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1.
Two-integral distribution functions for axisymmetric systems   总被引：1，自引：0，他引：1  
Some formulae are presented for finding two-integral distribution functions (DFs) which depend only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. They come from a combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of the system is required to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. The product of the density and its radial velocity dispersion can be also expressed as a sum of products of functions of the potential and of the radial coordinate. It can be further known that the density multiplied by its rotational velocity dispersion is equal to a sum of products of functions of the potential and of the radial coordinate minus the product of the density and the square of its mean rotational velocity. These formulae can be applied to the Binney and the Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also found under the assumption of the laws of the rotational velocity.  相似文献   

2.
Anisotropic distribution functions for spherical galaxies   总被引：1，自引：1，他引：0  
A method is presented for finding anisotropic distribution functions for stellar systems with known, spherically symmetric, densities, which depends only on the two classical integrals of the energy and the magnitude of the angular momentum. It requires the density to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum. The products of the density and its radial and transverse velocity dispersions can be also expressed as a sum of products of functions of the potential and of the radial coordinate. Several examples are given, including some of new anisotropic distribution functions. This device can be extended further to the related problem of finding two-integral distribution functions for axisymmetric galaxies.  相似文献   

3.
Prolate Jaffe models for galaxies   总被引：1，自引：0，他引：1  
We introduce a class of prolate Jaffe models for elliptical galaxies, which are a further extension of Jaffe's spherical models of axisymmetric elliptical systems, and study the properties of their densities, circular velocities, velocity dispersions and two-integral even distribution functions. The form of the potential allows the density to be expressed simply as a function of the potential and radial coordinate R . The models have finite total mass and their densities at large distances decay radially as r −4, except on the major axis, where the densities decay as r −3. It is known from Hunter's formulae that the velocity dispersions for prolate models can be expressed in terms of elementary functions of R and z , unlike those for the oblate Jaffe models recently given by Jiang, and that the prolate models have anisotropic velocity distributions. Thus the prolate models are easier to study than the oblate models. It is also found that the two-integral even distribution functions on the physical boundary of the galaxies increase monotonically with the relative energy, for the prolate models. Furthermore, numerical calculation shows that the two-integral even distribution functions generated from their densities are non-negative, even for very 'squeezed' prolate Jaffe models. However, the edge-on projected surface densities for these prolate models cannot be expressed as simply as for the oblate models.  相似文献   

4.
Flattened Jaffe models for galaxies   总被引：1，自引：0，他引：1  
This paper introduces a class of galactic models which extend Jaffe's spherical models to axisymmetric systems, and then studies the properties of their densities and two-integral even distribution functions. The models have finite total mass and finite densities which, at large distances, decay radially like r −4 except on the major axis, and like r −3 on the major axis. The more flattened the galaxy, the stronger is the dependence of the even distribution functions on the angular momenta of its stars. Their distribution functions can be obtained by using the maximum entropy principle or assuming the anisotropy of the models. In particular, some formulae analogous to those of Hunter & Qian are obtained to calculate two-integral odd distribution functions, and they can be applied to obtain the distribution functions under the assumption of anisotropy for the oblate models.  相似文献   

5.
Oscillation modes of relativistic slender tori   总被引：1，自引：0，他引：1  
Accretion flows with pressure gradients permit the existence of standing waves which may be responsible for observed quasi-periodic oscillations (QPO's) in X-ray binaries. We present a comprehensive treatment of the linear modes of a hydrodynamic, non-self-gravitating, polytropic slender torus, with arbitrary specific angular momentum distribution, orbiting in an arbitrary axisymmetric space–time with reflection symmetry. We discuss the physical nature of the modes, present general analytic expressions and illustrations for those which are low order, and show that they can be excited in numerical simulations of relativistic tori. The mode oscillation spectrum simplifies dramatically for near Keplerian angular momentum distributions, which appear to be generic in global simulations of the magnetorotational instability. We discuss our results in light of observations of high frequency QPO's, and point out the existence of a new pair of modes which can be in an approximate 3:2 ratio for arbitrary black hole spins and angular momentum distributions, provided the torus is radiation pressure dominated. This mode pair consists of the axisymmetric vertical epicyclic mode and the lowest order axisymmetric breathing mode.  相似文献   

6.
We apply the joint lensing and dynamics code for the analysis of early-type galaxies, 'Combined Algorithm for Unified Lensing and Dynamics ReconstructiON ( cauldron )', to a rotating N -body stellar system with dark matter halo which significantly violates the two major assumptions of the method, i.e. axial symmetry supported by a two-integral distribution function. The goal is to study how cauldron performs in an extreme case, and to determine which galaxy properties can still be robustly recovered. Three data sets, corresponding to orthogonal lines of sight, are generated from the N -body system and analysed with the identical procedure followed in the study of real lens galaxies, adopting an axisymmetric power-law total density distribution. We find that several global properties of the N -body system are recovered with remarkable accuracy, despite the fact that the adopted power-law model is too simple to account for the lack of symmetry of the true density distribution. In particular, the logarithmic slope of the total density distribution is robustly recovered to within less than 10 per cent (with the exception of the ill-constrained very inner regions), the inferred angle-averaged radial profile of the total mass closely follows the true distribution, and the dark matter fraction of the system (inside the effective radius) is correctly determined within ∼10 per cent of the total mass. Unless the line-of-sight direction is almost parallel to the total angular momentum vector of the system, reliably recovered quantities also include the angular momentum, the   V /σ  ratio and the anisotropy parameter δ. We conclude that the cauldron code can be safely and effectively applied to real early-type lens galaxies, also providing reliable information for the systems that depart significantly from the method's assumptions.  相似文献   

7.
We use two elementary solutions of the integral equation connecting the density of a stellar system with its two-integral distribution function in order to construct simple distribution functions. Applications for axisymmetric, disk-like and spherical systems are given.  相似文献   

8.
The phenomenon of negative viscosity-alpha in convectively unstable Keplerian accretion discs is discussed. The convection is considered as a random flow with an axisymmetric mesoscale pattern. Its correlation tensor is computed with a time-averaging procedure using Kley's 2D hydrocode. There is a distinct anisotropy between the turbulence intensities in the radial and azimuthal directions, i.e. the radial velocity rms dominates the azimuthal one. As a consequence, an extra term in the expression for the turbulent transport of angular momentum appears which does not vanish for rigid rotation ('Λ-effect'). It is negative ('inwards transport') and even seems to dominate the positive contribution of the eddy viscosity representing outwards transport of angular momentum. For a turbulence model close to that of the mixing-length theory, the rotational influence on the anisotropy of the turbulence intensities,     , and the covariance  〈 u ' R u ' φ 〉  – representing the angular momentum transport – is computed and compared with the accretion disc simulations. Indeed, the negative angular momentum transport can be explained with the observed dominance of the radial turbulence intensity. If, on the other hand, in turbulence fields the azimuthal intensity would dominate or the turbulence is even isotropic, then we always find a positive transport of the angular momentum.  相似文献   

9.
The Boltzmann moment equations are solved to determine the velocity ellipsoid in a particle disk near an isolated satellite resonance. In a coordinate frame which rotates with the pattern speed of the perturbation potential, the solutions are stationary functions of the azimuthal angle. From the velocity ellipsoid we obtain the stress tensor due to particle collisions and consequently, the viscous angular momentum flux. We show that the magnitude of the rate of deformation tensor in a perturbed particle disk is bounded from above by KΩ(1 + τ2)12, where Ω is the orbital angular velocity, τ is the optical depth, and K is a dimensionless constant of order unity. It is also found that in sufficiently perturbed regions there are ranges of azimuthal angle within which the radial component of the angular momentum flux is negative. It is even possible for the angular momentum luminosity, the radial flux integrated over azimuth, to be negative. These results are important for understanding sharp edges and the decay of density waves in planetary rings. They are also relevant to the damping of differential precession and eccentricity in narrow ringlets.  相似文献   

10.
We study the distribution function (DF) of dark matter particles in haloes of mass range  1014–1015 M⊙  . In the numerical part of this work we measure the DF for a sample of relaxed haloes formed in the simulation of a standard Λ cold dark matter (ΛCDM) model. The DF is expressed as a function of energy E and the absolute value of the angular momentum L , a form suitable for comparison with theoretical models. By proper scaling we obtain the results that do not depend on the virial mass of the haloes. We demonstrate that the DF can be separated into energy and angular momentum components and propose a phenomenological model of the DF in the form     . This formulation involves three parameters describing the anisotropy profile in terms of its asymptotic values (β0 and  β∞  ) and the scale of transition between them ( L 0). The energy part   f E ( E )  is obtained via inversion of the integral for spatial density. We provide a straightforward numerical scheme for this procedure as well as a simple analytical approximation for a typical halo formed in the simulation. The DF model is extensively compared with the simulations: using the model parameters obtained from fitting the anisotropy profile, we recover the DF from the simulation as well as the profiles of the dispersion and kurtosis of radial and tangential velocities. Finally, we show that our DF model reproduces the power-law behaviour of phase-space density   Q =ρ( r )/σ3( r )  .  相似文献   

11.
The instability of axisymmetric flows of ideal incompressible fluid with respect to infinitesimal perturbations with the nonconservation of angular momentum is investigated by numerically integrating the differential equations of hydrodynamics. The problem has been solved for two types of rotation profiles of an unperturbed flow: with zero and nonzero pressure gradients at the flow boundaries. Both rigid and free boundary conditions have been considered. The stability of axisymmetric flows with free boundaries is of great importance in disk accretion problems. Our calculations have revealed a crucial role of the flow pattern near the boundaries in the instability of the entire main flow. When the pressure gradient at the boundaries is zero, there is such a limiting scale of perturbations in azimuthal coordinate that longer-wavelength perturbations grow, while growing shorter-wavelength perturbations do not exit. In addition, for a fixed radial flow extent, there exists a nonzero minimum amplitude of the deviation of the angular velocity from the Keplerian one at which the instability vanishes. For a nonzero pressure gradient at the boundaries, the flow is unstable with respect to perturbations of any scale and at any small deviation of the angular velocity from the Keplerian one.  相似文献   

12.
A complete solution is given for a symmetric case of the problem of the planar central configurations of four bodies, when two bodies are on an axis of symmetry, and the other two bodies have equal masses and are situated symmetrically with respect to the axis of symmetry. The positions of the bodies on the axis of symmetry are described by angle coordinates with respect to the outside bodies. The solution is such, that giving the angle coordinates, the masses for which the given configuration is a central configuration, can be computed from simple analytical expressions of the angles. The central configurations can be described as one-parameter families, and these are discussed in detail in one convex and two concave cases. The derived formulae represent exact analytical solutions of the four-body problem.  相似文献   

13.
A magnetohydrodynamical theorem can be deduced for steady axisymmetric magnetized systems in situations which are usually accepted as representative in interstellar plasmas. It shows that for particular conditions detailed below, magnetic fields have either the direction of the symmetry axis of the system or have field lines encircling this axis of symmetry, but intermediate directions are excluded. Ring galaxies are probably of the second type, which would be important in explaining ring patterns. Our galaxy itself would be of this second type slightly separated from steady state, and the magnetic field radial distribution compatible with the gas content is estimated. The implication of this theorem in actual galaxies is difficult to analyze and gives rise to rather speculative discussions.  相似文献   

14.
Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.  相似文献   

15.
The problem triaxial satellite having a plane of dynamical symmetry in the restricted problem of three bodies has been studied. The first integrals are established and the general solution of the problem can be written in quadratures. The results show that the semi-major axis of the satellite orbit and its rotational angular momentum remain unchanged. The singular solution of this problem has been considered and the elements of satellite orbit can be determined.  相似文献   

16.
We investigate the stability of stationary integral solutions of an ideal irrotational fluid in a general static and spherically symmetric background, by studying the profile of the perturbation of the mass accretion rate. We consider low angular momentum axisymmetric accretion flows for three different accretion disk models and consider time dependent and radial linear perturbation of the mass accretion rate. First we show that the propagation of such perturbation can be determined by an effective 2 × 2 matrix, which has qualitatively similar acoustic causal properties as one obtains via the perturbation of the velocity potential. Next, using this matrix we analytically address the stability issues, for both standing and travelling wave configurations generated by the perturbation. Finally, based on this general formalism we briefly discuss the explicit example of the Schwarzschild spacetime and compare our results of stability with the existing literature, which instead address this problem via the perturbation of the velocity potential.  相似文献   

17.
We present a method for recovering the distribution functions of edge-on thin axisymmetric discs directly from their observable kinematic properties. The most generally observable properties of such a stellar system are the line-of-sight velocity distributions of the stars at different projected radii along the galaxy. If the gravitational potential is known, then the general two-integral distribution function can be reconstructed using the shapes of the high-velocity tails of these line-of-sight distributions. If the wrong gravitational potential is adopted, then a distribution function can still be constructed using this technique, but the low-velocity parts of the observed velocity distributions will not be reproduced by the derived dynamical model. Thus, the gravitational potential is also tightly constrained by the observed kinematics.  相似文献   

18.
夏一飞  萧耐园 《天文学报》2000,41(3):300-305
讨论了非刚体地球受迫章动奥波策项与简正模表达式中倾斜模的关系。结果表明天球历书极章动中倾斜振项对应于角动量极的章动,在球历书极章动与角动量极的章动奥波策项之和。同时还给出了岁差速率与自转极的章动奥波策项间的数学关系。  相似文献   

19.
We introduce a general mathematical framework to model the internal transport of angular momentum in a star hosting a close-in planetary/stellar companion. By assuming that the tidal and rotational distortions are small and that the deposit/extraction of angular momentum induced by stellar winds and tidal torques are redistributed solely by an effective eddy-viscosity that depends on the radial coordinate, we can formulate the model in a completely analytic way. It allows us to compute simultaneously the evolution of the orbit of the companion and of the spin and the radial differential rotation of the star. An illustrative application to the case of an F-type main-sequence star hosting a hot Jupiter is presented. The general relevance of our model to test more sophisticated numerical dynamical models and to study the internal rotation profile of exoplanet hosts, submitted to the combined effects of tides and stellar winds, by means of asteroseismology are discussed.  相似文献   

20.
We created a self-consistent triaxial stellar system through the cold disipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. The resulting system rotates slowly even though its total angular momentum is zero, i.e., it offers an example of figure rotation. The potential of the system was subsequently approximated with interpolating formulae yielding a smooth potential stationary in the rotating frame. The Lyapunov exponents could then be computed for a randomly selected sample of 3,472 of the bodies that make up the system, allowing the recognition of regular and partially and fully chaotic orbits. The regular orbits were Fourier analyzed and classified using their locations on the frequency map. A comparison with a similar non-rotating model showed that the fraction of chaotic orbits is slightly but significantly enhanced in the rotating model; alternatively, there are no significant differences between the corresponding fractions neither of partially and fully chaotic orbits nor of long axis tubes, short axis tubes, boxes and boxlets among the regular orbits. This is a reasonable result because the rotation causes a breaking of the symmetry that may increase chaotic effects, but the rotation velocity is probably too small to produce any other significant differences. The increase in the fraction of chaotic orbits in the rotating system seems to be due mainly to the effect of the Coriolis force, rather than the centrifugal force, in good agreement with the results of other investigations.  相似文献   

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