Magnetically self-regulated formation of early protoplanetary discs

The formation of protoplanetary discs during the collapse of molecular dense cores is significantly influenced by angular momentum transport, notably by the magnetic torque. In turn, the evolution of the magnetic field is determined by dynamical processes and non-ideal MHD effects such as ambipolar diffusion. Considering simple relations between various timescales characteristic of the magnetized collapse, we derive an expression for the early disc radius, r \simeq 18 \, {\rm AU} \, \left({\eta_{\rm AD} / 0.1 \, {\rm s}} \right)^{2/9} \left({B_z / 0.1\, {\rm G}} \right) ^{-4/9} \left({M / 0.1 \msol} \right) ^{1/3}, where $M$ is the total disc plus protostar mass, $\eta_\mathrm{AD}$ is the ambipolar diffusion coefficient and $B_z$ is the magnetic field in the inner part of the core. This is about significantly smaller than the discs that would form if angular momentum was conserved. The analytical predictions are confronted against a large sample of 3D, non-ideal MHD collapse calculations covering variations of a factor 100 in core mass, a factor 10 in the level of turbulence, a factor 5 in rotation, and magnetic mass-to-flux over critical mass-to-flux ratios 2 and 5. The disc radius estimates are found to agree with the numerical simulations within less than a factor 2. A striking prediction of our analysis is the weak dependence of circumstellar disc radii upon the various relevant quantities, suggesting weak variations among class-0 disc sizes. In some cases, we note the onset of large spiral arms beyond this radius.This research has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 247060 and no. 306483). We acknowledge financial support from ”Programme National de Physique Stellaire” (PNPS) of CNRS/INSU, France

On the Peak-to-Mean Envelope Power Ratio of Phase-Shifted Binary Codes

The peak-to-mean envelope power ratio (PMEPR) of a code employed in orthogonal frequency-division multiplexing (OFDM) systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift. Motivated by some previous designs of phase shifts using suboptimal methods, the following question is considered in this paper. For a given binary code, how much PMEPR reduction can be achieved when the phase shifts are taken from a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the achievable PMEPR is established, which is related to the covering radius of the binary code. Generally speaking, the achievable region of the PMEPR shrinks as the covering radius of the binary code decreases. The bound is then applied to some well understood codes, including nonredundant BPSK signaling, BCH codes and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated that most (presumably not optimal) phase-shift designs from the literature attain or approach our bound.Comment: minor revisions, accepted for IEEE Trans. Commun

Estimating Multidimensional Persistent Homology through a Finite Sampling

An exact computation of the persistent Betti numbers of a submanifold $X$ of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of $X$ is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of $X$ from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it

Smoothing Riemannian Metrics with Bounded Ricci Curvatures in Dimension Four, II

This note is a continuation of the author's paper \cite{Li}. We prove that if the metric $g$ of a 4-manifold has bounded Ricci curvature and the curvature has no local concentration everywhere, then it can be smoothed to a metric with bounded sectional curvature. Here we don't assume the bound for local Sobolev constant of $g$ and hence this smoothing result can be applied to the collapsing case.Comment: 17 page

Evolving turbulence and magnetic fields in galaxy clusters

We discuss, using simple analytical models and MHD simulations, the origin and parameters of turbulence and magnetic fields in galaxy clusters. Three physically distinct regimes can be identified in the evolution of cluster turbulence and magnetic fields. Firstly, the fluctuation dynamo will produce microgauss-strong, random magnetic fields during cluster formation and major mergers. Turbulent velocity of about 300 km/s can be maintained at scales 100-200 kpc. The magnetic field is intermittent, has a smaller scale of 20-30 kpc and average strength of 2 microgauss. Secondly, when major mergers end, turbulent speed and magnetic field undergo a power-law decay, decreasing in strength but increasing in scale by a factor of about two. Thirdly, smaller-mass subclusters and cluster galaxies produce turbulent wakes, with turbulent speeds and magnetic field strengths similar to those quoted above. The velocity scales are about 200 kpc and 10 kpc respectively, and the magnetic field scale is about 6 times smaller. Although these wakes may fill only a small fraction of the cluster volume, their area covering factor can be close to unity. So one can potentially reconcile observations that indicate the coexistence of turbulence with ordered filamentary gas structures, as in the Perseus cluster. Random Faraday rotation measure is estimated to be typically 100-200 rad/m^2, in agreement with observations. We predict detectable synchrotron polarization from cluster radio halos at wavelengths 3-6 cm, if observed at sufficiently high resolution (abridged).Comment: 20 pages, 9 figures, Replaced to match version accepted by MNRA