Earth's Oblateness Research Articles

A High-Accuracy and High-Efficiency Satellite Shadow Function Model for Oblate Earth

To achieve high-accuracy solar radiation pressure analysis, this study proposes a new high-accuracy and high-efficiency (HAHE) shadow function model considering the oblateness of the Earth, for the oblateness of the Earth has much influence in the calculation of the shadow factor. The sphere shadow function model has been improved in this study with an accurate shadow function that considers the Earth’s oblateness. Then, using the accurate shadow function model, a new HAHE shadow function model is proposed to calculate a simulated satellite orbit’s changes of shadow factor, which is analyzed and compared with the existing shadow function model and the accurate shadow function model. The results show that the difference between the HAHE shadow function and the accurate shadow function is tiny. Moreover, the HAHE shadow function is 113 times faster than the accurate shadow function model.

The increasingly dominant role of climate change on length of day variations

The melting of ice sheets and global glaciers results in sea-level rise, a pole-to-equator mass transport increasing Earth's oblateness and resulting in an increase in the length of day (LOD). Here, we use observations and reconstructions of mass variations at the Earth's surface since 1900 to show that the climate-induced LOD trend hovered between 0.3 and 1.0 ms/cy in the 20th century, but has accelerated to 1.33 [Formula: see text] 0.03 ms/cy since 2000. We further show that surface mass transport fully explains the accelerating trend in the Earth oblateness observed in the past three decades. We derive an independent measure of the decreasing LOD trend induced by Glacial Isostatic Adjustment (GIA) of [Formula: see text]0.80 [Formula: see text] 0.10 ms/cy, which provides a constraint for the mantle viscosity. The sum of this GIA rate and lunar tidal friction fully explains the secular LOD trend that is inferred from the eclipse record in the past three millennia prior to the onset of contemporary climate change. Projections of future climate warming under high emission scenarios suggest that the climate-induced LOD rate may reach 2.62 [Formula: see text] 0.79 ms/cy by 2100, overtaking lunar tidal friction as the single most important contributor to the long-term LOD variations.

Bi-circular model with test particle variable mass

This paper presents the effects of the perturbations on the motion of the test particle variable mass in the bi-circular Sun perturbed oblate Earth-Moon system where primary is taken as oblate Earth, secondary is taken as Moon and the third body is assumed as Sun. These three bodies are moving in the same plane around barycenter. The fourth and smallest body (test particle is assumed to be having variable mass with time) is moving in the space under the perturbations such as gravitational forces of the primaries, the solar radiation pressure, the coriolis and centrifugal forces. Then we determine the equations of motion, the locations of critical points, their stability and periodic orbits.

Trajectory Controllability of Artificial Satellite Under the Effect of Oblateness of Earth

Trajectory Controllability of Artificial Satellite Under the Effect of Oblateness of Earth

Perturbed low-thrust geostationary orbit transfer guidance via polynomial costate estimation

This paper proposes an optimal, robust, and efficient guidance scheme for the perturbed minimum-time low-thrust transfer toward the geostationary orbit. The Earth's oblateness perturbation and shadow are taken into account. It is difficult for a Lyapunov-based or trajectory-tracking guidance method to possess multiple characteristics at the same time, including high guidance optimality, robustness, and onboard computational efficiency. In this work, a concise relationship between the minimum-time transfer problem with orbital averaging and its optimal solution is identified, which reveals that the five averaged initial costates that dominate the optimal thrust direction can be approximately determined by only four initial modified equinoctial orbit elements after a coordinate transformation. Based on this relationship, the optimal averaged trajectories constituting the training dataset are randomly generated around a nominal averaged trajectory. Five polynomial regression models are trained on the training dataset and are regarded as the costate estimators. In the transfer, the spacecraft can obtain the real-time approximate optimal thrust direction by combining the costate estimations provided by the estimators with the current state at any time. Moreover, all these computations onboard are analytical. The simulation results show that the proposed guidance scheme possesses extremely high guidance optimality, robustness, and onboard computational efficiency.

Analytical propagation on relative motion of geostationary earth orbit satellites with orbital perturbations

AbstractAn accurate description of the relative motion of satellites is critical for space missions in the complex geostationary earth orbit space environment. Here, a new relative motion model is established considering major perturbations. To avoid the singularity of the classical orbital elements, a set of quasi‐non‐singular orbital elements are used to represent the relative motion states, which is further extended by considering the SRP coefficient difference. On the basis of existing classical model, an analytical model is derived by taking the Taylor expansion of the time derivative of the perturbation functions to the first order in the neighbourhood of the reference satellite, which contains the earth's oblateness perturbation, the third‐body perturbations and the solar radiation pressure perturbation. This method avoids solving the complex differential equations of relative motion directly and provides a general framework for including arbitrary perturbations into the model. The new model is verified in uncontrolled natural relative motion scenarios and position‐keeping control scenario, respectively. The results show that the proposed model has high accuracy and strong competitiveness.

Pluralizing measurement: Physical geodesy's measurement problem and its resolution

Derived measurements involve problems of coordination. Conducting them often requires detailed theoretical assumptions about their target, while such assumptions can lack sources of evidence that are independent from these very measurements. In this paper, I defend two claims about problems of coordination. I motivate both by a novel case study on a central measurement problem in the history of physical geodesy: the determination of the earth's ellipticity. First, I argue that the severity of problems of coordination varies according to scientists' predictive and experimental control over perturbations of the measurement process. Second, I identify a methodology by which scientists can solve hard problems of coordination and gradually increase their predictive control over perturbations. I dub this methodology ‘operational pluralism’ since it is driven by the introduction of alternative measurement operations that involve different physical indicators.

Application of elliptic integrals in marine navigation

AbstractIf the Earth's oblateness is neglected in marine navigation, then the sphere gives a relatively simple solution for course and distance between any two points. The navigation sphere where a span of one minute of arc is equal to nautical mile is used. The primary deficiency of this approach is the lack of a closed-form formula that takes the Earth's eccentricity into account. Considering the Earth as an oblate spheroid, i.e., a rotational ellipsoid with a small flattening, the problem of computing the length of the meridian arc leads to the understanding of elliptic integrals. In this paper, incomplete elliptic integrals of the first, second and third kind are used to find an arbitrary elliptical arc. The results prove an advantage of using geocentric latitude compared to geodetic and reduced latitude.

Analysis of the orbital evolution of space debris using a solar sail and natural forces

Since the launch of the first artificial satellite around the Earth up to the present date, various non-functional objects are orbiting the Earth, such as deactivated satellites, satellite components, rocket bodies, among others. These objects can collide with operational satellites or other debris, generating a cloud of smaller particles. Space agencies are concerned about the current scenario of the space environment. If we continue as we are the number of objects in orbit will make it difficult to operate safely in the space environment. So space exploration may become unsustainable (ESA (2020)). In this work, the orbital evolution of these objects that are located in the geostationary orbit (GEO) is analyzed. In the mathematical model we consider the main disturbing forces, such as the direct solar radiation pressure (SRP), the perturbation of the third body (Sun and Moon) and the Earth oblateness. The SRP is the most relevant perturbation for objects with larger area-mass ratio. Knowing this, the possibility of using a solar sail is considered to help to clean the space environment. The main natural environmental perturbations that act in the orbit of the debris are considered in the dynamics. Such forces acting in the solar sail can force the growth of the eccentricity of these objects in the GEO orbit. Several authors have presented models of the solar radiation pressure considering the single-averaged model. But, doing a literature research, we found that the authors consider the Earth around the Sun in a circular and inclined orbit. Our contribution to the SRP model is in developing a different approach from other authors, where we consider the Sun in an elliptical and inclined orbit, which is valid for other bodies in the solar system when the eccentricity cannot be neglected. The expression of the SRP is developed up to the second order. We found that the first-order term is much superior to the second-order term, so the quadrupole term can be neglected. Another contribution is the approach to identify the initial conditions of the perigee argument (g) and the longitude of the ascending node (h), where some values of the (g,h) plane contribute to amplify the eccentricity growth. In the numerical simulations we consider real data from space debris removed from the site Stuff in Space. The solar sail helps to clean up the space environment using a propulsion system that uses the Sun itself, a clean and abundant energy source, unlike chemical propellants, to contribute to the sustainability of space exploration.

A versatile method for target area coverage analysis with arbitrary satellite attitude maneuver paths

The agile Earth observation satellite (AEOS) is rapidly becoming the leading platform in Earth observation missions, as it implements attitude maneuvers and remote sensing simultaneously. However, present methods limit not only its remote sensing coverage to a small area during attitude maneuvering but also its operation to only certain orbit shapes. This paper proposes a new method, namely envelope curves hierarchical overlapping (ECHO), for the target area coverage analysis of an AEOS that implements attitude maneuvers during its mission. This method involves two stages. In the preparation stage, the satellite's instantaneous access area (IAA) is defined and analytically determined by solving the spatial equations of an oblate Earth and the sensor's view field. Then, the coverage belt of an AEOS is delineated by the envelope curves of a set of IAAs. In the manipulation stage, the coverage problem of a target area is solved by iteratively using the polygon clipping technique and hierarchically taking the intersection parts between the coverage belt and the target area. Numerical simulations demonstrate the practicability and high accuracy of the proposed method.

Prescribed-time time-varying sliding mode based integrated translation and rotation control for spacecraft formation flying

<p style='text-indent:20px;'>This paper studies the prescribed-time spacecraft formation flying problem. The coupled 6-degrees-of-freedom kinematics and dynamics for spacecraft are modeled on Lie group SE(3), where the gravity-gradient torque, the perturbing forces caused by Earth's oblateness and unknown external disturbances are all taken into account. The relative configuration is expressed by using exponential coordinates of SE(3). In order to regulate the relative motion between the leader spacecraft and follower spacecraft, a novel prescribed-time control law is given based on a prescribed-time time-varying high-gain sliding mode method. Numerical simulations verify the effectiveness of the proposed control scheme.</p>

Deployment strategies of a satellite constellation for polar ice monitoring

This research considers a constellation of 16 satellites equipped with SAR sensors and tailored to monitoring the polar ice evolution, with a suitable revisit time over the regions of interest. Satellite deployment includes three phases: (i) orbit injection, performed by the upper stage of the launch vehicle, (ii) orbit plane selection, and (iii) orbit phasing. This work is primarily focused on phase (ii). Carrier spacecraft are proposed as a valuable option to place the majority of satellites in their orbits. Two distinct strategies are proposed to complete this task. The first strategy is based on the use of chemical propulsion, combined with the perturbing action due to Earth oblateness. The second strategy considers the use of low-thrust electric propulsion, in conjunction with nonlinear orbit control. A comparison between these two approaches is drawn, in terms of deployment time and final mass ratio of the carrier. Orbit phasing concludes the constellation deployment, and is carried out by each satellite. A tradeoff is proven to exist between phasing time and propellant expenditure.

Re-entry prediction and demisability analysis for the atmospheric disposal of geosynchronous satellites

The paper presents a re-entry analysis of Geosynchronous Orbit (GSO) satellites on disposal trajectories that enhance the effects of the Earth oblateness and lunisolar perturbations. These types of trajectories can lead to a natural re-entry of the spacecraft within 20 years. An analysis was performed to characterise the entry conditions for these satellites and the risk they can pose for people on the ground if disposal via re-entry is used. The paper first proposes a methodology to interface the long-term propagation used to study the evolution of the disposal trajectories and the destructive re-entry simulations used to assess the spacecraft casualty risk. This is achieved by revisiting the concept of overshoot boundary. The paper also presents the demisability and casualty risk analysis for a representative spacecraft configuration, showing that the casualty risk is greater than the 10$^{-4}$ threshold and that further actions should be taken to improve the compliance of these satellites in case of disposal via re-entry is used.

Orbit and Attitude Control of a GEO Satellite in Station Keeping Flight Mode

In this study, the performance requirements influencing the orbital and attitude control system of a geostationary satellite in the station-keeping flight mode considering the coupling effect of both attitude and orbital motion is determined. Controlling and keeping the satellite in its orbital window have been done using a set of four thrusters located on one side of the satellite body, with considering the coupling effect of the attitude motion on orbital motion. The satellite’s orbital motion could be disturbed by the attitude motion in the allowable orbital window. The main factors conducting this behavior are derived utilizing the satellite attitude and orbital dynamic equations of motion. In the mathematical analysis of this study, the effects of environmental perturbations originating from the oblateness of Earth, third mass gravity like sun and moon, and solar radiation pressure on the satellite dynamic behavior are also considered. Afterward, the condition of using four installed thrusters on one side of the satellite and the reaction wheels in order to control the satellite orbital and attitude motion is investigated. To reduce the satellite attitude’s error, a proportional-derivative controller is employed to activate the reaction wheels properly. The satellite positions in north-south and east-west directions are controlled by a specific array of thrusters in order to maintain in its predefined orbital window. The required amount of velocity variations for a duration of one year via some simulation may demonstrate the effectiveness of the proposed approach in enhancing the orbital maintenance procedure of the satellite.

Relativistic positioning: including the influence of the gravitational action of the Sun and the Moon and the Earth\u2019s oblateness on Galileo satellites

Uncertainties in the satellite world lines lead to dominant positioning errors. In the present work, using the approach presented in Puchades and Sáez (Astrophys. Space Sci. 352, 307–320, 2014), a new analysis of these errors is developed inside a great region surrounding Earth. This analysis is performed in the framework of the so-called Relativistic Positioning Systems (RPS). Schwarzschild metric is used to describe the satellite orbits corresponding to the Galileo Satellites Constellation. Those orbits are circular with the Earth as their centre. They are defined as the nominal orbits. The satellite orbits are not circular due to the perturbations they have and to achieve a more realistic description such perturbations need to be taken into account. In Puchades and Sáez (Astrophys. Space Sci. 352, 307–320, 2014) perturbations of the nominal orbits were statistically simulated. Using the formula from Coll et al. (Class. Quantum Gravity. 27, 065013, 2010) a user location is determined with the four satellites proper times that the user receives and with the satellite world lines. This formula can be used with any satellite description, although photons need to travel in a Minkowskian space-time. For our purposes, the computation of the photon geodesics in Minkowski space-time is sufficient as demonstrated in Puchades and Sáez (Adv. Space Res. 57, 499–508, 2016). The difference of the user position determined with the nominal and the perturbed satellite orbits is computed. This difference is defined as the U-error. Now we compute the perturbed orbits of the satellites considering a metric that takes into account the gravitational effects of the Earth, the Moon and the Sun and also the Earth oblateness. A study of the satellite orbits in this new metric is first introduced. Then we compute the U-errors comparing the positions given with the Schwarzschild metric and the metric introduced here. A Runge-Kutta method is used to solve the satellite geodesic equations. Some improvements in the computation of the U-errors using both metrics are introduced with respect to our previous works. Conclusions and perspectives are also presented.

Dynamic Models of Satellite Relative Motion and their effects on Kalman Filter

In this paper we will study the effects of an increasingly complex Kalman filter state transition matrix on the accuracy of the estimation of a non-linear relative navigation system. The propagation model will be constant in all the cases. The measurements are generated using Xu-Wang model and then noise is added. Whereas the process non-linear vector function/propagation matrix of the Kalman filter will be modified. In the first scenario the Clohessy-Wiltshire equations are considered as the basic propagation model of the Kalman Filter. In the second scenario the eccentricity of the chief satellite’s orbit is included in the model of the Kalman Filter. In the 3rd case we are using the model presented by Xu-Wang which incorporates the oblateness of Earth into the Tschauner-Hempel model. The divergence of the Kalman Filter is studied and provides a guideline to design engineers.

Practical Optimization of Low-Thrust Minimum-Time Orbital Rendezvous in Sun-Synchronous Orbits

High-specific-impulse electric propulsion technology is promising for future space robotic debris removal in sun-synchronous orbits. Such a prospect involves solving a class of challenging problems of low-thrust orbital rendezvous between an active spacecraft and a free-flying debris. This study focuses on computing optimal low-thrust minimum-time many-revolution trajectories, considering the effects of the Earth oblateness perturbations and null thrust in Earth shadow. Firstly, a set of mean-element orbital dynamic equations of a chaser (spacecraft) and a target (debris) are derived by using the orbital averaging technique, and specifically a slow-changing state of the mean longitude difference is proposed to accommodate to the rendezvous problem. Subsequently, the corresponding optimal control problem is formulated based on the mean elements and their associated costate variables in terms of Pontryagin’s maximum principle, and a practical optimization procedure is adopted to find the specific initial costate variables, wherein the necessary conditions of the optimal solutions are all satisfied. Afterwards, the optimal control profile obtained in mean elements is then mapped into the counterpart that is employed by the osculating orbital dynamics. A simple correction strategy about the initialization of the mean elements, specifically the differential mean true longitude, is suggested, which is capable of minimizing the terminal orbital rendezvous errors for propagating orbital dynamics expressed by both mean and osculating elements. Finally, numerical examples are presented, and specifically, the terminal orbital rendezvous accuracy is verified by solving hundreds of rendezvous problems, demonstrating the effectiveness of the optimization method proposed in this article.

Tidal Geopotential Dependence on Earth Ellipticity and Seawater Density and Its Detection With the GRACE Follow‐On Laser Ranging Interferometer

AbstractOcean tides produce significant gravitational perturbations that affect near‐Earth orbiting spacecraft. The gravitational potential induced by tidal mass redistribution is routinely modeled for global gravity analysis and orbit determination, although generally by assuming a spherical Earth and a uniform seawater density. The inadequacy of these simplifications is here addressed. We have developed an accurate yet efficient algorithm to compute the ocean tidal geopotential, allowing for Earth's elliptical shape and variable seawater density. Using this new computation, we find that (1) the effect of ellipticity is several percent of the tide signal over mid to high‐latitude regions, which is comparable to elevation error in the state‐of‐the‐art ocean tide models; (2) the effect of seawater density variations on the potential is as large as 2–3 cm in water‐height equivalent, primarily in deep water where density increases 2%–3% from compressibility. Our analysis of new Gravity Recovery and Climate Experiment Follow‐On (GRACE‐FO) laser ranging interferometer measurements reveals evident errors when ellipticity and density variations are ignored. When accounted for, the GRACE‐FO residual tidal gravity perturbations are reduced by half, depending on the adopted tide model; only the remaining half likely represents actual model elevation error. The use of a spherical surface and a uniform seawater density is no longer tenable given the precision of gravity measurements from GRACE and GRACE‐FO satellites.

Self-Maintenance Orbits Using The perturbations Due to Air Drag and Due to Earth's Oblateness

Abstract The focus of this paper is the design of a self-maintenance orbit using two natural forces against each other. The effect of perturbations due to Earth's oblateness up to the third order on both the semi-major axis and eccentricity for a low Earth orbit satellite together with the perturbation due to air drag on the same orbital parameters were used, in order to create self-maintenance orbits. Numerical results were simulated for a low earth orbit satellite, which substantiates the applicability of the results.

Balanced Low Earth Satellite Orbits

The present work aims at constructing an atlas of the balanced Earth satellite orbits with respect to the secular and long periodic effects of Earth oblateness with the harmonics of the geopotential retained up to the 4th zonal harmonic. The variations of the elements are averaged over the fast and medium angles, thus retaining only the secular and long periodic terms. The models obtained cover the values of the semi-major axis from 1.1 to 2 Earth’s radii, although this is applicable only for 1.1 to 1.3 Earth’s radii due to the radiation belts. The atlas obtained is useful for different purposes, with those having the semi-major axis in this range particularly for remote sensing and meteorology.