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1.
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   

2.
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   

3.
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   

4.
We present a detailed investigation of the dramatic changes that occur in the \(\mathcal {L}_1\) halo family when radiation pressure is introduced into the Sun–Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. in Int J Bifurc Chaos 1:493–520, 1991) the families can be fully mapped out as the parameter \(\beta \) is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at \(\beta \approx 0.0387\) acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at \(\beta \approx 0.289\) , resulting in a third new family. The linear stability of the families changes rapidly at low values of \(\beta \) , with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter \(\beta \) and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of \(\beta \) there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.  相似文献   

5.
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   

6.
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   

7.
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8.
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   

9.
The number of equivalence classes of central configurations of $n \le 4$ bodies of positive mass is known to be finite, but it remains to be shown if this is true for $n \ge 5$ . By allowing one mass to be negative, Gareth Roberts constructed a continuum of inequivalent planar central configurations of $n = 5$ bodies. We reinterpret Roberts’ example and generalize the construction of his continuum to produce a family of continua of central configurations, each with a single negative mass. These new continua exist in even dimensional spaces $\mathbb R ^k$ for $k \ge 4$ .  相似文献   

10.
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   

11.
The quintessence dark energy model with a kinetic coupling to gravity within the Palatini formalism is studied in this paper. Two different coupling forms: $\hat{R}\partial^{\mu}\phi\partial_{\mu}\phi$ and $\hat {R}_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi$ are analyzed, respectively. We find that both the model with the $\hat{R}\partial^{\mu}\phi\partial_{\mu}\phi$ coupling and the one with the $\hat{R}_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi$ coupling can realize the phantom divide line crossing from phantom to quintessence at late time for its effective equation-of-state. Furthermore, the former can behave like phantom. These features are different from those found in the $\hat {R}\phi^{2}$ coupling case.  相似文献   

12.
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   

13.
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   

14.
The equation of motion of long periodic libration around the Lagrangian point $L_4$ L 4 in the restricted three-body problem is investigated. The range of validity of an approximate analytical solution in the tadpole region is determined by numerical integration. The predictions of the model of libration are tested on the Trojan asteroids of Jupiter. The long time evolution of the orbital eccentricity and the longitude of the perihelion of the Trojan asteroids, under the effect of the four giant planets, is also investigated and a slight dynamical asymmetry is shown between the two groups of Trojans at $L_4$ L 4 and $L_5$ L 5 .  相似文献   

15.
A linear analysis of the asymmetries in Stokes profiles of magnetic lines is performed. The asymmetries in the linear and circular polarization profiles are characterized by suitable quantities, \(\delta \tilde Q\) and \(\delta \tilde V\) , strictly related to observed profiles. The response functions of \(\delta \tilde Q\) and \(\delta \tilde V\) to velocity fields are introduced and computed for various configurations of the magnetic field vector in a Milne-Eddington atmosphere. Some conclusions are drawn as to the importance of the asymmetries in Stokes profiles for recovering the velocity gradients from observations.  相似文献   

16.
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   

17.
A statistical study is carried out on the photospheric magnetic nonpotentiality in solar active regions and its relationship with associated flares. We select 2173 photospheric vector magnetograms from 1106 active regions observed by the Solar Magnetic Field Telescope at Huairou Solar Observing Station, National Astronomical Observatories of China, in the period of 1988??C?2008, which covers most of the 22nd and 23rd solar cycles. We have computed the mean planar magnetic shear angle ( $\overline{\Delta\phi}$ ), mean shear angle of the vector magnetic field ( $\overline{\Delta\psi}$ ), mean absolute vertical current density ( $\overline{|J_{z}|}$ ), mean absolute current helicity density ( $\overline{|h_{\mathrm{c}}|}$ ), absolute twist parameter (|?? av|), mean free magnetic energy density ( $\overline{\rho_{\mathrm{free}}}$ ), effective distance of the longitudinal magnetic field (d E), and modified effective distance (d Em) of each photospheric vector magnetogram. Parameters $\overline{|h_{\mathrm{c}}|}$ , $\overline{\rho_{\mathrm{free}}}$ , and d Em show higher correlations with the evolution of the solar cycle. The Pearson linear correlation coefficients between these three parameters and the yearly mean sunspot number are all larger than 0.59. Parameters $\overline {\Delta\phi}$ , $\overline{\Delta\psi}$ , $\overline{|J_{z}|}$ , |?? av|, and d E show only weak correlations with the solar cycle, though the nonpotentiality and the complexity of active regions are greater in the activity maximum periods than in the minimum periods. All of the eight parameters show positive correlations with the flare productivity of active regions, and the combination of different nonpotentiality parameters may be effective in predicting the flaring probability of active regions.  相似文献   

18.
We analyzed 186 binary pulsars (BPSRs) in the magnetic field versus spin period (B-P) diagram, where their relations to the millisecond pulsars (MSPs) can be clearly shown. Generally, both BPSRs and MSPs are believed to be recycled and spun-up in binary accreting phases, and evolved below the spin-up line setting by the Eddington accretion rate ( $\dot{M}{\simeq}10^{18}~\mbox{g/s}$ ). It is noticed that most BPSRs are distributed around the spin-up line with mass accretion rate $\dot{M}=10^{16}~\mbox{g/s}$ and almost all MSP samples lie above the spin-up line with $\dot{M}\sim10^{15}~\mbox{g/s}$ . Thus, we calculate that a minimum accretion rate ( $\dot{M}\sim10^{15}~\mbox{g/s}$ ) is required for the MSP formation, and physical reasons for this are proposed. In the B-P diagram, the positions of BPSRs and their relations to the binary parameters, such as the companion mass, orbital period and eccentricity, are illustrated and discussed. In addition, for the seven BPSRs located above the limit spin-up line, possible causes are suggested.  相似文献   

19.
Using a new approach, we have obtained a formula for calculating the rotation period and radius of planets. In the ordinary gravitomagnetism the gravitational spin (S) orbit (L) coupling, $\vec{L}\cdot\vec{S}\propto L^{2}$ , while our model predicts that $\vec{L}\cdot\vec{S}\propto\frac{m}{M}L^{2}$ , where M and m are the central and orbiting masses, respectively. Hence, planets during their evolution exchange L and S until they reach a final stability at which MS∝mL, or $S\propto\frac{m^{2}}{v}$ , where v is the orbital velocity of the planet. Rotational properties of our planetary system and exoplanets are in agreement with our predictions. The radius (R) and rotational period (D) of tidally locked planet at a distance a from its star, are related by, $D^{2}\propto\sqrt{\frac{M}{m^{3}}}R^{3}$ and that $R\propto\sqrt{\frac {m}{M}}a$ .  相似文献   

20.
The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth–Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points \(L_1\) and \(L_2\) as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincaré sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions.  相似文献   

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