An efficient computational algorithm for the initial value problem associated with the universal Y functions

In the present paper, an efficient iterative method of arbitrary integer order of convergent ≥2 based on the homotopy continuation techniques for the solution of the initial value problem of space dynamics using the universal Y functions is presented. The method is of dynamic nature in the sense that, ongoing from one iterative scheme to the subsequent one, only additional instruction is needed. Most importantly, the method does not need any prior knowledge of the initial guess. This is a property which avoids the critical situations between divergent to very slow convergent solutions that may exist in other numerical methods which depend on initial guesses. A computational package for digital implementation of the method is given, together with numerical applications for elliptic, hyperbolic, and parabolic orbits. The accuracy of the results for all orbits is O(10^–16). (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Initial conditions and dynamics of triple systems

The dynamical evolution of triple systems with equal-mass components and zero initial velocities is studied. We consider two regions of initial conditions: a regionD of all possible configurations of triples and a circleR. The configurations are distributed uniformly within these regions. The calculations have been carried out until a time when escape or conditional escape (i.e. distant ejection) of one component takes place. The accuracy has been checked by doing time-reversed integration. Types of predictable and non-predictable systems are revealed. Averages for a number of evolution parameters are presented: the life-time, minimum perimeter during the last triple approach resulting in escape, semi-major axis and eccentricity of the final binary, and the smallest separation between the components during the evolution. It is shown that the statistical results for the regionsD andR do not differ significantly for the most part. Our results, which have been obtaned by a three-body regularization method, are in good agreement with previous work based on the RK4 integrator and Sundman's time smoothing.     

On the two-dimensional inverse problem of dynamics

The authors extend the deduction of the equations satisfied by the force fields from inertial to rotating frames, when the curves of a certain family are known to be solutions for the equations of motion. Then Drǎmbâ's equation is obtained as a consequence of this result. The works of Hadamard and Moiseev are proved to be closely related to the inverse problem of dynamics.     

Self-similar problem with increasing energy in radiative magneto-gas dynamics

In the present paper self-similar solutions have been investigated for the propagation of piston driven, radiative gas-dynamic shocks into an inhomogeneous ideal gas permeated by a current free azimuthal magnetic field for spherical symmetry. The effects of radiation flux and magnetic field together have been seen in the region of interest on the other flow variables. The total energy of the flow between the piston and the shock is taken to be dependent on the shock radius obeying a power law. The radiative pressure and energy have been neglected. This problem is more general than the others done so far. The word piston implies some means to drive plasma radially onwards.     

The extended phase space formulation of the Vinti problem

The Vinti problem, motion about an oblate spheroid, is formulated using the extended phase space method. The new independent variable, similar to the true anomaly, decouples the radius and latitude equations into two perturbed harmonic oscillators whose solutions toO(J ₂ ⁴ ) are obtained using Lindstedt's method. From these solutions and the solution to the Hamilton-Jacobi equation suitable angle variables, their canonical conjugates and the new Hamiltonian are obtained. The new Hamiltonian, accurate toO(J ₂ ⁴ ) is function of only the momenta.     

Resonant dynamics of Medium Earth Orbits: space debris issues

The Medium Earth Orbit region is the home of the navigation constellations. It is shown how the orbits of these constellations of satellites are strongly affected by the so-called inclination dependent luni-solar resonances. The analytical theory of these resonances is recalled and a large set of numerical integrations is used to investigate the stability of the orbits of the constellations over very long time spans. The stability issue is important in the definition of possible disposal strategies for the constellation spacecraft, after their end-of-life. Two possible disposal strategies are envisaged involving either stable or unstable orbits (from the eccentricity growth point of view). In particular it is shown how, to have disposal orbits with moderately short lifetime (of the orders of 40–50 years), a very large disposal maneuver would be required to rise the inital eccentricity to values above ~ 0.3.     

A problem of orbital dynamics,which is separable in KS-variables

The motion of a rocket (with constant thrust) in the gravitational field of a mass point is studied. As is well known, the two dimensional version of this problem is separable in Levi-Civitavariables. It is shown here that, for the three dimensional case, the KS-variables produce the separability. A closed solution is found by using elliptic functions. Equivalent problems are mentioned.     

The structure of the extended phase space of the Sitnikov problem

The extended phase space of the Sitnikov problem is studied by using a stroboscopic map and computing escape times. Comparisons of phase portraits and plots of escape times reveal the intrinsic connection between the geometry of the phase space and the dynamical behaviour of the system. Properties of the phase space are analysed both in the central regular region and far from it. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quasi-similar solution of the strong shock wave problem in non-ideal gas dynamics

The quasisimilar theory is used to investigate the solution of the blast wave problem with generalized geometries in a non-ideal gas satisfying the equation of state of the Van der Waals type. Here it is assumed that the distribution of normalized velocity, pressure and density are nearly similar in the narrow range of the shock strength. A comparison between approximate analytical solution and numerical solution of the problem is presented for the cylindrical geometry. The numerical solutions are presented for the generalized geometry in a non-ideal gas. It is also assessed as to how the non-idealness of the gas affects the behavior of the flow parameters.     

Navigation formula to A. Busemann's variation problem of space travel

The indicatrix of the variation problem is given (1) by the perturbation differential equation for the semi-major axis and eccentricity, and (2) by a relation between the momentum and eccentric anomaly. The Hamiltonian equations of an extremal solution, therefore, reduce to a navigation equation. The remaining perturbation equations for the longitude of perihelion and the mean longitude at epoch, finally, yield the time dependence that gives the most economic fuel consumption.     

Applying Lambert problem to association of radar-measured orbit tracks of space objects

Thousands of orbit tracks of space objects are collected by a radar each day,and many may be from uncatalogued objects.As such,it is an urgent demand to catalog...     

Solution of an inverse problem of the dynamics of a particle

The three-dimensional inverse problem of particle dynamics is studied here. The potentialU and the corresponding energyh are determined by the given family of possible trajectories. The classification of the solutions due to the geometry of the given family is obtained.     

Rigid body dynamics in the restricted ring problem of N + 1 bodies

The dynamic behavior of a small tri-axial body acted upon by the Newtonian forces of N major bodies of spherical symmetry which forma planar ring configuration is studied in this paper. The equations of the translational-rotational motion of the minor body are derived and its equilibrium states as well as their stability are investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Separable and partially separable systems in the light of the inverse problem of dynamics

Using a generalization of Joukovsky's formula, we determine three-dimensional families of curves that are orbits only in separable potentials and we note the importance of iso-energetic families of orbits. We also obtain more general families that are orbits of partially separable systems and we examine from this point of view the classical curvilinear coordinate systems.     

Long-term attitude dynamics of space debris in Sun-synchronous orbits: Cassini cycles and chaotic stabilization

Comprehensive analysis of space debris rotational dynamics is vital for active debris removal missions that require physical capture or detumbling of a target. We study the attitude motion of large space debris objects that admittedly pose an immediate danger to space operations in low Earth orbits. Particularly, we focus on Sun-synchronous orbits (SSO) with altitude range 600–800 km, where the density of space debris is maximal. Our mathematical model takes into account the gravity-gradient torque and the torque due to eddy currents induced by the interaction of conductive materials with the geomagnetic field. Using perturbation techniques and numerical methods, we examine the deceleration of the initial fast rotation and the subsequent transition to a relative equilibrium with respect to the local vertical. A better understanding of the latter phase is achieved owing to a more accurate model of the eddy-current torque than in most prior research. We show that SSO precession is also a crucial factor influencing the motion properties. One of its effects is manifested at the deceleration stage as oscillations of the angular momentum vector about the direction to the south celestial pole.     

Orbital dynamics of three-dimensional bars – II. Investigation of the parameter space

We investigate the orbital structure in a class of three-dimensional (3D) models of barred galaxies. We consider different values of the pattern speed, of the strength of the bar and of the parameters of the central bulge of the galactic model. The morphology of the stable orbits in the bar region is associated with the degree of folding of the x1 characteristic. This folding is larger for lower values of the pattern speed. The elongation of rectangular-like orbits belonging to x1 and to x1-originated families depends mainly on the pattern speed. A detailed investigation of the trees of bifurcating families in the various models shows that major building blocks of 3D bars can be supplied by families initially introduced as unstable in the system, but becoming stable at another energy interval. In some models without radial and vertical 2:1 resonances we find, except for the x1 and x1-originated families, also families related to the z -axis orbits, which support the bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis of the main bar. This is favoured in the slowly rotating bar case.