Laplace's problem in mathematical aspect

In this paper we consider the mathematical aspect of Laplace's problem pertaining to the occurrence probability of elliptical and hyperbolical orbits for comets. We show that, among other things, if we use arbitrary velocity distributions in Laplace's problem, we may obtain an arbitrarily small probability for elliptical orbits and a probability neighbouring around one for hyperbolic orbits, or conversely. Our theorems hold, even for the variable initial conditions. The aspect presented here permits us to describe easily studies of Laplace's problem.Dedicated to my teacher, Professor W. Orlicz, on the occasion of 80th birthday.     

Dynamical friction on globular clusters in core-triaxial galaxies: is it a cause of massive black hole accretion?

The present work extends and deepens previous examinations of the evolution of globular cluster orbits in elliptical galaxies, by means of numerical integrations of a wide set of orbits in five self-consistent triaxial galactic models characterized by a central core and different axial ratios. These models are valid and complete in the representation of regular orbits in elliptical galaxies. Dynamical friction is definitely shown to be an efficient cause of evolution for the globular cluster systems in elliptical galaxies of any mass or axial ratio. Moreover, our statistically significant sample of computed orbits confirms that the globular cluster orbital decay times are, at least for clusters moving on box orbits, much shorter than the age of the galaxies. Consequently, the mass carried into the innermost galactic region in the form of decayed globular clusters may have contributed significantly to feeding and accreting a compact object therein.     

The post-Newtonian relative orbit of the two-body system. Gravitational radiation

The average loss of energy over one period of the elliptical motion of the two-body system is given, within the quadrupole approximation, by using the relative motion in the post-Newtonian centre of the mass frame. More explicit formulae are derived for the elliptical orbits and detailed results are presented for the circular orbits assuming small orbital velocities compared to the velocity of light. On the other hand, using the defined Lagrangian we give the integrals of motion.     

On possibility of clustering of microparticles of orbital debris in circular and elliptical orbits

Estimation is made of the possibility of clustering of debris particles in circular and elliptical orbits around the Earth due to the change in drag, caused by quasi-periodic variations of the atmospheric density in the orbit. The estimations show that the collective behavior of particles has time to be manifested in highly elliptical orbits, where the relative change in the atmospheric density along the orbital path is greater and characteristic lifetimes of the particles are longer. However, in this case the limit distributions of the particles are not realized, because the clusters form and break down several times during the lifetime of the particles in the orbit.     

Modern view on Laplace's problem

We consider the Laplace's problem pertaining to occurrence probability of elliptical and hyperbolical orbits of comets, assuming various distributions of their velocities and variable initial conditions. Such approach seems to explain contradictory results obtained by different investigators and suggests that predominance of one type over the other type orbits, may result-among other-from assumed maximal velocity range of comets and their velocity distribution.     

Linearization Method for Optimizing Noncoplanar Flights with Low Thrust

Solar System Research - Problems on optimizing the noncoplanar flights with low thrust from elliptical to geosynchronous orbits for different control accelerations are solved. Recommendations for...     

Existence of quasi-periodic solutions to the three-body problem

A rigorous proof is given for the existence of quasi-periodic solutions with only two degrees of freedom to a planar three-body problem. The solution corresponds physically to the small bodies moving on different, nearly elliptical orbits about a large mass located at a focus. The perihelia of the two orbits are locked in such a way that the difference of the two perihelia has mean value zero.     

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.     

Improvement on the second-order solution of artificial satellite orbits

Three methods are proposed in this paper to lessen the complexity in the derivation and the resulting expressions of second-order analytical solutions of artificial satellite orbits while retaining the advantages of analytical solutions. Of the three, the one combining the theories of elliptical perturbation and of intermediate orbits is noteworthy for its simplicity. It can also be used with advantage in first-order and third- or higher order solutions.     

Stellar dynamics in razor-thin discs with massive nuclear black holes

The bifurcations of orbit-averaged dynamics are studied in a class of razor-thin discs with central black holes. The model used here consists of a perturbed harmonic oscillator Hamiltonian augmented with a GM r potential. Through a sequence of conformal and canonical transformations, we reduce the phase-space flows of the system to a set of non-linear differential equations on a sphere. Based on the critical points of the averaged system, we classify orbit families and reveal the existence of six types of periodic motions: circular , long - and short-axis elliptical , long - and short-axis radial and inclined radial orbits. Long-axis elliptical orbits and their surrounding tubes have significant features: whilst they keep stars away from the centre, they elongate in the same direction as the density profile. These properties are helpful in the construction of self-consistent equilibria.     

Analytical study of the swing-by maneuver in an elliptical system

The objective of the present paper is to derive a set of analytical equations that describe a swing-by maneuver realized in a system of primaries that are in elliptical orbits. The goal is to calculate the variations of energy, velocity and angular momentum as a function of the usual basic parameters that describe the swing-by maneuver, as done before for the case of circular orbits. In elliptical orbits the velocity of the secondary body is no longer constant, as in the circular case, but it varies with the position of the secondary body in its orbit. As a consequence, the variations of energy, velocity and angular momentum become functions of the magnitude and the angle between the velocity vector of the secondary body and the line connecting the primaries. The “patched-conics” approach is used to obtain these equations. The configurations that result in maximum gains and losses of energy for the spacecraft are shown next, and a comparison between the results obtained using the analytical equations and numerical simulations are made to validate the method developed here.     

Celestial-mechanical peculiarities of Uranus’s satellite system

We consider the structural peculiarities of Uranus’s satellite system associated with its separation into two groups: inner equatorial satellites moving in nearly circular orbits and distant irregular satellites with retrograde motion in highly elliptical orbits. The intermediate region is free from satellites in a wide range of semimajor axes. By analyzing the evolution of satellite orbits under the combined effect of solar attraction and Uranus’s oblateness, we offer a celestial-mechanical explanation for the absence of equatorial satellites in this region. M.L. Lidov’s studies during 1961–1963 have served as a basis for our analysis.     

Order and Chaos in Self-Consistent Galactic Models

We construct and compare two different self-consistent N-body equilibrium configurations of galactic models. The two systems have their origin in cosmological initial conditions selected so that the radial orbit instability appears in one model and gives an E5 type elliptical galaxy, but not in the other that gives an E1 type. We examine their phase spaces using uniformly distributed orbits of test particles in the resulting potential and compare with the distribution of the orbits of the real particles in the two systems. The main types of orbits in both cases are box, tube and chaotic orbits. One main conclusion is that the orbits of the test particles in the 3-dimensional potential are foliated in a way quite close to the foliation of invariant tori in a 2-dimensional potential. The real particles describe orbits having similar foliation. However, their distribution is far from being uniform. The difference between the two models of equilibrium is realized mainly by different balances of the populations of real particles in box and tube orbits.     

Elliptical multi-sun-synchronous orbits for Mars exploration

The multi-sun-synchronous orbits allow cycles of observation of the same area in which solar illumination repetitively changes according to the value of the orbit elements and returns to the initial condition after a temporal interval multiple of the repetition of observation. This paper generalizes the concept of multi-sun-synchronous orbits, whose classical sun-synchronous orbits represent particular solutions, taking into consideration the elliptical case. The feasibility of using this typology of orbits, referred to as elliptical periodic multi-sun-synchronous orbits, has been investigated for the exploration of Mars and particular solutions have been selected. Such solutions considerably reduce the manoeuvre of velocity variation at the end of the interplanetary transfer with respect to the case of a target circular orbit around Mars. They are based on the use of quasi-critical inclinations in order to minimize the apsidal line motion and thus reduce orbit maintenance costs. Moreover, in the case of high eccentricities, the argument of pericentre may be set in order to obtain, around the apocentre, a condition of quasi-synchronism with the planet (the footprint of the probe on the surface presents a small shift with respect to a fixed point on the Martian surface). The low altitude of pericentre allows observation of the planet at a higher spatial resolution, while the orbit arc around the apocentre may be used to observe Mars with a wide spatial coverage in quasi-stationary conditions. This latter characteristic is useful for analysing atmospheric and meteorological phenomena and it allows for most of the orbital period a link between a rover on the surface of Mars and a probe orbiting around the planet.     

The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid

This paper deals with the stationary solutions of the planar restricted three-body problem when the more massive primary is a source of radiation and the smaller primary is an oblate spheroid with its equatorial plane coincident with the plane of motion. The collinear equilibria have conditional retrograde elliptical periodic orbits around them in the linear sense, while the triangular points have long- or short-periodic retrograde elliptical orbits for the mass parameter 0 < _crit, the critical mass parameter, which decreases with the increase in oblateness and radiation force. Through special choice of initial conditions, retrograde elliptical periodic orbits exist for the case = _crit, whose eccentricity increases with oblateness and decreases with radiation force for non-zero oblateness.     

Stellar orbits in elliptical galaxies with relativistically active nuclei

In a model galaxy composed of a relativistically active nucleus, a main body, and a halo, all three components considered as homogeneous prolate ellipsoids, we explore the probable association of the internal characteristics of the nucleus and the observed orbits of the stars near the surface of the main body. Using the authors’ theoretical framework of post-Newtonian general relativistic galactic dynamics, proposed earlier, we prove that a fast-rotating and possibly expanding or contracting nucleus affects the distribution of the box-type orbits near the surface of the main body resulting in a flattening of the main body. The nuclear rotation always results in a flattening, and the contraction contributes less to the flattening than the expansion. However, the contributions of a rotating and changing nucleus are not additive. The study of the post-Newtonian effects in the nucleus on the stellar orbits in the main body, and the consequent modifications of the corresponding non-relativistic results, could in principle provide useful information concerning the kinematical and dynamical characteristics of the nuclei of the elliptical galaxies. The explanation (of at least the post-Newtonian part) of the flattening of elliptical galaxies attempted here seems to be the first theoretical one proposed in the literature.     

Controlled Solar Sail Transfers into Near-Sun Regions Combined with Planetary Gravity-Assist Flybys

Results of numerical simulations of 'local-optimal' (or 'instantaneously optimal') trajectories of a space probe with a flat solar sail which moves from the circular Earth orbit to near-Sun regions are presented. We examine planar (ecliptic) solar sail transfer with gravity-assist flybys of Earth, Venus and Mercury. Several complex control modes of the sail tilt orientation angle for near-Sun orbits and for some 'falling onto the Sun' trajectories are investigated. The numerical simulations are used to examine the flight duration of some sail missions and to investigate the evolution of osculating elliptical orbits.     

On the convergence of elliptic motion

The convergence of Lagrange series is studied on a part of the elliptical orbit for values of eccentricity exceeding the Laplace limit. The regions in the vicinity of the two apses of the orbit are identified in which the Lagrange series converge absolutely and uniformly for the values of the eccentricity greater than the Laplace limit. The obtained results are of practical interest for astronomy when studying motions of stellar bodies in orbits with high eccentricity. In particular, these series may be used to calculate the orbits of comets or asteroids with high eccentricity as they pass through the neighborhood of perihelion or to calculate the orbits of artificial satellites with high eccentricity “hanging” in the vicinity of apogee. In stellar dynamics, these series may be used in cases of close binary stars, many of which move in orbits with an eccentricity greater than the Laplace limit.     

Canonical theory of perturbations using eccentric anomaly as independent variable

It is shown in this paper how to build a canonical transformation of variables, so that the eccentric anomaly becomes the new independent variable. In the case of eccentric elliptical orbits it changes the equations of motion so, that they can be integrated analytically to any order of approximation comparatively easy.     

Planetary close encounters between Jupiter and about 3000 fictitious minor bodies

Single close encounters between Jupiter and about 3000 hypothetical minor bodies, initially on elliptical orbits, have been studied computing the evolution of the three-body system Sun-Jupiter-object, by means of a new numerical method of integration. The fictitious population processed contains almost all the orbits which allow a close approach to the planet. The efficiency of a single encounter in varying the orbital parameters of the objects resulted to be generally poor, as it is shown by the distributions of the orbital parameter variations. Collisions and ejections from the solar system on hyperbolic orbits are little numerous; some temporary satellite capture have been recognised. The results of this work show that any attempt to study the close encounter event by means of two distinct two-body problems is physically meaningless because the mid-range perturbations, disregarded in such cases, are very far from being negligible.