Solution of the Sitnikov Problem

A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincaré perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found.     

Curvilinear coordinate transformations for relative motion

Relative motion improvements have traditionally focused on inserting additional force models into existing formulations to achieve greater fidelity, or complex expansions to admit eccentric orbits for propagation. A simpler approach may be numerically integrating the two satellite positions and then converting to a modified equidistant cylindrical frame for comparison in a Hill’s-like frame. Recent works have introduced some approaches for this transformation within the Hill’s construct, and examined the accuracy of the transformation. Still others have introduced transformations as they apply to covariance operations. Each of these has some orbital or force model limitations and defines an approximate circular reference dimension. We develop a precise transformation between the Cartesian and curvilinear frame along the actual satellite orbit and test the results for various orbital classes. The transformation has wide applicability.     

A High Order Perturbation Analysis of the Sitnikov Problem

The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincaré–Lindstedt. The enormous amount of necessary computations were performed by extensive use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering an increasing number of algebraic terms originating from high order perturbation theory.     

On the covering of a Hill’s region by solutions in the restricted three-body problem

We consider two classical celestial-mechanical systems: the planar restricted circular three-body problem and its simplification, the Hill’s problem. Numerical and analytical analyses of the covering of a Hill’s region by solutions starting with zero velocity at its boundary are presented. We show that, in all considered cases, there always exists an area inside a Hill’s region that is uncovered by the solutions.     

RAPID: A fast,high resolution,flux-conservative algorithm designed for planet–disk interactions

We describe a newly developed hydrodynamic code for studying accretion disk processes. The numerical method uses a finite volume, non-linear, Total Variation Diminishing (TVD) scheme to capture shocks and control spurious oscillations. It is second-order accurate in time and space and makes use of a FARGO-type algorithm to alleviate Courant–Friedrichs–Lewy time step restrictions imposed by the rapidly rotating inner disk region. OpenMP directives are implemented enabling faster computations on shared-memory, multi-processor machines. The resulting code is simple, fast and memory efficient. We discuss the relevant details of the numerical method and provide results of the code’s performance on standard test problems. We also include a detailed examination of the code’s performance on planetary disk–planet interactions. We show that the results produced on the standard problem setup are consistent with a wide variety of other codes.     

Electron acoustic soliton energy of the Kadomtsev-Petviashvili equation in the Earth’s magnetotail region at critical ion density

The nonlinear properties of small amplitude electron-acoustic solitary waves (EAWs) in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili (KP) equation. At the critical ion density, the KP equation is not appropriate for describing the system. Hence, a new set of stretched coordinates is considered to derive the modified KP equation. Moreover, the solitary solution, soliton energy and the associated electric field at the critical ion density were computed. The present investigation can be of relevance to the electrostatic solitary structures observed in various space plasma environments, such as Earth’s magnetotail region.     

The circular restricted three-body problem in curvilinear coordinates

We derive the exact equations of motion for the circular restricted three-body problem in cylindrical curvilinear coordinates together with a number of useful analytical relations linking curvilinear coordinates and classical orbital elements. The equations of motion can be seen as a generalization of Hill’s problem after including all neglected nonlinear terms. As an application of the method, we obtain a new expression for the averaged third-body disturbing function including eccentricity and inclination terms. We employ the latter to study the dynamics of the guiding center for the problem of circular coorbital motion providing an extension of some results in the literature.     

New tortoise coordinate transformation and Hawking’s radiation in de Sitter space

Hawking’s radiation effect of Klein-Gordon equation, Dirac particles and Maxwell’s electromagnetic fields in the non-stationary rotating de Sitter cosmological space-time is investigated by using a new method of generalized tortoise coordinate transformation. It is found that the new transformation produces constant additional terms in the expressions of the surface gravities and the Hawking’s temperatures. If the constant terms are set to zero, then the surface gravities and Hawking’s temperatures will be equal to those obtained from the old generalized tortoise coordinate transformations. This shows that the new transformations are more reasonable. The Fermionic spectrum of Dirac particles displays a new spin-rotation coupling effect.     

A Hamiltonian approach to the planar optimization of mid-course corrections

Lawden’s primer vector theory gives a set of necessary conditions that characterize the optimality of a transfer orbit, defined accordingly to the possibility of adding mid-course corrections. In this paper a novel approach is proposed where, through a polar coordinates transformation, the primer vector components decouple. Furthermore, the case when transfer, departure and arrival orbits are coplanar is analyzed using a Hamiltonian approach. This procedure leads to approximate analytic solutions for the in-plane components of the primer vector. Moreover, the solution for the circular transfer case is proven to be the Hill’s solution. The novel procedure reduces the mathematical and computational complexity of the original case study. It is shown that the primer vector is independent of the semi-major axis of the transfer orbit. The case with a fixed transfer trajectory and variable initial and final thrust impulses is studied. The acquired related optimality maps are presented and analyzed and they express the likelihood of a set of trajectories to be optimal. Furthermore, it is presented which kind of requirements have to be fulfilled by a set of departure and arrival orbits to have the same profile of primer vector.     

A Review on Co-orbital Motion in Restricted and Planetary Three-body Problems

The 1:1 mean motion resonance may be referred to as the lowest order mean motion resonance in restricted or planetary three-body problems. The five well-known libration points of the circular restricted three-body problem are five equilibriums of the 1:1 resonance. Coorbital motion may take different shapes of trajectory. In case of small orbital eccentricities and inclinations, tadpole-shape and horseshoe-shape orbits are well-known. Other 1:1 libration modes different from the elementary ones can exist at moderate or large eccentricities and inclinations. Coorbital objects are not rare in our solar system, for example the Trojans asteroids and the coorbital satellite systems of Saturn. Recently, dozens of coorbital bodies have been identified among the near-Earth asteroids. These coorbital asteroids are believed to transit recurrently between different 1:1 libration modes mainly due to orbital precessions, planetary perturbations, and other possible effects. The Hamiltonian system and the Hill’s three-body problem are two effective approaches to study coorbital motions. To apply the perturbation theory to the Hamiltonian system, standard procedures involve the development of the disturbing function, averaging and normalization, theory of ideal resonance model, secular perturbation theory, etc. Global dynamics of coorbital motion can be revealed by the Hamiltonian approach with a suitable expansion. The Hill’s problem is particularly suitable for the studies on the relative motion of two coorbital bodies during their close encounter. The Hill’s equation derived from the circular restricted three-body problem is well known. However, the general Hill’s problem whose equation of motion takes exactly the same form applies to the non-restricted case where the mass of each body is non-negligible, namely the planetary case. The Hill’s problem can be transformed into a “canonical shape” so that the averaging principle can be applied to construct a secular perturbation theory. Besides the two analytical theories, numerical methods may be consulted, for example the approach of periodic orbit, the surface of section, and the computation of invariant manifolds carried by equilibriums or periodic orbits.     

The planar Hill problem with oblate primary

The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'. This revised version was published online in July 2006 with corrections to the Cover Date.     

The restricted three-body problem with logarithm potential

We present a study of the restricted three body problem with logarithm potential. We discuss equilibria, stability, Hill’s regions of motion and the families of periodic orbits near equilibria. Moreover, we show that equilibria and some periodic orbits continue in the logarithm three body problem.     

Numerical simulations of type III planetary migration – I. Disc model and convergence tests

We investigate the fast (type III) migration regime of high-mass protoplanets orbiting in protoplanetary discs. This type of migration is dominated by corotational torques. We study the details of flow structure in the planet's vicinity, the dependence of migration rate on the adopted disc model and the numerical convergence of models (independence of certain numerical parameters such as gravitational softening).
We use two-dimensional hydrodynamical simulations with adaptive mesh refinement, based on the flash code with improved time-stepping scheme. We perform global disc simulations with sufficient resolution close to the planet, which is allowed to freely move throughout the grid. We employ a new type of equation of state in which the gas temperature depends on both the distance to the star and planet, and a simplified correction for self-gravity of the circumplanetary gas.
We find that the migration rate in the type III migration regime depends strongly on the gas dynamics inside the Hill sphere (Roche lobe of the planet) which, in turn, is sensitive to the aspect ratio of the circumplanetary disc. Furthermore, corrections due to the gas self-gravity are necessary to reduce numerical artefacts that act against rapid planet migration. Reliable numerical studies of type III migration thus require consideration of both the thermal and the self-gravity corrections, as well as a sufficient spatial resolution and the calculation of disc–planet attraction both inside and outside the Hill sphere. With this proviso, we find type III migration to be a robust mode of migration, astrophysically promising because of a speed much faster than in the previously studied modes of migration.     

Quasi-satellite orbits in the general context of dynamics in the 1:1 mean motion resonance: perturbative treatment

Our investigation is motivated by the recent discovery of asteroids orbiting the Sun and simultaneously staying near one of the Solar System planets for a long time. This regime of motion is usually called the quasi-satellite regime, since even at the times of the closest approaches the distance between the asteroid and the planet is significantly larger than the region of space (the Hill’s sphere) in which the planet can hold its satellites. We explore the properties of the quasi-satellite regimes in the context of the spatial restricted circular three-body problem “Sun–planet–asteroid”. Via double numerical averaging, we construct evolutionary equations which describe the long-term behaviour of the orbital elements of an asteroid. Special attention is paid to possible transitions between the motion in a quasi-satellite orbit and the one in another type of orbits available in the 1:1 resonance. A rough classification of the corresponding evolutionary paths is given for an asteroid’s motion with a sufficiently small eccentricity and inclination.     

Stability of periodic solutions for Hill’s averaged problem with allowance for planetary oblateness

We analyze the stability of periodic solutions for Hill’s double-averaged problem by taking into account a central planet’s oblateness. They are generated by steady-state solutions that are stable in the linear approximation. By numerically calculating the monodromy matrix of variational equations, we plot its trace against the integral of the problem—an averaged perturbing function, for two model systems, [(Sun + Moon)-Earth-satellite] and (Sun-Uranus-satellite). We roughly estimate the ranges of values for the parameters of satellite orbits corresponding to periodic solutions of the evolutionary system that are stable in the linear approximation.     

Hierarchy of periodic solutions families of spatial Hill’s problem

Many modern space projects require the knowledge of orbits with certain properties. Most of these projects assume the motion of a space vehicle in the neighborhood of a celestial body, which in turn moves in the field of the Sun or another massive celestial body. A good approximation of this situation is Hill’s problem. This paper is devoted to the investigation of the families of spatial periodic solutions to the three-dimensional Hill’s problem. This problem is nonintegrable; therefore, periodic solutions are studied numerically. The Poincare theory of periodic solutions of the second kind is applied; either planar or vertical impact orbits are used as generating solutions.     

Small body surface gravity fields via spherical harmonic expansions

Conventional gravity field expressions are derived from Laplace’s equation, the result being the spherical harmonic gravity field. This gravity field is said to be the exterior spherical harmonic gravity field, as its convergence region is outside the Brillouin (i.e., circumscribing) sphere of the body. In contrast, there exists its counterpart called the interior spherical harmonic gravity field for which the convergence region lies within the interior Brillouin sphere that is not the same as the exterior Brillouin sphere. Thus, the exterior spherical harmonic gravity field cannot model the gravitation within the exterior Brillouin sphere except in some special cases, and the interior spherical harmonic gravity field cannot model the gravitation outside the interior Brillouin sphere. In this paper, we will discuss two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson’s equation. These two gravity fields are obtained by assuming the form of Helmholtz’s equation to Poisson’s equation. This method renders the gravitational potentials as functions of spherical Bessel functions and spherical harmonic coefficients. We refer to these gravity fields as the interior/exterior spherical Bessel gravity fields and study their characteristics. The interior spherical Bessel gravity field is investigated in detail for proximity operation purposes around small primitive bodies. Particularly, we apply the theory to asteroids Bennu (formerly 1999 RQ36) and Castalia to quantify its performance around both nearly spheroidal and contact-binary asteroids, respectively. Furthermore, comparisons between the exterior gravity field, interior gravity field, interior spherical Bessel gravity field, and polyhedral gravity field are made and recommendations are given in order to aid planning of proximity operations for future small body missions.     

La transformation de Lyapunov de l'equation de Hill et son interpretation dynamique

Résumé La transformation de Lyapunov transforme une équation de Hill en une autre qui occupe la même place dans la classification de Yakubovich.Soit (C) une solution périodique d'un système conservatif à deux degrés de liberté. D'après le principe de moindre action de Maupertuis (C) est l'image d'une géodésique ().Nous montrons que les équations aux variations au voisinage de (C) et de () sont réductibles à deux équations de Hill qui se correspondent par une transformation de Lyapunov.

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The Lyapunov transformation of Hill's equation and his dynamic interpretation

The Lyapunov transformation carries Hill's equationÿ+F(t)y=0,F(t+T)=F(t) into another one which belongs to the same class in Yakubovich's classification.Let (C) be a closed trajectory of a Lagrangian conservative system with two degrees of freedom. By the Principle of Least action, we know that (C) is the image of a geodesic () of a certain two-dimensional surface ().We show that the two Hill equations associated with (C) and () are related by a certain Lyapunov transformation.

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Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.