S. Sridhar

Counter-rotating stellar discs around a massive black hole: self-consistent, time-dependent dynamics

We formulate the collisionless Boltzmann equation for dense star clusters that lie within the radius of influence of a massive black hole in galactic nuclei. Our approach to these nearly Keplerian systems follows that of Sridhar & Touma: Delaunay canonical variables are used to describe stellar orbits and we average over the fast Keplerian orbital phases. The stellar distribution function (DF) evolves on the longer time-scale of precessional motions, whose dynamics is governed by a Hamiltonian, given by the orbit-averaged self-gravitational potential of the cluster. We specialize to razor-thin, planar discs and consider two counter-rotating (‘±’) populations of stars. To describe discs of small eccentricities, we expand the ± Hamiltonian to fourth order in the eccentricities, with coefficients that depend self-consistently on the ± DFs. We construct approximate ± dynamical invariants and use Jeans’ theorem to construct time-dependent ± DFs, which are completely described by their centroid coordinates and shape matrices. When the centroid eccentricities are larger than the dispersion in eccentricities, the ± centroids obey a set of four autonomous equations ordinary differential equations. We show that these can be cast as a two-degree-of-freedom Hamiltonian system which is non-linear, yet integrable. We study the linear instability of initially circular discs and derive a criterion for the counter-rotating instability. We then explore the rich non-linear dynamics of counter-rotating discs, with focus on the variety of steadily precessing eccentric configurations that are allowed. The stability and properties of these configurations are studied as functions of parameters such as the disc mass ratios and angular momentum.

Large-scale dynamo action due to α fluctuations in a linear shear flow

We present a model of large-scale dynamo action in a shear flow that has stochastic, zero-mean fluctuations of the a parameter. This is based on a minimal extension of the Kraichnan Moffatt model, to include a background linear shear and Galilean-invariant alpha-statistics. Using the firstorder smoothing approximation we derive a linear integro-differential equation for the largescale magnetic field, which is non-perturbative in the shearing rate S, and the alpha-correlation time r. The white-noise case, tau(alpha) = 0, is solved exactly, and it is concluded that the necessary condition for dynamo action is identical to the Kraichnan Moffatt model without shear; this is because white-noise does not allow for memory effects, whereas shear needs time to act. To explore memory effects we reduce the integro-differential equation to a partial differential equation, valid for slowly varying fields when is small but non-zero. Seeking exponential modal solutions, we solve the modal dispersion relation and obtain an explicit expression for the growth rate as a function of the six independent parameters of the problem. A non-zero r, gives rise to new physical scales, and dynamo action is completely different from the white-noise case; e.g. even weak a fluctuations can give rise to a dynamo. We argue that, at any wavenumber, both Moffatt drift and Shear always contribute to increasing the growth rate. Two examples are presented: (a) a Moffatt drift dynamo in the absence of shear and (b) a Shear dynamo in the absence of Moffatt drift.

Transport coefficients for the shear dynamo problem at small Reynolds numbers.

We build on the formulation developed in S. Sridhar and N. K. Singh [J. Fluid Mech. 664, 265 (2010)] and present a theory of the shear dynamo problem for small magnetic and fluid Reynolds numbers, but for arbitrary values of the shear parameter. Specializing to the case of a mean magnetic field that is slowly varying in time, explicit expressions for the transport coefficients α(il) and η(il) are derived. We prove that when the velocity field is nonhelical, the transport coefficient α(il) vanishes. We then consider forced, stochastic dynamics for the incompressible velocity field at low Reynolds number. An exact, explicit solution for the velocity field is derived, and the velocity spectrum tensor is calculated in terms of the Galilean-invariant forcing statistics. We consider forcing statistics that are nonhelical, isotropic, and delta correlated in time, and specialize to the case when the mean field is a function only of the spatial coordinate X(3) and time τ; this reduction is necessary for comparison with the numerical experiments of A. Brandenburg, K. H. Rädler, M. Rheinhardt, and P. J. Käpylä [Astrophys. J. 676, 740 (2008)]. Explicit expressions are derived for all four components of the magnetic diffusivity tensor η(il)(τ). These are used to prove that the shear-current effect cannot be responsible for dynamo action at small Re and Rm, but for all values of the shear parameter.

Stellar dynamics around a massive black hole – I. Secular collisionless theory

We present a theory in three parts, of the secular dynamics of a (Keplerian) stellar system of mass

The disruption of multiplanet systems through resonance with a binary orbit.

M

Slow m= 1 instabilities of softened gravity Keplerian discs

orbiting a black hole of mass

Unstable m=1 modes of counter-rotating Keplerian discs

M_\bullet \gg M

Recovery of transverse velocities of steadily rotating patterns in flat galaxies

. Here we describe the collisionless dynamics; Papers II and III are on the (collisional) theory of Resonant Relaxation. The mass ratio,

Secular instabilities of Keplerian stellar discs

\varepsilon = M/M_\bullet \ll 1

Deformation of the Galactic Centre stellar cusp due to the gravity of a growing gas disc

, is a natural small parameter implying a separation of time scales between the short Kepler orbital periods and the longer orbital precessional periods. The collisionless Boltzmann equation (CBE) for the stellar distribution function (DF) is averaged over the fast Kepler orbital phase using the method of multiple scales. The orbit-averaged system is described by a secular DF,