Orbital Evolution of the Sun–Jupiter–Saturn–Uranus–Neptune Four-Planet System on Long-Time Scales

The four-planet problem is solved by constructing an averaged semi-analytical theory of secondorder motion by planetary masses. A discussion is given of the results obtained by numerical integration of the averaged equations of motion for the Sun–Jupiter–Saturn–Uranus–Neptune system over a time interval of 10 Gyr. The integration is based on high-order Runge–Kutta and Everhart methods. The motion of the planets is almost periodic in nature. The eccentricities and inclinations of the planetary orbits remain small. Short-period perturbations remain small over the entire interval of integration. Conclusions are drawn about the resonant properties of the motion. Estimates are given for the accuracy of the numerical integration.     

Analytical Planetary solution VSOP2000

The analytical planetary solution VSOP2000 determines the planetary perturbations with the help of an iterative method from a solution developed till the third order of the disturbing masses. This solution is from 10 to 100 times more precise than the previous analytical solutions VSOP82 and VSOP87, at the level of some 0.1mas for Mercury, Venus and the Earth and some mas for the other planets over the timespan 1900–2000.With this solution, the relation between the Barycentric Coordinate Time (TCB) and the Geocentric Coordinate Time (TCG) is computed with an accuracy better than 0.1 nanosecond over the interval 1965–2015. We also determined the contribution to the Eulerian angles of the geodetic precession-nutation.     

Numerical Simulation of the Orbital Evolution of Near-Earth Asteroids Close to Mean Motion Resonances

The dynamics of near-Earth asteroids near mean motion resonances with the Earth or other planets is considered. The probability domains of the motion of some near-Earth asteroids close to low-order resonances are presented. The investigations have been carried out by means of a numerical integration of differential equations, taking into account the perturbations from the major planets and the Moon. For each investigated object an ensemble of 100 test particles with orbital elements nearby those of the nominal orbit has been constructed and its evolution has been retraced over the time interval (–3000, +3000 years). The initial set of orbits has been generated on the basis of probable variations of the initial orbital elements obtained from the least square analysis of observations.This revised version was published online in October 2005 with corrections to the Cover Date.     

The orbital evolution of NEA 30825 1900 TG1

The orbital evolution of the near-Earth asteroid (NEA) 30825 1990 TG1 has been studied by numerical integration of the equations of its motion over the 100 000-year time interval with allowance for perturbations from eight major planets and Pluto, and the variations in its osculating orbit over this time interval were determined. The numerical integrations were performed using two methods: the Bulirsch-Stoer method and the Everhart method. The comparative analysis of the two resulting orbital evolutions of motion is presented for the time interval examined. The evolution of the asteroid motion is qualitatively the same for both variants, but the rate of evolution of the orbital elements is different. Our research confirms the known fact that the application of different integrators to the study of the long-term evolution of the NEA orbit may lead to different evolution tracks.     

Model orbits of asteroid 3040 Kozai

The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.     

Implications from the motion of outer belt asteroids

A principal feature of the asteroidal distribution is the rapid truncation of its population outward from ～3.4 AU. This paper presents further evidence, based on the motion and distribution of certain minor planets with large semimajor axes, that this truncation cannot be strictly the result of gravitational perturbations of the major planets even acting over times of ～10⁹ years. The motion of other outer asteroids sets a probable upper limit of 0.081 on Jupiter's eccentricity.     

Planetary microlensing at high magnification

Simulations of planetary microlensing at high magnification that were carried out on a cluster computer are presented. It was found that the perturbations owing to two-thirds of all planets occur in the time interval  −0.5 t _FWHM,0.5 t _FWHM  with respect to the peak of the microlensing light curve, where   t _FWHM  is typically ∼14 h. This implies that only this restricted portion of the light curve need be intensively monitored for planets – a very significant practical advantage. Nearly all planetary detections in high-magnification events will not involve caustic crossings. We discuss the issues involved in determining the planetary parameters in high magnification events. Earth-mass planets may be detected with 1-m class telescopes if their projected orbital radii lie within about 1.5–2.5 au. Giant planets are detectable over a much larger region. For multiplanet systems the perturbations caused by individual planets can be separated under certain conditions. The size of the source star needs to be determined independently, but the presence of spots on the source star is likely to be negligible, as is the effect of planetary motion during an event.     

Constructive analytical solution of the evolution hill problem

A new analytical solution of the system of differential equations describing secular perturbations and long-period solar perturbations of mean orbits of outer satellites of giant planets was obtained. As distinct from other solutions, the solution constructed using von Zeipel’s method approximately takes into account, in the secular part of the perturbing function, the totality of fourth order with respect to the small parameter m of the ratio of the mean motions of the primary planet and the satellite. This enables us to describe more accurately the evolution of satellite orbits with large apocentric distances, which in the course of evolution may exceed the halved radius of the Hill sphere of the planet with respect to the Sun. Among these are the orbits of the two outermost Neptunian satellites N10 (Psamathe) and N13 (Neso). For these satellites, the parameter m amounts to 0.152 and 0.165, respectively. Different from a purely analytical solution, the proposed solution requires preliminary calculations for each satellite. More precisely, in doing so, we need to construct some simple functions to approximate more complex ones. This is why we use the phrase “constructive analytical.” To illustrate the solution, we compare it with the results of the numerical integration of the strict motion equations of the satellites N10 and N13 over time intervals 5–15 thousand years.     

New reductions of orbits based upon Chinese ancient cometary records

From 146 B.C. to 1760 A.D., 363 sets of cometary observations for a total 88 different comets were recorded in Chinese Ancient Records of Celestial Phenomena. According to those records, we reduced apparent positions and mean equatorial coordinates (epoch 2000.0) for all more than three times recorded comets. Taking into account the perturbations of all nine planets and using the numerical method of N-body problem, the orbits of correlative comets were calculated. For thirty different comets, new orbits are presented for the first time.     

Updated Numerical Ephemerides of the Galilean Satellites of Jupiter

New versions of the ephemerides for the Galilean satellites of Jupiter (Io, Europa, Ganymede, and Callisto) constructed by numerically integrating the equations of motion of the satellites are presented. The satellite motionmodel takes into account the non-sphericity of Jupiter, the mutual perturbations of the satellites, and the perturbations from the Sun and major planets. The initial satellite motion parameters have been improved based on all the available series of ground-based optical observations spanning the interval 1891-2017, spacecraft observations, and radar observations. As a result, the coefficients of the expansion of the satellite coordinates and velocities in terms of Chebyshev polynomials in the interval 1891- 2025 have been obtained. The root-mean-square errors of the observations and the graphs of comparison of the constructed ephemerides both with the observations and with Lainey's numerical ephemerides are presented. The constructed ephemerides are publicly accessible.     

On the detection of mutual perturbations as proof of planets around PSR1257+12

Unambiguous detection of the consequences of mutual perturbations of the hypothesized planets about the pulsar PSR1257+12 would be unassailable proof of their existence. Nearly all of the residuals in the times of arrival (TOA) of the pulses after subtraction of the TOA predicted from the best fit constant period model are accounted for by including the effects of two orbiting planets with constant orbital parameters. The nature and magnitude of additional residuals in the TOA due to the gravitational interactions between the planets are determined by numerically calculating the TOA residuals for the orbital motion including the perturbations and subtracting the TOA residuals from analytic expressions of the orbital motion with orbital parameters fixed at averaged values. The TOA residual differences so obtained oscillate with periods comparable to the orbital periods with the oscillations varying in amplitude as a function of epoch within any given observational period. The signature of the perturbations is thus a quasiperiodic modulation of the residual differences obtained after removal of the effects of the orbital motion with best fit, constant orbital parameters. The amplitudes of this modulation reach about 10sec for observational periods exceeding 1000 days for the minimum planetary masses with sini = 1, and they increase as 1 / sini for 1 / sini < 5, wherei is the inclination of the orbit plane to that of the sky. Greater accumulated phase differences between the effects of perturbed and unperturbed orbital motions are available in the times of zero values in the observed and predicted TOA residuals and these comprise a second signature of the perturbations. The perturbation signatures should become detectable as the observation interval approaches 1000 days.Paper presented at the Conference onPlanetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

The orbit of asteroid (99942) Apophis as determined from optical and radar observations

The results of improving the orbit accuracy for the asteroid Apophis and the circumstances of its approach to Earth in 2029 are described. Gravitational perturbations from all of the major planets and Pluto, Ceres, Pallas, and Vesta are taken into account in the equations of motion of the asteroid. Relativistic perturbations from the Sun and perturbations due to the oblateness of the Sun and Earth and due to the light pressure are also included in the model. Perturbations from the Earth and Moon are considered separately. The coordinates of the perturbing bodies are calculated using DE405. The phase correction and the gravitational deflection of light are taken into account. The numerical integration of the equations of motion and equations in variations is performed by the 15th-order Everhart method. The error of the numerical integration over the 2005–2029 interval, estimated using forward and backward computations, is not more than 3 × 10^?11 AU. Improved coordinates and velocities at epoch JD2454200.5 (April 10, 2007) were obtained applying the weighted leastsquares fit. For the period from March 15, 2004, to August 16, 2006, 989 optical and 7 radar observations were used. The resulting system represents the optical observations with an error of 0.37 (66 conditional equations were rejected). The residuals of the radar observations are an order, or more, smaller than their errors. The system of Apophis’ elements and the estimates of their precision obtained in this study are in perfect agreement with the results published by other authors. The minimum Apophis-Earth distance is about 38 200 km on April 13, 2029. This estimate agrees to within 20 km with those calculated based on other published systems of elements. The effect of some model components on the minimum distance is estimated.     

A new method of initial orbit determination

Up to now we have been dealing with the construction of entirely analytical planetary theories such as VSOP82 (Bretagnon, 1982) and TOP82 (Simon, 1983). These theories take into account the whole of the Newtonian perturbations of nine point masses: the Sun, the Earth-Moon barycentre, the planets Mercury, Venus, Mars, Jupiter, Saturn, Uranus and Neptune. They also take into account perturbations due to some minor planets, to the action of the Moon and the relativistic effects. The perturbations of these last three types are in a very simple way under analytical form but they considerably increase the computations when introduced in the numerical integration programs.In the present paper we thus study a solution in which the Newtonian perturbations for the ten point masses are treated through numerical integration, the other perturbations being analytically added.     

Toward the construction of a new precession-nutation theory of nonrigid Earth

The accuracy of the rigid Earth solution SMART97 is 2?μ as over the time interval (1968, 2023), accuracy showed by the comparison with a numerical integration using the positions of the Moon, the Sun, and the planets given by DE403. To obtain a nonrigid Earth solution, we use the transfer function of Mathews et al. (2000) and , to keep the precision of our rigid Earth solution in the computation of the geophysical effects, we apply this transfer function to the Earth's angular velocity vector in order to avoid the inherent approximations of the classical methods. Moreover the perturbations of the third component of the angular velocity vector are taken into account. Lastly, we take into account, in an iterative process, the second order perturbations due to the geophysical effects. The results are compared with the Herring solution (1996) published in the IERS Conventions.     

A Simple Method for Numerical Calculations of the Evolution of Orbits of Near-Earth Asteroids

A simple method for numerical integration of the equations of motion of small bodies of the Solar System is proposed, which is especially efficient in studying the orbits with small perihelion distances. The evolution of orbits of 121 numbered asteroids with perihelion distances q < 1.2 AU is investigated over the time interval of years 2000–2100 with allowance made for the gravitational influence of nine planets and three largest asteroids. The circumstances of close encounters of asteroids with the Earth and other terrestrial planets are presented.     

Coordinate Additional Perturbations to Mars Orbiters and Choice of Corresponding Coordinate System

Similar to the study of the related problems of Earth satellites, in the research of the motion of Mars orbiter especially for low-orbit satellites, it is more appropriate to choose an epoch Mars-centered and Mars-equator reference system, which indeed is called the Mars-centered celestial coordinate system. In this system, the xy-plane and the direction of the x-axis correspond to the mean equator and mean equinox. Similar to the precession and nutation of the Earth, the wiggling of instantaneous Mars equator causes the coordinate additional perturbations in this Mars coordinate system. The paper quotes a method which is similar to the one used in dealing with the coordinate additional perturbations of Earth. According to this method, based on the IAU2000 Mars orientation model and under the precondition of a certain accuracy, we are able to figure out the precession part of the change of Mars gravitation. This lays the foundation for further study of its influence on the Mars orbiter's orbit of precession and the solution of the corresponding coordinate additional perturbations. The obtained analytical solution is easy to use. Compared with the numerical solution with higher accuracy, the result shows that the accuracy of this analytical solution could satisfy the general requirements in use. Therefore, our result verifies that a unified coordinate system, the Mars-centered celestial system in which J2000.0 is chosen as its current initial epoch, could be applied to deal with the relative problems of Mars orbiters, especially for low-orbit satellites. It is different from the method we previously used in dealing with the corresponding problems of Earth satellites, where we adopted the instantaneous equator and epoch (J1950.0) mean equinox as xy-plane and the direction of x -axis. In contrast, the coordinate transformation brings heavy workload and certain inconvenience in relative former works in which the prior system is used. If adopting the unified coordinate system, the transformation could be simply avoided and the computation load could be decreased significantly.     

Limitations on outer planet mass determinations from their mutual perturbations

The limitations on determining the masses of the outer planets from their mutual perturbations are investigated based on the magnitudes of the periodic perturbations given in general theories and the accuracy of the observational data.     

Astrometric Observations of the Galilean Moons of Jupiter at the Pulkovo Normal Astrograph in 2016−2017

We present the results of observations of the Galilean moons of Jupiter carried out at the Normal Astrograph of the Pulkovo Observatory in 2016?2017. We obtained 761 positions of the Galilean moons of Jupiter in the system of the Gaia DR1 catalog (ICRF, J2000.0) and 854 differential coordinates of the satellites relative to each other. The mean errors in the satellites’ normal places and the corresponding root-mean-square deviations are ε_α = 0.0020′′, ε_δ = 0.0027′′, σ_α = 0.0546′′, and σ_δ = 0.0757′′. The equatorial coordinates of the moons are compared to the motion theories of planets and satellites. On average, the (O–C) residuals in the both coordinates relative to the motion theories are less than 0.031′′. The best agreement with observations is achieved by a combination of the EPM2015 and V. Lainey-V.2.0|V1.1 motion theories, which yields the average (O–C) residuals of approximately 0.02″. Peculiarities in the behavior of the (O–C) residuals and error values in Ganymede have been noticed.     

Orbital Evolution of Extrasolar Planets

The recent discovery of extrasolar planets and planetary systems has raised many new research problems for astronomers. It has become apparent that the newly discovered systems differ significantly from the Solar System. In particular, many massive planets of other stars, in contrast to Jupiter, have large orbital eccentricities. In the present paper, we investigate several dynamic implications of this finding. Numerical integration results show that the orbits of low-mass planets in such systems usually have large evolving eccentricities. If the motion remains regular and no close encounters occur, the orbital evolution can be described analytically by using secular perturbations of Laplace–Lagrange equations. In terms of the Lagrange variables, the trajectories are circles, and the semimajor axis remains constant. The loss of the regularity of motion is normally followed by a nonmonotone synchronous increase in the semimajor axis and eccentricity, and the orbit becomes similar to that of a large-period comet. Narrow resonance-related regions include more complex motions.     

Determining asteroid masses from perturbations on Mars

The orbit of Mars is perturbed more than 5 m, a value compatible with the accuracy of the Viking lander ranging data, by about three dozen asteroids. In addition to larger asteroids throughout the belt, significant perturbations of long period are generated by smaller objects near commensurabilities with Mars. The largest periodic terms induced by 1 Ceres and 2 Pallas have amplitudes of 0.8 and 0.2 km, respectively, both with 10-year periods. Due to a near commensurability, 4 Vesta causes a 5-km, 52-year term. While the Viking ranges will yield significant mass determinations for the largest three asteroids, and some of the smaller bodies should be detectable, it will be difficult to seperate the smaller bodies with useful accuracies. Accurate discrimination must await range data from future missions to Mars or other bodies in the neighborhood of the asteroid belt. The Viking ranges can also yield improved masses for the outer planets (except Pluto), an application which is being exploited by groups analyzing these data. Uncertainties in the asteroid masses limit the ultimate accuracy of the Viking determinations of both the long time scale motion of the system the inner four planets with respect to an inertial frame and the rate of change of the gravitational constant.