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"Gravitational fields."
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Satellite Gravimetry: A Review of Its Realization
Since Kepler, Newton and Huygens in the seventeenth century, geodesy has been concerned with determining the figure, orientation and gravitational field of the Earth. With the beginning of the space age in 1957, a new branch of geodesy was created, satellite geodesy. Only with satellites did geodesy become truly global. Oceans were no longer obstacles and the Earth as a whole could be observed and measured in consistent series of measurements. Of particular interest is the determination of the spatial structures and finally the temporal changes of the Earth's gravitational field. The knowledge of the gravitational field represents the natural bridge to the study of the physics of the Earth's interior, the circulation of our oceans and, more recently, the climate. Today, key findings on climate change are derived from the temporal changes in the gravitational field: on ice mass loss in Greenland and Antarctica, sea level rise and generally on changes in the global water cycle. This has only become possible with dedicated gravity satellite missions opening a method known as satellite gravimetry. In the first forty years of space age, satellite gravimetry was based on the analysis of the orbital motion of satellites. Due to the uneven distribution of observatories over the globe, the initially inaccurate measuring methods and the inadequacies of the evaluation models, the reconstruction of global models of the Earth's gravitational field was a great challenge. The transition from passive satellites for gravity field determination to satellites equipped with special sensor technology, which was initiated in the last decade of the twentieth century, brought decisive progress. In the chronological sequence of the launch of such new satellites, the history, mission objectives and measuring principles of the missions CHAMP, GRACE and GOCE flown since 2000 are outlined and essential scientific results of the individual missions are highlighted. The special features of the GRACE Follow-On Mission, which was launched in 2018, and the plans for a next generation of gravity field missions are also discussed.
Journal Article
Gravity Field and Internal Structure of Mercury from MESSENGER
Radio tracking of the MESSENGER spacecraft has provided a model of Mercury's gravity field. In the northern hemisphere, several large gravity anomalies, including candidate mass concentrations (mascons), exceed 100 mi Hi-Galileos (mgal). Mercury's northern hemisphere crust is thicker at low latitudes and thinner in the polar region and shows evidence for thinning beneath some impact basins. The low-degree gravity field, combined with planetary spin parameters, yields the moment of inertia CIMR² = 0.353 ± 0.017, where M and R are Mercury's mass and radius, and a ratio of the moment of inertia of Mercury's solid outer shell to that of the planet of CJC = 0.452 ± 0.035. A model for Mercury's radial density distribution consistent with these results includes a solid silicate crust and mantle overlying a solid iron-sulfide layer and an iron-rich liquid outer core and perhaps a solid inner core.
Journal Article
The Tides of Titan
We have detected in Cassini spacecraft data the signature of the periodic tidal stresses within Titan, driven by the eccentricity (e = 0.028) of its 16-day orbit around Saturn. Precise measurements of the acceleration of Cassini during six close flybys between 2006 and 2011 have revealed that Titan responds to the variable tidal field exerted by Saturn with periodic changes of its quadrupole gravity, at about 4% of the static value. Two independent determinations of the corresponding degree-2 Love number yield k 2 = 0.589 ± 0.150 and k 2 = 0.637 ± 0.224 (2σ). Such a large response to the tidal field requires that Titan's interior be deformable over time scales of the orbital period, in a way that is consistent with a global ocean at depth.
Journal Article
Accurate computation of gravitational curvature of a tesseroid
In recent years, the fundamental quantity of the gravitational field has been extended from gravitational potential, gravitational vector, and gravitational gradient tensor to gravitational curvature with its first measurement along the vertical direction in laboratory conditions. Previous studies numerically identified the near-zone and polar-region problems for gravitational curvature of a tesseroid, but these issues remain unresolved. In this contribution, we derive the new third-order central and single-sided difference formulas with one, two, and three arguments using the finite difference method. To solve these near-zone and polar-region problems, we apply a numerical approach combining the conditional split, finite difference, and double exponential rule based on these newly derived third-order difference formulas when the computation point is located below, inside, and outside the tesseroid. Numerical experiments with a spherical shell discretized into tesseroids reveal that the accuracy of gravitational curvature is about 4–8 digits in double precision. Numerical results confirm that when the computation point moves to the surface of the tesseroid, the relative and absolute errors of gravitational curvature do not change much, i.e., the near-zone problem can be adequately solved using the numerical approach in this study. When the latitude of the computation point increases, the relative and absolute errors of gravitational curvature do not increase, which solves the polar-region problem with this stable numerical approach. The provided Fortran codes at
https://github.com/xiaoledeng/xtessgc-xqtessgc
will help with potential applications for the gravitational field of different celestial bodies in geodesy, geophysics, and planetary sciences.
Journal Article
Translated spherical harmonics for semi-global gravitational field modeling: examples for Martian moon Phobos and asteroid 433 Eros
The gravitational field of a planetary body is most often modeled by an exterior spherical harmonic series, which is uniformly convergent outside the smallest mass-enclosing sphere centered at the origin of the coordinate system, known as the Brillouin sphere. The model can become unstable inside the spherical boundary. Rarely deliberated or emphasized is an obvious fact that the radius of the Brillouin sphere, which is the maximum radius coordinate of the body, changes with the origin. The sphere can thus be adjusted to fit a certain convex portion of irregular body shape via an appropriate coordinate translation, thereby maximizing the region of model stability above the body. We demonstrate that it is, while perhaps counterintuitive, rational to displace the coordinate origin from the center of figure, or even off the body entirely. We review concisely the theory and a method of spherical harmonic translation. We consider some textbook examples that illuminate the physical meaning and the practical advantage of the transformation, the discussion of which, as it turns out, is not so easily encountered. We provide seminormalized as well as fully normalized version of the algorithms, which are compact and easy to work with for low-degree applications. At little cost, the proposed approach enables the spherical harmonics to be comparable with the far more complicated ellipsoidal harmonics in performance in the case of two small objects, Phobos and 433 Eros.
Journal Article
Fixed-Magnetization Ising Model with a Slowly Varying Magnetic Field
The motivation for this paper is the analysis of the fixed-density Ising lattice gas in the presence of a gravitational field. This is seen as a particular instance of an Ising model with a slowly varying magnetic field in the fixed magnetization ensemble. We first characterize the typical magnetization profiles in the regime in which the contribution of the magnetic field competes with the bulk energy term. We then discuss in more detail the particular case of a gravitational field and the arising interfacial phenomena. In particular, we identify the macroscopic profile and propose several conjectures concerning the interface appearing in the phase coexistence regime. The latter are supported by explicit computations in an effective model. Finally, we state some conjectures concerning equilibrium crystal shapes in the presence of a gravitational field, when the latter contributes to the energy only to surface order.
Journal Article
Recursive Analytical Formulae of Gravitational Fields and Gradient Tensors for Polyhedral Bodies with Polynomial Density Contrasts of Arbitrary Non-negative Integer Orders
Exact computation of the gravitational field and gravitational gradient tensor for a general mass body is a core routine to model the density structure of the Earth. In this study, we report on the existence of closed-form solutions of the gravitational potential, gravitational field and gravitational gradient tensor for a general polyhedral mass body with a polynomial density function of arbitrary non-negative integer orders that can simultaneously vary in both horizontal and vertical directions. Our closed-form solutions of the gravitational potential and the gravitational field are singularity-free, which implies that the observation sites can have arbitrary geometric relationships with polyhedral mass source bodies. However, weak logarithmic singularities exist on the edges of polyhedra for the gravitational gradient tensor. A simple prismatic mass body with polynomial density contrast varying in the vertical direction and a complicated dodecahedral mass body with quartic-order density contrasts were tested to verify the accuracy of the newly derived closed-form solutions. For the gravitational potential, gravitational fields and gradient tensors, our closed-form solutions are in excellent agreement with previously published analytical solutions and Gaussian numerical quadrature solutions.
Journal Article
Information Storage in a Black Hole’s Gravitational Field
The key to resolving the black hole information loss paradox lies in clarifying the origin of black hole entropy and the mechanism by which black holes store information. By applying thermodynamic principles, we demonstrate that the entropy of a gravitational field is negative and proportional to the strength of the field, indicating that gravitational fields possess information storage capacity. For Schwarzschild black holes, we further demonstrate that information conventionally attributed to the black hole’s interior is in fact encoded within its external gravitational field. During black hole evaporation, the emitted particles transmit this information via gravitational correlations. This study advances our understanding of gravitational field entropy and provides valuable insights toward resolving the black hole information loss problem.
Journal Article
Optimized formulas of the gravitational field of a vertical cylindrical prism
Modeling the gravitational effects of the topography and other layers of the Earth is the basis not only for gravity corrections and reductions in geodesy but also for gravity inversions and interpretations in geophysics. Previous formulas of the Gravitational Potential (GP), Gravitational Vector (GV), and Gravitational Gradient Tensor (GGT) of a vertical cylindrical prism were derived from complex conversion relations, the result of which is relatively complicated. In this contribution, we are able to optimize such formulas through a particular geometrical relation between the computation point and integration point. The consistency between our newly derived formulas and previous formulas is confirmed analytically. By extending the cylindrical prism to a cylindrical shell, the analytical formulas of the GP, GV, and GGT of a cylindrical shell are derived when the computation point is located on the polar axis. Based on these analytical formulas, a cylindrical shell benchmark is put forward to evaluate the numerical properties of the cylindrical prism, that is to discrete a whole cylindrical shell into cylindrical prisms. For actual numerical calculations, we propose to approximate the cylindrical prism with a second-order 3D Taylor series expansion. Beyond the improved simplicity, our optimized formulas help to save computation time (particularly for the GGT up to 20%). Numerical results reveal that when the computation point's vertical distance changes, the relative and absolute errors are symmetric with respect to the center vertical distance of the cylindrical shell. Using the second-order 3D Taylor series expansion method provides sufficient computation precision, i.e. all relative errors of the GP, GV, and GGT are smaller than 10
−2
in the numerical experiments. The new expressions for the GP, GV, and GGT of a vertical cylindrical prism using the second-order 3D Taylor series expansion and a cylindrical shell are provided for practical applications of gravity forward modeling in the Python language at the GitHub website
https://www.github.com/xiaoledeng/optimized-formulas-of-gp-gv-ggt
.
Journal Article