Long-lived transient structure in collisionless self-gravitating systems

The evolution of self-gravitating systems, and long-range interacting systems more generally, from initial configurations far from dynamical equilibrium is often described as a simple two-phase process: a first phase of violent relaxation bringing it to a quasistationary state in a few dynamical times, followed by a slow adiabatic evolution driven by collisional processes. In this context the complex spatial structure evident, for example, in spiral galaxies is understood either in terms of instabilities of quasistationary states or as a result of dissipative nongravitational interactions. We illustrate here, using numerical simulations, that purely self-gravitating systems evolving from quite simple initial configurations can in fact give rise easily to structures of this kind, of which the lifetime can be large compared to the dynamical characteristic time but short compared to the collisional relaxation timescale. More specifically, for a broad range of nonspherical and nonuniform rotating initial conditions, gravitational relaxation gives rise quite generically to long-lived nonstationary structures of a rich variety, characterized by spiral-like arms, bars, and even ringlike structures in special cases. These structures are a feature of the intrinsically out-of-equilibrium nature of the system's collapse, associated with a part of the system's mass while the bulk is well virialized. They are characterized by predominantly radial motions in their outermost parts, but also incorporate an extended flattened region which rotates coherently about a well-virialized core of triaxial shape with an approximately isotropic velocity dispersion. We characterize the kinematical and dynamical properties of these complex velocity fields and we briefly discuss the possible relevance of these simple toy models to the observed structure of real galaxies, emphasizing the difference between dissipative and dissipationless disk formation.

Figure 1

(a) Gravitational radius as a function of time for different simulations. (b) Global virial ratio.

Figure 2

Energy distribution at $t=0$ and $t=30$ for simulations (a) A1, (b) B1, and (c) C1 at different times (see the legend).

Figure 3

Density profile for simulations (a) A1, (b) B1, and (c) C1 at different times (see the legend). The solid line corresponds to a decrease proportional to $r^{-4}$.

Figure 4

Density map of several snapshots of the different simulations at the indicated times. The rows, from top to bottom, are for A1, A2, B, B1a, B2, C1, and C2. The columns, from left to right, are for $t=0$, $t=2{\tau }_{\max }$, $t=5{\tau }_{\max }$, $t=10{\tau }_{\max }$, and $t=20{\tau }_{\max }$, where ${\tau }_{\max }$ is the time, determined approximately in each numerical simulation, at which the gravitational potential reaches its maximum value, which corresponds to an estimate of when the system reaches its greatest contraction. The color code is logarithmic in the density. The particle positions are projected on the plane orthogonal to the axis of initial rotation and the direction of this initial rotation is counterclockwise in these plots.

Figure 5

Energy profile for simulations (a) A1, (b) B1, and (c) C1 at different times (see the legend).

Figure 6

Velocity field for run A1. (a) Particles' velocity and its components averaged in radial shells, as a function of radius, for two different times (symbols denote $t_1=30$ and lines $t_2=60$). (b) Functions (i) $\beta \left(r\right)$ and (ii) $\zeta \left(r\right)$ (see definitions in the legend) at two different indicated times.

Figure 7

Same as in Fig. 6 but for run B1.

Figure 8

Same as in Fig. 6 but for run C1.

Figure 9

Spatial evolution of B1 (coarse grained on a grid) on the $x\text{−}y$ plane of the particles that form the bar or arm structures at $t=25$. The times are (a) $t=10$, (b) $t=5$, (c) $t=20$, and (d) $t=5$. We have plotted the coarse-grained distribution (see the text) with the corresponding velocity vector.

Figure 10

(a) Close-up of the central part of simulation B2 (randomly sampled). (b) Evolution of the density profile of B2. The position of the expanding ring at different times (see the legend) is indicated by an arrow.

Figure 11

Shape parameters for simulation A1: (a) internal particles and (b) external particles.