Luni-solar Perturbations Research Articles

Orbital lifetime design and optimization of CORBES based on lunisolar perturbation

In order to design the orbit for the COntellation of Radiation BElt Survey(CORBES) mission orbit that meets the mission requirements and deorbit constraints, this paper proposed a universal method to design optimal orbital parameters for Highly Elliptic Orbit(HEO). The impact of lunisolar perturbation on perigee altitude is evident, and once a portion of the orbital parameters is determined, the mathematical relationship between the amplitude of the long-period variation in perigee (referred to as “perigee variation”) and the orbital parameters to be designed can be described. The perigee variation exhibits a significant correlation with the orbital lifetime. The initial optimization of orbit parameters is conducted by considering the correlation between lunisolar perturbations and orbital lifetime. Based on this, the orbital parameter RAAN is designed using an improved Decimal Ant Colony Optimization(DACO), and the corresponding launch window is given. The simulation results and comparisons with previous methods validate the effectiveness and efficiency of the initial optimization and the improved DACO in providing optimal feasible solutions, taking into account the law of lunisolar perturbation on orbital lifetime. This method can be well applied to design and estimate the lifetime of orbits dominated by lunisolar perturbation, such as HEO and Geostationary Transfer Orbit(GTO).

Effects of lunisolar perturbations on TianQin constellation: An analytical model

Effects of lunisolar perturbations on TianQin constellation: An analytical model

Statistical Study of Uncontrolled Geostationary Satellites near an Unstable Equilibrium Point

The growth of the population of space debris in the geostationary ring and the resulting threat to active satellites require insight into the dynamics of uncontrolled objects in the region. A Monte Carlo simulation analyzed the sensitivity to initial conditions of the long-term evolution of geostationary spacecraft near an unstable point of the geopotential, where irregular behavior (e.g., transitions between long libration and continuous circulation) occurs. A statistical analysis unveiled sudden transitions from order to disorder, interspersed with intervals of smooth evolution. There is a periodicity of approximately half a century in the episodes of disorder, suggesting a connection with the precession of the orbital plane, due to Earth’s oblateness and lunisolar perturbations. The third-degree harmonics of the geopotential also play a vital role. They introduce an asymmetry between the unstable equilibrium points, enabling the long libration mode. The unpredictability occurs just in a small fraction of the precession cycle, when the inclination is close to zero. A simplified model including only gravity harmonics up to degree 3 and the Earth and moon in circular coplanar orbits is capable of reproducing most features of the high-fidelity simulation.

Semi-analytical investigations on the dynamics of BeiDou inclined geosynchronous satellite orbit

A semi-analytical investigation on the dynamics of BeiDou IGSO is conducted in this paper. First, a semi-analytical propagator is constructed based on the mean element method. The comparison of results of semi-analytical propagation with reference orbits shows that all orbital elements except the semi-major axis agree well within six centuries. For some example orbits, the semi-analytical propagation of the semi-major axis is accurate only within several decades, which is related to the chaotic motion of the one-to-one tesseral resonance. Compared with the numerical ephemeris of Moon, an improvement of accuracy in the semi-analytical propagation is observed when a simple analytical ephemeris of Moon is adopted. This phenomenon agrees with the doubly averaging process of the third-body perturbation. Second, the dynamics of BeiDou IGSO are investigated using the Hamiltonian approach with simplified models. The transition between different resonance regions of one-to-one tesseral resonance is investigated by the computation of one-dimensional maximal Lyapunov characteristic exponent (1-mLCE). The secular resonance from the lunisolar perturbation modeled by a 2-DOF system is systematically investigated using the surface of section. The secondary resonance related to the precession of lunar node is identified by the computation of 1-mLCE. In the end, the reentry disposal for BeiDou IGSO is investigated as an optimization problem. With IGSO3 as an example of reentry orbit, the sensitivity analysis is conducted to study its stability and predictability.

On the Dominant Lunisolar Perturbations for Long-Term Eccentricity Variation: The Case of Molniya Satellite Orbits

The aim of this work is to investigate the main dominant terms of lunisolar perturbations that affect the orbital eccentricity of a Molniya satellite in the long term. From a practical point of view, these variations are important in the context of space situational awareness—for instance, to model the long-term evolution of artificial debris in a highly elliptical orbit or to design a reentry end-of-life strategy for a satellite in a highly elliptical orbit. The study assumes a doubly averaged model including the Earth’s oblateness effect and the lunisolar perturbations up to the third-order expansion. The work presents three important novelties with respect to the literature. First, the perturbing terms are ranked according to their amplitudes and periods. Second, the perturbing bodies are not assumed to move on circular orbits. Third, the lunisolar effect on the precession of the argument of pericenter is analyzed and discussed. As an example of theoretical a application, we depict the phase space description associated with each dominant term, taken as isolated, and we show which terms can apply to the relevant dynamics in the same region.

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.

Semi-Analytical Estimates for the Orbital Stability of Earth\u2019s Satellites

Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. (i) We demonstrate the long-term stability of the semimajor axis within the framework of the J_2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining {mathcal {H}}_{J_2}. (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian {mathcal {H}}_mathrm{gls}), after a suitable reduction of the Hamiltonian to the Laplace plane. (iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the {mathcal {H}}_{J_2} and {mathcal {H}}_mathrm{gls} models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the {mathcal {H}}_{J_2} model is non-convex, but satisfies a ‘three-jet’ condition, while the {mathcal {H}}_mathrm{gls} model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.

Regularized luni-solar gravity dynamics on resident space objects

Resident space object population in highly elliptical high perigee altitude (> 600 km) orbits is significantly affected by luni-solar gravity. Using regularization, an analytical orbit theory with luni-solar gravity effects as third-body perturbations in terms of Kustaanheimo-Stiefel regular elements is developed. Numerical tests with different cases resulted in good accuracy for both short- and long-term orbit propagations. It is observed that the luni-solar perturbations affect the accuracy of the analytical solution seasonally. The analytical theory is tested with the observed orbital parameters of the few objects in highly elliptical orbits. The analytical evolution of osculating perigee altitude is found to be concurrent with observed data. Solar perturbation, when compared with lunar perturbation, is established to be dominant over such orbits.

Spectrophotometric and dynamical properties of the soviet/russian constellation of molniya satellites

We report on the extended observing campaign of the surviving Soviet/Russian spacecraft Molniya, carried out in the years 2014–17 at Mexican and Italian telescopes. Spectrophotometry and astrodynamical analysis have been carried out for all the 43 demised payloads still in uncontrolled HEO orbit at late 2017, in order to assess dynamical and reflectance properties of such a wide family of virtually identical orbiting objects, as well as their long-term orbit evolution, according to full historic Two-Line Elements (TLE) datasets in synergy with mathematical modeling to properly size up the prevailing effect of lunisolar perturbation and atmospheric drag.

Impact of orbital orientations and radii on TianQin constellation stability

TianQin is a proposed space-based gravitational-wave observatory mission to be deployed in high circular Earth orbits. The equilateral-triangle constellation, with a nearly fixed orientation, can be distorted primarily under the lunisolar perturbations. To accommodate science payload requirements, one must optimize the orbits to stabilize the configuration in terms of arm-length, relative velocity, and breathing angle variations. In this paper, we present an efficient optimization method and investigate how changing the two main design factors, i.e. the orbital orientation and radius, impacts the constellation stability through single-variable studies. Thereby, one can arrive at the ranges of the orbital parameters that are comparatively more stable, which may assist future refined orbit design.

How an aware usage of the long-term dynamics can improve the long-term situation in the LEO region

The ReDSHIFT H2020 European project has shown, among other findings, that passive disposal procedures can benefit from the exploitation of dynamical perturbations. One key aspect of the project was a study on the dynamical disposal of spacecraft at the end-of-life by exploiting natural perturbations and identifying stable and unstable regions in the phase space, where the objects could be moved to exploit either long term “graveyards” or, possibly and preferentially, faster escape routes (the so-called “de-orbiting highways”). In particular, for the Low Earth Orbit (LEO) region, a natural eccentricity growth can be leveraged in order to reenter to the Earth's atmosphere at a lower Δv-budget.The numerical cartography of the region has been proven from a theoretical perspective, on the basis of a frequency analysis focused on solar radiation pressure and lunisolar perturbations and on dynamical systems theory tools. In this work, we summarize the whole study concerning the dynamics that characterizes the LEO region in the long-term, by giving a comprehensive picture of the theoretical findings together with their possible exploitation for the debris mitigation. The efficiency of the “de-orbiting highways” is tested and validated with a “thought experiment” by means of long-term propagation of a population of objects stemming from a specific traffic launch. It is shown how the de-orbiting corridors could be very effective in removing the majority of objects from the high LEO region at the end-of-life, thus contributing to the stabilization of the space debris environment, in particular for high-altitude spacecraft.

Missions to Earth and Mars using a lunar launch facility

Many space agencies are working on solutions for manned operations on the Moon, both on its surface and in cislunar space. In such a scenario, with stable human presence, a launch facility would be needed and launch vehicles could take advantage of the benefits of the lunar environment. In the present paper missions to Earth (to a geostationary orbit) and to Mars (to an areostationary orbit), from a lunar launch facility, has been studied. The trajectories are analysed by using an ephemeris model, taking into account four attractive bodies (Earth, Moon, Sun and Mars). In the first phase of the two missions, the launcher places the spacecraft on a circular lunar orbit by using liquid, solid or hybrid propulsion systems. The latter solution is analysed assuming in situ propellant production using aluminum and oxygen, elements in which the moon is very rich. In the second phase of both the missions, a minimum energy trajectory is studied, which, exploiting the Luni-Solar perturbations, allows the spacecraft to escape the gravitational field of the Moon. In the mission to Earth, this manoeuver puts the spacecraft on a very high and inclined elliptical orbit. The last phase involves a circularization manoeuvre towards the GEO and transfers are studied for two different initial inclinations, using an electric engine. In the mission to Mars, the escape impulse is used to leave both the Earth-Moon and Sun-Earth systems. The interplanetary phase is analysed for several transfer final times and considering an electric engine with variable specific impulse. Finally, the performance of a lunar launch facility is compared to that of a traditional Earth facility, showing great advantages are obtained in terms of payload ratio.

Satellites’ orbital stability through normal forms

AbstractA powerful tool to investigate the stability of the orbits of natural and artificial bodies is represented by perturbation theory, which allows one to provide normal form estimates for nearly-integrable problems in Celestial Mechanics. In particular, we consider the orbital stability of point-mass satellites moving around the Earth. On the basis of the J2 model, we investigate the stability of the semimajor axis. Using a secular Hamiltonian model including also lunisolar perturbations, the so-called geolunisolar model, we study the stability of the other orbital elements, namely the eccentricity and the inclination. We finally discuss the applicability of Nekhoroshev’s theorem on the exponential stability of the action variables. To this end, we investigate the non-degeneracy properties of the J2 and geolunisolar models. We obtain that the J2 model satisfies a “three-jet” non-degeneracy condition, while the geolunisolar model is quasi-convex non-degenerate.

Luni-solar resonances and effect on long-term evolution of inclined geostationary transfer orbits

In this paper, a singly-averaged, semi-analytical orbital dynamic model in terms of Milankovitch elements is given for geostationary transfer orbits (GTOs), which includes the Earth's oblateness, luni-solar perturbations, atmospheric drag and solar radiation pressure (SRP). The orbital dynamic model is verified through a numerical simulation compared with the Semi-analytic Tool for End of Life Analysis software (STELA) by CNES. We find that if the area-to-mass ratio of the object is relatively large, the SRP will have a significant impact on the long-term evolution of GTOs. Then, the long-term dynamical evolution and orbital resonances of rocket upper stages in inclined geostationary transfer orbits (IGTOs) are studied. It has been found that the luni-solar secular resonances, which are inclination-dependent-only, play a leading role in the long-term evolution of IGTOs. By comparing and analyzing long-term dynamical evolution of orbits with different initial inclinations, the effects of luni-solar secular resonances are studied in details. Finally, a mitigation method for IGTO objects by utilizing luni-solar secular resonances is proposed, in which the eccentricity growth caused by the resonances is used for an early atmospheric re-entry.

A frequency portrait of Low Earth Orbits

In this work, we deepen and complement the analysis on the dynamics of Low Earth Orbits (LEO), carried out by the authors within the H2020 ReDSHIFT project, by characterising the evolution of the eccentricity of a large set of orbits in terms of the main frequency components. Decomposing the quasi-periodic time series of eccentricity of a given orbit by means of a numerical computation of Fourier transform, we link each frequency signature to the dynamical perturbation which originated it in order to build a frequency chart of the LEO region. We analyse and compare the effects on the eccentricity due to Solar radiation pressure, lunisolar perturbations and high-degree zonal harmonics of the geopotential both in the time and frequency domains. In particular, we identify the frequency signatures due to the dynamical resonances found in LEO, and we discuss the opportunity to exploit the corresponding growth of eccentricity in order to outline decommissioning strategies.

Long-Term Evolution of Highly-Elliptical Orbits: Luni-Solar Perturbation Effects for Stability and Re-entry

This paper investigates the long-term evolution of spacecraft in Highly Elliptical Orbits (HEOs). The single averaged disturbing potential due to luni-solar perturbations, zonal harmonics of the Earth gravity field is written in mean Keplerian elements. The double averaged potential is also derived in the Earth-centered equatorial system. Maps of long-term orbit evolution are constructed by measuring the maximum variation of the orbit eccentricity to identify conditions for quasi-frozen, long-lived libration orbits or initial orbit conditions that naturally evolve toward re-entry in the Earth’s atmosphere. The behavior of these long-term orbit maps is studied for increasing values of the initial orbit inclination and anomaly of the perigee with respect to the Moon’s orbital plane. In addition, to allow meeting specific mission constraints, quasi-frozen orbits can be selected as graveyard orbits for the end-of-life of HEO missions. On the opposite side, unstable conditions can be exploited to target Earth re-entry at the end-of-mission.

Natural highways for end-of-life solutions in the LEO region

We present the main findings of a dynamical mapping performed in the Low Earth Orbit region. The results were obtained by propagating an extended grid of initial conditions, considering two different epochs and area-to-mass ratios, by means of a singly averaged numerical propagator. It turns out that dynamical resonances associated with high-degree geopotential harmonics, lunisolar perturbations and solar radiation pressure can open natural deorbiting highways. For area-to-mass ratios typical of the orbiting intact objects, these corridors can be exploited only in combination with the action exerted by the atmospheric drag. For satellites equipped with an area augmentation device, we show the boundary of application of the drag, and where the solar radiation pressure can be exploited.

Analytical long-term evolution and perturbation compensation models for BeiDou MEO satellites

The current paper establishes the analytical models of the long-term evolution and perturbation compensation strategy for Medium Earth Orbits (MEO) shallow-resonant navigation constellation, with application to the Chinese BeiDou Navigation Satellite System (BDS). The long-term perturbation model for the relative motion is developed based on the Hamiltonian model, and the long-term evolution law is analyzed. The relationship between the control boundary of the constellation and the offset of the orbital elements is analyzed, and a general analytical method for calculating the offset of the orbit elements is proposed. The analytical model is further improved when the luni-solar perturbations are included. The long-term evolutions of the BDS MEO constellation within 10 years are illustrated, and the effectiveness of the proposed analytical perturbation compensation calculation approach is compared with the traditional numerical results. We found the fundamental reason for the nonlinear variations of the relative longitude of ascending node and the mean argument of latitude is the long-periodic variations of the orbital inclination due to the luni-solar perturbations. The proposed analytical approach can avoid the numerical iterations, and reveal the essential relationship between the orbital element offsets and the secular drifts of the constellation configuration. Moreover, there is no need for maintaining the BDS MEO constellation within 10 years while using the perturbation compensation method.

Long-term density evolution through semi-analytical and differential algebra techniques

This paper introduces and combines for the first time two techniques to allow long-term density propagation in astrodynamics. First, we introduce an efficient method for the propagation of phase space densities based on differential algebra (DA) techniques. Second, this DA density propagator is used in combination with a DA implementation of the averaged orbital dynamics through semi-analytical methods. This approach combines the power of orbit averaging with the efficiency of DA techniques. While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with averaged equations of motion yields a fast and accurate technique to propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics. To demonstrate the effectiveness of the proposed approach, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the effects of solar radiation pressure, the Earth’s oblateness and luni-solar perturbations. The method can propagate 10,000 random fragments and their density for 1 year within a few seconds on a common desktop PC.

Geostationary secular dynamics revisited: application to high area-to-mass ratio objects

The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of objects around a nominal solution, hereafter called the forced equilibrium solution, which shows a particularly strong dependence on the area-to-mass ratio. Here, we i) give theoretical estimates for the long-term behavior of such orbits, and ii) we examine the nature of the forced equilibrium itself. In the lowest approximation, the forced equilibrium implies motion with a constant non-zero average `forced eccentricity', as well as a constant non-zero average inclination, otherwise known in satellite dynamics as the inclination of the invariant `Laplace plane'. Using a higher order normal form, we demonstrate that this equilibrium actually represents not a point in phase space, but a trajectory taking place on a lower-dimensional torus. We give analytical expressions for this special trajectory, and we compare our results to those found by numerical orbit propagation. We finally discuss the use of proper elements, i.e., approximate integrals of motion for the GEO orbits.