Implicit single-sequence methods for integrating orbits

The solutions of \(\ddot x = F(x,t)\) , and also \(\dot x = F(x,t)\) , are developed in truncated series in timet whose coefficients are found empirically. The series ending in thet ⁶ term yields a position at a final prechosen time that is accurate through 9th order in the sequence size. This is achieved by using Gauss-Radau and Gauss-Lobatto spacings for the several substeps within each sequence. This timeseries method is the same in principle as implicit Runge-Kutta forms, including some not described previously. In some orders these methods are unconditionally stable (A-stable). In the time-series formulation the implicit system converges rapidly. For integrating a test orbit the method is found to be about twice as fast as high-order explicit Runge-Kutta-Nyström-Fehlberg methods at the same accuracies. Both the Cowell and the Encke equations are solved for the test orbit, the latter being 35% faster. It is shown that the Encke equations are particularly well-adapted to treating close encounters when used with a single-sequence integrator (such as this one) provided that the reference orbit is re-initialized at the start of each sequence. This use of Encke equations is compared with the use of regularized Cowell equations.     

Error propagation in the numerical solutions of the differential equations of orbital mechanics

The relationship between the eigen values of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigen value when linearized about the initial conditions. The real positive eigen value causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigen values and is numerically stable.     

Steady state populations of Apollo-Amor objects

The steady-state distribution of orbits of Apollo-Amor objects is calculated for a variety of possible sources. These include asteroids near the inner edge of the belt, cometary orbits similar to Encke, and hypothetical extinct cometary orbits with perihelia larger than that of Encke. In all but one case, the steady-state distributions are similar for all these sources, and predict Amor/Apollo ratios of 1.5 to 3. These ratios are lower than those predicted by work in which the effects of the ν₆ secular resonance were not considered. These results are in general agreement with observation, although the higher (～3) Amor/Apollo ratios found for many of the sources may turn out to be unacceptably high. The absolute number of Apollo-Amors observed is found to require an injection rate of ～15 objects/(10⁶ years). This rate is easily achieved if the present existence of Encke is assumed to be a reasonably probable event, and if Encke becomes a ～1-km-diameter Apollo object following exhaustion of its volatile material; best estimates of the injection rate from the asteroid belt [～1.5/(10⁶ years)] are too low. Hence a dominant cometary component is suggested. The predicted number of Apollo objects in small (q < 1.0 AU, a < 1.4 AU orbits is in agreement with observation. Predicted lunar and terrestrial cratering rates agree approximately with observation. An unexplained difference between the lunar and terrestrial results is probably caused by uncertainties in the scaling laws or crater counts used. This discrepancy precludes an exact test of these calculations using cratering data.     

Studies in the application of recurrence relations to special perturbation methods

The integration by recurrent power series of certain differential equations occurring in celestial mechanics is shown to be very much more efficient and accurate than that produced by classical one step methods. It is shown that for any such system of differential equations the machine time taken to carry out an integration is a minimum for a certain choice of the number of terms taken in the recurrent power series. In the two-body orbits considered this number is about 15. For the same accuracy criterion the power series is faster than the Runge-Kutta method of the fourth order by a factor which varies between 6 and 15 depending on the eccentricity of the orbit.

     

Nongravitational forces and meteoroid streams

The action of the solar electromagnetic radiation on the moving interplanetary dust particles in its more complete form than the special case known as the Poynting-Robertson effect is theoretically discussed in application to meteoroid stream of comet Encke.Normal and transversal components of the perturbing nongravitational force are used due to the action of the solar electromagnetic radiation. It is shown that the normal component of the force is negligible. However, transversal component is very important: it can probably completely explain all the observed meteoroid streams situated along the orbit of comet Encke (and, possibly, some asteroids) as the product of the comet Encke alone. Much shorter time is required for producing such a meteoroid stream than is a general conception.If the idea about the significance of the transversal component of the nongravitational force (may be, not produced by electromagnetic radiation) is correct, it may have important consequences for our understanding of ageing of comets, global evolution of the cometary (and, partially, asteroidal) system, and, of course, for a long-term evolution of small interplanetary particles.     

An Implementation of the Logarithmic Hamiltonian Method for Artificial Satellite Orbit Determination

We discuss the use of a recently discovered exact two-body leapfrog for accurate symplectic integration of perturbed two-body motion and for the computation of the state-transition matrix. We pay special attention to artificial satellite orbit determination and describe in detail the evaluation of the perturbing acceleration. Inclusion of air drag and other non-canonical forces are also discussed. The main advantage of this new formulation is conceptual simplicity, for easy programming and high accuracy for orbits with large eccentricity. The method has been evaluated in real artificial satellite orbit determinations.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the importance of the Poynting-Robertson effect on meteoroids

A small generalization of the equation of motion for the Poynting-Robertson effect is tested in order to find the significance of new terms. The test is made for dust particles ejected at perihelia of the orbit of the comet Encke. The particles are released at the speed of 40 m s^?1. Gravitational perturbations of planets, Poynting-Robertson effect and solar corpuscular radiation (solar wind) are considered. Other nongravitational effects may be represented by new terms in the suggested form of the nongravitational force. Various values of normal and transversal components of the perturbing nongravitational force are used. The final results of numerical integrations are compared with those obtained on the basis of the Poynting-Robertson effect.     

Intermediary LEO propagation including higher order zonal harmonics

Two new intermediary orbits of the artificial satellite problem are proposed. The analytical solutions include higher order effects of the geopotential, and are obtained by means of a torsion transformation applied to the quasi-Keplerian system resulting after the elimination of the parallax simplification, for the first intermediary, and after the elimination of the parallax and perigee simplifications, for the second one. The new intermediaries perform notably well for low Earth orbits propagation, are free from special functions, and result advantageous, both in accuracy and efficiency, when compared to the standard Cowell integration of the \(J_2\) problem, thus providing appealing alternatives for onboard, short-term, orbit propagation under limited computational resources.     

On the importance of the Poynting-Robertson effect on meteoroids

A small generalization of the equation of motion for the Poynting-Robertson effect is tested in order to find the significance of new terms. The test is made for dust particles ejected at perihelia of the orbit of the comet Encke. The particles are released at the speed of 40 m s^–1. Gravitational perturbations of planets, Poynting-Robertson effect and solar corpuscular radiation (solar wind) are considered. Other nongravitational effects may be represented by new terms in the suggested form of the nongravitational force. Various values of normal and transversal components of the perturbing nongravitational force are used. The final results of numerical integrations are compared with those obtained on the basis of the Poynting-Robertson effect.     

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the $J_{2}$ perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.     

Studies in the application of recurrence relations to special perturbation methods

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the integration of these equations by the Encke method and by a set of perturbational equations. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relations. It is shown that for very small perturbations the use of the perturbational equations is more efficient than the use of the Encke method. A discussion is also given of when Cowell's method is more efficient than either of these techniques.     

Numerical analysis of the symmetric methods

Aimed at the initial value problem of the particular second-order ordinary differential equations,y =f(x, y), the symmetric methods (Quinlan and Tremaine, 1990) and our methods (Xu and Zhang, 1994) have been compared in detail by integrating the artificial earth satellite orbits in this paper. In the end, we point out clearly that the integral accuracy of numerical integration of the satellite orbits by applying our methods is obviously higher than that by applying the same order formula of the symmetric methods when the integration time-interval is not greater than 12000 periods.     

The range of validity of the two-body approximation in models of terrestrial planet accumulation: II. Gravitational cross sections and runaway accretion

The validity of the two-body approximation in calculating encounters between planetesimals has been evaluated as a function of the ratio of unperturbed planetesimal velocity (with respect to a circular orbit) to mutual escape velocity when their surfaces are in contact (V/V_e). Impact rates as a function of V/V_e are calculated to within ～20% by numerical integration of the equations of motion. It is found that when V/V_e > 0.4, the two-body approximation is a good one. At low velocities (V/V_e < 0.1) two-body “collision-course” trajectories fail to lead to impacts. On the other hand, at these low velocities many impacts result from encounter trajectories with unperturbed separation distances far beyond the two-body gravitational radius. These two effects tend to cancel, and the resulting impact rates remain within a factor of ～3 of the two-body value in spite of these major differences in the nature of the impact trajectories. Therefore, on the average, the two-body approximation is useful well below the value of V/V_e for which it fails to describe individual encounters, and the required corrections are not large. As a consequence of this “anomalous gravitational focusing” planetesimals will continue to interact even when their orbits are noncrossing. This reduces the difficulty with premature isolation of planetesimal embryos during accumulation. Quantitatively, when 0.06 ? V/V_e ? 0.2, the impact rate varies approximately with the fifth power of the radius of the larger body, and is about a factor of 3 above that predicted using the conventional two-body gravitational cross-section formula. At lower values of V/V_e , the impact rate increases less rapidly. Finally, at the lowest values of V/V_e (<.02), the impact rate increases only in proportion to the geometric cross section, as a consequence of the swarm being essentially two dimensional for large unperturbed encounter distances. The gravitational enhancement in effective cross section is thereby limited to a value of about 3000. This leads to an optimal size for growth of planetesimals from a swarm of given eccentricity, and places a limit on the extent of runaway accretion.     

Superosculating intermediate orbits: Fourth-and fifth-order tangencies to trajectories of perturbed motion

The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion.     

Numerical stabilization of the differential equations of Keplerian motion

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.     

Computing secular motion under slowly rotating quadratic perturbation

We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.     

Symplectic Mapping for Satellites and Space Debris Including Nongravitational Forces

The paper presents an efficient algorithm for the study of satellite and space debris orbits on long time intervals. The averaged equations of motion are integrated by means of the implicit midpoint method. This approach is known as a symplectic mapping technique. The perturbing forces included in the mapping are: the geopotential, the atmospheric drag, lunisolar perturbations and the direct radiation pressure (without shadow effects). The influence of the atmosphere is approximated by simple methods for the estimation of integrals. The described mapping is valid for the wide range of orbits including the resonant and the eccentric ones; it can be helpful in practical and theoretical problems. The lifetime of GPS transfer orbits is discussed as an exemplary application.     

On the origin of the unusual orbit of Comet 2P/Encke

The orbit of Comet 2P/Encke is difficult to understand because it is decoupled from Jupiter—its aphelion distance is only 4.1 AU. We present a series of orbital integrations designed to determine whether the orbit of Comet 2P/Encke can simply be the result of gravitational interactions between Jupiter-family comets and the terrestrial planets. To accomplish this, we integrated the orbits of a large number of objects from the trans-neptunian region, through the realm of the giant planets, and into the inner Solar System. We find that at any one time, our model predicts that there should be roughly 12 objects in Encke-like orbits. However, it takes roughly 200 times longer to evolve onto an orbit like this than the typical cometary physical lifetime. Thus, we suggest that (i) 2P/Encke became dormant soon after it was kicked inward by Jupiter, (ii) it spent a significant amount of time inactive while rattling around the inner Solar System, and (iii) it only became active again as the ν₆ secular resonance drove down its perihelion distance.     

Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.