Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J ₂ is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J ₂-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J _2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.     

Analytical Approach to the Motion of a Lunar Artificial Satellite

The canonical equations of motion of an artificial lunar satellite are formulated including the effects of the asphericity of the Moon comprising the harmonics J ₂, J ₂₂, J ₃, J ₃₁, J ₄ andJ ₅, the oblateness of the Earth up to the second zonal harmonic, as well as the disturbing function due to the attractions of the Earth and of the Sun (terms are retained up to order 10^-6 for the higher orbits and 10^-8 for the lower orbits). This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of gravity-gradient torque on the rotational motion of A triaxial satellite in a precessing elliptic orbit

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the effects of gravity-gradient torque on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is free to precess and spin. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite to second order inn/0, wheren is the orbital mean motion of the center of mass and0 is a reference value of the magnitude of the satellite's rotational angular velocity. The differential equations derivable from the transformed Hamiltonian are integrable and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed and a comparison of theoretical results with observations is made.     

An analytic satellite theory using gravity and a dynamic atmosphere

An analytical solution is given for the motion of an artifical Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational effects of the zonal harmonicsJ ₂,J ₃, andJ ₄ are included, and the drag effects of any arbitrary dynamic atmosphere are included. By a dynamic atmosphere, we mean any of the modern empirical models which use various observed solar and geophysical parameters as inputs to produce a dynamically varying atmosphere model. The subtleties of using such an atmosphere model with an analytic theory are explored, and real world data is used to determine the optimum implementation. Performance is measured by predictions against real world satellites. As a point of reference, predictions against a special perturbations model are also given.     

Motion of artificial satellites in the set of Eulerian redundant parameters (III)

In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J ₃₆ . Numerical results proved the very high efficiency and flexibility of the developed equations.     

Long-Period Variations of the Eccentricity Vector Valid also for Near Circular Orbits around a Non-Spherical Body

This paper studies the long period variations of the eccentricity vector of the orbit of an artificial satellite, under the influence of the gravity field of a central body. We use modified orbital elements which are non-singular at zero eccentricity. We expand the long periodic part of the corresponding Lagrange equations as power series of the eccentricity. The coefficients characterizing the differential system depend on the zonal coefficients of the geopotential, and on initial semi-major axis, inclination, and eccentricity. The differential equations for the components of the eccentricity vector are then integrated analytically, with a definition of the period of the perigee based on the notion of “free eccentricity”, and which is also valid for circular orbits. The analytical solution is compared to a numerical integration. This study is a generalization of (Cook, Planet. Space Sci., 14, 1966): first, the coefficients involved in the differential equations depend on all zonal coefficients (and not only on the very first ones); second, our method applies to nearly circular orbits as well as to not too eccentric orbits. Except for the critical inclination, our solution is valid for all kinds of long period motions of the perigee, i.e., circulations or librations around an equilibrium point.     

Application of the integral variation method to satellite orbit prediction

The Integral Variation (IV) method is a technique to generate an approximate solution to initial value problems involving systems of first-order ordinary differential equations. The technique makes use of generalized Fourier expansions in terms of shifted orthogonal polynomials. The IV method is briefly described and then applied to the problem of near Earth satellite orbit prediction. In particular, we will solve the Lagrange planetary equations including the first three zonal harmonics and drag. This is a highly nonlinear system of six coupled first-order differential equations. Comparison with direct numerical integration shows that the IV method indeed provides accurate analytical approximations to the orbit prediction problem.Advanced Systems Studies; Bldg. 254EElectro-Optical Systems Laboratory; Bldg. 201.     

Long-term motion of resonant satellites with arbitrary eccentricity and inclination

A first-order, semi-analytical method for the long-term motion of resonant satellites is introduced. The method provides long-term solutions, valid for nearly all eccentricities and inclinations, and for all commensurability ratios. The method allows the inclusion of all zonal and tesseral harmonics of a nonspherical planet.We present here an application of the method to a synchronous satellite includingonly theJ ₂ andJ ₂₂ harmonics. Global, long-term solutions for this problem are given for arbitrary values of eccentricity, argument of perigee and inclination.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

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.     

Semi-analytic solution of a two-body problem with drag

An approximate semi-analytic solution of a two-body problem with drag is presented. The solution describesnon-lifting orbital motion in a central, inverse-square gravitational field. Drag deceleration is a non-linear function of velocity relative to a rotating atmosphere due to dynamic pressure and velocity-dependent drag coefficient. Neglected are aerodynamic lift, gravitational perturbations of the inverse-square field, and kinematic accelerations due to coordinate frame rotation at earth angular rate. With these simplifications, it is shown that (i) orbital motion occurs in an earth-fixed invariable plane defined by the radius and relative velocity vectors, and (ii) the simplified equations of motion are autonomous and independent of central angle measured in the invariable plane. Consequently, reduction of the differential equations from sixth to second-order is possible. Solutions for the radial and circumferential components of relative velocity are reduced to quadratures with respect to radial distance. Since the independent variable is radial distance, the solutions are singular at zero radial velocity (e. g., for circular orbits). General atmospheric density and drag coefficient models may be used to evaluate the velocity quadratures. The central angle and time variables are recovered from two additional quadratures involving the velocity quadratures. Theoretical results are compared with numerical simulation results.Presently affiliated with AVCO Systems Division, Wilmington, MA 01887, U.S.A.     

Effects of zonal harmonics on the out-of-plane equilibrium points in the generalized Robe's circular restricted three-body problem

This article examines the effects of the zonal harmonics on the out-of-plane equilibrium points of Robe's circular restricted three-body problem when the hydrostatic equilibrium shape of the first primary is an oblate spheroid, the shape of the second primary is an oblate spheroid with oblateness coefficients up to the second zonal harmonic, and the full buoyancy of the fluid is considered. It is observed that the size of the oblateness and the zonal harmonics affect the positions of the out-of-plane equilibrium points L₆ and L₇. It is also observed that these points within the possible region of motion are unstable.     

Resonance Caused by the Luni-Solar Attractions on a Satellite of the Oblate Earth

The present work deals with constructing a conditionally periodic solution for the motion of an Earth satellite taking into consideration the oblateness of the Earth and the Luni-Solar attractions. The oblatenessof the Earth is truncated beyond the second zonal harmonic J ₂. The resonance resulting from the commensurability between the mean motions of the satellite, the Moon, and the Sun is analyzed.     

Long-term orbit computations with KS uniformly regular canonical elements with oblateness

This paper concerns with the study of KS uniformly regular canonical elements with Earth's oblateness. These elements, ten in number, are all constant in the unperturbed motion and even in the perturbed motion, the substitution is straightforward and elementary due to the transformation laws being explicit and closed expression. By utilizing the recursion formulas of Legendre's polynomials, we are able to include any number of Earth's zonal harmonics J _n in the package and also economize the computations. A fixed step-size fourth-order Runge-Kutta-Gill method is employed for numerical integration of the canonical equations.Utilizing 5 test cases covering a large range of semimajor axis and eccentricity, we have carried out computations to study the effects of Earth's zonal harmonics (up to J ₃₆) and integration step-size variation. Bilinear relations and energy equation are used for checking the accuracies of numerical integration. From the application point of view, the package is utilized to study the behaviour of 900 km height near-circular sun-synchronous satellite orbit over a longer duration of 220 days time (nearly 3078 revolutions) and the necessity of including more number of Earth's zonal harmonic terms is noticed. The package is also used to study the effect of higher zonal harmonics on three 900 km height near-circular orbits with inclinations of 60, 63.2, and 65 degrees, by including Earth's zonal harmonics up to J ₂₄. The mean eccentricity (e _m) is found to have long-periods of 459.6, 6925.1 and 1077.6 days, respectively. Sharp changes in the variation of _m near the minima to e_m are noticed. The values of _m are found to be very near to +-90 degrees at the extrema of e_m. The same orbit is employed to study the effect of variation of inclination from 0 to 180 degrees on long-period (T) of eccentricity with J ₂ to J ₂₄ terms. T is found to increase rapidly as we proceed towards the critical inclinations.     

A combined theory for zonal harmonic and resonance perturbations of a near-circular orbit with applications to COSMOS 1603 (1984-106A)

Theory for the motion of a satellite in a near-circular orbit and perturbed by zonal and resonance terms in the Earth's gravity field is developed. Commensurability with respect to both primary and secondary terms is considered with the solution dependent on the depths of the resonances. The theory is applied to the motion of COSMOS 1603 (1984-106A) which approached 14 : 1 resonance in 1987. Values of lumped harmonics derived from least-squares analysis are in close agreement with previous studies of 1984-106A and global gravity field models. The theory is finally extended to incorporate the effects of air drag.     

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.     

An Analytical Solution of the Lagrange Equations Valid also for Very Low Eccentricities: Influence of a Central Potential

This paper presents an analytic solution of the equations of motion of an artificial satellite, obtained using non singular elements for eccentricity. The satellite is under the influence of the gravity field of a central body, expanded in spherical harmonics up to an arbitrary degree and order. We discuss in details the solution we give for the components of the eccentricity vector. For each element, we have divided the Lagrange equations into two parts: the first part is integrated exactly, and the second part is integrated with a perturbation method. The complete solution is the sum of the so-called “main” solution and of the so-called “complementary” solution. To test the accuracy of our method, we compare it to numerical integration and to the method developed in Kaula (Theory of Satellite Geodesy, Blaisdell publ. Co., New York. 1966), expressed in classical orbital elements. For eccentricities which are not very small, the two analytical methods are almost equivalent. For low eccentricities, our method is much more accurate.     

Earth albedo and the orbit of Lageos

The long-period perturbations in the orbit of Lageos satellite due to the earth's albedo have been found using a new analytical formalism. The earth is assumed to be a sphere whose surface diffusely reflects sunlight according to Lambert's law. Specular reflection is not considered. The formalism is based on spherical harmonics; it produces equations which hold regardless of whether the terminator is seen by the satellite or not. Specializing in the case of a realistic zonal albedo shows that Lageos' orbital semimajor axis changes periodically by only about a centimeter and the eccentricity by two parts in 10⁵. The longitude of the node increases secularly by about 6×10^–4 arc sec yr^–1. The effect considered here can explain neither the secular decay of 1.1 mm day^–1 in the semimajor axis nor the observed along-track variations in acceleration of order 2×10^–12 ms^–2.