Ellipticity Of Figure Research Articles

A re-examination of ellipticity corrections for seismic phases

SUMMARY The Earth’s ellipticity of figure has an effect on the traveltimes of seismic waves over teleseismic distances. Tables of ellipticity corrections and coefficients have been used by seismologists for several decades; however, due to the increasing variety and complexity of seismic phases in use, current tables of ellipticity coefficients are now outmoded and incomplete. We present a Python package, EllipticiPy, for the calculation of ellipticity corrections, which removes the dependence on pre-calculated coefficients at discrete source depths and epicentral distances. EllipticiPy also facilitates the calculation of ellipticity corrections on other planetary bodies. When applied to both Earth and Mars, the magnitudes of ellipticity corrections are of the order of single seconds and are significant for some seismic studies on Earth but remain negligible on Mars due to other greater sources of uncertainty.

Observation of Coriolis coupled modes below 1 mHz

We present observations of spectral energy at toroidal mode frequencies in vertical seismic recordings of the 1998 Balleny Islands earthquake. Since toroidal modes on a spherically symmetric, non-rotating Earth have horizontally polarized particle motion these observations call for an explanation. We first rule out local and instrumental effects as being responsible for the vertical-component signal of the toroidal modes 0T3 (0.59 mHz) and 0T4 (0.77 mHz). The global effects that we consider are general heterogeneous mantle structure, ellipticity of figure and rotation. We find that rotation through Coriolis coupling of low-order spheroidal and toroidal oscillations is the dominant mechanism.

IUE spectroscopy, visible-band photometry, and polarimetry of HD 47732 (V641 Monocerotis)

Several high- and low-dispersion IUE spectra are described for the hot, massive close binary HD 47732. The stellar silicon lines do not show theoretically expected strength, the weakness being attributed to incipicent emission. A weak stellar wind is measurable with an expansion velocity of about 500 km/s. There is difficulty in fitting a model atmosphere to all of the IUE continuum and visible band and IR measures. These difficulties are resolved by appealing to a 22,500 K model atmosphere and a distinctive interstellar extinction law. Numerous V band measures show the polarization to be variable, but not strongly in a phased-locked way. This suggests sporadic outbursts of gas feeding the shell. Several seasonal V band light curves are discussed, and the seasonal variations among them are attributed to variable developments of a systemic shell which scatters the photospheric light and dilutes to a variable degree the light variation due to ellipticity of figure of the component stars.

Tidal effects in pulsating stars. III - L = 1 /p 5/ modes in the ellipsoidal variable 14 Aurigae A

The noneclipsing single-lined spectroscopic binary 14 Aur A has a circular orbit of period 3.78858 days, and the A9 IV primary is a delta Scuti variable. Analysis of our 158 hours of blue light photometry obtained on 40 nights in 1972--1975 and of others' radial velocity measures provides an identification of three standing wave (in the rotating frame) pulsation modes as the tidal m=0, +- 1 substates belonging to an l=1 surface harmonic, with periods, P/sub 1,+1/=0./sup d/088088, P/sub 1,0/=0./sup d/096533, and P/sub 1,-1/=0./sup d/088243. These l=1 surface harmonics are probably p/sub 5/ global modes. The observed amplitudes A/sub l/m of these periods P/sub l/m, when combined with approximate theoretical visibility functions for the integrated flux variations (for l< or approx. =2, vertical-barmvertical-bar< or approx. =l) and with the assumption of energy equipartition among the observed substates, lead to an orbital inclination i=21./sup 0/2 ( +- 1./sup 0/4). The photometrically defined ellipticity of figure of the primary, the orbital velocity variation, published uvby..beta.. photometry and calibrations, pulsation theory, double-star theory, and stellar evolution theory all combine to yield these system characteristics: T/sub e,1/=7900 +- 110 K; M/sub 1//Msun=2.6 +- 0.7; /Rsun=5.2 +- 0.7; /Lsun=96 +- 26; Q/sub 1/=0./sup d/0124 +- more » 0./sup d/0011; M/sub 2//Msun=1.1 +- 0.3. The relative differences in the principal semiaxes are (b/sub 1/--b/sub 2/)/ =0.022 +- 0.004 and (b/sub 2/--b/sub 3/)/=0.025 +- 0.004 if rotation is synchronous with orbital motion. Both components of the binary are well detached from their respective Roche lobes. The relative amplitudes and phase of the pulsations in light and velocity agree satisfactorily with both theory and other observations. « less

The Effect of Small, Aspherical Perturbations on Travel Times and a Re-examination of the Corrections for Ellipticity

Summary We use the stationarity of the Fermat ray path to develop theoretical expressions that relate small, aspherical perturbations in velocity to small perturbations in travel times. For the Earth's ellipticity of figure we derive a compact expression for the perturbation in travel time. If the ellipticity is hydrostatic and is the dominant perturbation, then anomalies in travel times are linear constraints on the radial gradient of velocity. Otherwise, the anomalies are constraints on the (unknown) aspherical perturbations. We are led to an inverse problem in either case. Using recently derived models of the Earth we present calculations of the effect of ellipticity. If the effect of focal depth is neglected in the calculations, then mislocations in both epicentre and origin time can result. Differences between our calculations and those predicted by the approximate formula 6t = (h+H)f(A) are as large as 0-25 s for P at 90. Nowadays, seismologists are prone to attach significance to anomalies as small as 0.10 s. Consequently, we advocate that the effect of focal depth be considered and that traditional approximations be replaced by the more accurate calculations tabulated in this report. Theoretical development The effect of the Earth's ellipticity of figure on travel times was first studied more than 40 years ago by Comrie and by Gutenberg and Richter. Bullen (1965, p. 173176) has presented a concise theoretical derivation of the effect. Recent improvements in our knowledge of the mechanical structure of the Earth make it desirable not only to re-examine the effect of ellipticity, but also to consider other small, aspherical perturbations as well. We follow the derivation of Bullen A source is located a (r,,, ,!lo, &) in spherical polar co-ordinates, and a receiver at

Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

This present study focuses on three topics: (i) formulation of the first- order perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasi- degenerate perturbation calculation; and (iii) numerical results for both ordinary degenerate and quasi-degenerate multiplets using formulation (i). The Earth considered is a rotating, laterally-inhomogeneous Earth. To zeroth order, however, it is approximated by a spherically symmetric, non-rotating, elastic and isotropic sphere. Lateral perturbations and Earth's rotation are treated as perturbations. Examples of lateral perturbations are Earth's ellipticity of figure and lateral inhomogeneities. Because of the symmetry in the Earth model, unperturbed modes form multiplets with varying degrees of degeneracy. For almost all of these multiplets, the perturbation calculation need be carried out to first order. When one multiplet and another nearly coincide in their frequencies, first-order ordinary degenerate perturbation method must be replaced by first-order quasi-degenerate perturbation method. Since the latter is a more general form of the former, the first-order perturbation problem is completely solved. The perturbations considered in this study consist of Earth's rotation, ellipticity of figure, a continent-ocean discontinuity near the Earth's surface, and an artificially contrived discontinuity about the core-mantle interface. When the effect due to rotation becomes dominant, the first-order calculation becomes insufficient, and a variational method is posed. Such a method can be formulated from quasi-degenerate solutions. The numerical results show that ellipticity corrections are important and should always be included in the computations of the Earth's normal modes, that lateral inhomogeneities in the Earth's deep interior become an important perturbation for high-overtone multiplets, and that Coriolis coupling between a spheroidal and a toroidal multiplet overshadows all other types of coupling.

The Normal Modes of a Rotating, Elliptical Earth?11 Near-Resonance Multiplet Coupling

Certain of the Earth's poloidal and toroidal elastic-gravitational normal mode multiplets may be strongly coupled to another multiplet by the Earth's rotation and ellipticity of figure. This paper uses Rayleigh's variational principle to specify the selection rules for strong mode coupling and to examine the nature and effects of this mode coupling. It is found that some of the normal modes of a slowly rotating, slightly elliptical Earth model may consist of about half poloidal type motion and half toroidal type motion. This strong coupling can severely hinder the problem of proper mode identification and measurement of the angular frequencies of oscillation of the Earth's normal modes.

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.