Order Gravity Field Research Articles

Orbit Determination of High Orbit Spacecraft based on BDS-3 using Extended Kalman Filter

The positioning of highly elliptical orbit (HEO) spacecraft is very important in deep space exploration and other missions. The Beidou Navigation Satellite System (BDS-3), which was completed in 2020, has put forward new ideas for using GNSS technology to make up for the lack of earth-based TT&C systems. At the same time, the selection of the gravity field order and the perturbation forces must be considered when using dynamical methods for HEO spacecraft positioning. In this paper, simulation experiments are carried out based on the Two-Line Orbital Element (TLE) data of BDS. Two HEO satellites with different semimajor-axis (67509.6 km, 212857.3 km) are selected as the target spacecraft orbits. The single point positioning (SPP) method and Extended Kalman Filter (EKF) method are used for HEO spacecrafts positioning respectively. The experiments show that a 20×20 order gravity field model can be chosen for the orbital integral of HEO spacecraft orbits, and the integration error caused by solar gravity and lunar gravity should be taken into account with the change of orbital altitude. Adding Geostationary Orbit (GEO) and Inclined GeoSynchronous Orbit (IGSO) navigation satellites improved the visibility of these two orbits by 19.45% and 16.64%, respectively, and improved the GDOP value by 14.1% and 3.8%, respectively, compared to observing MEO satellites only. The use of the EKF method can effectively improve the positioning accuracy of the HEO spacecraft compared to the SPP method. For the spacecraft in HEO-1, the RMS value of the positioning errors of the EKF method was 67.82 m, and the accuracy of each direction was improved by 44.28%, 38.85%, and 38.62%, respectively. For the spacecraft in HEO-2, the positioning errors in the x direction of EKF method was within 12 km and within 2 km in the y and z directions. The accuracy of these three directions was improved by 84.91%, 79.24%, and 58.64%, respectively.

Periodic attitudes and bifurcations of a rigid spacecraft in the second degree and order gravity field of a uniformly rotating asteroid

In this work, periodic attitudes and bifurcations of periodic families are investigated for a rigid spacecraft moving on a stationary orbit around a uniformly rotating asteroid. Under the second degree and order gravity field of an asteroid, the dynamical model of attitude motion is formulated by truncating the integrals of inertia of the spacecraft at the second order. In this dynamical system, the equilibrium attitude has zero Euler angles. The linearised equations of attitude motion are utilised to study the stability of equilibrium attitude. It is found that there are three fundamental types of periodic attitude motions around a stable equilibrium attitude point. We explicitly present the linear solutions around a stable equilibrium attitude, which can be used to provide the initial guesses for computing the true periodic attitudes in the complete model. By means of a numerical approach, three fundamental families of periodic attitudes are studied, and their characteristic curves, distribution of eigenvalues, stability curves and stability distributions are determined. Interestingly, along the characteristic curves of the fundamental families, some critical points are found to exist, and these points correspond to tangent and period-doubling bifurcations. By means of a numerical approach, the bifurcated families of periodic attitudes are identified. The natural and bifurcated families constitute networks of periodic attitude families.

Dynamics of Haumea’s dust ring

The particle dynamics of the recently observed ring around dwarf planet Haumea is numerically investigated. The point mass gravitational force, a second degree and order gravity field, and the solar radiation pressure as the main perturbations are considered. The quasi-stationary state of the ring varies for different micron-sized grains and depends also on the spin-orbit resonances between the rotation rate of the main body and the orbital period of the dust particles. The simulations confirm the variable radial width of the ring observed during the transit ingress and egress. Results show that the micron sized grains, initially on circular orbits, become eccentric and form an apse-aligned ring at the observed radial distance near to the 3:1 spin-orbit resonance. It is also demonstrated that this coincidence is only apparent and independent of 3:1 resonance.

Application of Geoid Anomalies to the Tectonic Research in the East Asian Continental Margin

In this paper, we calculated multi-scale residual geoid anomalies with the method of geoid separation processing, according to EGM2008 ultra-high order gravity field model, remove-restore technique and Stokes integral. The East Asian continental margin was selected as the study area. The residual geoid anomalies have been calculated by programming. On the basis of residual geoid anomalies at various orders, the interlayer geoid anomalies at different depths were calculated to depict the spatial distribution characteristics of the residual geoid. Finally, we conducted a detailed geophysical interpretation for the study area according to the geoid anomalies in combination with other geophysical datasets. Four conclusions can be outlined as follows: 1) it is impracticable that geoid anomalies are used in the interpretation of the shallow objects due to the influence of the terrain; 2) the anomalies of residual geoid can reflect the intensity of small-scale mantle convection in the asthenosphere; 3) the interlayer geoid anomalies can reflect the magmatic activities associated with the mantle convection and mantle plume in different scales; 4) the study of the geoid may provide an approach for the research of the subduction zone, mantle convection and mantle plume.

Hamiltonian structure and relative equilibria of an axisymmetric rigid body in a second degree and order gravity field: cylindrical and generalized hyperbolic equilibria

The full dynamics of an axisymmetric rigid body in a uniformly rotating second degree and order gravity field are investigated, where orbit and attitude motions of the body are coupled through the gravity. Compared with the classical orbital dynamics with the body considered as a point mass, the full dynamics is a higher-precision model in close proximity of the central body where the gravitational orbit–attitude coupling is significant, such as a spacecraft about a small asteroid or an irregular-shaped natural satellite about a planet. The full dynamics are modeled by using the non-canonical Hamiltonian structure, in terms of variables expressed in the frame fixed with the central body. A Poisson reduction is carried out by means of the axial symmetry of the body, and a reduced system with lower dimension, as well as its non-canonical Hamiltonian structure and equations of motion, is obtained through the reduction process. With the second-order potential, three types of relative equilibria are found to be possible: cylindrical equilibria, generalized hyperbolic equilibria, and conic equilibria, which are counterparts to cylindrical equilibria, hyperbolic equilibria, and conic equilibria of an axisymmetric rigid body in a spherical gravity field, respectively. The geometrical properties and existence of the cylindrical equilibria and generalized hyperbolic equilibria are investigated in detail. It has been found that compared with the classical results in a spherical gravity field, the relative equilibria in this study are more complicated and diverse. The most significant difference is that the non-spherical gravity field enables the existence of non-Lagrangian hyperbolic equilibria, called generalized hyperbolic, which cannot exist in a spherical gravity.

Reduced dynamic and kinematic precise orbit determination for the Swarm mission from 4 years of GPS tracking

Precise science orbits for the first 4 years of the Swarm mission have been generated from onboard GPS measurements in a systematic reprocessing using refined models and processing techniques. Key enhancements relate to the introduction of macro-models for a more elaborate non-gravitational force modeling (solar radiation pressure, atmospheric drag and lift, earth albedo), as well as carrier phase ambiguity fixing. Validation using satellite laser ranging demonstrates a 30% improvement in the precision of the reduced dynamic orbits with resulting errors at the 0.5–1 cm level (1D RMS). A notable performance improvement is likewise achieved for the kinematic orbits, which benefit most from the ambiguity fixing and show a 50% error reduction in terms of SLR residuals while differences with respect to reduced dynamic ephemerides amount to only 1.7 cm (median of daily 3D RMS). Compared to the past kinematic science orbits based on float-ambiguity estimates, the new kinematic position solutions exhibit a factor of reduction of two to three in Allan deviation at time scales of 1000s and higher, and promise an improved recovery of low-degree and -order gravity field coefficients in Swarm gravity field analyses.

Analysis of equilibria in the doubly synchronous binary asteroid systems concerned with non-spherical shape

This paper investigates the equilibria and their stabilities in the doubly synchronous binary asteroid systems, which are modelled as the two tri-axial ellipsoids with various shape and system parameters. Particularly, the in uences of shape and system parameters on equilibria are discussed analytically. Firstly, the geometrical models of doubly synchronous binary asteroid systems are established. The dual second degree and order gravity field is employed to approximate the gravitational potential of the system. Six shape and system parameters are defined. Then, based on the linearized perturbation equations, the explicit expressions of the offsets of equilibria in doubly synchronous systems are derived, which clearly illustrate the relationship between the distribution of equilibria and the variations of shape parameters. Further, the approximate expressions are applied to estimate the offsets of equilibria due to parameter errors, respectively. Finally, in order to have a better insight into the equilibriaum structure, the stabilities of equilibria under different system parameters are investigated. In particular, critical regions of triangular equilibria are calculated and the role of the relative distance on the stability is discussed in detail. This study could provide a preliminary analysis of equilibria for the mission design in doubly synchronous binary asteroid systems.

Stabilization of coupled orbit–attitude dynamics about an asteroid utilizing Hamiltonian structure

The gravitationally coupled orbit–attitude dynamics, also called the full dynamics, in which the spacecraft is modeled as a rigid body, is a high-precision model for the motion in the close proximity of an asteroid. A feedback control law is proposed to stabilize relative equilibria of the coupled orbit–attitude motion in a uniformly rotating second degree and order gravity field by utilizing the Hamiltonian structure. The feedback control law is consisted of potential shaping and energy dissipation. The potential shaping makes the relative equilibrium a minimum of the modified Hamiltonian by modifying the potential artificially. With the energy-Casimir method, it is theoretically proved that an unstable relative equilibrium can always be stabilized in the Lyapunov sense by the potential shaping with sufficiently large feedback gains. Then, the energy dissipation leads the motion to converge to the relative equilibrium. The proposed stabilization control law has a simple form and is easy to implement autonomously, which can be attributed to the utilization of natural dynamical behaviors in the controller design.

AIUB-RL02: an improved time-series of monthly gravity fields from GRACE data

The new release AIUB-RL02 of monthly gravity models from GRACE GPS and K-Band range-rate data is based on reprocessed satellite orbits referring to the reference frame IGb08. The release is consistent with the IERS2010 conventions. Improvements with respect to its predecessor AIUB-RL01 include the use of reprocessed (RL02) GRACE observations, new atmosphere and ocean dealiasing products (RL05), an upgraded ocean tide model (EOT11A), and the interpolation of shallow ocean tides (admittances). The stochastic parametrization of AIUB-RL02 was adapted to include daily accelerometer scale factors, which drastically reduces spurious signal at the 161 day period in C20 and at other low degree and order gravity field coefficients. Moreover, the correlation between the noise in the monthly gravity models and solar activity is considerably reduced in the new release. The signal and the noise content of the new AIUB-RL02 monthly gravity fields are studied and calibrated errors are derived from their non-secular and non-seasonal variability. The short-period time-variable signal over the oceans, mostly representing noise, is reduced by 50% with respect to AIUB-RL01. Compared to the official GFZ-RL05a and CSR-RL05 monthly models, the AIUB-RL02 stands out by its low noise at high degrees, a fact emerging from the estimation of seasonal variations for selected river basins and of mass trends in polar regions. Two versions of the monthly AIUB-RL02 gravity models, with spherical harmonics resolution of degree and order 60 and 90, respectively, are available for the time period from March 2003 to March 2014 at the International Center for Global Earth Models (ICGEM) or from ftp://ftp.unibe.ch/aiub/GRAVITY/GRACE

Orbital dynamics and equilibrium points around an asteroid with gravitational orbit–attitude coupling perturbation

The strongly perturbed dynamical environment near asteroids has been a great challenge for the mission design. Besides the non-spherical gravity, solar radiation pressure, and solar tide, the orbital motion actually suffers from another perturbation caused by the gravitational orbit–attitude coupling of the spacecraft. This gravitational orbit–attitude coupling perturbation (GOACP) has its origin in the fact that the gravity acting on a non-spherical extended body, the real case of the spacecraft, is actually different from that acting on a point mass, the approximation of the spacecraft in the orbital dynamics. We intend to take into account GOACP besides the non-spherical gravity to improve the previous close-proximity orbital dynamics. GOACP depends on the spacecraft attitude, which is assumed to be controlled ideally with respect to the asteroid in this study. Then, we focus on the orbital motion perturbed by the non-spherical gravity and GOACP with the given attitude. This new orbital model can be called the attitude-restricted orbital dynamics, where restricted means that the orbital motion is studied as a restricted problem at a given attitude. In the present paper, equilibrium points of the attitude-restricted orbital dynamics in the second degree and order gravity field of a uniformly rotating asteroid are investigated. Two kinds of equilibria are obtained: on and off the asteroid equatorial principal axis. These equilibria are different from and more diverse than those in the classical orbital dynamics without GOACP. In the case of a large spacecraft, the off-axis equilibrium points can exist at an arbitrary longitude in the equatorial plane. These results are useful for close-proximity operations, such as the asteroid body-fixed hovering.

Relative equilibria of full dynamics of a rigid body with gravitational orbit-attitude coupling in a uniformly rotating second degree and order gravity field

The motion of a rigid body in a uniformly rotating second degree and order gravity field is a good model for the gravitationally coupled orbit-attitude motion of a spacecraft in the close proximity of an asteroid. The relative equilibria of this full dynamics model are investigated using geometric mechanics from a global point of view. Two types of relative equilibria are found based on the equilibrium conditions: one is the Lagrangian relative equilibria, at which the circular orbit of the rigid body is in the equatorial plane of the central body; the other is the non-Lagrangian relative equilibria, at which the circular orbit is parallel to but not in the equatorial plane of central body. The existences of the Lagrangian and non-Lagrangian relative equilibria are discussed numerically with respect to the parameters of the gravity field and the rigid body. The effect of the gravitational orbit-attitude coupling is especially assessed. The existence region of the Lagrangian relative equilibria is given on the plane of the system parameters. Numerical results suggest that the negative C20 with a small absolute value and a negative C22 with a large absolute value favor the existence of the non-Lagrangian relative equilibria. The effect of the gravitational orbit-attitude coupling of the rigid body on the existence of the non-Lagrangian relative equilibria can be positive or negative, which depends on the harmonics C20 and C22, and the angular velocity of the rotation of the gravity field.

Observer-based body-frame hovering control over a tumbling asteroid

Observer-based hovering control over a tumbling asteroid in the body-fixed frame is studied. An extended Kalman filter (EKF) is used to process range measurements from a small collection of ground stations, yielding estimates of the spacecraft state vector and the gravitational parameters of the asteroid assuming a second degree and order gravity field model. The estimated states are used in the optimal feedback control algorithm which consists of two alternatives: time-varying LQR or the Lyapunov–Floquet transformation (LFT) and time-invariant LQR. The closed-loop response of the system and the control effort required are investigated and compared for both control strategies.

Evolution of the Motion Around a Slowly Rotating Body

The motion of a satellite about a rotating triaxial body will be investigated, stressing on the case of slow rotation. The Hamiltonian of the problem will be formed including the zonal harmonic J2 and the leading tesseral harmonics C22 and S22. The small parameter of the problem is the spin rate (� ) of the primary. The solution proceeds through three canonical transformations to eliminate in succession; the short, intermediate and long-period terms. Thus secular and periodic terms are to be retained up to orders four and two respectively. To understand the dynamics of a spacecraft or a natural particle around a celestial body (a planet, an asteroid or even a comet) it's convenient to take into account the spin rate (� ) of the primary, since it is important for several applications especially for geodetic satellites and when dealing with communication satellites where there's commensur-ability between the satellite period and � , and a case of resonance arises (1-3). Considering a slowly rotating earth like planet, the addition of tesseral harmonics is necessary (4, 5) ana- lyzed the orbital dynamics about an asteroid and established that the major perturbations acting on the orbiter are due to the leading harmonics of the geopotential. In a subsequent work (6) he tacked the problem of secular motion in a 2 nd degree and order-gravity field with no rotation qualitatively. In this respect the main problem of artificial satellite theory is very useful; the major contributors to this subject were (7- 10) formed the Hamiltonian of the motion of an A.S. about a planet with an inhomogeneous gravitational field including the leading zonal and tesseral harmonics. He used the Whit- taker variables and then he normalized the Hamiltonian us- ing the method of elimination of the parallax developed by (11). Instead of obtaining an explicit solution of the problem he performed an exhaustion analysis of the problem. The method adopted by Palacian, through elegant, but is very difficult to include higher order terms and higher order grav- ity coefficients. In this paper the gravitational force exerted by an earth like planet on an artificial satellite will be considered, the Hamiltonian of the problem will be formed, in terms of the Delaunay variables, with the earth's spin ratetaken as a small parameter of O(1).The planet's potential will be considered up to the leading zonal and leading tesseral harmonics, an outline of the perturbation technique is given which is based on the Lie- Deprit - Kamel transform. The Hamiltonian is then normalized through three suc- cessive canonical transformations to eliminating succession of the short, intermediate, and long period terms. The proce- dure followed facilitates such including higher order terms and higher coefficients of the geopotential. 2. THE GEOPOTENTIAL

Analysis and implementation of in-plane stationkeeping of continuously perturbed Walker constellations

The stationkeeping of symmetric Walker constellations is analyzed by considering the perturbations arising from a high order and degree Earth gravity field and the solar radiation pressure. These perturbations act differently on each group of spacecraft flying in a given orbital plane, causing a differential drift effect that would disrupt the initial symmetry of the constellation. The analysis is based on the consideration of a fictitious set of rotating reference frames that move with the spacecraft in the mean sense, but drift at a rate equal to the average drift rate experienced by all the vehicles over an extended period. The frames are also allowed to experience the J 2 -precession such that each vehicle is allowed to drift in 3D relative to its frame. A two-impulse rendezvous maneuver is then constructed to bring each vehicle to the center of its frame as soon as a given tolerance deadband is about to be violated. This paper illustrates the computations associated with the stationkeeping of a generic Walker constellation by maneuvering each leading spacecraft within an orbit plane and calculating the associated velocity changes required for controlling the in-plane motions in an exacting sense, at least for the first series of maneuvers. The analysis can be easily extended to lower flying constellations, which experience additional perturbations due to drag.

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.

Orbital resonance widths in a uniformly rotating second degree and order gravity field

Aims. In this work I seek to derive analytical expressions for the widths of mean motion resonances in uniformly rotating second degree and order gravity fields. Methods. A Deprit perturbation series is used to derive Hamiltonians valid close to mean motion resonances. These expressions are reduced to the Hamiltonian of a pendulum through canonical transformations. Results. Analytical expressions for the orbital resonance widths in uniformly rotating second degree and order gravity field are derived and tested against numerical simulations. The widths of these mean motion resonances are independent of the rotation rate and the mass of the central body. The general stability of retrograde orbits are explained by the overlap criteria.

Numerical determination of stability regions for orbital motion in uniformly rotating second degree and order gravity fields

The stability of orbital motion about a uniformly rotating arbitrary second degree and order gravity field is investigated. A normalized form of the equations of motion are derived and analyzed. A numerical stability criteria is proposed and used to evaluate the stability of initially near-circular orbits in the equatorial plane of the body. Regions of stable and unstable motion are clearly delineated, and are seen to be strongly related to resonances between the mean motion and the body rotation rate.

Gravity field determination of a Comet Nucleus: Rosetta at P/Wirtanen

One of the prime objectives of the Rosetta Radio Science Investigations (RSI) experiment is the determination of the mass, the bulk density and the low degree and order gravity of the nucleus of comet P/Wirtanen, the target object of the international Rosetta mission. The RSI experiment will use the spacecraft's radio carrier frequencies at X-band (8.4 GHz) and S-band (2.3 GHz) in order to measure slight changes of the orbit velocity via the classical Doppler effect induced by the gravity attraction of the comet nucleus. Based on an estimate of the background Doppler noise, it is expected that a mass determination (assuming a representative radius of 700 m and a bulk density of 500 kg/m3) at an accuracy of 0.1% can be achieved if the spacecraft's orbit is iteratively reduced below 7 km altitude. The gravity field of degree and order two can be detected for reasonable tracking times below 5 km altitude. The major competing forces acting on the spacecraft are the radiation pressure and the gas mass flux from cometary activity. While the radiation pressure may be predicted, it is recommended to begin a gravity mapping campaign well before the onset of outgassing activity (>3.25 AU heliocentric distance). Radial acceleration by water outgassing is larger by orders of magnitude than the accelerations from the low degree and order gravity field and will mask the contributions from the gravity field.

Secular Motion in a 2nd Degree and Order-Gravity Field with no Rotation

The motion of a particle about a non-rotating 2nd degree and order-gravity field is investigated. Averaging conditions are applied to the particle motion and a qualitative analysis which reveals the general character of motion in this system is given. It is shown that the orbit plane will either be stationary or precess about the body's axis of minimum or maximum moment of inertia. It is also shown that the secular equations for this system can be integrated in terms of trigonometric, hyperbolic or elliptic functions. The explicit solutions are derived in all cases of interest.

Changes in Rotational Angular Momentum Due to Gravitational Interactions Between Two Finite Bodies

Interactions between an initially uniformly rotating body with a second degree and order gravity field and a sphere are analyzed. Explicit predictions of the change in rotational angular momentum of the non-spherical body are derived over one interaction (i.e. periapsis passage) between the bodies. The estimated changes are expressed in terms of trigonometric functions and generalized Hansen coefficients.