On measuring low-degree p-mode frequency splitting with full-disc integrated data

The standard method of measuring rotational splitting from solar full-disc oscillation data, based on maximum-likelihood fitting of multi-Lorentzian profiles to oscillation power spectra, systematically overestimates the splitting. One of the reasons is that the maximum likelihood estimators (MLE) become unbiased only asymptotically as the number of data tends to infinity; for a finite data set they are often biased, inducing a systematic error. In this paper we assess by Monte Carlo simulations the amount of systematic error in the splitting measurement, using artificially generated power spectra. The simulations are carried out for multiplets of degree     2 and 3 with various signal-to-noise ratios, linewidths and observing times. We address the possible use of non-MLE estimators that could provide a smaller or negligible systematic error. The implication for asteroseismology is also discussed.     

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.     

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.     

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.     

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)     

Rotation speed and stellar axis inclination from p modes: how CoRoT would see other suns

In the context of future space-based asteroseismic missions, we have studied the problem of extracting the rotation speed and the rotation-axis inclination of solar-like stars from the expected data. We have focused on slow rotators (at most twice solar rotation speed), first, because they constitute the most difficult case and, secondly, because some of the Convection Rotation and planetary Transits ( CoRoT ) main targets are expected to have slow rotation rates. Our study of the likelihood function has shown a correlation between the estimates of inclination of the rotation axis i and the rotational splitting δν of the star. By using the parameters, i and  δν^⋆=δν sin  i   , we propose and discuss new fitting strategies. Monte Carlo simulations have shown that we can extract a mean splitting and the rotation-axis inclination down to solar rotation rates. However, at the solar rotation rate we are not able to correctly recover the angle i , although we are still able to measure a correct  δν^⋆  with a dispersion less than 40 nHz.     

On the effect of error correlation on linear inversions

We have examined the effect on linear helioseismic inversions of correlations in data errors, taking an example from one-dimensional rotational splitting inversion. Artificial data with correlated errors were generated and then inverted with or without using the proper covariance matrix. The effects of using incorrect covariance matrices, on solutions as well as on trade-offs, are discussed. It is found that improper account of the correlations can be deleterious to the faithfulness of the inversions, and yields incorrect error estimates, which under some circumstances can lead to misleading inferences.     

Effects of a gap-filling method on p-mode parameters

The quality of helioseismological ground-based data strongly depends on the presence of a gap in the observational window. In order to address that problem in the case of full-disc low-degree p-mode velocity measurements, Fossat et al. proposed a gap-filling method called 'Repetitive music'. The autocorrelation function of the velocity signal shows a correlation of more than 70 per cent at about 4 h because of the quasi-periodicity of p-mode peaks in the Fourier spectrum. The method then consists of filling gaps of the velocity signal with data, when they exist, located 4 h before or after.
By using Monte Carlo simulations, we assess the effects of the gap-filling method on p-mode parameters and their errors. A way to remove the modulation in the power spectrum resulting from the gap-filling method is proposed; its effects on p-mode frequencies, linewidths, amplitudes and asymmetries are discussed as a function of frequency and signal-to-noise ratio.     

Rotation of the solar core from BiSON and LOWL frequency observations

Determination of the rotation of the solar core requires very accurate data on splittings for the low-degree modes which penetrate to the core, as well as for modes of higher degree to suppress the contributions from the rest of the Sun to the splittings of the low-degree modes. Here we combine low-degree data based on 32 months of observations with the BiSON network and data from the LOWL instrument. The data are analysed with a technique that specifically aims at obtaining an inference of rotation that is localized to the core. Our analysis provides what we believe is the most stringent constraint to date on the rotation of the deep solar interior.     

δScuti型变星BDS 1269 A的多重脉动频率激发

以同一参考系归化Percy及Rucinski在1976年10月9日至11月6日获得的BDS1269A的光电测光数据后,利用多重周期分析方法进行周期分析,在周期初值的选取中尽可能穷尽所有可能的组合,最后根据最小残差获得最佳组合.我们认为BDS 1269 A的最佳频率解为:f_1=6.22999周每天,f_2=6.50840周每天,f_3=11.77324周每天,f_4=9.7136周每天,f_5=11.33694周每天.反证实验证明这组频率是充分的.基于以上频率解,一个非径向脉动目转分裂模型得以建立,其预言的自转速度。v_6=30km s~(-1)与BDS 1269A是一颗低速自转恒星的观测结果相吻合. 1987年10月16日至24日,我们取得了BDS1269 A的459个新的V波段光电观测点.上述频率解拟合此数据结果很好,拟合残差与观测误差一致.频率解得到了一次验证.     

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.     

Measurements of Rotational Frequency Splitting of Low Angular Degree Modes

The rotational splitting of low-degree p modes is still a controversial issue. There are small but nevertheless real discrepancies between the different measurements of splittings obtained with the existing helioseismology experiments from ground (BISON, IRIS, GONG, LOWL) or from space (VIRGO, GOLF, MDI). I review the current status of rotational splitting in the field of low-degree helioseismology and how we could explain the remaining discrepancies between the various sets of splittings.     

A modified peak‐bagging technique for fitting low‐𝓁 solar p‐modes

We introduce a modified version of a standard power spectrum ‘peak‐bagging’ technique which is designed to gain some of the advantages that fitting the entire low‐degree p‐mode power spectrum simultaneously would bring, but without the problems involved in fitting a model incorporating many hundreds of parameters. Employing Monte‐Carlo simulations we show that by using this modified fitting code it is possible to determine the true background level in the vicinity of the p‐mode peaks. In addition to this we show how small biases in other mode parameters, which are related to inaccurate estimates of the true background, are also consequently removed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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°.     

Wavelet Cleaning of Solar Dynamic Radio Spectrograms

By applying the state-of-the-art mathematical apparatus, the wavelet transformation, we explore the possibillity of a dynamic cleaning of raw data obtained with the Chinese solar radio spectrographs over a wide wavelength range (from 0.7 to 7.6 GHz). We consider the problem of eliminating the interference caused by combination rates of data sampling (10-20ms), and the low-frequency interference (4-30s) caused by the receiving equipment changing its characteristics with time. It is shown that the best choice to reconstruct a signal suffering from amplitude, frequency and phase instabilities, is by means of wavelet transformation at both high and low frequencies. We analysed observational data which contained interferences of nonsolar origin such as instrumental effects and other man-made signals. A subsequent comparison of the reference data obtained with the acoustooptical receiver of the Siberian Solar Radio Telescope (SSRT) with the “cleaned” spectra confirms the correctness of this approach.     

Effect of stellar rotation on oscillation frequencies

We investigate whether the rotational splittings of β Cephei stars can give some clue about the existence of a differential rotation in latitude, and if they are contaminated by the cubic order effects of rotation on oscillation frequencies. We also study some properties of splitting asymmetries and axisymmetric mode frequencies which provide seismic constrains on the distortion of the star. We find that only non-perturbative methods are able to reproduce those two seismic characteristics within 0.01% error bars for stars when they rotate faster than 3.3%Ω _k . If error bars of 1% are acceptable, the threshold of validity of perturbative methods is extended to 10%Ω _k .     

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.     

YAOPBM – yet another peak bagging method

I present and discuss the fitting methodology I developed for very‐long time series (2088‐day‐long). This new method was first used to fit low degree modes, 𝓁 ≤ 25. That time series was also sub‐divided in somewhat shorter segments (728‐daylong) and also fitted for these low degrees, in order to measure changes with the solar activity level. I have recently extended the fitting in several “directions”: 1) to substantially higher degrees (𝓁 ≤ 125), 2) to shorter time series (364‐ and 182‐day‐long), and, 3) to additional 728‐day‐long segments, covering now some 10 years of observations. I present and discuss issues related to this expansion, namely problems at low frequencies affecting the f and p₁ modes, and the inadequacy of the leakage matrix at higher degrees. I also present some of the characteristics of the observed temporal changes in the resulting frequencies. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solar Radius Determination from Total Solar Eclipse Observations on 29 March 2006

Solar diameter measurements have been made nearly continuously through different techniques for more than three centuries. They were obtained mainly with ground-based instruments except for some recent estimates deduced from space observations. One of the main problems in such space data analysis is that, up to now, it has been difficult to obtain an absolute value owing to the absence of an internally calibrated system. Eclipse observations provide a unique opportunity to give an absolute angular scale to the measurements, leading to an absolute value of the solar diameter. However, the problem is complicated by the Moon limb, which presents asphericity because of the mountains. We present a determination of the solar diameter derived from the total solar eclipse observation in Turkey and Egypt on 29 March 2006. We found that the solar radius carried back to 1 AU was 959.22±0.04 arcsec at the time of the observations. The inspection of the compiled 19 modern eclipses data, with solar activity, shows that the radius changes are nonhomologous, an effect that may explain the discrepancies found in ground-based measurements and implies the role of the shallow subsurface layers (leptocline) of the Sun.