In this paper, we study the fractional Schrödinger equation in the Earth’s gravitational field. We firstly introduce a family of auxiliary functions to construct solutions to the fractional ...Schrödinger equation in the Planck length. These solutions include the particular solution obtained previously by using the classical “Fourier transform approach”. By analyzing the solutions, we find the transition phenomenon when the dimension of Lévy path changes from integer to non-integer: the energy changes from discrete to continuous and wave functions change from non-degenerate to degenerate. Then we study the effect of the minimal length on the fractional Schrödinger equation in the Earth’s gravitational field and the solutions. We find that the presence of the minimal length brings a perturbation to the Hamiltonian in equation but it does not change the transition phenomenon. Based on these result, we prove the existence of bound states in the continuum (BICs) for the fractional quantum system in the Earth’s gravitational field and compare our BICs with those previous ones. Moreover, we provide the energy characteristic of small mass particles.
The occurrence of subduction earthquakes usually leads to considerable localized mass migration changes. To improve the detection of earthquake subductions as well as the constraint of fault ...parameters, we derive expressions that describe changes of the gravitational curvatures (GC), i.e., the third-order derivatives of the Earth's gravitational potential, caused by a point dislocation while adopting a spherical symmetric Earth model. As a 3-D tensor matrix, the GC have twenty-seven components of those seven are independent. First, we investigate the dislocation Love numbers of the Earth's gravitational potential and derive the Green's functions of GC caused by four independent point sources in a spherical inhomogeneous Earth model. We then present the GC changes in a half-space Earth model. Furthermore, we conduct a sensitivity study by using three physical quantities that involve gravitation, gravitational gradients, and GC to compare their abilities in a seismic source depth detection. Our numerical results reveal that changes in the GC are more sensitive to a medium information about the field source compared to gravitation and gravitational gradients. This finding indicates that GC measurements could provide a more detailed information about slip fault parameters when considering a heterogeneous slip. Despite a widespread application of gravity gradients in Earth science, especially after launching the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission, measurements of the third-order derivatives of the gravitational potential have an enormous potential in the study of the solid Earth, although further work is needed in terms of instrument design and development.
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•Expressions describing changes of the third-order derivatives of the Earth’s gravitational potential caused by point dislocation are derived.•Changes in third-order derivatives of the Earth’s gravitational potential provide a more detailed information on slip fault parameters.
The most important high-resolution geopotential models such as EGM96 and EGM2008 have been released approximately once per decade. In light of the ability of modern satellite, airborne or terrestrial ...techniques to provide new data sets every year (e.g., in polar and ocean areas), these data can be readily included in existing models without waiting for a new release. In this article, we present a novel ellipsoidal approach for updating high-resolution models over the oceans with new gridded data. The problem is demonstrated using the EGM2008 model updated with DTU10 geoid and gravity grids that provide additional signal over the Arctic oceans. The result of the procedure are the ellipsoidal and the spherical harmonic coefficients up to degree 4320 and 4400, respectively. These coefficients represent the input data set to within 0.08mGal globally, with the largest differences located at the land–ocean boundaries, which is two orders of magnitude less than real accuracy of gravity data from satellite altimetry. Along with the harmonic coefficients a detailed map of the second vertical derivative of the anomalous potential (or vertical gravitational gradient) on 1arc-min grid is anticipated to improve or complement the original DTU10 geoid model. Finally, an optimized set of Jekeli’s functions is provided as they allow for computing oblate ellipsoidal harmonics up to a very high degree and order (>10,000) in terms of the hypergeometric formulation.
This paper offers the representation of the potential of planets by a sequence of non-harmonic continuous functions, which are determined by direct resolve of the inversion radius in the binomial ...series. The coefficients of such interpretation are, in essence, the quantities determined by the body's shape and its filling, generating the potential by the masses and, therefore, taking into account their features. In connection with this, there is a need to study the nature of these quantities (for example, the possible connection with the parameters of the external gravitational field) and to develop methods and means for their determination. There are introduced algorithms for finding these elements of such representation, investigated the character and structure, connected with the harmony in the middle of a bulk body, what allows to find the coefficients of the resolve of series by means of linear combinations of parameters of the external gravitational field of the celestial body (Stokes constant), which is confirmed by arithmetical experiment on a concrete example. On the contrary, according to the known values for the elements in this representation, it is possible to determine Stokes permanent gravitational fields of planetary bodies. This, in turn, allows us to determine the value of the potential of the force of attraction (gravity, if we take into account the rotating component) throughout the space, including at points that are close to the surface or are on it. Obviously, the advantage of this approach consists in the computational aspect, since such images are not harmonic functions of the external generating body, and therefore the proposed approximation should be considered as an approximation of the continuous function. But, given the slow convergence (or even difference) of approaching ball functions in areas close to the surface, such an approach can be considered as an additional tool (or even alternative) when studying the representation of a gravitational field by classical methods (for example, by means of ball functions). Also the recording of the attraction potential using binomial series can be considered as its analytic continuation in the middle of the generating body with the corresponding distribution of masses and used in the interpretation of planetary geodynamic processes, which consist precisely in the proposed areas.
In this paper, general analytical and computational technique for satellite-to-satellite visibility will be established firstly under the keplerian force, secondary under the effect of earth’s ...gravitational field (oblateness). The development is generally in the sense that the visibility conditions can be used whatever the types of the satellite orbits may be. Many data are taken to illustrate our technique.
In this paper, the GRACE Earth’s gravitational field complete up to degree and order 120 is recovered based on the combination of different inclinations using the energy conservation principle. The ...results show that because different inclinations of satellite are sensitive to the geopotential coefficients with different degrees
l and orders
m, the design of GRACE exploiting 89° inclination can effectively improve the accuracy of geopotential zonal harmonic coefficients. However, it is less sensitive to the geopotential tesseral harmonic coefficients. Accordingly, the second group of GRACE exploiting lower inclination is required to determine high-accurately the geopotential tesseral harmonic coefficients and cover the shortage of the single group of GRACE exploiting 89
o inclination. Two groups of GRACE individually exploiting 89°
+
(82°–84°) inclinations are the optimal combination of the Earth’s gravitational field recovery complete up to degree and order 120. In the degree 120, the joint accuracy of cumulative geoid height based on two groups of GRACE individually exploiting 89° and 83° inclinations is averagely two times higher than the accuracy of a group of GRACE exploiting 89° inclination.
► The new IRIM equation was established by introducing inter-satellite range into orbital position. ► The nine-point IRIM formula is preferable for improving the accuracy of the Earth's gravity field ...recovery. ► The new WHIGG-GEGM01S model complete up to degree and order 120 is produced using GRACE Level-1B data. ► The availability of new WHIGG-GEGM01S model is validated by the GPS/Levelling observations. ► The advantages and disadvantage of new IRIM method are compared with other gravity recovery methods.
New studies on the influence of inter-satellite range on the accuracy of the Earth's gravitational field recovery were carried out based on the Inter-satellite Range Interpolation Method (IRIM). Firstly, the IRIM observation equation by introducing the precise inter-satellite range into the Line-Of-Sight (LOS) component of the relative orbital position vector was established for the first time using the original satellite observations including the orbital position of the spaceborne Global Positioning System (GPS) receiver, the inter-satellite range of the K-Band Ranging (KBR) system and the non-conservative force of the accelerometer (ACC). Secondly, the nine-point IRIM formula is shown by a comparison of multipoint IRIM formulas to be preferable for effectively improving the accuracy of the Earth's gravitational field recovery. Thirdly, the WHIGG-GEGM01S (GRACE Earth Gravity Model developed by Wuhan Institute of Geodesy and Geophysics) global gravity field model, complete up to degree and order 120, is produced using the GRACE Level-1B data products, acquired during 2008-01-01 to 2008-12-31 and released by the American Jet Propulsion Laboratory (JPL). Cumulative geoid height and gravity anomaly errors are 1.098
×
10
−1
m and 1.741
×
10
−6
m/s
2 at degree 120, respectively. Finally, the availability of the WHIGG-GEGM01S model is validated by the GPS/Levelling observations in the U.S., Europe and Australia.
Gravity data observed on or reduced to the ellipsoid are preferably represented using ellipsoidal harmonics instead of spherical harmonics. Ellipsoidal harmonics, however, are difficult to use in ...practice because the computation of the associated Legendre functions of the second kind that occur in the ellipsoidal harmonic expansions is not straightforward. Jekeli’s renormalization simplifies the computation of the associated Legendre functions. We extended the direct computation of these functions—as well as that of their ratio—up to the second derivatives and minimized the number of required recurrences by a suitable hypergeometric transformation. Compared with the original Jekeli’s renormalization the associated Legendre differential equation is fulfilled up to much higher degrees and orders for our optimized recurrences. The derived functions were tested by comparing functionals of the gravitational potential computed with both ellipsoidal and spherical harmonic syntheses. As an input, the high resolution global gravity field model EGM2008 was used. The relative agreement we found between the results of ellipsoidal and spherical syntheses is 10
−14
, 10
−12
and 10
−8
for the potential and its first and second derivatives, respectively. Using the original renormalization, this agreement is 10
−12
, 10
−8
and 10
−5
, respectively. In addition, our optimized recurrences require less computation time as the number of required terms for the hypergeometric functions is less.
Dynamic resonance, arising from commensurate (orbital or rotational) periods of satellites or planets with each other, has been a strong force in the development of the solar system. The repetition ...of conditions over the commensurate periods can result in amplified long-term changes in the positions of the bodies involved. Such resonant phenomena driven by the commensurability between the mean motion of certain artificial Earth satellites and the Earth’s rotation originally contributed to the evaluation and assessment of the Stokes parameters (harmonic geopotential coefficients) that specify the Earth’s gravitational field. The technique constrains linear combinations of the harmonic coefficients that are of relevant resonant order (lumped coefficients). The attraction of the method eventually dwindled, but the very accurate orbits of CHAMP and GRACE have recently led to more general insights for commensurate orbits applied to satellite geodesy involving the best resolution for all coefficients, not just resonant ones. From the GRACE mission, we learnt how to explain and predict temporary decreases in the resolution and accuracy of derived geopotential parameters, due to passages through low-order commensurabilities, which lead to low-density ground-track patterns. For GOCE we suggest how to change a repeat orbit height slightly, to achieve the best feasible recovery of the field parameters derived from on-board gradiometric measurements by direct inversion from the measurements to the harmonic geopotential coefficients, not by the way of lumped coefficients. For orbiters of Mars, we have suggestions which orbits should be avoided. The slow rotation of Venus results in dense ground-tracks and excellent gravitational recovery for almost all orbiters.
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