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1.
Michel Hénon 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):87-100
We present five families of periodic solutions of Hill’s problem which are asymmetric with respect to the horizontal ξ axis. In one of these families, the orbits are symmetric with respect to the vertical η axis; in the four others, the orbits are without any symmetry. Each family consists of two branches, which are mirror images
of each other with respect to the ξ axis. These two branches are joined at a maximum of Γ, where the family of asymmetric periodic solutions intersects a family
of symmetric (with respect to the ξ axis) periodic solutions. Both branches can be continued into second species families for Γ → − ∞. 相似文献
2.
K. E. Papadakis 《Astrophysics and Space Science》2009,323(3):261-272
In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m
0=β
m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov
periodic orbits, which emanate from the collinear equilibrium points L
1 and L
2. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic
orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal
points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points.
All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating
homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented. 相似文献
3.
This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x
0 ∊ [−2,2],C ∊ [−2,5]) of the (x
0, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x
0 ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order
n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x
0, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D
i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families. 相似文献
4.
The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into
the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution
space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter
after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter),
which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain
(x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε =
0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce
the density parameter, say to ε = 10−6, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary.
The stability of all members of each and all families computed was calculated and presented in this paper for both the large
solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by
stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied
by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections.
All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received,
consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including
their stability and is available at request. It is concluded that approximation of the general solution of this system is
straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties. 相似文献
5.
Roman V. Baluev 《Celestial Mechanics and Dynamical Astronomy》2008,102(4):297-325
The full set of published radial velocity data (52 measurements from Keck + 58 ones from ELODIE + 17 ones from CORALIE) for
the star HD37124 is analysed. Two families of dynamically stable high-eccentricity orbital solutions for the planetary system
are found. In the first one, the outer planets c and d are trapped in the 2/1 mean-motion resonance. The second family of
solutions corresponds to the 5/2 mean-motion resonance between these planets. In both families, the planets are locked in
(or close to) an apsidal corotation resonance. In the case of the 2/1 MMR, it is an asymmetric apsidal corotation (with the
difference between the longitudes of periastra Δω ~ 60°), whereas in the case of the 5/2 MMR it is a symmetric antialigned one (Δω = 180°). It remains also possible that the two outer planets are not trapped in an orbital resonance. Then their orbital
eccentricities should be relatively small (less than, say, 0.15) and the ratio of their orbital periods is unlikely to exceed
2.3 − 2.5. 相似文献
6.
Maxwell’s ring-type configuration (i.e. an N-body model where the ν = Ν − 1 bodies have equal masses and are located at the vertices of a regular ν-gon while the N-th body with a different mass is located at the center of mass of the system) has attracted special attention during the
last 15 years and many aspects of it have been studied by considering Newtonian and post-Newtonian potentials (Mioc and Stavinschi
1998, 1999), homographic solutions (Arribas et al.
2007) and relative equilibrium solutions (Elmabsout 1996), etc. An equally interesting problem, known as the ring problem of (N + 1) bodies, deals with the dynamics of a small body in the combined force field produced by such a configuration. This is
the problem we are dealing with in the present paper and our aim is to investigate the variations in the dynamics of the small
body in the case that the central primary is also a radiating source and therefore acts on the particle with both gravitation
and radiation. Based on the general outlines of Radzievskii’s model, we study the permitted and the existing trapping regions
of the particle, its equilibrium locations and their parametric variations as well as the existence of focal points in the
zero-velocity diagrams. The distribution of the characteristic curves of families of planar symmetric periodic orbits and
their stability for various values of the radiation coefficient of the central body is additionally investigated. 相似文献
7.
The paper presents a variety of classes of interior solutions of Einstein–Maxwell field equations of general relativity for
a static, spherically symmetric distribution of the charged fluid with well behaved nature. These classes of solutions describe
perfect fluid balls with positively finite central pressure, positively finite central density; their ratio is less than one
and causality condition is obeyed at the center. The outmarch of pressure, density, pressure–density ratio and the adiabatic
speed of sound is monotonically decreasing for these solutions. Keeping in view of well behaved nature of these solutions,
two new classes of solutions are being studied extensively. Moreover, these classes of solutions give us wide range of constant
K for which the solutions are well behaved hence, suitable for modeling of super dense star. For solution (I1) the mass of a star is maximized with all degree of suitability and by assuming the surface density ρ
b
=2×1014 g/cm3 corresponding to K=1.19 and X=0.20, the maximum mass of the star comes out to be 2.5M
Θ with linear dimension 25.29 Km and central redshift 0.2802. It has been observed that with the increase of charge parameter K,
the mass of the star also increases. For n=4,5,6,7, the charged solutions are well behaved with their neutral counterparts however, for n=1,2,3, the charged solution are well behaved but their neutral counterparts are not well behaved. 相似文献
8.
We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed
regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions
of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on
the mass parameter β and the radiation factors q
i
, i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium
points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic
solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q
i
vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are
calculated. 相似文献
9.
The energy density of Vaidya-Tikekar isentropic superdense star is found to be decreasing away from the center, only if the
parameter K is negative. The most general exact solution for the star is derived for all negative values of K in terms of circular and inverse circular functions. Which can further be expressed in terms of algebraic functions for K = 2-(n/δ)2 < 0 (n being integer andδ = 1,2,3 4). The energy conditions 0 ≤ p ≤ αρc
2, (α = 1 or 1/3) and adiabatic sound speed conditiondp dρ ≤ c
2, when applied at the center and at the boundary, restricted the parameters K and α such that .18 < −K −2287 and.004 ≤ α ≤ .86. The maximum mass of the star satisfying the strong energy condition (SEC),
(α = 1/3) is found to be3.82 Mq· at K=−2/3, while the same for the weak energy condition (WEC), (α =1) is 4.57 M_
⊙ atK=−>5/2. In each case the surface density is assumed to be 2 × 1014 gm cm-3. The solutions corresponding to K>0 (in fact K>1) are also made meaningful by considering the hypersurfaces t= constant as 3-hyperboloid by replacing the parameter R
2 by −R2 in Vaidya-Tikekar formalism. The solutions for the later case are also expressible in terms of algebraic functions for K=2-(n/δ2 > 1 (n being integer or zero and δ =1,2,3 4). The cases for which 0 < K < 1 do not possess negative energy density gradient and therefore are incapable of representing any physically plausible
star model. In totality the article provides all the physically plausible exact solutions for the Buchdahl static perfect
fluid spheres.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
We consider a self consistent system of Bianchi Type-I cosmology and Binary Mixture of perfect fluid and dark energy. The
perfect fluid is taken to be obeying equations of state p
PF
=γρ
PF
with γ∈[0,1]. The dark energy is considered to be obeying a quintessence-like equation of state where the dark energy obeys equation
of state p
DE
=ωρ
DE
where ω∈[−1,0]. Exact solutions to the corresponding Einstein field equations are obtained. Some special cases are discussed and
studied. Further more power law models and exponential models are investigated. 相似文献
11.
We use the method of time – distance analysis to measure lifetimes of solar p modes in the range ℓ=100 − 600 and ν=3.0 − 4.5 mHz with data taken with the Taiwan Oscillation Network (TON). The lifetimes of p modes are determined by the changes in the amplitude and width of the cross-correlation function of a wave packet with the
number of skips. The amplitude of the cross-correlation function decreases exponentially with the number of skips as in previous
work. This decrease has been interpreted as the effect of the finite p-mode lifetime. In this study, we find that the width of the cross-correlation function increases with the number of skips.
We interpret this phenomenon as the effect of the dispersion of the wave packet. We include this effect in the determination
of the lifetime of the wave packet. The lifetime increases after the dispersion is taken into account. We also study the change
in lifetime between solar minimum and maximum. 相似文献
12.
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
13.
New doubly-symmetric families of comet-like periodic orbits in the spatial restricted (N + 1)-body problem 总被引：1，自引：0，他引：1
For any positive integer N ≥ 2 we prove the existence of a new family of periodic solutions for the spatial restricted (N +1)-body problem. In these solutions the infinitesimal particle is very far from the primaries. They have large inclinations
and some symmetries. In fact we extend results of Howison and Meyer (J. Diff. Equ. 163:174–197, 2000) from N = 2 to any positive integer N ≥ 2.
相似文献
14.
Assuming the time-dependent equation of state p=λ(t)ρ, five dimensional cosmological models with viscous fluid for an open universe (k=−1) and flat universe (k=0) are presented. Exact solutions in the context of the rest mass varying theory of gravity proposed by Wesson (Astron. Astrophys.
119, 145, 1983) are obtained. It is found that the phenomenon of isotropisation takes place in this theory, i.e. the mass scale factor A(t) which characterizes the rest mass of a typical particle is evolving with cosmic time just as the spatial scale factor R(t). It is further found that rest mass is approximately constant in the present universe. 相似文献
15.
When μ is smaller than Routh’s critical value μ
1 = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short
period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ
5), (μ
5, μ
4), (μ
4, μ
3), (μ
3, μ
2), (μ
2, μ
1), (μ
1, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ
2, μ
1), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points
was enriched. 相似文献
16.
17.
This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system.
The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and
Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation
of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for
the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the
dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation
of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood
of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the
vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov
family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a
very long time. It is remarkable that these regions seem to persist in the real system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
Based on the data on a spectral dependence of the geometric albedo of giant planet discs, we obtained depth variations in
the optical thickness τ
a
of the aerosol component and relative concentration γ of methane (Uranium, Neptune) lnτ
a
= −0.720 + 1.507Δlnp (for −2.2085 ≤ lnp ≤ −1.0018), lnτ
a
= +1.224 + 1.160Δlnp (for −1.0018 ≤ lnp ≤ −0.0595), lnτ
a
= +2.318 + 0.192Δlnp (for −0.0595 ≤ lnp), γ = 0.0027 for Jupiter; lnτ
a
= −0.846 + 1.598Δlnp (for −3.3619 ≤ lnp ≤ −2.0575), lnτ
a
= +1.238 + 1.342Δlnp (for −2.0575 ≤ lnp ≤ −1.2074), lnτ
a
= +2.379 + 0.722 (for −1.2074 ≤ lnp ≤ −0.6501), lnτ
a
= +2.781 + 0.326Δlnp (for 0.6501 ≤ lnp), γ = 0.0027 for Saturn; lnτ
a
= −2.694 + 0.087Δlnp (for +0.3685 ≤ lnp ≤ +1.2314), lnτ
a
= −2.619 + 7.341Δlnp (for +1.2314 ≤ lnp ≤ +1.7556), lnτ
a
= +1.229 + 0.956Δlnp (for +1.7556 ≤ lnp) for Uranium; lnτ
a
= −1.861 + 1.248Δlnp (for +0.3204 ≤ lnp ≤ +0.9051), lnτ
a
= −1.131 + 0.347Δlnp (for +0.9051 ≤ lnp) for Neptune; depth-averaged relative methane concentration lnγ = −9.982 + 2.676Δlnp(0.3584 ≤ lnp ≤ 1.5445); ln γ = −9.738 + 2.561Δlnp(0.3237 ≤ lnp ≤ 1.6156) and γ = 0.00382(lnp ≥ 1.6156); 0.00554(lnp ≥ 1.6156) for Uranium and Neptune, respectively (p is in bar). 相似文献
19.
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
20.
We study the simple periodic orbits of a particle that is subject to the gravitational action of the much bigger primary bodies
which form a regular polygonal configuration of (ν+1) bodies when ν=8. We investigate the distribution of the characteristic curves of the families and their evolution in the phase space of
the initial conditions, we describe various types of simple periodic orbits and we study their linear stability. Plots and
tables illustrate the obtained material and reveal many interesting aspects regarding particle dynamics in such a multi-body
system. 相似文献