# Modeling of a retarding potential analyzer and comparison with Express-A in-flight measurements

Lucas Nicolle, P Sarrailh, L Garrigues, S Hess, M Villemant

► To cite this version:

Lucas Nicolle, P Sarrailh, L Garrigues, S Hess, M Villemant. Modeling of a retarding potential analyzer and comparison with Express-A in-flight measurements. Frontiers in Physics, 2022. hal-05502937

HAL Id: hal-05502937

https://hal.science/hal-05502937v1

Submitted on 10 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Modeling of a retarding potential analyzer and comparison with Express-A in-flight measurements

1 L. Nicolle¹*, P. Sarrailh¹, L. Garrigues², S. Hess¹, M. Villemant¹

2 ¹ONERA, DPHY-CSE, Toulouse, France

3 ²LAPLACE, Université de Toulouse, CNRS, 31062 Toulouse, France

4 * Correspondence:

5 Lucas Nicolle

6 lucas.nicolle@onera.fr

7 Keywords: RPA (Retarding Potential Analyzer), Express-A, SPT-100, plasma thruster, in-

8 flight measurements, IEDF analysis, misoriented, tilted.

9 In-flight data from the Retarding Potential Analyzers (RPA) on the Russian geostationary
10 telecommunication satellite Express-A #3 are used to validate electric thruster plume simulations.

11 The RPA placed on the solar array have made it possible to measure ion properties, current densities
12 and energy distribution functions (IEDF), at different distances and field of view configurations with

13 regard to the thrusters. The present work shows that the conventional probes theory used to interpret
14 measurements is only valid if the probe is oriented toward the thruster. In most Express-A #3

15 measurements, this is not the case and it leads to incorrect interpretations.

16 In this paper, a new RPA probe model is presented. It enables the physics of a tilted probe to be
17 described more accurately, taking into account the view angle between the RPA entrance surface
18 normal relative to the incoming flux direction, the flux direction non-aligned with the collection
19 surface normal, the internal RPA electric fields and masking effects due to RPA walls. This probe
20 model is coupled with a simple model for ion transport from an electric thruster source to the probe
21 to allow a comparison with in-flight data.

22 The comparison between Express-A #3 RPA collected currents and calculations from the new model
23 discriminates true ion energy shift from probe misorientation effects. It also shows that highly tilted
24 RPA measurements previously never interpreted can be understood and analyzed.

25

26

27

28

29

30

31
32 **1 Introduction**

33 To understand and predict erosion and electrostatic discharges (ESD) on spacecraft in the presence of
34 a plasma thruster, it is necessary to properly represent the electrostatic coupling between the plume of
35 the thruster and the surfaces of the spacecraft. Several works have shown that this coupling has a
36 significant impact on the charging mechanism (1) and can increase the risk of ESD on solar arrays
37 (2,3). Erosion and contamination due to charge exchange ions created by the thruster device on
38 sensitive components such as solar cell interconnectors can also lead to premature termination of the
39 mission (4). As the demand for higher power plasma thrusters and higher solar array bus voltages
40 increases, these issues are magnified and become a limiting factor for power upscaling (5).

41 Plume models aim to represent the primary ion beam dynamics and the coupling with the
42 environment. Thus, a numerical plume simulation usually includes the transport of charged particles
43 and neutral atoms (using full Particle-In-Cell (PIC) (6) or hybrid fluid – PIC (1,7,8) approaches),
44 solves the Poisson equations and computes collisions between particles. To validate thruster plume
45 and spacecraft interaction models, experimental data are needed.

46 Although large ground-based vacuum chambers permit the study of plasma thruster far fields (9,10),
47 the conditions are different from space. The environment in the vacuum facility is denser (11),
48 leading to a greater charge exchange collision rate between the primary ion beam and the residual
49 Xe⁺ ions in the chamber. The study of the backflow (i.e., the Xe⁺ created from charge exchange
50 collisions) is therefore biased. Moreover, due to the presence of walls, the radial potential drop in the
51 vacuum chamber is also altered. In-flight experiments make it possible to have a representative
52 measurement for model comparisons.

53 An in-flight experiment has been conducted on SMART-1 (12), but few data are available in the
54 literature and the plasma diagnostic package position only makes it possible to study the backflow, at
55 a single point in space. On the other hand, the Russian geostationary telecommunication satellite
56 Express-A #3 (launched in June 2000) provides a large number of retarding potential analyzer (RPA)
57 measurements, at various locations with respect to the thruster firing axis, enabling a unique in-flight
58 experimental set-up to study the plasma plume. However, the RPA data processing is non-
59 conventional due to the oblique incidence of the plasma beam coming from the thruster. The probe
60 tilt-angle with respect to the plasma drift vector (i.e., the angle between the probe normal axis and the
61 probe - thruster axis) is within a range of 2 - 45° and, therefore, the RPA theory allowing the ion
62 energy distribution function (IEDF) to be obtained is no longer valid. Thus, Express-A RPA results
63 are under-exploited due to the lack of interpretability of the measurements when the probe tilt-angle
64 is significantly high.

65 In this paper, the set-up of the RPA probes onboard Express-A is presented. Then the modeling
66 approach for RPA measurements prediction is described, introducing a new RPA model. It makes it
67 possible to take into account the geometry of the RPA, an oblique incidence of the plasma beam with
68 respect to the probe normal axis, the effects of the axial electric field in the RPA on ions trajectories
69 and to consider grid transparency. Then, the coupling between the RPA model and a simple transport
70 model taking into input a thruster IEDF derived from a hybrid simulation code is detailed. Finally,
71 the comparison between the model and Express-A data is shown and discussed.

72

73
74 **2 Express-A mission and in-flight data**

75 Express-A #3 is equipped with 8 Fakel SPT-100, a well-documented Hall thruster used for station

76 keeping and two RPAs mounted on the solar panels (DRT1, DRT2), see Figure 1. The RPA position

77 allows a scan of ion current and ion energy distribution function (IEDF) as the solar arrays rotate.

78 This unique set-up is often compared to thruster plume simulation in space vacuum (13,14 and

79 references therein).

80 The spacecraft is also equipped with electric field and pressure sensors, along with torque

81 measurements induced by the thruster plume studied in other papers (8,15). However, in the scope of

82 this paper, only RPA data will be investigated.

83 RPA data have previously been used and compared with simulation (15,16). However, we

84 demonstrate that taking ion current measurements and IEDF without an RPA response model leads to

85 significant errors. This approach may be used in future work to obtain plume measurements on

86 unexploited Express-A data.

87

<figure><img src="image_2.png" /><figcaption>Figure 1. Express-A Thruster and RPA locations (17)</figcaption></figure>

88

89

90 On Express-A, the RPA collector surface normal is along +Z for DRT1 and along -Z for DRT2. The

91 DRT1 probe measures the ion current from the thruster unit TU4 and DRT2 measurements concern

92 the TU3. Note that each thruster unit TUX consists of two SPT-100 Hall thrusters, designated TX

93 and RTX.

94 To be able to analyze and interpret the RPA measurements without biases, the probe needs to be

95 oriented toward the studied ion source. However, regarding the position of the thruster units, TU3

96 and TU4, the probe tilt-angle $\beta_{RPA}$ (angle formed by the incoming ion flux and the RPA collector

97 surface normal) can be up to 45°, leading to a complicated measurement analysis.
98

99

100

101

Express-A RPA data have previously been used in two ways: first, as an ion current density probe and, second, to measure IEDF. Previous articles seem to take some precautions with IEDF analysis and only consider RPA data when the probe tilt-angle is close to 0° (15). But few cases are available. For ion current analysis, raw RPA data are directly used despite the varying tilt-angle $β_{RPA}$.

<figure><img src="image_3.png" /></figure>

102

103

<caption>Figure 2. a) Express-A firing angles for data configurations, b) Diagram for angle definition</caption>

104

105

106

107

108

This paper aims to study the effect of probe misorientation over Express-A RPA measurements (over 170 angle/distance configurations from 3.82m to 9.26m with an angle relative to thruster varying from 2° to 45°, see Figure 2) and to identify whether the data provide information about thruster IEDF only, or if more information can be extracted such as electrostatic effects or collision effects inside the plume.

109

To do so, a new RPA model is presented in section 3.

110

## 3 Transfer function model for RPA measurements prediction

111

112

113

114

115

116

117

This section introduces the model used in this paper. First, in subsection 3.1, the RPA theory is recalled, detailing the issue with tilted RPA measurements. Then, subsections 3.2 to 3.4 present the approach used to predict Express-A RPA measurements. To do so, the subsections successively focus on the modeling of the inside of the RPA, using a transfer function approach, then, on the input distribution function used to model the thruster ion source and finally, on the modeling of the ion motion from the thruster to the RPA. In subsection 3.5, the effect of the RPA tilt-angle on the probe measurements is analyzed using the model previously described.

118

### 3.1 Elementary theory of RPA

119

120

121

122

123

124

A retarding potential analyzer is an electrostatic hollow cylindrical probe consisting of three to four polarized grids. First, a screen grid is polarized to the probe floating body potential $φ_S$, restraining plasma perturbation. Then the cut-off grid, negatively polarized to $φ_{CO}$, repels the electrons and, finally, the analyzing grid is polarized to $φ_A$ sweeping from the floating potential to a chosen positive potential repelling all the ions. The latter acts like a high-pass energy filter for ions. The collector potential is also set to the probe floating body potential.
Let us consider a beam of singly charged ions of charge q and mass m whose distribution function along the axial z direction is noted f(vz) and an RPA with non-polarized grids. All the ions whose energy of the motion in the z direction is εz ≥ q(φA - φS) are collected by the RPA. We can easily calculate the collected current Ic = Sqτ ∫0⁺∞ vzf(vz)dvz = Sqτ/m ∫0⁺∞ f(√2εz/m)dεz, where S is the collection surface area, τ the product of grid transparencies and m the mass of an ion.

If a positive value of the analyzer potential φA is considered, the collected current becomes:

$$I_c(\phi_A) = \frac{Sq\tau}{m} \int_{q\phi_A}^{+\infty} f(\sqrt{2\epsilon_z/m}) d\epsilon_z = \frac{Sq\tau}{m} \cdot [F(\epsilon_z \rightarrow +\infty) - F(q\phi_A)] \quad [1]$$

The derivative of the collected current by the analyzing potential φA of Eq. 1 gives:

$$\begin{align*}
\frac{dI_c(\phi_A)}{d\phi_A} &= \frac{Sq\tau}{m} \frac{d(F(\epsilon_z \rightarrow +\infty) - F(q\phi_A))}{d\phi_A} = -\frac{Sq\tau}{m} \cdot \frac{d(F(q\phi_A))}{d(q\phi_A)} \cdot \frac{d(q\phi_A)}{d\phi_A} = -\frac{Sq^2\tau}{m} f\left(\sqrt{\frac{2q\phi_A}{m}}\right) \\
&= -\frac{Sq^2\tau}{m} f(v_z)
\end{align*}$$

$$\text{We finally obtain the proportionality relationship: } f(v_z) \propto -\frac{dI_c}{d\phi_A} \quad [2]$$

This equation relies on the initial hypothesis that if an ion has an axial energy εz ≥ q(φA - φS), it is collected but as soon as the ion axial energy is not high enough to get through the potential barrier imposed by the analyzer grid, it is repelled and therefore not collected. This hypothesis is valid when εz / εr ≫ 1, with εz and εr, respectively the axial and radial energy of the ion. For an un-tilted probe (i.e., directed toward the ion source), assuming that most of the ions are coming directly from the source in a straight line, this relationship holds. However, if the tilt-angle βRPA increases, this assumption is no longer valid. It must also be noted that the -dI/dφA curve is in fact a measurement of the axial component of the velocity. For a perfectly oriented probe, it corresponds to the velocity vector norm, but for a tilted-RPA, the velocity vector norm is obtained by performing a velocity shift such that the ion velocity is v = vz / cos²(βRPA).

Figure 3 shows the trajectory of an Xe⁺ entering the RPA with an energy ε = 250eV, leading to a velocity v = √2qe/mXe⁺ and a tilt-angle βRPA = 40°. Therefore, the energy along the axial and radial directions are:

$$\left\{
\begin{array}{l}
\epsilon_z = \frac{m_{Xe^+} \cdot v_z^2}{2q} = \frac{m_{Xe^+} \cdot (v \cdot \cos(\beta_{RPA}))^2}{2q} = \epsilon \cdot \cos^2(\beta_{RPA}) \approx 147\,\text{eV} \quad [3] \\
\epsilon_r = \epsilon \cdot \sin^2(\beta_{RPA}) \approx 103\,\text{eV}
\end{array}
\right.
$$
<figure><img src="image_4.png" /><figcaption>Figure 3. Trajectories of an ion $Xe^+$ (total energy of 250eV and a tilt-angle $\beta_{RPA}$ = 40°) in Express-A RPA for different values of analyzer grid potential. The grid potentials are $\phi_S$ = 0V, $\phi_{CO}$ = -30V and $\phi_A$ varies between 0V and 250V.</figcaption></figure>

This simple simulation of the motion of a single ion shows that, when $\epsilon_Z$ ≈ $\epsilon_r$, a significant part of the ion flux can reach the RPA walls before being collected. For example, in Figure 3, the ion has an axial energy $\epsilon_Z$ ≈ 147eV but it is intercepted by the walls as soon as the analyzer potential reaches $\phi_A$ ≈ 75V. As Eq. 2 states, any change in the ion current at the collector between two values of $\phi_A$ is interpreted as a presence of ions in this energy range, therefore in this example, the ion will be considered as an ion of 75eV of axial energy. This effect needs to be modeled to correct this probe measurement bias.

As $\beta_{RPA}$ increases, the proportion of the ion flux deviated toward the walls increases and is a function of the analyzer potential.

## 3.2 Modeling of the RPA: transfer function approach

The expression of the collected current introduced earlier can be completed by the introduction of a transfer function $H(\vec{v}, ...)$, dependent on the velocity vector and other parameters. The collected current in the RPA as a function of the IEDF at the probe entrance can be obtained. Considering $\overrightarrow{n_{RPA}}$ colinear to $-\vec{z}$, the collector surface $S$, the product of grid transparencies $\tau$, the axial velocity $v_z$, the velocity distribution function $f(v_x, v_y, v_z)$ and a transfer function $H$, the current at the collector can be expressed as:

$$I_C = S q \tau \iint_{-\infty}^{+\infty} \int_{0}^{+\infty} v_z f(v_x, v_y, v_z) H(v_x, v_y, v_z, ...) dv_x dv_y dv_z \quad [4]$$

where $H$ is defined as the product of different transfer functions representing different phenomena such as the effects of the axial electric fields on ion trajectories and the presence of walls. In principle, $H$ could also include space charge effects. Nevertheless, for typical conditions considered in this study: the electron temperature $T_e$ ~ 2eV and electron density $n_e$ ~ 10¹¹ m⁻³, the Debye length
176 $\lambda_D$ is approximately 0.03m, which is larger than the characteristic length of the probe, its diameter
177 $d_{RPA} \approx 0.02m$. Regarding the fact that $\lambda_D > d_{RPA}$, the hypothesis of no space charge effects can be
178 reasonably made in the RPA. Thus, $H = E_f \cdot W_f$ with $E_f$ the high-pass energy filter and $W_f$ the filter
179 induced by ion losses at the RPA walls.

180 The objective of the following approach is to express the collected current $I_c$ as a function of the ion
181 velocity distribution function at the source exit and other constant parameters related to the RPA
182 geometry such as the radius, length, grid transparency, grid potentials, etc.

183 Other RPA modeling approaches for example using PIC simulation, exist (18). However, the
184 duration time of the simulation is much larger than the pseudo-analytical approach due to the large
185 number of macro-particles needed to capture the shape of the energy distribution on the collector
186 surface with good accuracy. The misoriented RPA case has never been studied using PIC models. In
187 the end, the coupling between this fast analytical approach with a plasma-spacecraft solver (already
188 time-consuming due to the description of the charged particle transport and interactions with the
189 spacecraft) will be performed.

190 We will now detail the expression of $H$ in Eq. 4. First, the high pass energy filter (Eq. 5) is simply a
191 Heaviside step function derived from the conservation of energy, considering an initial velocity $\vec{v}$ and
192 a purely axial electric field in the RPA (derived from the electric potential). To be collected, the
193 initial ion energy divided by the charge must be higher than the potential barrier resulting from the
194 analyzer grid and the plasma potential. Otherwise, the incoming ion will be repelled and will not
195 reach the collector:

$$E_f(\|\vec{v}\|, \beta_{RPA}, \phi_A) = \begin{cases} 1 & \text{if } (\|\vec{v}\| \cdot \cos(\beta_{RPA}))^2 > \frac{2Zq}{m(\phi_A - \phi_{plasma})} \\ 0 & \text{otherwise} \end{cases} \quad [5]$$

197

198 In this paper, only singly charged ions are considered. Therefore, $Z = 1$.

199 Second, taking into account the effect of the wall filter is less straightforward. Considering only the
200 particles with $E_f = 1$, for a given $\vec{v}$, $\beta_{RPA}$ and $\phi_A$, due to a non-zero initial radial velocity, the
201 particles are shifted by a fixed value in the radial direction, regardless of their initial positions. This
202 shift $d_s$ can be computed with piecewise application of the fundamental principle of dynamics in
203 regions of iso-electric-field (i.e., between each inter-grid space). The integrated transfer function over
204 all the initial positions can be obtained (Eq. 6) by computing the intersection between two disks: the
205 entry disk, representing all the possible initial positions and the collector disk shifted by the distance
206 $d_s$, representing all the final positions, see Figure 4.
<figure><img src="image_5.png" /><figcaption>Figure 4. Integration over initial positions for a given velocity vector (axial kinetic energy greater than retarding potential energy).</figcaption></figure>

The surface of the intersection of the disks becomes the effective collection surface $S_{eff} = S_{entry} \cap shifted(S_{collector}, d_s)$ and the filter is defined as

$$W_f(v, \beta_{RPA}, \phi_A) = \frac{S_{eff}}{S_{entry}} \quad [6]$$

The transfer function $H$ is plotted in Figure 5 for several values of analyzer grid potential $\phi_A$. The left plot shows the 2D transfer function. The energy filter $E_f$ is responsible for the high pass filter above $\phi_A$. The right plot in Figure 5 shows the effect of $\phi_A$ on the value of $H$ for $\epsilon_s = 250eV$. The energy filter induces a cut-off tilt-angle above which no more current is collected and the wall filter leads to a decrease of $H$, as the analyzer potential increases, describing the ion deflection due to the increasing axial electrostatic field between the cut-off and the analyzer grids. The effect of the wall filter magnifies as $\beta_{RPA}$ increases.
<figure><img src="image_6.png" /><figcaption>Figure 5. Transfer function *H* of the RPA for Xe⁺ as a function of the kinetic energy and the RPA tilt-angle. Plotted for different values of the analyzer grid potential. The higher bound of the scale correspond to the grid transparency product *τ*. The *β*<sub>RPA</sub> dependency of the transfer function at 250eV is shown on the right. Parameters: *φ*<sub>CO</sub> = -30V.</figcaption></figure>

At this point, the IEDF at the thruster exit (i.e., the model input) need to be specified. The transfer function between the thruster and the RPA entrance is also needed. They are described respectively in sections 3.3 and 3.4.

## 3.3 Modeling of the thruster IEDF

The model presented in section 3.2 makes it possible to compute the transfer function of the RPA.

The IEDF at the RPA entrance depends on the IEDF at the thruster exit and how this distribution varies between the thruster and the RPA locations. This section presents the approach used to model the IEDF at the thruster exit.

Figure 6 illustrates an Xe⁺ ion current distribution function at the cathode line boundary fitted from calculations using a hybrid Hall thruster model for nominal conditions of the SPT-100 (19).
<figure><img src="image_7.png" /><figcaption>Figure 6. Radial ion current distribution function at the thruster cathode line boundary from hybrid model (19). The angle is $\alpha_T$, relative to the thruster surface normal (in degrees) and the radius represents the energy (in eV).</figcaption></figure>

As may be seen in Figure 6, the emitted current decreases with the angle, but the shape of the current energy distribution function between 0° and 45° evolves with very little change. Therefore, the IEDF can be approximated using separation of variables in spherical coordinates with $r, \theta, \phi$, respectively the radial position, the polar angle (equal to $\alpha_T$) and the azimuthal angle, such that $f_T(v_r, \theta) = f_{sm}(v_r) \cdot g(\theta)$. As the thruster is considered to emit ions homogeneously and with a symmetry axis along $z$ (independent of $\phi$), spherical coordinates are well suited. It allows simplifying the numerical integration; therefore, this coordinate system will be used in the rest of the paper.

The velocity distribution function is defined accurately by a fitted Gaussian distribution $f(v_r)$, with the velocity expressed in spherical coordinates and its parameters: $\mu = v_s = \sqrt{2q\epsilon_s/m_{Xe^+}}$, where $\epsilon_s \approx 250$eV and $\sigma = v_T = \sqrt{qT_i/m_{Xe^+}}$, with $T_i \approx 0.2$eV. It looks like a shifted Maxwellian distribution but in spherical coordinates. The $\alpha_T$-dependency is modeled by the function $g(\theta)$ also fitted on the hybrid model as a first approach. Figure 7 shows the fitted $f_{sm}(v_r)$ and $g(\theta)$ compared to the hybrid model.

$$f_T(v_r, \theta) = \begin{cases} f_T(v_r, \theta) = f_{sm}(v_r) \cdot g(\theta) \\ f_{sm}(v_r) = \left(\frac{m_{Xe^+}}{2\pi q T_i}\right)^{1/2} \cdot \exp\left(-\frac{(v_r - v_s)^2}{v_T^2}\right) \end{cases} \quad [7]$$
<figure><img src="image_8.png" /><figcaption>Figure 7. Top: normalized velocity distribution for various θ angles. Bottom: normalized angular distribution for various kinetic energies (log10). In black, the $f_{sm}(v_r)$ and $g(\theta)$ derived from the hybrid model fit.</figcaption></figure>

## 3.4 Ion transport model

To obtain the distribution function at the RPA location, the transport has to be modeled between the source and the RPA entrance. In this section, we detail the expression of the ion distribution function as a function of the position and the velocity $f(r, \theta, \phi, v_r, v_\theta, v_\phi)$.

First, we assume that there are no collisions and no plasma effects in the plume. This strong approximation will be discussed in section 4.3. Second, the RPAs on Express-A #3 are far from the thrusters (in-flight data, the distance is at least $r \sim 3.8m$) and the thruster exit and RPA are small devices (diameters of $d_T \sim 10cm$ and $d_{RPA} \sim 2cm$ respectively). It leads to $\frac{r}{d_{RPA}} \gg 1$ and $\frac{r}{d_T} \gg 1$.

Therefore, we have made the hypothesis that there is no geometrical factor due to the annular shape of the thruster neither due to the disk shape of the RPA entrance. They are treated respectively as a point source and a point target. These two hypotheses combined lead to a purely radial motion for ions. It implies that the ion density and ion current must follow an inverse square law. That also means that the $v_\theta$ and $v_\phi$ components of the velocity vector are always zero, so the distribution function is independent of these variables.

$$f(r, \theta, \phi, v_r) = \frac{r_0^2}{r^2} \cdot f(r_0, \alpha_T, \phi, v_r)$$

The hypothesis that the velocity distribution of the thruster is independent of the firing angle $\alpha_T$ has been made (section 3.3). Thus, the velocity distribution at any RPA location is independent of $\alpha_T$.

So, the shape of the velocity distribution function at the RPA location is always $f_{sm}(v_r)$. Moreover, the thruster distribution is independent of $\phi$, it is consequently the same for the distribution function in volume. So, the expression of the distribution function is:

$$f(r, \theta, v_r) = \frac{C}{r^2} \cdot f_T(v_r, \theta) = \frac{C}{r^2} \cdot g(\theta) \cdot f_{sm}(v_r)$$
278 With C, a constant such that the current of the thruster source $I_T$ is conserved:

$$279 \qquad I_T = 3.1 A = I(r) = \frac{qC}{r^2} \iint_{\Sigma} g(\theta) dS \cdot \int_{0}^{+\infty} v_r \cdot f_{sm}(v_r) dv_r$$

280 With, $dS = r^2 \sin(\theta) d\theta d\phi$, the surface element of $\Sigma$, the hemisphere collecting all the emitted ions.

281 Finally, the current collected by the RPA is expressed as:

$$282 \qquad I_c(r, \alpha_T, \beta_{RPA}, \phi_A) = S_{coll} \int_{0}^{+\infty} v_r \cdot \cos(\beta_{RPA}) \cdot f(r, \alpha_T, v_r) \cdot H(v_r, \alpha_T, \beta_{RPA}, \phi_A, \dots) dv_r \quad [8]$$

## 3.5 RPA measurements prediction: Effect of the RPA tilt-angle

284 As mentioned in section 2, RPA measurements can be analyzed in two interesting ways. First, the derivative of the collected current with respect to the analyzer potential $-dI_c/d\phi_A$ can give information about the ion IEDF and second, the absolute value of $I_{\phi_A=0}$ can give information about the ion current density. The effect of the RPA tilt-angle on both is analyzed in the following sections.

### 3.5.1 Effect of the RPA tilt-angle on $dI_c/d\phi_A$

289 In Figure 8, the effect of the RPA tilt-angle $\beta_{RPA}$ is studied. The predicted $-dI/d\phi_A$ curves shown are obtained with Eq. 8, using the transfer function described in section 3.2, the input distribution from section 3.3 and the transport model in section 3.4. The derivative of the collected current $-dI/d\phi_A$ has been normalized to allow a comparison with the IEDF. The light gray areas represent the IEDF that an RPA without walls should measure (such that the transfer function $H = E_f$). It

294 corresponds to a shifted Maxwellian distribution ($v_s = \sqrt{2q\epsilon_{shift}/m_{xe}^+}$ with $\epsilon_{shift} = \epsilon_s$).

295 $\cos^2(\beta_{RPA})$, as Eq. 3 state), with from left to right, $\beta_{RPA} = 40^\circ, 30^\circ, 20^\circ$. The blue area is the true input velocity distribution, $f_{sm}$.

<figure><img src="image_9.png" /><figcaption>Figure 8. a) $-dI/d\phi_A$ computed at different RPA tilt-angles, b) Model diagram. The input distribution is a shifted Maxwellian with $\epsilon_s = 250eV$ and $T_i = 0.2eV$.</figcaption></figure>

300 When the tilt-angle $\beta_{RPA}$ is near 0° (i.e., the perfectly oriented RPA case), the measure fits the injected IEDF, as expected. In terms of the transfer function, it means that $W_f(\phi_A)$ is constant. As
302 $\beta_{RPA}$ increases, the loss at walls effects described with $W_f(\phi_A)$ appear in the $-dI/d\phi_A$ curves. For

303 $\beta_{RPA} = 20^\circ$, the model computation of the RPA $-dI/d\phi_A$ curve is shifted, with a peak

304 corresponding to the expected value $\varepsilon_{peak} \approx \varepsilon_s \cdot \cos^2(\beta_{RPA}) \approx 221\,\text{eV}$. We can see that the shape of

305 the curve changes slightly at low energy due to the collection at walls described in $W_f$. The curve is

306 no longer symmetric compared to the case where $\beta_{RPA} = 0^\circ$.

307 Figure 5 shown that the effect of the tilt-angle is more pronounced as $\beta_{RPA}$ increases. At $\beta_{RPA} = 30^\circ$,

308 the value of $-dI/d\phi_A$ reaches 10% of the maximum value at $\phi_A \approx 125\,\text{V}$, but the peak is still around

309 the expected energy. However, for high $\beta_{RPA}$ values, as can be seen when $\beta_{RPA} = 40^\circ$, the predicted

310 peak is before the gray area peak ($\phi_A \sim 127\,\text{V}$ instead of 146V). It means that the loss of collected

311 ions due to the combined effects of wall collection and analyzer potential filtering is the strongest at

312 this energy. Thus, for $\beta_{RPA} > 30^\circ$, the energy peak cannot be predicted easily without modeling the

313 RPA transfer function.

314 For the study of the Express-A plasma thruster, the effect of the wall collection on the $-dI/d\phi_A$

315 curve is an issue. In the plume, charge exchange $Xe^+$ ions can be created and can be found at large

316 firing angle $\alpha_T$, with an energy of the order of the tens of eV (12). As Figure 2 shows, large $\alpha_T$ leads

317 to large tilt angle $\beta_{RPA}$. Therefore, it is important to model the transfer function accurately to

318 discriminate real low-energy ions from high-energy ions.

319

320 **3.5.2 Effect of the RPA tilt-angle on $I_{\phi_A=0}$**

321 It is possible to deduce the ion current at the RPA location from the RPA current measurement at

322 $\phi_A = 0\,\text{V}$, but only if the transfer function of the RPA is known. Figure 9 shows the collected current

323 for the shifted Maxwellian distribution $f_{sm}$ described in section 3.3. It shows that the decrease of 324 current as a function of the tilt-angle is much steeper than a simple cosine (as if it was a disk surface).

325 The wall effect on the measured ion current stays negligible while the tilt-angle $\beta_{RPA}$ is below $\approx 10^\circ$,

326 but it becomes quickly important. Indeed, at $\beta_{RPA} = 26^\circ$, the probe collects half the ion current

327 compared to a perfectly oriented RPA. For $\beta_{RPA} > 45^\circ$, no more current is collected. For the range of

328 probe tilt-angle of the Express-A RPA, the drop of collected current can be strong. It must be taken

329 into account to study the dependency of the emitted current to the firing angle $\alpha_T$.
<figure><img src="image_10.png" /><figcaption>Figure 9. Normalized collected current as a function of the RPA tilt-angle $\beta_{RPA}$ for $\phi_A = 0V$. The velocity distribution is sampled from a shifted Maxwellian with $T_i = 0.2eV$, $E_{shift} = 250eV$, $Xe^+$ ions.</figcaption></figure>

# 4 Comparisons between Express-A data and RPA modeling

## 4.1 Express-A data filtering

All the available RPA curves were processed using the following procedure. To analyze them, a data cleaning based on the following simple criterion was applied: the probe has to collect a current $I_c > 0A$ and data have to be acquired while the thruster is firing. The data selected in this paper have been processed to get a single I-V curve for each operational point. To obtain a single curve for each configuration, a first mean is computed among all the selected curves. To exclude extremum curves and to reduce the dispersion, when possible, the following processing was applied: if the RPA configuration has more than 10 acquisitions, only the 5 closest to the mean curve (computed by the residual sum of squares) are kept to compute the final averaged measurement. If the RPA configuration has between 5 and 9 curves, the two worst curves are omitted to compute the final averaged measurement and, finally, if the RPA configuration has less than 5 curves, all data are kept. An RPA provides two useful types of measurement for thruster characterization. The first is the shape of the $-dI/d\phi_A$ curve, which provides information on the ion energy distribution function. The second is the absolute value of the collected current when $\phi_A = 0V$, making it possible to obtain the current density at the RPA location, but only if the transfer function of the probe is known. The collected current can be plotted as a function of $\alpha_T$ to obtain information on the angular distribution of the emitted current of ions.

In the remainder of the section, we compare the introduced transfer function model to Express-A measurements on the two kinds of measurements from an RPA probe: on one hand, the study of the ion energy distribution function, and on the other hand, the analysis of the ion current at $\phi_A = 0V$.

## 4.2 Comparison between the model prediction of the $-dI/d\phi_A$ curve and Express-A RPA measurements
In this section, several curves obtained from Express data are shown in Figure 10. We have decided to focus on 4 configurations corresponding to the DRT1 (near probe) measurements with: RT4 firing with the solar array angle of (10.a) 45° and (10.c) 58°, and T4 firing with the solar array angle of (10.b) 75° and (10.d) 92° (see Figure 2). All probe tilt-angles $β_{RPA}$ are greater than 25°, making it possible to highlight cases where misorientation effects are significant (see Figure 8). For every configuration, 4 curves are shown: the Express data, the IEDF imposed by the model at the RPA entrance and the $-dI/dφ_A$ RPA curve prediction obtained from the transfer function model computation described in section 3.

<figure><img src="image_11.png" /><figcaption>Figure 10. Normalized $-dI/dV$ curves: processed Express data compared to the model and the raw IEDF applied by the simulation at the RPA location.</figcaption></figure>

In Figure 10, the firing angle $α_T$ and the distance stay approximately the same on each row, allowing to highlight tilt-angle effects. On the first raw, Figure 10.a and 10.b, the $-dI/dφ_A$ curves from Express data (in black) show an energy peak shifted from 200eV (for $β_{RPA}$ = 22.4°) to 160eV (for $β_{RPA}$ = 30.9°). The transfer function model, taking into account the tilt-angle and losses to the RPA walls also predicts an energy shift of the same order of magnitude, from 210eV to 180eV.

Comparing Figure 10.c and 10.d, the energy peak also shifts when $β_{RPA}$ increases. Thus, the peak shifts from 200eV ($β_{RPA}$ = 27.1°) to a peak at 150eV with a wider distribution ($β_{RPA}$ = 35.5°). Here, the model is closer to the energy peak value. In addition, the model manages to predict the widening of the shape of the $-dI/dφ_A$ curves at low energy when $β_{RPA}$ increases, also visible in the Express-A data.

It should be noted that there is good agreement between the simulation results and the in-flight measurements considering that all the model inputs are completely theoretical (the only input of the RPA model is the distribution function at the thruster exit and the latter is taken from the hybrid model). The sources of discrepancy between model and measurements can be due to approximations in our modeling approach, only primary singly charged ions are taken into account and no collision nor plasma effects are considered. It can also come from the considered input thruster IEDF or any Express-A measurement error.
## 4.3 Comparison between the model prediction of the absolute value of the collected current $I_{\phi_A=0}(\alpha_T)$ and Express-A measurements.

The previous section presented the bias due to a misoriented RPA on the shape of the IEDF. This section focuses on the consequences on the collected currents (in absolute values) and, therefore, on the measurement of the angular distribution function of the emitted ion current.

To fit plasma parameters in simulation from experimental data, Manzella (14) and Sang-Wook Kim (20) used a normalization to reconstruct the $\alpha_T$ dependency of the emitted current from probe collected currents. For Express, this is obtained by gathering all the collected current values for $\phi_A = 0V$ for all the data points. To eliminate the dependency on the varying distance between the thruster and the probe, the data are normalized 1m downstream of the thruster location using a $1/r^2$ decrease law. This gives a point cloud representing $I_{\phi_A} = f(\alpha_T)$. However, in this method, the effects of the tilt-angle on the collected current are not considered. This is a bias because the tilt-angle varies for each data point.

Figure 11 shows the comparison between the transfer function model as described in section 3 and the processed Express-A RPA measurements.

<figure><img src="image_12.png" /><figcaption>Figure 11. Collected current normalized at 1m from the thruster using a $1/r^2$ law as a function of the firing angle. Comparison of the processed Express-A data and model results (using input IEDF from hybrid Hall-thruster code). The near and far probes are respectively ‘DRT1’ and ‘DRT2’.</figcaption></figure>

Each point represents the collected current for a configuration (a unique set of thruster, probe and solar array rotation angle). These configurations are summarized in Figure 2. The RPA data are presented in black and the transfer function model predictions for a misoriented RPA, with loss at the RPA walls, are in red and blue. The model prediction splits into two curves as the firing angle $\alpha_T$ increases. This highlights the relationship between $\beta_{RPA}$ and $\alpha_T$ shown in Figure 2. Each thruster unit consists of two thrusters, one firing in the Y+ direction and the other in the Y- direction (see Figure 1). Therefore, the RPA tilt-angle is maximized when the firing thruster is on the opposite side of the RPA (T4 for near probe and RT3 for far probe), leading to the bottom branch. On the same principle, the probe tilt-angle is minimized when the thruster and the probe are on the same side (RT4 for near probe and T3 for far probe), leading to the top branch.
419 The comparison between the transfer function model prediction and the Express-A measurements
420 shows that the model overestimates the collected current at low angles (1 − 2 · 10²µA for Express-A
data vs. 10³µA for model prediction at α~2−3°). Regarding the α~T~ dependency of the current, we
421 can see that the model predicts a steeper decrease than the measured one. This leads to an
422 underestimation of the current at large angles (factor of 10 at α~T~ ≈ 45°).
423

424 These observations indicate that the beam at the RPA location seems too focalized. The beam
425 divergence is too low compared to what Express measurements show. The difference between
426 measurements and the model prediction can be attributed to several plume effects. First, electrostatic
427 effects and second, collisions. Considering electrostatic effects, ion transport from the thruster exit to
428 the probe is not purely radial due to plasma effects. In particular, a radial electric field builds up in
429 the plume due to electron pressure. High-velocity ions are then attracted outside of the main beam
430 modifying the velocity vector. These plasma effects can be seen in the measurements by comparing
431 near RPA measurements (~4m) to the far ones (~9m). Once normalized at 1m (Figure 11), we can
432 see in Figure 11 that the normalized RPA collected current at small angles (<15°) do not overlap for
433 near and far RPA (nearly a difference of a factor of 2), as it should be if the 1/r² law was correct.
434 This implies that divergence must build up in the plume. This was already mentioned and also
435 computed by Boyd (13).
436

436 Moreover, the results in Figure 11 only consider the primary ion beam. However, charge exchange
437 collisions (CEX) take place in the plume, fast ions can exchange their charge with a slow neutral,
438 creating slow ions. These slow ions are also attracted in the radial direction and may increase the
439 measured current at large α. The energy of CEX is relatively low (order of 50eV) and cannot be
440 discriminated from the main beam at large α (see Figure 10).
441

441 In addition to plume effects and collision, the gap between measurements and model could also be
due to the modeling of the thruster. Indeed, only singly charged ions are modeled, while a significant
442 proportion of the emitted current is doubly or triply charged (respectively 11 – 16% and 2 – 5%,
443 (21)). Their high charge leads to a stronger deviation due to electrostatic effects in the plume and
444 therefore to a higher current at large angle.
445

446
447
4.4 Comparison between the model prediction of the absolute value of the collected current
$I_{\phi A=0}(\alpha_T)$ and Express-A measurements with a modified thruster beam as input

448 Section 4.3 shows that the model is not able to predict the shape of the emitted current measured by
the RPA. To highlight tilt-angle effects in Express-A RPA measurements, it is necessary to change
the modeling approach. Figure 10 shows that the velocity distribution $f_{sm}(v_r)$ fitted from the hybrid
model gives good results. However, the measured angular distribution is far from the predicted one.
451 Therefore, to highlight tilt-angle effect, the function $g(\theta)$, describing the angular dependency of the
emitted current (Eq.7), is fitted from the measurements in Figure 12.
452
453
<figure><img src="image_13.png" /><figcaption>Figure 12. a) Fit of the ion angular distribution function $g(\theta)$ from Express-A measurements. b) Comparison between normalized ion angular distribution function from experimental fits and thruster hybrid model results. The ion energy distribution function is a shifted Maxwellian ($\epsilon_s = 250eV, T_i=0.2eV$).</figcaption></figure>

The experimental results of $g(\theta)$ in Figure 12.a are obtained by inverting the expression of the current collected by the RPA (Eq.8) such that:

$$g(\theta) = \frac{I_c(r, \theta, \beta_{RPA}, \phi_A = 0) \cdot r^2}{S_{coll} \cdot \cos(\beta_{RPA}) \cdot C \cdot \int_{0}^{+\infty} v_r \cdots f_{sm}(v_r) \cdot H(v_r, \theta, \beta_{RPA}, \phi_A, \dots) dv_r}$$

Then, a fit is performed using a fourth-order polynomial. The function is normalized to match Express-A measurements at a very low firing angle. Ideally (i.e., if the model assumptions were true), the Express-A data in Figure 12Figure 11.a should be on a unique curve. However, it is not the case. It can be due to dispersion in Express-A measurements on one side and strong model hypotheses on the other side. In Figure 11, the normalization at 1m shows that the normalized measurements of the near probe were twice the normalized measurements of the far probe, suggesting that divergence increases in the plume between the probes and invalidating the $1/r^2$ current law. However, it is not a certainty and could be also due to a partially blocked far probe. The only way to discriminate between these two causes would be to compute the current conservation. Unfortunately, it is not possible as the range of angle of the far probe does not permit the evaluation of the emitted current at high angles.

In conclusion, without taking into account the plasma effects in the plume, this approach is not able to give the $g(\theta)$ function at the thruster exit from the RPA measurements. But, the two fits can be used to study the effect of the RPA tilt-angle on the current measurements of Express-A.

The Figure 13 shows the comparison between the collected current normalized at 1m (for $\phi_A = 0V$) of the near and far probes to the value predicted by the model using the fits of $g(\theta)$ described previously.
480

<figure><img src="image_14.png" /></figure>

481

Figure 13. Collected current normalized at 1m from the thruster exit as a function of the firing angle. Comparison of the processed Express-A data and model results. a) near probe, b) far probe. The input IEDF is a shifted Maxwellian distribution for energy ($\epsilon_s$ = 250eV, $T_i$=0.2eV) and a fitted angular function $g(\theta)$.

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

First, the far probe results (Figure 13.b) show that Express-A data follow two distinct tendencies between $\alpha_T$ = 10° and 15°. In this range, the normalized current values while T3 is firing are above the measured values with RT3 on. This is predicted by the transfer function model as well. It is due to the RPA tilt-angle being different in the T3 vs. the RT3 configuration (the Figure 2 shown that in the range 10° < $\alpha_T$ < 15°, the tilt-angle $\beta_{RPA}$ in the RT3 case is 10° above the T3 case).

Second, the experimental data and model prediction for the near probe ‘DRT1’ are compared in Figure 13.a. As for the far probe, the predicted collected current is different depending on the firing thruster, due to the same tilt-angle effect. The prediction made with the T4 thruster configuration led to the bottom branch and the RT4 thruster configuration to the top one. The Express-A measurements are more dispersed with ‘DRT1’ than ‘DRT2’, however, the measurements for $\alpha_T$ > 20°, especially at very large angles ($\alpha_T$ > 40°) also split in two, as predicted. Moreover, although the measurements in the mid $\alpha_T$ range, between 10° and 35° are dispersed, the measured values when RT4 is firing seem generally above the cases where T4 is on. This is also in good agreement with the prediction.

To conclude, both far and near probe analysis demonstrate that effects of the tilt-angle are noticeable on Express-A RPA absolute current measurements and that plotting the current normalized at 1m as a function of the thruster firing angle $\alpha_T$ leads to a bias induced by the varying RPA tilt-angle. The predictions are in good agreement with the data, showing that the model performs well with a corrected $g(\theta)$ function. Doing so enables us to correct the RPA measurement and deduce the absolute value of the collected current at the probe location. Furthermore, as the collected current values can be corrected for the tilt-angle, the coupling of this model with a more advanced ion transport should allow deducing the angular distribution of the emitted current at the thruster exit.
510 **5 Conclusions**

511 The new model presented in this paper makes it possible to predict the RPA response for cases where
512 tilt-angle is significant. The new RPA model uses a transfer function formalism, describing the
513 motion of ions inside the RPA. It considers the RPA geometry (with boundary effects, grid
514 transparency, the field of view with respect to the source) and the ion deflection due to the axial
515 electrostatic fields between each grid. The space charge effects that can appear in dense plasma are
516 not taken into account in this model. To be able to compare this RPA model to Express-A
517 measurements, the plasma thruster has been modeled as a point source, using a realistic SPT-100
518 IEDF from existing simulation results, considering only primary singly charged ions. The ion
519 propagation model used between the thruster exit and the RPA entrance assumes no space charge
520 effects and no collision in the plume.

521 The analysis of the RPA transfer function suggests theoretically that tilt-angle effects are significant
522 in Express-A configurations. Comparing these predictions to the actual RPA measurements, it has
523 been shown that RPA misorientation effects are indeed visible on both $-dI/\phi_A$ curves and on the
524 absolute value of the collected current, affecting both the IEDF analysis and the analysis of angular
525 dependency of the collected ion current.

526 For IEDF analysis, the results from the model are in good agreement with the Express-A
527 measurements. It demonstrates that the probe misorientation induces two biases: first, the measured
528 ion energies are shifted toward lower values. Second, the $-dI/d\phi_A$ curves can become non-zero at
529 low energies due to the ions being deflected toward the RPA walls as the analyzer potential increases.
530 This must be taken into account as it can lead to under-estimation of primary ions energy and
531 overestimation of low energy ions density. Previously, the only Express-A $-dI/d\phi_A$ curves that
532 have been studied were when the tilt-angle was near 0°. This work shows that tilted-RPA
533 measurements can also be understood if this new RPA model is used.

534 Now switching to the angular dependency of the collected current $I_{\phi_A=0}(\alpha_T)$ analysis, the
535 comparison between the transfer function model and Express-A data shows that the model prediction
536 overestimates the collected current at low firing angles and underestimates it at large angles. The
537 reasons for these discrepancies can be attributed to plasma effects in the plume (electrostatic effects
538 and collisions). Indeed, the Express-A RPA measurements show that the measured thruster beam
539 divergence increases with the distance to the thruster, indicating that plasma effects created by the
540 electron pressure in the plume tend to widen the beam. To still be able to highlight tilt-angle effects
541 in $I_{\phi_A=0}(\alpha_T)$ measurements, the angular distribution of the emitted current has been fitted to match
542 Express-A measurements. It allowed to demonstrate that the tilt-angle effects are also visible on the
543 absolute value of the collected current and that the decrease of the measured current with respect to
544 the firing angle is much steeper than in reality. In previous work (13) results from modeling were
545 directly compared to raw Express-A tilted RPA measurements. It can induce an overestimation of
546 thruster beam divergence or an overestimation of plasma effects and collisions in the plume, leading
547 to erroneous conclusions about plasma properties.

548 The simple ion transport model has shown some limits and underlines the necessity to model plasma
549 effects in the plume to predict more accurately the shape of $I_{\phi_A=0}(\alpha_T)$. One solution would be to
550 integrate this model into software such as SPIS (or another plume model) solving plasma effects and
551 plasma-surface interactions so as to obtain more accurate results. This will fall within the scope of
552 future work.
553 The present probe model can also be applied in vacuum facilities where it is not always possible to
554 have a perfectly oriented RPA due to the shape of the chamber. This could be interesting to acquire
555 more precise backflow measurements. The distance between the plasma source and the RPA needs to
556 be large enough to consider a point source and point RPA. Moreover, the simplicity of this kind of
557 probe model (with respect to much more sophisticated codes using hybrid or Particle-In-Cell
558 methods) means it can be integrated into other numerical tools or used to solve the inverse problem.
559 For example, the probe model can be combined with an optimization code and a more advanced
560 plasma model to extract IEDF parameters using RPA data as input.

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582
583 **6 References**

584 1. Sarrailh P, Hess SL, Mateo-Velez J-C. "Simulations of plasma thruster effect on the electrostatic behavior of spacecrafts in GEO," in *AIAA SPACE 2015 Conference and Exposition*, Pasadena, CA, paper AIAA-2015-4640, doi:10.2514/6.2015-4640

587 2. K. H. Wright et al., "Electrostatic Discharge Testing of Multijunction Solar Array Coupons After Combined Space Environmental Exposures," in *IEEE Transactions on Plasma Science*, vol. 40, no. 2, pp. 334-344, Feb. 2012, doi:10.1109/TPS.2011.2174447

590 3. Likar JJ, Bogorad AL, Malko TR, Goodzeit NE, Galofaro JT, Mandell MJ. "Interaction of Charged Spacecraft with Electric Propulsion Plume: On Orbit Data and Ground Test Results," in *IEEE Trans Nucl Sci* (2006) **53**:3602–3606. doi:10.1109/TNS.2006.885107

593 4. Yan L, Wang P-Y, Ou Y-H, Kang X-L. "Numerical Study of Hall Thruster Plume and Sputtering Erosion," in *Journal of Applied Mathematics* (2012) **2012**:1–16. doi:10.1155/2012/327021

595 5. Reza M, Faraji F, Andreussi T. "Characterization of a high-power Hall thruster operation in direct drive," in *Acta Astronautica* (2021) **178**:392–405. doi:10.1016/j.actaastro.2020.09.008

597 6. Shan K, Chu Y, Li Q, Zheng L, Cao Y. "Numerical Simulation of Interaction between Hall Thruster CEX Ions and SMART-1 Spacecraft," in *Mathematical Problems in Engineering* (2015) **2015**:1–8. doi:10.1155/2015/418493

600 7. Muranaka T, Inanaga Y. "Development of a Numerical Tool for Hall Thruster Plume and Spacecraft Interaction Analysis," in *AEROSPACE TECHNOLOGY JAPAN* (2018) **16**:366–373. doi:10.2322/tastj.16.366

603 8. Mikellides IG, Jongeward, Gary A., Gardner BM, Katz I, Mandell MJ, Davis VA. "A Hall-Effect Thruster Plume and Spacecraft Interactions Modeling Package," in *27th International Electric Propulsion Conference* (2001), Pasadena, CA, paper IEPC-01-251.

606 9. Giono G, Gudmundsson JT, Ivchenko N, Mazouffre S, Dannenmayer K, Loubère D, Popelier L, Merino M, Olentšenko G. "Non-Maxwellian electron energy probability functions in the plume of a SPT-100 Hall thruster," in *Plasma Sources Sci Technol* (2017) **27**:015006. doi:10.1088/1361-6595/aaa06b

610 10. Kim S-W, Gallimore AD. "Plume Study of a 1.35-kW SPT-100 Using an ExB Probe," in *Journal of Spacecraft and Rockets* (2002) **39**:904–909. doi:10.2514/2.3897

612 11. Passaro A, Vicini A, Biagioni L. "Plasma Thruster Plume Simulation: Effect of Vacuum Chamber Environment," in *35th AIAA Plasmadynamics and Lasers Conference* (2004), Portland, OR, paper AIAA-2004-2357 doi:10.2514/6.2004-2357

615 12. Di Cara DM, Estublier D. "Smart-1: An analysis of flight data," in *Acta Astronautica* (2005) **57**:250–256. doi:10.1016/j.actaastro.2005.03.036

617 13. Boyd I. "Hall thruster far field plume modeling and comparison to Express flight data," in *40th AIAA Aerospace Sciences Meeting & Exhibit* (2002), Reno, NV, paper AIAA-2002-487 doi:10.2514/6.2002-487
620 14. Manzella D, Jankovsky R, Elliott F, Mikellides IG, Jongeward GA, Allen D. "Hall Thruster

621 Plume Measurements On-board the Russian Express Satellites," in *27th International Electric*

622 *Propulsion Conference* (2001), Pasadena, CA, paper IEPC-01-044. Available at:

623 http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=505965F4BFAC7A689E0E7A4C589B9A

624 DC?doi=10.1.1.618.6148&rep=rep1&type=pdf

625 15. Mikellides IG, Jongeward GA, Katz I, Manzella DH. "Plume Modeling of Stationary Plasma

626 Thrusters and Interactions with the Express-A Spacecraft," in *Journal of Spacecraft and Rockets*

627 (2002) **39**:894–903. doi:10.2514/2.3896

628 16. Boyd ID, Dressler RA. "Far field modeling of the plasma plume of a Hall thruster," in *Journal of*

629 *Applied Physics* (2002) **92**:1764–1774. doi:10.1063/1.1492014

630 17. Sitnikova N, Volkov D, Maximov I, Petrusevich V, Allen D. "Hall Effect Thruster Interactions

631 Data From the Russian Express-A2 and Express-A3 Satellites - Task 27A," (2003). Available at:

632 https://ntrs.nasa.gov/citations/20030062907

633 18. Chao CK, Su S-Y. "Charged particle motion inside the retarding potential analyzer," in *Physics of*

634 *Plasmas* (2000) **7**:101–107. doi:10.1063/1.873817

635 19. Garrigues L, Hagelaar GJM, Boniface C, Boeuf JP. "Anomalous conductivity and secondary

636 electron emission in Hall effect thrusters," in *Journal of Applied Physics* (2006) **100**:123301.

637 doi:10.1063/1.2401773

638 20. Kim S-W, Foster J, Gallimore A. "Very-near-field plume study of a 1.35 kW SPT-100," in *32nd*

639 *Joint Propulsion Conference and Exhibit* (1996), Lake Buena Vista, FL, paper AIAA-1996-2972

640 doi:10.2514/6.1996-2972

641 21. Kim, S.-W., "Experimental Investigations of Plasma Parameters and Species-Dependent Ion

642 Energy Distribution in the Plasma Exhaust Plume of a Hall Thruster," Ph.D. Dissertation, Aerospace

643 Engineering, University of Michigan, 1999.