The vector approach to the problem of physical libration of the Moon: the linearized problem     

B. P. Kondratyev 《Solar System Research》2011,45(1):60-74

A flexible and informative vector approach to the problem of physical libration of the rigid Moon has been developed in which three Euler differential equations are supplemented by 12 kinematic ones. A linearized system of equations can be split into an even and odd systems with respect to the reflection in the plane of the lunar equator, and rotational oscillations of the Moon are presented by superposition of librations in longitude and latitude. The former is described by three equations and consists of unrestricted oscillations with a period of T ₁ = 2.878 Julian years (amplitude of 1.855″) and forced oscillations with periods of T ₂ = 27.201 days (15.304″), one stellar year (0.008″), half a year (0.115″), and the third of a year (0.0003″) (five harmonics altogether). A zero frequency solution has also been obtained. The effect of the Sun on these oscillations is two orders of magnitude less than that of the Earth. The libration in latitude is presented by five equations and, at pertrubations from the Earth, is described by two harmonics of unrestricted oscillations (T ₅ ≈ 74.180 Julian years, T ₆ ≈ 27.347 days) and one harmonic of forced oscillations (T ₃ = 27.212 days). The motion of the true pole is presented by the same harmonics, with the maximum deviation from the Cassini pole being 45.3″. The fifth (zero) frequency yields a stationary solution with a conic precession of the rotation axis (previously unknown). The third Cassini law has been proved. The amplitudes of unrestricted oscillations have been determined from comparison with observations. For the ratio $ \frac{{\sin I}} {{\sin \left( {I + i} \right)}} \approx 0.2311 $ \frac{{\sin I}} {{\sin \left( {I + i} \right)}} \approx 0.2311 , the theory gives 0.2319, which confirms the adequacy of the approach. Some statements of the previous theory are revised. Poinsot’s method is shown to be irrelevant in describing librations of the Moon. The Moon does not have free (Euler) oscillations; it has oscillations with a period of T ₅ ≈ 74.180 Julian years rather than T ≈ 148.167 Julian years.  相似文献   

Conditionally periodic solutions near the libration points of the restricted circular three-body problem     

L. G. Luk’yanov 《Astronomy Letters》2012,38(3):201-211

Families of conditionally periodic solutions have been found by a slightly modified Lyapunov method of determining periodic solutions near the libration points of the restricted three-body problem. When the frequencies of free oscillations are commensurable, the solutions found are transformed into planar or spatial periodic solutions. The results are confirmed by numerically integrating the starting nonlinear differential equations of motion.  相似文献   

On quasi-periodic motions around the collinear libration points in the real Earth–Moon system     

X. Y. Hou  L. Liu 《Celestial Mechanics and Dynamical Astronomy》2011,110(1):71-98

Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L ₁ and L ₂, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.  相似文献   

Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies     

Elbaz I. Abouelmagd  S. M. El-Shaboury 《Astrophysics and Space Science》2012,341(2):331-341

In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ _c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ _c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease.  相似文献   

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

Ming-Jiang Zhang  Chang-Yin Zhao  Yong-Qing Xiong 《Astrophysics and Space Science》2012,337(1):107-113

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.  相似文献   

Orientation of the moon by numerical integration     

H. B. Papo 《Earth, Moon, and Planets》1973,8(4):539-545

The differential equations of rotational motion of the Moon are solved by numerical integration methods. Euler's dynamical equations transformed to a convenient form are treated by techniques analogous to ordinary orbit determination procedures. The proposed method is fully consistent with the ephemeris of the Moon and can utilize a variety of observational material for the solution of the selected parameters. The parameters are grouped into three distinct groups, namely:

--The physical libration angles of the Moon and their time rates at an arbitrary initial epoch.

--Physical constants featuring the principal moments of ineria of the Moon.

--Parameters associated with the particular observational material being used.

Examples are given of comparison between the proposed method and Eckhardt's 1970 model of the physical librations of the Moon. The merits of the new method are discussed in the light of conventional data sources like Earth-based or satellite-based photography as well as newly available data types like Laser ranging to retroreflectors on the Moon.  相似文献   

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude     

B. P. Kondratyev 《Solar System Research》2013,47(3):147-158

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.  相似文献   

A nonlinear analysis of the moon's physical libration in longitude     

Donald H. Eckhardt  Kenneth Dieter 《Earth, Moon, and Planets》1971,2(3):309-319

The Euler equations for the forced physical librations of the Moon have already been solved by using a digital computer to perform the semi-literal mathematical manipulations. Very near resonance, the computer solution for the physical libration in longitude is complemented by the solution of the appropriate Duffing equation with a dissipation term. Because of its apparent proximity to a resonant frequency, the term whose argument is 2 - twice the mean angular distance of the Moon's perigee from the ascending node of its orbit - is especially important. Its phase, which soon should be measurable, is related to the Moon's anelasticity. The term's frequency, in units of the sidereal month, increases as the semi-major axis of the Moon's orbit about the Earth increases. Using the Moon's mechanical ellipticity of Koziel and the rate of increase of the semi-major axis of MacDonald, it is estimated that the 2 term will cross the resonant frequency in 130 million years and, if the rate of energy dissipation is sufficiently low, a transient libration will be induced.  相似文献   

On the stabilization of the collinear libration points of the restricted circular three-body problem     

P. S. Krasilnikov  A. L. Kunitsyn 《Celestial Mechanics and Dynamical Astronomy》1977,15(1):41-51

The possibility of stabilizing the collinear libration points of the circular restricted three-body problem by using an additional jet acceleration (constant in magnitude) is investigated. Three stabilization laws are considered when the jet acceleration is either directed continuously to one of the primariesm ₁,m ₂ or is parallel to the line joining them. The solution of the problem formulated is based on the method of the driving forces structure analysis created by W. Thomson and P. Tait. It is shown that none of the stabilization laws mentioned ensures the existence of the isolated minimum of changed potential energy, and therefore the secular stability of the collinear libration points is impossible. In the 3rd and 4th paragraphs the possibility of a gyroscopic stabilization of these points is considered. It is shown that the gyroscopic stabilization of the external libration points is possible only when jet acceleration is either directed to the distant mass or is parallel to the line joining the primaries. The necessary and sufficient conditions of the gyroscopic stabilization are given. It is also shown that the internal libration points cannot be stabilized by any of the laws considered. For the Earth-Moon system the numerical data of time-existence of the satellite in the vicinity of the libration point situated near the Moon are given.  相似文献   

A new analytic approach to the Sitnikov problem     

J. Hagel 《Celestial Mechanics and Dynamical Astronomy》1992,53(3):267-292

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP  相似文献   

Accounting for the viscosity of the fluid core in the problem of the physical libration of the moon     

B. P. Kondratyev 《Solar System Research》2013,47(1):1-10

A two-component theoretical model of the physical libration of the Moon in longitude is constructed with account taken of the viscosity of the core. In the new version, a hydrodynamic problem of motion of a fluid filling a solid rotating shell is solved. It is found that surfaces of equal angular velocity are spherical, and a velocity field of the fluid core of the Moon is described by elementary functions. A distribution of the internal pressure in the core is found. An angular momentum exchange between the fluid core and solid mantle is described by a third-order differential equation with a right-hand side. The roots of a characteristic equation are studied and the stability of rotation is proved. A libration angle as a function of time is found using the derived solution of the differential equation. Limiting cases of infinitely large and infinitely small viscosity are considered and an effect of lag of a libration phase from a phase of action of an external moment of forces is ascertained. This makes it possible to estimate the viscosity and sizes of the lunar fluid core from data of observations.  相似文献   

Effect of the Earth’s compression on the physical libration of the Moon     

B. P. Kondratyev 《Solar System Research》2012,46(5):341-351

The effect of the Earth??s compression on the physical libration of the Moon is studied using a new vector method. The moment of gravitational forces exerted on the Moon by the oblate Earth is derived considering second order harmonics. The terms in the expression for this moment are arranged according to their order of magnitude. The contribution due to a spherically symmetric Earth proves to be greater by a factor of 1.34 × 10⁶ than a typical term allowing for the oblateness. A linearized Euler system of equations to describe the Moon??s rotation with allowance for external gravitational forces is given. A full solution of the differential equation describing the Moon??s libration in longitude is derived. This solution includes both arbitrary and forced oscillation harmonics that we studied earlier (perturbations due to a spherically symmetric Earth and the Sun) and new harmonics due to the Earth??s compression. We posed and solved the problem of spinorbital motion considering the orientation of the Earth??s rotation axis with regard to the axes of inertia of the Moon when it is at a random point in its orbit. The rotation axes of the Earth and the Moon are shown to become coplanar with each other when the orbiting Moon has an ecliptic longitude of L _? = 90° or L _? = 270°. The famous Cassini??s laws describing the motion of the Moon are supplemented by the rule for coplanarity when proper rotations in the Earth-Moon system are taken into account. When we consider the effect of the Earth??s compression on the Moon??s libration in longitude, a harmonic with an amplitude of 0.03?? and period of T ₈ = 9.300 Julian years appears. This amplitude exceeds the most noticeable harmonic due to the Sun by a factor of nearly 2.7. The effect of the Earth??s compression on the variation in spin angular velocity of the Moon proves to be negligible.  相似文献   

Indirect transfer to the Earth–Moon L1 libration point     

Mingtao Li  Jianhua Zheng 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):203-213

The Earth–Moon L1 libration point is proposed as a human gateway for space transportation system of the future. This paper studies indirect transfer using the perturbed stable manifold and lunar flyby to the Earth–Moon L1 libration point. Although traditional studies indicate that indirect transfer to the Earth–Moon L1 libration point does not save much fuel, this study shows that energy efficient indirect transfer using the perturbed stable manifold and lunar flyby could be constructed in an elegant way. The design process is given to construct indirect transfer to the Earth–Moon L1 libration point. Simulation results show that indirect transfer to the Earth–Moon L1 libration point saves about 420 m/s maneuver velocity compared to direct transfer, although the flight time is about 20 days longer.  相似文献   

Analytical theory of the libration of the Moon     

M. Moons 《Earth, Moon, and Planets》1982,27(3):257-284

This paper presents a new theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field. This field is represented through its third degree harmonics for the torque due to the Earth (second degree for the torque due to the Sun).The orbital motion of the Moon is described by the ELP 2000 solution (Chapront-Touzé, 1980) of the main problem of lunar theory.the physical libration variables are obtained as Poisson series and comparisons with the results of Eckhardt (Eckhardt, 1981) and Migus (Migus, 1980) are presented.  相似文献   

On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem     

Enric Castellà  Àngel Jorba 《Celestial Mechanics and Dynamical Astronomy》2000,76(1):35-54

This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system. The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a very long time. It is remarkable that these regions seem to persist in the real system. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem     

M. K. Ammar 《Astrophysics and Space Science》2008,313(4):393-408

In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L₃ is stable and some families of periodic orbits can be drawn around it.  相似文献   

Planetary perturbations on the Libration of the Moon     

M. Moons 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):263-273

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).  相似文献   

The stability of periodic orbits in the three-body problem     

John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1975,12(3):255-276

A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.This linear stability analysis is used to study the stability of a family of periodic motions of three bodies with equal masses, in a rotating frame of reference. This family represents motion such that two bodies revolve around each other and the third body revolves around this binary system in the same direction to a distance which varies along the members of the family. It was found that a large part of the family, corresponding to the case where the distance of the third body from the binary system is larger than the dimensions of the binary system, represents stable motion. The nonlinear effects to the linear stability analysis are studied by computing the intersections of several perturbed orbits with the surface of sectiony ₃=0. In some cases more than 1000 intersections are computed. These numerical results indicate that linear stability implies stability to all orders, and this is true for quite large perturbations.  相似文献   

Optimal low-thrust transfers between libration point orbits     

Yuan Ren  Pierpaolo Pergola  Elena Fantino  Bianca Thiere 《Celestial Mechanics and Dynamical Astronomy》2012,112(1):1-21

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.  相似文献   

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions     

Marco Giancotti  Mauro Pontani  Paolo Teofilatto 《Celestial Mechanics and Dynamical Astronomy》2014,120(3):249-268

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\) .  相似文献