Prediction of satellite orbits contraction due to diurnally varying oblate atmosphere and altitude-dependent scale height using KS canonical elements

A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. The important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.     

Contraction of satellite orbits using KS uniformly regular canonical elements in an oblate diurnally varying atmosphere

A new non-singular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed in terms of the Kustaanheimo and Stiefel (KS) uniformly regular canonical elements, by assuming the atmosphere to be oblate diurnally varying with constant density scale height. The series expansions include up to third-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. Numerical comparisons of the important orbital parameters semimajor axis and eccentricity up to 1000 revolutions, obtained with the present solution, with the third-order analytical theories of Swinerd and Boulton and in terms of the KS elements, with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

Analytical orbit predictions with air drag using KS uniformly regular canonical elements

A new nonsingular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed for long term motion in terms of the KS uniformly regular canonical elements by a series expansion method, by assuming the atmosphere to be symmetrically spherical with constant density scale height. The series expansions include up to third order terms in eccentricity. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. Numerical comparisons of the important orbital parameters semi major axis and eccentricity up to 1000 revolutions, obtained with the present solution, with KS elements analytical solution and Cook, King-Hele and Walker's theory with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

Analytical approach using KS elements to short-term orbit predictions including J 2

Analytical solutions using KS elements are derived. The perturbation considered is the Earth's zonal harmonic J ₂. The series expansions include terms of fourth power in the eccentricity. Only two of the nine KS element equations are integrated analytically due to the reasons of symmetry. The analytical solution is suitable for short-term orbit computations. Numerical studies show that reasonably good estimates of the orbital elements can be obtained in one step of 10 to 30 degrees of eccentric anomaly for near-Earth orbits of moderate eccentricity. For application purposes, the analytical solution can be effectively used for onboard computation in the navigation and guidance packages, where the modelling of J ₂ effect becomes necessary.     

Computation of elliptic Hansen coefficients and their derivatives

The problem of computation of elliptic Hansen coefficients and their derivatives is considered for constructing a motion theory of an artificial Earth satellite with large eccentricity. An algorithm for analytical and numerical computation of these coefficients and their derivatives is described. The recurrence relations for derivatives of the first and second order and initial values for recurrences are obtained. As an example, numerical values of some elliptic Hansen coefficients are given for the orbit with eccentricityk=0.74.     

Analytical short-term orbit predictions withJ 3 andJ 4 in terms of KS elements

Analytical theory for short-term orbit motion of satellite orbits with Earth's zonal harmonicsJ ₃ andJ ₄ is developed in terms of KS elements. Due to symmetry in KS element equations, only two of the nine equations are integrated analytically. The series expansions include terms of third power in the eccentricity. Numerical studies with two test cases reveal that orbital elements obtained from the analytical expressions match quite well with numerically integrated values during a revolution. Typically for an orbit with perigee height, eccentricity and inclination of 421.9 km, 0.17524 and 30 degrees, respectively, maximum differences of 27 and 25 cm in semimajor axis computation are noted withJ ₃ andJ ₄ term during a revolution. For application purposes, the analytical solutions can be used for accurate onboard computation of state vector in navigation and guidance packages.     

Grain abundance in the primordial atmosphere of the Earth

For models of planetary accumulation in the presence of solar nebular gas, the initial surface temperature of the Earth is controlled by the grain opacity of the atmosphere. The surface temperature in turn controls the quantity of neon dissolved and trapped within the interior of the Earth. To compare accumulation theory with observation calculations have been made of the grain opacity expected to be associated with accumulation in a gaseous nebula. There are two parameters that are in principle determined by the theory, but actually are at present uncertain: the mean eccentricity e of the planetesimal swarm, and the fraction ξ of the accretional energy that is expended in the release of grains into the atmosphere by ablation of the incoming planetesimal. It is found that if the eccentricity of swarm is low (10^?3), rather low values of ξ(10^?5) are required to match the observed neon data. In contrast higher values of ξ are required (10^?1) for the most probable case, intermediate eccentricity (10^?2). For the high eccentricity case (e ～ 0.1) ξ must be >10^?2. The results show that avoidance of excess trapped neon of solar composition places restrictive, but not necessarily impossible, conditions on the parameters of the accumulation theory.     

Resonant dynamics of Medium Earth Orbits: space debris issues

The Medium Earth Orbit region is the home of the navigation constellations. It is shown how the orbits of these constellations of satellites are strongly affected by the so-called inclination dependent luni-solar resonances. The analytical theory of these resonances is recalled and a large set of numerical integrations is used to investigate the stability of the orbits of the constellations over very long time spans. The stability issue is important in the definition of possible disposal strategies for the constellation spacecraft, after their end-of-life. Two possible disposal strategies are envisaged involving either stable or unstable orbits (from the eccentricity growth point of view). In particular it is shown how, to have disposal orbits with moderately short lifetime (of the orders of 40–50 years), a very large disposal maneuver would be required to rise the inital eccentricity to values above ~ 0.3.     

Reflection Properties of Gravito-MHD Waves in an Inhomogeneous Horizontal Magnetic Field

We derive the dispersion equation for gravito-magnetohydrodynamical (MHD) waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. Sound and Alfvén speeds are constant. Under these conditions, it is possible to derive analytically the equations for gravito-MHD waves. The high values of the viscous and magnetic Reynolds numbers in the solar atmosphere imply that the dissipative terms in the MHD equations are negligible, except in layers around the positions where the frequency of the MHD wave equals the local Alfvén or slow wave frequency. Outside these layers the MHD waves are accurately described by the equations of ideal MHD. We consider waves that propagate energy upward in the atmosphere. For the plane boundary, z=0, between two isothermal plasma regions with horizontal but different magnetic fields, we discuss the boundary conditions and derive the equations for the reflection and transmission coefficients. In the simpler case of a gravitationally stratified plasma without magnetic field, these coefficients describe the reflection and transmission properties of gravito-acoustic waves.     

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ ₂ throughJ ₉. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J ₂ only). In particular, we examine the manner in which the odd zonalJ ₃ breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J ₄+J ₂ ² =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.     

Comparison of Numerical Methods for the Integration of Natural Satellite Systems

We present a new implementation of the recurrent power series (RPS) method which we have developed for the integration of the system of N satellites orbiting a point-mass planet. This implementation is proved to be more efficient than previously developed implementations of the same method. Furthermore, its comparison with two of the most popular numerical integration methods: the 10th-order Gauss–Jackson backward difference method and the Runge–Kutta–NystrRKN12(10)17M shows that the RPS method is more than one order of magnitude better in accuracy than the other two. Various test problems with one up to four satellites are used, with initial conditions obtained from ephemerides of the saturnian satellite system. For each of the three methods we find the values of the user-specified parameters (such as the method's step-size (h or tolerance (TOL)) that minimize the global error in the satellites' coordinates while keeping the computer time within reasonable limits. While the optimal values of the step-sizes for the methods GJ and RKN are all very small (less than T/100, the ones that are suitable for the RPS method are within the range: T/13<h<T/6 (T being the period of the innermost satellite of the problem). Comparing the results obtained by the three methods for these step-sizes and for the various test problems we observe the superiority of the RPS method over GJ in terms of accuracy and over RKN both in accuracy and in speed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Spectral decomposition of geopotential,earth and ocean tidal perturbations in linear satellite theory

The spectra of geopotential, Earth and ocean tidal perturbations on a satellite can be obtained using Kaula's linear theory, or an extension thereof, as summations of terms depending on four indices l, m, p, q. In this work algorithms are presented that generate the equivalence classes induced by the composition rule of frequency on the set of all (l, m, p, q) combinations up to a maximum degree L and maximum value Q of the last index. These algorithms eliminate the need to search the set of frequencies when the linear theory is programmed on a computer.     

Mineralogical-petrological characterization of near-Earth asteroids

Studies of near-Earth asteroids are aimed at determining their dynamical and structural history. The mineralogy and petrology of 17 near-Earth asteroids are characterized using reflectance spectroscopy with ground-based telescopes as one method to address their major issues. Implications for the origin and evolution are discussed in a separate paper. Assuming the surfaces are composed of cosmically abundant materials, the presence of certain mineralogical species can be determined from diagnostic absorption features and spectral characteristics which have been studied under known laboratory conditions and understood in terms of crystal field theory. With one possible exception, the surface composition of near-Earth asteroids consists of common rock-forming minerals such as olivine, pyroxene, and phyllosilicates. Opaque components are present but cannot be mineralogically identified with existing experimental data. The spectrum of 2201 Oljato cannot be interpreted in terms of common rock-forming minerals. This spectrum was examined for cometary features because its high orbital eccentricity suggests a possible relation to comets. No common cometary features are identified in its spectrum. The predominance of mafic silicate absorption features in spectra of near-Earth asteroids compared to the majority of main-belt asteroids may be a primary compositional feature or may be the signature of relatively fresher asteroid material.     

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.     

Eccentricity evolution in hierarchical triple systems with eccentric outer binaries

We develop a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular, while the outer one is eccentric. We consider coplanar systems with well-separated components and comparable masses. The derivation of short-period terms is based on an expansion of the rate of change of the Runge–Lenz vector. Then, the short-period terms are combined with secular terms, obtained by means of canonical perturbation theory. The validity of the theoretical equations is tested by numerical integrations of the full equations of motion.     

Detailed prediction for the BYORP effect on binary near-Earth Asteroid (66391) 1999 KW4 and implications for the binary population

A previous theory by the authors for detailed modeling of the binary YORP effect is reviewed and expanded to accommodate doubly-synchronous binary systems, as well as a method for non-dimensionalizing the coefficients for application to binary systems where a shape model to compute its own coefficients is not available. The theory is also expanded to account for the effects of primary J₂ and the Sun’s 3rd body perturbation on the secular orbit evolution. The newly expanded theory is applied to the binary near-Earth Asteroid 1999 KW4, for which a detailed shape model is available. The result of simulation of the secular evolutionary equations shows that the KW4 orbit will be double in size in approximately 22,000 years, and will reach the Hill radius in approximately 54,000 years. The simulation also shows that the eccentricity will alternate growing and shrinking in magnitude, depending on the location of the solar node in the body-fixed frame. Therefore the eccentricity is not fixed to evolve in the opposite sign as the semi-major axis unless the circulation of the node (with a period of 500 years) is averaged out as well. The current orbit expansion rate for KW4 of 7 cm per year is shown to be detectable with observations of the mean anomaly which grows quadratically in time with an expanding orbit. Finally, the KW4 results are scaled for application to a number of other binary systems for which detailed shape models are not available. This application shows that the orbits considered can expand to their Hill radius in the range of 10⁴-10⁶ years. This implies rapid formation of binary systems is necessary to support the large percentage of binaries observed in the NEA population.     

Analysis of the orbit of 1970-87a (Cosmos 373)

Cosmos 373, 1970-87A, was launched on 20 October 1970 into an orbit inclined at 62.9° to the Equator, with an initial perigee height of 472 km. The orbit has been determined at 25 epochs covering a period of just over 4 yr using the RAE orbit refinement program PROP, with over 1500 observations. Observations from the Hewitt camera at Malvern were available for all 25 orbits.The main purpose of the orbit determination was to provide accurate values of the eccentricity for use in determining the odd zonal harmonics in the Earth's gravitational potential. The analysis has resulted in extremely accurate values of e with S.D.'s down to 0.000005 and has indicated an amplitude of the oscillation in eccentricity of 0.0085, equivalent to almost 60 km in perigee height—the largest yet recorded for any near-Earth orbit of high accuracy.     

Modelling close encounters with Öpik''s theory

The results obtained by numerical integration of the equations of motion of fictitious comets, in the restricted circular three-dimensional three-body problem, are compared with those obtained with Öpik's theory of close encounters, for an experimental set-up similar to that used by Froeschlé and Rickman (1980, 1981) to model both the infeed of comets from the trans-jovian region into the Jupiter family and their subsequent orbital evolution within the family. The distributions of perturbations in orbital energy E, eccentricity e and inclination i are well reproduced by Öpik's theory, as long as the comparison is made on the outcomes of encounters only up to a certain unperturbed distance b_max; several values of the latter are experimented with and it is found that, surprisingly, Öpik's theory seems to be still working reasonably well for values of b_max in excess of several times the Hill's radius of the planet.     

An averaging method to study the motion of lunar artificial satellites II: Averaging and Applications

On this, the second part of a two part study (Steichen, 1998) we further develop a semi-analytical theory for a lunar artificial satellite. This theory is obtained by averaging analytically the Hamiltonian function over period up to a month. The averaged equations are then numerically integrated. The solution is free from singularities at e = 0 and I = 0 and is not expanded in powers of these variables. In the last section, the analytic work is applied to characteristic examples to validate the method used. This revised version was published online in July 2006 with corrections to the Cover Date.     

Canonical Elements for Öpik Theory

The purpose of this paper is to find a set of canonical elements to use within the framework of Öpik theory of close encounters of a small body with a planet (Öpik, Interplanetary Encounters, 1976). Since the small body travels along a planetocentric hyperbola during the close approach and Öpik formulas are valid, without approximations, only at collision, we derive a set of canonical elements for hyperbolic collision orbits (eccentricity e → 1⁺, semi-major axis a fixed) and then we introduce the unperturbed velocity of the small body and the distance covered along the asymptote as a new canonically conjugate pair of orbital elements. An interesting result would be to get a canonical set containing the coordinates in the Target Plane (TP), useful for the analysis of the future encounters: in the last part we prove that this is not possible.