On a global expansion of the disturbing function in the planar elliptic restricted three-body problem

Starting with a simple Taylor-based expansion of the inverse of the distance between two bodies, we are able to obtain a series expansion of the disturbing function of the three-body problem (planar elliptic case) which is valid for all points of the phase space outside the immediate vicinity of the collision points. In particular, the expansion is valid for very high values of the eccentricity of the perturbed body. Furthermore, in the case of an interior mean-motion resonant configuration, the above-mentioned expression is easily averaged with respect to the synodic period, yielding once again a global expansion of (R) valid for very high eccentricities.Comparisons between these results and the numerically computed exact function are presented for various resonances and values of the eccentricity. Maximum errors are determined in each case and their origin is established. Lastly, we discuss the applicability of the present expansion to practical problems.     

On the stability of particular solutions of the singly averaged Hill problem

We consider the particular solutions of the evolutionary system of equations in elements that correspond to planar and spatial circular orbits of the singly averaged Hill problem. We analyze the stability of planar and spatial circular orbits to inclination and eccentricity, respectively. We construct the instability regions of both particular solutions in the plane of parameters of the problem.     

On the orbits and masses of the satellites of the Pluto-Charon system

Two small satellites of Pluto, S/2005 P1 (hereafter P1) and S/2005 P2 (hereafter P2), have recently been discovered outside the orbit of Charon, and their orbits are nearly circular and nearly coplanar with that of Charon. Because the mass ratio of Charon-Pluto is ∼0.1, the orbits of P2 and P1 are significantly non-Keplerian even if P2 and P1 have negligible masses. We present an analytic theory, with P2 and P1 treated as test particles, which shows that the motion can be represented by the superposition of the circular motion of a guiding center, the forced oscillations due to the non-axisymmetric components of the potential rotating at the mean motion of Pluto-Charon, the epicyclic motion, and the vertical motion. The analytic theory shows that the azimuthal periods of P2 and P1 are shorter than the Keplerian orbital periods, and this deviation from Kepler's third law is already detected in the unperturbed Keplerian fit of Buie and coworkers. In this analytic theory, the periapse and ascending node of each of the small satellites precess at nearly equal rates in opposite directions. From direct numerical orbit integrations, we show the increasing influence of the proximity of P2 and P1 to the 3:2 mean-motion commensurability on their orbital motion as their masses increase within the ranges allowed by the albedo uncertainties. If the geometric albedos of P2 and P1 are high and of order of that of Charon, the masses of P2 and P1 are sufficiently low that their orbits are well described by the analytic theory. The variation in the orbital radius of P2 due to the forced oscillations is comparable in magnitude to that due to the best-fit Keplerian eccentricity, and there is at present no evidence that P2 has any significant epicyclic eccentricity. However, the orbit of P1 has a significant epicyclic eccentricity, and the prograde precession of its longitude of periapse with a period of 5300 days should be easily detectable. If the albedos of P2 and P1 are as low as that of comets, the large inferred masses induce significant short-term variations in the epicyclic eccentricities and/or periapse longitudes on the 400-500-day timescales due to the proximity to the 3:2 commensurability. In fact, for the maximum inferred masses, P2 and P1 may be in the 3:2 mean-motion resonance, with the resonance variable involving the periapse longitude of P1 librating. Observations that sample the orbits of P2 and P1 well on the 400-500-day timescales should provide strong constraints on the masses of P2 and P1 in the near future.     

Long-Period Variations of the Eccentricity Vector Valid also for Near Circular Orbits around a Non-Spherical Body

This paper studies the long period variations of the eccentricity vector of the orbit of an artificial satellite, under the influence of the gravity field of a central body. We use modified orbital elements which are non-singular at zero eccentricity. We expand the long periodic part of the corresponding Lagrange equations as power series of the eccentricity. The coefficients characterizing the differential system depend on the zonal coefficients of the geopotential, and on initial semi-major axis, inclination, and eccentricity. The differential equations for the components of the eccentricity vector are then integrated analytically, with a definition of the period of the perigee based on the notion of “free eccentricity”, and which is also valid for circular orbits. The analytical solution is compared to a numerical integration. This study is a generalization of (Cook, Planet. Space Sci., 14, 1966): first, the coefficients involved in the differential equations depend on all zonal coefficients (and not only on the very first ones); second, our method applies to nearly circular orbits as well as to not too eccentric orbits. Except for the critical inclination, our solution is valid for all kinds of long period motions of the perigee, i.e., circulations or librations around an equilibrium point.     

On the calculation of perturbations in the radial position of satellites

With the aim of analyzing the errors in the radial position of satellites, we give in this paper the expressions for the radial perturbation which include the complete zeroth- and first-order terms in the orbital eccentricity. A simpler and faster method of calculating the perturbation is given.     

Celestial-mechanical peculiarities of Uranus’s satellite system

We consider the structural peculiarities of Uranus’s satellite system associated with its separation into two groups: inner equatorial satellites moving in nearly circular orbits and distant irregular satellites with retrograde motion in highly elliptical orbits. The intermediate region is free from satellites in a wide range of semimajor axes. By analyzing the evolution of satellite orbits under the combined effect of solar attraction and Uranus’s oblateness, we offer a celestial-mechanical explanation for the absence of equatorial satellites in this region. M.L. Lidov’s studies during 1961–1963 have served as a basis for our analysis.     

Numerical Stabilization of Orbital Motion

Mainly, the author focuses on Baumgarte's method and its applications in satellite, asteroid, stellar and planetary problems. In the paper arguments are given for the use of energy relations for stabilization in the elliptical two-body problem. Stabilizing properties of Baumgarte's equations and others are discussed. A simple approach is proposed for stabilizing the equations of almost circular motion. By using Baumgarte's technique, the author derives stabilized equations of perturbed restricted three-body problem. It is shown experimentally that stabilization in the problems mentioned above can raise the accuracy of numerical integration by several orders.     

关于修改港工钢筋混凝土规范小偏心受压构件正截面强度计算的建议

本文提出在小偏心受压构件正截面强度计算中，取消弯曲抗压强度ｆｃｍ（Ｒｗ），采用轴心抗压强度ｆｃ，这与构件的实际破坏形态相符。对小偏心受压构件正截面强度计算提出了设计建议。特别是对工程上大量应用的矩形截面对称配筋小偏压构件提示了简捷的设计方法，与以往的近似计算方法相比具有物理概念明确，精度高，方法简捷的特点。     

Periodicities in natural remanent magnetization and palaeoclimatic records detected by walsh spectra

The time series consisting of nonsinusoidal Natural Remanent Magnetization (NRM) and palaeoclimatic variations for the past two million years have been spectrally examined by using a new Walsh transform technique. The results show statistically significant periodicities (at 95% confidence level) of approximately 92,000 years; 43,000 years and 21,000 years in the ensemble spectra of the NRM intensity (mineralogic) variations. These NRM periodicities are remarkably close to the well-known Milankovitch cycles. The ensemble spectra of palaeoclimatic records reveal only 41,000 years statistically significant (95%) periodicity corresponding to the obliquity cycle. The study suggests that NRM variations in deep sea sediments probably are more sensitive recorders of palaeoclimatic memory than the oxygen isotope variations.     

Periodic accumulation and destruction of aeolian erg sequences in the Permian Cedar Mesa Sandstone, White Canyon, southern Utah, USA

The Permian Cedar Mesa Sandstone represents the product of at least 12 separate aeolian erg sequences, each bounded by regionally extensive deflationary supersurfaces. Facies analysis of strata in the White Canyon area of southern Utah indicates that the preserved sequences represent erg‐centre accumulations of mostly dry, though occasionally water table‐influenced aeolian systems. Each sequence records a systematic sedimentary evolution, enabling phases of aeolian sand sea construction, accumulation, deflation and destruction to be discerned and related to a series of underlying controls. Sand sea construction is signalled by a transition from damp sandsheet, ephemeral lake and palaeosol deposition, through a phase of dry sandsheet deposition, to the development of thin, chaotically arranged aeolian dune sets. The onset of the main phase of sand sea accumulation is reflected by an upward transition to larger‐scale, ordered sets which represent the preserved product of climbing trains of sinuous‐crested transverse dunes with original downwind wavelengths of 300–400 m. Regularly spaced reactivation surfaces indicate periodic shifts in wind direction, which probably occurred seasonally. Compound co‐sets of cross strata record the oblique migration of superimposed slipfaced dunes over larger, slipfaceless draa. Each aeolian sequence is capped by a regionally extensive supersurface characterized by abundant calcified rhizoliths and bioturbation and which represents the end product of a widespread deflation episode whereby the accumulation surface was lowered close to the level of the water table as the sand sea was progressively cannibalized by winds that were undersaturated with respect to their potential carrying capacity. Aeolian sequence generation is considered to be directly attributable to cyclical changes in climate and related changes in sea level of probable glacio‐eustatic origin that characterize many Permo‐Carboniferous age successions. Sand sea construction and accumulation occurred during phases of increased aridity and lowered sea level, the main sand supply being former shallow marine shelf sediments that lay to the north‐west. Sand sea deflation and destruction would have commenced at, or shortly after, the time of maximum aridity as the available sand supply became exhausted. Restricted episodes of non‐aeolian accumulation would have occurred during humid (interglacial) phases, accumulation and preservation being enabled by slow rises in the relative water table. Subsidence analysis within the Paradox Basin, together with comparisons to other similar age successions suggests that the climatic cycles responsible for generating the Cedar Mesa erg sequences could be the product of 413 000 years so‐called long eccentricity cycles. By contrast, annual advance cycles within the aeolian dune sets indicate that the sequences themselves could have accumulated in just a few hundred years and therefore imply that the vast majority of time represented by the Cedar Mesa succession was reserved for supersurface development.