Non-Continuation of Integrals of the Rotating Two-body Problem in Hill's Lunar Problem

We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem.     

Three-body planetary problem: study of KAM stability for the secular part of the Hamiltonian

The Hamiltonian representing the average over the mean-motion angles (i.e. the secular part) of the three-body planetary problem is considered. An efficient algorithm constructing invariant tori for the trajectories in phase space is provided. To give a possible practical application, we consider a toy-model including the main terms of the secular part of a hypothetic Sun-Jupiter-Saturn system having eccentricities and inclinations equal to 1/20 of the true ones. The scheme of a KAM proof of the stability of the model is sketched. The proof is “computer assisted”.     

Expansion of the Hamiltonian of the Two-Planetary Problem into the Poisson Series in All Elements: Application of the Poisson Series Processor

This paper is the third in a series of articles devoted to one of the basic problems of celestial mechanics: the study of the evolution of solar-type planetary systems. In the previous papers a brief review of the history and current state of the problem was given; the plan of the study was outlined; the Jacobi coordinates and the related osculating elements were introduced; the form of the Poisson expansion of the Hamiltonian in all elements was given; and the expansion coefficients for the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem were obtained (though with impure accuracy) by a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. In the present paper the expansion of the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem into the Poisson series in all elements is constructed with the help of the PSP Poisson series processor, which is capable of required accuracy.     

Numerical Integration of Regular Equations of the Planar Motion of Terrestrial Bodies

The parameters of L matrices are applied to the numerical integration of regular equations describing the motion of minor bodies in the Solar System. The problem of the optimal choice of the regularizing change of variables is formulated in the context of the numerical integration of the equations of motion using the Runge–Kutta–Fehlberg method. Arbitrary perturbations are taken into account. This problem is completely solved in the case of planar motion. The solution of the optimization problem reduces the amount of computations needed to determine the vector of perturbing accelerations. Results of numerical integrations are given.     

Preliminary Study on the Translunar Halo Orbits of the Real Earth–Moon System

The computation of translunar Halo orbits of the real Earth–Moon system (REMS) has been an open problem for a long time, but now, it is possible to compute Halo orbits of the REMS in a systematic way. In this paper, we describe the method used for the numerical computation of Halo orbits for a time span longer than 41 years. Halo orbits of the REMS are computed from quasi-periodic Halo orbits of the quasi-bicircular problem (QBCP). The QBCP is a model for the dynamics of a spacecraft in the Earth–Moon–Sun system. It is a Hamiltonian system with three degrees of freedom and depending periodically on time. In this model, Earth, Moon and Sun are moving in a self-consistent motion close to bicircular. The computed Halo orbits of the REMS are compared with the family of Halo orbits of the QBCP. The results show that the QBCP is a good model to understand the main features of the Halo family of the REMS.     

Stability analysis of C−N−O Nuclear reaction inside the stars

In view of the fact that the nuclear reactions inside the stars are very strong, we introduce the effects of convection and diffusion in the fundamental equation of the problem in modeling to the C–N–O nuclear reaction. The stability of the nuclear reaction inside the star is investigated, and the results of some interest are obtained.     

On the oscillations in Mercury's obliquity

Mercury's capture into the 3:2 spin–orbit resonance can be explained as a result of its chaotic orbital dynamics. One major objective of MESSENGER and BepiColombo spatial missions is to accurately measure Mercury's rotation and its obliquity in order to obtain constraints on internal structure of the planet. Analytical approaches at the first-order level using the Cassini state assumptions give the obliquity constant or quasi-constant. Which is the obliquity's dynamical behavior deriving from a complete spin–orbit motion of Mercury simultaneously integrated with planetary interactions? We have used our SONYR model (acronym of Spin–Orbit N-bodY Relativistic model) integrating the spin–orbit N-body problem applied to the Solar System (Sun and planets). For lack of current accurate observations or ephemerides of Mercury's rotation, and therefore for lack of valid initial conditions for a numerical integration, we have built an original method for finding the libration center of the spin–orbit system and, as a consequence, for avoiding arbitrary amplitudes in librations of the spin–orbit motion as well as in Mercury's obliquity. The method has been carried out in two cases: (1) the spin–orbit motion of Mercury in the 2-body problem case (Sun–Mercury) where an uniform precession of the Keplerian orbital plane is kinematically added at a fixed inclination (S_2K case), (2) the spin–orbit motion of Mercury in the N-body problem case (Sun and planets) (S_n case). We find that the remaining amplitude of the oscillations in the S_n case is one order of magnitude larger than in the S_2K case, namely 4 versus 0.4 arcseconds (peak-to-peak). The mean obliquity is also larger, namely 1.98 versus 1.80 arcminutes, for a difference of 10.8 arcseconds. These theoretical results are in a good agreement with recent radar observations but it is not excluded that it should be possible to push farther the convergence process by drawing nearer still more precisely to the libration center. We note that the dynamically driven spin precession, which occurs when the planetary interactions are included, is more complex than the purely kinematic case. Nevertheless, in such a N-body problem, we find that the 3:2 spin–orbit resonance is really combined to a synchronism where the spin and orbit poles on average precess at the same rate while the orbit inclination and the spin axis orientation on average decrease at the same rate. As a consequence and whether it would turn out that there exists an irreducible minimum of the oscillation amplitude, quasi-periodic oscillations found in Mercury's obliquity should be to geometrically understood as librations related to these synchronisms that both follow a Cassini state. Whatever the open question on the minimal amplitude in the obliquity's oscillations and in spite of the planetary interactions indirectly acting by the solar torque on Mercury's rotation, Mercury remains therefore in a stable equilibrium state that proceeds from a 2-body Cassini state.     

On the evolution of a system of intergalactic clouds

The presence of L _a -forests in the spectra of quasars is considered as proof of the fragmentary structure of the intergalactic medium. The masses of the clumps (of clouds) will increase as they merge together. Once the radial concentration of neutral hydrogen in a cloud attains its critical value, star formation begins, and the cloud turns into a galaxy. Certain physical properties of clouds are considered and a new approach to the investigation of the evolution of systems of clouds is proposed.Translated from Astrofizika, Vol. 37, No. 1, pp. 35–42, January–March, 1994.     

Does H− truly cool the solar chromosphere?

I examine the controversial problem of H^– radiative cooling in the solar chromosphere. I find, in agreement with Praderie and Thomas, that H^– is a substantial source of radiative heating in the outer atmosphere, especially when departures from LTE are important. The role of H^– as a chromospheric heating agent must be considered carefully before net radiative cooling rates can be assessed from empirical chromospheric models, or calculations of nonradiative heating, for example by acoustic waves, can be pursued meaningfully.     

Properties of New Coordinates for the General Three-Body Problem

This is an analysis of the features of the new coordinate system given by the principal axes of inertia, as determined by Euler angles, and twodistances related to the inertia principal moments and an auxiliar angleas coordinates, for studying the general three-body problem, interactingthrough gravitational forces.The reduction of order is performed in these new coordinates by using the angular velocity vector or the Euler angles.The Eulerian case of collinear motion is revisited from our own perspective.The value of the auxiliar angle is computed for the Sun–Earth–Moon system.     

Magnetosphere of barionic stars

Conclusions Our chief result is the proof that pulsars can possess a quasi-steady-state magnetosphere with temperature T10⁴–10⁶. The magnetosphere can be maintained in this state in its part nearest the star if the plasma is heated by radiation from the star (except for P 0531, for which such radiation is nearlyinsignificant). Plasma in the main part of the magnetosphere is maintained in such a hot state as a result of Joule heat due to drift currents. Radiation from the magnetosphere of P 0531 is found basically in the optical spectrum, though the intensity is several orders of magnitude less than the observed value, so that it does not correspond to the observed optical emission from the pulsar in the Crab nebula.Erevan State University. Translated from Astrofizika, Vol. 12, No. 2, pp. 339–349, April–June, 1976.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

A New Analysis of the Spectra Obtained by the <Emphasis Type="Italic">Venera</Emphasis> Missions in the Venusian Atmosphere. I. The Analysis of the Data Received from the <Emphasis Type="Italic">Venera-11</Emphasis> Probe at Altitudes Below 37 km in the 0.44–0.66 µm Wavelength Range

The processes of the solar radiation extinction in deep layers of the Venus atmosphere in a wavelength range from 0.44 to 0.66 µm have been considered. The spectra of the solar radiation scattered in the atmosphere of Venus at various altitudes above the planetary surface measured by the Venera-11 entry probe in December 1978 are used as observational data. The problem of the data analysis is solved by selecting an atmospheric model; the discrete-ordinate method is applied in calculations. For the altitude interval from 2–10 km to 36 km, the altitude and spectral dependencies of the volume coefficient of true absorption have been obtained. At altitudes of 3–19 km, the spectral dependence is close to the wavelength dependence of the absorption cross section of S₃ molecules, whence it follows that the mixing ratio of this sulfur allotrope increases with altitude from 0.03 to 0.1 ppbv.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 4, 2005, pp. 304–320.Original Russian Text Copyright © 2005 by Maiorov, Ignat’ev, Moroz, Zasova, Moshkin, Khatuntsev, Ekonomov.     

On the nature of the ice cap on the Tibetan Plateau during the late Quaternary

Considerable debate persists among scientists interested in the nature of the ice cap on the Tibetan Plateau during the late Quaternary. We examine the implications, on this problem, of the high resolution data that has recently become available from the Dunde ice cap in north Tibet. The observed −2% change in the δ¹⁸O of the ice formed at the Dunde ice cap during the Last Glacial Stage (LGS) suggests a limit in the range of 5–7°C on the reduction in annual surface air temperature over Tibet during the LGS. This then translates to an Equilibrium Line Altitude (ELA) lowering of 700–1200 m. Due to this lowering, ELA could have reached below the level of the surface of the plateau resulting in an extensive ice sheet formation during LGS.     

Computation of the elements of close eclipsing systems in the frequency-domain

The aim of the present paper will be to generalize the methods for computation of the elements of eclipsing binary systems in the frequency-domain, summarized in our recent Paper I (Kopal, 1981), to the case ofclose systems, in which photometric proximity effects become conspicuous and must be taken into account before the methods previously outlined in Paper I become directly applicable.Following a brief introduction to the subject given in Section 1, Section 2 summarizes (and comments upon) the difficulties previously encountered in separation of the photometric proximity and eclipse effects. In Section 3 we develop an alternative new approach to the problem by modulation of the light curves throughout the entire orbital cycle, intended to filter out proximity effects from the observed light changes and isolate those due to eclipses; while in Section 4 we shall present a numerical application of the new method to an analysis of the observed light changes of the eclipsing system W Ursae Maioris.In Section 5 we shall present a quantitative investigation of the photometric effects of distortion on the light changes of close eclipsing systems within eclipses-the most complicated part of the whole problem-with numerical application to the system of U Sagittae carried out in the concluding Section 6.Appendices 1–3 contain numerical data which should facilitate application of the methods developed and illustrated in Sections 3–4; while Appendix 4 will be reserved for a mathematical proof of certain expansions used in Section 5, which would have been too discursive for the main text.     

Solution of the Sitnikov Problem

A new analytic expression for the position of the infinitesimal body in the elliptic Sitnikov problem is presented. This solution is valid for small bounded oscillations in cases of moderate primary eccentricities. We first linearize the problem and obtain solution to this Hill's type equation. After that the lowest order nonlinear force is added to the problem. The final solution to the equation with nonlinear force included is obtained through first the use of a Courant and Snyder transformation followed by the Lindstedt–Poincaré perturbation method and again an application of Courant and Snyder transformation. The solution thus obtained is compared with existing solutions, and satisfactory agreement is found.     

Quaternionic Processor

A semi-analytical solution to the Kustaanheimo–Stiefel formulation of the perturbed Kepler problem is presented.     

Second-species solutions in the circular and elliptic restricted three-body problem

An analytical proof of the existence of some kinds of periodic orbits of second species of Poincaré, both in the Circular and Elliptic Restricted three-body problem, is given for small values of the mass parameter. The proof uses the asymptotic approximations for the solutions and the matching theory developed by Breakwell and Perko. In the paper their results are extended to the Elliptic problem and applied to prove the existence of second-species solutions generated by rectilinear ellipses in the Circular problem and nearly-rectilinear ones in the Elliptic case.     

The NLTE Formation of Iron Lines Used in Solar Polarimetry

We study in some detail one-dimensional NLTE effects in solar Fei lines. The lines selected are frequently used in solar polarimetry, and also in studies of line asymmetries and for abundance determinations. Our model atom for Fei–Feii–Feiii is realistic: it takes account of multiplet structure and it includes over 200 bound–bound and bound–free transitions in detail. We use very efficient iterative methods for the self-consistent solution of the kinetic and radiative transfer equations (Auer, Fabiani Bendicho, and Trujillo Bueno, 1994). We have applied these fast methods of solution because they are suitable for the investigation of 2D and 3D NLTE transfer effects with multilevel atoms, which constitutes the next step of our ongoing research project on the iron line formation problem.