Classical Orbital Elements Research Articles

A simplified analytical approach for determining eclipses of satellites occulted by a celestial body

The primary objective of this paper is to construct an analytical model for determining the total duration of eclipse events of satellites. The approach is based on the assumption that the trace formed in the orbital plane, cutting body’s shadow under the classical conical shadow model, can be described as an ellipse. This allows for the derivation of its parameters through the use of classical orbital elements and solar position in the body’s centric frame. Consequently, the problem of identifying satellite occultation events is simplified to searching for points of intersection between two ellipses: the satellite orbit and the shadow. The developed model has been demonstrated to be applicable for a wide range of inclinations, with the exception of cases where the orbital plane of the satellite's motion coincides with that of the sun's in the body-centric frame. It is shown that the shadow function constructed under the aforementioned assumptions can be simplified in the cylindrical shadow model. Several simplifications are provided in the research. The paper also considers the extension of the model to account for the non-spherical shape of celestial bodies and presents an algorithm for accounting for changes in eclipse duration caused by the heliocentric motion of celestial bodies. The model has been validated through numerous tests with a numerical algorithm and satellite real data, as well as with other analytical models.

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.

Approximations for Secular Variation Maxima of Classical Orbital Elements under Low Thrust

The reachability assessment of low-thrust spacecraft is of great significance for orbital transfer, because it can give a priori criteria for the challenging low-thrust trajectory design and optimization. This paper proposes an approximation method to obtain the variation maximum of each orbital element. Specifically, two steps organize the contribution of this study. First, combined with functional approximations, a set of analytical expressions for the variation maxima of orbital elements over one orbital revolution are derived. Second, the secular approximations for the variation maxima of the inclination and the right ascension of the ascending node are derived and expressed explicitly. An iterative algorithm is given to obtain the secular variation maxima of the other orbital elements the orbital elements other than the inclination and right ascension of the ascending node. Numerical simulations for approximating the variation maxima and a preliminary application in estimation of the velocity increment are given to demonstrate the efficiency and accuracy of the proposed method. Compared with the indirect method used alone for low-thrust trajectory optimization, the computation burden of the proposed method is reduced by over five orders of magnitude, and the computational accuracy is still high.

Inner third-body perturbations including the inclination and eccentricity of the perturbing body

ABSTRACT In the field of the orbital perturbations dealt with an approach based on the mean orbital elements theory, the outer third-body gravitational attraction has been widely investigated. On the contrary, since it represents a less common case in the Solar system, the inner third-body perturbation has only recently been considered. The aim of this paper is to provide a more rigorous formulation of the inner third-body perturbation using a double-averaged analytical model. The disturbing potential function of the inner third-body is expanded in Legendre polynomials up to the second order. Afterwards, it is averaged first with respect to the orbital period of the disturbing body and then with respect to the orbital period of the probe. This procedure eliminates the short periodic motion terms. By applying to the double-averaged disturbing potential, the Lagrange planetary equations, the equations which describe the long-term and the secular variations of the classical orbital elements have been obtained: they present an analogy with those related to the planetary oblateness. Lastly, several cases of inner third-body perturbation in the Solar system are discussed, with the conclusion that this is a disturbance of relevance for binary asteroidal systems.

Analytical spacecraft relative dynamics with gravitational, drag and third-body perturbations

There has been a growing interest in spacecraft formation flying for space science applications. Such missions will require an accurate and efficient dynamics model, on-board the flight computer, to calculate and control the desired relative motion. Hence, an analytical dynamics model which can be applied to eccentric orbits and includes perturbations can provide an increase in accuracy and efficiency. This paper develops an accurate analytical solution of relative motion between two spacecraft using classical orbital elements. The analytical solution is obtained by propagating the orbital elements forward in time, while taking into account gravitational field up to the fifth harmonic, third-body and drag secular and periodic perturbations, and calculating the relative motion in the local-vertical–local-horizontal reference frame at each time-step. The analytical solution was observed to accurately describe the relative motion when compared with a numerical simulator, yielding errors on the order of meters for separation distances on the order of hundreds of meters.

Performance Assessment of a Turn Around Ranging in Communication Satellite Orbit Determination

Satellite operators utilize a two-stations turn around ranging (TAR) system to reduce the ground station measurement system's complexity and cost while having the same or better orbit determination accuracy for communication satellites orbit determination recently. This study investigates two stations' performance, four-way ranging on communication satellite orbit determination, operational conformance, and cost. The observation data sets are collected using traditional single station tracking (SST) and the new method TAR. The computed results using the Monte Carlo method encourage the satellite operators to use a four-way ranging system to observe and measure required data sets. TAR performance is evaluated, taking SST as a reference. The six classical orbital elements (a, e, i, RAAN, AoP, and TA) are compared for large numbers of observation data. The SST and TAR results are very close to each other. The worst-case calculated Euclidian distance between SST and TAR is 1.893 km at the epoch below the 6 km success criteria. The TAR observation method is appropriate to collect data sets for precise orbit determination. This work result indicates that satellite operators should consider deploying TAR stations to collect two-station range data sets and compute the orbit for nominal north-south station-keeping maneuvers (NSSK) and east-west station-keeping (EWSK) maneuver operations. The TAR method is superior to SST in terms of accuracy, operational conformance, and costs.

Trajectory Planning for Optical Satellite’s Continuous Surveillance of Geostationary Spacecraft

This paper focuses on the trajectory planning that optical satellites approach the geostationary (GEO) spacecraft and the trajectory planning that the optical satellite flies around the GEO spacecraft for continuous surveillance of the GEO spacecraft The optical satellite approaches the GEO spacecraft with continuous thrust and flies around the GEO spacecraft without thrust. The observation angles of the optical satellite to the GEO spacecraft are less than 30 degrees throughout the entire fly-around mission. Two methods are used in fly-around trajectory planning: trajectory planning based on classical orbital elements and trajectory planning based on CW equations. The advantage of trajectory planning based on classical orbital elements is that the position of the optical satellite relative to the GEO spacecraft is clear in earthcentered-inertial frame and the calculation method of fly-around orbital elements is simple. If the orbital inclination of the GEO spacecraft is less than 0.02 degrees and the semi-major axis of the elliptical relative trajectory of the fly-around formation is greater than 60 km, fly-around trajectory planning based on classical orbital elements is more efficient and easier than that based on CW equations. Optimal control theory is used in the trajectory planning for optical satellites approaching GEO spacecraft and the trajectory optimization is a two-point boundary-value problem. Simulations have proved that this method is feasible, which is highly effective in engineering. This method can be used for space situational awareness and on-orbit servicing missions of GEO spacecraft.

Satellite Relative Motion Modeling and Estimation via Nodal Elements

In this paper, a new parameterization of the relative motion between two satellites orbiting a central body is presented. The parameterization is based on the nodal elements: a set of angles describing the orbit geometry with respect to the relative line of nodes. These are combined with classical orbital elements to yield a nonsingular relative motion description. The exact nonlinear, perturbed dynamic model resulting from the new parameterization is established. The proposed parameter set captures the fundamental Keplerian invariants, while retaining a simple relationship with local orbital coordinates. An angles-only relative navigation filter and a collision avoidance scheme are devised by exploiting these features. The navigation solution is validated on a case study of an asteroid flyby mission. It is shown that a collision can be detected early on in the estimation process, which allows one to issue a timely evasive maneuver.

Attitude control: A key factor during the design of low-thrust propulsion for CubeSats

Orbital capability is a decisive step forward for nanosatellites in general and CubeSats in particular. Although trajectory maneuvers and their implementation have been thoroughly studied for classical satellites, the high level of constraints on CubeSats in terms of mass, volume and power, makes the transition delicate. Orbit, attitude and power control subsystems available for this format limit too optimistic performance available in literature. To verify this hypothesis, we simulate trajectory maneuvers in Earth orbit with representative CubeSat hardware and software. A low-thrust trajectory solver based on classical orbital elements from the literature is adapted to our context. A home-made attitude control simulation tool is coupled to include both control and perturbative dynamics. Increases in time and propellant consumption of more than 100% are caused by thrust direction errors such as misalignments and attitude control limitations, sometimes leading to mission loss. These results highlight an important increase in complexity for the CubeSat format that is not covered by the usual approach. Such limitations should be considered from the very start of the design of a nanosatellite mission with trajectory modification requirements.

The circular restricted three-body problem in curvilinear coordinates

We derive the exact equations of motion for the circular restricted three-body problem in cylindrical curvilinear coordinates together with a number of useful analytical relations linking curvilinear coordinates and classical orbital elements. The equations of motion can be seen as a generalization of Hill’s problem after including all neglected nonlinear terms. As an application of the method, we obtain a new expression for the averaged third-body disturbing function including eccentricity and inclination terms. We employ the latter to study the dynamics of the guiding center for the problem of circular coorbital motion providing an extension of some results in the literature.

Second-Order Integral-Form Gauss’s Variational Equations Under Impulsive Control

Second-order integral-form Gauss’s variational equations (GVEs) under impulsive control are derived for both nonsingular orbital elements and classical orbital elements. These equations are used to estimate the changes of the orbital elements under assigned impulsive vector. These GVEs are obtained by solving the second-order partial derivatives of orbital elements with respect to velocity vector. In addition, using analytical series approximation in the second-order GVEs, the required impulsive control is obtained for specified orbital-element variations. Based on the proposed second-order GVEs, coplanar two-impulse maneuver problems are solved to perform corrections of three and four orbital elements, respectively. Compared with the classical linear GVEs, the proposed second-order GVEs are more accurate to estimate the orbital-element variations and to solve for the required impulsive control. Several numerical examples are provided to validate the accuracy gain of the second-order GVEs.

Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincar\'e mapping of perturbed Keplerian motion is a key topic in celestial mechanics and astrodynamics, e.g. to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits over many revolutions. For this, the method uses the high-order Taylor expansion of a Poincar\'e or stroboscopic map, which is accurate close to the expansion point. In this paper, we investigate the performance of the HOTM method using different element sets for building the high-order map. The element sets investigated are the classical orbital elements, modified equinoctial elements, Hill variables, cylindrical coordinates and Deprit's ideal elements. The performances of the different coordinate sets are tested by comparing the accuracy and efficiency of mapping low-Earth and highly-elliptical orbits perturbed by $J_2$ with numerical propagation. The accuracy of HOTM depends strongly on the choice of elements and type of orbit. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and centre manifolds of Poincar\'e maps.

Linearized relative motion equations through orbital element differences for general Keplerian orbits

A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits. This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rotating frame with respect to the orbital elements. The resulted solution is expressed in terms of two different sets of orbital elements. The first one is the classical orbital elements and the second one is the nonsingular orbital elements. Among of them, however, the semi-latus rectum and true anomaly are used due to their generality, rather than the semi-major axis and mean anomaly that are used in most references. This specific selection for orbital elements yields a new solution that is universally applicable to elliptic, parabolic and hyperbolic orbits. It is shown that the new orbital element-based relative motion equations are equivalent to the Tschauner-Hempel equations. A linear map between the initial orbital element differences and the integration constants associated with the solution of the Tschauner-Hempel equations is constructed. Finally, the presented solution is validated through comparison with a high-fidelity numerical orbit propagator. The numerical results demonstrate that the new solution is computationally effective; and the result is able to match the accuracy that is required for linear propagation of spacecraft relative motion over a broad range of Keplerian orbits.

Robust Positioning, Preliminary Orbit Determination, and Trajectory Prediction of Space Debris using In-Space Iterative-Bearing-Only Observations

This paper is concerned with the preliminary localisation, orbit determination and model-based path forecasting of space debris based on a robust procedure. In this work, an in-orbit observer utilises only relative bearing observations iteratively. To this end, the problem is first formulated in order to calculate the distance vector between the space debris and any orbiting observer. Afterwards, the obtained position vector is corrected through an Extended Kalman Filter (EKF) for shrinking the sensor and process errors and increasing robustness of the computations in the presence of uncertainties. After preliminary positioning, the related classical orbital elements are acquired via the predicted position and velocity vectors using a hybrid technique. Extensive simulations demonstrate the efficacy and robustness of the aforementioned method, and in particular it is verified that the proposed scheme is capable of producing a suitable solution for preliminary localisation and orbit determination of space debris based on the presented space-based observation, which is practical in phasing and chasing manoeuvres of any grabber space robot.

Analysis of relative motion in non-Keplerian orbits via modified equinoctial elements

This paper discusses a mathematical model to determine an analytical form of the equations describing the relative motion of two spacecraft that, using a suitable continuous-thrust propulsion system, track highly non-Keplerian orbits, whose orbital plane does not contain the primary's center-of-mass. The relative motion is described within a rotating reference frame via modified equinoctial elements, thus eliminating the singularities that arise when a set of classical orbital elements is used. In this sense, the paper completes and extends the recent analysis of the relative motion of two spacecraft in closed (either circular or elliptic) displaced orbits. When the eccentricity of the displaced orbits is sufficiently small, the method is able to calculate the approximate bounds of the two spacecraft relative distances using a semi-analytical approach and with a reduced computational effort. Some numerical simulation results provide an evidence of the effectiveness of the proposed method.

Invariant Manifold and Bounds of Relative Motion Between Heliocentric Displaced Orbits

This paper discusses a methodology for modeling the relative motion between heliocentric displaced orbits by using the Cartesian state variables in combination with a set of displaced orbital elements. Similar to classical Keplerian orbital elements, the newly defined set of displaced orbital elements has a clear physical meaning and provides an alternative approach to obtain a closed-form solution to the relative motion problem between displaced orbits, without linearizing or solving nonlinear equations. The invariant manifold of relative motion between two arbitrary displaced orbits is determined by coordinate transformations, thus obtaining a straightforward interpretation of the bounds, namely, maximum and minimum relative distances of three directional components. The extreme values of these bounds are then calculated from an analytical viewpoint, both for quasi-periodic orbits in the incommensurable case and periodic orbits in the 1:1 commensurable case. Moreover, in some degenerate cases, the extreme values of relative distance bounds can also be solved analytically. For each case, simulation examples are discussed to validate the correctness of the proposed method.

Trajectory Guidance Using Periodic Relative Orbital Motion

Relative motion of one satellite about another in circular orbit, where the two objects have the same semimajor axis, is periodic in the linearized approximation. A set of orbital elements, the geometric relative orbital elements, which are an exact geometric analogy to the classical orbital elements, can be defined. The relative orbit is manifestly seen to be an ellipse or circle in apocentral coordinates, analogous to perifocal coordinates in inertial motion and different from the local-vertical local-horizontal Cartesian coordinates customarily used for analysis of relative motion problems. These provide a way to do relative motion trajectory design and guidance that stands in contrast to the use of Cartesian coordinates. Algorithms are presented for computing the delta- and timing for impulsive maneuvers to change a single element: two that change the plane and one that changes the size of the relative orbit. The change in size is a two-maneuver transfer and uses the solution of the three-point periodic boundary value problem, also solved and presented here. This solution permits the direct computation, with no iteration or searching, of the periodic relative orbit that connects two points. Optimization to minimize fuel use can be done with a one-dimensional search.

The binary Asteroid 22 Kalliope: Linus orbit determination on the basis of speckle interferometric observations

In this paper we present the orbital elements of Linus satellite of 22 Kalliope asteroid. Orbital element determination is based on the speckle interferometry data obtained with the 6-m BTA telescope operated by SAO RAS. We processed 9 accurate positions of Linus orbiting around the main component of 22 Kalliope between 10 and 16 December, 2011. In order to determine the orbital elements of the Linus we have applied the direct geometric method. The formal errors are about 5mas. This accuracy makes it possible to study the variations of the Linus orbital elements influenced by different perturbations over the course of time. Estimates of six classical orbital elements, such as the semi-major axis of the Linus orbit a=1109±6km, eccentricity e=0.016±0.004, inclination i=101°±1° to the ecliptic plane and others, are presented in this work.

The unified ideal model of mean motion resonance of artificial satellites due to geopotential perturbations

From Kaula's Earth gravitational potential written in classical orbital elements, the unified ideal model of mean motion resonance of artificial satellites due to geopotential perturbations is developed in this paper first, through a suitable sequence of canonical transformations constructed by implicit functions. This unified ideal orbital resonance model is valid for all the commensurabilities between the rotational angular velocity of the Earth and the angular velocities of mean orbital motion of artificial satellites with arbitrary inclination and small eccentricity, and can be also transformed into Garfinkel's general expression of ideal resonance problem. Then 1/1 resonance of the 24-hour satellite with arbitrary inclination and small eccentricity is analyzed under the effect of harmonics of J (2) and J (22) of the geopotential, based on the unified ideal model of mean motion resonance. The analytical expressions of the libration period and libration half width of the 1/1 resonance of the 24-hour satellite with arbitrary inclination and small eccentricity are presented.

Near-Earth asteroid flyby trajectories from the Sun-Earth L2 for Chang’e-2’s extended flight

Driven by curiosity about possible flight options for the Chang’e-2 spacecraft after it remains at the Sun-Earth L2 point, effective approaches were developed for designing preliminary fuel-optimal near-Earth asteroid flyby trajectories. The approaches include the use of modified unstable manifolds, grid search of the manifolds’ parameters, and a two-impulse maneuver for orbital phase matching and z-axis bias change, and are demonstrated to be effective in asteroid target screening and trajectory optimization. Asteroid flybys are expected to be within a distance of 2 × 107 km from the Earth owing to the constrained Earth-spacecraft communication range. In this case, the spacecraft’s orbital motion is significantly affected by the gravities of both the Sun and the Earth, and therefore, the concept of the “heliocentric oscillating-Kepler orbit” is proposed, because the classical orbital elements of the flyby trajectories referenced in the heliocentric inertial frame oscillate significantly with respect to time. The analysis and results presented in this study show that, among the asteroids whose orbits are the most accurately predicted, “Toutatis”, “2005 NZ6”, or “2010 CL19” might be encountered by Chang’e-2 in late 2012 or 2013 with total impulses less than 100m/s.