Subtleties in spherical harmonic synthesis of the gravity field

2020 ◽  
Author(s):  
Ropesh Goyal ◽  
Sten Claessens ◽  
Will Featherstone ◽  
Onkar Dikshit
Keyword(s):  
Gravity Field ◽  
Spherical Harmonic ◽  
Gravitational Constant ◽  
Angular Rate ◽  
Physical Geodesy ◽  
Major Axis ◽  
Gravity Potential ◽  
Semi Major Axis ◽  
The Earth ◽  
Spherical Harmonic Synthesis

<p>Spherical harmonic synthesis (SHS) can be used to compute various gravity functions (e.g., geoid undulations, height anomalies, deflections of vertical, gravity disturbances, gravity anomalies, etc.) using the 4pi fully normalised Stokes coefficients from the many freely available Global Geopotential Models (GGMs).  This requires a normal ellipsoid and its gravity field, which are defined by four parameters comprising (i) the second-degree even zonal Stokes coefficient (J2) (aka dynamic form factor), (ii) the product of the mass of the Earth and universal gravitational constant (GM) (aka geocentric gravitational constant), (iii) the Earth’s angular rate of rotation (ω), and (iv) the length of the semi-major axis (a). GGMs are also accompanied by numerical values for GM and a, which are not necessarily identical to those of the normal ellipsoid.  In addition, the value of W<sub>0,</sub> the potential of the geoid from a GGM, needs to be defined for the SHS of many gravity functions. W<sub>0</sub> may not be identical to U<sub>0</sub>, the potential on the surface of the normal ellipsoid, which follows from the four defining parameters of the normal ellipsoid.  If W<sub>0</sub> and U<sub>0</sub> are equal and if the normal ellipsoid and GGM use the same value for GM, then some terms cancel when computing the disturbing gravity potential.  However, this is not always the case, which results in a zero-degree term (bias) when the masses and potentials are different.  There is also a latitude-dependent term when the geometries of the GGM and normal ellipsoids differ.  We demonstrate these effects for some GGMs, some values of W<sub>0</sub>, and the GRS80, WGS84 and TOPEX/Poseidon ellipsoids and comment on its omission from some public domain codes and services (isGraflab.m, harmonic_synth.f and ICGEM).  In terms of geoid heights, the effect of neglecting these parameters can reach nearly one metre, which is significant when one goal of modern physical geodesy is to compute the geoid with centimetric accuracy.  It is also important to clarify these effects for all (non-specialist) users of GGMs.</p>

Differential geometry and curvatures of equipotential surfaces in the realization of the World Height System

2020 ◽  
Author(s):  
Petr Holota ◽  
Otakar Nesvadba
Keyword(s):  
Differential Geometry ◽  
Gravity Field ◽  
Physical Geodesy ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
The Earth ◽  
The World ◽  
Height System ◽  
Equipotential Surfaces ◽  
Gravity Field Models

<p>The notion of an equipotential surface of the Earth’s gravity potential is of key importance for vertical datum definition. The aim of this contribution is to focus on differential geometry properties of equipotential surfaces and their relation to parameters of Earth’s gravity field models. The discussion mainly rests on the use of Weingarten’s theorem that has an important role in the theory of surfaces and in parallel an essential tie to Brun’s equation (for gravity gradient) well known in physical geodesy. Also Christoffel’s theorem and its use will be mentioned. These considerations are of constructive nature and their content will be demonstrated for high degree and order gravity field models. The results will be interpreted globally and also in merging segments expressing regional and local features of the gravity field of the Earth. They may contribute to the knowledge important for the realization of the World Height System.</p>

scholarly journals Some Special Types of Orbits around Jupiter

Aerospace ◽  
2021 ◽  
Vol 8 (7) ◽  
pp. 183
Author(s):  
Yongjie Liu ◽  
Yu Jiang ◽  
Hengnian Li ◽  
Hui Zhang
Keyword(s):  
Gravity Model ◽  
Small Perturbation ◽  
Equilibrium Points ◽  
Major Axis ◽  
Semi Major Axis ◽  
Ground Track ◽  
The Earth ◽  
Degenerate Equilibrium ◽  
The Mean ◽  
Element Theory

This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination, and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufficiently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits.

scholarly journals Accretion of the Moon from non-canonical discs

2014 ◽  
Vol 372 (2024) ◽  
pp. 20130256 ◽  
Author(s):  
J. Salmon ◽  
R. M Canup
Keyword(s):  
Angular Momentum ◽  
Rotation Period ◽  
Major Axis ◽  
The Moon ◽  
Large Excess ◽  
Semi Major Axis ◽  
Earth’S Mantle ◽  
The Earth ◽  
Post Impact ◽  
Impact Simulations

Impacts that leave the Earth–Moon system with a large excess in angular momentum have recently been advocated as a means of generating a protolunar disc with a composition that is nearly identical to that of the Earth's mantle. We here investigate the accretion of the Moon from discs generated by such ‘non-canonical’ impacts, which are typically more compact than discs produced by canonical impacts and have a higher fraction of their mass initially located inside the Roche limit. Our model predicts a similar overall accretional history for both canonical and non-canonical discs, with the Moon forming in three consecutive steps over hundreds of years. However, we find that, to yield a lunar-mass Moon, the more compact non-canonical discs must initially be more massive than implied by prior estimates, and only a few of the discs produced by impact simulations to date appear to meet this condition. Non-canonical impacts require that capture of the Moon into the evection resonance with the Sun reduced the Earth–Moon angular momentum by a factor of 2 or more. We find that the Moon's semi-major axis at the end of its accretion is approximately 7 R ⊕ , which is comparable to the location of the evection resonance for a post-impact Earth with a 2.5 h rotation period in the absence of a disc. Thus, the dynamics of the Moon's assembly may directly affect its ability to be captured into the resonance.

Using spacecraft range data and radar observations for the improvement of the orbital elements of planets and parameters of Mars rotation

1996 ◽  
Vol 172 ◽  
pp. 45-48
Author(s):  
E.V. Pitjeva
Keyword(s):  
Major Axis ◽  
Astronomical Unit ◽  
Range Data ◽  
Orbital Elements ◽  
Semi Major Axis ◽  
The Earth ◽  
Electromagnetic Signals ◽  
Least Squares Estimates ◽  
Radar Ranging ◽  
Made In

The extremely precise Viking (1972–1982) and Mariner data (1971–1972) were processed simultaneously with the radar-ranging observations of Mars made in Goldstone, Haystack and Arecibo in 1971–1973 for the improvement of the orbital elements of Mars and Earth and parameters of Mars rotation. Reduction of measurements included relativistic corrections, effects of propagation of electromagnetic signals in the Earth troposphere and in the solar corona, corrections for topography of the Mars surface. The precision of the least squares estimates is rather high, for example formal standard deviations of semi-major axis of Mars and Earth and the Astronomical Unit were 1–2 m.

scholarly journals High-frequency analysis of Earth gravity field models based on terrestrial gravity and GPS/levelling data: a case study in Greece

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
T. D. Papanikolaou ◽  
N. Papadopoulos
Keyword(s):  
Gravity Field ◽  
Spherical Harmonic ◽  
Gravity Models ◽  
Earth Gravity Field ◽  
Earth Gravity ◽  
Gravity Field Recovery ◽  
High Degree ◽  
Spherical Harmonic Synthesis ◽  
Gravity Field Models ◽  
Terrestrial Data

AbstractThe present study aims at the validation of global gravity field models through numerical investigation in gravity field functionals based on spherical harmonic synthesis of the geopotential models and the analysis of terrestrial data. We examine gravity models produced according to the latest approaches for gravity field recovery based on the principles of the Gravity field and steadystate Ocean Circulation Explorer (GOCE) and Gravity Recovery And Climate Experiment (GRACE) satellite missions. Furthermore, we evaluate the overall spectrum of the ultra-high degree combined gravity models EGM2008 and EIGEN-6C3stat. The terrestrial data consist of gravity and collocated GPS/levelling data in the overall Hellenic region. The software presented here implements the algorithm of spherical harmonic synthesis in a degree-wise cumulative sense. This approach may quantify the bandlimited performance of the individual models by monitoring the degree-wise computed functionals against the terrestrial data. The degree-wise analysis performed yields insight in the short-wavelengths of the Earth gravity field as these are expressed by the high degree harmonics.

scholarly journals A new formulation of Stokes’ approach in determining the global gravimetric geoid

2013 ◽  
Vol 56 (3) ◽  
Author(s):  
Wen-Bin Shen
Keyword(s):  
Potential Function ◽  
Normal Gravity ◽  
Major Axis ◽  
Gravity Potential ◽  
Semi Major Axis ◽  
Disturbing Potential ◽  
Fundamental Parameters ◽  
Ellipsoidal Surface ◽  
New Formulation ◽  
Two Parameters

<p>According to Stokes’ approach, given the gravity anomaly on the geoid as the boundary, a disturbing potential function satisfying some boundary conditions should be solved. The basic requirement is that the disturbing potential function should be harmonic in the region outside the geoid. However, since the normal gravity potential is not defined inside the reference ellipsoid (taking the WGS84 ellipsoid as an example), when the geoid is below the ellipsoidal surface, the disturbing potential function is not harmonic in the whole region outside the geoid, and is not defined on the whole geoid itself. These are theoretical difficulties in Stokes’ approach. To remove these difficulties from Stokes’ approach, this study provides a new formulation of Stokes’ approach. An inner ellipsoid with four fundamental parameters is chosen, two of which, the geocentric gravitational constant and the rotational angular velocity, coincide with the corresponding parameters of the WGS84 ellipsoid. The other two parameters, the semi-major axis and flattening, are different from the corresponding ones of the ellipsoid. Then, the normal gravity potential generated by the inner ellipsoid is determined, by requiring that it holds a constant on the surface of the inner ellipsoid or on the surface of the ellipsoid. With this new formulation, the disturbing potential function is harmonic in the whole region outside the geoid, and the difficulties in Stokes’ approach disappear. The new formulation proposed in this study is also adequate for analogous geodetic boundary-value problems.</p>

Determination of upper-atmosphere air-density profile from satellite observations

1959 ◽  
Vol 252 (1268) ◽  
pp. 28-34 ◽  
Keyword(s):  
Density Profile ◽  
Satellite Orbit ◽  
Major Axis ◽  
Scale Height ◽  
Semi Major Axis ◽  
Relative Variation ◽  
Air Density ◽  
The Earth ◽  
Point Data

The theory previously developed for the changes in the perigee distance and semi-major axis of a satellite orbit due to air drag is extended to enable the air-density profile (i. e. its relative variation with height) to be derived from the motion of the orbit’s perigee. The solution is first obtained in terms of the change in perigee distance and then in terms of the change in the radius of the earth at the sub-perigee point. Data are analyzed by the two methods, leading to 39 (± 9) and 36 (± 15) km for the scale height in the 180 and 220 km altitude regions.

Optical clocks for gravity field observation and further geodetic applications

2020 ◽  
Author(s):  
Hu Wu ◽  
Jürgen Müller ◽  
Annike Knabe
Keyword(s):  
Gravity Field ◽  
Rapid Development ◽  
Satellite Orbit ◽  
Equipotential Surface ◽  
Gravity Potential ◽  
Height Difference ◽  
The Earth ◽  
Temporal Gravity ◽  
Reference Clock

&lt;p&gt;In the past three decades, optical clocks and frequency transfer techniques have experienced a rapid development. They are approaching a fractional frequency uncertainty of 1.0x10&lt;sup&gt;-18&lt;/sup&gt;, corresponding to about 1.0 cm in height. This makes them promising to realize &amp;#8220;relativistic geodesy&amp;#8221;, and it opens a new door to directly obtain gravity potential values by the comparison of clock frequencies. Clocks are thus considered as a novel candidate for determining the Earth&amp;#8217;s gravity field. We propose to use a spaceborne clock to obtain gravity potential values along a satellite orbit through its comparison with reference clocks on ground or with a co-orbital clock. The sensitivity of clock measurements is mapped to gravity field coefficients through closed-loop simulations.&lt;/p&gt;&lt;p&gt;In addition, clocks are investigated for other geodetic applications. Since they are powerful in providing the height difference between distant sites, clocks can be applied for the unification of local/regional height systems, by estimating the offsets between different height datums and the systematic errors within levelling networks. In some regions like Greenland, clocks might be a complementary tool to GRACE(-FO) for detecting temporal gravity signals. They can be operated at locations of interest and continuously track changes w.r.t. reference clock stations. The resulting time-series of gravity potential values reveal the temporal gravity signals at these points. Moreover, as the equipotential surface at a high satellite altitude is more regular than that on the Earth&amp;#8217;s surface, a couple of clocks in geostationary orbits can realize a space-based reference for the determination of physical heights at any point on the Earth through clock comparisons.&lt;/p&gt;&lt;p&gt;We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) under Germany&amp;#8217;s Excellence Strategy EXC-2123/1 (Project-ID: 390837967).&lt;/p&gt;

Fundamental Notions in Relativistic Geodesy - physics of a timelike Killing vector field

2020 ◽  
Author(s):  
Dennis Philipp ◽  
Claus Laemmerzahl ◽  
Eva Hackmann ◽  
Volker Perlick ◽  
Dirk Puetzfeld ◽  
...  
Keyword(s):  
Gravity Field ◽  
Vector Fields ◽  
Dimensional Space ◽  
Killing Vector ◽  
Three Dimensional ◽  
Scalar Function ◽  
Gravity Potential ◽  
Static Case ◽  
The Earth ◽  
Timelike Killing Vector

&lt;p&gt;The Earth&amp;#8217;s geoid is one of the most important fundamental concepts to provide a gravity field- related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions within Einstein&amp;#8217;s theory of General Relativity is inevitable.&lt;span&gt;&amp;#160;&lt;/span&gt;&lt;/p&gt;&lt;p&gt;Building on the theoretical construction of isochronometric surfaces we define a relativistic gravity potential as a generalization of known (post-)Newtonian notions. It exists for any stationary configuration and rigidly co-rotating observers; it is the same as realized by local plumb lines and determined by the norm of a timelike Killing vector. In a second step, we define the relativistic geoid in terms of this gravity potential in direct analogy to the Newtonian understanding. In the respective limits, it allows to recover well-known results.&amp;#160;Comparing the Earth&amp;#8217;s Newtonian geoid to its relativistic generalization is a very subtle problem. However, an isometric embedding into Euclidean three-dimensional space can solve it and allows an intrinsic comparison. We show that the leading-order differences are at the mm-level.&lt;span&gt;&amp;#160;&lt;/span&gt;In the next step, the framework is extended to generalize the normal gravity field as well. We argue that an exact spacetime can be constructed, which allows to recover the Newtonian result in the weak-field limit. Moreover, we comment on the relativistic definition of chronometric height and related concepts.&lt;/p&gt;&lt;p&gt;In a stationary spacetime related to the rotating Earth, the aforementioned gravity potential is of course not enough to cover all information on the gravitational field. To obtain more insight, a second scalar function can be constructed, which is genuinely related to gravitomagnetic contributions and vanishes in the static case. Using the kinematic decomposition of an isometric observer congruence, we suggest a potential related to the twist of the worldlines therein. Whilst the first potential is related to clock comparison and the acceleration of freely falling corner cubes, the twist potential is related to the outcome of Sagnac interferometric measurements. The combination of both potentials allows to determine the Earth&amp;#8217;s geoid and equip this surface with coordinates in an operational way. Therefore, relativistic geodesy is intimately related to the physics of timelike Killing vector fields.&lt;/p&gt;

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