Gravitational field of the homogeneous rotational ellipsoidal body: a simple derivation and applications

Pohánka, V.

doi:10.2478/v10126-011-0005-0

Cited by 17 publications

(12 citation statements)

References 1 publication

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“…In this case, we should note that the shape of an equipotential surface can be an ellipsoid only for a homogenous body. This fact is known as the Hamy‐Pizzetty theorem (Hamy, ; Moritz, ; Pohánka, ; Rambaux et al, ). However, we find that the accuracy of the Tricarico () approximation using ellipsoids of revolution to represent equipotential surfaces is on the order of 10 m for a Ceres‐like multilayer body.…”

Section: Methodsmentioning

confidence: 90%

Constraints on Ceres' Internal Structure and Evolution From Its Shape and Gravity Measured by the Dawn Spacecraft

et al. 2017

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Ceres is the largest body in the asteroid belt with a radius of approximately 470 km. In part due to its large mass, Ceres more closely approaches hydrostatic equilibrium than major asteroids. Pre‐Dawn mission shape observations of Ceres revealed a shape consistent with a hydrostatic ellipsoid of revolution. The Dawn spacecraft Framing Camera has been imaging Ceres since March 2015, which has led to high‐resolution shape models of the dwarf planet, while the gravity field has been globally determined to a spherical harmonic degree 14 (equivalent to a spatial wavelength of 211 km) and locally to 18 (a wavelength of 164 km). We use these shape and gravity models to constrain Ceres' internal structure. We find a negative correlation and admittance between topography and gravity at degree 2 and order 2. Low admittances between spherical harmonic degrees 3 and 16 are well explained by Airy isostatic compensation mechanism. Different models of isostasy give crustal densities between 1,200 and 1,400 kg/m3 with our preferred model giving a crustal density of 1,287+70−87 kg/m3. The mantle density is constrained to be 2,434+5−8 kg/m3. We compute isostatic gravity anomaly and find evidence for mascon‐like structures in the two biggest basins. The topographic power spectrum of Ceres and its latitude dependence suggest that viscous relaxation occurred at the long wavelengths (>246 km). Our density constraints combined with finite element modeling of viscous relaxation suggests that the rheology and density of the shallow surface are most consistent with a rock, ice, salt and clathrate mixture.

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Section: Methodsmentioning

confidence: 90%

Constraints on Ceres' Internal Structure and Evolution From Its Shape and Gravity Measured by the Dawn Spacecraft

et al. 2017

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“…Self gravity is calculated analytically using the expressions for ellipsoids of revolution derived in Pohánka (2011). We apply zero pressure and zero traction boundary conditions to the outer surface of the asteroid.…”

Section: Viscous Relaxation Modelmentioning

confidence: 99%

Efficient early global relaxation of asteroid Vesta

Hager

Ermakov

et al. 2014

Icarus

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The asteroid Vesta is a differentiated planetesimal from the accretion phase of solar system formation. Although its present-day shape is dominated by a non-hydrostatic fossil equatorial bulge and two large, mostly unrelaxed impact basins, Vesta may have been able to approach hydrostatic equilibrium during a brief early period of intense interior heating. We use a finite element viscoplastic flow model coupled to a 1D conductive cooling model to calculate the expected rate of relaxation throughout Vesta's early history. We find that, given sufficient non-hydrostaticity, the early elastic lithosphere of Vesta experienced extensive brittle failure due to self-gravity, thereby allowing relaxation to a more hydrostatic figure. Soon after its accretion, Vesta reached a closely hydrostatic figure with <2 km non-hydrostatic topography at degree-2, which, once scaled, is similar to the maximum disequilibrium of the hydrostatic asteroid Ceres. Vesta was able to support the modern observed amplitude of non-hydrostatic topography only >40-200 My after formation, depending on the assumed depth of megaregolith. The Veneneia and Rheasilvia giant impacts, which generated most non-hydrostatic topography, must have therefore occurred >40-200 My after formation. Based on crater retention ages, topography, and relation to known impact generated features, we identify a large region in the northern hemisphere that likely represents relic hydrostatic terrain from early Vesta. The long-wavelength figure of this terrain suggests that, before the two late giant impacts, Vesta had a rotation period of 5.02 hr (6.3% faster than present) while its spin axis was offset by 3.0• from that of the present. The evolution of Vesta's figure shows that the hydrostaticity of small bodies depends strongly on its age and specific impact history and that a single body may embody both hydrostatic and non-hydrostatic terrains and epochs.

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“…After substituting this form of the operator to the left side of (27), and applying a standard procedure of finding a Green function [68], we get the Green function, G(x, x ′ ), in the ellipsoidal coordinates. It is represented in the form of expansion with respect to the ellipsoidal harmonics [69,70] G(x,…”

Section: Green's Function Of the Poisson Equation In The Ellipsoidmentioning

confidence: 99%

“…where the terms standing in the right hand side of this formula have been provided above in sections IV and V. As we consider the multipolar expansion of V outside the body, we need only the external solutions which are where m N = M N /α, and the constant Newtonian mass, M N , is given by (70). After replacing (127) -(130) in (126) and reducing similar terms, the scalar potential takes on the following form,…”

Section: A Mass Multipole Momentsmentioning

confidence: 99%

Normal gravity field in relativistic geodesy

2018

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Modern geodesy is subject to a dramatic change from the Newtonian paradigm to Einstein's theory of general relativity. This is motivated by the ongoing advance in development of quantum sensors for applications in geodesy including quantum gravimeters and gradientometers, atomic clocks and fiber optics for making ultra-precise measurements of the geoid and multipolar structure of the Earth's gravitational field. At the same time, Very Long Baseline Interferometry, Satellite Laser Ranging and Global Navigation Satellite System have achieved an unprecedented level of accuracy in measuring 3-d coordinates of the reference points of the International Terrestrial Reference Frame and the world height system. The main geodetic reference standard to which gravimetric measurements of the of Earth's gravitational field are referred, is a normal gravity field represented in the Newtonian gravity by the field of a uniformly rotating, homogeneous Maclaurin ellipsoid which mass and quadrupole momentum are equal to the total mass and (tide-free) quadrupole moment of the Earth gravitational field. The present paper extends the concept of the normal gravity field from the Newtonian theory to the realm of general relativity. We focus our attention on the calculation of the post-Newtonian approximation of the normal field that is sufficient for current and near-future practical applications. We show that in general relativity the level surface of homogeneous and uniformly rotating fluid is no longer described by the Maclaurin ellipsoid in the most general case but represents an axisymmetric spheroid of the fourth order with respect to the geodetic Cartesian coordinates. At the same time, admitting a post-Newtonian inhomogeneity of the mass density in the form of concentric elliptical shells allows to preserve the level surface of the fluid as an exact ellipsoid of rotation. We parametrize the mass density distribution and the level surface with two parameters which are intrinsically connected to the existence of the residual gauge freedom, and derive the post-Newtonian normal gravity field of the rotating spheroid both inside and outside of the rotating fluid body. The normal gravity field is given, similarly to the Newtonian gravity, in a closed form by a finite number of the ellipsoidal harmonics. We employ transformation from the ellipsoidal to spherical coordinates to deduce a more conventional post-Newtonian multipolar expansion of scalar and vector gravitational potentials of the rotating spheroid. We compare these expansions with that of the normal gravity field generated by the Kerr metric and demonstrate that the Kerr metric has a fairly limited application in relativistic geodesy as it does not match the normal gravity field of the Maclaurin ellipsoid already in the Newtonian limit. We derive the post-Newtonian generalization of the Somigliana formula for the normal gravity field measured on the surface of the rotating spheroid and employed in practical work for measuring the Earth gravitational field anomalies. Finally, ...

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Gravitational field of the homogeneous rotational ellipsoidal body: a simple derivation and applications

Cited by 17 publications

References 1 publication

Constraints on Ceres' Internal Structure and Evolution From Its Shape and Gravity Measured by the Dawn Spacecraft

Constraints on Ceres' Internal Structure and Evolution From Its Shape and Gravity Measured by the Dawn Spacecraft

Efficient early global relaxation of asteroid Vesta

Normal gravity field in relativistic geodesy

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