The precession of eccentric discs in close binaries

We consider the precession rates of eccentric discs in close binaries, and compare theoretical predictions with the results of numerical disc simulations and with observed superhump periods. A simple dynamical model for precession is found to be inadequate. For mass ratios less than approximately 1/4 a linear dynamical model does provide an upper limit for disc precession rates. Theory suggests that pressure forces have a significant retrograde impact upon the precession rate (Lubow 1992). We find that the disc precession rates for three systems with accurately known mass ratios are significantly slower than predicted by the dynamical theory, and we attribute the difference to pressure forces. By assuming that pressure forces of similar magnitude occur in all superhumping systems, we obtain an improved fit to superhump observations.Comment: 6 pages to appear in MNRAS (accepted

SPH simulations of irradiation-driven warped accretion discs and the long periods in X-ray binaries

We present three dimensional smoothed particle hydrodynamics (SPH) calculations of irradiation-driven warping of accretion discs. Initially unwarped planar discs are unstable to the radiation reaction when the disc is illuminated by a central radiation source. The disc warps and tilts and precesses slowly in a retrograde direction; its shape continuously flexes in response to the changing orientation of the Roche potential. We simulate ten systems: eight X-ray binaries, one cataclysmic variable (CV), and a `generic' low mass X-ray binary (LMXB). We adopt system parameters from observations and tune a single parameter: our model X-ray luminosity ($L_{*}$) to reproduce the observed or inferred super-orbital periods. Without exception, across a wide range of parameter space, we find an astonishingly good match between the observed $L_{X}$ and the model $L_{*}$. We conclude irradiation-driven warping is the mechanism underlying the long periods in X-ray binaries. Our Her X-1 simulation simultaneously reproduces the observed $L_{X}$, the "main-" and "short-high" X-ray states and the orbital inclination. Our simulations of SS 433 give a maximum warp angle of $18.6^{\circ}$, a good match to the cone traced by the jets, but this angle is reached only in the outer disc. In all cases, the overall disc tilt is less than \degrees{13} and the maximum disc warp is less than and or equal to \degrees{21}.Comment: 17 pages, 14 figures, shorter abstract (24 lines limit

Superhumps in Low-Mass X-Ray Binaries

We propose a mechanism for the superhump modulations observed in optical photometry of at least two black hole X-ray transients (SXTs). As in extreme mass-ratio cataclysmic variables (CVs), superhumps are assumed to result from the presence of the 3:1 orbital resonance in the accretion disc. This causes the disc to become non-axisymmetric and precess. However the mechanism for superhump luminosity variations in low mass X-ray binaries (LMXBs) must differ from that in CVs, where it is attributed to a tidally-driven modulation of the disc's viscous dissipation, varying on the beat between the orbital and disc precession period. By contrast in LMXBs, tidal dissipation in the outer accretion disc is negligible: the optical emission is overwhelming dominated by reprocessing of intercepted central X-rays. Thus a different origin for the superhump modulation is required. Recent observations and numerical simulations indicate that in an extreme mass-ratio system the disc area changes on the superhump period. We deduce that the superhumps observed in SXTs arise from a modulation of the reprocessed flux by the changing area. Therefore, unlike the situation in CVs, where the superhump amplitude is inclination-independent, superhumps should be best seen in low-inclination LMXBs, whereas an orbital modulation from the heated face of the secondary star should be more prominent at high inclinations. Modulation at the disc precession period (10s of days) may indicate disc asymmetries such as warping. We comment on the orbital period determinations of LMXBs, and the possibility and significance of possible permanent superhump LMXBs.Comment: 6 pages, 1 encapsulated figure. MNRAS in press; replaced to correct typographical error

Theoretical Analysis of STM Experiments at Rutile TiO_2 Surfaces

A first-principles atomic orbital-based electronic structure method is used to investigate the low index surfaces of rutile Titanium Dioxide. The method is relatively cheap in computational terms, making it attractive for the study of oxide surfaces, many of which undergo large reconstructions, and may be governed by the presence of Oxygen vacancy defects. Calculated surface charge densities are presented for low-index surfaces of TiO$_2$, and the relation of these results to experimental STM images is discussed. Atomic resolution images at these surfaces tend to be produced at positive bias, probing states which largely consist of unoccupied Ti 3$d$ bands, with a small contribution from O 2$p$. These experiments are particularly interesting since the O atoms tend to sit up to 1 angstrom above the Ti atoms, so providing a play-off between electronic and geometric structure in image formation.Comment: 9 pages, Revtex, 3 postscript figures, accepted by Surf. Scienc

Comprehensive simulations of superhumps

(Abridged) We use 3D SPH calculations with higher resolution, as well as with more realistic viscosity and sound-speed prescriptions than previous work to examine the eccentric instability which underlies the superhump phenomenon in semi-detached binaries. We illustrate the importance of the two-armed spiral mode in the generation of superhumps. Differential motions in the fluid disc cause converging flows which lead to strong spiral shocks once each superhump cycle. The dissipation associated with these shocks powers the superhump. We compare 2D and 3D results, and conclude that 3D simulations are necessary to faithfully simulate the disc dynamics. We ran our simulations for unprecedented durations, so that an eccentric equilibrium is established except at high mass ratios where the growth rate of the instability is very low. Our improved simulations give a closer match to the observed relationship between superhump period excess and binary mass ratio than previous numerical work. The observed black hole X-ray transient superhumpers appear to have systematically lower disc precession rates than the cataclysmic variables. This could be due to higher disc temperatures and thicknesses. The modulation in total viscous dissipation on the superhump period is overwhelmingly from the region of the disc within the 3:1 resonance radius. As the eccentric instability develops, the viscous torques are enhanced, and the disc consequently adjusts to a new equilibrium state, as suggested in the thermal-tidal instability model. We quantify this enhancement in the viscosity, which is ~10 per cent for q=0.08. We characterise the eccentricity distributions in our accretion discs, and show that the entire body of the disc partakes in the eccentricity.Comment: 18 pages (mn2e LaTeX), 14 figures, 5 tables, Accepted for publication in MNRA

On the number of sides necessary for polygonal approximation of black-and-white figures in a plane

A bound on the number of extreme points or sides necessary to approximate a convex planar figure by an enclosing polygon is described. This number is found to be proportional to the fourth root of the figure's area divided by the square of a maximum Euclidean distance approximation parameter.An extension of this bound, preserving its fourth root quality, is made to general planar figures. This is done by decomposing the general figure into nearly convex sets defined by inflection points, cusps, and multiple windings.A procedure for performing actual encoding of this type is described. Comparisons of parsimony are made with contemporary figure encoding schemes

Non-invasive estimation of left atrial dominant frequency in atrial fibrillation from different electrode sites: Insight from body surface potential mapping

© 2014, CardioFront LLC. All rights reserved. The dominant driving sources of atrial fibrillation are often found in the left atrium, but the expression of left atrial activation on the body surface is poorly understood. Using body surface potential mapping and simultaneous invasive measurements of left atrial activation our aim was to describe the expression of the left atrial dominant fibrillation frequency across the body surface. 20 patients in atrial fibrillation were studied. The spatial distributions of the dominant atrial fibrillation frequency across anterior and posterior sites on the body surface were quantified. Their relationship with invasive left atrial dominant fibrillation frequency was assessed by linear regression analysis, and the coefficient of determination was calculated for each body surface site. The correlation between intracardiac and body surface dominant frequency was significantly higher with posterior compared with anterior sites (coefficient of determination 67±8% vs 48±2%,

Biological pattern formation on two-dimensional domains: A nonlinear bifurcation analysis

A tissue interaction model for skin organ pattern formation is presented. Possible spatially patterned solutions on rectangular domains are investigated. Linear stability analysis suggests that the model can exhibit pattern formation. A weakly nonlinear two-dimensional perturbation analysis is then carried out. This demonstrates that when bifurcation occurs via a simple eigenvalue, patterns such as rolls, squares, and rhombi can be supported by the model equations. Our nonlinear analysis shows that more complex patterns are also possible if bifurcation occurs via a double eigenvalue. Surprisingly, hexagonal patterns could not develop from a primary bifurcation