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1.
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
2.
Linda Dimare 《Celestial Mechanics and Dynamical Astronomy》2010,107(4):427-449
We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know
that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin
and Negrini, J Differ Equ 190:539–558, 2003): this result has been obtained by the use of the Poincaré-Melnikov theory. Here
we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation
of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the
perturbing centre in
\mathbbR2{\mathbb{R}^2} , we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use
of a general result of Bolotin and MacKay (Celest Mech Dyn Astron 77:49–75, 77:49–75, 2000; Celest Mech Dyn Astron 94(4):433–449,
2006). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic
orbits of the regularised problem passing through the third centre. 相似文献
3.
We use a three dimensional generalization of Szebehely’s invariant relation obtained by us (Makó and Szenkovits, Celest. Mech.
Dyn. Astron. 90, 51, 2004) in the elliptic restricted three-body problem, to establish more accurate criterion of the Hill stability. By
using this criterion, the Hill stability of four extrasolar planets (γ Cephei Ab, Gliese 86 Ab, HD 41004 Ab and HD 41004 Bb) is investigated. 相似文献
4.
G. F. Gronchi D. Farnocchia L. Dimare 《Celestial Mechanics and Dynamical Astronomy》2011,110(3):257-270
The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of
a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this
problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This
paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299–318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace–Lenz vector
in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half. 相似文献
5.
Michael Efroimsky 《Celestial Mechanics and Dynamical Astronomy》2006,96(3-4):259-288
We continue the study undertaken in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)], this opinion had stemmed from an inexact interpretation of a correct result by Goldreich [Astron. J. 70, 5–9 (1965)]. Although Goldreich [Astron. J. 70, 5–9 (1965)] mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition had been persistently ignored for forty years. It was explained in Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)] that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich [Astron. J. 70, 5–9 (1965)], remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in , where is the planetary equator’s total precession rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander. We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator.By “precession,” in its most general sense, we mean any change of the direction of the spin axis of the planet—from its long-term variations down to nutations down to the Chandler wobble and polar wander. 相似文献
6.
Masatoshi Hirabayashi Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2013,117(3):245-262
Recursive computation of mutual potential, force, and torque between two polyhedra is studied. Based on formulations by Werner and Scheeres (Celest Mech Dyn Astron 91:337–349, 2005) and Fahnestock and Scheeres (Celest Mech Dyn Astron 96:317–339, 2006) who applied the Legendre polynomial expansion to gravity interactions and expressed each order term by a shape-dependent part and a shape-independent part, this paper generalizes the computation of each order term, giving recursive relations of the shape-dependent part. To consider the potential, force, and torque, we introduce three tensors. This method is applicable to any multi-body systems. Finally, we implement this recursive computation to simulate the dynamics of a two rigid-body system that consists of two equal-sized parallelepipeds. 相似文献
7.
Michael Nauenberg 《Celestial Mechanics and Dynamical Astronomy》2007,97(1):1-15
Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects
the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates
with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped
orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron.
78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family
of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight
orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for
n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the
figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given
numerically, and some examples are given of nearby stable orbits which bifurcate from these families. 相似文献
8.
9.
Report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements: 2003 总被引：1，自引：1，他引：0
P. K. Seidelmann B. A. Archinal M. F. A’Hearn D. P. Cruikshank J. L. Hilton H. U. Keller J. Oberst J. L. Simon P. Stooke D. J. Tholen P. C. Thomas 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):203-215
Every three years the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the
directions of the north poles of rotation and the prime meridians of the planets, satellites, and asteroids. This report introduces
a system of cartographic coordinates for asteroids and comets. A topographic reference surface for Mars is recommended. Tables
for the rotational elements of the planets and satellites and size and shape of the planets and satellites are not included,
since there were no changes to the values. They are available in the previous report (Celest. Mech. Dyn. Astron., 82, 83–110, 2002), a version of which is also available on a web site. 相似文献
10.
We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries
is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile
coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by
taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem. 相似文献
11.
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
12.
The purpose of this work is to evaluate the effect of deformation inertia on tide dynamics, particularly within the context of the tide response equations proposed independently by Boué et al. (Celest Mech Dyn Astron 126:31–60, 2016) and Ragazzo and Ruiz (Celest Mech Dyn Astron 128(1):19–59, 2017). The singular limit as the inertia tends to zero is analyzed, and equations for the small inertia regime are proposed. The analysis of Love numbers shows that, independently of the rheology, deformation inertia can be neglected if the tide-forcing frequency is much smaller than the frequency of small oscillations of an ideal body made of a perfect (inviscid) fluid with the same inertial and gravitational properties of the original body. Finally, numerical integration of the full set of equations, which couples tide, spin and orbit, is used to evaluate the effect of inertia on the overall motion. The results are consistent with those obtained from the Love number analysis. The conclusion is that, from the point of view of orbital evolution of celestial bodies, deformation inertia can be safely neglected. (Exceptions may occur when a higher-order harmonic of the tide forcing has a high amplitude.) 相似文献
13.
Alexei V. Tsygvintsev 《Celestial Mechanics and Dynamical Astronomy》2007,99(1):23-29
We consider the Newtonian planar three-body problem with positive masses m
1, m
2, m
3. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian
orbit besides three exceptional cases ∑m
i
m
j
/(∑m
k
)2 = 1/3, 23/33, 2/32 where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis
of the three-body problem started in papers [Tsygvintsev, A.: Journal für die reine und angewandte Mathematik N 537, 127–149
(2001a); Celest. Mech. Dyn. Astron. 86(3), 237–247 (2003)] and based on the Morales–Ramis–Ziglin approach. 相似文献
14.
Rocío Isabel Páez Christos Efthymiopoulos 《Celestial Mechanics and Dynamical Astronomy》2018,130(2):20
One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced ‘basic Hamiltonian model’ \(H_\mathrm{b}\) for Trojan dynamics (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez et al. in Celest Mech Dyn Astron 126:519, 2016): we show that the inner border of the secondary resonance of lowermost order, as defined by \(H_\mathrm{b}\), provides a good estimation of the region in phase space for which the orbits remain regular regardless of the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an ‘asymmetric expansion’ of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed. 相似文献
15.
Jacques Laskar 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):351-356
Recently, Breiter et al. [Celest. Mech. Dyn. Astron., 2004, 88, 153–161] reported the computation of Hansen coefficients X
k
γ ,m
for non-integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b
s
(m), and generalized Laplace coefficients b
s,r
(m) [Laskar and Robutel, 1995, Celest. Mech. Dyn. Astron., 62, 193–217] that do not require s,r to be integers. In particular, the coefficients X
0
γ ,m
have very simple expressions in terms of the usual Laplace coefficients b
γ +2
(m), and all their properties derive easily from the known properties of the Laplace coefficients. 相似文献
16.
Yoshio Kubo 《Celestial Mechanics and Dynamical Astronomy》2011,110(2):143-168
Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational
axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the
Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation
is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate
Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The
CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that
in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both
for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear
further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the
Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive
an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation
(in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin
axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the
motion of the outer bodies is taken into account. 相似文献
17.
Yoshio Kubo 《Celestial Mechanics and Dynamical Astronomy》2009,105(4):261-274
We calculate the so-called convective term, which shows up in the expression for the angular velocity of the elastic Earth,
within the Andoyer formalism. The term emerges due to the fact that the elasticity-caused perturbation depends not only on
the instantaneous orientation of the Earth but also on its instantaneous angular velocity. We demonstrate that this term makes
a considerable contribution into the overall angular velocity. At the same time the convective term turns out to be automatically
included into the correction to the nutation series due to the elasticity, if the series is defined by the perturbation of
the figure axis (and not of the rotational axis) in accordance with the current IAU resolution. Hence it is not necessary
to take the effect of the convective term into consideration in the perturbation of the elastic Earth as far as the nutation
is related to the motion of the figure axis. 相似文献
18.
Alessandra Celletti Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2017,127(1):1-18
We revisit the relegation algorithm by Deprit et al. (Celest. Mech. Dyn. Astron. 79:157–182, 2001) in the light of the rigorous Nekhoroshev’s like theory. This relatively recent algorithm is nowadays widely used for implementing closed form analytic perturbation theories, as it generalises the classical Birkhoff normalisation algorithm. The algorithm, here briefly explained by means of Lie transformations, has been so far introduced and used in a formal way, i.e. without providing any rigorous convergence or asymptotic estimates. The overall aim of this paper is to find such quantitative estimates and to show how the results about stability over exponentially long times can be recovered in a simple and effective way, at least in the non-resonant case. 相似文献
19.
K. Karami 《Astrophysics and Space Science》2009,319(1):37-44
Here the effect of rotation up to third order in the angular velocity of a star on the p, f and g modes is investigated. To
do this, the third-order perturbation formalism presented by Soufi et al. (Astron. Astrophys. 334:911, 1998) and revised by Karami (Chin. J. Astron. Astrophys. 8:285, 2008), was used. I quantify by numerical calculations the effect of rotation on the oscillation frequencies of a uniformly rotating
β-Cephei star with 12 M
⊙. For an equatorial velocity of 90 km s−1, it is found that the second- and third-order corrections for (l,m)=(5,−4), for instance, are of order of 0.07% of the frequency for radial order n=−3 and reaches up to 0.6% for n=−20. 相似文献
20.
We revisit a set of symplectic variables introduced by Andre Deprit (Celest Mech 30, 181–195, 1983), which allows for a complete symplectic reduction in rotation invariant Hamiltonian systems, generalizing to arbitrary dimension
Jacobi’s reduction of the nodes. In particular, we introduce an action-angle version of Deprit’s variables, connected to the
Delaunay variables, and give a new hierarchical proof of the symplectic character of Deprit’s variables. 相似文献