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1.
The 2-D lattice theory of Flower Constellations, generalizing Harmonic Flower Constellations (the symmetric subset of Flower Constellations) as well as the Walker/ Mozhaev constellations, is presented here. This theory is a new general framework to design symmetric constellations using a $2\times 2$ 2 × 2 lattice matrix of integers or by its minimal representation, the Hermite normal form. From a geometrical point of view, the phasing of satellites is represented by a regular pattern (lattice) on a two-Dimensional torus. The 2-D lattice theory of Flower Constellations does not require any compatibility condition and uses a minimum set of integer parameters whose meaning are explored throughout the paper. This general minimum-parametrization framework allows us to obtain all symmetric distribution of satellites. Due to the $J_2$ J 2 effect this design framework is meant for circular orbits and for elliptical orbits at critical inclination, or to design elliptical constellations for the unperturbed Keplerian case.  相似文献   

2.
The restricted three-body problem (R3BP) possesses the property that some classes of doubly asymptotic (i.e., homoclinic or heteroclinic) orbits are limit members of families of periodic orbits, this phenomenon has been known as the “blue sky catastrophe” termination principle. A similar case occurs in the restricted four body problem for the collinear equilibrium point $L_{2}$ L 2 . In the restricted four body problem with primaries in a triangle relative equilibrium, we show that the same phenomenon observed in the R3BP occurs. We prove that there exists a critical value of the mass parameter $\mu _{b}$ μ b such that for $\mu =\mu _{b}$ μ = μ b a Hamiltonian Hopf bifurcation takes place. Moreover we show that for $\mu >\mu _{b}$ μ > μ b the stable and unstable manifolds of $L_{2}$ L 2 intersect transversally and the spectrum corresponds to a complex saddle. This proves that Henrard’s theorem applies at least for $\mu $ μ close to $\mu _{b}$ μ b . In particular there exists a family of periodic orbits having the homoclinic orbit as a limit.  相似文献   

3.
We consider a two-planet system migrating under the influence of dissipative forces that mimic the effects of gas-driven (Type II) migration. It has been shown that, in the planar case, migration leads to resonant capture after an evolution that forces the system to follow families of periodic orbits. Starting with planets that differ slightly from a coplanar configuration, capture can, also, occur and, additionally, excitation of planetary inclinations has been observed in some cases. We show that excitation of inclinations occurs, when the planar families of periodic orbits, which are followed during the initial stages of planetary migration, become vertically unstable. At these points, vertical critical orbits may give rise to generating stable families of \(3D\) periodic orbits, which drive the evolution of the migrating planets to non-coplanar motion. We have computed and present here the vertical critical orbits of the \(2/1\) and \(3/1\) resonances, for various values of the planetary mass ratio. Moreover, we determine the limiting values of eccentricity for which the “inclination resonance” occurs.  相似文献   

4.
We explore the long-term stability of Earth Trojans by using a chaos indicator, the Frequency Map Analysis. We find that there is an extended stability region at low eccentricity and for inclinations lower than about $50^{\circ }$ even if the most stable orbits are found at $i \le 40^{\circ }$ . This region is not limited in libration amplitude, contrary to what found for Trojan orbits around outer planets. We also investigate how the stability properties are affected by the tidal force of the Earth–Moon system and by the Yarkovsky force. The tidal field of the Earth–Moon system reduces the stability of the Earth Trojans at high inclinations while the Yarkovsky force, at least for bodies larger than 10 m in diameter, does not seem to strongly influence the long-term stability. Earth Trojan orbits with the lowest diffusion rate survive on timescales of the order of $10^9$  years but their evolution is chaotic. Their behaviour is similar to that of Mars Trojans even if Earth Trojans appear to have shorter lifetimes.  相似文献   

5.
This paper examines the design of transfers from the Sun–Earth libration orbits, at the \(L_{1}\) and \(L_{2}\) points, towards the Moon using natural dynamics in order to assess the feasibility of future disposal or lifetime extension operations. With an eye to the probably small quantity of propellant left when its operational life has ended, the spacecraft leaves the libration point orbit on an unstable invariant manifold to bring itself closer to the Earth and Moon. The total trajectory is modeled in the coupled circular restricted three-body problem, and some preliminary study of the use of solar radiation pressure is also provided. The concept of survivability and event maps is introduced to obtain suitable conditions that can be targeted such that the spacecraft impacts, or is weakly captured by, the Moon. Weak capture at the Moon is studied by method of these maps. Some results for planar Lyapunov orbits at \(L_{1}\) and \(L_{2}\) are given, as well as some results for the operational orbit of SOHO.  相似文献   

6.
Using γ-ray data detected by Fermi Large Area Telescope (LAT) and multi-wave band data for 35 TeV blazars sample, we have studied the possible correlations between different broad band spectral indices ( $\alpha_{\rm r.ir}$ , $\alpha_{\rm{r.o}}$ , $\alpha_{\rm r.x}$ , $\alpha_{\rm r.\gamma}$ , $\alpha_{\rm{ir.o}}$ , $\alpha_{\rm ir.x}$ , $\alpha_{\rm ir.\gamma}$ , $\alpha_{\rm o.x}$ , $\alpha_{\rm o.\gamma}$ , $\alpha_{\rm r.x}$ , $\alpha_{\rm x.\gamma}$ ) in all states (average/high/low). Our results are as follows: (1) For our TeV blazars sample, the strong positive correlations were found between $\alpha_{\rm r.ir}$ and $\alpha_{\rm{r.o}}$ , between $\alpha_{\rm r.ir}$ and $\alpha_{\rm r.x}$ , between $\alpha_{\rm r.ir}$ and $\alpha_{\rm r.\gamma}$ in all states (average/high/low); (2) For our TeV blazars sample, the strong anti-correlations were found between $\alpha_{\rm r.ir}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm ir.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm o.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\mathrm{ir.o}}$ and $\alpha_{\rm o.\gamma}$ , between $\alpha_{\rm r.x}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\rm ir.x}$ and $\alpha_{\rm x.\gamma}$ in all states (average/high/low). The results suggest that the synchrotron self-Compton radiation (SSC) is the main mechanism of high energy γ-ray emission and the inverse Compton scattering of circum-nuclear dust is likely to be a important complementary mechanism for TeV blazars. Our results also show that the possible correlations vary from state to state in the same pair of indices, Which suggest that there may exist differences in the emitting process and in the location of the emitting region for different states.  相似文献   

7.
It is now recognised that the traditional method of calculating the LSR fails. We find an improved estimate of the LSR by making use of the larger and more accurate database provided by XHIP and repeating our preferred analysis from Francis and Anderson (New Astron 14:615–629, 2009a). We confirm an unexpected high value of $U_0$ by calculating the mean for stars with orbits sufficiently inclined to the galactic plane that they do not participate in bulk streaming motions. Our best estimate of the solar motion with respect to the LSR $(U_0, V_0, W_0) = (14.1\, \pm \, 1.1, 14.6\, \pm \, 0.4, 6.9\, \pm \, 0.1)$ km s $^{-1}$ .  相似文献   

8.
We consider a class of Hamiltonian systems with two degrees of freedom with singularities. This class includes several symmetric subproblems of the $n$ -body problem where the singularities are due to collisions involving two or more bodies. “Schubart-like” periodic orbits having two collisions in one period, are present in most of these subproblems. The purpose of this paper is to study the existence of families of such a periodic orbits in a general setting. The blow up techniques of total collision and infinity are applied to our class of Hamiltonian system. This allows us to derive sufficient conditions to ensure the existence of families of double symmetric “Schubart-like” periodic orbits having many singularities. The orbits in the family can be parametrized by the number of singularities in one period. The results are applied to some subproblems of the gravitational $n$ -body problem.  相似文献   

9.
We examine the stability of the triangular Lagrange points L 4 and L 5 for secondary masses larger than the Gascheau??s value ${\mu_{\rm G}= (1-\sqrt{23/27}/2)= 0.0385208\ldots}$ (also known as the Routh value) in the restricted, planar circular three-body problem. Above that limit the triangular Lagrange points are linearly unstable. Here we show that between??? G and ${\mu \approx 0.039}$ , the L 4 and L 5 points are globally stable in the sense that a particle released at those points at zero velocity (in the corotating frame) remains in the vicinity of those points for an indefinite time. We also show that there exists a family of stable periodic orbits surrounding L 4 or L 5 for ${\mu \ge \mu_G}$ . We show that??? G is actually the first value of a series ${\mu_0 (=\mu_G), \mu_1,\ldots, \mu_i,\ldots}$ corresponding to successive period doublings of the orbits, which exhibit ${1, 2, \ldots, 2^i,\ldots}$ cycles around L 4 or L 5. Those orbits follow a Feigenbaum cascade leading to disappearance into chaos at a value ${\mu_\infty = 0.0463004\ldots}$ which generalizes Gascheau??s work.  相似文献   

10.
We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler’s equation $E-e\sin (E)=M$ for any $e\in [0,1)$ and $M\in [0,\pi ]$ . Our solution is guaranteed, via Smale’s $\alpha $ -theory, to converge to the actual solution $E$ through Newton’s method at quadratic speed, i.e. the $n$ -th iteration produces a value $E_n$ such that $|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|$ . The formula provided for $\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$ , where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $[0,1)\times [0,\pi ]$ if only rational functions are allowed in each branch.  相似文献   

11.
We study the secular dynamics of lunar orbiters, in the framework of high-degree gravity models. To achieve a global view of the dynamics, we apply a frequency analysis (FA) technique which is based on Prony’s method. This allows for an extensive exploration of the eccentricity ( $e$ )—inclination ( $i$ ) space, based on short-term integrations ( $\sim $ 8 months) over relatively high-resolution grids of initial conditions. Different gravity models are considered: 3rd, 7th and 10th degree in the spherical harmonics expansion, with the main perturbations from the Earth being added. Since the dynamics is mostly regular, each orbit is characterised by a few parameters, whose values are given by the spectral decomposition of the orbital elements time series. The resulting frequency and amplitude maps in ( $e_0,i_0$ ) are used to identify the dominant perturbations and deduce the “minimum complexity” model necessary to capture the essential features of the long-term dynamics. We find that the 7th degree zonal harmonic ( $J_7$ term) is of profound importance at low altitudes as, depending on the initial secular phases, it can lead to collision with the Moon’s surface within a few months. The 3rd-degree non-axisymmetric terms are enough to describe the deviations from the 1 degree-of-freedom zonal problem; their main effect is to modify the equilibrium value of the argument of periselenium, $\omega $ , with respect to the “frozen” solution ( $\omega =\pm 90^{\circ }, \forall \Omega $ , where $\Omega $ is the nodal longitude). Finally, we show that using FA on a fine grid of initial conditions, set around a suitably chosen ‘first guess’, one can compute an accurate approximation of the initial conditions of a periodic orbit.  相似文献   

12.
In this paper we study the periodic orbits of the Hamiltonian system with the Armburster-Guckenheimer-Kim potential and its $\mathcal{C}^{1}$ non-integrability in the sense of Liouville-Arnold.  相似文献   

13.
We investigate the dynamics of two satellites with masses $\mu _s$ and $\mu '_s$ orbiting a massive central planet in a common plane, near a first order mean motion resonance $m+1{:}m$ (m integer). We consider only the resonant terms of first order in eccentricity in the disturbing potential of the satellites, plus the secular terms causing the orbital apsidal precessions. We obtain a two-degrees-of-freedom system, associated with the two critical resonant angles $\phi = (m+1)\lambda ' -m\lambda - \varpi $ and $\phi '= (m+1)\lambda ' -m\lambda - \varpi '$ , where $\lambda $ and $\varpi $ are the mean longitude and longitude of periapsis of $\mu _s$ , respectively, and where the primed quantities apply to $\mu '_s$ . We consider the special case where $\mu _s \rightarrow 0$ (restricted problem). The symmetry between the two angles $\phi $ and $\phi '$ is then broken, leading to two different kinds of resonances, classically referred to as corotation eccentric resonance (CER) and Lindblad eccentric Resonance (LER), respectively. We write the four reduced equations of motion near the CER and LER, that form what we call the CoraLin model. This model depends upon only two dimensionless parameters that control the dynamics of the system: the distance $D$ between the CER and LER, and a forcing parameter $\epsilon _L$ that includes both the mass and the orbital eccentricity of the disturbing satellite. Three regimes are found: for $D=0$ the system is integrable, for $D$ of order unity, it exhibits prominent chaotic regions, while for $D$ large compared to 2, the behavior of the system is regular and can be qualitatively described using simple adiabatic invariant arguments. We apply this model to three recently discovered small Saturnian satellites dynamically linked to Mimas through first order mean motion resonances: Aegaeon, Methone and Anthe. Poincaré surfaces of section reveal the dynamical structure of each orbit, and their proximity to chaotic regions. This work may be useful to explore various scenarii of resonant capture for those satellites.  相似文献   

14.
The object of study is the geodesic structure of a \(z=2\) Lifshitz black hole in 3+1 space–time dimensions, which is an exact solution to the Einstein-scalar-Maxwell theory. The motion of massless and massive particles in this background is researched using the standard Lagrangian procedure. Analytical expressions are obtained for radial and angular motions of the test particles, where the polar trajectories are given in terms of the \(\wp \) -Weierstraß elliptic function. It will be demonstrated that an external observer can see that photons with radial motion arrive at spatial infinity in a finite coordinate time. For particles with non-vanished angular momentum, the motion is studied on the invariant plane \(\phi = \pi /2\) and, it is shown that bounded orbits are not allowed for this space–time on this plane. These results are consistent with other recent studies on Lifshitz black holes.  相似文献   

15.
In this paper we give a short analytical proof of the inequalities proved by Albouy–Moeckel through computer algebra, in the cases $n=5$ and $n=6$ . These inequalities guarantee that, in the $n$ -body problem, the family of mass vectors making a given collinear configuration a central configuration is 2-dimensional. The induction techniques here may be used to prove the inequalities for general $n$ with more subtle estimation but currently the inequalities still remains unproved for $n\ge 7$ .  相似文献   

16.
The classic $F$ and $G$ Taylor series of Keplerian motion are extended to solve the Stark problem and to use the generalized Sundman transformation. Exact recursion formulas for the series coefficients are derived, and the method is implemented to high order via a symbolic manipulator. The results lead to fast and accurate propagation models with efficient discretizations. The new $F$ and $G$ Stark series solutions are compared to the Modern Taylor Series (MTS) and 8th order Runge–Kutta–Fehlberg (RKF8) solutions. In terms of runtime, the $F$ and $G$ approach is shown to compare favorably to the MTS method up to order 20, and both Taylor series methods enjoy approximate order of magnitude speedups compared to RKF8 implementations. Actual runtime is shown to vary with eccentricity, perturbation size, prescribed accuracy, and the Sundman power law. The method and results are valid for both the Stark and the Kepler problems. The effects of the generalized Sundman transformation on the accuracy of the propagation are analyzed. The Taylor series solutions are shown to be exceptionally efficient when the unity power law from the classic Sundman transformation is applied. An example low-thrust trajectory propagation demonstrates the utility of the $F$ and $G$ Stark series solutions.  相似文献   

17.
In a previous paper, Hayliet al. (1983), two families of periodic orbits in the three-dimensional potential $$U = \frac{1}{2}(Ax^2 + By^2 + Cz^2 ) - \varepsilon xz^2 - nyz^2 $$ with \(\sqrt A :\sqrt B :\sqrt C = 6:4:3\) and ?=0.5 were described. It was found empirically that the characteristic curves of the two families intersect in the space (x0, y0, η) for |η|?0.2. This property is demonstrated in the present paper by writing explicitely the Poincaré mapping and by giving an approximation directly comparable with the numerical results obtained in Hayliet al. (1983). It is thus shown that one family bifurcates off the other.  相似文献   

18.
We examine the possibility that the observed cosmic-ray protons are of primary extragalactic origin. The present \(\bar p\) data are consistent with a primary extragalactic component having \(\bar p\) /p?3.2±0.7 x 10-4 independent of energy. Following the suggestion that most extragalactic cosmic rays are from active galaxies, we propose that most of the observed \(\bar p\) 's are alos from the same sites. This would imply the possibility of destroying the corresponding \(\bar \alpha \) 'sat the source, thus leading to a flux ratio \(\bar \alpha \) /α< \(\bar p\) /p. We further predict an estimate for \(\bar \alpha \) α～10-5, within the range of future cosmic-ray detectors. the cosmological implications of this proposal are discussed.  相似文献   

19.
We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitudes, is expanded in power series of eccentricities and inclinations. The model, which is valid in the entire co-orbital region, possesses an integrable approximation modeling the planar and quasi-circular motions. First, focusing on the fixed points of this approximation, we highlight relations linking the eigenvectors of the associated linearized differential system and the existence of certain remarkable orbits like the elliptic Eulerian Lagrangian configurations, the anti-Lagrange (Giuppone et al. in MNRAS 407:390–398, 2010) orbits and some second sort orbits discovered by Poincaré. Then, the variational equation is studied in the vicinity of any quasi-circular periodic solution. The fundamental frequencies of the trajectory are deduced and possible occurrence of low order resonances are discussed. Finally, with the help of the construction of a Birkhoff normal form, we prove that the elliptic Lagrangian equilateral configurations and the anti-Lagrange orbits bifurcate from the same fixed point $L_4$ L 4 .  相似文献   

20.
This paper deals with the existence of libration points and their linear stability when the more massive primary is radiating and the smaller is an oblate spheroid. Our study includes the effects of oblateness of $\bar{J}_{2i}$ (i=1,2) with respect to the smaller primary in the restricted three-body problem. Under combining the perturbed forces that were mentioned before, the collinear points remain unstable and the triangular points are stable for 0<μ<μ c , and unstable in the range $\mu_{c} \le\mu\le\frac{1}{2}$ , where $\mu_{c} \in(0,\frac{1}{2})$ , it is also observed that for these points the range of stability will decrease. The relations for periodic orbits around five libration points with their semimajor, semiminor axes, eccentricities, the frequencies of orbits and periods are found, furthermore for the orbits around the triangular points the orientation and the coefficients of long and short periodic terms also are found in the range 0<μ<μ c .  相似文献   

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