Short-period effects of the planetary perturbations on the Sun–Earth Lagrangian point L₃

Abstract

Context. The Lagrangian point L3 of the Sun–Earth system, and its Lyapunov orbits, have been proposed to perform station-keeping, although L3 is only rigorously defined for the extremely simplified model represented by the reduced Sun–Earth–spacecraft system. As in L3 the planetary perturbations (mainly from Jupiter and Venus) are stronger than Earth’s attraction, it is necessary to understand whether or not the dynamics close to L3 persist under such a strong perturbation, allowing for a definition of dynamical substitutes for models that are more realistic than the circular restricted three-body problem. Aims. In this paper we address the problem of the existence of motions that remain close to L3 for a time-span which is relevant for space missions in a model of the Solar System compatible with the precision of JPL digital ephemerides. Methods. First, we computed analytically the main short-period effects of planetary perturbations in a simplified model of the Solar System with the orbits of all the planets co-planar and circular. We then applied the Fast Lyapunov Indicator method in order to find dynamical substitutes that exist for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Results. We find that the original system is conjugate by a canonical transformation to an averaged system that has an equilibrium close to L3: even if Venus and Jupiter each move the position of this equilibrium by about 218 and 176 km, respectively, in opposite directions, in the model where both the planets are included, their effects almost perfectly compensate for one another, leaving a displacement of about 40 km only. This equilibrium is then mapped in the original system to a quasi-periodic dynamical substitute; the contributions of each planet to the amplitude of this quasi-periodic libration around L3 do not compensate for one another, and sum to about 10 000 km. The Fast Lyapunov Indicator method allowed us to find orbits of any amplitude bigger than this one (up to 0.03 AU) for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Conclusions. Using a combination of the Hamiltonian averaging method with a new implementation of the Fast Lyapunov Indicator method we find orbits useful for astrodynamics originating at the Sun–Earth Lagrangian point L3 for a realistic model of the Solar System. In particular, this usage of the chaos indicator provides an innovative application of dynamical systems theory to astrodynamics, where the short-period perturbations represent a relevant part of the model.

Forced Resonance Orbit Analysis of Binary Asteroid System

“…Note, Equation ( 24) is the first-order approximation of the system instead of a linear solution. Affected by the system's nonlinearity, the variations of an, and φn (n = 1, 2) in Equation ( 24) strictly follow Equation (23). However, the amplitudes of the linear solution can be chosen as any value.…”

“…It is known that steady-state motions occur when D2a1=D2a2=0, D2φ1=D2φ2=0, which corresponds to the singular points of Equation (23). It is clear that a1=0 must be the steadystate motion for the long period motion (i.e., motion relevant to ω1).…”

“…Takei et.al [22] applied planer model of SRP to Hayabusa2's stationkeeping operation in the proximity of the asteroid Ryugu. The station-keeping process is similar to the one done by Scantamburlo and Guzzo [23]. The last type of orbit involves the Sun-terminator orbit [24], quasi-terminator orbit [25] and its alternating orbit near the asteroid utilizing the solar sail spacecraft [26][27][28].…”

On the analytical construction of halo orbits and halo tubes in the elliptic restricted three-body problem

“…Due to their importance for space-flight dynamics, many efforts to use halo orbits in models more complicated than the CRTBP have been done in the literature [11,13,21,22]. For example, in the real Solar System the eccentricity of the orbits of the planet identified as the secondary body, as well as the perturbations from the other planets, limit the study to look for orbits with features similar to the orbits identified in the approximation of the CRTBP (see for example [20,3,17,33,32,23,34]).…”

Transits close to the Lagrangian solutions $L_1,L_2$ in the Elliptic Restricted Three-body Problem

“…These solutions transit close to the Lagrangian points L 1 , L 2 of the CRTBP. But, the use for small bodies of the Solar System requires to consider hierarchical extensions of the model, from the CRTBP to the the full N planetary problem, which are still subject of study [2,24,25,40].…”