Consecutive collision orbits in the limiting case μ=0 of the elliptic restricted problem

A timing condition for consecutive collision orbits in the planar, circular three-body problem has been extended to the elliptic restricted problem for =0. The expression developed relates eccentric anomalies at the time of collision. Some families of solutions are presented.     

Time-varying transformations for Hill–Clohessy–Wiltshire solutions in elliptic orbits

The relative motion of chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be approximated by linearized time-periodic equations of motion. The linear time-invariant Hill–Clohessy–Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Two Lyapunov–Floquet transformations and an integral-preserving transformation are here presented which relate the linearized time-varying equations of relative motion to the Hill–Clohessy–Wiltshire equations in a one-to-one manner through time-varying coordinate transformations. These transformations allow the Hill–Clohessy–Wiltshire equations to describe the linearized relative motion for elliptic chief satellites.     

A model of neutron-star–white-dwarf collision for fast radio bursts

Fast radio bursts (FRBs) with unknown origin emit a huge luminosity (about 1 Jy at 1 GHz) with a duration of milliseconds or less at extragalactic distances estimated from their large dispersion measure (DM). We propose herein a scenario for a collision between a neutron star (NS) and a white dwarf (WD) as the progenitor of the FRBs by considering the burst duration scaling to the collision time and the radio luminosity proportional to the kinetic energy of the collision. The relations among the observed flux density, pulse width, and the DM are derived from the model and compared with the statistical results from the observed FRBs. Although the sample is quite small, we tentatively report a nearly inverse-square correlation between the observed peak flux density and the DM excess, which is an consequence of the assumption that the DM excess (i.e. that not due to our Galaxy) is dominated by the intergalactic medium. We also tentatively note a correlation among the duration of the FRB and the DM excess (possibly interpreted as due to the broadening of the signal in the intergalactic medium) and a correlation among the duration of the FRB and the flux density (shorter burst should be brighter), both roughly in agreement with the proposed model.     

Approximate analytic method for high-apogee twelve-hour orbits of artificial Earth’s satellites

We propose an approach to the study of the evolution of high-apogee twelve-hour orbits of artificial Earth’s satellites. We describe parameters of the motion model used for the artificial Earth’s satellite such that the principal gravitational perturbations of the Moon and Sun, nonsphericity of the Earth, and perturbations from the light pressure force are approximately taken into account. To solve the system of averaged equations describing the evolution of the orbit parameters of an artificial satellite, we use both numeric and analytic methods. To select initial parameters of the twelve-hour orbit, we assume that the path of the satellite along the surface of the Earth is stable. Results obtained by the analytic method and by the numerical integration of the evolving system are compared. For intervals of several years, we obtain estimates of oscillation periods and amplitudes for orbital elements. To verify the results and estimate the precision of the method, we use the numerical integration of rigorous (not averaged) equations of motion of the artificial satellite: they take into account forces acting on the satellite substantially more completely and precisely. The described method can be applied not only to the investigation of orbit evolutions of artificial satellites of the Earth; it can be applied to the investigation of the orbit evolution for other planets of the Solar system provided that the corresponding research problem will arise in the future and the considered special class of resonance orbits of satellites will be used for that purpose.     

Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun–Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.     

Symmetric periodic orbits in Størmer's problem

The hypothesis on the -ray burst generation in the process of the collapse of surpermassive bodies in the nuclei of active galaxies is considered. It is shown that -ray burst properties observed may be interpreted within the frames of the given model. A statistical test for choosing a hypotheses on -ray burst nature is discussed.     

The influence of the correlations between an asteroid’s orbital parameters on the estimation of the probability of planetary collision by the Monte Carlo method

The probability of an asteroid colliding with a planet can be estimated by the Monte Carlo method, in particular, through the statistical simulation of the possible initial conditions for the motion of an asteroid based on the probability density distribution set by the respective covariance matrix to be further projected with the orbital model onto the supposed time point of the collision. Hence, the collision probability is calculated as the ratio between the number of projected (virtual) asteroids striking the planet and their total number. The main problem is that different elements of the initial conditions (orbit or state vector) are correlated and, therefore, cannot be simulated independently. These correlations are reflected in the nondiagonal covariance matrix of the solution. The matrix is diagonalized by an orthogonal transformation. In the uncertainty domain constructed from the diagonal matrix elements, the initial values for each of the six orbital elements are simulated independently from the other elements, but with the accounting for their normal distribution. The program for calculating the normal distribution is based on the central limit theorem. Each sample of the initial values for the six orbital elements is transferred to the initial reference frame using an inverse transformation. Then, numerical integration is used to track the asteroid’s motion along the respective orbit to predict a possible impact event. Asteroids 99942 Apophis and 2007 WD5 are used as examples to show that disregarding the correlations when diagonalizing the covariance matrix to set the initial conditions may seriously distort the collision probability estimates. The paper gives the probabilities of the collisions of Apophis with the Earth and asteroid 2007 WD5 with Mars calculated by the author from observation sets showing nonzero collision probabilities. The author’s estimates are compared to those calculated by NASA.     

Evolution characteristics of Jupiter’s outer satellites orbits

A systematic study is made of the long-period evolution of all distant satellite orbits in the system of Jupiter. We determine the extreme eccentricities and inclinations, as well as the circulation periods of the pericenter arguments and of the longitudes of the nodes. Moreover, we perform a comparative analysis of the research methods employed—analytical, numerical, and numerical—analytical methods.     

Families of double symmetric “Schubart-like” periodic orbits

We consider a class of Hamiltonian systems with two degrees of freedom with singularities. This class includes several symmetric subproblems of the $n$ -body problem where the singularities are due to collisions involving two or more bodies. “Schubart-like” periodic orbits having two collisions in one period, are present in most of these subproblems. The purpose of this paper is to study the existence of families of such a periodic orbits in a general setting. The blow up techniques of total collision and infinity are applied to our class of Hamiltonian system. This allows us to derive sufficient conditions to ensure the existence of families of double symmetric “Schubart-like” periodic orbits having many singularities. The orbits in the family can be parametrized by the number of singularities in one period. The results are applied to some subproblems of the gravitational $n$ -body problem.     

On the stability of circular ‘asteroid’ orbits in anN-planetary system

The restricted problem of the motion of a point of negligible mass (asteroid) in anN-planetary system is considered. It is assumed that all the planets move about the central body (Sun) along circular orbits in the same plane and the mean motions of the asteroid and the planets are incommensurable. The asteroid orbit evolution is described as a first approximation by secular equations with the perturbing function averaged by the mean longitudes of the asteroid and the planets. For small values of the asteroid orbit eccentricity an expression for the secular part of the perturbing function has been obtained. This expression holds for the arbitrary values of the asteroid orbit semiaxis which are different from those of the planet orbit radii. The stability of the asteroid circular orbits in a linear approximation with respect to the eccentricity is studied. The critical inclinations for a Solar system model are calculated.     

Oscillatory orbits in the planar three‐body problem with equal masses

In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario to prove their existence is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics of Earth‐crossing asteroids: the protected Toro orbits

Some asteroids in Earth‐crossing orbits avoid close approaches by entering in a mean motion resonance whenever the distance between the two orbits is small. These orbits are ‘Toro class’ according to the classification of (Milani et al., 1989). This protection mechanism can be understood by a semi‐averaged model, in which the fast variables are removed and the dynamical variables are the critical argument and the semimajor axis, with dependence upon a slow parameter. The adiabatic invariant theory can be applied to this model and accounts for all the qualitative features of the orbits in this class, including the onset of the libration when the orbit distance is small. Because of the neglected perturbations by the other planets, this theory is approximate and the adiabatic invariant is conserved only with low accuracy moreover, the Toro state can be terminated by a close approach to another planet (typically Venus). “Would you tell me, please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the Cat. Alice in Wonderland, L. Carroll     

Periodic orbits of the generalized Friedmann–Robertson–Walker Hamiltonian systems

The averaging theory of first order is applied to study a generalization of the Friedmann–Robertson–Walker Hamiltonian systems with three parameters. We provide sufficient conditions on the three parameters of the generalized system to guarantee the existence of continuous families of periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter.     

Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

Behaviour of a new type of Runge–Kutta methods when integrating satellite orbits

Recently, González, Martín and Farto have developed new numerical methods (RKGM methods) of Runge–Kutta type and fixed step size for the numerical integration of perturbed oscillators. Moreover, it seems natural to study the behaviour of these new methods for the accurate integration of orbital problems after the application of linearizing transformation, such us KS or BF due to the fact that in these variables, the structure of the problem is of the form of perturbed oscillators, for which the methods constructed are indicated. In this paper, we check the efficiency of these new methods when integrating the satellite problem. The RKGM methods show a very good behaviour when they compete with other, classical and special, methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic orbits accumulating onto elliptic tori for the (<Emphasis Type="Italic">N</Emphasis> + 1)-body problem

We prove the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori for (an “outer solar-system” model of) the planar (N + 1)-body problem.      

Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body

In this paper, we have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when more massive body is a source of radiation and the smaller primary is an oblate body. We have determined periodic orbits for fixed values of μ, σ and different values of p and h (μ mass ratio of the two primaries, σ oblate parameter, p radiation parameter and h energy constant). These orbits have been determined by giving displacements along the tangent and normal to the mobile co-ordinates as defined by Karimov and Sokolsky (in Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of radiation pressure on the periodic orbits by taking some fixed values of μ and σ.