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1.
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   

2.
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   

3.
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   

4.
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   

5.
We have investigated the resonances due to the perturbations of a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it’s equatorial ellipticity. The resonances at the points resulting from (i) the commensurability between \(\dot{\theta}_{0}\) (steady-state orbital angular rate of the satellite) and \(\dot{\theta}_{m}\) (angular velocity of the moon around the earth) and (ii) the commensurability between \(\dot{\theta}_{0}\) and \(\dot{\psi}_{0}\) (steady-state regression rate of the synchronous satellite) are analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure as given in Brown and Shook (Planetary Theory, Cambridge University Press, Cambridge, 1933). We have observed that as θ m (0°≤θ m ≤45°) and ψ (0°≤ψ≤135°) increase, the amplitude decreases and the time period also decreases. We have also shown the effect of ψ on amplitude and time period for 0°≤Γ≤45°, where Γ is the angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the satellite on the plane of the equator.  相似文献   

6.
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   

7.
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   

8.
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   

9.
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   

10.
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   

11.
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   

12.
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   

13.
We present estimates of the size of the analytic domain of stability for co-orbital motions obtained by a high order normal form in the framework of the elliptic restricted three body problem. As a demonstration example, we consider the motion of a Trojan body in an extrasolar planetary system with a giant planet of mass parameter $\mu =0.005$ μ = 0.005 and eccentricity $e^{\prime }=0.1$ e ′ = 0.1 . The analysis contains three basic steps: (i) derivation of an accurate expansion of the Hamiltonian, (ii) computation of the normal form up to an optimal order (in the Nekhoroshev sense), and (iii) computation of the optimal size of the remainder at various values of the action integrals (proper elements) of motion. We explain our choice of variables as well as the method used to expand the Hamiltonian so as to ensure a precise model. We then compute the normal form up to the normalisation order $r=50$ r = 50 by use of a computer-algebraic program. We finally estimate the size $||R||$ | | R | | of the remainder as a function of the normalization order, and compute the optimal normalization order at which the remainder becomes minimum. It is found that the optimal value $\log (||R_{opt}||)$ log ( | | R o p t | | ) can serve in order to construct a stability map for the domain of co-orbital motion using only series. This is compared to the stability map found by a purely numerical approach based on chaotic indicators.  相似文献   

14.
Theoretical estimates of the rates of radial pulsation period change in Galactic Cepheids with initial masses 5.5 M ⊙ ≤ M ZAMS ≤ 13 M ⊙, chemical composition X = 0.7, Z = 0.02 and periods 1.5 day ≤ Π ≤ 100 day are obtained from consistent stellar evolution and nonlinear stellar pulsation computations. Pulsational instability was investigated for three crossings of the instability strip by the evolutionary track in the HR diagram. The first crossing occurs at the post-main sequence helium core gravitational contraction stage which proceeds in the Kelvin-Helmholtz timescale whereas the second and the third crossings take place at the evolutionary stage of thermonuclear core helium burning. During each crossing of the instability strip the period of radial pulsations is a quadratic function of the stellar evolution time. Theoretical rates of the pulsation period change agree with observations but the scatter of observational estimates of \(\dot \Pi\) noticeably exceeds the width of the band \(\left( {\delta \log \left| {\dot \Pi } \right| \leqslant 0.6} \right)\) confining evolutionary tracks in the period-period change rate diagram. One of the causes of the large scatter with very high values of \(\dot \Pi\) in Cepheids with increasing periods might be the stars that cross the instability strip for the first time. Their fraction ranges from 2% for M ZAMS = 5.5 M ⊙ to 9% for M ZAMS = 13 M ⊙ and variables α UMi and IX Cas seem to belong to such objects.  相似文献   

15.
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   

16.
In this work, we explore a sample of 362 flat-spectrum radio quasars (FSRQs) to investigate the jet formation. We find that the fundamental plane for our FSRQs can be expressed as $L_{\rm 5~GHz}\propto M_{\rm bh}^{-0.19}L_{\rm 2~keV}^{1.08}$ . We also find that the 5?GHz luminosities are tightly related to both black hole mass and Eddington ratio, which is established as $L_{\rm 5~GHz}\propto M_{\rm bh}^{0.67}(L_{\rm bol}/ L_{\rm EDD})^{1.32}$ .  相似文献   

17.
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   

18.
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   

19.
The planar problem of three bodies is described by means of Murnaghan's symmetric variables (the sidesa j of the triangle and an ignorable angle), which directly allow for the elimination of the nodes. Then Lemaitre's regularized variables \(\alpha _j = \sqrt {(\alpha ^2 - \alpha _j )}\) , where \(\alpha ^2 = \tfrac{1}{2}(a_1 + a_2 + a_3 )\) , as well as their canonically conjugated momenta are introduced. By finally applying McGehee's scaling transformation \(\alpha _j = r^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \tilde \alpha _j\) , wherer 2 is the moment of inertia a system of 7 differential equations (with 2 first integrals) for the 5-dimensional triple collision manifold \(T\) is obtained. Moreover, the zero angular momentum solutions form a 4-dimensional invariant submanifold \(N \subset T\) represented by 6 differential equations with polynomial right-hand sides. The manifold \(N\) is of the topological typeS 2×S 2 with 12 points removed, and it contains all 5 restpoint (each one in 8 copies). The flow on \(T\) is gradient-like with a Lyapounov function stationary in the 40 restpoints. These variables are well suited for numerical studies of planar triple collision.  相似文献   

20.
In November 2004 radar delay measurements of near-Earth asteroid (3908) Nyx obtained at the Arecibo radio telescope turned out to be \(7.5\sigma \) away from the orbital prediction. We prove that this discrepancy was caused by a poor astrometric treatment and an incomplete dynamical model, which did not account for nongravitational perturbations. To improve the astrometric treatment, we remove known star catalog biases, apply suitable weights to the observations, and use an aggressive outlier rejection scheme. The main issue related to the dynamical model is having not accounted for the Yarkovsky effect. Including the Yarkovsky perturbation in the model makes the orbital prediction and the radar measurements statistically consistent by both reducing the offset and increasing the prediction uncertainty to a more realistic level. This analysis shows the sensitivity of high precision predictions to the astrometric treatment and the Yarkovsky effect. By using the full observational dataset we obtain a \(5\sigma \) detection of the Yarkovsky effect acting on Nyx corresponding to an orbital drift \(da/dt = (142 \pm 29)\)  m/year. In turn, we derive constraints on thermal inertia and bulk density. In particular, we find that the bulk density of Nyx is around 1 g/cm \(^3\) , possibly less. To make sure that our results are not corrupted by an asteroid impact or a close approach with a perturbing asteroid not included in our dynamical model, we show that the astrometry provides no convincing evidence of an impulsive variation of Nyx’s velocity while crossing the main belt region.  相似文献   

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