Dynamical theory of collisionless relaxation

In his theory of violent relaxation, Lynden-Bell gave a rigorous derivation of the equilibrium distribution, but only a qualitative discussion of the manner in which equilibrium is attained Here we present a fully explicit dynamical theory of collisionless relaxation towards Lynden-Bel equilibrium.The analysis proceeds from the coarse-graining in phase space of the collisionless Boltzmann equation the mesh size being determined by the precision of the observational data. The theoretical developmen leads to a kinetic equation generalizing that obtained by Kadomtsev and Pogutse in the rather differen context of homogeneous plasma turbulence. The collision integral differs from the classical Fokker Planck type essentially by the appearance of products of three distribution functions. It drives th systems towards the Lynden-Bell equilibrium state, on a time-scale which is inversely proportional to th coarse-graining mesh and, in the non-degenerate limit, to the fine-grained phase density. Owing to th various approximations introduced, the theory does not, however, describe the violent relaxation proces itself, but rather its late quiescent phases.     

A numerical comparison of theories of violent relaxation

Using N -body simulations with a large set of massless test particles, we compare the predictions of two theories of violent relaxation, the well-known Lynden-Bell theory and the more recent theory by Nakamura. We derive 'weakened' versions of both the theories in which we use the whole equilibrium coarse-grained distribution function     as a constraint instead of the total energy constraint. We use these weakened theories to construct expressions for the conditional probability   K_i (τ)  that a test particle initially at the phase-space coordinate τ would end-up in the i th macro-cell at equilibrium. We show that the logarithm of the ratio   R_ij (τ) ≡ K_i (τ)/ K_j (τ)  is directly proportional to the initial phase-space density   f ₀(τ)  for the Lynden-Bell theory and inversely proportional to   f ₀(τ)  for the Nakamura theory. We then measure   R_ij (τ)  using a set of N -body simulations of a system undergoing a gravitational collapse to check the validity of the two theories of violent relaxation. We find that both the theories are at odds with the numerical results, both qualitatively and quantitatively.     

Scaling up tides in numerical models of galaxy and halo formation

The purpose of this article is to show that when dynamically cold, dissipationless self-gravitating systems collapse, their evolution is a strong function of the symmetry in the initial distribution. We explore with a set of pressureless homogeneous fluids the time evolution of ellipsoidal distributions and map the depth of potential achieved during relaxation as function of initial ellipsoid axis ratios. We then perform a series of N -body numerical simulations and contrast their evolution with the fluid solutions. We verify an analytic relation between collapse factor and particle number N in spherical symmetry, such that  ∝ N ^1/3  . We sought a similar relation for axisymmetric configurations, and found an empirical scaling relation such that  ∝ N ^1/6  in these cases. We then show that when mass distributions do not respect spherical or axial symmetry, the ensuing gravitational collapse deepens with increasing particle number N but only slowly: 86 per cent of triaxial configurations may collapse by a factor of no more than 40 as   N →∞  . For   N ≈10⁵  and larger, violent relaxation develops fully under the Lin–Mestel–Shu instability such that numerical N -body solutions now resolve the different initial morphologies adequately.     

Gravitational energy of spherically-symmetric static systems

Two definitions of gravitational energy, Einstein's pseudo-tensor expression of energy in curved space-time and Lynden-Bell and Katz's (LBK) definition, are shown to give equivalent results, as applied to spherically-symetric static systems in the presence of arbitary sources. The conjectured expression for energy density of field plus matter for general static space times, given by Katz, Lynden-Bell, and Israel (KLBI), is applied to a system with internal and external Schwarzschild metric, and also to a vacuum dominated space-time with the De Sitter metric. The physical contents of the KLBI-expression is discussed by analyzing the Newtonian limit in a space filled with matter.     

A global model for violent relaxation

A global model for violent relaxation in dynamical systems is developed on basis of the virial equation and a second equation which describes the approach to equilibrium of the potential energy. It is shown under very general arguments, that the central region of the system must shed mass to an extended envelope in the collapse episode. Specific upper and lower limits for the mass in the core can be calculated.Comparison with numerical experiments agrees with the predictions of the model in a satisfactory fashion.     

Degenerate equilibrium states of collisionless stellar systems

We study spherically symmetrical equilibrium states of collisionless stellar systems confined to a spherical box. These equilibrium states correspond to the statistics introduced by Lynden-Bell in his theory of 'violent relaxation', and are described by a Fermi–Dirac distribution function. We compute the corresponding equilibrium diagram and show that a global entropy maximum exists for any accessible control parameter. This equilibrium state shows a pronounced separation between a degenerate core and a halo. We therefore check that degeneracy is able to stop the gravitational collapse (of a collisionless system), and we propose a simple model for the 'core–halo' structure. We also discuss the relevance of our study for real galaxies or other astrophysical systems such as massive neutrinos.     

Numerical experiments on Lynden-Bell's statistics

We performed computer experiments on 13 different initial configurations of one-dimensional self-gravitating systems. The three most and the three least violently relaxed systems were compared with the predictions of Lynden-Bell's statistical mechanics. The agreement between the experimental results and the theoretical predictions became worse as the relaxation became more violent. While all six systems were theoretically nondegenerate, the violent systems invariably flung out a halo that took most of the energy, leaving behind a low-energy degenerate core.     

Integrated-light VRI imaging photometry of globular clusters in the Magellanic Clouds

We compare the conclusions reached using the coarse-graining technique employed by Henriksen for a one degree of freedom (per particle) collisionless system to those presented in a paper by Binney based on an exact one degree of freedom model. We find agreement in detail, but in addition we show that the isolated 1D system is self-similar and therefore unrelaxed. Fine graining of this system recovers much less prominent wave-like structure than in a spherically symmetric isotropic 3D system. The rate of central flattening is also reduced in the 1D system. We take this to be evidence that relaxation of collisionless systems proceeds ultimately by way of short wavelength Landau damping. N -body systems, both real and simulated, can be trapped in an incompletely relaxed state because of a break in the cascade of energy to small scales. This may be due to the rapid dissipation of the small-scale oscillations in an isolated system to the existence of conserved quantities such as angular momentum, or to the failure in simulations to resolve sub-Jeans length scales. Such a partially relaxed state appears to be the Navarro, Frenk and White (NFW) state and is to be expected especially in young systems. The NFW core is shown to be isolated. In non-isolated systems, continuing coarse-grained relaxation should be towards a density core in solid body rotation.     

Simulations of dense planetary rings: IV. Spinning self-gravitating particles with size distribution

Previous self-gravitating simulations of dense planetary rings are extended to include particle spins. Both identical particles as well as systems with a modest range of particle sizes are examined. For a ring of identical particles, we find that mutual impact velocity is always close to the escape velocity of the particles, even if the total rms velocity dispersion of the system is much larger, due to collective motions associated to wakes induced by near-gravitational instability or by viscous overstability. As a result, the spin velocity (i.e., the product of the particle radius and the spin frequency) maintained by mutual impacts is also of the order of the escape velocity, provided that friction is significant. For the size distribution case, smaller particles have larger impact velocities and thus larger spin velocities, particularly in optically thick rings, since small particles move rather freely between wakes. Nevertheless, the maximum ratio of spin velocities between the smallest and largest particles, as well as the ratio for translational velocities, stays below about 5 regardless of the width of the size distribution. Particle spin state is one of the important factors affecting the temperature difference between the lit and unlit face of Saturn's rings. Our results suggest that, to good accuracy, the spin frequency is inversely proportional to the particle size. Therefore, the mixing ratio of fast rotators to slow rotators on the scale of the thermal relaxation time increases with the width of the particle size distribution. This will offer means to constrain the particle size distribution with the systematic thermal infrared observations carried by the Cassini probe.     

Dynamical relaxation and the orbits of low-mass extrasolar planets

We consider the evolution of a system containing a population of massive planets formed rapidly through a fragmentation process occurring on a scale on the order of 100 au and a lower mass planet that assembles in a disc on a much longer time-scale. During the formation phase, the inner planet is kept on a circular orbit owing to tidal interaction with the disc, while the outer planets undergo dynamical relaxation. Interaction with the massive planets left in the system after the inner planet forms may increase the eccentricity of the inner orbit to high values, producing systems similar to those observed.     

The flare process in a topologically non-equilibrium magnetohydrodynamical system

The existing models for solar flares fail to treat in an appropriate manner the energy release mechanisms based on a step-like transformation of the magnetic field energy from magnetic field to energy of the hydrodynamical motion of the medium and dissipation of these motions through shock waves into heat.In considering an example of the relaxation process in a topologically non-equilibrium magnetohydrodynamical system resulting from the merging of two magnetic loops that possess balanced longitudinal currents, this paper suggests one such energy release mechanism. Due to a certain degree of universality for different topologically non-equilibrium systems, a variety of characteristics of the relaxation process obtained may form the basis for constructing a model of solar flares based on a step-like transformation of the magnetic field energy in topologically non-equilibrium magnetohydrodynamical systems.     

Exact density–potential pairs from the holomorphic Coulomb field

We show how the complex-shift method developed by Appell to study the gravitational field of a point mass (and used in electrodynamics by, among others, Newman, Carter, Lynden-Bell, and Kaiser to determine some remarkable properties of the electromagnetic field of rotating charged configurations) can be extended to obtain new and explicit density–potential pairs for self-gravitating systems departing significantly from spherical symmetry. The rotational properties of two axisymmetric baroclinic gaseous configurations derived with the proposed method are illustrated.     

The chemical composition of the halo and evolutionary problems

The chemical composition of the Milky-Way halo is studied on the basis of a sample of metal-poor (logarithmic metallicity less than –1) globular clusters. The histogram obtained may be interpreted in the terms of the classical galaxy formation theory of Eggen, Lynden-Bell and Sandage. Interestingly enough, this is in a rough agreement with the data on metallicity of high-redshift damped Ly-alpha and Ly-alpha forest systems. This may serve the important purpose of constraining the nature and the formation timescale of MACHOs discovered through gravitational microlensing experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linked twist maps in hamiltonian systems

Linked twist maps of ergodic theory are modified to provide a model for chaotic orbits in ‘double bowl’ potentials in conservative systems, e.g. direct ‘exchange’ orbits in the planar restricted problem of three bodies. The maps are typically ergodic (and mixing), and provide a clue to the nature of relaxation or diffusion in chaotic Hamiltonian systems.     

Galaxy secular mass flow rate determination using the potential-density phase shift approach: Application to six nearby spiral galaxies

Using the potential-density phase shift approach developed by the present authors in earlier publications, we estimate the magnitude of radial mass accretion/excretion rates across the disks of six nearby spiral galaxies (NGC 628, NGC 3351, NGC 3627, NGC 4321, NGC 4736, and NGC 5194) having a range of Hubble types. Our goal is to examine these rates in the context of bulge building and secular morphological evolution along the Hubble sequence. Stellar surface density maps of the sample galaxies are derived from SINGS 3.6 μm and SDSS i-band images using colors as an indicator of mass-to-light ratios. Corresponding molecular and atomic gas surface densities are derived from published CO (1-0) and HI interferometric observations of the BIMA SONG, THINGS, and VIVA surveys. The mass flow rate calculations utilize a volume-type torque integral to calculate the angular momentum exchange rate between the basic state disk matter and what we assume to be density wave modes in the observed galaxies. This volume-type integral contains the contributions from both the gravitational surface torque couple and the advective surface torque couple at the nonlinear, quasi-steady state of the wave modes, in sharp contrast to its behavior in the linear regime, where it contains only the contribution from the gravitational surface torque couple used by Lynden-Bell & Kalnajs in 1972. The potential-density phase shift approach yields angular momentum transport rates several times higher than those estimated using the Lynden-Bell and Kalnajs approach. And unlike Lynden-Bell and Kalnajs, whose approach predicts zero mass redistribution across the majority of the disk surface (apart from the isolated locations of wave-particle resonances) for quasi-steady waves, the current approach leads to predictions of significant mass redistribution induced by the quasi-steady density wave modes, enough for the morphological types of disks to evolve substantially within its lifetime. This difference with the earlier conclusions of Lynden-Bell and Kalnajs reflects the dominant role played by collisionless shocks in the secular evolution of galaxies containing extremely non-linear, quasi-steady density wave modes, thus enabling significant morphological transformation along the Hubble sequence during a Hubble time. We show for the first time also, using observational data, that stellar mass accretion/excretion is just as important, and oftentimes much more important, than the corresponding accretion/excretion processes in the gaseous component, with the latter being what had been emphasized in most of the previous secular evolution studies.     

On the 'coarse-grained' evolution of collisionless stellar systems

We have suggested in a previous article that the coarse-grained evolution of a collisionless stellar system could be viewed as a diffusion process in velocity space compensated by an appropriate friction. Using a quasi-linear theory, we calculate the diffusion coefficient associated with this evolution. This provides a new self-consistent relaxation equation for f , the locally averaged distribution function. This equation bears some resemblance to the conventional Fokker–Planck equation of collisional systems but the friction term is non-linear in f (accounting for degeneracy effects) and the relaxation time is much smaller (in agreement with the concept of 'violent relaxation'). Under the condition that the diffusion current vanishes identically at equilibrium, we recover Lynden-Bell's distribution function; but if we allow stars to escape from the system at a constant rate, we can derive a truncated model which coincides with Lynden-Bell's solution in the core but provides a depletion of high-energy stars in the halo. This distribution function has a finite mass and is the generalization of the Michie–King model to the case of (possibly degenerate) collisionless stellar systems.     

DoH-functions always increase during violent relaxation?

Recent work on the violent relaxation of collisionless stellar systems has been based on the notion of a wide class of entropy functions. A theorem concerning entropy increase has been proved. We draw attention to some underlying assumptions that have been ignored in the applications of this theorem to stellar dynamical problems. Once these are taken into account, the use of this theorem is at best heuristic. We present a simple counter-example.     

Relaxation times in strictly disk systems

It is shown that the time of relaxation by particle encounters of self-gravitating systems in the plane interacting by 1/r ² forces is of the same order of magnitude as the mean orbit time. There-fore such a system does not have a Vlasov limit for large numbers of particles, unless appeal is made to some non-zero thickness of the disk. The relevance of this results to numerical experiments on galactic structure is discussed.     

Alfvén Wave Phase Mixing as a Source of Fast Magnetosonic Waves

The nonlinear excitation of fast magnetosonic waves by phase mixing Alfvén waves in a cold plasma with a smooth inhomogeneity of density across a uniform magnetic field is considered. If initially fast waves are absent from the system, then nonlinearity leads to their excitation by transversal gradients in the Alfvén wave. The efficiency of the nonlinear Alfvén–fast magnetosonic wave coupling is strongly increased by the inhomogeneity of the medium. The fast waves, permanently generated by Alfvén wave phase mixing, are refracted from the region with transversal gradients of the Alfvén speed. This nonlinear process suggests a mechanism of indirect plasma heating by phase mixing through the excitation of obliquely propagating fast waves.     

Fluctuations in finite-N equilibrium stellar systems

Gravitational amplification of Poisson noise in stellar systems is important on large scales. For example, it increases the dipole noise power by roughly a factor of 6 and the quadrupole noise by 50 per cent for a King model profile. The dipole noise is amplified by a factor of 15 for the core-free Hernquist model. The predictions are computed by summing over the wakes caused by each star in the system — the dressed-particle formalism of Rostoker & Rosenbluth — and are demonstrated by N -body simulation.   This result implies that a collisionless N -body simulation is impossible; the fluctuation noise which causes relaxation is an intrinic part of self-gravity. In other words, eliminating two-body scattering at interparticle scales does not eliminate relaxation altogether.   Applied to dark matter haloes of disc galaxies, particle numbers of at least 10⁶ will be necessary to suppress this noise at a level that does not dominate or significantly affect the disc response. Conversely, haloes are most likely far from phase-mixed equilibrium and the resulting noise spectrum may seed or excite observed structure such as warps, spiral arms and bars. For example, discreteness noise in the halo, similar to that caused by a population of 10⁶-M⊙ black holes, can produce observable warping and possibly excite or seed other disc structure.