The possible motions of a satellite about an oblate planet

The regions of motions of a satellite for given values of energy and angular momentum about polar axes are shown. Special attention is paid to the circular equatorial orbits which have been shown to be Hill stable. The anomalistic and the nodal period for the motions near to the circular equatorial orbits have been found.     

Yukawa-type potential effects in the anomalistic period of celestial bodies

Several contemporary modified models of gravity predict the existence of a non-Newtonian Yukawa-type correction to the classical gravitational potential. We study the motion of a secondary celestial body under the influence of the corrected gravitational force of a primary. We derive two equations to approximate the periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period) under the influence of the non-Newtonian radial acceleration. Kinematically, this influence produces apsidal motion. We performed numerical estimations for Mercury, for the companion star of the pulsar PSR 1913+16, and for the extrasolar Planet b of the star HD 80606. We also considered the case of the artificial Earth satellite GRACE-A, but the results present a low degree of reliability from a practical standpoint.     

Long-periodic and secular perturbations to the orbit of a spherical satellite due to direct solar radiation pressure

A theory is developed for the perturbations to the orbit of a spherically symmetric satellite which accounts for the changes in the perigee and nodal positions and the variations of the Sun-Earth distance and direction over an orbital revolution. The theory is semi-analytical, the equations of motion being integrated with respect to time over the sunlit period of each orbital revolution. Long-periodic and short-periodic perturbations may be treated separately, and this is important for long-term analyses in terms of mean elements where short-period terms are averaged or omitted.     

Effects in the anomalistic period of celestial bodies due to a logarithmic correction to the Newtonian gravitational potential

We study the motion of a secondary celestial body under the influence of the logarithmic corrected gravitational force of a primary one. This kind of correction was introduced by Fabris and Campos (Gen. Relativ. Gravit. 41(1):93, 2009). We derive two equations to compute the rate of change of the anomalistic period with respect to the eccentric anomaly and its total variation over one revolution. In a kinematical sense, this influence produces an apsidal motion. We perform numerical estimations for some celestial bodies. We also compare our results to those obtained by considering a Yukawa correction.     

Perturbations in the Nodal Period of Artificial Satellites Caused by the Terrestrial Infrared Radiation Pressure

Considering the terrestrial infrared radiaton pressure as a disturbing factor, its influence on the nodal period of an artificial satellite moving in a quasi-circular orbit is estimated.     

Some useful results on initial node locations for near-equatorial circular satellite orbits

Perturbation theory based on Lie transforms is used to obtain a second-order long period solution for inclination and right ascension of ascending node, of near-equatorial circular satellite orbits. The solution includes the average effects of the Earth's oblateness and the luni-solar perturbations. Three algorithms, useful in mission analysis, are then given. The first algorithm finds the initial node location that results in a decrease of inclination to zero and it also finds the corresponding time to arrive at this zero inclination. The second algorithm determines the initial nodal band that maintains the orbital inclination below a specified value for a given time interval. The third algorithm obtains the initial node location that maximizes the time in which the satellite can be maintained within a given inclination tolerance without the use of any active control and it also obtains the corresponding maximum time. The results of the first and the third algorithms are given for 24-h near-equatorial circular satellite orbits and are cast in simple closed forms.     

Perturbations of the orbit of Explorer 19 due to solar radiation

The orbit of the balloon satellite, Explorer 19, is analysed to determine the effects of direct solar radiation pressure over one revolution of the satellite (111 min) for MJD 42822 and MJD 42966. At the earlier date, the satellite entered the Earth's shadow, presenting an opportunity to examine the effectiveness of two different shadow models. The reflectivity of the surface of the satellite was estimated from analysis of the variations in orbital eccentricity over a period of 236 days.Although many of the parameters associated with the shape and orientation of the satellite are unknown, the theory for a non-spherical satellite is applied using trial and error methods to determine the parameters of best fit. The paper concludes with an examination of the perturbations in orbital eccentricity and inclination due to incident, specularly reflected, and diffusely reflected radiation.     

A method for averaging the perturbing function in Hill’s problem

A modified method for averaging the perturbing function in Hill’s problem is suggested. The averaging is performed in the revolution period of the satellite over the mean anomaly of its motion with a full allowance for a variation in the position of the perturbing body. At its fixed position, the semimajor axis of the satellite orbit during the revolution of the satellite is constant in view of the evolution equations, while the remaining orbital elements undergo secular and long-period perturbations. Therefore, when the motion of the perturbing body is taken into account, the semimajor axis of the satellite orbit undergoes the strongest perturbations. The suggested approach generalizes the averaging method in which only the linear (in time) term is included in the perturbing function. This method requires no expansion in powers of time. The described method is illustrated by calculating the perturbations of the semimajor axes for two distant satellites of Saturn, S/2000 S 1 and S/2000 S5. An approximate analytic solution is compared with the results of numerical integration of the averaged system of equations of motion for these satellites.     

Perturbations in the Anomalistic Period of Artificial Satellites Caused by the Direct Solar Radiation Pressure Centre for Astronomy and Space Sciences,Satellite Tracking Station No. 1132

Approximate formulae for the evaluation of the influence of direct solar radiation pressure (neglecting the POYNTING -ROBERTSON effect) on the anomalistic period of artificial satellites with quasi-circular and completely illuminated orbits are derived.     

Some investigations into the atmospheric drag problem

This paper calls into question the validity of the well-known formulae for the perturbations in the Keplerian elements, over one revolution of an orbit, for the motion of a drag-perturbed artificial satellite. These formulae are derived from Gauss's form of the planetary equations, by averaging over a single revolution of the orbit, and using the eccentric anomaly as the independent variable.It is shown that for light balloon-type satellites in near-circular orbits neither the eccentric anomaly nor the true longitude is a suitable choice of independent variable for the averaging procedure. Under these circumstances, it would seem that simple formulae for the variations in the elements cannot be derived from Gauss's equations.     

Significant high number commensurabilities in the main lunar problem II: The occurrence of Saros-like near periodicities

The effect of changes in the Moon's semimajor axis and the Earth's orbital eccentricity on the occurrence of Saros-like cycles is examined. The Earth-Moon-Sun dynamical system exhibits such cycles for only 25 to 30% of the time interval between –5×10⁷ to +5×10⁷ years. Not only has the present Saros the smallest period during this time, but it also has one of the longest durations and the period closest to an integral number of anomalistic years, thus making it one of the most efficient Saros-like cycles for reversing solar perturbations in the main lunar problem. During the lifetime of a Saros-like cycle, variations of the Earth's orbital eccentricity cause frequent disappearances and reappearances of the cycle.     

Perturbations of a close-Earth satellite due to sunlight diffusely reflected from the Earth

A semianalytic method has been developed to calculate the radiation-pressure perturbations of a close-Earth satellite due to sunlight reflected from the Earth. The assumptions made are that the satellite is spherically symmetric and that the solar radiation is reflected from the Earth according to Lambert's Law with uniform albedo. By using expressions for the components of the radiation-pressure force due to Lochry, the expressions for the perturbations of the elements were developed into series in the true anomalyv. The perturbations within a given revolution can be obtained analytically by integrating with respect tov while holding all slowly varying quantities constant. The long-range perturbations are then obtained by accumulating the net perturbations at the end of each revolution.     

High-precision repeat-groundtrack orbit design and maintenance for Earth observation missions

The focus of this paper is the design and station keeping of repeat-groundtrack orbits for Sun-synchronous satellites. A method to compute the semimajor axis of the orbit is presented together with a station-keeping strategy to compensate for the perturbation due to the atmospheric drag. The results show that the nodal period converges gradually with the increase of the order used in the zonal perturbations up to \(J_{15}\). A differential correction algorithm is performed to obtain the nominal semimajor axis of the reference orbit from the inputs of the desired nodal period, eccentricity, inclination and argument of perigee. To keep the satellite in the proximity of the repeat-groundtrack condition, a practical orbit maintenance strategy is proposed in the presence of errors in the orbital measurements and control, as well as in the estimation of the semimajor axis decay rate. The performance of the maintenance strategy is assessed via the Monte Carlo simulation and the validation in a high fidelity model. Numerical simulations substantiate the validity of proposed mean-elements-based orbit maintenance strategy for repeat-groundtrack orbits.     

On the formation of the rings of saturn

It is shown in this paper that a satellite which revolves round a primary in a circular orbit in tied revolution and which spirals within Roche's limit must form by its disintegration a ring of ellipse-like cross-section which is more than 11 times larger than the original spherical diameter of the disintegrating body. The individual particles have elliptic orbits with small eccentricities and revolve in different planes at small inclinations to each other. By inelastic collisions of particles in differently inclined orbits the original considerable thickness of the ring is very greatly reduced.

Applied to the system of Saturn it is assumed that the Rings A and B are formed by disintegration of two different satellites. It can be shown that the satellite A had an original diameter of 1721 km and a density of 0.95 g/cm³, the satellite B a diameter of 2463 km and a density of 1.95.

Thus the hypothesis of G. P. Kuiper, that the rings of Saturn are composed of H²O-snow contaminated by silicate dust (as Jupiter III), seems to fit very well.     

Ringlet–satellite interactions

We study the interaction of a satellite and a nearby ringlet on eccentric and inclined orbits. Secular torques originate from mean motion resonances and the secular interaction potential which represents the m  = 1 global modes of the ring. The torques act on the relative eccentricity and inclination. The resonances damp the relative eccentricity. The inclination instability owing to the resonances is turned off by a finite differential eccentricity of the order of 0.27 for nearly coplanar systems. The secular potential torque damps the eccentricity and inclination and does not affect the relative semi-major axis; also, it suppresses the inclination instability that persists at small differential eccentricities. The damping of the relative eccentricity and inclination forces an initially circular and planar small mass ringlet to reach the eccentricity and inclination of the satellite. When the planet is oblate, the interaction of the satellite damps the proper precession of a small mass ringlet so that it precesses at the satellite's rate independently of their relative distance. The oblateness of the primary modifies the long-term eccentricity and inclination magnitudes and introduces a constant shift in the apsidal and nodal lines of the ringlet with respect to those of the satellite. These results are applied to Saturn's F-ring, which orbits between the moons Prometheus and Pandora.     

Sur la stabilite de certains mouvements particuliers d'un satellite artificiel de la terre soumis aux efforts aerodynamiques

L'auteur considère le mouvement d'un satellite artificiel de la terre évoluant suffisamment près de celle-ci pour qu'on ait à tenir compte des efforts aérodynamiques et en admettant que l'air est un fluide parfait incompressible en mouvement irrotationnel.Il met le problème en équations et, dans le cas d'un satellite de révolution, démontre l'existence de mouvements particuliers où le centre de gravité du satellite a un mouvement circulaire uniforme le satellite tournant uniformément autour de son axe perpendiculaire au plan du cercle. Il donne des conditions suffisantes de stabilité et d'instabilité de ces mouvements au moyen de la méthode de Liapounoff.

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The author considers the motion of an artificial satellite of the Earth revolving sufficiently near it so that the aerodynamic forces can be taken into account. It is supposed that air is a perfect incompressible fluid in irrotational motion.The problem is posed in the form of equations and proves the existence of particular motions in the case of a satellite of revolution when the centre of gravity of the satellite has uniform circular motion, the satellite revolving uniformly around its axis, perpendicular to the plan of the circle.Sufficient conditions for stability and instability of particular motions are given by the method of Liapunov.

     

On the determination of the long period tidal perturbations in the elements of artificial earth satellites

In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk ₂ andk ₃ from pure constants are not taken into consideration, then the perturbations caused by the variability ofk ₂ andk ₃ across the parallels will be misinterpreted as the perturbations caused byk ₄...-terms, and the spurious values ofk ₄... will be deduced. It is extremely doubtful, however, that the real effects caused byk ₄,k ₅,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon. The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.     

Neutral composition changes during a period of increasing magnetic activity

Neutral composition data obtained by the gas analyzer aboard the satellite ESRO 4 are investigated for a period of increasing magnetic activity. The prominent feature observed is the development of localized disturbance zones in the high-and mid-latitude regions which show a significant enhancement in argon and nitrogen densities and a simultaneous decrease in helium densities. The behavior of oxygen within these zones is complex, and both increases and decreases are observed. Both the latitudinal extent and the magnitude of the disturbed regions increase with growing magnetic activity. In contrast the low-latitude region exhibits moderate enhancements of all constituents. Using the AE index as an indicator of magnetic activity, we find that at higher latitudes the atmospheric response time is of the order of one orbital revolution or less. Comparisons with other observations and with a theoretical model by Mayr and Volland (1974) show good agreement.     

Normal modes of synchronous rotation

The dynamics of synchronous rotation and physical librations are revisited in order to establish a conceptually simple and general theoretical framework applicable to a variety of problems. Our motivation comes from disagreements between the results of numerical simulations and those of previous theoretical studies, and also because different theoretical studies disagree on basic features of the dynamics. We approach the problem by decomposing the orientation matrix of the body into perfectly synchronous rotation and deviation from the equilibrium state. The normal modes of the linearized equations are computed in the case of a circular satellite orbit, yielding both the periods and the eigenspaces of three librations. Libration in longitude decouples from the other two, vertical modes. There is a fast vertical mode with a period very close to the average rotational period. It corresponds to tilting the body around a horizontal axis while retaining nearly principal-axis rotation. In the inertial frame, this mode appears as nutation and free precession. The other vertical mode, a slow one, is the free wobble. The effects of the nodal precession of the orbit are investigated from the point of view of Cassini states. We test our theory using numerical simulations of the full equations of the dynamics and discuss the disagreements among our study and previous ones. The numerical simulations also reveal that in the case of eccentric orbits large departures from principal-axis rotation are possible due to a resonance between free precession and wobble. We also revisit the history of the Moon's rotational state and show that it switched from one Cassini state to another when it was at 46.2 Earth radii. This number disagrees with the value 34.2 derived in a previous study.