Analytical study of the powered Swing-By maneuver for elliptical systems and analysis of its efficiency

Analytical equations describing the velocity and energy variation of a spacecraft in a Powered Swing-By maneuver in an elliptic system are presented. The spacecraft motion is limited to the orbital plane of the primaries. In addition to gravity, the spacecraft suffers the effect of an impulsive maneuver applied when it passes by the periapsis of its orbit around the secondary body of the system. This impulsive maneuver is defined by its magnitude \(\delta V\) and the angle that defines the direction of the impulse with respect to the velocity of the spacecraft (\(\alpha\)). The maneuver occurs in a system of main bodies that are in elliptical orbits, where the velocity of the secondary body varies according to its position in the orbit following the rules of an elliptical orbit. The equations are dependent on this velocity. The study is done using the “patched-conics approximation”, which is a method of simplifying the calculations of the trajectory of a spacecraft traveling around more than one celestial body. Solutions for the velocity and energy variations as a function of the parameters that define the maneuver are presented. An analysis of the efficiency of the powered Swing-By maneuver is also made, comparing it with the pure gravity Swing-by maneuver with the addition of an impulse applied outside the sphere of influence of the secondary body. After a general study, the techniques developed here are applied to the systems Sun-Mercury and Sun-Mars, which are real and important systems with large eccentricity. This problem is highly nonlinear and the dynamics very complex, but very reach in applications.     

The Hill stability of inclined small mass binary systems in three-body systems with special application to triple star systems, extrasolar planetary systems and Binary Kuiper Belt systems

The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable.     

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star–planet–moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.     

The Hill stability of inclined bound triple star and planetary systems

The dynamical stability of a triple system composed of a binary or planetary system and a bound third body moving on a orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of planetary systems against disruption, component exchange or capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects. These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the third body initially increases the stability of the system then decreases stability as the eccentricity reaches higher values.The Hill stability criterion is applied to extrasolar planetary systems to determine the critical distances at which planets of the same mass as the observed extrasolar planet moving on a circular orbit could remain on a stable orbit. It was found that these distances were sufficiently short suggesting that the presence of further as yet unobserved stable extrasolar planets in observed systems was very likely.     

A “Spring–mass” model of tethered satellite systems: properties of planar periodic motions

This paper is devoted to the dynamics in a central gravity field of two point masses connected by a massless tether (the so called “spring–mass” model of tethered satellite systems). Only the motions with straight strained tether are studied, while the case of “slack” tether is not considered. It is assumed that the distance between the point masses is substantially smaller than the distance between the system’s center of mass and the field center. This assumption allows us to treat the motion of the center of mass as an unperturbed Keplerian one, so to focus our study on attitude dynamics. A particular attention is given to the family of planar periodic motions in which the center of mass moves on an elliptic orbit, and the point masses never leave the orbital plane. If the eccentricity tends to zero, the corresponding family admits as a limit case the relative equilibrium in which the tether is elongated along the line joining the center of mass with the field center. We study the bifurcations and the stability of these planar periodic motions with respect to in-plane and out-of-plane perturbations. Our results show that the stable motions take place if the eccentricity of the orbit is sufficiently small.     

The Hill stability of a binary or planetary system during encounters with a third inclined body

The dynamical interaction of a binary or planetary system and a third body moving on a parabolic orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in binary star disruption and exchange, in extrasolar planetary system disruption, exchange and capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. This makes exchange or disruption of the component masses more likely as does increasing the eccentricity of the binary.
The stability criteria are applied to determine possible disruption and capture distances for currently known extrasolar planetary systems.     

A minimal dynamical model for tidal synchronization and orbit circularization

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.     

Dust torus produced by particles ejected in the apsidal points

Dust complexes in the Solar System are produced and maintained by different physical processes. However, one often requires equations of the same type to be studied and solved in order to mathematically describe these processes. We have analytically found earlier a shape of the complex formed when particles are ejected in a single-ejection event from a parent body orbiting a central body in a circular orbit. In the present paper, we consider a parent body moving in an arbitrary elliptical orbit and ejecting particles at the pericenter or apocenter. For illustration purposes, the theoretical results are applied to the Geminid meteoroid stream. The comparison with the results obtained by other authors shows good agreement.     

Reconstruction of A Non-Stationary Potential of Gravitating System on Given Evolving Orbits

We study the problem of the reconstruction of a non-stationary space symmetrical regular planar potential of the gravitating system on a family of evolving types of orbits being used in the dynamics of stationary stellar systems. An application of such an inverse problem to the dynamical evolution of stellar systems with variable masses is given. The general form of the evolving orbit which we use when writing out the differential equations for non-stationary potential may also be interpreted as an osculating orbit of the perturbed Keplerian motion. In this case we are making an additional transformation of the basic equation of the problem and demonstrating an appropriate example of the construction of a non-stationary potential of a gravitating system. In connection with the stellar dynamical character of our inverse problem, we also give a generalized form of its basic equation in a rotating coordinate system.     

The 1/1 resonance in extrasolar planetary systems

We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits.     

Stability of generalized elliptic restricted four body problem with radiation and oblateness effects

This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter.     

Secular orbit variation due to solar radiation effects: a detailed model for BYORP

A detailed derivation of the effect of solar radiation pressure on the orbit of a body about a primary orbiting the Sun is given. The result is a set of secular equations that can be used for long-term predictions of changes in the orbit. Solar radiation pressure is modeled as a Fourier series in the body’s rotation state, where the coefficients are based on the shape and radiation properties of the body as parameters. In this work, the assumption is made that the body is in a synchronous orbit about the primary and rotates at a constant rate. This model is used to write explicit variational equations of the energy, eccentricity vector, and angular momentum vector for an orbiting body. Given that the effect of the solar radiation pressure and the orbit are periodic functions, they are readily averaged over an orbit. Furthermore, the equations can be averaged again over the orbit of the primary about the Sun to give secular equations for long-term prediction. This methodology is applied to both circular and elliptical orbits, and the full equations for secular changes to the orbit in both cases are presented. These results can be applied to natural systems, such as the binary asteroid system 1999 KW4, to predict their evolution due to the Binary YORP effect, or to artificial Earth orbiting, nadir-pointing satellites to enable more precise models for their orbital evolution.     

Exploiting Unstable Periodic Orbits of a Chaotic Invariant Set for Spacecraft Control

The purpose of this work is to show that chaos control techniques (OGY, in special) can be used to efficiently keep a spacecraft around another body performing elaborate orbits. We consider a satellite and a spacecraft moving initially in coplanar and circular orbits, with slightly different radii, around a heavy central planet. The spacecraft, which is the inner body, has a slightly larger angular velocity than the satellite so that, after some time, they eventually go to a situation in which the distance between them becomes sufficiently small, so that they start to interact with one another. This situation is called as an encounter. In previous work we have shown that this scenario is a typical situation of a chaotic scattering for some well-defined range of parameters. Considering this scenario, we first show how it is possible to find the unstable periodic orbits that are located in the chaotic invariant set. From the set of unstable periodic orbits, we select the ones that can be combined to provide the desired elaborate orbit. Then, chaos control technique based on the OGY method is used to keep the spacecraft in the desired orbit. Finally, we analyze the results and make considerations regarding a realistic scenario of space exploration.     

Exchange orbits: an interesting case of co-orbital motion

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.     

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.     

第三体摄动分析解的一种表达式

在太阳系中,大行星、小行星和卫星（包括自然卫星和人造卫星）等对应的运动问题,都可以处理成受摄二体问题,而摄动源又多为第三体,作为第三体的摄动天体,有的比运动天体离中心天体近,有的则相反,前者称为内摄内体,全者则称为外摄天体,对一个具体的运动天体,可以同时出现这两个摄动天体,但是,只要运动天体与摄动天体的轨道都建立在以中心天体（质心）为坐标原点的同一坐标系内,那么在一定条件下（即除运动天体与摄动天体     

The Parabolic Three-Body Problem

In this communication we present an analytical model for the restricted three-body problem, in the case where the perturber is in a parabolic orbit with respect to the central mass. The equations of motion are derived explicitly using the so-called Global Expansion of the disturbing function, and are valid for any eccentricity of the massless body, as well as in the case where both secondary masses have crossing orbits. Integrating the equations of motion over the complete passage of the perturber through the system, we are then able to construct a first-order algebraic mapping for the change in semimajor axis, eccentricity and inclination of the perturbed body.Comparisons with numerical solutions of the exact equations show that the map yields precise results, as long as the minimum distance between both bodies is not too small. Finally, we discuss several possible applications of this model, including the evolution of asteroidal satellites due to background bodies, and simulations of passing stars on extra-solar planets.     

A note on the tidally induced potential of a satellite in eccentric orbit

A planetary body moving on an eccentric orbit around the primary is subject to a periodic perturbing potential, affecting its internal mass distribution. In a previous paper (Rappaport et al., 1997, Icarus 126, 313), we have calculated the periodic modulation of the gravity coefficients of degree 2, for a body on a synchronous orbit. Here, the previous analysis is extended by considering also non-synchronous orbits, and by properly accounting for the apparent motion of the primary due to the non uniform motion along the elliptical orbit. The cases of Titan and Mercury are briefly discussed.     

On the attitude dynamics of perturbed triaxial rigid bodies

Attitude dynamics of perturbed triaxial rigid bodies is a rather involved problem, due to the presence of elliptic functions even in the Euler equations for the free rotation of a triaxial rigid body. With the solution of the Euler–Poinsot problem, that will be taken as the unperturbed part, we expand the perturbation in Fourier series, which coefficients are rational functions of the Jacobian nome. These series converge very fast, and thus, with only few terms a good approximation is obtained. Once the expansion is performed, it is possible to apply to it a Lie-transformation. An application to a tri-axial rigid body moving in a Keplerian orbit is made.