Pierfrancesco Di Cintio

Aims. We investigated the interplay between radial velocity anisotropy and internal rotation in self-gravitating systems characterised by initial spherical symmetry and a moderate density cusp. We studied the stability properties of such configurations to disentangle the impact of anisotropy, rotation, and density structure on the formation of triaxial stellar systems. Methods. We performed a set of collisionless N -body simulations starting from initial equilibria with a phase space distribution function of the class introduced by Osipkov and Merritt, modified to impose a net global angular momentum with the Lynden–Bell daemon protocol. We analysed the growth of the density and phase space distribution modes, as well as the axial ratios and triaxiality index of the final configurations. Results. We find that internal rotation has a mitigating effect on the strength of the radial orbit instability in radially anisotropic models with a Fridman–Polyachenko–Shukhman anisotropy index close to the consistency limit, while it slightly hastens the onset of the instability in marginally stable models. From the analysis of the growth rate of the density modes, we also find that an inner bar, which for maximally rotating initial conditions declines after its initial growth, is formed even in systems with initial profiles with a central density cusp.

The largest Lyapunov exponent is widely used to diagnose chaos in gravitational dynamics; however, in mixed phase spaces and finite-N systems, it does not always provide a complete description of the orbital complexity and phase-space transport. Entropy-based diagnostics could offer a complementary perspective. We investigate whether trajectory-based information entropy can provide a useful diagnostic of chaos in gravitational systems and how it relates to the largest Lyapunov exponent as a function of orbital energy and of the number of degrees of freedom. We computed the largest Lyapunov exponent and a coarse-grained Shannon entropy for ensembles of trajectories in the Hénon-Heiles potential and for test-particle orbits in live N-body realisations of a Plummer model. We compared the dependence of both quantities on the orbital energy as well as on the particle number for the N-body case in particular. In the Hénon-Heiles system, the Shannon entropy follows the transition from weak to widespread chaos and exhibits an energy dependence that closely mirrors that of the largest Lyapunov exponent. For test-particle orbits in live N-body potentials, both diagnostics indicate stronger chaos for more tightly bound trajectories. However, their dependence on N differs: the largest Lyapunov exponent remains nearly constant over the explored range of particle numbers, whereas the Shannon entropy decreases monotonically as N increases. These results show that the Shannon information entropy could serve as a complement to the largest Lyapunov exponent and could also better capture changes in global phase-space mixing, especially in systems where the leading Lyapunov exponent alone is not sufficiently informative. It provides a promising alternative for diagnosing chaos when tangent-space dynamics is unavailable, prohibitively expensive, or slowly convergent. In particular, because this entropy measure relies only on time series of phase-space coordinates, it is naturally suited to systems with densely sampled trajectories, such as minor bodies in the Solar System.

Context. Interactions between stellar-mass black holes (BHs) and the accretion disks of supermassive BHs in active galactic nuclei (AGN) constitute a promising channel for the formation of gravitational wave sources. The efficiency of this process depends critically on how embedded BHs evolve under the influence of gaseous drag. Previous studies have assumed laminar disk conditions, leading to idealized configurations with BHs on circular, coplanar orbits. However, AGN disks are expected to be turbulent, and the impact of turbulence on BH orbital evolution remains largely unexplored. Aims. We investigate how AGN disk turbulence affects the orbital dynamics of a stellar-mass BH initially located at a migration trap, focusing on the long-term behavior of eccentricity and inclination in the quasi-embedded regime. Methods. We developed a semi-analytical framework in which turbulence is modeled as a stochastic velocity field acting through a modified drag force. We integrated the resulting stochastic differential equations both in Cartesian coordinates and in orbital elements using a linearized perturbative approach and compared these results with full numerical simulations. Results. Eccentricity and inclination evolve toward steady-state Rayleigh distributions, with variances determined by the local disk properties and the ratio of the gas damping rate to the orbital frequency. The analytical predictions agree well with the numerical simulations. We provide closed-form expressions for the variances in both the fast and slow damping regimes. These results are directly applicable to Monte Carlo population models and can serve as physically motivated initial conditions for hydrodynamical simulations. Conclusions. Turbulent forcing prevents full circularization and alignment of BH orbits in AGN disks, even in the presence of strong gas drag. This has important implications for BH merger and binary formation rates, which are sensitive to the residual eccentricity and inclination. Our results highlight the need to account for turbulence-induced stochastic heating when modeling the dynamical evolution of compact objects in AGN environments.

Context. Dynamical friction is an important phenomenon in stellar dynamics (and plasma physics) resulting in the slowing down of a massive test particle upon many two-body scatters with background particles. Chandrasekhar’s original formulation, developed for idealized infinite and homogeneous systems, has been found to be sufficiently accurate even in models of finite extent and radially dependent density profiles. However, in some cases N-body simulations have evidenced a breakdown of Chandrasekhar’s formalism. In particular, in the case of cored stellar systems, the analytical predictions underestimate the rate of in-fall of the test particle. Moreover, the orbital decay appears to stop at a critical radius where in principle the effect of dynamical friction should still be non-negligible. Aims. Several explanations for such discrepancy have been proposed so far. In spite of this, it remains unclear whether the origin is a finite N effect or an effect arising from the resonance (or near-resonance) of the orbits of the test and field particles, which is independent of N, such as dynamical buoyancy. Here we aim to shed some light on this issue with tailored numerical experiments. Methods. We performed ad hoc simulations of a massive tracer initially placed on a low-eccentricity orbit in spherical equilibrium models with increasing resolution. We used an N-body code for which the self-consistent interaction among the background particles can be substituted with the effect of the static smooth potential of the system’s continuum limit, so that the higher-order contributions to the dynamical friction arising from the formation of a wake can be neglected if needed. Results. We find that, contrary to what has been reported in previous literature, a suppression of dynamical friction happens in both cuspy and cored models. When neglecting the interaction among field particles, we observe in both cases a clear N^−1/2 scaling of the radius at which dynamical friction ceases to be effective. This hints at a granularity-induced origin of the so-called core-stalling of the massive tracer in cored models.

Aims. Violent relaxation is often regarded as the mechanism that leads stellar systems to collisionless meta equilibrium via rapid changes in the collective potential. Methods. We investigate the role of chaotic instabilities on single particle orbits in leading to nearly invariant phase-space distributions, aiming at disentangling their role from that of the chaos induced by collective oscillations in the self-consistent potential. Results. We explore, as a function of the system’s size (i.e. number of particles N), the chaoticity in terms of the largest Lyapunov exponent of test trajectories in a simplified model of gravitational cold collapse, mimicking an N-body calculation via a time-dependent smooth potential and a noise-friction process accounting for the discreteness effects. A new numerical method to evaluate effective Lyapunov exponents for stochastic models is presented and tested. Conclusions. We find that the evolution of the phase-space of independent trajectories reproduces rather well what is observed in self-consistent N-body simulations of dissipationless collapses. The chaoticity of test orbits rapidly decreases with N for particles that remain weakly bounded in the model potential, while it decreases with different power laws for more bound orbits, consistently with what was observed in previous self-consistent N-body simulations. The largest Lyapunov exponents of ensembles of orbits starting from initial conditions uniformly sampling the accessible phase-space are somewhat constant for N ≲ 10⁹, while decreases towards the continuum limit with a power-law trend. Moreover, our numerical results appear to confirm the trend of a specific formulation of dynamical entropy and its relation with Lyapunov timescales.

Context. Recent numerical results seem to suggest that, in certain regimes of typical particle velocities, when the post-Newtonian (PN) force terms are included, the gravitational N-body problem (for 3 ≤ N ≲ 10³) is intrinsically less chaotic than its classical counterpart, which exhibits a slightly larger maximal Lyapunov exponent Λ_max. Aims. In this work, we explore the dynamics of wildly chaotic, regular and nearly regular configurations of the three-body problem with and without the PN corrective terms, with the aim being to shed light on the behaviour of the Lyapunov spectra under the effect of the PN corrections. Methods. Because the interaction of the tangent-space dynamics in gravitating systems – which is needed to evaluate the Lyapunov exponents – becomes rapidly computationally heavy due to the complexity of the higher-order force derivatives involving multiple powers of v/c, we introduce a technique to compute a proxy of the Lyapunov spectrum based on the time-dependent diagonalization of the inertia tensor of a cluster of trajectories in phase-space. In addition, we also compare the dynamical entropy of the classical and relativistic cases. Results. We find that, for a broad range of orbital configurations, the relativistic three-body problem has a smaller Λ_max than its classical counterpart starting with the exact same initial conditions. However, the other (positive) Lyapunov exponents can be either lower or larger than the corresponding classical ones, thus suggesting that the relativistic precession effectively reduces chaos only along one (or a few) directions in phase-space. As a general trend, the dynamical entropy of the relativistic simulations as a function of the rescaled speed of light falls below the classical value over a broad range of values. Conclusions. We observe that analyses based solely on Λ_max could lead to misleading conclusions regarding the chaoticity of systems with small (and possibly large) N.

Aims. The Fornax dwarf spheroidal galaxy (dSph) represents a challenge for some globular cluster (GC) formation models, because an exceptionally high fraction of its stellar mass is locked in its GC system. In order to shed light on our understanding of GC formation, we aim to constrain the amount of stellar mass that Fornax has lost via tidal interaction with the Milky Way (MW). Methods. Exploiting the flexibility of effective multi-component N-body simulations and relying on state-of-the-art estimates of Fornax’s orbital parameters, we study the evolution of the mass distribution of the Fornax dSph in observationally justified orbits in the gravitational potential of the MW over 12 Gyr. Results. We find that, though the dark-matter mass loss can be substantial, the fraction of stellar mass lost by Fornax to the MW is always negligible, even in the most eccentric orbit considered. Conclusions. We conclude that stellar-mass loss due to tidal stripping is not a plausible explanation for the unexpectedly high stellar mass of the GC system of the Fornax dSph and we discuss quantitatively the implications for GC formation scenarios.