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.     

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.     

Analysis of MDI High-Degree Mode Frequencies and their Rotational Splittings

We present a detailed analysis of solar acoustic mode frequencies and their rotational splittings for modes with degree up to 900. They were obtained by applying spherical harmonic decomposition to full-disk solar images observed by the Michelson Doppler Imager onboard the Solar and Heliospheric Observatory spacecraft. Global helioseismology analysis of high-degree modes is complicated by the fact that the individual modes cannot be isolated, which has limited so far the use of high-degree data for structure inversion of the near-surface layers (r>0.97R _⊙). In this work, we took great care to recover the actual mode characteristics using a physically motivated model which included a complete leakage matrix. We included in our analysis the following instrumental characteristics: the correct instantaneous image scale, the radial and non-radial image distortions, the effective position angle of the solar rotation axis, and a correction to the Carrington elements. We also present variations of the mode frequencies caused by the solar activity cycle. We have analyzed seven observational periods from 1999 to 2005 and correlated their frequency shift with four different solar indices. The frequency shift scaled by the relative mode inertia is a function of frequency alone and follows a simple power law, where the exponent obtained for the p modes is twice the value obtained for the f modes. The different solar indices present the same result.     

Internal rotation of the Sun as inferred from GONG observations

Helioseismology is a direct and most informative method of studying the structure and dynamics of the Sun. Determining the internal differential rotation of the Sun requires that the frequencies of its eigentones be estimated with a high accuracy, which is possible only on the basis of continuous long-term observations. The longest quasi-continuous series of data have been obtained by the Global Oscillation Network Group (GONG). The parameters of each individual mode of solar acoustic oscillations with low spherical degrees l=0, 1, 2, 3, 4, 5, 6 are determined by using 1260-day-long series of GONG observations. The mean frequency splitting by rotation for the modes of each radial order n is calculated by using all possible combinations between the eigenfrequencies in multiplets. As a result, it has become possible to statistically estimate the splitting and its measurement errors for the modes of each radial order. The mean splitting for each given degree l=1–6 is presented under the assumption of its independence of oscillation frequency, which holds for the achieved accuracy. The frequencies and splittings for the modes with low spherical degrees l, together with the MDI group results for higher degrees l, are used to invert the radial profile of solar angular velocity. Using the SOLA method to solve the inverse problem of restoring the rotation profile has yielded solutions sensitive to the deepest stellar interiors. Our results indicate that the solar core rotates faster than the surface, and there may be a local minimum in angular velocity at its boundary.     

Propagating Linear Waves in Convectively Unstable Stellar Models: A Perturbative Approach

Linear time-domain simulations of acoustic oscillations are unstable in the stellar convection zone. To overcome this problem it is customary to compute the oscillations of a stabilized background stellar model. The stabilization affects the result, however. Here we propose to use a perturbative approach (running the simulation twice) to approximately recover the acoustic wave field while preserving seismic reciprocity. To test the method we considered a 1D standard solar model. We found that the mode frequencies of the (unstable) standard solar model are well approximated by the perturbative approach within 1 μHz for low-degree modes with frequencies near 3 mHz. We also show that the perturbative approach is appropriate for correcting rotational-frequency kernels. Finally, we comment that the method can be generalized to wave propagation in 3D magnetized stellar interiors because the magnetic fields have stabilizing effects on convection.     

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)     

Variability of Solar Five-Minute Oscillations in the Corona as Observed by the Extreme Ultraviolet Spectrophotometer (ESP) on the Solar Dynamics Observatory/Extreme Ultraviolet Variability Experiment (SDO/EVE)

Solar five-minute oscillations have been detected in the power spectra of two six-day time intervals from soft X-ray measurements of the Sun observed as a star using the Extreme Ultraviolet Spectrophotometer (ESP) onboard the Solar Dynamics Observatory (SDO)/Extreme Ultraviolet Variability Experiment (EVE). The frequencies of the largest amplitude peaks were found to match the known low-degree (?=0?–?3) modes of global acoustic oscillations within 3.7 μHz and can be explained by a leakage of the global modes into the corona. Due to the strong variability of the solar atmosphere between the photosphere and the corona, the frequencies and amplitudes of the coronal oscillations are likely to vary with time. We investigated the variations in the power spectra for individual days and their association with changes of solar activity, e.g. with the mean level of the EUV irradiance, and its short-term variations caused by evolving active regions. Our analysis of samples of one-day oscillation power spectra for a 49-day period of low and intermediate solar activity showed little correlation with the mean EUV irradiance and the short-term variability of the irradiance. We suggest that some other changes in the solar atmosphere, e.g., magnetic fields and/or inter-network configuration may affect the mode leakage to the corona.     

No Evidence Supporting Flare-Driven High-Frequency Global Oscillations

The underlying physics that generates the excitations in the global low-frequency (<?5.3?mHz) solar acoustic power spectrum is a well-known process that is attributed to solar convection; however, a definitive explanation as to what causes excitations in the high-frequency regime (>?5.3?mHz) has yet to be found. Karoff and Kjeldsen (Astrophys. J. 678, 73??C?76, 2008) concluded that there is a correlation between solar flares and the global high-frequency solar acoustic waves. We have used Global Oscillation Network Group (GONG) helioseismic data in an attempt to verify the Karoff and Kjeldsen (2008) results as well as compare the post-flare acoustic power spectrum to the pre-flare acoustic power spectrum for 31 solar flares. Among the 31 flares analyzed, we observe that a decrease in acoustic power after the solar flare is just as likely as an increase. Furthermore, while we do observe variations in acoustic power that are most likely associated with the usual p-modes associated with solar convection, these variations do not show any significant temporal association with flares. We find no evidence that consistently supports flare-driven high-frequency waves.     

Optimal Masks for Low-Degree Solar Acoustic Modes

We suggest a solution to an important problem in observational helioseismology of the separation of lines of solar acoustic (p) modes of low angular degree in oscillation power spectra by constructing optimal masks for Doppler images of the Sun. Accurate measurements of oscillation frequencies of low-degree modes are essential for the determination of the structure and rotation of the solar core. However, these measurements for a particular mode are often affected by leakage of other p-modes arising when the Doppler images are projected on to spherical harmonic masks. The leakage results in overlapping peaks corresponding to different oscillation modes in the power spectra. In this Letter, we present a method for calculating optimal masks for a given (target) mode by minimizing the signals of other modes appearing in its vicinity. We apply this method to time series of 2 yr obtained from the Michelson Doppler Imager instrument on board the Solar and Heliospheric Observatory space mission and demonstrate its ability to reduce efficiently the mode leakage.     

Further Evidence Suggestive of a Solar Influence on Nuclear Decay Rates

Recent analyses of nuclear decay data show evidence of variations suggestive of a solar influence. Analyses of datasets acquired at the Brookhaven National Laboratory (BNL) and at the Physikalisch-Technische Bundesanstalt (PTB) both show evidence of an annual periodicity and of periodicities with sidereal frequencies in the neighborhood of 12.25 year^?1 (at a significance level that we have estimated to be 10^?17). It is notable that this implied rotation rate is lower than that attributed to the solar radiative zone, suggestive of a slowly rotating solar core. This leads us to hypothesize that there may be an ??inner tachocline?? separating the core from the radiative zone, analogous to the ??outer tachocline?? that separates the radiative zone from the convection zone. The Rieger periodicity (which has a period of about 154 days, corresponding to a frequency of 2.37 year^?1) may be attributed to an r-mode oscillation with spherical-harmonic indices l=3,m=1, located in the outer tachocline. This suggests that we may test the hypothesis of a solar influence on nuclear decay rates by searching BNL and PTB data for evidence of a ??Rieger-like?? r-mode oscillation, with l=3,m=1, in the inner tachocline. The appropriate search band for such an oscillation is estimated to be 2.00??C?2.28 year^?1. We find, in both datasets, strong evidence of a periodicity at 2.11 year^?1. We estimate that the probability of obtaining these results by chance is 10^?12.     

Uncovering the Bias in Low-Degree p-Mode Linewidth Fitting

Obtaining reliable estimates of linewidths in the power spectra of low-degree p modes is problematic at low frequency. In this regime, the mode coherence time increases with decreasing frequency, often causing the modes to be unresolved in relatively long duration spectra. The signal-to-noise ratio is also less favourable at low frequency, resulting in fits to power spectra underestimating the true linewidth of the p modes owing to the tails of the Lorentzian peaks becoming dominated by the background noise. We use a numerical simulation approach to assess the effect of this bias on the fitted widths of p-mode peaks and calculate observational duration limits required to obtain an unbiased estimate of the p-mode linewidth as a function of frequency. This is done in four different cases, where the precision of the artificial data is set at 0.25, 0.50, 0.75, and 1.00 m?s^?1 by adding random?scatter to increase the sample standard deviation per 40-second measurement. In all cases, the observational duration required to accurately obtain width estimates increases beyond that required for sufficient spectral resolution below a certain threshold frequency. For modes at ≈?1500 μHz, with an amplitude of approximately ten times the background, observations of up to 972 days are required to obtain an unbiased estimate of the linewidth. This is equivalent to ≈?18 times the coherence time of the corresponding p modes.     

Global Effects of Local Sound-Speed Perturbations in the Sun: A Theoretical Study

We study the effect of localized sound-speed perturbations on global mode frequencies by applying techniques of global helioseismology to numerical simulations of the solar acoustic wave field. Extending the method of realization-noise subtraction (e.g., Hanasoge, Duvall, and Couvidat, Astrophys. J. 664, 1234, 2007) to global modes and exploiting the luxury of full spherical coverage, we are able to achieve very highly resolved frequency differences that are then used to study sensitivities and the signatures of the thermal asphericities. We find that i) global modes are almost twice as sensitive to sound-speed perturbations at the bottom of the convection zone in comparison to anomalies well inside the radiative interior (r?0.55R _⊙), ii) the m degeneracy is lifted ever so slightly, as seen in the a coefficients, and iii) modes that propagate in the vicinity of the perturbations show small amplitude shifts. Through comparisons with error estimates obtained from Michelson Doppler Imager (MDI; Scherrer et al., Solar Phys. 162, 129, 1995) observations, we find that the frequency differences are detectable with a sufficiently long time series (70?–?642 days).     

Kinematic Properties of Slow ICMEs and an Interpretation of a Modified Drag Equation for Fast and Moderate ICMEs

We report kinematic properties of slow interplanetary coronal mass ejections (ICMEs) identified by SOHO/LASCO, interplanetary scintillation, and in situ observations and propose a modified equation for the ICME motion. We identified seven ICMEs between 2010 and 2011 and compared them with 39 events reported in our previous work. We examined 15 fast (V _SOHO?V _bg>500 km?s^?1), 25 moderate (0 km?s^?1≤V _SOHO?V _bg≤500 km?s^?1), and 6 slow (V _SOHO?V _bg<0 km?s^?1) ICMEs, where V _SOHO and V _bg are the initial speed of ICMEs and the speed of the background solar wind. For slow ICMEs, we found the following results: i) They accelerate toward the speed of the background solar wind during their propagation and reach their final speed by 0.34±0.03 AU. ii) The acceleration ends when they reach 479±126 km?s^?1; this is close to the typical speed of the solar wind during the period of this study. iii) When γ ₁ and γ ₂ are assumed to be constants, a quadratic equation for the acceleration a=?γ ₂(V?V _bg)|V?V _bg| is more appropriate than a linear one a=?γ ₁(V?V _bg), where V is the propagation speed of ICMEs, while the latter gives a smaller χ ² value than the former. For the motion of the fast and moderate ICMEs, we found a modified drag equation a=?2.07×10^?12(V?V _bg)|V?V _bg|?4.84×10^?6(V?V _bg). From the viewpoint of fluid dynamics, we interpret this equation as indicating that ICMEs with 0 km?s^?1≤V?V _bg≤2300 km?s^?1 are controlled mainly by the hydrodynamic Stokes drag force, while the aerodynamic drag force is a predominant factor for the propagation of ICME with V?V _bg>2300 km?s^?1.     

Low-Degree Low-Order Solar p Modes As Seen By GOLF On board SOHO

Data recovered from the GOLF experiment on board the ESA/NASA SOHO spacecraft have been used to analyze the low-order low-degree solar velocity acoustic-mode spectrum below =1.5 mHz (i.e., 1n9,l2). Various techniques (periodogram, RLAvCS, homomorphic-deconvolution and RLSCSA) have been used and compared to avoid possible biases due to a given analysis method. In this work, the acoustic resonance modes sensitive to the solar central region are studied. Comparing results from the different analysis techniques, 10 modes below 1.5 mHz have been identified.     

On the MOND external field effect in the solar system

In the framework of the MOdified Newtonian Dynamics (MOND), the internal dynamics of a gravitating system s embedded in a larger one S is affected by the external background field E of S even if it is constant and uniform, thus implying a violation of the Strong Equivalence Principle: it is the so-called External Field Effect (EFE). In the case of the solar system, E would be A _cen≈10^?10 m?s^?2 because of its motion through the Milky Way: it is orders of magnitude smaller than the main Newtonian monopole terms for the planets. We address here the following questions in a purely phenomenological manner: are the Sun’s planets affected by an EFE as large as 10^?10 m?s^?2? Can it be assumed that its effect is negligible for them because of its relatively small size? Does E induce vanishing net orbital effects because of its constancy over typical solar system’s planetary orbital periods? It turns out that a constant and uniform acceleration, treated perturbatively, does induce non-vanishing long-period orbital effects on the longitude of the pericenter ? of a test particle. In the case of the inner planets of the solar system and with E≈10^?10 m?s^?2, they are 4–6 orders of magnitude larger than the present-day upper bounds on the non-standard perihelion precessions \(\Delta\dot{\varpi}\) recently obtained with by E.V. Pitjeva with the EPM ephemerides in the Solar System Barycentric frame. The upper limits on the components of E are E _x ≤1×10^?15 m?s^?2, E _y ≤2×10^?16 m?s^?2, E _z ≤3×10^?14 m?s^?2. This result is in agreement with the violation of the Strong Equivalence Principle by MOND. Our analysis also holds for any other exotic modification of the current laws of gravity yielding a constant and uniform extra-acceleration. If and when other corrections \(\Delta\dot{\varpi}\) to the usual perihelion precessions will be independently estimated with different ephemerides it will be possible to repeat such a test.     

Stability of the Triangular Points Under Combined Effects of Radiation and Oblateness in the Restricted Three-Body Problem

This paper deals with the existence of triangular points and their linear stability when the primaries are oblate spheroid and sources of radiation considering the effect of oblateness up to 10^?6 of main terms in the restricted three-body problem; we see that the locations of the triangular points are affected by the oblateness of the primaries and solar radiation pressure. It is further seen that these points are stable for 0 ≤ μ ≤μ _c ; and unstable for μ _c  ≤ μ ≤1/2; where μ _c is the critical mass value depending on terms which involve parameters that characterize the oblateness and radiation repulsive forces such that $ \mu_{c} \in (0,1/2) $ ; in addition to this an algorithm has been constructed to calculate the critical mass value.     

First View of the Solar Core from Golf Acoustic Modes

After 8 months of nearly continuous measurements the GOLF instrument, aboard SOHO, has detected acoustic mode frequencies of more than 100 modes, extending from 1.4 mHz to 4.9 mHz. In this paper, we compare these results with the best available predictions coming from solar models. To verify the quality of the data, we examine the asymptotic seismic parameters; this confirms the improvements achieved in solar models during the last decade. Using the GOLF set of frequencies for l=0, 1, 2, 3 combined with the LOWL second year data set for l > 3 we then carry out inversions to infer properties of the solar core. This largely confirms the previous results down to around 0.1 R⊙, while there remain differences, even closer to the centre, where the present study shows an extreme sensitivity of the inversion results to the values of the frequencies. We finally consider physical processes which may influence directly or indirectly the solar core structure.     

Radial Speed Evolution of Interplanetary Coronal Mass Ejections During Solar Cycle 23

We report radial-speed evolution of interplanetary coronal mass ejections (ICMEs) detected by the Large Angle and Spectrometric Coronagraph onboard the Solar and Heliospheric Observatory (SOHO/LASCO), interplanetary scintillation (IPS) at 327 MHz, and in-situ observations. We analyze solar-wind disturbance factor (g-value) data derived from IPS observations during 1997?–?2009 covering nearly the whole period of Solar Cycle 23. By comparing observations from SOHO/LASCO, IPS, and in situ, we identify 39 ICMEs that could be analyzed carefully. Here, we define two speeds [V _SOHO and V _bg], which are the initial speed of the ICME and the speed of the background solar wind, respectively. Examinations of these speeds yield the following results: i) Fast ICMEs (with V _SOHO?V _bg>500 km?s^?1) rapidly decelerate, moderate ICMEs (with 0 km?s^?1≤V _SOHO?V _bg≤500 km?s^?1) show either gradually decelerating or uniform motion, and slow ICMEs (with V _SOHO?V _bg<0 km?s^?1) accelerate. The radial speeds converge on the speed of the background solar wind during their outward propagation. We subsequently find; ii) both the acceleration and the deceleration are nearly complete by 0.79±0.04 AU, and those are ended when the ICMEs reach a 480±21 km?s^?1. iii) For ICMEs with (V _SOHO?V _bg)≥0 km?s^?1, i.e. fast and moderate ICMEs, a linear equation a=?γ ₁(V?V _bg) with γ ₁=6.58±0.23×10^?6 s^?1 is more appropriate than a quadratic equation a=?γ ₂(V?V _bg)|V?V _bg| to describe their kinematics, where γ ₁ and γ ₂ are coefficients, and a and V are the acceleration and speed of ICMEs, respectively, because the χ ² for the linear equation satisfies the statistical significance level of 0.05, while the quadratic one does not. These results support the assumption that the radial motion of ICMEs is governed by a drag force due to interaction with the background solar wind. These findings also suggest that ICMEs propagating faster than the background solar wind are controlled mainly by the hydrodynamic Stokes drag.     

STRUCTURE AND ROTATION OF THE SOLAR INTERIOR: INITIAL RESULTS FROM THE MDI MEDIUM-L PROGRAM

The medium-l program of the Michelson Doppler Imager instrument on board SOHO provides continuous observations of oscillation modes of angular degree, l, from 0 to 300. The data for the program are partly processed on board because only about 3% of MDI observations can be transmitted continuously to the ground. The on-board data processing, the main component of which is Gaussian-weighted binning, has been optimized to reduce the negative influence of spatial aliasing of the high-degree oscillation modes. The data processing is completed in a data analysis pipeline at the SOI Stanford Support Center to determine the mean multiplet frequencies and splitting coefficients. The initial results show that the noise in the medium-l oscillation power spectrum is substantially lower than in ground-based measurements. This enables us to detect lower amplitude modes and, thus, to extend the range of measured mode frequencies. This is important for inferring the Sun's internal structure and rotation. The MDI observations also reveal the asymmetry of oscillation spectral lines. The line asymmetries agree with the theory of mode excitation by acoustic sources localized in the upper convective boundary layer. The sound-speed profile inferred from the mean frequencies gives evidence for a sharp variation at the edge of the energy-generating core. The results also confirm the previous finding by the GONG (Gough et al., 1996) that, in a thin layer just beneath the convection zone, helium appears to be less abundant than predicted by theory. Inverting the multiplet frequency splittings from MDI, we detect significant rotational shear in this thin layer. This layer is likely to be the place where the solar dynamo operates. In order to understand how the Sun works, it is extremely important to observe the evolution of this transition layer throughout the 11-year activity cycle.     

First Results on it p Modes from GOLF Experiment

The GOLF experiment on the SOHO mission aims to study the internal structure of the Sun by measuring the spectrum of global oscillations in the frequency range 10^-7 to 10^-2 Hz. Here we present the results of the analysis of the first 8 months of data. Special emphasis is put into the frequency determination of the p modes, as well as the splitting in the multiplets due to rotation. For both, we show that the improvement in S/N level with respect to the ground-based networks and other experiments is essential in achieving a very low-degree frequency table with small errors ∼ 2 parts in 10^-5). On the other hand, the splitting found seems to favour a solar core which does not rotate slower than its surface. The line widths do agree with theoretical expectations and other observations.