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1.
The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this
purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see
Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function,
we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to
regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered
in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced,
in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described
in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically,
for the Earth-Moon binary system. 相似文献
2.
M. Šidlichovský 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):167-185
An adiabatic approximation for the non-planar, circular, restricted 3BP is presented for the external resonance 4/7. It can
be used as a model for resonant Kuiper belt objects. The Hamiltonian is truncated at the fourth order in eccentricities and
inclinations. After averaging, we have a system of two degrees of freedom with two frequencies. Numerical calculations show
that the ratio of these frequencies is ~102. Having introduced suitable canonical variables, we used the adiabatic approach introduced by Wisdom in a different context.
We left slow variables frozen and after solving the pendulum problem for fast variables, we used the averaged effect of fast
variables on slow variables. In this way we obtained the guiding trajectories for slow variables as contour lines of adiabatic
invariant. We discuss the existence of a chaotic region which is formed by trajectories crossing a critical curve which corresponds
to the separatrix of fast pendulum motion, where the assumption of sharp division between fast and slow frequencies is not
correct and the adiabatic theory fails. The model works well for e ~ 0.1 and can be used for finding the chaotic regions, but for e~ 0.17 it becomes unsatisfactory due to truncation and bad convergence of the Laplace expansion. Qualitatively it can, however,
help us to understand how the protective mechanism works as the interplay of mean motion and Kozai–Lidov resonance. 相似文献
3.
Two basic problems of dynamics, one of which was tackled in the extensive work of Z. Kopal (see e.g. Kopal, 1978, Dynamics of Close Binary Systems, D. Reidel Publication, Dordrecht, Holland.), are presented with their approximate general solutions. The ‘penetration’ into
the space of solution of these non-integrable autonomous and conservative systems is achieved by application of ‘The Last Geometric Theorem of Poincaré’ (Birkhoff, 1913, Am. Math. Soc. (rev. edn. 1966)) and the calculation of sub-sets of ‘solutions précieuses’ that are covering densely the spaces of all solutions
(non-periodic and periodic) of these problems. The treated problems are: 1. The two-dimensional Duffing problem, 2. The restricted
problem around the Roche limit. The approximate general solutions are developed by applying known techniques by means of which
all solutions re-entering after one, two, three, etc, revolutions are, first, located and then calculated with precision.
The properties of these general solutions, such as the morphology of their constituent periodic solutions and their stability
for both problems are discussed. Calculations of Poincaré sections verify the presence of chaos, but this does not bear on
the computability of the general solutions of the problems treated. The procedure applied seems efficient and sufficient for
developing approximate general solutions of conservative and autonomous dynamical systems that fulfil the PoincaréBirkhoff
theorems. The same procedure does not apply to the sub-set of unbounded solutions of these problems. 相似文献
4.
V. S. Geroyannis 《Astrophysics and Space Science》1985,114(1):175-189
In this paper the ‘class of near homoaxial rotations’ is defined, being suitable for treatment of problems of nonuniform rotation
of stars. This class is represented by a proper form of the so-called ‘velocity tensor’, the latter describing efficiently
the motion of a deformable finite material continuum in the common frame. The ‘class of particular near homoaxial rotations’
is then defined, characterized by simple transformation equations of the velocity tensor in two noninertial frames; namely,
in a ‘frame rotating uniformly’ relative to the common frame, and in a ‘frame rotating nonuniformly’ relative to it. A sufficient
condition is also derived so that a near homoaxial rotation be reducible to a particular one. ‘Preferred frames’ are then
defined in the sense that they preserve a near homoaxial rotation in its class when referring thismotion; that is, such frames
keep invariant the intertial class of the motion. Finally, a method is proposed for constructing a nonuniformly rotating preferred
frame, to which a near homoaxial rotation is referred simply as ‘radial distortion’. 相似文献
5.
6.
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
7.
Any dynamical system can be put in generalized canonical form through the introduction of a set of auxiliary ‘conjugate’ variables
or momenta and solved by perturbation theory based on Lie series. The application of Hori's method for generalized canonical
system leads to a new canonical transformation — the Mathieu transformation — defined by the solution of the Hori auxiliary
system. This new transformation simplifies the algorithm since the inversion of the solution of the Hori auxiliary system
is no longer necessary. In this paper, we wish to show some peculiarities of this technique. 相似文献
8.
Some asteroids in Earth‐crossing orbits avoid close approaches by entering in a mean motion resonance whenever the distance between the two orbits is small. These orbits are ‘Toro class’ according to the classification of (Milani et al., 1989). This protection mechanism can be understood by a semi‐averaged model, in which the fast variables are removed and the dynamical variables are the critical argument and the semimajor axis, with dependence upon a slow parameter. The adiabatic invariant theory can be applied to this model and accounts for all the qualitative features of the orbits in this class, including the onset of the libration when the orbit distance is small. Because of the neglected perturbations by the other planets, this theory is approximate and the adiabatic invariant is conserved only with low accuracy moreover, the Toro state can be terminated by a close approach to another planet (typically Venus). “Would you tell me, please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the Cat. Alice in Wonderland, L. Carroll 相似文献
9.
Michael Efroimsky 《Celestial Mechanics and Dynamical Astronomy》2005,91(1-2):75-108
It was believed until very recently that a near-equatorial satellite would always keep up with the planet’s equator (with
oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich [Astronomy & Astrophysics (2004) Vol. 415, pp. 1187–1199], this misconception originated from a wrong interpretation of a (mathematically correct)
result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating
elements will endow the planetary equations with some extra terms caused by the planet’s obliquity change. Some of these terms
will be non-trivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations
of a planet’s obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical
formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely
slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital
averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However,
it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations
will resonate with the satellite’s orbital frequency and, thereby, may instigate a secular drift. A detailed study of this
process will be offered in a subsequent publication, while here we work out the required mathematical formalism and point
out the key aspects of the dynamics.
In this article, as well as in (Efroimsky 2004), we use the word ‘‘precession’’ in its most general sense which embraces the
entire spectrum of changes of the spin-axis orientation -- from the long-term variations down to the Chandler Wobble down
to nutations and to the polar wonder. 相似文献
10.
We analyze the out-of-plane librations of a tethered satellite system that is nominally rotating in the orbit plane. To isolate
the librational dynamics, the system is modeled as two point masses connected by a rigid rod with the system mass center constrained
to an unperturbed circular orbit. For small out-of-plane librations, the in-plane motion is unaffected by the out-of-plane
librations and a solution for the in-plane motion is determined in terms of Jacobi elliptic functions. This solution is used
in the linearized equation for the out-of-plane librations, resulting in a Hill’s equation. Floquet theory is used to analyze
the Hill’s equation, and we show that the out-of-plane librations are unstable for certain ranges of in-plane spin rate. For
relatively high in-plane spin rates, the out-of-plane librations are stable, and the Hill’s equation can be approximated by
a Mathieu’s equation. Approximate solutions to the Mathieu’s equation are determined, and we analyze the dominant characteristics
of the out-of-plane librations for high in-plane spin rates. The results obtained from the analysis of the linearized equations
of motion are compared to numerical simulations of the nonlinear equations of motion, as well as numerical simulations of
a more realistic system model that accounts for tether flexibility. The instabilities discovered from the linear analysis
are present in both the nonlinear system and the more realistic system model. The approximate solutions for the out-of-plane
librations compare well to the nonlinear system for relatively high in-plane rotation rates, and also capture the significant
qualitative behavior of the flexible system. 相似文献
11.
The increasing number and variety of extrasolar planets illustrates the importance of characterizing planetary perturbations. Planetary orbits are typically described by physically intuitive orbital elements. Here, we explicitly express the equations of motion of the unaveraged perturbed two-body problem in terms of planetary orbital elements by using a generalized form of Gauss’ equations. We consider a varied set of position and velocity-dependent perturbations, and also derive relevant specific cases of the equations: when they are averaged over fast variables (the “adiabatic” approximation), and in the prograde and retrograde planar cases. In each instance, we delineate the properties of the equations. As brief demonstrations of potential applications, we consider the effect of Galactic tides. We measure the effect on the widest-known exoplanet orbit, Sedna-like objects, and distant scattered disk objects, particularly with regard to where the adiabatic approximation breaks down. The Mathematica code which can help derive the equations of motion for a user-defined perturbation is freely available upon request. 相似文献
12.
C. Polymilis G. Rowlands A. N. Yannacopoulos 《Celestial Mechanics and Dynamical Astronomy》1997,68(4):273-281
A simple method for the determination of the position of singularities in the complex time domain for dynamical systems which
are described by ordinary differential equations is presented. The method is designed for integrable separable systems whose
solutions are not expressible in closed form. A direct consequence of this method is that it ‘closes’ the phase space. Simple
physical meaning is given to the singularity position.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
13.
We present an analysis of the behaviour of the ‘coarse-grained’ (‘mesoscopic’) rank partitioning of the mean energy of collections
of particles composing virialized dark matter halos in a Λ-CDM cosmological simulation. We find evidence that rank preservation
depends on halo mass, in the sense that more massive halos show more rank preservation than less massive ones. We find that
the most massive halos obey Arnold’s theorem (on the ordering of the characteristic frequencies of the system) more frequently
than less massive halos. This method may be useful to evaluate the coarse-graining level (minimum number of particles per
energy cell) necessary to reasonably measure signatures of ‘mesoscopic’ rank orderings in a gravitational system. 相似文献
14.
Enrique Guzmán 《Astrophysics and Space Science》1997,253(1):7-12
By solving a Wheeler-De Witt ‘extended’ equation in the Brans-Dicke theory, we have found that the probability distribution
predicts: i) An initial value for the Brans-Dicke scalar field φ ∼ ρ1/2_VAC in the beginning of the inflation, where ρVAC is the vacuum density energy (this gives a planck mass ∼ ρ1/4_VAC) ii) Large values for the Brans-Dicke parameter w. On the other hand it is shown that by taking into account the dynamical
behaviour of φ and the matter scalar field σ we can formulate a ‘creation boundary condition’ where in the ‘beginning’ of
the Universe (R =0, ‘nothing’ for some authors) we have a dynamical σ already ‘created’. This could be the energetic mechanism
which makes Universe tunnels the potential barrier to evolve classically after. Besides we have found the possibility of a
cosmological uncertainty principle.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
15.
In the present paper the equations of the orbital motion of the major planets and the Moon and the equations of the three–axial
rigid Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and
lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and
lunar evolution). Hence, the theory of the Earth’s rotation can be presented by means of the series in powers of the evolutionary
variables with quasi-periodic coefficients with respect to the planetary–lunar mean longitudes. This form of the Earth’s rotation
problem is compatible with the general planetary theory involving the separation of the short–period and long–period variables
and avoiding the appearance of the non–physical secular terms. 相似文献
16.
Hasmukh K. Tank 《Astrophysics and Space Science》2011,336(2):341-343
This letter points out that the values of ‘critical-acceleration’ of MOND, and the ‘accelerated-expansion’ of the universe
are just two of the fourteen strikingly equal values of accelerations recurring in different physical situations. Some of
them could be explained by a new law of equality of potential-energy and energy-of-mass of reasonably-independent systems
(Tank in Astrophys. Space Sci. 330:203–205, 2010; Tank in Adv. Stud. Theor. Phys. 5:45–55, 2011). This new conservation-law, of equality of potential-energy, energy-of-mass and ‘kinetic-energy’ may be a clue to understand
MOND, and the ‘accelerated-expansion’ of the universe. Alternative expressions for the cosmological red-shift, the ‘critical-acceleration’
of MOND and Newton’s law of universal gravitation are also presented for comparison of three different accelerations. 相似文献
17.
François Farago Jacques Laskar Jocelyn Couetdic 《Celestial Mechanics and Dynamical Astronomy》2009,104(3):291-306
Exploring the global dynamics of a planetary system involves computing integrations for an entire subset of its parameter
space. This becomes time-consuming in presence of a planet close to the central star, and in practice this planet will be
very often omitted. We derive for this problem an averaged Hamiltonian and the associated equations of motion that allow us
to include the average interaction of the fast planet. We demonstrate the application of these equations in the case of the μ Arae system where the ratio of the two fastest periods exceeds 30. In this case, the effect of the inner planet is limited
because the planet’s mass is one order of magnitude below the other planetary masses. When the inner planet is massive, considering
its averaged interaction with the rest of the system becomes even more crucial. 相似文献
18.
Martin J. Rees 《Journal of Astrophysics and Astronomy》1984,5(4):331-348
The evidence for unseen mass (which is briefly reviewed) suggests that the cosmological density parameter Ω is at least 0.1–0.2.
An Einstein-de-Sitter ‘flat’ universe with Ω = 1 — which is appealing for theoretical reasons — can only be reconciled with
the data if the galaxies are more ‘clumped’ than the overall mass distribution, and are poor tracers of the unseen mass even
on scales of several Mpc. Possible forms for the unseen mass are discussed; and feedback processes are outlined whereby galaxy
formation can be suppressed in underdense regions. 相似文献
19.
A. V. Kolesnichenko 《Solar System Research》2010,44(4):334-347
This paper considers the modern approach to the thermodynamic modeling of developed turbulent flows of a compressible fluid
based on the systematic application of the formalism of extended irreversible thermodynamics (EIT) that goes beyond the local
equilibrium hypothesis, which is an inseparable attribute of classical nonequilibrium thermodynamics (CNT). In addition to
the classical thermodynamic variables, EIT introduces new state parameters—dissipative flows and the means to obtain the respective
evolutionary equations consistent with the second law of thermodynamics. The paper presents a detailed discussion of a number
of physical and mathematical postulates and assumptions used to build a thermodynamic model of turbulence. A turbulized liquid
is treated as an indiscrete continuum consisting of two thermodynamic sub-systems: an averaged motion subsystem and a turbulent
chaos subsystem, where turbulent chaos is understood as a conglomerate of small-scale vortex bodies. Under the above formalism,
this representation enables the construction of new models of continual mechanics to derive cause-and-effect differential
equations for turbulent heat and impulse transfer, which describe, together with the averaged conservations laws, turbulent
flows with transverse shear. Unlike gradient (noncausal) relationships for turbulent flows, these differential equations can
be used to investigate both hereditary phenomena, i.e., phenomena with history or memory, and nonlocal and nonlinear effects.
Thus, within EIT, the second-order turbulence models underlying the so-called invariant modeling of developed turbulence get
a thermodynamic explanation. Since shear turbulent flows are widespread in nature, one can expect the given modification of
the earlier developed thermodynamic approach to developed turbulence modeling (see Kolesnichenko, 1980; 1998; 2002–2004; Kolesnichenko
and Marov, 1985; Kolesnichenko and Marov, 2009) to be used in research on a broad class of dissipative phenomena in various
astro- and geophysical applications. In particular, a major application of the proposed approach is the reconstruction of
the processes in the preplanetary circumsolar disk, which might help solve the fundamental problems of stellar-planetary cosmogony. 相似文献
20.
Viviane Pierrard Milan Maksimovic Joseph Lemaire 《Astrophysics and Space Science》2001,277(1-2):195-200
Electron velocity distribution functions (VDF) observed in the low speed solar wind flow are generally characterized by ‘core’
and ‘halo’ electrons. In the high speed solar wind, a third population of ‘strahl’ electrons is generally observed. New collisional
models based on the solution of the Fokker-Planck equation can be used to determine the importance of the different electron
populations as a function of the radial distance. Typical electron velocity distribution functions observed at 1 AU from the
Sun are used as boundary conditions for the high speed solar wind and for the low speed solar wind. Taking into account the
effects of external forces and Coulomb collisions with a background plasma, suprathermal tails are found to be present in
the electron VDF at low altitudes in the corona when they exist at large radial distances.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献