Periodic orbits of the Hill problem with radiation and oblateness

The modification of Hill’s problem where the primary is radiating and the secondary is an oblate spheroid is considered. The evolution of the network of the basic families of planar periodic orbits for various values of the parameters of the problem is studied. For specific values of the parameters these families are determined accurately together with their stability properties. The stability of retrograde satellites in an appropriate space of initial conditions is also determined by means of surface of section portraits of the Poíncare map and higher order resonances are studied. Simple asymmetric periodic orbits of the problem are also determined.     

Some special orbits in the two-body problem with radiation pressure

The motion has been studied of a particle in a gravitational field perturbed by radiation pressure. By combining the formulation in the physical space variables with the KS variables we obtained explicit evidence for the existence of a surface of stable circular orbits with centers on an axis through the primary body. Furthermore, the effects of a sharp shadow on the two-dimensional unstable parabolic orbits were investigated. It was found that they do not survive the introduction of a shadow.     

On the periodically evolving orbits in the singly averaged Hill problem

We continue to analyze the periodic solutions of the singly averaged Hill problem. We have numerically constructed the families of solutions that correspond to periodically evolving satellite orbits for arbitrary initial values of their eccentricities and inclinations to the plane of motion of the perturbing body. The solutions obtained are compared with the numerical solutions of the rigorous (nonaveraged) equations of the restricted circular three-body problem. In particular, we have constructed a periodically evolving orbit for which the well-known Lidov-Kozai mechanism manifests itself, just as in the doubly averaged problem.     

Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure

The backbone of the analysis in most dynamical systems is the study of periodic motions, since they greatly assist us to understand the structure of all possible motions. In this paper, we deal with the photogravitational version of the rectilinear restricted four-body problem and we investigate the dynamical behaviour of a small particle that is subjected to both the gravitational attraction and the radiation pressure of three bodies much bigger than the particle, the primaries. These bodies are always in syzygy and two of them have equal masses and are located at equal distances from the third primary. We study the effect of radiation on the distribution of the periodic orbits, their stability, as well as the evolution of the families and their main features.     

Combined solar radiation pressure and drag effects on the orbits of artificial satellites

A semi analytical theory is proposed to study the joint effects of direct solar radiation pressure and atmospheric drag on the orbit of an artificial Earth satellite. Making the solar radiation pressure equal to zero the problem is reduced to one already solved by Brouwer and Hori. The solutions are not equivalent, however, since in the Brouwer and Hori theory one has spurious Poisson terms.     

Long-periodic and secular perturbations to the orbits of Explorer 19 and Lageos due to direct solar radiation pressure

A theory for the long-term variations in the orbit of a spherically symmetric satellite due to direct solar radiation pressure is tested using two satellite orbit analyses. The first of these analyses is in terms of mean elements for the balloon satellite Explorer 19. The results are compared with the expected theoretical variations with short-period terms omitted. The second analysis utilises satellite laser ranging observations of the geodetic satellite, Lageos. A novel long-term analysis technique is developed primarily for laser ranging studies. The technique is tested along with the solar radiation pressure perturbation theory by comparing the results from the theory and the analysis.     

Asymptotic orbits and terminations of families in the Copenhagen problem

The physico-chemical origin of the hydrogenated carbon clusters (cumulenes, PAHs, graphite or amorphous carbon) in space is still an open question. We have worked out a numerical simulation code in order to build up planar (graphite-like) carbon clusters. We assume that hydrogen atoms can fix on the carbon skeleton following a random process allowing forH ₂ formation. The structures we have found are very complex. In a given cluster, several molecular entities can simultaneously be present: (sp ²) carbon chains, rings or compact formations (aromatic structures or small PAHs). We argue that these very contorted hydrogenated structures could be ubiquitous in the interstellar medium, in carbon-rich circumstellar regions and PNe.     

Satellite orbits perturbed by direct solar radiation pressure: General expansion of the disturbing function

An expression is derived for the solar radiation pressure disturbing function on an Earth satellite orbit which takes into account the variation of the solar radiation flux with distance from the Sun's centre and the absorption of radiation by the satellite. This expression is then expanded in terms of the Keplerian elements of the satellite and solar orbits using Kaula's method. The Kaula inclination functions are replaced by an equivalent set of modified Allan inclination functions.The resulting expression reduces to the form commonly used in solar radiation pressure perturbation studies (e.g. Aksnes, 1976), when certain terms are neglected. If, as happens quite often in practice, a satellite's orbit is in near-resonsnce with certain of these neglected terms, these near-resonant terms can cause changes in the satellite's orbital elements comparable to those produced by the largest term in Aksnes's expression. A new expression for the solar radiation pressure disturbing function expansion is suggested for use in future studies of satellite orbits perturbed by solar radiation pressure.     

Periodic orbits in the Chermnykh-like restricted problem of oblate bodies with radiation

In this paper, the periodic orbits around triangular points in the range of linear stability of the restricted three body problem, when the smaller primary and the test particle have the shape of an oblate spheroid and the larger primary is a radiation emitter with the allowance for the gravitational potential from the belt, is studied. It is observed that the orbits around these points are elliptical and have long and short periodic orbits. The period, orientation, eccentricities, the semi-major and semi-minor axis of the elliptic orbits are found. The study includes some numerical examples in the case of the Sun-Earth and Sun-Jupiter systems.     

The dynamics of the elliptic Hill problem: periodic orbits and stability regions

The motion of a satellite around a planet can be studied by the Hill model, which is a modification of the restricted three body problem pertaining to motion of a satellite around a planet. Although the dynamics of the circular Hill model has been extensively studied in the literature, only few results about the dynamics of the elliptic model were known up to now, namely the equations of motion and few unstable families of periodic orbits. In the present study we extend these results by computing a large set of families of periodic orbits and their linear stability and classify them according to their resonance condition. Although most of them are unstable, we were able to find a considerable number of stable ones. By computing appropriate maps of dynamical stability, we study the effect of the planetary eccentricity on the stability of satellite orbits. We see that, even for large values of the planetary eccentricity, regular orbits can be found in the vicinity of stable periodic orbits. The majority of irregular orbits are escape orbits.     

To the problem of satellite's perturbed motion under the influence of solar radiation pressure

A possibility of developing the analytical theory of perturbed motion for a balloon-satellite influenced by solar radiation pressure force is analysed here on the basis of the limit case modification of the two fixed centers problem whose force-field is a superposition of the Newtonian central field and a homogeneous one. Such an approach enables us in the intermediate orbit already to take into account the effect of a constant force, all coordinates of a satellite being expressed as functions of some monotonically increasing variable by means of inversion of elliptic quadratures. The relations between canonical constants of the intermediate orbit and a quasikeplerian elements coinciding in the absence of solar radiation pressure with keplerian ones are derived. The numerical results and illustrating the perturbations in the radius-vector of the intermediate orbit of a balloon-satellite of the Echo-I type are given.

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On the evolution of families of periodic orbits approaching a small primary

A network of families of periodic orbits is obtained approximately for the case =0.1 of the restricted problem using a direct grid search. Only such orbits of the third body are considered that cross the synodical line of the primaries outside the smaller of the two, perpendicularly, and in the direction of rotation of the system.     

Perturbations of a spheroidal satellite due to direct solar radiation pressure

The orbital accelerations of certain balloon satellites exhibit marked oscillations caused by solar radiation impinging on the surface of the satellites, which, once spherical, have assumed a spheroidal shape producing a component of force at right-angles to the Sun-satellite direction. Given the characteristics and orientation of the satellite, the equations of force are determined by the formulae of Lucas. Otherwise the phase-angle and magnitude of the right-angle force are determined by trial and error, or best-fit techniques. Using a variation of the approach developed by Aksnes, a semi-analytical algorithm is presented for evaluating the perturbations of the Keplerian elements by direct solar radiation pressure on a spheroidal satellite. The perturbations are obtained by summing over the sunlit part of each orbit and allow for a linear variation in the phase-angle. The algorithm is used to determine the orbital accelerations of 1963-30D due to direct solar radiation pressure, and these results are compared to the observed values over two separate periods of the satellite's lifetime.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

The effect of solar radiation pressure on the Lagrangian points in the elliptic restricted three-body problem

In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied, within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived. For large values of the force ratio β, we found that at β=0.12, the collinear point L₃ is stable and some families of periodic orbits can be drawn around it.     

Bifurcation points and intersections of families of periodic orbits in the three-dimensional restricted three-body problem

Intersections of families of three-dimensional periodic orbits which define bifurcation points are studied. The existence conditions for bifurcation points are discussed and an algorithm for the numerical continuation of such points is developed. Two sequences of bifurcation points are given concerning the family of periodic orbits which starts and terminates at the triangular equilibrium pointsL ₄,L ₅. On these sequences two trifurcation points are identified forµ = 0.124214 andµ = 0.399335. The caseµ = 0.5 is studied in particular and it is found that the space families originating at the equilibrium pointsL ₂,L ₃,L ₄,L ₅ terminate on the same planar orbitm _1v of the familym.     

The two-body problem with radiation pressure in a rotating reference frame

We study a perturbed Newtonian two-body problem, in which the perturbation is due to a force field of constant magnitude but rotating direction. By considering this system as a perturbation of the non-rotating case a Melnikov-type analysis allows us to show the existence of horseshoes in the level sets of the Hamiltonian and the subsequent sensitive dependence on initial conditions and non-integrability. We discuss the consequences of these results for a particular planar restricted three-body problem.Supported by a grant from the Royal Swedish Academy of Sciences and AFOSR NM 91-0329.