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1.
In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.  相似文献   

2.
A comparison is drawn between the expansion of the potential in spherical harmonics on the one hand and in ellipsoidal harmonics on the other, with the objective of associating the spherical and ellipsoidal gravity coefficients of the Earth's potential.For this purpose the properties of orthogonality of the Lamé functions of the first kind have been tailored to this subject of investigation and become instrumental in establishing the mathematical expressions which relate the two classes of gravity coefficients to each other. In deriving the elements of the transition matrices elliptic integrals have been encountered whose reduction to the three kinds of canonical elliptic integrals is discussed.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.  相似文献   

3.
The model of Hidalgo (J. Geophys. Res. 108(A8):1320, 2003) has meant one more step in the understanding of the magnetic cloud events. We have modified a physical assumption over the y-components of the current density, have developed an algorithm which is able to resolve the elliptic integrals and have incorporated constrains over the parameters. This article is a new contribution for the improvement of the fitting procedure.  相似文献   

4.
By use of a new canonical transformation procedure, a third-order intermediary for planetary motion is developed. The intermediary contains all contributions that arise from the assumption of circular, coplanar orbits for the disturbing masses. The results are expressible in terms of elliptic integrals of the first, second, and third kinds.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.  相似文献   

5.
In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofF will be established for anyx i ; withn, N positive integers 1 and | i | fori=1, 2,...n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As a by-product of the analyses is an integral which we may call a complete elliptic integral of thenth kind, in which the known complete elliptic integrals (1st, 2nd and 3rd kinds) are special cases of it.  相似文献   

6.
Two new analytical solutions for Poinsot motion in terms of Euler parameters are derived. The first solution is a straightforward ‘universal’ (no branches) time series practical for short time motion calculations or as a basis for analytical continuation. The second, more involved solution is also universal but is not restricted to short times; it is in terms of circular, hyperbolic, and elliptic functions and elliptic integrals.  相似文献   

7.
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, K(m) and E(m), for the standard domain of the elliptic parameter, 0 < m < 1. For the case 0 < m < 0.9, the method utilizes 10 pairs of approximate polynomials of the order of 9–19 obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, K(1 − m) and E(1 − m), are first computed by a pair of the approximate polynomials and then transformed to K(m) and E(m) by means of Jacobi’s nome, q, and Legendre’s identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody’s Chebyshev polynomial approximation of Hastings type and Innes’ formulation based on q-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, sn(u|m), cn(u|m), and dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, u, after its domain is reduced to the standard range, 0 ≤ u < K(m)/4, with the help of the new method to compute K(m). The new procedure is 25–70% faster than the methods based on the Gauss transformation such as Bulirsch’s algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of K(m) is not taken into account.  相似文献   

8.
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.  相似文献   

9.
The local character of self-gravity along with the number of spatial dimensions are critical issues when computing the potential and forces inside massive systems like stars and disks. This appears from the discretisation scale where each cell of the numerical grid is a self-interacting body in itself. There is apparently no closed-form expression yet giving the potential of a three-dimensional homogeneous cylindrical or spherical cell, in contrast with the Cartesian case. By using Green’s theorem, we show that the potential integral for such polar-type 3D sectors—initially, a volume integral with singular kernel—can be converted into a regular line-integral running over the lateral contour, thereby generalising a formula already known under axial symmetry. It therefore is a step towards the obtention of another potential/density pair. The new kernel is a finite function of the cell’s shape (with the simplest form in cylindrical geometry), and mixes incomplete elliptic integrals, inverse trigonometric and hyperbolic functions. The contour integral is easy to compute; it is valid in the whole physical space, exterior and interior to the sector itself and works in fact for a wide variety of shapes of astrophysical interest (e.g. sectors of tori or flared discs). This result is suited to easily providing reference solutions, and to reconstructing potential and forces in inhomogeneous systems by superposition. The contour integrals for the 3 components of the acceleration vector are explicitely given.  相似文献   

10.
A fundamentally new approach to an elliptic Gaussian ring has been developed. It has been ascertained that it can be produced from a uniform plane elliptic disk by mass balayage into an elementary homothetic layer with the center of homothety at an ellipse focus. An advantage of new interpretation is in the fact that the spatial potential of a Gaussian ring is expressed in terms of the potential of a uniform elliptic disk, well-known in the finite form, and its derivatives. A general formula for the potential of a two-dimensional homothetic layer has been derived with this purpose. As a result, the potential of a Gaussian ring is represent-able in a simple analytical form in terms of standard complete elliptic integrals in both elliptic and Cartesian coordinates. The mass asymmetry along the ring is considered explicitly. The potential formulas are verified numerically and have no singular points at ellipse foci. Particular cases are considered; the 3D potential surface and system of equipotentials are constructed. Knowledge of the potential extends the range of application of a Gaussian ring in the problem of calculation of secular perturbations in celestial mechanics.  相似文献   

11.
The gravitational potential due to uniform disks and rings   总被引：1，自引：0，他引：1  
The gravitational potential due to thin uniform disks and rings is obtained in closed form in terms of complete elliptic integrals.  相似文献   

12.
This paper deals with the existence and the stability of the libration points in the restricted three-body problem when the smaller primary is an ellipsoid. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the collinear and non collinear libration points and their stability. This is observed that there exist five collinear libration points and the non collinear libration points are lying on the arc of the unit circle whose centre is the bigger primary. Further observed that the libration points either collinear or non-collinear all are unstable.  相似文献   

13.
The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.  相似文献   

14.
We present an efficient,robust computational method for modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies.With appropriate mathematical treatments,the apparent numerical difficulties associated with singularities in computing elliptic integrals are completely removed.Using a boundary element discretization procedure,the governing equations are transformed into a linear algebra matrix equation that can be solved by straightforward Gauss elimination in one step without further ...  相似文献   

15.
The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves.  相似文献   

16.
An analytical solution in terms of elliptic integrals is found for the problem connected to the isolthermal collapse of a gaseous sphere. In this way the time-dependence of spatial density distribution is promptly obtained in each point inside the sphere itself. An application for interstellar clouds is given.  相似文献   

17.
Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.  相似文献   

18.
An analytic solution for theJ 2 perturbed equatorial orbit is obtained in terms of elliptic functions and integrals. The necessary equations for computing the position and velocity vectors, and the time are given in terms of known functions. The perturbed periapsis and apoapsis distances are determined from the roots of a characteristic cubic.  相似文献   

19.
A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail.  相似文献   

20.
The differential equations of planetary theory are solved analytically to first order for the two-dimensional case, using only Jacobian elliptic functions and the elliptic integrals of the first and second kind. This choice of functions leads to several new features potentially of importance for planetary theory. The first of these is that the solutions do not require the expansion of the reciprocal of the distance between two planets, even for those variables which depend on two angular arguments. A second result is that the solution is free from small divisors with the exception of two special resonances. In fact, not only are the solutions for resonant orbits free from small divisors, the perturbations for all variables are expressible in closed form. A subset of the resonant orbits maintains this form and in addition has the remarkable feature that the first order perturbations are purely periodic; they contain no secular terms. A solution for the 13 resonance case is given as an example.  相似文献   

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