Collisionless Systems Research Articles

DYNAMICS OF SELF-GRAVITY WAKES IN DENSE PLANETARY RINGS. I. PITCH ANGLE

We investigate the dynamics of self-gravity wakes in dense planetary rings. In particular, we examine how the pitch angle of self-gravity wakes depend on ring parameters using N-body simulations. We calculate the pitch angles using the two-dimensional autocorrelation function of the ring surface density. We obtain the pitch angles for the inner and outer parts of the autocorrelation function separately. We confirm that the pitch angles are 15 to 30 degrees for reasonable ring parameters, which are consistent with previous studies. We find that the inner pitch angle increases with the Saturnicentric distance, while it barely depends on the optical depth and the restitution coefficient of ring particles. The increase of the inner pitch angle with the Saturnicentric distance is consistent with the observations of the A ring. The outer pitch angle does not have the clear dependence on any ring parameters and is about 10 - 15 degrees. This value is consistent with the pitch angle of spiral arms in collisionless systems.

NON-UNIVERSALITY OF DARK-MATTER HALOS: CUSPS, CORES, AND THE CENTRAL POTENTIAL

Dark-matter halos grown in cosmological simulations appear to have central NFW-like density cusps with mean values of $d\log\rho/d\log r \approx -1$, and some dispersion, which is generally parametrized by the varying index $\alpha$ in the Einasto density profile fitting function. Non-universality in profile shapes is also seen in observed galaxy clusters and possibly dwarf galaxies. Here we show that non-universality, at any given mass scale, is an intrinsic property of DARKexp, a theoretically derived model for collisionless self-gravitating systems. We demonstrate that DARKexp - which has only one shape parameter, $\phi_0$ - fits the dispersion in profile shapes of massive simulated halos as well as observed clusters very well. DARKexp also allows for cored dark-matter profiles, such as those found for dwarf spheroidal galaxies. We provide approximate analytical relations between DARKexp $\phi_0$, Einasto $\alpha$, or the central logarithmic slope in the Dehnen-Tremaine analytical $\gamma$-models. The range in halo parameters reflects a substantial variation in the binding energies per unit mass of dark-matter halos.

The formation of the smooth halo component

AbstractThe detection and characterization of debris in the integral-of-motion space is a promising avenue to uncover the hierarchical formation of the Milky Way. Yet, the fact that the integrals do not remain constant during the assembly process adds considerable complexity to this approach. Indeed, in time-dependent potentials tidal substructures tend to be effaced from the integral-of-motion space through an orbital diffusion process, which naturally leads to the formation of a ‘smooth’ stellar halo. In this talk I will introduce a new probability theory that describes the evolution of collisionless systems subject to a time-dependent potential. The new theory can be used to reconstruct the hierarchical assembly of our Galaxy through modelling the observed distribution of accreted stars in the integral-of-motion space.

Radially anisotropic systems with forces: equilibrium states

We continue the study of collisionless systems governed by additive$r^{-{\it\alpha}}$interparticle forces by focusing on the influence of the force exponent${\it\alpha}$on radial orbital anisotropy. In this preparatory work, we construct the radially anisotropic Osipkov–Merritt phase-space distribution functions for self-consistent spherical Hernquist models with$r^{-{\it\alpha}}$forces and$1\leqslant {\it\alpha}<3$. The resulting systems are isotropic at the centre and increasingly dominated by radial orbits at radii larger than the anisotropy radius$r_{a}$. For radially anisotropic models we determine the minimum value of the anisotropy radius$r_{ac}$as a function of${\it\alpha}$for phase-space consistency (such that the phase-space distribution function is nowhere negative for$r_{a}\geqslant r_{ac}$). We find that$r_{ac}$decreases for decreasing${\it\alpha}$, and that the amount of kinetic energy that can be stored in the radial direction relative to that stored in the tangential directions for marginally consistent models increases for decreasing${\it\alpha}$. In particular, we find that isotropic systems are consistent in the explored range of${\it\alpha}$. By means of direct$N$-body simulations, we finally verify that the isotropic systems are also stable.

A probability theory for non-equilibrium gravitational systems

This paper uses dynamical invariants to describe the evolution of collisionless systems subject to time-dependent gravitational forces without resorting to maximum-entropy probabilities. We show that collisionless relaxation can be viewed as a special type of diffusion process in the integral-of-motion space. In time-varying potentials with a fixed spatial symmetry the diffusion coefficients are closely related to virial quantities, such as the specific moment of inertia, the virial factor and the mean kinetic and potential energy of microcanonical particle ensembles. The non-equilibrium distribution function is found by convolving the initial distribution function with the Green function that solves Einstein's equation for freely diffusing particles. Such a convolution also yields a natural solution to the Fokker–Planck equations in the energy space. Our mathematical formalism can be generalized to potentials with a time-varying symmetry, where diffusion extends over multiple dimensions of the integral-of-motion space. The new probability theory is in many ways analogous to stochastic calculus, with two significant differences: (i) the equations of motion that govern the trajectories of particles are fully deterministic, and (ii) the diffusion coefficients can be derived self-consistently from microcanonical phase-space averages without relying on ergodicity assumptions. For illustration we follow the cold collapse of N-body models in a time-dependent logarithmic potential. Comparison between the analytical and numerical results shows excellent agreement in regions where the potential evolution does not depart too strongly from the adiabatic regime.

The redistribution of matter in the cores of galaxy clusters

We present cosmological N-body resimulations of the assembly of the Brightest Cluster Galaxies (BCGs) in rich clusters. At z = 2 we populate dark matter subhalos with self-gravitating stellar systems whose abundance and structure match observed high-redshift galaxies. By z = 0, mergers have built much larger galaxies at cluster centre. Their dark matter density profiles are shallower than in corresponding darkmatter-only simulations, but their total mass density profiles (stars + dark matter) are quite similar. Differences are found only at radii where the effects of central black holes may be significant. Dark matter density slopes shallower than = 1.0 occur for r/r200 < 0.015, close to the half-light radii of the BCGs. Our experiments support earlier suggestions that NFW-like profiles are an attractor for the hierarchical growth of structure in collisionless systems – total mass density profiles asymptote to the solution found in dark-matter-only simulations over the radial range where mergers produce significant mixing between stars and dark matter. Simulated dark matter fractions are substantially higher in BCGs than in field ellipticals, reaching 80% within the half-light radius. We also estimate that supermassive black hole mergers should create BCG cores as large as rc � 3kpc. The good agreement of all these properties with recent observational studies of BCG structure suggests that dissipational processes have not played a dominant role in the assembly of the observed systems.

Secular resonant dressed orbital diffusion – II. Application to an isolated self-similar tepid galactic disc

The main orbital signatures of the secular evolution of an isolated self-gravitating stellar Mestel disc are recovered using a dressed Fokker-Planck formalism in angle-action variables. The shot-noise-driven formation of narrow ridges of resonant orbits is recovered in the WKB limit of tightly wound transient spirals, for a tepid Toomre-stable tapered disc. The relative effect of the bulge, the halo, the disc temperature and the spectral properties of the shot noise are investigated in turn. For such galactic discs all elements seem to impact the locus and direction of the ridge. For instance, when the halo mass is decreased, we observe a transition between a regime of heating in the inner regions of the disc through the inner Lindblad resonance to a regime of radial migration of quasi-circular orbits via the corotation resonance in the outer part of the disc. The dressed secular formalism captures both the nature of collisionless systems (via their natural frequencies and susceptibility), and their nurture via the structure of the external perturbing power spectrum. Hence it provides the ideal framework in which to study their long term evolution.

Clarifying the Role of Thermodynamics in Self-gravitating Dark Matter Systems

Astrophysicists currently favor the idea that the vast majority of mass in the uni- verse exists in a collisionless form that interacts only through gravity. This paradigm has been successful in allowing computer simulations of cosmological volumes to re- produce structures similar to those observed. Our work aims to better understand the physics at work in collisionless systems, in general. Specifically, we investigate how well thermodynamics-based approaches to understanding equilibrium structures agree with results of computer simulations. Using a suite of N -body simulations with differ- ing initial conditions, we examine the density and velocity profiles of self-gravitating collisionless systems. Statistical comparisons between simulated equilibrium struc- tures and two thermodynamics-based models indicate the relative appropriateness of the models. We find that no single model can describe systems resulting from the entire range of initial conditions investigated here. Our major result is that these thermodynamics-based models can successfully reproduce equilibria that arise in gen- tly evolving systems. However, the failure of thermodynamics-based models to de- scribe the equilibria of more violent evolutions suggests that they must retain some memory of their initial conditions.

Vlasov simulations of kinetic Alfvén waves at proton kinetic scales

Kinetic Alfvén waves represent an important subject in space plasma physics, since they are thought to play a crucial role in the development of the turbulent energy cascade in the solar wind plasma at short wavelengths (of the order of the proton gyro radius ρp and/or inertial length dp and beyond). A full understanding of the physical mechanisms which govern the kinetic plasma dynamics at these scales can provide important clues on the problem of the turbulent dissipation and heating in collisionless systems. In this paper, hybrid Vlasov-Maxwell simulations are employed to analyze in detail the features of the kinetic Alfvén waves at proton kinetic scales, in typical conditions of the solar wind environment (proton plasma beta βp = 1). In particular, linear and nonlinear regimes of propagation of these fluctuations have been investigated in a single-wave situation, focusing on the physical processes of collisionless Landau damping and wave-particle resonant interaction. Interestingly, since for wavelengths close to dp and βp ≃ 1 (for which ρp ≃ dp) the kinetic Alfvén waves have small phase speed compared to the proton thermal velocity, wave-particle interaction processes produce significant deformations in the core of the particle velocity distribution, appearing as phase space vortices and resulting in flat-top velocity profiles. Moreover, as the Eulerian hybrid Vlasov-Maxwell algorithm allows for a clean almost noise-free description of the velocity space, three-dimensional plots of the proton velocity distribution help to emphasize how the plasma departs from the Maxwellian configuration of thermodynamic equilibrium due to nonlinear kinetic effects.

The effects of a non-zero cosmological constant on the Veltmann models

The Veltmann models, which include the Plummer and Hernquist models as special cases, are studied in the presence of a cosmological constant. Physically relevant quantities such as the velocity dispersion profiles and the anisotropy parameter are computed through the use of the self-consistent approach. The cutoff radii for these models and the mass contained within this volume are also calculated. It is shown that the inclusion of a cosmological constant leads to many observable quantities such as the surface density, dispersion profiles and the anisotropy parameter becoming increasingly modified. In some scenarios, they are easily distinguished from the case where the cosmological constant is absent, as a result of their non-monotonic behaviour. The effects of neighbouring gravitational systems on the central system are also studied, and compared against the effects arising from the cosmological constant. Consequently, it is suggested that the effects of a cosmological constant can prove to be quite important when modelling dilute collisionless systems.

Jeans analysis in modified gravity

MOdified Gravity (MOG) is a covariant modification of Einstein's general relativity. This theory is one of the current alternatives to dark matter models. We describe dynamics of collisionless self-gravitating systems in the context of MOG. By studying the weak field approximation of this theory, we derive the equations governing the dynamics of the self-gravitating systems. More specifically, we consider the Jeans instability for self-gravitating fluid and stellar systems, and derive new Jeans mass limit $\tilde{M}_J$ and wave-number $\tilde{k}_J$. Furthermore, considering the gravitational instability in star forming regions, we show that MOG has not a significant difference with general relativity on this astrophysical scale. However, at larger scales such as intergalactic space MOG may lead to different galaxy and structure formation processes.

Energy conversion and magnetic reconnection in space plasmas: Role of nonlinear kinetic processes and structures

Dynamics of plasma systems in space involves processes of large-scale energy conversion. Like in conventional gas dynamics, the conversion can occur on shocks. However, in collisionless magnetized plasma systems, quite different nonlinear structures may be responsible for energy conversion. Those are anisotropic kinetic current sheets. It is demonstrated that observations at the Earth’s magnetopause provide evidence of long-term existence of such structures.

Stirring N-body systems – II. Necessary conditions for the dark matter attractor

We study the evolution of the phase-space of collisionless N-body systems under repeated stirrings or perturbations, which has been shown to lead to a convergence towards a limited group of end states. This so-called attractor was previously shown to be independent of the initial system and environmental conditions. However the fundamental reason for its appearance is still unclear. It has been suggested that the origin of the attractor may be either radial infall (RI), the radial orbit instability (ROI), or energy exchange which, for instance, happens during violent relaxation. Here we examine the effects of a set of controlled perturbations, referred to as `kicks', which act in addition to the standard collisionless dynamics by allowing pre-specified instantaneous perturbations in phase-space. We first demonstrate that the attractor persists in the absence of RI and ROI by forcing the system to expand. We then consider radial velocity kicks in a rigid potential and isotropic velocity kicks, since there are no energy exchanges in these two recipes of kicks. We find that these kicks do not lead to the attractor, indicating that the energy exchange in a dynamic potential is indeed the physical mechanism responsible for the attractor.

Transition from gas to plasma kinetic equilibria in gravitating axisymmetric structures

The problem of the transition from gas to plasma in gravitating axisymmetric structures is addressed under the assumption of having initial and final states realized by kinetic Maxwellian-like equilibria. In astrophysics, the theory applies to accretion-disc scenarios around compact objects. A formulation based on non-relativistic kinetic theory for collisionless systems is adopted. Equilibrium solutions for the kinetic distribution functions describing the initial neutral matter and the resulting plasma state are constructed in terms of single-particle invariants and expressed by generalized Maxwellian distributions. The final plasma configuration is related to the initial gas distribution by the introduction of appropriate functional constraints. Qualitative aspects of the solution are investigated and physical properties of the system are pointed out. In particular, the admitted functional dependences of the fluid fields carried by the corresponding equilibrium distributions are determined. Then, the plasma is proved to violate the condition of quasi-neutrality, implying a net charge separation between ions and electrons. This result is shown to be independent of the precise realization of the plasma distribution function, while a physical mechanism able to support a non-neutral equilibrium state is proposed.

STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. IV. DISTRIBUTION OF ANGULAR MOMENTUM

It has been shown in previous work that DARKexp, which is a theoretically derived, maximum entropy, one shape parameter model for isotropic collisionless systems, provides very good fits to simulated and observed dark-matter halos. Specifically, it fits the energy distribution, N(E), and the density profiles, including the central cusp. Here, we extend DARKexp N(E) to include the distribution in angular momentum, L^2, for spherically symmetric systems. First, we argue, based on theoretical, semi-analytical, and simulation results, that while dark-matter halos are relaxed in energy, they are not nearly as relaxed in angular momentum, which precludes using maximum entropy to uniquely derive N(E,L^2). Instead, we require that when integrating N(E,L^2) over squared angular momenta one retrieves the DARKexp N(E). Starting with a general expression for N(E,L^2) we show how the distribution of particles in L^2 is related to the shape of the velocity distribution function, VDF, and velocity anisotropy profile, \beta(r). We then demonstrate that astrophysically realistic halos, as judged by the VDF shape and \beta(r), must have linear or convex distributions in L^2, for each separate energy bin. The distribution in energy of the most bound particles must be nearly flat, and become more tilted in favor of radial orbits for less bound particles. These results are consistent with numerical simulations and represent an important step towards deriving the full distribution function for spherically symmetric dark-matter halos.

Oscillations and stability of polytropic filaments

We study the oscillations and stability of self-gravitating cylindrically symmetric fluid systems and collisionless systems. This is done by studying small perturbations to the equilibrium system and finding the normal modes, using methods similar to those used in astroseismology. We find that there is a single sequence of purely radial modes that become unstable if the adiabatic exponent is less than 1. Nonradial modes can be divided into p modes, which are stable and pressure-driven, and g modes, which are are gravity driven. The g modes become unstable if the adiabatic exponent is greater than the polytrope index. These modes are analogous to the modes of a spherical star, but their behavior is somewhat different because a cylindrical geometry has less symmetry than a spherical geometry. This implies that perturbations are classified by a radial quantum number, an azimuthal quantum number, and wavelength in the z direction, which can become arbitrarily large. We find that decreasing this wavelength increases the frequency of stable modes and increases the growth rate of unstable modes. We use use variational arguments to demonstrate that filaments of collisionless matter with ergodic distribution functions are stable to purely radial perturbations, and that filaments with ergodic power-law distribution functions are stable to all perturbations.

KINETIC THEORY FOR GRAVITATING SYSTEMS OF COLLISIONLESS NEUTRAL MATTER

The study of the statistical properties of collisionless systems formed by neutral particles subject to gravitational field represents an intriguing theoretical issue. In astrophysics, it directly relates to the description of collisionless gravitating dark matter (DM) halos. Structures of this type are expected to be characterized by intrinsically non-Maxwellian kinetic distribution functions (KDFs) and to exhibit temperature anisotropy, i.e. an anisotropy in the directional particle velocity dispersions. In this paper, a theoretical analysis of the issue is proposed, based on the kinetic theory developed in the framework of the Vlasov–Poisson description for nonrelativistic DM systems at equilibrium. By implementing the method of invariants, explicit solutions for the equilibrium KDFs are constructed and expressed through generalized Gaussian distributions. A perturbative theory is developed which allows them to be cast in terms of Chapman–Enskog series representations and to evaluate analytically the corresponding fluid fields. The conditions for the occurrence of temperature anisotropy are investigated for different physical and geometrical configurations. It is shown that this feature can arise at equilibrium due to specifically-kinetic effects associated with phase-space conservation laws in the presence of a nonuniform gravitational field.

Radially anisotropic models of collisionless spherical stellar systems without central singularity

A new class of the simplest equilibrium two-parameter distribution functions for spherical stellar systems with a radially anisotropic stellar velocity distribution is investigated. The models under consideration are a less singular counterpart of the so-called generalized polytropes, which in the past were among the most popular models in works on the equilibrium and stability of gravitating systems. In contrast to the well-known generalized polytropes, the proposed models have finite density and potential at the center. The absence of a singularity is necessary for a proper consideration of the radial orbit instability, which is the most important instability of spherical stellar systems. The main observed parameters of the proposed models (potential, density, anisotropy) are compared with those in well-known equilibrium models.

Excitation of poloidal standing Alfvén waves through drift resonance wave‐particle interaction

Drift‐resonance wave‐particle interaction is a fundamental collisionless plasma process studied extensively in theory. Using cross‐spectral analysis of electric field, magnetic field, and ion flux data from the Van Allen Probe (Radiation Belt Storm Probes) spacecraft, we present direct evidence identifying the generation of a fundamental mode standing poloidal wave through drift‐resonance interactions in the inner magnetosphere. Intense azimuthal electric field (Eφ) oscillations as large as 10mV/m are observed, associated with radial magnetic field (Br) oscillations in the dawn‐noon sector near but south of the magnetic equator at L∼5. The observed wave period, Eφ/Br ratio and the 90° phase lag between Br and Eφ are all consistent with fundamental mode standing Poloidal waves. Phase shifts between particle fluxes and wave electric fields clearly demonstrate a drift resonance with ∼90 keV ring current ions. The estimated earthward gradient of ion phase space density provides a free energy source for wave generation through the drift‐resonance instability. A similar drift‐resonance process should occur ubiquitously in collisionless plasma systems. One specific example is the “fishbone” instability in fusion plasma devices. In addition, our observations have important implications for the long‐standing mysterious origin of Giant Pulsations.

Influence of the dissipation mechanism on collisionless magnetic reconnection in symmetric and asymmetric current layers

Numerical studies implementing different versions of the collisionless Ohm's law have shown a reconnection rate insensitive to the nature of the non-ideal mechanism occurring at the X line, as soon as the Hall effect is operating. Consequently, the dissipation mechanism occurring in the vicinity of the reconnection site in collisionless systems is usually thought not to have a dynamical role beyond the violation of the frozen-in condition. The interpretation of recent studies has, however, led to the opposite conclusion that the electron scale dissipative processes play an important dynamical role in preventing an elongation of the electron layer from throttling the reconnection rate. This work re-visits this topic with a new approach. Instead of focusing on the extensively studied symmetric configuration, we aim to investigate whether the macroscopic properties of collisionless reconnection are affected by the dissipation physics in asymmetric configurations, for which the effect of the Hall physics is substantially modified. Because it includes all the physical scales a priori important for collisionless reconnection (Hall and ion kinetic physics) and also because it allows one to change the nature of the non-ideal electron scale physics, we use a (two dimensional) hybrid model. The effects of numerical, resistive, and hyper-resistive dissipation are studied. In a first part, we perform simulations of symmetric reconnection with different non-ideal electron physics. We show that the model captures the already known properties of collisionless reconnection. In a second part, we focus on an asymmetric configuration where the magnetic field strength and the density are both asymmetric. Our results show that contrary to symmetric reconnection, the asymmetric model evolution strongly depends on the nature of the mechanism which breaks the field line connectivity. The dissipation occurring at the X line plays an important role in preventing the electron current layer from elongating and forming plasmoids.