Lagrange Planetary Equations Research Articles

Orbital dynamics in the vicinity of asteroid 4660 Nereus

Spacecrafts operating in close proximity to small celestial bodies within our Solar System, such as asteroids, face dynamic environments that differ significantly from those encountered in the near-Earth space. Understanding the characteristics of orbital dynamics in the vicinity of asteroids is essential for the successful execution of asteroid exploration missions. Focusing on the asteroid 4660 Nereus, this paper studies the equilibrium points of it and the potentially natural, long-lifetime orbits subject to perturbations from both the asteroid's non-spherical gravitational field and the solar radiation pressure (SRP). Using the semi-analytical method based on averaged Lagrange planetary equations (LPEs), heliotropic and terminator orbits are found more likely to have long lifetimes. It is noteworthy that the conventional near-equatorial heliotropic orbits around Nereus struggle to maintain a long lifetime. In contrast, orbits characterized by a specific inclination and geometry where the apocenter approximately points toward the Sun tend to offer prolonged lifetimes. Moreover, the impact of variations in SRP, gravitational parameter, and pole direction on feasibility of long-term orbit solutions has been explored, resulting in an enhanced comprehension of the intricate orbital dynamics. This study contributes valuable insights for the design of exploration orbits in the vicinity of the Nereus asteroid.

Frozen orbits with inner planar perturbing body up to triakontadipole level of approximation

In the exploration of celestial bodies, including planets, moons, and asteroids, it is useful to identify stable orbits in which the orbital elements remain, on average, constant (referred to as frozen orbits). This paper aims to investigate the feasibility of frozen orbits for a small body, such as a probe, orbiting a main body under the perturbation due to the gravitational attraction of an inner third-body, on a circular (or slightly eccentric) orbit coplanar with the main body equator, by means of an approach based on mean element theory. The disturbing potential has been developed in Legendre polynomials up to order l=5 (triakontadipole level of approximation). Thus, the double-averaged potential function and the associated constants of motion have been determined up to the same order. Then, by applying the Lagrange Planetary Equations, the secular and long-term orbital element variations have been obtained. Therefore, frozen orbital solutions have been evaluated, and the effects on them of the addition of main body oblateness perturbation have been considered. Furthermore, the procedure has been repeated, and the frozen solutions are re-evaluated in the case of a disturbing body with small eccentricity up to order l=4 (hexadecapole level of approximation). These solutions offer significant advantages in exploring bodies subject to a notable perturbation from an inner third-body, such as binary asteroid systems.

Balancing Conditions for the Relativistic Correction Using Lorentz Acceleration

In this work, the average effects of the Lorentz acceleration on the charged spacecraft’s orbit are studied encounter with relativistic correction. The relativistic correction as function of the orbital elements, and may be time, is formulated. Lagrange planetary equations are used to calculate the perturbations due to considered perturbing forces. The needed conditions to neutralize the effects of the relativistic corrections, using Lorentz acceleration, are derived. Numerical examples for different kinds of orbits are applied.

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.

Balancing the effects of solar radiation pressure on the orbital elements of a spacecraft using Lorentz force

In this work, orbits of Lorentz spacecrafts and satellites are investigated under the perturbation of solar radiation pressure. An attempt is made to control the perturbation of the solar radiation pressure using the effect of Lorentz force that affects an electrically charged spacecraft. The charge per unit mass is the controlling parameter in this process. The redial, transverse and normal components of the mentioned forces are constructed. The Lagrange planetary equations for perturbations in the Keplerian orbital elements are formulated. The formula describing the charge per unit mass, as function of the physical parameter of the problem as well as the orbital elements, were derived. The effects of the combined forces are analytically and numerically studied.

Dynamics of a Particle in 3:1 Tesseral Resonance with the Dwarf Planet Haumea

The dynamics of a particle in 3:1 tesseral resonance with the dwarf planet Haumea is analysed. This resonance, three rotations of the primary per orbital period of the particle, is located inside the region where Haumea’s ring was observed. Thus, determining the effect of this resonance on a particle’s orbit reveals its relationship to the orbits that follow the particles of the ring. To analyse the effect, we propose four models of anisotropy; two of them are a reduced representation of the distribution of the mass of Haumea that we use to determine the centre of the resonance by means of the Hamiltonian formulation. After this, we analyse the effects of the four models on the resonance orbit by using the Lagrange planetary equations technique. The results show that the resonance centre has a high eccentricity value, meaning that a particle in 3:1 resonance with Haumea does not remain confined to the region that we consider to be the ring region.

Heliotropic orbits at asteroid 99942 Apophis: Considering solar radiation pressure and zonal gravity perturbations

Heliotropic orbits are ideal for low-altitude science orbits around asteroids since they can maintain long lifetimes by balancing perturbations from the asteroid's non-spherical gravitational field and the direct solar radiation pressure. In this study, Lagrange Planetary Equations and averaged disturbing potentials are applied to investigate the region around the asteroid Apophis where long-lifetime maintenance-free heliotropic orbits may exist. Based on both singly-averaged and constrained doubly-averaged disturbing potentials, the equations of motion in the orbital elements are derived, considering perturbations from the J2, J3, J4 term and solar radiation pressure including simple shadowing. The possible existence zones of heliotropic orbits at the asteroid Apophis (99942) are obtained with a semi-analytical method, applying the heliotropic and frozen conditions to the motion equations. In addition, consideration of the J4 term is found to diminish the maximum of a, e, i of the possible existence zones. The analysis in this paper helps us to quickly determine the distribution of possibly long-lifetime heliotropic orbits in the a-e-i space at a low computational cost.

Frozen Orbits Under Radiation Pressure and Zonal Gravity Perturbations

The orbital motion of spacecraft around asteroids is strongly disturbed primarily because of solar radiation pressure (SRP) and higher-order gravitational forces. To achieve stable orbits in such an environment, the implementation of frozen orbits is a promising approach. This research derives semi-analytical solutions of frozen orbits subject to SRP and zonal gravity up to the fourth order based on singly averaged Lagrange planetary equations. Moreover, the stability of frozen orbits is characterized analytically by introducing linearized variational equations, revealing the complex eigenstructures of the proposed frozen orbits. These analytical theories identify several different types of stable frozen orbits, which are further validated via high-fidelity numerical simulations. Consequently, this paper demonstrates the feasibility and intriguing dynamic characteristics of frozen orbits around asteroids.

Higher-order effects in the dynamics of hierarchical triple systems: Quadrupole-squared terms

We analyze the secular evolution of hierarchical triple systems to second-order in the quadrupolar perturbation induced on the inner binary by the distant third body. The Newtonian three-body equations of motion, expanded in powers of the ratio of semimajor axes $a/A$, become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, $O[(a/A)^3]$, octupole, $O[(a/A)^4]$, and so on. In the Lagrange planetary equations for the evolution of the instantaneous orbital elements, second-order effects arise from obtaining the first-order solution for each element, consisting of a constant (or slowly varying) piece and an oscillatory perturbative piece, and reinserting it back into the equations to obtain a second-order solution. After an average over the two orbital timescales to obtain long-term evolutions, these second-order quadrupole ($Q^2$) terms would be expected to produce effects of order $(a/A)^6$. However we find that the orbital average actually enhances the second-order terms by a factor of the ratio of the outer to the inner orbital periods, $ \sim (A/a)^{3/2}$. For systems with a low-mass third body, the $Q^2$ effects are small, but for systems with a comparable-mass or very massive third body, such as a Sun-Jupiter system orbiting a solar-mass star, or a $100 \, M_\odot$ binary system orbiting a $10^6 \, M_\odot$ massive black hole, the $Q^2$ effects can completely suppress flips of the inner orbit from prograde to retrograde and back that occur in the first-order solutions. These results are in complete agreement with those of Luo, Katz and Dong, derived using a "Corrected Double-Averaging" method.

Theoretical investigation of the perturbed artificial satellite problem using oblate continued fraction potential

In this research article, a new idea concerning the application of concept of the continued fractions to the expansion of the geopotential is treated. The perturbed motion of the satellite in the oblate gravitational field of the Earth using a potential of a continued fractions procedure is studied. To compare with the usual perturbation theory, the continued fraction series is truncated beyond the third term. The first oblateness term of the potential is retained. The equations of motion are constructed using the Lagrange planetary equations. The integrals and averages required for solving the problem is outlined as well as the different cases are highlighted. The perturbations in all orbital elements due to considered model of continued fractions are evaluated. The new findings of the manuscript contrast with classical oblate potential field, the semi-major axis, eccentricity, and inclination, nonzero averaged terms in perturbations are obtained. Extra nonzero averaged perturbing terms in the argument of perigee and ascending node are revealed in addition to the classical perturbations.

Relativistic three-body effects in hierarchical triples

The hierarchical three-body problem has many applications in relativistic astrophysics, and can play an important role in the formation of the binary black hole mergers detected by LIGO/Virgo. However, many studies have only included relativistic corrections responsible for the precession of pericenter of the inner and outer binaries, neglecting relativistic interactions between the three bodies. We revisit this problem and develop a fully consistent derivation of the secular three-body problem to first post-Newtonian order. We start with the Einstein-Infeld-Hoffman equations for a three-body system and expand the accelerations as a power series in the ratio of the semi-major axes of the inner ($a_1$) and outer ($a_2$) binary. We then perform a post-Keplerian, two-parameter expansion of the single-orbit-averaged Lagrange planetary equations in $\delta = v^2/c^2$ and $\epsilon = a_1/a_2$ using the method of multiple scales. Using this method, we derive previously-indentified secular effects at $\delta \epsilon^{5/2}$ order that arise directly from the equations of motion. We also calculate new secular effects through $\delta \epsilon^4$ order that can lead to eccentricity growth over many Lidov-Kozai cycles when the tertiary is much more massive than the inner binary. In such cases, inclusion of these effects can substantially alter the evolution of three-body systems as compared to an analysis in which they are neglected. Careful analysis of post-Newtonian three-body effects will be important to understand the formation and properties of coalescing binaries that form via three-body dynamical processes.

Using an integral index to search for orbits around oblate spheroids

An integral index is used in this manuscript to search for the least gravitationally perturbed orbits around an oblate spheroid. Although there are missions where perturbations are desired, such as Sun-synchronous orbits, near Keplerian orbits can be useful in some cases during the whole mission or partially, helping to keep the oscillations of the orbital parameters in the minimum possible level, which can be interesting to observe celestial bodies. These orbits are also good candidates to require a lower number of station-keeping maneuvers, helping to simplify the logistic of the mission. The index used is available in the literature and it is based on the integration of the accelerations suffered by a spacecraft over time. An oblate spheroid is used to represent the approximate shape of non-spherical bodies because it has a closed equation for the potential, which makes it ideal for the analysis proposed, and because it is a shape similar to what is found for several objects in the small bodies population. The Lagrange planetary equations are also used to map orbits that have a minimum rate of variation in their orbital elements, and compared with the results obtained with the integral index. The results show a very good agreement between the index and the variations of the orbital elements of the spacecraft, in particular in terms of locating the least perturbed orbits to place the spacecraft.

Semi-analytical orbital dynamics around the primary of a binary asteroid system

ABSTRACT The orbital dynamics in the vicinity of a binary asteroid system has been studied extensively, motivated by the special dynamical environment and possible exploration missions. Equilibrium points, periodic orbits, and invariant manifolds have been investigated in many studies based on the model of the Restricted Full Three Body Problem (RF3BP). In this paper, a new semi-analytical orbital dynamical model around the primary of a binary system is developed as a perturbed two-body problem. The solution includes the effect of the primary's oblateness and the secondary's third-body gravity. The semi-analytical dynamical model, also denoted as the averaged model, is obtained by using the averaging process and Lagrange planetary equations in terms of the Milankovitch orbital elements. This semi-analytical model enables much faster orbital propagations than the non-averaged counterpart, and is particularly useful in orbital stability analysis and the design of long-term passively stable orbits and orbits with specific performance, e.g. frozen orbits. The applicability of the semi-analytical model is then discussed. Two parameters describing relative magnitudes of both perturbations w.r.t. the primary's point mass gravity and the third parameter related to the orbital period ratio w.r.t. the secondary are defined to provide indicators for the validity of the averaged model. The validity boundaries in terms of the three parameters are given based on numerical simulations, by comparing with the full orbital model. The application to a real binary system, 2003 YT1, has shown that the averaged solution has a high precision in the long-term orbital propagation.

Frozen Orbits Construction for a Lunar Solar Sail

Frozen orbit is an attractive option for orbital design owing to its characteristics (its argument of pericenter and eccentricity are kept constant on an average). Solar sails are attractive solutions for massive and expensive missions. However, the solar radiation pressure effect represents an additional force on the solar sail that may greatly affect its orbital behavior in the long run. Thus, this force must be included as a perturbation force in the dynamical model for more accuracy. This study shows the calculations of initial conditions for a lunar solar sail frozen orbit. The disturbing function of the problem was developed to include the lunar gravitational field that is characterized by uneven mass distribution, third body perturbation, and the effect of solar radiation. An averaging technique was used to reduce the dynamical problem to a long period system. Lagrange planetary equations were utilized to formulate the rate of change of the argument of pericenter and eccentricity. Using the reduced system, frozen orbits for the Moon sail orbiter were constructed. The resulting frozen orbits are shown by two 3Dsurface (semimajor, eccentricity, inclination) figures. To simplify the analysis, we showed inclination–eccentricity contours for different values of semi-major axis, argument of pericenter, and values of sail lightness number.

Pericenter advance in general relativity: comparison of approaches at high post-Newtonian orders

The advance of the pericenter of the orbit of a test body around a massive body in general relativity can be calculated in a number of ways. In one method, one studies the geodesic equation in the exact Schwarzschild geometry and finds the angle between pericenters as an integral of a certain radial function between turning points of the orbit. In another method, one describes the orbit using osculating orbit elements, and analyzes the ‘Lagrange planetary equations’ that give the evolution of the elements under the perturbing effects of post-Newtonian (PN) corrections to the motion. After separating the perturbations into periodic and secular effects, one obtains an equation for the secular rate of change of the pericenter angle. While the different methods agree on the leading post-Newtonian contribution to the advance, they do not agree on the higher-order PN corrections. We show that this disagreement is illusory. When the orbital variables in each case are expressed in terms of the invariant energy and angular momentum of the orbit and when account is taken of a subtle difference in the meaning of ‘pericenter advance’ between the two methods, we show to the third post-Newtonian order that the different methods actually agree perfectly.

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from $$J_2$$ up to $$J_6$$ , and the tesseral harmonics $$\overline{C}_{22}$$ that is of the same magnitude than zonal $$J_2$$ . In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.

Orbital flips in hierarchical triple systems: Relativistic effects and third-body effects to hexadecapole order

We analyze the secular evolution of hierarchical triple systems in the post-Newtonian approximation to general relativity. We expand the Newtonian three-body equations of motion in powers of the ratio $a/A$, where $a$ and $A$ are the semimajor axis of the inner binary's orbit and of the orbit of the third body relative to the center of mass of the inner binary, respectively. The leading order "quadrupole" terms, of order $(a/A)^3$ relative to the $1/a^2$ acceleration within the inner binary, are responsible for the well-known Kozai-Lidov oscillations of orbital inclination and eccentricity. The octupole terms, of order $(a/A)^4$ have been shown to allow the inner orbit to "flip" from prograde relative to the outer orbit to retrograde and back, and to permit excursions to very large eccentricities. We carry the expansion of the equations of motion to hexadecapole order, corresponding to contributions of order $(a/A)^5$. We also include the leading orbital effects of post-Newtonian theory, namely the pericenter precessions of the inner and outer orbits. Using the Lagrange planetary equations for the orbit elements of both binaries, we average over orbital timescales, obtain the equations for the secular evolution of the elements through hexadecapole order, and employ them to analyze cases of astrophysical interest. We find that, for the most part, the orbital flips found at octupole order are robust against both relativistic and hexadecapole perturbations. We show that, for equal-mass inner binaries, where the octupole terms vanish, the hexadecapole contributions can alone generate orbital flips and excursions to very large eccentricities.

A vectorial approach to determine frozen orbital conditions

Taking into consideration a probe moving in an elliptical orbit around a celestial body, the possibility of determining conditions which lead to constant values on average of all the orbit elements has been investigated here, considering the influence of the planetary oblateness and the long-term effects deriving from the attraction of several perturbing bodies. To this end, three equations describing the variation of orbit eccentricity, apsidal line and angular momentum unit vector have been first retrieved, starting from a vectorial expression of the Lagrange planetary equations and considering for the third-body perturbation the gravity-gradient approximation, and then exploited to demonstrate the feasibility of achieving the above-mentioned goal. The study has led to the determination of two families of solutions at constant mean orbit elements, both characterised by a co-planarity condition between the eccentricity vector, the angular momentum and a vector resulting from the combination of the orbital poles of the perturbing bodies. As a practical case, the problem of a probe orbiting the Moon has been faced, taking into account the temporal evolution of the perturbing poles of the Sun and Earth, and frozen solutions at argument of pericentre 0 $$^{\circ }$$ or 180 $$^{\circ }$$ have been found.

Global optimization of fuel consumption in J2 rendezvous using interval analysis

This paper addresses an open-time Lambert problem under first-order gravitational perturbations with unfixed parking time and transfer time. The perturbations are compensated by introducing its analytical solutions derived from Lagrange's planetary equations into Lambert problem. A drift vector of aim position correction is defined to reduce the aim position bias caused by the perturbations. The first purpose of optimization is to find sufficiently small intervals involving the global optimal parking time, transfer time, drift vector and velocity increment. The second is to determine the global solution or the solution close to it in these intervals. Interval analysis and a double-deck gradient-based method with GA estimating the initial range of drift vector are utilized to obtain the sufficiently small intervals including the global minimum velocity increment and the global minimum solution or one sufficiently close to it in these intervals.

Heliotropic Orbits with Zonal Gravity and Shadow Perturbations: Application at Bennu

Heliotropic orbits provide long-lifetime low-altitude orbits in the presence of large and solar radiation pressure perturbations. Formal inclusion of high-degree zonal gravity harmonics and simple shadowing provides a more realistic model to initiate the search for heliotropic orbits at irregular primitive bodies like Bennu, which is the target of the OSIRIS-Rex mission. The constrained, doubly averaged potential and the Lagrange planetary equations yield a single equation to enforce the heliotropic constraint. The equation is solved for inclinations across a range of semimajor axes and eccentricities, providing a surface of potential solutions. The fast process allows for Monte Carlo simulations to assess the likelihood of a heliotropic orbit existing in the presence of parameter uncertainty. The existence of heliotropic orbits is shown to be reasonably robust to uncertainty in the solar radiation pressure acceleration and reference gravity parameters for Bennu.