Theoretical Eigenfrequencies Research Articles

Do solar models with weakly interacting massive particles reproduce the Stanford seismic data?

The discrepancy between theoretical eigenfrequencies of standard solar models and the frequencies of solar modes of degree between 2 and 5 measured at Stanford is degree-independent for cyclic frequencies above about 2 mHz. Below that frequency the discrepancy for dotriacontapole modes diverges from that of the modes of lower degree. The differences between eigenfrequencies of a simple solar model containing a cloud of weakly interacting particles in its core and of one without do not reproduce this behaviour.

Jovian seismology

Standing acoustic waves, with periods between about 4.5 and 9 min, may be trapped in a wave duct beneath Jupiter's tropopause. Detection of these oscillations by observations of Doppler shifting of infrared and ultraviolet absorption lines would offer a new important method for probing the giant planet's deep atmosphere and interior. Information would be revealed on Jupiter's thermal and density structure and the depth to which its zonal winds penetrate. Standing oscillations in the molecular hydrogen envelope are modeled and their theoretical eigenfrequencies are presented as they might appear in actual data analysis. Several forcing functions for wave generation are considered. These include coupling with turbulent and convective motions, thermal overstability due to radiative transfer, effects of wave propagation in a saturated atmosphere, and consequences of ortho- to parahydrogen conversion. Altjough the forcing mechanisms couple well with the acoustic waves, allowing for possible maintenance of the oscillations, the contribution they make to velocity amplitudes is very small, between 1.0 and 0.1 m sec −1. This implies that the Doppler shifting caused by the waves may be unresolvable except, perhaps, by methods of superposing time records of oscillations to enhance acoustic signals and diminish random noise.

Deviations from the normal mode spectrum of asymptotic theory

The work of Hill (1985a) on the low-degree 5 min eigenfrequency spectrum of the Sun based on differential radius observations combined with Doppler shift and total irradiance observations has been extended to include the work of Harvey and Duvall (1984) and Libbrecht and Zirin (1986). The differences between eigenfrequencies obtained in this analysis are compared to the predictions of asymptotic theory, and the deviations between observation and theory are observed to be ≈4 times larger than expected based on estimated accuracy of eigenfrequency determinations. These deviations are tested for departures from predictions of asymptotic theory which are quasi-periodic as a function of radial ordern and degreel. It is observed that the superposition of the seven distributions of frequency differences vsn obtained forl=0, ..., 6 generates an overdetermined quasi-periodic function when Δn/Δl=0.420±0.018 and Δv/Δl=−0.35 μHz in the superposition process. The probability that this quasi-periodic parent function is obtained from seven independent random distributions is estimated to be ≈1.2×10−7. Numerical experiments performed with theoretical eigenfrequency spectra demonstrate that the existence of a quasiperiodic behaviour in the superposed spectrum of frequency differences is physically plausible and that the parameters used in the superposition process are consistent with theory. One significant theoretical quasiperiodic behaviour is obtained for Δn/Δl=0.399. By comparing the properties of the observed quasi-periodic behaviour with those obtained in the numerical experiments, we infer that the location of the region which leads to the greatest departure from asymptotic theory predictions is 0.757±0.002 solar radii, which suggests that this region is connected with the transition zone between the radiative interior and the convection zone.

Theoretical eigenfrequencies of solar oscillations of low harmonic degree l in five-minute range

We have calculated eigenfrequencies of radial and nonradial p-mode oscillations with low harmonic index l (l = 0, 1, 2, 3, and 4) for a standard solar model with normal composition and appoximately the correct age. It is found that theoretical eigenfrequencies calculated for our standard model agree approximately with observed peaks in the power spectra for the full-disk five minute oscillation of the Sun (Claverie et al., 1980; Grec et al., 1983; Scherrer et al., 1983) in agreement with other recent works (Christensen-Dalsgaard and Gough, 1980; Scuflaire et al., 1981). However, there still remains a slight discrepancy between theory and observations in such a sense that the theoretical eigenfrequencies are slightly lower than observations (see Figure 1). Eigenfrequencies of nonradial p-modes with high degree l for the same model are also calculated for comparison with observations of the conventional five-minute oscillation with shorter horizontal wavelength. It is found that theoretical eigenfrequencies lie slightly above the observed ridges in the diagnostic (k, ω)-diagram, which is in accord with Ulrich and Rhodes (1977) and Berthomieu et al. (1980) because our standard model has a slightly shallow convective zone. It remains to be seen whether improvement in the equilibrium model can remove this small discrepancy, concurrently with a better agreement between theory and observation for the whole-disk oscillations of low degree. A full account of this work will be found in Shibahashi and Osaki (1981).

Theoretical Eigenfrequencies of Solar Oscillations of Low Harmonic Degree l in Five-Minute Range

AbstractWe have calculated eigenfrequencies of radial and nonradial p-mode oscillations with low harmonic index l (l = 0, 1, 2, 3, and 4) for a standard solar model with normal composition and appoximately the correct age. It is found that theoretical eigenfrequencies calculated for our standard model agree approximately with observed peaks in the power spectra for the full-disk five minute oscillation of the Sun (Claverie et al., 1980; Grec et al., 1983; Scherrer et al., 1983) in agreement with other recent works (Christensen-Dalsgaard and Gough, 1980; Scuflaire et al., 1981). However, there still remains a slight discrepancy between theory and observations in such a sense that the theoretical eigenfrequencies are slightly lower than observations (see Figure 1).

The effect of uncertainty in the eigendata of the pure host lattice on Greens-function calculations

The effect of the uncertainty in the eigendata of KI on gapmode calculations using the Greens-function technique has been examined. Certain eigenfrequencies calculated with the Shell-model could be compared with experimental singularities in the induced acoustic spectrum of KI containing impurities. Changing the theoretical eigenfrequencies to the experimental values for insertion in the isotopic substitution model, which was used as a testcase for Greens-function calculations, resulted in an appreciable shift (−0.3 cm −1) in a gapmode frequency.

The Normal Modes of a Rotating, Elliptical Earth

Summary It is possible to calculate precisely the theoretical eigen-frequencies of any Earth model which is non-rotating, spherically symmetric, and which has an isotropic static stress field and an isotropic dynamic stress-strain relation. In this paper Rayleigh's principle is used to provide a formalism which allows the approximate computation of the normal mode eigen-frequencies of any Earth model which is slowly rotating and slightly aspherical and anisotropic. This formalism is used to compute, correct to second order, the effects of the Earth's angular rotation, and correct to first order, the effects of the Earth's ellipticity of figure on the normal mode eigenfrequencies. It is found that for an arbitrary poloidal or toroidal multiplet, the central (m= 0) member of the multiplet is shifted slightly in frequency and that the other members of the multiplet are split apart asymmetrically by the effects of the Earth's rotation and ellipticity. The results may be used to make a preliminary correction for rotation and ellipticity to the Earth's raw normal mode data.