Mutual Collisions Research Articles

The strength of outgassed porous dust aggregates

Context. Outgassing of dust-ice aggregates plays an important role on the surfaces of cometary nuclei as well as for snow-line crossings in protoplanetary disks. Aims. To assess the stability of desiccated dust aggregates, we measured the tensile strength of silica dust samples over a wide range of volume filling factors. Methods. We produced these silica dust samples over a wide range of volume filling factors by gently evaporating dust-ice mixtures with various dust-to-ice mass ratios under vacuum conditions. The tensile strengths of these samples were then measured using the standardized Brazilian disk test. Experiments were performed in a vacuum and at room temperature but were also compared to measurements in air at room temperature and in a vacuum at elevated temperatures. Results. For spherical amorphous silica dust, we find no influence of the environmental conditions (air, vacuum, or heating) on the measured tensile strength. However, for angular crystalline silica dust we see a strong increase in tensile strength in a vacuum compared to air and an even higher increase when the samples are heated in a vacuum. For the spherical silica dust samples, we find a characteristic increase in the tensile strength with decreasing particle size. The tensile strength of samples with identical particle sizes increases strongly with an increasing volume filling factor. Extrapolation of our data to a volume filling factor of 0.1 (90% porosity) shows that a tensile strength as low as 1 Pa can be reached. Conclusions. Numerical simulations show that evaporating water ice in the subsurface layers of comets can reach gas pressures of ~1 Pa. Thus, a desiccated dust layer with a 10% volume filling factor should be detachable and released into the cometary coma. Using a relation between the tensile strength and the critical fragmentation energy, we predict the break-up speed of dust aggregates in mutual collisions as a function of the volume filling factor. Furthermore, we discuss the susceptibility of the aggregates to ram pressure. These values are relevant for protoplanetary disk research and for meteoroids entering planetary atmospheres.

Properties of outer solar system pebbles during planetesimal formation from meteor observations

Observations of proto-planetary disks, as well as theoretical modeling, suggest that in the late stages of accretion leading up to the formation of planetesimals, particles grew to pebbles the size of 1-mm to tens of cm, depending on the location and ambient conditions in the disk. That is the same size range that dominates the present-day comet and primitive asteroid mass loss. Meteoroids that size cause visible meteors on Earth. Here, we hypothesize that the size distribution and the physical and chemical properties of young meteoroid streams still contain information about the conditions in the solar nebula during these late stages of accretion towards planetesimal formation. If so, they constrain where long-period (Oort Cloud) comets, Jupiter-family (Scattered Disk Kuiper Belt) comets, and primitive asteroids (Asteroid Belt) formed. From video and visual observations of 47 young meteor showers, we find that freshly ejected meteoroids from long-period comets tend to have low bulk density and are distributed with equal surface area per log-mass interval (magnitude distribution index χ ∼ 1.85), suggesting gentle accretion conditions. Jupiter-family comets, on the other hand, mostly produce meteoroids twice as dense and distributed with a steeper χ ∼ 2.15 or even χ ∼ 2.5, which implies that those pebbles grew from particles fragmenting in a collisional cascade or by catastrophic collisions, respectively. Some primitive asteroids show χ > 2.5, with most mass in small particles, indicating an even more aggressive fragmentation by processes other than mutual collisions. Both comet populations contain an admixture of compact materials that are sometimes sodium-poor, but Jupiter-family comets show a higher percentage (∼8% on average) than long-period comet showers (∼4%) and a wider range of percentages among comets. While there are exceptions in both groups, the implication is that most long-period comets formed under gentle particle growth conditions, possibly near the 30 AU edge of the Trans Neptunian Disk, while most Jupiter family comets formed closer to the Sun where pebbles reached or passed the fragmentation barrier, and primitive asteroids formed in the region where the cores of the giant planets formed. This is possible if the Scattered Disk represents all objects scattered by Neptune during its migration, while the present-day outer Oort cloud formed only during and after the time of the planet instability, well after the Sun had moved away from sibling stars.

Applications of Fast Magnetic Reconnection Models to the Atmospheres of the Sun and Protoplanetary Disks

Partially ionized plasmas consist of charged and neutral particles whose mutual collisions modify magnetic reconnection compared with the fully ionized case. The collisions alter the rate and locations of the magnetic dissipation heating and the distribution of energies among the particles accelerated into the nonthermal tail. We examine the collisional regimes for the onset of fast reconnection in two environments: the partially ionized layers of the solar atmosphere, and the protoplanetary disks that are the birthplaces for planets around young stars. In both these environments, magnetic nulls readily develop into resistive current sheets in the regime where the charged and neutral particles are fully coupled by collisions, but the current sheets quickly break down under the ideal tearing instability. The current sheets collapse repeatedly, forming magnetic islands at successively smaller scales, until they enter a collisionally decoupled regime where the magnetic energy is rapidly turned into heat and charged-particle kinetic energy. Small-scale, decoupled fast reconnection in the solar atmosphere may lead to preferential heating and energization of ions and electrons that escape into the corona. In protoplanetary disks such reconnection causes localized heating in the atmospheric layers that produce much of the infrared atomic and molecular line emission observed with the Spitzer and James Webb Space Telescopes.

Massive Debris Disks May Hinder Secular Stirring by Planetary Companions: An Analytic Proof of Concept

Debris disks or exo-Kuiper belts, detected through their thermal or scattered emission from their dusty components, are ubiquitous around main-sequence stars. Since dust grains are short-lived, their sustained presence is thought to require dynamical excitation, i.e., “stirring,” of a massive reservoir of large planetesimals, such that mutual collisions are violent enough to continually supply fresh dust. Several mechanisms have been proposed to explain debris disk stirring, with the commonly accepted being long-term, secular planet–debris disk interactions. However, while effective, existing planet-stirring models are rudimentary; namely, they ignore the (self-)gravity of the disk, treating it as a massless reservoir of planetesimals. Here, using a simple analytical model, we investigate the secular interactions between eccentric planets and massive, external debris disks. We demonstrate that the disk gravity drives fast apsidal precession of both planetesimal and planetary orbits, which, depending on the system parameters, may well exceed the planet-induced precession rate of planetesimals. This results in strong suppression of planetesimal eccentricities and thus relative collisional velocities throughout the disk, often by more than an order of magnitude when compared to massless disk models. We thus show that massive debris disks may hinder secular stirring by eccentric planets orbiting near, e.g., the disk’s inner edge, provided the disk is more massive than the planet. We provide simple analytic formulae to describe these effects. Finally, we show that these findings have important implications for planet inferences in debris-bearing systems, as well as for constraining the total masses of debris disks (as done for β Pic).

Deep learning‐based space debris detection for space situational awareness: A feasibility study applied to the radar processing

AbstractThe increasing number of space objects (SO), debris, and constellation of satellites in Low Earth Orbit poses a significant threat to the sustainability and safety of space operations, which must be carefully and efficiently addressed to avoid mutual collisions. The space situational awareness is currently addressed by an ensemble of radar and radio‐telescopes that detect and track SO. However, a large part of space debris is composed of very small and tiny metallic objects, very difficult to detect. The authors demonstrate the benefits of using deep learning (DL) architectures for small space object detection by radar observations. TIRA radio telescope has been simulated to generate range‐Doppler maps, then used as inputs for object detection exploiting You‐Only‐Look‐Once (YOLO) frameworks. The results demonstrate that the object detection by using YOLO algorithms outperform conventional target detection approaches, thus indicating the potential benefits of using DL techniques for space surveillance applications.

Dynamic modeling and motion prediction of two projectiles launched successively underwater

The technology of successive underwater launching of multiple projectiles is a growing trend in military development. However, the process is complex, and the wake field formed after the launch of the first projectile can significantly impact the trajectories and attitudes of the subsequent projectiles. This can potentially lead to mutual collisions or destabilization of motion among the projectiles, resulting in mission failures. In this paper, a mathematical model incorporating unknown parameters is proposed to effectively describe the underwater motion of projectiles. To refine this model, we employed a system identification method that utilizes experimentally validated numerical simulation data to solve for the unknown parameters. The model is then used to predict the trajectory and attitude changes of the projectiles across various launching conditions. The results indicate that to ensure a successful launch mission, certain adjustments must be made. These adjustments include increasing the launching velocity within acceptable limits, extending the launch time interval, increasing the spatial distances between projectiles, and reducing the platform velocity.

A Crater Chronology for the Jupiter’s Trojan Asteroids

We present a new crater chronology for Jupiter’s Trojan asteroids. This tool can be used to interpret the collisional history of the bodies observed by NASA’s Lucy mission. The Lucy mission will visit a total of six Trojan asteroids: Eurybates, Polymele, Orus, Leucus, and the near-equal-mass binary Patroclus–Menoetius. In addition, Eurybates and Polymele each have a small satellite. Here we present a prediction of Trojan cratering based on current models of how the solar system and the objects themselves evolved. We give particular emphasis to the time lapsed since their implantation into stable regions near Jupiter’s Lagrangian L4 and L5 points. We find that cratering on Trojans is generally dominated by mutual collisions, with the exception of a short period of time (∼10 Myr) after implantation, in which cometary impacts may have been significant. For adopted crater scaling laws, we find that the overall spatial density of craters on Trojans is significantly lower than that of Main Belt asteroids on surfaces with similar formation ages. We also discuss specific predictions for similar-sized Eurybates and Orus, and the binary system Patroclus–Menoetius.

New Insights into Variations in Enceladus Plume Particle Launch Velocities from Cassini-VIMS Spectral Data

Enceladus’s plume consists mainly of a mixture of water vapor and solid ice particles that may originate from a subsurface ocean. The physical processes underlying Enceladus’s plume particle dynamics are still being debated, and quantifying the particles’ size distribution and launch velocities can help constrain these processes. Cassini’s Visual and Infrared Mapping Spectrometer observed the Enceladus plume over a wavelength range of 0.9–5.0 μm for a significant fraction of Enceladus’s orbital period on three dates in the summer of 2017. We find that the relative brightness of the plume on these different dates varies with wavelength, implying that the particle size distribution in the plume changes over time. These observations also enable us to study how the particles’ launch velocities vary with time and observed wavelength. We find that the typical launch velocity of particles remains between 140 and 148 m s−1 at wavelengths between 1.2 and 3.7 μm. This may not be consistent with prior models where particles are only accelerated by interactions with the vent walls and gas and could imply that mutual particle collisions close to the vent are more important than previously recognized.

Resonant collisions among multi-breathers in the Mel’nikov system

We study the resonant collisions among multi-breathers in the Mel’nikov system, which is the Kadomtsev–Petviashvili (KP) equation coupled to the nonlinear Schrödinger equation. The two-breather resonant collisions are classified into two different types: (i) the emergence of a Y–shaped pattern when the breathers are oblique on each other; and (ii) the occurrence of fission of one breather into two breathers or fusion of two breathers into a single one when the involving breathers are parallel to each other. The resonant collision of three breathers generates more complicated collision scenarios. They are divided into separate and mutual resonant collisions depending on whether the whole breathers are involved in the collision process. On this basis, each collision mode demonstrates two distinct forms: (i) diverse web–shaped waveforms with three to five infinitely long breathers when three breathers are obliquely intersecting, where some waveforms display a polygonal pattern under certain parametric constraints; and (ii) the combination of plentiful fission or fusion processes when the breathers are parallel to each other. The parameter constraints are discussed in detail for the classification of resonant collision scenarios.

Planetesimal Growth in Evolving Protoplanetary Disks: Constraints from the Pebble Supply

In the core accretion model, planetesimals grow by mutual collisions and engulfing millimeter-to-centimeter particles, i.e., pebbles. Pebble accretion can significantly increase the accretion efficiency and help explain the presence of planets on wide orbits. However, the pebble supply is typically parameterized as a coherent pebble mass flux, sometimes being constant in space and time. Here we solve the dust advection and diffusion within viciously evolving protoplanetary disks to determine the pebble supply self-consistently. The pebbles are then accreted by planetesimals interacting with the gas disk via gas drag and gravitational torque. The pebble supply is variable with space and decays with time quickly, with a pebble flux below 10 M ⊕ Myr−1 after 1 Myr in our models. As a result, only when massive planetesimals (>0.01 M ⊕) are luckily produced by the streaming instability or the disk has low viscosity (α ∼ 0.0001) can the herd of planetesimals grow over a Mars mass within 2 Myr. By then, planetesimals only capture pebbles about 50 times their mass and as little as 10 times beyond 20 au due to limited pebble supply. Further studies considering multiple dust species in various disk conditions are warranted to fully assess the realistic pebble supply and its influence on planetesimal growth.

An Analytical Theory for the Growth from Planetesimals to Planets by Polydisperse Pebble Accretion

Pebble accretion is recognized as a significant accelerator of planet formation. Yet only formulae for single-sized (monodisperse) distribution have been derived in the literature. These can lead to significant underestimates for Bondi accretion, for which the best accreted pebble size may not be the one that dominates the mass distribution. We derive in this paper the polydisperse theory of pebble accretion. We consider a power-law distribution in pebble radius, and we find the resulting surface and volume number density distribution functions. We derive also the exact monodisperse analytical pebble accretion rate for which 3D accretion and 2D accretion are limits. In addition, we find analytical solutions to the polydisperse 2D Hill and 3D Bondi limits. We integrate the polydisperse pebble accretion numerically for the MRN distribution, finding a slight decrease (by an exact factor 3/7) in the Hill regime compared to the monodisperse case. In contrast, in the Bondi regime, we find accretion rates 1–2 orders of magnitude higher compared to monodisperse, also extending the onset of pebble accretion to 1–2 orders of magnitude lower in mass. We find megayear timescales, within the disk lifetime, for Bondi accretion on top of planetary seeds of masses 10−6 to 10−4 M ⊕, over a significant range of the parameter space. This mass range overlaps with the high-mass end of the planetesimal initial mass function, and thus pebble accretion is possible directly following formation by streaming instability. This alleviates the need for mutual planetesimal collisions as a major contribution to planetary growth.

Dual-Species All-Optical Magnetometer Based on a Cs-K Hybrid Vapor Cell

We build and characterize a dual-species all-optical magnetometer based on a Cs-K hybrid vapor cell with a paraffin coating. Using the amplitude-modulated Bell-Bloom scheme and Faraday-rotation measurement, we show that the spin polarization of Cs and K atoms can be generated and detected independently. The noise floor of the Cs channel is $35\phantom{\rule{0.2em}{0ex}}\mathrm{fT}/\sqrt{\mathrm{Hz}}$, whereas that of the K channel is $260\phantom{\rule{0.2em}{0ex}}\mathrm{fT}/\sqrt{\mathrm{Hz}}$, which is affected mainly by the magnetic resonance broadening in the presence of mutual spin-exchange collisions. With use of the proposed device, in situ magnetic field stabilization is realized by use of (i) the K channel to monitor and compensate for the magnetic field noise and (ii) the Cs channel to read out the magnetic field after active stabilization. The low-frequency magnetic field noise can be suppressed by a factor of more than $1000$ in the presence of magnetic field gradients. When applied to magnetic field stabilization, such in situ measurement is achieved more compactly and is more accurate than ex situ measurement that is realized with two spatially separated vapor cells. In addition, the proposed scheme could also be applied in a practical ground-surveying magnetometer based on dual working species.

Vector breathers in the Manakov system

AbstractWe study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two‐component extension of the one‐dimensional focusing nonlinear Schrödinger equation—the Manakov system. With the dressing method, we generate the multibreather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovačič, Nonlinearity 28(9), 3101, (2015)], the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two‐component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then, we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two‐breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type I breathers participate in the development of modulation instability from small‐amplitude perturbations withing the superregular scenario, while the breathers of types II and III, belonging to the stable branch of the dispersion law, are not involved in this process.

Growing the seeds of pebble accretion through planetesimal accretion

We explore the growth of planetary embryos by planetesimal accretion up to and beyond the point at which pebble accretion becomes efficient at the so-called Hill-transition mass. Both the transition mass and the characteristic mass of planetesimals that formed by the streaming instability increase with increasing distance from the star. We developed a model for the growth of a large planetesimal (embryo) embedded in a population of smaller planetesimals formed in a filament by the streaming instability. The model includes in a self-consistent way the collisional mass growth of the embryo, the fragmentation of the planetesimals, the velocity evolution of all involved bodies, and the viscous spreading of the filament. We find that the embryo accretes all available material in the filament during the lifetime of the protoplanetary disc only in the inner regions of the disc. In contrast, we find little or no growth in the outer parts of the disc beyond 5-10 AU. Overall, our results demonstrate very long timescales for collisional growth of planetesimals in the regions of the protoplanetary disc in which giant planets form. This means that in order to form giant planets in cold orbits, pebble accretion must act directly on the largest bodies present in the initial mass function of planetesimals with little or no help from mutual collisions.

Management of breather interactions

We propose theoretically and confirm experimentally a general approach to manage multiple nonlinear interactions of coherent solitary wave structures on an unstable background---breathers. It allows adjusting the initial positions and phases of more than two moving breathers to observe various desired wave states at controllable moments of wave evolution. Our theoretical framework relies on exact multibreather solutions to the one-dimensional focusing nonlinear Schr\"odinger equation and asymptotic expressions describing shifts of breather positions and phases acquired by them in mutual collisions. As proof-of-principle, we consider a couple of separated pairs of breathers initially synchronized in a small-amplitude patterns; meanwhile, our approach can be generalized to other breather types and wave states. We obtain an explicit expression for the separation interval between the pairs so that the interactions of the breathers from the neighboring patterns lead to the formation of an extreme amplitude wave or recurrence to the initial small-amplitude state. Experiments are carried out on a light wave platform with a nearly conservative optical fiber system, which accurately reproduces the predicted dynamics and proves the viability of our nonlinear wave theory.

Self-Collision Avoidance Control of Dual-Arm Multi-Link Robot Using Neural Network Approach

The problem of mutual collisions of manipulators of a dual-arm multi-link robot (so-called self-collisions) arises during the performance of a cooperative technological operation. Self-collisions can lead to non-fulfillment of the technological operation or even to the failure of the manipulators. In this regard, it is necessary to develop a method for online detection and avoidance of self-collisions of manipulators. The article presents a method for detecting and avoiding self-collisions of multi-link manipulators using an artificial neural network by the example of the dual-arm robot SAR-401. A comparative analysis is carried out and the architecture of an artificial neural network for self-collisions avoidance control of dual-arm robot manipulators is proposed. The novelty of the proposed approach lies in the fact that it is an alternative to the generally accepted methods of detecting self-collisions based on the numerical solution of inverse kinematics problems for manipulators in the form of nonlinear optimization problems. Experimental results performed based on MATLAB model, the simulator of the robot SAR-401 and on the real robot itself confirmed the applicability and effectiveness of the proposed approach. It is shown that the detection of possible self-collisions using the proposed method based on an artificial neural network is performed approximately 10 times faster than approaches based on the numerical solution of the inverse kinematics problem while maintaining the specified accuracy.

The 3D Direct Simulation Monte Carlo Study of Europa’s Gas Plume

Europa has been spotted as having water outgassing activities by space- and ground-based telescopes as well as reanalysis of the Galileo data. We adopt a 3D Direct Simulation Monte Carlo (DSMC) model to investigate the observed plume characteristics of Europa assuming that supersonic expansion originated from the subsurface vent. With a parametric study of the total gas production rate and initial gas bulk velocity, the gas number density, temperature and velocity information of the outgassing plumes from various case studies were derived. Our results show that the plume gases experience acceleration through mutual collisions and adiabatic cooling when exiting from the surface. The central part of the plume with relatively large gas production rates (1029 and 1030 H2O s−1) was found to sustain thermal equilibrium and near continuum condition. Column density maps integrated along two different viewing angles are presented to demonstrate the importance of the projection effect on remote sensing diagnostics. Finally, the density profiles at different altitudes are provided to prepare for observations of Europa’s plumes including upcoming spacecraft missions such as JUICE and Europa Clipper.

A robotic surgical tool with continuum wrist, kinematically optimized curved stem, and collision avoidance kinematics for single port procedure

Robot-assisted minimally invasive key-hole surgeries have thrived in recent decades, primarily robotic MPL (Multi-Port Laparoscopy) and SPL (Single Port Laparoscopy). In robotic MPL, surgical manipulators are mainly equipped with straight-stemmed tools with distal wrists. For robotic SPL, when these MPL tools and manipulators are used via a single entry, they often suffer from mutual collisions. On the other hand, dedicated SPL manipulators/tools usually lack adequate payload capabilities due to the slim multi-joint designs. To facilitate SPL, a SORST (Stem-Optimized Robotic Surgical Tool) with a continuum wrist and a kinematically optimized rigid curved stem is proposed. The SORST's curved but rigid stem can help reduce possible collisions and maintain the tool's payload capability. In this curved-stem tool, the tendon tension might be difficult to properly maintain, due to the fact that the tendon may get in contact with the guiding cannula with uncertain friction. Hence, the continuum wrist is introduced for enhancing distal dexterity. Furthermore, an instantaneous inverse kinematics framework is proposed to simultaneously avoid mutual collisions and satisfy the remote-center-of-motion constraint, joint range/velocity limits, and the desired task space twist. Extensive experimental verifications were conducted to demonstrate the efficacy.

MAIN TECHNOLOGICAL FACTORS DETERMINING THE EFFICIENCY AND QUALITY OF THE VIBRATION PROCESS

The factors that determine the efficiency and quality of vibration treatment are indicated. Characteristic cases of interaction of abrasive granules with the processed surface are noted. The influence of the hardness of the processed part material and the shape of its surface, as well as the influence of chemically active solutions on the efficiency and quality of vibration processing, is substantiated. The characteristics of abrasive granules and their volume ratio with the processed parts are given. It is indicated that the underestimation of the possibilities of vibration processing technologies is explained by their insufficient studies. It has been established that vibration processing, depending on the characteristics and composition of the processing medium, is a mechanical or mechanochemical removal of the smallest particles of metal or its oxides and plastic deformation of microroughness due to mutual collisions of the medium granules with the processed surface, caused by vibrations of the reservoir in which the processing medium and, the processed parts are placed. It is noted that, according to the classification, vibration treatment refers to mechanical processing methods and, in particular, to the group of mechanical-chemical processing methods or to combined methods when chemically active solutions are introduced into the working medium, It is also noted that vibration treatment refers to dynamic, and for technological purposes – to dimensionless processing methods, according to the type of tool used - to the group of processing methods with a free abrasive. It has been established that the efficiency of vibration processing depends on the oscillation modes of the vibrating machine, the mass of the processed parts and abrasive granules, the hardness of the parts material and the shape of their treated surfaces, the characteristics of the abrasive medium, the volume ratio of the parts and abrasive granules, as well as on the composition of the chemically active solution. The characteristic cases of interaction of abrasive granules with the processed surface are given. The situations of the highest processing productivity for performing the operations of vibration grinding, vibration polishing, washing and descaling have been established. It is noted how the hardness of the processed part and the shape of their surface affects the performance and quality of vibration processing operations. The characteristics of the working medium, which affects the efficiency and quality of vibration treatment, are given, including the influence of grain size and hardness of the material of abrasive granules. The volume ratios of abrasive and processed parts are considered. The types of actions on the vibration treatment processes are given.

Spraying of Composite Liquid Fuels Based on Types of Coal Preparation Waste: Current Problems and Achievements: Review

This article discusses the atomization of composite liquid fuels. A large group of injectors is considered. A comparative analysis of the atomization characteristics (droplet sizes and velocities, jet opening angles) and the influence of the fuel characteristics (density, viscosity, component composition) and the process parameters (the ratio of the fuel–air mass flow rates, the features of the jet formation) has been carried out. Finally, the most effective types of injectors, which provide for the necessary characteristics of fuel atomization for its combustion, have been determined. The most favorable conditions for the applicability of each type of atomization have been formulated. Possible mechanisms of secondary fragmentation of droplets of composite fuels have been analyzed: those resulting from mutual collisions of droplets in the flux and from the interaction with a solid surface as well as those resulting from thermal overheating in the presence of a phase boundary or a large gradient of component volatility. A conclusion is made about the need of using a synergistic effect of primary and secondary atomization of fuel suspension droplets.