共查询到20条相似文献,搜索用时 15 毫秒
1.
K. T. Alfriend R. Dasenbrock H. Pickard A. Deprit 《Celestial Mechanics and Dynamical Astronomy》1977,16(4):441-458
The Vinti problem, motion about an oblate spheroid, is formulated using the extended phase space method. The new independent variable, similar to the true anomaly, decouples the radius and latitude equations into two perturbed harmonic oscillators whose solutions toO(J
2
4
) are obtained using Lindstedt's method. From these solutions and the solution to the Hamilton-Jacobi equation suitable angle variables, their canonical conjugates and the new Hamiltonian are obtained. The new Hamiltonian, accurate toO(J
2
4
) is function of only the momenta. 相似文献
2.
Mohammed Adel Sharaf Mervat El-Sayed Awad Samiha Al-Sayed Abdullah Najmuldeen 《Earth, Moon, and Planets》1992,56(2):141-164
In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J
36
. Numerical results proved the very high efficiency and flexibility of the developed equations. 相似文献
3.
Theory of the motion of an artificial Earth satellite 总被引:1,自引:0,他引:1
Felix R. Hoots 《Celestial Mechanics and Dynamical Astronomy》1981,23(4):307-363
An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ
4, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions. 相似文献
4.
Bernard De Saedeleer 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):239-268
This paper is a contribution to the Theory of the Artificial Satellite, within the frame of the Lie Transform as canonical
perturbation technique (elimination of the short period terms). We consider the perturbation by any zonal harmonic J
n
(n ≥ 2) of the primary on the satellite, what we call here the complete zonal problem of the artificial satellite. This is quite useful for primaries with symmetry of revolution. We give an analytical formula to compute directly the first
order averaged Hamiltonian. The computation is carried out in closed form for all terms, avoiding therefore tedious expansions
in the eccentricity or in any anomaly; this feature makes the averaging process, not only valid for all kind of elliptic trajectories
but at the same time it yields the averaged Hamiltonian in a very short and compact way. The formula allows us to now skip
the averaging process, which means an asymptotic gain of a factor 3n/2 regarding the computational cost of the n
th
zonal. Our analytical formulae have been widely checked, by comparison on one hand with published works (Brouwer, 1959) (which
contained results for particular zonal harmonics, let’s say typically from J
2 to J
8), and on the other hand with the results of 3 symbolic manipulation software, among which the MM (standing for ‘Moon’s series
Manipulator’), which has already been used and described in (De Saedeleer B., 2004). Additionally, the first order generator
associated with this transformation is given into the same closed form, and has also been validated. 相似文献
5.
Mohamed Radwan 《Astrophysics and Space Science》2003,283(2):137-154
The canonical equations of motion of an artificial lunar satellite are formulated including the effects of the asphericity
of the Moon comprising the harmonics J
2, J
22, J
3, J
31, J
4 andJ
5, the oblateness of the Earth up to the second zonal harmonic, as well as the disturbing function due to the attractions of
the Earth and of the Sun (terms are retained up to order 10-6 for the higher orbits and 10-8 for the lower orbits).
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
Mohamad Radwan 《Astrophysics and Space Science》2002,282(3):551-562
The present work deals with constructing a conditionally periodic solution for the motion of an Earth satellite taking into
consideration the oblateness of the Earth and the Luni-Solar attractions. The oblatenessof the Earth is truncated beyond the
second zonal harmonic J
2. The resonance resulting from the commensurability between the mean motions of the satellite, the Moon, and the Sun is analyzed. 相似文献
7.
This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on
a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling
of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian
is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation,
and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and
perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an
example, closed-form solutions for J2-invariant orbits are obtained. 相似文献
8.
We expand both parts, the principal and indirect, of the Hamiltonian function up to the third order in the masses for the four major planets Jupiter-Saturn-Uranus-Neptune. Accordingly we write down the secular terms ofF
1,F
2,F
3 and the critical terms ofF
1,F
2 in terms of the canonical variables of H. Poincaré neglecting powers higher than the second inH, K, P, Q. 相似文献
9.
I. Stellmacher 《Celestial Mechanics and Dynamical Astronomy》1982,28(4):351-365
Résumé On développe une méthode de construction d'orbites périoldiques dans un système d'axes tournants, pour un satellite gravitant autour d'un sphéroide. Les orbites sont quasi circulaires,i est l'inclinaison sur le plan équatorial de la planète. Pour les petites inclinaisons, la solution est donnée jusqu'aux termes enJ
2
2
etJ
4.Ce modèle peut être appliqué aux satellites de Saturne. Des valeurs observées des longitudes des noeuds ascendants de Mimas et Téthys, on donne une estimation des valeurs deJ
2 etJ
4 du potentiel de Saturne. La valeur deJ
2 est très sensible aux valeurs adoptées pour le rayon équatorial de la planète.
Construction of periodic orbits of satellites in a moving system of axes, I
We give an algorithm for the construction of periodic orbits in a rotating frame for the cases of satellites moving around an oblate planet.The orbits are near to the circular case; the asymptotic developments of the periodic solutions are completely calculated for the termsJ 2 andJ 4 of the potential. The solutions for small inclinations are given up toJ 2 2 .The families of solutions depend on three parameters: the semi-major axis, the inclination of the generating orbit and the initial position on this orbit.These solutions can be applied to the motion of the Saturnian satellites. From the observed longitudes of the ascending nodes of Mimas and Tethys, we estimate the valuesJ 2 andJ 4 of the Saturnian potential, the value ofJ 2 very strongly depends on the adopted value of the planet's equatorial diameter.相似文献
10.
Peter Musen 《Celestial Mechanics and Dynamical Astronomy》1971,4(3-4):378-396
This paper derives the contributionF
2
* by the great inequality to the secular disturbing function of the principal planets. Andoyer's expansion of the planetary disturbing function and von Zeipel's method of eliminating the periodic terms is employed; thereby, the corrected secular disturbing function for the planetary system is derived. An earlier solution suggested by Hill is based on Leverrier's equations for the variation of elements of Jupiter and Saturn and on the semi-empirical adjustment of the coefficients in the secular disturbing function. Nowadays there are several modern methods of eliminating periodic terms from the Hamiltonian and deriving a purely secular disturbing function. Von Zeipel's method is especially suitable. The conclusion is drawn that the canonicity of the equations for the secular variation of the heliocentric elements can be preserved if there be retained, in the secular disturbing function, terms only of the second and fourth order relative to the eccentricity and inclinations.The Krylov-Bogolubov method is suggested for eliminating periodic terms, if it is desired to include the secular perturbations of the fifth and higher order in the heliocentric elements. The additional part of the secular disturbing functionF
2
* derived in this paper can be included in existing theories of the secular effects of principal planets. A better approach would be to preserve the homogeneity of the theory and rederive all the secular perturbations of principal planets using Andoyer's symbolism, including the part produced by the great inequality. 相似文献
11.
Osman M. Kamel 《Earth, Moon, and Planets》1988,40(2):119-147
We shall establish a second order - with respect to a small parameter which is of the order of planetary masses - Uranus-Neptune canonical planetary theory. The construction will be through the Hori-Lie perturbation theory. We perform the elliptic expansions by hand, taking into account powers 0, 1, 2 of the eccentricity-inclination. Only the principal part of the planetary Hamiltonian will be taken into consideration. Our theory will be expressed in terms of the canonical variables of Henri Poincaré, referring the planetary coordinates to the Jacobi-Radau system of origin. Only U- N critical terms will be assumed as the periodic terms. 相似文献
12.
We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S
1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J
2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J
2-Hamiltonian is a collective Hamiltonian of the T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J
2 effect as geometric phases. 相似文献
13.
Bin Kang Cheng 《Celestial Mechanics and Dynamical Astronomy》1979,19(1):31-41
In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given. 相似文献
14.
Mercè Ollé Joan R. Pacha Jordi Villanueva 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):87-107
The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that
takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic
orbits and the 3D nearby tori are also described. 相似文献
15.
Shannon L. Coffey André Deprit Etienne Deprit 《Celestial Mechanics and Dynamical Astronomy》1994,59(1):37-72
We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ
2 throughJ
9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J
2 only). In particular, we examine the manner in which the odd zonalJ
3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J
4+J
2
2
=0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions. 相似文献
16.
For a Hamiltonian that can be separated into N+1(N\geq 2) integrable parts, four algorithms can be built for a symplectic integrator. This research compares these algorithms for the
first and second order integrators. We found that they have similar local truncation errors represented by error Hamiltonian
but rather different numerical stability. When the computation of the main part of the Hamiltonian, H
0, is not expensive, we recommend to use S
* type algorithm, which cuts the calculation of the H
0 system into several small time steps as Malhotra(1991) did. As to the order of the N+1 parts in one step calculation, we found that from the large to small would get a slower error accumulation.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
17.
Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J 2 perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J 2 Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J 2 problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator. 相似文献
18.
A second order atmospheric drag theory based on the usage of TD88 model is constructed. It is developed to the second order
in terms of TD88 small parameters K
n,j
. The short periodic perturbations, of all orbital elements, are evaluated. The secular perturbations of the semi-major axis
and of the eccentricity are obtained. The theory is applied to determine the lifetime of the satellites ROHINI (1980 62A),
and to predict the lifetime of the microsatellite MIMOSA. The secular perturbations of the nodal longitude and of the argument
of perigee due to the Earth’s gravity are taken into account up to the second order in Earth’s oblateness. 相似文献
19.
The possibility of using a generalized perfect resonance for the study of libration motions of asteroids near the (p+ q)/p-type commensurabilities of the mean motions of asteroids and Jupiter is considered. Based on the equations of the planar circular restricted three-body problem, the libration-motion equations are derived and their solutions for the intermediate Hamiltonian, as well as a solution taking into account perturbations of the order O(m
3/2), are determined. 相似文献
20.
In this paper we first emphasize why it is important to know the successive zonal harmonics of the Sun's figure with high
accuracy: mainly fundamental astrometry, helioseismology, planetary motions and relativistic effects. Then we briefly comment
why the Sun appears oblate, going back to primitive definitions in order to underline some discrepancies in theories and to
emphasize again the relevant hypotheses. We propose a new theoretical approach entirely based on an expansion in terms of
Legendre's functions, including the differential rotation of the Sun at the surface. This permits linking the two first spherical
harmonic coefficients (J
2 and J
4) with the geometric parameters that can be measured on the Sun (equatorial and polar radii). We emphasize the difficulties
in inferring gravitational oblateness from visual measurements of the geometric oblateness, and more generally a dynamical
flattening. Results are given for different observed rotational laws. It is shown that the surface oblateness is surely upper
bounded by 11 milliarcsecond. As a consequence of the observed surface and sub-surface differential rotation laws, we deduce
a measure of the two first gravitational harmonics, the quadrupole and the octopole moment of the Sun: J
2=−(6.13±2.52)×10−7 if all observed data are taken into account, and respectively, J
2=−(6.84±3.75)×10−7 if only sunspot data are considered, and J
2=−(3.49±1.86)×10−7 in the case of helioseismic data alone. The value deduced from all available data for the octopole is: J
4=(2.8±2.1)×10−12. These values are compared to some others found in the literature.
Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005238718479 相似文献