T. Sekii

Helioseismic Studies of Differential Rotation in the Solar Envelope by the Solar Oscillations Investigation Using the Michelson Doppler Imager

The splitting of the frequencies of the global resonant acoustic modes of the Sun by large-scale flows and rotation permits study of the variation of angular velocity Ω with both radius and latitude within the turbulent convection zone and the deeper radiative interior. The nearly uninterrupted Doppler imaging observations, provided by the Solar Oscillations Investigation (SOI) using the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO) spacecraft positioned at the L1 Lagrangian point in continuous sunlight, yield oscillation power spectra with very high signal-to-noise ratios that allow frequency splittings to be determined with exceptional accuracy. This paper reports on joint helioseismic analyses of solar rotation in the convection zone and in the outer part of the radiative core. Inversions have been obtained for a medium-l mode set (involving modes of angular degree l extending to about 250) obtained from the first 144 day interval of SOI-MDI observations in 1996. Drawing inferences about the solar internal rotation from the splitting data is a subtle process. By applying more than one inversion technique to the data, we get some indication of what are the more robust and less robust features of our inversion solutions. Here we have used seven different inversion methods. To test the reliability and sensitivity of these methods, we have performed a set of controlled experiments utilizing artificial data. This gives us some confidence in the inferences we can draw from the real solar data. The inversions of SOI-MDI data have confirmed that the decrease of Ω with latitude seen at the surface extends with little radial variation through much of the convection zone, at the base of which is an adjustment layer, called the tachocline, leading to nearly uniform rotation deeper in the radiative interior. A prominent rotational shearing layer in which Ω increases just below the surface is discernible at low to mid latitudes. Using the new data, we have also been able to study the solar rotation closer to the poles than has been achieved in previous investigations. The data have revealed that the angular velocity is distinctly lower at high latitudes than the values previously extrapolated from measurements at lower latitudes based on surface Doppler observations and helioseismology. Furthermore, we have found some evidence near latitudes of 75° of a submerged polar jet which is rotating more rapidly than its immediate surroundings. Superposed on the relatively smooth latitudinal variation in Ω are alternating zonal bands of slightly faster and slower rotation, each extending some 10° to 15° in latitude. These relatively weak banded flows have been followed by inversion to a depth of about 5% of the solar radius and appear to coincide with the evolving pattern of torsional oscillations reported from earlier surface Doppler studies.

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.

Observational Upper Limits to Low-Degree Solar g-Modes

Observations made by the Michelson Doppler Imager (MDI) and Variability of solar IRradiance and Gravity Oscillations (VIRGO) on the Solar and Heliospheric Observatory (SOHO) and by the ground-based Birmingham Solar Oscillations Network (BiSON) and Global Oscillations Network Group (GONG) have been used in a concerted effort to search for solar gravity oscillations. All spectra are dominated by solar noise in the frequency region from 100 to 1000 μHz, where g-modes are expected to be found. Several methods have been used in an effort to extract any g-mode signal present. These include (1) the correlation of data—both full-disk and imaged (with different spatial-mask properties)—collected over different time intervals from the same instrument, (2) the correlation of near-contemporaneous data from different instruments, and (3) the extraction—through the application of complex filtering techniques—of the coherent part of data collected at different heights in the solar atmosphere. The detection limit is set by the loss of coherence caused by the temporal evolution and the motion (e.g., rotation) of superficial structures. Although we cannot identify any g-mode signature, we have nevertheless set a firm upper limit to the amplitudes of the modes: at 200 μHz, they are below 10 mm s-1 in velocity, and below 0.5 parts per million in intensity. The velocity limit corresponds very approximately to a peak-to-peak vertical displacement of δR/R☉ = 2.3 × 10-8 at the solar surface. These levels which are much lower than prior claims, are consistent with theoretical predictions.

Prospects for Measuring Differential Rotation in White Dwarfs through Asteroseismology

We examine the potential of asteroseismology for exploring the internal rotation of white dwarf stars. Data from global observing campaigns have revealed a wealth of frequencies, some of which show the signature of rotational splitting. Tools developed for helioseismology to use many solar p-mode frequencies for inversion of the rotation rate with depth are adapted to the case of more limited numbers of modes of low degree. We find that the small number of available modes in white dwarfs, coupled with the similarity between the rotational-splitting kernels of the modes, renders direct inversion unstable. Accordingly, we adopt what we consider to be plausible functional forms for the differential rotation profile; this is sufficiently restrictive to enable us to carry out a useful calibration. We show examples of this technique for PG 1159 stars and pulsating DB white dwarfs. Published frequency splittings for white dwarfs are currently not accurate enough for meaningful inversions; reanalysis of existing data can provide splittings of sufficient accuracy when the frequencies of individual peaks are extracted via least-squares fitting or multipeak decompositions. We find that, when mode trapping is evident in the period spacing of g modes, the measured splittings can constrain dΩ/dr.

FIRST RESULTS FROM VIRGO, THE EXPERIMENT FOR HELIOSEISMOLOGY AND SOLAR IRRADIANCE MONITORING ON SOHO

First results from the VIRGO experiment (Variability of solar IRradiance and Gravity Oscillations) on the ESA/NASA Mission SOHO (Solar and Heliospheric Observatory) are reported. The observations started mid-January 1996 for the radiometers and sunphotometers and near the end of March for the luminosity oscillation imager. The performance of all the instruments is very good, and the time series of the first 4 - 6 months are evaluated in terms of solar irradiance variability, solar background noise characteristics and -mode oscillations. The solar irradiance is modulated by the passage of active regions across the disk, but not all of the modulation is straightforwardly explained in terms of sunspot flux blocking and facular enhancement. Helioseismic inversions of the observed -mode frequencies are more-or-less in agreement with the latest standard solar models. The comparison of VIRGO results with earlier ones shows evidence that magnetic activity plays a significant role in the dynamics of the oscillations beyond its modulation of the resonant frequencies. Moreover, by comparing the amplitudes of different components of -mode multiplets, each of which are influenced differently by spatial inhomogeneity, we have found that activity enhances excitation.

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.

Phase inversion of one-dimensional artificial seismic data

Oscillations of an inhomogeneous one-dimensional loop have been simulated for the purpose of examining the effect of excitation and damping on the sound-speed inversion based on phase analysis. It has been demonstrated that the procedure is robust against the realization noise arising from frequent, stochastic excitation of weakly damped waves, but that strong damping can spoil the inversion.