Constraining the masses of planets in protoplanetary discs from the presence or absence of vortices – comparison with ALMA observations

ABSTRACT A massive planet in a protoplanetary disc will open a gap in the disc material. A steep gap edge can be hydrodynamically unstable, which results in the formation of vortices that can act as tracers for the presence of planets in observational results. However, in a viscous disc, the potential formation of these vortices is dependent on the time-scale over which the massive planet accretes mass and with a sufficiently long time-scale it is possible for no vortices to form. Hence, there is a connection between the presence of vortices and the growth time-scale of the planet and it may therefore be possible to exclude a planetary interpretation of observed structure from the absence of vortices. We have investigated the effect of the planet growth time-scale on vortex formation for a range of planet masses and viscosities and have found an approximate relation between the planet mass, viscosity and planet growth time-scale for which vortices are not formed within the disc. We then interpret these results in the light of recent observations. We have also found that planets do not need to be close to a Jupiter mass to form vortices in the disc if these discs have low viscosity, as these can be caused by planets as small as a few Neptune masses.

Design of Pressurized Vaulted Rectangular Conduits Using the Rough Model Method (Part 2)

The problem of calculating the diameter of a pressurized vaulted rectangular conduit is solved explicitly through two methods which are based on an referential rough model whose geometric and hydraulic characteristics are well defined. The first method provides a highly reliable approximate relation which gives very accurate results for all practical applications. The second method consists in introducing into the calculations a rectangular rough model whose geometric and hydraulic properties were judiciously chosen. The theoretical development leads to an equation of third degree whose resolution has been made possible by the hyperbolic and trigonometric functions. Practical examples are presented to explain the procedure of calculation.

Numerical Lifetime Prediction of Polymer Pipes Taking into Account Residual Stress

In this paper a methodology for assessment of residual stress effects on crack behaviour in the polymer pipe is developed. For simplicitys sake, a linear distribution of residual stresses across the pipe wall is assumed. Linear elastic fracture mechanics is used for the fracture mechanics analysis of the cracked pipe. An approximate relation for the stress intensity factor estimation for a crack in a polymer pipe, with residual stress taken into account is suggested and discussed. The methodology presented can be helpful for a rapid lifetime estimation of polyolefin pipelines.

Approximate Bisimulation and Optimization of Software Programs Based on Symbolic-Numeric Computation

To achieve behavior and structure optimization for a type of software program whose data exchange processes are represented by nonlinear polynomial systems, this paper establishes a novel formal description called a nonlinear polynomial transition system to represent the behavior and structure of the software program. Then, the notion of bisimulation for software programs is proposed based on the equivalence relation of corresponding nonlinear polynomial systems in their nonlinear polynomial transition systems. However, the exact equivalence is too strict in application. To enhance the flexibility of the relation among the different software systems, the notion of approximate bisimulation within a controllable error range and the calculation algorithm of approximate bisimulation based on symbolic-numeric computation are given. In this calculation, an approximate relation is represented as a MAX function that is resolved with the full filled method. At the same time, the actual error is calculable. An example on a multithreading program indicates that the approximate bisimulation relation is feasible and effective in behavior and structure optimization.

CALCULATION OF VOLUME AS FUNCTIONS OF TEMPERATURE AND PRESSURE IN ICE I CLOSE TO THE MELTING POINT

We calculate in this study the volume of ice I as functions of temperature and pressure close to the melting point by analyzing the experimental data for the thermal expansivity. Using an approximate relation, the temperature dependence of the volume is calculated at 202.4 MPa from the thermal expansivity of ice I. The pressure dependence of the volume is also calculated at 252.3 K from the isothermal compressibility of ice I close to the melting point. The volume calculated here as functions of temperature and pressure shows critical behavior close to the melting point in ice I, which can be tested by the experimental measurements.

EFFECTIVE SLIP LENGTH OF NANOSCALE MIXED-SLIP SURFACES

We present an approximate relation for the effective slip length for flows over mixed-slip surfaces with patterning at the nanoscale, whose minimum slip length is greater than the pattern length scale.