Poisson Series Research Articles

Harmonic development of tide-generating potential of terrestrial planets

The aim of this study is to obtain high-accurate harmonic developments of the tide-generating potential (TGP) of Mercury, Venus and Mars. The planets’ TGP values have been first calculated on the base of DE/LE-406 numerical planetary/lunar ephemerides over a long period of time and then processed by a new spectral analysis method. According to this method the development is directly made to Poisson series where both amplitudes and arguments of the series’ terms are high-degree polynomials of time. A new harmonic development of Mars TGP is made over the time period 1900 AD–2100 AD and includes 767 second-order Poisson series’ terms of minimum amplitude equal to 10−7 m2 s−2. Analogous series composing both Mercury and Venus TGP harmonic models are built over the time period 1000 AD–3000 AD and include 1,061 and 693 terms, respectively. A modification of the standard HW95 format for representation of the terrestrial planets’ TGP is proposed. The number of terms in the planets’ TGP models transformed to the modified HW95 format is 650 for Mercury, 422 for Venus, and 480 for Mars. The quality of the new developments of the terrestrial planets’ TGP is better than that of the similar developments obtained earlier.

Semi-analytical integration algorithms based on the use of several kinds of anomalies as temporal variable

One of the main problems in celestial mechanics is the management of long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system. In this work we shall develop a software package that is suitable for managing these objects. This package includes algorithms to obtain the inverse of the distance based on an iterative method, a set of integration algorithms according to several sets of temporal variables. This paper contains a comparative study on the use of the true, eccentric, and elliptic anomalies in semi-analytical methods on celestial mechanics.

Computational tools to construct semi-analytical planetary theories

The main aim of this paper is the construction of a set of computational tools in order to improve the management of large expansions used in celestial mechanics. This package involves an iterative algorithm to develop the inverse of the distance according to an appropriate set of temporal variables, a new Poisson series processor, and an iterative integration method to obtain the solution of Lagrange planetary equations. The method has been applied to compute a set of examples using several kinds of anomalies in order to test the performance of the algorithms.

An improved algorithm to compute circular functions of Poisson series

The efficiency in the computation of circular functions, such as cos(u) or sin(u), where u is a Poisson series, is important to derive accurate solutions of many problems of Celestial Mechanics, for instance, the orbital or rotational perturbed motion of natural or artificial bodies, since expansions in terms of Legendre functions and multiple Fourier series appear almost everywhere. Therefore, it is worth searching for alternative algorithms with lower computational cost. In this article, we propose a method based on the idea of elimination, which was originally applied to solve numerical problems, mainly in the case of matrix functions. Our comparisons with the traditional Taylor expansion prove that this new method can be more efficient when applied to compute the sine and cosine of a Poisson series.

Long-term harmonic development of lunar ephemeris

Aims. The aim of this study is to develop new analytical series representing lunar coordinates to accuracy compatible with the accuracy of the modern numerical ephemeris of the Moon. Methods. An improved method of spectral analysis of tabulated function is used to make harmonic development of the latest long-term numerical ephemeris of the Moon LE-406 which covers a six thousand-year interval. A feature of the method is that the development is made directly to Poisson series where both amplitudes and arguments of the series’ terms are high-degree polynomials of time. Results. The new analytical development includes 42 270 Poisson series’ terms of minimal amplitude equivalent to 1 cm and is valid over 1500–2500. A simplified version of the development includes 7952 series’ terms of minimal amplitude equivalent to 1 m and is valid over 3000BC–3000AD. Over 1500–2500 the maximum difference between lunar coordinates calculated by means of the new analytical series and numerical ephemeris LE-406 is 3.2 m in geocentric distance, 0. �� 0056 in ecliptic longitude, and 0. �� 0018 in ecliptic latitude. This is 9–70 times better than the accuracy of the latest analytical theory of lunar motion ELP/MPP02, and the number of terms in the new development is less than that in ELP/MPP02.

Bernard Katz, quantal transmitter release and the foundations of presynaptic physiology

Bernard Katz, quantal transmitter release and the foundations of presynaptic physiology

S-power series: an alternative to Poisson expansions for representing analytic functions

Morin and Goldman [Computer Aided Geometric Design 17 (2000) 813] have recently presented a remarkable new framework, based on employing Poisson series, for describing analytic functions in CAD. We compare this Poisson formulation with s-power series, modified Newton series that can be regarded as the two-point analogue of Taylor expansions. Such s-power series yield, over finite intervals, better approximations for CAD purposes, as they are polynomial and hence expressible in the Bernstein–Bézier standard, can be pieced together in a smooth Hermitian spline and, in general, display better convergence.

Behaviour of a two-planetary system on a cosmogonic time-scale

The orbital evolution of planetary systems similar to our Solar one represents one of the most important problems of Celestial Mechanics. In the present work we use Jacobian coordinates, introduce two systems of osculating elements, construct the Hamiltonian expansions in Poisson series for all the elements for the planetary three-body problem (including the problem Sun–Jupiter–Saturn). Further we construct the averaged Hamiltonian by the Hori–Deprit method with accuracy up to second order with respect to the small parameter, the generating function, the change of variables formulae, and the right-hand sides of the averaged equations. The averaged equations for the Sun–Jupiter–Saturn system are integrated numerically over a time span of 10 Gyr. The Liapunov Time turns out to be 14 Myr (Jupiter) and 10 Myr (Saturn).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

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.

The lunar theory ELP revisited. Introduction of new planetary perturbations

On the basis of the semi-analytical theory ELP, a new solution has been built that makes use of the planetary perturbations MPP01 constructed by P. Bidart. This new solution, called ELP/MPP02, is an analytical solution that contains all the perturbations to represent the lunar motion. The level of truncation for the series is the centimeter. This limit induces analytical developments of a reasonable dimension. ELP/MPP02 is compared to the JPL ephemerides DE405 and DE406. After fitting the constants and the reference frame to DE405, over a few centuries, numerical comparisons with DE405 show a significant improvement in precision, in particular for the radius vector. Over the long range of several millennia, comparisons with DE406 show an improvement in the long periodic perturbations due to planets. On the time interval of one century around J2000 we added to ELP/MPP02 numerical complements based on the differences with DE405 in such a way that ELP/MPP02+ has the precision of the numerical integration. Using this new ephemeris to analyse LLR data provided since 1970, we build the solution ELP/MPP02(LLR) fitted to LLR observations. Various approximations have been also tested on the residuals DE406 - ELP/MPP02. We first determined corrections to the secular variations of the mean longitude, longitudes of the node and perigee under a simple polynomial form; they reduce the differences between ELP/MPP02 and DE406 to less than 3 arcsec in longitude and latitude and 2 km in distance over the whole time interval of DE406 [ -3000; +3000] . Next, the differences themselves between the two solutions are approximated over 2 millennia with Poisson series in a pseudo analytical form similar to ELP/MPP02. We reduce the residuals to less than 0.03 arcsec in longitude and latitude and 30 m in distance.

Risk of radiogenic thyroid cancer in the population of the Bryansk and Oryol regions of Russia after the Chernobyl accident (1991–1998)

The manuscript presents results of the radiogenic thyroid cancer risk analysis in the Bryansk and Oryol regions among children and adolescents at exposure (0–17 years of age). A total of 170 cases of thyroid cancer were diagnosed from 1991 to 1998. Of these, 106 cases were in the Bryansk and 64 in the Oryol region. The size of the exposed population under study is 374 447 persons in the Bryansk region and 207 592 persons in the Oryol region (data of the 1989 census). The mean thyroid dose from incorporated isotopes of 131I for children and adolescents at exposure is 0.071 Gy in the Bryansk region and 0.013 Gy in the Oryol region. The method of maximum likelihood for nonstationary Poisson series of events was used for risk calculation. The analysis focuses on the relationship between thyroid cancer incidence and thyroid dose due to incorporated iodine radioisotopes. The calculations were based on personal data about disease cases and demographic and dose information for the population points in the regions under study. Statistically significant excess of thyroid cancer incidence above the spontaneous rate was obtained for children and adolescents of the Bryansk region. The excess relative risk per unit dose of 1 Gy (ERR 1Gy) with 95% confidence intervals is 11.9 (7.2, 16.6). For children and adolescents of the Oryol region between 1991 and 1998 statistically significant excess risk was not detected: 6.5 (−20.2, 30.2). The value of the standardized ratio of spontaneous thyroid cancer incidence in the region to the general incidence in Russia is 3.5 with 95% confidence intervals (2.8, 4.2) for the Bryansk region. For the Oryol region this ration is 5.3 (4.0, 6.6). The excess of spontaneous incidence observed in the study areas is primarily due to regional differences in incidence and registration features (screening effect). The attributive risk of induction of radiogenic cancers among children and adolescents in the Bryansk region is 46%. This mean that one of every two thyroid cancers detected in the Bryansk region between 1991 and 1998 are radiation induced.

The Expansion of the Hamiltonian of the Two-Planetary Problem into a Poisson Series in All Elements: Estimation and Direct Calculation of Coefficients

This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper.

Ptolemaic transformation in Keplerian problem

We propose the Ptolemaic transformation: a canonical change of variables reducing the Keplerian motion to the form of a perturbed Hamiltonian problem. As a solution of the unperturbed case, the Ptolemaic variables define an intermediary orbit, accurate up to the first power of eccentricity, like in the kinematic model of Claudius Ptolemy. In order to normalize the perturbed Hamiltonian we modify the recurrent Lie series algorithm of HoriuuMersman. The modified algorithm accounts for the loss of a term's order during the evaluation of a Poisson bracket, and thus can be also applied in resonance problems. The normalized Hamiltonian consists of a single Keplerian term; the mean Ptolemaic variables occur to be trivial, linear functions of the Delaunay actions and angles. The generator of the transformation may serve to expand various functions in Poisson series of eccentricity and mean anomaly.

A New Symbolic Processor for the Earth Rotation Theory

In this paper we present a new symbolic processor specially suited for the Earth rotation theory. This processor works with a more general kind of Poisson series called Kinoshita series, which has resulted to be very useful in the Earth rotation theory. Its structure is adapted for dealing with the more general analytical expressions that appear in the Earth rotation theory. This new algebraic processor has been successfully used for computing different contributions to the nutation series of the rigid Earth.

Algebraic and Symbolic Manipulation of Poisson Series

Efficiency in handling Poisson series is essential to obtain high-accuracy analytical theories in celestial mechanics and non-linear dynamics in general. A good knowledge of the mathematical structure of these objects is fundamental to create data structures to store and handle efficiently its equivalent computational object. In this paper we analyse the mathematical, symbolic and computational structure of Poisson series.

Communications between the Poisson series processor PSPC and general scientific software

Special efficient Poisson series processors (PSP) have been created. Poison series appear frequently in problems of non-linear dynamics and celestial mechanics and their size makes their manipulation by means of general computer algebra systems (CAS) inefficient, but the characteristics of the problems suggest the use of general CAS with other general scientific software like compilers, libraries of compiled programs or word processors. To combine these specialized and general tools we need to communicate between them. In this paper we describe the tools to share information between PSPC, our own PSP, and other general software.

The Expansion of the Hamiltonian of the Planetary Problem into the Poisson Series in All Keplerian Elements (Theory)

The study of the evolution of planetary systems, primarily of the Solar System, is one of the basic problems of celestial mechanics. The stability of motion of giant planets on cosmogonic time scales was established by numerical and analytical methods, but the question about the evolution of orbits of terrestrial planets and arbitrary solar-type planetary systems remained open. This work initiates a series of papers allowing one to advance in solving the problem of the evolution of the solar-type planetary systems on cosmogonic time scales by using powerful analytical tools. In the first paper of this series, we choose the optimum reference system and obtain the Poisson series expansion of the Hamiltonian of the problem in all Keplerian elements. We propose to use the integral representation of the corresponding coefficients or the Poisson processor means instead of conventionally addressing any possible special functions. This approach extremely simplifies the algorithm. The next paper of this series deals with the calculation of the expansion coefficients.

A New Echeloned Poisson Series Processor (EPSP)

A specialized Echeloned Poisson Series Processor (EPSP) is proposed. It is a typical software for the implementation of analytical algorithms of Celestial Mechanics. EPSP is designed for manipulating long polynomial-trigonometric series with literal divisors. The coefficients of these echeloned series are the rational or floating-point numbers. The Keplerian processor and analytical generator of special celestial mechanics functions based on the EPSP are also developed.

Solving problems symbolically by using Poisson Series Processors

article Free Access Share on Solving problems symbolically by using Poisson Series Processors Authors: J. F. San Juan Departamento de Matemhticas y Computación, Universidad de la Rioja, 26004 Logroño, Spain Departamento de Matemhticas y Computación, Universidad de la Rioja, 26004 Logroño, SpainView Profile , A. Abad Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, SpainView Profile , S. Serrano Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, SpainView Profile , A. Gavín Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, SpainView Profile Authors Info & Claims ACM SIGSAM BulletinVolume 33Issue 3Sept.1999 pp 18–19https://doi.org/10.1145/347127.347317Published:01 September 1999Publication History 0citation39DownloadsMetricsTotal Citations0Total Downloads39Last 12 Months3Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Stochastic resonance as an inherent property of rate-modulated random series of events

At the molecular level, processes in physics and biology are intrinsically random. Quite often they can be described as random series of elementary events, Poisson trains, whose event generation rate is a function of an input driving parameter. We analyze the transduction of weak signals by such trains in the presence of additive input noise in terms of transfer coefficient and output signal-to-noise ratio. We show that the noise-facilitated signal transduction, stochastic resonance (SR), is an inherent property of parameter-dependent Poisson trains with a nonlinear relationship between the event generation rate, r( V( t)), and the driving parameter, V( t). We obtain a sufficient condition for the SR onset that shows that for the input noise with a sufficiently broad bandwidth, SR can be observed in a wide variety of such systems. If this relationship is represented by r( V( t)) ∝ 1 + Σ 1 3 r n V n ( t), then the system is capable of SR if 6r 3 r 1 − r 2 > 0 Addition of small noise with a broad bandwidth to the system input increases the output signal quality. To highlight the mechanism of noise-facilitated signal transduction in such systems, we also analyze the “linearizing” effect of the additive noise on signal transduction in rate-modulated Poisson series.