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1.
This paper investigates the periodic orbits around the triangular equilibrium points for 0<μ<μ c , where μ c is the critical mass value, under the combined influence of small perturbations in the Coriolis and the centrifugal forces respectively, together with the effects of oblateness and radiation pressures of the primaries. It is found that the perturbing forces affect the period, orientation and the eccentricities of the long and short periodic orbits. 相似文献
2.
《New Astronomy》2021
The aim of this research is to show the significant effects of albedo on the existence of out-of-plane equilibria in the elliptic restricted three-body problem under an oblate primary model. We computed out-of-plane equilibria numerically and graphically for different values of the parameters μ, α, e, k and σ where μ, α, e, k and σ are mass parameter, albedo factor, eccentricity, ratio of the luminosity of smaller primary to luminosity of bigger primary considered as constant and oblateness factor due to smaller primary, respectively. Further, we examined the stability of out-of-plane equilibria and found that these equilibria are unstable in linear sense for all parameters μ, α, e, k and σ. Finally, the three-dimensional periodic orbits are analyzed for different values of albedo factor α and oblateness factor σ. 相似文献
3.
We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries
is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile
coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by
taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem. 相似文献
4.
We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter and fixed oblateness coefficent A1 = 0.005, as well as for varying A1 and fixed = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits. 相似文献
5.
We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries. 相似文献
6.
This paper studies the motion of an infinitesimal mass in the framework of the restricted three-body problem (R3BP) under the assumption that the primaries of the system are radiating-oblate spheroids, enclosed by a circular cluster of material points. It examines the effects of radiation and oblateness up to J 4 of the primaries and the potential created by the circular cluster, on the linear stability of the liberation locations of the infinitesimal mass. The liberation points are found to be stable for 0<μ<μ c and unstable for $\mu_{c}\le\mu\le\frac{1}{2}$ , where μ c is the critical mass value depending on terms which involve parameters that characterize the oblateness, radiation forces and the circular cluster of material points. The oblateness up to J 4 of the primaries and the gravitational potential from the circular cluster of material points have stabilizing propensities, while the radiation of the primaries and the oblateness up to J 2 of the primaries have destabilizing tendencies. The combined effect of these perturbations on the stability of the triangular liberation points is that, it has stabilizing propensity. 相似文献
7.
We consider the motion of a test particle around a triaxial primary and an oblate companion orbiting each other in elliptic orbits about their common barycenter in the neighborhood of triangular libration points. The positions and stability of these points are influenced by the triaxiality and oblateness of the primary and secondary, and by the semi-major axis and eccentricity of the orbits. The triangular points are stable for 0<μ<μ c ; where μ is the mass ratio (μ≤1/2) and μ c is the critical mass value influenced by the eccentricity, oblateness, semi major axis and triaxiality factors. The size of the region of stability increases with decreasing values of triaxiality and oblateness. An application of the results obtain to double neutron star binaries results show that the positions and stability of the triangular points of PSR J1518+4904, PSR B1534+12, PSR B1914+16 and PSR B2127+11c are affected by the parameters in the systems’ dynamics. 相似文献
8.
We have investigated an improved version of the classic restricted three-body problem where both primaries are considered oblate and are enclosed by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effect on the number and on the linear stability of the equilibrium locations of the small particle due to both, the primaries’ oblateness and the potential created by the circular cluster. We have drawn the zero-velocity surfaces and we have found that in addition to the usual five Lagrangian equilibrium points of the classic restricted three-body problem, there exist two new collinear points L n1,L n2 due to the potential from the circular cluster of material points. Numerical investigations reveal that with the increase in the mass of the circular cluster of material points, L n2 comes nearer to the more massive primary, while L n1 moves away from it. Owing to oblateness of the bodies, L n1 comes nearer to the more massive primary, while L n2 moves towards the less massive primary. The collinear equilibrium points remain unstable, while the triangular points are stable for 0<μ<μ c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ c is the critical mass ratio influenced by oblateness of the primaries and the potential from the circular cluster of material points. The oblateness and the circular cluster of material points have destabilizing tendency. 相似文献
9.
Binary systems hosting astrophysical compact objects such as white dwarfs and/or neutron stars provide excellent test beds for studying the impact of the oblateness of the main bodies in the restricted three-body problem (R3BP). The case is investigated when the primary bodies are non-luminous, non-spherical (oblate) bodies and the third body of infinitesimal mass is also an oblate spheroid. The existence of extra solar planets orbiting these systems constitutes a three-body problem which makes them excellent models for this axisymmetric ER3BP. The positions of the equilibrium points are affected by the oblateness parameters of the three-bodies; this is shown for double neutron star binaries. The triangular points are stable for 0<μ<μ c ; where μ is the mass ratio (μ≤1/2) and μ c is the critical mass value influenced by the eccentricity, semi major axis and oblateness factors. The size of the region of stability increases with decreasing values of the oblateness. The oblateness of the system’s bodies does not affect the nature of the stability of the collinear points since they remain unstable. Due to the almost equal masses of the primaries, our study shows that even the triangular points of these systems are unstable. 相似文献
10.
We have studied a modified version of the classical restricted three-body problem (CR3BP) where both primaries are considered as oblate spheroids and are surrounded by a homogeneous circular planar cluster of material points centered at the mass center of the system. In this dynamical model we have examined the effects of oblateness of both primaries up to zonal harmonic J 4; together with gravitational potential from the circular cluster of material points on the existence and linear stability of the triangular equilibrium points. It is found that, the triangular points are stable for 0<μ<μ c and unstable for $\mu_{c} \le \mu \le \frac{1}{2}$ , where μ c is the critical mass ratio affected by the oblateness up to J 4 of the primaries and potential from the circular cluster of material points. The coefficient J 4 has stabilizing tendency, while J 2 and the potential from the circular cluster of material points have destabilizing tendency. A practical application of this model could be the study of the motion of a dust particle near oblate bodies surrounded by a circular cluster of material points. 相似文献
11.
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
12.
We have examined the effects of oblateness up to J 4 of the less massive primary and gravitational potential from a circum-binary belt on the linear stability of triangular equilibrium points in the circular restricted three-body problem, when the more massive primary emits electromagnetic radiation impinging on the other bodies of the system. Using analytical and numerical methods, we have found the triangular equilibrium points and examined their linear stability. The triangular equilibrium points move towards the line joining the primaries in the presence of any of these perturbations, except in the presence of oblateness up to J 4 where the points move away from the line joining the primaries. It is observed that the triangular points are stable for 0 < μ < μ c and unstable for \(\mu_{\mathrm{c}} \le \mu \le \frac {1}{2},\) where μ c is the critical mass ratio affected by the oblateness up to J 4 of the less massive primary, electromagnetic radiation of the more massive primary and potential from the belt, all of which have destabilizing tendencies, except the coefficient J4 and the potential from the belt. A practical application of this model could be the study of motion of a dust particle near a radiating star and an oblate body surrounded by a belt. 相似文献
13.
Elbaz I. Abouelmagd 《Earth, Moon, and Planets》2013,110(3-4):143-155
This paper deals with the existence of triangular points and their linear stability when the primaries are oblate spheroid and sources of radiation considering the effect of oblateness up to 10?6 of main terms in the restricted three-body problem; we see that the locations of the triangular points are affected by the oblateness of the primaries and solar radiation pressure. It is further seen that these points are stable for 0 ≤ μ ≤μ c ; and unstable for μ c ≤ μ ≤1/2; where μ c is the critical mass value depending on terms which involve parameters that characterize the oblateness and radiation repulsive forces such that $ \mu_{c} \in (0,1/2) $ ; in addition to this an algorithm has been constructed to calculate the critical mass value. 相似文献
14.
《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
15.
In this paper, we extend the basic model of the restricted four-body problem introducing two bigger dominant primaries m 1 and m 2 as oblate spheroids when masses of the two primary bodies (m 2 and m 3) are equal. The aim of this study is to investigate the use of zero velocity surfaces and the Poincaré surfaces of section to determine the possible allowed boundary regions and the stability orbit of the equilibrium points. According to different values of Jacobi constant C, we can determine boundary region where the particle can move in possible permitted zones. The stability regions of the equilibrium points expanded due to presence of oblateness coefficient and various values of C, whereas for certain range of t (100≤t≤200), orbits form a shape of cote’s spiral. For different values of oblateness parameters A 1 (0<A 1<1) and A 2 (0<A 2<1), we obtain two collinear and six non-collinear equilibrium points. The non-collinear equilibrium points are stable when the mass parameter μ lies in the interval (0.0190637,0.647603). However, basins of attraction are constructed with the help of Newton Raphson method to demonstrate the convergence as well as divergence region of the equilibrium points. The nature of basins of attraction of the equilibrium points are less effected in presence and absence of oblateness coefficients A 1 and A 2 respectively in the proposed model. 相似文献
16.
We locate and examine the stability of the ‘out of plane’ equilibrium points, L 6,7 of an infinitesimal body in the field of stellar-oblate binary systems moving in elliptic orbits around their common center of mass. Their positions and stability depend on the oblateness as well as radiation coefficients of the primaries and the eccentricity of their orbits. A numerical application of this problem for the systems: Gamma Leporis and Altair are given. 相似文献
17.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them. 相似文献
18.
N. I. Vishnu Namboodiri D. Sudheer Reddy Ram Krishan Sharma 《Astrophysics and Space Science》2008,318(3-4):161-168
In the three-dimensional restricted three-body problem, by considering the more massive primary as an oblate spheroid with its equatorial plane coincident with the plane of motion as well as source of radiation, it is found that the collinear point L 1 comes nearer to the primaries with the increase in oblateness and radiation pressure, while L 2 and L 3 move away from the more massive primary with the increase in oblateness and come nearer to it with the increase in radiation pressure. It is noted that the angular frequency s 1 at L 1 increases with oblateness as well as with radiation pressure. s 2 increases with oblateness and decreases with radiation pressure and s 3 decreases with oblateness and increases with radiation pressure. A study on the norms of the characteristic roots λ and s at L 1, L 2 and L 3 is carried out. It is established that for certain oblateness and radiation pressure parameters there is a one-to-one commensurability at the collinear points L 2, L 3 between the planar angular frequencies (s 2,3) and the corresponding angular frequency (s z ) in the z-direction, and that at L 1 no such commensurability exists. At L 2 and L 3, the value of oblateness parameter providing the commensurability decreases with the increase in the radiation pressure. However, the commensurable angular frequencies and eccentricity of the periodic orbits decrease at L 2 and increase at L 3, with the increase in the radiation pressure. 相似文献
19.
Antonis D. Pinotsis 《Planetary and Space Science》2009,57(12):1389-1404
By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L4 and L5 are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L4 and L5, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem. 相似文献
20.
In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease. 相似文献