共查询到20条相似文献，搜索用时 15 毫秒
1.
We consider a restricted three-body problem where the primaries are moving in an elliptic collision orbit and the infinitesimal
mass moves in a three dimensional space. This paper is devoted to prove analytically the existence of several families of
symmetric periodic solutions as continuation of Keplerian circular orbits. In our approach the perturbing parameter is related
with the energy of the primaries. 相似文献
2.
Peter J. Shelus 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):483-489
Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios. 相似文献
3.
Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem. 相似文献
4.
R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):369-378
This is a numerical study of orbits in the elliptic restricted three-body problem concerning the dependence of the critical orbits on the eccentricity of the primaries. They are defined as being the separatrix between stable and unstable single periodic orbits. As our results are adapted to the existence of planetary orbits in double stars we concentrated first on the P-orbits (defined to surround both primaries). Due to the complexity of the elliptic problem there is no analytical approach possible. Using the results of some 300 integrated orbits for 103 to 3. 103 periods of the primaries we established lower and upper bounds for the critical orbits for different values of the eccentricity. 相似文献
5.
Kyriaki I. Antoniadou Anne-Sophie Libert 《Celestial Mechanics and Dynamical Astronomy》2018,130(6):41
We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted three-body problem, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances, and thus, new families continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consists of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long-time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far. 相似文献
6.
The rectilinear elliptic restricted three-body problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity \(e'=1\), but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (AIAA J 7:1003–1009, 1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter \(\mu =0.5\) (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke’s computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to \(\mu \) and \(e'<1\). Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449. 相似文献
7.
The Sitnikov configuration is a special case of the restricted three-body problem where the two primaries are of equal masses
and the third body of a negligible mass moves along a straight line perpendicular to the orbital plane of the primaries and
passes through their center of mass. It may serve as a toy model in dynamical astronomy, and can be used to study the three-dimensional
orbits in more applicable cases of the classical three-body problem. The present paper concerns the straight-line oscillations
of the Sitnikov family of the photogravitational circular restricted three-body problem as well as the associated families
of three-dimensional periodic orbits. From the stability analysis of the Sitnikov family and by using appropriate correctors
we have computed accurately 49 critical orbits at which families of 3D periodic orbits of the same period bifurcate. All these
families have been computed in both cases of equal and non-equal primaries, and consist entirely of unstable orbits. They
all terminate with coplanar periodic orbits. We have also found 35 critical orbits at which period doubling bifurcations occur.
Several families of 3D periodic orbits bifurcating at these critical Sitnikov orbits have also been given. These families
contain stable parts and close upon themselves containing no coplanar orbits. 相似文献
8.
P. Delibaltas 《Celestial Mechanics and Dynamical Astronomy》1983,29(2):191-204
In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits. 相似文献
9.
《New Astronomy》2020
This paper deals with the stability analysis of the triangular equilibrium points for the generalized problem of the photogravitational restricted three body where both the primaries are radiating. The problem is generalized in the sense that the eccentricity of the orbits and the oblateness due to both the primaries and infinitesimal are considered. The stability in the case of linear resonance are analyzed based on the Floquet’s theory for finding the characteristic exponent for a system containing periodic coefficients. It was found that the critical value of μ for the stability boundary for parametric excitation is dependent on the oblateness of the primaries as well as infinitesimal. 相似文献
10.
Stability of generalized elliptic restricted four body problem with radiation and oblateness effects
《New Astronomy》2020
This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter. 相似文献
11.
We consider an elliptic restricted four-body system including three primaries and a massless particle. The orbits of the primaries are elliptic, and the massless particle moves under the mutual gravitational attraction. From the dynamic equations, a quasi-integral is obtained, which is similar to the Jacobi integral in the circular restricted three-body problem (CRTBP). The energy constant \(C\) determines the topology of zero velocity surfaces, which bifurcate at the equilibrium point. We define the concept of Hill stability in this problem, and a criterion for stability is deduced. If the actual energy constant \(C_{\mathrm{ac}}\ ( {>} 0 ) \) is bigger than or equal to the critical energy constant \(C_{\mathrm{cr}}\), the particle will be Hill stable. The critical energy constant is determined by the mass and orbits of the primaries. The criterion provides a way to capture an asteroid into the Earth–Moon system. 相似文献
12.
We study two and three-dimensional resonant periodic orbits, usingthe model of the restricted three-body problem with the Sun andNeptune as primaries. The position and the stability character ofthe periodic orbits determine the structure of the phase space andthis will provide useful information on the stability and longterm evolution of trans-Neptunian objects. The circular planarmodel is used as the starting point. Families of periodic orbitsare computed at the exterior resonances 1/2, 2/3 and 3/4 withNeptune and these are used as a guide to select the energy levelsfor the computation of the Poincaré maps, so that all basicresonances are included in the study. Using the circular planarmodel as the basic model, we extend our study to more realisticmodels by considering an elliptic orbit of Neptune and introducingthe inclination of the orbit. Families of symmetric periodicorbits of the planar elliptic restricted three-body problem andthe three-dimensional problem are found. All these orbitsbifurcate from the families of periodic orbits of the planarcircular problem. The stability of all orbits is studied. Althoughthe resonant structure in the circular problem is similar for allresonances, the situation changes if the eccentricity of Neptuneor the inclination of the orbit is taken into account. All theseresults are combined to explain why in some resonances there aremany bodies and other resonances are empty. 相似文献
13.
Ashraf Hamdy Mostafa K. Ahmed Mohamad Radwan Fawzy Ahmed Abd El-Salam 《Earth, Moon, and Planets》2003,93(4):261-273
The objective of the present work is to develope explicit analytical expressions for the small amplitude orbits of the infinitesimal
mass about the equilibrium points in the elliptic restricted three body problem. To handle this dynamical problem, the Hamiltonian
for the elliptic problem is formed with the true anomaly and then with the eccentric anomaly as independent variables. The
origin is then transformed to a fixed point and the Hamiltonian is developed up to O(4) in the eccentricity, e, (which plays the role of the small parameter of the problem) of the primaries. The integration of the model problem under
consideration is undertaken by means of a perturbation technique based on Lie series developments, which leads to the solution
of the canonical equations of motion. 相似文献
14.
Maria Gousidou-Koutita 《Earth, Moon, and Planets》1985,32(1):21-45
The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem. 相似文献
15.
K. E. Papadakis 《Earth, Moon, and Planets》2008,103(1-2):25-42
This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L 1, L 2 and L 3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations. 相似文献
16.
We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003. 相似文献
17.
We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system. 相似文献
18.
John D. Hadjidemetriou 《Astrophysics and Space Science》1976,40(1):201-224
Several families of the planar general three-body problem for fixed values of the three masses are found, in a rotating frame of reference, where the mass of two of the bodies is small compared to the mass of the third body. These families were obtained by the continuation of a degenerate family of periodic orbits of three bodies where two of the bodies have zero masses and describe circular orbits around a third body with finite mass, in the same direction.The above families represent planetary systems with the body with the large mass representing the Sun and the two small bodies representing two planets or comets. One section of a family is shown to represent the Jupiter family of comets and also a model for the Sun-Jupiter-Saturn system is found.The stability analysis revealed that stability exists for small masses and small eccentricities of the two planets. Planetary systems with relatively large masses and eccentricities are proved to be unstable. In particular, the Jupiter family of comets, for small masses of the two small bodies, and the Sun-Jupiter-Saturn system are proved to be stable. Also, it was shown that resonances are not necessarily associated with instabilities. 相似文献
19.
Reference periodic orbits are determined accurately for the planets of the solar system, using a restricted problem model with the Sun and Jupiter as the two primaries. The prediction is verified that stability of the planetary orbits should imply stability of those reference orbits that are simple-periodic. 相似文献
20.
The elliptic restricted problem of three bodies with unit eccentricity of the primaries is used to generate a family of periodic orbits in the general problem of three bodies. The parameter of the family is the mass of one of the participating bodies. This varies from zero to a termination value. The mass ratio of the primaries of the unperturbed problem (three to five) is maintained throughout the generation of the family. In this way an asymmetry is introduced generalizing the Copenhagen elliptic problem as the generating model. All members of the family experience a close approach and a collision between the primaries during half of the period of the orbit, therefore, the family is classified as Class Two. 相似文献