Multiple Disk Gaps and Rings Generated by a Single Super-Earth: II. Spacings, Depths, and Number of Gaps, with Application to Real Systems

ALMA has found multiple dust gaps and rings in a number of protoplanetary disks in continuum emission at millimeter wavelengths. The origin of such structures is in debate. Recently, we documented how one super-Earth planet can open multiple (up to five) dust gaps in a disk with low viscosity ($\alpha\lesssim10^{-4}$). In this paper, we examine how the positions, depths, and total number of gaps opened by one planet depend on input parameters, and apply our results to real systems. Gap locations (equivalently, spacings) are the easiest metric to use when making comparisons between theory and observations, as positions can be robustly measured. We fit the locations of gaps empirically as functions of planet mass and disk aspect ratio. We find that the locations of the double gaps in HL Tau and TW Hya, and of all three gaps in HD 163296, are consistent with being opened by a sub-Saturn mass planet. This scenario predicts the locations of other gaps in HL Tau and TW Hya, some of which appear consistent with current observations. We also show how the Rossby wave instability may develop at the edges of several gaps and result in multiple dusty vortices, all caused by one planet. A planet as low in mass as Mars may produce multiple dust gaps in the terrestrial planet forming region.Comment: 16 pages; ApJ accepte

Multiple Disk Gaps and Rings Generated by a Single Super-Earth

We investigate the observational signatures of super-Earths (i.e., Earth-to-Neptune mass planets) in their natal disks of gas and dust. Combining two-fluid global hydrodynamics simulations with a radiative transfer code, we calculate the distributions of gas and of sub-mm-sized dust in a disk perturbed by a super-Earth, synthesizing images in near-infrared scattered light and the mm-wave thermal continuum for direct comparison with observations. In low viscosity gas ($\alpha\lesssim10^{-4}$), a super-Earth opens two annular gaps to either side of its orbit by the action of Lindblad torques. This double gap and its associated gas pressure gradients cause dust particles to be dragged by gas into three rings: one ring sandwiched between the two gaps, and two rings located at the gap edges farthest from the planet. Depending on system parameters, additional rings may manifest for a single planet. A double gap located at tens of AUs from a host star in Taurus can be detected in the dust continuum by the Atacama Large Millimeter Array (ALMA) at an angular resolution of ~0".03 after two hours of integration. Ring and gap features persist in a variety of background disk profiles, last for thousands of orbits, and change their relative positions and dimensions depending on the speed and direction of planet migration. Candidate double gaps have been observed by ALMA in systems like HL Tau (D5 and D6) and TW Hya (at 37 and 43 AU); we submit that each double gap is carved by one super-Earth in nearly inviscid gas.Comment: 23 pages, 1 table, 14 figures, ApJ accepte

Robustness of Network of Networks with Interdependent and Interconnected links

Robustness of network of networks (NON) has been studied only for dependency coupling (J.X. Gao et. al., Nature Physics, 2012) and only for connectivity coupling (E.A. Leicht and R.M. D Souza, arxiv:0907.0894). The case of network of n networks with both interdependent and interconnected links is more complicated, and also more closely to real-life coupled network systems. Here we develop a framework to study analytically and numerically the robustness of this system. For the case of starlike network of n ER networks, we find that the system undergoes from second order to first order phase transition as coupling strength q increases. We find that increasing intra-connectivity links or inter-connectivity links can increase the robustness of the system, while the interdependency links decrease its robustness. Especially when q=1, we find exact analytical solutions of the giant component and the first order transition point. Understanding the robustness of network of networks with interdependent and interconnected links is helpful to design resilient infrastructures

Percolation on interacting networks with feedback-dependency links

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here we develop a mathematical framework and study numerically and analytically percolation of interacting networks with feedback-dependency links. We find that when nodes of between networks are lowly connected, the system undergoes from second order transition through hybrid order transition to first order transition as coupling strength increases. And, as average degree of each inter-network increases, first order region becomes smaller and second-order region becomes larger but hybrid order region almost keep constant. Especially, the results implies that average degree \bar{k} between intra-networks has a little influence on robustness of system for weak coupling strength, but for strong coupling strength corresponding to first order transition system become robust as \bar{k} increases. However, when average degree k of inter-network is increased, the system become robust for all coupling strength. Additionally, when nodes of between networks are highly connected, the hybrid order region disappears and the system first order region becomes larger and secondorder region becomes smaller. Moreover, we find that the existence of feedback dependency links between interconnecting networks makes the system extremely vulnerable by comparing non-feedback condition for the same parameters.First author draf

Simple spatial scaling rules behind complex cities

Although most of wealth and innovation have been the result of human interaction and cooperation, we are not yet able to quantitatively predict the spatial distributions of three main elements of cities: population, roads, and socioeconomic interactions. By a simple model mainly based on spatial attraction and matching growth mechanisms, we reveal that the spatial scaling rules of these three elements are in a consistent framework, which allows us to use any single observation to infer the others. All numerical and theoretical results are consistent with empirical data from ten representative cities. In addition, our model can also provide a general explanation of the origins of the universal super- and sub-linear aggregate scaling laws and accurately predict kilometre-level socioeconomic activity. Our work opens a new avenue for uncovering the evolution of cities in terms of the interplay among urban elements, and it has a broad range of applications.This work is supported by the National Natural Science Foundation of China under Grant Nos. 61673070, 61773069, 71731002 and the Fundamental Research Funds for the Central Universities with the Grant No. 2015KJJCB13, and also partially supported by NSF Grants PHY-1505000, CMMI-1125290, CHE-1213217, DTRA Grant HDTRA1-14-1-0017, DOE Grant DE-AC07-05Id14517. J.Z. acknowledges discussions with Prof. Bettencourt of the Santa Fe Institute, Dr. Lingfei Wu of Arizona State University, and Profs. Yougui Wang and Qinghua Chen of Beijing Normal University. R.L. acknowledges helpful discussions with and comments from Dr. Remi Louf in CASA, University College London, Dr. Longfeng Zhao from Huazhong (Central China) Normal University, and selfless help from Prof. Yougui Wang. R.L. is also supported by the Chinese Scholarship Council. (61673070 - National Natural Science Foundation of China; 61773069 - National Natural Science Foundation of China; 71731002 - National Natural Science Foundation of China; 2015KJJCB13 - Fundamental Research Funds for the Central Universities; PHY-1505000 - NSF; CMMI-1125290 - NSF; CHE-1213217 - NSF; HDTRA1-14-1-0017 - DTRA Grant; DE-AC07-05Id14517 - DOE; Chinese Scholarship Council)Published versio

Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application

The limited penetrable horizontal visibility algorithm is a new time analysis tool and is a further development of the horizontal visibility algorithm. We present some exact results on the topological properties of the limited penetrable horizontal visibility graph associated with random series. We show that the random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution $P(k)\sim exp[-\lambda (k-2\rho-2)], \lambda = ln[(2\rho+3)/(2\rho+2)],\rho=0,1,2,...,k=2\rho+2,2\rho+3,...$, independent of the probability distribution from which the series was generated. We deduce the exact expressions of the mean degree and the clustering coefficient and demonstrate the long distance visibility property. Numerical simulations confirm the accuracy of our theoretical results. We then examine several deterministic chaotic series (a logistic map, the H$\acute{e}$non map, the Lorentz system, and an energy price chaotic system) and a real crude oil price series to test our results. The empirical results show that the limited penetrable horizontal visibility algorithm is direct, has a low computational cost when discriminating chaos from uncorrelated randomness, and is able to measure the global evolution characteristics of the real time series.Comment: 23 pages, 12 figure