Sun-Jupiter System Research Articles

Lagrange points and regionally conserved quantities

Lagrange points are the equilibrium points within a restricted three-body system, epitomized by the Trojan asteroids near the L4 and L5 points of the Sun–Jupiter system. They also play a crucial role in some space missions, including the James Webb Space Telescope which is located at the Sun–Earth L2 point. While the existence of five Lagrange points is a well-known feature of the restricted three-body problem, the equations describing the precise location of all five points are not extensively documented. This work presents a derivation of all Lagrange points using polar coordinates and a new normalization scheme that offers a simpler interpretation of solutions compared to prior analyses. A subtle issue concerning the treatment of angular momentum in the potential formulation of this problem is addressed and resolved. The supplementary material to this work contains additional mathematical details and discussion.

Influence of non-gravitational forces on the co-orbital motion

ABSTRACT In the Solar system, there exist many non-gravitational perturbations for co-orbital objects, such as the solar radiation pressure, Yarkovsky effect, and so forth. Their effects play important roles in the dynamics of co-orbital objects as they lead to long-term perturbations accumulating. The motivation of this paper is to investigate the general mechanism of the non-gravitational force on the co-orbital motion in the circular restricted three-body problem. We propose an effective method for perturbed co-orbital motions by analysing the locus of the co-orbital objects in a two-dimensional map. Several expressions derived uncover how the non-gravitational force acts on orbital parameters. Taking the Sun–Jupiter system as an example, we implement numerical computations to demonstrate the validity of our results. Numerical computation shows that most of loci of co-orbital motions are in agreement with our conclusions. Some interesting phenomena of perturbed co-orbital motion, such as the co-orbital transition and escape, are found and explained. The results obtained from this paper provide an efficient approach to analyse the evolution of perturbed co-orbital motions.

Utilizing a control technique for orbital maintenance near [formula omitted] point and Lyapunov exponents

In present article, we have explored the orbit maintenance strategy around an artificial equilibrium point in the Sun–Jupiter system. Two control strategies, proportional control and proportional derivative control laws are discussed for orbit maintenance. The maximum Lyapunov exponent is used to check the chaotic behavior of planetary orbit. We anticipate that the recently established Lyapunov exponent criteria may be used to study motion of planets in generalized stellar binary systems. This research is being guided by the conclusions of recent observations as well as those that are expected from existing and upcoming planet search missions.

Relegation-free closed-form perturbation theory and the domain of secular motions in the restricted three-body problem

We propose a closed-form (i.e., without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (Celest Mech Dyn Astron 134(2):1–36, 2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method’s accuracy in terms of reproducibility of the orbital elements’ variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series’ remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle’s fast degree of freedom provides a valid approximation of the true dynamics.

Asymmetry in the number of L4 and L5 Jupiter Trojans driven by jumping Jupiter

Context. More than 10 000 Jupiter Trojans have been detected so far. They are moving around the L4 and L5 triangular Lagrangian points of the Sun-Jupiter system and their distributions can provide important clues about the early evolution of the Solar System. Aims. The number asymmetry of the L4 and L5 Jupiter Trojans is a longstanding problem. We aim to test a new mechanism in order to explain this anomalous feature by invoking the jumping-Jupiter scenario. Methods. First, we introduce the orbital evolution of Jupiter caused by the giant planet instability in the early Solar System. In this scenario, Jupiter could undergo an outward migration at a very high speed. We then investigate how such a jump changes the numbers of the L4 (N4) and L5 (N5) Trojans. Results. The outward migration of Jupiter can distort the co-orbital orbits near the Lagrangian points, resulting in L4 Trojans being more stable than the L5 ones. We find that this mechanism could potentially explain the unbiased number asymmetry of N4/N5 ~ 1.6 for the known Jupiter Trojans. The uncertainties of the system parameters, such as Jupiter’s eccentricity and inclination as well as the inclination distribution of Jupiter Trojans, are also taken into account and our results about the L4/L5 asymmetry have been further validated. However, the resonant amplitudes of the simulated Trojans are excited to higher values compared to the current population. A possible solution is that collisions among the Trojans may reduce their resonant amplitudes.

Post-Newtonian effects on some characteristic time-scales of transiting exoplanets

ABSTRACT Some measurable characteristic time-scales {ttrn} of transiting exoplanets are investigated in order to check preliminarily if their cumulative shifts over the years induced by the post-Newtonian (pN) gravitoelectric (Schwarzschild) and gravitomagnetic (Lense–Thirring) components of the stellar gravitational field are, at least in principle, measurable. Both the primary (planet in front of the star) and the secondary (planet behind the star) transits are considered along with their associated characteristic time intervals: the total transit duration tD, the ingress/egress transit duration $\tau$, the full-width at half maximum primary transit duration tH, and also the time of conjunction tcj. For each of them, the net changes per orbit $\left\langle \Delta t_D\right\rangle ,\, \left\langle \Delta \tau\right\rangle ,\, \left\langle \Delta t_H\right\rangle ,\, \left\langle \Delta t_\mathrm{cj}\right\rangle$ induced by the aforementioned pN accelerations are analytically obtained; also the Newtonian effect of the star’s quadrupole mass moment $J_2^\star$ is worked out. They are calculated for a fictitious Sun-Jupiter system in an edge-on elliptical orbit, and the results are compared with the present-day experimental accuracies for the HD 286123 b exoplanet. Its pN gravitoelectric shift $\left\langle \Delta t_\mathrm{cj}^\mathrm{1pN}\right\rangle$ may become measurable, at least in principle, at a ≃8 × 10−5 level of (formal) relative accuracy after about 30 yr of continuous monitoring corresponding to about 1000 transits. Systematics like, e.g. confusing time standards, neglecting star-spots, neglecting clouds, additional planets in the system, etc. would likely deteriorate the actual accuracy. The method presented is general enough to be applied also to modified models of gravity.

Analysis of Resonance Transition Periodic Orbits in the Circular Restricted Three-Body Problem

Resonance transition periodic orbits exist in the chaotic regions where the 1:1 resonance overlaps with nearby interior or exterior resonances in the circular restricted three-body problem (CRTBP). The resonance transition periodic orbits have important applications for tour missions between the interior and the exterior regions of the system. In this work, following the increase of the mass parameter μ in the CRTBP model, we investigate the breakup of the first-order resonant periodic families and their recombination with the resonance transition periodic families. In this process, we can describe in detail how the 1:1 resonance gradually overlaps with nearby first-order resonances with increasing strength of the secondary’s perturbation. Utilizing the continuation method, features of the resonance transition periodic families are discussed and characterized. Finally, an efficient approach to finding these orbits is proposed and some example resonance transition periodic orbits in the Sun–Jupiter system are presented.

Artificial Controlling of the Collinear Liberation Points Using Lorentz Force in the Restricted Three-Body Problem

This work studies the possibility of generating artificial collinear liberation points for the planar circular restricted three-body problem using Lorentz force affecting a charged spacecraft due to the magnetic field of a planet. It is considered to be a magnetic dipole inclined by angle α with the spin axis of the planet. The acceleration components for Lorentz force are first derived in an inertial planet-center coordinate system. Then, they are transformed into the rotating coordinate system of the three-body system, with the planet naturally the smaller primary in a planet-Sun system. The equations for the liberation points are derived including the charge per unit mass as the controlling parameter. Finally, the values of the charge per unit mass required for controlling the collinear liberation point positions are derived. A numerical application for the Sun-Jupiter system is introduced and the relation between the position of the artificial liberation point and the charge per unit mass is presented graphically.

Artificial Triangular Points by Lorentz Force in the Restricted Three-Body Problem

The restricted three body problem was outlined. The acceleration due to planetary magnetic field, in terms of space craft’s orbital elements, was analysed. The conditions for calculating the liberation points including the mutual gravitational attraction and the effect of Lorentz acceleration were derived for the case of circular planer restricted three bodies. The stability of the solution for the artificial Lorentz triangular liberation points was studied. Finally, numerical investigation for the case of Sun-Jupiter system was calculated as case study. The results show the ability of changing the position of the triangular liberation points by an order from 10−7 to 10−6 for the dimensionless x, y coordinates and distance r from Jupiter. This is equivalent to about hundreds of Kilometers which is considerable.

Controlling the libration point orbits for CRTBP with non-ideal solar sail and albedo effect

Libration point orbits around collinear points present numerous properties which are worthwhile for space missions. Because orbits around these points are exponentially unstable, station-keeping strategies aim for periodic motion.This paper considers the problem of Lagrangian point stabilisation in the Sun-Jupiter system with non-ideal solar sail and albedo effect. Energy shaping and dissipation injection are used to stabilize the Lagrangian points, which are originally unstable. Through shaping of Hamiltonian (energy) and dissipation injection at the designed equilibrium point, we obtained that a closed-loop system is stable and asymptotic stable respectively. We discover that the orbits are stable under port- Hamiltonian control using best fit results.

Station-keeping error analysis for halo orbits around Libration point L1 using linear control logic

In this article, we perform the computation of halo orbits around libration point L1 in the Sun–Jupiter system. Third-order approximation, Lindstedt–Poincare method is used to select the initial guess for libration point orbits. Furthermore, Newton’s differential correction scheme is utilized to correct the initial guess obtained from an analytical approach. Halo orbits are exceptionally unstable, the station-keeping maneuver is required to maintain the nominal halo orbit in the vicinity of collinear libration points. We have applied linear control logic in the non-linear equation of motion for the station-keeping error analysis of halo orbits and also computed the control input for halo orbits. Numerical results illustrate the viability of the proposed control technique for the exact station-keeping error at libration point L1.

Capture of interstellar objects: a source of long-period comets

ABSTRACT We simulate the passage through the Sun–Jupiter system of interstellar objects (ISOs) similar to 1I/‘Oumuamua or 2I/Borisov. Capture of such objects is rare and overwhelmingly from low incoming speeds on to orbits akin to those of known long-period comets. This suggests that some of these comets could be of extrasolar origin, in particular inactive ones. Assuming ISOs follow the local stellar velocity distribution, we infer a volume capture rate of $0.051\, \mathrm{au}^3\, \mathrm{yr}^{-1}$. Current estimates for orbital lifetimes and space densities then imply steady-state captured populations of ∼102 comets and ∼105 ‘Oumuamua-like rocks, of which 0.033 per cent are within 6 au at any time.

Lissajous motion near Lagrangian point $$ L_{2} $$ L 2 in radial solar sail

An attempt was made to study the dynamics close to the collinear libration point $$ L_{2} $$ of the radial solar sail circular-restricted three-body problem (RSCRTBP) in the Sun–Jupiter System, where the third massless body is a solar sail. We analyse the qausi-periodic (Lissajous solutions) orbits about the libration point $$ L_{2} $$ . The Lindstedt–Poincare approximation for the qausi-periodic orbits was used for numerical simulations. We utilized linear quadratic regulator (LQR) to stabilize the full nonlinear model, and linear state-feedback controller was designed to stabilize the trajectory.

Implications of Captured Interstellar Objects for Panspermia and Extraterrestrial Life

Abstract We estimate the capture rate of interstellar objects by means of three-body gravitational interactions. We apply this model to the Sun–Jupiter system and the Alpha Centauri A&B binary system, and find that the radius of the largest captured object is a few tens of km and Earth-sized, respectively. We explore the implications of our model for the transfer of life by means of rocky material. The interstellar comets captured by the “fishing net” of the solar system can be potentially distinguished by their differing orbital trajectories and ratios of oxygen isotopes through high-resolution spectroscopy of water vapor in their tails.

Orbital dynamics in the post-Newtonian planar circular restricted Sun–Jupiter system

The theory of the post-Newtonian (PN) planar circular restricted three-body problem is used for numerically investigating the orbital dynamics of a test particle (e.g. a comet, asteroid, meteor or spacecraft) in the planar Sun–Jupiter system with a scattering region around Jupiter. For determining the orbital properties of the test particle, we classify large sets of initial conditions of orbits for several values of the Jacobi constant in all possible Hill region configurations. The initial conditions are classified into three main categories: (i) bounded, (ii) escaping and (iii) collisional. Using the smaller alignment index (SALI) chaos indicator, we further classify bounded orbits into regular, sticky or chaotic. In order to get a spherical view of the dynamics of the system, the grids of the initial conditions of the orbits are defined on different types of two-dimensional planes. We locate the different types of basins and we also relate them with the corresponding spatial distributions of the escape and collision time. Our thorough analysis exposes the high complexity of the orbital dynamics and exhibits an appreciable difference between the final states of the orbits in the classical and PN approaches. Furthermore, our numerical results reveal a strong dependence of the properties of the considered basins with the Jacobi constant, along with a remarkable presence of fractal basin boundaries. Our outcomes are compared with the earlier ones regarding other planetary systems.

The First Planetary Microlensing Event with Two Microlensed Source Stars

Abstract We present the analysis of the microlensing event MOA-2010-BLG-117, and show that the light curve can only be explained by the gravitational lensing of a binary source star system by a star with a Jupiter-mass ratio planet. It was necessary to modify standard microlensing modeling methods to find the correct light curve solution for this binary source, binary-lens event. We are able to measure a strong microlensing parallax signal, which yields the masses of the host star, M * = 0.58 ± 0.11 M ⊙, and planet, m p = 0.54 ± 0.10M Jup, at a projected star–planet separation of a ⊥ = 2.42 ± 0.26 au, corresponding to a semimajor axis of au. Thus, the system resembles a half-scale model of the Sun–Jupiter system with a half-Jupiter0mass planet orbiting a half-solar-mass star at very roughly half of Jupiter’s orbital distance from the Sun. The source stars are slightly evolved, and by requiring them to lie on the same isochrone, we can constrain the source to lie in the near side of the bulge at a distance of D S = 6.9 ± 0.7 kpc, which implies a distance to the planetary lens system of D L = 3.5 ± 0.4 kpc. The ability to model unusual planetary microlensing events, like this one, will be necessary to extract precise statistical information from the planned large exoplanet microlensing surveys, such as the WFIRST microlensing survey.

Dynamical spreading of small bodies in 1:1 resonance with planets by the diurnal Yarkovsky effect

A simple model is introduced to describe the inherent dynamics of Trojans in the presence of the diurnal Yarkovsky effect. For different spin statuses, the orbital elements of the Trojans (mainly semimajor axis, eccentricity and inclination) undergo different variations. The variation rate is generally very small, but the total variation of the semimajor axis or the orbit eccentricity over the age of the Solar system may be large enough to send small Trojans out of the regular region (or, vice versa, to capture small bodies in the regular region). In order to demonstrate the analytical analysis, we first carry out numerical simulations in a simple model, and then generalize these to two ‘real’ systems, namely the Sun–Jupiter system and the Sun–Earth system. In the Sun–Jupiter system, where the motion of Trojans is regular, the Yarkovsky effect gradually alters the libration width or the orbit eccentricity, forcing the Trojan to move from regular regionsto chaotic regions, where chaos may eventually cause it to escape. In the Sun–Earth system, where the motion of Trojans is generally chaotic, our limited numerical simulations indicate that the Yarkovsky effect is negligible for Trojans of 100 m in size, and even for larger ones. The Yarkovsky effect on small bodies captured in other 1:1 resonance orbits is also briefly discussed.

Orbits of Minor Bodies of the Solar System in the Circular Restricted Three-Body Problem

A brief introduction to the circular restricted three-body problem (CR3BP) is given where a third body of negligible mass moves under the combined gravitational- centrifugal potential of two co-rotating massive bodies restricted to circular orbits. The equipotential contours of a variety of two body systems in the solar system are presented along with interesting orbits of Trojans, Hildas, Thule in the Sun-Jupiter system, the libration of Pluto in the Sun-Neptune system and choreographic orbits.

Horseshoe orbits in the Earth–Moon system

Horseshoe orbits in the restricted three-body problem have been mostly considered in the Sun–Jupiter system and, in recent years, in the Sun–Earth system. Here, these orbits have been used to find asteroids that have orbits of this kind. We have built a planar family of horseshoe orbits in the Earth–Moon system and determined the points of planar and 1/1 vertical resonances on this family. We have presented examples of orbits generated by these spatial families.

원형 제한 3체 문제의 불변위상공간을 이용한 행성간 궤적설계 기초 연구

This paper represents a trajectory design and analysis technique which uses invariant manifolds of the circular restricted three body problem. Instead of the classical patched conic method based on 2-body problem, the equation of motion and dynamical behavior of spacecraft in the circular restricted 3-body problem are introduced, and the characteristics of Lyapunov orbits near libration points and their invariant manifolds are covered in this paper. The trajectories from/to Lyapunov orbits are numerically generated with invariant manifolds in the Earth-moon system. The trajectories in the Sun-Jupiter system are also analyzed with various initial conditions in the boundary surface. These methods can be effectively applied to interplanetary trajectory designs.