Celestial Mechanics and Dynamical Astronomy

In this paper we study a particular four-body problem: three bodies revolve around their center of mass in circular orbits under the influence of their mutual gravitational attraction, while a fourth body moves in the plane defined by the three bodies but non influencing their motion. The linear stability of the eight equilibrium points is studied, and it is found that it depends on the values of the masses.

In the first part of this paper [Marchal, Yoshida, Sun Yi-Sui 1985] we have analyzed three-body systems satisfying the condition r≦kR where k is a suitable constant, r the mutual distance of the two masses of the “binary” and R the distance between the center of mass of the binary and the “third mass”. That condition r≦kR puts limits on the acceleration of the third mass and these limits allow us to determine the corresponding “escape velocities”. In this second part we look for initial conditions under which the inequality r≦kR will remain forever satisfied and we develop the corresponding tests of escape and their applications. This leads to a major improvement of the knowledge of the nature of three-body motions especially in the vicinity of triple close approaches. The region of bounded motions is much smaller than was generally expected and numerical computations of particular solutions show that we approach very near to the true limit.

We study near-Lagrange collinear configurations in the three-dimensional three-body problem, with arbitrary gravitating masses. A general method, previously developed for Coulombian systems, which provides a unique formalism for treating few-body systems close to the breakup threshold, has been employed to study the motion of three bodies both for bounded and unbounded configurations. The dependence of the triple-escape function on the small total energy E has been evaluated, as well as rovibronic configurations for bounded motion. An approximate characteristic constant is found for symmetrical systems, /, where and are the vertical libration mode and rotational mode angular frequencies, respectively, and is the threshold exponent. For equal-mass systems this constant acquires the value 1.05288.

In 1950 Brouwer and van Woerkom published a secular theory of the variations of the planetary elements in analytical form. In the present paper we provide a graphical representation of this theory in the form of element plots for a time span of ten million years.

We describe the implementation of Aarseth's NBODY2 code on a HP 1000 computer. We use the Extended Memory Array (EMA) feature with this code in order to investigate problems that include several hundreds of bodies, but the use of EMA requires some care in order to avoid large increases in computing time. The Vector Instruction Set (VIS) feature, a group of arithmetic subroutines that operates on arrays of floating point numbers and significantly reduces the computing time, turned out to be of little value for this application. We present the computing times demanded by two different problems for a variety of programs, including EMA and VIS. Finally, we present mass loss and mass accretion results for several simulations of galaxy-galaxy encounters performed with our implementation of the NBODY2 code.

Theory for the motion of a satellite in a near-circular orbit and perturbed by zonal and resonance terms in the Earth's gravity field is developed. Commensurability with respect to both primary and secondary terms is considered with the solution dependent on the depths of the resonances. The theory is applied to the motion of COSMOS 1603 (1984-106A) which approached 14 : 1 resonance in 1987. Values of lumped harmonics derived from least-squares analysis are in close agreement with previous studies of 1984-106A and global gravity field models. The theory is finally extended to incorporate the effects of air drag.

The resonance of GEOS-II (1968-002A) with 13th-order terms of the geopotential is analyzed. The odd-degree geopotential coefficients (13, 13), (15, 13), and (17, 13) given by Yionoulis most accurately model the resonance effects on GEOS-II of any of the published sets of 13th-order coefficients. However, this set is not adequate for precision orbit determination; additional even-degree coefficients are required.Values ofC 14,13(=0.5710–21) andS 14,13(=6.510–21) to be used with the odd-degree set of Yionoulis were obtained from an analysis of the observed along-track position variation of GEOS-II. These coefficients, when used with those of Yionoulis, yield greatly improved fits to the data and orbital prediction capability. However, further refinement is possible because the small effects of the remaining even-degree resonant terms were not modeled.The composite coefficientsC 13,13(=1.710–20) andS 13,13(=+2.710–20) were obtained under the assumption that the (13, 13) spherical harmonic of the geopotential is responsible for all of the observed along-track variation of GEOS-II due to resonance. The good agreement of these deliberately composite values with some published values ofC 13,13 andS 13,13 suggests that some of the published values may also be composite to some extent.

Without Abstract

In this paper an analytical model, suitable for a global description of the dynamics in a secular resonance of order 1, is derived from the general perturbation study developed in a previous paper (Morbidelli and Henrard (1991)). Such a model is then used to study the secular resonances 6, 5 and 16, and pictures illustrating the secular motion are obtained. The peculiarities of the 5 resonances are discussed in detail. The results are compared with those obtained by the theories of Yoshikawa and Nakai-Kinoshita. Some numerical simulations performed by Ch. Froeschl and H. Scholl are discussed in the light of the new theoretical results. New numerical experiments on the 6 resonance are also presented.

We review theoretical and numerical results obtained for secular resonant motion in the asteroidal belt. William's theory (1969) yields the locations of the principal secular resonances 5, 6, and 16 in the asteroidal belt. Theories by Nakai and Kinoshita (1985) and by Yoshikawa (1987) allow us to model the basic features of orbital evolution at the secular resonances 16 and 6, respectively. No theory is available for the secular resonance v5. Numerical experiments by Froeschl and Scholl yield quantitative and new qualitative results for orbital evolutions at the three principal secular resonances 5, 6, and 16. These experiments indicate possible chaotic motion due to overlapping resonances. A secular resonance may overlap with another secular resonance or with a mean motion resonance. The role of the secular resonances as possible sources of meteorites is discussed.

A new formula has been derived for geopotential expressed in terms of orbital elements. The summation sequence was changed so that the terms of the same frequencies would be grouped and the generalized lumped coefficients were derived. The proposed formula has the same form for both odd and evenl-m. Applying Hori's perturbation method, new formulae were derived for tesseral harmonic perturbations in nonsingular orbital elements:l+g, h, e cosg,e sing, L, andH. We show the possibility of effective application of the derived formulae to the calculation of orbits of very low satellites taking into account the coefficients of tesseral harmonics of the Earth's gravitational field up to high orders and degrees. As an example the perturbations up to the order and degree of 90 for the orbit of GRM satellites were calculated. The calculations were carried out on an IBM AT personal computer.

Numerical orbit integrations have been conducted to characterize the types of trajectories in the one-dimensional Newtonian three-body problem with equal masses and negative energy. Essentially three different types of motions were found to exist. They may be classified according to the duration of the bound three-body state. There are zero-lifetime predictable trajectories, finite lifetime apparently chaotic orbits, and infinite lifetime quasi-periodic motions. The quasi-periodic orbits are confined to the neighbourhood of Schubart's stable periodic orbit. For all other trajectories the final state is of the type binary + single particle in both directions of time. The boundaries of the different orbit-type regions seem to be sharp. We present statistical results for the binding energies and for the duration of the bound three-body state. Properties of individual orbits are also summarized in the form of various graphical maps in a two-dimensional grid of parameters defining the orbit.

A theory for the long-term variations in the orbit of a spherically symmetric satellite due to direct solar radiation pressure is tested using two satellite orbit analyses. The first of these analyses is in terms of mean elements for the balloon satellite Explorer 19. The results are compared with the expected theoretical variations with short-period terms omitted. The second analysis utilises satellite laser ranging observations of the geodetic satellite, Lageos. A novel long-term analysis technique is developed primarily for laser ranging studies. The technique is tested along with the solar radiation pressure perturbation theory by comparing the results from the theory and the analysis.

The orbit of the balloon satellite, Explorer 19, is analysed to determine the effects of direct solar radiation pressure over one revolution of the satellite (111 min) for MJD 42822 and MJD 42966. At the earlier date, the satellite entered the Earth's shadow, presenting an opportunity to examine the effectiveness of two different shadow models. The reflectivity of the surface of the satellite was estimated from analysis of the variations in orbital eccentricity over a period of 236 days. Although many of the parameters associated with the shape and orientation of the satellite are unknown, the theory for a non-spherical satellite is applied using trial and error methods to determine the parameters of best fit. The paper concludes with an examination of the perturbations in orbital eccentricity and inclination due to incident, specularly reflected, and diffusely reflected radiation.

About 60,000 observations of lunar occultations made during 1955–1980 are analysed using recently-developed semi-analytical solution ELP2000-82 for the Moon's position in order to determine the constants in the lunar theory and to investigate the tidal term in the Moon's mean longitude and the motions of the perigee and node of the lunar orbit. The equinox correction and systematic correction to the fundamental star catalogue and the correction to the datum of the lunar-profile in Watts' charts are also investigated. It is confirmed that the occultation observations do not have inconsistent tidal term with the modern observations and the observed mean motions of the perigee and node coincide with the theoretical ones within the error of observations. Some of the values of the constants in the lunar theory and the equinox correction to the fundamental catalogue FK5 obtained in this paper are significantly different from the values obtained using the Brown's theory. The reason of the difference is almost attributed to the deficiencies in the Brown's theory. The obtained correction to the datum of the lunar-profile in Watts' charts is almost consistent with the results by earlier investigators.

A definitive orbit of Comet 1961 V (Wilson-Hubbard), based on 84 observations, is given.

The new analysis of radar observations of inner planets for the time span 1964–1989 is described. The residuals show that Mercury topography is an important source of systematic errors which have not been taken into account up to now. The longitudinal and latitudinal variations of heights of Mercury surface were found and an approximate map of equatorial zone |[(G)\dot]/G = (0.47 ±0.47) 10 - 11 yr - 1\dot G/G = (0.47 \pm 0.47) \times 10^{ - 11} yr^{ - 1} . The correction to Mercury perihelion motion: D[(p)\dot] = - 0".017 ±0".052 cy - 1\Delta \dot \pi = - 0''.017 \pm 0''.052 cy^{ - 1} and linear combination of the parameters of PPN formalism: u = (2 + 2g- b)/3 = 0.9995 ±0.0013\upsilon = (2 + 2\gamma - \beta )/3 = 0.9995 \pm 0.0013 were determined; they are in a good agreement with General Relativity predictions. The obtained values J2 = ( - 0.13 ±0.41) 10 - 6J_2 = ( - 0.13 \pm 0.41) \times 10^{ - 6} .

We construct a spacecraft transfer with low cost and moderate flight time from the Earth to the Moon. The motion of the spacecraft is modeled by the planar circular restricted three-body problem including a perturbation due to the solar gravitation. Our approach is to reduce computation of optimal transfers to a non-linear boundary value problem. Using a computer software called AUTO, we solve it and continue its solutions numerically to obtain the optimal transfers. Our result also shows that the use of the solar gravitation can further lower the transfer cost drastically.

An analytic model is presented which gives upper atmospheric densities as a function of the exospheric temperature and the altitude. The densities produced are identical to those produced by Jacchia's 1970 models (1970) for altitudes between 90 and 125 km and closely approximate Jacchia's values for altitudes greater than 125 km.

Criticism of Deprit and Rom's (1970) extension of Brouwer's (1959) first-order satellite theory to the third order in J sub 2. It is noted that Deprit and Rom's theory suffers from the drawback that the perturbations of all orders are expressed as infinite power series in the eccentricity e. It is demonstrated that Deprit and Rom's reported failure to extend Brouwer's theory to the second order in a closed form by means of Lie transforms is due to an oversight and is not caused by difficulties inherent in the problem or in the particular perturbation method used.

The developments in planetary theories during the period 1973–1976 are reviewed. Emphasis is placed on the efforts to prepare new general theories, on the new techniques of numerical integration and their application, and on the studies of minor planets, particularly of those with periods nearly commensurable with the period of Jupiter.

Analysis of high precision 2-day mean orbital elements of several U.S. Navy navigation satellites is presented. The combined analytical-numerical method for the computation of the elements from the Defense Mapping Agency precision ephemerides is described. The precision of the semimajor axis and the inclination orbital elements are determined to be 2 cm and 0.016 respectively. The 28th degree and 27th order terms of the geopotential field are determined with the mean elements.

Closed-form expressions are derived for the density of the atmosphere as a function of altitude and exospheric temperature. The numerical model used as a reference is the Jacchia 1977 Static Density Model. Unlike previous work, the present method of derivation is independent of the particular mathematical form of the reference model. In fact, many intermediate tuning parameters provide the flexibility necessary to adjust these solutions optimally to all numerical models of the Jacchia type. Some other improvements to existing analytic Jacchia models are made, namely increased numerical efficiency, smaller deviation from the reference model and continuity of the first derivative at all altitudes. The model can also be expanded for improved accuracy. Numerical results and comparisons with the Jacchia 1977 model are made. Potential numerical difficulties are identified and methods given to remove them.

In 1979 the Seventeenth General Assembly of the International Astronomical Union (IAU) in Montreal, Canada, adopted the 1979 IAU Theory of Nutation upon the recommendation of this Working Group. Subsequently the International Union of Geodesy and Geophysics (IUGG) passed a resolution requesting that this action be reconsidered in favor of a theory based on a different Earth model. As a consequence of that reconsideration the 1980 IAU Theory of Nutation was adopted. The details of that theory and the history of its adoption are described here in the Final Report of the IAU Working Group on Nutation. A summary of these events and the essence of our recommendations is provided first while the body of the report discusses these matters in greater detail. The theory itself is contained in Table I.

The long-term systematic errors of the analytical theories IAU 2000 and IAU 2006 of the Earth’s precession-nutational motion are studied making use of the VLBI data of 1984–2007. Several independent methods give indubitable evidence of the significant quadratic error dϕ=(23±3)mas/cy 2 in the IAU 2000 residuals of the precessional angle ϕ while the adopted value of the secular decrease e ˙=-7·9×10 -9 /cy of the Earth’s ellipticity e (derived from Satellite Laser Ranging data) should manifest itself in the residuals of ϕ as the negative quadratic trend dϕ e ˙ ≈-8mas/cy 2 . The problem with the precession of the IAU 2006 theory adopted as a new international standard and based on the precession model P03 (Capitaine et al., Astron Astrophys 432:355-367, 2005) appears to be even more serious because the above mentioned quadratic term dϕ e ˙ has already been incorporated into the P03 precession. Our analysis of the VLBI data demonstrates that the quadratic trend of the IAU 2006 residuals dϕ does amount to the expected value (30·0±3)mas/cy 2 . It means, first, that the theoretical precession rate of IAU 2006 should be augmented by the large secular correction dp s =2dϕ≈(60·0±6)mas/cy 2 and, second, that the available VLBI data have potentiality of estimating the rate e ˙. And indeed, processing these data by the numerical theory ERA of the Earth’s rotation [G. A. Krasinsky, ibid. 96, 169–217 (2006; Zbl 1116.86002); G. A. Krasinsky and M. V. Vasilyev, ibid. 96, 219–237 (2006; Zbl 1116.86003)] yields the estimate e ˙=-(14±4)×10 -9 /cy statistically in accordance with the satellite-based e ˙. On the other hand, applying IAU 2000/2006 models, the positive value e ˙=(27±4)×10 -9 /cy is found which is incompatible with the SLR estimate and, evidently, has no physical meaning. The large and steadily increasing error of the precession motion of the IAU 2006 theory makes the task of replacing IAU 2006 by a more accurate model be most pressing.

This report incorporates revisions to the tables giving the directions of the north poles of rotation and the prime meridians of the planets and satellites since the last report. The tables giving the sizes and shapes of the planets and satellites have also been revised.

In this paper we study the time evolution of the non-gravitational forces (NGF) along with the observational bias, during the 1986 return of comet Halley. First, evidence is presented which shows that a NGF model with constant coefficients along the radial and transverse directions (A1 andA2), and a constant coefficient radial-only observation bias model fails to represent the observations within the conservative optical measurement noise of one second of arc. Second, we present a stochastic approach to the problem, where coefficients for the radial, transverse and normal components of both the NGF and the observation bias, are allowed to vary in time as three statistically independent Gauss-Markov processes. Finally, an orbit is estimated with this model, to fit observations made during the last apparition, employing a stochastic optimal smoother. Results are given in terms of time histories of the coefficient estimates along with their smoother computed uncertainties. A plot of the observation residuals is also included, showing a uniform and unbiased behavior. The analysis of the results confirms some of the assumptions often made when modelling cometary motion, but questions others. In particular, the normal bias coefficient shows an unexpected pre-perihelion peak (−1200±250 km) which proves that a radial-only observation bias model may lead to biased orbital estimates and unrealistic computed uncertainties.

Every three years the IAU/IAG/COSPAR Working Group on Cartographic Coordinates and Rotational Elements of the Planets and Satellites revises tables giving the directions of the north poles of rotation and the prime meridians of the planets and satellites. Also presented are revised tables giving their sizes and shapes.

The three authors sent independently their comments to the editor, who proposed to join them in the present note. These comments concern the paper by S. Kasperczuk where the object was to find integrable Hamiltonians of the "Yang-Mills" type It=1 2 1 4 1~ 22 g(Pl "q- p2 q_ alq2 q_ a2q2) q_ lq4 q_ ~a3q 2 ~_ ~"4qlq2" Kasperczuk found three integrable cases, and cites two of them, (A) al = a2, a3 = a4 = 1, (B) al = a2, a3 = 1, a4 = 3 as known and presents as new (C) a2 = 4al, a3 = 16, a4 = 6. S. Tompaidis pointed out a misprint in equations 2.20 (a missing factor of q2 in the first term). The integral for case (C) should read

Every three years the IAU/IAG/COSPAR Working Group on Cartographic Coordinates and Rotational Elements of the Planets and Stallites revises tables giving the directions of the north poles of rotation and the prime meridians of the planets, satellites, and asteroids. Also presented are revised tables giving their sizes and shapes.

Systematic reductions of nineteenth century observations to the system of the FK4 are discussed. Reductions made on a nightly basis are described and compared with the results obtained through the use of conventional tables. The series of observations made at the Paris Observatory from 1837 to 1881 was used to compare the two methods, and a combined system of 24 000 FK4, FK4 Sup and AGK 3R positions and proper motions provided the reference stars. The results show that for Uranus the mean error of a single observation in right ascension is 1..33 when tables are used for the reductions, and 1.12 when nightly reductions are made, while in declination the corresponding mean errors are 0.88 and 0.80. The observations of Neptune show an even greater difference between the two methods; the mean errors for the tabular and nightly reductions are 1.57 and 1.09 in right ascension and 0.88 and 0.75 in declination. Secular rates in the (0–C)'s of Uranus of –0.029/year in right ascension and 0.030/year in declination are present when the observations are reduced with tables. These rates are reduced to –0.007/year and +0.015/year, respectively, when nightly reductions are made.

Every three years the IAU/IAG Working Group on cartographic coordinates and rotational elements of the planets and satellites revises tables giving the directions of the north poles of rotation and the prime meridians of the planets, satellites, and asteroids. Also presented are revised tables giving their sizes and shapes. Changes since the previous report are summarized in the Appendix.

Binaries in the Kuiper-belt are unlike all other known binaries in the Solar System. Both their physical and orbital properties are highly unusual and, because these objects are thought to be relics dating back to the earliest days of the Solar System, understanding how they formed may provide valuable insight into the conditions which then prevailed. A number of different mechanisms for the formation of Kuiper-belt binaries (KBBs) have been proposed including; two-body collisions inside the Hill sphere of a larger body; strong dynamical friction; exchange reactions; and chaos assisted capture. So far, no clear consensus has emerged as to which of these mechanisms (if any) can best explain the observed population of KBBs. Indeed, the recent characterization of the mutual orbit of the symmetric (i.e., roughly equal mass) KBB 2001 QW322 has only served to complicate the picture because its orbit does not seem readily explicable by any of the available models. The binary 2001 QW322 stands out even among the already unusual population of KBBs for the following reasons: its mutual orbit is extremely large (≈105 km or about 30% of the Hill sphere radius), retrograde, it is inclined ≈120° from the ecliptic and has very low eccentricity, i.e., e ≤0.4 (and possibly e ≤ 0.05). Here we propose a hybrid formation mechanism for this object which combines aspects of several of the mechanisms already proposed. Initially two objects are temporarily trapped in a long-living chaotic orbit that lies close to a retrograde periodic orbit in the three-dimensional Hill problem. This is followed by capture through gravitational scattering with a small intruder object. Finally, weak dynamical friction gradually switches the original orbit “adiabatically” into a large, almost circular, retrograde orbit similar to that actually observed. KeywordsKuiper-belt binary-Binaries-Formation mechanisms-Chaos-assisted capture-2001 QW322 -Irregular satellites-FLI maps

Every three years the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the north poles of rotation and the prime meridians of the planets, satellites, and asteroids. This report introduces a system of cartographic coordinates for asteroids and comets. A topographic reference surface for Mars is recommended. Tables for the rotational elements of the planets and satellites and size and shape of the planets and satellites are not included, since there were no changes to the values. They are available in the previous report (Celest. Mech. Dyn. Astron., 82, 83–110, 2002), a version of which is also available on a web site.

Every three years the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the poles of rotation and the prime meridians of the planets, satellites, minor planets, and comets. This report introduces improved values for the pole and rotation rate of Pluto, Charon, and Phoebe, the pole of Jupiter, the sizes and shapes of Saturn satellites and Charon, and the poles, rotation rates, and sizes of some minor planets and comets. A high precision realization for the pole and rotation rate of the Moon is provided. The expression for the Sun’s rotation has been changed to be consistent with the planets and to account for light travel time

Every three years the IAU Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the poles of rotation and the prime meridians of the planets, satellites, minor planets, and comets. This report takes into account the IAUWorking Group for Planetary System Nomenclature (WGPSN) and the IAU Committee on Small Body Nomenclature (CSBN) definition of dwarf planets, introduces improved values for the pole and rotation rate of Mercury, returns the rotation rate of Jupiter to a previous value, introduces improved values for the rotation of five satellites of Saturn, and adds the equatorial radius of the Sun for comparison. It also adds or updates size and shape information for the Earth, Mars’ satellites Deimos and Phobos, the four Galilean satellites of Jupiter, and 22 satellites of Saturn. Pole, rotation, and size information has been added for the asteroids (21) Lutetia, (511) Davida, and (2867) Šteins. Pole and rotation information has been added for (2) Pallas and (21) Lutetia. Pole and rotation and mean radius information has been added for (1) Ceres. Pole information has been updated for (4) Vesta. The high precision realization for the pole and rotation rate of the Moon is updated. Alternative orientation models for Mars, Jupiter, and Saturn are noted. The Working Group also reaffirms that once an observable feature at a defined longitude is chosen, a longitude definition origin should not change except under unusual circumstances. It is also noted that alternative coordinate systems may exist for various (e.g. dynamical) purposes, but specific cartographic coordinate system information continues to be recommended for each body. The Working Group elaborates on its purpose, and also announces its plans to occasionally provide limited updates to its recommendations via its website, in order to address community needs for some updates more often than every 3 years. Brief recommendations are also made to the general planetary community regarding the need for controlled products, and improved or consensus rotation models for Mars, Jupiter, and Saturn.

Every three years the IAU Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the poles of rotation and the prime meridians of the planets, satellites, minor planets, and comets. This report takes into account the IAU Working Group for Planetary System Nomenclature (WGPSN) and the IAU Committee on Small Body Nomenclature (CSBN) definition of dwarf planets, introduces improved values for the pole and rotation rate of Mercury, returns the rotation rate of Jupiter to a previous value, introduces improved values for the rotation of five satellites of Saturn, and adds the equatorial radius of the Sun for comparison. It also adds or updates size and shape information for the Earth, Mars’ satellites Deimos and Phobos, the four Galilean satellites of Jupiter, and 22 satellites of Saturn. Pole, rotation, and size information has been added for the asteroids (21) Lutetia, (511) Davida, and (2867) Šteins. Pole and rotation information has been added for (2) Pallas and (21) Lutetia. Pole and rotation and mean radius information has been added for (1) Ceres. Pole information has been updated for (4) Vesta. The high precision realization for the pole and rotation rate of the Moon is updated. Alternative orientation models for Mars, Jupiter, and Saturn are noted. The Working Group also reaffirms that once an observable feature at a defined longitude is chosen, a longitude definition origin should not change except under unusual circumstances. It is also noted that alternative coordinate systems may exist for various (e.g. dynamical) purposes, but specific cartographic coordinate system information continues to be recommended for each body. The Working Group elaborates on its purpose, and also announces its plans to occasionally provide limited updates to its recommendations via its website, in order to address community needs for some updates more often than every 3 years. Brief recommendations are also made to the general planetary community regarding the need for controlled products, and improved or consensus rotation models for Mars, Jupiter, and Saturn.

In the 2006–2009 triennium, the International Astronomical Union (IAU) Working Group on Numerical Standards for Fundamental Astronomy determined a list of Current Best Estimates (CBEs). The IAU 2009 Resolution B2 adopted these CBEs as the IAU (2009) System of Astronomical Constants. Additional work continues to define the process of updating the CBEs and creating a standard electronic document. KeywordsNumerical standards–Fundamental Astronomy–Fundamental constants

A major program at the University of Calgary's Rothney Astrophysical Observatory (RAO) is the study of eclipsing close-binary stars, which, because of circularization effects, are all cases of the restricted three-body problem. The physical properties of such systems are derived from modeling of the light and radial velocity curves. Uniqueness questions notwithstanding, careful use of the technique is found to yield reliable elements of the orbit and parameters of the component stars, even in the face of light curve perturbations. This work requires a greatly enlarged data base over previous work, and more complex modeling procedures, necessitating the use of a supercomputer. A version of the generalized synthetic light curve program of Wilson (1979) with star spot simulations has been adapted to and optimized for the University of Calgary's Cyber 205 supercomputer and further improvements are underway. With the use of personal computer graphics software, results have been transformed into three-dimensional, rotating models which help visualize the overcontact and perturbation conditions.

Ignoring luni-solar perturbations, analytical solution for longitude deviation of the stroboscopic mean center of 24-h satellites ground trace in the neighborhood of a prespecified station is obtained. The initial semimajor axis for two-maneuver east-west stationkeeping is then deduced. Finally, the luni-solar long and short period effects on this initial semimajor axis are discussed.

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 44 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.

Records of major geologic events of the past about 260 Myr including: biologic extinction events, ocean-anoxic and black-shale events, major changes in sea level, major evaporite (salt) deposits, continental flood-basalt eruptions, first-order discontinuities in sea-floor spreading, and major mountain building events, have been aggregated and analyzed with moving-window and spectral techniques that facilitate recognition of clustering and possible cyclicity. Significant clustering of events suggests a model in which changes in rates and directions of sea-floor spreading ('ridge jumps') are associated with episodic rifting, volcanism, mountain building, global sea level and changes in the composition of the earth's atmosphere via the carbon cycle. The geologic data formally show a statistically significant underlying periodicity of 26.6 Myr for the Mesozoic and Cenozoic. Phase information suggests that the most recent maximum of the cycle occurred with the last 9 Myr, and may be close to the present time.

An ephemeris has been obtained for Explorer 28 (IMP 3) which agrees well with 2 years of radio observations and with SAO observations a year later. This ephemeris is generated over the 3 year lifetime by a numerical integration method utilizing a set of initial conditions, at launch and without requiring further differential correction. Because highly eccentric orbits are difficult to compute with acceptable accuracy and because a long continuous arc has been obtained which compares with actual data to a known precision, this ephemeris may be used as a standard for computing highly eccentric orbits in the Earth-Moon system. Orbit improvement was used to obtain the initial conditions which generated the ephemeris. This improvement was based on correcting the energy by adjusting the semimajor axis to match computed times of perigee passage with the observed. This procedure may generate errors in semimajor axis to compensate for model errors in the energy; however this compensation error is also implicit in orbit determination itself.

In this paper the problem of two, and thus, after a generalization, of an arbitrary finite number, of rigid bodies is considered. We show that the Newton-Euler equations of motion are Hamiltonian with respect to a certain non-canonical structure. The system possesses natural symmetries. Using them we shown how to perform reduction of the number of degrees of freedom. We prove that on every stage of this process equations of motion are Hamiltonian and we give explicite form corresponding of non-canonical Poisson bracket. We also discuss practical consequences of the reduction. We prove the existence of 36 non-Lagrangean relative equilibria for two generic rigid bodies. Finally, we demonstrate that our approach allows to simplify the general form of the mutual potential of two rigid bodies.

In this paper we analyse the relations between a previously described oblate Jaffe model for an ellipsoidal galaxy and the observed quantities for NGC 2974, and obtain the length and velocity scales for a relevant elliptical galaxy model. We then derive the finite total mass of the model from these scales, and finally find a good fit of an isotropic oblate Jaffe model by using the Gauss-Hermite fit parameters and the observed ellipticity of the galaxy NGC 2974. The model is also used to predict the total luminous mass of NGC 2974, assuming that the influence of dark matter in this galaxy on the image, ellipticity and Gauss-Hermite fit parameters of this galaxy is negligible within the central region, of radius 0.5R e.

We numerically study the bifurcations of two nonlinear maps, with the same linear part, which depend on a parameter namely the Hnon quadratic map and the so called beam-beam map. Many families of periodic orbits which bifurcate from the central family, are studied. Each family undergoes a sequence of period doubling bifurcations in the quadratic map, But the behavior of the beam-beam map is completely different. Inverse bifurcations occur in both maps. But some families of the same type which bifurcate inversely in the quadratic map do not bifurcate inversely in the beam-beam map, even though both maps have common linear part.

We study the stability domain of generic 2D area-preserving polynomialdiffeomorphisms. The starting point of our analysis is the study of thedistribution of stable and unstable fixed points. We show that the locationof fixed points and their stability type are linked to the degree of thepolynomial map. These results are based on a classification Theorem forplane automorphisms by Friedland and Milnor. Then we discuss the problem ofdetermining the domain in phase space where stable motion occurs. We showthat the boundary of the stability domain is given by the invariantmanifolds emanating from the outermost unstable fixed point of low period(one or two). This fact extends previous results obtained for reversiblearea-preserving polynomial maps of the plane. This analysis is based onanalytical arguments and is supported by the results of numericalsimulations.