S. V. Vorontsov

The ratio of small to large separations of acoustic oscillations as a diagnostic of the interior of solar-like stars

By considering stellar models with the same interior structure but different outer layers we demonstrate that the ratio of the small to large separations of acoustic oscillations in solar-like stars is essentially independent of the structure of the outer layers, and is determined solely by the interior structure. Defining the scaled Eulerian pressure perturbation ψ� (ω,t) = rp � /(ρc) 1/2 we define the internal phase shift δ� (ω,t) through the relation ωψ/(dψ/dt) = tan(ωt − π�/ 2 + δ� ). The δ� are almost independent of acoustic radius t = � dr/c outside the stellar core and can be determined as a continuous functions of ω from partial wave solutions for the interior - that is solutions of the oscillation equations for any ω that satisfy the Laplace boundary condition at a sufficiently large acoustic radius tf outside the stellar core. If the ω are eigenfrequencies then they satisfy the Eigenfrequency Equation ωT = (n + �/ 2)π + α(ω) − δ� (ω )w hereα(ω )i s theindependent surface phase shift (Roxburgh & Vorontsov 2000). Using this result we show that the ratio of small to large separations is determined to high accuracy solely by the internal phase shifts δ� (ω) and hence by the interior structure alone. The error in this result is estimated and shown to be smaller than that associated with the errors in the determination of the frequencies (≈0.1-0.3 µHz) from the upcoming space missions MOST, COROT and Eddington.

The Seismic Structure of the Sun

Global Oscillation Network Group data reveal that the internal structure of the sun can be well represented by a calibrated standard model. However, immediately beneath the convection zone and at the edge of the energy-generating core, the sound-speed variation is somewhat smoother in the sun than it is in the model. This could be a consequence of chemical inhomogeneity that is too severe in the model, perhaps owing to inaccurate modeling of gravitational settling or to neglected macroscopic motion that may be present in the sun. Accurate knowledge of the suns structure enables inferences to be made about the physics that controls the sun; for example, through the opacity, the equation of state, or wave motion. Those inferences can then be used elsewhere in astrophysics.

On the use of the ratio of small to large separations in asteroseismic model fitting

Aims. We aim to show that model fitting by searching for a best fit of observed and model separation ratios at the same radial orders n is in principle incorrect, and to show that a correct procedure is to compare the model ratios interpolated to the observed frequencies. Methods. We compare models with different interior structures and outer layers, relate the separation ratios to phase shift differences, conduct model fitting experiments using separation ratios, and relate phase shift differences to internal phase shifts. Results. We show that the separation ratios of stellar models with the same interior structure, but different outer layers, are not the same when compared at the same radial order n , but are the same when evaluated at the same frequencies by interpolation. The separation ratios trace the phase shift differences as a function of frequency, not of n , and it is the phase shift differences which are determined by the interior structure. We give examples from model fitting where the ratios at the same n values agree but the models have different interior structure, and where the ratios agree when interpolated to the same frequencies and the models have the same interior structure. The correct procedure is to compare observed ratios with model values interpolated to the observed frequencies.

Digital image deblurring with SOR

We address the numerical performance of the successive overrelaxation technique (SOR) in the restoration of astronomical images. Local analysis of the convergence rates reveals resonant properties, with convergence enhancement at a spatial frequency which is determined by the SOR relaxation parameter τ. The analysis can serve as a guide for the practical choice of the relaxation parameter(s), which governs the regularization properties of the SOR algorithm. One particular prediction is that in typical implementations, fast image deblurring requires deep underrelaxation (τ 1). This theoretical result allows us to better understand the productive properties of underrelaxation, which have been discovered in earlier work. Comparison of SOR in artificial inversions with conjugate gradients and related methods (GMRES) indicates that a solution of similar or better quality may be obtained in a comparable or smaller number of iterations. Restricting the solution with non-negativity constraint (+SOR) enhances both the quality of the solutions and the convergence rate. Theoretical analysis of the convergence properties of +SOR, however, remains a challenge which cannot be addressed by the simple analysis implemented in this paper.

MODELING SOLAR OSCILLATION POWER SPECTRA. I. ADAPTIVE RESPONSE FUNCTION FOR DOPPLER VELOCITY MEASUREMENTS

Improving the accuracy and resolution of helioseismic inversions calls for more accurate modeling of the observational p-mode power spectra from which the solar oscillation frequencies are traditionally measured. We present a new technique of calculating the response function (leakage matrix) for Doppler velocity measurements that is based largely on an analytical description of the relevant instrumental and physical effects. The computational efficiency of the new approach allows us to implement the response function in an adaptive manner: i.e., the compensation for instrumental or optical distortions of unknown magnitude can be performed as a part of the spectral fitting procedure.

A New Way to Model the Solar Oscillation ℓ−ν Power Spectrum

We present a new approach for the precise and accurate forward modeling of the solar oscillation ℓ−ν power spectrum. The approach is designed to provide the basis for a streamlined solar seismic inversion without measurements of the p-mode frequencies. The new strategy represents a paradigm change in how information is extracted from the oscillation spectrum. It also represents a step toward the ideal case of inferring the Suns properties directly from the raw observations.

Spatiotemporal fragmentation and the uncertainties in the solar rotation law

Analyses of recent helioseismic data indicate that the dynamical regimes at the base of the convection zone can be different from those observed at the top, having either significantly shorter periods or non–periodic behaviour. Recently spatiotemporal fragmentation/bifurcation has been proposed as a dynamical mechanism to account for the multi-mode behaviour that is possibly observed in the solar convection zone, without requiring separate physical mechanisms with different time scales at different depths. Here we study the robustness of this mechanism with respect to changes to the zero order rotation profile, motivated by the uncertainties of and differences between the various reductions of the helioseimic data. We find that spatiotemporal fragmentation is a common feature of the reductions investigated.

MODELING SOLAR OSCILLATION POWER SPECTRA. II. PARAMETRIC MODEL OF SPECTRAL LINES OBSERVED IN DOPPLER-VELOCITY MEASUREMENTS

We describe a global parametric model for the observed power spectra of solar oscillations of intermediate and low degree. A physically motivated parameterization is used as a substitute for a direct description of mode excitation and damping as these mechanisms remain poorly understood. The model is targeted at the accurate fitting of power spectra coming from Doppler-velocity measurements and uses an adaptive response function that accounts for both the vertical and horizontal components of the velocity field on the solar surface and for possible instrumental and observational distortions. The model is continuous in frequency, can easily be adapted to intensity measurements, and extends naturally to the analysis of high-frequency pseudomodes (interference peaks at frequencies above the atmospheric acoustic cutoff).

Helioseismic measurements in the solar envelope using group velocities of surface waves

At intermediate and high degree l, solar p- and f modes can be considered as surface waves. Using variational principle, we derive an integral expression for the group velocities of the surface waves in terms of adiabatic eigenfunctions of normal modes, and address the benefits of using group-velocity measurements as a supplementary diagnostic tool in solar seismology. The principal advantage of using group velocities, when compared with direct analysis of the oscillation frequencies, comes from their smaller sensitivity to the uncertainties in the near-photospheric layers. We address some numerical examples where group velocities are used to reveal inconsistencies between the solar models and the seismic data. Further, we implement the group-velocity measurements to the calibration of the specific entropy, helium abundance Y and heavy-element abundance Z in the adiabatically-stratified part of the solar convective envelope, using different recent versions of the equation of state. The results are in close agreement with our earlier measurements based on more sophisticated analysis of the solar oscillation frequencies (Vorontsov et al. 2013, MNRAS 430, 1636). These results bring further support to the downward revision of the solar heavy-element abundances in recent spectroscopic measurements.

Helioseismic Inversions and the Equation of State

Techniques of helioseismic inversions targeted at the diagnostics of the equation of state are reviewed, with particular emphasis on the uniqueness of the solutions. Measurement of the adiabatic exponent deep in the solar convective envelope with p‐mode frequencies currently available from SOHO MDI data is presented. Further efforts in the field of global helioseismic inversions are briefly discussed.