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1.
The nonlinear stability of the equilibrium points in the restricted three-body problem with variable mass has been studied. It is found that, in the nonlinear sense, the collinear points are unstable for all mass ratios and the triangular points are stable in the range of linear stability except for three mass ratios, which depend upon β, the constant due to the variation in mass governed by Jeans’ law.  相似文献   

2.
This study investigates the nonlinear stability of the triangular equilibrium points when the bigger primary is an oblate spheroid and the infinitesimal body varies (decreases) it’s mass in accordance with Jeans’ law. It is found that these points are stable for all mass ratios in the range of linear stability except for three mass ratios depending upon oblateness coefficient A and β, a constant due to the variation in mass governed by Jeans’ law.  相似文献   

3.
The existence and stability of a test particle around the equilibrium points in the restricted three-body problem is generalized to include the effect of variations in oblateness of the first primary, small perturbations ϵ and ϵ′ given in the Coriolis and centrifugal forces α and β respectively, and radiation pressure of the second primary; in the case when the primaries vary their masses with time in accordance with the combined Meshcherskii law. For the autonomized system, we use a numerical evidence to compute the positions of the collinear points L 2κ , which exist for 0<κ<∞, where κ is a constant of a particular integral of the Gylden-Meshcherskii problem; oblateness of the first primary; radiation pressure of the second primary; the mass parameter ν and small perturbation in the centrifugal force. Real out of plane equilibrium points exist only for κ>1, provided the abscissae x < \fracn(k-1)b\xi<\frac{\nu(\kappa-1)}{\beta}. In the case of the triangular points, it is seen that these points exist for ϵ′<κ<∞ and are affected by the oblateness term, radiation pressure and the mass parameter. The linear stability of these equilibrium points is examined. It is seen that the collinear points L 2κ are stable for very small κ and the involved parameters, while the out of plane equilibrium points are unstable. The conditional stability of the triangular points depends on all the system parameters. Further, it is seen in the case of the triangular points, that the stabilizing or destabilizing behavior of the oblateness coefficient is controlled by κ, while those of the small perturbations depends on κ and whether these perturbations are positive or negative. However, the destabilizing behavior of the radiation pressure remains unaltered but grows weak or strong with increase or decrease in κ. This study reveals that oblateness coefficient can exhibit a stabilizing tendency in a certain range of κ, as against the findings of the RTBP with constant masses. Interestingly, in the region of stable motion, these parameters are void for k = \frac43\kappa=\frac{4}{3}. The decrease, increase or non existence in the region of stability of the triangular points depends on κ, oblateness of the first primary, small perturbations and the radiation pressure of the second body, as it is seen that the increasing region of stability becomes decreasing, while the decreasing region becomes increasing due to the inclusion of oblateness of the first primary.  相似文献   

4.
This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L 4 and L 5 naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.  相似文献   

5.
The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter), which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain (x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε = 0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce the density parameter, say to ε = 10−6, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary. The stability of all members of each and all families computed was calculated and presented in this paper for both the large solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections. All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received, consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including their stability and is available at request. It is concluded that approximation of the general solution of this system is straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties.  相似文献   

6.
The effect of small perturbations and in the coriolis and the centrifugal forces respectively on the stability of the triangular points in the restricted problem of three bodies with variable mass has been studied. It is found that the range of stability of triangular points increases or decreases depending upon whether the perturbation point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line J8–J9=0 where J8 and J9 depend upon , the constant due to the variation in mass governed by Jeans' law.  相似文献   

7.
This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L i (i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable.  相似文献   

8.
In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J 2 and J 4. After that, we show that the triangular points are stable for 0<μ<μ c and unstable when , where μ c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J 4 for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems.  相似文献   

9.
This paper examines the effect of a constant κ of a particular integral of the Gylden-Meshcherskii problem on the stability of the triangular points in the restricted three-body problem under the influence of small perturbations in the Coriolis and centrifugal forces, together with the effects of radiation pressure of the bigger primary, when the masses of the primaries vary in accordance with the unified Meshcherskii law. The triangular points of the autonomized system are found to be conditionally stable due to κ. We observed further that the stabilizing or destabilizing tendency of the Coriolis and centrifugal forces is controlled by κ, while the destabilizing effects of the radiation pressure remain unchanged but can be made strong or weak due to κ. The condition that the region of stability is increasing, decreasing or does not exist depend on this constant. The motion around the triangular points L 4,5 varying with time is studied using the Lyapunov Characteristic Numbers, and are found to be generally unstable.  相似文献   

10.
The location and the stability of the libration points in the restricted problem have been studied when small perturbation and are given to the Coriolis and the centrifugal forces respectively. It is seen that the pointsL 4 andL 5 form nearly equilateral triangles with the primaries and the pointsL 1,L 2,L 3 remain collinear. It is further observed that for the pointsL 4 andL 5, the range of stability increases or decreases depending upon whether the point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line 36-19=0 and the stability of the collinear points is not influenced by the perturbations and they remain unstable.  相似文献   

11.
We consider a system of nonlinear spinor and a Bianchi type I gravitational fields in presence of viscous fluid. The nonlinear term in the spinor field Lagrangian is chosen to be λ F, with λ being a self-coupling constant and F being a function of the invariants I an J constructed from bilinear spinor forms S and P. Self-consistent solutions to the spinor and BI gravitational field equations are obtained in terms of τ, where τ is the volume scale of BI universe. System of equations for τ and ε, where ε is the energy of the viscous fluid, is deduced. This system is solved numerically for some special cases.   相似文献   

12.
In a recent paper, published in Astrophys. Space Sci. (337:107, 2012) (hereafter paper ZZX) and entitled “On the triangular libration points in photogravitational restricted three-body problem with variable mass”, the authors study the location and stability of the generalized Lagrange libration points L 4 and L 5. However their study is flawed in two aspects. First they fail to write correctly the equations of motion of the variable mass problem. Second they attribute a variable mass to the third body of the restricted three-body model, a fact that is not compatible with the assumptions used in deriving the mathematical formulation of this model.  相似文献   

13.
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   

14.
When μ is smaller than Routh’s critical value μ 1 = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ 5), (μ 5, μ 4), (μ 4, μ 3), (μ 3, μ 2), (μ 2, μ 1), (μ 1, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ 2, μ 1), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points was enriched.  相似文献   

15.
We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L 3, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2
, at which the triangular equilibria disappear by coalescing with the inner collinear equilibrium point L 1. We also compute the horizontal and the vertical stability of these families for the angular velocity parameter ω under consideration. Series of horizontal–critical periodic orbits of the short-Trojan families with the angular velocity ω and the mass ratio μ as parameters, are given.  相似文献   

16.
We calculate the expression forΔ -s in terms of true anomalies and classical orbital elements, referring to a common fixed plane and working up to power four of eccentricities and tangents of inclinations. We obtained two final results: the first wheny ′>y, the second wheny>y ′.  相似文献   

17.
We present new radio observations of molecular lines in the region of high mass star formation, namely G122.0-7.1. A large-scale map of the emission observed in the 12CO (J = 1−0) and 13CO (J = 1−0) lines covers the area of 15′ × 9′, revealing two dense regions. The molecular bipolar outflows have been resolved in ASO1 region. It is associated with the known candidate YSO nearby IRAS 0042 + 5530. Also, a new dense region has been discovered in the North-Western part of the G122.0-7.1 at a distance of 5′ from IRAS 0042 + 5530. Its position is close to the peak of 4850 MHz emission. The text was submitted by the authors in English.  相似文献   

18.
A number of empirical relationships are shown to indicate an increase in the abundance of phosphorus logε(P) with height in the atmosphere of HR 1512. These include: (a) a correlation of logε(P) with the observed equivalent width Wobs of PII lines; (b) a correlation of logε(P) with the wavelength of the lines; (c) a systematic divergence in the values of logε(P) for lines with different excitation potentials El; in particular, lines with lower El correspond on the average to higher abundances logε(P); and, (d) a distinct dependence of logε(P) on the average geometrical height of formation, Hf. In addition, assuming that logε(P) is constant in the star's atmosphere leads to a systematic discrepancy between the theoretical equivalent widths Wth and the observed values Wobs. By a trial and error method we have found a distribution of the phosphorus abundance logε(P) with height H such that the systematic difference between Wth and Wobs vanishes. It turned out, however, that a simpler, step distribution of logε(P) yields equally good agreement between Wth and Wobs. Although the solution is nonunique, both distributions have some features in common, specifically: (1) a sharp rise in logε(P) occurs in the same range of heights H corresponding to optical thicknesses τ 5000 ≈ 10−2–10−3, i.e., stratification of phosphorus takes place in rather high layers of the atmosphere of HR 1512, and (2) the upper bound, logε up(P) = 8.9, is the same in both cases, so that in the region of the rise, logε(P) increases by 3.4 dex. A comparison with available data for HgMn, Am, and Ap type stars shows that similar sharp changes in the abundances logε of several elements occur in other CP stars, at the same optical depths or in even higher layers of their atmospheres. __________ Translated from Astrofizika, Vol. 51, No. 2, pp. 239–253 (May 2008).  相似文献   

19.
The nonlinear ion-acoustic double layers (IADLs) in a warm magnetoplasma with positive-negative ions and nonthermal electrons are investigated. For this purpose, the hydrodynamic equations for the positive-negative ions, nonthermal electron density distribution, and the Poisson equation are used to derive a modified Zakharov–Kuznetsov (MZK) equation, in the small amplitude regime. It is found that compressive and rarefactive IADLs strongly depend on the mass and density ratios of the negative-to-positive ions as well as the nonthermal electron parameter. Also, it is shown that there are one critical value for the density ratio of the negative-to-positive ions (ν), the ratio between unperturbed electron-to-positive ion density (μ), and the nonthermal electron parameter (β), which decide the existence of positive and negative IADLs. The present study is applied to examine the small amplitude nonlinear IADL excitations for the (H+, O2-)(\mathrm{H}^{+}, \mathrm{O}_{2}^{-}) and (H+,H−) plasmas, where they are found in the D- and F-regions of the Earth’s ionosphere. This investigation should be helpful in understanding the salient features of the nonlinear IADLs in either space or laboratory plasmas where two distinct groups of ions and non-Boltzmann distributed electrons are present.  相似文献   

20.
In the problem of 2+2 bodies in the Robe’s setup, one of the primaries of mass m*1m^{*}_{1} is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ 1. The second primary is a mass point m 2 outside the shell. The third and the fourth bodies (of mass m 3 and m 4 respectively) are small solid spheres of density ρ 3 and ρ 4 respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal. We assume m 2 is describing a circle around m*1m^{*}_{1}. The masses m 3 and m 4 mutually attract each other, do not influence the motion of m*1m^{*}_{1} and m 2 but are influenced by them. We also assume masses m 3 and m 4 are moving in the plane of motion of mass m 2. In the paper, the equations of motion, equilibrium solutions, linear stability of m 3 and m 4 are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are unstable for all values of the mass parameters μ,μ 3,μ 4. There exist an infinite number of non collinear equilibrium solutions each for m 3 and m 4, lying on circles of radii λ,λ′ respectively (if the densities of m 3 and m 4 are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ,μ 3,μ 4, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon.  相似文献   

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