Influence of Low-Degree High-Order p-Mode Splittings on the Solar Rotation Profile

The solar rotation profile is well constrained down to about 0.25R _⊙ thanks to the study of acoustic modes. Since the radius of the inner turning point of a resonant acoustic mode is inversely proportional to the ratio of its frequency to its degree, only the low-degree p modes reach the core. The higher the order of these modes, the deeper they penetrate into the Sun and thus they carry more diagnostic information on the inner regions. Unfortunately, the estimates of frequency splittings at high frequency from Sun-as-a-star measurements have higher observational errors because of mode blending, resulting in weaker constraints on the rotation profile in the inner core. Therefore inversions for the solar internal rotation use only modes below 2.4 mHz for ?≤3. In the work presented here, we used an 11.5-year-long time series to compute the rotational frequency splittings for modes ?≤3 using velocities measured with the GOLF instrument. We carried out a theoretical study of the influence of the low-degree modes in the region from 2 to 3.5 mHz on the inferred rotation profile as a function of their error bars.     

Rigid rotation of the solar core? On the reliable extraction of low-ℓ rotational p-mode splittings from full-disc observations of the Sun

We present low-ℓ rotational p-mode splittings from the analysis of 8 yr of observations made by the Birmingham Solar-Oscillations Network (BiSON) of the full solar disc. These data are presented in the light of a thorough investigation of the fitting techniques used to extract them. Particular attention is paid to both the origin and magnitude of bias present in these estimates. An extensive Monte Carlo strategy has been adopted to facilitate this study – in all, several thousand complete, artificial proxies of the 96-month data set have been generated to test the analysis of real 'full-disc' data. These simulations allow for an assessment of any complications in the analysis which might arise from variations in the properties of the p modes over the 11-yr solar activity cycle.
The use of such an extended data set affords greater precision in the splittings, and by implication the rotation rate inferred from these data, and reduces bias inherent in the analysis, thereby giving a more accurate determination of the rotation. The grand, weighted sidereal average of the BiSON set is     , a value consistent with that expected were the deep radiative interior     to rotate at the same frequency, and in the same 'rigid' manner, as the more precisely and accurately studied outer part of the radiative zone.     

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.     

Peak Bagging for Solar-like Stars

The identification of the low-degree p modes in other stars is the challenge of future asteroseismology space missions such as COROT, MONS, MOST or Eddington. The identification is based on a priori knowledge of the characteristics of the modes. We shall review the most common assumptions needed for the identification such as basic stellar structure, visibilities, rotational splittings or linewidths. We shall describe a few tools needed for facilitating the identification. As soon as modes are properly identified, the peakbagging of the mode characteristics can be done using Maximum Likelihood Estimation. We give examples of the whole process using solar data and hare-and-hound exercises performed in the frame work of the COROT project.     

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.     

About the rotation of the solar radiative interior

In the modern era of helioseismology we have a wealth of high-quality data available, e.g., more than 6 years of data collected by the various instruments on board the SOHO mission, and an even more extensive ground-based set of observations covering a full solar cycle. Thanks to this effort a detailed picture of the internal rotation of the Sun has been constructed. In this paper we present some of the actions that should be done to improve our knowledge of the inner rotation profile discussed during the workshop organized at Saclay on June 2003 on this topic. In particular we will concentrate on the extraction of the rotational frequency splittings of low- and medium-degree modes and their influence on the rotation of deeper layers. Furthermore, for the first time a full set of individual |m|-component rotational splittings is computed for modes ℓ≤4 and 1<ν<2 mHz, opening new studies on the latitudinal dependence of the rotation rate in the radiative interior. It will also be shown that these splittings have the footprints of the differential rotation of the convective zone which can be extremely useful to study the differential rotation of other stars where only these low-degree modes will be available.     

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.     

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.     

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.     

Are There Pulsationally Unstable Low-degree p1 Modes in the Sun?

By using a non-local time-dependent theory of stellar convection, the solar non-adiabatic pulsations of the low- and intermediate-degree (l < 25) modes are calculated. The results show that the non-radial p1 modes of l = 1–5 are pulsationally unstable. However, the adjacent g, f, p2-p5 modes and the p1 modes of l > 5 are stable. From the analysis of the diagram of integrated work it is discovered that the excitation of oscillations comes from the radiation zone beneath the convective region. Whether the sun possesses unstable low-degree p1 modes is of signi?cant importance for clarifying the excitation mechanism of solar ?ve-minute oscillations.     

Variations of the low‐degree solar p‐mode frequencies for 10 years of GOLF observations

We experiment with a method of measuring the frequency of solar p modes, intended to extend the passband for the variations of the frequency spectrum as high as possible. So far this passband is limited to a fraction of μ Hz for the classical analysis based on numerical fits of a theoretical line profile to a power spectrum averaged over periods lasting at least several weeks. This limit for the present analysis can be shifted to the mHz range, corresponding to some of the “5 min” oscillations, but in this range we use a lower resolution which allows us to separate odd and even p modes. We show an example of the results for long term variations and apply this analysis to search for a modulation of the p‐mode frequency spectrum by asymptotic series of solar g modes. A faint signal is found in the analysis of 10 years of GOLF data. This very preliminary result possibly indicates the detection of a small number of g modes of degree l = 1. A tentative determination of an observational value of the parameter P₀ follows. P₀ is the scaling factor of the asymptotic series of g modes and is a key data for solar core physics. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the solar rotation and activity

The interaction between differential rotation and magnetic fields in the solar convection zone was recently modelled by Brun (2004). One consequence of that model is that the Maxwell stresses can oppose the Reynolds stresses, and thus contribute to the transport of the angular momentum towards the solar poles, leading to a reduced differential rotation. So, when magnetic fields are weaker, a more pronounced differential rotation can be expected, yielding a higher rotation velocity at low latitudes taken on the average. This hypothesis is consistent with the behaviour of the solar rotation during the Maunder minimum. In this work we search for similar signatures of the relationship between the solar activity and rotation determined tracing sunspot groups and coronal bright points. We use the extended Greenwich data set (1878–1981) and a series of full-disc solar images taken at 28.4 nm with the EIT instrument on the SOHO spacecraft (1998–2000). We investigate the dependence of the solar rotation on the solar activity (described by the relative sunspot number) and the interplanetary magnetic field (calculated from the interdiurnal variability index). Possible rotational signatures of two weak solar activity cycles at the beginning of the 20th century (Gleissberg minimum) are discussed. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

On model predictions of the power spectral density of radial solar p modes

We investigate the frequency dependence of the power spectral density of low-degree solar p modes by comparing measurements with the results of a stochastic-excitation model. In the past it was common practice to use the total power in such investigations. Using the maximum of the power spectral density instead provides a direct comparison with the measured mode heights in the observed power spectrum. This method permits a more careful calibration of the adjustable parameters in the excitation model, a model which we present here, for the first time, in a format that precisely and unambiguously relates the amplitudes of the modes of oscillation to the Reynolds stress in the equilibrium model. We find that errors in the theory of the linear mode damping rates, particularly at low frequency, have a dramatic impact on the predictions of the mode heights in the spectral density, whereas parameter changes in the stochastic excitation model, within a plausible domain of parameter space, have a comparatively small effect.     

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.     

Helioseismic search for magnetic field in the solar interior

The observed splittings of solar oscillation frequencies can be utilized to study possible large-scale magnetic fields present in the solar interior. Using the GONG data on frequency splittings an attempt is made to infer the strength of magnetic fields inside the Sun.     

New aspects of Doppler imaging in Sun-as-a-star observations

Birmingham Solar Oscillations Network (BiSON) instruments use resonant scattering spectrometers to make unresolved Doppler velocity observations of the Sun. Unresolved measurements are not homogenous across the solar disc and so the observed data do not represent a uniform average over the entire surface. The influence on the inhomogeneity of the solar rotation and limb darkening has been considered previously and is well understood. Here, we consider a further effect that originates from the instrumentation itself. The intensity of light observed from a particular region on the solar disc is dependent on the distance between that region on the image of the solar disc formed in the instrument and the detector. The majority of BiSON instruments have two detectors positioned on opposite sides of the image of the solar disc and the observations made by each detector are weighted towards differing regions of the disc. Therefore, the visibility and amplitudes of the solar oscillations and the realization of the solar noise observed by each detector will differ. We find that the modelled bias is sensitive to many different parameters such as the width of solar absorption lines, the strength of the magnetic field in the resonant scattering spectrometer, the orientation of the Sun's rotation axis, the size of the image observed by the instrument and the optical depth in the vapour cell. We find that the modelled results best match the observations when the optical depth at the centre of the vapour cell is 0.55. The inhomogeneous weighting means that a 'velocity offset' is introduced into unresolved Doppler velocity observations of the Sun, which varies with time, and so has an impact on the long-term stability of the observations.