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1.
Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J 2 is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J 2-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J 2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.  相似文献   

2.
In a previous work we studied the effects of (I) the J 2 and C 22 terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J 2 + J 3’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.  相似文献   

3.
This paper concerns with the study of KS uniformly regular canonical elements with Earth's oblateness. These elements, ten in number, are all constant in the unperturbed motion and even in the perturbed motion, the substitution is straightforward and elementary due to the transformation laws being explicit and closed expression. By utilizing the recursion formulas of Legendre's polynomials, we are able to include any number of Earth's zonal harmonics J n in the package and also economize the computations. A fixed step-size fourth-order Runge-Kutta-Gill method is employed for numerical integration of the canonical equations.Utilizing 5 test cases covering a large range of semimajor axis and eccentricity, we have carried out computations to study the effects of Earth's zonal harmonics (up to J 36) and integration step-size variation. Bilinear relations and energy equation are used for checking the accuracies of numerical integration. From the application point of view, the package is utilized to study the behaviour of 900 km height near-circular sun-synchronous satellite orbit over a longer duration of 220 days time (nearly 3078 revolutions) and the necessity of including more number of Earth's zonal harmonic terms is noticed. The package is also used to study the effect of higher zonal harmonics on three 900 km height near-circular orbits with inclinations of 60, 63.2, and 65 degrees, by including Earth's zonal harmonics up to J 24. The mean eccentricity (e m) is found to have long-periods of 459.6, 6925.1 and 1077.6 days, respectively. Sharp changes in the variation of m near the minima to em are noticed. The values of m are found to be very near to +-90 degrees at the extrema of em. The same orbit is employed to study the effect of variation of inclination from 0 to 180 degrees on long-period (T) of eccentricity with J 2 to J 24 terms. T is found to increase rapidly as we proceed towards the critical inclinations.  相似文献   

4.
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   

5.
Rozelot  J.P.  Godier  S.  Lefebvre  S. 《Solar physics》2001,198(2):223-240
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   

6.
Theory of the motion of an artificial Earth satellite   总被引：1，自引：0，他引：1  
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   

7.
The order of magnitude of the error is investigated for a first-order von Zeipel theory of satellite orbits in an axisymmetric force field, i.e., first-order long period and short-period effects are included along with second order secular rates. The treatment is valid for zero eccentricity and/or inclination. In the case where initial position and velocity vectors are known, the in-track position error over time intervals of order 1/J 2 is kept at 0(J 2 2), like the other position errors and velocity errors, by calibration of the mean motion with the aid of the energy integral. The results are specifically applicable to accuracy comparisons of the Brouwer orbit prediction method with numerical integration. A modified calibration is presented for the general asymmetric force field which includes tesseral harmonics.  相似文献   

8.
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   

9.
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.
The non-spherical gravitational potential of the planet Mars is sig- nificantly different from that of the Earth. The magnitudes of Mars’ tesseral harmonic coefficients are basically ten times larger than the corresponding val- ues of the Earth. Especially, the magnitude of its second degree and order tesseral harmonic coefficient J2,2 is nearly 40 times that of the Earth, and approaches to the one tenth of its second zonal harmonic coefficient J2. For a low-orbit Mars probe, if the required accuracy of orbit prediction of 1-day arc length is within 500 m (equivalent to the order of magnitude of 10−4 standard unit), then the coupled terms of J2 with the tesseral harmonics, and even those of the tesseral harmonics themselves, which are negligible for the Earth satellites, should be considered when the analytical perturbation solution of its orbit is built. In this paper, the analytical solutions of the coupled terms are presented. The anal- ysis and numerical verification indicate that the effect of the above-mentioned coupled perturbation on the orbit may exceed 10−4 in the along-track direc- tion. The conclusion is that the solutions of Earth satellites cannot be simply used without any modification when dealing with the analytical perturbation solutions of Mars-orbiting satellites, and that the effect of the coupled terms of Mars's non-spherical gravitational potential discussed in this paper should be taken into consideration.  相似文献   

11.
The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J 2 is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J 2 and J 4 is analyzed. Besides, when given certain values of J 2 and J 4, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.  相似文献   

12.
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   

13.
We study the problem of critical inclination orbits for artificial lunar satellites, when in the lunar potential we include, besides the Keplerian term, the J 2 and C 22 terms and lunar rotation. We show that, at the fixed points of the 1-D averaged Hamiltonian, the inclination and the argument of pericenter do not remain both constant at the same time, as is the case when only the J 2 term is taken into account. Instead, there exist quasi-critical solutions, for which the argument of pericenter librates around a constant value. These solutions are represented by smooth curves in phase space, which determine the dependence of the quasi-critical inclination on the initial nodal phase. The amplitude of libration of both argument of pericenter and inclination would be quite large for a non-rotating Moon, but is reduced to <0°.1 for both quantities, when a uniform rotation of the Moon is taken into account. The values of J 2, C 22 and the rotation rate strongly affect the quasi-critical inclination and the libration amplitude of the argument of pericenter. Examples for other celestial bodies are given, showing the dependence of the results on J 2, C 22 and rotation rate.  相似文献   

14.
Perturbation techniques based on Lie transforms as suggested by Deprit were used as the theoretical foundation for programming the analytical solution of the Main Problem in Satellite Theory (all gravitational harmonics being zero exceptJ 2). The collection of formulas necessary and sufficient to construct an ephemeris is given in the exposition. Short and long period displacements, as well as the secular terms, have been obtained up to the third order inJ 2 as power series of the eccentricity. They result from two successive completely canonical transformations which it has been found convenient not to compose into a unique transformation. Division by the eccentricity appears nowhere in the developments-neither explicitly nor implicitly. The determination of the constants of motion from the initial conditions has been given an elementary solution that is both complete and explicit without being iterative. The program was developed by Rom from MAO's package of subroutines forMechanizedAlgebraicOperations. Reliability tests have been run in two instances: in-track errors for ANNA 1B are only 20 cm after 210 days in orbit, while for RELAY II, they are 2.4 m, even after 350 days in orbit.  相似文献   

15.
We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ 2 problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ 2 andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ 2, as large as |J 2|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.  相似文献   

16.
Analytical theory for short-term orbit motion of satellite orbits with Earth's zonal harmonicsJ 3 andJ 4 is developed in terms of KS elements. Due to symmetry in KS element equations, only two of the nine equations are integrated analytically. The series expansions include terms of third power in the eccentricity. Numerical studies with two test cases reveal that orbital elements obtained from the analytical expressions match quite well with numerically integrated values during a revolution. Typically for an orbit with perigee height, eccentricity and inclination of 421.9 km, 0.17524 and 30 degrees, respectively, maximum differences of 27 and 25 cm in semimajor axis computation are noted withJ 3 andJ 4 term during a revolution. For application purposes, the analytical solutions can be used for accurate onboard computation of state vector in navigation and guidance packages.  相似文献   

17.
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   

18.
The interior structure of Jupiter serves as a benchmark for an entire astrophysical class of liquid–metallic hydrogen-rich objects with masses ranging from ~0.1M J to ~80M J (1M J = Jupiter mass = 1.9e30 g), comprising hydrogen-rich giant planets (mass < 13M J) and brown dwarfs (mass > 13M J but ~ < 80M J), the so-called substellar objects (SSOs). Formation of giant planets may involve nucleated collapse of nebular gas onto a solid, dense core of mass ~0.04M J rather than a stellar-like gravitational instability. Thus, detection of a primordial core in Jupiter is a prime objective for understanding the mode of origin of extrasolar giant planets and other SSOs. A basic method for core detection makes use of direct modeling of Jupiter’s external gravitational potential terms in response to rotational and tidal perturbations, and is highly sensitive to the thermodynamics of hydrogen at multi-megabar pressures. The present-day core masses of Jupiter and Saturn may be larger than their primordial core masses due to sedimentation of elements heavier than hydrogen. We show that there is a significant contribution of such sedimented mass to Saturn’s core mass. The sedimentation contribution to Jupiter’s core mass will be smaller and could be zero.  相似文献   

19.
W. Landgraf 《Solar physics》1992,142(2):403-406
From astrometric observations of minor planet (1566) Icarus from 1949 to 1987 were made solutions for improved orbital elements of Icarus and the quadrupole moment of the Sun. The formal result was J2 = -0.6±5.8 &d 10–6. From this we can conclude that J 2 is very probably less than 2 · 10–-5.  相似文献   

20.
Analytical solutions using KS elements are derived. The perturbation considered is the Earth's zonal harmonic J 2. The series expansions include terms of fourth power in the eccentricity. Only two of the nine KS element equations are integrated analytically due to the reasons of symmetry. The analytical solution is suitable for short-term orbit computations. Numerical studies show that reasonably good estimates of the orbital elements can be obtained in one step of 10 to 30 degrees of eccentric anomaly for near-Earth orbits of moderate eccentricity. For application purposes, the analytical solution can be effectively used for onboard computation in the navigation and guidance packages, where the modelling of J 2 effect becomes necessary.  相似文献   

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