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1.
We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on the mass parameter β and the radiation factors q i , i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q i vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are calculated.  相似文献   

2.
We use the global construction which was made in [6, 7] of the secular systems of the planar three-body problem, with regularized double inner collisions. These normal forms describe the slow deformations of the Keplerian ellipses which each of the bodies would describe if it underwent the universal attraction of only one fictitious other body. They are parametrized by the masses and the semi-major axes of the bodies and are completely integrable on a fixed transversally Cantor set of the parameter space. We study this global integrable dynamics reduced by the symmetry of rotation and determine its bifurcation diagram when the semi-major axes ratio is small enough. In particular it is shown that there are some new secular hyperbolic or elliptic singularities, some of which do not belong to the subset of aligned ellipses. The bifurcation diagram may be used to prove the existence of some new families of 2-, 3- or 4-frequency quasiperiodic motions in the planar three-body problem [7], as well as some drift orbits in the planar n-body problem [8].  相似文献   

3.
In this short paper we show that making turbulence two‐rather than three‐dimensional may increase the effective turbulent viscosity by about 40 %. Dimensionless hydrodynamical viscosity parameters up to αmax = 0.25 Mt2 may be obtained in this approach, which are in better agreement with the observational data on non‐stationary accretion than the values obtained in numerical simulations. However, the α ‐parameter values known from observations are still several times higher (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

4.
The confining curves in the general three-body problem are studied; the role of the integralc 2 h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio ε=m 3/m 2 both the positions of the collinear equilibrium points and thec 2 h integral are expanded in power series of ε. In this way the relationship is found between the confining curves resulting from thec 2 h integral in the general problem, and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of ε the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual, system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small ε, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be ‘unstable’, i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that given for the most important three-body subsystems of the Solar System essentially the same information on ‘stability’ as the full computation of thec 2 h integral and of the bifurcation values.  相似文献   

5.
The formulation of the tensor virial equations is generalized to unrelaxed configurations, where virial equilibrium does not coincide with dynamical (or hydrostatic) equilibrium. Homeoidally striated, Jacobi ellipsoids, which generalize classical Jacobi ellipsoids, are studied in detail. Further investigation is devoted to the generation of sequences of virial equilibrium configurations where the anisotropy parameters are left unchanged, including both flattened and elongated, triaxial configurations, and the determination of the related bifurcation points. An application is made to dark matter haloes hosting giant galaxies (M ≈ 1012 m⊙), with regard to assigned initial and final configuration, following and generalizing to many respects a procedure conceived by Thuan & Gott (1975). The dependence of the limiting axis ratios, below which no configuration is allowed for the sequence under consideration, on the change in mass, total energy, and angular momentum, during the evolution, is illustrated in some representative situations. The dependence of the axis ratios, ε31 and ε21, on a parameter, related to the initial conditions of the density perturbation, is analysed in connection with a few special cases. The same is done for the rotation parameters. Within the range of the rotation parameter, λ, deduced from high‐resolution numerical simulations, the shape of dark matter haloes is mainly decided by the amount of anisotropy in residual velocity distribution. On the other hand, the contribution of rotation has only a minor effect on the meridional plane, and no effect on the equatorial plane, as bifurcation points occur for larger values of λ. To this respect, dark matter haloes are found to resemble giant elliptical galaxies. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

6.
Using the extended Poincaré-Lighthill-Kuo (PLK) reductive perturbation method, which incorporates the phase-shift variations, it is shown that common features on propagation and head-on collisions of ion-acoustic waves exist for a magnetized plasmas of different inertial-less particle distributions. For instance it is remarked that, the soliton amplitude is always independent of magnetic field strength while strictly depends on its angle regarding the propagation direction. Both types of solitons (compressive or rarefactive) are shown to exist which are defined through the critical angle γ=π/2 or other critical values depending on plasma fractional parameters. These critical plasma parameter values also define the sign of head-on collision phase shift. Furthermore, it is proved that for a given set of plasma parameters there is always a relative angle of propagation regarding to that of the magnetic-field for which the soliton width is maximum. Current findings apply to a wide range of magnetized plasmas including those containing background dust ingredients or two-temperature inertial-less particles and may be used to study laboratory or astrophysical magnetoplasmas.  相似文献   

7.
For 14 values of the mass parameter μ (from 0.0010 to 0.0150) the non-periodic Transtrojan orbits (around L4 and L5) are investigated, which on the plane restricted problem of three bodies pass the point situated opposite to the body μ (looking from the main mass) with zero velocity in the rotating coordinate system. Results: The Transtrojan state contains a finite number of ”︁double-librations„ (around L4 and L5); this number decreases with growing value of the mass parameter. Above a value of mass parameter between 0.010 and 0.015 no further double-libration takes place. Certain topological properties of the Transtrojan state are found; e.g. this state has a phase of narrowing and a phase of widening of the single librations; thereby the amplitudes of the librations fluctuate in a characteristic manner.  相似文献   

8.
The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J 2 is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J 2 and J 4 is analyzed. Besides, when given certain values of J 2 and J 4, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.  相似文献   

9.
Dynamical behaviour of a small binary with equal components, each of mass m, is considered under attraction of a heavy body of mass M. Differential equations of the general three-body problem are integrated numerically using the code by S. J. Aarseth (Aarseth, Zare 1974) for mass ratios m/M within 10−11–10−4 range. The direct and retrograde orbits of light bodies about each other are considered which lie either in the plane of moving their center of mass or in the plane perpendicular to it. It is shown numerically that the critical separation between the binary components which leads to disruption of binary is proportional to (m/M)1/3. The criterion can be used for studying (in the first approximation) the motion of double stars and binary asteroids or computing the parameters of magnetic monopol and antimonopol pairs.  相似文献   

10.
We study the non-linear stability of the equilibria corresponding to the motion of a particle orbiting around a finite straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by elongated celestial bodies. By means of the Arnold's theorem for non-definite quadratic forms we determine the orbital stability of the equilibria, for all values of the parameter k of the problem, resonant cases included.  相似文献   

11.
The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.  相似文献   

12.
We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ 2 throughJ 9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J 2 only). In particular, we examine the manner in which the odd zonalJ 3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J 4+J 2 2 =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.  相似文献   

13.
Starting from a general solution for the birth functionB(m, t) of stars described in detail in Casusoet al. (1989), we have obtained a first-order analytical approximation to this function as a function of metallicityZ. Using this, we obtained a fit to the observational curve compiled by Tinsley (1980) for the cumulative function of stars with metallicity lower than a given value in the solar neighbourhood. In addition, using the same expression, with its numerical fit to previous data, we obtain a good fit to the differential distributions of stars at low metallicity given in the review by Pagel (1987), given a bifurcation in the birth function at low values ofZ, which would correspond to two distinct epochs of onset of star formation. The analysis gives an infall of gas towards the solar neighbourhood up to the epoch of metallicityZ=6.7×10–3 with a correspondingly increased star formation rate, which subsequently stabilized, and another similar inflow up toZ=1.2×10–3, followed again by a steady star formation rate for largerZ. Although the assumptions made are still relatively crude, and the numbers should be considered tentative, the flexibility of the model in handling the problem is that we wish to show here.  相似文献   

14.
We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ 2 problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ 2 andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ 2, as large as |J 2|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.  相似文献   

15.
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   

16.
Linear and nonlinear properties of magnetoconvection in rotating inviscid fluid are investigated. A time dependent one dimensional nonlinear Landau-Gonzburg equation is derived and analyzed at a supercritical pitchfork bifurcation corresponding to a region0 < R m < R * m (R m =R * m corresponds to Takens-Bogdanov bifurcation point). This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

17.
Grassmugg  Maria  Hanslmeier  Arnold  Laback  Otto 《Solar physics》1997,174(1-2):437-441
In this contribution we present a nonlinear dynamo model, described by an infinite dimensional system of differential equations, whose solutions depend on the essential parameter D, the dynamo number. The solutions and the bifurcation points of the system are determined with the help of a new developed computer code. We show that, depending on D, stationary, oscillatory and chaotic solutions, which are characterized by Lyapunov exponents, result. We find that the solar dynamo may operate either in the chaotic or in the stable limit cycle domain, depending on the characteristic value of the dynamo number or the motion of the convection zone.  相似文献   

18.
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19.
The generation of magnetic field in a homogeneous, electrically conducting fluid – as required for the dynamo generation of the fields of many astrophysical bodies – is normally a threshold process; the dynamo mechanism, applicable to such bodies in unmagnetised environments, requires motions of sufficient strength to overcome the innate magnetic diffusion. In the presence of an ambient field, however, the critical nature of the field generation process is relaxed. Motions can distort and amplify the ambient field for all amplitudes of flow. For motions with appropriate geometries, an internal ‘dynamo‐like’ field of appreciable strength can be generated, even for relatively weak flows. At least a minority of planets, moons and other bodies exist within significant external astrophysical fields. For these bodies, the ambient field problem is more relevant than the classical dynamo problem, yet it remains relatively little studied. In this paper we consider the effect of an axial ambient field on a spherical mean‐field α 2ω dynamo model, through nonlinear calculations with α ‐quenching feedback. Ambient fields of varying strengths, and both stationary and oscillatory in time, are imposed. Particular focus is placed on the effects of these fields on the equatorial symmetry and the time dependence of the preferred solutions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

20.
Forgács-dajka  E.  Petrovay  K. 《Solar physics》2001,203(2):195-210
Helioseismic measurements indicate that the solar tachocline is very thin, its full thickness not exceeding 4% of the solar radius. The mechanism that inhibits differential rotation to propagate from the convective zone to deeper into the radiative zone is not known, though several propositions have been made. In this paper we demonstrate by numerical models and analytic estimates that the tachocline can be confined to its observed thickness by a poloidal magnetic field B p of about one kilogauss, penetrating below the convective zone and oscillating with a period of 22 years, if the tachocline region is turbulent with a diffusivity of η∼1010 cm2 s−1 (for a turbulent magnetic Prandtl number of unity). We also show that a similar confinement may be produced for other pairs of the parameter values (B p, η). The assumption of the dynamo field penetrating into the tachocline is consistent whenever η≳109 cm2 s−1. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1013389631585  相似文献   

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