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1.
The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10−3. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions. 相似文献
2.
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
3.
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
4.
5.
Edelman Colette 《Celestial Mechanics and Dynamical Astronomy》1993,55(4):369-386
The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS 1 andS 2 changing the eccentricity of a solution. The family of symmetries,S 1, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S 2, is related to a possibleaffinity of the configurations space. Via a symmetry of theS 2 family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS 1.S 2, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC 2 H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:
- The two families of symmetriesS 1 andS 2 are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
- TheS 1 family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
- TheS 2 family of symmetries may regularize multiple collisions among central solutions of the same type.
6.
This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees
of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double
oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed
by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions.
The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy
of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space
of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical
integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented
here conservation of the energy constant between 10-9 and 10-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the
number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions
were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter
any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the
boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that
investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas
of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of
sections.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
Antonis D. Pinotsis 《Planetary and Space Science》2009,57(12):1389-1404
By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L4 and L5 are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L4 and L5, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem. 相似文献
8.
For the equation describing plane oscillations and rotations of a satellite, we consider families of symmetric generalized
periodic solutions with integral rotation number p. We give new confirmations of the hypothesis: there are only four classes of these families with topologically different
structures, namely, the classes of families of periodic solutions with p≥ 1, p= 0, p=−1, and p≤−2. Besides, we demonstrate that the vertices of cusps of these families are placed on some analytical curves, and the same
is true for the multiple intersections of these families with other families. 相似文献
9.
O. Ragos K. E. Papadakis C. G. Zagouras 《Celestial Mechanics and Dynamical Astronomy》1997,67(4):251-274
The regions of quasi-periodic motion around non-symmetric periodic orbits in the vicinity of the triangular equilibrium points
are studied numerically. First, for a value of the mass parameter less than Routh's critical value, the stability regions
determined by quasi-periodic motion are examined around the existing families of short (Ls
4) and long (Ll
4) period solutions. Then, for two values of μ greater than the Routh value, the unified family Lsl
4, to which, in these cases, Ls
4 and Ll
4 merge, is considered. It is found that such regions surround in general the linearly stable segments of the corresponding
families and become smaller as the mass ratio increases.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
《New Astronomy》2021
This report presents i) various characteristic features in photoionization cross sections (σPI) of Cl II + hν → Cl III + e for many fine structure levels of Cl II, 392 in total with n ≤ 10 and l ≤ 9, ii) comparison of features with those observed in an experiment carried out at the Advanced Light Source at Lawrence Berkeley National Lab, and iii) partial photoionization cross sections of the ground level for ionization leaving the core ion in to various excited levels for applications in plasma modeling. The features correspond to resonant structures, the shape of the background, and their interference effect in σPI of this near neutral ion Cl II with 16 electrons. σPI for the 5 levels of the ground configuration 3s23p4(3P0,1,2, 1D2, 1S0) of Cl II show regions of narrow Rydberg resonances at and near threshold energies, and resonant structures at higher energies in contrast to typical smooth decrease in the background. Various other features in σPI of levels of excited equivalent electron states and broad Seaton resonances in single valence electron excited levels are illustrated with examples. Comparison of calculated σPI of the 15 lowest levels with the combined features of the measured photoionization spectrum shows excellent agreement by reproducing and thus identifying them to the levels that they belong to. The calculations were carried out in relativistic Breit-Pauli R-matrix (BPRM) method using a close coupling wave function expansion of 45 levels up to 4s of the core ion Cl III. These levels were optimized using a set of 12 configurations going up to orbital 5s, 3s23p3, 3s3p4, 3p5, 3s23p23d, 3s23p24s, 3s23p24p, 3s23p24d, 3s23p24f, 3s23p25s, 3s3p33d, 3s23p3d2, 3p43d producing 283 levels of Cl III. The autoionizing resonances are delineated with a fine energy mesh to observe the fine structure effects. The present results will provide high precision parameters for various applications involving this less studied ion. 相似文献
11.
We describe global bifurcations from the libration points of non-stationary periodic solutions of the restricted three body problem. We show that the only admissible continua of non-stationary periodic solutions of the planar restricted three body problem, bifurcating from the libration points, can be the short-period families bifurcating from the Lagrange equilibria L
4, L
5. We classify admissible continua and show that there are possible exactly six admissible continua of non-stationary periodic solutions of the planar restricted three body problem. We also characterize admissible continua of non-stationary periodic solutions of the spatial restricted three body problem. Moreover, we combine our results with the Déprit and Henrard conjectures (see [8]), concerning families of periodic solutions of the planar restricted three body problem, and show that they can be formulated in a stronger way. As the main tool we use degree theory for SO(2)-equivariant gradient maps defined by the second author in [25].This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
12.
The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL
1,L
2 andL
3 are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a
v
|=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given. 相似文献
13.
The main focus of this paper is calculation of the diameters of asteroids belonging to five families (Vesta, Eos, Eunomia, Koronis, and Themis). To do that, we used the HCM algorithm applied for a data set containing 292,003 numbered asteroids, and a numerical procedure for choosing the crucial parameter of the HCM, called “the cutting velocity” vcut. It was established with a precision as high as 1 m s?1. Thereafter, we used the WISE (Wide‐field Infrared Survey Explorer) catalog to set a range of albedo for the largest members of each family considered. The albedo data were supported by the data concerning color classification (SDSS MOC4). The asteroids with albedo out of this range were classified as interlopers and were therefore disqualified as family members. Sizes were calculated for the asteroids with albedo within the acceptable range. For the other asteroids (those chosen by means of the HCM, but with albedo not listed in the WISE), the value of albedo of the largest member of the family was adopted. Results are given in a set of figures showing the families on the planes (a, e), (a, i), (e, i). Diameters and volumes of the asteroids that are the individual members of a family were calculated on the basis of their known or assumed albedo and on their absolute magnitude. Volumes of the parent bodies of the families were found on the basis of the cumulative volume distribution of these families. We also studied the secular resonances of the family members. We have shown that the locations of members of the considered asteroid families are related to the lines of secular resonances z1, z2, and z3 with Saturn. 相似文献
14.
We consider the circular planar restricted three-body problem with the mass parameter μ = 5 × 10?5. Two families of periodic solutions are calculated: family c, starting from the collinear fixed point L 1, and the initial part of familyi, which begins by direct circular orbits of an infinitely small radius around the body of bigger mass. The calculated families are very close to the generating ones, which we described earlier. In particular, the existence of the predicted zigzag structure of characteristics of family iis verified. New properties of the planar and vertical traces are discovered. 相似文献
15.
A. Peton 《Astrophysics and Space Science》1979,64(2):433-442
In a static gravitational field the paths of light are curved, as noticed by H. Weyl. This property can bea priori stated for aV 3 Riemannian manifold: through any two points ofV 3 it is possible to draw two families of curves, the straight lines of Euclidean geometry and the photon trajectoriesz. We can perform a fibration of the Galilean space-time in an original way, by taking thez-trajectories of the photons as the base, the isochronic surfaces as fibres, and ‘the equal length time on az trajectory to reach a given point’ as the equivalence relation. The straight lines of Euclidean geometry can then carry the classical mechanics timet, and thez trajectories can carry the optics time t. These times are related by dt=F(x,t) dt. If we class the Universe as a pseudo-Riemannian manifold of normal hyperbolic typeC ∞, the time t determined above can be taken as the time coordinate inV 4. Under these conditions we have \(d\overline s ^2 \) =F 2 \(d\overline s ^2 \) , where \(d\overline s ^2 \) is the metric of the Riemannian manifold, conforming to the metric ds 2 and allowing t as the cosmic time. We can then use the results previously achieved by the author (Peton, 1979) and write: 1 +Z G =F(A s,t s,)/F(Aos,t o) wherez G denotes the shift of the spectral lines due to the metric. In the case of relative motion betweenO andS, we have $${\text{1 + z' = (1 + }}z_{\text{G}} {\text{)(1 + }}\beta _{\text{r}} {\text{)(1 }} - {\text{ }}\beta ^2 {\text{)}}^{ - 1/2} $$ The Doppler-Fizeau effect therefore appears as a result of the application of the Fermat principle. 相似文献
16.
B. Khalesseh 《Astrophysics and Space Science》1998,260(3):299-307
New radial velocity measurements of the Algol-type eclipsing binary AI Dra, based on Reticon observations, are presented.
The velocity measures themselves are based on fitting theoretical profiles, generated by a physical model of the binary, to
the observed cross-correlation function (ccf). Such profiles match this function very well, much better in fact that Gaussian
profiles which are generally used. Measuring the ccf's with Gaussian profiles yields following results: mp sin3 i=2.55± 0.05m⊙, ms sin3 i = 1.14 ± 0.03m⊙, (ap + as) sin i=7.34 ±0.05R⊙, and mp/ms =2.23± 0.05. Where as measuring the ccf's with theoretical profiles yields a mass ratio of 2.33 and following results:
mp sin3 i=2.84± 0.05m⊙, ms sin3 i=1.22 ± 0.03m⊙, (ap +as) sin i=7.56± 0.05R⊙.
The system comprising a semi-detached configuration. From the solution of a previously published light curved and combining
it's results with the spectroscopic orbit, one can lead to the following physical parameters: mp =2.99m⊙, ms =1.28m⊙, > Tp < =9600 K, > Ts < =5400 K, > Rp < =2.35R⊙, > Rs < =2.12R⊙. The system comprising an AO primary and a secondary of G2 spectral type.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
17.
Recent calculations of electron impact excitation rates in He-like Alxii are used to derive the theoretical electron temperature and density sensitive emission line ratios G ( = (f + i)/r and R ( = f/i, where f, i, and r are the forbidden 1s 2 1 S ? 1s2s 3 S, intercombination 1s 2 1 S ? 1s2p 3 P and resonance 1s 2 1 S ? 1s2p 1 P transitions, respectively. These ratios are found to be significantly different from earlier calculations, and are in much better agreement with X-ray spectral data for two solar flares obtained with the SMM and P78-1 satellites. 相似文献
18.
In this paper, following the increase of the mass ratio μ, the vertical stability curves of the long and the short period families were studied, and the vertical bifurcation families
from these two families were computed. It is found that these vertical bifurcation families connect the long and short period
families with the spatial periodic family emanating from the equilateral equilibrium points. The evolution details of these
vertical bifurcation families were carefully studied and they are found to be similar to the planar bifurcation families connecting
the long period family with the short period family in the planar case. 相似文献
19.
Asymptotic motion near the collinear equilibrium points of the photogravitational restricted three-body problem is considered.
In particular, non-symmetric homoclinic solutions are numerically explored. These orbits are connected with periodic ones.
We have computed numerically the families containing these orbits and have found that they terminate at both ends by asymptotically
approaching simple periodic solutions belonging to the Lyapunov family emanating from L3. 相似文献
20.
We calculate the expected counting rate of a flat micrometeoroid detector of finite sensitivity passing in hyperbolic orbit near a planet. We assume that the distribution of particle sizes, s, can be expressed as a power law spectrum of index p, i.e. dN(s) = Cs?pds, and also that the particles encounter the sphere of influence of the planet with a certain speed v∞. The results of the calculations are then compared with the results returned by Pioneer 10 in its flyby of Jupiter. The observed increase in impact rate near Jupiter can be completely explained in terms of gravitational “focusing” of particles which are in heliocentric orbits; i.e., they are not in orbit about Jupiter. The absolute concentration of particles near the orbit of Jupiter is of the same order as at 1 AU: the exact ratio being a function of particle speed and spectral index. Data from one flyby are insufficient to determine a unique value for both the spectral index, p, and the particle velocity, v∞, but limits can be set. For reasonable encounter speeds (corresponding to eccentricities and inclinations of dust particles experienced near the Earth), the particles near Jupiter are characterized by a spectrum of index p ～ 3. The spectral index which best fits the data increases with increasing encounter speeds. 相似文献