Maria Choi and John T. Yim

Glenn Research Center, Cleveland, Ohio

Development Status of a Three-Dimensional 

Electron Fluid Model for Hall Thruster 

Plume Simulations

NASA/TM—2018-219962

October 2018

AIAA–2018–4808

https://ntrs.nasa.gov/search.jsp?R=20180007354 2018-12-10T08:11:00+00:00Z
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated 

to the advancement of aeronautics and space science. 

The NASA Scientific and Technical Information (STI) 

Program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Officer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI Program provides access 

to the NASA Technical Report Server—Registered 

(NTRS Reg) and NASA Technical Report Server—

Public (NTRS) thus providing one of the largest 

collections of aeronautical and space science STI in 

the world. Results are published in both non-NASA 

channels and by NASA in the NASA STI Report 

Series, which includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of significant 

scientific and technical data and information 

deemed to be of continuing reference value. 

NASA counter-part of peer-reviewed formal 

professional papers, but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or of 

specialized interest, e.g., “quick-release” reports, 

working papers, and bibliographies that contain 

minimal annotation. Does not contain extensive 

analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientific and technical 

conferences, symposia, seminars, or other 

meetings sponsored or co-sponsored by NASA.

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific and 

technical material pertinent to NASA’s mission.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question to help@sti.nasa.gov

• Fax your question to the NASA STI 

Information Desk at 757-864-6500

• Telephone the NASA STI Information Desk at

757-864-9658

• Write to:

NASA STI Program

Mail Stop 148

NASA Langley Research Center

Hampton, VA 23681-2199
Maria Choi and John T. Yim

Glenn Research Center, Cleveland, Ohio

Development Status of a Three-Dimensional 

Electron Fluid Model for Hall Thruster 

Plume Simulations

NASA/TM—2018-219962

October 2018

AIAA–2018–4808

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

Propulsion and Energy Forum

sponsored by the American Institute of Aeronautics and Astronautics

Cincinnati, Ohio, July 9–11, 2018
Acknowledgments

The authors would like to like to thank the Space Technology Mission Directorate through the Solar Electric Propulsion 

Technology Demonstration Mission Project for funding this project for the Advanced Electric Propulsion System.

Available from

Trade names and trademarks are used in this report for identification 

only. Their usage does not constitute an official endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 

NASA STI Program

Mail Stop 148

NASA Langley Research Center

Hampton, VA 23681-2199

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22161

703-605-6000

This report is available in electronic form at http://www.sti.nasa.gov/ and http://ntrs.nasa.gov/
NASA/TM—2018-219962 1

Development Status of a Three-Dimensional Electron Fluid 

Model for Hall Thruster Plume Simulations

Maria Choi and John T. Yim

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Summary

A three-dimensional (3-D) electron fluid model has been developed as a stepping stone to fully 

describe the electron current flow across magnetic fields inside a vacuum chamber and to provide electron 

flux to solar arrays for a spacecraft surface charging model. The physical and numerical models of the 

electric potential solver, including finite-volume formulation and the treatment of boundary conditions, 

are presented in this paper. Verification tests of the model are presented.

1.0 Introduction

While electric propulsion (EP) plume models need to accurately predict on-orbit operations of EP 

thrusters, the models are most often validated through comparisons with experimental data measured in 

ground-based vacuum facilities. Predicting the behavior of a flight system requires a deep understanding 

of both component- and system-level interactions, including thruster, cathode, and facility effects, such as

background pressure, electrical, and carbon backsputter effects. In order to better understand the testing 

environment of ground vacuum chambers, which is absent in space, it is critical to understand both the 

motion of heavy particles in the plume and the movement of bulk electron flows. While the background 

pressure effects have been extensively studied in the past (Refs. 1 to 6), there is still limited understanding 

of how electrons in the plume of EP thrusters travel through and interact with the metallic conductive 

walls of vacuum chambers. Recent studies have suggested that the presence of conducting walls provides 

alternate pathways for electrons to travel from the cathode and serve as a recombination site (Refs. 7 and 

8). This electrical facility effect can become more significant for higher power thrusters operating in 

vacuum facilities. Since this effect is absent in orbit and simulations are most often validated with the data 

in these facilities, the transport of electrons in a vacuum chamber needs to be better understood and 

simulated with more detailed physics.

A conventional way to simulate electrons in the plume of EP thrusters is to assume the Boltzmann 

relation. This method is simple and useful for isothermal, collisionless, and unmagnetized regions, but has 

major limitations for Hall thruster plume simulations. First, the electron temperature gradient in the plume is 

not negligible as it can vary by more than an order of magnitude from the near field to the far field. Second, 

the magnetic field strength is still strong in the near-field plume as the field leaks into the plume. In order to 

improve the fidelity of electron model, early work by Boyd and Yim (Ref. 9) demonstrated the viability of 

using the generalized Ohm’s law and steady-state current conservation equation approach to solve for 

electron current flows for a Hall thruster plume, assuming negligible magnetic fields. This work was further 

demonstrated on a cluster of Hall thruster plumes by Cai and Boyd (Ref. 10). In both cases, the authors 

demonstrated the difference in plume structures between their approach and the traditional Boltzmann 

relation approximation used in plume codes, which enforces exactly zero electron current everywhere in the 

plume such as the Coliseum tool described by Brieda and VanGilder (Ref. 11). The work in References 9

and 10 was further improved by Choi and Boyd (Ref. 12) by including a full-electron mobility tensor to 
NASA/TM—2018-219962 2

account for anisotropies in electron transport in a vacuum chamber due to magnetic field effects. This study

used a two-dimensional (2-D) axisymmetric unstructured mesh and showed that the magnetic field effects 

are non-negligible in the near-field plumes of the anode and cathode.

The present work utilizes a similar formulation by Choi and Boyd (Ref. 12) in a three-dimensional 

(3-D) structured cell-centered finite volume method. In this paper, the full electron mobility tensor, μ , 

from Reference 12, and a full representation of the electron current density, je, in 3-D Cartesian grids is 

developed and presented to better understand the electron transport and current flow in a vacuum 

chamber. Using the generalized Ohm’s law and steady-state current conservation equation, the electric 

potential is derived. This new electron model has the capability to model electron transport across a 

complex magnetic field topology in the plume, which includes a magnetic field separatrix and a purely 

axial component along the cathode centerline axis and a purely radial component near the discharge 

channel exit, as shown in Figure 1. A detailed description of the physical and numerical models of the 

electric potential solver, including finite-volume formulation and treatment of boundary conditions, is

provided in the paper. Before simulating a full thruster, the model is verified by using the method of 

manufactured solutions and a mock Hall thruster test case. The accuracy of the new model and effect of 

the magnetic field in the near-field plume will be discussed.

This model is currently being implemented into a hybrid particle-in-cell (PIC) and fluid framework, 

known as the Thermophysics Universal Research Framework (TURF), developed by the Air Force 

Research Laboratory (AFRL) at Edwards Air Force Base (Ref. 13). The present model in TURF 

framework will ultimately be used to pursue integration modeling of both the ground facility and flight 

operations. More specifically, the model should be able to (1) map out the electrostatic potential and the 

electron current flows throughout the chamber volume of a Hall thruster firing in a vacuum chamber and 

(2) provide the electron current flux to surfaces, which is required for their surface charging model (Ref. 

13) that was recently implemented.

Figure 1.—Complex magnetic field lines in a Hall thruster 

(Ref. 12), where Dth is thruster diameter, R is radial 

displacement, and Z is axial displacement.
NASA/TM—2018-219962 3

2.0 Physical and Numerical Models

For a fluid description of the electron plasma, the generalized Ohm’s law and steady-state current 

conservation equation are derived as

( ) 1 e ee e e

e

P

en

  =μ × +σ + ∇    

j jB E (1)

∇⋅ + = (j j e i) 0 (2)

where je and ji are the electron and ion current densities, respectively, B is the magnetic field, E is the 

electric field, σe is the electron conductivity, ne is the electron number density, Pe is the electron pressure 

assuming ideal gas, and μe is the electron mobility defined as

e

e ce

q

m

μ =

ν

(3)

with q being the elementary charge, me the electron mass, and νce the total collision frequency of the 

electron fluid.

Introducing plasma potential φ in relation to the electric field, –∇φ = E, Equation (1) can be rewritten as

1 e e e

e

P

en

  μ = σ −∇φ + ∇    

j (4)

where μ is the electron mobility tensor and Bx, By, and Bz are the value of the magnetic field in the x-, y-, 

and z-directions,

1

1

1

ex ez

e x e y

ez ey

B B

B B

B B

  −μ μ  

μ = μ −μ

−μ μ  

(5)

Combining Equation (4) with Equation (3) obtains the following equation to solve for the plasma 

potential:

( ) 1 1 e e e i

e

P

en

− −   σ ∇ ⋅ μ σ ∇φ = ∇ ⋅ μ ∇ + ∇ ⋅    

j (6)

where −1 μ is the inverse of the electron mobility tensor,

22 2 2

1 2 22 2

2 2 2 2 2 2

1

1 1

1 1

ex exy ez exz ey

exy ez ey eyz ex

e exz ey eyz ex ey

B BB B BB B

BB B B BB B

B BB B BB B B

−

  +μ μ +μ μ +μ  

μ = μ −μ +μ μ +μ

+ μ μ +μ μ −μ +μ  

(7)

A cell-centered finite-volume method is used to discretize the governing equation in Equation (6)

inside a control volume, V,
NASA/TM—2018-219962 4

( ) ( ) 1 1 e e e i VV V e

dV P dV dV

en

− −     σ   ∇ ⋅ μ σ ∇φ = ∇ ⋅ μ ∇ + ∇ ⋅           ∫∫ ∫ j (8)

The volume integrals are transformed as surface integrals using Green’s theorem. Then, the surface 

integral is approximated as the sum of all fluxes along all faces. In 3-D, this operation for the left-hand 

term of Equation (8) will be

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Allfaces

1

11 1 1

1 1

ˆ

ˆˆ ˆˆ

ˆ ˆ

q q

p

enw s en ws

b f b f

ndS

ndS ndS ndS ndS

ndS ndS

−

−− − −

− −

σμ ∇φ ⋅

= σμ ∇φ ⋅ + σμ ∇φ ⋅ + σμ ∇φ ⋅ + σμ ∇φ ⋅

+ σμ ∇φ ⋅ + σμ ∇φ ⋅

∑

(9)

where subscripts e, w, n, s, f, and b represent the east, west, north, south, front, and back faces, 

repectively. The final discretized form of this equation turns out to be identical to a cell-centered finitedifferencing discretization and has 27 stencils in 3-D (Figure 2(a)). After discretization, the equation 

becomes a linear system of equations:

ab c d e h i j F φ + φ + φ + φ + φ + + φ +φ + φ + = ijk i jk i jk j ik j ik + − + − 1, 1, 1, 1,   i j i j i j ++ +− −+ 1 1 1 1 1 1 ijk (10)

where Fijk is the right-hand side (RHS) term at location ijk and all coefficients in front of φ’s (i.e., a, b, 

c...) represent flux terms (i.e., −1 σμ ) at faces such as e, w, n, s, f, and b as indicated in the red boxes in 

Figure 2(b).

The electron pressure is assumed to be isotropic and follows the ideal gas law. The ion current density 

information in each control volume is acquired through the PIC method, assuming quasi-neutrality. The 

plasma potential is determined by inverting the matrix. After the plasma potential is calculated, the 

electron current density in Equation (4) can be solved:

1 1 ee e

e

P

en

−   = μ σ −∇φ + ∇    

j (11)

where the central differencing is used to calculate the derivatives.

Figure 2.—Electron fluid model with 27-point stencils. (a) 3-D representation. (b) Cell centers and faces.
NASA/TM—2018-219962 5

3.0 Boundary Conditions

In a Hall thruster plume simulation, two types of conditions are typically used: (1) direct value (e.g., 

all metallic and ground surfaces including the chamber wall) and (2) gradient or flux to the surface (e.g., 

zero-gradient potential such as a dielectric thruster body), which are the Dirichlet and Neumann boundary 

conditions, respectively. Although various options of treating these boundary conditions for this model 

are discussed in this section, not all of these boundary conditions were presented.

3.1 Dirichlet Boundary Condition

There are multiple ways to implement the Dirichlet boundary condition for the plasma potential. The 

simplest way is to have the surface on the cell center using a ghost cell (e.g., φ0), as shown in Figure 3(a).

Then, the known potential value can be incorporated as a boundary condition by moving the boundary 

term to the RHS of the equation for the interior cells. This method requires modifying all interior cells 

that have a boundary cell adjacent to them and will result in a matrix with size (Nx × Ny × Nz) by (Nx × Ny

× Nz), where Nx, Ny, and Nz are the number of cells in x-, y-, and z-directions. Instead of this incorporation, 

the boundary terms can be added in the matrix as additional rows, which preserves the matrix shape of 

interior cells but increases the matrix to [(Nx + 2) × (Ny + 2) × (Nx + 2)] by [(Nx + 2) × (Ny + 2) × (Nx + 2)].

When the Dirichlet condition is imposed on a cell face instead of a cell center, as shown in Figure 3(b),

either the finite-volume approach or finite-difference approach can be taken. For the finite-volume 

approach, forward differencing can be at the boundary surface (e.g., 1

2 x = in Figure 3(b)) used as

1 1

2

1

2 2

x x

φ −φ

∂φ = ∂   ∆

   

(12)

For the finite-difference approach, the ghost cell, φ0, is used and its value is replaced by a known 

potential value at the wall (i.e., 1

2

φ ), which is approximated using Taylor’s expansion. In this study, both 

first- and second-order extrapolations are tested and defined as

0 11

2

φ = φ −φ 2 (13)

and

0 11 2

2

8 1 2

3 3 φ = φ − φ+ σ (14)

respectively. When approximating the flux at the boundary cell face, ( ) 1

2

σμ−1 , a second-order 

extrapolation is used:

( ) ( ) ( ) 1 1

1 1 2 1

2

3

2

− −

− σμ − σμ

σμ = (15)
NASA/TM—2018-219962 6

Figure 3.—Two types of Dirichlet conditions. (a) A solid surface lying on cell center. (b) A solid surface lying on 

cell face.

Figure 4.—Neumann boundary 

condition using ghost cells.

With a corner cell, as shown in Figure 4, the corner ghost cell, φ00, has to be approximated. The 

simplest approach would be to use distance-weighted averaging. The two- and three-point averaging are 

as shown here:

10 01

00

y x

x y

φ ∆ +φ ∆ φ = ∆ +∆ (16)

10 01 11

00

y r xy x

x yr

φ ∆ +φ ∆ +φ ∆ ∆ φ = ∆ +∆ + (17)

where 2 2 r xy = ∆ +∆ . When ∆x = ∆y, the two-point distance averaging becomes φ = φ +φ 00 10 01 0.5( ), 

and substituting the first- and second-order extrapolations gives the following:

00 1 1 11 1 1

2 2

φ =φ +φ −φ (18)

and

00 1 1 11 12 21 ( ) 1 1

2 2

1 8 1 4

2 3 3

   

φ = φ +φ − φ + φ +φ          

(19)

3.2 Neumann Boundary Condition

As illustrated in Figure 5, the Neumann boundary condition is where the normal component of the 

potential gradient, or the electric field, is known at a cell face. In this 1-D case (Figure 5(a)), the ghost 

cell, φ0, is replaced by the known gradient at the cell face as φ0 = φ1 + gw∆x, which is then used to modify 

the matrix. For a zero-gradient condition, a mirror condition can be used by setting φ0 = φ1.
NASA/TM—2018-219962 7

Figure 5.—Neumann boundary conditions. (a) One-dimensional. (b) Two-dimensional.

For a 3-D case, the treatment of this condition is slightly more involved due to mixed derivative 

terms. Figure 5(b) illustrates this condition in 2-D with a solid surface boundary on the left side of the 

domain, where φi – 1, j+ 1 and φi – 1, j+ 1 in addition to φij having to be replaced in terms of the interior cells. 

The surface integral of the west-side face with a zero-gradient condition (i.e., gw 0

x

∂φ = ≠

∂ ) for this case 

in 3-D yields

( ) 11 12 13

,1 ,1 ,1 ,1

11 12 13

( ) ( )

2 2

w w w

w

ij ij ik ik w

A A

xyz

g y z

y z

+ − + −

  ∂φ ∂φ ∂φ σμ∇φ ⋅ = σμ + σμ + σμ     ∂∂∂

  φ −φ φ −φ = σμ + σμ   + σμ ∆ ∆

  ∆ ∆

(20)

where gw∆x terms were cancelled out, and A is the area. Integrals of all other remaining faces that include

φi – 1,j +1 and φi – 1,j – 1 should also account for the gradient terms appropriately.

4.0 Verification Tests

Before implementing this model into a Hall thruster plume code, the model and algorithm have to be 

tested and verified. In this section, the method of manufactured solutions and a Hall thruster-like test case 

are used to verify the model and algorithm.

4.1 Method of Manufactured Solutions

In order to verify the numerical model, the method of manufactured solutions is used. In this study, 

the new model is tested using various types of functions as manufactured (exact) solutions, including 

polynomial, sinusoidal, and exponential solutions, or some combination of these. The manufactured 

solution is substituted into the governing equation being solved numerically, and the source term in the 

RHS is acquired that satisfies this solution. As an example, one of the manufactured solutions tried is 

shown here:

( ) 222

exact 6

K φ = ++ xyz (21)

where K = 3, and the RHS is
NASA/TM—2018-219962 8

( 11 22 33 ) 3 e

K F = σ μ +μ +μ (22)

The numerical and exact solutions are compared in Figure 6. The numerical solution calculated using the 

new model qualitatively reproduces the exact solution very well.

For more quantitative study, a grid convergence study can be performed using the L2-norm error:

( ) 2

2

exact L

1

k

k

N

h k

k

e d

= Ω

= φ −φ Ω ∑ ∫ (23)

where φh is the numerical value of the plasma potential and φexact is the true solution. The convergence 

study is performed as shown in Figure 7, which confirms the second-order accuracy, as indicated by the 

slope of the linear plot (≈–2).

Figure 6.—Qualitative comparison of solutions. (a) Numerical solution. 

(b) Exact (manufactured) solution φ = 0.5(x2 + y2 + z2).

Figure 7.—Grid convergence study of the current model.
NASA/TM—2018-219962 9

4.2 A Hall Thruster-Like Test Case

Before implementing this model into a Hall thruster plume simulation code, a Hall thruster-like test

case was constructed by Dragnea, Hara, and Boyd using a similar 2-D axisymmetric hybrid model

(Ref. 14). Figure 8 shows the strength of the magnetic fields applied throughout the computational 

domain. The field is radial only. Fixed plasma potentials of 300 and 0 V at the left (anode-like) and the 

right (cathode-like) boundaries, respectively, and zero-gradient conditions were applied at the top and the 

bottom boundaries (Figure 9(a)). Constant electron temperature (Te = 25 eV) and electron number density 

(ne = 1×1017 m–3

) were used. Since the current model is in 3-D, infinitely long boundaries were assumed 

in the z-direction. Figure 9 shows the plasma potential calculated using (a) 2-D axisymmetric model 

solution that serves as a true solution and (b) 3-D finite-volume model developed in this study.

Using the plasma potential fields calculated, the electron current densities in Equation (4) can be 

determined. The finite-difference scheme is used to perform this operation, and the result is shown in 

Figure 10.

Figure 8.—Hall thruster-like test setup showing magnetic field 

strength.

Figure 9.—Calculated plasma potential. (a) Using two-dimensional axisymmetric finite element model from Ref. 12. 

(b) Using three-dimensional rectangular finite volume model in this study with constant values in z-direction.
NASA/TM—2018-219962 10

Figure 10.—Electron current density streamline.

5.0 Conclusions

A three-dimensional (3-D) electron fluid model has been developed as a stepping stone to ultimately 

better understand the electrical facility effects in a conducting vacuum chamber and to provide electron 

flux to solar arrays for spacecraft surface charging model in Thermophysics Universal Research 

Framework. This unique formulation can include a representative magnetic field throughout an entire 

vacuum chamber domain for future Hall thruster plume simulations. This paper described the derivation 

and verification of the plasma potential solver. While not all of them were used, various types of 

boundary condition treatments were derived and presented. Ongoing activities include studying the 

effects of various boundary condition types and conducting further verification tests with various 

magnetic field strengths and topologies.

In the near future, this model, combined with a steady-state and/or transient electron energy equation,

will be implemented in the research framework and used to simulate a Hall thruster plume in both a full 

vacuum chamber at Glenn and on-orbit with spacecraft geometry.
NASA/TM—2018-219962 11

Appendix—Nomenclature

1-D one-dimensional

2-D two-dimensional

3-D three-dimensional

AFRL Air Force Research Laboratory

EP electric propulsion

PIC particle-in-cell

RHS right-hand side

TURF Thermophysics Universal Research Framework

Symbols

A area

B magnetic field

Bx value of magnetic field in x-direction

By value of magnetic field in y-direction

Bz value of magnetic field in z-direction

Dth thruster diameter

e electron charge

E electric fields

F right-hand side term

gw flux

je electron current density

ji electron ion density

K constant coefficient

me electron mass

n̂ normal vector

ne electron number density

Nx number of cells in the x-direction

Ny number of cells in the y-direction

Nz number of cells in the z-direction

Pe electron pressure

q elementary charge

R radial displacement

S surface

Te electron temperature

V cell volume

Z axial displacement

φ plasma potential

φ0, φ00 ghost cells

φexact true solution

φh numerical solution

φw wall potential

μe electron mobility

μ full electron mobility tensor
NASA/TM—2018-219962 12

−1 μ inverse of electron mobility tensor

νce total collision frequency of electron fluid

σe electron conductivity

Subscripts

b back face

e east face

f front face

i index in x-direction

j index in y-direction

k index in z-direction

n north face

s south face

w west face
NASA/TM—2018-219962 13

References

1. Walker, Mitchell L.R.; Hofer, Richard R.; and Gallimore, Alec D.: The Effects of Nude Faraday 

Probe Design and Vacuum Facility Backpressure on the Measured Ion Current Density Profile of Hall 

Thruster Plumes. AIAA 2002–4253, 2002.

2. Walker, Mitchell L.R, et al.: Effect of Backpressure on Ion Current Density Measurements in Hall 

Thruster Plumes. J. Propul. Power, vol. 21, no. 3, 2005, pp. 408–415.

3. Walker, Mitchell Louis Ronald: Effects of Facility Backpressure on the Performance and Plume of a 

Hall Thruster. Ph.D. Dissertation, Univ. of Michigan, 2005.

4. Reid, Bryan Michael: The Influence of Neutral Flow Rate in the Operation of Hall Thrusters. Ph.D. 

Dissertation, Univ. of Michigan, 2009.

5. Kamhawi, Hani, et al.: Investigation of the Effects of Facility Background Pressure on the 

Performance and Voltage-Current Characteristics of the High Voltage Hall Accelerator. AIAA 2014–

3707, 2014.

6. Huang, Wensheng, et al.: Effect of Background Pressure on the Plasma Oscillation Characteristics of 

the HiVHAc Hall Thruster. AIAA 2014–3708, 2014.

7. Frieman, Jason D., et al.: Electrical Facility Effects on Hall Thruster Cathode Coupling: Performance 

and Plume Properties. J. Propul. Power, vol. 32, no. 1, 2016, pp. 251–264.

8. Frieman, Jason D., et al.: Role of a Conducting Vacuum Chamber in the Hall Effect Thruster 

Electrical Circuit. J. Propul. Power, vol. 30, no. 6, 2014, pp. 1471–1479.

9. Boyd, Iain D.; and Yim, John T.: Hall Thruster Plume Simulation Using a Detailed Hybrid Model. 

AIAA 2004–3952, 2004.

10. Cai, Chunpei; and Boyd, Iain D.: 3D Simulation of Plume Flows From a Cluster of Plasma Thrusters. 

AIAA 2005–4462, 2005.

11. Brieda, Lubos; and VanGilder, Douglas: Multi-Domain Plasma Expansion Simulations Using a 

Particle-in-Cell Method. AIAA 2006–5023, 2006.

12. Choi, M.; and Boyd, I.D.: Hybrid Simulation of Magnetic Field Effects in the Plume of a Hall 

Thruster. Presented at Space Propulsion, Rome, 2016.

13. Araki, Samuel J.; and Barrie, Alexander C.: Electric Propulsion Plume Simulation Coupled With 

Spacecraft Charging. AIAA 2018–4906, 2018.

14. Dragnea, Horatiu C.; Hara, Kentaro; and Boyd, Iain D.: Development of a 2D Axisymmetric Electron 

Fluid Model in Hall Thrusters. AIAA 2017–4632, 2017.