Three-dimensional calculations of high- and low-mass planets embedded in protoplanetary discs

We analyse the non-linear, three-dimensional response of a gaseous, viscous protoplanetary disc to the presence of a planet of mass ranging from 1 Earth mass (1 M_⊕) to 1 Jupiter mass (1 M_J) by using the zeus hydrodynamics code. We determine the gas flow pattern, and the accretion and migration rates of the planet. The planet is assumed to be in a fixed circular orbit about the central star. It is also assumed to be able to accrete gas without expansion on the scale of its Roche radius. Only planets with masses   M _p≳ 0.1 M_J  produce significant perturbations in the surface density of the disc. The flow within the Roche lobe of the planet is fully three-dimensional. Gas streams generally enter the Roche lobe close to the disc mid-plane, but produce much weaker shocks than the streams in two-dimensional models. The streams supply material to a circumplanetary disc that rotates in the same sense as the orbit of the planet. Much of the mass supply to the circumplanetary disc comes from non-coplanar flow. The accretion rate peaks with a planet mass of approximately 0.1 M_J and is highly efficient, occurring at the local viscous rate. The migration time-scales for planets of mass less than 0.1 M_J, based on torques from disc material outside the Roche lobes of the planets, are in excellent agreement with the linear theory of type I (non-gap) migration for three-dimensional discs. The transition from type I to type II (gap) migration is smooth, with changes in migration times of about a factor of 2. Starting with a core which can undergo runaway growth, a planet can gain up to a few M_J with little migration. Planets with final masses of the order of 10 M_J would undergo large migration, which makes formation and survival difficult.     

Non-axisymmetric instabilities in shocked adiabatic accretion flows

We investigate the linear stability of a shocked accretion flow on to a black hole in the adiabatic limit. Our linear analyses and numerical calculations show that, despite the post-shock deceleration, the shock is generally unstable to non-axisymmetric perturbations. The simulation results of Molteni, Tóth & Kuznetsov can be well explained by our linear eigenmodes. The mechanism of this instability is confirmed to be based on the cycle of acoustic waves between the corotation radius and the shock. We obtain an analytical formula to calculate the oscillation period from the physical parameters of the flow. We argue that the quasi-periodic oscillation should be a common phenomenon in accretion flows with angular momentum.     

Non-linear fluid dynamics of eccentric discs

A new theory of eccentric accretion discs is presented. Starting from the basic fluid-dynamical equations in three dimensions, I derive the fundamental set of one-dimensional equations that describe how the mass, angular momentum and eccentricity vector of a thin disc evolve as a result of internal stresses and external forcing. The analysis is asymptotically exact in the limit of a thin disc, and allows for slowly varying eccentricities of arbitrary magnitude. The theory is worked out in detail for a Maxwellian viscoelastic model of the turbulent stress in an accretion disc. This generalizes the conventional alpha viscosity model to account for the non-zero relaxation time of the turbulence, and is physically motivated by a consideration of the nature of magnetohydrodynamic turbulence. It is confirmed that circular discs are typically viscously unstable to eccentric perturbations, as found by Lyubarskij, Postnov & Prokhorov, if the conventional alpha viscosity model is adopted. However, the instability can usually be suppressed by introducing a sufficient relaxation time and/or bulk viscosity. It is then shown that an initially uniformly eccentric disc does not retain its eccentricity as had been suggested by previous analyses. The evolutionary equations should be useful in many applications, including understanding the origin of planetary eccentricities and testing theories of quasi-periodic oscillations in X-ray binaries.     

Two-fluid Dynamics in Clusters of Galaxies

We develop a theoretical formulation for the large-scale dynamics of galaxy clusters involving two spherical ‘isothermal fluids‘ coupled by their mutual gravity and derive asymptotic similarity solutions analytically. One of the fluids roughly approximates the massive dark matter halo, while the other describes the hot gas, the relatively small mass contribution from the galaxies being subsumed in the gas. By properly choosing the self-similar variables, it is possible to consistently transform the set of time-dependent two-fluid equations of spherical symmetry with serf-gravity into a set of coupled nonlinear ordinary differential equations (ODEs). We focus on the analytical analysis and discuss applications of the solutions to galaxy clusters.     

Model dependence of transonic properties of accretion flows around black holes

We analytically study how the behaviour of accretion flows changes when the flow model is varied. We study the transonic properties of the conical flow, a flow of constant height and a flow in vertical equilibrium, and show that all these models are basically identical, provided that the polytropic constant is suitably changed from one model to another. We show that this behaviour is extendible even when standing shocks are produced in the flow. The parameter space where shocks are produced remains roughly identical in all these models when the same transformation among the polytropic indices is used. We present applications of these findings.