Stability of binary asteroids

The restricted problem of 2+2 bodies is applied to the study of the stability and dynamics of binary asteroids in the solar system. Numerical investigation of the behavior of the orbital elements and the maximal Lyapunov characteristic number of binary asteroids reveal extensive regions where bounded quasiperiodic motion is possible. These regions are compared to the bounded regions which are predicted by the classical restricted problem of three bodies. Regions of bounded chaotic solutions are also found.     

Integrable approximation of J 2-perturbed relative orbits

Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J ₂ perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J ₂ Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J ₂ problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.     

Bounded relative motion under zonal harmonics perturbations

The problem of finding natural bounded relative trajectories between the different units of a distributed space system is of great interest to the astrodynamics community. This is because most popular initialization methods still fail to establish long-term bounded relative motion when gravitational perturbations are involved. Recent numerical searches based on dynamical systems theory and ergodic maps have demonstrated that bounded relative trajectories not only exist but may extend up to hundreds of kilometers, i.e., well beyond the reach of currently available techniques. To remedy this, we introduce a novel approach that relies on neither linearized equations nor mean-to-osculating orbit element mappings. The proposed algorithm applies to rotationally symmetric bodies and is based on a numerical method for computing quasi-periodic invariant tori via stroboscopic maps, including extra constraints to fix the average of the nodal period and RAAN drift between two consecutive equatorial plane crossings of the quasi-periodic solutions. In this way, bounded relative trajectories of arbitrary size can be found with great accuracy as long as these are allowed by the natural dynamics and the physical constraints of the system (e.g., the surface of the gravitational attractor). This holds under any number of zonal harmonics perturbations and for arbitrary time intervals as demonstrated by numerical simulations about an Earth-like planet and the highly oblate primary of the binary asteroid (66391) 1999 KW4.     

Green's functions of the induction equation on regions with boundary. I. General properties and a construction principle

The evolution of a magnetic field is considered which pervades an electrically conducting fluid and its non-conducting surroundings under the influence of electromotive forces due to internal motion and other causes. The governing equations -- among which the induction equation of magnetohydrodynamics is the most prominent -- pose an initial value problem for the magnetic flux density. Properties of this initial value problem and of the solving Green's function are reviewed and a general construction principle for the Green's function is given. Detailed treatment of cases where the fluid occupies a sphere or an evenly bounded half-space are presented in two subsequent papers     

Planar periodic orbits in exterior resonances with Neptune

In the framework of the planar restricted three-body problem we study a considerable number of resonances associated to the basic dynamical features of Kuiper belt and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is an indication of the existence of asymmetric ones only in a few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies, the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem, the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals.     

A new analytic approach to the Sitnikov problem

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP     

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J ₂ is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J ₂-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J _2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.     

Linearization in special cases of perturbed Keplerian motions

This paper treats linearization of problems of motion in three-dimensions in not-central force fields, when the problems are reducible to the one-dimensional case by using integrals of the motion. Linearizing transformations of the independent variable are found to solve such problems if the motion is bounded, and explicit forms of the regularizing functions, corresponding to more common potentials, are given. An application is presented to the integration of a radial intermediary orbit that arises in the analytical study of the theory of artificial satellites.     

Analytical solutions for J 2-perturbed unbounded equatorial orbits

While solutions for bounded orbits about oblate spheroidal planets have been presented before, similar solutions for unbounded motion are scarce. This paper develops solutions for unbounded motion in the equatorial plane of an oblate spheroidal planet, while taking into account only the J ₂ harmonic in the gravitational potential. Two cases are distinguished: A pseudo-parabolic motion, obtained for zero total specific energy, and a pseudo-hyperbolic motion, characterized by positive total specific energy. The solutions to the equations of motion are expressed using elliptic integrals. The pseudo-parabolic motion unveils a new orbit, termed herein the fish orbit, which has not been observed thus far in the perturbed two-body problem. The pseudo-hyperbolic solutions show that significant differences exist between the Keplerian flyby and the flyby performed under the the J ₂ zonal harmonic. Numerical simulations are used to quantify these differences.     

Sitnikov restricted four-body problem with radiation pressure

An analytical study of the elliptic Sitnikov restricted four-body problem when all the primaries as source of same radiation pressure is presented. We find a solution, which is valid for small bounded oscillations in case of moderate eccentricity of the primary. We have linearized the equation of motion to obtain the Hill’s type equation. Using the Courant and Snyder transformation, Hill’s equation transformed into harmonic oscillator type equation. We have used the Lindstedt-Poincare perturbation method and again we have applied the Courant and Snyder transformation to obtain the final result.     

Planar heliocentric roto-translatory motion of a spacecraft with a solar sail of complex shape

A complete treatment of the general motion of rotation and translation of a solar-sail spacecraft is proposed for the non-flat sail of complex shape. The planar heliocentric roto-translatory motion is considered, orbit-rotational coupling in the problem of altitude and orbital sail motion is investigated for the two-folding sail formed by two unequal reflective rectangular plates oriented at a right angle. The problem of orbit-rotational coupling is essentially a planar one: both sail plates are orthogonal to the orbital plane. The possibility of the non-controlled interplanetary transfer with such two-folding sail at its passive radiational orientation is established analytically from point of view of orbit-rotational coupling. Optimal geometric proportions of this sail are found at minimum-time interplanetary transfers.     

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.     

Chaotic motion of comets in near-parabolic orbit: Mapping approaches

There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assumes that there is a cometary cloud at a distance 10³ – 10⁵ AU from the Sun and that perturbing forces from planets or stars make orbits of some of these comets become of near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they stay in the Solar System forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighbourhood of any separatrix i.e. the trajectory with zero frequency of the unperturbed motion of an Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study on near-parabolic motion is very important. Petrosky's elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion (Petrosky, 1988). In this paper we derive a similar mapping directly and discuss its dynamical characters.     

On the Evolution of Balloon Satellite Motions in a Plane Restricted Three-Body Problem with Light Pressure

We investigate the evolution of high Earth satellite orbits under gravitational perturbations from the Sun and light pressure forces, without the Earth shadow effect. We present the disturbing function of the problem provided that the satellite is a sphere. The mean value of the disturbing function in the absence of resonances between the mean unperturbed motion of the satellite and the mean motion of the Sun has also been obtained. The semimajor axis of the satellite orbit and the mean value of the disturbing function are shown to be integrals of the averaged osculating equations. TheHill version of the problem, whereby the distance to the satellite is much smaller than the Earth–Sun distance, has been studied in detail: we have constructed the phase portraits of the oscillations at various parameters and described three types of quasiperiodic satellite trajectories—librational and rotational trajectories as well as Earth collision trajectories. Numerical simulations of trajectories have allowed the additional effects caused by light pressure to be described: the displacement of the bounded trajectory of the satellite as a whole relative to the trajectory of the classical three-body problem into a region more distant from the Sun.     

Diffuse reflection and transmission of solar radiation by a finite atmosphere bounded by a reflecting surface

The problem of diffuse reflection and transmission of solar radiation through a planetary atmosphere bounded from below by a reflecting surface is solved. The solution method based on rewriting the solution of the proposed problem in terms of the well known standard problem solution, where the planetary surface does not reflect. The solution of the standard problem can be found elsewhere or as we did by using the maximum entropy method. Numerical results for the angular radiation intensity and for the reflection and transmission coefficients are presented and compared with those obtained by Chandrasekhar's method.     

Equivalent problems in rigid body dynamics — I

The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order.     

Out-of-plane motion about libration points: Nonlinearity and eccentricity effects

Out-of-plane motion about libration points is studied within the framework of the elliptic restricted three-body problem. Nonlinear motion in the circular restricted problem is given to third order in the out-of-plane amplitudeA _z by Jacobi elliptic functions. Linear motion in the elliptic problem is studied using Mathieu's and Hill's equations. Additional terms needed for a complete third-order theory are found using Lindsted's method. This theory is constructed for the case of collinear libration points; for the case of triangular points, a third-order nonlinear solution is given separately in terms of Jacobi elliptic functions.