Rigorous estimates for the series expansions of Hamiltonian perturbation theory

In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the small parameter . For any order of normalization, we give a lower bound _* ^r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed , the orderr of normalization, one gets for the remainder a greater bound of the formAe ^–(*₁/) ^a , with positive constantsA,a and ₁ ^* exponential estimate of Nekhoroshev's type.     

A unified theory of the formation of galactic bars

We give arguments for a basically unified formation mechanism of slow (Lynden-Bell) and fast (common) galactic bars. This mechanism is based on an instability that is akin to the well-known instability of radial orbits and is produced by the mutual attraction and alignment of precessing stellar orbits (so far, only the formation of slow bars has been explained in this way). We present a general theory of the low-frequency modes in a disk that consists of orbits precessing at different angular velocities. The problem of determining these modes is reduced to integral equations of moderately complex structure. The characteristic pattern angular velocities Ω_p of the low-frequency modes are of the order of the mean orbital precession angular velocity \(\bar \Omega _{pr}\). Bar modes are also among the low-frequency modes; while \(\Omega _p \approx \bar \Omega _{pr}\) for slow bars, Ω_p for fast bars can appreciably exceed even the maximum orbital precession angular velocity in the disk Ω _pr ^max (however, it remains of the order of these precession angular velocities). The possibility of such an excess of Ω_p over Ω _pr ^max is associated with the effect of “repelling” orbits. The latter tend to move in a direction opposite to the direction in which they are pushed. We analyze the pattern of orbital precession in potentials typical of galactic disks. We note that the maximum radius of an “attracting” circular orbit r_c can serve as a reasonable estimate of the bar length l_b. Such an estimate is in good agreement with the available results of N-body simulations.     

Period–luminosity relations for type II Cepheids and their application

JHK _s magnitudes corrected to mean intensity are estimated for Large Magellanic Cloud (LMC) type II Cepheids in the OGLE-III survey the third phase of the Optical Gravitational Lensing Experiment (OGLE). Period–luminosity (PL) relations are derived in JHK _s as well as in a reddening-free VI parameter. Within the uncertainties, the BL Her stars  ( P < 4 d)  and the W Vir stars (   P = 4  to 20 d) are colinear in these PL relations. The slopes of the infrared relations agree with those found previously for type II Cepheids in globular clusters within the uncertainties. Using the pulsation parallaxes of V553 Cen and SW Tau, the data lead to an LMC modulus uncorrected for any metallicity effects of  18.46 ± 0.10  mag. The type II Cepheids in the second-parameter globular cluster, NGC 6441, show a PL( VI ) relation of the same slope as that in the LMC, and this leads to a cluster distance modulus of  15.46 ± 0.11  mag, confirming the hypothesis that the RR Lyrae variables in this cluster are overluminous for their metallicity. It is suggested that the Galactic variable κ Pavonis is a member of the peculiar W Vir class found by the OGLE-III group in the LMC. Low-resolution spectra of OGLE-III type II Cepheids with   P > 20  d (RV Tau stars) show that a high proportion have TiO bands; only one has been found showing C₂. The LMC RV Tau stars, as a group, are not colinear with the shorter period type II Cepheids in the infrared PL relations in marked contrast to such stars in globular clusters. Other differences between LMC, globular cluster and Galactic field type II Cepheids are noted in period distribution and infrared colours.     

Spin-orbit coupling: A unified theory of orbital and rotational resonances

The dynamics of the spin-orbit interaction of a sphereM ₈ and a rotating asymmetrical rigid bodyM _a are examined. No restrictions are imposed on the masses, on the orientation of the rotation axis to the orbit plane, or on the orbit eccentricity. The zonal potential harmonics ofM _a induce a precession of the spin axis as well as a precession of the orbit plane, the net effect being a uniform precession of the node on an invariant plane normal to the constant total angular momentum of the system. In general, the effect of the tesseral harmonics is to induce short-period perturbations of small amplitude in both the orbital and spin motions. Resonances are shown to exist whenever the orbital and rotational periods are commensurable. In any resonant state a single coordinate is found to represent both orbital and spin perturbations; and the system may be described as trapped in a localized potential well. The resultant spin and orbit librations are in phase with a common period. The relative amplitudes of the spin/orbit modes are determined by the characteristic parameter =M _a M _s a ² /3(M _a +M _s )C, wherea is the semimajor axis of the orbit, andC is the moment of inertia ofM _a about the rotation axis. When ga1, the solutions reduce to those for pureorbital resonance, in whichM _s librates in an appropriate reference frame while the rotation rate of the asymmetrical body remains constant. In the opposite extreme of 1, the solutions are appropriate to purerotational resonance, in which the orbital motion is unperturbed but the spin ofM _a librates. In each of these special cases the equations developed herein on the basis of a single theory are in agreement with those previously determined from separate theories of spin and orbital resonances.     

A unified treatment of some expansion procedures in perturbation theory: Lie series,Faà di Bruno operators,and Arbogast's Rule

The problem of expanding the transform of a function under a near-identity change of coordinates is reviewed. A common derivation is given for Musen's Faà di Bruno operators and Kamel's Lie series, and both are related to Arbogast's Rule for composing power series.     

A FORTRAN-based Poisson series processor and its applications in celestial mechanics

The present article describes the design and applications of our Poisson series subroutine package developed and maintained since 1968. The programs are written in standard FORTRAN-77 and are almost independent of the wordlength of the particular computer. The system has no restriction on the number of polynomial and angular variables and the storage allocations are completely automatic, invisible to the user.The nucleus of the system consists of about 20 basic traffic subroutines that handle the terms of the different series. Besides these subroutines, we have a number of I/O routines as well as arithmetic subroutines and a large number of Celestial Mechanics applications such as the classical expansions of the Kepler Problem and several expansions of Disturbing Functions. A preprocessor has also been built, allowing the user to write code in a high-level language, such as Jefferys' Trigman, and then translate it in our call-statements.The system was developed on several different computers: first on the Univac 1108 at the Jet Propulsion Laboratory in 1968, then the IBM 360-91 at UCLA, Los Angeles, California and finally the CDC 6600 and CYBER 170/750 at the University of Texas, Austin, Texas. The latest version is entirely geared towards the IBM-PC and compatibles.     

Present status of projective unified field theory and its physical predictions

In this article we shortly present the basic equations of our 5–dimensional Projective Unified Field Theory and their physical interpretation. Our main intention is to confront this theory with physical experience. For this reason we treat the Einstein effects (including the exact solutions for a central body) in detail. Further we once more discuss the first order effects on the quasi–Newtonian level with respect to the equivalence principle.     

A dynamic theory of type III solar radio bursts

All four large EUV bursts (peak 10–1030 Å flux enhancements 2 ergs cm^–2 s^–1 at 1 AU as deduced from sudden frequency deviations), for which there were available concurrent white light observations of at least fair quality, were detected as white light flares. The rise times and maxima of the white light emissions coincided with rise times and maxima of the EUV bursts. The frequency of strong EUV bursts suggests that white light flares may occur at the rate of five or six per year near sunspot maximum. All of the white light flare areas coincided with intense bright areas of the H flares. These small areas appeared to be sources of high velocity ejecta in H. The white light flares occurred as several knots or patches of 2 to 15 arc-sec diameter, with bright cores perhaps less than 2 arc-sec diameter (1500 km). They preferred the outer penumbral borders of strong sunspots within 10 arc-sec of a longitudinal neutral line in the magnetic field. The peak continuum flux enhancement over the 3500–6500 Å wavelength range is about the same order of magnitude as the peak 10–1030 Å flux enhancement.     

Development of planetary ephemerides EPM and their applications

This paper outlines the progress in development of the numerical planet ephemerides EPM—Ephemerides of Planets and the Moon. EPM was first created in the 1970s in support of Russian space flight missions and constantly improved at IAA RAS. Comparison between various available EPM ephemerides (EPM2004, EPM2008, EPM2011) is shown. The first results of the updated EPM2013 version which takes into account the two-dimensional annulus of small asteroids are presented. Currently two main factors drive the progress of planet ephemerides: dynamical models of planet motion and observational data, with the crucial role of spacecraft ranging. EPM ephemerides are the basis for the Russian Astronomical and Nautical Astronomical Yearbooks, are planned to use in the GLONASS and LUNA-RESOURCE programs, and are being used for determination of physical parameters: masses of asteroids, planet rotation parameters and topography, the \(GM_\odot \) and its secular variation, the PPN parameters, and the upper limit on the mass of dark matter in the Solar System. The files containing polynomial approximation for EPM ephemerides (EPM2004, EPM2008, EPM2011) along with TT–TDB and ephemerides of Ceres, Pallas, Vesta, Eris, Haumea, Makemake, and Sedna are available from ftp://quasar.ipa.nw.ru/incoming/EPM/. Files are provided in IAA’s binary and ASCII formats, as well as in the SPK format.     

二次多项式模型和灰色理论模型在接收机钟差预报中的应用和比较

为了更好地对接收机钟差作科学、准确的预报,基于实验室解算出的接收机钟差数据,分别采用二次多项式模型和灰色理论模型对接收机钟差进行了预报和分析.预报结果及其比较表明,对于需要用少量接收机钟差数据建立预报模型时,灰色理论模型预报不论是单点预报还是多点(30个点)预报,其预报精度都优于二次多项式预报模型,可以在实际工作中应用.     

A new theory for the formation of the maria and cayley type lunar regions

A new liquefaction theory for the origin of the flat marial and Cayley areas on the lunar surface is described. It is supposed that the flat terrain in these areas resulted from periods in the development of the Moon when these regions, although not liquid, had a sufficiently low viscosity for the surfaces to relax more or less completely to a level form. To account for this low viscosity a model is developed in which, within these regions and for relatively short periods in the early history of the Moon, preferentially high temperatures were maintained close to the lunar surface. The paper examines in some detail the possibility that these high temperatures may have resulted from instabilities in the lunar heat flow pattern caused by the presence of a surface layer of very low thermal conductivity produced by the debris of early meteorite impacts.A comparison is made between current models for the formation of the lunar surface and the theory here proposed: the advantages of the latter are enumerated and discussed.Normally at Queen Mary College, University of London, England.     

Duality rotations and type D solutions of Einstein-Born-Infeld theory

Within the non-linear electrodynamics of Born-Infeld type, constrained by the condition that admits the freedom of the duality rotations the explicit type D solutions, which generalize the charged Taub-NUT metric with cosmological constant, are constructed. The obtained type D solution exhausts all solutions within the considered class, assumed that the real eigenvectors of the electromagnetic field are aligned along the geodesic and shear-free principa null directions.     

A unified model for the evolution of galaxies and quasars

We incorporate a simple scheme for the growth of supermassive black holes into semi-analytic models that follow the formation and evolution of galaxies in a cold dark matter-dominated Universe. We assume that supermassive black holes are formed and fuelled during major mergers. If two galaxies of comparable mass merge, their central black holes coalesce and a few per cent of the gas in the merger remnant is accreted by the new black hole over a time-scale of a few times 10⁷ yr. With these simple assumptions, our model not only fits many aspects of the observed evolution of galaxies, but also reproduces quantitatively the observed relation between bulge luminosity and black hole mass in nearby galaxies, the strong evolution of the quasar population with redshift, and the relation between the luminosities of nearby quasars and those of their host galaxies. The strong decline in the number density of quasars from z ∼2 to z =0 is a result of the combination of three effects: (i) a decrease in the merging rate; (ii) a decrease in the amount of cold gas available to fuel black holes, and (iii) an increase in the time-scale for gas accretion. The predicted decline in the total content of cold gas in galaxies is consistent with that inferred from observations of damped Ly α systems. Our results strongly suggest that the evolution of supermassive black holes, quasars and starburst galaxies is inextricably linked to the hierarchical build-up of galaxies.     

A unified state model of orbital trajectory and attitude dynamics

The trajectory and attitude dynamics of an orbital spacecraft are defined by a unified state model, which enables efficient and rapid machine computation for mission analysis, orbit determination and prediction, satellite geodesy and reentry analysis. The state variables are momenta — a general form for attitude, and a parametric form for orbital motion. The orbital parameters are the velocity state characteristics of the orbital hodograph. The coordinate variables are sets of four Euler parameters, which define the rotation transformation by the quaternion algebra. The unified state model possesses many analytical properties which are invaluable for dynamical system synthesis, numerical analysis and machine solution: regularization, unified matrix algebra, state graphs and transforms. The analytic partials of position and velocity with the state and coordinate variables are presented, as well as representative perturbation functions such as air drag, gravitational potential harmonics, and propulsion thrust.     

Formation of binary systems and protoplanetary discs: A unified approach

We develop a new thermodynamic approach to the problem of the last stages of star formation, when a collapsing fragment evolves adiabatically into its final state: single protostar, surrounded or not by protoplanetary disc, or binary system. In this context, we point out the crucial role of the angular momentum transfer: a very efficient mechanism tends to form double stars with small mass secondaries, while a total decoupling yields twin binaries. Intermediate assumptions allow the birth of both kinds of binary systems, as well as the formation of not very massive protoplanetary discs. Discs of larger mass, which would be required to produce protoplanetary systems as a consequence of dynamical instabilities, do not form under any circumstances. A representation of the outcomes as functions of the corresponding initial conditions on the usual – plane gives well definite regions for single stars, protoplanetary discs, unbalanced systems and twin binaries. On this ground, a preliminary estimate of the percentage of stars surrounded by planetary systems is possible. A particular numerical simulation confirms the bimodality of the mass ratio distribution as well as the main features of the – plane partition. A few suggestions about non-adiabatic effects are also given. Our thermodynamic approach, supported by the numerical one and by the analysis of the observational statistics, allow to define a first unitary sketch for the formation of binary systems and protoplanetary discs.