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1.
This paper is a sequel to an earlier article of the same title. The two formal analytical solutions of the Ideal Resonance Problem developed respectively by Garfinkel and Jupp are here compared, atsecond-order in the appropriate small parameter, with numerical integrations; the second-order circulation solution for Jupp's theory being presented for the first time. It transpires that throughout most of the deep resonance regime the second-mentioned solution provides greater accuracy. In addition, it is demonstrated that the first solution is not appropriate when general initial values of the variables are prescribed. 相似文献
2.
The second-order solution of the Ideal Resonance Problem, obtained by Henrard and Wauthier (1988), is developed further to fourth order applying the same method. The solutions for the critical argument and the momentum are expressed in terms of elementary functions depending on the time variable of the pendulum as independent variable. This variable is related to the original time variable through a Kepler-equation. An explicit solution is given for this equation in terms of elliptic integrals and functions. The fourth-order formal solution is compared with numerical solutions obtained from direct numerical integrations of the equations of motion for two specific Hamiltonians. 相似文献
3.
Paul E. Nacozy 《Astrophysics and Space Science》1971,14(1):40-51
The numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes. 相似文献
4.
A. G. Mavraganis 《Astrophysics and Space Science》1975,37(2):365-369
The matrizants of periodic solutions in Störmer's problem, because of their symplectic character, can be transformed by means of multiplication with constant matrices into symmetric ones. As a result the six bilinear relations between their elements, existing on account of the symplectic property, are replaced by 14 linear and simple forms. This fact is very useful in numerical integrations where these relations are used as criteria of accuracy. 相似文献
5.
Lei Chen Shengmiao Wu Feng Yuan 《Monthly notices of the Royal Astronomical Society》2009,398(4):1900-1904
We consider a thin accretion disc warped due to the Bardeen–Petterson effect, presenting both analytical and numerical solutions for the situation in which the two viscosity coefficients vary with radius as a power law, with the two power-law indices not necessarily equal. The analytical solutions are compared with numerical ones, showing that our new analytical solution is more accurate than the previous one, which overestimated the inclination change in the outer disc. Our new analytical solution is appropriate for moderately warped discs, while for extremely misaligned discs only a numerical solution is appropriate. 相似文献
6.
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. 相似文献
7.
This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees
of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double
oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed
by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions.
The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy
of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space
of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical
integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented
here conservation of the energy constant between 10-9 and 10-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the
number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions
were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter
any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the
boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that
investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas
of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of
sections.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
Anna M Nobili 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):293-304
Modern computer technology allows dynamical astronomers to investigate the long term stability of real systems as thoroughly as ever. However, the process is not straightforward and new problems need to be solved. This work deals with only one such problem: the construction-from the numerical integration- of a secular perturbation theory that is able to describe the dynamical behavior of the system. The discussion refers to the outer planets and is based on the knowledge acquired by the author during her participation in project LONGSTOP. A digital filter is used in order to reduce the output and eliminate short periodic terms. Filtering uncovers long term variations in the semimajor axes. From the filtered output a secular perturbation theory is constructed in the assumption that the solution is regular, as secular perturbation theories can only be constructed for regular solutions. If we succeed, this means that the solution is indeed regular for the computed span of time; if not-and this can be established in a rigorous way-it has to be concluded a posteriori that the solution is not regular. The LONGSTOP 1A and 1B integrations show well that as the timespan of the integration increases it is possible to detect the non-regular behavior of the solution. This happens in the eccentricity of Saturn at the 10–4 level. 相似文献
9.
10.
Marc Fouchard Christiane Froeschlé John J. Matese Giovanni Valsecchi 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):229-262
Different models of the action of the galactic tide are compared. Each model is a substitute for direct numerical integrations
allowing a drastic decrease of the computation time. The models are built using two different techniques, (i) averaging of
the fast variable (the mean anomaly) over one cometary period and (ii) fixing the comet in its aphelion direction. Moreover,
we consider two different formalisms (Lagrangian and Hamiltonian) and also two different sets of variables. As expected, we
find that the model results are independent of the formalism and the set of variables considered, and are highly accurate,
whereas mathematical technique leads to poor results. In order to further reduce the computation time, mappings are built
from the development of the solution of the models. We show that for these mappings, the set of variables giving the most
accurate results is strongly dependent on the cometary eccentricity, e, and semimajor axis, a. 相似文献
11.
New high-precision, semianalytical and numerical solutions to the problem of the rotational motion of the Moon are obtained, for use in the long 418.9-year time frame. The dynamics of the rotational motion of the Moon is studied numerically using the Rodrigues-Hamilton parameters, relative to the fixed ecliptic for the epoch J2000. The results of the numerical solution to the problem under study are compared with a compiled semianalytical theory of Moon rotation (SMR). The initial conditions for the numerical integration have been taken from the SMR. The comparative discrepancies derived from the comparison between the numerical solutions and the SMR do not exceed 1.5″ on the time-scale of 418.9 yr. The investigation of the comparative discrepancies between the numerical and semianalytical solutions is performed using the least squares and spectral analysis methods in the Newtonian case. All the periodic terms describing the behavior of the comparative discrepancies are interpreted as the corrections to the semianalytical SMR theory. As a result, the series are constructed to describe the rotation of the Moon (MRS2010) in the time interval under study. The numerical solution for the Moon’s rotation has been obtained anew, with new initial conditions calculated using MRS2010. The discrepancies between the new numerical solution and MRS2010 do not exceed 20 arc milliseconds on the time-scale of 418.9 years. The results of the comparison suggest that that the MRS2010 series describe the rotation of the Moon more correctly than the SMR series. 相似文献
12.
R. Broucke J. D. Anderson L. Blitzer E. Davoust H. Lass 《Celestial Mechanics and Dynamical Astronomy》1981,24(1):63-82
The article describes the solutions near Lagrange's circular collinear configuration in the planar problem of three bodies with three finite masses. The article begins with a detailed review of the properties of Lagrange's collinear solution. Lagrange's quintic equation is derived and several expressions are given for the angular velocity of the rotating frame.The equations of motion are then linearized near the circular collinear solution, and the characteristic equation is also derived in detail. The different types of roots and their corresponding solutions are discussed. The special case of two equal outer masses receives special attention, as well as the special case of two small outer masses.Finally, the fundamental family of periodic solutions is extended by numerical integration all the wap up to and past a binary collision orbit. The stability and the bifurcations of this family are briefly enumerated. 相似文献
13.
Nader Haghighipour † 《Monthly notices of the Royal Astronomical Society》2000,316(4):845-855
An analytical treatment of the evolutionary dynamics of a three-body planetary system subject to dynamical friction with an interplanetary medium is presented. The analysis presented here is in connection with the results of numerical integrations of such systems recently published by Haghighipour. Using the method of partial averaging near a resonance, the dynamics of a restricted, circular, planar three-body system, with the inner body more massive, is studied and the time variation of quantities such as the orbital angular momentum and the eccentricity of the outer planet, which were previously obtained from numerical integrations, is analytically verified. 相似文献
14.
15.
Ahmed Kamel Donald Ekman Richard Tibbitts 《Celestial Mechanics and Dynamical Astronomy》1973,8(1):129-148
A closed form solution, for longitude and semimajor axis deviations in the neighborhood of a prespecified station, is obtained for nearly synchronous satellites. The model use includes the important terms in Earth's zonal and tesseral harmonics as well as the luni-solar perturbations. The initial semimajor axis for two-maneuver east-west stationkeeping is then deduced. Due to the luni-solar effects, it is found that the initial semimajor axis deviation from synchronous orbit value is highly dependent on the initial position of the satellite relative to the Moon and the Sun. Verifications of the results by means of numerical integrations are also included. 相似文献
16.
G. A. Wilkins 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):368-368
At present the fundamental lunar ephemeris is based on Brown's theory of the motion of the Moon with improvements based on the bypassing of Brown's Tables, the removal of the great empirical term, the substitution of the relevant constants of the IAU system of astronomical constants and the retransformation of Brown's series in rectangular coordinates to spherical coordinates. Even so this ephemeris does not represent adequately the recent range and range-rate radio observations, and it will be inadequate for use in the analysis of laser observations of corner reflectors on the Moon. Numerical integrations for these purposes have already been made at the Jet Propulsion Laboratory, but improved theoretical developments are also required; new solutions of the main problem are in hand elsewhere. Work at H.M. Nautical Almanac Office is aimed at obtaining improved values of the constants of the lunar orbit by a rediscussion of occultation observations made since 1943 and at the redevelopment of the series for the planetary perturbations using more precise theories of the motion of the Sun and planets. The techniques and preliminary results of exploratory numerical integrations were briefly described.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, 17–23 August, 1969. 相似文献
17.
Similarity solutions, using Planck's diffusion approximation, for propagation of strong plane shock waves in an optically thin grey atmosphere of uniform density are obtained. Solutions for the ratio of specific heats 4/3 and several values of the radiation parameter are compared with those in the absence of any radiation. Excerpts from the results of integrations for different values of the ratio of specific heats are also given in Tables I-III. 相似文献
18.
The use of precise numerical integrations and Fourier analysis techniques allowed us an investigation of the regular motions of asteroids near the 3:2 resonance with Jupiter (Hildas). The results are shown and compared to similar results previously obtained with analytical models. 相似文献
19.
《New Astronomy》2020
The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time. 相似文献
20.
T. N. Krishnamurti H. S. Bedi G. Rohaly M. Fulakeza D. Oosterhof K. Ingles 《Global and Planetary Change》1995,10(1-4)
The results of several seasonal integrations with an atmospheric global circulation model with prescribed “perfect” sea surface temperatures are presented. These experiments illustrate the results of seasonal simulations for the years 1987 and 1988. These were a dry and a wet monsson year, respectively, when compared to the average. The integrations cover the period from June 1 through August 31 for both years and were carried out at two horizontal resolutions, T42 and T106, of a global model containing two different parameterizations of surface hydrology. The seasonal differences of the motion fields, divergent circulations and rainfall distributions for these respective experiments are compared with the corresponding observed fields.The sensitivity of seasonal simulations to the initial state is explored with integrations starting on two successive dates. In these experiments we diagnose differences of the simulated time mean states from residue free budgets of the complete vorticity equation. 相似文献