Equatorial Ellipticity Research Articles

Decadal Variations in Equatorial Ellipticity and Principal Axis of the Earth from Satellite Laser Ranging/GRACE

Decadal Variations in Equatorial Ellipticity and Principal Axis of the Earth from Satellite Laser Ranging/GRACE

On Clairaut's theory and its extension for planetary hydrostatic equilibrium derived using gravitational multipole formalism

SUMMARY Clairaut's theory that relates the Earth's oblate figure and internal ellipticity to its gravity under rotational-hydrostatic equilibrium has reigned classical geodesy over the centuries. In this paper, we (i) derive from first principles the classical Clairaut's theory for the polar oblateness of a rotating planet under axi-symmetric rotational-hydrostatic equilibrium and (ii) extend the development to the triaxial case for the equatorial ellipticity of a tidally locked synchronous-rotating moon under rotational-tidal-hydrostatic equilibrium. Typical derivations of the classical Clairaut's theory presented in the literature being rather laborious even to first order, we instead exploit two concise forms of methodology: the gravitational multipole formalism on the physics side, and the Jacobian determinant for the Clairaut coordinate transformation on the mathematics side. The outcome is a logical and straightforward derivation of Clairaut's theory to first order in its entirety, encompassing all the equations and related formulas in geodesy bearing Clairaut's name. That further allows a natural extension to a tidally locked moon. In particular it is demonstrated that the same Clairaut's differential equation applies to both cases governing both the polar oblateness and the equatorial ellipticity.

Analysis of the Effect of Earth’s Equatorial Ellipticity on the Existence and Stability of the Lagrangian Points in the Earth–Moon–Sun System

Analysis of the Effect of Earth’s Equatorial Ellipticity on the Existence and Stability of the Lagrangian Points in the Earth–Moon–Sun System

Inferring neutron star properties with continuous gravitational waves

ABSTRACT Detection of continuous gravitational waves from rapidly spinning neutron stars opens up the possibility of examining their internal physics. We develop a framework that leverages a future continuous gravitational wave detection to infer a neutron star’s moment of inertia, equatorial ellipticity, and the component of the magnetic dipole moment perpendicular to its rotation axis. We assume that the neutron star loses rotational kinetic energy through both gravitational wave and electromagnetic radiation, and that the distance to the neutron star can be measured, but do not assume electromagnetic pulsations are observable or a particular neutron star equation of state. We use the Fisher information matrix and Monte Carlo simulations to estimate errors in the inferred parameters, assuming a population of gravitational-wave-emitting neutron stars consistent with the typical parameter domains of continuous gravitational wave searches. After an observation time of 1 yr, the inferred errors for many neutron stars are limited chiefly by the error in the distance to the star. The techniques developed here will be useful if continuous gravitational waves are detected from a radio, X-ray, or gamma-ray pulsar, or else from a compact object with known distance, such as a supernova remnant.

Resonant curves of geo-centric satellite under the gravitational effect of the Earth-Moon-Sun system including Earth’s equatorial ellipticity and resistive force using unperturbed solutions

This research article presents the effect of the Earth’s equatorial ellipticity and the resistive force on a geocentric satellite’s resonant curves. The resonance occurs in the motion of the satellite due to the frequencies, the angular velocity of the satellite, the rate of change of Earth’s equatorial ellipticity parameter (γ̇), the steady-state angular velocity of the Moon around the Earth (ϑ0̇), and the steady-state angular velocity of barycenter (Θ̇) around the Sun. We express the equations of motion of the satellite in a spherical coordinate system by using the Earth’s potential. By applying the unperturbed solution, we reduced the system of equations into a second-order differential equation and obtained its solution. From the solution it is observed that the resonance appears between the frequencies δs0̇,γ̇,Θ0̇ and ϑ0̇ at ten resonant points. We analyze the resonant curves corresponding to the resonant points and show the effect of resistive force and latitude of a satellite. Also, We show the effect of eccentricity (e) and semi-major axis (a) on the oscillatory amplitudes (Ai), for i=1,2,3…21.

Resonant Curve Due to Perturbations in Geo-Synchronous Satellite Including the Earth’s Equatorial Ellipticity and Resistive Force

Resonant Curve Due to Perturbations in Geo-Synchronous Satellite Including the Earth’s Equatorial Ellipticity and Resistive Force

Stability analysis of Lagrangian points of geo-synchronous satellite including the resistive force and earth’s equatorial ellipticity

In this paper, we have investigated the existence of the Lagrangian points and their stability in the problem of geo-synchronous satellite including the effect of resistive force and earth’s equatorial ellipticity parameter γ. Equations of motion of satellite are expressed in spherical coordinates (r,θ,ϕ) including the terms of earth’s equatorial ellipticity parameter γ and the resistive force. It is observed that there exists two collinear and two non-collinear Lagrangian points for different values of γ. It is found that the effect of γ on the position of the Lagrangian points is very small. Zero velocity curves are also drawn at different values of Jacobi constant for different values of γ. Finally, stability of Lagrangian point is discussed. We have found that all the Lagrangian points are unstable for different values of γ.

Analysis of Resonant Curve and Phase Portrait Due to Earth’s Equatorial Ellipticity In the Earth-Moon System Using Perturbation Technique

In this paper, the resonance due to equatorial ellipticity of Earth, spin rate of Earth and angular velocity of bary-center are investigated. Using spherical coordinate system, we have determined equations of motion of the moon. By using perturbations relative to moon, equations of motion are simplified to second order ordinary differential equation containing the equatorial ellipticity parameter of the Earth. Two types of resonant curves due to the frequencies (a) , and (b) are analyzed. It is observed that the amplitudes of the oscillation are different at different resonant points due to different size of the bandwidth. Finally, the phase portrait of the orbits of the moon and the phase spaces are analyzed by using Poincare section with periodicity in time t for the unpertubed system.

Effect of Earth’s Equatorial Ellipticity on the Resonant Curve and Phase Portrait of Geo-centric Satellite Under the Gravitational Effect of the Earth–Moon–Sun System by Using Unperturbed Solution

Effect of Earth’s Equatorial Ellipticity on the Resonant Curve and Phase Portrait of Geo-centric Satellite Under the Gravitational Effect of the Earth–Moon–Sun System by Using Unperturbed Solution

Analysis of resonant curves and phase portrait in the earth–moon system by using its unperturbed solution and earth’s equatorial ellipticity

In this research article, we have investigated resonant curves due to the rate of change of earth’s equatorial ellipticity parameter (γ̇), steady-state value of the angular velocity of the moon (θ̇mo), and angular velocity of barycenter (α0̇) in the Earth-Moon system. Equations of motion of the moon are determined in a spherical coordinate system with the help of the gravitational potential of the earth. By using the unperturbed solution, equations of motion of the moon reduced into the second-order differential equation. From the solution it is observed that resonance occurs due to the frequencies γ̇, θ̇m0, and α0̇ at the resonant points θ̇m0=2γ̇, 3θ̇m0=2γ̇, θ̇m0=γ̇, θ̇m0=α̇0. Finally, we have analyzed the phase portrait and phase space by method of Poincaré section when the system is free from forces.

Diving below the Spin-down Limit: Constraints on Gravitational Waves from the Energetic Young Pulsar PSR J0537-6910

Abstract We present a search for quasi-monochromatic gravitational-wave signals from the young, energetic X-ray pulsar PSR J0537−6910 using data from the second and third observing runs of LIGO and Virgo. The search is enabled by a contemporaneous timing ephemeris obtained using Neutron star Interior Composition Explorer (NICER) data. The NICER ephemeris has also been extended through 2020 October and includes three new glitches. PSR J0537−6910 has the largest spin-down luminosity of any pulsar and exhibits fRequent and strong glitches. Analyses of its long-term and interglitch braking indices provide intriguing evidence that its spin-down energy budget may include gravitational-wave emission from a time-varying mass quadrupole moment. Its 62 Hz rotation frequency also puts its possible gravitational-wave emission in the most sensitive band of the LIGO/Virgo detectors. Motivated by these considerations, we search for gravitational-wave emission at both once and twice the rotation frequency from PSR J0537−6910. We find no signal, however, and report upper limits. Assuming a rigidly rotating triaxial star, our constraints reach below the gravitational-wave spin-down limit for this star for the first time by more than a factor of 2 and limit gravitational waves from the l = m = 2 mode to account for less than 14% of the spin-down energy budget. The fiducial equatorial ellipticity is constrained to less than about 3 ×10−5, which is the third best constraint for any young pulsar.

Einstein@Home All-sky Search for Continuous Gravitational Waves in LIGO O2 Public Data

Abstract We conduct an all-sky search for continuous gravitational waves in the LIGO O2 data from the Hanford and Livingston detectors. We search for nearly monochromatic signals with frequency 20.0 Hz ≤ f ≤ 585.15 Hz and spin-down Hz s−1. We deploy the search on the Einstein@Home volunteer-computing project and follow-up the waveforms associated with the most significant results with eight further search stages, reaching the best sensitivity ever achieved by an all-sky survey up to 500 Hz. Six of the inspected waveforms pass all the stages but they are all associated with hardware injections, which are fake signals simulated at the LIGO detector for validation purposes. We recover all these fake signals with consistent parameters. No other waveform survives, so we find no evidence of a continuous gravitational wave signal at the detectability level of our search. We constrain the h 0 amplitude of continuous gravitational waves at the detector as a function of the signal frequency, in half-Hz bins. The most constraining upper limit at 163.0 Hz is h 0 = 1.3 × 10−25, at the 90% confidence level. Our results exclude neutron stars rotating faster than 5 ms with equatorial ellipticities larger than 10−7 closer than 100 pc. These are deformations that neutron star crusts could easily support, according to some models.

Resonant curve of geo-synchronous satellite including effect of earth’s equatorial ellipticity and resistive force using perturbations technique

In this paper, we have investigated the effect of earth’s equatorial ellipticity parameter and resistive force on the resonant curve of a geo-synchronous satellite using perturbations technique. Equations of motion of the satellite are determined in spherical coordinates (r,θ,ϕ) with the help of gravitational potential of the earth. Furthermore, the equations of motion of the satellite are simplified in the form of second order differential equation by using perturbations technique. It is observed from the solution that resonance occurs due to the frequencies θ˙E (angular rate of rotation of the earth) and γ˙ (where γ be the measured from minor axis of the earth’s equatorial ellipse to the projection of the satellite). Finally, we study the effect of the earth’s equatorial ellipticity parameter and the resistive force on the resonant curve with the help of different graphs. It is observed that amplitude A decreases when the magnitude of the resistive force increases.

Gravitational-wave Constraints on the Equatorial Ellipticity of Millisecond Pulsars

Abstract We present a search for continuous gravitational waves from five radio pulsars, comprising three recycled pulsars (PSR J0437−4715, PSR J0711−6830, and PSR J0737−3039A) and two young pulsars: the Crab pulsar (J0534+2200) and the Vela pulsar (J0835−4510). We use data from the third observing run of Advanced LIGO and Virgo combined with data from their first and second observing runs. For the first time, we are able to match (for PSR J0437−4715) or surpass (for PSR J0711−6830) the indirect limits on gravitational-wave emission from recycled pulsars inferred from their observed spin-downs, and constrain their equatorial ellipticities to be less than 10−8. For each of the five pulsars, we perform targeted searches that assume a tight coupling between the gravitational-wave and electromagnetic signal phase evolution. We also present constraints on PSR J0711−6830, the Crab pulsar, and the Vela pulsar from a search that relaxes this assumption, allowing the gravitational-wave signal to vary from the electromagnetic expectation within a narrow band of frequencies and frequency derivatives.

Pressure torque of torsional Alfvén modes acting on an ellipsoidal mantle

SUMMARY We investigate the pressure torque between the fluid core and the solid mantle arising from magnetohydrodynamic modes in a rapidly rotating planetary core. A 2-D reduced model of the core fluid dynamics is developed to account for the non-spherical core–mantle boundary. The simplification of such a quasi-geostrophic model rests on the assumption of invariance of the equatorial components of the fluid velocity along the rotation axis. We use this model to investigate and quantify the axial torques of linear modes, focusing on the torsional Alfvén modes (TM) in an ellipsoid. We verify that the periods of these modes do not depend on the rotation frequency. Furthermore, they possess angular momentum resulting in a net pressure torque acting on the mantle. This torque scales linearly with the equatorial ellipticity. We estimate that for the TM calculated here topographic coupling to the mantle is too weak to account for the variations in the Earth’s length-of-day.

Formulation of a Triaxial Three‐Layered Earth Rotation: Theory and Rotational Normal Mode Solutions

AbstractIn this study, we formulated a triaxial three‐layered anelastic Earth rotation theory considering various core‐mantle couplings, including the pressure and gravitational couplings acting on the elastic inner core by the outer core and mantle, and the viscoelectromagnetic couplings between the outer core and mantle and between the outer and inner cores. With this formulation, using the eigenvalue method, we solved four rotational normal modes, including the Chandler Wobble (CW), Free Core Nutation (FCN), Free Inner Core Nutation (FICN), and the Inner Core Wobble (ICW). The triaxiality of the actual Earth has notable effects on rotational normal modes, that is, 0.005, 0.0, 0.071, and 1.174 d (mean solar days), respectively, for the CW, FCN, FICN, and ICW. In the case that the triaxiality is bigger, for instance, if the Earth's equatorial dynamical ellipticity were , which is 1 order of magnitude larger than the presently determined ellipticity ( ), the CW and ICW periods will be lengthened by 3.579 and 94.912 d, and the FICN period will be decreased by 0.591 d. Based on our present knowledge, the numerical solutions of the four normal modes under the new rotation frame suggest that the periods of CW, FCN, FICN, and ICW are, respectively, 432.2, 429.9, 934.0, and 2,718.7 d. Our theory for Earth can be extended to other planets in our solar system.

A 6-year westward rotary motion in the Earth: Detection and possible MICG coupling mechanism

Seeking geophysical explanations for the periodic ∼six-year oscillation (SYO) previously found in the Earth's length-of-day variation (ΔLOD), we analyze the global GPS displacement and geomagnetic data using the array processing technique of OSE (Optimal Sequence Estimation), and find clear evidences of the 6-yr signals which manifest as a westward rotary propagating wave of the sectoral spherical-harmonic pattern of degree-2 order-2 (Y22). Based on the period, the spatial pattern, and the amplitudes and the estimated phases that exhibit consistent synchronicity among the three datasets, we propose the following scenario: The mantle-inner core gravitational (MICG) coupling gives rise to a 6-yr axial torsional libration of the inner core controlled by the sectoral Y22 density anomalies, or the equatorial ellipticities, in the inner core and the (lower) mantle, the angular momentum exchange between which manifests as the SYO in ΔLOD. It forces into action a pressure wave-2 cyclic in 6 yr through the fluid outer core, which in turn produces the corresponding GPS and geomagnetic variations. Our findings provide insight into the dynamics of the deep Earth, and corroborate a positive density anomaly for the lower-mantle Large Low-Shear-Velocity Provinces.

Dynamics of axial torsional libration under the mantle‐inner core gravitational interaction

AbstractThe aims of this paper are (i) formulating the dynamics of the mantle‐inner core gravitational (MICG) interaction in terms of the spherical‐harmonic multipoles of mass density. The modeled MICG system is composed of two concentric rigid bodies (mantle and inner core) of near‐spherical but otherwise heterogeneous configuration, with a fluid outer core in between playing a passive role. We derive the general equation of motion for the vector rotation but only focus on the polar component that describes the MICG axial torsional libration. The torsion constant and hence the square of the natural frequency of the libration is proportional to the product of the equatorial ellipticities of the mantle and inner‐core geoid embodied in their multipoles (of two different types) of degree 2 and order 2 (such as the Large Low‐Shear‐Velocity Provinces above the core‐mantle boundary) and (ii) studying the geophysical implications upon equating the said MICG libration to the steady 6 year oscillation that are observed in the Earth's spin rate or the length‐of‐day variation (ΔLOD). In particular, the MICG torsion constant is found to be = CIC σz2 ≈ 6.5 × 1019 N m, while the inner core's (BIC − AIC) ≈ 1.08 × 1031 kg m2 gives the inner core triaxiality (BIC − AIC)/CIC ≈ 1.8 × 10−4, about 8 times the whole‐Earth value. It is also asserted that the required inner‐core ellipticity amounts to no more than ~140 m in geoid height, much smaller than the sensitivity required for the seismic wave travel time to resolve the variation of the inner core.

Convection in deformed bodies: The effect of equatorial ellipticity on convective behavior

Tidal interactions between bodies such as hot jupiter planets and their host stars are likely to result in non-spherical geometries. These elliptical instabilities may have interesting effects on interior fluid convective patterns, which in turn could influence the nature of the magnetic dynamo within these planets. Simulations of thermal convection in the 2D rotating equatorial plane are conducted to determine to first order the effect of equatorial eccentricity on convection for varying density contrasts with differing convective vigor and rotation rates. This survey is conducted in two dimensions in order to simulate a broad range of eccentricities and to maximize the parameter space explored. The location of the three regimes documented in previous work (Evonuk and Samuel, 2012), dipolar flow, transitional flow, and differential flow, are found to be offset by the introduction of equatorial eccentricity to the system. The introduction of equatorial eccentricity changes the fluid behavior such that bodies with high amounts of deformation are likely to have weaker differential flows shifting their behavior towards transitional and dipolar flow structures. A scaling law based on the convective Rossby number, density contrast, and the eccentricity of the equatorial plane can therefore provide a way to estimate which regime a given body lies in.

Perturbations of a geo-centric synchronous satellite with resonance

We have investigated the resonances due to the perturbations of a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it’s equatorial ellipticity. The resonances at the points resulting from (i) the commensurability between $\dot{\theta}_{0}$ (steady-state orbital angular rate of the satellite) and $\dot{\theta}_{m}$ (angular velocity of the moon around the earth) and (ii) the commensurability between $\dot{\theta}_{0}$ and $\dot{\psi}_{0}$ (steady-state regression rate of the synchronous satellite) are analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure as given in Brown and Shook (Planetary Theory, Cambridge University Press, Cambridge, 1933). We have observed that as θ m (0∘≤θ m ≤45∘) and ψ (0∘≤ψ≤135∘) increase, the amplitude decreases and the time period also decreases. We have also shown the effect of ψ on amplitude and time period for 0∘≤Γ≤45∘, where Γ is the angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the satellite on the plane of the equator.