CORIOLIS力和离心力摄动对Robe问题平动点位置和线性稳定性的影响

研究了Coriolis力和离心力摄动对Robe限制性三体问题主要平动点位置及线性稳定性的影响，给出了Robe限制性三体问题主要平动点的摄动位置和Coriolis力和离心力摄动对主要平动点位置和线性稳定的影响量级．改进了Shrivastava的结果．     

受摄限制性三体问题平动点线性稳定性的判断条件及在Robe问题的应用

给出了受摄限制性三体问题平动点线性稳定性的一个判断条件．条件只与平动点切映像的特征方程系数有关，使用方便．用判断条件，讨论了Robe问题平动点在阻力摄动下的线性稳定性，得到了Hallan等给出的Robe问题平动点在阻力摄动下的线性稳定范围．并改进了Giordanoc等的结果．     

Conditions for the linear stability of libration points and effects of drag on the linear stability of triangular libration points in the restricted three-body problem

A number of criteria for linear stability of libration points in the perturbed restricted three-body problem are presented. The criteria involve only the coefficients of the characteristic equation of the tangent map of the libration points and can be easily applied. With these criteria the effect of drag on the linear stability of the triangular libration points in the classical restricted three-body problem is investigated. Some of Murray et al.'s results are improved.     

平动点线性稳定条件及阻力对限制性三体问题三角平动点线性稳定性的影响

给出了受摄限制性三体问题平动点线性稳定性的一些判断条件，条件只与相应的平动点切映像的特征方程系数有关，使用方便，用这些判断条件，讨论了一些阻力对经典限制性三体问题三角平动点线性稳定性的影响，改进了Murray等的一些结果。     

A criterion for the linear stability of theequilibrium points in the perturbed restricted three-body problem and its application in Robe''s problem

A criterion for the linear stability of the equilibrium points in the perturbed restricted three-body problem is given. This criterion is related only to the coefficients of the characteristic equation of the tangent map of an equilibrium point, and this is convenient to use. With this criterion, we have discussed the linear stability of the equilibrium points in the Robe problem under the perturbation of a drag force, derived the linearly stable region of the equilibrium point in the perturbed Robe's problem with the drag given by Hallen et al., and improved as well the results obtained by Giordano et al.     

Robe’s circular restricted three-body problem under oblate primaries with perturbations in Coriolis and centrifugal forces

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid, and the shape of the second primary is also an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.     

Perturbed Robe’s circular restricted three-body problem under an Oblate Primary

This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid. The problem is perturbed in the sense that small perturbations are given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. And a special case where the density of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable.     

有摄圆型限制性三体问题平动点稳定的充要条件及应用

摘要给出了一个判断有摄圆型限制性三体问题平动点稳定性的充要条件．该条件只依赖于平动点变分方程的特征方程系数的一个简单关系，使用很方便．用所得到的条件，讨论了任意外力摄动对经典圆型限制性三体问题三角平动点稳定性的影响和惯性阻力摄动对Robe圆型限制性三体问题主要平动点的稳定性的影响．     

Existence of equilibrium points and their linear stability in the generalized photogravitational Chermnykh-like problem with power-law profile

We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods, we have found equilibrium points and examined their linear stability. We have also found the zero velocity surface for the present model. In addition to five equilibrium points there exists a new equilibrium point on the line joining the two primaries. It is found that L ₁ and L ₃ are stable for some values of inner and outer radius of the disk while other collinear points are unstable, but L ₄ is conditionally stable for mass ratio less than that of Routh’s critical value. Lastly, we have studied the effects of radiation pressure, oblateness and mass of the disk on the motion and stability of equilibrium points.     

Einstein-Maxwell field equations in isotropic coordinates: an application to neutron star and quark star

The photogravitational restricted three bodies within the framework of the post-Newtonian approximation is carried out. The mass of the primaries are assumed changed under the effect of continuous radiation process and oblateness effects of the two primaries. New perturbed locations of the triangular points are computed. In order to introduce a semi-analytical view, A Mathematica program is constructed so as to draw the locations of triangular points versus the whole range of the mass ratio μ taking into account the photo-gravitational effects, the relativistic corrections and/or oblateness effects. All the obtained figures are analyzed.     

Instability of Libration Points and Resonance Phenomena in the Photogravitational Elliptic Restricted Three-Body Problem

The stability of collinear and triangular libration points is investigated in the photogravitational elliptic restricted three-body problem, in which two primary bodies emit light energy simultaneously. The conditions of stability of the collinear and triangular libration points are obtained based on a linearized set of equations of perturbed motion for various values of the eccentricity of the Keplerian orbits and the mass ratio of the primary bodies. The maximal numerical value is defined for the eccentricity at which a stable libration point can still exist. It is demonstrated how the parametric resonance causes an instability of collinear and triangular libration points; the evolution of the origination of the instability zones is traced. The minimal eccentricity value is found at which zones of instability of triangular libration points arise.     

The photogravitational Hill problem with oblateness: equilibrium points and Lyapunov families

We introduce a new version of Hill’s problem that incorporates the effects of radiation of the primary and oblateness of the secondary and study the basic dynamical features of this new model-problem. This formulation is more appropriate for some astronomical applications as an approximation to the corresponding restricted three-body problem. We use iterative methods for deriving approximate expressions of the equilibrium point locations and study their stability properties by using a linear stability analysis. All equilibrium points are unstable. We also employ singular perturbations methods for obtaining approximate expressions of the Lyapunov families emanating from equilibrium points, in both coplanar and spatial case, and numerical techniques for their continuation.     

Motion around the out-of-plane equilibrium points of the perturbed restricted three-body problem

This paper studies the motion of an infinitesimal body near the out-of-plane equilibrium points, L _6,7, in the perturbed restricted three-body problem. The problem is perturbed in the sense that the primaries of the system are oblate spheroids as well as sources of radiation and small perturbations are give to the Coriolis and centrifugal forces. It locates the positions and examines the stability of L _6,7 with a particular application to the binary system Struve 2398. It is observed that their positions are affected by the radiation, oblateness and a small perturbation in the centrifugal force, but is unaffected by that of the Coriolis force. They are also found to be unstable.     

A Hill Problem with Oblate Primaries and Effect of Oblateness on Hill Stability of Orbits

We introduce a new version of Hill's problem to include the effect of oblateness of the primaries, and briefly discuss its equilibrium points and zero velocity curves. As a first application we use this to study Hill stability of direct orbits around the small primary. This can be employed to study the stability of a planet's moon perturbed by an oblate Sun, or of a star's planet perturbed by a distant disk-shaped galaxy. Oblateness of the `Sun' is found to decrese the maximum distance of Hill stable direct `moon' orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bounds on the ideal MHD stability of line-tied 2-D coronal magnetic fields

A tractable method for investigating the linear stability of line-tied 2-D coronal magnetic fields is introduced. It is based on the Bernstein et al. (1958) energy principle and can be applied to non-isothermal equilibria with gravity, having a translational invariance. The perturbed potential energy integral is manipulated to produce either necessary conditions for stability to localized modes or sufficient conditions for stability to global modes. Each condition only requires the solution of a set of ordinary differential equations, integrated along the magnetic field lines. The tests are employed to two different classes of equilibria. A linear force-free field is shown to be completely stable, regardless of the shear. The role of pressure gradients, footpoint displacements, line-tying and stratification on an isothermal magneto-hydrostatic equilibrium is assessed.     

On the Stability of Equilibrium Points in the Relativistic Restricted Three-Body Problem

The equilibrium points of the relativistic restricted three-body problem are considered. The stability of the triangular points is determined and contrary to recent results of other authors a region of linear stability in the parameter space is obtained. The positions of the collinear points are approximated by series by expansions and their stability is similarly determined. It is found that these are always unstable.This revised version was published online in October 2005 with corrections to the Cover Date.     

The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness

This paper deals with the existence of libration points and their linear stability when the more massive primary is radiating and the smaller is an oblate spheroid. Our study includes the effects of oblateness of $\bar{J}_{2i}$ (i=1,2) with respect to the smaller primary in the restricted three-body problem. Under combining the perturbed forces that were mentioned before, the collinear points remain unstable and the triangular points are stable for 0<μ<μ _c , and unstable in the range $\mu_{c} \le\mu\le\frac{1}{2}$ , where $\mu_{c} \in(0,\frac{1}{2})$ , it is also observed that for these points the range of stability will decrease. The relations for periodic orbits around five libration points with their semimajor, semiminor axes, eccentricities, the frequencies of orbits and periods are found, furthermore for the orbits around the triangular points the orientation and the coefficients of long and short periodic terms also are found in the range 0<μ<μ _c .     

The effect of photogravitational force and oblateness in the perturbed restricted three-body problem

The model of restricted three-body is generalized to include the effects of the oblateness, the radiation pressure and fictitious forces. The positions of libration points, their stability, the critical mass ratio and periodic orbits emanating from these points are analyzed under the influence of these effects. The results obtained are more generalized. In addition the locations of the out of plane equilibrium points are studied. We also observe that there is no explicit effect for the perturbation of Coriolis force on the positions of the out of plane equilibrium points. It is worth mentioning that this model can be degraded into 128 special cases.     

The stability of periodic orbits in the three-body problem

A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.This linear stability analysis is used to study the stability of a family of periodic motions of three bodies with equal masses, in a rotating frame of reference. This family represents motion such that two bodies revolve around each other and the third body revolves around this binary system in the same direction to a distance which varies along the members of the family. It was found that a large part of the family, corresponding to the case where the distance of the third body from the binary system is larger than the dimensions of the binary system, represents stable motion. The nonlinear effects to the linear stability analysis are studied by computing the intersections of several perturbed orbits with the surface of sectiony ₃=0. In some cases more than 1000 intersections are computed. These numerical results indicate that linear stability implies stability to all orders, and this is true for quite large perturbations.