On the accuracy of the orbit of asteroid (99942) APOPHIS at the time of its encounter with the Earth in 2029

We estimate the effect of trajectory measurement errors on the orbital parameters of asteroid Apophis determined from improvements. For this purpose, based on all of the optical and radar observations available to date, we have computed a nominal orbit of the asteroid. The scatter ellipsoid of the initial conditions of motion has been obtained by two methods. In the first, universally accepted method, the scatter ellipsoid is calculated by assuming a linear dependence of the errors in the parameters being determined on observational errors. In the second method, the scatter region of the orbital parameters around the nominal-orbit parameters is determined by the Monte Carlo method. We show that the region determined by the latter method at the initial epoch differs only slightly from the scatter ellipsoid for the linear approximation. We estimate the sizes of the projections of the corresponding regions onto the target plane at the time of the closest encounter of the asteroid with the Earth in 2029. The projections are approximated by ellipses. Our computations have shown that the ellipse has the following sizes: 389.6 km for the semimajor axis and 16.4 km for the semiminor axis in the linear case and 330.0 and 11.1 km, respectively, in the nonlinear case.     

Determination of an intermediate perturbed orbit from three position vectors

We suggest a new approach to solving the problem of finding the orbit of a celestial body from its three spatial position vectors and the corresponding times. It allows most of the perturbations in the motion of a celestial body to be taken into account. The approach is based on the theory of intermediate orbits that we developed previously. We construct the orbit the motion along which is a combination of two motions: the motion of a fictitious attracting center whose mass varies according to Mestschersky’s first law and the motion relative to the fictitious center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden-Mestschersky problem. The constructed orbit has such parameters that their limiting values at any reference epoch define a superosculating intermediate orbit with a fourth-order tangency. We have performed a numerical analysis to estimate the accuracy of approximating the perturbed motion of two minor planets, 145 Adeona and 4179 Toutatis, by the orbits computed using two-position procedures (the classical Gauss method and the method that we suggested previously), a three-position procedure based on the Herrick-Gibbs equation, and the new method. Comparison of the results obtained suggests that the latter method has an advantage.     

How precise is the orbit of asteroid (99942) Apophis and how probable is its collision with the Earth in 2036–2037?

The nearest in time close approach of potentially hazardous asteroid (99942) Apophis with the Earth will take place on April 13, 2029, when the minimum distance of the asteroid from the Earth’s center will be as small as 38 000 km. Such a close approach will result in substantial transformation of the asteroid’s orbit. The value of the perturbations depends on the minimum distance between the bodies during the approach. Among possible transformations of the orbit are those which result in new dangerous approaches and even in probable Apophis collisions with the Earth starting from 2036. At present, at least four solutions are known for the Apophis orbit which were obtained using all radar and most of available optical observations. The procedures of assigning weights to conditional equations and the models of the asteroid’s motion have differed to some extent when finding these solutions. Of considerable interest is the comparison of the found orbital parameters with the estimates of their accuracy, since small distinctions in their values result in considerable distinctions in the forecast of Apophis’ motion after 2029 and beyond. It is shown in the paper that the estimates of the probability of an Apophis collision with the Earth in 2036 differ by some orders of magnitude, according to various solutions. The influence of factors which were disregarded in the models of motion even more increases the uncertainty in forecasting the motion after 2029. More accurate forecasting can be achieved as a result of additional optical and, to a greater extent, a series of radar observations in 2013 and then in 2020–2021, and/or as a result of processing radio signals of the transmitter delivered to the Apophis surface or to the orbit of its artificial satellite, as it was proposed in a number of papers.     

A Method for the Determination of an Intermediate Orbit from Three Positions of the Small Body on the Celestial Sphere

A new method is suggested for finding the preliminary orbit from three complete measurements of the angular coordinates of a celestial body developed by analogy with the classic Lagrange–Gauss method. The proposed method uses the intermediate orbit that we had constructed in an earlier paper based on two position vectors and the corresponding time interval. This intermediate orbit allows for most of the perturbations in the motion of the body. Using the orbital motion of asteroid 1566 Icarus as an example, we compare the results obtained by applying the classic and the new method. The comparison shows the new method to be highly efficient for studying perturbed motion. It is especially efficient if applied to high-precision observational data covering short orbital arcs.     

A method of intermediate orbit determination based on four positions of a minor body on the celestial sphere

A new method of computing the preliminary orbit of a celestial body based on four pairs of angle measurements has been suggested. The method makes use of preliminary orbit previously constructed by the author based on two position vectors and a corresponding time interval, taking into account the main part of the perturbations in the motion of the body under study. Using the example of constructing the orbit of the minor planet 1383 Limburgia, the results obtained using a four-position procedure of the Gaussian type based on the solution of a two-body problem have been compared with those of the new method. The comparison showed the new method to be highly efficient for perturbed motion studies. It is especially advantageous in the case of high-accuracy observation data on small orbital arcs.     

Astrometric study of the triple star ADS 9173

An approximate orbit of the wide visual binary star ADS 9173 A(Bb) with a period of ～6000 yr has been determined for the first time by the method of apparent motion parameters. The orbit was computed using a short (1982–2004) arc of photographic observations obtained with the 26-inch Pulkovo Observatory refractor and the Hipparcos parallax. Agreement of the new orbit with the observations from the WDS catalog beginning in 1832 serves as a check. The errors in the orbital elements are large, but the orientation elements of the orbital plane (i and Ω) were estimated reliably. Component B has an invisible spectroscopic companion with a period of 4.9 yr. An astrometric orbit of Bb consistent with radial velocity measurements was determined from the residuals to the relative orbital motion of A(Bb). The orbital planes are nearly coplanar. If the mass of component B is taken in accordance with the mass—luminosity relation, 1.5 M _⊙, and the parallax is 0.″021, then the mass of the secondary component is no less than 0.5M _⊙. Component A may also be a long-period binary system.     

Orbits of new Hipparcos binaries: III

We present apparent orbits and fundamental parameters of three pairs of early M-type dwarfs. The orbital elements are determined from speckle interferometric observations at the 6-m BTA telescope of the Special Astrophysical Observatory. The orbits of two pairs, HIP39402 and HIP 104565 are built for the first time. The orbit of HIP106972 is revised using new observational data obtained in 2007–2008. The periods of motion and semimajor axes of all the three binaries have very similar values, namely 13 years and 5.5–6 AU, respectively. The dynamical total masses of the systems, obtained from the orbital elements are determined with quite large errors of 25–40%, which is due to the parallax errors.     

Approximate analytic method for high-apogee twelve-hour orbits of artificial Earth’s satellites

We propose an approach to the study of the evolution of high-apogee twelve-hour orbits of artificial Earth’s satellites. We describe parameters of the motion model used for the artificial Earth’s satellite such that the principal gravitational perturbations of the Moon and Sun, nonsphericity of the Earth, and perturbations from the light pressure force are approximately taken into account. To solve the system of averaged equations describing the evolution of the orbit parameters of an artificial satellite, we use both numeric and analytic methods. To select initial parameters of the twelve-hour orbit, we assume that the path of the satellite along the surface of the Earth is stable. Results obtained by the analytic method and by the numerical integration of the evolving system are compared. For intervals of several years, we obtain estimates of oscillation periods and amplitudes for orbital elements. To verify the results and estimate the precision of the method, we use the numerical integration of rigorous (not averaged) equations of motion of the artificial satellite: they take into account forces acting on the satellite substantially more completely and precisely. The described method can be applied not only to the investigation of orbit evolutions of artificial satellites of the Earth; it can be applied to the investigation of the orbit evolution for other planets of the Solar system provided that the corresponding research problem will arise in the future and the considered special class of resonance orbits of satellites will be used for that purpose.     

A numerical-analytical method for studying the orbital evolution of distant planetary satellites

We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods.     

On the Laplacian orbit determination of asteroids

A modified Laplacian technique is described for initial orbit determination of asteroids from CCD observations and its applications for orbit determination of the main belt asteroids and near Earth asteroids. The proposed modification is based on a simultaneous improvement of both the orbital elements and the derivatives of spherical coordinates in frames of Laplace's method. It provides an orbit which represents the used observations with the residuals comparable with errors of these observations. The improved values of the derivatives might be used as ephemeris parameters for identification of newly discovered objects.     

Determination of the orbits of inner Jupiter satellites

Some problems in determining the orbits of inner satellites associated with the complex behavior of the target function, which is strongly ravine and which possesses multiple minima in the case of the satellite orbit is determined based on fragmentary observations distributed over a rather long time interval, are studied. These peculiarities of the inverse problems are considered by the example of the dynamics of the inner Jupiter satellites: Amalthea, Thebe, Adrastea, and Metis. Numerical models of the satellite motions whose parameters were determined based on ground-based observations available at the moment to date have been constructed. A composite approach has been proposed for the effective search for minima of the target function. The approach allows one to obtain the respective evaluations of the orbital parameters only for several tens of iterations even in the case of very rough initial approximations. If two groups of observations are available (Adrastea), a formal minimization of the target function is shown to give a solution set, which is the best solution from the point of view of representation of the orbital motion, which is impossible to choose. Other estimates are given characterizing the specific nature of the inverse problems.     

The effect of various solar ephemerides on equator and equinox solutions from observations of the Sun with the reversible transit circle at Cape observatory from 1907 to 1959

Observations of the Sun were made with the Cape reversible transit circle from 1907 to 1959. We have made least squares solutions for six unknowns viz., equator and equinox corrections and corrections to earth orbital parameters including the ephemeris mean longitude of the Sun, the mean obliquity of the ecliptic, the mean longitude of perihelion, and the mean eccentricity of the earth's orbit based on Newcomb's, DE102, and DE200 Ephemerides for each of six catalogs of observations made during that period. The six unknowns are also determined simultaneously for the six catalogs taken together. The six catalogs are absolute, in that methods of observation and reduction were adopted in such a way as to produce a system of results not closely dependent on the adopted system of assumed clock and azimuth star positions.The observed equator and equinox corrections from a comparison of DE200 with the Cape Sun observations referred to an improved FK4 system are –0.07±0.01 arcsec and –0.20±0.04 arcsec, respectively, at the mean epoch of observation, 1933.02. The time rate of change of the equator correction was not significant. The time rate of change of the observed equinox is –1.02±0.30 arcsec per century.The observed equinox correction of the DE102 at 1933.02 is –0.41±0.04 arcsec, which is 0.5 arcsec less than the NEWCOMB (Herget) equinox correction. This confirms the result based on Washington Sun observations.     

Nonlinear methods of statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations

Determination of orbital parameters from observations is formally a nonlinear inverse problem for solving which evidently nonlinear methods are required. Meanwhile, an accompanying stage in solving the inverse problem is the evaluation of parametric accuracy to which, however, linear methods are conventionally applied. This is quite justified if parametric errors caused by observation errors are rather small, otherwise this is not at all since the nonlinearity of the inverse problem can be considerable to influence on the evaluations of parametric accuracy especially when the observations are very few. With the advent of quick-operating and multiprocessor computers, recently one tends to employ statistic simulation of virtual parameter values for investigating uncertainties in orbits determined from observations. In the paper are just discussed the methods designed specially for nonlinear statistic simulation of virtual parameter values. Their efficiency is investigated in application to estimating uncertainties in the orbit of Jovian satellite S/2003 J04 whose orbital parameters are ill-determined owing to scanty available observations. Indices of nonlinearity are introduced for making decision in the choice between linear and nonlinear methods.     

Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J₂-invariant orbits are obtained.     

A Review on Co-orbital Motion in Restricted and Planetary Three-body Problems

The 1:1 mean motion resonance may be referred to as the lowest order mean motion resonance in restricted or planetary three-body problems. The five well-known libration points of the circular restricted three-body problem are five equilibriums of the 1:1 resonance. Coorbital motion may take different shapes of trajectory. In case of small orbital eccentricities and inclinations, tadpole-shape and horseshoe-shape orbits are well-known. Other 1:1 libration modes different from the elementary ones can exist at moderate or large eccentricities and inclinations. Coorbital objects are not rare in our solar system, for example the Trojans asteroids and the coorbital satellite systems of Saturn. Recently, dozens of coorbital bodies have been identified among the near-Earth asteroids. These coorbital asteroids are believed to transit recurrently between different 1:1 libration modes mainly due to orbital precessions, planetary perturbations, and other possible effects. The Hamiltonian system and the Hill’s three-body problem are two effective approaches to study coorbital motions. To apply the perturbation theory to the Hamiltonian system, standard procedures involve the development of the disturbing function, averaging and normalization, theory of ideal resonance model, secular perturbation theory, etc. Global dynamics of coorbital motion can be revealed by the Hamiltonian approach with a suitable expansion. The Hill’s problem is particularly suitable for the studies on the relative motion of two coorbital bodies during their close encounter. The Hill’s equation derived from the circular restricted three-body problem is well known. However, the general Hill’s problem whose equation of motion takes exactly the same form applies to the non-restricted case where the mass of each body is non-negligible, namely the planetary case. The Hill’s problem can be transformed into a “canonical shape” so that the averaging principle can be applied to construct a secular perturbation theory. Besides the two analytical theories, numerical methods may be consulted, for example the approach of periodic orbit, the surface of section, and the computation of invariant manifolds carried by equilibriums or periodic orbits.     

Comparing analytical and numerical approaches to meteoroid orbit determination using Hayabusa telemetry

Fireball networks establish the trajectories of meteoritic material passing through Earth's atmosphere, from which they can derive pre‐entry orbits. Triangulated atmospheric trajectory data require different orbit determination methods to those applied to observational data beyond the Earth's sphere of influence, such as telescopic observations of asteroids. Currently, the vast majority of fireball networks determine and publish orbital data using an analytical approach, with little flexibility to include orbital perturbations. Here, we present a novel numerical technique for determining meteoroid orbits from fireball network data and compare it to previously established methods. The re‐entry of the Hayabusa spacecraft, with its known pre‐Earth orbit, provides a unique opportunity to perform this comparison as it was observed by fireball network cameras. As initial sightings of the Hayabusa spacecraft and capsule were made at different altitudes, we are able to quantify the atmosphere's influence on the determined pre‐Earth orbit. Considering these trajectories independently, we found the orbits determined by the novel numerical approach to align closer to JAXA's telemetry in both cases. Using simulations, we determine the atmospheric perturbation to become significant at ~90 km—higher than the first observations of typical meteorite dropping events. Using further simulations, we find the most substantial differences between techniques to occur at both low entry velocities and Moon passing trajectories. These regions of comparative divergence demonstrate the need for perturbation inclusion within the chosen orbit determination algorithm.     

Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.     

Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.