East-west stationkeeping requirements of nearly synchronous satellites due to Earth's triaxiality and luni-solar effects

A closed form solution, for longitude and semimajor axis deviations in the neighborhood of a prespecified station, is obtained for nearly synchronous satellites. The model use includes the important terms in Earth's zonal and tesseral harmonics as well as the luni-solar perturbations. The initial semimajor axis for two-maneuver east-west stationkeeping is then deduced. Due to the luni-solar effects, it is found that the initial semimajor axis deviation from synchronous orbit value is highly dependent on the initial position of the satellite relative to the Moon and the Sun. Verifications of the results by means of numerical integrations are also included.     

Ephemeris of a highly eccentric orbit: Explorer 28

An ephemeris has been obtained for Explorer 28 (IMP 3) which agrees well with 2 years of radio observations and with SAO observations a year later. This ephemeris is generated over the 3 year lifetime by a numerical integration method utilizing a set of initial conditions, at launch and without requiring further differential correction. Because highly eccentric orbits are difficult to compute with acceptable accuracy and because a long continuous arc has been obtained which compares with actual data to a known precision, this ephemeris may be used as a standard for computing highly eccentric orbits in the Earth-Moon system.Orbit improvement was used to obtain the initial conditions which generated the ephemeris. This improvement was based on correcting the energy by adjusting the semimajor axis to match computed times of perigee passage with the observed. This procedure may generate errors in semimajor axis to compensate for model errors in the energy; however this compensation error is also implicit in orbit determination itself.     

On the long-period evolution of the sun-synchronous orbits

The dynamic evolution of sun-synchronous orbits at a time interval of 20 years is considered. The numerical motion simulation has been carried out using the Celestial Mechanics software package developed at the Institute of Astronomy of the University of Bern. The dependence of the dynamic evolution on the initial value of the ascending node longitude is examined for two families of sun-synchronous orbits with altitudes of 751 and 1191 km. Variations of the semimajor axis and orbit inclination are obtained depending on the initial value of the ascending node longitude. Recommendations on the selection of orbits, in which spent sun-synchronous satellites can be moved, are formulated. Minimal changes of elements over a time interval of 20 years have been observed for orbits in which at the initial time the angle between the orbit ascending node and the direction of the Sun measured along the equator have been close to 90° or 270°. In this case, the semimajor axis of the orbit is not experiencing secular perturbations arising from the satellite’s passage through the Earth’s shadow.     

Optimal two-impulse rendezvous using constrained multiple-revolution Lambert solutions

A solution to the fixed-time minimum-fuel two-impulse rendezvous problem for the general non-coplanar elliptical orbits is provided. The optimal transfer orbit is obtained using the constrained multiple-revolution Lambert solution. Constraints consist of lower bound for perigee altitude and upper bound for apogee altitude. The optimal time-free two-impulse transfer problem between two fixed endpoints implies finding the roots of an eighth order polynomial, which is done using a numerical iterative technique. The set of feasible solutions is determined by using the constraints conditions to solve for the short-path and long-path orbits semimajor axis ranges. Then, by comparing the optimal time-free solution with the feasible solutions, the optimal semimajor axis for the two fixed-endpoints transfer is identified. Based on the proposed solution procedure for the optimal two fixed-endpoints transfer, a contour of the minimum cost for different initial and final coasting parameters is obtained. Finally, a numerical optimization algorithm (e.g., evolutionary algorithm) can be used to solve this global minimization problem. A numerical example is provided to show how to apply the proposed technique.     

Sensitivity study of high eccentricity orbits for Mars gravity recovery

By linear perturbation theory, a sensitivity study is presented to calculate the contribution of the Mars gravity field to the orbital perturbations in velocity for spacecrafts in both low eccentricity Mars orbits and high eccentricity orbits(HEOs). In order to improve the solution of some low degree/order gravity coefficients, a method of choosing an appropriate semimajor axis is often used to calculate an expected orbital resonance, which will significantly amplify the magnitude of the position and velocity perturbations produced by certain gravity coefficients. We can then assess to what degree/order gravity coefficients can be recovered from the tracking data of the spacecraft. However, this existing method can only be applied to a low eccentricity orbit, and is not valid for an HEO. A new approach to choosing an appropriate semimajor axis is proposed here to analyze an orbital resonance. This approach can be applied to both low eccentricity orbits and HEOs. This small adjustment in the semimajor axis can improve the precision of gravity field coefficients and does not affect other scientific objectives.     

Bounded orbits in the elliptic restricted three-body problem

Special solutions of the planar rectilinear elliptic restricted 3-body problem are investigated for the limiting case e=1. Numerical integration is performed for primaries of equal masses. Starting values which define circular orbit solutions lead to bounded solutions if the initial radius a₀ is larger than 3.74 in units of the primaries' semimajor axis a. A comparison with the Eulerian two-fixedcentre problem is presented in order to understand qualitatively the characteristic features of bounded orbits and the transition to escape orbits.     

On the applicability of Benford law to exoplanetary and asteroid data

In this article we investigate the masses, orbital periods, semimajor axis, eccentricities and radii of the existing exoplanets by comparing the first and second digit probabilities with Benford laws’s predictions. It is found that the masses, orbital periods and semimajor axis conform to Benfordós law quite well, but radii fail. It is also investigated the first digits occurrence corresponding to a given order of magnitude. We introduce a top function which can estimate all the probabilities of the first digit order-to-order.     

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.     

A simple explanation of some characteristics of the asteroidal belt based on the restricted three-body problem

A simple explanation of some characteristics of the present distribution of number vs. semimajor axis and number vs. eccentricity of the asteroids is obtained in the framework of the restricted three-body problem, choosing suitable initial conditions. We compare the results with the numerical ones by Lecar and Franklin (1973), and we try to explain the depletion of the zone beyond 3.2 A.U. as a possible effect of a primeval collisional process.     

Orbital stability of systems of closely-spaced planets

An investigation of the stability of systems of 1 M_⊕ (Earth-mass) bodies orbiting a Sun-like star has been conducted for virtual times reaching 10 billion years. For the majority of the tests, a symplectic integrator with a fixed timestep of between 1 and 10 days was employed; however, smaller timesteps and a Bulirsch-Stoer integrator were also selectively utilized to increase confidence in the results. In most cases, the planets were started on initially coplanar, circular orbits, and the longitudinal initial positions of neighboring planets were widely separated. The ratio of the semimajor axes of consecutive planets in each system was approximately uniform (so the spacing between consecutive planets increased slowly in terms of distance from the star). The stability time for a system was taken to be the time at which the orbits of two or more planets crossed. Our results show that, for a given class of system (e.g., three 1 M_⊕ planets), orbit crossing times vary with planetary spacing approximately as a power law over a wide range of separation in semimajor axis. Chaos tests indicate that deviations from this power law persist for changed initial longitudes and also for small but non-trivial changes in orbital spacing. We find that the stability time increases more rapidly at large initial orbital separations than the power-law dependence predicted from moderate initial orbital separations. Systems of five planets are less stable than systems of three planets for a specified semimajor axis spacing. Furthermore, systems of less massive planets can be packed more closely, being about as stable as 1 M_⊕ planets when the radial separation between planets is scaled using the mutual Hill radius. Finally, systems with retrograde planets can be packed substantially more closely than prograde systems with equal numbers of planets.     

Orbital stability of systems of closely-spaced planets,II: configurations with coorbital planets

We numerically investigate the stability of systems of 1 \({{\rm M}_{\oplus}}\) planets orbiting a solar-mass star. The systems studied have either 2 or 42 planets per occupied semimajor axis, for a total of 6, 10, 126, or 210 planets, and the planets were started on coplanar, circular orbits with the semimajor axes of the innermost planets at 1 AU. For systems with two planets per occupied orbit, the longitudinal initial locations of planets on a given orbit were separated by either 60° (Trojan planets) or 180°. With 42 planets per semimajor axis, initial longitudes were uniformly spaced. The ratio of the semimajor axes of consecutive coorbital groups in each system was approximately uniform. The instability time for a system was taken to be the first time at which the orbits of two planets with different initial orbital distances crossed. Simulations spanned virtual times of up to 1 × 10⁸, 5 × 10⁵, and 2 × 10⁵ years for the 6- and 10-planet, 126-planet, and 210-planet systems, respectively. Our results show that, for a given class of system (e.g., five pairs of Trojan planets orbiting in the same direction), the relationship between orbit crossing times and planetary spacing is well fit by the functional form log(t _c /t ₀) = b β + c, where t _c is the crossing time, t ₀ = 1 year, β is the separation in initial orbital semimajor axis (in terms of the mutual Hill radii of the planets), and b and c are fitting constants. The same functional form was observed in the previous studies of single planets on nested orbits (Smith and Lissauer 2009). Pairs of Trojan planets are more stable than pairs initially separated by 180°. Systems with retrograde planets (i.e., some planets orbiting in the opposite sense from others) can be packed substantially more closely than can systems with all planets orbiting in the same sense. To have the same characteristic lifetime, systems with 2 or 42 planets per orbit typically need to have about 1.5 or 2 times the orbital separation as orbits occupied by single planets, respectively.     

Dynamical evolution of the Oort cloud: II. A theoretical approach

We investigate the dynamical evolution of a cloud of comets created by stellar perturbations. We first show the respective advantages of numerical simulations and of studies of more theoretical character. Then we investigate the probability distribution of the velocity changes imparted to comets by passing stars. This distribution is shown to be different from a Maxwellian distribution, mainly because of pronounced tails. The number of fairly large impulses is thus more important than it would be in the case of a Maxwellian distribution. Finally we estimate the probability for a comet to be ejected from the Solar System. About 10% of the cloud population is lost through this mechanism over the age of the Solar System. Taking advantage of the velocity change distribution, we study the random walk of semimajor axes of comets as a function of time. We derive the probability that a comet is lost into interstellar space as a function of its initial semimajor axis.     

On Stable Chaos in the Asteroid Belt

The twenty most chaotic objects found among first hundred of numbered asteroids are studied. Lyapunov time calculated with and without inner planets indicates that for eleven of those asteroids the strongest chaotic effect results from the resonances with Mars. The filtered semimajor axis displays an abrupt variation only when a close approach to Mars takes place. The study of the behaviour of the critical argument for candidate resonances can reveal which is responsible for the semimajor axis variation. We have determined these resonances for the asteroids in question. For the asteroids chaotic even without the inner planets we have determined the most important resonances with Jupiter, or three-body resonances. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tidal evolution of close binary asteroid systems

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than 5 times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent. This work has possible implications for the determination of the Roche limit and for spin-state alteration during close flybys.     

Analysis of the Hipparcos Measurements of HD 10697: A Mass Determination of a Brown Dwarf Secondary

HD 10697 is a nearby main-sequence star around which a planet candidate has recently been discovered by means of radial velocity measurements (Vogt et al.). The stellar orbit has a period of about 3 yr, the secondary minimum mass is 6.35 Jupiter masses (MJ), and the minimum semimajor axis is 0.36 mas. Using the Hipparcos data of HD 10697 together with the spectroscopic elements of Vogt et al., we found a semimajor axis of 2.1+/-0.7 mas, implying a mass of 38+/-13 MJ for the unseen companion. We therefore suggest that the secondary of HD 10697 is probably a brown dwarf, orbiting around its parent star at a distance of 2 AU.     

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.     

Effect of the high-order resonances on the orbital evolution of objects near geostationary orbit

The area-to-mass ratio of high-orbit space objects is estimated on the basis of positional observations from the SBG telescope at the Kourovka astronomical observatory of the Ural Federal University. The properties of regions where high-order resonances are located are analyzed. The position and sizes of the resonance zones depending on the area-to-mass ratio of objects are determined on the basis of numerical modeling. It is shown that a system transits through the high-order resonances due to secular perturbations of the semimajor axis under the Poynting-Robertson effect. The high-order resonances weakly influence the formation of the stochastic trajectories. The stochastic properties are mostly manifested in evolution of the semimajor axis of the orbit.     

On the Origin of the 1999 Leonid Storm as Deduced from Photographic Observations

Photographic multi-station observations of 18 Leonid meteorsobtained by the Spanish Photographic Meteor Network are presented. For each meteoroidthe radiant position, trajectory data and orbital parameters are discussed and compared totheoretical radiant positions and orbital elements of particles ejected from 55P/Tempel–Tuttle in 1899.We discuss the role of mean velocity imprecision in the dispersion of some orbital parameters,specially the semimajor axis. Finally, by applying the dust trail theory we have adjusted the1999 Leonidstorm orbits to a defined semimajor axis value to test the quality of photographic observations.     

Eccentric Extrasolar Planets: The Jumping Jupiter Model

Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 10⁸ y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.     

Theoretical models for wide binary evolution

Summary Wide binary evolution under the influence of stars, giant molecular clouds, and the galactic tidal field is reviewed. Recent results show that binary lifetimes ata≳0.1 pc are determined by an interplay between relatively gentle evolution due to stellar encounters and catastrophic collisions with cloud subclumps. It is emphasized that binary semimajor axis distributions may not be understood from lifetimes alone. To determine the theoretical distribution binary ‘birthrate functions’ must be convolved with probability distributions for the evolution of an ensemble of binaries from given initial separations. Simple models show no sharp breaks below the tidal cut-off imposed by the Galaxy.