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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.
Rabindra Nath Das 《Astrophysics and Space Science》2010,326(1):91-103
In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness
can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled
singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems
using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral
equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N
1- and N
2- in the complex plane for both nonconservative and conservative cases. The N
1- and N
2-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral
equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case. 相似文献
3.
Alberto Escapa 《Celestial Mechanics and Dynamical Astronomy》2011,110(2):99-142
We explore the evolution of the angular velocity of an elastic Earth model, within the Hamiltonian formalism. The evolution
of the rotation state of the Earth is caused by the tidal deformation exerted by the Moon and the Sun. It can be demonstrated
that the tidal perturbation to spin depends not only upon the instantaneous orientation of the Earth, but also upon its instantaneous
angular velocity. Parameterizing the orientation of the Earth figure axis with the three Euler angles, and introducing the
canonical momenta conjugated to these, one can then show that the tidal perturbation depends both upon the angles and the
momenta. This circumstance complicates the integration of the rotational motion. Specifically, when the integration is carried
out in terms of the canonical Andoyer variables (which are the rotational analogues to the orbital Delaunay variables), one
should keep in mind the following subtlety: under the said kind of perturbations, the functional dependence of the angular
velocity upon the Andoyer elements differs from the unperturbed dependence (Efroimsky in Proceedings of Journées 2004: Systèmes
de référence spatio-temporels. l’Observatoire de Paris, pp 74–81, 2005; Efroimsky and Escapa in Celest. Mech. Dyn. Astron. 98:251–283, 2007). This happens because, under angular velocity dependent perturbations, the requirement for the Andoyer elements to be canonical
comes into a contradiction with the requirement for these elements to be osculating, a situation that parallels a similar
antinomy in orbital dynamics. Under the said perturbations, the expression for the angular velocity acquires an additional
contribution, the so called convective term. Hence, the time variation induced on the angular velocity by the tidal deformation
contains two parts. The first one comes from the direct terms, caused by the action of the elastic perturbation on the torque-free
expressions of the angular velocity. The second one arises from the convective terms. We compute the variations of the angular
velocity through the approach developed in Getino and Ferrándiz (Celest. Mech. Dyn. Astron. 61:117–180, 1995), but considering the contribution of the convective terms. Specifically, we derive analytical formulas that determine the
elastic perturbations of the directional angles of the angular velocity with respect to a non-rotating reference system, and
also of its Cartesian components relative to the Tisserand reference system of the Earth. The perturbation of the directional
angles of the angular velocity turns out to be different from the evolution law found in Kubo (Celest. Mech. Dyn. Astron.
105:261–274, 2009), where it was stated that the evolution of the angular velocity vector mimics that of the figure axis. We investigate comprehensively
the source of this discrepancy, concluding that the difference between our results and those obtained in Ibid. stems from an oversimplification made by Kubo when computing the direct terms. Namely, in his computations Kubo disregarded
the motion of the tide raising bodies with respect to a non-rotating reference system when compared with the Earth rotational
motion. We demonstrate that, from a numerical perspective, the convective part provides the principal contribution to the
variation of the directional angles and of length of day. In the case of the x and y components in the Tisserand system, the convective contribution is of the same order of magnitude as the direct one. Finally,
we show that the approximation employed in Kubo (Ibid.) leads to significant numerical differences at the level of a hundred micro-arcsecond. 相似文献
4.
The present paper addresses the existence of J
2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincaré
techniques in the J
2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J
2 invariant relative orbits from the dynamical system point of view. A new type of Poincaré mapping is defined from different
central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum H
z
and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods T
d
and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincaré mapping is surjective because it compresses
the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration
of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure
to perform the ergodic representation of E- and H
z
-contour maps on the space of T
d
–ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may
achieve non-unique variables from a single image, which makes the generation of J
2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be
defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J
2 invariant relative orbits to different invariant parameters E and H
z
depending on the E- and H
z
-contour map, which will hold J
2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies,
a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present
work, which is beyond the scope of the linear conditions of J
2 invariant relative orbits. Therefore, the existence of J
2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system
point of view. 相似文献
5.
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
6.
In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time
and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular
velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”.
The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional
dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint
are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the
elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and
it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer
elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.)
The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations
render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations
depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations
noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial
frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating,
extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating
variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise
instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity
is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even
though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity
relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional
angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered
a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity
that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as
long as they are rightly identified as the formulae for the inertial angular velocity). 相似文献
7.
Edward Liverts Evgeny Griv David Eichler Michael Gedalin 《Astrophysics and Space Science》2000,274(1-2):315-320
An improved linear stability theory of small-amplitude oscillations of a self-gravitating, infinitesimally thin gaseous disk
of spiral galaxies has been developed by Bertin, Lau, Lin, Mark, Morozov, Polyachenko, and others in the approximation of
moderately tightly wound gravity perturbations. In this regime, the generalized Lin–Shu type dispersion relation was also
found by including higher order terms in the small parameter 1/kr for wavenumber k and radius r. It was shown that in the differentially rotating disks for nonaxisymmetric (spiral)perturbations Toomre's modified critical
Q-parameter is larger than the standard one: the fact that the spiral perturbations in the nonuniformly rotating system are
more unstable than the axisymmetric ones is taken into account in this modified local stability criterion. We use hydrodynamical
simulations to test the validity of the modified local criterion.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
Ahmad Rami El-Nabulsi 《Astrophysics and Space Science》2010,325(2):163-175
We processed the data on radial velocities and HI line widths for 1678 flat edge-on spirals from the Revised Flat Galaxy Catalogue.
We obtained the parameters of the multipole components of the large-scale velocity field of collective non-Hubble galaxy motion
as well as the parameters of the generalized Tully–Fisher relation in the “HI line width—linear diameter” version. All the
calculations were performed independently in the framework of three models, where the multipole decomposition of the galaxy
velocity field was limited to a dipole, a quadrupole, and an octopole term. We showed that both the quadrupole and the octopole
components are statistically significant.
On the basis of the compiled list of peculiar velocities of 1623 galaxies we obtained estimations of cosmological parameters
Ω
m
and σ
8. These estimations were obtained in both graphical form and as a constraint of the value S
8=(Ω
m
/0.3)0.35
σ
8=0.91±0.05. 相似文献
9.
H. Claes 《Celestial Mechanics and Dynamical Astronomy》1980,21(2):193-198
The problem of A.T.E.A.S. is treated, for the zonal perturbations, in its Hamiltonian form. The method consists in eliminating angular variables from the Hamiltonian function. Nearly identity canonical transformations are used, first to remove short periodic terms, second to remove long periodic terms. The general solution, up toJ
2
3
, is represented by the generators of the transformations and by the mean motions of averaged variables, known up toJ
2
4
. Open expressions in the eccentricity are avoided as far as possible. It permits to obtain a closed second order theory with closed third order mean motions.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978. 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.
José Edmundo Mansilla 《Celestial Mechanics and Dynamical Astronomy》2006,94(3):249-269
Results are obtained about formal stability and instability of Hamiltonian systems with three degrees of freedom, two equal
frequencies and the matrix of the linear part is not diagonalizable, in terms of the coefficients of the development in Taylor
series of the Hamiltonian of the system. The results are applied to the study of stability of the Lagrangian solutions of
the Three Body-Problem in the case in which the center of mass is over the curve ρ*, on the border of the region of linear stability of Routh. The curve ρ* is divided symmetrically in three arcs in such a way that if the center of mass of the three particles lies on the central
arc, the Lagrangian solution is unstable in the sense of Liapunov (in finite order), while if the center of mass determines
one point that lies on one of the other two arcs of ρ*, then the Lagrangian solution is formally stable. 相似文献
12.
Recent cosmological observations of large-scale structures (red shift of type Ia supernovae) confirm that the universe is
currently expanding at an accelerating rate and its dominant component is dark energy. This has stimulated the development
of the theory of gravity and led to many alternative variants, including tensor-scalar ones. This paper deals with the role
of conformal transformations in the Jordan-Brans-Dicke theory. Variants of intrinsic, conformally coupled, and Einstein representations
are examined. In the Einstein representation an exact analytic solution for the standard cosmological model is obtained. It
is expressed in terms of the relative energy contributions of ordinary matter Ω
m
, the scalar field Ω
CK
, and a term ΩΛ related to the cosmological constant Λ . Information on the evolution of the universe for the case with a minimally coupled
scalar field is given in the form of graphs. 相似文献
13.
M. Šidlichovský 《Celestial Mechanics and Dynamical Astronomy》1990,49(2):177-196
A nonlinear theory of secular resonances is developed. Both terms corresponding to secular resonances 5 and 6 are taken into account in the Hamiltonian. The simple overlap criterion is applied and the condition for the overlap of these resonances is found. It is shown that in given approximation the value p = (1 - e2)1/2(1 - cosI) is an integral of motion, where the mean eccentricity e and mean inclination I are obtained by eliminating short-period perturbations as well as the nonresonant terms from the planets. The overlap criterion yields a critical value of parameter p depending on the semi-major axis a of the asteroid. For p greater than the critical value, resonance overlap occurs and chaotic motion has to be expected. A mapping is presented for fast calculation of the trajectories. The results are illustrated by level curves in surfaces of section method. 相似文献
14.
We study the influence of mutual planetary perturbations on the process of eccentricity excitation by jet acceleration suggested by Namouni (Astron. J. 130, 280–294). Modeling the jet’s action by a constant-direction acceleration, we solve the linear secular equations of the combined planetary perturbations and the jet acceleration of the host star for a two-planet system. The effects of the acceleration’s strength, relative mass ratio and the relative distance of the two planets are investigated. The model is applied to the extrasolar planetary systems of HD 108874, 47 Uma, and HD 12661. 相似文献
15.
Hans Rickman Marc Fouchard Giovanni B. Valsecchi Christiane Froeschlé 《Earth, Moon, and Planets》2005,97(3-4):411-434
We investigate different approximate methods of computing the perturbations on the orbits of Oort cloud comets caused by passing stars, by checking them against an accurate numerical integration using Everhart’s RA15 code. The scenario under study is the one relevant for long-term simulations of the cloud’s response to a predefined set of stellar passages. Our sample of stellar encounters simulates those experienced by the Solar System currently, but extrapolated over a time of 1010 years. We measure the errors of perihelion distance perturbations for high-eccentricity orbits introduced by several estimators – including the classical impulse approximation and Dybczyński’s (1994, Celest. Mech. Dynam. Astron. 58, 1330–1338) method – and we study how they depend on the encounter parameters (approach distance and relative velocity). We introduce a sequential variant of Dybczyński’s approach, cutting the encounter into several steps whereby the heliocentric motion of the comet is taken into account. For the scenario at hand this is found to offer an efficient means to obtain accurate results for practically any domain of the parameter space. 相似文献
16.
N. D. Caranicolas 《Journal of Astrophysics and Astronomy》2001,22(4):309-319
We present a map for the study of resonant motion in a potential made up of two harmonic oscillators with quartic perturbing
terms. This potential can be considered to describe motion in the central parts of non-rotating elliptical galaxies. The map
is based on the averaged Hamiltonian. Adding on a semi-empirical basis suitable terms in the unperturbed averaged Hamiltonian,
corresponding to the 1:1 resonant case, we are able to construct a map describing motion in several resonant cases. The map
is used in order to find thex − p
x
Poincare phase plane for each resonance. Comparing the results of the map, with those obtained by numerical integration of
the equation of motion, we observe, that the map describes satisfactorily the broad features of orbits in all studied cases
for regular motion. There are cases where the map describes satisfactorily the properties of the chaotic orbits as well. 相似文献
17.
We study the effects of a non-singular gravitational potential on satellite orbits by deriving the corresponding time rates
of change of its orbital elements. This is achieved by expanding the non-singular potential into power series up to second
order. This series contains three terms, the first been the Newtonian potential and the other two, here R
1 (first order term) and R
2 (second order term), express deviations of the singular potential from the Newtonian. These deviations from the Newtonian
potential are taken as disturbing potential terms in the Lagrange planetary equations that provide the time rates of change
of the orbital elements of a satellite in a non-singular gravitational field. We split these effects into secular, low and
high frequency components and we evaluate them numerically using the low Earth orbiting mission Gravity Recovery and Climate
Experiment (GRACE). We show that the secular effect of the second-order disturbing term R
2 on the perigee and the mean anomaly are 4″.307×10−9/a, and −2″.533×10−15/a, respectively. These effects are far too small and most likely cannot easily be observed with today’s technology. Numerical
evaluation of the low and high frequency effects of the disturbing term R
2 on low Earth orbiters like GRACE are very small and undetectable by current observational means. 相似文献
18.
Classical trans-Neptunian objects (TNOs) are believed to represent the most dynamically pristine population in the trans-Neptunian
belt (TNB) offering unprecedented clues about the formation of our Solar System. The long term dynamical evolution of classical
TNOs was investigated using extensive simulations. We followed the evolution of more than 17000 particles with a wide range
of initial conditions taking into account the perturbations from the four giant planets for 4 Gyr. The evolution of objects
in the classical region is dependent on both their inclination and semimajor axes, with the inner (a<45 AU) and outer regions (a>45 AU) evolving differently. The reason is the influence of overlapping secular resonances with Uranus and Neptune (40–42 AU)
and the 5:3 (a∼
∼42.3 AU), 7:4 (a∼
∼43.7 AU), 9:5 (a∼
∼44.5 AU) and 11:6 (a∼
∼ 45.0 AU) mean motion resonances strongly sculpting the inner region, while in the outer region only the 2:1 mean motion
resonance (a∼
∼47.7 AU) causes important perturbations. In particular, we found: (a) A substantial erosion of low-i bodies (i<10°) in the inner region caused by the secular resonances, except those objects that remained protected inside mean motion
resonances which survived for billion of years; (b) An optimal stable region located at 45 AU<a<47 AU, q>40 AU and i>5° free of major perturbations; (c) Better defined boundaries for the classical region: 42–47.5 AU (q>38 AU) for cold classical TNOs and 40–47.5 AU (q>35 AU) for hot ones, with i=4.5° as the best threshold to distinguish between both populations; (d) The high inclination TNOs seen in the 40–42 AU region
reflect their initial conditions. Therefore they should be classified as hot classical TNOs. Lastly, we report a good match
between our results and observations, indicating that the former can provide explanations and predictions for the orbital
structure in the classical region. 相似文献
19.
R. A. Krikorian 《Astrophysics》1999,42(3):338-343
The field-to-particle method of H. P. Robertson as applied by Noonan, in order to obtain the general relativistic equations
describing the trajectory of a photon in a refractive medium, is compared with Synge’s general relativistic Hamiltonian theory
of waves and rays. For a photon in vacuum it is known that both approaches yield the same equation for the trajectory, i.e.,
a null geodesic. However for a photon in a medium, in contradistinction to the Hamiltonian theory, the field-to-particle method
(a) yields equations of the photon trajectory valid only in a nondispersive medium, (b) the time component u0
of the tangent to the ray remains an undetermined quantity, (c) agreement with the Hamiltonian theory is achieved by substituting
into Noonan’s equations the Hamiltonian expression for u
0.
Published in Astrofizika, Vol. 42, No. 3, pp. 449–455, July–September, 1999. 相似文献
20.
This work derives the linearized equations of motion, the Lagrangian density, the Hamiltonian density, and the canonical angular
momentum density for general perturbations [∝ exp (imφ) with m = 0, ± 1, ...] of a geometrically thin self-gravitating, homentropic fluid disk including the pressure. The theory is applied
to “eccentric,” m = ± 1 perturbations of a geometrically thin Keplerian disk. We find m = 1 modes at low frequencies relative to the Keplerian frequency. Further, it is shown that these modes can have negative
energy and negative angular momentum. The radial propagation of these low-frequency m = 1 modes can transport angular momentum away from the inner region of a disk and thus increase the rate of mass accretion.
Depending on the radial boundary conditions there can be discrete low-frequency, negative-energy, m = 1 modes. 相似文献