Chapter 10 - Introduction to orbital perturbations

This paper deals with the effects of electromagnetic forces on the orbital motion of a spacecraft. The electrostatic charge which a spacecraft generates on its surface in the Earth's magnetic field will be subject to a perturbative Lorentz force. A model incorporating all Lorentz forces as a function of orbital elements has been developed on the basis of magnetic and electric fields. This Lorentz force can be used to modify or perturb the spacecraft's orbits. Lagrange's planetary equations in the Gauss variational form are derived using the Lorentz force as a perturbation to a Keplerian orbit. Our approach incorporates orbital inclination and the true anomaly. The numerical results of Lagrange's planetary equations for some operational satellites show that the perturbation in the orbital elements of the spacecraft is a second order perturbation for a certain value of charge. The effect of the Lorentz force due to its magnetic component is three times that of the Lorentz force due to its electric component. In addition, the numerical results confirm that the strong effects are due to the Lorentz force in a polar orbit, which is consistent with realistic physical phenomena that occur in polar orbits. The results confirm that the magnitude of the Lorentz force depends on the amount of charge. This means that we can use artificial charging to create a force to control the attitude and orbital motion of a spacecraft.

The motion of a close artificial satellite of the Moon is considered. The principal perturbations taken into account are caused by the nonsphericity of the Moon and the attraction of the Earth and the Sun. To begin with, the expansions of the disturbing functions due to the nonsphericity of the primary body and the action of the disturbing mass-point body have been derived. The second expansion is produced in terms of the Keplerian elements of a satellite and the spherical coordinates of the disturbing body. Both expansions are valid for an arbitrary reference plane. The motion of a satellite of the Moon is studied in the selenocentric coordinate system referred to the Lunar equator and rotating with respect to the fixed ecliptic system. However, the coordinate exes in the equatorial plane are chosen so that the angular speed of rotation of the system is small. The motion of the satellite is described by means of the contact elements which enable one to utilize the conventional Lagrange's planetary equations and may be regarded as the generalization of the notion of the osculating elements to the case of the disturbing function depending not only o the coordinates and the time but on the velocities as well. Two methods are proposed to represent the motion of Lunar satellites over long intervals of time: the von Zeipel method and the Euler method of analytical integration with application of the variation-of-elements technique at every step of integration. The second method is exposed in great detail.

The problem of a weak gravitational wave impinging upon a nonrelativistic bound system of two point masses is considered. The geodesic equation for each mass is expanded in terms of two small parameters, v/c and dimensionless wave amplitude, in a manner similar to the post-Newtonian expansion; the geodesic equations are resolved into orbital and center-of-mass equations of motion. The effect of the wave on the orbit is determined by using Lagrange's planetary equations to calculate the time evolution of the orbital elements. The gauge properties of the solutions and, in particular, the gauge invariance of the secular effects are discussed.

Circumplanetary Dust Dynamics: Effects of Solar Gravity, Radiation Pressure, Planetary Oblateness, and Electromagnetism

Two semi-analytic solutions for a perturbed two-body problem known as Lagrange planetary equations (LPE) were compared to a numerical integration of the equation of motion with same perturbation force. To avoid the critical conditions inherited from the configuration of LPE, non-singular orbital elements (EOE) had been introduced. In this study, two types of orbital elements, classical Keplerian orbital elements (COE) and EOE were used for the solution of the LPE. The effectiveness of EOE and the discrepancy between EOE and COE were investigated by using several near critical conditions. The near one revolution, one day, and seven days evolutions of each orbital element described in LPE with COE and EOE were analyzed by comparing it with the directly converted orbital elements from the numerically integrated state vector in Cartesian coordinate. As a result, LPE with EOE has an advantage in long term calculation over LPE with COE in case of relatively small eccentricity.

The use of solar radiation pressure for reconfiguration of spacecraft formations near one of the collinear sun-Earth libration points is examined. Specifically, it is considered whether spacecraft that are equipped with sun-facing thermal shields can exploit the solar radiation pressure force that acts on them to perform gradual relative maneuvering. Small shield off-pointing angles give rise to differential solar radiation pressure forces that are essentially perpendicular to the sun line: The effect of these small differential forces on the relative motion of the spacecraft in the formation is, therefore, quite different from that of the total (down-sun) solar force on the absolute orbital motion. First, the linearized equations of relative motion for libration point formations under differential solar radiation forcing are set up, with the sun shield off-pointing angle taken as the control input variable. The range of validity of these equations is examined. The effect of solar radiation pressure on the absolute motion is also analyzed, and the station-keeping issue is avoided by shifting the original libration point to a new equilibrium point. These equations are then used to study the existence and properties of solution trajectories that take the spacecraft from a given initial formation geometry to a desired final one. When the vector control produced by solar radiation pressure is reduced to a scalar one, the existence and properties of time-optimal trajectories based on bang-bang control theory are analyzed. Comparisons made between these two types of transfer trajectories show that the former possess simpler steering of sun shields, whereas the latter have a shorter transfer time and more reachable points in the target orbit.

In this research article, a new idea concerning the application of concept of the continued fractions to the expansion of the geopotential is treated. The perturbed motion of the satellite in the oblate gravitational field of the Earth using a potential of a continued fractions procedure is studied. To compare with the usual perturbation theory, the continued fraction series is truncated beyond the third term. The first oblateness term of the potential is retained. The equations of motion are constructed using the Lagrange planetary equations. The integrals and averages required for solving the problem is outlined as well as the different cases are highlighted. The perturbations in all orbital elements due to considered model of continued fractions are evaluated. The new findings of the manuscript contrast with classical oblate potential field, the semi-major axis, eccentricity, and inclination, nonzero averaged terms in perturbations are obtained. Extra nonzero averaged perturbing terms in the argument of perigee and ascending node are revealed in addition to the classical perturbations.

In this work, the Hamiltonian of the four-body problem is considered under the effects of solar radiation pressure. The equations of motion of the infinitesimal body are obtained in the Hamiltonian canonical form. The libration points and the corresponding Jacobi constants are obtained with different values of the solar radiation pressure coefficient. The motion and its stability about each point are studied. A family of periodic orbits under the effects of the gravitational forces of the primaries and the solar radiation pressure are obtained depending on the pure numerical method. This purpose is applied to the Sun-Earth-Moon-Space craft system, and the results obtained are in a good agreement with the previous work such as (Kumari and Papadouris, 2013).

This paper presents the results of a numerical evaluation of the natural lifetime reduction in low Earth orbit, due to dynamical perturbations. The study considers two values for the area-to-mass ratio, a nominal ratio which resembles a typical value of spacecraft in orbit today, and an enhanced ratio which covers the surface augmentation. The results were obtained with two orbit propagators, one of a semi-analytical nature and the second one using non-averaged equations of motion. The simulations for both propagators were set up similarly to allow comparison. They both use the solar radiation pressure and the secular terms of the geopotential (J2,J4 and J6). The atmospheric drag was turned on and off in both propagators to alternatively study the eccentricity build up and the residual lifetime. The non-averaging case also covers a validation with the full 6 × 6 geopotential. The results confirm the findings in previous publications, that is, the possibility for de-orbiting from altitudes above the residual atmosphere if a solar sail is deployed at the end-of-life, due to the combined effect of solar radiation pressure and the oblateness of the Earth. At near polar inclinations, shadowing effects can be exploited to the same end. The results obtained with the full, non-averaging propagator revealed additional de-orbiting corridors associated with solar radiation pressure which were not found by previous work on space debris mitigation. The results of both tools are compared for specific initial conditions. For nominal values of area-to-mass ratio, instead, it is confirmed that this resonance effect is negligible.The paper then puts the findings in the perspective of the current satellite catalogue. It identifies space missions which are currently close to a resonance corridor and shows the orbit evolution within the resonances with a significantly shorter residual orbital lifetime. The paper finishes with a discussion on the exploitation of these effects with regards to the long-term simulation of the space debris environment and a flux and collision probability comparison.

The advance of the pericenter of the orbit of a test body around a massive body in general relativity can be calculated in a number of ways. In one method, one studies the geodesic equation in the exact Schwarzschild geometry and finds the angle between pericenters as an integral of a certain radial function between turning points of the orbit. In another method, one describes the orbit using osculating orbit elements, and analyzes the ‘Lagrange planetary equations’ that give the evolution of the elements under the perturbing effects of post-Newtonian (PN) corrections to the motion. After separating the perturbations into periodic and secular effects, one obtains an equation for the secular rate of change of the pericenter angle. While the different methods agree on the leading post-Newtonian contribution to the advance, they do not agree on the higher-order PN corrections. We show that this disagreement is illusory. When the orbital variables in each case are expressed in terms of the invariant energy and angular momentum of the orbit and when account is taken of a subtle difference in the meaning of ‘pericenter advance’ between the two methods, we show to the third post-Newtonian order that the different methods actually agree perfectly.

The focus of this paper is on the exploration of the $$a$$ and $$g$$ - $$g'$$ families of planar symmetric periodic orbits around minor bodies under the effect of solar radiation pressure (SRP). An extended Hill problem with SRP allows the study of spacecraft trajectories in the vicinity of asteroids orbiting the Sun. The evolution of the $$a$$ and $$g$$ - $$g'$$ families is presented with SRP increasing from the classical Hill problem to levels characteristic of current and future planned missions to minor bodies, as well as one extreme case with very large SRP for a small asteroid. In addition, the implications of considering a spherical body are analysed, in terms of trajectories colliding with the asteroid and eclipses, which limits the feasibility of various family branches. Finally, the influence of SRP on the linear stability of feasible orbits is calculated.

This article reports an investigation of the effect of solar radiation pressure on the motion of an artificial satellite. The theory has been applied to the orbit of the Vanguard I satellite, and is found to produce significant perturbations in the perigee height of that satellite. In the case of a satellite with a large ratio of area to mass the major terms introduced by solar radiation pressure can reduce the perigee distance at the rate of 1 to 2 km per day, so that the lifetime of the satellite will become considerably shorter than it would be without this effect.

Cable-connected satellites in elliptical orbits are subject to nonlinear effects from solar radiation pressure and Earth's magnetic field, which can impact their stability and performance. This literature review provides a comprehensive overview of the current state of research on these nonlinear effects. We examine the individual and combined effects of solar radiation pressure and Earth's magnetic field on the stability of cable-connected satellites in elliptical orbits, including nonlinear dynamics, chaos, and bifurcations. Our review highlights the key findings, methodologies, and limitations of existing studies, and identifies areas for future research. The results of this review can inform the design and operation of cable-connected satellites in elliptical orbits, ensuring their stability and performance in the presence of these nonlinear effects.

The OSIRIS cameras onboard ESA’s Rosetta spacecraft observed comet 9P/Tempel 1 near-continuously from 28 June until 14 July 2005. The Narrow Angle Camera (NAC) monitored the cometary dust, while the wide angle camera (WAC) observed through filters sensitive to emissions of OH, CN, OI, Na, and the associated continuum.The brightness of the dust increased by a factor of 4.5 within 30 minutes, measured in a circular aperture of 3000 km radius centered on the nucleus. The brightening is likely caused by ejection of material from the crater, decreasing optical depth of the ejecta, and the evaporation of icy grains in the ejecta cloud. The dust moved with a typical velocity of 160 ms-1, suggesting acceleration of the dust by gas in the inner cometary coma. Images of the dust cloud taken several days after the impact show the effect of solar radiation pressure. The size distribution derived from an analysis of the radiation pressure is comparable to that commonly observed in cometary comae.The increase of the intensity of the OH emission due to the impact corresponds to a water production of (1.5–3)× 1032 molecules. The dust production depends on the uncertain amount of large dust grains ejected by the impact, but the dust/gas ratio is likely to be larger than one.KeywordsLight CurveRadiation PressureDust CloudSolar Radiation PressureDeep ImpactThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Drift-free solar sail formations in elliptical Sun-synchronous orbits