A uniform solution of the Lambert problem

A uniform method (with respect to the type of the conic section) for solving the Lambert boundary value problem is presented. It is derived by using the KS-transformation and consists in solving one nonlinear equation. It can be used either in the ordinaryx-frame (physical space) as well as in the KS-coordinates (regularized space).     

The problem of clock synchronization: A relativistic approach

The problem of synchronization of the Earth-based clocks has been discussed in the framework of General Relativity Theory. The synchronization is considered as the transformation of the observers' proper time scales to the coordinate time scale of local inertial geocentric reference system, which is single for all the observers. The formulas for the relativistic corrections occurring in some methods of Earth-based clock synchronization (transported clock, duplex communication via geostationary satellite and meteor-burst link, LASSO experiments) have been derived enabling one to attain the accuracy of 0.1 ns.     

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

The uniform,regular differential equations of the KS transformed perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973     

The G-dwarf problem: A tentative analytical approach,relaxing instantaneous recycling

Starting from a general solution for the birth functionB(m, t) of stars described in detail in Casusoet al. (1989), we have obtained a first-order analytical approximation to this function as a function of metallicityZ. Using this, we obtained a fit to the observational curve compiled by Tinsley (1980) for the cumulative function of stars with metallicity lower than a given value in the solar neighbourhood. In addition, using the same expression, with its numerical fit to previous data, we obtain a good fit to the differential distributions of stars at low metallicity given in the review by Pagel (1987), given a bifurcation in the birth function at low values ofZ, which would correspond to two distinct epochs of onset of star formation. The analysis gives an infall of gas towards the solar neighbourhood up to the epoch of metallicityZ=6.7×10^–3 with a correspondingly increased star formation rate, which subsequently stabilized, and another similar inflow up toZ=1.2×10^–3, followed again by a steady star formation rate for largerZ. Although the assumptions made are still relatively crude, and the numbers should be considered tentative, the flexibility of the model in handling the problem is that we wish to show here.     

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...     

The vector approach to the problem of physical libration of the Moon: the linearized problem

A flexible and informative vector approach to the problem of physical libration of the rigid Moon has been developed in which three Euler differential equations are supplemented by 12 kinematic ones. A linearized system of equations can be split into an even and odd systems with respect to the reflection in the plane of the lunar equator, and rotational oscillations of the Moon are presented by superposition of librations in longitude and latitude. The former is described by three equations and consists of unrestricted oscillations with a period of T ₁ = 2.878 Julian years (amplitude of 1.855″) and forced oscillations with periods of T ₂ = 27.201 days (15.304″), one stellar year (0.008″), half a year (0.115″), and the third of a year (0.0003″) (five harmonics altogether). A zero frequency solution has also been obtained. The effect of the Sun on these oscillations is two orders of magnitude less than that of the Earth. The libration in latitude is presented by five equations and, at pertrubations from the Earth, is described by two harmonics of unrestricted oscillations (T ₅ ≈ 74.180 Julian years, T ₆ ≈ 27.347 days) and one harmonic of forced oscillations (T ₃ = 27.212 days). The motion of the true pole is presented by the same harmonics, with the maximum deviation from the Cassini pole being 45.3″. The fifth (zero) frequency yields a stationary solution with a conic precession of the rotation axis (previously unknown). The third Cassini law has been proved. The amplitudes of unrestricted oscillations have been determined from comparison with observations. For the ratio $ \frac{{\sin I}} {{\sin \left( {I + i} \right)}} \approx 0.2311 $ \frac{{\sin I}} {{\sin \left( {I + i} \right)}} \approx 0.2311 , the theory gives 0.2319, which confirms the adequacy of the approach. Some statements of the previous theory are revised. Poinsot’s method is shown to be irrelevant in describing librations of the Moon. The Moon does not have free (Euler) oscillations; it has oscillations with a period of T ₅ ≈ 74.180 Julian years rather than T ≈ 148.167 Julian years.     

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     

Radial variation of type-III source parameters

The parameters of type-III sources have been observed to vary as powers of the distance of the source from the Sun. Here, the values of the observed exponents are reviewed and theoretical relationships between them are discussed and extended. It is shown that 11 observed exponents can be derived from a four-element subset. A least-squares fit is carried out by varying these four exponents and it is shown that the results are consistent with observaton to within the observational uncertainties. Best-fit expressions are given for the plasma density and temperature, the solar wind speed, beam velocity and density, frequency drift rate, peak Langmuir fields, brightness temperature, volume emissivity, beam duration and burst decay time.     

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.     

Solar cycle variation of heliosphepic parameters

This paper makes a statistical analysis of the solar cycle variation of heliospheric quantities observed at 1 AU. Two kinds of solar cycle variation with different characters have been identified, i.e. the sunspot and coronal-hole cycles, the latter is characterized by the coronal structure lifetime L_C. The kinetic and internal energy parameters of solar wind particles follow the coronal-hole cycle, reaching a maximum at 1973 or 1974. Certain parameter combinations involving IMF quantities are found to be the looked-for heliospheric quantities that follow the sunspot cycle. Among them the ratio of magnetic to kinetic energy density μ_k and the ratio of magnetic to thermal pressure μ_p show a positive correlation with the sunspot number R while the plasma parameter and the Alfvenic Mach number M_a correlate negatively with the sunspot number. The solar wind temperature T, velocity V as well as the adiabatic sound speed C_s vary basically with the coronal-hole cycle as compared to the Alfvenic speed C_a following the sunspot cycle, while the magnetic sonic speed C_as possesses a dual nature. The implication of the above results is that the solar winds observed during the sunspot maximum and the coronal-hole maximum years differ basically in their characters. The former, on the average, is a stream with a low speed, temperature, and density, but under highly magnetic control; while the latter is one with high speed, temperature and density, but under weakly magnetic control. The output of the mass, kinetic and thermal energy fluxes in the former is much less than in the latter.     

The Kepler problem: A unifying view

This paper is an attempt to bring unity in the study of the classical Kepler problem by combining, through simple vectorial and quaternionic techniques, its two peculiar aspects: the determination of the constants of the motion and the regularization at the origin.Research supported by the Consiglio Nazionale delle Ricerche of Italy (C.N.R.-G.N.F.M.).     

The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem

By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω₁, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.     

The critical inclination problem: A simple treatment

A short analysis is presented in the hope of clarifying the situation.     

Triple close approach in the three-body problem: A limit for the bounded orbits

Using the results of Sundman and Birkhoff as also the studies on the generalized Hill's curves, we develop a new method for computing a lower bound of the moment of inertia for the bounded orbits in the general three-body problem.Thus we obtain much higher values than in the classical results. We show for instance, that when the integral of the energy goes to zero, this lower bound goes to the minimum moment of inertia of the corresponding parabolic Euler motion of the same angular momentum: it is then the greatest lower bound.

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Résumé En utilisant les résultats de Sundman et de Birkhoff ainsi que les études sur les courbes de Hill généralisées, on développe une nouvelle méthode pour calculer un minorant du moment d'inertie pour les orbites bornées dans le problème des trois corps.On obtient ainsi des valeurs beaucoup plus élevées que dans les résultats classiques. On montre en particulier que lorsque I'intégrale de I'énergie tend vers zéro ce minorant tend vers le minimum du moment d'inertie du mouvement parabolique d'Euler correspondant de même moment cinétique: c'est alors la limite inférieure.

     

The outbursts of the comet 29P/Schwassmann‐Wachmann 1: A new approach to the old problem

As far as outbursts activity is concerned, the 29P/Schwassmann‐Wachmann 1 is the exceptional comet. This Centaur object shows quasi‐regular flares with periodicities of 50 days (Trigo‐Rodriguez et al. 2010). In the introductory part of the presented paper the most well‐known hypotheses which try to explain this cometary behaviour are reviewed. The second, actual part of this paper presents the new model for the outburst activity of this comet. The model is based on the idea of Ipatov (2012), according to which there are large cavities below a considerable fraction of the comet's surface containing material under high gas pressure. In favourite conditions the surface layers over the cavities are thrown away and the interior of these cavities is exposed. Consequently, an outburst of the comet's brightness may be observed. The main characteristics of an outburst of this comet, the brightness jump, is calculated. Numerical simulations were carried out for wide range of possible cometary parameters. The obtained results are in good agreement with the observations. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Planetary close encounters: geometry of approach and post-encounter orbital parameters

Öpik's assumptions on the geometry of particle trajectories leading to and through planetary close encounters are used to compute the distribution of changes in heliocentric orbital elements that result from such encounters for a range of initial heliocentric orbits. Behaviour at encounter is assumed to follow two-body (particle—planet) gravitational scattering, while before and after encounter particle motion is only governed by the force of the Sun. Derivation of these distributions allows precise analysis of the probability of various outcomes in terms of the physical characteristics of the bodies involved. For example, they allow an explanation and prediction of the asymmetry of the extreme energy perturbations for different initial orbits. The formulae derived here may be applied to problems including the original accumulation of planets and satellites, and the continuing evolution of populations of small bodies, such as asteroids and comets.