Analytical lunar libration tables

We have developed a theory of the rotation of the Moon, for the purpose of obtaining libration series explicitly dependent upon lunar gravitational field model parameters. A nonlinear model is used in which the rigid Moon, whose motion in space is that of the main problem of lunar theory, and whose gravity potential is represented through its third degree harmonics, is torqued by the Earth and Sun. The analytical series are then obtained as Poisson series. Numerical comparisons with Eckhardt's solution are briefly exposed.     

Analytical extension of lunar libration tables

Tables of lunar physical libration defining the analytical dependence upon the parameters of the lunar gravitational field are presented. The tables are obtained on the framework of the main problem in lunar libration by integration of the Hamilton equations reduced to the harmonic oscillator equations.The variables of physical libration have been obtained in the form of Poisson series. The distinguishing feature of the tables is that these series are the analytical extension of semianalytical solution computed for a number of dynamical parameters LURE2.A comparison with the Eckhardt's solution is briefly presented. The previously revealed disagreement of the mean inclination of lunar equator to ecliptic with that in Eckhardt's solution 500 has been maintained.     

Lunar baselines and libration from differential Vlbi observations of alseps

Differential very-long-baseline interferometric observations of signals from Apollo Lunar Surface Experiment Package telemetry transmitters will yield the relative projected positions of the transmitters with uncertainty of only 1-3 m, set mainly by uncertainty of the lunar ephemeris. Noise and systematic instrumental errors which in the past affected similar observations have been reduced to the equivalent of a few centimeters on the lunar surface by the development of a new type of differential receiver. Continued observations should yield a determination of the motion of the Moon about its center of mass with uncertainty less than 1 s of selenocentric arc. Improvements (by other means) in our knowledge of the Moon's orbital motion would allow a further order-of-magnitude refinement in the libration and relative position results obtainable by differential VLBI.Communication presented at the conference on Lunar Dynamics and Observational Coordinate Systems held January 15–17, 1973 at the Lunar Science Institute, Houston, Tex. U.S.A.     

A theory of libration

It is shown here that many problems of libration in celestial mechanics can be reduced to a perturbation of anintermediary defined by the Hamiltonian $$F = B\left( y \right) + 2\mu ^2 A\left( y \right)f\left( x \right).$$ This generalization of the Ideal Resonance Problem, with a periodic functionf(x) replacing sin² x, is solved here toO(μ ²) by an algorithm that is essentially the same as the one used in the original formulation. The solution is of the formx=x(u), u=u(t), y=y(x), with the functionx(u) commonly involving the inversion of a hyperelliptic integralu(x), evaluated by quadrature. Libration may be simple or multiple, depending on the nature of the functionf(x) and on the initial conditions. Double libration is illustrated here by the horseshoe-shaped orbits enclosing two libration centers.     

Lunar Beagle and Lunar Astrobiology

The study of the elements and molecules of astrobiological interest on the Moon can be made with the Gas Analysis Package (GAP) and associated instruments developed for the Beagle 2 Mars Express Payload. The permanently shadowed polar regions of the Moon may offer a unique location for the “cold-trapping” of the light elements (i.e. H, C, N, O, etc.) and their simple compounds. Studies of the returned lunar samples have shown that lunar materials have undergone irradiation with the solar wind and adsorb volatiles from possible cometary and micrometeoroid impacts. The Beagle 2’s analytical instrument package including the sample processing facility and the GAP mass spectrometer can provide vital isotopic information that can distinguish whether the lunar volatiles are indigenous to the moon, solar wind derived, cometary in origin or from meteoroids impacting on the Moon. As future Lunar Landers are being considered, the suite of instruments developed for the Mars Beagle 2 lander can be consider as the baseline for any lunar volatile or resource instrument package.     

The high-eccentricity libration theory revisted

Using the local asymmetric expansion of the disturbing function for the planar elliptic restricted three-body problem up to degree 1 ine ₁ we develop a small amplitude libration theory. We review the laws that characterize the mean-motion resonances of asteroids with Jupiter and we obtain other new laws. Special attention is paid to the second forced mode whose equations are reformulated and new consequences of this component are discussed. An analytical expression for the trajectories in the phase space is obtained. The predictions are compared with numerical experiments which confirm the results.     

Counterglow from the Earth-Moon libration points

Results from the OSO-6 Rutgers Zodiacal Light Analyzer experiment show photometric perturbations above the background in the anti-Sun line of sight. Sixteen successive lunations were examined, and the accumulated perturbations show a maximum value in the direction of the L₄ and L₅ Earth-Moon libration points. This is interpreted as a counterglow from a cloud of particles at the libration points. The average brightness of these libration clouds is 20 S₁₀ Vis. The average angular size of the libration clouds is approximately 6 degrees. Their position varies from one lunation to the next, within an ellipsoidal zone centered on the libration point direction, with its semi-major axis, of approximately 6 degrees, nominally in the ecliptic and its semi-minor axis, of approximately 2 degrees perpendicular to the ecliptic. The position of these clouds with respect to the Lagrangian L₄ and L₅ points, is towards the Moon in the northern summer and away from the Moon in the northern winter.     

Long-period forcing of Mercury's libration in longitude

Planetary perturbations of Mercury's orbit lead to forced librations in longitude in addition to the 88-day forced libration induced by Mercury's orbital motion. The forced librations are a combination of many periods, but 5.93 and 5.66 years dominate. These two periods result from the perturbations by Jupiter and Venus respectively, and they lead to a 125-year modulation of the libration amplitude corresponding to the beat frequency. Other periods are also identified with Jupiter and Venus perturbations as well as with those of the Earth, and these and other periods in the perturbations cause several arc second fluctuations in the libration extremes. The maxima of these extremes are about 30″ above and the minima about 7″ above the superposed ∼^60″, 88-day libration during the 125-year modulation. Knowledge of the nature of the long-period forced librations is important for the interpretation of the details of Mercury's rotation state to be obtained from radar and spacecraft observations. We show that the measurement of the 88-day libration amplitude for the purposes of determining Mercury's core properties is not compromised by the additional librations, because of the latter's small amplitude and long period. If the free libration in longitude has an amplitude that is large compared with that of the forced libration, its ∼10-year period will dominate the libration spectrum with the 88-day forced libration and the long-period librations from the orbital perturbations superposed. If the free libration has an amplitude that is comparable to those of the long-period forced libration, it will be revealed by erratic amplitude, period and phase on the likely time span of a series of observations. However, a significant free libration component is not expected because of relatively rapid damping.     

Physical libration of the moon in longitude

The application of modern computing techniques in the study of the physical libration of the Moon in longitude brings into a new perspective this problem that has been debated so much in the past.     

Low-energy Lunar Trajectories with Lunar Flybystwo

The low-energy lunar trajectories with lunar flybys are investigated based on the Sun-Earth-Moon bicircular problem (BCP). The characteristics of the distribution of trajectories in the phase space are summarized. Using the invariant manifolds in the BCP system, the low-energy lunar trajectories with lunar flybys are sought. Then, take time as an augmented dimension in the phase space of a nonautonomous system, we present the state space map and reveal the distribution of these lunar trajectories in the phase space. Consequently, we find that the low-energy lunar trajectories exist as families, and that the every moment in the Sun-Earth-Moon synodic period can be the departure date. Finally, we analyse the velocity increment, transfer duration, and system energy for the different trajectory families, and obtain the velocity-impulse optimal family and the transfer-duration optimal family, respectively.     

Non-linear stability around the triangular libration points

The configuration space around the triangular libration points in the Earth-Moon system is partitioned according to the stability of the motion. The regions aroundL ₄ andL ₅ are established where particles placed with zero initial velocity will librate. The complexity of the partitioning is revealed.     

Quasi-periodic orbits about the translunar libration point

Analytical solutions for quasi-periodic orbits about the translunar libration point are obtained by using the method of Lindstedt-Poincaré and computerized algebraic manipulations. The solutions include the effects of nonlinearities, lunar orbital eccentricity, and the Sun's gravitational field. For a small-amplitude orbit, the orbital path as viewed from the Earth traces out a Lissajous figure. This is due to a small difference in the fundamental frequencies of the in-plane and out-of-plane oscillations. However, when the amplitude of the in-plane oscillation is greater than 32 379 km, there is a corresponding value of the out-of-plane amplitude that will produce a path where the fundamental frequencies are equal. This synchronized trajectory describes a halo orbit of the Moon.     

The free precession and libration of Mercury

An analysis based on the direct torque equations including tidal dissipation and a viscous core-mantle coupling is used to determine the damping time scales of O(¹⁰⁵) years for free precession of the spin about the Cassini state and free libration in longitude for Mercury. The core-mantle coupling dominates the damping over the tides by one to two orders of magnitude for the plausible parameters chosen. The short damping times compared with the age of the Solar System means we must find recent or on-going excitation mechanisms if such free motions are found by the current radar experiments or the future measurement by the MESSENGER and BepiColombo spacecraft that will orbit Mercury. We also show that the average precession rate is increased by about 30% over that obtained from the traditional precession constant because of a spin-orbit resonance induced contribution by the C₂₂ term in the expansion of the gravitational field. The C₂₂ contribution also causes the path of the spin during the precession to be slightly elliptical with a variation in the precession rate that is a maximum when the obliquity is a minimum. An observable free precession will compromise the determination of obliquity of the Cassini state and hence of C/MMR² for Mercury, but a detected free libration will not compromise the determination of the forced libration amplitude and thus the verification of a liquid core.     

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.     

Numerical orbits near the triangular lunar libration points

The behavior of a test particle placed at a triangular libration point of the Earth-Moon system is calculated using Newton's equations for the four-body problem, with arbitrarily chosen initial conditions. If the orbits of the massive bodies have their real eccentricities, then the test particle leaves the vicinity of the libration point in three years, much faster than if the orbits were circular. Very small particles are affected by solar radiation pressure, and may leave even faster.     

Resonant forcing of Mercury's libration in longitude

The period of free libration of Mercury's longitude about the position it would have had if it were rotating uniformly at 1.5 times its orbital mean motion is close to resonance with Jupiter's orbital period. The Jupiter perturbations of Mercury's orbit thereby lead to amplitudes of libration at the 11.86 year period that may exceed the amplitude of the 88 day forced libration determined by radar. Mercury's libration in longitude may be thus dominated by only two periods of 88 days and 11.86 years, where other periods from the planetary perturbations of the orbit have much smaller amplitudes.     

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.