Orbital Motion Theory Research Articles

Simulation of dust grain charging under hypervelocity impact plasma environment

Dust plasma readily forms during hypervelocity impact, which serves as a source of plasma macroscopic charge separation and strong electromagnetic fields. In this study, we examine the dynamic evolution of surface charging of aluminum dust grains with micrometer or submicrometer sizes in a hypervelocity impact plasma environment based on the theory of orbital motion limited. As dust grains traverse the expanding plasma, plasma density and temperature decrease with increasing distance from the impact point. This leads to longer relaxation times for charging equilibrium (ranging from picoseconds to microseconds) and reduced equilibrium charges. The model incorporates thermionic and secondary electron emission effects on dust grain charging processes while also examining the impacts of five heating and cooling mechanisms on the thermal equilibrium temperatures of dust grains. Near the impact point, thermal equilibrium temperatures exceed aluminum's boiling point, which results in phase transition ablation processes. As dust grain temperatures increase, thermionic emission currents may dominate charging dynamics and influence final equilibrium charge numbers. High-temperature dust grains tend to acquire positive charges. Moreover, we observe that the radius of dust grains considerably affects charging processes, and smaller grain radii correspond to low equilibrium charges and longer relaxation times.

Beyond analytic approximations with machine learning inference of plasma parameters and confidence intervals

Machine learning techniques are used to construct models capable of inferring plasma state variables from non-emissive (LP) and emissive (EP) cylindrical Langmuir probes under conditions in which standard analytic theories are not applicable. Synthetic data sets, consisting of plasma parameters and probe characteristics computed kinetically in the orbital motion theory framework, are used to train and test regression models to infer electron densities, temperatures, and plasma potentials. Model skill metrics are introduced to determine uncertainty margins on inferred parameters, when models are applied to test sets not involved in the model optimization process. The different scalings and transformations required to obtain optimal accuracy are described in each case considered for both LPs and EPs. Excellent inferences are made for all three parameters considered from LP characteristics, but owing to the strong dependence on the plasma potential, and weak dependences on electron temperature and density with EPs, only plasma potential inferences are reported with acceptable accuracy for this type of probe. Our findings demonstrate that the combination of kinetic simulations and machine learning techniques is a promising and practical way to infer plasma parameters efficiently from cylindrical probes, under conditions beyond, and more general than those under which commonly used analytic approximations are valid.

Parametrization of current–voltage characteristics and operation domains of cylindrical emissive probes in collisionless Maxwellian plasmas at rest

Important progress has recently been made on the Orbital Motion Theory for cylindrical emissive probes immersed at rest in collisionless and Maxwellian plasmas. However, due to the computational cost of its numerical algorithm, only solutions for specific values of the physical parameters were found, thus preventing its direct application to the interpretation of experimental current–voltage characteristics ( curves). In this work, and thanks to an analytical analysis of a Jacobian matrix appearing in the algorithm, the computational cost was reduced by a factor in the order of , where is the number of grid points. This achievement, together with the implementation of parallel programming, allowed to construct a database with more than 18 000 curves for a broad range of physical parameters, including the emission level, the probe radius-to-Debye length, and the ion-to-electron temperature ratios. The boundaries in parameter space of the operational regimes of emissive probes, covering both orbital motion limited (OML) and space charge limited (SCL) transitions were computed. A novel OML/non-OML transition for emissive probes operating at low bias was found. The numerical results were used to propose useful analytical laws for the SCL boundary happening at negative bias, the reduction of the emitted electron current due to SCL effects, and the floating potential of emissive probes. The applications of the results to the modeling of low work function tethers and three experimental methods for measuring the plasma potential, i.e. the separation point, the inflection point, and the floating potential techniques, were discussed. The formation of an inverse sheath for strong emission was investigated.

Current-Voltage and Floating-Potential characteristics of cylindrical emissive probes from a full-kinetic model based on the orbital motion theory

To model the sheath structure around an emissive probe with cylindrical geometry, the Orbital-Motion theory takes advantage of three conserved quantities (distribution function, transverse energy, and angular momentum) to transform the stationary Vlasov-Poisson system into a single integro-differential equation. For a stationary collisionless unmagnetized plasma, this equation describes self-consistently the probe characteristics. By solving such an equation numerically, parametric analyses for the current-voltage (IV) and floating-potential (FP) characteristics can be performed, which show that: (a) for strong emission, the space-charge effects increase with probe radius; (b) the probe can float at a positive potential relative to the plasma; (c) a smaller probe radius is preferred for the FP method to determine the plasma potential; (d) the work function of the emitting material and the plasma-ion properties do not influence the reliability of the floating-potential method. Analytical analysis demonstrates that the inflection point of an IV curve for non-emitting probes occurs at the plasma potential. The flat potential is not a self-consistent solution for emissive probes.

Kinetic features of collisionless sheaths around polarized cylindrical emitters from the orbital motion theory

The kinetic features of the sheath around a cylindrical emitter immersed in collisionless plasma at rest are analysed. After finding self-consistently the electric potential by applying the Orbital Motion Theory to the Vlasov-Poisson system, the local distribution functions are reconstructed and the radial profiles of important macroscopic quantities (plasma densities, currents, and temperatures) are then computed. It is found that there can only be three kinds of holes that are bound by three different boundaries—two related to the constraints from orbital effects and the other due to the electric potential barrier. The results are presented for three regimes: negative probe bias with monotonic and non-monotonic potential and positive probe bias with non-monotonic potential. To understand the variation of macroscopic-quantity radial profiles, three diagrams are presented for kinetic features: the ϵl-diagram for the integration domains of the two orbital invariants, the effective potential, and the local ...

Orbital motion theory and operational regimes for cylindrical emissive probes

A full-kinetic model based on the orbital-motion theory for cylindrical emissive probes (EPs) is presented. The conservation of the distribution function, the energy, and the angular momentum for cylindrical probes immersed in collisionless and stationary plasmas is used to write the Vlasov-Poisson system as a single integro-differential equation. It describes self-consistently the electrostatic potential profile and, consequently, the current-voltage (I-V) probe characteristics. Its numerical solutions are used to identify different EP operational regimes, including orbital-motion-limited (OML)/non-OML current collection and monotonic/non-monotonic potential, in the parametric domain of probe bias and emission level. The most important features of the potential and density profiles are presented and compared with common approximations in the literature. Conventional methods to measure plasma potential with EPs are briefly revisited. A direct application of the model is to estimate plasma parameters by fitting I-V measurements to the theoretical results.

Theory of physical libration of the Moon with the liquid core: Forced librations

For a two-layer model of the Moon that consists of a solid nonspherical mantle and an ellipsoidal homogeneous liquid core, a theory of forced librations under the effect of gravitational Earth’s moments has been developed. The motion of the Moon over its orbit has been described by the high-accuracy theory of DE/LE-4 orbital motion. Tables have been constructed that present forced librations of the Moon caused by the second harmonic of its force function, in the neighborhood of its motion according to the generalized Cassini laws. Disturbances of the first-order with respect to dynamic compressions of the Moon and its core are obtained in analytical form for Andoyer variables and Poincare variables and for the projection of the angular velocity vector of Moon’s mantle rotation and the Poincare coordinate system (relative to which core’s liquid accomplishes simple motion) on its major central axes of inertia, as well as for the classical variables in the Moon libration theory, etc. Constructed tables of the forced librations theory give the amplitudes and periods of librations and combinations of arguments of the orbital motion theory that correspond to libration parameters. The interpretation of basic variations has been given and a comparison with the previous theories has been carried out, in particular with the modern empirical theory constructed based on the laser observation data.

Experimental radial motion to orbital motion transition in cylindrical Langmuir probes in low pressure plasmas

This article shows a transition in the behavior of the positive ions movement around a cylindrical Langmuir probe which we have experimentally observed in a low pressure plasma. In the case of helium plasma, depending on the plasma conditions, the ion current collected by the probe behaves as predicted by a radial motion theory, by an orbital motion theory, or by none of them when the transition between the two behaviors takes place. In the case of argon and neon plasmas, the ion current is well described by radial motion theories. The knowledge of the positive ions behavior is essential to diagnose the plasma parameters by using the ion saturation zone of the current–voltage characteristic curve. The use of this zone is one of the less intrusive in probe plasma diagnostics methods providing local information about the plasma parameters, since the charge drained from the plasma is very low.

Evaluation of plasma density in RF CCP discharges from ion current to Langmuir probe: experiment and numerical simulation

Experimental measurements of current-voltage relationship in RF CCP discharge in argon at 81 MHz have been performed by cylindrical Langmuir probes technique. Two different probe radii have been used: 50 and 250 μm. The high plasma density 1010–1011 cm-3 has been estimated at specific input power under study. The experimental data on nonmonotonic behavior of probe current with pressure were observed firstly for conditions of RF discharge plasmas. To analyze the probe measurements the fast numerical model for ion current collected by a cylindrical probe has been developed. This model is based on the particle-in-cell with Monte-Carlo collision method for ions motion and Boltzmann relation for electrons. The features of probe data at studied conditions were discussed. The comparative analysis of different collisionless approaches for plasma density calculation from ion probe current is done. It is shown that in general collisionless theories underestimate the plasma density value. For correct evaluation of plasma density experimental I-V probe measurement should be supplied by the numerical simulation. It was demonstrated that the collisionless analytical theory of orbital motion can formally give correct results on plasma density at some plasma conditions even when ion collisions take place. The physical reasons of this accidental validity are explained.

Exact orbital motion theory of the shielding potential around an emitting, spherical body

A kinetic theory for the equilibrium of an unmagnetized plasma consisting of electrons and ions surrounding a spherical body emitting electrons (due to thermionic emission, photoemission, or secondary emission) is presented. The theory is valid for positively charged bodies, neglects collisions of the plasma particles, and is formulated for profiles of the shielding potential presenting an attractive well. Particle-in-cell simulations are shown to be in good agreement with the theory. An approximated criterion is derived to determine the presence of the potential well.

Experimental determination of dust-particle charge in a discharge plasma at elevated pressures.

The charge of dust particles is determined experimentally in a bulk dc discharge plasma in the pressure range 20-100 Pa. The charge is obtained by two independent methods: one based on an analysis of the particle motion in a stable particle flow and another on an analysis of the transition of the flow to an unstable regime. Molecular-dynamics simulations of the particle charging for conditions similar to those of the experiment are also performed. The results of both experimental methods and the simulations demonstrate good agreement. The charge obtained is several times smaller than predicted by the collisionless orbital motion theory, and thus the results serve as an experimental indication that ion-neutral collisions significantly affect particle charging.

Hooke, Newton, and the Trials of Historical Examination

Nauenberg replies: Ofer Gal writes that he discovered, to his surprise, that “a closed curve orbit created by force had simply been inconceivable” to Isaac Newton before Newton finally learned about the idea from Robert Hooke. But ample historical evidence indicates that Gal’s opinion is incorrect. For example, in a cryptic remark written in his notebook 15 years before his 1679 correspondence with Hooke, Newton stated that “if the body moved in an Ellipsis, then the force in each point … may be found.” 1 1. J. Herivel, The Background to Newton’s Principia, Clarendon Press, Oxford, England (1965), p. 130. In another manuscript, composed before his appointment as the Lucasian chair in mathematics in 1669 at Cambridge University, Newton found that “the force of gravity [at Earth’s surface] is 4000 times and more greater than the endeavor of the Moon to recede from the Earth.” 2 2. Ref. 1, p. 196. The discrepancy between Newton’s figure and the correct value of approximately 3600 (according to the inverse square law) resulted from the erroneous estimate that he used for Earth’s radius.Apparently, Newton did not discover his error until shortly before starting to write the Principia. By applying Kepler’s third law—that for the “primary planets the cubes of their distances from the Sun are reciprocally as the square of the number of revolutions in a given time”—Newton had found that his apparently failed assumption that Earth’s gravitational force satisfies the inverse square law did apply to the gravitational force of the Sun. He wrote that “the endeavours [of the planets] of receding from the Sun will be reciprocally as the squares of the distances from the Sun.” 3 3. Ref. 1, p. 197. By insisting that Newton did not develop a “sophisticated mathematical theory of orbital motion” before 1684, Gal indicates that he cannot understand the subtle mathematical results about orbital dynamics that Newton had exposed in his 13 December 1679 letter to Hooke. Acknowledging those results, however, would invalidate Gal’s arguments of what Newton learned about orbital dynamics from his correspondence with Hooke.REFERENCESSection:ChooseTop of pageREFERENCES <<1. J. Herivel, The Background to Newton’s Principia, Clarendon Press, Oxford, England (1965), p. 130. Google Scholar2. Ref. 1, p. 196. Google Scholar3. Ref. 1, p. 197. Google Scholar© 2004 American Institute of Physics.

Hooke and Newton: ‘Divining’ Planetary Motions

Hooke and Newton: ‘Divining’ Planetary Motions

The floating potential of spherical probes and dust grains. II: Orbital motion theory

Probe theory is generally used to find the potential of dust particles immersed in plasma. The orbital motion limited theory (OML) is often used to find the potential at the probe surface, but the assumptions underlying this theory are usually not valid in the case of dust and the more general orbital motion (OM) theory is much harder to calculate. Solutions are given for the OM theory in a range of cases applicable to dust. It is shown that the surface potential the full theory gives reduces to the OML result for small probes. Commonly in dusty plasmas the OML surface potential is used, with the surrounding distribution given by Debye–Hückel, or Yukawa theory. This form, however, neglects ion depletion due to the absorption of particles on the probe surface. In this paper a new analytical solution to the system is given which is applicable to small probes and dust. This new expression is equivalent to Yukawa form, but takes ion absorption into account.

Theory of the near-probe layer in electronegative gases

Experimental data on an investigation of a plasma, containing negative ions, in the active discharge phase and in the afterglow stage are reported. The applicability of the orbital-motion theory to probe investigations in electronegative gases is discussed.

The collection of positive ions by spherical and cylindricalprobes in an electronegative plasma

The collection of positive ions by spherical and cylindrical probes is investigated for electronegative plasmas in collisionless, unmagnetized and non-flowing regimes as a function of the ratio of negative ion density to positive ion density. The pre-sheath is given by a solution assuming quasineutrality. The potential in the sheath and the current-voltage characteristics are obtained by solving Poisson's equation with the initial value given by the pre-sheath solution. The calculations are mostly based on the radial motion theory but the orbital motion theory is also considered to take the finite ion temperature effect into account. Particularly, the thin sheath model is considered for the latter case. A method for diagnostics in negative ion plasmas is suggested.

Applicability of langmuir’s probe theory for measuring an ion-ion plasma

On the basis of published works devoted to probe measurements in an ion-ion plasma, we consider the possibility of using the theory of orbital motion to find the parameters of a low-temperature ion-ion plasma. We discuss the problem of overestimating the temperature of the ions, which is determined by the method of taking logarithms of the second derivative of the volt-ampere characteristic. A comparison is made with results of measurements performed by the theory of orbital motion for cylindrical probes.

Probe theory - the orbital motion approach

A review is given of the orbital motion limited (O.M.L.) theory of cylindrical and spherical Langmuir probes. In many cases the O.M.L. theory is invalid and the second orbital motion theory presented here, that of Bohm, Burhop and Massey, shows that the concept of an absorption radius must be introduced. The extensive numerical extensions to this theory are briefly discussed.Recent experimental work is referred to in which measurements of ion currents to cylindrical probes showed better agreement with the ABR (radial-motion) theory than with the calculations of Laframboise. It is suggested that the mean free path must be greater than the probe radius by a large factor if the calculations of the Laframboise are to apply; the numerical value depends on the ratio of probe potential to ion temperature.

Ion currents to cylindrical Langmuir probes in RF plasmas

The positive ion current collected by cylindrical Langmuir probes has been measured in an RF argon plasma. The current-voltage characteristics were measured using an RF compensation technique whereby the probe was forced to follow the RF fluctuation in the local plasma potential. The I-V curves from probes of different radii and material show an ion current which is always greater than that predicted by the orbital motion theory and that agrees well with the radial motion theory. This is because the ions travel to the probe from a distance which is limited by the vessel geometry or by the collisional mean free path. The role of the possible collisions, ion-ion and ion-neutral, in the ion trajectories is analysed. It is suggested that the criterion for the application of the orbital motion theory should be tested whenever this theory is used for the determination of the charged particle density.

The astronomical work of Carl Friedrich Gauss (1777–1855)

Gauss's interest in astronomy dates from his student-days in Göttingen, and was stimulated by his reading of Franz Xavier von Zach's Monatliche Correspondenz… where he first read about Giuseppe Piazzi's discovery of the minor planet Ceres on 1 January 1801. He quickly produced a theory of orbital motion which enabled that faint star-like object to be rediscovered by von Zach and others after it emerged from the rays of the Sun. Von Zach continued to supply him with the observations of contemporary European astronomers from which he was able to improve his theory to such an extent that he could detect the effects of planetary perturbations in distorting the orbit from an elliptical form. To cope with the complexities which these introduced into the calculations of Ceres and more especially the other minor planet Pallas, discovered by Wilhelm Olbers in 1802, Gauss developed a new and more rigorous numerical approach by making use of his mathematical theory of interpolation and his method of least-squares analysis, which was embodied in his famous Theoria motus of 1809. His laborious researches on the theory of Pallas's motion, in which he enlisted the help of several former students, provided the framework of a new mathematical formulation of the problem whose solution can now be easily effected thanks to modern computational techniques. Up to the time of his appointment as Director of the Göttingen Observatory in 1807, Gauss had little opportunity for engaging himself in practical astronomical work. His first systematic observations were concerned with re-establishing the latitude of of that observatory, which had been well-determined by Tobias Mayer more than fifty years earlier. However, he found a small but not negligible discrepancy between results obtained independently from stellar and solar observations, as well as irregularities among later measurements of polar altitudes (made at the new observatory completed in 1816), which he was never able to explain, despite repeated attempts to do so using different instruments and observational techniques. Similar anomalies were also detected by a number of other astronomers at around this time. These may have been associated--at any rate, partially--with the phenomenon identified later in the century as a “variation of latitude” due to minor periodic fluctuations in the Earth's axis of rotation produced by meteorological and geological factors.