Three-dimensional subsatellite motion under air drag and oblateness perturbations

The three-dimensional relative motion of a subsatellite with respect to a reference station in an elliptical orbit is studied. A general theory based on the variation of the relative elements, i.e. the instantaneous differences between the orbital parameters of the subsatellite and those of the station, is formulated in order to incorporate arbitrary perturbing forces acting on both satellites. The loss of precision inherent in the subtraction of almost identical quantities is avoided by the consistent use of difference variables. In the absence of perturbations exact analytical representations can be obtained for the relative state parameters. The influences of air drag and Earth's oblateness on the relative motion trajectories are investigated and illustrated graphically for a number of cases.     

考虑地球扁率摄动影响的初轨计算方法

在二体问题意义下的短弧定轨,Laplace型方法是最主要最典型的一种初轨计算方法。若测角资料达到10^-4-10^-5精度(相当于2″—20″之间),那么要使定轨精度达到与其相应的程度,地球非球形引力位中的扁率项摄动应该考虑,在此前提下,同样可以采用相应的Laplace型定轨方法。即给出这种严格包含扁率摄动的初轨计算方法的原理和具体计算过程以及计算实例,除采用多资料定轨方法外,这种方法也是提高初轨计算精度的一种途径,它同样可用于多资料的情况,这种方法对于大扁率主天体(即中心天体)的卫星定轨将更有实用价值。     

Radius and limb topography of Mercury obtained from images acquired during the MESSENGER flybys

Analysis of images obtained by the MESSENGER spacecraft during its three flybys of Mercury yields a new estimate for the planet's mean radius of 2439.25±0.69 km, in agreement with results from Mariner 10 and Earth-based observations, as well as with MESSENGER altimeter and occultation data. The mean equatorial radius and polar radius are identical to within error, suggesting that rotational oblateness is negligible when compared with other sources of topography. This result is consistent with the small gravitational oblateness of the planet. Minor differences in radius obtained at different locations reflect regional variations in topography. Residual topography along three limb profiles has a dynamic range of 7.4 km and a root-mean-square roughness of 0.8 km over hemispherical scales. Following MESSENGER's entry into orbit about Mercury in March 2011, we expect considerable additional improvements to our knowledge of Mercury's size and shape.     

Analytical solutions to the four post-Newtonian effects in a near-Earth satellite orbit

On the basis of the results by Huang et al. (1990), this paper further discusses and analyses the four post-Newtonian effects in a near-Earth satellite orbit: the Schwarzschild solution, the post-Newtonian effects of the geodesic precession, the Lense-Thirring precession and the oblateness of the Earth. A full analytical solution to the effects including their direct perturbations and mixed perturbations due to the Newtonian oblateness (J ₂) perturbation and the Schwarzschild solution is obtained using the quasi-mean orbital element method analogous to the Kozai's mean orbital element one. Some perturbation properties of the post-Newtonian effects are revealed. The results obtained not only can provide a sound scientific basis for the precise determination of a man-made satellite orbit but also is suitable for similar mechanics systems, such as the motions of planets, asteroids and natural satellites.     

Motion of a space station. I

In this first part of the work we develop the equations of motion of a triaxial space station in orbit around the oblate Earth. A first order solution of the problem is presented and the method of complete integration of the system is outlined up to second order of approximation. The zero order part of the Hamiltonian includes both the kinetic and potential energy (Earth's Newtonian attraction) of the station, while the motion in the vicinity of a specific configuration is assumed.The solution leads to deviations on the attitude introduced by the oblateness of the Earth. Such attitude is an exact solution of the equations of the station when its center of mass moves in an elliptic Keplerian orbit.The explicit expressions of the complete solution, discussion of other possible effects on the motion and numerical comparisons will be presented in the second part of the work.     

Qualitative features of the evolution of some polar satellite orbits

Two special cases of the problem of the secular perturbations in the orbital elements of a satellite with a negligible mass produced by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun are considered. These cases are among the integrable ones in the general nonintegrable evolution problem. The first case is realized when the plane of the satellite orbit and the rotation axis of the planet lie in its orbital plane. The second case is realized when the plane of the satellite orbit is orthogonal to the line of intersection between the equatorial and orbital planes of the planet. The corresponding particular solutions correspond to those polar satellite orbits for which the main qualitative features of the evolution of the eccentricity and pericenter argument are described here. Families of integral curves have been constructed in the phase plane of these elements for the satellite systems of Jupiter, Saturn, and Uranus.     

Constructive-analytical solution of the problem of the secular evolution of polar satellite orbits

The well-known twice-averaged Hill problem is considered by taking into account the oblateness of the central body. This problem has several integrable cases that have been studied qualitatively by many scientists, beginning with M.L. Lidov and Y. Kozai. However, no rigorous analytical solution can be obtained in these cases due to the complexity of the integrals. This paper is devoted to studying the case where the equatorial plane of the central body coincides with the plane of its orbital motion relative to the perturbing body, while the satellite itself moves in a polar orbit. A more detailed qualitative study is performed, and an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of the eccentricity and pericenter argument of the satellite orbit is proposed. The methodical accuracy for the polar orbits of lunar satellites has been estimated by comparison with the numerical solution of the system.     

Numerical Determination of Homoclinic and Heteroclinic Orbits at Collinear Equilibria in the Restricted Three-Body Problem with Oblateness

Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular, homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections.     

Study of the Earth's oblateness and rotating atmosphere effects upon an artificial satellite in terms of KS-regular variables

This paper is concerned with an orbit prediction with the KS-regular variables. Perturbations due to the gravitational oblateness of the Earth, and the rotating atmosphere are considered. Application of the method for the perturbed motion are illustrated by numerical and graphical examples for a test case of the orbit.     

Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem

We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries.     

Minimum control for spacecraft formations in a J2 perturbed environment

Large ΔV amounts are often required to maintain the relative geometry which is needed to implement a formation flying concept. A wise use of the orbital environment makes the orbit keeping phase easier, reducing the overall propellant consumption. A first important step in this direction is the selection of formation configurations and orbits which, while still satisfying the mission requirements, are less subject to perturbations resulting naturally in closed relative motion. Within this frame, a number of studies have been recently carried out in order to identify possible sets of invariant relative orbits under the effects of the Earth oblateness, a conservative force commonly referred to as J2 which is also the most important perturbation for Low Earth Orbit. These efforts clearly marked the difficulties connected with achieving genuine periodic relative motion under J2 effect, but they also showed the existence of a set of conditions on the orbital parameters which allow for quasi-periodic J2 trajectories. This paper presents these particular trajectories, by means of deeper theoretical explanations, showing the dependency of the shape of the relative configurations on the orbital inclination. Since the quasi-periodic trajectories cannot be written analytically, and moreover, they are very sensitive with respect to the initial conditions, difficulties arise when trying to exploit these paths as reference for the control of a formation. This paper proposes a novel approach to find, from the actual quasi periodic natural trajectories, minimal control periodic reference trajectories. Next, it evaluates quantitatively the amount of propellant which is needed to control a spacecraft formation along these paths. The choice of Hill’s classical circular projected configuration as a nominal trajectory is considered as a comparison, showing the clear advantages of the proposed guidance design, which assumes low-perturbed periodic reference orbits as nominal trajectories.     

Analysis of the orbit of 1970-114F in its last 20 days

The satellite 1970-114F, the final-stage rocket of the Molniya 1S communications satellite, decayed in the atmosphere on 3 March 1973. During the last 20 days of its life the orbit suffered exceptionally rapid decay, with the apogee height decreasing from 7000 to 1000 km while the perigee height remained near 110 km. About 650 observations, made by visual observers in Britain and by U.S. Navy sensors, have been used with the PROP6 orbit refinement program to determine orbits at 14 epochs. Although the decay rate was more than ten times greater than in any previous orbit determination with PROP, good orbits were obtained, the standard deviation in inclination being less than 0.002° on eight orbits.The combination of high drag and good accuracy allows three techniques in orbital analysis to be successfully applied for the first time. Since zonal winds have little effect on the orbit, the changes in inclination are analysed to determine meridional winds near perigee, at heights of 110–120 km, latitudes of 63–65°S, and 6–12 hr LT. The changes in right ascension of the node are also successfully analysed for the same purpose. The two methods agree in indicating a south-to-north wind of 40 ± 30 m/sec from 11 to 21 February, a geomagnetically quiet period, and a south-to-north wind averaging 150 ± 30 m/sec from 22 February to 3 March, a geomagnetically disturbed period. Thirdly, the changes in the argument of perigee are analysed to determine atmospheric oblateness, which is found to be equal to the Earth's oblateness, to within ±20%. Lastly, the drag coefficient in transition flow is evaluated and found to be 0.85 ± 0.20.     

Halo orbit transfer trajectory design using invariant manifold in the Sun-Earth system accounting radiation pressure and oblateness

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.     

On the Coplanar Integrable Case of the Twice-Averaged Hill Problem with Central Body Oblateness

The twice-averaged Hill problem with the oblateness of the central planet is considered in the case where its equatorial plane coincides with the plane of its orbital motion relative to the perturbing body. A qualitative study of this so-called coplanar integrable case was begun by Y. Kozai in 1963 and continued by M.L. Lidov and M.V. Yarskaya in 1974. However, no rigorous analytical solution of the problem can be obtained due to the complexity of the integrals. In this paper we obtain some quantitative evolution characteristics and propose an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of satellite orbit elements. The methodical accuracy has been estimated for several orbits of artificial lunar satellites by comparison with the numerical solution of the evolution system.     

The rotational motion of an earth orbiting gyroscope according to the Einstein theory of general relativity

The classical Poisson equations of rotational motion are used to study the attitude motions of an Earth orbiting, rapidly spinning gyroscope perturbed by the effects of general relativity (Einstein theory). The center of mass of the gyroscope is assumed to move about a rotating oblate Earth in an evolving elliptic orbit which includes all first-order oblateness effects produced by the Earth.A method of averaging is used to obtain a transformation of variables, for the nonresonance case, which significantly simplifies the Poisson differential equations of motion of the gyroscope. Longterm solutions are obtained by an exact analytical integration of the simplified transformed equations. These solutions may be used to predict both the orientation of the gyroscope and the motion of its rotational angular momentum vector as viewed from its center of mass. The results are valid for all eccentricities and all inclinations not near the critical inclination.This paper represents a part of the author's Ph. D. dissertation for the Mathematics Department, Auburn University.     

Measures of basins of attraction in spin-orbit dynamics

We consider a dissipative spin-orbit model where it is assumed that the orbit of the satellite is Keplerian, the obliquity is zero, and the dissipative effects depend linearly on the relative angular velocity. The measure of the basins of attraction associated to periodic and quasi-periodic attractors is numerically investigated. The results depend on the interaction among the physically relevant parameters, namely, the orbital eccentricity, the equatorial oblateness and the dissipative constant. In particular, it appears that, for astronomically relevant parameter values, for low eccentricities (as in the Moon’s case) about 96% of the initial data belong to the basin of attraction of the 1/1 spin-orbit resonance; for larger values of the eccentricities higher order spin-orbit resonances and quasi-periodic attractors become dominant providing a mechanism for explaining the observed state of Mercury into the 3/2 resonance.     

Exoplanet's Figure and Its Interior

Along with the development of the observing technology, the observation and study on the exoplanets’ oblateness and apsidal precession have achieved significant progress. The oblateness of an exoplanet is determined by its interior density profile and rotation period. Between its Love number k₂ and core size exists obviously a negative correlation. So oblateness and k₂ can well constrain its interior structure. Starting from the Lane-Emden equation, the planet models based on different polytropic indices are built. Then the flattening factors are obtained by solving the Wavre's integro-differential equation. The result shows that the smaller the polytropic index, the faster the rotation, and the larger the oblateness. We have selected 469 exoplanets, which have simultaneously the observed or estimated values of radius, mass, and orbit period from the NASA (National Aeronautics and Space Administration) Exoplanet Archive, and calculated their flattening factors under the two assumptions: tidal locking and fixed rotation period of 10.55 hours. The result shows that the flattening factors are too small to be detected under the tidal locking assumption, and that 28% of exoplanets have the flattening factors larger than 0.1 under the fixed rotation period of 10.55 hours. The Love numbers under the different polytropic models are solved by the Zharkov's approach, and the relation between k₂ and core size is discussed.     

W-shaped occultation signatures: Inference of entwined particle orbits in charged planetary ringlets

Observed W-shaped occultation signatures of certain narrow ringlets in the ring systems of Saturn and Uranus imply a concentration of material near their inner and outer radial edges. A model is proposed where edge bunching is a natural consequence of particles in entwined elliptical orbits, with the same particles alternately defining both edges. While such orbits cross over in radius, collisions would not occur if they have small inclinations, the same fixed argument of periapse ω, and other parameters whereby the particles would “fly in formation” along compressed helical paths relative to the core of the ringlet, which is taken to be a circle in the equatorial plane. For this model to match the observed ring thickness and ringlet widths, orbit inclinations i must be much smaller than their eccentricities e, which themselves would be very small compared to unity. Thus, the meridional cross section of the resultant torus would be a very thin ellipse of thickness proportional to i∣cos ω∣, tilted slightly from the equatorial plane by (i/e)∣sin ω∣ radians. However, gravitational perturbations due to the oblateness of the planet would cause a secular change in ω so that this cross section would collapse periodically to a tilted line, and collisions would then occur. If this collapse could be prevented, the torus could remain in a continuous state of nearly zero viscosity. Stabilization against collapse appears possible due to several remarkable characteristics that are added to the model when the particles are electrically charged. First, because of inherent features of the torus structure, a weak electric force could counter the key effect of the vastly larger oblateness force. Second, because the electric perturbation also affects i, there is a large region in ω,i space where stability against cross-sectional collapse is automatic. For this region, the thickness of the elliptical cross section would expand and contract in concert with the way that the major axis of the ellipse rocks back and forth relative to the equatorial plane. The period of these “rocking and breathing” changes would be from 1 to 3 weeks for a torus in the C ring of Saturn, for example. The electric effects could change considerably without driving the parameters of the torus from the stable domain where cross-sectional collapse does not occur. While specialized and in several important ways still incomplete, the proposed model could account for the W-shaped patterns and explain how very dense ringlets might endure without energy loss due to collisions. It also appears to be capable of explaining the observed sorting of particles by size within a ringlet. Several characteristics of the model suggest definitive tests of its applicability, including its prediction that a nonsymmetrical W-shaped occultation signature could be reversed a half orbit away, and that grazing solar illumination of tilted ringlets might cast shadows that change with time in a prescribed way.     

A three-dimensional solution for the orbit of the asteroidal satellite of 22 Kalliope

We carried out new observations of the binary asteroid 22 Kalliope (S2/2001) with the Shane 3-m telescope of the Lick observatory in October and November 2001. With a FWHM (full width at half maximum) of 0″.2, Kalliope (apparent size of about 0″.15) was not resolved but it was possible to separate the secondary from its primary whose apparent separation was of the order of 0″.7 with a magnitude difference of 3.22±0.20. As each set of observations spanned a few days of time, they are well distributed along the secondary's orbit, enabling us to accurately estimate its orbit.The satellite orbits 22 Kalliope in a prograde manner with respect to Kalliope's rotational spin (which is in a retrograde sense relative to its orbit around the Sun), on a highly inclined (i=19.8±2.0 with respect to the equator of 22 Kalliope) and moderately eccentric orbit (e=0.07±0.02) with an orbital period of 3.58±0.08 days. The semi-major axis is 1020±40 km. Using Kalliope's diameter as determined from IRAS data, the asteroid's bulk density is about 2.03±0.16 g cm⁻³, suggestive of a highly porous body with a porosity of 70% considering that the grain density of its meteoritic analog is of ∼7.4 g cm⁻³. This suggests a rubble pile, rather than solid, body. The measured nodal precession rate of the secondary's orbit seems to be much higher than expected from Kalliope's oblateness, assuming a homogeneous body (constant density). This suggests that Kalliope may be 60% more elongated or 35% larger than presently believed or/and that its internal structure is highly inhomogeneous with a denser outer shell.     

Sun-synchronous orbits near critical inclination

The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.