Density models for the upper atmosphere

Modeling the effects of atmospheric drag is one of the more important problems associated with the determination of the orbit of a near-earth satellite. Errors in the drag model can lead to significant errors in the determination and prediction of the satellite motion. The uncertainty in the drag acceleration can be attributed to three separate effects: (a) errors in the atmospheric density model, (b) errors in the ballistic coefficient, and (c) errors in the satellite relative velocity. In a number of contemporary satellite missions, the requirements for performing the orbit determination and predictions in near real-time has placed an emphasis on density model computation time as well as the model accuracy. In this investigation, a comparison is made of three contemporary atmospheric density models which are candidates for meeting the current orbit computation requirements. The models considered are the analytic Jacchia-Roberts model, the modified Harris-Priester model, and the USSR Cosmos satellite derived density model. The computational characteristics of each of the models are compared and a modification to the modified Harris-Priester model is proposed which improves its ability to represent the diurnal variation in the atmospheric density.This investigation was supported by the NASA Goddard Spaceflight Center under contract NAS5-20946 and Contract NSG 5154.     

Elliptical orbit with non-central force field in celestial mechanics

Equations describing planar motion along a constant elliptical orbit in non-central force field are derived in a simple way and discussed. The angular moment along the elliptical orbit is not a constant of motion. The problem appears to be related to the problem of planar motion with drag force. Relations with related works are discussed. In presence of drag force, conditions for the force field to be derived from the gradient of a function are discussed, and -elation of this function with the potential function is indicated.     

Variational and symplectic integrators for satellite relative orbit propagation including drag

Orbit propagation algorithms for satellite relative motion relying on Runge–Kutta integrators are non-symplectic—a situation that leads to incorrect global behavior and degraded accuracy. Thus, attempts have been made to apply symplectic methods to integrate satellite relative motion. However, so far all these symplectic propagation schemes have not taken into account the effect of atmospheric drag. In this paper, drag-generalized symplectic and variational algorithms for satellite relative orbit propagation are developed in different reference frames, and numerical simulations with and without the effect of atmospheric drag are presented. It is also shown that high-order versions of the newly-developed variational and symplectic propagators are more accurate and are significantly faster than Runge–Kutta-based integrators, even in the presence of atmospheric drag.     

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.     

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.     

The orbital-thermal evolution and global expansion of Ganymede

The tectonically and cryovolcanically resurfaced terrains of Ganymede attest to the satellite's turbulent geologic history. Yet, the ultimate cause of its geologic violence remains unknown. One plausible scenario suggests that the Galilean satellites passed through one or more Laplace-like resonances before evolving into the current Laplace resonance. Passage through such a resonance can excite Ganymede's eccentricity, leading to tidal dissipation within the ice shell. To evaluate the effects of resonance passage on Ganymede's thermal history we model the coupled orbital-thermal evolution of Ganymede both with and without passage through a Laplace-like resonance. In the absence of tidal dissipation, radiogenic heating alone is capable of creating large internal oceans within Ganymede if the ice grain size is 1 mm or greater. For larger grain sizes, oceans will exist into the present epoch. The inclusion of tidal dissipation significantly alters Ganymede's thermal history, and for some parameters (e.g. ice grain size, tidal Q of Jupiter) a thin ice shell (5 to 20 km) can be maintained throughout the period of resonance passage. The pulse of tidal heating that accompanies Laplace-like resonance capture can cause up to 2.5% volumetric expansion of the satellite and contemporaneous formation of near surface partial melt. The presence of a thin ice shell and high satellite orbital eccentricity would generate moderate diurnal tidal stresses in Ganymede's ice shell. Larger stresses result if the ice shell rotates non-synchronously. The combined effects of satellite expansion, its associated tensile stress, rapid formation of near surface partial melt, and tidal stress due to an eccentric orbit may be responsible for creating Ganymede's unique surface features.     

Long-term effects of main-body's obliquity on satellite formation perturbed by third-body gravity in elliptical and inclined orbit

A new non-simplified model of formation flying is derived in the presence of an oblate mainbody and third-body perturbation.In the proposed model,considering the perturbation of the thirdbody in an inclined orbit,the effect of obliquity(axial tilt) of the main-body is becoming important and has been propounded in the absolute motion of a reference satellite and the relative motion of a follower satellite.From a new point of view,J2 perturbed relative motion equations and considering a disturbing body in an elliptic inclined three dimensional orbit,are derived using Lagrangian mechanics based on accurate introduced perturbed reference satellite motion.To validate the accuracy of the model presented in this study,an auxiliary model was constructed as the Main-body Center based Relative Motion(MCRM) model.Finally,the importance of the main-body's obliquity is demonstrated by several examples related to the Earth-Moon system in relative motion and lunar satellite formation keeping.The main-body's obliquity has a remarkable effect on formation keeping in the examined in-track and projected circular orbit(PCO) formations.     

The two-body problem with drag and radiation pressure

The Kepler problem including radiation pressure and drag is treated. The equation of the orbit is derived and the scalar and vector integrals of motion are obtained by direct operation on the vector form of the equation of motion.     

Gas-surface interactions and satellite drag coefficients

Information on gas-surface interactions in orbit has accumulated during the past 35 years. The important role played by atomic oxygen adsorbed on satellite surfaces has been revealed by the analysis of data from orbiting mass spectrometers and pressure gauges. Data from satellites of special design have yielded information on the energy accommodation and angular distributions of molecules reemitted from satellite surfaces. Consequently, it is now possible to calculate satellite drag coefficients from basic physical principles, utilizing parameters of gas-surface interactions measured in orbit. The results of such calculations are given. They show the drag coefficients of four satellites of different compact shapes in low-earth orbit with perigee altitudes in the range from about 150 to 300 km, where energy accommodation coefficients and diffuse angular distributions have been measured. The calculations are based on Sentman's analysis of drag forces in free-molecular flow. His model incorporates the random thermal motion of the incident molecules, and assumes that all molecules are diffusely reemitted The uncertainty caused by the assumption of diffuse reemission is estimated by using Schamberg's model of gas-surface interaction, which can take into account a quasi-specular component of the reemission. Such a quasi-specular component is likely to become more important at higher altitudes as the amount of adsorbed atomic oxygen decreases. A method of deducing accommodation coefficients and angular distributions at higher altitudes by comparing the simultaneous orbital decay of satellites of different shapes at a number of altitudes is suggested. The purpose is to improve thermospheric measurements and models, which are significantly affected by the choice of drag coefficients.     

Models of Motion in a Central Force Field with Quadratic Drag

Some general properties for the motion of a particle in a central force field with a general power law drag are derived. Exact and approximate solutions of the equations of motion are found in various cases. Emphasis is placed on inverse square gravitation and drag that varies with the square of the speed and inversely with the distance from the center of attraction. For this model two results stand out. The first is a particular solution in closed form that demonstrates the decay of an initially circular orbit under drag. The second, found from an approximation of the equations of motion when the radial speed is small compared with the tangential speed, demonstrates the decay of an initially elliptic orbit that is not highly eccentric. Formulas for calculation of the time of flight are presented for the two principal results.     

Attitude stability of a spinning symmetric satellite in a planar periodic orbit

This paper contains an analysis of the attitude stability of a spinning axisymmetric satellite whose mass center moves in any known planar periodic orbit of the restricted three-body problem while the spin axis remains normal to the orbit plane. A procedure based on Floquet theory is developed for constructing attitude instability charts, and examples of these are presented for two stable periodic orbits of the Earth-Moon system—one direct and one retrograde. The physical significance of these instability predictions is then explored by means of numerical integration of the full nonlinear equations of motion. Finally, an analysis based on averaging is performed, leading to approximate instability charts and indicating a possible connection between certain orbital-attitude resonance conditions and unstable attitude motions.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

First-order analytic propagation of satellites in the exponential atmosphere of an oblate planet

The paper offers the fully analytic solution to the motion of a satellite orbiting under the influence of the two major perturbations, due to the oblateness and the atmospheric drag. The solution is presented in a time-explicit form, and takes into account an exponential distribution of the atmospheric density, an assumption that is reasonably close to reality. The approach involves two essential steps. The first one concerns a new approximate mathematical model that admits a closed-form solution with respect to a set of new variables. The second step is the determination of an infinitesimal contact transformation that allows to navigate between the new and the original variables. This contact transformation is obtained in exact form, and afterwards a Taylor series approximation is proposed in order to make all the computations explicit. The aforementioned transformation accommodates both perturbations, improving the accuracy of the orbit predictions by one order of magnitude with respect to the case when the atmospheric drag is absent from the transformation. Numerical simulations are performed for a low Earth orbit starting at an altitude of 350 km, and they show that the incorporation of drag terms into the contact transformation generates an error reduction by a factor of 7 in the position vector. The proposed method aims at improving the accuracy of analytic orbit propagation and transforming it into a viable alternative to the computationally intensive numerical methods.     

Mass density of the upper atmosphere derived from Starlette’s Precise Orbit Determination with Satellite Laser Ranging

The atmospheric mass density of the upper atmosphere from the spherical Starlette satellite’s Precise Orbit Determination is first derived with Satellite Laser Ranging measurements at 815 to 1115 km during strong solar and geomagnetic activities. Starlette’s orbit is determined using the improved orbit determination techniques combining optimum parameters with a precise empirical drag application to a gravity field. MSIS-86 and NRLMSISE-00 atmospheric density models are compared with the Starlette drag-derived atmospheric density of the upper atmosphere. It is found that the variation in the Starlette’s drag coefficient above 800 km corresponds well with the level of geomagnetic activity. This represents that the satellite orbit is mainly perturbed by the Joule heating from geomagnetic activity at the upper atmosphere. This result concludes that MSIS empirical models strongly underestimate the mass density of the upper atmosphere as compared to the Starlette drag-derived atmospheric density during the geomagnetic storms. We suggest that the atmospheric density models should be analyzed with higher altitude acceleration data for a better understanding of long-term solar and geomagnetic effects.     

Orbital perturbations due to radiation pressure for a spacecraft of complex shape

We analyze the perturbations due to solar radiation pressure on the orbit of a high artificial satellite. The latter is modelled in a simplified way (axisymmetric body plus despun antenna emitting a radio beam), which seems suitable to describe the main effects for existing telecommunication satellites. We use the regularized general perturbation equations, by expressing the force in the moving Gauss' reference frame and by expanding the results in terms of some small parameters, referring both to the orbit (small eccentricity and inclination) and to the spacecraft's attitude. Some interesting results are derived, which assess the relative importance of different physical effects and of different parts of the spacecraft in determining the long-term evolution of the orbital elements.     

Motion of Pasiphae’s perijove and node

We investigated the motion of the perijove and ascending node of the 8th satellite of Jupiter, Pasiphae. The main perturbations by the Sun on the satellite permitted to use an intermediate orbit obtained by approximated solutions of differential equations previously transformed by the Von Zeipel method. The orbit is a non-Keplerian ellipse. The secular motion of the ascending node, argument of perijove, and essential periodic perturbations were taken into account. Using our theory we showed that the inclination and eccentricity of Pasiphae can acquire values by which the orbit becomes a librating one; but, within Pasiphae’s observation period, the motion of its perijove is circulating. Taking into account the results of our previous works on Pasiphae motion, we can conclude that the mean motion of the ascending node is similar for different values of the satellite inclination and eccentricity. But the mean motion of the perijove strongly depends on the orbit inclination and eccentricity, according to the Lidov–Kozai mechanism.     

Satellite motion in a Manev potential with drag

In this paper, we consider a satellite orbiting in a Manev gravitational potential under the influence of an atmospheric drag force that varies with the square of velocity. Using an exponential atmosphere that varies with the orbital altitude of the satellite, we examine a circular orbit scenario. In particular, we derive expressions for the change in satellite radial distance as a function of the drag force parameters and obtain numerical results. The Manev potential is an alternative to the Newtonian potential that has a wide variety of applications, in astronomy, astrophysics, space dynamics, classical physics, mechanics, and even atomic physics.     

Analytical direct planetary perturbations on the orbital motion of satellites

An analytical expansion of the disturbing function arising from direct planetary perturbations on the motion of satellites is derived. As a Fourier series, it allows the investigation of the secular effects of these direct perturbations, as well as of every argument present in the perturbation. In particular, we construct an analytical model describing the evection resonance between the longitude of pericenter of the satellite orbit and the longitude of a planet, and study briefly its dynamic. The expansion developed in this paper is valid in the case of planar and circular planetary orbits, but not limited in eccentricity or inclination of the satellite orbit.     

Relativistic effects for near-earth satellite orbit determination

The relativistic formulations for the equations which describe the motion of a near-Earth satellite are compared for two commonly used coordinate reference systems (RS). The discussion describes the transformation between the solar system barycentric RS and both the non-inertial and inertial geocentric RSs. A relativistic correction for the Earth's geopotential expressed in the solar system barycentric RS and the effect of geodesic precession on the satellite orbit in the geocentric RS are derived in detail. The effect of the definition of coordinate time on scale is also examined. A long-arc solution using 3 years of laser range measurements of the motion of the Lageos satellite is used to demonstrate that the effects of relativity formulated in the geocentric RS and in the solar system barycentric RS are equivalent to a high degree of accuracy.     

On the true circular orbit of a satellite

The osculating orbit of a planetary satellite moving in the equatorial plane of the central body under the influence of a rotational symmetric perturbation force is elliptical in first order approximation even if the true orbit is always circular. The satellite motion is influenced by a resonance effect due to this perturbing force. An inclined true satellite orbit cannot be circular.