G. Chabrier, F. Douchin, A. Y. Potekhin
We briefly examine the properties of dense plasmas characteristic of the atmospheres of neutron stars and of the interior of massive white dwarfs. These astrophysical bodies are natural laboratories to study respectively the problem of pressure ionization of hydrogen in a strong magnetic field and the crystallization of the quantum one-component-plasma at finite temperature.
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The term ``dynamo'' means different things to the laboratory fusion plasma and astrophysical plasma communities. To alleviate the resulting confusion and to facilitate interdisciplinary progress, we pinpoint conceptual differences and similarities between laboratory plasma dynamos and astrophysical dynamos. We can divide dynamos into three types: 1. magnetically dominated helical dynamos which sustain a large scale magnetic field against resistive decay and drive the magnetic geometry toward the lowest energy state, 2. flow-driven helical dynamos which amplify or sustain large scale magnetic f…
Three known examples of coherent emission in radio astronomical sources are reviewed: plasma emission, electron cyclotron maser emission (ECME) and pulsar radio emission.
This article illustrates how very small deviations from the Maxwellian exponential tail, while leaving unchanged bulk quantities, can yield dramatic effects on fusion reaction rates and discuss several mechanisms that can cause such deviations.
Dynamical generation of macroscale magnetic fields and fast flows in a four-component astrophysical plasma
本研究探讨了四组分天体物理尘埃等离子体中宏观磁场与流的产生机制，指出动能主导时适用直流发电机机制，磁能主导时适用反向/直流发电机机制，并发现Alfvén Mach数受密度与螺旋度影响。
The ratio of ion to electron heating due to the dissipation of Alfvenic turbulence in astrophysical plasmas is calculated based on a cascade model for turbulence in weakly collisional plasmas. Conditions for validity of this model are discussed, a prescription for the turbulent heating is presented, and it is applied to predict turbulent heating in accretion disks and the interstellar medium.
Through relating a self-generated magnetic field to the regular motion of free electrons that is responsible for the magnetic field generation in astrophysical plasmas, we solve the Boltzmann kinetic equation in the presence of the self-generated magnetic fields to obtain a steady-state, collisional invariant analytic solution of the equation.
We review the plasma phenomenology in the astrophysical sources which show appreciable radio emissions, namely Radio-Jets from Pulsars, Microquasars, Quasars and Radio-Active Galaxies. A description of their basic features is presented, then we discuss in some details the links between their morphology and the mechanisms that lead to the different radio-emissions, investigating especially the role played by the plasma configurations surrounding compact objects (Neutron Stars, Black Holes). For the sake of completeness, we briefly mention observational techniques and detectors, whose structure …
For degenerate astrophysical electron-positron plasmas we have considered dynamics of electromagnetic beams carrying angular momentum. It is found for arbitrary level of degeneracy such a beam having the power exceeding a certain critical value breaks up into many filaments, eventually leading to the formation of stable spatial solitons keeping zero field in the center of the structure.
IR spectra gained from laboratory investigations on analogue materials are an important tool for the interpretation of astronomical IR data and the understanding of the observed phenomena in DISM. Dust particles grown in argon/acetylene low pressure plasma are proposed for a new analog. We test the validity of proposed analog according to the criteria from the literature. We also present preliminary results of nitrogen band at 4.62 \$mu$m, which has been identified in dense cold dust.
Two distinct routes lead to the creation of multi--scale equilibrium structures in dense degenerate plasmas, often met in astrophysical conditions. By analyzing an e-p-i plasma consisting of degenerate electrons and positrons with a small contamination of mobile classical ions, we show the creation of a new macro scale $L_{\rm{macro}}$ (controlled by ion concentration). The temperature and degeneracy enhancement effective inertia of bulk e-p components also makes the effective skin depths larger (much larger) than the standard skin depth. The emergence of these intermediate and macro scales le…
We briefly review analytic approximations of thermodynamic functions of fully ionized nonideal electron-ion plasmas, applicable in a wide range of plasma parameters, including the domains of nondegenerate and degenerate, nonrelativistic and relativistic electrons, weakly and strongly coupled Coulomb liquids, classical and quantum Coulomb crystals. We present improvements to previously published approximations. Our code for calculation of thermodynamic functions based on the reviewed approximations is made publicly available.
We discuss the current understanding of the most basic properties of astrophysical MHD turbulence and trace the origins of the modern views and theoretical uncertainties to the ideas set forth in 1950s and 1960s by Iroshnikov, Kraichnan, Batchelor, Schlueter and Biermann. Universal aspects of the theory are emphasised. Two main astrophysical applications are touched upon: turbulence in the solar wind and in clusters of galaxies. These are, in a certain (very approximate) sense, two ``pure'' cases of small-scale turbulence, where theoretical models of the two main regimes of MHD turbulence -- w…
A laboratory simulation of astrophysical processes is one of the intensively developed areas of plasma physics. A new series of experiments has been launched recently on the Plasma Focus type facility in NRC Kurchatov Institute. The main goal is to study the mechanisms of the jet stabilization, due to which it can propagate at distances much greater than their transverse dimensions. The experiments with stationary gas filling revealed regimes in which a narrowly collimated plasma jet was formed, the head of which was no wider than several centimeters at jet propagation distances of up to 100 c…
We develop a two dimensional, self-consistent, compressible fluid model to study evolution of Alfvenic modes in partially ionized astrophysical and space plasmas. The partially ionized plasma consists mainly of electrons, ions and significant neutral atoms. The nonlinear interactions amongst these species take place predominantly through direct collision or charge exchange processes. Our model uniquely describe the interaction processes between two distinctly evolving fluids. In our model, the electrons and ions are described by a single fluid compressible magnetohydrodynamic (MHD) model and a…
The constraint imposed by magnetic helicity conservation on the alpha effect is considered for both magnetically and flow dominated self-organizing plasmas. Direct numerical simulations are presented for a dominant contribution to the alpha effect, which can be cast in the functional form of a total divergence of an averaged helicity flux, called the helicity-flux-driven alpha ( H$α$) effect. Direct numerical simulations of the H$α$ effect are prese nted for two examples---the magnetically dominated toroidal plasma unstable to tearing modes, and the flow-dominated accretion disk.
Nonlinear Coupling of Electromagnetic and Spin-Electron-Acoustic Waves in Spin-polarized Degenerate Relativistic Astrophysical Plasma
研究有限振幅电磁波在部分自旋极化简并等离子体中传播引发的不稳定性，发现其源于电磁波与纵向自旋电子声波的相互作用，且不稳定性随自旋极化增强而增加。
There are many astrophysical and laboratory scenarios where kinetic effects play an important role. These range from astrophysical shocks and plasma shell collisions, to high intensity laser-plasma interactions, with applications to fast ignition and particle acceleration. Further understanding of these scenarios requires detailed numerical modelling, but fully relativistic kinetic codes are computationally intensive, and the goal of one-to-one direct modelling of such scenarios and direct comparison with experimental results is still difficult to achieve. In this paper we discuss the issues i…
Electron Acceleration via Lower-Hybrid Drift Instability in Astrophysical Plasmas: Dependence on Plasma Beta and Suprathermal Electron Distributions
本研究拓展低杂漂移不稳定性（LHDI）电子加速研究至更广天体等离子体环境，通过准线性模型确定等离子体β与超热电子谱斜率的加速临界值，发现超热电子可显著提升高β等离子体中的LHDI电子加速效率。
Certain classes of astrophysical objects, namely magnetars and central engines of supernovae and gamma-ray bursts (GRBs), are characterized by extreme physical conditions not encountered elsewhere in the Universe. In particular, they possess magnetic fields that exceed the critical quantum field of 44 teragauss. Figuring out how these complex ultra-magnetized systems work requires understanding various plasma processes, both small-scale kinetic and large-scale magnetohydrodynamic (MHD). However, an ultra-strong magnetic field modifies the underlying physics to such an extent that many relevant…
Electromotive field in space and astrophysical plasmas
本文综述了空间与天体物理等离子体中电动势场的理论认识与航天器观测，阐明其在闭合湍流磁流体力学方程组中的关键作用，并简要介绍其在地球磁层、日球层激波、星际介质和相对论性喷流等天体物理场景中的应用。
Turbulence is a ubiquitous phenomenon in space and astrophysical plasmas, driving a cascade of energy from large to small scales and strongly influencing the plasma heating resulting from the dissipation of the turbulence. Modern theories of plasma turbulence are based on the fundamental concept that the turbulent cascade of energy is caused by the nonlinear interaction between counterpropagating Alfven waves, yet this interaction has never been observationally or experimentally verified. We present here the first experimental measurement in a laboratory plasma of the nonlinear interaction bet…
Photoionization and its inverse, electron-ion recombination, are key processes that influence many astrophysical plasmas (and gasses), and the diagnostics that we use to analyse the plasmas. In this review we provide a brief overview of the importance of photoionization and recombination in astrophysics. We highlight how the data needed for spectral analyses, and the required accuracy, varies considerably in different astrophysical environments. We then discuss photoionization processes, highlighting resonances in their cross-sections. Next we discuss radiative recombination, and low and high …
Strong gradients in plasma flows play a major role in space and astrophysical plasmas. A typical situation is that a static plasma equilibrium is surrounded by a plasma flow, which can lead to strong plasma flow gradients at the separatrices between field lines with different magnetic topologies, e.g., planetary magnetospheres, helmet streamers in the solar corona, or at the boundary between the heliosphere and interstellar medium. Within this work we make a first step to understand the influence of these flows towards the occurrence of current sheets in a stationary state situation. We concen…
Magnetic reconnection is a fundamental process in a plasma that facilitates the release of energy stored in the magnetic field by permitting a change in the magnetic topology. In this article we present a review of the current state of understanding of magnetic reconnection. We discuss theoretical results regarding the formation of current sheets in complex 3D magnetic fields, and describe the fundamental differences between reconnection in two and three dimensions. We go on to outline recent developments in modelling of reconnection with kinetic theory, as well as in the MHD framework where a…
The possibility of formation of quasibound states and ordered structures in dense plasma is investigated. The effective potentials of dense plasma are used. On the basis of these models the condition of the ordered structures formation in the system is obtained. It is shown that in the dense classical plasma the quasibound states form whereas the ordered structures form in the dense semiclassical plasma.
Nonlinear wave--coupling is studied in a multi-species degenerate astrophysical plasma consisting of two electron species (at different temperatures): a highly degenerate main component plus a smaller classical relativistic flow immersed in a static neutralizing ion background. It is shown that the high frequency electromagnetic (HF EM) waves, through their strong nonlinear interactions with the electron--acoustic waves (sustained by a multi-electron component (degenerate) plasma surrounding a compact astrophysical object) can scatter to lower frequencies so that the radiation observed faraway…
We investigate the rates for multielectron recombination within a dense plasma environment in local thermodynamic equilibrium (LTE). We find that these multielectron recombination rates can be high within dense plasmas, and they should be treated in the simulations of the plasmas created by intense radiation, in particular for plasmas created by intense VUV radiation from free-electron-laser (FEL) or for modelling the inertial confinement fusion (ICF) plasmas.
A heuristic approach for collisionless perpendicular diffusion of energetic particles is presented. Analytic forms for the corresponding diffusion coefficient are derived. The heuristic approach presented here explains the parameter $a^2$ used in previous theories in order to achieve agreement with simulations and its relation to collisionless Rechester & Rosenbluth diffusion. The obtained results are highly relevant for applications because previously used formulas are altered significantly in certain situations.
Macroscopic plasma polarization, which is created by gravitation and other mass-acting (inertial) forces in massive astrophysical objects is under discussion. Non-ideality effect due to strong Coulomb interaction of charged particles is introduced into consideration as a new source of such polarization. Simplified situation of totally equilibrium isothermal star without relativistic effects and influence of magnetic field is considered. The study is based on variational approach combined with "local density approximation". It leads to two local forms of thermodynamic equilibrium conditions: co…
The coupling between a rapidly expanding cloud of ionized debris and an ambient magnetized plasma is revisited with a hybrid (kinetic ion/fluid electron) simulation code that allows a study over a wide range of plasma parameters. Over a specified range of hypothetical conditions, simple scaling laws in terms of the total debris mass and explosion speed are derived and verified for the maximal size of the debris cloud and the fraction of debris that free-streams from the burst along the magnetic field. The amount of debris that escapes from the burst with minimal coupling to the background magn…
The values of electrical conductivity of plasma of stars with a magnetic field or moving in the magnetic field of the other component in a binary system could be of significant interest, since they are useful for the study of thermal evolution of such objects, cooling, nuclear burning of accreted matter, and the investigation of their magnetic fields. So, on the basis of numerically calculated values for the dense plasma conductivity in an external HF electric field, we determine the HF characteristics of astrophysical plasmas under extreme conditions. The examined range of frequencies covers …
How the turbulent energy cascade develops below the magnetohydrodynamic scales in space and astrophysical plasmas is a major open question. Here, we measure the power spectrum of magnetic fluctuations in Parker Solar Probe's observations close to the Sun and in state-of-the-art numerical simulations of plasma turbulence. Both reveal a power-law behavior with a slope compatible with $-11/3$ at scales smaller than the ion characteristic scales, steeper than what is typically observed in the solar wind and in the Earth's magnetosheath. We explain such behavior by developing a simple two-fluid mod…
The plasma astrophysical S factor for the 3He(D, p)4He fusion reaction was measured for the first time at temperatures of few keV, using the interaction of intense ultrafast laser pulses with molecular deuterium clusters mixed with 3He atoms. Different proportions of D2 and 3He or CD4 and 3He were mixed in the gas jet target in order to allow the measurement of the cross-section for the 3He(D, p)4He reaction. The yield of 14.7 MeV protons from the 3He(D, p)4He reaction was measured in order to extract the astrophysical S factor at low energies. Our result is in agreement with other S factor pa…
We have shown the generation/amplification of fast macro-scale plasma flows in the degenerate two-fluid astrophysical systems with initial turbulent (micro--scale) magnetic/velocity fields due to the Unified Reverse Dynamo/Dynamo mechanism. This process is simultaneous with and complementary to the micro-scale unified dynamo. It is found that the generation of macro--scale flows is an essential consequence of the magneto-fluid coupling; the generation of macro--scale fast flows and magnetic fields are simultaneous, they grow proportionately. The resulting dynamical flow acceleration is directl…
Resonant instabilities mediated by drag and electrostatic interactions in laboratory and astrophysical dusty plasmas
本文将共振阻力不稳定性中的缺陷特征值共振机制推广为共振不稳定性框架，揭示双流、丝状电离与声学RDI的物理共性及其实验室研究价值。
A theory for the ion acoustic wave damping in dense plasmas and warm dense matter, accounting for the Umklapp process, is presented. A higher decay rate compared to the prediction from the Landau damping theory is predicted for high-Z dense plasmas where the electron density ranges from $10^{21}$ to $ 10^{24} \mathrm{cm^{-3}}$ and the electron temperature is moderately higher than the Fermi energy.
A sufficient condition for maser instability in a weakly magnetized relativistic plasma with an isotropic particle distribution function is given. The maser growth rates and polarizations are computed starting from the exact dielectric permittivity tensor of a magnetized plasma. For very weak magnetic fields, our results confirm the approximate validity of the 'standard maser theory', which is based on the Einstein coefficients method, with one significant exception. For inclined propagation and realistic (small but finite) field, the growth rates of the two (nearly circular) polarizations dif…
A refined cascade model for kinetic turbulence in weakly collisional astrophysical plasmas is presented that includes both the transition between weak and strong turbulence and the effect of nonlocal interactions on the nonlinear transfer of energy. The model describes the transition between weak and strong MHD turbulence and the complementary transition from strong kinetic Alfven wave (KAW) turbulence to weak dissipating KAW turbulence, a new regime of weak turbulence in which the effects of shearing by large scale motions and kinetic dissipation play an important role. The inclusion of the e…
Fermi Dirac free electron model is applied to very dense plasmas with medium or low temperatures. While Boltzmann statistics can lead to very high densities of ionized electrons, only at very high temperatures, Fermi Dirac statistics can support the high densities of ionized electrons at medium or low temperatures due to the high degeneracies obtained in this model. Since very dense plasmas may be obtained at low temperatures the corresponding black body radiation with the plasma luminosity will be quite small. On the other hand gravitational effects might be quite large due to the high densit…
Here we discuss the possibility of employment of ultrarelativistic electron and proton bunches for generation of high plasma wakefields in dense plasmas due to the Cherenkov resonance plasma-bunch interaction. We estimate the maximum amplitude of such a wake and minimum system length at which the maximum amplitude can be generated at the given bunch parameters.
We have shown the simultaneous generation of macro-scale fast flows and strong magnetic fields in the 2-temperature relativistic electron-ion plasmas of astrophysical objects due to Unified Reverse Dynamo/Dynamo mechanism. The resulting dynamical magnetic field amplification and/or flow acceleration is directly proportional to the initial turbulent kinetic/magnetic (magnetic) energy; the process is very sensitive to relativistically hot electron-ion fraction temperature and magneto-fluid coupling. It is shown, that for realistic physical parameters of White Dwarfs accreting hot astrophysical f…
Magnetic reconnection is a fundamental process in a laboratory, magnetospheric, solar and astrophysical plasma, whereby magnetic energy is converted into heat, bulk kinetic energy and fast particle energy. Its nature in two dimensions is much better understood than in three dimensions (3D), where its character is completely different and has many diverse aspects that are currently being explored. Here we focus on the magnetohydrodynamics of 3D reconnection in the plasma environment of the solar system, especially solar flares. The theory of reconnection at null points, separators and quasi-sep…
Flow-Regulated Suprathermal Particle Acceleration in Weakly Collisional Astrophysical Plasmas
利用一维Fokker-Planck框架研究弱碰撞等离子体中超热粒子群的形成，引入系统性速度空间漂移项描述流驱动的净能量化，发现速度空间扩散主导超热尾形成，而流漂移调控其效率和谱特性。
A local turbulence model is developed to study energy cascades in the interstellar medium (ISM) based on self-consistent two-dimensional fluid simulations. The model describes a partially ionized magnetofluid interstellar medium (ISM) that couples a neutral hydrogen fluid with a plasma primarily through charge exchange interactions. Charge exchange interactions are ubiquitous in warm ISM plasma and the strength of the interaction depends largely on the relative speed between the plasma and the neutral fluid. Unlike small length-scale linear collisional dissipation in a single fluid, charge exc…
Small-scale turbulent dynamo is responsible for the amplification of magnetic fields on scales smaller than the driving scale of turbulence in diverse astrophysical media. Most earlier dynamo theories concern the kinematic regime and small-scale magnetic field amplification. Here we review our recent progress in developing the theories for the nonlinear dynamo and the dynamo regime in a partially ionized plasma. The importance of reconnection diffusion of magnetic fields is identified for both the nonlinear dynamo and magnetic field amplification during gravitational contraction. For the dynam…
The decays of the Langmuir waves in dense plasmas are computed using the dielectric function theory widely used in the solid state physics. Four cases are considered: a classical plasma, a Maxwellian plasma, a degenerate quantum plasma, and a partially degenerate plasma. The result is considerably different from the conventional Landau damping theory.
A self-consistent determination of the spectral function and the self-energy of electrons in a hot and dense plasma is reported. The self-energy is determined within the approximation of the screened potential. It is shown, that the quasi-particle concept is not an adequate concept for hot and dense plasmas, since the width of the spectral function has to be considered. As an example, the solar core plasma is discussed. An effective quasi-particle picture is introduced and results for the solar core plasma as well as for ICF plasmas are presented.
We present a model for the rate of temperature relaxation between electrons and ions in plasmas. The model includes self-consistently the effects of particle screening, electron degeneracy and correlations between electrons and ions. We successfully validate the model over a wide range of plasma coupling against molecular-dynamics simulations of classical plasma of like-charged electrons and ions. We present calculations of the relaxation rates in dense hydrogen and show that, while electron-ion correlation effects are indispensable in classical, like-charged plasmas at any density and tempera…
Effect of superthermal electrons on the Quantum electron acoustic double layers in dense astrophysical plasmas
本文研究致密天体等离子体中超热电子对电子声双层的影响，通过QHD模型和KdV方程发现kappa指数比密度比更主导地控制双层的振幅、宽度和极性。
We use particle-in-cell (PIC) simulations to study the nonlinear evolution of ion velocity space instabilities in an idealized problem in which a background velocity shear continuously amplifies the magnetic field. We simulate the astrophysically relevant regime where the shear timescale is long compared to the ion cyclotron period, and the plasma beta is ~ 1-100. The background field amplification in our calculation is meant to mimic processes such as turbulent fluctuations or MHD-scale instabilities. The field amplification continuously drives a pressure anisotropy with the perpendicular pre…
We present a theoretical investigation of the excitation of multiple electrostatic wakefields by the ponderomotive force of a short electromagnetic pulse propagating through a dense plasma. It is found that the inclusion of the quantum statistical pressure and quantum electron tunneling effects can qualitatively change the classical behavior of the wakefield. In addition to the well known plasma oscillation wakefield, with a wavelength of the order of the electron skin depth, which in a dense plasma is of the order of several nanometers, wakefields in dense plasmas with a shorter wavelength ar…
Propagation of right-hand circularly polarized laser into highly magnetized over-dense collisional plasma is analytically studied from basic equations. Highly magnetized plasma with density of 10^23 cm^-3 is efficiently heated from 100 eV to 1 keV within depth of around 20 μ{m} in one nanosecond by Nd:YAG laser with moderate irradiance of around 10^14 Wcm^-2.
When two plasmas collide, their interaction can be mediated by collisionless plasma instabilities or binary collisions between particles of each shell. By comparing the maximum growth rate of the collisionless instabilities with the collision frequency between particles of the shells, we determine the critical density separating the collisionless formation from the collisional formation of the resulting shock waves. This critical density is also the density beyond which the shock downstream is field free, as plasma instabilities do not have time to develop electromagnetic patterns. We further …
A local turbulence model is developed to study energy cascades in the heliosheath and outer heliosphere (OH) based on self-consistent two-dimensional fluid simulations. The model describes a partially ionized magnetofluid OH that couples a neutral hydrogen fluid with a plasma primarily through charge-exchange interactions. Charge-exchange interactions are ubiquitous in warm heliospheric plasma, and the strength of the interaction depends largely on the relative speed between the plasma and the neutral fluid. Unlike small-length scale linear collisional dissipation in a single fluid, charge-exc…
Heat flux suppression in collisionless plasmas for a large range of plasma $β$ is explored using two-dimensional particle-in-cell simulations with a strong, sustained thermal gradient. We find that a transition takes place between whistler-dominated (high-$β$) and double-layer-dominated (low-$β$) heat flux suppression. Whistlers saturate at small amplitude in the low beta limit and are unable to effectively suppress the heat flux. Electrostatic double layers suppress the heat flux to a mostly constant factor of the free streaming value once this transition happens. The double layer physics is …
We describe Pegasus, a new hybrid-kinetic particle-in-cell code tailored for the study of astrophysical plasma dynamics. The code incorporates an energy-conserving particle integrator into a stable, second-order--accurate, three-stage predictor-predictor-corrector integration algorithm. The constrained transport method is used to enforce the divergence-free constraint on the magnetic field. A delta-f scheme is included to facilitate a reduced-noise study of systems in which only small departures from an initial distribution function are anticipated. The effects of rotation and shear are implem…
Past and ongoing research activities at the Heidelberg heavy-ion storage-ring TSR are reviewed which aim at providing accurate absolute rate coefficients and cross sections of atomic collision processes for applications in astrophysics and magnetically confined fusion. In particular, dielectronic recombination and electron impact ionization of iron ions are discussed as well as dielectronic recombination of tungsten ions.
Variational Formulation of a Hybrid Kinetic and Gyrokinetic Model for Astrophysical and Laboratory Plasmas
该研究针对太阳风电磁湍流，提出含全动理学离子与 gyrokinetic 电子的拉格朗日模型，采用一致数学形式，通过高阶李变换微扰方法与变分法实现高效描述高频波和动理学电子效应的模型。
This letter discusses rotating magnetohydrodynamics (MHD) of a thin layer of astrophysical plasma. To describe a thin plasma layer with a free surface in a vertical external magnetic field we use the shallow water ap- proximation. The presence of a vertical magnetic field essentially changed the wave processes dynamics in astrophysical plasma compared to the neu- tral uid and plasma layer in a thoroidal magnetic field. In present case thre are three-waves nonlinear interactions. Using the asymptotic mul- tiscale we deduced nonlinear wave packets interaction equations: three magneto-Poincare wa…
It is shown that a small contamination of a relativistically hot electron component can induce a new scale (for structure formation) to a system consisting of an ion-degenerate electron plasma. Mathematically expression of this additional scale length is the increase in the index of quasi-equilibrium Beltrami-Bernoulli states that have been invoked to model several astrophysical systems of interest. The two species of electrons, due to different origin of their relativistic effective masses, behave as two distinct components (each with its own conserved helicity) and add to the richness of the…
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