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1.
We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.  相似文献   

2.
The inclinations of exoplanets detected via radial velocity method are essentially unknown. We aim to provide estimations of the ranges of mutual inclinations that are compatible with the long-term stability of the system. Focusing on the skeleton of an extrasolar system, i.e. considering only the two most massive planets, we study the Hamiltonian of the three-body problem after the reduction of the angular momentum. Such a Hamiltonian is expanded both in Poincaré canonical variables and in the small parameter \(D_2\), which represents the normalised angular momentum deficit. The value of the mutual inclination is deduced from \(D_2\) and, thanks to the use of interval arithmetic, we are able to consider open sets of initial conditions instead of single values. Looking at the convergence radius of the Kolmogorov normal form, we develop a reverse KAM approach in order to estimate the ranges of mutual inclinations that are compatible with the long-term stability in a KAM sense. Our method is successfully applied to the extrasolar systems HD 141399, HD 143761 and HD 40307.  相似文献   

3.
We study the secular evolution of several exoplanetary systems by extending the Laplace-Lagrange theory to order two in the masses. Using an expansion of the Hamiltonian in the Poincaré canonical variables, we determine the fundamental frequencies of the motion and compute analytically the long-term evolution of the Keplerian elements. Our study clearly shows that, for systems close to a mean-motion resonance, the second order approximation describes their secular evolution more accurately than the usually adopted first order one. Moreover, this approach takes into account the influence of the mean anomalies on the secular dynamics. Finally, we set up a simple criterion that is useful to discriminate between three different categories of planetary systems: (i) secular systems (HD 11964, HD 74156, HD 134987, HD 163607, HD 12661 and HD 147018); (ii) systems near a mean-motion resonance (HD 11506, HD 177830, HD 9446, HD 169830 and $\upsilon $ υ  Andromedae); (iii) systems really close to or in a mean-motion resonance (HD 108874, HD 128311 and HD 183263).  相似文献   

4.
The high performance photometric data obtained with space mission CoRoT offer the opportunity to efficiently constrain our models for the stellar interior of solar-like pulsating stars. On the occasion of the analysis of the oscillations of solar-like pulsator HD 49385, a G0-type star in an advanced stage of evolution, we revisit the phenomenon of the avoided crossings. Christensen-Dalgaard proposed a simple analogy to describe an avoided crossing between two modes. We here present an extension of this analogy to the case of n modes, and show that it should lead, in certain cases, to a characteristic behavior of the eigenfrequencies, significantly different from the n=2 case. This type of behavior seems to be observed in HD 49385, from which we infer that the star should be in a Post Main Sequence phase.  相似文献   

5.
On the basis of a high-order (order 12) expansion of the perturbative potential in powers of the eccentricities and the inclinations, we analyze the secular interactions of two non-coplanar planets which are not in mean-motion resonance. The model is based on the planetary three-body problem which can be reduced to two degrees of freedom by the well-known elimination of the nodes [Jacobi, C.G.J., 1842. Astron. Nachr. XX, 81-102]. We introduce non-singular canonical variables which bring forward the symmetries of the problem. The main dynamical features depend on the location and stability of the equilibria which are easily found with our analytical model. We find that there exists an equilibrium when both eccentricities are zero. When the mutual inclination is small, this equilibrium is stable, but for larger mutual inclination it becomes unstable, generating a large chaotic zone and, by bifurcation, two regular regions, the so-called Kozai resonances. This analytical study which depends on only two parameters (the ratio of the semi-major axes and the mass ratio of the planets) makes possible a large survey of the problem and enables us to identify and quantify its main dynamical features, periodic orbits, regular and chaotic zones, etc. The results of our analytical model are illustrated and confirmed by numerical integrations.  相似文献   

6.
In this paper, we present a formalism designed to model tidal interaction with a viscoelastic body made of Maxwell material. Our approach remains regular for any spin rate and orientation, and for any orbital configuration including high eccentricities and close encounters. The method is to integrate simultaneously the rotation and the position of the planet as well as its deformation. We provide the equations of motion both in the body frame and in the inertial frame. With this study, we generalize preexisting models to the spatial case and to arbitrary multipole orders using a formalism taken from quantum theory. We also provide the vectorial expression of the secular tidal torque expanded in Fourier series. Applying this model to close-in exoplanets, we observe that if the relaxation time is longer than the revolution period, the phase space of the system is characterized by the presence of several spin-orbit resonances, even in the circular case. As the system evolves, the planet spin can visit different spin-orbit configurations. The obliquity is decreasing along most of these resonances, but we observe a case where the planet tilt is instead growing. These conclusions derived from the secular torque are successfully tested with numerical integrations of the instantaneous equations of motion on HD 80606 b. Our formalism is also well adapted to close-in super-Earths in multiplanet systems which are known to have non-zero mutual inclinations.  相似文献   

7.
Accurate estimation of cratering asymmetry on the Moon is crucial for understanding Moon evolution history. Early studies of cratering asymmetry have omitted the contributions of high lunar obliquity and inclination. Here, we include lunar obliquity and inclination as new controlling variables to derive the cratering rate spatial variation as a function of longitude and latitude. With examining the influence of lunar obliquity and inclination on the asteroids population encountered by the Moon, we then have derived general formulas of the cratering rate spatial variation based on the crater scaling law. Our formulas with addition of lunar obliquity and inclination can reproduce the lunar cratering rate asymmetry at the current Earth-Moon distance and predict the apex/ant-apex ratio and the pole/equator ratio of this lunar cratering rate to be 1.36 and 0.87, respectively. The apex/ant-apex ratio is decreasing as the obliquity and inclination increasing. Combining with the evolution of lunar obliquity and inclination, our model shows that the apex/ant-apex ratio does not monotonically decrease with Earth-Moon distance and hence the influences of obliquity and inclination are not negligible on evolution of apex/ant-apex ratio. This model is generalizable to other planets and moons, especially for different spin-orbit resonances.  相似文献   

8.
The triple systems display several interesting effects; and are also important for understanding the origin and evolution of binaries. Namely mutual inclination of two orbits, changes of observed inclination, and problems with the third light will be discussed in this contribution.  相似文献   

9.
We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50?–60?, which may be related with the existence of real planetary systems.  相似文献   

10.
The aim of the present investigation will be to determine the explicit forms of differential equations which govern secular perturbations of the orbital elements of close binary systems in the plane of the orbit (i.e., of the semi-major axisA, eccentricitye, and longitude of the periastron ), arising from the lag of dynamical tides due to viscosity of stellar material. The results obtained are exact for any value of orbital eccentricity comprised between 0e<1; and include the effects produced by the second, third and fourth-harmonic dynamical tides, as well as by axial rotation with arbitrary inclination of the equator to the orbital plane.In Section 2 following brief introductory remarks the variational equations of the problem of plane motion will be set up in terms of the rectangular componentsR, S, W of disturbing accelerations with respect to a revolving system of coordinates. The explicit form of these coefficients will be established in Section 3 to the degree of accuracy to which squares and higher powers of quantities of the order of superficial distortion can be ignored. Section 4 will be devoted to a derivation of the explicit form of the variational equations for the case of a perturbing function arising from axial rotation; and in Section 5 we shall derive variational equations which govern the perturbation of orbital elements caused by lagging dynamical tides.Numerical integrations of these equations, which govern the tidal evolution of close binary systems prompted by viscous friction at constant mass, are being postponed for subsequent investigations.Prepared at the Lunar Science Institute, Houston, Texas, under the joint support of the Universities Space Research Association, Charlottesville, Virginia, and the National Aeronautics and Space Administration Manned Spacecraft Center, Houston, Texas, under Contract No. NSR 09-051-001. This paper constitutes Lunar Science Institute Contribution no. 100.Normally at the Department of Astronomy, University of Manchester, England.  相似文献   

11.
Planetary, stellar and galactic physics often rely on the general restricted gravitational $N$ -body problem to model the motion of a small-mass object under the influence of much more massive objects. Here, I formulate the general restricted problem entirely and specifically in terms of the commonly used orbital elements of semimajor axis, eccentricity, inclination, longitude of ascending node, argument of pericentre, and true anomaly, without any assumptions about their magnitudes. I derive the equations of motion in the general, unaveraged case, as well as specific cases, with respect to both a bodycentric and barycentric origin. I then reduce the equations to three-body systems, and present compact singly- and doubly-averaged expressions which can be readily applied to systems of interest. This method recovers classic Lidov–Kozai and Laplace–Lagrange theory in the test particle limit to any order, but with fewer assumptions, and reveals a complete analytic solution for the averaged planetary pericentre precession in coplanar circular circumbinary systems to at least the first three nonzero orders in semimajor axis ratio. Finally, I show how the unaveraged equations may be used to express resonant angle evolution in an explicit manner that is not subject to expansions of eccentricity and inclination about small nor any other values.  相似文献   

12.
We summarize results from deep spectroscopic observations of the HD 209458 planetary system, carried out with the Hubble Space Telescope—Cosmic Origins Spectrograph. Orbitally resolved observations are used to show that hot gas emission lines, arising only in the stellar atmosphere, are not variable, while lower ionizations species found in the upper atmosphere of the hot Jupiter HD 209458b absorb stellar photons during transit. For both C II and Si III, we find mean transit attenuation of ～8%. The firm detection of silicon is in direct conflict with previous low-resolution studies, which we attribute to long-term variability in the system. We also use these observations to search for auroral emission from the planet, detecting a statistically significant emission feature at 1582 Å that is consistent with H2 photoexcited by stellar O I photons.  相似文献   

13.
The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60° of mutual planetary inclination, but in most families, the stability does not exceed 20°–30°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.  相似文献   

14.
In this paper we extend the theory of close encounters of a giant planet on a parabolic orbit with a central star developed in our previous work (Ivanov and Papaloizou in MNRAS 347:437, 2004; MNRAS 376:682, 2007) to include the effects of tides induced on the central star. Stellar rotation and orbits with arbitrary inclination to the stellar rotation axis are considered. We obtain results both from an analytic treatment that incorporates first order corrections to normal mode frequencies arising from stellar rotation and numerical treatments that are in satisfactory agreement over the parameter space of interest. These results are applied to the initial phase of the tidal circularisation problem. We find that both tides induced in the star and planet can lead to a significant decrease of the orbital semi-major axis for orbits having periastron distances smaller than 5?C6 stellar radii with tides in the star being much stronger for retrograde orbits compared to prograde orbits. Assuming that combined action of dynamic and quasi-static tides could lead to the total circularisation of orbits this corresponds to observed periods up to 4?C5 days. We use the simple Skumanich law to characterise the rotational history of the star supposing that the star has its rotational period equal to one month at the age of 5 Gyr. The strength of tidal interactions is characterised by circularisation time scale, t ev , which is defined as a typical time scale of evolution of the planet??s semi-major axis due to tides. This is considered as a function of orbital period P obs , which the planet obtains after the process of tidal circularisation has been completed. We find that the ratio of the initial circularisation time scales corresponding to prograde and retrograde orbits, respectively, is of order 1.5?C2 for a planet of one Jupiter mass having P obs ~ 4 days. The ratio grows with the mass of the planet, being of order five for a five Jupiter mass planet with the same P orb . Note, however, this result might change for more realistic stellar rotation histories. Thus, the effect of stellar rotation may provide a bias in the formation of planetary systems having planets on close orbits around their host stars, as a consequence of planet?Cplanet scattering, which favours systems with retrograde orbits. The results reported in the paper may also be applied to the problem of tidal capture of stars in young stellar clusters.  相似文献   

15.
We have made a Monte Carlo simulation of the intergalactic absorption in order to model the Lyman continuum absorption, which is required to estimate the escape fraction of the Lyman continuum from distant galaxies. To input into the simulation, we derive an empirical distribution function of the intergalactic absorbers which reproduces recent observational statistics of the Lyman α forest, Lyman limit systems (LLSs) and damped Lyman α systems (DLAs) simultaneously. In particular, we assume a common functional form of the number evolution along the redshift for all types of absorbers. The Lyman series transmissions in our simulation reproduce the observed redshift evolution of the transmissions excellently, and the Lyman continuum transmission also agrees with an observed estimation which is still quite rare in the literature. The probability distribution of the Lyman α opacity in our simulation is lognormal with a tail towards a large opacity. This tail is produced by DLAs. The probability distribution of the Lyman continuum opacity in our simulation also shows a broad tail towards a large opacity. This tail is produced by LLSs. Because of the rarity of LLSs, we have a chance to have a clean line of sight in the Lyman continuum even for   z ∼ 4  with a probability of about 20 per cent. Our simulation expects a good correlation between the Lyman continuum opacity and the Lyman α opacity, which may be useful to estimate the former from the latter for an individual line of sight.  相似文献   

16.
Trojan asteroids undergo very large perturbations because of their resonance with Jupiter. Fortunately the secular evolution of quasi circular orbits remains simple—if we neglect the small short period perturbations. That study is done in the approximation of the three dimensional circular restricted three-body problem, with a small mass ratio μ—that is about 0.001 in the Sun Jupiter case. The Trojan asteroids can be defined as celestial bodies that have a “mean longitude”, M + ω + Ω, always different from that of Jupiter. In the vicinity of any circular Trojan orbit exists a set of “quasi-circular orbits” with the following properties: (A) Orbits of that set remain in that set with an eccentricity that remains of the order of the mass ratio μ. (B) The relative variations of the semi-major axis and the inclination remain of the order of ${\sqrt{\mu}}$ . (C) There exist corresponding “quasi integrals” the main terms of which have long-term relative variations of the order of μ only. For instance the product c(1 – cos i) where c is the modulus of the angular momentum and i the inclination. (D) The large perturbations affect essentially the difference “mean longitude of the Trojan asteroid minus mean longitude of Jupiter”. That difference can have very large perturbations that are characteristics of the “horseshoes orbit”. For small inclinations it is well known that this difference has two stable points near ±60° (Lagange equilibrium points L4 and L5) and an unstable point at 180° (L3). The stable longitude differences are function of the inclination and reach 180° for an inclination of 145°41′. Beyond that inclination only one equilibrium remains: a stable difference at 180°.  相似文献   

17.
The late-stage formation of giant planetary systems is rich in interesting dynamical mechanisms. Previous simulations of three giant planets initially on quasi-circular and quasi-coplanar orbits in the gas disc have shown that highly mutually inclined configurations can be formed, despite the strong eccentricity and inclination damping exerted by the disc. Much attention has been directed to inclination-type resonance, asking for large eccentricities to be acquired during the migration of the planets. Here we show that inclination excitation is also present at small to moderate eccentricities in two-planet systems that have previously experienced an ejection or a merging and are close to resonant commensurabilities at the end of the gas phase. We perform a dynamical analysis of these planetary systems, guided by the computation of planar families of periodic orbits and the bifurcation of families of spatial periodic orbits. We show that inclination excitation at small to moderate eccentricities can be produced by (temporary) capture in inclination-type resonance and the possible proximity of the non-coplanar systems to spatial periodic orbits contributes to maintaining their mutual inclination over long periods of time.  相似文献   

18.
Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 108 y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.  相似文献   

19.
The eccentricities of the barium stars   总被引：3，自引：0，他引：3  
We investigate the eccentricities of barium (Ba  ii ) stars formed via a stellar wind accretion model. We carry out a series of Monte Carlo simulations using a rapid binary evolution algorithm, which incorporates full tidal evolution, mass loss and accretion, and nucleosynthesis and dredge-up on the thermally pulsing asymptotic giant branch. We follow the enhancement of barium in the envelope of the accreting main-sequence companion and dilution into its convective envelope once the star ascends the giant branch.
The observed eccentricities of Ba  ii stars are significantly smaller than those of an equivalent set of normal red giants but are nevertheless non-zero. We show that such a distribution of eccentricities is consistent with a wind accretion model for Ba  ii star production with weak viscous tidal dissipation in the convective envelopes of giant stars. We successfully model the distribution of orbital periods and the number of observed Ba  ii stars. The actual distribution of eccentricities is quite sensitive to the strength of the tides, so that we are able to confirm that this strength is close to, but less than, what is expected theoretically and found with alternative observational tests. Two systems – one very short-period but eccentric, and one long-period and highly eccentric – still lie outside the envelope of our models, and so require a more exotic formation mechanism. All our models, even those which were a good fit to the observed distributions, overproduced the number of high-period barium stars, a problem that could not be solved by some combination of the three parameters: tidal strength, tidal enhancement and wind accretion efficiency.  相似文献   

20.
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