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1.
Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.  相似文献   

2.
D. Vokrouhlický  W.F. Bottke 《Icarus》2005,175(2):419-434
In this paper, we show that Asteroid (433) Eros is currently residing in a spin-orbit resonance, with its spin axis undergoing a small-amplitude libration about the Cassini state 2 of the proper mode in the nonsingular orbital element sinI/2exp(?Ω), where I the orbital inclination and Ω the longitude of the node. The period of this libration is ?53.4 kyr. By excluding these libration wiggles, we find that Eros' pole precesses with the proper orbital plane in inertial space with a period of ?61.4 kyr. Eros' resonant state forces its obliquity to oscillate with a period of ?53.4 kyr between ?76° and ?89.5°. The observed value of ?89° places it near the latter extreme of this cycle. We have used these results to probe Eros' past orbit and spin evolution. Our computations suggest that Eros is unlikely to have achieved its current spin state by solar and planetary gravitational perturbations alone. We hypothesize that some dissipative process such as thermal torques (e.g., the so-called YORP effect) may be needed in our model to obtain a more satisfactory match with data. A detailed study of this problem is left for future work.  相似文献   

3.
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   

4.
The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.  相似文献   

5.
An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 105 revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.  相似文献   

6.
The author's earlier solution for Trojan asteroids is developed further. It is shown that depending on the amplitude of libration around the Lagrangian point L4, there is a critical inclination which determines the sign of the variation of the ascending node. If the orbital inclination of a Trojan is smaller than the critical one, then the ascending node decreases and otherwise it increases. The variation of the eccentricity and of the longitude of the perihelion has also a dependence on the critical inclination.  相似文献   

7.
For a satellite in a nominally circular orbit at arbitrary inclination whose mean motion is commensurable with the Earth's rotation, the dependence of gravity on longitude leads to a resonant variation in eccentricity as well as the long-period oscillation in longitude. Provided forces capable of processing perigee are present, it is shown that the change in eccentricity for a satellite captured in librational resonance is not secular but periodic.

There are corresponding resonance effects for a satellite in a nominally equatorial but eccentric orbit. Here the commensurability condition is that the longitudes of the apses shall be nearly repetitive relative to the rotating Earth. There will be a long-period oscillation in longitude which can take the form of either a libration (trapped) or a circulation (free), and there will also be an oscillation of the orbital plane having the same period as the precession of perigee relative to inertial space.  相似文献   


8.
Epimetheus, a small moon of Saturn, has a rotational libration (an oscillation about synchronous rotation) of 5.9°±1.2°, placing Epimetheus in the company of Earth’s Moon and Mars’ Phobos as the only natural satellites for which forced rotational libration has been detected. The forced libration is caused by the satellite’s slightly eccentric orbit and non-spherical shape.Detection of a moon’s forced libration allows us to probe its interior by comparing the measured amplitude to that predicted by a shape model assuming constant density. A discrepancy between the two would indicate internal density asymmetries. For Epimetheus, the uncertainties in the shape model are large enough to account for the measured libration amplitude. For Janus, on the other hand, although we cannot rule out synchronous rotation, a permanent offset of several degrees between Janus’ minimum moment of inertia (long axis) and the equilibrium sub-Saturn point may indicate that Janus does have modest internal density asymmetries.The rotation states of Janus and Epimetheus experience a perturbation every 4 years, as the two moons “swap” orbits. The sudden change in the orbital periods produces a free libration about synchronous rotation that is subsequently damped by internal friction. We calculate that this free libration is small in amplitude (<0.1°) and decays quickly (a few weeks, at most), and is thus below the current limits for detection using Cassini images.  相似文献   

9.
The acceleration of the mean lunar longitude has a small effect on the periods of most terms in a Fourier expansion of the longitude. There are several planetary perturbation terms that have small amplitudes, but whose periods are close to the resonant period of the lunar libration in longitude. Some of these terms are moving toward resonance, some are moving away from resonance, and the periods of those terms that do not include the Delaunay variables in their arguments are not moving. Because of its acceleration of longitude, the Moon is receding from the Earth, so the magnitude of the restoring torque that the Earth exerts on the rotating Moon is gradually attenuating; thus resonance itself is moving, but at a much slower rate than the periods of the accelerating planetary perturbations. There are five planetary perturbation terms from the ELP-2000 Ephemeris (with amplitudes of 0.00001 or greater) that have passed through resonance in the past two million years. One of them is of special interest because it appears to be the excitation source of a supposed free libration in longitude that has been detected by the lunar laser ranging experiment. The amplitude of the term is only 0.00021 but it could be the source of the 1 amplitude free libration term if the viscoelastic properties of the Moon are similar to those of the Earth.  相似文献   

10.
The Moon’s physical librations and determination of their free modes   总被引：2，自引：0，他引：2  
The Lunar Laser Ranging experiment has been active since 1969 when Apollo astronauts placed the first retroreflector on the Moon. The data accuracy of a few centimeters over recent decades, joined to a new numerically integrated ephemeris, DE421, encourages a new analysis of the lunar physical librations of that ephemeris, and especially the detection of three modes of free physical librations (longitude, latitude, and wobble modes). This analysis was performed by iterating a frequency analysis and linear least-squares fit of the wide spectrum of DE421 lunar physical librations. From this analysis we identified and estimated about 130–140 terms in the angular series of latitude librations and polar coordinates, and 89 terms in the longitude angle. In this determination, we found the non-negligible amplitudes of the three modes of free physical libration. The determined amplitudes reach 1.296′′ in longitude (after correction of two close forcing terms), 0.032′′ in latitude and 8.183′′ × 3.306′′ for the wobble, with the respective periods of 1056.13 days, 8822.88 days (referred to the moving node), and 27257.27 days. The presence of such terms despite damping suggests the existence of some source of stimulation acting in geologically recent times.  相似文献   

11.
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.  相似文献   

12.
Variations in diurnal tidal stress due to Europa’s eccentric orbit have been considered as the driver of strike-slip motion along pre-existing faults, but obliquity and physical libration have not been taken into account. The first objective of this work is to examine the effects of obliquity on the predicted global pattern of fault slip directions based on a tidal-tectonic formation model. Our second objective is to test the hypothesis that incorporating obliquity can reconcile theory and observations without requiring polar wander, which was previously invoked to explain the mismatch found between the slip directions of 192 faults on Europa and the global pattern predicted using the eccentricity-only model. We compute predictions for individual, observed faults at their current latitude, longitude, and azimuth with four different tidal models: eccentricity only, eccentricity plus obliquity, eccentricity plus physical libration, and a combination of all three effects. We then determine whether longitude migration, presumably due to non-synchronous rotation, is indicated in observed faults by repeating the comparisons with and without obliquity, this time also allowing longitude translation. We find that a tidal model including an obliquity of 1.2°, along with longitude migration, can predict the slip directions of all observed features in the survey. However, all but four faults can be fit with only 1° of obliquity so the value we find may represent the maximum departure from a lower time-averaged obliquity value. Adding physical libration to the obliquity model improves the accuracy of predictions at the current locations of the faults, but fails to predict the slip directions of six faults and requires additional degrees of freedom. The obliquity model with longitude migration is therefore our preferred model. Although the polar wander interpretation cannot be ruled out from these results alone, the obliquity model accounts for all observations with a value consistent with theoretical expectations and cycloid modeling.  相似文献   

13.
By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω1, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.  相似文献   

14.
Physical librations of the Moon are small cyclic perturbations with periods of one month and longer, and amplitudes of 100 arc seconds or less. These cause the selenographic axes fixed in the true Moon to have a different orientation than similar axes fixed in the mean Moon.Physical librations have two types of effects of present interest. If the orbital elements of a lunar satellite are referred to selenographic axes in the true Moon as it rotates and librates, then the librations cause changes in the orientation angles (node, inclination, and periapsis argument of the satellite) large enough that long-period perturbation theory cannot be used without compensation for such geometrical effects. As a second effect, the gravitational potential of the Moon is actually wobbled in inertial space, a condition not included in the potential expression used in perturbation theory.This paper gives data on the magnitude of the physical librations, the geometrical effects on the orbital elements and the equivalent changes in the coefficients in the potential. It is shown that geometrical effects can be accommodated either by using an inertial axes system or by compensating for the lunar librations and precession when the selenographic axes are used. Further, it is shown that physical effects are small and negligible for all but the most exacting endeavors.  相似文献   

15.
Abstract— The newly discovered asteroid 2003 YN107 is currently a quasi‐satellite of the Earth, making a satellite‐like orbit of high inclination with apparent period of one year. The term quasi‐satellite is used since these large orbits are not completely closed, but rather perturbed portions of the asteroid's orbit around the Sun. Due to its extremely Earth‐like orbit, this asteroid is influenced by Earth's gravity to remain within 0.1 AU of the Earth for approximately 10 years (1997 to 2006). Prior to this, it had been on a horseshoe orbit closely following Earth's orbit for several hundred years. It will re‐enter such an orbit, and make one final libration of 123 years, after which it will have a close interaction with the Earth and transition to a circulating orbit. Chaotic effects limit our ability to determine the origin or fate of this object.  相似文献   

16.
Exact differential equations relating the perturbations to satellite orbital elements by the motion of the Earth's equatorial plane are derived, and they are solved to second order in precession. The system proposed in a previous paper (Kozai, 1960), in which the inclination and the argument of perigee are referred to the equator of date and the longitude of the ascending node is measured from a fixed point along a fixed plane and then along the equator of date, can still be recommended for precise studies of satellite motion even when the second-order perturbations are taken into account.  相似文献   

17.
In this paper we present an investigation on the tidal evolution of a system of three bodies: the Earth, the Moon and the Sun. Equations are derived including dissipation in the planet caused by the tidal interaction between the planet and the satellite and between the planet and the sun. Dissipation within the Moon is included as well. The set of differential equations obtained is valid as long as the solar disturbances dominate the perturbations on the satellite's motion due to the oblateness of the planet, namelya/R e greater than 15, and closer than that point equations derived in a preceding paper are used.The result shows the Moon was closer to the Earth in the past than now with an inclination to the ecliptic greater than today, whereas the obliquity was smaller. Toward the past, the inclination to the Earth's equator begins decreasing to 12° fora/R e=12 and suddenly grows. During the first stage the results are weakly dependant on the magnitude of the dissipation within the satellite, whereas the distance of the closest approach and the prior history are strongly dependent on that dissipation. In particular, the crossing of the Roche limit can be avoided.  相似文献   

18.
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.  相似文献   

19.
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   

20.
The shaking of Mercury’s orbit by the planets forces librations in longitude in addition to those at harmonics of the orbital period that have been used to detect Mercury’s molten core. We extend the analytical formulation of Peale et al. (Peale, S.J., Margot, J.L., Yseboodt, M. [2009]. Icarus 199, 1-8) in order to provide a convenient means of determining the amplitudes and phases of the forced librations without resorting to numerical calculations. We derive an explicit relation between the amplitude of each forced libration and the moment of inertia parameter (B-A)/Cm. Far from resonance with the free libration period, the libration amplitudes are directly proportional to (B-A)/Cm. Librations with periods close to the free libration period of ∼12 years may have measurable (∼arcsec) amplitudes. If the free libration period is sufficiently close to Jupiter’s orbital period of 11.86 years, the amplitude of the forced libration at Jupiter’s period could exceed the 35 arcsec amplitude of the 88-day forced libration. We also show that the planetary perturbations of the mean anomaly and the longitude of pericenter of Mercury’s orbit completely determine the libration amplitudes.While these signatures do not affect spin rate at a detectable level (as currently measured by Earth-based radar), they have a much larger impact on rotational phase (affecting imaging, altimetry, and gravity sensors). Therefore, it may be important to consider planetary perturbations when interpreting future spacecraft observations of the librations.  相似文献   

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