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1.
In this paper, an efficient iterative method of arbitrary positive integer order of convergence 2 will be established for the two-body universal initial value problem. The method is of dynamic nature in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. Moreover, which is the most important, the method does not need any a priori knowledge of the initial guess. A property which avoids the critical situations between divergent to very slow convergent solutions, that may exist in other numerical methods which depend on initial guess. Some applications of the method are also given. 相似文献
2.
In this paper, an efficient iterative method of arbitrary integer order of convergence ≥ 2 has been established for solving
the hyperbolic form of Kepler’s equation. The method is of a dynamic nature in the sense that, moving from one iterative scheme
to the subsequent one, only additional instruction is needed. Most importantly, the method does not need any prior knowledge
of the initial guess. A property which avoids the critical situations between divergent and very slow convergent solutions
that may exist in other numerical methods which depend on initial guess. Computational Package for digital implementation
of the method is given and is applied to many case studies. 相似文献
3.
Xiaodong Liu Hexi Baoyin Xingrui Ma 《Celestial Mechanics and Dynamical Astronomy》2011,109(3):303-320
Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity
and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their
basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied
to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately,
and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations
in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough
results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian
frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show
some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J
3 has the same sign as J
2; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of
the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at
90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide
much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the
monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the
more complex models. 相似文献
4.
In this paper, we consider a statistical method for distance determination of stellar groups. The method depends on the assumption
that the members of the group scatter around a mean absolute magnitude in Gaussian distribution. The mean apparent magnitude
of the members is then expressed by frequency function, so as to correct for observational incompleteness at the faint end.
The problem reduces to the solution of a highly transcendental equation for a given magnitude parameter α. For the computational developments of the problem, continued fraction by the Top-Down algorithm was developed and applied
for the evaluation of the error function erf(z). The distance equation Λ(y) = 0 was solved by an iterative method of second order of convergence using homotopy continuation technique. This technique
does not need any prior knowledge of the initial guess, a property which avoids the critical situations between divergent
and very slow convergent solutions, that may exist in the applications of other iterative methods depending on initial guess. 相似文献
5.
Aimed at the initial value problem of the particular second-order ordinary differential equations,y
=f(x, y), the symmetric methods (Quinlan and Tremaine, 1990) and our methods (Xu and Zhang, 1994) have been compared in detail by integrating the artificial earth satellite orbits in this paper. In the end, we point out clearly that the integral accuracy of numerical integration of the satellite orbits by applying our methods is obviously higher than that by applying the same order formula of the symmetric methods when the integration time-interval is not greater than 12000 periods. 相似文献
6.
In /1/, we have discussed the question whether Kopal's iterative method of solving eclipsing binary orbits is always convergent. In this paper, we show, by means of an example, that, even when the solution is convergent, it may not be the true solution but a false one. Through calculations, we empirically identify the circumstances in which Kopal's method will either be divergent or lead to a false solution. We also make an improvement on the optimization procedure in /1/. 相似文献
7.
I. A. Robin 《Celestial Mechanics and Dynamical Astronomy》1981,23(1):97-106
We show that the procedure employed in the circular restricted problem, of tracing families of three-dimensional periodic orbits from vertical self-resonant orbits belonging to plane families, can also be applied in the elliptic problem. A method of determining series of vertical bifurcation orbits in the planar elliptic restricted problem is described, and one such series consisting of vertical-critical orbits (a
v=+1) is given for the entire range (0,1/2) of the mass parameter . The initial segments of the families of three-dimensional orbits which bifurcate from two of the orbits belonging to this series are also given. 相似文献
8.
Alexander Lopatnikov Leonid Marochnik Lev Mukhin Roald Sagdeev Daniel Usikov Georgy Zaslavsky 《Astrophysics and Space Science》1991,186(2):225-243
This paper considers the evolution of a flat svarm of cometary bodies (under the effect of the passage of stars), initially moving in one direction along the circular orbits with radii 1.4×104<r<2×104 AU and along elliptic orbits with semi-major axes 5×103<a<1×104 AU and with perihelia within 50<q<100 AU. Numerical simulation shows that the original flat belt of comets is thermalizing. Its root-mean-squarez-coordinate grows withr. A cometary cloud forms with a dense flattened inner core and a rarefied halo (the Oort cloud proper). The value =N
core/N
halo varies within a wide range (up to the order of magnitude) depending on the model used (N
core andN
halo are the numbers of comets in the core and the halo, respectively).The hypothesis of a massive Oort cloud (Marochniket al., 1988) implies that the Oort cloud should have a large angular momentum. This paper employs numerical simulation to calculate Oort cloud models to which the initially flat located at the periphery of the solar nebula rotating cometary swarms is evolving in time. The loss of the initial angular momentum over the time of the Oort cloud evolution is not large. 相似文献
9.
We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L
1 and L
2 in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth
system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points
inside L
1 and beyond L
2 where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial
halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion
systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum
amplitude of halo orbits about artificial equilibrium points inside L
1 decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L
2 in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first
increases and then decreases after the thrust acceleration exceeds 0.415 mm/s2. Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times
and require a reasonably small low-thrust acceleration 0.0593 mm/s2. 相似文献
10.
One of the possible early states of the Earth-Moon system was a system of several large satellites around the Earth. The dynamical
evolution of coplanar three-body systems is studied; a planet (Earth) and two massive satellites (proto-moons) with geocentric
orbits of slightly different radii. Such configurations may arise in multiple satellite systems receding from a planet due
to tidal friction. The numerical integration of the equations of motion shows that initially circular Keplerian orbits are
soon transformed into disturbed elliptic orbits which are intersecting. The life-time of such a coplanar system between two
probable physical collisions of satellites is roughly from one day to one year for satellite systems with radii less than
20R⊕, and may reach 100 yr for three-dimensional systems. This time-scale is short in comparison with the duration of the removal
of satellites due to tides raised on the planet, which is estimated as 106–108 yr for the same orbital dimensions. Therefore, the life-time of a system of several proto-moons is mainly determined by their
tidal interactions with the Earth. For conditions which we have considered, the most probable result of the evolution was
coalescence of satellites as the consequence of the collisions. 相似文献
11.
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
12.
A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when
a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic
solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of
these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of
all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist
very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections,
that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion. 相似文献
13.
N. Caranicolas 《Astrophysics and Space Science》1982,87(1-2):237-245
We study the families of periodic orbits in a time-independent two-dimensional potential field symmetric with respect to both axes. By numerical calculations we find characteristic curves of several families of periodic orbits when the ratio of the unperturbed frequencies isA
1/2/B
1/2=2/1. There are two groups of characteristic curves: (a) The basic characteristic and the characteristics which bifurcate from it. (b) The characteristics which start from the boundary line and the axisx=0. 相似文献
14.
The purpose of this paper is to extend the study of the so called p-q resonant orbits of the planar restricted three-body problem to the spatial case. The p-q resonant orbits are solutions of the restricted three-body problem which have consecutive close encounters with the smaller primary. If E, M and P denote the larger primary, the smaller one and the infinitesimal body, respectively, then p and q are the number of revolutions that P gives around M and M around E, respectively, between two consecutive close approaches. For fixed values of p and q and suitable initial conditions on a sphere of radius around the smaller primary, we will derive expressions for the final position and velocity on this sphere for the orbits under consideration. 相似文献
15.
We consider a model that describes the evolution of distant satellite orbits and that refines the solution of the doubly averaged
Hill problem. Generally speaking, such a refinement was performed previously by J. Kovalevsky and A.A. Orlov in terms of Zeipel’s
method by constructing a solution of the third order with respect to the small parameter m, the ratio of the mean motions of the planet and the satellite. The analytical solution suggested here differs from the solutions
obtained by these authors and is closest in form to the general solution of the doubly averaged problem (∼m
2). We have performed a qualitative analysis of the evolutionary equations and conditions for the intersection of satellite
orbits with the surface of a spherical planet with a finite radius. Using the suggested solution, we have obtained improved
analytical time dependences of the elements of evolving orbits for a number of distant satellites of giant planets compared
to the solution of the doubly averaged Hill problem and, thus, achieved their better agreement with the results of our numerical
integration of the rigorous equations of perturbed motion for satellites. 相似文献
16.
Xiangyuan Zeng Hexi Baoyin Junfeng Li Shengping Gong 《Celestial Mechanics and Dynamical Astronomy》2011,111(4):415-430
A new concept of three dimensional non-Keplerian trajectories with double angular momentum reversal is investigated with high
performance solar sails. The main discussion of this paper is about such 3D solar inverse orbits with inner constraints. The
problem is addressed in a time optimal control framework solved by an indirect method. Two typical solar inverse orbits have
been achieved and presented in a 3D non-dimensional dynamic model in the Heliocentric Inertial Frame. Starting from the Earth
orbit ecliptic plane, a sailcraft in the inverse orbit exhibits a butterfly shape trajectory. As such, the new orbits are
symmetrical with respect to a plane which contains the Sun-perihelion line. The relation of the sail attitude angles between
the two symmetrical parts of the orbits are used to reduce the simulation effort. The quasi-heliostationary property at its
aphelia is demonstrated with variation of the orbital radius. Evolutions of the orbital velocity and optimal sail orientations
are also outlined and discussed to benefit future design work. As is suited for space observation guaranteed by its butterfly
shape, the inverse orbits are thoroughly studied in terms of the concerned parameters. The discussion of the parametric influence
is ranked in order as perihelion distance r
E
, required maximum position z
max, perihelion position z
f
and the sail lightness number β. Suitable ranges of each parameter are adopted to illustrate the orbital variation trend. Through numerical simulations the
features of such inverse orbits are further emphasized to provide an initial reference for future researchers. 相似文献
17.
Nonlinear dynamical analysis and the control problem for a displaced orbit above a planet are discussed. It is indicated that
there are two equilibria for the system, one hyperbolic (saddle) and one elliptic (center), except for the degenerate h
z
max, a saddle-node bifurcation point. Motions near the equilibria for the nonresonance case are investigated by means of the
Birkhoff normal form and dynamical system techniques. The Kolmogorov–Arnold–Moser (KAM) torus filled with quasiperiodic trajectories
is measured in the τ 1 and τ 2 directions, and a rough algorithm for calculating τ 1 and τ 2 is proposed. A general iterative algorithm to generate periodic Lyapunov orbits is also presented. Transitions in the neck
region are demonstrated, respectively, in the nonresonance, resonance, and degradation cases. One of the important contributions
of the paper is to derive necessary and sufficiency conditions for stability of the motion near the equilibria. Another contribution
is to demonstrate numerically that the critical KAM torus of nontransition is filled with the (1,1)-homoclinic orbits of the
Lyapunov orbit. 相似文献
18.
Yijun Lian Gerard Gómez Josep J. Masdemont Guojian Tang 《Celestial Mechanics and Dynamical Astronomy》2013,115(2):185-211
In this paper we study the dynamics of a massless particle around the L 1,2 libration points of the Earth–Moon system in a full Solar System gravitational model. The study is based on the analysis of the quasi-periodic solutions around the two collinear equilibrium points. For the analysis and computation of the quasi-periodic orbits, a new iterative algorithm is introduced which is a combination of a multiple shooting method with a refined Fourier analysis of the orbits computed with the multiple shooting. Using as initial seeds for the algorithm the libration point orbits of Circular Restricted Three Body Problem, determined by Lindstedt-Poincaré methods, the procedure is able to refine them in the Solar System force-field model for large time-spans, that cover most of the relevant Sun–Earth–Moon periods. 相似文献
19.
R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):71-80
We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e
primaries
0.66) and also a range of initial conditions for the distance of the 3
rd
body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities. 相似文献
20.
Yu. D. Medvedev 《Solar System Research》2011,45(5):386-391
Observations at the first opposition are used to determine the orbits of 16 near-Earth asteroids with two or more observed
oppositions. The orbits are improved by the differential method. This paper considers two modifications of the improvement
technique, which are compared to the classical method based on the principle of the least square method (LSM). The first modification
uses the principle of least absolute deviations (LAD). In the second modification, the differences O - C (between the observed and calculated positions) are transformed to fit into a new coordinate system whereby the axes go parallel
and perpendicular to the asteroid’s apparent path (the modified differential method (MDM)). The orbits determined from one
opposition by the classical LSM, LAD, and MDM are compared to a more accurate orbit calculated by the LSM from all the available
oppositions. The calculations show that in 13 cases out of 16, the asteroid orbits calculated by LAD are more accurate than
those calculated by the classical LSM. The improvement by the modified differential method, which includes the O - C transformation, does not produce any perceptible increase in accuracy when compared to the orbits calculated by the classical
method. 相似文献