Resonant thickening of self-gravitating discs: imposed or self-induced orbital diffusion in the tightly wound limit

The secular thickening of a self-gravitating stellar galactic disc is investigated using the dressed collisionless Fokker-Planck equation and the inhomogeneous multicomponent Balescu-Lenard equation. The thick WKB limits for the diffusion fluxes are found using the epicyclic approximation, while assuming that only radially tightly wound transient spirals are sustained by the disc. This yields simple quadratures for the drift and diffusion coefficients, providing a clear understanding of the positions of maximum vertical orbital diffusion within the disc, induced by fluctuations either external or due to the finite number of particles. These thick limits also offer a consistent derivation of a thick disc Toomre parameter, which is shown to be exponentially boosted by the ratio of the vertical to radial scale heights. Dressed potential fluctuations within the disc statistically induce a vertical bending of a subset of resonant orbits, triggering the corresponding increase in vertical velocity dispersion. When applied to a tepid stable tapered disc perturbed by shot noise, these two frameworks reproduce qualitatively the formation of ridges of resonant orbits towards larger vertical actions, as found in direct numerical simulations, but overestimates the time-scale involved in their appearance. Swing amplification is likely needed to resolve this discrepancy, as demonstrated in the case of razor-thin discs. Other sources of thickening are also investigated, such as fading sequences of slowing bars, or the joint evolution of a population of giant molecular clouds within the disc.Comment: 31 pages, 19 figure

Wavefunction Collapse and Random Walk

Wavefunction collapse models modify Schrodinger's equation so that it describes the rapid evolution of a superposition of macroscopically distinguishable states to one of them. This provides a phenomenological basis for a physical resolution to the so-called "measurement problem." Such models have experimentally testable differences from standard quantum theory. The most well developed such model at present is the Continuous Spontaneous Localization (CSL) model in which a fluctuating classical field interacts with particles to cause collapse. One "side effect" of this interaction is that the field imparts momentum to particles, causing a small blob of matter to undergo random walk. Here we explore this in order to supply predictions which could be experimentally tested. We examine the translational diffusion of a sphere and a disc, and the rotational diffusion of a disc, according to CSL. For example, we find that a disc of radius 2 cdot 10^{-5} cm and thickness 0.5 cdot 10^{-5} cm diffuses through 2 pi rad in about 70sec (this assumes the "standard" CSL parameter values). The comparable rms diffusion of standard quantum theory is smaller than this by a factor 10^-3. At the reported pressure of < 5 cdot10^{-17} Torr, achieved at 4.2^{circ} K, the mean time between air molecule collisions with the disc is approximately 45min (and the diffusion caused by photon collisons is utterly negligible). This is ample time for observation of the putative CSL diffusion over a wide range of parameters. This encourages consideration of how such an experiment may actually be performed, and the paper closes with some thoughts on this subjectComment: 27 pages, 2 figure

On the diffusive propagation of warps in thin accretion discs

In this paper we revisit the issue of the propagation of warps in thin and viscous accretion discs. In this regime warps are know to propagate diffusively, with a diffusion coefficient approximately inversely proportional to the disc viscosity. Previous numerical investigations of this problem (Lodato & Pringle 2007) did not find a good agreement between the numerical results and the predictions of the analytic theories of warp propagation, both in the linear and in the non-linear case. Here, we take advantage of a new, low-memory and highly efficient SPH code to run a large set of very high resolution simulations (up to 20 million SPH particles) of warp propagation, implementing an isotropic disc viscosity in different ways, to investigate the origin of the discrepancy between the theory and the numerical results. Our new and improved analysis now shows a remarkable agreement with the analytic theory both in the linear and in the non-linear regime, in terms of warp diffusion coefficient and precession rate. It is worth noting that the resulting diffusion coefficient is inversely proportional to the disc viscosity only for small amplitude warps and small values of the disc $\alpha$ coefficient ($\alpha < 0.1$). For non-linear warps, the diffusion coefficient is a function of both radius and time, and is significantly smaller than the standard value. Warped accretion discs are present in many contexts, from protostellar discs to accretion discs around supermassive black holes. In all such cases, the exact value of the warp diffusion coefficient may strongly affect the evolution of the system and therefore its careful evaluation is critical in order to correctly estimate the system dynamics (abridged).Comment: 16 pages, 14 figures. Accepted to MNRAS. Movies and additional figures can be found at http://users.monash.edu.au/~dprice/pubs/warp/index.htm

Propagating mass accretion rate fluctuations in X-ray binaries under the influence of viscous diffusion

Many statistical properties of X-ray aperiodic variability from accreting compact objects can be explained by the propagating fluctuations model applied to the accretion disc. The mass accretion rate fluctuations originate from variability of viscosity, which arises at every radius and causes local fluctuations of the density. The fluctuations diffuse through the disc and result in local variability of the mass accretion rate, which modulates the X-ray flux from the inner disc in the case of black holes, or from the surface in the case of neutron stars. A key role in the theoretical explanation of fast variability belongs to the description of the diffusion process. The propagation and evolution of the fluctuations is described by the diffusion equation, which can be solved by the method of Green functions. We implement Green functions in order to accurately describe the propagation of fluctuations in the disc. For the first time we consider both forward and backward propagation. We show that (i) viscous diffusion efficiently suppress variability at time scales shorter than the viscous time, (ii) local fluctuations of viscosity affect the mass accretion rate variability both in the inner and the outer parts of accretion disc, (iii) propagating fluctuations give rise not only to hard time lags as previously shown, but also produce soft lags at high frequency similar to those routinely attributed to reprocessing, (iv) deviation from the linear rms-flux relation is predicted for the case of very large initial perturbations. Our model naturally predicts bumpy power spectra.Comment: 20 pages, 17 figures, accepted for publication in MNRA

Transport of magnetic flux and the vertical structure of accretion discs: II. Vertical profile of the diffusion coefficients

We investigate the radial transport of magnetic flux in a thin accretion disc, the turbulence being modelled by effective diffusion coefficients (viscosity and resistivity). Both turbulent diffusion and advection by the accretion flow contribute to flux transport, and they are likely to act in opposition. We study the consequences of the vertical variation of the diffusion coefficients, due to a varying strength of the turbulence. For this purpose, we consider three different vertical profiles of these coefficients. The first one is aimed at mimicking the turbulent stress profile observed in numerical simulations of MHD turbulence in stratified discs. This enables us to confirm the robustness of the main result of Paper I obtained for uniform diffusion coefficients that, for weak magnetic fields, the contribution of the accretion flow to the transport velocity of magnetic flux is much larger than the transport velocity of mass. We then consider the presence of a dead zone around the equatorial plane, where the physical resistivity is high while the turbulent viscosity is low. We find that it amplifies the previous effect: weak magnetic fields can be advected orders of magnitude faster than mass, for dead zones with a large vertical extension. The ratio of advection to diffusion, determining the maximum inclination of the field at the surface of the disc, is however not much affected. Finally, we study the effect of a non-turbulent layer at the surface of the disc, which has been suggested as a way to reduce the diffusion of the magnetic flux. We find that the reduction of the diffusion requires the conducting layer to extend below the height at which the magnetic pressure equals the thermal pressure. As a consequence, if the absence of turbulence is caused by the large-scale magnetic field, the highly conducting layer is inefficient at reducing the diffusion.Comment: 15 pages, 12 figures, accepted for publication in MNRA

The evolution of a supermassive binary caused by an accretion disc

The interaction of a massive binary and a non-self-gravitating circumbinary accretion disc is considered. The shape of the stationary twisted disc produced by the binary is calculated. It is shown that the inner part of the disc must lie in the binary orbital plane for any value of viscosity. When the inner disc midplane is aligned with the binary orbital plane on the scales of interest and it rotates in the same sense as the binary, the modification of the disc structure and the rate of decay of the binary orbit, assumed circular, due to tidal exchange of angular momentum with the disc, are calculated. It is shown that the modified disc structure is well described by a self-similar solution of the non-linear diffusion equation governing the evolution of the disc surface density. The calculated time scale for decay of the binary orbit is always smaller than the "accretion" time $t_{acc}=m/{\dot M}$ ($m$ is the mass of the secondary component, and $\dot M$ is the disc accretion rate), and is determined by ratio of secondary mass $m$, assumed to be much smaller than the primary mass, the disc mass inside the initial binary orbit, and the form of viscosity in the disc.Comment: to be published in MNRA

Application of hpDGFEM to mechanisms at channel microband electrodes

We extend our earlier work (Harriman et al., Oxford University Computing Laboratory Technical Report NA04/19) on hp-DGFEM for disc electrodes to the case of reaction mechanisms to the increasingly popular channel microband electrode configuration. We present results for the simple E reaction mechanism (convection-diffusion equation), for the ECE and EC2E reaction mechanisms (linear and nonlinear systems of reaction-convection- diffusion equations, respectively) and for the DISP1 and DISP2 reaction mechanisms (linear and nonlinear coupled systems of reaction-convection-diffusion equations, respectively). In all cases we demonstrate excellent agreement with previous results using relatively coarse meshes and without the need for streamline-diffusion stabilisation, even at high flow rates

Density waves in the shearing sheet III. Disc heating

The problem of dynamical heating of galactic discs by spiral density waves is discussed using the shearing sheet model. The secular evolution of the disc is described quantitatively by a diffusion equation for the distribution function of stars in the space spanned by integrals of motion of the stars, in particular the radial action integral and an integral related to the angular momentum. Specifically, disc heating by a succession of transient, `swing amplified' density waves is studied. It is shown that such density waves lead predominantly to diffusion of stars in radial action space. The stochastical changes of angular momenta of the stars and the corresponding stochastic changes of the guiding centre radii of the stellar orbits induced by this process are much smaller.Comment: 6 pages, 1 figure, accepted by MNRA