定点在日-地(月)系L1点附近的探测器的发射及维持

在限制性三体问题中共线平动点附近的运动虽然是不稳定的,但可以是有条件稳定的,该动力学特征使得一些有特殊目的的探测器只需消耗较少的能量即可定点在这些点附近(如ISEE-3、SOHO).以日-地(月)系的L1点为例,根据其附近的运动特征,探讨定点探测器的发射与轨道控制问题,给出了相应的数值模拟结果,为工程上的实现提供理论依据.     

Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth–Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.     

Bifurcating families around collinear libration points

The planar and the vertical Lyapunov families are two basic periodic families around the collinear libration points. The stability curves of these two families are given first, and then periodic families bifurcating from them are explored in detail. Several properties of these bifurcating families are found. This study follows a series of the authors’ publications on periodic families around the libration points in the restricted three-body problem.     

A note on the dynamics around the Lagrange collinear points of the Earth–Moon system in a complete Solar System model

In this paper we study the dynamics of a massless particle around the L _1,2 libration points of the Earth–Moon system in a full Solar System gravitational model. The study is based on the analysis of the quasi-periodic solutions around the two collinear equilibrium points. For the analysis and computation of the quasi-periodic orbits, a new iterative algorithm is introduced which is a combination of a multiple shooting method with a refined Fourier analysis of the orbits computed with the multiple shooting. Using as initial seeds for the algorithm the libration point orbits of Circular Restricted Three Body Problem, determined by Lindstedt-Poincaré methods, the procedure is able to refine them in the Solar System force-field model for large time-spans, that cover most of the relevant Sun–Earth–Moon periods.     

On the theoretical possibility of the libration cloud

In the present paper, the problem of whether the interplanetary matter has a tendency to accumulate around the Lagrangian libration pointsL ₄ andL ₅, is examined statistically. It is concluded that: (1) If the particles are initially assumed to be distributed uniformly, they keep the uniformity ever after around the libration points. (2) If the particles receive random stochastic perturbations, their distribution tends to become uniform even if initially they have non-uniform distributions. (3) If the particles mutually collide inelastically, they have a tendency to avoid the regions near the libration points. Therefore, the interplanetary matter will not tend to accumulate near the libration points. Even if the observations of the libration cloud so far reported are confirmed, the clouds are likely to be but temporary objects.     

Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.     

On the Transfer and Control of Space Probes Around the L1 Point of the Sun-Earth+Moon System

The motion around the collinear libration points in the restricted three body problem is unstable. But there exist conditionally stable periodic orbits around these points. Special-purpose space probes located in the vicinity of these points (e.g., ISEE-3, SOHO) can benefit from this dynamical property, in regard to maintaining the orbit in position and the energy required of placing the probe in position. As an example, we study in this paper the launch and orbital control of a space probe around the L1 libration point in the system consisting of the Sun and the Earth-Moon. We present some theoretical and numerical simulations’ results, which may serve as a basis for the realization of such a space probe in future.     

Latitudinal librations of Mercury with a fluid core

In the framework of the space missions to Mercury, an accurate model of rotation is needed. Librations around the 3:2 spin-orbit resonance as well as latitudinal librations have to be predicted with the best possible accuracy. In this paper, we use a Hamiltonian analysis and numerical integrations to study the librations of Mercury, both in longitude and latitude. Due to the proximity of the period of the free libration in longitude to the orbital period of Jupiter, the 88-day and 11.86-year contributions dominate Mercury’s libration in longitude (with the Hermean parameters chosen). The amplitude of the libration in latitude is much smaller (under 1 arcsec) and should not be detected by the space missions. Nevertheless, we point out that this amplitude could be much larger (up to several tens of arcsec) if the free period related to the libration in latitude approaches the period of the Jupiter-Saturn Great Inequality (883 years). Given the large uncertainties on the planetary parameters, this new resonant forcing on Mercury’s libration in latitude should be borne in mind.     

Lunar Gravity-Assist Maneuver As a Way of Reducing the Orbit Amplitude in the Spectrum–Röntgen–Gamma Project

Spectrum–Röntgen–Gamma (SRG) is a space observatory designed to observe astrophysical objects in the X-ray range of the electromagnetic spectrum. SRG is planned to be launched in 2019 by a Proton-M launch vehicle with a DM3 upper stage. The spacecraft will be delivered to an orbit around the Sun–Earth collinear libration point L2 located at a distance of ~1.5 million km from the Earth. Although the SRG launch scheme has already been determined at present, in this paper we consider an alternative spacecraft transfer scenario using a lunar gravity-assist maneuver. The proposed scenario allows a oneimpulse transfer from a low Earth orbit to a small-amplitude orbit around the libration point to be performed while fulfilling the technical constraints and the scientific requirements of the mission.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

The scattering map in the planar restricted three body problem

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the information already obtained for the homoclinic orbits. An example application to Astrodynamics is also proposed.     

On the theoretical possibility of the libration cloud

We show within the framework of the restricted three body problem that

1. Only in the immediate neighbourhood of the Lagrangian pointsL ₄ andL ₅ the distribution of a cloud of particles tends to become uniform under the influence of random stochastic perturbations. No consequences can be derived from this fact for a tendency of dispersion of clouds librating at arbitrary distances around the Lagrangian points.
2. From an elementary inspection of the Jacobi integral we cannot conclude that mutual completely inelastic collisions tend to drive the particles away from the vicinity of the libration points.

Finally the motion of a particle within the libration cloud, approximated as a resisting medium, is briefly examined.     

Librations of the Galilean satellites: The influence of global internal liquid layers

The four Galilean satellites are thought to harbor one or even two global internal liquid layers beneath their surface layer. The iron core of Io and Ganymede is most likely (partially) liquid and also the core of Europa may be liquid. Furthermore, there are strong indications for the existence of a subsurface ocean in Europa, Ganymede, and Callisto. Here, we investigate whether libration observations can be used to prove the existence of these liquid layers and to constrain the thickness of the overlying solid layers. For Io, the presence of a small liquid core increases the libration of the mantle by a few percent with respect to an entirely solid Io and mantle libration observations could be used to determine the mantle thickness with a precision of several tens of kilometers given that the libration amplitude can be measured with a precision of 1 m. For Europa, Ganymede, and Callisto, the presence of a water ocean close to the surface increases by at least an order of magnitude the ice shell libration amplitude with respect to an entirely solid satellite. The shell libration depends essentially on the shell thickness and to a minor extent on the density difference between the ocean and the ice shell. The possible presence of a liquid core inside Europa and Ganymede has no noticeable influence on their shell libration. For a precision of several meters on the libration measurements, in agreement with the expected accuracy with the NASA/ESA EJSM orbiter mission to Europa and Ganymede, an error on the shell thickness of a few tens kilometers is expected. Therefore, libration measurements can be used to detect liquid layers such as Io’s core or water subsurface oceans in Europa, Ganymede, and Callisto and to constrain the thickness of the overlying solid surface layers.     

Planetary perturbations on the Libration of the Moon

A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p₁, p₂ and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983).     

Existence and stability of the non-collinear libration points in the restricted three body problem when both the primaries are ellipsoid

This paper deals with the existence and stability of the non-collinear libration points in the restricted three-body problem when both the primaries are ellipsoid with equal mass and identical in shape. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the existence and stability of the non-collinear libration points. This is observed that the non-collinear libration points exist only in the interval 52^°<φ<90^° and form an isosceles triangle with the primaries. Further we observed that the non collinear libration points are unstable in 52^°<φ<90^°.     

On the triangular libration points in photogravitational restricted three-body problem with variable mass

This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L ₄ and L ₅ naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.     

Tides in a body librating about a spin–orbit resonance: generalisation of the Darwin–Kaula theory

The Darwin-Kaula theory of bodily tides is intended for celestial bodies rotating without libration. We demonstrate that this theory, in its customary form, is inapplicable to a librating body. Specifically, in the presence of libration in longitude, the actual spectrum of Fourier tidal modes differs from the conventional spectrum rendered by the Darwin–Kaula theory for a nonlibrating celestial object. This necessitates derivation of formulae for the tidal torque and the tidal heating rate, that are applicable under libration. We derive the tidal spectrum for longitudinal forced libration with one and two main frequencies, generalisation to more main frequencies being straightforward. (By main frequencies we understand those emerging due to the triaxiality of the librating body.) Separately, we consider a case of free libration at one frequency (once again, generalisation to more frequencies being straightforward). We also calculate the tidal torque. This torque provides correction to the triaxiality-caused physical libration. Our theory is not self-consistent: we assume that the tidal torque is much smaller than the permanent-triaxiality-caused torque, so the additional libration due to tides is much weaker than the main libration due to the permanent triaxiality. Finally, we calculate the tidal dissipation rate in a body experiencing forced libration at the main mode, or free libration at one frequency, or superimposed forced and free librations.     

High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system

High-order analytical solutions of invariant manifolds, associated with Lissajous and halo orbits in the elliptic restricted three-body problem (ERTBP), are constructed in this paper. The equations of motion of ERTBP in the pulsating synodic coordinate system have five equilibrium points, and the three collinear libration points as well as the associated center manifolds are unstable. In our calculation, the general solutions of the invariant manifolds associated with Lissajous and halo orbits around collinear libration points are expressed as power series of five parameters: the orbital eccentricity, two amplitudes corresponding to the hyperbolic manifolds, and two amplitudes corresponding to the center manifolds. The analytical solutions up to arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the center and invariant manifolds, transit and non-transit trajectories in ERTBP are all parameterized. Since the circular restricted three-body problem (CRTBP) is a particular case of ERTBP when the eccentricity is zero, the general solutions constructed in this paper can be reduced to describe the dynamics around the collinear libration points in CRTBP naturally. In order to check the validity of the series expansions constructed, the practical convergence of the series expansions up to different orders is studied.     

A dynamical equivalent to the equilateral libration points of the earth-moon system

Consider the Earth-Moon-particle system as a Restricted Three Body Problem. There are two equilateral libration points. In the actual world system, those points are no longer relative equilibrium points mainly due to the effect of the Sun and to the noncircular motion of the Moon around the Earth. In this paper we present the problem as a perturbation of the RTBP and we look for the dynamical equivalent of L _4,5. It turns out to be a quasiperiodic orbit. It is obtained for a simplified model but the procedure to obtain it is general and can be carried out with an additional computational effort.