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

Abstract

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 ∼ 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 nonsingular 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.

The Lynden-Bell bar formation mechanism in simple and realistic galactic models

“…Using the standard technique of finding stationary points of the Hamiltonian, we show that orientation of orbits is governed by the signs of the precession rate Q, of the LBderivative 3 of the precession rate P, and of the orbital responsiveness to the bar potential b. This is in accordance with our previous works based on the matrix methods of the linear perturbation theory that show the importance of the sign of the precession rate for radial-orbit (Polyachenko et al 2010a, 2017 and loss cone instabilities (Polyachenko et al 2007(Polyachenko et al , 2008(Polyachenko et al , 2010b. These new results extend the theory of bar formation by Lynden-Bell (1979) that classifies all disc orbits using only one of these three parameters -the sign of P. A majority of orbits consist of so-called 'normal' orbits characterised by the negative sign of P. It tends to align in the direction perpendicular to the bar.…”

The evolution of a primordial binary black hole due to interaction with cold dark matter and the formation rate of gravitational wave events

“…These perturbations are amplified in the halo as a results of various potential instabilities, e.g., the instability of radial orbits, see e.g. [43]. Gravitational field associated with the amplified perturbations causes the torque large enough to explain the obtained values of angular momentum.…”

“…The anisotropy itself may be originated from the action of the radial orbit instability (see e.g. [43]) with initial perturbations provided by the action of binary's quadrupole gravitation field. In this picture the smaller is d 0 the smaller would be the anisotropy and this is indeed observed in the simulations.…”

l = 1: Weinberg’s weakly damped mode in an N-body model of a spherical stellar system

“…For example, by use of such methods, much attention has been paid to the case of radially anisotropic systems, where the radial orbit instability leads to the presence of bars, which are of obvious interest from the point of view of the morphology of galaxies (e.g. Polyachenko 1989;Allen, Palmer, & Papaloizou 1990;Carpintero & Muzzio 1995;Cincotta, Nunez, & Muzzio 1996;Trenti & Bertin 2006;MacMillan, Widrow, & Henriksen 2006;Buyle et al 2007;Bellovary et al 2008;Barnes, Lanzel, & Williams 2009;Maréchal & Perez 2010;Gajda, Lokas, & Wojtak 2015;Polyachenko & Shukhman 2017).…”