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1.
The secular terms of the first-order planetary Hamiltonian is determined, by two methods, in terms of the variables of H. Poincaré, neglecting powers higher than the second in the eccentricity-inclination. 相似文献
2.
Osman M. Kamel 《Earth, Moon, and Planets》1988,43(3):271-273
A method to calculate secular terms of the two parts of the planetary disturbing function— when it is expressed in terms of the true anomalies or the eccentric anomalies instead of the mean anomalies - is described. Also an alternative method is outlined. 相似文献
3.
A solution of the Uranus-Neptune planetary canonical equations of motion through the Von Zeipel technique is presented. A unique determinging function which depends upon mixed canonical variables, reduces the 12 critical terms of the Hamiltonian to the set of its secular terms. The Poincaré canonical variables are used. We refer to a common fixed plane, and apply the Jacobi-Radau set of origins. In our expansion we neglected terms of power higher than the fourth with respect to the eccentricities and sines of the inclinations. 相似文献
4.
Osman M. Kamel 《Earth, Moon, and Planets》1992,59(2):111-140
We eliminate the 1:2 critical terms — after a previous elimination of the short period terms — in the Hamiltonian of a first order U-N theory. We take into account terms of degree 0, 1, 2, 3, 4 in the eccentricity-inclination. We apply for this elimination the Hori-Lie technique through the Poincaré canonical variables and the Jacobi coordinates. The purely principal first order secular U-N Hamiltonian admits a complete solution. We obtained the U-N equations of motion generated by the principal first order long period U-N Hamiltonian which will be solved later. This part III is closely related to the two previous papers (Kamel, 1982, 1983). 相似文献
5.
We eliminate by the method of von Zeipel the short-period terms in a first order-with respect to planetary masses—general planetary Uranus-Neptune theory. We exclude in the expansion terms of eccentricities and sines of inclinations higher than the third power.Our variables are the Poincaré canonical variables. We use the Jacobi-Radau set of origins, and we refer the planes of the osculating ellipses to a common fixed plane, the longitudes to a common origin. The short-periodic terms arising from the indirect and principal parts of the disturbing functions, are eliminated separately. The Fourier series of the principal part of the disturbing function, is reduced to the sum of only the first three terms. 相似文献
6.
Jean Meffroy 《Astrophysics and Space Science》1973,25(2):271-354
- 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 γj 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.
- 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.
- 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+qα3/2;p, q relative integers, or products of such divisors appear inA,..., H.
- 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.
7.
Celestial Mechanics and Dynamical Astronomy - When on searches for a planetary theory valid over 1 million years, one can leave in the solution the short period terms whose amplitude are small, and... 相似文献
8.
V. A. Brumberg L. S. Evdokimova V. I. Skripnichenko 《Celestial Mechanics and Dynamical Astronomy》1975,11(1):131-138
Based on a general planetary theory, the secular perturbations in the motion of the eight major planets (excluding Pluto) have been derived in polynomial form. The results are presented in the tables. The linear terms of second order with respect to the planetary masses and the nonlinear terms of first order up to the fifth (and partly seventh) degree with respect to eccentricities and inclinations were taken into account in the right-hand members of the secular system. Calculations were carried out by computer with the use of a system that performed analytic operations on power series with complex coefficients.
qA ( ). . ( ) . .相似文献
9.
Osman M. Kamel 《Earth, Moon, and Planets》1982,26(1):47-60
We review in this part the outline of a third-order general planetary theory established through Von Zeipel's method and in terms of Poincaré's canonical variables We consider our system to consist of the Sun as the primary body, one disturbed planet, and one disturbing planet. 相似文献
10.
All the necessary formulae for constructing a general solution for the motion of a planet, in rectangular coordinates, at the first order of the disturbing masses, in purely literal form in eccentricities and inclinations, are given. The authors present the transformation formulae in the two-body problem which give the correspondence between the constants of integration introduced in the theory and the classical keplerian elements. The practical elaboration of the algorithm and some partial results for the couple of planets Jupiter and Saturn are described. 相似文献
11.
Osman M. Kamel 《Earth, Moon, and Planets》1989,44(3):275-289
We explain how the first step of Hori-Lie procedure is applied in general planetary theory to eliminate short-period terms. We extend the investigation to the third-order planetary theory. We solved the canonical equations of motion for secular and periodic perturbations by this method, and obtained the first integrals of the system of canonical equations. Also we showed the relation between the determining function in the sense of Hori and the determining function in the sense of Von Zeipel. 相似文献
12.
Osman M. Kamel 《Earth, Moon, and Planets》1990,49(3):269-276
In this part we present the complete solution of the planetary canonical equations of motion by the method of G. Hori through successive changes of canonical variables using the Lie series. Thus, we can eliminate the long or critical terms of the planetary perturbing function, in our general planetary theory. In our formulas, we neglect perturbation terms of order higher than the third with respect to planetary masses. 相似文献
13.
Sergei A. Klioner 《Celestial Mechanics and Dynamical Astronomy》1996,66(4):345-363
Intermediate orbit for general planetary theory is constructed in the form of multivariate Fourier series with numerical coefficients. The structure and efficiency of the derived series are illustrated by giving various statistical properties of the coefficients.The ability of the recently proposed elliptic function approach to compress the Fourier series representing the intermediate orbit is investigated. Our results confirm that when mutual perturbations of a pair of planets are considered the elliptic function approach is quite efficient and allows one to compress the series substantially. However, when perturbations of three or more planets are under study the elliptic function approach does not give any advantages. 相似文献
14.
In the present paper the equations of the translatory motion of the major planets and the Moon and the Poisson equations of
the Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar
orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar
evolution). Hence, the theory of the Earth’s rotation is presented by means of the series in powers of the evolutionary variables
with quasi-periodic coefficients. 相似文献
15.
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
16.
Osman M. Kamel 《Earth, Moon, and Planets》1982,27(4):407-415
An outline for the elimination of the critical terms of a first order Uranus-Neptune theory is presented with a stress on the application of Hori's procedure to the problem. 相似文献
17.
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
18.
We expand the principal part of the planetary disturbing function, by Smart's method, using Taylor's theorem. In our expansion we neglect terms of degree higher than the fourth in the eccentricities and tangents of the inclinations.Now at the JPL Pasadena, California. 相似文献
19.
Center of mass-center of figure offsets are known for the Earth, Moon, Mars and Venus. Such an offset requires a density distribution asymmetric about the center of mass. Observational evidence indicates that the terrestrial, lunar and Martian offsets result from crusts of variable thickness rather than lateral density inhomogeneities and that the thickness variations are more likely caused by internal convection than impact.Paper dedicated to Professor Harold C. Urey on the occasion of his 80th birthday on 29 April, 1973. 相似文献
20.
Pavel Voráček 《Astrophysics and Space Science》1981,74(2):497-499
In this paper two examples are given, showing that the existence of black holes in the Universe violates, in its consequences, the principle of causality. The solution presented is based on the idea that the primordial black holes have zero-mass-energy and consequently zero-radius of the event horizon. Despite the existence of the surface of last influence, gravitational collapse does not produce black holes during a finite time interval as measured by an external observer. The only singularity, possible to accept (if any), is the initial and final cosmological singularity. 相似文献