A Study on Hall Effect Thruster Exhaust Plume Simulation

DEGREE PROJECT IN ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2019

A study on Hall Effect Thruster exhaust plume simulation

ANDRIUS ŠUKYS

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

ROYAL INSTITUTE OF TECHNOLOGY EF233X DEGREE PROJECT IN SPACE TECHNOLOGY

A study on Hall Effect Thruster exhaust plume simulation

Author: Supervisors: Andrius ŠUKYS Nickolay IVCHENKO Dejan PETKOW

Examiner: Tomas KARLSSON A thesis submitted in fulﬁllment of the requirements for the degree of master’s in aerospace engineering in the

Department of Space and Plasma Physics School of Electrical Engineering and Computer Science

December, 2019

Contents

List of FiguresIV

List of TablesV

NomenclatureVI

Abstract XIV

Abstrakt/SammanfattningXV

Acknowledgement XVI

Author’s contribution XVII

1 Introduction1 1.1 Hall Effect Thruster...... 1 1.1.1 Working principle...... 2 1.2 Motivation...... 3 1.3 Objective...... 4 1.4 Limitations...... 4 1.5 Outline...... 4

2 Theoretical Background5 2.1 Tsiolkovsky equation...... 5 2.2 Plasma Characteristics...... 7 2.2.1 Debye length...... 7 2.2.2 Larmor radius...... 8 2.2.3 Magnetization...... 9 2.2.4 Gyro frequency...... 11 2.2.5 Plasma frequency...... 11

I CONTENTS

2.3 Field equations...... 12 2.3.1 Maxwell‘s equations...... 12 2.4 Dilute gas assumption...... 13 2.4.1 Mean free path perspective...... 14 2.5 Particle-particle interactions...... 14 2.5.1 Elastic collisions...... 14 2.5.2 Inelastic collisions...... 16 2.6 Particle-surface interactions...... 18 2.6.1 Neutralization...... 18 2.6.2 Reﬂection/Scattering...... 19 2.6.3 Secondary electron emission...... 19 2.6.4 Sputtering...... 19

3 Electric Propulsion 22 3.1 Types...... 22 3.1.1 Electrothermal thrusters...... 23 3.1.2 Electromagnetic thrusters...... 23 3.1.3 Electrostatic thrusters...... 24 3.2 Spacecraft-plume interaction...... 25 3.2.1 Physical interactions...... 25 3.2.2 Mechanical interactions...... 26

4 Hall Effect Thruster (HET) 27 4.1 Types...... 27 4.1.1 Stationary Plasma Thruster (SPT)...... 27 4.1.2 Thruster with Anode Layer (TAL)...... 29 4.1.3 External discharge Hall thruster (XPT)...... 30 4.1.4 Magnetically Shielded...... 31 4.1.5 Cylindrical Hall Thruster...... 32 4.2 Thruster Performance...... 34 4.2.1 Anode efﬁciency...... 35 4.3 Thruster lifetime...... 35 4.3.1 Erosion rate...... 35 4.4 Neutralizer...... 36 4.4.1 Working principle...... 36 4.4.2 Neutralizer cathode KN-3B...... 38 4.5 SPT-100B...... 38

5 Set-Up, Simulation and Results 41 5.1 VSTRAP...... 41

II CONTENTS

5.1.1 Boundary Element Method...... 42 5.1.2 Fast Multipole Method...... 42 5.1.3 Direct Simulation Monte Carlo...... 42 5.1.4 Fokker Planck...... 43 5.1.5 Particle Pusher...... 43 5.1.6 Particle Wall Interaction...... 45 5.2 Inflow Conditions...... 46 5.2.1 Exit Inflow...... 46 5.2.2 Cathode Inflow...... 50 5.3 Potential Simulation...... 51 5.3.1 Geometry...... 51 5.3.2 Boundary Conditions...... 53 5.3.3 Results...... 55 5.4 DSMC Simulation...... 57 5.4.1 Geometry...... 57 5.4.2 Simulation parameters...... 58 5.4.3 Simulation...... 59 5.4.4 DSMC Validation...... 60 5.5 Plasma Simulation...... 64 5.5.1 Far field versus Near field...... 64 5.5.2 Boundary Conditions...... 64 5.5.3 Inflow and simulation parameters...... 65 5.5.4 Simulation time...... 66

6 Discussion 68 6.1 Outlook...... 69 6.1.1 Simulation...... 69 6.1.2 Hardware...... 70 6.1.3 Software...... 70

7 Conclusion 72

Bibliography 74

III List of Figures

1.1 Hall Effect thruster operating on Xenon [1]...... 2

2.1 Sketch for Tsiolkovsky equation derivation (adapted from [2])...... 6 2.2 Positively charged particle moving in a uniform vertical magnetic ﬁeld [3]...... 9

4.1 Schematic of SPT [3]...... 29 4.2 Schematic of TAL [3]...... 30 4.3 External discharge thruster [4]...... 31 4.4 Magnetically shielded HET thruster [5]...... 32 4.5 CHT with cusp magnetic ﬁeld design [6]...... 33 4.6 HETs with cusp magnetic ﬁeld design...... 33 4.7 Nested channel Hall Thruster X3 [7]...... 34 4.8 Hollow cathode geometry [3]...... 36 4.9 SPT-100B thruster...... 39

5.1 Illustration of specular scattering...... 45 5.2 Simulation domain...... 53 5.3 Boundary conditions...... 54 5.4 Potential distribution inside the channel...... 55 5.5 Potential distribution inside the domain...... 56 5.6 Comparison of potentials with [8]...... 56 5.7 Simulation Domain...... 57 5.8 Particle number evolution towards the steady-state... 60 5.9 Mean particle collision per iteration...... 61 5.10 Direct plots at steady-state...... 62 5.11 Derived plots at steady-state...... 63 5.12 BCs for plasma simulation...... 65

IV List of Tables

2.1 Fitting parameters for CEX cross sections [9]...... 17 2.2 Fitting parameters for secondary electron yield data [3].. 20 2.3 Material model ﬁtting coefﬁcients for sputtering yield [10]. 21

3.1 Comparison of electric thrusters...... 25

4.1 Cathode neutralizer KN-3 parameters...... 38 4.2 HET SPT-100B operational properties...... 40 4.3 Dimmensions of SPT-100B...... 40

5.1 Inﬂow conditions...... 51 5.2 Boundary Conditions...... 54 5.3 Boundary Conditions...... 65 5.4 Hardware estimations...... 67

V Nomenclature

Note: All quantities and equations are in SI units, unless stated other- wise Physical Constants −12 −1 0 Vacuum permittivity 8,854 · 10 F m −7 −1 µ0 Vacuum permeability 4π · 10 H m −1 c0 Speed of light 299 792 458 m s e Elementary charge 1,602 · 10−19 C g Gravitational Constant 6,673 · 10−11 N m2 kg−2 h Plank Constant 6,626 · 10−34 J s −23 2 −2 −1 kB Boltzmann constant 1,381 · 10 m kg s K −31 me Electron mass 9,109 · 10 kg −25 mi Xenon ion mass 2,180 · 10 kg 23 −1 NA Avogadro number 6,022 · 10 mol

VI NOMENCLATURE

Acronyms and Abbreviations AFRL Air Force Research Laboratory AR Aerojet Rocketdyne BC Boundary Condition BEM Boundary Element Method CEX Charge-Exchange CHT Cylindrical Hall Thruster CPU Central Processing Unit CP Chemical Propulsion CSP Cross Section Pre-processor DSMC Direct Simulation Monte Carlo EDB Experimental Design Bureau EMI Electromagnetic Interference EP Electric Propulsion FMM Fast Multipole Method FP Fokker Planck GPU Graphics Processing Unit GRC Glenn Research Center HET Hall Effect Thruster JPL Jet Propulsion Laboratory LeRC Lewis Research Center MEX Momentum-Exchange MPDT Magnetoplasmadynamic Thruster MS Magnetic Shielding NASA The National Aeronautics and Space Administra- tion NHT Nested channel Hall Thruster PPT Pulsed Plasma Thruster PP Particle Pusher

VII NOMENCLATURE

PWI Particle Wall Interaction SEE Secondary Electron Emission SPT Stationary Plasma Thruster STEX Space Technology Experiment TAL Thruster with Anode Layer UM University of Michigan VSTRAP Versatile Software Tool for Rareﬁed Plasmas WRT With Respect To XHT External discharge Hall Thruster

VIII NOMENCLATURE

Variables χ Angle of electron deﬂection ∆F Force increment

∆mP Change of propellant

∆pp Change of particle momentum ∆v Change of velocity or velocity increment δ Mean distance between particles

δMe Electron magnetization parameter

δMi Ion magnetization parameter

δM Magnetization parameter ∆W Work function m˙ a Anode mass flow m˙ c Cathode mass flow m˙ P Propellant mass flow m˙ Xe+ Xenon single charged ion mass flow from anode m˙ Xe2+ Xenon double charged ion mass flow from anode m˙ Xea Xenon neutral gas mass flow from anode m˙ Xec Xenon neutral gas mass flow from cathode ˙ Np Particle flow in particles per second ˙ Ne Electron particle flow from anode ˙ NXe+ Xenon single charged ion particle flow from anode ˙ NXe2+ Xenon double charged ion particle flow from anode ˙ NXea Xenon neutral gas particle flow from anode ˙ NXec Xenon neutral gas particle flow from cathode

η+ Fraction of singly charged ions

η2+ Fraction of doubly charged ions

ηa Anode efﬁciency

ηT Total thruster efﬁciency

IX NOMENCLATURE

Γ(x) Gamma function κ Angle of collision

λDe Debye length ∇ Short-hand notation for a differentiation called Nabla, del or gradient

σCEX CEX collision cross-section B Magnetic field vector j plasma current density in vector form v Velocity vector A Richardson coefficient a, b Fitting coefficients for CEX or secondary electron yield

Aco Area of the cathode oriﬁce

Aex Area of the thruster exit

Ako Area of the keeper orifice a(t) Acceleration function b0, b1, b2, b3 and k Material model fitting coefficients for sputtering yield d Mean particle diameter dw Wall thickness

Eb Binding energy

Ec Electric ﬁeld strength at cathode surface

Ee Electron energy

Eiz Ionisation energy

Ei Ion energy

Ep Particle energy F Thrust

FC Coulomb force

Fc Centripetal force

X NOMENCLATURE

FL Lorentz force fc CEX collision rate or frequency fgi Ion gyro frequency fpi Ion plasma frequency

Ic Cathode current

Id Discharge current

Ie Electron beam current

Ii Ion beam current

Isp Speciﬁc impulse j Number of solvers that produce ∆v k Number of solvers that produce ∆F m Mass of spacecraft mf Mass at the end mP Mass of propellant mp Mass of particle ms Mass at the start ne Electron number density ni Ion number density nn Neutral number density

NIn Number of injected particles per iteration

Nlife Time particle stays in the simulation domain nXe+ Xenon single charged ion particle density at anode nXe2+ Xenon double charged ion particle density at anode nXec Xenon neutral particle number density at cathode pp Particle momentum phii Ion ﬂux q Charge of a particle rLe Electron Larmor radius

XI NOMENCLATURE

rLi Ion Larmor radius rL Larmor radius

Te Electron temperature in K tc Mean collision time

Tem Emitter temperature in K

TeV Electron temperature in eV v Spacecraft velocity v⊥ Perpendicular velocity ve Electron drift velocity vf Velocity at the end vi Ion drift velocity vn Neutral drift velocity vp Particle velocity vs Velocity at the start

Vb Beam voltage/potential

Vcg Cathode to ground voltage

Vd Discharge voltage vex Exhaust velocity vrel Relative velocity between ion and neutral vth,e Electron drift velocity vth Thermal velocity v(t) Velocity function

Wp Particle weight wge Electron gyro frequency wpe Electron plasma frequency α Incident angle of projectile normal to the surface in degrees v¯ex Average exhaust velocity v¯n Average velocity of neutral atoms

XII NOMENCLATURE

ηu Propellant utilization/ionization coefﬁcient ρ Plasma charge density or material density B Magnetic ﬁeld strength jt Thermionic current density

LD Domain length

Ld Discharge channel length ne Electron number density at cathode nXea Xenon neutral particle number density at anode r0 Closest approach or impact parameter R Erosion rate r Radius of circular orbit of a moving charged particle S Sputtering coefﬁcient Y Sputtering yield E Electric ﬁeld vector

XIII Abstract

The results of the attempt to simulate a 1,5 kW-class Stationary Plasma Thruster (SPT) type Hall Effect thruster exhaust plumes are presented. This work focuses on the simulation of the near field region since a charged particle can directly affect the other systems of the spacecraft, such as solar panels, camera etc. Hence, it is important to know if it is possible to simulate this process and what level of accuracy can be achieved. The simulation is carried out using Versatile Software Tool for Rarefied Plasmas (VSTRAP), a 3D kinetic solver currently under development in SPARC Industries sarl. The SPT-100B thruster and its cathode KN-3B both basic flight models developed by EDB Fakel were used as a reference for geometry modelling. The modelling was done using gmsh, an open-source geometry and mesh generation software. The study shows that potential and DSMC simulations give realistic results. However, with the computers available today, particle-based plasma simulations are not possible.

XIV Abstrakt/Sammanfattning

Resultaten av försöket att simulera en 1,5 kW-klass Stationär Plasma Thruster (SPT) typ Hall Effekt thruster avgaserna presenteras. Det här arbetet fokuserar främst på närfältsområdet då det en laddad par- tikel direkt kan påverka rymd-farkostens system, såsom solpaneler, kamera och etc. Det är således viktigt att veta om det är möjligt att simulera denna process och vilken nivå av noggrannhet som kan upp- nås. Simuleringen utförs med hjälp av Versatile Software Tool for Rar- efied Plasmas (VSTRAP), en 3D-kinetisk lösare som för närvarande är under utveckling i SPARC Industries sarl. SPT-100B motor och dess ka- tod KN-3B, båda grundläggande flygmodeller som utvecklats av EDB Fakel, användes som referens för geometrimodellering. Modellering gjordes med hjälp av gmsh, ett open-source program för geometri- och mesh-generering. Studien visar att potentiella och DSMC-simuleringar ger realistiska resultat. Men med de datorer som finns idag är det inte partikelbaserade plasmasimuleringar

XV Acknowledgement

The thesis work was done in SPARC Industries sarl in Luxembourg. I am grateful to people who helped me out throughout my thesis work. First of all, I want to thank the SPARC Industries team for allow- ing me to do my master thesis at their company. Big thanks to Sander Rowette, Leo Basov and Fa Zhu for helping me with my work throughout the whole period of my stay. I also wish to thank, Dejan Petkow and Nickolay Ivchenko, for being my supervisors. The comments received from Anita Kullen helped me a lot to improve my report. Furthermore, the help from Susanna Lyne gave me the ability to proofread my report further.

XVI Author’s contribution

My contributions to this thesis are the following: 1. I did literature review on plasma physics, plasma simulation principles and electric propulsion technologies with a focus on Hall Effect thrusters. 2. I selected a test case to be EDB Fakel’s SPT-100 thruster. 3. For this particular thruster, I have chosen boundary conditions and calculated inﬂow conditions. 4. Based on the template, I created simulation input ﬁles for every simulation. 5. I executed VSTRAP package for potential and DSMC simulations and post-processed its results. 6. I estimated plasma simulation time and since it exceeded my stay at the company I looked for the solutions that would reduce simulations costs. 7. I collected the main results of this work in the report.

XVII Chapter 1

Introduction

Since the invention in the 1960s and ﬁrst ﬂight (SPT type) in 1971 [11], Hall Effect Thrusters have been extensively used as electric propulsion in the many Soviet Union and later Russian telecommunication satellites. From the late 1990s, Hall Effect Thrusters started gaining more and more research interest in the United States and Europe. Now, more than 300 SPTs have been used on-board of geostationary satellites. A PPS1350 Hall Effect Thruster jointly developed by EDB Fakel (Russia) and Snecma-Group SAFRAN (France) was with success used for primary propulsion on the SMART-1 spacecraft [12], manufactured by Swedish Space Corporation for a moon orbiting mission [13]. The mission began on 27th September 2003 and ended by deliberately crashing the spacecraft on the moon’s surface on 3rd September 2006.

1.1 Hall Effect Thruster

Hall Effect Thruster (HET) is an important electric propulsion technology for certain low thrust applications requiring low thrusts, such as satellite position keeping and orbit transfer [14]. Some researchers [15, 16] suggested a method for space debris removal which exploits high-speed focused ion beam. Therefore, for debris removal, electric propulsion systems such as Hall Effect and Ion thruster can be used. HET is a simple device comprised of a channel with an anode at the bottom, a magnetic circuit and a cathode placed usually outside of the channel. The working principle is described in the following section.

1 CHAPTER 1. INTRODUCTION

Figure 1.1: Hall Effect thruster operating on Xenon [1].

1.1.1 Working principle

The geometry of a HET can include either one large or few smaller magnetic coils surrounding the whole thruster, and an internal magnetic coil. Few authors [17, 18, 19, 20] investigated a possibility to substitute electromagnetic circuit with permanent magnets to reduce the power consumption and the weight of a thruster. An incandescent cathode is commonly placed outside of the accelerating channel and an anode is installed at the bottom of the discharge channel. Commonly in HET designs anode has an orifice through which majority of the neutral propellant gas (typically Xenon or other noble gas, due to their high mass and non-reacting properties) is injected. Rest of the propellant (usually around 10 % of the anode mass flow [21]) is injected through externally located hollow cathode. The basic process of operation begins with the release of electrons from the cathode. The electrons are created by thermionic emission where the escape energy for the electrons is supplied by heating the cathode to a very high temperature. The electrons are accelerated towards the anode by the cathode and anode potential difference. A magnetic circuit creates a radial magnetic field, which reaches its maximum around the exit plane (on the order of 1 · 10−2 T [21]). Magnetic field traps electrons which then move azimuthally. This orbiting elec-

2 CHAPTER 1. INTRODUCTION

tron motion creates a potential difference, perpendicular both to the flow of electrons and the applied magnetic field. The magnetic field is perpendicular to the flow of current and thus causes resistance in terms of Lorentz force. This force helps efficiently ionize and focus the propellant injected into the channel from the anode region. Ions are accelerated by the static electric field. Some of the electrons pass through to the anode and get absorbed. The rest of the electrons, that are generated by the Hollow cathode, turn the positively charged ion cloud into a quasi-neutral plasma, which prevents particles from inter- acting with other spacecraft units e.g. solar panels, shielding, camera etc.

1.2 Motivation

Since the experimental analysis of HETs has a high cost, there is substantial interest in the development of accurate simulation tools. Such tools could help to understand the effects of the plasma inside the thruster as well as in the plume. The properties of plasma and its constituents such as particle number density, particle temperature and their velocity could help to make better design considerations. Cur- rently, the work on such tool, namely, VSTRAP, is in progress in SPARC Industries sarl. Furthermore, there is widespread research interest to develop an understanding of thruster-spacecraft interaction phenomena. It is known that spacecraft can be affected by three processes: electromagnetic interference (EMI); particle impingement; and radiant heating. Electro- magnetic interference will interfere with communication signals and affect instrument performance. Whereas particle impingement and radiant heating can cause physical damage to spacecraft hardware. The simulation of HET plumes allows the testing of different operating conditions and removes the inﬂuence of the experimental facilities. However, computer codes need to be veriﬁed by experimental data. The goal of the present study is to assess the ability to simulate exhaust plumes accurately.

3 CHAPTER 1. INTRODUCTION

1.3 Objective

The goal of this thesis work is the validation of VSTRAP solver. It is done by comparing simulation results with the results from the experimental thruster analysis.

1.4 Limitations

There is limited available information about SPT-100 operational parameters, geometry and the experimental setup. In VSTRAP plasma simulated using macro-particle interactions. Therefore, to achieve high accuracy user needs a higher number of macro-particles and ﬁner mesh which leads to large simulation costs. Furthermore, the hardware available at SPARC Industries sarl imposes limits on mesh size and macro- particle number. So, lowers the accuracy of the results.

1.5 Outline

The outline of this thesis is as follows: Chapter 2 introduces the necessary theoretical background behind the HETs; Chapter 3 gives a summary of Electric Propulsion technology. Then Chapter 4 discusses the HETs in detail and sets into the context of the current research trends. Chapter 5 presents all the necessary preparation steps together with the description of the simulations and their results. Chapter 6 discusses the simulation results and gives guidelines for future work. Finally, Chapter 7 gives a summary and conclusion of the thesis works.

4 Chapter 2

Theoretical Background

This chapter gives a brief description of the principles behind spacecraft propulsion. Also, gives a summary of plasma physics needed to understand processes happening inside the HET chamber and in the plume.

2.1 Tsiolkovsky equation

There are different types of propulsion systems (electric, chemical), but in most cases, the spacecraft is propelled by ejecting a mass from a spacecraft. For example, in HETs the mass has a form of energetic charged particles (plasma). One starts by writing an equation of motion for the spacecraft using the Newton’s 3rd Law of motion (For every action there is an equal and opposite reaction) based on the sketch in ﬁgure 2.1. This results in conservation of linear momentum and one gets following expression:

(m + mP )(v + ∆v) + ∆mP (v − vex ) − mv = 0 , where m is mass of spacecraft, mP is mass of propellant, v is spacecraft velocity, vex is exhaust velocity, and ∆mP is amount of propellant used to change of velocity by increment ∆v . It can then further simpliﬁed to: (m − ∆mP )∆v − ∆mP vex = 0 expressing it per unit time and by moving the thrust generated by

5 CHAPTER 2. THEORETICAL BACKGROUND

Figure 2.1: Sketch for Tsiolkovsky equation derivation (adapted from [2]). propulsion system to the right side, it reads as:

dv dm m = v P dt ex dt which is equivalent to 2nd Newton’s law of motion1. In space environment one can neglect drag forces. The propellant mass ﬂow can be expressed as a rate of drop of spacecraft mass per unit time, because the rate of change for both is the same.

dv dm dm dm dm m = F = v P = P = − = −v dt ex dt dt dt ex dt By assuming constant exhaust velocity, multiplying both sides by dt and separating variables one can integrate as:

Z vf Z mf dm dv = −vex vs ms m

Integration of left side from starting, vs, to ﬁnal, vf , velocities and integration of right side from starting, ms, to ﬁnal mf , masses as following:

Z vf Z mf dm ms dv = vf − vs = ∆v = −vex = vex ln vs ms m mf leads to: ms ∆v = vex ln (2.1) mf

1Force equals mass times acceleration or F = ma.

6 CHAPTER 2. THEORETICAL BACKGROUND

which is a famous Tsiolkovsky equation. Spacecraft propulsion efficiency is measured by calculating its specific impulse, which is denoted as Isp and has units of seconds. Specific impulse describes the thrust efficiency, i.e. how effectively the propellant, thrown out of the back of the spacecraft, is converted into thrust. As seen in equation 2.2, it is defined as the ratio between the thrust, F , and the rate of propellant consumption m˙ P multiplied by gravitational constant g. Also, it can be expressed as the ratio between the average effective exhaust velocity, v¯ex, and the acceleration of gravity, g. The higher the specific impulse, the more thrust is achieved for the fuel that is burnt by the propulsion system. Or, in other words, specific impulse determines the amount of fuel necessary to achieve the desired thrust.

F v¯ex Isp = = (2.2) ˙mpg g

2.2 Plasma Characteristics

Plasma is a collection of various charged particles that move in response to electric and magnetic ﬁelds, self-generated or applied. Elec- trostatic interactions in plasmas are described by Coulomb force [22] which is a long-range force and decays slowly as r−2. This means that the particles in plasma experience collective behaviour meaning that each particle interacts with a large number of other particles. Conse- quently, one can see a macroscopic plasma response towards an external force. On the scales of larger than Debye, length plasma tends to be "quasi- neutral" [22], which means that the ion and electron number densities are nearly equal (ni ≈ ne). However, near the boundaries of a plasma volume, particle densities are not equal and quasi-neutrality disappears. The most important plasma parameters are Debye length; Larmor radius; magnetization; gyro and plasma frequencies. These parameters will be described in the following subsections.

2.2.1 Debye length

The electron Debye length, λDe, is the characteristic length scale in a plasma. It is a solution of a one dimensional (1D) Poisson equation

7 CHAPTER 2. THEORETICAL BACKGROUND

with the distance from a charge at which the Coulomb potential decays as 1/e and is the crucial parameter, which helps to determine if plasma conditions prevail. A simpliﬁed expression, which excludes ions due to their small effect is expressed as: r 0kB Te λDe = 2 (2.3) ne e

3 where ne is number of electrons per m , Te is electron temperature in K, kB is Boltzmann constant, e is elementary charge and 0 is vacuum permittivity. For the typical values of SPT-100B plume: electron temperature of 8 eV and electron density of 10 · 1017 m−3, Debye length is around 6,64 · 10−8 m. This very small value indicates that the Debye length is small with respect to features of interest in the plume region and thus it can be assumed that plume is quasi-neutral everywhere except in a thin sheath region near the solid surfaces.

2.2.2 Larmor radius

The importance of the magnetic field in the plume region is measured by the Larmor radius. The Larmor radius describes the movement of charged particle (ion) of mass, mi, in a uniform magnetic field with velocity in one dimension as depicted in figure 2.2. The particle will be affected by Lorentz force:

FL = q(E + v × B) (2.4) where E is electric field vector, B is magnetic field vector, v is velocity vector and q is charge of a particle. In the absence of electric field E the Lorentz force will cause charged particle to move in circular orbit in the v⊥ × B direction and will experience a corresponding centripetal force such that: 2 mv⊥ Fc = qv⊥ × B = (2.5) r where r is the radius of circular orbit of a moving charged particle and v⊥ is the perpendicular velocity. One can find radius r, which is Larmor radius, rL: 2 mv⊥ rL = (2.6) qB

8 CHAPTER 2. THEORETICAL BACKGROUND

Figure 2.2: Positively charged particle moving in a uniform vertical magnetic ﬁeld [3]. where B is magnetic ﬁeld strength. When species temperatures are similar, the electron Larmor radius is distinctly smaller than the ion Larmor radius: r me rLe ∼ rLi (2.7) mi where rLe and rLi are electron and ion Larmor radius respectively, me is electron mass, and mi is ion mass. Based on the book by Goebel and Katz [3], the Larmor radius for electrons can be expressed as: r 1 8 me rLe = TeV (2.8) B π q where TeV electron temperature in eV and for ions: r 1 2mi rLi = Vb (2.9) B q where Vb is ion beam potential in eV.

2.2.3 Magnetization

According to Fitzpatrick [23], the plasma is magnetized when the magnetic field is strong enough to significantly alter trajectories of electrons and ions. One of the magnetized plasma features is anisotropy, i.e. hav- ing a different response to forces in different directions (perpendicular or parallel to the direction of magnetic field). A plasma is magnetized

9 CHAPTER 2. THEORETICAL BACKGROUND

if its characteristic length scale L or mean free path l is large compared to the Larmor radius and this can be expressed by magnetization parameter: rL rL δM ≡ ≡ . (2.10) L l Depending on plasma conditions, plasmas can be [24]: 1. Strongly magnetized:

(a) Strongly electron magnetized if δMe 1: For example, the electron Larmor radius for typical values of SPT-100B, electron temperature of 25 eV [3] and a maximum radial magnetic ﬁeld strength in exhaust region of 20 mT [21] is 0,95 mm which is much smaller than typical channel width of 15 mmof SPT-100. Thus, the magnetization parameters are equal to 0,06 In comparison, one can do similar calculation for ions using formula 2.9 and it yields Larmor radius equal to 1,43 m and magnetization parameter 95,25 So, one can conclude that in the exhaust region only electrons are magnetized, and ions are not.

(b) Strongly ion magnetized if δMi 1: Example would be a plasma of Hydrogen fusion with Lar- mor radii of rLe ≈ 2 µm and rLi ≈ 200 µm, while the typical length scale from 1 m to 10 m. So, in a hydrogen fusion plasma both electrons and ions are strongly magnetized. 2. Un-magnetized or weakly magnetized if the electron Larmor radius is larger or equal to length scale or mean free path is δMe > 1: For example, a plume of a small plasma thruster in low earth orbit (LEO) with electron velocity of around 10 · 106 m s−1 and magnetic ﬁeld is around 3 · 10−5 T. Strongly electron magnetized plasma is an interesting case, but, with the term "magnetized" is referred to plasma, where all particle species are magnetized. This state is generally achieved when the ion magnetization parameter is way less than one:

rLi δMi ≡ 1 (2.11) L In some papers such regime is deﬁned as partially magnetized [25, 26] or this type of plasma is called Hall plasma [27, 28]. Thus, for the purpose of this work it will be referred to as partially magnetized.

10 CHAPTER 2. THEORETICAL BACKGROUND

2.2.4 Gyro frequency

By Goebel and Katz [3] and Chen [29] is referred as cyclotron frequency and describes how many gyrations (pseudo-circular motions) the particle can have per second. In Hz and for ions it can be found by:

ωgi |q|B fgi = = (2.12) 2π 2πmi

For SPT-100B, which has maximum magnetic ﬁeld of 20 mT, single ionized Xenon ion has charge equal to e and double ionized is 2 e respectively. Thus, the gyro frequency for single and double charged ions equal to about 234 Hz and 568 Hz, respectively. And similarly, for the electrons: ωge eB fge = = (2.13) 2π 2πme which gives electron gyro frequency of around 56 MHz.

2.2.5 Plasma frequency

The plasma frequency is one of the fundamental parameters of a plasma. And the inverse of this value is approximately the minimum time required for the plasma to react to changes in its boundaries (described by the Debye sphere) or in the applied potentials. The electron plasma frequency in Hertz can be calculated as: s 2 ne e fpe = 2 (2.14) 4 π 0me

At the exit plane of SPT-100B, where electron number density , ne, is between 10 × 1017 m−3 and 10 × 1018 m−3 [30], the electron plasma frequency is between 284 MHz and 898 MHz. And similar formula also exists for ions: s 2 ni e fpi = 2 , (2.15) 4π 0mi which provides an approximate time scale for ion movement in the plasma. The assumed "quasi-neutrality" prevails and thus ni ≈ ne, which gives that Xenon ion plasma frequency is between 0,6 MHz and 1,8 MHz.

11 CHAPTER 2. THEORETICAL BACKGROUND

2.3 Field equations

2.3.1 Maxwell‘s equations

Electric propulsion plasmas are governed by Maxwell‘s equations formulated for the conditions of a low pressure and static electric field ∂E ( ∂t = 0). The plasma contains a similar or same amount of positive and negative charges and thus is quasi-neutral. Maxwell‘s equations for these conditions can be formulated as: 1. Gauss’ law for electric field that describes the behaviour of electric field in presence of electric charges: ρ ∇ · E = (2.16) 0

2. Gauss’ law for magnetism which shows that the divergence of the magnetic ﬁeld is always zero:

∇ · B = 0 (2.17)

3. Faraday’s Law which shows that changing in time magnetic ﬁeld produces electric ﬁeld that circulates around it:

∂B ∇ × E = − (2.18) ∂t

4. Ampere’s Law shows that a ﬂowing electric current creates a magnetic ﬁeld: ∇ × B = µ0 j (2.19)

where ρ is the plasma charge density, j is the plasma current density in vector form, 0 is vacuum permittivity, c0 is vacuum speed of light and µ0 is permeability of free space. Also, here ∇ (called Nabla, del or gradient) is a short-hand notation for a differentiation [31] in form:

∂ ∂ ∂ ∇ = , , (2.20) ∂x ∂y ∂z

If no dynamic magnetic ﬁeld is present (∂B/∂t = 0), the electric ﬁeld

12 CHAPTER 2. THEORETICAL BACKGROUND

(E) is expressed as the gradient of the electric potential:

E = −∇φ, (2.21) negative sign in this equation comes from the assumption that the electric ﬁeld always point in the direction of ion acceleration.

2.4 Dilute gas assumption

As stated by Kaviany [32], DSMC is used for weakly (partially) ionized gases (plasma in HETs is type of partially ionized gas[33]) with the Poisson equation in order to include effects of electric ﬁeld. DSMC is based on dilute gas assumption and the kinetic theory. DSMC applies these properties of dilute gas [34]: 1. Molecules move freely without interaction for time scales on the order of the local mean collision time. 2. the impact parameters and initial orientations of colliding molecules are random. 3. There are an enormous number of molecules and only a small fraction need to be simulated to obtain an accurate molecular description of the ﬂow. As stated in [35], the dilute gas assumption is valid when:

δ d (2.22) where δ is mean distance between particles and d is mean particle diameter. This assumption is valid when:

δ ≥ 7 (2.23) d The 1/n is the volume occupied by molecule. Therefore, one can also ﬁnd δ by: − 1 δ = n 3 (2.24)

13 CHAPTER 2. THEORETICAL BACKGROUND

2.4.1 Mean free path perspective

Again, as stated in [35] the mean free path compares to δ and d as:

l δ d (2.25)

2.5 Particle-particle interactions

When an electron collides with the atoms, two types of the process can occur. It is either elastic scattering in which primarily the electron momentum is exchanged or inelastic processes such as excitation and ionization. For ions colliding with the atoms, the main processes are either the elastic scattering, in which momentum and energy are exchanged or Charge-Exchange (CEX). The total momentum and energy of the colliding particles after the collision are equal to that before the collision and thus the total energy (sum of kinetic, internal and potential energies) and momentum are conserved. Electrons and fully-stripped ions possess only kinetic energy. Atoms and partially stripped ions have internal energy level structures and can be excited, de-excited, or ionized, corresponding to changes in internal energy. In the case of Hall effect thrusters, the most important interactions that must be considered in modelling are elastic Momentum-Exchange (MEX) and CEX collisions; ionization and excitation of neutrals. This and the following section describe the most common particle interactions that can happen inside the HET chamber and in the near ﬁeld plume and are based on books by Chen [29] and Lieberman and Licht- enberg [36].

2.5.1 Elastic collisions

Elastic is the type of collision, where the internal energies of the in- teracting particles do not change and the sum of kinetic energies is conserved. Although the total kinetic energy is conserved, kinetic energy is exchanged between particles. However, a super-elastic type of collisions can occur in which an excited atom can be de-excited by a collision, increasing the sum of kinetic energies.

14 CHAPTER 2. THEORETICAL BACKGROUND

Coulomb collisions

Coulomb collisions are the main mechanism in which particles interact with each other in plasmas. This mechanism is very important for the description of plasma diffusion, mobility and resistivity. An elastic collision is the scattering between pairs of numerous particle species: electron-electron, electron-ion and ion-ion. The probability of this process to happen can be measured by the particle species cross-sections. When an electron collides with an ion, the electron is gradually deflected by the long-range Coulomb field of the ion. The electrons are deflected by angle χ by a Coulomb force

q 2 FC = − 2 , (2.26) 4π0r which is inverse square law, and thus it has the highest impact when the electron is closer to ion and this can be expressed as time, τ:

r0 τ = , (2.27) ve where, r0 is closest approach or impact parameter, ve is electron drift velocity. For large-angle collisions (κ > 90°) the change in particle momentum, pp, is of order of pp itself. Thus, the change in momentum is: 2 ∼ ∼ q ∆pp = mpvp = , (2.28) 4π0r0ve where mp is particle mass and the impact parameter can be found by:

q 2 r0 = 2 . (2.29) 4π0me vq

The cross section , σ is then approximated by:

4 2 q σ = πr0 = 2 2 4 . (2.30) 16π0me ve

The collision frequency vei is, therefore:

4 ne q vei = ne σv = 2 2 3 (2.31) 16π0me ve

15 CHAPTER 2. THEORETICAL BACKGROUND

and the resistivity, ρ, is:

2 me e ρ = 2 vei = 2 3 (2.32) ne e 16π0me ve

2.5.2 Inelastic collisions

If the sum of kinetic energies is not conserved, then the collision is inelastic. Most inelastic collisions involve excitation or ionization, such that the sum of kinetic energies after collision is less than that before collision.

Excitation

When electron collides with neutral atom it transfers some of its kinetic energy to the neutral atom. If the electron energy is lower than energy 3 for ionization (Ee ≈ 4 Eiz [36]) , the neutral atom is excited, transitioned from lowest energy state to higher energy state. For Xenon the process can be expressed as:

Xe + e− −→ Xe∗ + e−

Ionization

It is a process when the neutrally charged atom loses an electron and becomes positively charged. In hot plasmas, the ions can have multiple charges, but in relatively cold, a plasma typical for HET, majority of the ions have a single charge. For a typical HET, the ionization occurs by electron collision with neutral:

Xe + e− −→ Xe+ + 2e−

The probability for ionization to happen is expressed by particle ionization cross-section

CEX

When a fast-positive ion collides with a neutral with a thermal velocity, the neutral can capture an electron from the ion. This reaction can be

16 CHAPTER 2. THEORETICAL BACKGROUND

written as:

Xe+(fast) + Xe(slow) −→ Xe(fast) + Xe+(slow)

The result is the creation of a slow ion and a fast neutral. The cross section for charge transfer is large at low collision energies, making this an important process in weakly ionized plasmas. The CEX collision frequency is expressed as:

fc = nn vrel σCEX (2.33)

3 where nn is neutral particle number density per m is relative velocity −1 between ion and neutral in m s and σCEX is charge-exchange collision 2 cross-section in m . The ion energy , Ei, based model fit for CEX cross- section that is commonly used was derived by Miller et al. [9] as looks like: σCEX = a − b log Ei (2.34) where a and b are fitting coefficients derived from experimental data and for Xenon are shown in table 2.1.

Table 2.1: Fitting parameters for CEX cross sections [9].

Collision a, m2 b, m2 Xe+, Xe 1,71 · 10−18 1,18 · 10−19 Xe++, Xe 1,03 · 10−18 7,7 · 10−20

Collisions between neutrals

Mean free path. Mean free path is a distance, which particle travels before the collision and is found: 1 l = (2.35) nn σnn where nn is number density of neutral particles and σnn is collision cross section found by: ¯ 2 σnn = πd (2.36) where d¯ is the distance between centres of collision pairs when they collide.

17 CHAPTER 2. THEORETICAL BACKGROUND

Mean collision time Mean collision time describes how much time on average passes between the neutral particle collisions. It is found by: l tc = (2.37) ¯vn

Mean collision frequency Describes how many collisions will happen per second and it is found as inverse of mean collision time:

1 ¯vn fc = = (2.38) tc l where l is mean free path and v¯n is mean velocity of neutrals. Mean neutral is assumed to be like mean thermal velocity found by: r kbT ¯vn = vth = (2.39) mn

2.6 Particle-surface interactions

The charged particles, whether they are in the centre of the chamber or near the walls, always follow the field lines and therefore can interact with the surface if these field lines leak into the walls. The charged particles can be absorbed, scattered or neutralized as well as generate secondary electrons or sputter the material from the wall. Since the neutral particles can move freely without being guided by magnetic or electric fields, before they get ionized, they interact with the wall only by scattering.

2.6.1 Neutralization

On collision with the wall, the ion becomes neutral by recovering electrons in a three-body reaction:

e− + A+ + S −→ A + S where e− is electron, A+ is ion and S surface. This reaction is fast, and ions are immediately neutralized at the surface. Then the neutral scatters and moves freely in the plasma until it becomes re-ionized

18 CHAPTER 2. THEORETICAL BACKGROUND

in contact with other electrons or ions. This process repeats until the particle is taken out of the system, for instance, by being absorbed by the wall. This phenomenon can be also named as recycling process.

2.6.2 Reﬂection/Scattering

Neutrals and Ions can undergo two types of scattering. Specular scattering is accomplished by reversing the radial component of velocity when a particle reaches the channel boundaries. In diffuse scattering, the particle thermalizes at the wall temperature and is re-emitted with a random angle and speed (usually randomly chosen from a one-way Maxwellian velocity distribution function). Electrons can scatter in three main ways, namely secondary electron emission (SEE), inelastic back-scattering and elastic reflection. If the energies of electrons are less than 50 eV, the secondary electrons and in- elastically back-scattered electrons cannot be distinguished while they both have low energies (from 2 eV to 5 eV), but elastically reflected electrons remain confined at the similar energy to the primary electron. [37]. The elastic reflection yield is in general higher at low energies for dielectrics such as borosil, a material used in the SPT-100B thruster.

2.6.3 Secondary electron emission

Ion and electron bombardment of the HET walls, which are made of common insulator materials, such as boron nitride, at the energies characteristic of Hall thrusters produces a signiﬁcant number of secondary electrons. From [3] expression for the secondary electron yield from electron bombardment of materials is:

b γ = Γ (2 + b)aTeV (2.40) where the electron temperature is in electron volts, Γ(x) is the gamma function, and the coefﬁcients a and b are found from ﬁts to the data. The values for certain HET materials are summarized in table 2.2.

2.6.4 Sputtering

When the particle hits the thruster wall and its energy Ep is greater than the binding energy Eb of separate atoms forming the thrusters wall,

19 CHAPTER 2. THEORETICAL BACKGROUND

Table 2.2: Fitting parameters for secondary electron yield data [3].

a b Γ(2 + b)

Alumina (Al2O3) 0,145 0,650 1,49 Boron Nitride (BN) 0,150 0,549 1,38 Borosil (BNSiO2) 0,123 0,528 1,36 Stainless steel 0,040 0,610 1,44 surface sputtering, the process of knocking out the particles from the wall, begins. The main characteristic of the sputtering is its coefﬁcient and from [21] for simple materials:

Np δV S = = (2.41) Ni q where Np particles knocked out of the surface and Ni number of ions incident onto it. It could be also expressed as ratio between the sputtered material volume δV and incidence of total charge q.

Sputtering Yield

M. Gamero-Castano [10] used Yamamura’s and Tawara’s [38] sputtering model together with experimental data reported by Garnier et al. [39] to obtain an semi-empirical expression for the sputtering yield for borosil (BNSiO2), material used in SPT-100B [40] (In paper called BGP-10), in units of mm3 C−1:

r !2,5 2 3 p Eth Y = k b0 + b1α + b2α + b3α Ei 1 − (2.42) Ei where α is the incident angle of the projectile normal to the surface expressed in degrees, the value Eth represents the estimated threshold energy in eV for sputtering. k, b1, b2, b3 are the ﬁtting coefﬁcients and for borosil is taken from Gamero-Castano et al. [10] is summarized in table 2.3.

20 CHAPTER 2. THEORETICAL BACKGROUND

Table 2.3: Material model ﬁtting coefﬁcients for sputtering yield [10].

Material Borosil (BNSiO2)

Eth, eV 58, 6 −3 b0 9,9 · 10

b1 0 −6 b2 6,04 · 10 −8 b3 −4,75 · 10 k 1,00

21 Chapter 3

Electric Propulsion

The Electric Propulsion (EP) was independently envisioned by Robert Goddard in 1906 and Konstantin Tsiolkovsky in 1911 as written in his- torical review by Choueiri [41]. EP is a technology that uses electricity to accelerate propellant to high exhaust velocities and achieve thrust with low consumption of propellant. Even though a thrust achieved by EP is low compared to Chemical Propulsion (CP), EP generally has a very high speciﬁc impulse (Isp). In the next section, the different types of EP are shortly described.

3.1 Types

As stated in books by Goebel and Katz [3] and Jahn and Choueiri [42], electric thrusters are commonly grouped into electrothermal, electrostatic, or electromagnetic.

Electrothermal The propellant is heated by electrical process and then accelerated thermodynamically through a converging-diverging nozzle.

Electrostatic The propellant is accelerated by electrostatic forces applied to ionized particles.

Electromagnetic The propellant is accelerated under the combined action of electric and magnetic ﬁelds. The examples of common Electric

22 CHAPTER 3. ELECTRIC PROPULSION

thrusters are shortly described in the following.

3.1.1 Electrothermal thrusters

Arcjet

An arcjet is an electrothermal thruster that applies a high current arc to heat the propellant that passes through the feed system. In this type of thruster, the propellant is weakly ionized and thus plasma effects can be neglected in the exhaust. The Isp is not larger than 700 s for easily stored propellants due to limits of thermal heating.

Resistojet

A resistojet is also electrothermal thruster, wherein the propellant is heated by a solid surface, such as the chamber wall or a heater coil. The Isp is not larger than 500 s due to limits of this type of thermal heating.

3.1.2 Electromagnetic thrusters

Pulsed Plasma Thruster

A pulsed plasma thruster (PPT) is an electromagnetic thruster that uses a pulsed discharge. This pulsed discharge ionizes a fraction of a solid propellant introduced into a plasma arc. Electromagnetic effects caused by pulsed discharge accelerate the ions to high exit velocity. The thrust level is determined by the pulse repetition rate.

Magnetoplasmadynamic Thruster

Magnetoplasmadynamic thrusters (MPDT) use a very high current arc for ionization of large fraction of the propellant. This charged propellant is accelerated by electromagnetic forces that occur in the plasma discharge. These thrusters operate at very high powers to generate enough force for high speciﬁc impulse and high thrust.

23 CHAPTER 3. ELECTRIC PROPULSION

Variable Speciﬁc Impulse Magnetoplasma Rocket (VASIMR®)

VASIMR® is an electromagnetic thruster for spacecraft propulsion currently under development at Ad Astra Rocket Company in USA [43]. It uses radio waves to ionize and heat a propellant. As mention in the article by Charles Bolden, VASIMR® technology could be the break- through technology that could let us reach Mars in as little as 39 days [44].

3.1.3 Electrostatic thrusters

Colloid thrusters

Colloid thrusters extract the ions or charged droplets from conductive liquids fed through small needles. These ions or charged droplets are accelerated by electrostatic forces. Field Emission Electric Propulsion (FEEP) thrusters apply ﬁeld emission process to transport liquid metals (typically indium or caesium) along with needles and extracting ions from the sharp tip. These thrusters produce very low thrust and thus used for precise control of spacecraft attitude or position.

Ion thruster

Ion thrusters achieve ionization of large fraction of the propellant by employing different techniques for plasma generation. These thrusters use biased grids to electrostatically accelerate ions, extracted from the plasma to high velocities. The applied voltages are very high and can exceed 10 000 V. Ion thrusters are highly efﬁcient (from 60 % to >80 %) and have the very high speciﬁc impulse (from 2000 s to over 10 000 s) in comparison to other thruster types. Furthermore, due to low ion beam divergence, they can be used for debris removal as mentioned by Bombardelli and Pelaez [15].

HET

In this type of electrostatic thruster (Jahn and Choueiri [42] classifies HET as electromagnetic due to the magnetic circuit) the plasma is generated by a cross-field discharge created by the Hall effect. An Electric field, perpendicular to the magnetic field, accelerates ions to high ex-

24 CHAPTER 3. ELECTRIC PROPULSION

Table 3.1: Comparison of electric thrusters.

Thruster Isp, s P, kW η, % Source Arcjet 500 − 2000 0.3 − 100 up to 55 [45] Resistojet 300 0, 5 − 1 65 − 90 [33] PPT 850 − 1200 < 0, 2 7 − 13 [33] MPD 2000 − 5000 200 − 1000 50 [45] VASIMR® 800 − 3500 200 10 − 54 [46] Colloid 400 − 2300 0, 005 − 0, 0015 > 80 [47] Ion 1000 − 4000 0, 3 − 5 50 [45] HET 500 − 3000 0, 5 − 100 50 [45,7] haust velocities. Hall thruster efﬁciency and speciﬁc impulse are lower compared to ion thrusters, but the thrust at the same power is higher. Furthermore, the design of the HET device is much simpler and requires fewer power supplies to operate.

Now the previously described thrusters are compared in a table 3.1 by three main characteristics: speciﬁc impulse, power and efﬁciency. Every EP thruster in this section has its advantages and would be interesting and worthwhile to study in detail. However, the focus of this thesis is the HET type.

3.2 Spacecraft-plume interaction

Integration of EP, e.g. HET, into the spacecraft requires knowledge on how the thruster plumes interact with the spacecraft surfaces and its payloads. The plume consists of electrons, high energy source ions, and low energy CEX ions. These particles can interact with and subse- quently damage the spacecraft. HET plumes have a wider divergence compared to other EP and thus can have a higher level of impact. These interactions can be grouped into physical, mechanical and electrical.

3.2.1 Physical interactions

CEX ions tend to be accelerated radially to 10s or 100s of eV before impacting on the surfaces of the spacecraft, and they are a major cause of physical processes for erosion and contamination [48].

25 CHAPTER 3. ELECTRIC PROPULSION

1. Erosion of surface material by Sputtering. When a highly energetic particle (ion or neutral) hits the surface of a spacecraft, it can cause erosion of the walls by extracting particles from the surface. This process depends on particle energy and incidence angle of the incoming particle. The process occurs when the energy of the incoming particle is larger than the binding energy of the surface particle. Surface erosion weakens mechanical and thermal properties of the spacecraft subsystems and leads to their malfunction or failure. For example, solar arrays have a sputtering threshold of 100 eV. The data of sputtering yield, energy and incidence angle is found from experiments. 2. Contamination. The surface can be contaminated either by plume impingement or the material which origin is propellant. Hall effect thruster is mainly contaminated by the ions in the plume. 3. Deposition. The eroded material from discharge chamber ceramic walls of HETs can be deposited on the sensitive spacecraft surfaces such as solar panels, optical systems etc.

3.2.2 Mechanical interactions

By its name, the most intuitive interactions, since the electrons and ions can be directly attracted by the positively and negatively charged spacecraft surfaces respectively or guided by electrical or magnetic fields that cross spacecraft surfaces. These interactions can create forces and torques, that directly affect the attitude of the spacecraft. These forces must be counteracted by the attitude control system of the spacecraft. Plume impingement on solar arrays can also result in a loss in thrust and perturbation in torque, causing a change in spacecraft attitude. Besides, sensitive optical instruments can be interfered by plume optical emission and the electromagnetic field in the plume field can distort communication signals.

26 Chapter 4

Hall Effect Thruster (HET)

In this chapter, the different types of HETs and the motivation for the selection of SPT-100B are described. Part of this chapter is dedicated to explaining how the performance and life of a thruster are evaluated.

4.1 Types

On the internet one can ﬁnd different types of Hall Effect thrusters [3]. This chapter gives a detailed description of SPT and TAL (thruster with anode layer) since both have been successfully ﬂown on-board of the spacecraft [49, 50] and long history of laboratory testing. Other types will be discussed shortly since they are mainly developed and tested only in laboratories.

4.1.1 Stationary Plasma Thruster (SPT)

In this type of thruster seen in figure 4.1, especially in-flight models, the plasma chamber is made of a dielectric material such as borosil or boron-nitride. In the paper by [37] the experimental testing of laboratory thrusters with channel walls made of graphite (C), silicon carbide (SiC) and alumina (Al2O3) is described. The choice of borosil or boron- nitride is backed up by the fact that these materials have low sputtering yield, low Secondary Electron Emission [37], good thermal and mechanical properties. Secondary electron emission from the ceramic wall in SPTs is much more prolific and has a significant impact on the thruster physics. The exchange of high energy primary electrons with

27 CHAPTER 4. HALL EFFECT THRUSTER (HET)

low-energy secondary electrons affects the electron energy distribution within the exit plane and thus effectively lowers the average electron temperature. Furthermore, the electron emission to the walls strongly affects the resulting wall sheath potential, which controls the electron energy dynamics but also the energy and angle at which the ions travel towards the wall. The magnetic system creates a quasi-radial magnetic ﬁeld with convex magnetic force lines. Magnetic ﬁeld strength is highest at the outlet and is chosen such that electron-Larmor radius, rLe, is much smaller than the discharge channel length, Ld:

rLe Ld rLi (4.1) where rLi is ion Larmor radius. All these factors not only affect the performance but also the lifetime of the thruster since erosion of the channel wall by ion bombardment is a primary concern in SPTs. In this type of thruster, one can distinguish three main zones: near the anode, ionization and acceleration. In the near-anode region, the ionization is low (around 10 % [21]) and the electrons are the carriers of a current. In the ionization zone (degree of ionization can be between 95 % to 98 % [21]), the current is a combination of current generated by electrons travelling to the anode and current generated by ions going to the SPT outlet. Finally, in the acceleration zone, which is between the ionization zone and the channel outlet, the current in the channel is carried predominantly by ions. Since in SPTs most of the ions are singly charged, the approximate equality is used in calculations:

Ie (Near anode zone) ≈ Ie+Ii (Ionization zone) ≈ Ii (Acceleration zone)

28 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Figure 4.1: Schematic of SPT [3].

4.1.2 Thruster with Anode Layer (TAL)

The TAL thruster name comes from the fact that the ionization and acceleration zones occur within a thin layer close to the anode. In this thruster, channel walls are made of conducting metal as seen in ﬁgure 4.2 and has a negative potential (same as cathode) in order to repel electrons in the ionization region and thus reduce the electron- power losses. Another feature of TAL is that the ion acceleration zone is considerably shorter. Thus, the thruster is usually smaller compared to SPT. However, electron energy is not regulated by channel wall losses and thus the temperature of electrons in the channel is larger. Furthermore, the secondary electron emission of metallic walls is com- paratively low in contrast with dielectric walls. Due to the potential bias, low secondary electron emission, and large electron temperatures the large sheath potentials can be measured at the channel walls [33]. These sheath potentials provide signiﬁcant amount of energy to the ions leads to wall erosion due to ion bombardment (highest impact on thruster life). Since anode is near the high-temperature plasma, a high amount of energy is lost, and it is the primary mechanism for energy losses in this type of device. This type of thruster, namely D-55, has been successfully tested at NASA LeRC [51] and used on Space Technology Experiment (STEX) spacecraft [52].

29 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Figure 4.2: Schematic of TAL [3].

4.1.3 External discharge Hall thruster (XPT)

Karadag et al. [4] designed and tested a new concept of a HET is an external discharge thruster (XPT). The main feature of XPT is that it produces and sustains a plasma beam fully outside of a non-anodic cavity. In ﬁgure 4.3 XPTs schematic (a) and cross section (b) are shown. The XPT has annual geometry and it includes three main components: an anode 1, permanent magnets and a cathode. The walls of a thruster are made of Boron Nitride (BN). It has large magnetic ﬁeld strength of around 0,12 T (at anode centreline). The major advantage in this design is that the wall erosion is eliminated due to external plasma generation. Another pro of this thruster is that it is lightweight and small and thus it could be suitable for cube/nanosatellite applications using a condensable propellant2 feed system.

1which also is a gas feed 2solid propellant such as bismuth, zinc and magnesium.

30 CHAPTER 4. HALL EFFECT THRUSTER (HET)

(a) Schematic. (b) Cross section view

Figure 4.3: External discharge thruster [4].

4.1.4 Magnetically Shielded

The Hall thruster lifetime is limited by the degradation of thruster surfaces, which is caused by the flux of energetic ions. Since the magnetic field affects the plasma properties inside the thruster channel, non-standard magnetic configurations have been proposed in recent years to substantially increase the thruster lifetime. One of them is the thruster jointly developed by NASA Jet Propulsion Lab (JPL), NASA Glenn Research Center (GRC) and University of Michigan (UM) has a magnetic shielding [5]. Magnetic shielding (MS) is a concept which would eliminate the failure of thruster due to erosion of walls. The schematic is shown in figure 4.4. In Magnetic Shielding the magnetic circuit is designed in such way that the magnetic lines generated doesn’t go through thruster walls and thus achieving at the walls high sheath potential and low electron temperature. In this way, the erosion rate is decreased, and thruster lifetime is increased.

31 CHAPTER 4. HALL EFFECT THRUSTER (HET)

(b) Operation.

(a) magnetically shielded conﬁguration

Figure 4.4: Magnetically shielded HET thruster [5].

4.1.5 Cylindrical Hall Thruster

The motivation to develop a Cylindrical Hall Thruster (CHT) arises from difficulties with scaling down3 of a conventional hall thruster. The problems as stated in [6] include the low efficiency of conventional low power thruster (6 % for 100 − 200 W), which arises from large axial electron current, which is increased by electron-wall collisions. The solution could be the design of cylindrical Hall thruster with cusp magnetic field. The cylindrical hall thruster, as shown in figure 4.5, consists of cylindrical boron nitride ceramic channel, ring-shaped anode 4, neutralizer cathode, the magnetic core and magnetized sources. What distinguishes this thruster from conventional annular end Hall thrusters is the cylindrical configuration with an enhanced radial component of the magnetic field. Also, the use of ceramic walls reduces the defocusing effect of the electron pressure and all the advantages of closed electron trajectories are retained.

3Making the thruster smaller and lower power. 4Also could be a gas distributor.

32 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Figure 4.5: CHT with cusp magnetic ﬁeld design [6]

Cusp magnetic ﬁeld

Moscow State Technical University of Radio Engineering, Electronics and Automation (MSTUMIREA) developed a SPT-ATON A3 thruster which is of SPT type and incorporates the cusp magnetic ﬁeld design and cylindrical anode [53]. Harbin Institute of Technology designed an P100 model, which follows similar design principle [54].

(a) SPT-ATON A3 [55] (b) P100 [54].

Figure 4.6: HETs with cusp magnetic ﬁeld design.

33 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Nested channel Hall Thruster

The motivation for developing Nested channel Hall Thruster (NHT) is to increase power to values over 100 kW and maintain an acceptable mass and size. UM in collaboration with the Air Force Research Lab- oratory (AFRL) and NASA developed an X3, which was successfully implemented and tested [56]. The X3 has three discharge channels and each of them features inner and outer electromagnet that are controlled separately. The X3 has one centrally mounted cathode. It can effectively operate on both xenon and krypton propellant with discharge voltage from 200 V to 800 V and discharge currents up to 250 A. NASA consid- ers NHTs a promising technology and is funding it through NextSTEP program, which is investing in the technologies that will be necessary for future crewed missions to Mars. The X3 NHT is part of XR-100 system that is being developed by a team led by Aerojet Rocketdyne (AR) and includes the UM, NASA GRC, and the NASA JPL.

(b) Firing in all channel mode. (a) Mounted in Vacuum Test Facility

Figure 4.7: Nested channel Hall Thruster X3 [7].

4.2 Thruster Performance

The beam current and ion energy cannot be directly measured in Hall thrusters; thus, an alternative expression is derived. According to [3]

34 CHAPTER 4. HALL EFFECT THRUSTER (HET)

the total efﬁciency is deﬁned as:

F 2 ηT = (4.2) 2 ˙mP Pin where Pin is input power in W and m˙ P is total propellant mass ﬂow in g s−1.

4.2.1 Anode efﬁciency

If the anode mass flow and discharge power are known, the efficiency of a Hall thruster can be expressed in terms of the anode efficiency:

F 2 ηa = (4.3) 2 ˙ma Pd −1 where m˙ a a propellant fraction coming from anode in g s and Pd is the discharge power in W.

4.3 Thruster lifetime

The life of the thruster is mainly determined by the channel structure (surface properties and chemical composition of the walls), magnetic ﬁeld shape and plasma properties on Debye length and electron- Larmor radius scales. The channel wall’s primary purpose is to pro- tect the thrusters magnetic circuit from the harsh environment of the plasma discharge. Thus, it is important to understand wall damaging mechanisms such as erosion and sputtering.

4.3.1 Erosion rate

According to [3] the erosion rate R can be expressed by the rate of change of wall thickness, dw. The equation here has different letters for the same values than in: [3]

∂dw φiW R = = Y (Ei) (4.4) ∂t ρZieNA where φi is the ion ﬂux, W is the atomic weight, ρ is the material density, Zi is charge number, NA is Avogadro number, and Y is the sputtering

35 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Figure 4.8: Hollow cathode geometry [3]. yield of the material, which is dependent on the ion type and energy Ei in electron-Volts.

4.4 Neutralizer

In Hall thrusters, a neutralizer is a cathode which emits electrons to ionize the propellant and create plasma inside the thruster. The secondary function of the electrons that are coming from the cathode is to neutralize (here it gets its name), the ion beam coming from the thruster. In HETs, the hollow cathode design is most commonly used, and it is depicted in ﬁgure 4.8. The working principle is described in the next section.

4.4.1 Working principle

The hollow cathode consists of a hollow refractory tube with an ori- ﬁce plate at the top. Inside the tube, there is a cylinder insert that is pushed against the oriﬁce plate. This insert is an active electron emitter which can be made of several different materials such as Lanthanum Hexaboride (LaB6), Tungsten (W ) or Molybdenum (Mo). The cathode tube is wrapped in a heater, which is used to raise the insert temperature. The increase in temperature will cause the thermionic emission of electrons (discharge of electrons). These electrons ionize the gas,

36 CHAPTER 4. HALL EFFECT THRUSTER (HET)

injected through the cathode, and form a cathode plasma from which the discharge-current electrons are extracted into the thruster plasma. Hollow cathodes are normally enclosed in a keeper, another positively biased electrode. The function of a keeper is turning on the cathode discharge, maintaining the cathode temperature and operation during any temporary interruption of the discharge or beam current. The keeper also protects the cathode oriﬁce plate and external heater from, life-limiting, high-energy ion bombardment.

Richardson-Dushman equation

Thermionic emission in hollow cathodes are described by Richardson- Dushman equation, which relates the current density of thermionic emission, jt, to the Richardson work function b0 = eφ and temperature Tem of the emitting material [57]:

2 −b0/kB Tem jt = ATem e (4.5)

−2 −2 where A = 120 A K m is the Richardson coefﬁcient for LaB6 and φ = 2,9 V is the potential for LaB6 [58].

The Schottky effect

The Schottky effect can be significant inside hollow cathodes, where the plasma density is very high and there is a presence of the electric field. In the presence of strong electric fields at the surface of the cathode, the potential barrier that must be overcome by the electrons in the material’s conduction band is reduced, which results in effectively reduced work function. The modified Richardson-Dushman equation reads out as: 2 −(b0−∆W )/kB Tem jt = ATem e (4.6) where ∆W is work function for Schottky effect and can be expressed as: s 3 e Ec ∆W = (4.7) 4π0 where, Ec is the electric field strength at the cathode surface.

37 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Table 4.1: Cathode neutralizer KN-3 parameters.

Property Value Source Propellant Xenon Overall dimensions, mm 70 × 25 × 24 [63] Voltage, V 20 Nominal discharge current, A 4, 5 Xenon start mass ﬂow, mg s−1 0, 35...0, 50 [60] Start power consumption, W 80...95 Start time, s 160 Characteristic potential on channel axis, V 18 [62]

4.4.2 Neutralizer cathode KN-3B

The KN-3B cathode, in Russian literature called cathode compensator, is used in SPT-100B model [59, 60]. This neutralizer was patented in United States [61]. Thermal emitter of this cathode is made of highly emissive material called Lanthanum Hexaboride (LaB6). Due to the lack of technical data for KN-3B, the potential values are assumed to be like K-2 model [62]. The value of potential on cathode channel axis is approximated from ﬁg. 3 in [62] to be 18 V at discharge current of 18 A and mass ﬂow of 0,4 mg s. The technical parameters of the cathode are summarized in table 4.1.

4.5 SPT-100B

SPT-100B model designed and manufactured by EDB Fakel has been selected for the simulation using VSTRAP, since it was successfully tested in orbit as described by Pidgeon et. al [49]. It this flight model of SPT-100 Hall effect thruster is axis-symmetric and equipped with two lanthanum hexaboride (LaB6) cathodes (KN-3B). In the previous experimental studies [64, 65, 66] only one cathode is used and other is installed for redundancy in case the first one fails. The acceleration channel of the thruster has a 100 mm outer diameter, a 70 mm inner diameter, and a channel depth of 21 mm, and a channel width of 15 mm. For its nominal xenon operating condition, the thruster has been char- acterized to have a thrust of 83 m N with a specific impulse of 1600 s, yielding an anode efficiency near 45 %. The operational performance

38 CHAPTER 4. HALL EFFECT THRUSTER (HET)

is summarized in table 4.2 and thruster geometry in table 4.3. The un- available geometrical values were approximated using available data.

(b) SPT-100 ﬂight model (SPT-100B) [64] (a) Schematic [67].

Figure 4.9: SPT-100B thruster.

39 CHAPTER 4. HALL EFFECT THRUSTER (HET)

Table 4.2: HET SPT-100B operational properties.

Property Value Source Overall dimensions, mm 225 × 150 × 125 Discharge Power, W 1 350 Discharge Voltage, V 300 Discharge Current, A 4, 5 Thrust, mN 83 [70] Specific Impulse, s 1 600 Mass, kg 3, 6 Total impulse, MN s 2, 6 Operating cycles Not less than 5 000 Propellant Xenon [67] Plume divergence, ° ∼ 45 [40, 70] Total efficiency % 45 [68] Background pressure, mPa 2, 27 Anode mass flow, mg s−1 5, 16 Anode potential, V 300 [1] Anode current, A 4, 24 Cathode mass flow, mg s−1 0, 396

Table 4.3: Dimmensions of SPT-100B.

Type Value source Thruster W × H × L, mm 225 × 150 × 125 [70] Channel depth, mm 21 [64, 71] Inner channel radius, mm 35 Outer channel radius, mm 50 [67] Channel width, mm 15 Thruster width w/o cathode, mm 135 N/A Cathode keeper oriﬁce diameter, mm 6 N/A Cathode keeper diameter, mm 24 N/A From Cathode to thruster centreline, mm 145 [72] Cathode angle, ◦ 45 N/A

40 Chapter 5

Set-Up, Simulation and Results

This chapter has ﬁve sections. Firstly VSTRAP, a solver used in simulation is described. Second section shows calculation and selection of inﬂow conditions. Then on third section description and results of potential simulation is presented. Forth section talks about DSMC simulation, its results and validation. Finally, the last section describes the plasma simulation and its results.

5.1 VSTRAP

VSTRAP is a tool for kinetic simulation of a rarefied plasma and it deals with the behaviour of charged particles under the influence of electric fields. Thus, the strength of the electric field affecting particle must be known in order to calculate the correct particle trajectory. The governing equation that is approximately solved by the VSTRAP code is the Boltzmann equation as:

Z +∞ Z 4π ∂fα ∂fα Fα ∂fα 0 0 + vα + = [f(vα)f(vβ) − f(vα)f(vβ)]gσDdΩdv ∂t ∂x mα ∂vα −∞ 0 (5.1) which describes the evolution of the velocity distribution function by f(vα) of species α scattered by a background species β. In order to increase efficiency, VSTRAP was designed to include individual kinetic solvers that can later be superimposed to find the overall solution. Thus, only the solvers that are required for the specific simulation can be selected. Further in this section, the solvers that are used in SPT-

41 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

100B simulations are shortly described.

5.1.1 Boundary Element Method

The Boundary Element Method (BEM) is a numerical tool used to solve boundary value problems for Partial Differential Equations (PDEs). In VSTRAP, this tool is used to calculate the distribution of the electric ﬁeld and potential in the domain based on assigned Boundary Condi- tions (BCs). This calculated electric ﬁeld governs the charged particle trajectories. The strong asset of BEM in comparison to other methods such as Finite Element (FE) is that BEM only requires the boundary of the domain to be discretised by mesh, thus making the solution much faster. The drawback of BEM comes in the form of singularities at the boundary, which are caused by the edges and corners in the geometry of the simulation domain [73].

5.1.2 Fast Multipole Method

The Fast Multipole Method (FMM) is used to calculate electric forces between charged particles. Direct summation of electric forces between particles requires N 21 operations, which inherently limits the system size on any computer. Thus, the FMM solver is used since it reduces this time-scaling to a N log N dependency by grouping macro-particles together using an octree. An octree is a tree data structure in which each cell is divided into eight equal volume elements.

5.1.3 Direct Simulation Monte Carlo

The Direct Simulation Monte Carlo (DSMC) technique is a stochastic particle-based method for simulating rareﬁed gas and plasma. The method was pioneered by Bird [74] in the 1960s and since become one of the most accepted methods for solving gas ﬂows in the nonequilibrium regime described by Knudsen number. In a DSMC, the large number of real gas atoms or molecules are represented by a single particle. This approach reduces the requirements of computational power. Each of these particles is moving freely according to its velocity and the local time step in 3D space. These particles can also interact

1where N is the number of simulated particles.

42 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

with other particles and the boundaries of the domain. Collisions between particles are handled in a stochastic manner after all particle movements have taken place. In this way, an evolving simulation emu- lates the physics of a real gas. The main input for the DSMC algorithm is cross-section data, i.e. data which represents a probability for a given particle-particle interaction. Such data is available in various forms such as a collection of data points or parametrized equations, both derived from theoretical calculations and experimental measurements. Thus, cross-section data sets can vary in accuracy and completeness. For a VSTRAP [75] a tool called Cross Section Pre-processor (CSP) was developed. Cross-section data for various interactions can be pro- vided by the user. The cross-section data is then processed by CSP to minimize the number of data points that are written in an output ﬁle which is read by the VSTRAP code. Moreover, an additional output is generated which allows veriﬁcation of the collision algorithms implemented in VSTRAP by comparison of the reference rate constants (computed by CSP) with the simulated rate constants (computed by VSTRAP).

5.1.4 Fokker Planck

The Fokker Planck (FP) is a Monte Carlo based code that solves direct Coulomb collisions. The FP code models the plasma chemistry including re-population of electrons. Coulomb collisions are formally described by FP operator. Takizuka and Abe [76] proposed a solution approximating the FP operator but with the necessity to calculate all collisions one by one. This fact puts a restraint on the time step requiring it to be smaller than the relaxation time. Nanbu [77] found a way to increase the time step by combining several small-angle collisions into one big angle collision. The so derived scattering angle is referred to by Nanbu as the cumulative scattering angle. In VSTRAP, FP was implemented based on Nanbu [77] with improvements for application to Lorentz collisions by Dimits et al. [78].

5.1.5 Particle Pusher

VSTRAP solvers are heterogeneous and produce either a velocity increment, ∆v, or a force increment, ∆F. Thus, in order to calculate

43 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

new particle position and velocity after each time-step, a multi-step approach must be applied. Therefore, the Particle Pusher (PP) is divided into two parts. The ﬁrst part calculates the velocity change of every particle due to all solvers that produce velocity increments ∆v as:

0 v (t) = v(t) + ∆vtot (5.2) where total velocity increment is introduced as:

j X ∆vtot = ∆vi (5.3) i=1 and j is number of solvers that outputs ∆v. In the second part the Verlet velocity algorithm is implemented. This algorithm calculates new particle positions and velocities. It has four steps: 1. Find acceleration from force increment, ∆F:

k 1 X a(t) = ∆Fi (5.4) k i=1

2. Find new position after time step ∆t:

0 1 x(t + ∆t) = x(t) + v (t)∆t + a(t)∆t 2 (5.5) 2

3. Find new acceleration after time step ∆t:

x(t + ∆t) − 2x(t) + x(t − ∆t) a(t + ∆t) = (5.6) ∆t 2

4. Find new velocity after time step ∆t:

0 1 v(t + ∆t) = v (t) + (a(t + ∆t) + a(t))∆t (5.7) 2 where k is the total number of solvers that outputs ∆F. Particle pusher works only together with other solvers.

44 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Figure 5.1: Illustration of specular scattering.

5.1.6 Particle Wall Interaction

In VSTRAP Particle Wall Interaction (PWI) solver four types of particle- wall interactions are implemented 2: 1. Specular Scattering. It is a mirror-like reflection, shown in figure 5.1, of particles colliding with the wall and the algorithm, can be divided into five steps: (a) Calculate the coordinates of a point where the particle hits the wall. (b) Calculate vector of incidence. (c) Calculate vector of reflection. (d) Calculate the new position of the particle. (e) Set new velocity of the particle. 2. Electron Absorption When an electron collides with the wall, the electron is removed from the simulation domain. 3. Neutralization When ion collides with the wall, it is replaced by its neutral atom and it is reflected in the same manner as in

2A is Absorption; N is Neutralization; SEE is Secondary electron emission; S is Scattering;

45 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

specular scattering. 4. Secondary Electron Emission At the moment, this sub-solver can only simulate two types of electron emission after particle hits the wall: (a) Electron Induced Electron Emission When high energy electron hits the wall it can hit out an electron from the wall surface. (b) Ion Induced Electron Emission When high energy ion with hits the wall it can also hit out an electron from the wall surface.

5.2 Inﬂow Conditions

In the input file for VSTRAP code, the following inflow parameters must be given as input for each particle species: temperature, drift velocity, number density and particle weight. In this work, there are two inflow surfaces: thruster exit and keeper orifice. Thus, in this section, the selection and calculation of these parameters for each of the surfaces are described.

5.2.1 Exit Inﬂow

One starts by calculating particle number densities at thruster exit plane, which are found by: ˙ Np np = (5.8) vpAex ˙ where np is particle number density, Np is particle ﬂow in particles per second, vp is particle drift velocity, Aex is exit area and p is particle species 3. Exit area can be found as:

2 2 Aex = π(Rout − Rin ) (5.9) where, Rout and Rin are channel outer and inner radii respectively.

3Expressed as n for neutrals and i for ions. These further are expressed as particle species (Xe) or its ions .

46 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Total mass ﬂow rate from anode 4 is assumed, due to high level of ionization, to contain only singly and doubly charged Xenon ions. The mass ﬂow rate of different species for anode can be expressed as:

˙mXe+ = ˙ma ηu η+ , ˙mXe2 + = ˙ma ηu η2 + (5.10) where η+ and η2+ are fraction of singly and doubly charged ions taken as 0, 89 and 0, 11 respectively from measurements by King and Gal- limore [79]. And ηu is propellant utilization coefﬁcient. Zhurin et al. [80] claims that according to experimental data the utilization coefﬁ- cient for xenon propellant ηu ≈ 1. It can be found from the formula found in the same paper as:

ni vi ηu = (5.11) (ni vi + nn vn ) where vi and vn are velocities of ions and neutrals respectively. From experiments vi vn, nn ni and for xenon propellant:

vi nn (5.12) vn ni which in agreement that ηu ≈ 1. However, the utilization coefﬁcient is taken as 0, 95, referring to [33] and [81]. The neutrals originate both from the anode and the cathode. Neutral mass ﬂow from the anode is found as:

˙mXea = ˙ma (1 − ηu ) (5.13)

Furthermore, mass ﬂow is expressed as particle ﬂow in following way: ˙m ˙m + ˙m 2 + ˙ Xea ˙ Xe ˙ Xe NXe = , NXe+ = and NXe2 + = (5.14) mi mi mi and the values for neutral, singly ionized and doubly ionized particles are 1,18 · 1018 s−1, 1,79 · 1019 s−1 and 4,48 · 1018 s−1 correspondingly. Based on [67], the neutral particle drift velocity is assumed to be 300 m s−1. The ion velocity is found using the following energy-based expression from [80]: r r 2qVb 2Ei vi = = (5.15) mi mi

4mass of the gas that is ejected from the anode per unit of time

47 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

where i in vi represents the ionized species and Vb is the net voltage through which the ion is accelerated. It is obtained approximately by:

Vb ≈ Vd − Vcg (5.16)

Vd is discharge voltage taken as 300 V from [67] and Vcg is cathode to −1 ground voltage taken as 20 V . It is found that vXe+ ≈ 20 290 m s and −1 vXe2+ ≈ 28 690 m s . By inserting the particle flow and velocity values into equation 5.8 one gets number densities for neutral, singly ionized and doubly ionized particles as 4,902 · 1017 m−3, 1,942 · 1017 m−3 and 3,531 · 1016 m−3 respectively. In the book by Goebel and Katz [3] it is stated that the ratio between electron and ion temperatures in the channel is approximately 10. Also, in the book one can find a rule of thumb, which agrees with obser- vations, that electron temperature can be approximated as 1/10th of beam voltage: Te ≈ 0 , 01Vb (5.17) and thus, the ion temperature can be as high as 2,4 eV in the ionization zone. However, when ions travel towards the exit their temperature decrease and the minimum electron temperature in this area is 8 eV as found in figure 4 in the paper by Kim et al. [82]. Thus, the ion temperature is estimated to be in the range of 0,8 eV. Also, singly and doubly charged ions should have different temperatures. However, due to the lack of information on how to find the difference in temperatures, the temperatures are assumed to be the same. Furthermore, temperature of neutral atoms is assumed to be equal to 500 K [80].

Velocity distribution

Particles in plasma obey the 0th law of thermodynamics (If a body C, be in thermal equilibrium with two other bodies, A and B, then A and B are in thermal equilibrium with one another [83]) and the plasma tends to evolve towards the thermal equilibrium. Thus, one can make use of Boltzmann distribution function for energy:

− E f (E) = Ae kB T (5.18)

48 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

where E is kinetic energy of a particles in three dimensions expressed as: m E = v 2 + v 2 + v 2 (5.19) 2 x y z The above function 5.18 must be normalized by finding normalization coefficient A, so the probability would be equal to 1. One can find A by constructing for every dimension a Gaussian integral in the form:

∞ 2 ∞ mv2 ∞ 2 Z mvx Z y Z mvz −1 − 2k T − 2k T − 2k T A = e B dvx e B dvy e B dvz (5.20) −∞ −∞ −∞

One can make use of the deﬁnite integral of form:

∞ Z 2 e−x dx = π (5.21) −∞ with substitution equal to:

r m x = vx (5.22) 2kT for each spatial dimension. Then, Integration and substitution yields normalization coefﬁcient as:

m 3 /2 A = (5.23) 2 πkB T

By inserting energy equation and normalization coefﬁcient into Boltz- mann distribution function 5.18 one gets a normalized Maxwellian5 velocity distribution function as:

3 /2 m m 2 2 2 − 2k T vx +vy +vz f (vx , vy , vz ) = e B (5.24) 2 πkB T

Further integration in polar spherical coordinates gives us Maxwellian speed distribution:

3 /2 mv2 m 2 − f (v) = 4 π v e 2kB T (5.25) 2 πkB T 5Maxwellian-Boltzmann

49 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

5.2.2 Cathode Inﬂow

Based on [62, 59] Xenon gas temperature, TXe, and electron temperature , Te, leaving the neutralizer cathode is chosen to be 2000 K and 58 000 K respectively. To calculate drift velocity reference density inside 21 −3 the cathode ne = 1 · 10 m [62] is taken and used in formula from [33]: Id ve = (5.26) ne eAco where Aco is cathode oriﬁce area found by:

2 Aco = πrco (5.27) where rco is cathode oriﬁce radius equal to 0,25 mm. Thus, the electron −1 drift velocity is ve = 143 000 m s . Furthermore, Xenon neutrals and electrons from cathode also have thermal velocity components, vth,Xe and vth,e, which are already calculated in VSTRAP using: r 3kB Te vth,e = (5.28) me

−1 For the previous values it is found vth,Xe = 568 m s and vth,e = 1,3 · 106 m s−1. Neutral particle ﬂow from cathode is found as:

˙mXe,c NXe,c = (5.29) mi where m˙ Xe,c is neutral mass ﬂow from cathode assumed to be equal:

˙mXe,c = ˙mc (5.30)

Inserting the values into equations 5.29 and 5.30 one gets cathode −7 −1 neutral mass flow m˙ Xe,c = 4 × 10 g s and neutral particle flow ˙ 18 −1 NXe,c = 1,83 · 10 s To find neutral particle density formula 5.8 is applied, but instead of thruster exit area cathode exit area is used, which is equal to keeper orifice area, Ako, found by:

2 Ako = πRko (5.31) where Rko is keeper oriﬁce radius. For electrons different formula is used since electrons are injected not from cathode oriﬁce but from

50 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

keeper oriﬁce and it looks like:

Id ne = (5.32) ve eAko The number densities for Xenon neutral atoms and electrons leaving 19 −3 18 −3 cathode are nXe,c = 2,157 · 10 m and ne = 6,944 · 10 m correspondingly. Further, the inﬂow condition, as calculated before in this and previous subsections, are summarized in table 5.1. Since the number of physical particles is very large and computation of them would require quite a large amount of computer resources, VSTRAP uses macro- particles. The certain number of physical particles are substituted with one macro-particle, which is then introduced into the system. The number is called particle weight and will be deﬁned in the following sections together with parameters for each simulation individually.

Table 5.1: Inﬂow conditions

Species Number density, m−3 Temp., K Velocity, m s−1 17 Xea 9,545 · 10 773 300 Xe+ 2,153 · 1017 9 284 20 290 Xe++ 3,794 · 1016 9 284 28 690 19 Xec 2,157 · 10 2 000 300 18 ec 6,944 · 10 58 000 143 000

5.3 Potential Simulation

It is the ﬁrst simulation performed to get potential distribution and check if it could be used for plasma simulation as a background. Here the description of the geometry, BCs and results are presented.

5.3.1 Geometry

Based on the summary in sections 4.4 and 4.5 the thruster geometry is shown in ﬁgure 5.2 was modelled using gmsh, a ﬁnite element based tool with Graphical User Interface (GUI) to create, edit and visualize

51 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

both 2D and 3D geometry and its mesh. The coordinates are following: Z coordinate is parallel to thruster exhaust; Y coordinate is on the cathode symmetry plane and is perpendicular to Z; X coordinate is perpendicular to Z-Y plane. The thruster has two cathodes, but since in most experiments [84, 66, 70] only one is working, it was decided to model only one cathode of similar dimensions. Furthermore, cathodes are positioned at certain angle WRT two planes (X-Z and X-Y). How- ever, to make the geometry simpler, the cathode is positioned at 45 ◦ WRT X-Y plane. The geometry has dimensions presented in table 4.3. The thrusters magnetic circuit is not included in simulation domain. The simulation domain size is approximately 2 × 2 × 1m. The mesh sizes have been selected to be different for different surfaces. The applied mesh is shown in ﬁgure 5.2.

Fine mesh A fine mesh was assigned to anode, chamber and keeper orifice inflow surfaces. It is done to achieve more accurate potential distribution at these locations.

Coarse mesh A coarse mesh was assigned the walls of the thruster and keeper and the domain outer surfaces. It is done in order to make simulation faster.

52 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

(a) Isometric view. (b) side view.

Figure 5.2: Simulation domain

5.3.2 Boundary Conditions

BCs presented in figure 5.3 and table 5.2 below used in the potential simulation. In this simulation only BEM solver were used and thus there is no inflow of particles. Two types of BCs are used: Dirichlet and Neumann. Dirichlet specifies the values that a solution will take along the boundary of the domain. Neumann condition specifies the rate at which the solution changes within the boundary of the domain. The anode, cathode, keeper and domain boundaries were assigned to be of Dirichlet type and have 300 V, 18 V and last two has 0 V potential respectively. Other boundaries were assigned to be Neumann and was set to equal to 0. The cathode boundary was assigned to have potential of hollow cathode KN-3B orifice axis, as stated in section 4.4.2. This potential was selected to be equal to 18 V. So, if one assumes that 90 % of cathode power is used to produce electric field and rest is used by heater, one

53 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Figure 5.3: Boundary conditions can ﬁnd the cathode potential:

0 .9Pc Vc = (5.33) Ic where Pc is cathode power taken as 95 W and cathode current Ic taken as discharge current 4,5 A, since the SPT-100B circuit is connected in series. It gives the value of Vc to be around 19 V. Thus, the potential of 18 V is a good approximation. The anode boundary potential was taken from thruster speciﬁcation, which was summarized in table 4.2.

Table 5.2: Boundary Conditions

Anode Walls & Domain Cathode Keeper Type Dirichlet Neumann Dirichlet Dirichlet φ 300 V [53] - 18 V 0 V dφ dn - 0 - -

54 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

(a) 3 mm from anode. (b) 21 mm from anode.

Figure 5.4: Potential distribution inside the channel.

5.3.3 Results

Here the results of the potential simulation are presented. In the next page, one can find plots of the simulation without plasma and it shows potential distribution. In figure 5.4a one can see the potential distribution near the anode. It is considered to be homogeneous. The yellowish blueish patches at the outer edge come from the singularities as described in subsection 5.1.1, which are caused by the geometry. In figure 5.4b one can notice an asymmetry between upper and lower parts. This asymmetry comes from the cathode which is placed above thruster exit and has a potential of 18 V. In figure 5.5 the potential distribution plotted on the cut at the Z-Y symmetry plane of the thruster. Since this simulation does not include plasma the potential values are expected to be lower. In comparison to plasma simulation results by Pérez-Grande et al. [8] (shown in figure 5.6b), here the values in potential simulation are smaller. In the potential simulation one can see that at the exit the potential has the value of around 120 V and in the simulation by Pérez-Grande et al. [8] it has around 200 V. Also, the gradient of distribution here is similar but larger, thus the values from anode to exhaust decrease faster. Fur- thermore, in Perez-Grande simulation the potential at the exit is almost symmetric while in potential simulation has a shift towards the centre of the thruster. This is because Perez-Grande et al. do not include the

55 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Figure 5.5: Potential distribution inside the domain.

(a) Figure 5.5 zoomed in. (b) Top graph from ﬁgure 10 in [8].

Figure 5.6: Comparison of potentials with [8]. cathode boundary and simulates only a quarter of the thruster. On the contrary, the model here has the whole thruster and a cathode is placed at its approximate location.

56 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

5.4 DSMC Simulation

5.4.1 Geometry

(a) Isometric view. (b) side view.

Figure 5.7: Simulation Domain

The geometry of simulation domain here is almost the same with the one described in subsection 5.3.1. There are two differences.The ﬁrst difference is that in this geometry the thruster chamber is removed, since the main interest is in simulation of the thruster plume. Another difference is that domain size was made to be 400 × 400 × 200 mm. The mesh sizes in this simulation are also different for different surfaces to decrease simulation time. Also, the mesh size must be chosen such that the particles do not move more than one cell per time-step [85]. The applied mesh is shown in ﬁgure 5.7:

Coarse mesh Coarse mesh was assigned for the thruster, keeper and domain walls6, due to limitations in computational power and lower

6Note: the central wall surface has ﬁne mesh due to limitations in gmsh 4.0.

57 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

requirements for accuracy at these locations.

Fine mesh Fine mesh was assigned to exit and keeper oriﬁce inﬂow surfaces. It is done to more accurately introduce the particles into the system, or in other words, particles will be less likely to be introduced from the same point. Also, this will give us a higher accuracy of the results.

5.4.2 Simulation parameters

This section presents the selection of simulation parameters: time-step, number of iterations and particle weight. The minimum time-step for DSMC has to be signiﬁcantly smaller than the mean collision time [85] found by equation 2.37 and can be expressed as: l ∆t < (5.34) ¯vn In the case, where the cell size is smaller than the mean free path it should be modiﬁed to: ∆x ∆t < (5.35) ¯vn where ∆x is the minimum cell size. Neutral number density for the equation is taken the largest value, which is cathode number density. The collision cross-section is found using equation 2.36 with the distance between atom centres taken as double Van Der Waals radius from [86]. From equation 2.37 it is found that mean collision time is equal to 2,635 · 10−4 s. The minimum cell size was found from the geometry to be approximately 0,75 mm. From equation 5.35 it is found that the time-step should be less than 2,5 · 10−6 s. The number of iterations is equal to the number of iterations required to reach a steady-state, a state when there is no change in the system. It is expressed as the number of iterations needed for a particle to leave the system twice in the form: LD Nit = (5.36) ¯vn ∆t and it is equal to 6666. But We choose 15 000 iterations in order to have more data for neutral background averaging: The particle weight was selected by trying different values, so the

58 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

number of particles at a steady-state does not exceed 10 · 106. Also, the weight of particles must be the same for every particle, since DSMC cannot handle collisions between particles of different weight. The following simulation parameters are chosen: 1. Timestep: 1 · 10−7 s 2. Iterations: 15 000 3. Particle weight: 1 · 1010 The number of macro-particles at steady-state can be approximated by: ˙ NpLD Nss = NIn Nlife = (5.37) vpWp where NIn is number of injected particles per iteration, Nlife is time ˙ particle stays in the simulation domain, Np is particle ﬂow in particles per second, vp is particle velocity and Wp is particle weight. With the data for DSMC simulation there should be around 198 000 neutral macro-particles at steady-state.

5.4.3 Simulation

In order to generate a neutral background, a DSMC simulation is performed. Only neutral Xenon atoms are injected and only DSMC, PWI and PP solvers are used. PWI solver is used to account for possible collisions with thruster surfaces. PP is used to move macro-particles at every time-step. Inflow conditions are taken from table 5.1 in section 5.2. Simulation was run for 15 000 iterations and in figure 5.8 one can see particle number evolution per iteration. The simulation reached steady-state at about 10000th iteration. And at steady-state, there is about 130 000 macro-particles. This number is lower than the approximation because thermal velocity was not included in the equation 5.37. Only the data from every 20th iteration is saved and thus for averaging there are 250 neutral particle state files. Later, the averaged neutral particle state will be loaded in plasma simulation as a neutral background.

59 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Figure 5.8: Particle number evolution towards the steady-state

5.4.4 DSMC Validation

To do a plasma simulation, it is necessary to know if the use of DSMC for neutral background generation is valid. Thus, some of the parameters from DSMC simulation must be checked if they agree with the assumptions. Figure 5.9 shows average particle collisions per iteration. For best results, this number should be less than one. In the graph, one can see that a few iterations this number exceeds one, which means that each particle collides on average more than once. This can be ﬁxed by choosing smaller time-step. The number density distribution and temperature, shown in ﬁgures 5.10a and 5.10b respectively, are plotted directly from simulation results after averaging the steady-state. In these plots, one can see that highest number density and temperature are near both exits and decrease further. This behaviour is expected since particles are introduced from a small surface compared to the domain. Thus, there will be more particles near these surfaces. And their temperature will be higher since they are spaced closer and thus there are lower heat losses. The

60 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Figure 5.9: Mean particle collision per iteration next plots are derived from these two. Figure 5.11c shows mean collision frequency and the inverse of it is the amount of time in seconds must pass until one collision happens. This frequency is found by equation 2.38. The frequency governed by two main properties: particle number density and particle cross- section. Particle cross-section was found by interpolating temperature data and was found to be almost constant. The larger collision frequencies are more often particles collide. This can be seen near thruster and cathode exits since here particle number density is very high. Lowest values are just above the cathode since here there are so little particles. Figure 5.11a shows the velocity distribution, which was derived from neutral temperature using equation 2.39. One can see that particles have the highest velocities at thruster and cathode exits. This is expected since here they have the highest amount of energy. Figure 5.11d represents the average mean free path at steady-state. As was mentioned before, the mean free path tells how far particle travels before it collides with another particle. In the ﬁgure, one can see that lowest values are near the thruster and cathode exits and thus in this region particles collide more often than in the far region, because in this region particle number density is highest.

61 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

(a) Number density of neutrals (b) Temperature of neutrals

Figure 5.10: Direct plots at steady-state

Furthermore, the plasma should agree with the dilute gas assumption described in section 2.4. Thus, the ratio δ/d was plotted in the Z-Y plane as in ﬁgure 5.11b. In this ﬁgure, one can see the ratio plotted in the cut in the middle of the domain (Z-Y plane). The lowest values are at thruster and cathode exits due to higher particle density in this region and thus on average particles are closer to each other. Highest values are in the area above the cathode since particles have a very low chance to travel there. And in this location, there are a lot fewer particles and on average they are more distant from each other.

62 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

δ (a) Velocity of neutrals (b) Ratio d

Figure 5.11: Derived plots at steady-state

63 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

5.5 Plasma Simulation

This is the ﬁnal simulation. The neutral particle background generated in DSMC simulation described in section 5.4 is imported and then used in this plasma simulation and then ions and electrons are injected together.

5.5.1 Far ﬁeld versus Near ﬁeld

The measurements of hall thruster’s performance parameters are done on either the near [82] or the far-ﬁeld [87] of the exit plume. This is a general division and each region has different physical qualities [48].

The far field The far-field starts a few thruster radii away from exit plane and is defined as the region of the plume which is largely inde- pendent of the initial characteristics of the discharge. In this region, particle collisions and the influence of thruster and neutralizer become negligible. Even though the experimental measurements of this region are difficult, the modelling and simulation are easy.

The near field The near field is defined as the region whose characteristics are dominated by the initial discharge conditions. These include large densities of neutrals and electrons, which leads to the creation of secondary plasma due to CEX. This region is influenced by neutralizer, residual electric fields, magnetic fields, and inhomogeneities in the 3D plume. Thus, it is hard to model and simulate but easy to measure in the laboratory. This work will focus on near field exhaust plume.

5.5.2 Boundary Conditions

Figure 5.12 represents the locations of BCs summarized in table 5.3: The BCs are almost the same to ones described in section 5.3.2 except in this simulation there are fewer surfaces/boundaries. Here, the thruster chamber was replaced with exit surface which was assigned to have Dirichlet boundary conditions and have the potential of 250 V, which was approximated from ﬁgure 9 in [67].

64 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

Table 5.3: Boundary Conditions

Exit Walls & Domain Cathode Keeper Type Dirichlet Neumann Dirichlet Dirichlet φ 250 V [67] 0 V 18 V 0 V dφ dn 0 0 0 0 Inﬂow NXe+ NXe++ - Ne− Process - N/SEE/S & A - N/SEE/S

Figure 5.12: BCs for plasma simulation

PWI at the Walls

As was described in section 5.1.6, VSTRAP has PWI solver and thus one can assign different processes to different surfaces. These PWI processes are also included in table 5.3.

5.5.3 Inﬂow and simulation parameters

The inﬂow conditions taken from table 5.1 in section 5.2. In this simulation, electron particles and singly and doubly charged Xenon atoms introduced into the system. For simulation to be stable requires a time-

65 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

step on the order of the inverse of the plasma frequency [88]: 1 ∆t ≤ (5.38) wp

The time-step is selected based on electron plasma frequency. Also, to account for plasma oscillations this time-step should be expressed as [89]: 0 .07 ∆t ≤ (5.39) wpe Number of iterations is found similarly to formula 5.36, but instead of average neutral velocity minimum ion velocity is used:

LD Nit = (5.40) vi,min ∆t and equals to around 1 · 106. With the particle weight as in DSMC simulation, the code will not inject any particles. Thus, the particle weight should be lowered. For the best results, it should be decreased by the same factor as the time-step. The following simulation parameters are chosen: 1. Timestep: 10−11 s 2. Iterations: 1 · 106 3. Weight: 106 The number of macro-particles can be found in the same way as for the DSMC simulation. It is found that there should be around 1,91 · 108 singly charged Xenon, 1,67 · 107 doubly charged Xenon and 1,82 · 106 electron macro-particles.

5.5.4 Simulation time

This subsection gives a rough estimate of simulation time using estimations from [90] scaled to our hardware. The table 5.4 shows the values used in our calculation.

66 CHAPTER 5. SET-UP, SIMULATION AND RESULTS

R8000 SGI Power Challenge AMD Ryzen 7 2700x (16 threads) Speed 75 MHz 3,7 GHz Collision of particles 40 µs 5,068 · 10−8 s Moving particle 34 µs 4,307 · 10−8 s

Table 5.4: Hardware estimations

The simulation time until convergence (for the 1 · 106 iterations) was approximated using simple Matlab routine: clc close all t1=4.307e-8;% time needed to move particle t2=5.068e-8;% time required fora collision eta=0.43;% percentage of collided particles n_s=3;% solvers that solve particle collisions N=linspace(0,2.077e8,1e6);% Approximate number of ... particles in each iteration sum=0; speedup=3.58; for i=1:length(N) tsum(i)=N(i)*t1+eta*n_s*N(i)*t2; sum=sum+tsum(i); end sum;% Simulation time in seconds. formatSpec ='Simulation will take approximately %4.2f days'; sprintf(formatSpec,(days(seconds(sum)))% Simulation time ... in seconds.

Thus, the simulation for the baseline code could take approximately 130 days. However, in VSTRAP optimization technique of vectorization was to some extend implemented. As stated in [91], this technique could increase the speed of particle pushing 3,6 times. With this technique the simulation could take around 93 days.

67 Chapter 6

Discussion

Since the experimental testing of plasma technologies and studying plasma phenomena is very difficult and expensive, it is important to develop tools that will remove the need for such. One of such tools, namely VSTRAP is being developed at SPARC industries sarl. VSTRAP is being designed for plasma simulations and consists of 6 solvers, which can work separately. Thus, the goal of this thesis work was to validate VSTRAP package using a test case of SPT-100 Hall Effect Thruster. The solvers were tested in three stages: potential simulation (BEM solver), DSCM simulation (DSMC solver) and finally plasma simulation (all solvers). However, due to high simulation cost, e.g. simulation time, plasma simulation was not performed. Therefore, in this work, 2 solvers could be validated: BEM and DSCM. The BEM solver was validated with potential simulation as seen in section 5.3. The purpose of the potential simulation was to see if the achieved data could be loaded as a potential background into plasma simulation so the simulation costs could be reduced. The noticeable outcome of the analysis is that the results reflect applied BCs. However, it was found that the values at the exhaust plane are much lower than the expected value of the difference between anode and cathode potentials. Thus, the potential distribution, which results from the potential simulation, cannot be used in the plasma simulation. The reason is that the potential and electric field generated by the particles inside the plasma has a substantial effect on the results as shown in measurements by Grimaud et al. [92] and simulations by Adam et al. [93] and Pérez-Grande et al. [8]. To account for potential and electric field generated by charged particles VSTRAP uses FMM solver, which eventually

68 CHAPTER 6. DISCUSSION

used in plasma simulation together with FP solver, which accounts for collisions of charged particles. The DSMC solver was validated with DSMC simulation. The DSMC simulation aimed to determine if DSMC solver is correctly implemented and it can simulate collisions between neutral particles. The validation was done by examining how well results agree with the dilute gas assumption. The result was that the ratio of the mean distance between particles and particle diameter is much higher than the lowest value required for the dilute gas assumption to be valid. Therefore, it was concluded that the use of DSMC is valid. Moreover, the simulation should be rerun with smaller time-step to get rid of or reduce the anomalies found in ﬁgure 5.9. The plasma simulation was not performed due to the long simulation time of several months as was estimated in section 5.5.4. This long time can be expected since electrons are also simulated as particles. Electrons move very fast and thus we need very small time-step to be able to track their interactions. Ions move very slow compared to electrons and thus to reach steady-state one needs a lot of iterations. Macro-particles are injected at each iteration therefore there will be a lot of macro-particles at steady-state. It can be concluded that for this kind of simulations one needs a super-computer, which could do the tasks a lot faster. However, before choosing to use a super-computer one should consider applying some techniques which are described in the following section.

6.1 Outlook

This section gives some ideas on what can be done to reduce simulation costs. This section is divided into three subsections, which addresses different aspects: simulation, software and hardware.

6.1.1 Simulation

The simplest method to reduce simulation costs is to select a smaller simulation domain and coarser mesh size. Choice of smaller domain reduces the number of time-steps required to reach steady-state and results in shorter simulation time. Since the simulation of particle collisions (DSMC and FP solvers) and particle-wall interactions (PWI

69 CHAPTER 6. DISCUSSION

solver) takes more time, this reduction is not very signiﬁcant. Choice of coarser mesh will also reduce simulation time. However, this will lower the accuracy of the results. According to [88, 94] computational time can also be reduced by linearly scaling down the simulation domain and input parameters. The [94] mentions two methods of scaling, the ﬁrst of which will have greater computational advantages. The main idea of this method is to reduce the simulation domain in such a way that the main plasma characteristics remain unchanged. This would reduce the number of iterations required to reach steady-state as well as the number of cells required to cover the simulation domain (if the same size of the cell is used). Furthermore, there will require fewer real particles which will allow the use of smaller particle weight and this would increase the accuracy of the results.

6.1.2 Hardware

The current simulation used an AMD Ryzen 7 2700x Central Processing Unit (CPU) with 8 cores. Even though the software is to some extent parallelized and can use all 8 cores (all 16 threads), current CPU is not enough to make simulation fast and puts a constraint on the number of macro-particles in the system. The solutions could be following the use of multiple CPUs; instead of CPU use a Graphics Processing Unit (GPU) with CUDA architecture; or GPU and CPU hybrid system. A GPU with CUDA architecture could be a good choice, since according to Fatemi et al. [95] and Stantchev et al. [96] has a better performance and price ratio compared to CPUs. Since it is known that CPUs are better at performing complex task and GPUs are better at basic math operations [97], the best solution would be to optimize the software for heterogeneous cluster systems[98, 91], which have GPUs, CPUs and other processing units ( e.g. Xeon Phi coprocessors) working simultaneously.

6.1.3 Software

The VSTRAP solver is written in C++, which makes it quite efﬁcient. However, the code must be optimized. Surmin et al. [91] showed that optimization techniques such as improvement of data locality, en-

70 CHAPTER 6. DISCUSSION

hanced efﬁciency of parallelization1 and vectorization2 could speed up the CPUs, GPUs and Xeon Phi coprocessors by a factor. When implemented into VSTRAP, the techniques mentioned above will help to reduce simulation costs and some of them will help to increase the limit of macro-particles that can be simulated and thus increase the achievable accuracy of the results.

1Make use of all CPUs or GPUs threads simultaneously 2i.e. instead of injecting particles one by one inject them all at the same time

71 Chapter 7

Conclusion

The study aimed to simulate the exhaust plumes of Hall Effect Thruster. The SPT-100 thruster was chosen as a test case since this thruster has been studied extensively in both simulations and experiments. The VSTRAP package is being developed for commercial purpose and the user should see it as a black box. VSTRAP consists of 6 different solvers and the user can know what each solver does, but the user cannot know how it works and how the software is designed. At the current stage of development, inside VSTRAP the geometry cannot be directly created as well as input parameters chosen. Thus, the user needs additional packages. Gmsh for geometry modelling and simple notepad to write input files in XML format 1. The main benefit of these packages that they are free and can be easily downloaded and installed on both Linux and Windows. However, the user will have to learn how to use them taking into account that they could have some flaws 2. Another thing is that geometry cannot directly be imported into VSTRAP since it also has to be converted into XML format using the company’s converter. To validate VSTRAP package three simulations were planned: potential simulation, DSMC simulation and plasma simulation. Based on the results of potential and DSMC simulations it is certain that BEM and DSMC solvers work and give reasonable values. As found in subsection 5.5.4, for current software and hardware to complete plasma

1Notepad++ with XML plugin is a great choice since it can help to ﬁnd errors in XML ﬁles 2e.g. Geometry created with the newest version of gmsh might not be compatible with VSTRAP

72 CHAPTER 7. CONCLUSION

simulation will require several months of simulation. Consequently, plasma simulation was not performed and thus the validation of the VSTRAP solver is incomplete. However, although the plasma simulation was not performed, the user could use section 5.5 as a guide for further study. In the previous chapter, some techniques were mentioned that could make the plasma simulation possible without the need for a hardware upgrade. Therefore, one could proceed to plasma simulation and investigate if the upgrade for hardware is necessary. Finally, even if the VSTRAP software is still in an early stage of development, it promises to become a powerful and versatile tool which will let the user create different models and perform complex plasma simulations. One of the applications would be the testing of a thruster under development. The results from simulations can contribute to the further estimation of thruster parameters and development of thruster geometry. It clearly shows that with the increase of computational capabilities and new optimization techniques these simulations could eventually replace the need for experimental thruster testing. Thus, making the development of plasma technologies cheaper and much faster.

73 Bibliography

[1] Natalia Macdonald-Tenenbaum, Quinn Pratt, Michael Nakles, Nickolas Pilgram, and William Hargus. Ion velocity distributions in an SPT-100 Hall thruster. In 35th International Electric Propulsion Conference, pages 1–16, 2017. [2] Samil Korkmaz. Tsiolkovsky rocket equation — Wikipedia, the free encyclopedia. https://en.wikipedia.org/w/index. php?title=Tsiolkovsky_rocket_equation, 2018. [Online; Accessed on December 1st 2018]. [3] Dan M. Goebel and Ira Katz. Fundamentals of Electric Propulsion: Ion and Hall Thrusters. John Wiley & Sons, Inc., 2008. ISBN 9780470429273. doi: 10.1002/9780470436448. [4] Burak Karadag, Shinatora Cho, Ikkoh Funaki, Yushi Hamada, and Kimiya Komurasaki. External Discharge Plasma Thruster. Journal of Propulsion and Power, 34(4):1094–1096, 2018. ISSN 0748-4658. doi: 10.2514/1.B36900. [5] Richard R. Hofer, Sarah E. Cusson, Robert B. Lobbia, and Alec D. Gallimore. The H9 Magnetically Shielded Hall Thruster. 35th International Electric Propulsion Conference, pages 2017–232, 2017. URL https://iepc2017.org/sites/default/files/ speaker-papers/iepc-2017-232_hofer_r05.pdf. [6] Y. Raitses and N. J. Fisch. Parametric investigations of a noncon- ventional Hall thruster. Physics of Plasmas, 8(5 II):2579–2586, 2001. ISSN 1070664X. doi: 10.1063/1.1355318. [7] Scott James Hall. Characterization of a 100-kW Class Nested-Channel. PhD thesis, University of Michigan, 2017. URL http://hdl. handle.net/2027.42/144053.

74 BIBLIOGRAPHY

[8] Daniel Pérez-Grande, Pablo Fajardo, and Eduardo Ahedo. Evalu- ation of erosion reduction mechanisms in hall effect thrusters. In Joint Conference of 30th International Symposium on Space Technology and Science, 34th International Electric Propulsion Conference,(Hyogo- Kobe), 2015. [9] J. Scott Miller, Steve H. Pullins, Dale J. Levandier, Yu Hui Chiu, and Rainer A. Dressler. Xenon charge exchange cross sections for electrostatic thruster models. Journal of Applied Physics, 2002. ISSN 00218979. doi: 10.1063/1.1426246. [10] M. Gamero-Castaño and I. Katz. Estimation of Hall Thruster Ero- sion Using HPHall. In Proceedings of the International Electric Propul- sion Conference 2005 (IEPC05), pages 1–10, 2005. [11] A. I. Morozov. The conceptual development of stationary plasma thrusters. Plasma Physics Reports, 29(3):235–250, 2003. ISSN 1063- 780X. doi: 10.1134/1.1561119. URL http://link.springer. com/10.1134/1.1561119. [12] G. D. Racca, G. P. Whitcomb, and B. H. Foing. The SMART-1 mission. ESA BULLETIN-EUROPEAN SPACE AGENCY, 1998. [13] Denis Estublier, Giorgio Saccoccia, and Jose Gonzalez Del Amo. Electric propulsion on SMART-1. European Space Agency Bulletin, 2007. ISSN 03764265. [14] Robert Jankovsky. Grc - overview of hall thrusters. https:// www.grc.nasa.gov/www/hall/overview/overview.htm, February 2007. [Online; Accessed on December 1st 2018]. [15] Claudio Bombardelli and Jesus Pelaez. Ion Beam Shepherd for Contactless Space Debris Removal. Journal of Guidance, Control, and Dynamics, 2011. ISSN 0731-5090. doi: 10.2514/1.51832. [16] Shoji Kitamura, Yukio Hayakawa, and Satomi Kawamoto. A reor- biter for large GEO debris objects using ion beam irradiation. Acta Astronautica, 2014. ISSN 00945765. doi: 10.1016/j.actaastro.2013. 07.037. [17] Yevgeny Raitses, Enrique Merino, and Nathaniel J. Fisch. Cylin- drical Hall thrusters with permanent magnets. Journal of Applied Physics, 108(9), 2010. ISSN 00218979. doi: 10.1063/1.3499694.

75 BIBLIOGRAPHY

[18] Kurt A. Polzin, Yevgeny Raitses, Jean Carlos Gayoso, and Nathaniel J. Fisch. Comparisons in Performance of Elec- tromagnet and Permanent-Magnet Cylindrical Hall-Effect Thrusters. 46th AIAA/ASME/SAE/ASEE Joint Propulsion Con- ference, pages 1–7, 2010. doi: 10.2514/6.2010-6695. URL https://ntrs.nasa.gov/archive/nasa/casi.ntrs. nasa.gov/20100035731.pdf. [19] Yongjie Ding, Hong Li, Hezhi Sun, Liqiu Wei, Boyang Jia, Hongbo Su, Wuji Peng, Peng Li, and Daren Yu. A 200-W permanent magnet Hall thruster discharge with graphite channel wall. Physics Letters A, 1:6–9, 2018. ISSN 03759601. doi: 10.1016/j. physleta.2018.08.017. URL https://linkinghub.elsevier. com/retrieve/pii/S0375960118308892. [20] Ludovico Lorello, Igor Levchenko, Kateryna Bazaka, Michael Kei- dar, Luxiang Xu, S. Huang, J. W.M. Lim, and Shuyan Xu. Hall Thrusters with Permanent Magnets: Current Solutions and Per- spectives. IEEE Transactions on Plasma Science, 46(2):239–251, 2018. ISSN 00933813. doi: 10.1109/TPS.2017.2772332. [21] A. I. Morozov and V. V Savelyev. Fundamentals of Stationary Plasma Thruster Theory. In B. B. Kadomtsev and V. D. Shafranov, editors, Reviews of Plasma Physics, pages 203–391. Springer US, Boston, MA, 2000. ISBN 978-1-4615-4309-1. doi: 10.1007/978-1- 4615-4309-1{\_}2. URL https://doi.org/10.1007/978-1- 4615-4309-1_2. [22] Alexander Piel. Plasma physics: an introduction to laboratory, space, and fusion plasmas. Springer, 2017. [23] Richard Fitzpatrick. Magnetized plasmas. https://farside. ph.utexas.edu/teaching/plasma/lectures1/node10. html, March 2011. [Online; Accessed on December 8th 2018]. [24] Manuel Martinez-Sanchez and Paulo Lozano. 16.55 Ionized Gases. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA, Fall 2014. URL https: //ocw.mit.edu. [Online; Accessed on December 8th 2018].

76 BIBLIOGRAPHY

[25] Winston Frias, Andrei I. Smolyakov, Igor D. Kaganovich, and Yevgeny Raitses. Long wavelength gradient drift instability in Hall plasma devices. I. Fluid theory. Physics of Plasmas, 19(7), 2012. doi: 10.1063/1.4736997. URL https://doi.org/10.1063/1. 4736997. [26] Winston Frias, Andrei I. Smolyakov, Igor D. Kaganovich, and Yevgeny Raitses. Long wavelength gradient drift instability in Hall plasma devices. II. Applications. Physics of Plasmas, 20(5):0–9, 2013. ISSN 1070664X. doi: 10.1063/1.4804281. [27] A. I. Smolyakov, O. Chapurin, W. Frias, O. Koshkarov, I. Ro- madanov, T. Tang, M. Umansky, Y. Raitses, I. D. Kaganovich, and V. P. Lakhin. Fluid theory and simulations of instabilities, tur- bulent transport and coherent structures in partially-magnetized plasmas of e × B discharges. Plasma Physics and Controlled Fusion, 59(1), 2017. ISSN 13616587. doi: 10.1088/0741-3335/59/1/014041. [28] Salomon Janhunen, Andrei Smolyakov, Oleksandr Chapurin, Dmytro Sydorenko, Igor Kaganovich, and Yevgeni Raitses. Non- linear structures and anomalous transport in partially magnetized e × B plasmas. Physics of Plasmas, 25(1), 2018. ISSN 10897674. doi: 10.1063/1.5001206. [29] Francis F. Chen. Introduction to Plasma Physics and Controlled Fusion. Springer International Publishing, 3 edition, 2016. ISBN 978-3-319- 22309-4. doi: 10.1007/978-3-319-22309-4. [30] Damiano Pagano, Giovanni Coduti, Simone Scaranzin, Gian- franco Meniconi, Fabrizio Scortecci, and Niccola Kutufa. Perfor- mance of Plume Characterization of the SPT100-B Thruster. Pre- sented at Joint Conference of 30th International Symposium on Space Technology and Science, 34th International Electric Propulsion Confer- ence and 6th Nano-satellite Symposium, pages 1–15, 2015. [31] Richard P Feynman, Robert B Leighton, and Matthew Sands. The Feynman lectures on physics, Vol. I: The new millennium edition: mainly mechanics, radiation, and heat, volume 1. Basic books, 2011. [32] Massoud Kaviany. Heat transfer physics. Cambridge University Press, 2014.

77 BIBLIOGRAPHY

[33] Dan M. Goebel and Ira Katz. Fundamentals of Electric Propulsion: Ion and Hall Thrusters. California Institute of Technology, 2008. ISBN 9780470429273. doi: 10.1002/9780470436448. [34] Iain D Boyd and Thomas E Schwartzentruber. Nonequilibrium Gas Dynamics and Molecular Simulation. Cambridge University Press, 2017. [35] Ehsan Roohi. Introduction to DSMC. Lecture 3: GAS Dynam- ics. http://e.roohi.profcms.um.ac.ir/imagesm/1019/ stories/Powerpoints/dsmc2.pdf, December 2011. [Online; Accessed on April 23rd 2019]. [36] Michael A. Lieberman and Allan J. Lichtenberg. Principles of Plasma Discharges and Materials Processing: Second Edition. John Wiley & Sons, Inc., 2 edition, 2005. ISBN 0471720011. doi: 10.1002/0471724254. [37] S. Barral, K. Makowski, Z. Peradzy´nski, N. Gascon, and M. Dudeck. Wall material effects in stationary plasma thrusters. II. Near-wall and in-wall conductivity. Physics of Plasmas, 10(10): 4137–4152, 2003. ISSN 1070664X. doi: 10.1063/1.1611881. [38] Yasunori Yamamura and Hiro Tawara. Energy dependence of ion- induced sputtering yields from monatomic solids at normal incidence. Atomic Data and Nuclear Data Tables, 1996. ISSN 0092640X. doi: 10.1006/adnd.1996.0005. [39] Y. Garnier, V. Viel, J. F. Roussel, and J. Bernard. Low-energy xenon ion sputtering of ceramics investigated for stationary plasma thrusters. Journal of Vacuum Science & Technology A: Vacuum, Sur- faces, and Films, 1999. doi: 10.1116/1.582050. [40] O. A. Mitrofanova, R. Yu. Gnizdor, V. M. Murashko, A. I. Koryakin, and A. N. Nesterenko. New Generation of SPT-100. 32nd International Electric Propulsion Conference, pages 1–7, 2011. URL http://erps.spacegrant.org/ uploads/images/images/iepc_articledownload_1988- 2007/2011index/IEPC-2011-041.pdf.

78 BIBLIOGRAPHY

[41] Edgar Y. Choueiri. A Critical History of Electric Propulsion: The First 50 Years (1906-1956). Journal of Propulsion and Power, 20(2): 1–21, 2004. doi: 10.2514/1.9245. URL https://doi.org/10. 2514/1.9245. [42] Robert G. Jahn and Edgar Y. Choueiri. Electric Propulsion. Encyclopedia of Physical Science and Technology, pages 125– 141, 1 2003. doi: 10.1016/B0-12-227410-5/00201-5. URL https://www.sciencedirect.com/science/article/ pii/B0122274105002015. [43] Our engine | ad astra rocket. http://www.adastrarocket. com/aarc/VASIMR, 2018. [Online; Accessed on December 2nd 2018]. [44] Eric Berger. Nasa’s longshot bet on a revolutionary rocket may be about to pay off | ars technica. https://arstechnica. com/science/2017/02/nasas-longshot-bet-on-a- revolutionary-rocket-may-be-about-to-pay-off/, February 2017. [Online; Accessed on November 30th 2018]. [45] L Garrigues and P Coche. Electric propulsion: comparisons between different concepts. Plasma Physics and Controlled Fusion, 53 (12):124011, 2011. [46] Leonard Cassady, Benjamin Longmier, Chris Olsen, Maxwell Bal- lenger, Greg McCaskill, Andrew Illin, Mark Carter, Tim Glover, Jared Squire, Franklin Chang Diaz, et al. Vasimr performance results. In 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, page 6772, 2010. [47] Busek electrospray thrusters. http://www.busek.com/ technologies__espray.htm, -. [Online; Accessed on March 15th 2019]. [48] M Merino. Modeling the physics of plasma thruster plumes in electric propulsion. Technical report, ESAC, 2017. URL https://www.cosmos.esa.int/documents/ 13611/1287426/170126_Merino.pdf.

79 BIBLIOGRAPHY

[49] David Pidgeon, Ron Corey, Birgit Sauer, and Michael Day. Two Years of On-Orbit Performance of SPT-100 Electric Propulsion. In 24th AIAA International Communications Satellite Systems Conference, pages 1–11, 2006. ISBN 978-1-62410-027-7. doi: 10.2514/6.2006- 5353. URL http://arc.aiaa.org/doi/10.2514/6.2006- 5353. [50] R. Lynn, M. Osborn, John M. Sankovic, and L. Caveny. Electric Propulsion Demonstration Module (EPDM) Flight Hall Thruster System. In Proceedings of the International Electric Propulsion Con- ference 1997 (IEPC97), 1997. URL http://erps.spacegrant. org/uploads/images/images/iepc_articledownload_ 1988-2007/1997index/7100.pdf. [51] John Sankovic, Thomas Haag, and David Manzella. Operating characteristics of the russian d-55 thruster with anode layer. In Presented at the 30th Joint Propulsion Conference, Indianapolis, IN, 27-30 Jun. 1994; sponsored by AIAA, ASME, SAE, and ASEE, Joint Propulsion Conferences. American Institute of Aeronautics and Astronautics, Jun 1994. doi: 10.2514/6.1994-3011. URL https: //doi.org/10.2514/6.1994-3011. 0. [52] STEX - eoPortal Directory - Satellite Missions, Feb 2019. URL https://directory.eoportal.org/web/eoportal/ satellite-missions/s/stex. [Online; Accessed on 8th Feb. 2019]. [53] A. I. Bugrova, A. M. Bishaev, A. V. Desyatskov, M. V. Kozint- seva, A. S. Lipatov, and M. Dudeck. Experimental investigations of a krypton stationary plasma thruster. International Jour- nal of Aerospace Engineering, 2013:1–8, 2013. ISSN 16875966. doi: 10.1155/2013/686132. [54] Jinwen Liu, Hong Li, Xu Zhang, Yongjie Ding, Liqiu Wei, Jianzhi Li, Daren Yu, and Xiaogang Wang. On matching the anode ring with the magnetic ﬁeld in an ATON-type Hall effect thruster. Jour- nal of Applied Physics, 123(22):223301, 2018. ISSN 0021-8979. doi: 10.1063/1.5026486.

80 BIBLIOGRAPHY

[55] A. I. Bugrova, A. S. Lipatov, A. I. Morozov, and L. V. Solomatina. Global characteristics of an ATON stationary plasma thruster operating with krypton and xenon. Plasma Physics Reports, 28(12): 1032–1037, 2002. ISSN 1063-780X. doi: 10.1134/1.1528234. [56] Scott J. Hall, Roland E. Florenz, Alec Gallimore, Hani Kamhawi, Daniel L. Brown, James E. Polk, Dan M. Goebel, and Richard R. Hofer. Implementation and Initial Validation of a 100-kW Class Nested-channel Hall Thruster. 50th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, pages 1–15, 2014. doi: 10.2514/6.2014-3815. URL http://arc.aiaa.org/doi/10.2514/6.2014-3815. [57] Saul Dushman. Electron Emission from Metals as a Function of Temperature. Physical Review, 1923. ISSN 0031899X. doi: 10.1103/ PhysRev.21.623. [58] Ioannis G. Mikellides, Ira Katz, Dan M. Goebel, James E. Polk, and Kristina K. Jameson. Plasma processes inside dispenser hollow cathodes. Physics of Plasmas, 2006. ISSN 1070664X. doi: 10.1063/1. 2208292. [59] B Archipov, L Krochak, and N Maslennikov. Numerical and experimental researches of thermal processes in cathode-neutralizers for electrical plasma thrusters. In 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Joint Propulsion Confer- ences. American Institute of Aeronautics and Astronautics, 7 1998. doi: doi:10.2514/6.1998-3479. URL https://doi.org/ 10.2514/6.1998-3479. [60] Vjacheslav M. Murashko, Aleksandr I. Koryakin, Aleksandr N. Nesterenko, Sergey V. Olotin, Anatoliy I. Oranskiy, Andriy V. Loyan, Mikola M. Koshelev, Volodimir I. Bilokon, and Sergiy Yu. Nesterenko. Russian Flight Hall Thrusters SPT-70 & SPT-100 After Cathode Change Start During 20-25 ms. In International Electric Propulsion Conference 2007, pages 1–4, Florence, 2007. URL http://erps.spacegrant.org/ uploads/images/images/iepc_articledownload_1988- 2007/2007index/IEPC-2007-062.pdf. [61] Boris A. Arkhipov, Yuriy M. Gorbachev, Viktor A. Ivanov, Kon- stantin N. Kozubsky, and Georgy A. Komarov. PLASMA COM- PENSATION CATHODE, 1994.

81 BIBLIOGRAPHY

[62] B. A. Arkhipov and K. N. Kozubsky. The Development of the Cathodes-Compensators for Stationary Plasma Thrusters (SPT) in USSR. 22nd International Electric Propulsion Conference, pages 91–023, 1991. [63] B. A. Arhipov, S. S. Kudriavtsev, N. A. Maslennikov, Vjacheslav M Murashko, and I. A. Turgeneva. The development investigation of the cathode - compensator of stationary plasma thrusters for discharge currents up to 50 A. 24th International Electric Propulsion Conference, 100:1–6, 1995. URL http://erps.spacegrant.org/ uploads/images/images/iepc_articledownload_1988- 2007/1995index/IEPC1995-230.pdf. [64] Michael R. Nakles, William A. Hargus, Jorge J. Delgado, and Ronald L. Corey. A Performance Comparison of Xenon and Kryp- ton Propellant on an SPT-100 Hall Thruster. 32nd International Elec- tric Propulison Conference, pages 1–12, 2011. doi: 10.2514/6.2012- 4116. [65] Michael R. Nakles, William A. Hargus, Jorge J. Delgado, and Ronald L. Corey. A 205 Hour Krypton Propellant Life Test of the SPT-100 Operating at 2 kW. 33rd International Electric Prop- ulison Conference, pages 1–21, 2013. URL www.iepc2013.org/ get?id=347. [66] C. E. Garner, J. R. Brophy, and J. E. Polk. Cyclic Endurance Test of a SPT-100 Stationary Plasma Thruster. 30th Joint Propulsion Conference, Indianapolis, IN, 1994. doi: 10.2514/6.1994-2856. [67] Jean Pierre Boeuf. Tutorial: Physics and modeling of Hall thrusters. Journal of Applied Physics, 2017. ISSN 10897550. doi: 10.1063/1.4972269. [68] EDB Fakel. Stationary plasma thrusters. https://fakel- russia.com/images/gallery/produczia/fakel_spd_ en_print.pdf, November 2018. [Online; Accessed on March 18th 2019]. [69] Yevgeny Raitses and Igor D. Kaganovich. Kinetic Effects of Electron-Induced Secondary Electron Emission on Plasma-Wall Interactions Effects of Electron-Induced Secondary Electron, 2013.

82 BIBLIOGRAPHY

[70] R. Gnizdor, A. Komarov, O. Mitrofanova, P. Saevets, and D. Seme- nenko. High-impulse SPT-100D thruster with discharge power of 1.0. . . 3.0 kW. 35th International Electric Propulsion Conference, pages 2017–40, 2017. URL https://iepc2017.org/sites/ default/files/speaker-papers/iepc-2017-40_0.pdf. [71] Vladimir Kim, Garry Popov, Vyacheslav Kozlov, Alexander Skryl- nikov, and Dmitry Grdlichko. Investigation of SPT Performance and Particularities of its Operation with Kr and Kr/Xe Mixtures *+. Society, pages 15–19, 2001. [72] Ioannis G. Mikellides, Gary A. Jongeward, Ira Katz, and David H. Manzella. Plume modeling of stationary plasma thrusters and interactions with the express-a spacecraft. Journal of Spacecraft and Rockets, 39(6):894–903, 2002. doi: 10.2514/2.3896. [73] Martin Costabel. Principles of boundary element methods. Techn. Hochsch., Fachbereich Mathematik, 1986. [74] G. A. Bird. Monte Carlo Simulation of Gas Flows. Annual Review of Fluid Mechanics, 10(1):11–31, 1978. doi: 10.1146/annurev.fl.10. 010178.000303. [75] Leo Basov, Dejan Petkow, and Sander Rowette. Modeling & Veri- fication Report. Technical report, SPARC Industries sarl, Esch-sur- Alzette, 2018. [76] Tomonor Takizuka and Hirotada Abe. A binary collision model for plasma simulation with a particle code. Journal of computational physics, 25(3):205–219, 1977. [77] K Nanbu. Theory of cumulative small-angle collisions in plasmas. Physical Review E, 55(4):4642, 1997. [78] Andris M Dimits, Chiaming Wang, Russel Caflisch, Bruce I Cohen, and Yanghong Huang. Understanding the accuracy of nanbu’s numerical coulomb collision operator. Journal of Computational Physics, 228(13):4881–4892, 2009. [79] Lyon B. King and Alec D. Gallimore. Mass Spectral Measurements in the Plume of an SPT-100 Hall Thruster. Journal of Propulsion and Power, 2000. ISSN 0748-4658. doi: 10.2514/2.5681.

83 BIBLIOGRAPHY

[80] V. V. Zhurin, H. R. Kaufman, and R. S. Robinson. Physics of closed drift thrusters. Plasma Sources Science and Technology, 8(1), 1999. ISSN 09630252. doi: 10.1088/0963-0252/8/1/021. [81] AI Morozov and VV Savelyev. Fundamentals of stationary plasma thruster theory. In Reviews of plasma physics, pages 203–391. Springer, 2000. [82] Sang-Wook Kim, John E. Foster, and Alec D. Gallimore. Very-near- field plume study of a 1.35 kW SPT-100. In ASME, SAE, and ASEE, Joint Propulsion Conference and Exhibit, 32nd, Lake Buena Vista, FL, July 1-3, 1996, 1996. ISBN 9780000000002. doi: 10.2514/6.1996- 2972. [83] Max Planck. The theory of heat radiation, trans. M. Masius, P. Blakiston’s Son & Co, Philadelphia, 1914. [84] John M. Sankovic, John A. Hamley, and Thomas W. Haag. Per- formance Evaluation of the Russian SFT Thruster at NASA LeRC. 23rd International Electric Propulsion Conference, pages 1–25, 1993. URL https://ntrs.nasa.gov/archive/nasa/casi. ntrs.nasa.gov/19940019157.pdf. [85] George Karniadakis, Ali Beskok, and Narayan Aluru. Microflows and nanoflows: fundamentals and simulation, volume 29. Springer Science & Business Media, 2006. [86] Theodore Gray. Technical data for the element xenon in the peri- odic table. https://periodictable.com/Elements/054/ data.wt.html, 2014. [Online; Accessed on May 25th 2019]. [87] Gabriel Giono, Stéphane Mazouffre, Dimitry Loubere, Lara Pope- lier, Chrisophe Theoroude, Kathe Dannenmayer, Fabien Marguet, Jon Tomas Gudmundsson, Nickolay Ivchenko, Georgi Olentsenko, and Mario Merino. Experimental Determination of the Plasma Properties in the Far-plume of an SPT-100 Hall Thruster. In 35th International Electric Propulsion Conference, 2017. [88] Rodrigo A Miranda, Alexandre A Martins, and José L Ferreira. Particle-in-cell numerical simulations of a cylindrical hall thruster with permanent magnets. In Journal of Physics: Conference Series, volume 911, page 012021. IOP Publishing, 2017.

84 BIBLIOGRAPHY

[89] James Joseph Szabo. Fully kinetic numerical modeling of a plasma thruster. PhD thesis, Massachusetts Institute of Technology, 2001. [90] Marc A Rieffel. A method for estimating the computational requirements of dsmc simulations. Journal of Computational Physics, 149(1):95–113, 1999. [91] I. A. Surmin, S. I. Bastrakov, E. S. Eﬁmenko, A. A. Gonoskov, A. V. Korzhimanov, and I. B. Meyerov. Particle-in-cell laser-plasma simulation on xeon phi coprocessors. Computer Physics Communica- tions, 202:204–210, 2016. [92] L Grimaud, A Pétin, J Vaudolon, and S Mazouffre. Perturbations induced by electrostatic probe in the discharge of hall thrusters. Review of Scientiﬁc Instruments, 87(4):043506, 2016. [93] J. C. Adam, J. P. Boeuf, N. Dubuit, M. Dudeck, L. Garrigues, D. Gresillon, A. Heron, G. J.M. Hagelaar, V. Kulaev, N. Lemoine, S. Mazouffre, J. Perez Luna, V. Pisarev, and S. Tsikata. Physics, simulation and diagnostics of Hall effect thrusters. Plasma Physics and Controlled Fusion, 50(12), 2008. ISSN 07413335. doi: 10.1088/0741- 3335/50/12/124041. [94] Francesco Taccogna, Savino Longo, Mario Capitelli, and Ralf Schneider. Self-similarity in hall plasma discharges: Applications to particle models. Physics of Plasmas, 12(5):053502, 2005. [95] Shahab Fatemi, Andrew R Poppe, Gregory T Delory, and William M Farrell. Amitis: A 3d gpu-based hybrid-pic model for space and plasma physics. In Journal of Physics: Conference Series, volume 837, page 012017. IOP Publishing, 2017. [96] George Stantchev, William Dorland, and Nail Gumerov. Fast parallel particle-to-grid interpolation for plasma pic simulations on the gpu. Journal of Parallel and Distributed Computing, 68(10):1339– 1349, 2008. [97] Alexander Fox. The difference between a cpu and a gpu - make tech easier. https://www.maketecheasier.com/ difference-between-cpu-and-gpu/, Jan 2017. [Online; Ac- cessed on March 15th 2019].

85 BIBLIOGRAPHY

[98] Sergey Bastrakov, R Donchenko, Arkady Gonoskov, Evgeny Eﬁ- menko, A Malyshev, Iosif Meyerov, and Igor Surmin. Particle-in- cell plasma simulation on heterogeneous cluster systems. Journal of Computational Science, 3(6):474–479, 2012.

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