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.     

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.     

A Multi-Angle Averaging Theorem Applied to the J2 Problem

A particular multi-angle averaging theorem for systems admitting a finite Fourier expansion of the field is presented, together with its application to the problem of motion around an oblate planet (the J₂ problem), in harmonic oscillator formulation. This method of approximate integration has the advantage of working with (close on) directly measurable elements. This revised version was published online in July 2006 with corrections to the Cover Date.     

Recent developments of integrating the gravitational problem ofN-bodies

This paper discusses the formulation and the numerical integration of large systems of differential equations occurring in the gravitational problem ofn-bodies.Different forms of the pertinent differential equations of motion are presented, and various regularizing and smoothing transformations are compared. Details regarding the effectiveness and the efficiency of the Kustaanheimo-Stiefel and of other methods are discussed. In particular, a method is described in which some of the phase variables are treated in the regularized system and others in the ordinary system. This mixed method of numerical regularization offers some advantages.Several numerical integration techniques are compared. A high order Runge-Kutta method, Steffensen's method, and a finite difference method are investigated, especially with regard to their adaptability to regularization.The role of integrals and integral invariants is displayed in controlling the accuracy of the numerical integration.Numerical results are described with 5, 25 and 500 bodies participating. These examples compare the various integration techniques, several regularization methods and different logics in treating binaries.     

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.     

Quaternions and the rotation of a rigid body

The orientation of an arbitrary rigid body is specified in terms of a quaternion based upon a set of four Euler parameters. A corresponding set of four generalized angular momentum variables is derived (another quaternion) and then used to replace the usual three-component angular velocity vector to specify the rate by which the orientation of the body with respect to an inertial frame changes. The use of these two quaternions, coordinates and conjugate moments, naturally leads to a formulation of rigid-body rotational dynamics in terms of a system of eight coupled first-order differential equations involving the four Euler parameters and the four conjugate momenta. The equations are formally simple, easy to handle and free of singularities. Furthermore, integration is fast, since only arithmetic operations are involved.     

A Time-Transformed Leapfrog Scheme

We present a time-transformed leapfrog scheme combined with the extrapolation method to construct an integrator for orbits in N-body systems with large mass ratios. The basic idea can be used to transform any second-order differential equation into a form which may allow more efficient numerical integration. When applied to gravitating few-body systems this formulation permits extremely close two-body encounters to be considered without significant loss of accuracy. The new scheme has been implemented in a direct N-body code for simulations of super-massive binaries in galactic nuclei. In this context relativistic effects may also be included.     

The symplectic character of periodic orbits in the three dipole problem

In this work we reveal for the first time that in the three dipole problem only asymmetric periodic orbits exist.For these periodic orbits — planar and three dimensional — of a charged particle moving under the influence of the electromagnetic field of the three dipoles we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. Also we study the properties of the symplectic matrix and we give the relations there are among the variations of a periodic solution. These relations have been used to check the accuracy of numerical integration of equations of first order variations.     

Proton collisional excitation in the ground configuration of Fe+12, II

Improved values of the proton impact excitation cross sections at coronal energies for all the Fe⁺¹² ground configuration (3s ²3p ²) transitions are presented. These were obtained by direct computer integration of the Schroedinger equation (with the states expressed in intermediate coupling) resulting from the semi-classical Coulomb excitation theory formulation of the process. Comparison is made with previous results. The associated rate constants at coronal temperatures are given and compared with the corresponding electron impact excitation rate constants. Some cross-section values for the Fe⁺¹³ 3s ²3p ² P _1/2² P _3/2 excitation are also presented.     

Star Cluster Simulations: the State of the Art

This paper concentrates on four key tools for performing star cluster simulations developed during the last decade which are sufficient to handle all the relevant dynamical aspects. First we discuss briefly the Hermite integration scheme which is simple to use and highly efficient for advancing the single particles. The main numerical challenge is in dealing with weakly and strongly perturbed hard binaries. A new treatment of the classical Kustaanheimo-Stiefel two-body regularization has proved to be more accurate for studying binaries than previous algorithms based on divided differences or Hermite integration. This formulation employs a Taylor series expansion combined with the Stumpff functions, still with one force evaluation per step, which gives exact solutions for unperturbed motion and is at least comparable to the polynomial methods for large perturbations. Strong interactions between hard binaries and single stars or other binaries are studied by chain regularization which ensures a non-biased outcome for chaotic motions. A new semi-analytical stability criterion for hierarchical systems has been adopted and the long-term effects on the inner binary are now treated by averaging techniques for cases of interest. These modifications describe consistent changes of the orbital variables due to large Kozai cycles and tidal dissipation. The range of astrophysical processes which can now be considered by N-body simulations include tidal capture, circularization, mass transfer by Roche-lobe overflow as well as physical collisions, where the masses and radii of individual stars are modelled by synthetic stellar evolution. This revised version was published online in July 2006 with corrections to the Cover Date.     

The equilibrium of a rotating body of arbitrary density

This paper extends Clairaut's theory of rotational equilibrium to third order terms in a small parameter and is meant to be a sequel to a 1962 publication by the author bearing on the same topic. It has been feasible to obtain the Clairaut equation, which governs the deformation of the equipotential surfaces within a rapidly rotating mass in hydrostatic equilibrium, as an ordinary differential equation. This has been achieved by eliminating the two integral terms which appeared in the original formulation. It is expected that the numerical integration of this newly obtained equation will contribute toward a more precise solution of certain geophysical problems — e.g., the determination of the geoid to an accuracy of ±1 m, and the correction to the travel-time of seismic waves; it should also assist in some planetary questions like the determination of the exterior shape for the rapidly rotating planets Jupiter and Saturn.     

Bandlimited implicit Runge–Kutta integration for Astrodynamics

We describe a new method for numerical integration, dubbed bandlimited collocation implicit Runge–Kutta (BLC-IRK), and compare its efficiency in propagating orbits to existing techniques commonly used in Astrodynamics. The BLC-IRK scheme uses generalized Gaussian quadratures for bandlimited functions. This new method allows us to use significantly fewer force function evaluations than explicit Runge–Kutta schemes. In particular, we use a low-fidelity force model for most of the iterations, thus minimizing the number of high-fidelity force model evaluations. We also investigate the dense output capability of the new scheme, quantifying its accuracy for Earth orbits. We demonstrate that this numerical integration technique is faster than explicit methods of Dormand and Prince 5(4) and 8(7), Runge–Kutta–Fehlberg 7(8), and approaches the efficiency of the 8th-order Gauss–Jackson multistep method. We anticipate a significant acceleration of the scheme in a multiprocessor environment.     

Volume integrals of the products of spherical harmonics and their application to viscous dissipation phenomena in fluids

The equations governing the conversion of kinetic energy into heat in moving viscous media are formulated as volume integrals of products of spherical harmonics. Although the formulation of the fundamental equations is classical, difficulties in the integration of certain products of generalized spherical harmonics over a sphere have permitted heretofore the treatment of only two cases. The closed, form evaluation of eight fundamental types of definite integrals of the product of spherical harmonics, some of them new, or at least missing in the literature, makes possible for the first time the evaluation of these volume integrals in closed form for arbitrary order and index. Explicit details are given for the rates of energy dissipation produced by viscous motions characterized by spheroidal as well as toroidal symmetry.     

GRB 060418 and 060607A: the medium surrounding the progenitor and the weak reverse shock emission

We provide a remedy for a recently published formulation of the Voigt function by reformulating the function into a single proper integral with a damped sine integrand. The present formulation clears up concerns highlighted about the original formulation. The reduction of the Voigt function to a single proper integral enables the use of algorithms available in the literature and included in many software packages to integrate the function and to evaluate the line profile with relative simplicity and superior accuracy. Evidence of the usefulness and superior accuracy of the new formulation is provided.     

On the energy integral for first post-Newtonian approximation

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials \(W^j (j=1,2,3)\), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.     

The line integral approach to radarclinometry

Radarclinometry, the invention of which has been previously reported, is a technique for deriving a topographic map from a single radar image by using the dependence upon terrain-surface orientation of the integrated signal of an individual image pixel. The radiometric calibration required for precise operation and testing does not yet exist, but the imminence of important applications justifies parallel, rather than serial, development of radarclinometry and radiometrically calibrated radar. The present investigation reports three developmental advances: (1) The solid angle of integration of back-scattered specific intensity constituting a pixel signal is more accurately accounted for in its dependence on surface orientation than in previous work. (2) The local curvature hypothesis, which removes the requirement of a ground-truth profile as a boundary condition and enables the formulation of the theory in terms of a line integral, has been expanded to include the three possibilities of Local Cylindricity, Local Biaxial Ellipsoidal Hyperbolicity, and Least-Squares Local Sphericity. (3) The theory is integrated in the cross-ground-range direction, which is ill-conditioned compared to the ground-range direction, whereas the original formulation was based on enforced isotropy in the two-dimensional power spectrum of the topography. It was found necessary to prohibit the hypothesis of Local Biaxial Ellipsoidal Hyperbolicity in the cross-range stepping, for reasons not completely clear. Variation in the proportioning between curvature assumptions had produced topographic maps that are in good mutual agreement but not realistic in appearance. They are severely banded parallel to the ground-range direction, most especially at small radar zenith angles. Numerical experimentation with the falsification of topography through incorrect decalibration as performed on a Gaussian hill suggests that the banding and its exaggeration at high radar incidence angles could easily be due to our lack of radiometric calibration.     

Magnetic field aligned modeling for the upper F region and for the plasmasphere

Existing empirical models, e.g., the IRI and the PRIME model, have shortcomings for the upper-most F region and usually have no realistic formulation for the plasmasphere. These shortcomings can be overcome by replacing purely height oriented modeling by magnetic field aligned approaches.A magnetic field approximation is presented which uses dipole field lines with apexes above the dip equator. Modeling along these field lines can be based on diffusive equilibrium. For a single ion plasma (e.g., an H+ plasma) the integrations which are necessary to model along the field lines in a realistic way can be carried out by means of series expansions. For a multiple ion plasma and in case of arbitrary dependence of electron and ion temperatures on the coordinates one has to apply numerical integration.The principles of joining a field aligned model to a height oriented one are discussed including a method to cross the dip equator in a consistent way.A practical example is presented with a plasmasphere model added to the global model NeUoG which was developed at the University of Graz. The future development aims at replacing all of the topside F region of the model by a magnetic field aligned approach.     

Orbit and uncertainty propagation: a comparison of Gauss–Legendre-, Dormand–Prince-, and Chebyshev–Picard-based approaches

We present a new variable-step Gauss–Legendre implicit-Runge–Kutta-based approach for orbit and uncertainty propagation, VGL-IRK, which includes adaptive step-size error control and which collectively, rather than individually, propagates nearby sigma points or states. The performance of VGL-IRK is compared to a professional (variable-step) implementation of Dormand–Prince 8(7) (DP8) and to a fixed-step, optimally-tuned, implementation of modified Chebyshev–Picard iteration (MCPI). Both nearly-circular and highly-elliptic orbits are considered using high-fidelity gravity models and realistic integration tolerances. VGL-IRK is shown to be up to eleven times faster than DP8 and up to 45 times faster than MCPI (for the same accuracy), in a serial computing environment. Parallelization of VGL-IRK and MCPI is also discussed.     

Dynamical evolution of NEAs: Close encounters, secular perturbations and resonances

We discuss the main mechanisms affecting the dynamical evolution of Near-Earth Asteroids (NEAs) by analyzing the results of three numerical integrations over 1 Myr of the NEA (4179) Toutatis. In the first integration the only perturbing planet is the Earth. So the evolution is dominated by close encounters and looks like a random walk in semimajor axis and a correlated random walk in eccentricity, keeping almost constant the perihelion distance and the Tisserand invariant. In the second integration Jupiter and Saturn are present instead of the Earth, and the 3/1 (mean motion) and v ₆ (secular) resonances substantially change the eccentricity but not the semimajor axis. The third, most realistic, integration including all the three planets together shows a complex interplay of effects, with close encounters switching the orbit between different resonant states and no approximate conservation of the Tisserand invariant. This shows that simplified 3-body or 4-body models cannot be used to predict the typical evolution patterns and time scales of NEAs, and in particular that resonances provide some fast-track dynamical routes from low-eccentricity to very eccentric, planet-crossing orbits.On leave from the Department of Mathematics, University of Pisa, Via Buonarroti 2, 56127 Pisa, Italy, thanks to the G. Colombo fellowships of the European Space Agency.