Stability criteria in many-body systems

The stability parameters developed and discussed in the first paper of this series (Walkeret al., 1980) are used to determine empirically, by means of numerical integration experiment, regions of stability for corotational, coplanar, hierarchical three-body systems. The initially circular case of these systems is studied: the components of the close binary are taken to move initially in circular orbits with respect to their common mass-centre, the third mass initially moving in a circular orbit with respect to the same mass-centre such that its orbit lies wholly outside those of the former two masses. The stability of these systems is then studied by reference to the empirical stability parameters and the initial ratio of the semi-major axes of the orbit of the close binary to that of the third mass about the binary's mass-centre, which is less than unity. For given values of the stability parameters it is determined how the stability of a system is affected by changes in the ratio of the semi-major axes. It is found that an upper limit to this ratio exists which determines the region of stability for such systems. It is also found possible, in the region of instability, to predict how unstable a system will be i.e. crudely speaking, the number of orbits it may be expected to execute before some gross instability sets in. The effect commensurabilities in mean motion have on the stability of these systems is also considered. It is generally found that these commensurabilities enhance the stability of these systems. The predictive powers of the method are then tested: using many test cases it is seen how accurately the stability or instability of a system may be predicted.     

On the stability of the lagrangian points in the spatial restricted problem of three bodies

The problem of stability of the Lagrangian equilibrium point of the circular restricted problem of three bodies is investigated in the light of Nekhoroshev-like theory. Looking for stability over a time interval of the order of the estimated age of the universe, we find a physically relevant stability region. An application of the method to the Sun-Jupiter and the Earth-Moon systems is made. Moreover, we try to compare the size of our stability region with that of the region where the Trojan asteroids are actually found; the result in such case is negative, thus leaving open the problem of the stability of these asteroids.     

Convective stability in magnetic stars

Dynamical stability of a static axisymmetrical magnetic star with respect to high-order modes of oscillation is investigated by means of the energy method, neglecting the Eulerian perturbation of gravity. The magnetic field is assumed to be continuous across the surface of the star and its first-order spatial derivatives, but it may have both toroidal and poloidal components.The second variation of the potential energy is written in a way which, in the case of apurely toroidal field, and for axisymmetrical and non-axisymmetrical modes, yields Tayler's local stability criteria which are necessary and sufficient conditions for convective stability, and in the case of ageneral field yields a single local stability criterion, which is a sufficient condition for convective stability.     

The Hill stability of inclined bound triple star and planetary systems

The dynamical stability of a triple system composed of a binary or planetary system and a bound third body moving on a orbit inclined to the system is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of planetary systems against disruption, component exchange or capture. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects. These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the third body initially increases the stability of the system then decreases stability as the eccentricity reaches higher values.The Hill stability criterion is applied to extrasolar planetary systems to determine the critical distances at which planets of the same mass as the observed extrasolar planet moving on a circular orbit could remain on a stable orbit. It was found that these distances were sufficiently short suggesting that the presence of further as yet unobserved stable extrasolar planets in observed systems was very likely.     

Stability analysis of non-steady mhd-equilibria

Following the work of Bernsteinet al. (1958), Frieman and Rotenberg (1960) and Unno (1968) a formalism is developed which allows to examine the adiabatic stability of a perfectly conducting, rotating and self-gravitating plasma in non-steady equilibrium. Using this method the stability of a plasma in a dynamical phase of its evolution can be predicted. Global stability investigations are carried out which are based on a variation of the total energy of the system and, in general, lead to sufficient conditions for stability. The formalism is applied to the stability of a horizontal magnetic field in a medium stratified by a gravitational field.     

The Hill stability of inclined small mass binary systems in three-body systems with special application to triple star systems, extrasolar planetary systems and Binary Kuiper Belt systems

The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable.     

Connection between Hill stability and weak stability in the elliptic restricted three-body problem

This paper provides a study on the connection between Hill stability and weak stability in the framework of the spatial elliptic restricted three-body problem. We determine a necessary condition for weak stability by giving an upper and a lower bound of qualitative measure of the Hill stability. The sufficient condition for weak stability and the symmetry of weak stable regions around the planets of the Solar System is also investigated.     

Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem,I

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure. It is found that radiation pressure of the larger body for solar system cases exerts only a small quantitative influence on the stability regions.     

脉冲星对氢原子钟的频率驾驭算法研究

氢原子钟具有较高的短期稳定度,将其作为主钟可在短期内产生高精度的本地时间信号.但氢钟存在频率漂移现象,导致其长期稳定度较差,从而影响本地时间的准确性.毫秒脉冲星自转高度稳定,借助于其长期稳定度高的特性,可定期实现对氢原子钟的频率驾驭,并对实时信号加以控制.首先分析了国际脉冲星计时阵(International Pulsar Timing Array,IPTA)第二批发布数据中四颗毫秒脉冲星的稳定度随时间的变化,同时采用哈达玛方差分析了中国科学院国家授时中心(National Time Service Center,NTSC)一台氢钟的频率稳定性能,最终给出了利用脉冲星驾驭氢原子钟频率的方法.     

Thermal stability of red giants

The thermal stability of stars with a condensed core and an extended convective envelope (red giants) is discussed. It is shown that the thernal stability is determined by the way the opacity of the photosphere depends on density and temperature. In particular, the H^? opacity in red giants' photospheres ensures their stability.     

Discussion of simple stability criteria for stars with central helium burning

The validity of simple stability criteria for stars with central He-burning used in earlier papers is investigated. The determinant of secular stability for σ=0 is related to quantities defined as far as possible at the surface of the He-core and at the bottom of the envelope. This relation is used to discuss the validity of two intuitive criteria for secular stability.     

非平衡动力学中恒星结构稳定性研究

采用非平衡动力学方法研究由CNO循环决定的恒星结构稳定性,本文得出:CNO循环自身是稳定的;扩散效应不会改变处于稳定的恒星结构稳定性;对流效应对其稳定性有重要影响,波数小的扰动更容易激发不稳定性;巨大的温度梯度也能激发不稳定性。本文给出了稳定性条件和恒星结构保持稳定所允许的最小扰动波数。     

Stability of the classical type of relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J ₂ problem is generalized to that of a rigid body in a J ₂ gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J ₂ and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.     

Gravitational stability of finitely conducting two-component plasma through porous medium

The gravitational stability of magnetized self-gravitating two-component plasma of finite conductivity flowing through porous medium is studied. Effect of magnetic field, porosity, viscosity, finite conductivity, and neutral gas friction is considered on the stability of the system. Dispersion relations are derived from linearized equations using normal mode analysis. Longitudinal and transverse wave propagations are discussed. On the basis of Hurwitz criterion, the stability of the system is discussed. It is found that Jeans's criterion determines the stability of the system. Jeans's expression depends on the sonic speeds in both the components. For transverse wave propagation in perfectly conducting plasma. Jeans's expression is modified due to magnetic field and porosity but in case of finitely conducting plasma the Jeans's expression remains unaltered. Collisional frequency, viscosity, permeability of the porous medium have damping effect.     

Stability of the solar tachocline with magnetic fields

The stability of magnetic fields in the solar tachocline is investigated. We present stability limits for higher azimuthal wave numbers and results on the dependence of the stability on the location of toroidal magnetic fields in latitude. While the dependence of the wave number with the largest growth rate on the magnetic field strength and the magnetic Prandtl number is small, the dependence on the magnetic Reynolds number Rm indicates that lowest azimuthal modes are excited for very high Rm. Upon varying the latitudinal position of the magnetic field belts, we find slightly lower stability limits for high latitudes, and very large stability limits at latitudes below 10°, with little dependence on latitude in between. An increase of the maximum possible field was achieved by adding a poloidal field. The upper limit for the toroidal field which can be stored in the radiative tachocline is then 1000 G, compared to about 100 G for a purely toroidal field as was found in an earlier work. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the effect of turbulent anisotropy on pulsation stability of stars

Within the framework of a non-local time-dependent stellar convection theory, we study in detail the effect of turbulent anisotropy on stellar pulsation stability. The results show that anisotropy has no substantial influence on pulsation stability of g modes and low-order(radial order n_r 5) p modes.The effect of turbulent anisotropy increases as the radial order increases. When turbulent anisotropy is neglected, most high-order(n r 5) p modes of all low-temperature stars become unstable. Fortunately,within a wide range of the anisotropic parameter c_3, stellar pulsation stability is not sensitive to the specific value of c_3. Therefore it is safe to say that calibration errors of the convective parameter c_3 do not cause any uncertainty in the calculation of stellar pulsation stability.     

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.     

Investigation of equilibria of a satellite subjected to gravitational and aerodynamic torques

Attitude motion of a satellite subjected to gravitational and aerodynamic torques in a circular orbit is investigated. In special case, when the center of pressure of aerodynamic forces is located on one of the principal central axes of inertia of the satellite, all equilibrium orientations are determined. Necessary and (or) sufficient conditions of stability are obtained for each equilibrium orientation. Evolution of domains where stability conditions take place is investigated. All bifurcation values of parameters corresponding to qualitative change of domains of stability are determined.     

The structure and stability of coronal magnetic fields

The high degree of symmetry often assumed in studies of the structure and stability of coronal magnetic field configurations is restrictive and can yield misleading results. We have therefore developed fully three-dimensional numerical methods for constructing force-free equilibria and for examining their stability properties, which make no assumptions about symmetry. A test of the stability analysis has been performed by applying it to the Gold-Hoyle twisted flux tube, which is known to be kink-unstable if the helical field makes more than about one turn between the line-tying end-plates. Our preliminary result is that the critical number of turns is about 1.1, in good agreement with the previous best estimate. However, we find that the growth rate, which has not been discussed previously, is orders of magnitude smaller than expected, even when the flux tube is twisted far beyond the stability limit.     

Attitude stability of artificial satellites subject to gravity gradient torque

The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient torque, using a canonical formulation, and Andoyer’s variables to describe the rotational motion. The stability criteria employed requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined through a numerical study of the Hamilton’s equations of motion and linear study of their stability. Subsequently a canonical linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized quadratic Hamiltonian. A semi-analytic process of normalization based on Lie–Hori algorithm is applied to obtain the Hamiltonian normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko’s theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability requirements scenario.