Radial Orbit Instability Research Articles

New dipole instabilities in spherical stellar systems

ABSTRACT Spherical stellar systems have weakly damped response modes. The dipole modes are seiche modes. The quadrupole are zero pattern-speed prolate modes, the stable precursors to the radial orbit instability (ROI). We demonstrate that small wiggles in the distribution function (DF) can destabilize the dipole modes and describe the newly identified instabilities in NFW-like dark-matter (DM) haloes and other power-law spherical systems. The modes were identified in N-body simulations using multivariate singular spectrum analysis and corroborated using linear-response theory. The new mode peaks inside the half-mass radius but has a pattern speed typical of an outer-halo orbit. As it grows, the radial angle of the eccentric orbits that make up the mode correlates and loses angular momentum by a resonant couple to outer-halo orbits. This leads to an unsteady pattern with a density enhancement that swings from one side of the halo to another along a diameter, like the orbits that comprise the instability. In this way, the dipole mode is similar to the ROI. Since the DF found in Nature is unlikely to be smooth and isotropic with df(E)/dE < 0 necessary for Antonov stability, these modes may be ubiquitous albeit slowly growing. Haloes that are less extended than NFW, such as the Hernquist model, tend to be stable to this dipole instability. We present the critical stability exponents for one- and two-power models. These different critical outer power-law exponents illustrate that the gravitational coupling between the inner and outer DM halo depends on the global shape of density profile.

Discreteness effects, N-body chaos and the onset of radial-orbit instability

ABSTRACT We study the stability of a family of spherical equilibrium models of self-gravitating systems, the so-called γ models with Osipkov–Merritt velocity anisotropy, by means of N-body simulations. In particular, we analyse the effect of self-consistent N-body chaos on the onset of radial-orbit instability. We find that degree of chaoticity of the system associated with its largest Lyapunov exponent Λmax has no appreciable relation with the stability of the model for fixed density profile and different values of radial velocity anisotropy. However, by studying the distribution of the Lyapunov exponents λm of the individual particles in the single-particle phase space, we find that more anisotropic systems have a larger fraction of orbits with larger λm.

Noise, friction and the radial-orbit instability in anisotropic stellar systems: stochastic N–body simulations

AbstractBy means of numerical simulations we study the radial-orbit instability in anisotropic self-gravitating N–body systems under the effect of noise. We find that the presence of additive or multiplicative noise has a different effect on the onset of the instability, depending on the initial value of the orbital anisotropy.

A new class of galaxy models with a central BH – I. The spherical case

ABSTRACTThe dynamical properties of spherically symmetric galaxy models, where a Jaffe stellar density profile is embedded in a total mass density decreasing as r−3 at large radii, are presented. The orbital structure of the stellar component is described by the Osipkov–Merritt anisotropy; the dark matter halo is isotropic, and a black hole is added at the centre of the galaxy. First, the conditions for a nowhere negative and monotonically decreasing dark matter halo density profile are derived; this profile can be made asymptotically coincident with an NFW profile at the centre and large radii. Then, the minimum value of the anisotropy radius for phase-space consistency is derived as a function of the galaxy parameters. The Jeans equations for the stellar component are solved analytically; the projected velocity dispersion at the centre and large radii is also obtained, for generic values of the anisotropy radius. Finally, analytical expressions for the terms entering the Virial Theorem are derived, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, built following an approach already adopted in our previous works, can be a useful starting point for a more advanced modelling of the dynamics of elliptical galaxies, and can be easily implemented in numerical simulations requiring a realistic dynamical model of a galaxy.

The kinematics of star clusters undergoing gas expulsion in Newtonian and Milgromian dynamics

We study the kinematics of stars in clusters undergoing gas expulsion in standard Newtonian dynamics and also in Milgromian dynamics (MOND). Gas expulsion can explain the observed line-of-sight (LoS) velocity dispersion profile of NGC 2419 in Newtonian dynamics. For a given star formation efficiency (SFE), the shapes of the velocity dispersion profiles, which are normalised by the velocity dispersion at the projected half-mass radius, are almost indistinguishable for different SFE models in Newtonian dynamics. The velocity dispersion of a star cluster in the outer halo of a galaxy can indeed have a strong radial anisotropy in Newtonian dynamics after gas expulsion. MOND displays several different properties from Newtonian dynamics. In particular, the slope of the central velocity dispersion profile is less steep in MOND for the same SFE. Moreover, for a given SFE, more massive embedded cluster models result in more rapidly declining central velocity dispersion profiles for the final star clusters, while less massive embedded cluster models lead to flatter velocity dispersion profiles for the final products. The onset of the radial-orbit instability in post-gas-expulsion MOND models is discussed. SFEs as low as a few percent, typical of molecular clouds, lead to surviving ultra-diffuse objects. Gas expulsion alone is unlikely the physical mechanism for the observed velocity dispersion profile of NGC 2419 in MOND.

Linear stability of stellar rotating spheres

AbstractRecent observations of globular clusters encourage to revise some aspects of the traditional paradigm, in which they were considered to be isotropic in velocity space and non-rotating. However, the theory of collisionless spheroids with some kinematic richness has seldom been studied. We present here a further step in this direction, owing to new results regarding the linear stability of rotating Plummer spheres, with varying anisotropy in velocity space and total amount of angular momentum. We extend the well-known radial orbit instability to rotating systems, and discover a new regime of instability in fast rotating, tangentially anisotropic systems.

A linear stability study of stellar rotating spheres

AbstractRecent observations of globular clusters imposed major revisions to the previous paradigm, in which they were considered to be isotropic in velocity space and non-rotating. However, the theory of collisionless spheroids with some kinematic richness has seldom been studied. We present here a first step in this direction, owing to new results regarding the linear stability of rotating Plummer spheres, with varying anisotropy in velocity space and total amount of angular momentum. We extend the well-known radial orbit instability to rotating systems, and discover a new regime of instability in fast rotating, tangentially anisotropic systems.

The Delay Time Distribution of Tidal Disruption Flares

Recent observations suggest that stellar tidal disruption events (TDE) are strongly overrepresented in rare, post-starburst galaxies. Several dynamical mechanisms have been proposed to elevate their TDE rates, ranging from central stellar overdensities to the presence of supermassive black hole (SMBH) binaries. Another such mechanism, introduced here, is a radial velocity anisotropy in the nuclear star cluster produced during the starburst. These, and other, dynamical hypotheses can be disentangled by comparing observations to theoretical predictions for the TDE delay time distribution (DTD). We show that SMBH binaries are a less plausible solution for the post-starburst preference, as they can only reproduce the observed DTD with extensive fine-tuning. The overdensity hypothesis produces a reasonable match to the observed DTD (based on the limited data currently available), provided that the initial stellar density profile created during the starburst, $\rho(r)$, is exceptional in both steepness and normalization. In particular, explaining the post-starburst preference requires $\rho \propto r^{-\gamma}$ with $\gamma \gtrsim 2.5$, i.e. much steeper than the classic Bahcall-Wolf equilibrium profile of $\gamma = 7/4$. For "ultrasteep" density cusps ($\gamma \ge 9/4$), we show that the TDE rate decays with time measured since the starburst as $\dot{N} \propto t^{-(4\gamma-9)/(2\gamma-3)} / \ln t$. Radial anisotropies also represent a promising explanation, provided that initial anisotropy parameters of $\beta_0 \approx 0.5$ are sustainable against the radial orbit instability. TDE rates in initially anisotropic cusps will decay roughly as $\dot{N} \propto t^{-\beta_0}$. As the sample of TDEs with well-studied host galaxies grows, the DTD will become a powerful tool for constraining the exceptional dynamical properties of post-starburst galactic nuclei.

Numerical studies of instability of generalized polytropic models of stellar disks

The distribution function for generalized polytropic models has been used to construct a series of numerical models of anisotropic disks. It has been shown with the help of simulation that such systems are unstable with respect to bar formation at any degree of radial elongation of star orbits. The result is completely at variance with the conclusions of earlier works, where similar models were studied. The pattern speeds and amplitudes of the forming bar were found, and the initial distributions of orbit precession rates were calculated. Such systems have been shown to fulfill all conditions for the onset of radial orbit instability, responsible for the formation of a slow bar.

Two-component Jaffe models with a central black hole – I. The spherical case

Dynamical properties of spherically symmetric galaxy models where both the stellar and total mass density distributions are described by the Jaffe (1983) profile (with different scale-lenghts and masses), are presented. The orbital structure of the stellar component is described by Osipkov--Merritt anisotropy, and a black hole (BH) is added at the center of the galaxy; the dark matter halo is isotropic. First, the conditions required to have a nowhere negative and monothonically decreasing dark matter halo density profile, are derived. We then show that the phase-space distribution function can be recovered by using the Lambert-Euler $W$ function, while in absence of the central BH only elementary functions appears in the integrand of the inversion formula. The minimum value of the anisotropy radius for consistency is derived in terms of the galaxy parameters. The Jeans equations for the stellar component are solved analytically, and the projected velocity dispersion at the center and at large radii are also obtained analytically for generic values of the anisotropy radius. Finally, the relevant global quantities entering the Virial Theorem are computed analytically, and the fiducial anisotropy limit required to prevent the onset of Radial Orbit Instability is determined as a function of the galaxy parameters. The presented models, even though highly idealized, represent a substantial generalization of the models presentd in Ciotti et al. (2009), and can be useful as starting point for more advanced modeling the dynamics and the mass distribution of elliptical galaxies.

Particle number dependence in the non-linear evolution of N-body self-gravitating systems

Simulations of purely self-gravitating N-body systems are often used in astrophysics and cosmology to study the collisionless limit of such systems. Their results for macroscopic quantities should then converge well for sufficiently large N. Using a study of the evolution from a simple space of spherical initial conditions – including a region characterized by so-called ‘radial orbit instability’ – we illustrate that the values of N at which such convergence is obtained can vary enormously. In the family of initial conditions we study, good convergence can be obtained up to a few dynamical times with N ∼ 103 – just large enough to suppress two body relaxation – for certain initial conditions, while in other cases such convergence is not attained at this time even in our largest simulations with N ∼ 105. The qualitative difference is due to the stability properties of fluctuations introduced by the N-body discretisation, of which the initial amplitude depends on N. We discuss briefly why the crucial role which such fluctuations can potentially play in the evolution of the N body system could, in particular, constitute a serious problem in cosmological simulations of dark matter.

Radial orbit instability in systems of highly eccentric orbits: Antonov problem reviewed

Stationary stellar systems with radially elongated orbits are subject to radial orbit instability -- an important phenomenon that structures galaxies. Antonov (1973) presented a formal proof of the instability for spherical systems in the limit of purely radial orbits. However, such spheres have highly inhomogeneous density distributions with singularity $\sim 1/r^2$, resulting in an inconsistency in the proof. The proof can be refined, if one considers an orbital distribution close to purely radial, but not entirely radial, which allows to avoid the central singularity. For this purpose we employ non-singular analogs of generalised polytropes elaborated recently in our work in order to derive and solve new integral equations adopted for calculation of unstable eigenmodes in systems with nearly radial orbits. In addition, we establish a link between our and Antonov's approaches and uncover the meaning of infinite entities in the purely radial case. Maximum growth rates tend to infinity as the system becomes more and more radially anisotropic. The instability takes place both for even and odd spherical harmonics, with all unstable modes developing rapidly, i.e. having eigenfrequencies comparable to or greater than typical orbital frequencies. This invalidates orbital approximation in the case of systems with all orbits very close to purely radial.

Radially anisotropic systems with r−α forces – II: radial-orbit instability

We continue to investigate the dynamics of collisionless systems of particles interacting via additive $r^{-\alpha}$ interparticle forces. Here we focus on the dependence of the radial-orbit instability on the force exponent $\alpha$. By means of direct $N$-body simulations we study the stability of equilibrium radially anisotropic Osipkov-Merritt spherical models with Hernquist density profile and with $1\leq\alpha<3$. We determine, as a function of $\alpha$, the minimum value for stability of the anisotropy radius $r_{as}$ and of the maximum value of the associated stability indicator $\xi_s$. We find that, for decreasing $\alpha$, $r_{as}$ decreases and $\xi_s$ increases, i.e. longer-range forces are more robust against radial-orbit instability. The isotropic systems are found to be stable for all the explored values of $\alpha$. The end products of unstable systems are all markedly triaxial with minor-to-major axial ratio $>0.3$, so they are never flatter than an E7 system.

Simulations of slow bars in anisotropic disk systems

The instability of anisotropic disk systems with elongated stellar orbits has been investigated. N-body generalized polytropic models of stellar disks have been constructed. They are shown to be unstable with respect to the bar formation at any degree of anisotropy. This result differs from the results of the studies of such models by other authors. The bar pattern speed and amplitude have been found. The initial distribution of precession rates and the adiabatic invariants of stellar orbits have been calculated. A bar is shown to be formed in such systems due to the radial orbit instability.

THE EFFECTS OF ANGULAR MOMENTUM ON HALO PROFILES

ABSTRACT The near universality of DM halo density profiles provided by N-body simulations proved to be robust against changes in total mass density, power spectrum, and some forms of initial velocity dispersion. Here we study the effects of coherently spinning up an isolated DM-only progenitor on halo structure. Halos with spins within several standard deviations of the simulated mean (λ ≲ 0.20) produce profiles with negligible deviations from the universal form. Only when the spin becomes quite large (λ ≳ 0.20) do departures become evident. The angular momentum distribution also exhibits a near universal form, which is also independent of halo spin up to λ ≲ 0.20. A correlation between these epidemic profiles and the presence of a strong bar in the virialized halo is also observed. These bar structures bear resemblance to the radial orbit instability in the rotationless limit.

A class of spherical, truncated, anisotropic models for application to globular clusters

Recently, a class of non-truncated radially-anisotropic models (the so-called $f^{(\nu)}$-models), originally constructed in the context of violent relaxation and modeling of elliptical galaxies, has been found to possess interesting qualities in relation to observed and simulated globular clusters. In view of new applications to globular clusters, we improve this class of models along two directions. To make them more suitable for the description of small stellar systems hosted by galaxies, we introduce a 'tidal' truncation (by means of a procedure that guarantees full continuity of the distribution function). The new $f_T^{(\nu)}$-models are shown to provide a better fit to the observed photometric and spectroscopic profiles for a sample of 13 globular clusters studied earlier by means of non-truncated models; interestingly, the best-fit models also perform better with respect to the radial-orbit instability. Then we design a flexible but simple two-component family of truncated models, to study the separate issues of mass segregation and of multiple populations. We do not aim at a fully realistic description of globular clusters, to compete with the description currently obtained by means of dedicated simulations. The goal here is to try to identify the simplest models, that is, those with the smallest number of free parameters, but still able to provide a reasonable description for clusters that are evidently beyond the reach of one-component models: with this tool we aim at identifying the key factors that characterize mass segregation or the presence of multiple populations. To reduce the relevant parameter space, we formulate a few physical arguments (based on recent observations and simulations). A first application to two well-studied globular clusters is briefly described and discussed.

On one classical problem in the radial orbit instability theory

Antonov’s classical problem of stability of a collisionless sphere with a purely radial motion of stars is considered as a limit of the problem in which stars move in nearly radial orbits. We provide the proper limiting equations that take into account the singularity in the density distribution at the sphere center and give their solutions. We show that there is instability for even and odd spherical harmonics, with all unstable modes being not slow. The growth rates of aperiodic even modes increase indefinitely when approaching purely radial models. The physics of the radial orbit instability is discussed.

Self-consistent triaxial models

We present self-consistent triaxial stellar systems that have analytic distribution functions (DFs) expressed in terms of the actions. These provide triaxial density profiles with cores or cusps at the centre. They are the first self-consistent triaxial models with analytic DFs suitable for modelling giant ellipticals and dark haloes. Specifically, we study triaxial models that reproduce the Hernquist profile from Williams & Evans (2015), as well as flattened isochrones of the form proposed by Binney (2014). We explore the kinematics and orbital structure of these models in some detail. The models typically become more radially anisotropic on moving outwards, have velocity ellipsoids aligned in Cartesian coordinates in the centre and aligned in spherical polar coordinates in the outer parts. In projection, the ellipticity of the isophotes and the position angle of the major axis of our models generally changes with radius. So, a natural application is to elliptical galaxies that exhibit isophote twisting. As triaxial St\"ackel models do not show isophote twists, our DFs are the first to generate mass density distributions that do exhibit this phenomenon, typically with a gradient of $\approx 10^\circ$/effective radius, which is comparable to the data. Triaxiality is a natural consequence of models that are susceptible to the radial orbit instability. We show how a family of spherical models with anisotropy profiles that transition from isotropic at the centre to radially anisotropic becomes unstable when the outer anisotropy is made sufficiently radial. Models with a larger outer anisotropy can be constructed but are found to be triaxial. We argue that the onset of the radial orbit instability can be identified with the transition point when adiabatic relaxation yields strongly triaxial rather than weakly spherical endpoints.

On the nature of the radial orbit instability in spherically symmetric collisionless stellar systems

We consider a two-parametric family of radially anisotropic models with non-singular density distribution in the centre. If highly eccentric orbits are locked near the centre, the characteristic growth rate of the instability is much less than the Jeans and dynamic frequencies of the stars (slow modes). The instability occurs only for even spherical harmonics and the perturbations are purely growing (aperiodic). On the contrary, if all orbits nearly reach the outer radius of the sphere, both even and odd harmonics are unstable. Unstable odd modes oscillate having characteristic frequencies of the order of the dynamical frequencies (fast modes). Unstable even harmonics contain a single aperiodic mode and several oscillatory modes, the aperiodic mode being the most unstable. The question of the nature of the radial orbit instability (ROI) is revisited. Two main interpretations of ROI were suggested in the literature. The first one refers to the classical Jeans instability associated with the lack of velocity dispersion of stars in the transverse direction. The second one refers to Lynden-Bell's orbital approach to bar formation in disc galaxies, which implies slowness and bi-symmetry of the perturbation. Oscillatory modes, odd spherical harmonics modes, and non-slow modes found in one of the models show that the orbital interpretation is not the only possible.

Radial orbit instability in dwarf dark matter haloes

Using N-body simulations we study the phenomenon of radial orbit instability occurring in dark matter haloes of the size of a dwarf galaxy. We carried out simulations of seven spherical models, with the same standard NFW (Navarro, Frenk & White) density profile but different anisotropy profiles of particle orbits. Four of them underwent instability: two with a constant positive anisotropy, one with an anisotropic core and an isotropic envelope and one with a very small isotropic core and an anisotropic envelope. Haloes affected by the instability become approximately axisymmetric and prolate, with the profile of the shortest-to-longest axis ratio increasing with radius. The lower limit for the central value of this axis ratio is 0.3 for an NFW halo. The density profiles of the haloes did not change significantly, whereas the velocity distributions became axisymmetric. The angular momentum modulus rose due to large-amplitude oscillations of its components perpendicular to the symmetry axis of the halo. We also studied orbits of individual particles assigning them to classical orbit families in triaxial potentials. We find that the membership of a given particle in a family depends on its initial angular momentum modulus and its components along the principal axes of matter distribution.