Solution of canonical equations which result from elimination of short-periodic terms in second-order planetary theory     

Jean Meffroy 《Earth, Moon, and Planets》1982,26(2):143-169

We solve the first order non-linear differential equation and we calculate the two quadratures to which are reduced the canonical differential equations resulting from the elimination of the short period terms in a second order planetary theory carried out through Hori's method and slow Delaunay canonical variables when powers of eccentricities and the sines of semi-inclinations which are >3 are neglected and the eccentricity of the disturbing planet is identically equal to zero. The procedure can be extended to the case when the eccentricity of the disturbing planet is not identically equal to zero. In this latter general case, we calculatedthe two quadratures expressing angular slow Delaunay canonical variable ₁ of the disturbed planet and angular slow Delaunay canonical variable ₂ of the disturbing planet in terms of timet.  相似文献   

The elimination of the critical terms of a first order Uranus-Neptune theory by Hori's method     

Osman M. Kamel 《Earth, Moon, and Planets》1983,28(3):221-245

In this part we determine the value ofS ₁, and in terms of the canonical variables of H. Poincaré. A complete solution of the auxiliary system of equations generated by the Hamiltonian is presented.  相似文献   

Lagrange variational equations from Hori's method for canonical systems     

Wagner Sessin 《Celestial Mechanics and Dynamical Astronomy》1983,29(4):361-366

Hori, in his method for canonical systems, introduces a parameter through an auxiliary system of differential equations. The solutions of this system depend on the parameter and constants of integration. In this paper, Lagrange variational equations for the study of the time dependence of this parameter and of these constants are derived. These variational equations determine how the solutions of the auxiliary system will vary when higher order perturbations are considered. A set of Jacobi's canonical variables may be associated to the constants and parameter of the auxiliary system that reduces Lagrange variational equations to a canonical form.  相似文献   

On the radial intermediaries and the time transformation in satellite theory     

R. Cid  S. Ferrer  M. L. Sein-Echaluce 《Celestial Mechanics and Dynamical Astronomy》1986,38(2):191-205

Taking advantage of the radial intermediaries and the regularization and linearization methods, the zonal Earth satellite theory is studied in the polar nodal canonical set of variables (, , ,R, ¡,N).The variable is eliminated in the first order of the Hamiltonian by applying Deprit's method. Then, the elimination of the perigee is carried out by another canonical transformation. As a consequence, a new radial intermediary, which contains all theJ _2n(n1) harmonics, is given. A comparison with the previous radial intermediaries of Cid and Lahulla, Deprit and Alfriend and Coffey is made.Finally, a regularizing transformation which allows us to linearize part of the radial intermediary is proposed, and an analytical study of this process is presented.  相似文献   

Canonical solution of the two critical argument problem     

Jack Wisdom 《Celestial Mechanics and Dynamical Astronomy》1986,38(2):175-180

The solution by Sessin and Ferraz-Mello (Celes. Mech. 32, 307–332) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 21 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem.  相似文献   

On nonclassical canonical systems     

G. Scheifele 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):296-310

Generalizations in the canonical theory of dynamics are made; at first transformations which augment the number of canonical variables, and secondly differential transformations of the independent variable are outlined. This is applied to the perturbed two-body problem. The results are canonical systems using independent variables other than time. This leads to Delaunay-similar sets of 8 canonical elements when the Jacobian equation is separable. The application of the theory to the KS-transformation yields a completely regular canonical system in a 10-dimensional phase-space, using the eccentric anomaly as independent variable. Subsequently sets of 10 regular canonical elements are introduced.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.  相似文献   

The main problem of lunar theory solved by the method of Brown     

Dieter S. Schmidt 《Celestial Mechanics and Dynamical Astronomy》1982,26(1):75-75

Brown's method for solving the main problem of lunar theory has been adapted for the computation by machine. The computations are carried out with the help of an algebraic processor called POLYPAK, which can manipulate power series in several real or complex variables. Brown's result have been recovered and refined first and the solution, in Cartesian coordinates to the sixth order, has been compared to the work of Eckert (see Gutzwiller, 1979). The solution has then been expanded to include most terms through order nine. This order is necessary to get an accuracy of 0.00001 for the terms in longitude and latitude and of 0.000001 for the terms in the sine parallax. A preliminary comparison with the theories of Chapront and Henrard (1980) indicates that the solution has an accuracy which is close to the one desired. For details see Schmidt (1980).The next step in developing a complete lunar theory requires the computation of the partial derivatives of the solution with respect to the primary parameters. Since Brown's method gives a semianalytical solution only the derivative with respect ton, the mean motion of the Moon, is difficult to compute. It is possible to find this derivative with one quadrature from the other derivatives if one takes into account that the Jacobian has to be a symplectic matrix when a canonical set of primary parameters is used.The mean motions of the perigee and node often exhibit the largest discrepancies among the different theories. Therefore it is not too surprising that also their derivatives show significant differences. It is hoped by providing another independent computation of the derivatives by a different method their accuracy can be improved.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   

Delaunay normalisations     

André Deprit 《Celestial Mechanics and Dynamical Astronomy》1982,26(1):9-21

Too many terms are generated by a Delaunay normalisation when the perturbation is developed in powers of the eccentricity. Ways of bypassing the expansion are discussed. There are: (i) Brouwer's method of implicit variables; (ii) the preparation by canonical transformations; and (iii) the application of representation theory for Lie algebras. Illustrations of the techniques are drawn from the main problem of satellite theory and from the (1–1) resonance at the triangular equilibrium in the restricted problem of three bodies.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   

The statistical approach to the study of planetary surfaces     

R. Bianchi  A. Coradini  M. Fulchignoni 《Earth, Moon, and Planets》1980,22(3):293-304

A review of the statistical methods and their application to the planetary sciences is presented. Univariate and multivariate methods are used, with different sophistication levels, to the search for the relationships between samples and/or variables belonging to data coming from space missions. Some numerical expressions, which summarize the information in mathematical terms, such as the mean, the variance, and the coefficient of variation are the best known and most widely used to perform a preliminar analysis of a data set. Techniques of regression and trend analysis allow us to compare the reciprocal behaviour of different variables, each treated as an univariate one. Making inferences about the statistical parameters permits the determination of the range of the parameter values which are consistent with the information contained in the samples. Multivariate analysis is devoted to the study of multidimensional (i.e., of several variables) distributions of the samples. The analysis of the variance is a multivariate technique useful in investigating whether or not two or more groups of multivariate observations come from populations with the same mean value. The discriminant analysis provides a function for classifying multivariate observations into groups. One of the simpler factorial techniques is the principal-components method in which the original variables are associated in a typical form that can be interpreted in terms of a genetic process. TheR-mode analysis is useful in identifying the number of important variables. TheQ-mode analysis permits us to group the observations providing clusters of samples. TheG-mode method, where the original multivariate distribution is transformed in a univariate quasi-Gaussian distribution, can be applied to identify homogenous group of samples and to classify new samples on the basis of the general groups.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.  相似文献   

Third-order Uranus-Neptune theory by Hori's method     

Osman M. Kamel 《Earth, Moon, and Planets》1989,45(2):127-137

We construct a U-N secular canonical planetary theory of the third order with respect to planetary masses. The Hori-Lie procedure is adopted to solve the problem. Expansions have been carried out by hand, neglecting powers higher than the second with respect to the eccentricity-inclination. We take into account the principal as well as the indirect part of the planetary disturbing function. The theory is expressed in terms of the Poincaré canonical variables, referring to the Jacobi-Radau set of origins. We assume that the 1:2 U-N critical terms and its multiples are the only periodic terms.  相似文献   

On The Ideal Resonance Problem     

Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):359-359

The Ideal Resonance Problem in its normal form is defined by the Hamiltonian (1) $$F = A (y) + 2B (y) sin^2 x$$ with (2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?^1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by (3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance, (4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration ₁/n₂ of the natural frequencies of the motion generates a sequence. (5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form.  相似文献   

The precession and nutation of deformable bodies,III     

Zdenêk Kopal 《Astrophysics and Space Science》1969,4(4):427-458

In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides.In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the tidal breathing in eccentric binary systems are investigated.In Section 3, we consider the angular velocity of rotation about theZ-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles and , and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form.The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.  相似文献   

On a new form for the differential equations of relative motion of the three-body problem     

Carlos A. Altavista 《Celestial Mechanics and Dynamical Astronomy》1972,6(2):208-213

A new method for the development of the disturbing function of the three-body problem is outlined in this paper. A special process is devised to get the distance between two planetsP ₁ andP ₂ in terms of their heliocentric distances. It is then shown that the differential equations of relative motion of this problem can be brought in an homogeneous set of differential equations.  相似文献   

Elimination of the critical terms in a first-order general planetary theory     

Kamel  Osman M. 《Earth, Moon, and Planets》1981,24(2):137-155

We construct a first order canonical general planetary theory, assuming the solar system to be composed of 8 planets excluding Pluto, referring to common fixed plane and applying the Jacobi-Radau set of origins. We eliminated by von Zeipel's method the 2:5 and 1:2 critical terms of Jupiter-Saturn and Uranus-Neptune inequalities. Our variables are those of Poincaré, and we expanded up to power three in the eccentricities and sines of the inclinations.  相似文献   

On the elimination of short-period terms in second-order general planetary theory investigated by Hori's method     

Jean Meffroy 《Astrophysics and Space Science》1973,25(2):271-354

1. The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiese _j and sines γ_j of the semi inclinations are respectively replaced by the Jacques Henrard variablesE _j ,J _j which lead to formulas remarkably simple.F is reduced to the sumF ₀+F ₁ of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F ₁ is not calculated beyond its terms of degree 3 inE _j ,E _j ,J _j , the determining functionS ₂ of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′ _j ,E _j ,J _j and the expressions of slow Delaunay canonical variables of the disturbed planetP ₁ and the disturbing planetP ₂ in terms of the new slow Delaunay canonical variables ofP ₁ andP ₂ which result from the elimination of the short period terms ofF ₁ being therefore reduced to their terms of degree <1 in theE′ _j ,E′ _j ,J′ _j . Calculation of the principal partF _1m ofF ₁ is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP ₁ andP ₂. Eccentricitye ₂ of the disturbed planetP ₂ is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF ₁ are written down only for the disturbed planetP ₁.
2. Small divisors in 1/E′ ₁ and 1/E′ ₁ ² appear in the longitude ?₁ of perihelia ofP ₁. No small divisors appear in the other five slow Delaunay variables ofP ₁. The only Jacques Henrard variables which appear in the longitude Ω₁ of the ascending node ofP ₁ are the J _j′ j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX ₁,Y ₁,Z ₁, λ₁. The solving of the ten canonical equations ofP ₁ andP ₂ in the slow Delaunay canonical variablesX′ _j ,Y′ ₁,Z′ _j ,λ′ _j ,ω′ _j ,Ω′ _j resulting from the elimination of the short period terms ofF ₁ reduces to that of four canonical equations inZ′ _j ,©′ _j and to six quadratures three of them expressing theX′ _j ,Y′ ₁ are constants and the three others expressingλ′ _j ,?′ _j as functions of timet. Solving of the four canonical equations inZ′ _j ,Ω′ _j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′ ₁ is then constant, so is the Jacques Henrard variableE′ ₁. If the eccentricitye ₂ ofP ₂ is no more assumed to be zero, additive small divisors inE′ ₂/E′ ² ₁ appear in longitude ?′₁ of perihelia ofP ₁ and the solving of the twelve canonical equations ofP ₁ andP ₂ inX′ _j ,Y′ _j ,Z′ _j ,λ′ _j ,?′ _j ,Ω′ _j is reduced to that of eight canonical equations inY′ _j ,?′ _j ,Z′ _j ,Ω′ _j and to four quadratures expressingX′ _j are constants andλ′ _j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′ _j ,?′ _j and the other one inZ′ _j ,Ω′ _j . Each of those two systems is identical to the system inZ′ _j ,Ω′ _j corresponding toe ₂=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee ₂=0.
3. Expressions ofX ₁,Y ₁,Z ₁,λ ₁,? ₁,Ω ₁ as functions ofX′ _j ,Y′ ₁,Z′ _j ,λ′ _j ,?′ ₁,Ω′ _j ;j=1, 2 are sums of sines and cosines of the multiples ofλ′ _j ,?′ ₁,Ω′ _j for the terms arising from the indirect partF _1j ofF ₁, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF _1m ofF ₁, coefficients of those sums and Fourier series having one of the eight forms: $$A,{\text{ }}\frac{B}{{E'}},{\text{ }}\frac{C}{{E'^2 }},{\text{ }}D\frac{{j'^{2_1 } }}{{E'^{2_1 } }},{\text{ }}E\frac{{j'^{2_2 } }}{{E'^{2_1 } }},{\text{ }}F\frac{{j'^{_1 } j'^2 }}{{E'^{2_1 } }},{\text{ }}G\frac{{j'^2 }}{{j'^{_1 } }},{\text{ }}H\frac{{j'^{22} }}{{j'^{2_1 } }}{\text{.}}$$ A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF _1j is easily carried out; that of theA,..., H arising from the terms ofF _1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′ _j ,?′ _j ,Ω _ij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα^3/2;p, q relative integers, or products of such divisors appear inA,..., H.
4. the method extends to the case whenF ₁ is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F ₁ being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S ₂ has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP ₁ andP ₂ in terms of the slow Delaunay canonical variables ofP ₁ andP ₂ resulting from the elimination of the short period terms ofF ₁ have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables.

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Computer implementation of a new approach to the ideal resonance problem     

R. A. Howland 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):207-211

A new approach to the librational solution of the Ideal Resonance Problem has been devised--one in which a non-canonical transformation is applied to the classical Hamiltonian to bring it to the form of the simple harmonic oscillator. Although the traditional form of the canonical equations of motion no longer holds, a quasi-canonical form is retained in this single-degree-of-freedom system, with the customary equations being multiplied by a non-constant factor. While this makes the resulting system amenable to traditional transformation techniques, it must then be integrated directly. Singularities of the transformation in the circulation region limit application of the method to the librational region of motion.Computer-assisted algebra has been used in all three stages of the solution to fourth order of this problem: using a general-purpose FORTRAN program for the quadratic analytical solution of Hamiltonians in action-angle variables, the initial transformation is carried out by direct substitution and the resulting Hamiltonian transformed to eliminate angular variables. The resulting system of differential equations, requiring the expected elliptic functions as part of their solution, is currently in the process of being integrated using the LISP-based REDUCE software, by programming the required recursive rules for elliptic integration.Basic theory of this approach and the computer implementation of all these techniques is described. Extension to higher order of the solution is also discussed.  相似文献   

The loss of magnetic flux from pre-main sequence stars     

David Moss 《Astrophysics and Space Science》1983,97(1):213-216

For the case ofn planets, we derive Lagrange's secular planetary equations in terms of the Poincaré canonical variables, using the Jacobi-Radau set of origins, and referring to a common fixed plane.  相似文献   

An accelerated elimination technique for the solution of perturbed Hamiltonian systems     

Robert A. Howland Jr 《Celestial Mechanics and Dynamical Astronomy》1977,15(3):327-352

A new technique is developed for the formal solution of non-degenerate or degenerate Hamiltonian systems under periodic perturbation through continually accelerated elimination of the periodic terms. Special features of the method are the ability to eliminate both short-period and longperiod variables simultaneously and the attainment of [formal] quadratic convergence for non-degenerate systems and nearly quadratic convergence in degenerate cases. The technique utilizes Lie transforms and is based on an approach due to Kolmogorov and Arnol'd.  相似文献   

Global modes constituting the solar magnetic cycle     

M. H. Gokhale  J. Javaraiah 《Solar physics》1995,156(1):157-177

The sunspot occurrence probability defined in Paper I is used to determine the Legendre-Fourier (LF) terms in the rate of emergence of toroidal magnetic flux,Q(, t), above the photosphere per unit latitude interval, per unit time. Assuming that the magnetic flux tubes whose emergence yields solar activity are produced by interference of global MHD waves in the Sun, we determine how the amplitudes and phases of the LF terms in the toroidal magnetic fieldB , representing the waves, will be related to those of the LF terms inQ(, t). The set of LF terms in Q that represents the set of waves whose interference produces most of the observed sunspot activity is {l = 1, 3, , 13;v =nv _*,n = 1, 3, 5}, wherev _* = 1/21.4 yr^–1. However, among the shapes of sunspot cycles modeled using various sets of the computed LF terms the best agreement with the observed shape, for each cycle, is given by the set {l = 3 orl = 3, 5; andn = 1, 3 orn = 1, 3, 5}. The sets of terms: {l = 1, 3, 5, 7;n = 1}, {l = 1, 3, 5, 7;n = 3}, {l = 9, 11, 13, 15;n = 1} and {l = 9, 11, 13, 15;n = 3} seem to represent four modes of global MHD oscillation. Correlations between the amplitudes (and phases) of LF terms in different modes suggest possible existence of cascade of energy from constituent MHD waves of lowerl andn to those of higherl andn. The spectrum of the MHD waves trapped in the Sun may be maintained by the combined effect of this energy cascade and the loss of energy in the form of the emerging flux tubes. The primary energy input into the spectrum may be occurring in the mode {l = 1, 3, 5, 7;n = 1). As expected from the above phenomenological model, the size of a sunspot cycle and its excess over the previous cycle are well correlated (e.g., 90%) to the phase-changes of the two most dominant oscillation modes during the previous one or two cycles. These correlations may provide a physical basis to forecast the cycle sizes.  相似文献   

On the elimination of the critical terms of a first order theory of Jupiter perturbed by Saturn carried out through Hori's method and Poincaré canonical variables     

Jean Meffroy 《Earth, Moon, and Planets》1978,19(1):3-60

The elimination of the critical terms inside the Hamiltonian of a first order theory of Jupiter perturbed by Saturn is carried out through the Poincaré canonical variables and the Hori's procedure. Powers of the eccentricities and the sines of inclinations which are>3 are neglected. The Poincaré variablesL ₁,H ₁,P ₁, ₁,K ₁,Q ₁ of Jupiter which result from a previous elimination of the short period terms are expressed in terms of the Poincaré canonical variablesL _u ,H _u ,P _u , _u ,Q _u ;u=1, 2; index 1 Jupiter, index 2 Saturn resulting from the elimination of the short period and critical terms. The differential equations inH _u ,P _u ,K _u ,Q _u are solved through the method of Lagrange and the analytical expressions ofL ₁,H ₁,P ₁, ₁,K ₁,Q ₁ as functions of timet are finally obtained.  相似文献