Significant high number commensurabilities in the main lunar problem II: The occurrence of Saros-like near periodicities

The effect of changes in the Moon's semimajor axis and the Earth's orbital eccentricity on the occurrence of Saros-like cycles is examined. The Earth-Moon-Sun dynamical system exhibits such cycles for only 25 to 30% of the time interval between –5×10⁷ to +5×10⁷ years. Not only has the present Saros the smallest period during this time, but it also has one of the longest durations and the period closest to an integral number of anomalistic years, thus making it one of the most efficient Saros-like cycles for reversing solar perturbations in the main lunar problem. During the lifetime of a Saros-like cycle, variations of the Earth's orbital eccentricity cause frequent disappearances and reappearances of the cycle.     

The use of the saros in lunar dynamical studies

It is shown that the near-periodicity in the Earth-Moon-Sun system demonstrated by the possibility of using the Saros to predict eclipses, suggests that the Saros can also be used in a fast and accurate method of special perturbations which can be applied for long term study of the evolution of the Moon's orbit.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April 1973.     

The arrangement in mean elements space of the periodic orbits close to that of the Moon

In a simplified model of the Earth-Moon-Sun system based on the restricted circular 3-dimensional 3-body problem, it is possible to find numerically a set of 8 periodic orbits whose time evolutions closely resemble that of the Moon's orbit. These orbits have a period of 223 synodic months (i.e. the period of the Saros cycle known for more than two millennia as a means of predicting eclipses), and are characterized by a secular rotation of the argument of perigee . Periodic orbits of longer durations exhibiting this last feature are very abundant in Earth-Moon-Sun dynamical models. Their arrangement in the space of the mean orbital elements- for various values of the lunar mean motion is presented.     

Dynamical History of the Earth–Moon System

Differential equations describing the tidal evolution of the earth's rotation and of the lunar orbital motion are presented in a simple close form. The equations differ in form for orbits fixed to the terrestrial equator and for orbits with the nodes precessing along the ecliptic due to solar perturbations. Analytical considerations show that if the contemporary lunar orbit were equatorial the evolution would develop from an unstable geosynchronous orbit of the period about 4.42 h (in the past) to a stable geosynchronous orbit of the period about 44.8 days (in the future). It is also demonstrated that at the contemporary epoch the orbital plane of the fictitious equatorial moon would be unstable in the Liapunov's sense, being asymptotically stable at early stages of the evolution. Evolution of the currently near-ecliptical lunar orbit and of the terrestrial rotation is traced backward in time by numerical integration of the evolutional equations. It is confirmed that about 1.8 billion years ago a critical phase of the evolution took place when the equatorial inclination of the moon reached small values and the moon was in a near vicinity of the earth. Before the critical epoch t _cr two types of the evolution are possible, which at present cannot be unambiguously distinguished with the help of the purely dynamical considerations. In the scenario that seems to be the most realistic from the physical point of view, the evolution also has started from a geosynchronous equatorial lunar orbit of the period 4.19 h. At t < t _cr the lunar orbit has been fixed to the precessing terrestrial equator by strong perturbations from the earth's flattening and by tidal effects; at the critical epoch the solar perturbations begin to dominate and transfer the moon to its contemporary near-ecliptical orbit which evolves now to the stable geosynchronous state. Probably this scenario is in favour of the Darwin's hypothesis about originating the moon by its separation from the earth. Too much short time scale of the evolution in this model might be enlarged if the dissipative Q factor had somewhat larger values in the past than in the present epoch. Values of the length of day and the length of month, estimated from paleontological data, are confronted with the results of the developed model.     

Luni-solar perturbations of an Earth satellite

Luni-solar perturbations of the orbit of an artificial Earth satellite are given by modifying the analytical theory of an artificial lunar satellite derived by the author in recent papers. Expressions for the first-order changes, both secular and periodic, in the elements of the geocentric Keplerian orbit of the earth satellite are given, the moon's geocentric orbit, including solar perturbations in it, being found by using Brown's lunar theory.The effects of Sun and Moon on the satellite orbit are described to a high order of accuracy so that the theory may be used for distant earth satellites.     

日月轨道计算对TLE深空编目目标预报精度的影响

双行根数(Two Line Element, TLE)作为一类广泛使用的空间物体编目数据, 其预报精度和误差特性是TLE编目 在空间碎片研究中所要关注的问题之一. TLE编目需要配合SGP4/SDP4 (Simplified General Perturbations 4/Simplified Deep Space 4)模型进行轨道预报, 对深空物体来说, 主要考虑带谐项$J_2$、$J_3$、$J_4$摄动、 第三体日月摄动和特殊轨道共振问题修正等. 其中, SGP4/SDP4模型第三体摄动计算时, 对日月轨道近似采用了长期进动根数和 简单平运动的方式, 在外推10d时存在约2$^\circ$--3${^\circ     

Spectral photometry of selected areas of the lunar surface during maximum solar activity

Photoelectric observations have been carried out during maximum solar activity in order to investigate the variation of the brightness of the lunar surface with the solar cycle and to detect the possiblity of existing colour anomaly. The brightness data, using a wide-passband filter, do not show any evidence of variation with the solar cycle. No colour anomaly is found for the investigated lunar regions, and the doubt that has been felt by some investigators about the nonexistence of certain colour indices' variation with phase is not confirmed.     

New Challenges to Trajectory Design by the Use of Electric Propulsion and Other New Means of Wandering in the Solar System

The design of spacecraft trajectories is a crucial part of a space mission design. Often the mission goal is tightly related to the spacecraft trajectory. A geostationary orbit is indeed mandatory for a stationary equatorial position. Visiting a solar system planet implies that a proper trajectory is used to bring the spacecraft from Earth to the vicinity of the planet. The first planetary missions were based on conventional trajectories obtained with chemical engine rockets. The manoeuvres could be considered 'impulsive' and clear limitations to the possible missions were set by the energy required to reach certain orbits. The gravity-assist trajectories opened a new way of wandering through the solar system, by exploiting the gravitational field of some planets. The advent of other propulsion techniques, as electric or ion propulsion and solar sail, opened a new dimension to the planetary trajectory, while at the same time posing new challenges. These 'low thrust' propulsion techniques cannot be considered 'impulsive' anymore and require for their study mathematical techniques which are substantially different from before. The optimisation of such trajectories is also a new field of flight dynamics, which involves complex treatments especially in multi-revolution cases as in a lunar transfer trajectory. One advantage of these trajectories is that they allow to explore regions of space where different bodies gravitationally compete with each other. We can exploit therefore these gravitational perturbations to save fuel or reduce time of flight. The SMART-1 spacecraft, first European mission to the Moon, will test for the first time all these techniques. The paper is a summary report on various activities conducted by the project team in these areas.     

On the problem of lunar time

New achievements in lunar investigations by spacecrafts provide the reasons why some new international scale of special lunar time nowdays is under discussion. However, the introduction of special time scales for other celestial bodies would turn to be a complex affair. To document this point of view it is shown that, both for making precise astrometrical calculations and for creating a lunar calendar, it should be reasonable to make use of the generally accepted scales of terrestrial time; there is no real need to introduce any special lunar time unit. For calendar needs it is suggested to introduce a local scale of time based on count of the lunations and usual mean solar time units being used within the every lunation.     

Two-dimensional guiding-center model of the solar wind-moon interaction

This paper presents a continued study of the two-dimensional guiding-center model of the solar wind interaction with the Moon. The characteristics theory and the computational method are discussed. The magnetic permeability of plasma is (1 + /2)^–1 in the solar wind flow upstream of the Moon, and it changes to 1 in the void region of the lunar wake. The gradual change of the magnetic permeability in the penumbral region from the interplanetary condition to the void condition is explained as the source of field perturbations in the lunar wake. Perturbations of the magnetic field propagate as magnetoacoustic waves in a frame of reference moving with the plasma flow. Computer solutions were obtained to show that (i) the two principal perturbations of the magnetic field in the lunar wake (the umbral increase and the penumbral decrease) are confined to a region bounded by a Mach cone tangent to the lunar body, and (ii) the penumbral increases occur outside the lunar Mach cone. Computer solutions are also used to identify the source of field perturbations and to simulate the solar wind-moon interaction under varying interplanetary conditions.     

The motion of charged dust particles in interplanetary space—I. The zodiacal dust cloud

The problem of electromagnetic perturbations of charged dust particle orbits in interplanetary space has been re-examined in the light of our better understanding of the large scale spatial and temporal interplanetary plasma and field topology. Using both analytical and numerical solutions for particle propagation it was shown that: (1) stochastic variations induced by electromagnetic forces are unimportant for the zodiacal dust cloud except for the lowest masses, (2) systemetic variations in orbit inclinations are unimportant if orbital radii are larger than 10 a.u. This is due to the solar cycle variation in magnetic polarity which tends to cancel out systematic effects, (3) systematic variations in orbital parameters (inclination, longitude of ascending node, longitude of perihel) induced by electromagnetic forces inside 1 a.u. tend to shift the plane of symmetry of the zodiacal dust cloud somewhat towards the solar magnetic equatorial plane, (4) inside 0.3 a.u. there is a possibility that dust particles may enter a region of “magnetically resonant” orbits for some time. Changes in orbit parameters are then correspondingly enhanced, (5) the observed similarity of the plane of symmetry of zodiacal light with the solar equatorial plane may be the effect of the interaction of charged interplanetary dust particles with the interplanetary magnetic field. Numerical orbit calculation of dust particles show that one of the results of this interaction is the rotation of the orbit plane about the solar rotational axis.     

A theoretical evaluation of the lunar tidal variations in the ionospheric F2-layer

Time-varying solutions of the full continuity equation for electrons in the F2-region are obtained. The effects of production, loss, diffusion and electrodynamic ‘E × B’ drift are taken into account. The ‘E × B’ drift term consists of a solar and a lunar component. The solar component of drift is assumed diurnal with 14.6m/sec maximum upward speed at mid-day. The lunar component is assumed sinusoidal with period of half lunar day and amplitude one tenth of the solar drift; the phase is assumed to remain constant in lunar time, in accordance with Chapman's phase law.The results show that the lunar variations in the F2-region are markedly dependent on solar time and latitude. It is also shown that the average semi-diurnal lunar variations in N_mF2 and h_mF2 at any particular lunar time are almost opposite in phase to each other (i.e. out of phase by 6 hr) in the magnetic equatorial zone, and out of phase by 2 hr at moderate latitudes. The phase of δh_mF2 is 10 hr at low latitudes and 9 hr at moderate latitudes. The phase of δN_mF2 is 4 hr at low latitudes and 11 lunar hr at moderate latitudes.The results also show that the phase of the lunar semi-monthly oscillations in N_mF2 undergoes a rapid shift of about 5 lunar hr in going from 8 to 12° and the so called phase reversal occurs at about 10° lat at which the amplitude of N_mF2. becomes extremely small.These and other results are in good agreement with observations. Thus it is shown that the main features of the observed lunar tidal variations of the F2-region within 20° of the magnetic equator can be explained satisfactorily by the superposition of a small lunar drift on a large solar drift.     

Analytic short period lunar and solar perturbations of artificial satellites

The short period luni-solar theory of Kozai is generalized for arbitrary obliquity of the ecliptic and inclination of the moon's orbit to the ecliptic. Analytic first order lunar perturbations to the elements are derived. The theory is illustrated by an application to the communication satellite Intelsat 3F3.Presently at the Department of Environmental Sciences, University of Tel Aviv, Ramat Aviv, Israel.     

Search for Phobos and Deimos gas/dust tori using in situ observations from Mars Global Surveyor MAG/ER

More than 490 elliptical aerobraking and science phasing orbits made by Mars Global Surveyor (MGS) in 1997 and 1998 provide unprecedented coverage of the solar wind in the vicinity of the orbits of the martian moons Phobos and Deimos. We have performed a comprehensive survey of magnetic field perturbations in the solar wind to search for possible signatures of solar wind interaction with dust or gas escaping from the moons. A total of 1246 solar wind disturbance events were identified and their distribution was examined relative to Phobos, the Phobos orbit, and the Deimos orbit. We find that the spatial distribution of solar wind perturbations does not increase near or downstream of Phobos, Phobos’ orbit, or Deimos’ orbit, which would have been expected if there is significant outgassing or dust escape from the martian moons. Of the 1246 magnetic field perturbation events found in the MGS data set, 11 events were found within 2000 km of the Phobos orbit, while three events were found within 2000 km of the Deimos orbit. These events were analyzed in detail and found to likely have other causes than outgassing/dust escape from the martian moons. Thus we conclude that the amount of gas/dust escaping the martian moons is not significant enough to induce detectable magnetic field perturbations in the solar wind. In essence we have not found any clear evidence in the MGS magnetic field data for outgassing or dust escape from the martian moons.     

Direct imaging of extra-solar planets with stationary occultations viewed by a space telescope

The feasibility of detecting planets outside the solar system through imaging at optical wavelengths by a telescope in space is considered. The “black” limb of the moon can be used as an occulting edge to greatly reduce the background light from the planet's star. With this technique and if certain other technical requirements can be realized, a hypothetical Jupiter-Sun system could be detected at a distance of 10 pc. For this system, a signal-to-noise ratio of 9 could be achieved in less than 20 min with a 2.4-m telescope in space. For diffraction limited optics, the required integration time is inversely proportional to the fifth power of the telescope aperture. An orbit for the telescope is described that could achieve a stationary lunar occultation of any star that would last nearly two hours, providing six times more integration time than required by the hypothetical Jupiter-Sun example.     

On the perturbations of a close-Earth satellite due to lunar inequalities

The lunar disturbing function for a close-Earth satellite is expressed as a sum of products of harmonics of the satellite's position and harmonics of the Moon's position, and the latter are expanded about a rotating and precessing elliptic orbit inclined to the ecliptic. The deviations of the Moon from this approximate orbit are computed from Brown's lunar theory andthe perturbations in satellite orbital elements due to these inequalities are derived. Numerical calculations indicate that several perturbations in the position of the satellite's node and perigee have magnitudes on the order of one meter.The author is supported in part by a National Science Foundation Graduate Fellowship.     

External perturbations on distant planetary orbits and objects in the Kuiper disk

We investigate the potential importance of molecular cloud and stellar perturbations on the orbits of Pluto and more distant (hypothetical) planets up to 500 AU from the Sun. It is found that stellar and molecular cloud-core perturbations are of roughly equal importance. It also is found that the likelihood of substantial perturbations on Pluto is not insignificant, and that numerous substantial stellar and molecular cloud perturbations are likely to have influenced the orbits of any planets beyond 200 AU. These perturbations may contribute to a prevalence of moderate eccentricities and inclinations for planets beyond the orbit of Neptune, and may be a characteristic of distant planetary orbits in other solar systems. Given the recent discovery of chaotic behavior in Pluto's orbit (Sussman and Wisdom 1988), the effects of external perturbations on the long-term stability of Pluto's orbit warrant continued study.     

The sodium tail of the Moon

During the few days centered about new Moon, the lunar surface is optically hidden from Earth-based observers. However, the Moon still offers an observable: an extended sodium tail. The lunar sodium tail is the escaping “hot” component of a coma-like exosphere of sodium generated by photon-stimulated desorption, solar wind sputtering and meteoroid impact. Neutral sodium atoms escaping lunar gravity experience solar radiation pressure that drives them into the anti-solar direction forming a comet-like tail. During new Moon time, the geometry of the Sun, Moon and Earth is such that the anti-sunward sodium flux is perturbed by the terrestrial gravitational field resulting in its focusing into a dense core that extends beyond the Earth. An all-sky camera situated at the El Leoncito Observatory (CASLEO) in Argentina has been successfully imaging this tail through a sodium filter at each lunation since April 2006. This paper reports on the results of the brightness of the lunar sodium tail spanning 31 lunations between April 2006 and September 2008. Brightness variability trends are compared with both sporadic and shower meteor activity, solar wind proton energy flux and solar near ultra violet (NUV) patterns for possible correlations. Results suggest minimal variability in the brightness of the observed lunar sodium tail, generally uncorrelated with any single source, yet consistent with a multi-year period of minimal solar activity and non-intense meteoric fluxes.     

On the history of the lunar orbit

A frequency-dependent model of tidal friction is used in the determination of the time rate of change of the lunar orbital elements and the angular velocity of the Earth. The variational equations consider eccentricity, the solar tide on the Earth, Earth oblateness, and higher-order terms in the Earth's tidal potential. A linearized solution of the equations governing the precission of the Earth's rotational angular momentum and the lunar ascending node is found. This allows the analytical averaging of the variational equations over the period of relative precession which, though large, is necessarily small in comparison to the time step of the numerical integrator that yields the system history over geological time. Results for this history are presented and are identified as consistent with origin of the Moon by capture. This model may be applied to any planet-satellite system where evolution under tidal friction is of interest.     

A dynamical investigation of the conjecture that Mercury is an escaped satellite of Venus

The possibility that Mercury might once have been satellite of a Venus, suggested by a number of anomalies, is investigated by a series of numerical computer experiments. Tidal interaction between Mercury and Venus would result in the escape of Mercury into a solar orbit. Only two escape orbits are possible, one exterior and one interior to the Venus orbit. For the interior orbit, subsequent encounters are sufficiently distant to avoid recapture or large perturbations. The perihelion distance of Mercury tends to decrease, while the orientation of perihelion librates for the first few thousand revolutions. If dynamical evolution or nonconservative forces were large enough in the early solar system, the present semimajor axes could have resulted. The theoretical minimum quadrupole moment of the inclined rotating Sun would rotate the orbital planes out of coplanarity. Secular perturbations by the other planets would evolve the eccentricity and inclination of Mercury's orbit through a range of possible configurations, including the present orbit. Thus the conjecture that Mercury is an escaped satellite of Venus remains viable, and is rendered more attractive by our failure to disprove it dynamically.