SOHO spacecraft have allowed the detection of low degree p‐modes with increasingly lower order n. In particular, the GOLF experiment is currently able to unambiguously identify low degree modes with frequencies as low as 1.3 mHz. The detection of p‐modes with very low frequency (i.e., low n), is difficult due to the low signal‐to‐noise ratio in this spectral region and its contamination by solar signals that are not of acoustic origin. To address this problem without using any theoretical a priory, we propose a methodology that relies only on the inversion of observed values to define a spectral window for the expected locations of these low frequency modes. The application of this method to 2920‐day‐long GOLF observations is presented and its results discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Seismic signatures of strange stars with crust

We study acoustic oscillations (eigenfrequencies, velocity distributions, damping times) of normal crusts of strange stars. These oscillations are very specific because of huge density jump at the interface between the normal crust and the strange matter core. The oscillation problem is shown to be self-similar. For a low (but non-zero) multipolarity l , the fundamental mode (without radial nodes) has a frequency of ∼300 Hz and mostly horizontal oscillation velocity; other pressure modes have frequencies ≳20 kHz and almost radial oscillation velocities. The latter modes are similar to radial oscillations (having approximately the same frequencies and radial velocity profiles). The oscillation spectrum of strange stars with crust differs from the spectrum of neutron stars. If detected, acoustic oscillations would allow one to discriminate between strange stars with crust and neutron stars and constrain the mass and radius of the star.     

On helioseismic tests of basic physics

An important goal of helioseismology is to provide information about the basic physics and parameters that determine the structure of the solar interior. Here we discuss the procedures applied in such analyses, using as an example attempts to obtain significant constraints on the value of Newton's gravitational constant G from helioseismology. The analysis is based on complete direct and inverse helioseismic analysis of a set of accurate observed acoustic frequencies. We confirm, as found by previous investigations based on different approaches, that the actual level of precision of the helioseismic inferences does not allow us to constrain G with a precision better than that which can be reached with direct experimental measurements. The conclusion emphasizes the importance in helioseismic inferences of considering not only the accuracy with which solar oscillations are measured, but also the effect of uncertainties in other aspects of the model computation and helioseismic analysis.