Modelling of plasma thruster plumes for spacecraft plume ...

Modelling of plasma thruster plumes for spacecraftplume-impingement analysis

Nuno Joao Machado Loureiro

Dissertacao para a obtencao de Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Doutor Joao SeixasOrientador: Doutor Jorge LoureiroCo-Orientador: Doutor Nickolay IvchenkoVogal: Doutor David Resendes

October 2010

Acknowledgements

First of all I would like to express my gratitude to all the Space Physics group at EADS Astrium Toulousefor their support and orientation during these six months I spent in France. In particular I thank MatiasWartelski, Cristophe Theroude, Guilhem Chanteperdrix and Guillaume Scremin for all the help they gaveme and all the interesting discussions we had. You taught me a lot about space engineering.

A special thanks goes to Dr. Nickolay Ivchenko from KTH, without whom this thesis would not havebeen possible. I am glad he encouraged me to apply to the Astrium internship and I thank the support andfruitful comments/suggestions he made to all preliminary version of this document.

I am grateful to Prof. Jorge Loureiro from IST and Prof. Lars Blomberg from KTH for having agreedto supervise my work and for making important suggestions on this thesis.

I would also like to thank my friends that accompanied me during the last five years of my life. I’ve had somany great moments with you and I hope we still have many more to come.

Thank you Zita for all the excellent time we spent together and for all the motivation you gave me dur-ing these years. You really helped me a lot!

Above all I want to thank my family for the total and unconditional support they have always given meand to whom I cannot ever thank enough. Thank you for always being there for me!

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Resumo

As partıculas ejectadas por um propulsor electrico a bordo de um veıculo espacial formam um plasma emtorno do veıculo espacial que pode interagir com as suas superfıcies e componentes. As interaccoes podemdanificar estes componentes e superfıcies e por isso tem uma grande importancia nas fases de planeamento edesenho do veıculo espacial, onde a sua modelizacao deve ser realizada o mais rigorasamente possıvel. Um dosprincipais pontos que deve ser considerado nessa modelizacao e a expansao do plasma a partir do propulsor.Conhecendo o desenvolvimento do plasma e possıvel fazer o estudo da sua interaccao com a geometria doveıculo espacial e do efeito que esta pode ter na deposicao de carga nas suas superfıcies.

O trabalho realizado nesta tese consistiu na implementacao de modelos de plasmas de propulsores electricosno programa SPIS, um programa livre que permite fazer simulacoes 3D de interaccoes de plasma com umveıculo espacial. Dois tipos de propulsores foram modelizados: propulsor efeito de Hall e o propulsor ionico.Foram lancadas varias simulacoes com ambos os tipos de propulsores e os resultados foram analisados demodo a avaliar os modelos implementados. Foram ainda realizadas outras evolucoes ao programa de modoa melhorar as partes de processamento e pos-processamento das simulacoes, permitindo uma melhor com-paracao entre os resultados obtidos e as medicoes feitas experimentalmente.

Para testar o modelo do propulsor de efeito de Hall foi utilisado um propulsor do tipo SPT100. Os resul-tados para estas simulacoes foram bastante proximos dos resultados experimentais em termos de densidadede corrente, densidade de ioes e perfil de energia do plasma. O modelo do propulsor ionico foi testado como propulsor T5 e para este caso os resultados obtidos encontram-se ainda longe dos medidos experimental-mente. Neste sentido varias sugestoes foram feitas com vista a melhorar as simulacoes e os seus resultados,tendo sido deixadas como ideias para trabalho futuro.

Palavras-chave: Interaccao de plasma com veıculos espaciais, propulsao electrica espacial, softwareSPIS, modelizacao de propulsores electricos

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Abstract

When an electric thruster’s plasma plume impinges on spacecraft surfaces and instruments several interactionissues occur that may damage the satellite’s components and have critical impact on spacecraft design andperformances. Modelling, as precisely as possible, the interactions between the thruster and the satellitesurfaces is then of extreme importance. One of the key points of the analysis is the modelling of the plasmaexpansion from the thruster and its interaction with the geometry of the spacecraft in which some surfacescan be electrically charged.

The work in this thesis consisted of the implementation of thruster plume models and evolutions in theSPIS software, which is an open-source software that allows performing 3D simulations of spacecraft-plasmainteractions. The Hall effect thruster (HET) and the ion thruster were modelled. Simulations with both typeswere launched and the results were discussed in order to evaluate the implemented models. Also evolutionsto the simulation method and post-processing tools were developed in order to obtain plume characteristicsfrom the simulation that could be comparable with experimental data.

The HET model was tested using one of the most common thrusters of this type, the SPT100. Thesimulation results showed a good agreement with the experimental data in terms of the ion current density,ion particle density and energy profiles of the plasma plume. The ion thruster was tested with the T5 thrustermodel. The results for the T5 thruster simulations were not as close to the experimental data as expectedand so suggestions were made to improve these results as future work.

Keywords: Spacecraft-plasma interactions, electric space propulsion, SPIS software, electric thrustersmodelling

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Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiResumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Contents vii

List of Tables ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Figures xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 Introduction 11.1 Thesis background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Work and thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Overview of the Electric Propulsion Systems 52.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Chemical propulsion and its limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Types of Electric Propulsion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Spacecraft-Plume Interaction Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 The AISEPS Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Project description and SPIS software 153.1 SPIS software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Plasma Simulation 194.1 3D Particle-in-Cell Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 The PIC-MCC approach with the SPIS software . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 CEX collision in the SPIS software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Main Evolutions to the thruster models in the SPIS software 255.1 Multiple Particle Sources and Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Default Source Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Hall Thruster Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Ion Thruster Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.5 Plasma Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.6 The Poisson Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.7 The assumption of a quasi-neutral plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.8 Quality of simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Results with new thruster models 356.1 The SPT100 model simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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6.2 The T5 model simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7 Conclusions and future work 47

Bibliography 49

viii

List of Tables

4.1 Cross section parameters for CEX collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.1 Input Parameters for the SPT-100 Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 SPT100 simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 Input Parameters for the T5 Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.4 T5 simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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List of Figures

2.1 Thrust as a function of specific impulse, Isp for the typical spacecraft propulsion technologies.Adapted from [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Ion thruster model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Hall effect thruster model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Diagram of the AISEPS Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Simulation volume with mesh on the Gmsh software (left) and a zoom-in on the thrusters region(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Object structure of the SPIS software. Adapted from [10] . . . . . . . . . . . . . . . . . . . . . . 17

4.1 Typical cycle of a PICcode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Weighting of density to the grid points by an interpolation function fd. Each grid point i acquires

a fraction of the density of the superparticle ni = fd × n. . . . . . . . . . . . . . . . . . . . . . . 224.3 Typical cycle of a PIC(-MCC) code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4 Representation of a CEX collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.1 Diagram depicting the implemented model for a HET. Different angles are generated for differentpositions in the thrusters exit. Adapted from [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Diagram depicting the implemented model for a Ion Thruster. Different angles are generated fordifferent positions in the thrusters exit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.1 Number of ejected Xe+ and Xe2+ superparticles per cell . . . . . . . . . . . . . . . . . . . . . . . 376.2 Number of CEX Xe+ and Xe2+ superparticles per cell . . . . . . . . . . . . . . . . . . . . . . . . 376.3 Plasma potential involving the SPT100 thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4 Simulation results and experimental data of the dependence of the current density on the angle

from the thruster axis, at a distance of 1m from the SPT100 thruster . . . . . . . . . . . . . . . 396.5 Simulation results and experimental data of the dependence of the current density on the angle

from thruster axis, at a distance of 60cm from the SPT100 thruster . . . . . . . . . . . . . . . . 396.6 Simulation results and experimental data of the dependence of the energy per charge of Xe+ and

Xe2+ ejected from the SPT100 on the angle from the thruster axis at 50cm and 1m from thethruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.7 Simulation results and experimental data of the dependence of total particle density on the anglefrom the thruster axis at 50cm and 1m from the SPT100 thruster . . . . . . . . . . . . . . . . . . 41

6.8 Plasma potential in the region close to the thruster’s exit . . . . . . . . . . . . . . . . . . . . . . 416.9 Dependence of the current density on the angle from the thruster axis at 60cm from the thruster

for two different values of background pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.10 Ratio of Debye length over cell size in the simulation volume . . . . . . . . . . . . . . . . . . . . 436.11 Dependence of current density on the angle from the thruster axis at 100cm from the thruster.

Comparison of simulations using the quasi-neutral assumption and the Poisson solver . . . . . . 436.12 Simulation results of the ion density profile of the T5 thruster . . . . . . . . . . . . . . . . . . . . 456.13 Simulation results (left) and experimental data (right) of the ion density profile as a function of

horizontal distance to the thruster’s axis and axial distance to the T5 thruster . . . . . . . . . . 45

xi

Chapter 1

Introduction

1.1 Thesis background

This thesis was developed in the ambit of the Erasmus Program where the host university KTH (RoyalInstitute of Technolgy) in Stockholm, Sweden proposed a six-month internship at EADS Astrium SatellitesSAS located in Toulouse, France. Astrium is a filial of the EADS (European Aeronautic Defense and Space)group and is a well-established company in the space industry, placed in five European countries - France,Spain, Netherlands, England and Germany. It is specialized in civil and military space systems and is dividedin three main sections: Astrium Space Transportation, Astrium Satellites and Astrium Services.

This internship took place at EADS Astrium Satellites on the MOS - ASG42 (Modelisation, Outils etSimulation) department. The department is divided in five teams: Framework, Mission, Image treatment,Image quality and Space physics. For 15 years this department has been in charge of the development ofseveral simulation tools funded both internally and by space agencies (ESA, CNES, DGA,...). Many softwaretools for space applications have been developed and commercialized for more than 10 years (Thermica,DOSRAD,...). All these tools have been put together into one software named SYSTEMA that is distributedall over the world and to many of the biggest aerospace companies (Boeing, Alcatel, Northrop Grummar,Melco) and agencies (ESA, NASA).

In particular, the work for this thesis was carried out with the Space physics team who is in charge, amongother subjects, of the electric propulsion analysis. These analysis imply the study of different phenomenasuch as the spacecraft space environment, thermal behavior of spacecraft structures and spacecraft surfaceerosion and contamination due to particle collision. This team has also developed several numerical tools inorder to model several aspects of electric propulsion such as Pionic and Cionic.

With the objective of performing more detailed analysis of spacecraft charging due to electric propulsionsystems, the Space physics team decided to implement several developments to an already existing software(see Chapter 3) in order to introduce models of plasma thruster plumes.

1.2 Motivation

Rocket propulsion has been studied for more than a century, having a history full of great discoveries andachievements as well as setbacks and fiascos. Arguably, it is said that Rocket Science first begun in 1903, whenthe Russian physicist and astronautic theorist Konstantin Tsiolkovsky first derived the Rocket Equation fromthe well known Newton’s laws of dynamics. This derivation considers that the acceleration of a spacecraftin free space is obtained only by the discharge of propellant mass and follows from the conservation of thespacecraft’s total momentum and its exhaust stream. Mathematically this concept looks as easy as it seems:

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FTotal = 0⇔ d(mv)

dt= mv + v m = 0

(1.1)

⇒ m v = ve m

where m is the mass of the spacecraft, v its acceleration vector, m is the rate at which the mass of thespacecraft is changing due to the exhausted propellant and ve the velocity vector of the exhausted jet inrelation to the spacecraft (and thus the positive sign in the second expression). By considering the exhaustvelocity constant over a period of thrust, an integration of the last expression can be performed and a relationof the speed increment, ∆v, and the amount of exhausted propellant can be found:

∆v = ve lnm0

mf(1.2)

with m0 the initial total mass of the spacecraft and mf the mass after the considered time of acceleration.This is similar as writing

mf

m0= e−∆v/ve (1.3)

where it can be understood that the higher the exhaust velocity the more mass a spacecraft can deliever.Although the simplest case was taken for the derivation of the Rocket Equation, the inclusion of externalforces would still maintain the same relation given by 1.3 changing only the variable ∆v which would theninclude all the aspects of a particular mission (drag, gravitational forces, ...).

An important quantity that needs to be defined when considering space missions is the specific impulse,Isp of a thruster. This quantity describes the efficiency of a rocket by representing the change in momentumgained by a spacecraft per unit of propellant used, which can be seen by the following expression:

Isp =T

m g0=veg0

(1.4)

where g0 is the acceleration of gravity at Earth sea level (9.8 ms−2) and T = ve m is the thrust produced bythe engine. Considering the above expression it can be realized that specific impulsed is measured in seconds.

As will be explained in more detail in Chapter 2, one of the limits of chemical propulsion (CP) is its lowexhaust velocity compared to electric propulsion (EP). For a typical chemical thruster these values are in arange of several hundreds to few thousands meters per second compared to EP where values one order ofmagnitude greater than these are normally achieved. Using (1.4) it can be seen that a low exhaust speed willcorrespond to a low specific impulse and so to a low efficiency of the thruster, since the higher the specificimpulse, the less propellant is needed to gain a given amount of momentum.

In order to escape from Earth’s gravitational fields large values of thrust are needed and so, for thesecases, CP is clearly the best and maybe the only solution at the moment. However, for long missions such asinterplanetary missions for example, very high efficiencies of the thrusters are demanded in order to assurethat the objective for a spacecraft is achieved consuming the minimum possible amount of propellant. Thehigh thrust values of the CP are also reflected in a high value of mass flow, suggesting this may not bethe best option for such types of missions and opening the way for electric propulsion. Another type ofmissions where EP can be extremely useful are the ones where a very high precision is required. In thesetypes of missions very short and precise pulses are needed in order to control the attitude of a satellite.The Laser Interferometer Space Antenna (LISA) satellite[1], to be launched in 2025, is a good example ofsuch a mission, where very accurate electric thrusters are required in order to maintain the relative positionof three satellites. These satellites should guarantee that their relative position is not affected by externalforces, like the solar radiation pressure, by counteracting them with the generated thrust. Since the valuesof these types of forces can be very small the thrust to counteract them will also have to be small and veryprecise, something that can not easily be done with chemical propulsion. For these cases and many othersimilar cases, EP has already been applied and their good results can also serve as a motivation to increase

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the research and further application of this technology.

EP has revealed itself as a very good solution to some of the problems imposed by the chemical propulsion.However, when analyzing the EP technology one should also have in mind some of its weaknesses such as theones pointed out below:

• The values of thrust density (thrust per unit exhaust area) that can be reached at the moment with EPare much lower than the ones provided with CP, requiring much longer periods of thrust to produce adesired change in velocity or trajectory.

• Since the source of electrical energy is independent of the propellant it must be provided by other typesof sources on the spacecraft, like solar panels or nuclear sources. These sources are very suitable forlong missions, providing an almost unlimited amount of energy. However they require some types ofsystems that can increase the complexity of the spacecraft compared to the use of CP.

• Since the main types of particles ejected by an electric thruster are charged, charging problems onthe spacecraft’s surfaces can appear if some of the particles bombard the spacecraft’s components andare adsorbed. This issue is very serious from the perspective of the threat it poses for spacecraft andeven entire missions since it can generate various complications from single upset events and damage ofelectronics onboard to major failures originated by arcing discharges due to differential charged partsof the spacecraft. Charging problems in EP are added to the ones already known and studied for CP.Some of the problems are induced forces and torques on the spacecraft due to interaction of the plumewith its components as well as the erosion of some parts of the spacecraft. In the case of EP the erosioncan be even greater than in the case of CP since, as stated above, the ions leaving the thruster have amuch greater speed than the atoms leaving chemical thrusters.

Despite the fact that EP has already more than half a century of existence and hundreds of spacecraft haveflown electric thrusters [2], there is still a lack of knowledge on the charging subject due to the ejectedparticles. This is mainly due to the fact that a complete understanding of the ejected plasma plume is stillmissing, because measurements in this types of environments are extremely difficult to perform. A solutionto overcome this issue is the plasma plume simulation and the plasma interaction with the spacecraft’ssurfaces, allowing the study of the interaction on different positions of the spacecraft. This can be of extremeimportance when trying to find the best configuration for different components of the spacecraft, for examplethe position of the thrusters in relation to the solar panels.

1.3 Work and thesis structure

This thesis is devoted to the study of the plasma plume near the exit of an electric propulsion thruster andcontributes to the development of a robust software tool to better estimate the behaviour and the influence ofthe plasma on the spacecraft and on its surrounding environment. When completed, the software is intendedto be used both as an engineering design tool for contamination analysis and also as a research tool.This thesis was performed in three main steps, where each step served as the basis for the following:

1. Takeover of the SPIS software - In this first step the structure of the software was studied, by un-derstanding the tools which were included in it to simulate a thruster’s plume and identifying thelimitations. The methods of plasma simulation for electric propulsion systems were also studied withinthe duration of this first phase. This first stage took about the first two months of the project.

2. Development of new models in the software - The implementation of new tools in the SPIS software(which allowed for a more realistic simulation of the thruster’s plume) was performed after researchon the models used to simulate different thrusters. With a duration of three months this phase alsoencompasses the evolutions performed in cooperation with the SPIS developing team.

3. Simulations and data comparison - This last phase was performed during the last one and a half monthat Astrium and allowed the validation of the new implementations by comparing the simulation results

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with experimental data and other simulations. The simulations performed were tuned to give the bestresults in the less amount of simulation time possible, as expected from an engineering tool.

Divided in seven Chapters, this document is structured as follows: in Chapter 2 the concept of EP isintroduced by briefly explaining its history and main events. A description of the electric propulsion (EP)systems is presented as well as a comparison with CP systems. It also presents a description of the projectwhere this thesis is included. This description gives a general idea of the importance of the project and thedevelopment of a software capable of describing a plasma thruster. During this project several developmentsto the SPIS software were performed. Several developments were done individually at Astrium and otherswere discussed with the SPIS developing team and performed by them. Comments on how the evolutionswere chosen and who performed them are also addressed in this Chapter.Chapter 3 describes the SPIS project and software, explaining its main purposes and giving a general ideaon how it works. The pre-processing, the simulation and the post-processing parts are described.Chapter 4 includes theoretical and numeric considerations on plasma simulation, describing how a full 3D PICsimulation is performed. The Monte-Carlo collisions (MCC) method is described and the implementation ofthis method in SPIS is explained.In Chapter 5 the evolutions to the software performed during the project are presented. Physical details aregiven when describing the implemented models to simulate the thrusters and a description on how it wasimplemented. Evolutions performed in cooperation with the SPIS developing team are also described andjustifications for each evolution made to the software are presented.Chapter 6 presents the main results of the evolutions in a way to validate the new code. These results arecompared to the correspondent experimental data. The main differences and similarities are presented andcommented. In this Chapter a comment on the limitations of the SPIS software is also made as well as onhow this can affect the results.Chapter 7 concludes this thesis by summarizing the work and leaving some ideas for a continuation of theproject.

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Chapter 2

Overview of the Electric Propulsion Systems

2.1 Historical background

Electric Space Propulsion was first described scientifically by Robert Goddard in 1906. By conductingexperiments with discharge tubes, Goddard noted that the tube walls remained relatively cool when chargedparticles were accelerated to great velocities by electric fields within the tube. He realized that no knownmaterial could be capable of tolerating the temperatures needed to accelerate gas particles at similar speedsthrough conventional propulsion methods, i.e., chemical propulsion. With the help of his students, Goddardconducted several experiments where gases of Mercury and Cesium were exposed to hot tungsten surfacesand achieved very high temperatures. These experiments are now considered the first electric propulsionexperiments with ion sources.

Several years passed without further significant evolutions in the field, until 1929 when Hermann Oberthenvisioned the utilization of the technology to achieve very high exhaust velocity propulsion in space andpredicted it could carry a 150-ton payload into space. He also predicted that thrust would likely be smallbut steady and allowing the operation of a thruster for a long period of time.

Around 1947, Wernher von Braun encouraged Ernst Stuhlinger to review Oberth’s work on electricpropulsion with a famous saying[3]:

Professor Oberth has been right with so many of his early proposals; I wouldn’t be a bitsurprised if one day we flew to Mars electrically!

Following von Braun’s suggestion, Stuhlinger presented the first comprehensive study of the major compo-nents associated with an electrically propelled spacecraft in 1954. In this study he showed for the first timea relation between the desired speed increment for a spacecraft, the optimum exhaust speed and the specificmass of the power unit. This last parameter, usually called alpha, represents the power supply mass dividedby the source output power. It is a fundamental parameter to identify the best operating characteristics ofan EP system. Stuhlinger also noted that a propellant with large mass-to-charge ratios (e.g., Mercury andXenon) are desirable to minimize the size of the ion engine for a constant thrust level.

Many other scientists started to study the subject of EP and many contributions were made during theyears that followed Stuhlinger’s first considerations. These studies and research finally led to the acceptanceof EP as a viable propulsion technology in 1957 (the year Sputnik was launched). From this point theresearch on EP stopped being a matter of whether it was a worthwhile technology or not and started beingdirected to solving technical challenges that were impeding EP’s implementation. During the next coupleof years almost every large rocket and aircraft company created an active EP program, including mostof the companies still operating today, such as Lockheed (now Lockheed-Martin), Rockwell (now part ofBoeing) and Aerojet-General (now Aerojet). The National Advisory Committee for Aeronautics, NACA (thepredecessor of NASA) also established EP programs in three different laboratories, promoting the researchand development of this technology in the USA.

5

With all these programs established, both in the USA and other places in the world (mainly in the USSR),significant developments to EP started to occur. Between 1970 and 1990 the United States focused mainlyon the research of the Mercury (and later Xenon) Ion thrusters, hydrazine resistojets and arcjets. The USSRon the other hand, put great efforts into a program where the development of the hall thruster technologywas the main objective.

These new technologies were flown for the first time with the PanAmSat 5 probe (launched by the UScompany Hughes (now Boeing) and became the first ion thruster in space) and Meteor-1 probe (the SPT-60was launched by the USSR in 1972 and became the first hall thruster in space [2]).

Further tests were performed with both types of technologies and in 1998 NASA launched the first deepspace mission DS1 with an ion thruster on-board. ESA adopted the Russian technology of Hall thruster andin 2003, in cooperation with the Swedish Space Corporation (SSC), launched Smart1 carrying a PPS1350.For future missions, space agencies are planning to combine different types of technologies into the samespacecraft in order to perform different operations. A combination of Hall effect thrusters supplementedwith hydrazine arcjets (and maybe ion thrusters) is being considered as an efficient solution and the bestpropulsion choice for stationkeeping and orbit raising of large commercial space vehicles.

2.2 Chemical propulsion and its limitations

Chemical propulsion can be summarized as a reaction force between two different propellants initially storedin separated containers that react in a combustion chamber producing a hot gas to be expelled through anozzle. This expulsion generates a reaction on the spacecraft which equals the force created by the gas,allowing the spacecraft to move. This technology allows very high values of thrust and power but is limitedto considerably low values of exhaust speed when compared to EP.Since the gas exhaust is created by thermodynamic expansion its velocity is bounded by a relation with thetemperature of the nozzle, the average molecular mass of the exhaust gas and its inner and outer pressuresin the chamber [4]:

vmaxe =

√√√√( 2 k

k − 1

)(RTcM

)(1−

(PePc

)((k−1)/k))

(2.1)

where k is the specific heat ratio, R is the universal gas constant (8,314.51 J/kg mol K), Tc is the combustiontemperature, M is the average molecular mass of the exhaust gases, Pc is the combustion chamber pressure,and Pe is the pressure at the nozzle’s exit. This is a limitation to all thrusters that work by thermodynamicexpansion and so the above expression is also applied to heat-exchanger nuclear rockets and electro-thermalrockets (see Section 2.3).

There is also another limitation of chemical rocket engines, which usually brings its maximum speed toeven lower values. As explained above, CP is based on chemical reactions which are limited by the energystored in each of the chemical reactants, the oxidizer and the fuel. The velocity is then proportional to thereaction enthalpy of the mixture, ∆Hf , and is given by:

vmaxe ∼√

∆Hf (2.2)

A mixture of oxygen/hydrogen, for example, will develop a typical exhaust velocity of the order ofve ∼ 4km/s. These are generally the higher values that can be obtain with this type of technology andthe most common chemical thrusters have a typical exhaust speed that is usually below ve = 2.5 km/s.Improvements can be made on some types of chemical thrusters that could take the exhaust speed limit toa higher value (for example with the use of tri-propellant systems) but these are still far from the typicalvalues that EP can produce, as will be seen later on.

These restrictions to the maximum allowed velocity on CP systems led to the classification of this technol-ogy as energy limited. Electric propulsion systems, on the other hand, are said to be power limited becausetheir performances mainly depend on the rate at which energy from the external source can be supplied tothe propellant. They provide, virtually, no limit to the exhaust velocity of the gas as long as the neededamount of power to accelerate the ions is provided.

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Fig 2.1 shows the relation between thrust and specific impulse for different types of thruster technologies.It can be seen that EP presents values of Isp of approximately 10 to 15 times greater than CP but that, onthe other hand, the values of thrust are much lower. The best chemical propulsion system have thrust valuesalmost one million times higher than EP. These low thrust values imply that EP is not useful in fields of

Figure 2.1: Thrust as a function of specific impulse, Isp, for the typical spacecraft propulsion technologies.Adapted from [5]

strong gravitational gradients (takeoff and landing of a spacecraft, for example) but is the best for a longspace flight mission due to its high exhaust velocity and consequently highly efficient thrusters, as explainedin Chapter 1.2.

2.3 Types of Electric Propulsion systems

Taking into account the many designs and concepts of electric thrusters available today, EP technology canbe sub-divided in three fundamental types: electrothermal, electrostatic, and electromagnetic. In this Sectionthe three types will be described and a more detailed view of the types of thruster modelled in this thesis(Ion thruster and the Hall Effect thruster) is given.

Electrothermal systems

In the electrothermal category the propellant is electrically heated and then, like in the chemical propulsioncases, it expands thermodynamically achieving supersonic speeds when passing through a nozzle. Two mainsystems can be included in this category:

Resistojet

These are the simplest type of EP devices. The technology is based on radiation heat exchange, where apropellant flows over an ohmically heated refractory-metal surface which can consist of different types, suchas coils of heated wires, heated hollow tubes or heated cylinders. The heated propellant is then directed andexpelled through a nozzle which, like in the CP, accelerates it to higher velocities. The available materialslimit the maximum gas temperature of a resistojet and materials like tungsten, tantalum and platinum havebeen used with these systems. The resistojet technology has specific impulse values similar to CP systems

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but much lower thrust values. However these systems have the advantage of being less complex than thechemical ones. Virtually any propellant can be used with this type of technology which made resistojets tobe proposed for manned long-duration deep space mission, where the spacecraft’s waste products like H2Oor CO2 could be used as propellants [6]. These devices have already been implemented in several missionssuch as the Satcom 1-R, Meteor 3-1, and Iridium spacecrafts.

Arcjet

An arc discharge is formed from the tip of a central cathode to an anode that makes part of a nozzle thataccelerates the propellant. The discharge will directly heat the propellant stream to temperatures muchhigher than the wall temperatures, overcoming the main limitation of the resistojet technology. The simpledesign of this technology hides, however, a very complicated theory where the instability nature of the formedarcs may form pinches and wiggles. The stabilization and control of the arc is normally achieved by applyingan external electric field. These systems have thrust values similar to the resistojets but may achieve valuesthree times greater. The typical propellants used with arcjets are hydrazine and ammonia and they have beentested in several communication satellites. The Telstar-4 series used the arcjet technology with hydrazinepropellant and the US Air Force launched an arcjet program (ESEX) to test the ammonia option [7].

Even though the results of this technology look very promising, its development has suffered many delays.This is mainly caused by the difficulty of providing such high power in space (needed to create the highvoltage arc discharge) and to the lifetime-limiting problems of electrode erosion by the heated propellant.

Electrostatic systems

Electrostatic (and electromagnetic) systems overcome the fundamental thermal limitations of the electrother-mal thrusters by directly accelerating ions of the propellant. In these devices the propellant gas is initiallyionized, creating both ions and electrons which form what is called a plasma. The plasma is then a collectionof the various charged particles that are free to move in response to fields that are applied to it. The plasmabeam is then accelerated by a suitable electric field and ions are ejected from the thruster chamber. Giventhe orientation of the electric field, the ejected beam is mainly positive and needs to be neutralized by anequal flux of free electrons in order to maintain the charge of the spacecraft.

One of the objectives with this technology is to achieve high Isp values and maintain a thrust density ashigh as possible. For a given exhaust velocity, ions with a higher mass-to-charge ratio yield a higher thrustdensity. This is the reason why massive ions like Cesium, Mercury and Xenon are preferred to less massiveones. Nowadays Xenon is the most popular type of propellant used for electrostatic and electromagneticdevices due to its low ionization energy, inert nature, low erosion and reasonably high atomic structure.

Field Emission Electric Propulsion (FEEP)

FEEP combines the possibility of generating very low thrust with very high Isp even when compared tothe other EP systems. It uses a liquid propellant, usually Cesium, which is first directly ionized by a highelectric field concentrated at the lips of a capillary slit. Then the ionized propellant is accelerated to veryhigh velocities and ejected in the exit part of the thruster. Values of specific impulse for a FEEP can reach10000s and generated thrust values range from µN to mN. The high precision of these systems makes themvery attractable for drag-free satellite applications, like the LISA mission described in Chapter 1.2.

Ion Thruster

The basic elements of this thruster are a cylindrical chamber containing a centerline cathode emitting elec-trons, a surrounding anode shell and three ring magnets efficiently disposed in the chamber to generateazimuthal and radial magnetic field. This configuration of the magnetic field constrains the electrons togyrate within the chamber for a sufficient time to ionize the injected propellant gas and to direct it to theextractor and accelerator grids, see Fig. 2.2. The disposition of the magnet rings yields a highly divergentand doubly cusped pattern inside the chamber, which is implemented in a way to optimize the ionization

8

discharge and the ion extraction. By tuning the magnitude of the field along with anode-cathode differentialvoltage, the production of single charged ions is maximized and the production of doubly-charged ions mini-mized. Minimizing the production of doubly-charged ions is fundamental because these particles are out offocus in the accelerator gap and so they would tend to erode the grids by sputtering.

Along with the magnetic field inside the thruster’s chamber, an electric field is created by a system of gridsthat allows the acceleration of the created ions. The typical ion thruster contains two electrodes, also calledions optics, in the downstream part of the chamber; the upstream electrode (called the screen grid) is chargedhighly positive, and the downstream electrode (called the accelerator grid) is charged highly negative. Thegreater this voltage difference between the two grids is, the faster the positive ions move toward the negativecharge. Each grid contains thousands of small perforations and both are exactly aligned with each other.The high positive potential inside the chamber and the negative potential of the accelerator’s grid generatesa high attraction of ions towards the accelerator grid, making them pass first through the screen grid. Thisallows the focusing of the ion stream into an array of thousand of ion jets that pass through the upstreamgrid with minimum impingement. This stream of ion jets is usually know as the ion beam.

The constant emission of positive ions would charge negatively the spacecraft. In order to avoid thisand associated problems, a separate cathode emits electrons assuring that the same positive and negativecharge is ejected. Given its function the cathode is called a neutralizer. The electrons emitted by the cathodewould backscatter directly to the discharge plasma due to the high difference potential. However, the nega-tive potential of the downstream grid prevents this to happen and so giving a double function on the thruster.

Ion thrusters have been developed and tested for many decades and have been used in several spacemission. For example, NASA has developed the NASA Solar Electric Propulsion Technology ApplicationReadiness (NSTAR) thruster which was used for the propulsion of the Deep Space One probe and is nowcurrently working on a higher efficiency ion thruster called HiPEP [8]. In Europe, Giessen University inGermany and EADS Astrium have developed several radio frequency ion thrusters (RIT) which have beenused in ESA’s missions. The Advanced Relay and TEchnology MISsion (Artemis) is a telecommunicationsatellite that serves as an example of such missions, where the RITA-10 model is successfully used. Morerecently the UK company QinetiQ developed the T5 and T6. The first was launched with the ESA’s Gravityfield and steady-state Ocean Circulation Explorer (GOCE) satellite and the T6 thruster, which constitutesan evolution of T5 and is baselined for the BepiColomb mission.

Figure 2.2: Ion thruster model

9

Electromagnetic systems

In electromagnetic devices there is an interaction of an electric and magnetic fields within a plasma whichwill create the spacecraft’s acceleration in a similar way to the one of a electrostatic device. As commentedbefore Xenon is the most used type of propellant on these types of thrusters. However, due to short supplyand higher prices new solutions are now being considered. Bismuth, for example, has been presented as anattractive alternative to Xenon due to its larger atomic mass and electron-impact ionization cross-section.This element also as a much more attractive price since it is more abundant than Xenon. For now the use ofthis propellant has been tested on Hall effect thrusters but can be extended to the electrostatic technologyin the future as well [9].

Magnetoplasmadynamic (MPD)

This type of devices have been under development since the late 1960’s both in Russia and United States. Twomain types of thruster can be distinguished: self field and applied field. In the first case the MPD thrusterproduces an arc current between a central cathode and an annular peripheral anode. This arc current ionizesthe propellant that is ejected from the upstream part of the thruster and induces an azimuthal magneticfield. A J×B Lorentz force is generated that will accelerate and eject the particles from the plasma formed.

The self-induced magnetic field is only significant at very-high power and so another model of thesetypes of thrusters was developed where an external applied magnetic field is applied in order to enhance theacceleration process.

MPD scan a wide range of operating parameters, ranging from the mN to almost the kN scale for thrustand from 1000s to 10000s for Isp. The main problems of these devices are the high cathode erosion, whichlimits their life time to a few hundred hours at best. These problems are the main reason for their littleimplementation. An example of an used device of this type is EPEX which flown onboard the Space FlyerUnit - Mission One (SFU-1) in 1995.

Hall Effect Thruster (HET)

The Hall effect thruster, represented in Fig. 2.3 consists of a ring-shaped channel with an interior anode, amagnetic circuit that generates a primarily radial magnetic field across the channel and an external hollowcathode. The performance of these thrusters is mainly dependent on the channel structure and the magneticfield shape. The tuning of these two characteristics lead to a more efficient thruster with a prolonged lifetime.

Even though the efficiency and Isp of the HET can be lower than the ion thrusters, they can producehigher thrust-to-power ratios, and so being very attractive due to their low power demanding when comparingto the electrostatic family. For these devices most of the propellant gas is ejected from the anode located inthe upstream part of the thruster’s channel. The rest of the gas is released from the exterior hollow cathodeas well as electrons which are also ejected at the same time. Usually the electrons would go directly from thecathode to the anode inside the chamber due to the potential difference between the two elements. However,the radial magnetic field created by the coils in the thruster prevents the electrons from directly moving tothe anode. Instead the electrons start spiraling along the magnetic field gaining a drift velocity of the order ofv = (E×B)/|B|2. Given the radial profile of the magnetic field (B) and the axial profile of the electric field(E) due to the potential anode-cathode difference, one can notice that the drift velocity will keep electrons ina circular motion around the center of the thruster. This azimuthal drift of the electrons around the channelis reminiscent of the hall current and gives the name to the thruster. The effect is also the reason why someauthors call it a closed drift thruster.

The Xe particles of the gas released from the anode will eventually collide with the trapped electrons inthe cross electric and magnetic fields, producing charged ions. The axial mobility of the electrons is highlyreduced by the radial magnetic field, which permits a discharge voltage to be distributed along the channelaxis in the formed quasi-neutral plasma. This results in an electric field in the axial direction that acceleratesand ejects the ions in the downstream direction, forming the thruster’s plume. The electrons on the otherhand diffuse in the direction of the anode and the thruster’s walls by collisions and electrostatic fluctuations.

10

In the downstream part of the thruster, and like in the ion thruster description, the cathode also has thefunction of neutralizing the plasma, preventing the spacecraft from getting negatively charged

Each of the indicated EP thrusters has its advantages and would be interesting and worthwhile to studyin detail. However, this thesis will focus on the Hall effect thruster and the Ion thruster, which, as seen, arecharacterized by an acceleration and ejection of charged particles forming a plasma plume.This plasma involves the spacecraft and interacts directly and indirectly with it, leading to several issues thatneed to be studied in detail to fully understand the impact on the spacecraft’s mission.

Figure 2.3: Hall effect thruster model

2.4 Spacecraft-Plume Interaction Issues

Up to now the electrical propulsion systems have been described as well as their value in space missions.It has been seen that a plasma and a neutral environment are created around the spacecraft. When thesetypes of environments are considered, different types of collisions should be accounted for, since they canmodify the original state of the plasma. Charge-exchange (CEX), Coulomb and recombination collisions areexamples of the type of interactions that can be found between particles in these environments.

Charge exchange is a very important phenomenon as far as plume-spacecraft interactions are considered.The ions ejected by the thruster have a very high energy and their trajectory is more or less straight; thusthey do not interact directly with spacecraft surfaces, which are never positioned in front of a thruster. Onthe contrary, slow ions generated by CEX collisions, where a slow (thermal) neutral gives an electron to afast ion, are deviated by the electric field surrounding the spacecraft creating a backflow. Many of these slowions will therefore impinge on the spacecraft surfaces.

Many more effects can be found on plasma interaction with spacecrafts and in this section these effectsare presented in three different categories: physical, mechanical and electrical effects.

Physical effects

• Erosion-Sputtering - This phenomenon occurs when particles, neutrals or ions, with enough energyextract material from an exposed surface. It depends on the energy and angular distributions of theincoming particles as well as the material of the surfaces of impact.On a spacecraft, solar arrays are the most critical part subject to this effect, having a sputteringthreshold of 30eV for Xe+. The computation of eroded surfaces is mostly based on data provided

11

by experimental campaigns for the sputtering yields for different materials, ion energies and incidentangles.

Surface erosion can be the cause of mechanical weakening or thermal properties modification which canrepresent malfunctions and failures of subsystems onboard a spacecraft.

• Contamination - It is possible to distinguish two kinds of contamination processes: direct and indirectcontamination. On a spacecraft the first type of contamination occurs when the particles deposited inthe material are originated from the propellant or the impurities ejected by the thrusters. On the otherhand the indirect contamination may occur by sputtered particles originated by the impingement of aplume on a surface.

Hall effect thrusters are mainly contaminated by slow ions created by CEX which represent almost allthe population of the thruster’s plume backflow.

Mechanical effects

• Forces and torques - These effects can be created by ions coming directly from the thruster hittingthe spacecraft surfaces. Many problems can result from situations of this kind including destabilizationof a spacecraft due to constant movement of its center of gravity and creation of forces counteractingthe created thrust which must be studied and considered for a propellant budget of a mission.

Electrical effects

• Spacecraft charging - Depending on the energy of the charged particle interacting with the spacecrafttwo types of charging can occur: surface and internal charging.For moderate energies of the particles (typically below 10keV) surface charging takes place, where thecharged particles that move freely in the plasma are trapped on the surfaces of the spacecraft. Thistype of charging involves large currents contrary to internal charging where very energetic particles butinvolving small currents cross the outer layers of the spacecraft and are deposited on internal parts ofthe spacecraft.

Charging of a spacecraft creates an electric field that can cause various disorders to the spacecraft’ssystems and that affects the environment surrounding the spacecraft. The electric field can induceacceleration on particles in the direction of the spacecraft’s surface increasing the degree of erosion andcontamination with the consequences outlined above. Spacecraft charging can also eventually lead toa powerful electrostatic discharge.

• Electrostatic discharges can result either from a high electrostatic field between two objects or fromdirect contact transfer and are normally induced either by surface or internal charging. In both casesit can cause serious damages to the spacecraft by degrading its material properties leading to similarproblems as the ones stated above for erosion.

At the moment these issues represent one of the biggest concerns regarding the electric propulsion subjectand many projects have been launched in order to better understand their negative effects on missions and totry to minimize them. In particular the European Space Agency has been interested in studying these issuesproposing projects of different natures from theoretical analysis to experimental and simulation projects.

2.5 The AISEPS Project

The AISEPS project, for Assessment of the Interaction between Spacecraft and Electrical Propulsion Systems,is a project aiming to develop a system engineering tool that can be used by the European space communityfor modeling electric thrusters plumes characteristics in all spacecraft’s configurations.This project is funded by the European Space Agency (ESA) and proposed as a response to a European

12

Space Research and Technology Centre (ESTEC) invitation to tender (ITT). A consortium of differentcompanies was set up in order to perform this project and is composed of EADS Astrium Satellites SAS(France) as a primer contractor, Austrian Institute of Technology - AIT (Austria), EADS Astrium SpaceTransportation GmbH (Germany) and Giessen University (Germany) as subcontractors. A diagram depictingthe organization of the different companies and their main role in their main role in the project is presentedin Fig. 2.4

Figure 2.4: Diagram of the AISEPS Project

The total project is intended to last eighteen months with several milestones within its duration. Fourmain tasks have been planned for Astrium SAS, in the scope of this project which are expected ready andfully operational at the end of the 18 months:

1. Adapt and develop SPIS modeling kernel, SPIS-NUM, (see section 3.1) to take into account missingphysical models for thruster plumes. The plume models to implement are suggested by AIT and createdwith AIT support;

2. Develop a Graphical User Interface allowing to account for real thrusters and reducing the number ofparameters to be set by the user, in the frame of the analysis of the interaction between an electricpropulsion system and its geometrical environment (spacecraft or vacuum chamber)

3. Test and validate the system tool against measurements in order to check the representativeness of thetool and evaluate its accuracy. The simulation tool will also be compared to experimental data to begenerate during this project by the Astrium Space Transportation and Giessen University.

4. Application of the software to three different systems in order to demonstrate its applicability to concreteoperational scenarios.

Task one is the main priority for this thesis and would correspond to the first milestone for Astrium SASto be presented in the end of October 2010. This task comprises the evolution and development of differentmodules in the SPIS software to better model the thruster’s plume. Different thrusters are taken into accountand simulations are run in order to validate these evolutions, comprising also part of the third task describedabove.In order to achieve the proposed third task and also in the scope of the first one described above, all typesof modifications needed to better compare the results to existing data should also be implemented in theSPIS software. These may include evolutions of simulation tools or post-processing options missing from thecurrent version of the SPIS software.

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Chapter 3

Project description and SPIS software

3.1 SPIS software

SPIS, for Spacecraft Plasma Interaction System, is a software being developed by ONERA, Artenum andUniversity Paris 7 - Denis Diderot with the support and funding of the ESA/ESTEC. The SPIS softwaredevelopment started in December 2002 and is included in a project with the same name, having as mainobjectives the creation and development of a software capable of modelling and simulating spacecraft-plasmainteraction and spacecraft charging.

Aiming for a future community based development, SPIS has been developed in an Open Source approach,already allowing the members of the Spacecraft Plasma Interactions Network in Europe (SPINE) to freelyaccess to the software and its source. Future developments of SPIS can include most of the challenges ofspacecraft interactions like the environment of electric thruster systems, solar arrays plasma interactions, andaccurate calibration of scientific plasma instruments. It is intended to be the most accurate, adaptable andextensible simulation tool to be used in many different industrial and scientific applications.[10]

SPIS is a very modular tool divided in two main parts, a simulation kernel (NUM) and a GraphicalUser Interface (UI). The first consists of a JAVA based Object Oriented library to simulate an electrostatic3D unstructured Particle-In-Cell (PIC) plasma model and is integrated in the UI, a framework capable ofperforming the pre-processing, the computation and post-processing. This last part is entirely implementedusing several Jython script modules and integrates very different external tools for the different steps of thesimulation; such as CAD, meshers and visualization libraries (VTK). Both modules are fully multi-platform,fully tested and working in Linux, Windows and Mac operating systems.

On the scope of the AISEPS project and as stated in Section 2.5, evolutions need to be made to thecode to better model the different plasma plumes of existing thrusters. These evolutions are to be madeeither on the Astrium side alone or as a cooperation of Astrium and the SPIS developing team depending onthe specificity of the evolutions. Modules directly related to the thrusters’ models and considered intrinsicto the project AISEPS are fully developed by Astrium while other modules of a more general nature thatcan be helpful to all the SPINE community are proposed by Astrium to the SPIS developing team and areimplemented by the last.

A brief description on how the SPIS software works is given below. Three main phases of a simulationcan be identified: the pre-processing, the simulation and the post-processing phase. Very general steps aredescribed as a way to better understand the view of a SPIS’s user.

Pre-processing phase

When launching a SPIS simulation a set of steps need to be followed before launching the numerical kernelwhere the ”real” calculation is running. This is mainly done in the SPIS UI environment.

15

The first step is to use a software to generate a volume and introduce a grid on this volume. This allowsthe introduction of different properties of our simulation on the grid’s nodes, limiting the number of pointsto which the simulation properties (plasma potential, current density, energy) are attributed.

3D finite element mesh generator

The software used to generate the simulation volume and its mesh is Gmsh. This is a 3D finite elementgrid generator with a build-in CAD engine and post-processor software. The objects wanted for the volumeare designed in the software and the mesh’s size is attributed to each of the main introduced points. Thecalculation volume is then filled with a 3D mesh where the number of cells will depend on the size of themesh chosen at each point. Fig. 3.1 shows a simulation volume in Gmsh where the attributed mesh can beseen.

Z

XY

Figure 3.1: Simulation volume with mesh on the Gmsh software (left) and a zoom-in on the thrusters region(right)

Gmsh also allows the user to select which surfaces and volumes he wants to consider with similar properties.These are called physical groups and each group may be composed of one or several surfaces and one or severalvolumes.

Physical group identification and introduction of boundary conditions

After designing the desired objects with Gmsh and defining the physical groups, the user may start using theSPIS software environment. The following step is the physical group identification where the user needs toidentify each of the groups and introduce certain boundary conditions. As examples of boundary conditionsthe user may introduce an imposed surface potential (Dirichlet condition), or an imposed current to be ejectedfrom a surface. With this part finished all the group identifications and boundary conditions are put togetherwith the mesh and saved on file.

Global parameters definition

In this step the user selects the main parameters to be used in the simulation. These mainly comprise:

• Simulation properties: the simulation’s duration and the time step used can be defined;

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• Ejection models: the type of distribution to use, the densification of each superparticle (more on thislater) and the type of ejected particle can be selected;

• Volume interaction model: the background pressure, the properties of each collision such as the energyof the background population and the cross section can be defined;

• Solver type: The Poisson solver can be selected as well as the quasi-neutrality with constant or variabletemperature

Some of the indicated properties include already evolutions made during the work for this thesis. Withall the parameters and volume defined the simulation is ready to be launched. By selecting the option ”UI toNUM”, the SPIS software will write all the properties to a couple of files which will be loaded by the kernelpart of the software and used for the simulation.

Simulation phase

The general structure of the SPIS software is shown in Fig. 3.2.

Figure 3.2: Object structure of the SPIS software. Adapted from [10]

The simulation consists of two main parts: the spacecraft circuit and the plasma solver. The spacecraftcircuit consists of a cycle where an electric circuit is solved based on the collected and emitted currentsfrom and to the plasma and on an user defined spacecraft circuit where resistances and capacitances can bedefined to the different components. The plasma solver includes two different cycles and applies the 3D-PICmethod (which will be described in detail in the next Chapter) to calculate the plasma evolution. In generalthe represented Field part of the plasma solver will resolve the plasma fields by using a Poisson solver orby assuming a quasi-neutral plasma and using simpler methods. The Matter part will generate and moveparticles taking into account the calculate field on first part of the plasma cycle. Several simulation timesteps may be defined for each cycle which may be very useful when different types of particles (electrons,ions, background neutrals) are included in the simulation and may have completely different velocities.

At defined simulation time steps the results can be export to the UI module of the SPIS software andpost-processing of the results can be done. At the end of the simulation an average of the results is computedand also exported to the UI module. This average can be defined by the user and in the simulations performedfor this thesis the results were averaged from 0.6×ttot to ttot, where ttot is the total duration of the simulation.

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Post-processing phase

After the simulation is run the results are sent back to the UI and the user may export them as .vtk files.The analysis of the results is done using a 3D visualization tool like Cassandra or Paraview. These toolsgive the possibility of knowing each exported property on each cell of the simulation volume hence knowingthe plasma profile in the volume. They also allow the interpolation of results between cell nodes which maygive a good approximation of the results in the interior of a mesh cell. In Chater 6 the simulation results areanalyzed and more details on the post-processing part are given.

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Chapter 4

Plasma Simulation

Self-consistent numerical models for plasma modelling can generally be divided in three main categories: thekinetic, the fluid and the fluid-kinetic, also known as hybrid descriptions.

A classical, complete description of a plasma would, in principle, be possible to achieve by applying thebasic Newton’s law to all the particles of the plasma and solve all the possible interactions between them.This approach, however, is totally impracticable given the huge amount of data it would need to handle.Even just storing the information of all the particles at each time would already be beyond the capacitiesof the best existing devices and performing the interactions among them would be completely impossible.Using this microscopic description is then simply fictional but it offers a good starting point to the plasmamodelling approaches used today.

By applying statistical concepts one can pass from the microscopic theory described above to the kinetictheory. Considering a statistical ensemble of particles, the different populations’ microscopic information canbe included and can be averaged to give statistical, kinetic equations. With this approach the precise locationand information of each individual particle is lost but it is included in a distribution function f ≡ fi(x,vi, t),with velocity vi in the location x at time t. The distribution function of a specific type of particle i can beintroduced as:

fi(x,vi, t)dxdv = dN(x,vi, t) (4.1)

where dx ≡ dxdydz and dv ≡ dvxdvydvz. This function represents the number of particles in the element ofvolume dV = dxdv in the velocity phase space and it can be written:

∞∫−∞

fi(x,vi, t)dV = N(t) (4.2)

Several equations can result from the kinetic approach, depending on the type of interactions considered. TheBoltzmann equation is a general equation that translates mathematically the kinetic description by taking aset of S simultaneous equations, one for each species on a multicomponent medium. Each equation gives theevolution of the distribution function fi ≡ fi(x,vi, t), with i representing a particular species:

dfidt

=

(∂fi∂t

)col

(4.3)

This equation describes the evolution of each distribution function caused by the collisions with othertypes of particles, as can be noticed by the index col on the RHS of the equation. The LHS of the (4.3) canbe written explicitly as:

dfidt

=∂fi∂t

+∂x

∂t

∂fi∂x

+∂vi∂t

∂fi∂vi

=∂fi∂t

+ vi∂fi∂x

+Fimi

∂fi∂vi

(4.4)

where the symbol Fi represents the external forces and mi the mass of the particle type considered.The RHS can also be rewritten as:

19

(∂fi∂t

)col

=

S∑j=1

∫ ∫(f ′if

′j − fifj) gij dσij dvj (4.5)

where i and j represent two different typs of particles, gij ≡ |vi − vj | and dσij is the differential cross-section of i− j. It was also used the notation f ′i ≡ f ′i(x,v′i, t) where the symbol ′ indicates the post-collisionsituation.Putting together the above equations, the Boltzmann equation is obtained:

∂fi∂t

+ vi∂fi∂x

+Fimi

∂fi∂vi

=

S∑j=1

∫ ∫(f ′if

′j − fifj) gij dσij dvj (4.6)

The kinetic approach leads to the most accurate results of all three categories referred above, however itis also the more computationally demanding and so the other methods can be useful when large calculationsneed to be performed.

The fluid description is less complex than the kinetic one but also less accurate. This reduction ofcomplexity is achieved by taking a macroscopic view of the plasma where the quantities describing it areobtained by computing the velocity moments of the Boltzmann equation (4.6). The different orders ofmoments of the equation give the different useful quantities: density for the 0th order, mean velocity forthe 1st order and mean kinetic energy for the 2nd order moment. The derivation of these equations isstraightforward and will not be detailed here. However, it can easily be found in several Plasma Physicsbooks (see [11], for example).

As explained, using the kinetic approach for all the particles (electrons, ions, neutrals) can be too timeconsuming and too demanding for normal computer resources. This situation is generally overcome bycombining the two different methods into one, the hybrid model. The selection of which distribution shouldbe given to a certain type of particles mainly depends on their mass and, consequently, their velocity. Usuallyfast particles like electrons require very small time steps in order to have a travel distance for each time stepcomparable to the size of the cell used in the simulation volume. This can make the code computationallyimpracticable and so many codes that simulate EP plasmas use the hybrid approach, where fast particleslike electrons are modelled as a fluid and slower particles like ions and neutrals are model as particles withthe kinetic description, (see for example [12][13]). This method requires less computational capabilities andcan still be sufficiently accurate for some types of calculations, and so it can sometimes be the best optionto take for a simulation.

4.1 3D Particle-in-Cell Simulations

The Particle-in-Cell (PIC) method represents by itself a simulator for the non-collision Boltzmann’s equation,also known as Vlasov’s equation. By considering the RHS of (4.6) equals to zero, the Vlasov’s equation canbe written as:

∂fi∂t

+ vi∂fi∂x

+Fimi

∂fi∂vi

= 0 (4.7)

In a general PIC method scheme with the kinetic model, a plasma is simulated using the so-calledsuperparticles. A superparticle is a representation of a defined quantity of real particles, containing anaverage information of the mass and velocity of the represented particles. The number of superparticlesto be simulated is generally a parameter controlled by the user. The user as the goal of reaching theminimum number of superparticles in each cell in order to sample macroscopic properties by averaging overtheir microscopic quantities. The weight given to a superparticle is defined as the number of real particlesit represents. Assuming that all superparticles have the same weight W (which is not always true) it iscalculated by the following expression:

W =#PhysicalParticles

#SimulatedParticles(4.8)

20

In figure 4.1 a typical cycle of a PIC code is represented. The SPIS software uses an unstructured meshgrid that is adjusted to the surfaces and volumes where the simulation is to be performed as was seen in Fig.3.1.

Figure 4.1: Typical cycle of a PIC simulation

At the beginning of the simulation, a set of initial conditions is loaded into the simulator. These initialconditions are macroscopic quantities that allow the generation of superparticles. According to the macro-scopic properties superparticle are given a set of initial microscopic quantities: mass, position and velocity.Also the initial electric and magnetic fields and boundary conditions are defined prior to the start of the code.The latter three are normally directly introduced into the mesh nodes. By knowing the initial conditions oneis able to start the simulation loop. Considering the initial position of the superparticles and their initialvelocity vector, the following position is calculated and the superparticles are moved.

After moving all the superparticles, each one of them deposits its physical density and current on nodes ofits current mesh cell with a linear weighing process. This process considers the position of the superparticlein the cell where it is contained and attributes to each of the cell’s node, a fraction of the density and currentas a function of the distance to that node. This function will give a higher density to the closer nodes tothe particle and a lower density to ones further away. Doing this for all particles allows the code to havethe quantities always in the mesh grid. One can also see that this quantity is conserved when consideringall the nodes by summing the density attributed to each node by a superparticle and realizing it is the sameas the one contained in the same superparticle. In Fig.4.2 this method is represented for a 2D case with arectangular grid.

Taking the last accumulated quantities in the grid, the electric potential can be retrieved on each nodeby solving the Poisson’s equation:

∇2φ+ρ

ε0= 0 (4.9)

where φ corresponds to the potential and ρ to the density. ε0 is the permittivity in vacuum. There aredifferent forms of solving the Poisson’s equation and it is even possible to obtain the fields in much simplerways by making the assumption of plasma quasi-neutrality. This topic will be discussed in Section 5.7

Knowing the potential on each node allows calculation of the fields. Once the electric and magnetic fieldshave been solved, the total force exerted on each superparticle is calculated and used to determine the newvelocity for each particle v∗i and, with this, the new position. Here the cycle re-starts and the steps describedare repeated until it reaches the final time chosen by the user.

21

Figure 4.2: Weighting of density to the grid points by an interpolation function fd. Each grid point i acquiresa fraction of the density of the superparticle ni = fd × n.

4.2 The PIC-MCC approach with the SPIS software

The method used in the SPIS software is called PIC-MCC from Particle-In-Cell with Monte-Carlo Collision(MCC) and is equivalent to solving the following modified Boltzmann equation:

∂fi∂t

+ vi∂fi∂x

+Fimi

∂fi∂vi

=

∫ ∫(f ′iG

′g − fiGg) gig dσig dvg (4.10)

whereGg is the distribution introduced for the background particles and the index g is referring to particlesof the background gas. By this equation it can be understood that only collisions with the backgroundneutrals are considered. Furthermore, with the SPIS software, only the charge-exchange collisions (CEX) areconsidered, neglecting all the elastic collisions between the different particle types. This will be discussed inthe next Section. The introduction of Monte Carlo collisions in the PIC code will, of course, affect the waythe cycle evolves. Fig. 4.3 shows an extension of the standard PIC simulation presented in Fig. 4.1.

Figure 4.3: Typical cycle of a PIC-MCC simulation

22

When simulating a thruster in the SPIS software the superparticles are ejected from a surface witha defined velocity and position, see Section 5.3. These constitute the refered initial conditions and thesimulation loop takes them as a starting point. At each timestep new particles are injected in the simulationvolume and considered by the code in addition to the ones already being simulated.For the initial conditionsin the SPIS software superparticles are introduced with a random position on the ejection’s surface. As canbe noticed in the figure depicting the PIC-MCC simulation, the collisions between particles are introducedas another ”module” in the code between the calculation of the superparticles’ positions and the attributionof their characteristics to the mesh grid. The rest of the loop is followed as described before for the standardPIC simulations.

4.3 CEX collision in the SPIS software

A CEX collision is depicted in Fig. 4.4. When a fast ion collides with a neutral with a thermal velocitythe two particles exchange an electron. For the most part of the simulation codes the initial momentum ofeach particle stays unchanged (which is not completely true in reality, where only the total momentum mustalways be conserved but the momentum of each particle may change).

Figure 4.4: Representation of a CEX collisions

Two types of CEX collisions will be considered in SPIS:

Xefast+ + Xeslow

CEX−→ Xefast + Xeslow+

Xefast2+ + Xeslow

CEX−→ Xefast + Xeslow2+

Usually when simulating CEX collisions the charge of the particle is conserved and it is the velocity thatchanges. This has the same final effect and is in general simpler to perform on a simulation code. To simulatethe collisions, the fast ion acquires the initial thermal velocity of the slow neutral and to the neutral theinitial velocity of the ion is given. By doing this the change of electron is simulated without changing theproperties of charge of each particle.

In the SPIS software the distribution of the initial particles is considered constant before and after thecollision and the result of the CEX collision is just the creation of a new slow ion. If the amount of collision(and so the amount of created slow ions) is very small compared to the amount of existing fast ions, thenthe effect of not erasing the initial particles after the collisions can be neglected. However if the amountof generated slow ions is comparable to the amount of fast ions, then this method may introduce very bigerrors in the results. The same reasoning can be done to the distribution of neutrals, which is also consideredconstant before and after the collision, i.e., the initial slow neutral is not deleted and the new fast neutral isnot created. This is equivalent as saying that for the CEX collisions in SPIS the background gas is alwaysassumed in equilibrium and undisturbed, then maintaining its initial distribution: G = G′.In the SPIS software, the user may choose whether he wants to use the CEX collisions in the simulation ornot. For the simulations with the thruster models the CEX collisions were used where the collision rate ofan ion in the background density of neutrals, n2, during a time step dt is given by [10] :

23

dp = n2 ∆v σ(∆v)dt (4.11)

where σ is the CEX cross section and ∆v the relative velocity between the two particles, given in ms−1. TheCEX cross section values used correspond to the most recent measurements performed by Miller et al.[14]and are modeled by:

σ(∆v) = a − b ln ∆v (4.12)

where the two constant parameters a and b are derived by fitting experimental curves. The parameters aand b for both types of collisions are presented in Table 4.1.

Type of collision Parameter a (m2) Parameter b (m2)

(Xe+ , Xe) 1.71× 10−18 1.18× 10−19

(Xe2+ , Xe) 1.03× 10−18 7.7× 10−20

Table 4.1: Cross section parameters for CEX collisions

24

Chapter 5

Main Evolutions to the thruster models in the

SPIS software

As described in section 3.1 the SPIS software aims at developing a toolkit for modelling of spacecraft-plasmainteractions and spacecraft charging. However, until the start of this project, the options to simulate realisticthruster plumes had not yet been properly implemented in the software; only very basic plasma sources wereavailable. Default options in SPIS software included two main methods to simulate the ejection of particlesfrom a thruster with only one type of particles for each surface of ejection.

5.1 Multiple Particle Sources and Interactions

Experimental evidence[16] indicates that around 10%-20% of the ions ejected by a Hall Thruster are doubly-charged. This number is characteristic of each type of thruster and should not be neglected when simulationsare performed.

At the beginning of the project only one type of particles could be ejected from the same surface and so,taking the above values into consideration it was proposed that a new way of ejecting particles should becreated and that it would include a user defined percentage of each species. This option was considered verygeneral and important to all the SPIS community and it was requested to the SPIS developing team to extendthe implementation of the particle’s ejection function and include the possibility of ejecting other types ofparticles in the same surface. This was performed by the SPIS developing team, following the methodologyadopted for the AISEPS project for code evolutions, see Section 2.5.

Since new particle types were created it was also considered that their interactions with neutral particlesshould not be neglected and so CEX collision were also implemented for this new population, following theoriginal CEX method described before 4.3.

5.2 Default Source Distributions

Two main methods to eject particles were available in SPIS at the beginning of this project: Maxwellian andAxisymmetric sources.

Maxwellian distribution

The first method is the simplest, giving the possibility to eject particles with a mean velocity and a thermalvelocity which follows a Maxwellian distribution. By defining a temperature T and the type of ions to eject,a thermal velocity can be calculated using the expression

25

vTh

= (kB T/mi)1/2

(5.1)

where the index i refers to the type of particle to eject. Superparticles created with this method are givena velocity where each component follows a Maxwellian distribution in 3D:

f(v) = exp(−v2/v2

Th

)= exp

{−mv2

2kBT

}(5.2)

This function ejects particles with an energy dispersion around a mean drift velocity < v >= 0 m/s. Howeverthe user may want to use a defined mean velocity for the ejection of the particles, as normally happens in athruster for example. In this case, the SPIS software allows the user to define a variable which relates thetemperature of the particles and their mean velocity. This quantity, is called Mach number and defined asM = v0/vTh

, gives the possibility to define a velocity in the direction perpendicular to the thruster, v0, and toadd it to the same component of velocity calculated with (5.1). The other two components stay unchanged,with the velocity calculated before. In mathematical terms changing the perpendicular component would beequivalent to shifting the Maxwellian distribution by v0 = v

Th×M :

f(v) = exp

{−m (v − v0)2

2kBT

}(5.3)

Axisymmetric distribution

This second method included in the SPIS software gives a better control of the particles distribution thanthe above explained Maxwellian method. It allows the user to define the flux, mean energy and temperatureof the ejected particles as a function of their angle of ejection, which is considered between the radial velocityvector and the radius of the thruster. These quantities are defined in a table which is saved as a .txt file andloaded when this distribution is selected.

The value of the flux to be set corresponds to the accumulated flux from the vertical of the thrusterto the given angle and is introduced as a normalized quantity. In this method a random angle is gener-ated following the given distribution of the flux, i.e., an angle where the flux has a higher value will have ahigher probability to be generated. After generating the angle of ejection for the considered superparticle,its position is randomly created on the surface of the thruster and its mean energy and temperature areobtained by interpolation of the values set in the table. With these two quantities the mean and thermalvelocities are calculated and used to generate the final velocity with a similar method to the one explainedabove, following the Maxwellian distribution of (5.3). This method allows then to set a flux, velocity andtemperature distribution to the particles which cannot be obtained with the simple Maxwellian method.

Although useful for ideal cases, neither of the two methods above can accurately simulate a real thrusterand so new models had to be developed as a way to obtain more precise results. The configuration of HallEffect Thrusters and Ion Thrusters (see section2.3) as well as previous publications on these types of thrusters[17] suggest that each type of particle is ejected with a constant speed in the whole thruster’s surface andthat the ejection angle of the particles changes as a function of the position on this surface. With the originaldistributions these considerations could not be simulated, since both the positions and angles are randomlyobtained. To implement and test this new type of distribution, the source code of the SPIS software had tobe changed and new methods had to be introduced. A brief description of this new model implementation isgiven in the next section.

5.3 Hall Thruster Model

Tajmar et al. [17] have proposed a mathematical model to describe the plasma plume ejected from a Hallthruster. Following AIT suggestions (see Section 2.5) this model was implemented in the SPIS software.Taking into account measurements data[18] and theoretical analysis[19] of a HET it was suggested that thecreation of ions in the channel of the thruster was greater near the inner wall. This bigger production implied

26

that different angles would have to be assumed for the particles depending on their position of formation, sothat they wouldn’t collide with the walls of the channel before leaving it. Fig. 5.1 pictures this difference inthe angles of ejection, where the absolute value for the inner angle is smaller than for the outer angle.

Figure 5.1: Diagram depicting the implemented model for a HET. Different angles are generated for differentpositions in the thrusters exit. Adapted from [17]

The variation of the angle of ejection is assumed linear as suggested by Manzella [20] and can be expressedby:

α(r) = αi + (r −Ri)αo − αiRo −Ri

(5.4)

where the parameters αi and αo correspond to the inner and outer angle of ejection, Ri and Ro to the innerand outer radius of the thruster’s exit, as shown in Fig. 5.1 and the parameter r is the radial position wherethe particle is ejected. As an example, using the input parameters for the SPT100 model (which will be fullydiscussed in Section 6.1) the angles of ejection are αi = −12◦ and αo = 40◦ and the inner and outer radiusare respectively Ri = 28mm and Ro = 50mm . With these values one can calculate the position on thethruster where an ejected particle will have a vertical direction in relation to the thruster, r(α0) = 33.07mm.This corresponds to 20% of the width of the thruster’s ejection ring and means that a particle created furtherway than this value will already be directed outwards.

By assuming this linear variation for the angle of ejection, the number of particles ejected per time step(or, equivalently, the massflow of each species) and their correspondent velocities are the two parameters leftthat are needed to simulate the thruster. The first is given by the mass flow rate, i.e. the quantity of massof the gas to be ejected by the thruster per unit of time. In this model only the ejection from the anode isconsidered. However for a real thruster the quantity of mass ejected by the cathode (where both electronsand neutrals are ejected) also has an influence and this has to be considered when modeling the anode. Thetotal mass flow rate is assumed constant and can be written as:

mTηu ηa = m+

i + m2+i (5.5)

where the mT

is the total mass flow rate and the variables ηu and ηa correspond to the ionization efficiencyand the percentage of propellant directed through the anode, respectively. This means that only a percentageof the total mass is ejected by the anode and only a part of this fraction is ionized (since some neutrals arealways released).

The thrust is one of the intrinsic characteristics of a thruster and is normally measured for each case.This is then a parameter that is introduced in the simulation for each thruster. This quantity can be writtenin the following way:

T = (m+i v

+i + m2+

i v2+i )

(sinαaverαaver

)2

(5.6)

where αaver is the average angle of ejection and is given by αaver = (αleft + αright)/2. Here the thrustincludes only the single and doubly charged particles, neglecting the effects of the ejection of neutrals due to

27

their low percentage compared to other particles and also because their ejection velocity is much lower thanthe one considered for the accelerated ion particles (typically neutrals are ejected just by thermal agitationsince they are not accelerated by the electric field). Considering the two expressions above and realizing thatwhen applying the same electric field to the two different charged particles one has

v2+i =

√2 v+

i (5.7)

the expressions can be combine to a form where only known parameters are used. Assuming a percentage ηpof doubly charged ions, a relation of mass flows for the two types of particles can be found:

n2+i = ηp n

+i ⇔

m2+i

A√

2 v+i

= ηpm+i

A v+i

(5.8)

⇒ m2+i = ηp

√2 m+

i (5.9)

Combining equations (5.5), (5.6), (5.7) and (5.9) the final expressions for velocity and mass flow of eachtype of particle are obtained:

v+i =

T (1 +√

2 ηp)

mTηu ηa(1 + 2 ηp) (sinαaver/αaver)

2 (5.10)

m+i =

T

v+i (1 + 2 ηp) (sinαaver/αaver)

2 (5.11)

The distribution of ions along the anode is assumed to be homogeneous, both densities and temperatureof ions can be assumed to be radially uniform. As indicated in (5.5) an ionization efficiency ηu must beconsidered due to the impossibility to ionize all the neutral gas injected in the thruster chamber. Giventhe importance of the CEX interactions discussed in Section 4.3, one can understand the interest to alsomodel the neutrals leaving the thruster. Ideally the ejection of neutrals should also be modeled by thedescribed kinetic approach (see Section 4) and, since this option was not implemented in SPIS softwareat the beginning of this thesis, it was also developed during this project. However this tool was not fullytested and so the simulations presented on this thesis used analytical descriptions of the neutral population.Two types of neutrals density distributions where considered: background neutrals (directly related to thebackground pressure) and ejected neutrals from the thruster (given by a fraction of the number of ejectedions). Contrarily to the first distribution where the neutrals density is introduced as constant in the volumeof simulation, the ejected neutrals follow a distribution depending on the radial distance to the thruster, r,and the angle to the thruster’s axis, θ. This density distribution is then given by [10]:

n(r, θ) =1

(8π)1/2

cos θ

r2

mN S

vTh(5.12)

where mN is the total flux of neutrals, S the surface of ejection and vTh the thermal velocity at which theneutrals are ejected. The angle θ in the expression assumes values in the interval

[−π2 ,

π2

].

The possibility to combine these two types of distributions was also implemented during this internshipby ONERA as a request by our part. This allowed to simulate the CEX interactions with both the neutralsejected by the thruster and also the neutrals of the background.

Implementing the HET distribution in SPIS

The implementation in the SPIS software of the method described above is performed in the following way:

• First the mass flow rate for the two types of particles (Xe+ and Xe2+) is calculated by using (5.11)and (5.9) and the number of particles to ejected at each time step is easily obtained by dividing themass flow rate by the mass of the particle and multiplying by the time step:

NXe+ =m+Xe

mXedt and NXe2+ =

m2+Xe

mXedt (5.13)

28

• After finding the number of physical particles to be ejected, the number of superparticles is calculatedby considering a densification for each superparticle. This value is usually introduced by the user andallows to control the weight (number of physical particles represented by each superparticle) and thenumber of superparticles.

• For each superparticle a random position on the surface of ejection is generated and an angle accordingto this position is calculated by a linear interpolation of the divergence angles of the inner and outeredges, as expressed in (5.4).

• After calculating the angle, it is necessary to find the velocity for the particle. The absolute value ofthe velocity is calculated using equations (5.10) for Xe+ and (5.7) Xe2+. This vector has only theradial and normal component with respect to the thruster.

• A thermal velocity following a Maxwellian distribution, given by (5.2), is also generated and this newvector is added to the previous calculated velocity. As a result the generated velocity vector is thenformed by 3 components where two of them are the sum of the ejected and thermal velocities and thethird component is given exclusively by the thermal velocity.

• With this calculated velocity for each superparticle the code then moves them by the distance corre-sponding to r = vtotal × dt where vtotal = vi + vth and starts the original PIC-MCC code described inSection 4.2

5.4 Ion Thruster Model

Another model implemented in the SPIS software was the Ion thruster plume model. This model wassuggested by AIT, who modified the HET plume model described above to implement the configuration ofthe ion thruster. The first big difference that can be noticed between the two types of thrusters (see Section2.3) is that the Ion thruster is not ring shaped but instead has a completely circular surface of injection. Byassuming the same model of the HET only the outer divergence angle αo is defined and αi is taken as 0◦

in expression (5.4). Fig. 5.2 depicts the ejection angle for the Ion thruster model. According to the model

Figure 5.2: Diagram depicting the implemented model for a Ion Thruster. Different angles are generated fordifferent positions in the thrusters exit.

proposed by AIT the velocity vector (both speed and direction) and mass flow rate for each type of particlesare calculated by the same expressions presented for the HET (equations (5.10), (5.11), (5.7) and (5.9)). Thespeed of the ejected particles is still assumed constant. However, in contrast to the HET model, experimentalresults with Ion Thrusters by de Boer[21] suggest that the current density profile is Gaussian. Therefore,instead of the uniform ion density distribution used in the HET plume model, a Gaussian ion density profile

29

should be used at the exit plane. The ejection of superparticles is still performed with an uniform distributionon the surface of ejection, however since the number of physical particles ejected is always represented by theweight of a superparticle, it is the weight that changes with the radial distance in order to follow a Gaussianprofile. By doing so even considering an uniform distribution of superparticles, the correspondent numberof physical particles is changing and the correct amount is being ejected in each place. This is done byrandomly choosing a position on the surface of ejection and then attributing a weight to the superparticleby considering the radial Gaussian distribution:

G(r) = K exp

[− r2

2R02 σ2

](5.14)

It is then necessary to find the value for the constant K. This is done by equalizing the total numberof particles to be ejected in the whole thruster to the integrating of the above Gaussian distribution in thesurface area of ejection: ∫ 2π

0

∫ R0

0

K exp

[− r2

2R02 σ2

]r dr dθ = N (5.15)

By using the substitution method and considering

y2 =r2

2R02 σ2

⇔ r =√

2R0 σ y ⇒ dr =√

2Ro σ dy

the integral can be written as

∫ 2π

0

∫ 1/√

2σ

0

2KR02σ2 exp

[−y2

]y dy dθ = N (5.16)

and the constant K can be found to be:

4π R02 σ2K

[−1

2exp

[−y2

]]1/√

2σ

0

= N ⇔

K =N

2π R02 σ2

exp[1/2σ2

]exp [1/2σ2]− 1

(5.17)

Replacing the constant in (5.14), the radial gaussian distribution can be explicitly written as:

G(r) =N

2π R02 σ2

exp[1/2σ2

]exp [1/2σ2]− 1

exp

[− r2

2R02 σ2

](5.18)

The equation (5.18) represents the number of physical particles ejected per unit area, and it is a gaussianfunction of the radial distance r to the center of the thruster. The final weight of each superparticle can thenbe calculated by:

W =G(r) . A

# Superparticles(5.19)

where A is the total emission surface.

Implementing the Ion Thruster distribution in SPIS

The implementation of this model is similar to the implementation of the HET described before. However,the weight of each superparticle is not constant as before but depends on the position where the particle isgenerated. Therefore the steps followed for the previous model where the attribution of weight precedes thegeneration of an ejection position must be reversed and so, for the ion thruster model, the ejection positionis firstly generated and then the weight is given according to its position. The following steps are performedin the same way and only the parameters used change in relation to the ones used for the HET model(forexample the inner angle is fixed as αi = 0◦).

30

5.5 Plasma Properties

Understanding the environment surrounding a spacecraft is one of the critical aspects to determine thecharging and erosion a spacecraft is exposed to. Probes provide different types of characteristics of the en-vironment and allow to characterize it both when tests are being carried on ground (in a vacuum chamber)or in space. The main quantities that are usually provided in the bibliography for experimental tests arethe number/charge density, the current density, the potential of the plasma and the energy of the particles.These measurements are performed either by using probes that scans in a circle within a fixed distance fromthe thruster or in a line parallel to the thruster’s exit.

At the beginning of this project, the only quantities calculated and exported for post-processing by SPISwhere the charge density and plasma potential on each node of the mesh grid. These characteristics areexported to files with the .vtk extension which can then be analysed using the post-processing software likeCassandra or Paraview. Several simulations were run to understand the behaviour of the plasma with thesequantities but it was soon realized that to comply with the future objective of the AISEPS project, studyingthe interactions of the plasma with the spacecraft, it would be crucial to understand how the profile of theplasma was in terms of current density and energy. The implementation of these characteristics was discussedby Astrium and the SPIS development team and it was understood that they could also be important fordifferent projects with the SPIS software. It was then agreed that both ONERA and ARTENUM wouldtogether develop this post-processing tools in order to export the values on the grid nodes as files with theVTK extension. These tools were very useful to compare the results with experimental data and the mainresults can be found in the following Chapter.

5.6 The Poisson Solver

As described in Section 4.1 the calculation of the plasma potential comprises one of the steps of a PICsimulation. Different ways of doing these calculations can be used, which will affect both the degree of com-plexity of the code and the quality of the results. At the beginning of this project the Poisson solver wasthe only method implemented in SPIS. A detailed description on how this solver is implemented in the SPISsoftware can be found in [22]. This method runs in two different parts where in the first part of the codethe electron density at each node is calculated and in the second part this density is used to find the potential.

The calculation of the electron number density is made using the well known Boltzmann relation:

ne = nref exp

(e φ

kB Te

)(5.20)

In this equation e, kB and Te represent, as usually, the electron charge, the Boltzmann constant and theelectron temperature. The parameter φ is the old potential on the same node where the electron densityis being calculated and nref is a reference density corresponding to the density obtained when the potentialequals zero. In the case of the simulation performed in this thesis, the value of nref was calculated byreplacing the known values of density and potential in the ejections surface and solving the above equation.The potential of the ions surface of ejection (known as φref) was set to a constant value by a Dirichletboundary condition. On the other hand, the density of ions could also be obtained from the known exitconditions of ejection velocity (vi), area of ejection (A) and total ejected current (I). This calculation couldbe performed by the following expression:

ni =I

eA vi(5.21)

Taking the ions density equal to the electrons density at the thruster’s exit and using the values of Te andφref presented in Table 6.1, (5.20) was solved and the value of nref was found.

When the electron density on each node is calculated the new potential can be computed. This is doneby solving the Poisson’s equation:

∇2φ+ρ

ε0= 0 (5.22)

31

which can be written explicitly as a function of the ion and electron density:

∇2φ+ρi + ρeε0

= 0⇔

(5.23)

⇔ ∇2φ+ρiε0−ρerefε0

exp

(e φ

kB Te

)= 0

This is a second order differential equation that can be quite complex to solve by numerical methods.Finding its solution implies the storage of density and potential on different matrices and solving this systemwith these matrices requires the usage of linearizing methods which are computationally expensive. Afterfinding the potential values on each node, the derivation of this quantity allows the calculation of the electricfield which is then used to move the particles. The usage of simpler methods to find the potential wherediscussed with the AIT team involved in this project and it was found useful to have a method where thePoisson’s solver could be skipped in order to reduce the computational time of each simulation. This methodis described in the next section.

5.7 The assumption of a quasi-neutral plasma

By assuming that the plasma is quasi-neutral (ni ≈ ne) in the whole domain of the simulation the calculationof the potential can be performed without solving the Poisson’s equation. Two different methods were imple-mented in the SPIS software during the project to calculate the potential with the quasi-neutral assumption;both isothermal and non-isothermal plasmas were considered. Since this is an assumption, one should beable to identify whether it is valid or not in the domain of our simulation. This is done by considering theDebye length λD, i.e., the characteristic length over which charges are neutralized:

λD =

√ε0 kB Tene e2

(5.24)

Values of electron density on plasma thruster plumes usually range between ne ≈ 1014 − 1018m−3 andhave electron temperatures of 4eV. Plugging these values into equation (5.24) gives typical values for theDebye length of the order of λD ≈ 10−3−10−5m. These relatively small values indicate that the Debye lengthis small with respect to features of interest in the plume region. However in some parts of the plume, forexample sheath regions near the satellite’s surface the density can be very small increasing a lot the value ofthe Debye length and making it useful to consider the Poisson’s solver. The approximation of quasi-neutralityshould then be checked in every part of the simulation volume and a method to perform this analysis wasalso implemented in the SPIS software (see Section 5.8).

Isothermal plasma and the Boltzmann relation

In the case of an isothermal plasma the Boltzmann relation presented in (5.20) can be inverted in order tothe potential:

φ =kB Tee

ln

(ninref

)(5.25)

This approach was implemented in SPIS in order to reduce computation time relatively to the Poisson’ssolver.

Variable temperature in the plasma

Experimental results by [23] and [24] suggest that a variable electron temperature should be considered whensimulating plasma thruster plumes. Even though the more common simulation methods for quasi-neutralplasmas solve the Boltzmann relation to find the plasma potential, the assumption of a non-isothermal plasma

32

demands a new expression to calculate the new potential. This expression can be deduced by assuming anadiabatic variation of the temperature, given by:

kB Te ne−γ+1 = C1 (5.26)

where C1 can be given by the reference electron temperature and density:

C1 = kB Teref neref−γ+1 (5.27)

Solving in order to the electron temperature:

Te = Teref

(neneref

)γ−1

(5.28)

Starting from the momentum balance equation:

∇(ne kB Te) = e ne∇φ (5.29)

and replacing Te by the obtain quantity in (5.28) one can find:

∇(kB Terefneγ

nerefγ−1

) = e ne∇φ⇔

⇔ ∇φ =kB Terefe neref

γ−1

∇neγ

ne(5.30)

Integrating the equation on both sides the expression to calculate the potential in a non-isothermal plasmacan be found:

φ =kB γ Terefe (γ − 1)

((ninref

)γ−1

− 1

)(5.31)

In this equation the constant neref includes all the integration constants which should be considered bythe user when introducing the correspondent value.

5.8 Quality of simulations

In order to verify the quality of the simulation results two other functionalities were added to the exportproperties of the SPIS software:

• Verification of the plasma’s quasi-neutrality validity

• Verification of the simulation result quality by analyzing the number of superparticles per cell

The first tool allows the validation of the plasma’s quasi-neutrality assumptions by comparing the Debyelength with the average cell size. As explained before, a plasma can be considered quasi-neutral outsidea sphere with a radius equal to the Debye length. It is then important to guarantee that the mesh cellconsidered in the volume of the calculation is not smaller than the Debye length. The ratio

r =λDλCell

where λD is the Debye length and λC is the cell width is calculated and exported as a .vtk. The cell width isthe average width of the lines of the considered tetrahedron cell. In Section 6.1 the validity of quasi-neutrality

33

is studied by using this quantity.

The second implemented tool, also exported as a .vtk file, allows the user to see the number of superparticlesin every volume cell. This is very useful since it can give an idea of whether the results were produced withonly few superparticles (and so subjected to a larger error) or with a lot of them (giving a more reliableresult). For the simulations performed the result of this tool were analyzed and are presented in Chapter 6

New version of the SPIS software

The evolutions described in this Chapter are being integrated in the actual version of the SPIS software andwill be include on the next released version. This version, as usual, will be open-source and will be availableto the public through the SPIS website [25].

34

Chapter 6

Results with new thruster models

As described in Chapter 5, evolutions were performed to the SPIS code and new models were added toproperly simulate a plasma plume of Hall Effect Thrusters and Ion Thrusters. In this Chapter simulationsresults with SPIS are presented and comparison with experimental data and other simulations are performed.The ParaView software was used to analyze the results provided by SPIS, by interpolating values in pointswhich did not belong to the mesh grid. ParaView[26] is an open-source, multi-platform data analysis andvisualization application. It allows an interactive 3D data exploration of data and permits exporting largeamount of data to be used with other data-analysis software.

Simulations were run to validate the newly implemented thrusters’ models and the assumptions made whenconsidering them. As it was already referred (see Section 2.5), the models to be analyzed were suggested byAIT.

6.1 The SPT100 model simulations

In this Section the results for a simulation launched with the SPT100 model are presented. The SPT100 is themost used and studied HET and has been tested in many different situations and experimental conditions.The main parameters used were taken from [27]. Table 6.1 presents these parameters; some of them aredirectly used by the code and others are taken and plugged into equations (5.10) and (5.11) to find theejection speed and mass flow rate of each particle type.

Parameter Value for Simulation

Outer Radius (Ro) 50mmInner Radius (Ri) 28mm

Thrust (T) 84mNMass Flow Rate (mT ) 5.6mg/s

Doubly-Charged Percentage (ηp) 10%Ionization Efficiency (ηu) 95%

Anode Split (ηa) 90%Neutral Temperature 1000K

Inner angle (αi) −12◦

Outer angle (αo) 40◦

Electron Temperature (Te) 4eVIon Beam Temperature (Ti) 3eV

Reference Potential (φref ) 20V

Table 6.1: Input Parameters for the SPT-100 Thruster

35

Apart from these parameters there are other options that need to be considered prior to the launch of asimulation. The background pressure, the Poisson’s vs quasi-neutrality solver, the number of mesh cells andthe volume of the simulation (and consequently the size of each cell) and the duration of each simulation arechosen before the simulation and can have a big influence not only in the simulation run time but also in theresults of the simulation.

For every simulation the equilibrium state of the plasma was checked and guaranteed by comparing theresults of current density, number density and number of superparticles of the final time step with a timestep corresponding to half of the duration. If the profile of these quantities didn’t change more than 5% fromone time step to the other the simulation was considered to have reached the equilibrium state. The finalresults presented here correspond to the averaged values from 0.6× ttot to ttot, where ttot is the total time ofthe simulation.

It is also important to note that even though an anode split is considered in the initial parameters, thecathode was not simulated. The cathode also ejects neutral particles which will collide with the ejected ionsfrom the anode, creating new slow ions. This introduces an asymmetry in the results due to an higher valueof number density, current density and plasma potential on the cathode’s side. This alone is already a causeof error in the simulation results, which, with the simulation performed with the SPIS software, are expectedto have a symmetric profile.

Table 6.2 presents the main values considered for this simulation apart from the ones already presentedin Table 6.1, which are fixed for every SPT100 model.

Parameter Value for Simulation

Simulation duration (ttot) 0.004 sBackground pressure 2.9×10−4 Pa

Numerical Solver Quasi-neutrality with Te = variableVolume Sphere with radius r = 1.2m

Number of tetrahedra cells 97569

Table 6.2: SPT100 simulation parameters

Several simulations were run before the one presented here in order to tune the best time which wouldgive an equilibrium state. The background pressure value was chosen to be equal to one of the values usedby Manzella and Sankovic in [28] so that the comparisons could have the same initial parameters. The useof a spherical volume with a radius of 1.2 meter was chosen in order to be able to compare the simulationresults with the given experimental data, where most of the measurements are presented at 1 meter distancefrom the thruster. The center of the thruster corresponded to the center of the simulation volume.

Fig 6.1 was taken from the post-processing analysis of the simulation with Paraview and show the numberof ejected superparticles in each cell of the volume as seen from a cut perpendicular to the thruster’s exitand centered in the thruster’s center. As can be noticed the limit of the plume for these particles is around40◦ with respect to the thruster’s axis which corresponds to the outer angle of ejection used in the modeland presented in Table 6.1. In this figure zones in red represent 100 or more superparticles per cell while thedark blue represents 0 or 1 superparticles per cell.

Regarding the particles produced by CEX collisions it can be seen in Fig. 6.2 (where the cuts were per-formed as described above for the ejected particles) that for angles close to 120◦ with respect to the thruster’saxis there are around 10 superparticles per cell, which are enough to accept the results in these nodes. Forangles higher than 120◦ the number of superparticles starts to decrease and around 150◦ there is only onesuperparticle per cell. In this region the results should be analyzed carefully and some more simulationsshould be run with more superparticles in order to obtain higher values for these cells. Since most of theexperimental data ranges from angles of −120◦ to 120◦ or even less, this simulation seems to be enough tocompare with them.

36

Figure 6.1: Number of ejected Xe+ (left) and Xe2+ (right) superparticles per cell

Figure 6.2: Number of CEX Xe+ (left) and Xe2+ (right) superparticles per cell

The reason to have CEX particles in the backflow but not ejected particles is justified by both the profileof the electric field and the energy differences between the two types of particles (ejected and CEX). Theelectric field around the thruster can be understood by studying the plasma potential on a cut perpendicularto the thruster as shown in Fig. 6.3. It is directed from higher potential regions (which can be found in theregion in front of the thruster) to lower potential regions (found in the great angle regions). The ions willthen be accelerated towards the big angles. However the very energetic ejected ions (which,for example, havean energy close to 200eV, for the for the case of the Xe+) will be much less affected by the electric field thanthe CEX particles (which have an energy of 0.0086 eV that corresponds to the neutral temperature of thebackground 1000K).

37

Figure 6.3: Plasma potential involving the SPT100 thruster

When studying the charging effects of a plasma plume on a spacecraft (which is the main objective ofthe AISEPS by using the SPIS software) the current density profile is the most important quantity to un-derstand. In Fig. 6.4 the current density profile at one meter from the thruster is considered and plottedagainst the angular distance from the thruster’s axis. It also shows how the measurements were performed:a circle was introduced with a radius of one meter and the points belonging to the circle and located betweentwo nodes of the mesh grid were given an interpolated value of the values on the two nodes. Again thered colour in the figure represents values equal or greater than the value presented in the scale (20Am−2)and blue represents values equal or lower than the one in the scale (0.001Am−2). The simulation resultswere plotted against the experimental data from Manzella and Sankovic [28], which is considered by manyengineers and researchers as the best reference for current density measurements of the HET. Analyzing theresults obtained one can see that the two curves present a good fit for angles greater than 10◦ but that theyare clearly below on axis measurements. On axis the values reach a relative error to the measurement dataof 60.8%, decreasing then to errors around 1% in the region from 10◦ to 30◦. The linear scale presented inthe article is very convenient to compare the largest current density values near the axis but it gets verydifficult to compare the results for larger angles. Also the asymmetric profile of the current density which, asstated before, results from the collisions between the ejected ions and the neutrals ejected from the cathode,makes it quite difficult to understand the differences between the simulation results and the experimental ones.

To compare these smaller values at larger angles the plot presented in the same article by Manzella andSankovic for current density at 60 cm from the thruster was used. The pressure value of P = 2.2×10−6Torr= 2.9×10−4Pa was chosen, which corresponded to the same value used in the simulation. The results arepresented in Fig. 6.5 and the relative error on axis can be found to be 28%. For angles between 30◦ and 60◦

the relative error can reach values close to 100% but decreasing again to 20%-30% for angles higher than 60◦.Again, for the angles between 10◦ and 30◦ the two lines match very well, with errors of 1% for the negativeangles and 25% for the positive. This difference is again created by the neutrals ejection from the cathodeas commented before, which give an assymetry of the results on the experimental data.

38

- 8 0 - 4 0 0 4 0 8 00

4

8

1 2

1 6

2 0

E x p e r i m e n t a l r e s u l t s f r o m M a n z e l l a a n d S a n k o v i c S P I S s i m u l a t i o n w i t h Q u a s i - n e u t r a l i t y a n d T e = v a r

Curre

nt de

nsity

(A/m2 )

A n g l e ( D e g )

Figure 6.4: Dependence of the current density on the angle from the thruster axis, at a distance of 1mfrom the SPT100 thruster: plot of SPIS simulation results and experimental data (left) and image of thepost-processing analysis in Paraview(right)

- 1 2 0 - 9 0 - 6 0 - 3 0 0 3 0 6 0 9 0 1 2 00 . 0 1

0 . 1

1

1 0

Curre

nt de

nsity

(A/m2 )

A n g l e ( D e g ) S P I S s i m u l a t i o n w i t h Q u a s i - n e u t r a l i t y a n d T e = v a r E x p e r i m e n t a l r e s u l t s f r o m M a n z e l l a a n d S a n k o v i c

Figure 6.5: Simulation results and experimental data of the dependence of the current density on the anglefrom thruster axis, at a distance of 60cm from the SPT100 thruster

Since the results did not match the experimental curves as expected, further analysis was performed inorder to understand what could be the source of error. Current density is obtained by the multiplication ofion density and velocity:

J(r,t) = q n(r, t) v(r,t) = ρ(r, t)v(r, t) (6.1)

By analyzing individually each of these quantities one should be able to find why the results do not matchthe experimental data. To analyze the velocity profile one can analogously study the energy of the particles.The energy of each type of particles is analyzed and compared with experimental measurements by Kim etal. [29]. Fig. 6.6 presents the beam energy per charge of ion (Xe+ and Xe2+) at a radial distance of 1m and

39

50cm from the thruster. Only the energy of the ejected ions was measured and presented in [29] and one cannotice that it fits well to the obtained energy values with the SPIS simulation. For the case of the resultsat a distance of 1 meter, a relative error of 15.1% was found between the average value of Xe+ obtainedexperimentally and the one obtained with SPIS as well as a relative error of 7.8% for the Xe2+. Identicallyfor the case of 50 cm, the relative error for Xe+ is was 16.7% and for Xe2+ was 9.5%.

- 1 8 0 - 1 2 0 - 6 0 0 6 0 1 2 0 1 8 0

1

1 0

1 0 0

Ener

gy/q

(eV)

a n g l e ( d e g ) E x p e r i m e n t a l r e s u l t s f o r X e + b y K i m a n d G a l l i m o r c E x p e r i m e n t a l r e s u l t s f o r X e 2 + b y K i m a n d G a l l i m o r c S P I S X e + ( C E X ) S P I S X e + ( E j e c t e d ) S P I S X e 2 + ( E j e c t e d ) S P I S X e 2 + ( C E X )

- 1 8 0 - 1 2 0 - 6 0 0 6 0 1 2 0 1 8 0

1

1 0

1 0 0

E x p e r i m e n t a l r e s u l t s f o r X e + b y K i m a n d G a l l i m o r c E x p e r i m e n t a l r e s u l t s f o r X e 2 + b y K i m a n d G a l l i m o r c S P I S X e + ( C E X ) S P I S X e + ( E j e c t e d ) S P I S X e 2 + ( E j e c t e d ) S P I S X e 2 + ( C E X )

Ener

gy/q

(eV)

a n g l e ( d e g )

Figure 6.6: Simulation results and experimental data of the dependence of the energy per charge of Xe+ andXe2+ ejected from the SPT100 on the angle from the thruster axis at 50cm (left) and 1m(right) from thethruster

As referred before the electric field created by the potential difference in different regions of the plasmaplume will be particularly influent in the movement of CEX ions but not that much in the movement of thehigh energetic ejected ions. From the energy plots this difference can also be perceived. Taking the anglesclose to zero, it can be realized that the energy of ejected particles at 50cm and 1m from the thruster does notchange (or has a negligible change when compared to its energy value) which indicates that these particlesare not very accelerated. On the other hand the energy of the CEX ions does increase from 50cm (wherethe energy of the particles is around 0.43eV ) to 1m (where the energy is around 0.67eV) proving that theelectric field does have an influence on this types of particles.

The other physical quantity that should be analyzed in order to understand the discrepancy of the valuesof current density obtained above is the particle number density. Fig. 6.7 presents the dependence of totalion number density (Xe+ and Xe2+) on the angle from the thruster axis at 50cm and 1m. Since the energyprofile was quite similar to experimental values it would be expected a depression of ion number density onaxis in order to explain the lower values in the same region for current density. This is indeed what can beobserved on the presented plots where the relative error to the experimental data on axis at 50cm is 47.7%and at 1m is 69.3%. Once again the values around the angles 30◦ and 50◦ are the ones where the error isgreater, reaching a maximum of 66.0% at 40◦. Unfortunately the experimental data only goes to angles of60◦ where the relative error of the simulation results is 19.7% and so the bigger angles cannot be comparedin terms of current density.

40

0 3 0 6 0 9 00 . 0

0 . 5

1 . 0

1 . 5

2 . 0Pa

rticle

dens

ity (x

1016

A/m

2 )

A n g l e ( D e g ) S P I S s i m u l a t i o n w i t h Q u a s i - n e u t r a l i t y a n d T e = v a r E x p e r i m e n t a l r e s u l t s f r o m K i n g a t 5 0 c m f r o m t h e t h r u s t e r

0 3 0 6 0 9 00

2

4

6

8

1 0

S P I S s i m u l a t i o n w i t h Q u a s i - n e u t r a l i t y a n d T e = v a r E x p e r i m e n t a l r e s u l t s f r o m K i n g a t 1 0 0 c m f r o m t h e t h r u s t e r

Partic

le de

nsity

(x10

15 A

/m2 )

A n g l e ( D e g )

Figure 6.7: Simulation results and experimental data of the dependence of total particle density on the anglefrom the thruster axis at 50cm (left) and 1m(right) from the SPT100 thruster

These results prove that the biggest source of error comes from the profile of density. The reason to havethis difference on the results was not completely understood and further investigation has to be performedto explain it. One of the possible sources of errors can come from the inner angle of the proposed model.The inner angle has an influence on the potential on the axis region, since it will produce a high potentialregion at around 6cm from the thruster’s exit. This can be seen in Fig. 6.8 where the thruster’s exit regionis shown. A cut was performed in a direction perpendicular to the thruster and the potential is shown forthis region.

Figure 6.8: Plasma potential in the region close to the thruster’s exit

This high potential region may have a direct influence on the distribution of particles above it and maybe repelling some of the particles that would go to this axial region. Even though, as seen before, the electric

41

field created by the potential difference is not sufficient to change the trajectory of the fast ions, this canaffect the slow ions since they are readily deviated when created near the exit region of the thruster. Oneof the tests to perform may then be the study of the influence of the inner angle on the axial distribution ofions.

Effect of pressure on the current density profile

When the background pressure increases and the temperature and volume is kept constant, the densityof background neutrals increases. Considering (4.11) one can see that this will increase the probability ofions colliding with the background neutrals and, consequently, also increase the production of more CEXions with a thermal velocity. This production makes one expected an increase of ion density on for greatangles, since the particles will be deviated by the electric field, as commented before. Fig. 6.9 shows thecomparison of two current density profiles which correspond to two different background pressures. For bothsimulations the parameters used were the same as the ones presented in Tables 6.1 and 6.2 except for thevalue of the background pressure of the second simulation, which was increased from P = 2.9 × 10−4Pa toP = 8.4× 10−3Pa.

- 1 8 0 - 1 2 0 - 6 0 0 6 0 1 2 0 1 8 0

0 . 1

1

1 0

Curre

nt de

nsity

(A/m2 )

A n g l e ( D e g ) S P I S s i m u l a t i o n w i t h b a c k g r o u n d p r e s s u r e = 8 . 4 e - 3 P a S P I S s i m u l a t i o n w i t h b a c k g r o u n d p r e s s u r e = 2 . 9 E - 4 P a

Figure 6.9: Dependence of the current density on the angle from the thruster axis at 60cm from the thrusterfor two different values of background pressure

Analyzing the figure it can be seen that, as expected, the value of current density increases with theincrease of pressure and that the greatest increase occurs for big angles. However, when comparing to theexperimental data from Manzella and Sankovic [28], the ion density increase is quite big. The justificationfor this fact was already discussed when the CEX collision model was described in Section 4.3: when thebackground density is too big, the amount of collisions will increase so much that the number of createdparticles is not neglected when comparing with the number of ejected ones. Since in the SPIS software theinitial population is unchanged and slow particles are created (instead of changing the properties of the initialparticles), the total number of particles will increase more than it would in reality, giving bigger values ofcurrent density as can be noticed in the Fig. 6.9.

Validation of the quasi-neutrality tool

In Section 5.7 the implementation of the quasi-neutrality assumption in the SPIS software was discussed. Itwas also referred in Section 5.8 that an option to analyze the quality of this assumption was implemented.Fig. 6.10 shows the ratio of Debye length over cell size for each cell in the simulation volume. To use thequasi-neutrality assumptions one should guarantee that the cell size is not smaller than the Debye length,i.e. that the ratio is smaller than one. From the figure, it can be seen that the quasi-neutrality assumption

42

Figure 6.10: Ratio of Debye length over cell size in the simulation volume

may be applied in the most part of the simulation volume. Only a small region behind the thrust has a cellsize close to the local Debye length (even if it is still twice smaller) and so, for a more accurate simulationthe Poisson solver should be used.

In Fig. 6.11 the results of a simulation using the quasi-neutrality and the Poisson solver are presented.Analyzing the figure it can be noticed a good agreement of both quasi-neutrality and Poisson curves. The

- 1 8 0 - 1 2 0 - 6 0 0 6 0 1 2 0 1 8 00 . 0 1

0 . 1

1

1 0

S P I S s i m u l a t i o n w i t h P o i s s o n s o l v e r S P I S s i m u l a t i o n w i t h Q u a s i - n e u t r a l i t y a n d T e = v a r

Curre

nt de

nsity

(A/m2 )

A n g l e ( D e g )

Figure 6.11: Dependence of current density on the angle from the thruster axis at 100cm from the thruster.Comparison of simulations using the quasi-neutral assumption and the Poisson solver

biggest difference between the two curves start for angles around 90◦ which, by analyzing the Fig. 6.10,correspond to the regions where the ration λD/CellSize is greater. The assumption of quasi-neutrality forthese types of simulations can then be considered as a good option to take.

43

6.2 The T5 model simulations

The T5 model is a type of Ion thruster that was developed by QinetiQ and that was launched in 2009 onboardthe GOCE satellite. This thruster was simulated using the new implemented ion thruster model in the SPISsoftware. The AIT group is still working on the best parameters for this model, thus the simulations launchedand the parameters here presented will not be the final ones. Table 6.3 presents the main parameters usedin SPIS for the simulation of the T5 thruster:

Parameter Value for Simulation

Outer Radius (Ro) 50mmThrust (T) 18mN

Mass Flow Rate (mT ) 0.677mg/sDoubly-Charged Percentage (ηp) 5%

Ionization Efficiency (ηu) 77.6%Anode Split (ηa) 14.8%

Neutral Temperature 500KOuter angle (αo) 12◦

Electron Temperature (Te) 3eVIon Beam Temperature (Ti) 0.008eV

Reference Potential (φref ) 16VGaussian FWHM 0.35

Table 6.3: Input Parameters for the T5 Thruster

Together with these parameters, Table 6.4 presents the options used for the simulations presented on thisSection.

Parameter Value for Simulation

Simulation duration (ttot) 0.006 sBackground pressure 3× 10−4 Pa

Numerical Solver Quasi-neutrality with Te = variableVolume Sphere with radius r = 0.7m

Number of tetrahedra cells 55873

Table 6.4: T5 simulation parameters

Once again the equilibrium of the simulation was assured by comparing the results of the last time stepwith the results corresponding to half the duration of the simulation. Also the results presented in this partcorrespond to the averaged values from 0.6× ttot to ttot, where ttot is the total time of the simulation.

The background pressure value was selected in agreement with the tests performed by de Boer [30] withwhom the results will be compared.

Fig 6.12 presents the ejected ion density distribution, on a cut performed on the simulation volume, per-pendicular to the thruster’s exit and passing in its center. The white lines presented are the regions where themeasurements of ion density were performed by de Boer. The CEX ions were not analyzed in this simulationand so it can already be noticed that for the biggest distances from the thruster’s axis (d = ±0.15cm) thevalue of ion density for the three closest lines will be zero.

44

Figure 6.12: Simulation results of the ion density profile of the T5 thruster

Fig. 6.13 presents the profile of ion density as a function of the horizontal and axial distances to thethruster for both the simulated results and the experimental values (as presented by de Boer). The articlepresented the UK-10 ion thruster results which was the precedent thruster of the T5 and has similar properties.

- 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5

0

5

1 0

1 5

2 0

2 5

3 0 A x i a l D i s t a n c e f r o m t h e t h r u s t e r 7 . 5 c m 1 0 c m 1 5 c m 2 5 c m 3 5 c m 4 8 c m 6 0 c m

Partic

le de

nsity

(x10

15 m

-3)

h o r i z o n t a l d i s t a n c e t o a x i s ( m )- 0 . 1 5 - 0 . 1 0 - 0 . 0 5 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5

- 0 . 20 . 00 . 20 . 40 . 60 . 81 . 01 . 21 . 41 . 61 . 82 . 0 A x i a l D i s t a n c e f r o m t h e t h r u s t e r

7 . 5 c m 1 0 c m 1 5 c m 2 5 c m 3 5 c m 4 8 c m 6 0 c m

Partic

le de

nsity

(x10

15 m

-3)

h o r i z o n t a l d i s t a n c e t o a x i s ( m )

Figure 6.13: Simulation results (left) and experimental data (right) of the ion density profile as a function ofhorizontal distance to the thruster’s axis and axial distance to the T5 thruster

Analyzing the figure it can be seen that, even though the profile of the curves seems correct, the valuesobtained for the ion density profile are one order of magnitude higher than the experimental data. To explainthese facts two reasons may be suggested. Either the model was not well implemented, for instance theconstant used for the Gaussian distribution (see (5.18)) was incorrectly calculated, or the parameters usedin the model still do not correspond to the real thruster characteristics.

Also from the figure, it can be noticed that for the experimental data, the current density values atdistances of 10cm and 15cm to the thruster’s surface of ejection are higher than at 7.5cm. Regarding the

45

simulation results, the values of current density always decrease with the distance and the value at 7.5cm isthe highest of all the values presented. According to de Boer’s article this is explained by the inward curva-ture of the thruster’s accelerator grid. The T5 model was implemented in SPIS with a cylindrical geometryand the particles are ejected from the surface of the cylinder. This difference in the geometry explains whythe current density profile of the first 3 curves is different from the experimental results and the rest of thepresented values are similar.

The results with this model were far from what it was expected, hence it is predictable that the valuesof current density are also incorrect. Experimental results of the current density and energy profiles aroundthe T5 thruster could not be found and so no comparisons were performed for these quantities. Furtherinvestigation of experimental studies performed for the T5 thruster needs to be done in order to obtainexperimental data to compare with the simulations.

6.3 Conclusions

Several results were presented for both types of thrusters modeled with the SPIS software (HET and ionthruster). The results for the SPT100 thruster are already very satisfactory, giving a profile of ion den-sity, energy and current density close to the experimental measurements. This model still requires furtherinvestigation (mainly to understand the differences for the values close to the thruster’s axis) but it is pre-dictable that the differences of the simulated and experimental values can be overcome by the adjustment ofthe thruster’s parameters. The AIT team has already released the parameters for several other HET modelswhich will be implemented and validated with the SPIS software, for instance the PPS1350 and the PPS5000.These thrusters’ models will be implemented after the improvement of the SPT100.

Regarding the ion thruster model, and the T5 model in particular, the results are still far from theexperimental data suggesting that a more profound analysis has to be performed in order to obtain the correctvalues. However less time was invested in this model, hence it may also turn out to be an easy adjustmentof the model that will give the correct values. Regarding the ion thruster models to be implemented, T6 isanother thruster to be modelled on the AISEPS thrusters list.

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Chapter 7

Conclusions and future work

The work presented in this thesis was mainly focused on the implementation of new thruster models in SPISsoftware and the development of new tools that allow these models to be analysed. These new simulationtools were sometimes developed in cooperation with the SPIS developing team. Regarding the simulationprocess the main evolutions consisted on:

• Neutrals ejection from the same surface where ions were ejected,

• CEX interactions of two different types of main particles with background (and ejected) particles

• Quasineutrality simulations with variable and constant electron temperature to avoid long simulationswith conventional Poisson’s solver.

• Post-processing evaluation of the quasi-neutrality assumption by comparing the Debye length and thecell size

• Post-processing evaluation of the quality of the results by analyzing the number of simulated particlein a mesh cell.

Two types of thrusters were modelled: the Hall effect thruster and the ion thruster and a detaileddescription of their models and implementations was given in Chapter 5.

Furthermore simulations were launched with the implemented models and the results were comparedwith experimental data. To test the Hall effect thruster the SPT100 thruster was implemented. The resultsshowed a good agreement with the experimental data of the plasma energy profile. The ion density profilepresents lower values than experimental data in the thruster’s axis region but fits well for almost all the otherangles. Consequently the current density profile has good agreement with the experimental data for most ofthe angles, also presenting a lower value near thruster’s axis. This could mainly be due to the value of theinner angle of ejection considered in the simulation and further studies of this quantity may give the goodmatching results in the regions where the value is still low.

It was shown that the background pressure (which leads to an increase of background density) had aninfluence on the current density profile due to the increase of collisions between the ejected ions and the neu-trals. The profile was found to increase more than what was expected. This is explained by the Monte-Carlomethod used, where the initial population is unchanged after the collision and a new slow ion is created. Inorder to solve this issue for high pressures, the implementation of a variation of the Monte-Carlo collisionsmethod could be used where the fast ion involved would be eliminated. To perform an even more accuratesimulation the Direct Simulation Monte Carlo method could be used where all the particles would be fol-lowed and changed. This, however, is much more computationally expensive and must be consider beforeperforming the evolutions.

The SPT100 results also allowed to observe the fact that the thruster’s cathode has a considerable influ-ence in the current density profile creating an asymmetry that can be important when studying the position

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of eventual spacecraft components. The implementation of the cathode where the correspondent amount ofneutrals would be ejected is also one of the improvements that will be considered for the future work.

Regarding the T5 thruster, the results obtained clearly indicate that its implementation is not yet perfect.Further analysis should be performed on this model to understand why the results have such high values whencompared with the experimental data. Two suggestions are left for future work. The first suggestion is toverify the implementation of the model where a constant factor of the Gaussian distribution may incorrectlylead to very high values of ejected particles (and consequently high values of density). The study of the usedparameters may also be an important point in order to better understand the results. Another suggestion isto implement the curved geometry of T5 as described in Chapter 7 since it has a direct impact on the resultsas it was understood from the figures presented.

Project continuation

The continuation of the AISEPS project at Astrium will be carried out by another internship which will lastuntil the end of the project. The models developed for this thesis will be tuned in order to obtain simulationresults closer to the experimental data and new thrusters parameters will be introduced to simulate differentmodels (PPS1350, PPS5000 and T6 for example). Testing with a real project (satellite and thruster system)shall be performed and the simulation charging results of the spacecraft should be compared to in-flight data.

Regarding the prospects after the end of the AISEPS project, it shall be possible to use SPIS as a plume-spacecraft interaction simulator and perform engineering analysis for future space missions. SPIS softwareis in constant development and in the last 6 months it has had one of the highest rate of evolutions, withmany new options implemented and fully functional for the whole community. The continuation of the SPISproject may converge to the creation of a complete tool for the study of spacecraft-plasma interactions. Thismay then provide a better understanding physics of electric propulsion and avoid many of the interactionissues known today.

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