Fourier Analysis of Stochastic Sampling Strategies for Assessing Bias and Variance in Integration

Each pixel in a photorealistic, computer generated picture is calculated by approximately integrating all the light arriving at the pixel, from the virtual scene. A common strategy to calculate these high-dimensional integrals is to average the estimates at stochastically sampled locations. The strategy with which the sampled locations are chosen is of utmost importance in deciding the quality of the approximation, and hence rendered image. We derive connections between the spectral properties of stochastic sampling patterns and the first and second order statistics of estimates of integration using the samples. Our equations provide insight into the assessment of stochastic sampling strategies for integration. We show that the amplitude of the expected Fourier spectrum of sampling patterns is a useful indicator of the bias when used in numerical integration. We deduce that estimator variance is directly dependent on the variance of the sampling spectrum over multiple realizations of the sampling pattern. We then analyse Gaussian jittered sampling, a simple variant of jittered sampling, that allows a smooth trade-off of bias for variance in uniform (regular grid) sampling. We verify our predictions using spectral measurement, quantitative integration experiments and qualitative comparisons of rendered images.</jats:p

Variable line profiles due to non-axisymmetric patterns in an accretion disc around a rotating black hole

We have explored spectral line profiles due to spiral patterns in accretion discs around black holes. A parametrization was employed for the shape and emissivity of spiral waves, which can be produced by non-axisymmetric perturbations affecting the disc density and ionization structure. The effects of the light-travel time, energy shift, and gravitational focusing near to a rotating black hole were taken into account. A high-resolution ray-tracing code was used to follow the time variations of the synthetic line profile. A variety of expected spectral features were examined and the scheme applied to a broad iron line observed in MCG-6-30-15.Comment: 11 pages (LaTeX), 7 figures (EPS); PASJ accepte

Catch me if you can: is there a runaway-mass black hole in the Orion Nebula Cluster?

We investigate the dynamical evolution of the Orion Nebula Cluster (ONC) by means of direct N-body integrations. A large fraction of residual gas was probably expelled when the ONC formed, so we assume that the ONC was much more compact when it formed compared to its current size, in agreement with the embedded cluster radius-mass relation from Marks & Kroupa (2012). Hence, we assume that few-body relaxation played an important role during the initial phase of evolution of the ONC. In particular, three body interactions among OB stars likely led to their ejection from the cluster and, at the same time, to the formation of a massive object via runaway physical stellar collisions. The resulting depletion of the high mass end of the stellar mass function in the cluster is one of the important points where our models fit the observational data. We speculate that the runaway-mass star may have collapsed directly into a massive black hole (Mbh > 100Msun). Such a dark object could explain the large velocity dispersion of the four Trapezium stars observed in the ONC core. We further show that the putative massive black hole is likely to be a member of a binary system with appr. 70 per cent probability. In such a case, it could be detected either due to short periods of enhanced accretion of stellar winds from the secondary star during pericentre passages, or through a measurement of the motion of the secondary whose velocity would exceed 10 km/s along the whole orbit.Comment: 10 pages, 6 figures, accepted by Ap

The warped young stellar disc in the Galactic Centre

Within the central parsec of the Galaxy, several tens of young stars orbiting a central supermassive black hole are observed. A subset of these stars forms a coherently rotating disc. Other observations reveal a massive molecular torus which lies at a radius ~1.5pc from the centre. In this paper we consider the gravitational influence of the molecular torus upon the stars of the stellar disc. We derive an analytical formula for the rate of precession of individual stellar orbits and we show that it is highly sensitive upon the orbital semi-major axis and inclination with respect to the plane of the torus as well as on the mass of the torus. Assuming that both the stellar disc and the molecular torus are stable on the time-scale >6Myr, we constrain the mass of the torus and its inclination with respect to the young stellar disc. We further suggest that all young stars observed in the Galactic Centre may have a common origin in a single coherently rotating structure with an opening angle <5deg, which was partially destroyed (warped) during its lifetime by the gravitational influence of the molecular torus.Comment: 5 pages, 3 figures, accepted for publication in A&