Jacobi Integral Research Articles

Large-step neural network for learning the symplectic evolution from partitioned data

ABSTRACT In this study, we focus on learning Hamiltonian systems, which involves predicting the coordinate ($\boldsymbol q$) and momentum ($\boldsymbol p$) variables generated by a symplectic mapping. Based on Chen & Tao (2021), the symplectic mapping is represented by a generating function. To extend the prediction time period, we develop a new learning scheme by splitting the time series ($\boldsymbol q_i$, $\boldsymbol p_i$) into several partitions. We then train a large-step neural network (LSNN) to approximate the generating function between the first partition (i.e. the initial condition) and each one of the remaining partitions. This partition approach makes our LSNN effectively suppress the accumulative error when predicting the system evolution. Then we train the LSNN to learn the motions of the 2:3 resonant Kuiper belt objects for a long time period of 25 000 yr. The results show that there are two significant improvements over the neural network constructed in our previous work: (1) the conservation of the Jacobi integral and (2) the highly accurate predictions of the orbital evolution. Overall, we propose that the designed LSNN has the potential to considerably improve predictions of the long-term evolution of more general Hamiltonian systems.

Quasi-conservative Integration Method for Restricted Three-body Problem

The simplest restricted three-body problem, in which two massive points and a massless point particle attract one another according to Newton’s law of inverse squares, has pulsating Hill’s regions where the massless particle moves inside the closed regions surrounding only one of the massive points. Until now, no numerical integrator is known to maintain these regions, making it challenging to reproduce the phenomenon of gravitational capture of massless particles. In this article, we propose a second-order integrator that preserves Hill’s regions to accurately simulate this phenomenon. Our integrator is based on a logarithmic Hamiltonian leapfrog method developed by Mikkola and Tanikawa and features a parameter that is adjusted to preserve a second-order approximation of an invariant integration relation of this restricted three-body problem. We analytically and numerically clarify that this integrator has the following properties: (i) it retains the collinear and triangular Lagrangian solutions regardless of the eccentricity of the relative orbit of the two massive points, (ii) it has the same Hill stability criterion for satellite-type motion of the massless point particle as the original problem, and (iii) it conserves the Jacobi integral for zero eccentricity.

Capacity of Sun-driven lunar swingby sequences and their application in asteroid retrieval

For deep-space mission design, the gravity of the Sun and the Moon can be first considered and utilized. Their gravity can provide the energy change for launching spacecraft and retrieving spacecraft as well as asteroids. Regarding an asteroid retrieval mission, it can lead to the mitigation of asteroid hazards and an easy exploration and exploitation of the asteroid. This paper discusses the application of the Sun-driven lunar swingby sequence for asteroid missions. Characterizing the capacity of this technique is not only interesting in terms of dynamic insights but also non-trivial for trajectory design. The capacity of a Sun-driven lunar swingby sequence is elucidated in this paper with the help of the "Swingby-Jacobi" graph. The capacity can be represented by a range of the Jacobi integral that encloses around 660 asteroids currently cataloged. To facilitate trajectory design, a database of Sun-perturbed Moon-to-Moon transfers, including multi-revolution cases is generated and employed. Massive trajectory options for spacecraft launch and asteroid capture can then be explored and optimized. Finally, a number of asteroid flyby, rendezvous, sample-return, and retrieval mission options enabled by the proposed technique are obtained.

Bulirsh–Stoer algorithm in the planar restricted three-body problem

The applicability of the Bulirsh–Stoer algorithm for solving the planar restricted three-body problem is investigated. Variations in the value of the Jacobi integral are considered as the main parameter. Massive calculations were carried out with a small step in the parameter characterizing the ratio of the masses of a planet and star (the mass parameter). It is shown that violations of the Jacobi integral occur inside the planetary chaotic region. This fact can be used to determine chaotic region boundaries, as well as the boundaries of a stable structure which coorbital with the planet. The average dependences of the value of the Jacobi integral on the mass parameter, which determine the boundaries of the chaotic zone and the coorbital ring, are derived. Estimates were obtained for the maximum relative change in the Jacobi integral for different values of the accretion radius. It is shown that the value of the accretion radius corresponding to the average radius of known exoplanets does not cause significant changes in the value of the Jacobi integral. The dependences of the clearing time of the chaotic zone for different accretion radii are also given with and without taking into account the coorbital structure.

Motion of Stars in Layered Inhomogeneous Elliptical Galaxies

The problem of the spatial motion of a passively gravitating body (PGB) in the gravitational field of a layered inhomogeneous elliptical galaxy (LIEG) is considered on the basis of the previously developed model. It is assumed that a LIEG consists of baryonic mass (BM) and dark matter (DM), which have different laws of density distribution. A star or the center of mass of a globular cluster is taken as the PGB, the motion of which considers the BM and DM attraction. To obtain accurate results, the BM and DM attraction potentials are not expanded in a series, but their exact expressions are taken. An analogue of the Jacobi integral is found, the region of the possible motion of the PGB is determined, and the zero-velocity surfaces are constructed. The stationary solutions (libration points) are found to be stable in the sense of Lyapunov. The results are applied to the elliptical galaxies NGC 4472 (M 49), NGC 4697, and NGC 4374 (M 84).

Approximating orbits in a rotating gravity field with oblateness and ellipticity perturbations

This paper explores the problem of analytically approximating the orbital state for a subset of orbits in a rotating potential with oblateness and ellipticity perturbations. This is done by isolating approximate differential equations for the orbit radius and other elements. The conservation of the Jacobi integral is used to make the problem solvable to first order in the perturbations. The solutions are characterized as small deviations from an unperturbed circular orbit. The approximations are developed for near-circular orbits with initial mean motion $$n_{0}$$ around a body with rotation rate c. The approximations are shown to be valid for values of angular rate ratio $$\varGamma = c/n_{0} > 1$$ , with accuracy decreasing as $$\varGamma \rightarrow 1$$ , and singularities at and near $$\varGamma = 1$$ . Extensions of the methodology to eccentric orbits are discussed, with commentary on the challenges of obtaining generally valid solutions for both near-circular and eccentric orbits.

A splitting of collinear libration points in circular restricted three-body problem by an artificial electrostatic field

This paper focuses on the study of a new type of a planar, circular restricted three-body problem with an attractive artificial electrostatic field (E-field) at collinear libration points. For instance, this attractive field can be generated by an orbiting spacecraft located at the Mars-Phobos L1 libration point and an electrostatic capsule launched from Phobos. The feasibility of the proposed retrieval system is discussed from the aspect of local space weather Debye length. The attractive E-field splits the collinear libration point into two new collinear points, and the greater the E-field potential and the Debye length the greater the distance between the new libration points and the “old” original libration point. The new equilibrium positions caused by the action of the E-field have been found, and an instability of these new libration points has been proven. A new Jacobi integral in analytical form is obtained and equations of motion are derived for the restricted problem of three bodies taking into account the E-field. A numerical simulation shows the impact of the E-field potential on capsule capture in the small vicinity of the Mars-Phobos L1 libration point. This work expands the classic three-body problem filling with new content. The obtained results can be applied, for example, to study an opportunity of delivering the Phobos samples using Coulomb interaction of bodies in space.

The optimized approach to get the fundamental property of the gravity assists maneuvers from the Jacobi integral

The design of interplanetary trajectories using a series of gravity assists maneuvers begins with the ballistic mission design. It is reasonable to construct the corresponding initial approximation using the patched conics method within the model of the circular restricted three-body problem. Such a construction requires the calculation of the “transfer parameter” Vinf - the dimensionless asymptotic velocity of the spacecraft relative the target planet, when switching from heliocentric arcs to planetocentric segments and vice versa. In the circular restricted three-body problem model, Vinf can be calculated using the Jacobi integral J (or using it’s analogue - the Tisserand parameter TTiss and the basic property of the Jacobi integral for the gravity assists maneuvers within the framework of the circular restricted three body problem: const J ≈ TTISS ≈ 3 – Vinf Vinf . According to this property, the J value does not change during the multiple gravity assists maneuvers that preserve the Jacobi integral constant are performed. This fact is known in astrodynamics but it is classically derived in a rather cumbersome method. In this study, a shorter method for its obtaining is proposed. The modifications of the representation of the Jacobi integral in the circular restricted three-body problem for the various configurations of three bodies are presented.

Optimal spacecraft asymptotic velocity for the high inclined orbits formation using gravity assists in the planetary systems

The inclination changing of the celestial bodies’ orbits is one of the most energy-consuming procedures. Nevertheless, for a number of promising space projects, leaving the main plane of the planetary or satellite system is fundamentally necessary and demanded. In this case only the possibility of realizing the formation of maximally inclined spacecraft (SC) orbits using gravity assist maneuvers (GM) is remained. However, the dynamic possibilities of GM using are also limited. The maximum possible value of the SC orbital inclination (which gives the extreme point inclination pole – IncPole) depends on the modulus V inf of the SC asymptotic velocity on the flyby hyperbola and is directly proportional to it (GM geometric limitation). However, the magnitude of the change in inclination at one GM is inversely proportional to V inf (GM dynamic limitation). As a result, a compromise value of V inf is existing, which varies for each specific case of the flyby planet. Since in order to achieve a significant inclination a whole series of GMs may be required, in addition, each GM must ensure the “resonance” of the orbital periods of the planet and SC after the GM in order to guarantee their new meeting. According the Jacobi integral in the restricted three body problem the V inf is an invariant during in all interplanetary flights using GMs. It’s possible to construct the series of increasing GM in form of “jumps” along the resonance levels from the initial inclination to the maximum possible point IncPole. This point may not belong to any isoline of one of the main resonances itself in the general case. This reduces the GMs effectiveness and necessarily increases the mission time of flight. A heuristic consideration consists in choosing (in the “resonant tuning”) the such value of the design V inf, which ensures the localization of the inclination pole in the vicinity of some resonance curves between the orbital periods of the spacecraft and the planet (resonant ). Thus it is possible to get to the IncPole by jumping along the resonance curve thru the minimum amount of GM. The paper describes the formulas and gives its estimates for the Solar system and for the planets satellite systems.

Minimal Velocity Surface in a Restricted Circular Three-Body Problem

In a restricted circular three-body problem, the concept of the minimum velocity surface $$\mathscr{S}$$ is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of the Hill surface requires the occurrence of the Jacobi integral. The minimum velocity surface, apart from the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of motion of the main bodies. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system over the longitude of the main bodies. The properties of $$\mathscr{S}$$ are investigated. We highlight the most significant issues. The set of possible motions of the zero-mass body bounded by surface $$\mathscr{S}$$ is compact. As an example, surfaces $$\mathscr{S}$$ for four small moons of Pluto are considered within the averaged Pluto–Charon–small satellite problem. In all four cases, $$\mathscr{S}$$ is a topological torus with a small cross section, having a circumference in the plane of motion of the main bodies as the center line.

A Universal Property of the Jacobi Integral for Gravity Assists in the Solar System

This paper shows that the standard and cumbersome method of traditional justification of constancy of an asymptotic velocity in gravity assists, in the model of the circular restricted three-body problem (RTBP), which is applied in the modern astrodynamics, can be fundamentally simplified. Updated forms of the Jacobi integral are presented, which allow one, among other things, to reveal a transparent relationship between the Jacobi integral and the method of conjugate conic sections in the RTBP.

Applicability of the Bulirsch-Stoer algorithm in the circular restricted three-body problem

AbstractThe dynamics of massless planetesimals in the gravitational field of a star with a planet in a circular orbit is considered. The invariant of this problem is the Jacobi integral. Preserving the value of the Jacobi integral can be a test for numerical algorithms solving the equation of motion. The invariant changes for particles in the planetary chaotic zone due to numerical errors that occur during close encounters with the planet. The limiting distances from the planet, upon reaching which the value of the Jacobi integral changes, are determined for Bulirsch-Stoer algorithm. It is shown that violation of the Jacobi integral can be used to define the boundaries of the planetary chaotic zone.

Two-maneuver transfers from the collinear L2 point to the triangular L4 point in the planar Earth-Moon system

China’s Chang’E-4 (CE-4) probe has started its exploration on the far side of the Moon thanks to the relay satellite---Queqiao---which is around the Earth-Moon collinear libration point L2 and provides continuous communication links between the lander and ground stations. The triangular libration points of the Earth-Moon system, L4 and L5, have long been considered as potential locations for future space applications. As a possible extension of Queqiao or other future missions around L2, the work presented here contributes to constructing transfers from the collinear libration point (L2) to the triangular libration points. Taking the L4 point as an example, two types of two-maneuver transfers from a planar Lyapunov orbit around L2 to the periodic orbit around L4 are investigated in the planar Earth-Moon system and compared with each other, including the type-I transfer utilizing invariant manifolds along with powered lunar gravity assist and the type-II transfer using the Jacobi integral. Our studies indicate that type-I transfer saves more Δv cost than type-II transfer. For the cases studied by us, the minimum Δv cost is about 0.1636 km/s for type-I transfer, with a corresponding transfer time of 169.84 days. For type-II transfer and the cases studied by us, the corresponding values are 0.4745 km/s and 137.19 days. At the end of the paper, the type-I transfer is generalized (with some modifications) to the ephemeris model and some example transfer trajectories are given. Although only the transfer from L2 to L4 is studied in this paper, the same orbit design strategy can be generalized to the transfer to the triangular libration point L5, and the transfers from the collinear libration point L1 to triangular libration points.

Minimal velocity surface in the restricted circular Three-Body-Problem

In the framework of the restricted circular Three-Body-Problem, the concept of the minimum velocity surface S is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of Hill surface requires occurrence of the Jacobi integral. The minimum velocity surface, other than the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of the main bodies motion. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system by longitude of the main bodies. Properties of S are investigated. Here is the most significant. The set of possible motions of the zero-mass body bounded by the surface S is compact. As an example the surfaces S for four small moons of Pluto are considered in the framework of the averaged problem Pluto — Charon — small satellite. In all four cases, S represents a topological torus with small cross section, having a circumference in the plane of motion of the main bodies as the center line.

Low thrust Earth–Moon transfer trajectories via lunar capture set

A cislunar cargo spacecraft with low-thrust propulsion traveling between the Earth and the Moon is essential for sustainable, long-term manned lunar exploration. In low-thrust Earth–Moon transfer (LTEMT), lunar capture is the primary prerequisite for spacecraft subject to the circular restricted three-body model. Therefore, this study identifies sufficient conditions for lunar capture, which are determined by the Jacobi integral and Hill’s region. This paper proposes a guidance scheme that includes thrust direction, thrust efficiency, and a five-stage flight control sequence based on the variation of the Jacobi integral. The LTEMT problem is then converted to an initial value problem of a differential equation with three parameters. Lunar capture set theories (LCSTs), which are convenient for identifying lunar capture sets, are presented and proved according to the continuous properties of the ordinary differential equation. Finally, the solutions of the LTEMT trajectories departing from a geosynchronous orbit with an altitude of approximately 35,827 km are discussed for different thrust accelerations and cut-off values of the thrust efficiency. The robustness is analyzed assuming that navigation and switching time errors are present to demonstrate the adaptability of this method. The results reveal that the proposed guidance scheme and LCSTs can provide technical support for future cislunar cargo missions.

Main Property of the Jacobi Integral for Gravity Assist Maneuvers in the Solar System

In the context of the circular restricted three-body problem (RTBP), the search for the spacecraft’s asymptotic velocity $${{V}_{\infty }}$$ for gravity assist maneuvers (GMs) can be performed using the Jacobi integral $$J$$ and the main property of the Jacobi integral constant for GMs, $${\text{const}} = J \approx 3 - V_{\infty }^{2}.$$ It is shown in the paper that the standard cumbersome way of traditional derivation of this property that is used in modern astrodynamics can be significantly simplified.

Solving procedure for 3D motions near libration points in CR3BP

In a novel approach for solving the equations of the Circular Restricted Three-Body Problem (CR3BP) first formulated in Ershkov (Acta Mech. 228(7):2719–2723, 2017a), we apply in this communication a procedure for solving the Euler-Poisson equations for the momentum equations of the CR3BP near the libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution in the vicinity of libration points. A new and elegant ansatz is suggested in this publication, whereby, in solving, the momentum equation is reduced to a system of three linear ordinary differential equations of first order in regard to the three components of the velocity of the infinitesimal mass $m$ (dependent on time $t$ ). Under this premise, a proper elegant partial solution has been obtained due to the invariant dependence between temporary components of the solution. We conclude that the system of CR3BP equations does not have the analytical presentation of the solution (in quadratures) even in the vicinity of the libration points except of the generalized Jacobi integral.

A method based on Jacobi Integral variational equation for computing Earth-Moon trajectories in the four-body problem

This work proposes an alternative method for solving the two-point boundary value problem concerning to Earth-Moon bi-impulsive trajectories in the dynamics of the planar bi-circular restricted four-body problem, which describes the motion of a space vehicle subjected to the gravitational attraction of Earth, Moon and Sun. Initially, the space vehicle is at a circular low Earth orbit (LEO) with prescribed altitude. After applying the first impulsive velocity increment, the space vehicle is inserted into a transfer trajectory. The second velocity increment is applied to decelerate and circularize the movement of the space vehicle at a circular low Moon orbit (LMO) with prescribed altitude. To solve this problem, a new two-point boundary value problem (TPBVP) is formulated, which includes an unknown value of the Jacobi integral at the departure time, and, a prescribed value at the arrival time. Since the Jacobi integral is not a first integral for the four-body problem, it is taken as additional state variable, and, its variational equation is added to the system of differential equation in the description of the dynamics of the space vehicle. Taking into account the boundary conditions, expressions for the velocity increments are deduced from the Jacobi integral computed at the initial and final times. Based on this new TPBVP, a numerical procedure is proposed to obtain different families of Earth-Moon trajectories with decreasingly fuel consumption.

Effect of variation of charge in the circular restricted three-body problem with variable masses

In the present paper, we are concerned by some investigation on circular restricted three-body problem (CR3BP), where we assume that the primaries have variable masses and variable charges. Among the principal tools used in the present study, we cite the well known Meshcherskii transformation. We have derived the equations of motion and Jacobi integral which differ by variation constant k and charge q from the classical restricted three-body problem. More exactly, in this paper, we have drawn the equilibrium points, the zero-velocity curves, the periodic orbits, the surfaces and the basins of attraction for the different values of charge. We have found one equilibrium point when the charge is q=0.4 and three equilibrium points when its value is q=0.501. We also have drawn the periodic orbits for these two values of charge and found that they are periodic. We have also plotted the zero-velocity surfaces for these two values of charges and found a tremendous variation in these two surfaces. We notice that the Poincaré surfaces of section are shifting away from the origin, when we increase the value of charge. We also got different surfaces for the motion of infinitesimal body, with respect to the variations of charge. The basins of attraction have been drawn for these two values of charge by using Newton-Raphson iterative method. We also noticed that by increasing the values of charge, the basins of attraction are shrinking. For the stability of the equilibrium points that we have studied, we found that, among them, one is stable and three others are unstable.

Основное свойство интеграла Якоби для гравитационных маневров в Солнечной системе

Основное свойство интеграла Якоби для гравитационных маневров в Солнечной системе