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1.
Sedna is the first inner Oort cloud object to be discovered. Its dynamical origin remains unclear, and a possible mechanism is considered here. We investigate the parameter space of a hypothetical solar companion which could adiabatically detach the perihelion of a Neptune-dominated TNO with a Sedna-like semimajor axis. Demanding that the TNO’s maximum value of osculating perihelion exceed Sedna’s observed value of 76 AU, we find that the companion’s mass and orbital parameters (m c , a c , q c , Q c , i c ) are restricted to $$m_c>rapprox 5\hskip.25em\hbox{M}_{\rm J}\left(\frac{Q_c}{7850\hbox{ AU}} \frac{q_c}{7850\hbox{ AU}}\right)^{3/2}$$ during the epoch of strongest perturbations. The ecliptic inclination of the companion should be in the range $45{\deg}\lessapprox i_c\lessapprox 135{\deg}$ if the TNO is to retain a small inclination while its perihelion is increased. We also consider the circumstances where the minimum value of osculating perihelion would pass the object to the dynamical dominance of Saturn and Jupiter, if allowed. It has previously been argued that an overpopulated band of outer Oort cloud comets with an anomalous distribution of orbital elements could be produced by a solar companion with present parameter values $$m_c\approx 5\hskip.25em\hbox{M}_{\rm J}\left(\frac{9000\hbox{ AU}}{a_c}\right)^{1/2}.$$ If the same hypothetical object is responsible for both observations, then it is likely recorded in the IRAS and possibly the 2MASS databases.  相似文献   

2.
We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.  相似文献   

3.
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   

4.
The radio tracking apparatus of the New Horizons spacecraft, currently traveling to the Pluto system where its arrival is scheduled for July 2015, should be able to reach an accuracy of 10 m (range) and 0.1  $\text{ mm } \text{ s }^{-1}$ mm s ? 1 (range-rate) over distances up to 50 au. This should allow to effectively constrain the location of a putative trans-Plutonian massive object, dubbed Planet X (PX) hereafter, whose existence has recently been postulated for a variety of reasons connected with, e.g., the architecture of the Kuiper belt and the cometary flux from the Oort cloud. Traditional scenarios involve a rock-ice planetoid with $m_\mathrm{X}\approx 0.7\,m_{\oplus }$ m X ≈ 0.7 m ⊕ at some 100–200 au, or a Jovian body with $m_\mathrm{X}\lesssim 5\,m_\mathrm{J}$ m X ? 5 m J at about 10,000–20,000 au; as a result of our preliminary sensitivity analysis, they should be detectable by New Horizons since they would impact its range at a km level or so over a time span 6 years long. Conversely, range residuals statistically compatible with zero having an amplitude of 10 m would imply that PX, if it exists, could not be located at less than about 4,500 au ( $m_\mathrm{X}=0.7\,m_{\oplus }$ m X = 0.7 m ⊕ ) or 60,000 au ( $m_\mathrm{X}=5\,m_\mathrm{J}$ m X = 5 m J ), thus making a direct detection quite demanding with the present-day technologies. As a consequence, it would be appropriate to rename such a remote body as Thelisto. Also fundamental physics would benefit from this analysis since certain subtle effects predicted by MOND for the deep Newtonian regions of our Solar System are just equivalent to those of a distant pointlike mass.  相似文献   

5.
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   

6.
The study of the reservoirs of small bodies in the Solar System can help us to refine our theory of the origin and evolution of the whole planetary system we live in. In this contribution, we introduce a simulation of the evolution of an initial proto-planetary disc (PPD) for 2 Gyr period, in which 10,038 studied test particles (TPs), representing the disc, are perturbed by four giant planets in their current orbits and having their current masses. In addition, Galactic-tide and stellar perturbations are considered. The outer Oort cloud (OC) appears to be formed within about 0.5 Gyr. At 2 Gyr, only 0.14% of all TPs reside in the outer OC, according to our simulation. This is the largest discrepancy with the results of previous authors. The TPs in the outer OC originate from almost all regions of the PPD. Specifically, 14, 7, 29, 21, and 29% of all TPs, that are the members of the outer OC at 2 Gyr, originate in the Jupiter, Saturn, Uranus, Neptune, and Edgeworth-Kuiper-belt regions, respectively. The largest number of the TPs, 6,669 (66.4%), was ejected into the interstellar space. Besides other results, we found a dominance of high galactic inclinations of outer-OC orbits. Obviously, this is a consequence of the action of Galactic tide.  相似文献   

7.
The recent observation of the absorption of radiation belts in the vicinity of Saturn's bright rings and historical observations of the ring system make the following related results apparent:
  • - The gaps in the rings are caused by the presence of at least 6 small, extremely dense and probably electrically charged ‘sweeper’ moons which effectively sweep the ring matter clean from the gaps. This is known due to the fading of the inner ring edges whereas the outer edges are well defined. Their orbital periods will differ from the expected Keplerian periods if the moons and Saturn do possess electric fields.
  • - Absorption of radiation belts near the rings (of Jupiter also) implies that the ring particles themselves are not absorbing the radiation but the small moons are. This is consistent with the observed radiation belt absorption near the outer Saturnian moons.
  • - If electric fields of the sweeper moons cause the ring edge fading as observed (and not simply gravitational), then Saturn itself must maintain an electric field in its vicinity by way of a sizeable proton wind to affect the uneven ring edge fading and will be surrounded by an H+ cloud at least to approximately the A-ring. this is consistent with the detection of an H+ cloud surrounding Saturn (Weiseret al., 1977, p. 755). The other possibility is that these moons are extremely dense and have very large internal magnetic fields.
  • - Because of their location, these moons must be captured and if very dense as believed, may be core remnants of a nova.
    •   相似文献   

    8.
    Two types of trial three-layer models have been constructed for the satellites Io and Europa. In the models of the first type (Io1 and E1), the cores are assumed to consist of eutectic Fe-FeS melt with the densities ρ 1 = 5.15 g cm?3 (Io1) and 5.2 g cm?3 (E1). In the models of the second type (Io3 and E3), the cores consist of FeS with an admixture of nickel and have the density ρ 1 = 4.6 g cm?3. The approach used here differs from that used previously both in chosen model chemical composition of these satellites and in boundary conditions imposed on the models. The most important question to be answered by modeling the internal structure of the Galilean satellites is that of the condensate composition at the formation epoch of Jupiter’s system. Jupiter’s core and the Galilean satellites were formed from the condensate. Ganymede and Callisto were formed fairly far from Jupiter in zones with temperatures below the water condensation temperature, water was entirely incorporated into their bodies, and their modeling showed the mass ratio of the icy (I) component to the rock (R) component in them to be I/R ～ 1. The R composition must be clarified by modeling Io and Europa. The models of the second type (Io3 and E3), in which the satellite cores consist of FeS, yield 25.2 (Io3) and 22.8 (E3) for the core masses (in weight %). In discussing the R composition, we note that, theoretically, the material of which the FeS+Ni core can consist in the R accounts for ～25.4% of the satellite mass. In this case, such an important parameter as the mantle silicate iron saturation is Fe# = 0.265. The Io3 and E3 models agree well with this theoretical prediction. The models of the first and second types differ markedly in core radius; thus, in principle, the R composition in the formation zone of Jupiter’s system can be clarified by geophysical studies. Another problem studied here is that of the error made in modeling Io and Europa using the Radau-Darvin formula when passing from the Love number k 2 to the nondimensional polar moment of inertia $\bar C$ . For Io, the Radau-Darvin formula underestimates the true value of $\bar C$ by one and a half units in the third decimal digit. For Europa, this effect is approximately a factor of 3 smaller, which roughly corresponds to a ratio of the small parameters for the satellites under consideration α Io/α Europa ～ 3.4. In modeling the internal structure of the satellites, the core radius depends strongly on both the mean moment of inertia I* and k 2. Therefore, the above discrepancy in $\bar C$ for Io is appreciable.  相似文献   

    9.
    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   

    10.
    We re-examine the formation of the inner Oort comet cloud while the Sun was in its birth cluster with the aid of numerical simulations. This work is a continuation of an earlier study (Brasser, R., Duncan, M.J., Levison, H.F. [2006]. Icarus 184, 59–82) with several substantial modifications. First, the system consisting of stars, planets and comets is treated self-consistently in our N-body simulations, rather than approximating the stellar encounters with the outer Solar System as hyperbolic fly-bys. Second, we have included the expulsion of the cluster gas, a feature that was absent previously. Third, we have used several models for the initial conditions and density profile of the cluster – either a Hernquist or Plummer potential – and chose other parameters based on the latest observations of embedded clusters from the literature. These other parameters result in the stars being on radial orbits and the cluster collapses. Similar to previous studies, in our simulations the inner Oort cloud is formed from comets being scattered by Jupiter and Saturn and having their pericentres decoupled from the planets by perturbations from the cluster gas and other stars. We find that all inner Oort clouds formed in these clusters have an inner edge ranging from 100 AU to a few hundred AU, and an outer edge at over 100,000 AU, with little variation in these values for all clusters. All inner Oort clouds formed are consistent with the existence of (90377) Sedna, an inner Oort cloud dwarf planetoid, at the inner edge of the cloud: Sedna tends to be at the innermost 2% for Plummer models, while it is 5% for Hernquist models. We emphasise that the existence of Sedna is a generic outcome. We define a ‘concentration radius’ for the inner Oort cloud and find that its value increases with increasing number of stars in the cluster, ranging from 600 AU to 1500 AU for Hernquist clusters and from 1500 AU to 4000 AU for Plummer clusters. The increasing trend implies that small star clusters form more compact inner Oort clouds than large clusters. We are unable to constrain the number of stars that resided in the cluster since most clusters yield inner Oort clouds that could be compatible with the current structure of the outer Solar System. The typical formation efficiency of the inner Oort cloud is 1.5%, significantly lower than previous estimates. We attribute this to the more violent dynamics that the Sun experiences as it rushes through the centre of the cluster during the latter’s initial phase of violent relaxation.  相似文献   

    11.
    Long-period (LP) comets, Halley-type (HT) comets, and even some comets of the Jupiter family, probably come from the Oort cloud, a huge reservoir of icy bodies that surrounds the solar system. Therefore, these comets become important probes to learn about the distant Oort cloud population. We review the fundamental dynamical properties of LP comets, and what is our current understanding of the dynamical mechanisms that bring these bodies from the distant Oort cloud region to the inner planetary region. Most new comets have original reciprocal semimajor axes in the range2 × 10-5 < 1/aorig < 5 × 10-5AU-1. Yet, this cannot be taken to represent the actual space distribution of Oort cloud comets, but only the region in the energy space in which external perturbers have the greatest efficiency in bringing comets to the inner planetary region. The flux of Oort cloud comets in the outer planetary region is found to be at least several tens times greater than the flux in the inner planetary region. The sharp decrease closer to the Sun is due to the powerful gravitational fields of Jupiter and Saturn that prevent most Oort cloud comets from reaching the Earth’s neighborhood (they act as a dynamical barrier). A small fraction of ～10-2 Oort cloud comets become Halley type (orbital periods P < 200 yr), and some of them can reach short-period orbits with P < 20 yr. We analyze whether we can distinguish the latter, very ‘old” LP comets, from comets of the Jupier family coming from the Edgeworth-Kuiper belt.  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.
    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   

    14.
    In this paper, an efficient algorithm is established for computing the maximum (minimum) angular separation ρ max(ρ min), the corresponding apparent position angles ( $\theta|_{\rho_{\rm max}}$ , $\theta|_{\rho_{\rm min}}$ ) and the individual masses of visual binary systems. The algorithm uses Reed’s formulae (1984) for the masses, and a technique of one-dimensional unconstrained minimization, together with the solution of Kepler’s equation for $(\rho_{\rm max}, \theta|_{\rho_{\rm max}})$ and $(\rho_{\rm min}, \theta|_{\rho_{\rm min}})$ . Iterative schemes of quadratic coverage up to any positive integer order are developed for the solution of Kepler’s equation. A sample of 110 systems is selected from the Sixth Catalog of Orbits (Hartkopf et al. 2001). Numerical studies are included and some important results are as follows: (1) there is no dependence between ρ max and the spectral type and (2) a minor modification of Giannuzzi’s (1989) formula for the upper limits of ρ max functions of spectral type of the primary.  相似文献   

    15.
    In this paper, we apply the ideas presented by one of us (Prentice, 1978a, b) for the development of the proto-solar cloud into a system of Laplacian rings to the development of the protoplanetary clouds which ultimately led to Jupiter, Saturn and Uranus. We show that if one accepts this scenario — especially the idea of supersonic turbulence in the proto-planetary clouds — one can satisfactorily explain, on the basis of fixing a single adjustable parameter, both the geometric precession of the orbital radii of the regular satellite systems of these three planets and the chemical composition and mass distribution of these satellites. We suggest that thermal stirring in the proto-planetary cloud in the vicinity of the surface of the planet may be responsible for the smaller masses of some of the inner satellites as well as for the formation of the rocky rings of Uranus. The icy rings of Saturn are suggested to be the product of condensation processes in a continuous gaseous disc within the Roche limit of the planet.  相似文献   

    16.
    Jupiter’s satellites are subject to strong tidal forces which result in variations of the gravitational potential and deformations of the satellites’ surfaces on the diurnal tidal cycle. Such variations are described by the Love numbers \(k_2\) and \(h_2\) for the tide-induced potential variation due to internal mass redistribution and the radial surface displacement, respectively. The phase-lags \( \phi _{k_2}\) and \( \phi _{h_2}\) of these complex numbers contain information about the rheological and dissipative states of the satellites. Starting from interior structure models and assuming a Maxwell rheology to compute the tidal deformation, we calculate the phase-lags in application to Ganymede and Europa. For both satellites we assume a decoupling of the outer ice-shell from the deep interior by a liquid subsurface water ocean. We show that, in this case, the phase-lag difference \(\varDelta \phi = \phi _{k_2}- \phi _{h_2}\) can provide information on the rheological and thermal state of the deep interiors if the viscosities of the deeper layers are small. In case of Ganymede, phase-lag differences can reach values of a few degrees for high-pressure ice viscosities \({<}10^{14}\) Pa s and would indicate a highly dissipative state of the deep interior. In this case \(\varDelta \phi \) is dominated by dissipation in the high-pressure ice layer rather than dissipation within the ice-I shell. These phase lags would be detectable from spacecraft in orbit around the satellite. For Europa \(\varDelta \phi \) could reach values exceeding \(20^\circ \) and phase-lag measurements could help distinguish between (1) a hot dissipative silicate mantle which would in thermal equilibrium correspond to a very thin outer ice-I shell and (2) a cold deep interior implying that dissipation would mainly occur in a thick (several tens of km) outer ice-I shell. These measurements are highly relevant for ESA’s Jupiter Icy Moons Explorer (JUICE) and NASA’s Europa Multiple Flyby Mission, both targeted for the Jupiter system.  相似文献   

    17.
    We present new approaches to the simulation of stellar collapse, the formation of black holes, and explosive core-collapse supernova nucleosynthesis that build upon open-source codes and microphysics. We discuss the new spherically-symmetric general-relativistic (GR) collapse code GR1D that is endowed with an approximate 1.5D treatment of rotation, comes with multiple nuclear equations of state, and handles neutrinos with a multi-species leakage scheme. Results from a first set of spinning black hole formation simulations are presented. We go on to discuss the derivative code GR1D+N which is tuned for calculations of explosive nucleosynthesis and includes a NSE/non-NSE equation of state treatment, and a nuclear reaction network. We present sample results showing GR1D+N??s performance in reproducing previous results with thermal-bomb-driven explosions. Finally, we introduce the 3?+?1 GR Zelmani core collapse simulation package and present first results obtained in its application to the 3D modeling of failing core-collapse supernovae.  相似文献   

    18.
    A study of circumnuclear star-forming regions (CNSFRs) in several early-type spirals has been carried out in order to investigate their main properties: stellar and gas kinematics, dynamical masses, ionising stellar masses, chemical abundances and other properties of the ionised gas. Both high resolution (R～20,000) and moderate resolution (R～5000) have been used. In some cases, these regions (about 100–150 pc in size) are composed of several individual star clusters with sizes between 1.5 and 4.9 pc, estimated from Hubble Space Telescope images. Stellar and gas velocity dispersions are found to differ by about 20 to 30 km?s?1, with the Hβ emission lines being narrower than both the stellar lines and the [Oiii]λ5007 Å lines. The twice ionised oxygen, on the other hand, shows velocity dispersions comparable to those of stars. We have applied the virial theorem to estimate dynamical masses of the clusters, assuming that the systems are gravitationally bounded and spherically symmetric, and using previously measured sizes. The measured values of the stellar velocity dispersions yield dynamical masses of the order of 107 to 108 M⊙ for the full CNSFRs. We obtain oxygen abundances which are comparable to those found in high-metallicity disc Hii regions from direct measurements of electron temperatures and consistent with solar values within the errors. The region with the highest oxygen abundance is R3+R4 in NGC3504, 12+log(O/H)=8.85, about 1.5 times solar. The derived N/O ratios are, on average, larger than those found in high-metallicity disc Hii regions, and they do not seem to follow the trend of N/O vs. O/H which marks the secondary behaviour of nitrogen. On the other hand, the S/O ratios span a very narrow range—between 0.6 and 0.8 times solar. Compared to high-metallicity disc Hii regions, CNSFRs show values of the O23 and the N2 parameters whose distributions are shifted to lower and higher values, respectively. Hence, even though their derived oxygen and sulphur abundances are similar, higher values would in principle be obtained for the CNSFRs if pure empirical methods were used to estimate abundances. CNSFRs also exhibit lower ionisation parameters than their disc counterparts, as derived from [Sii]/[Siii]. Their ionisation structure also seems to be different, with CNSFRs showing radiation-field properties more similar to Hii galaxies than to disc high-metallicity Hii regions.  相似文献   

    19.
    We analyze the evolution of coronal plasma upflows from the edges of AR 10978, which has the best limb-to-limb data coverage with Hinode’s EUV Imaging Spectrometer (EIS). We find that the observed evolution is largely due to the solar rotation progressively changing the viewpoint of nearly stationary flows. From the systematic changes in the upflow regions as a function of distance from disc center, we deduce their 3D geometrical properties as inclination and angular spread in three coronal lines (Si vii, Fe xii, and Fe xv). In agreement with magnetic extrapolations, we find that the flows are thin, fan-like structures rooted in quasi separatrix layers (QSLs). The fans are tilted away from the AR center. The highest plasma velocities in these three spectral lines have similar magnitudes and their heights increase with temperature. The spatial location and extent of the upflow regions in the Si vii, Fe xii, and Fe xv lines are different owing to i) temperature stratification and ii) line of sight integration of the spectral profiles with significantly different backgrounds. We conclude that we sample the same flows at different temperatures. Further, we find that the evolution of line widths during the disc passage is compatible with a broad range of velocities in the flows. Everything considered, our results are compatible with the AR upflows originating from reconnections along QSLs between over-pressure AR loops and neighboring under-pressure loops. The flows are driven along magnetic field lines by a pressure gradient in a stratified atmosphere. Our interpretation of the above results is that, at any given time, we observe the superposition of flows created by successive reconnections, leading to a broad velocity distribution.  相似文献   

    20.
    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   

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