Solving linearized equations of the N-body problem using the Lie-integration method

Several integration schemes exist to solve the equations of motion of the N -body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s, this method was applied to solve the equations of motion of the N -body problem by giving the recurrence formulae for the calculation of the Lie-terms. The aim of this work is to present the recurrence formulae for the linearized equations of motion of N -body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step-size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30–40 per cent faster than other integration methods.     

发生在1～2GHz的Ⅲ型爆发

对北京天文台动态频谱仪1996～1999年观测到的68群Ⅲ型爆发作了统计分析,并对这些事件的频率漂移、持续时间、偏振和带宽的基本特性作了定性分析.     

A new simple method for the analytical solution of Kepler's equation

A new simple method for the closed-form solution of nonlinear algebraic and transcendental equations through integral formulae is proposed. This method is applied to the solution of the famous Kepler equation in the two-body problem for elliptic orbits. The resulting formulae are quite elementary and, beyond their analytical interest, they can also provide quite accurate numerical results by using Gausstype quadrature rules.     

Tesseral harmonic perturbations in radial transverse and binormal components

The formulae for the perturbations in radial, transverse and binormal components of the Earth artificial satellite motion have been derived. Perturbations due to the tesseral part of the geopotential are considered. The geopotential expressed in terms of the orbital elements has the form proposed by Wnuk (1988). The formulae for the perturbations have been obtained using the Hori (1966) method. They can be effectively applied in calculation of the perturbations in the components including the coefficients of the high order and degree tesseral harmonics. The derived formulae reveal no singularities at zero eccentricity.     

Satellite prediction formulae for Vinti's model

For Vinti's dynamical problem, there is proposed a new form of solution wherein all three coordinates are expressed in terms of one independent variable. The formulae for the three co-ordinates are clear generalizations of the corresponding formulae for the Kepler problem while the independent variable corresponds to the true anomaly. The solution is completed by the relation connecting the independent variable with time: the latter is a generalization of the well known Kepler time-angle relationship. From the form and method of solution the main qualitative features of the motion are readily derived.     

Some investigations into the atmospheric drag problem

This paper calls into question the validity of the well-known formulae for the perturbations in the Keplerian elements, over one revolution of an orbit, for the motion of a drag-perturbed artificial satellite. These formulae are derived from Gauss's form of the planetary equations, by averaging over a single revolution of the orbit, and using the eccentric anomaly as the independent variable.It is shown that for light balloon-type satellites in near-circular orbits neither the eccentric anomaly nor the true longitude is a suitable choice of independent variable for the averaging procedure. Under these circumstances, it would seem that simple formulae for the variations in the elements cannot be derived from Gauss's equations.     

Magnetic field estimation in microwave radio sources

Eliminating the termN L, useful formulae for the magnetic field estimation in microwave radio sources are presented. Applications of these formulae to observed solar microwave radio bursts are shown.     

Analytical solutions for saturated P Cygni type profiles

Analytical formulae for single P Cygni type saturated resonance line profiles in stellar winds have been derived. The limbdarkening and presence of underlying intrinsic atmospheric profile have been ignored. The Sobolev approximation for radiative transfer has been used and the general velocity law has been specified by widely used β parameter. The analytical formulae for the saturated resonance line profiles can be found for cases when 2β is an integer. The formulae for 2β = 1,2, 3 and 4 have been found by us. Also the formulae for calculating the line profiles in the cases of external and internal sharp truncation (cutoff) of the scattering shell have been given. Some characteristic line profiles have been presented. It has been shown that the turbulence-generated isotropic dominant backscattering of radiation in stellar winds generates wide dark plateaux in the blue wings of spectral lines, and the slopes of plateaux are shaped by turbulence. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the numerical evaluation of theJ-integrals

This is a third paper dealing with the numerical evaluation of the light changes exhibited by close binary systems; for previous communications, Lanzano, 1976a, b.TheJ-integrals which were introduced by Kopal for the purpose of this numerical evaluation can be expanded in terms of the Appell hypergeometric series of the first kind. This relationship has been instrumental in establishing a number of recursion formulae for theJ-integrals applicable to the case of an annular eclipse. It was found that some recursion formulae hold both for the case of a partial and annular eclipse.The appropriate use of these recursion formulae should facilitate any numerical computation for eclipsing binaries.     

Displaced mass,depth, diameter,and effects of oblique trajectories for impact craters formed in dense crystalline rocks

Empirical formulae are presented for calculating the displaced mass, depth, diameter, and effects of oblique trajectories for impact craters formed in dense crystalline rocks. The formulae are applicable to craters with diameters from approximately 10^?3–10³ cm that require, respectively, impact kinetic energies of approximately 10 to 10¹⁶ ergs for their formation. The experimental results are in poor agreement with Öpik’s theoretical calculations and raise questions on the validity of his theoretical model.     

The vertical structure of atmospheric oscillations formulated by classical tidal theory

From the equations of classical tidal theory with Newtonian cooling (Chapman and Lindzen, Atmospheric Tides: thermal and gravitational, Reidel, 1970), formulae are obtained for wind, temperature and pressure oscillations generated by thermal, gravitational and lower-boundary excitations of given frequency. The analysis is an extension of that of Butler and Small (Proc. R. Soc. Lond.A274, 91, 1963) who formulated solutions of the vertical structure equation in terms of two independent solutions of the homogeneous equation and derived expressions for surface pressure oscillations. A comprehensive formulation is presented for wind, temperature and pressure oscillations as functions of height with the above-mentioned sources of excitation and an upper-boundary radiation condition. The formulae obtained are applied at the surface leading to evaluations of the surface oscillation weighting function W_p(z) which weights the thermal excitation at height z according to its differential contribution to the surface oscillation. The formulae are shown to simplify at heights above a region of excitation and evaluations are undertaken of the thermal response weighting function W_t(z) which weights the thermal excitation at height z according to its differential contribution to the oscillation at any height above the region of thermal excitation. Computational procedures are described for obtaining two independent solutions of the homogeneous equation and results are presented for an adopted profile of atmospheric scale height. The problem of deriving the surface pressure oscillation due to a tidal potential is briefly reviewed and results are presented as an example of the application of formulae that have been derived.     

Molecular distribution in the comae of H2O-comets: An analytic model

Density distribution in cometary comae resulting from photodissociation, ionization and ion-molecule reaction of H₂O is investigated in an analytic manner. It is assumed that particles expand isotropically around the nucleus, and that each species has its own constant radial velocity. Formulae for the density distribution of photochemical products are presented throught the coma, and approximate formulae are given for the distribution of ion-molecule reaction products in the inner coma. Characterictics of the density profile are discussed on the basis of these analytic formulae.     

New consideration of atmospheric refraction in laser ranging data

In this paper we reconsider the formulae of tropospheric refraction correction for the Satellite Laser Range technique. From the expansion of the complementary error function, a new continued fraction form of the mapping function at optical frequencies is derived. The correction terms related to the operation frequency of the laser beam are considered in both the zenith delay and the mapping function. The correction for low-elevation satellites is briefly reviewed. The theoretical accuracy of the new mapping function has been analysed via the ray tracing integrals under the standard atmospheric profile. With respect to the radiosonde data, the deviations of the new mapping function are investigated in an elevation range down to near 1°, which is comparable with the results of the Marini–Murray formulae .     

On the history of some explicit formulae for solving Kepler's equation

In this paper we examine, in their historical context, some approximate solutions for Kepler's equation. These explicit formulae, obtained by Trembley, Pacassi, Fergola, and Horrebow, had not a great diffusion and were thus often rediscovered by other astronomers. We will prove that the formulae are equivalent and, moreover, we will give an evaluation of the error. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elliptic integrals for cosmological constant cosmologies

Following the discussion of some general properties and analytical formulae for cosmological models with non-zero cosmological constant, we show how the elliptic integrals for comoving distance and light travel times as function of redshift can be expressed through Legendre integrals of the first and third kind, for which standard implementations are readily available. Observational properties are then illustrated for selected but typical models using the previously derived formulae.     

Accretion and erosion on the surface of the Moon

By considering the accumulation of dust within a lunar crater (formed by the effects of accretion and erosion), formulae are derived that give (a) the radius of the crater floor and (b) the depth of the crater, as functions of the age of the crater. Comparison of these formulae with the observed relationships provides a means of dating the craters in an absolute sense.     

On the potential–density relation for Stäckel discs

The general form of the surface density of an infinitely thin disc is given that generates a Sta¨ckel potential in the disc only, using formulae for the potential of elliptic and hyperbolic strings. This is useful for problems in which a simple form for the potential is important, while the corresponding surface density need only be known to check (numerically) that it is positive. A simple potential with a positive surface density is given. Also, formulae are given to calculate the surface density of such a Sta¨ckel disc, in the case in which the rotation curve is given and all the mass is concentrated in the disc.     

Fast computation of satellite gravitational gradient

The computation of the Earth's potential function at high order and degree with the method of reference [1], causes overflow most of the time. The normalized method [2–6] can eliminate the overflows, but leads to formulae much more involved than those in reference [1]; besides, the programming is more complex and the computer time required larger. The method presented in this paper has the following features: each component of the satellite gravitational gradient can be computed; the formulae are short and easy to be programme; the method is much quicker than the normalization method and can be carried out with a micro‐computer, without overflow even in the case of Earth's spherical harmonics of order and degree as high as 1025 or higher. This method satisfies the present demand to compute satellite gravitational gradient with high accuracy. Furthermore, we present formulae for the fast computation, without overflow, of the gravitational gradient corresponding to Earth's spherical harmonics up to order and degree of 3170 × 3170 or higher. This revised version was published online in July 2006 with corrections to the Cover Date.