Modelling an Ion Thruster for a Small Spacecraft in Very Low Earth Orbit

A gridded ion thruster running on two different propellants (Xenon and Iodine) and sited within a 3U CubeSat satellite, is modelled in a Low-Earth-Orbit (altitude 400 km), using a Particle-in-Cell approach. Charged exchange collisions cause a plume backflow, which results in erosion or contamination of external spacecraft surfaces. In particular, this research is motivated by evaluating the risks to solar panel arrays from plume backflow. At low altitudes there is a wide range of ambient species able to interact electrostatically with the ion plume. In particular one must consider the plasma potential of the incoming particle flux at the solid boundaries as well as a sheath comparison and the temperature of the solar panel surfaces. This enables evaluation of the effect of temperature on the charged exchange region. A Particle-in-Cell is used, with a hybrid approach where electrons are treated as a fluid and the propellant as kinetic particles. Our results compare surface ablation for two propellants, focusing on solar panel arrays which are considered to be especially vulnerable to the backflow of ions from the plume expansion. It is found that the flux of incoming particles increases for lower satellite surface temperatures. Furthermore, Xenon results in having a lower overall effect on the sensitivity of the particle flux in comparison to Iodine.


Introduction
The working principles of electric propulsion devices were derived by Robert Goddard in 1906 [1].Between 1929 and 1933 came the first realisations of the importance of using a large atomic mass propellant [2] and of beam neutralization.The typical configuration of a gridded ion thruster used today was conceived around 1954 by Ernst Stuhlinger [1].
The main technical and scientific development in the past years has been aimed at optimising efficiency [2].Many recent electric propulsion systems are for small satellites [3] due to their increasing importance in creating mega constellations, such as the Iridium constellation [4] and the OneWeb project [2].
By 2019, more than 500 active spacecrafts have used an electric propulsion device [5] for orbit raising, orbit control and for atmospheric drag compensation, especially in Low-Earth-Orbits.Applications for deep space exploration [2], debris avoidance, and end-of life maneuvering [6] have also been undertaken.Precise manoeuvrability is the main attraction of electric propulsion and ion thrusters, which in addition offer low propellant consumption and high specific impulse when compared with traditional chemical thrusters [7].The two most used and studied propellants were mercury and caesium [1].However, currently the most commonly used propellant for ion thrusters is the inert gas Xenon [2].Xenon is a rare resource and this has motivated exploration of alternatives [8].
Important elemental characteristics when considering a propellant are its atomic mass, ionization energy, boiling point and its ionization cross-section.In this work we compare the standard propellant, Xenon, with a novel alternative: Iodine.We model a series of solar array configurations in a CubeSat, first in vacuum conditions and then in a Low Earth Orbit environment.A range of satellite surface temperatures enables us to evaluate the effect the surface temperature has on the incoming flux of propellant particles and the consequences this may have.
This paper starts by introducing the plume physics of an ion thruster, as well as an evaluation of the two propellants being compared in this paper (Xenon and Iodine) in section 2. Section 3 describes the numerical model used throughout this paper.The results of two solar panel configurations as well as simulations across a range of surface temperatures corresponding to that of the environment at an altitude of 400km are presented in section 4.

Background
The integration of ion thrusters to spacecraft present system engineering challenges.One of these challenges is the effect the plume of the thruster has on the spacecraft and thus designers often seek to minimize the interactions between both of these.Often, there is a trade-off between a configuration that aids the thruster's efficiency and one that would preserve the life of other components in the spacecraft system such as solar arrays [1].

Plume Physics
The plume of an ion thruster is made up of of ions, neutrals and electrons [1].These ions and neutrals arise from the ionised and non-ionised propellant particles, which are expelled from the thruster.The electrons are emitted by a neutralising cathode in order to keep the plume quasi-neutral.The ionised propellant can be divided into two categories: fast moving ions and slow-moving ions.Fast-moving ions carry high energies imparted by the accelerator grids.Slow-moving ions are created by charged exchange collisions between the fast-moving ions and neutral particles which initially leave the thruster at thermal velocity [9].Slow moving ions are produced in a region surrounding the thruster exit that will be henceforth be denoted as the charge exchange or CEX region.A diagram illustrating the plume is shown in figure 1.
Figure 1: The different plume regions surrounding a spacecraft (the colours of the different regions are: red the core of the plume at the thruster exit, blue the plume expansion and green the charge exchange).
Although ion thrusters typically have high mass utilization efficiency, CEX collisions are the dominant collision process near the thruster exit region [10].Other types of collision that occur at the plume consist of ionisation, and recombination.Recombination is particularly important for interactions in high-density discharges [11].There are also momentum exchange collisions (MEX) where two particles interact, and the faster particle transfers a part of its momentum to the slower moving particle.The two main collision types investigated throughout this paper are CEX and MEX collisions.Ionisation and recombination are beyond the scope of this research.In the CEX region, there is backflow of slow-moving ions towards vulnerable parts of the satellite body that is caused by the electric field developing a radial component.These ions are more susceptible to electrostatic forces due to their lower velocity and therefore some displace backwards towards the satellite.This produces "charged exchange wings" [9].Two forms of damage may be caused: sputtering (erosion and deformation) and deposition on the spacecraft surfaces.In the case of solar panels, external erosion is caused by the incoming particle flux [1].Sputtering is caused by a momentum transfer between the incoming particles and the surface of the spacecraft.Over long timescales, these charged exchange wings can expand and wrap around the spacecraft to form a plasma sheath.If the charged particle density is high enough, a full envelopment, known as "plasma cut-off" can occur [1].The presence of these charged particles can cause the spacecraft surfaces to become excessively charged, affecting the sensors, cause communication blackouts or destructive arcing [12].This is a particular concern for alternative propellants such as Iodine that are not noble gases, since they are more likely to deposit and react with surfaces than propellants such as Xenon.

Propellant Model
The desired properties for a propellant are for it to have a low ionisation energy, large collision cross-section, low reactivity, high densities and low cost [13].The characteristics as a propellant of Xenon and Iodine are shown in table 1.
Table 1: Comparison between the elemental characteristics of Xenon and Iodine [14].

Properties Xenon Iodine
Atomic mass (g/mol) 131.3 126.9First ionisation potential (eV ) 12.1 10.5 Peak cross-section (1 × 10 −16 cm 2 ) 4.8 6.0 Solid density (g/cm 3 ) 1.6 4.9 A key advantage of Iodine is that it is stored as a solid state and it can easily be brought into a gaseous state by sublimation with only a small amount of electrical power [14].This is especially important for small satellites such as CubeSats as Iodine does not require the engineering complexity of a pressure vessel [2].However, Iodine is highly reactive and all of the internal components of the thruster and spacecraft must be covered with a protective layer.External metal surfaces are usually coated with a polymer film to protect the surfaces from corrosion [2].
The dominant ion species are I + and the dimer I + 2 [8].The fraction of these two species is dictated by the applied generator power.Multiply charged ions such as I 2+ are rare and of negligible population in the thrusters plume [15].The presence of I + 2 increases the propulsive performance due to its greater atomic mass.Overall it has been shown that Iodine has similar mechanical performance to that of Xenon [15].

Modelling
The physical development and testing of ion thrusters needs expensive facilities for simulating space and orbital conditions.Numerical simulations of ion thrusters and electric propulsion devices complement experimental testing by supplying data on the thruster performance, enabling better understanding of the system.
The electric and magnetic fields present in electric propulsion devices can be described by the Maxwell equations [1].These equations take into account a vacuum in which charge ρ = s qn (q is the charge of the species Cm −3 , and n is the number density #m −3 ) and current densities The assumption of a static magnetic field ( δ ⃗ B δt = 0) is adopted as the plasma number density in the plume of the thruster is very low [1].This leads to the assumption that the current density is sufficiently low for the self-induced magnetic fields to have a negligible effect.For a typical ion thruster of diameter 10cm, the number density is around 1 × 10 15 m −3 .
The Knudsen number implies free molecular flow, as a particle is more likely to collide with a boundary rather than interacting with another particle.The open-source Particle-In-Cell code, Starfish [16] used in this study models the electric field using ⃗ E = −∇ϕ as it is an electrostatics problem.The particles motions and acceleration are obtained from Lorentz force is the velocity in ms −1 and ⃗ B is the magnetic field in T ).Particle-in-Cell codes have been used significantly in the research area of plume expansions in the field of electric propulsion devices since the early 2000s [17].

Lagrangian -Particle-in-Cell modelling
A Particle-in-Cell code models plasma as a gas containing "particles" which can be ions, electrons and neutrals.These particles follow their own Maxwellian velocity distribution function [11].In the method the charge density is calculated first; then the electric potential (from the Poisson equation where the electric field is obtained from the potential gradient).The particles velocities and positions are then updated following Newton's second law.Lastly, a sampling of N simulation particles is performed from the Maxwellian velocity distribution function in order to inject new particles into the domain.The procedure above is iteratively repeated.The Courant-Friedrichs-Lewy (CFL) conditions must be satisfied in order for the electric field to be computed as continuous [18].This means that the time-step must be smaller than the highest plasma frequency ∆t < 1 τmax (s) and small enough for the particles to not displace more than one cell length per iteration ∆t < ∆x vmax (s) [11].Although fully kinetic simulations are successful at modelling particle distributions, it is also computationally expensive.To speed up the simulations and still be able to obtain much resolution as possible throughout the runs, a hybrid solver is used in which the electrons are described through a fluid approach.Modelling electrons as fluids using the Boltzmann relationship to calculate the electron number density also allows larger time-steps [11].The hybrid solver flags nodes as quasi-neutral (Boltzmann relationship (n e = n 0 exp e(ϕ−ϕ 0 ) D < ∆x∆y, where the local Debye length (λ D in m).Otherwise, the Poisson solver (ε 0 ∇ 2 ϕ = −e(n i − n e )) is used.This allows to capture potential drops in non-neutral areas, typically outside of the core of the plume, giving more detail around the spacecraft [9].Let T e be the electron temperature in eV and e the elementary charge.

Particle-Particle Interaction
Particle collisions are modelled using Monte-Carlo-Collision (MCC) and Direct-Simulation-Monte-Carlo (DSMC) algorithms.MCC models charged exchange collisions between the present ions in the domain and the neutral particles.It treats the source as an individual particle colliding against a "cloud" of neutral particles.MCC is computationally faster than DSMC, but cannot be used for low number density scenarios as energy is not conserved due to a particle colliding against a "cloud" [11].However, MCC is applicable in this study as the density of the target particles in the thruster plume (where CEX collisions typically occur) is several orders of magnitude higher than that of the source particle.Furthermore, the collision frequency is low [2].The stochastic particle-based DSMC method is used for a source particle interacting with another individual target particle [2].The particles in the domain are paired and the probability of these particles colliding is calculated.Where a collision is bound to take place, the respective particle pair have their velocities altered [19].This allows DSMC to conserve energy and momentum during a collision (unlike MCC).The DSMC model is used here to model momentum exchange collisions (MEX).The modelled collisions occurring in the ion thruster plume and their interaction with the ambient species in the orbital environment are shown in table 2. The Variable-Hard-Sphere (VHS) is used to calculate the interaction cross-sections σ between Xe−Xe + , Xe−A, Xe−A + , Xe + −A and Xe + −A + (eq.1 from [19]).Due to the lack of available information for Iodine, eq.2 is used for computing the interactions between I 2 − I + 2 and I 2 − I + [20] [21].Also the collision cross-section between the Iodine propellant and the ambient species are modelled via a Hard-Sphere (HS) approach σ = π(r 1 + r 2 ) 2 , where r is the corresponding atomic radius (m).Unlike VHS, the HS approach is not relative to the particles velocity.It is a suitable assumption as the collisions between the Iodine particles and the ambient species often occur at high angles, causing isotropic particle scattering [18].A CEX collision has a probability of 0.5 when also taking into account the possibility of a MEX collision taking place for the same particles undergoing that collision [22].Note that d is the molecular diameter (m), and the viscosity is ω(kgm −1 s −1 ). (2)

Particle-Surface Interaction
The particles which interact with a surface are modelled as a diffuse reflection [23].The two main parameters investigated are the coefficient of restitution and the coefficient of thermal accommodation.The coefficient of restitution is calculated via, α rest = |v f | |v i | taking into account the flux of incoming energy that is lost when particles come into contact with a surface [11].The model used here assumes that the surface is at rest, and its temperature is not updated from the particle to surface interaction.The coefficient of thermal accommodation α acc on the other hand defines the fraction of incoming particles which would discard their previous known velocities, and would instead carry a velocity corresponding to that of the thermal velocity of the surface.The coefficient of thermal accommodation is of vital importance when calculating spacecraft drag or perform orbit predictions [24].A semiempirical model eq.3 [24] is used to calculate the thermal accommodation coefficient for certain ambient species in Low-Earth-Orbit.In this model, K is equal to T n being the model fitting parameter, and P is the partial pressure for the particle.These contribute to generating drag around the spacecraft, such as atomic oxygen [25].This model is based on the Langmuir adsorption isotherm, which is stated to agree with atomic oxygen data within 3% for orbits below 500km [24].

Results
Two solar panel configurations for a 3U CubeSat are evaluated in this section.The first solar panel configuration is that of a typical mounting of a 30cm×10cm solar panel onto the surface of the satellite (figure 2a) [26].The other configuration follows that of a solar sailing spacecraft as shown in figure 2b.Although the solar sailing set-up is not as common as the surface of the CubeSat placement, it is of interest comparing these configurations as solar sailing may increase in use over the next decades [27].The satellite is modelled at an altitude of 400km with an integral BIT-3 ion thruster.The BIT-3 thruster has been tested with Xenon and Iodine, hence, the set-up parameters are used throughout this study as a baseline model [15].The thruster set-up parameters used are shown in table 3.
(b) Solar sailing configuration for solar panels [27].All the surface potentials are set to 0V , including the solar panels [25].The CubeSat surface is modelled as an Aluminium 6 series, with a density of 2700kg/m −3 , and a material molecular weight of 27.The solar panels are modelled as Gallium with a density of 5320kgm −3 and a material molecular weight of 145.The mesh spacing is 5 × 10 −3 m. with a time-step of 5 × 10 −8 s in order to meet the CFL conditions.The simulations are performed at the University of Bristol's HPC Blue Pebble supercomputer [28], and each run took between 60-90 hours, depending on the case.A total of 60,000 iterations are completed for each run in this section.
Figure 3: ZR computational domain for both solar panel set-ups, where grey represents the satellite body, red is the virtual inlet, green is the symmetry boundary, and blue a zero Neumann boundary condition.

Vacuum Case
To compare both solar panel configurations, a vacuum case is set-up where the thruster uses Xenon propellant.This allows observation of the effect that the plume back-flow has on the solar panels without the influence of a LEO environment.Xenon is adopted here as it has the best available data to model the collisions in detail, so as to provide a benchmark before modelling the same satellite in Low-Earth-Orbit.The ionised Xenon number density and the velocity of the ionised particles are illustrated in figure 4 and 5.
The ionised Xenon number density plot in figure 4 for the solar sailing configuration shows a greater density of particles near the solar panel and satellite surface in comparison to the solar panel mounted on the CubeSats surface.This suggests that a greater number of Xe + particles will come into contact with the solar panel and CubeSat surfaces which may lead to a higher surface pitting and thus, cause structural/material damage.It must be noted that outside the solar panel region, the number density is similar for both set-ups, including the core of the plume.
When evaluating the velocity of the particles in figure 5, the particles carry a significantly lower velocity for the solar sailing configuration between the CubeSat and solar panel region.This decrease in velocity increases the probability of a momentum exchange collision occurring between two Xe + particles and thus, leads to a bigger expansion area -as seen in figure 4 with the number density in that specific region.Furthermore, this causes an increase in pressure around the solar panel and overall CubeSat for the solar sailing configuration (figure 6).This is due to the increase in interactions between particle-particle and particle-surface.Additionally, the flux of incoming Xe + particles that the solar panel experiences, is shown in figure 6.The solar sailing configuration has a higher maximum flux that the solar panel boundary experiences.The significantly higher level of flux for the solar sailing configuration suggests that the solar panels may experience a greater degree of particles impacting in a concentrated area and thus, resulting in a greater degree of pitting of the solar panels in certain regions causing a greater loss in their efficiency.

Low Earth Orbit
The 3U CubeSat is modelled at an altitude of 400km.The ambient species present which are simulated at this altitude are O, O + , N 2 , O 2 , He, He + , H, H + , N , N + and Ar.The number density and temperature of the ambient species is obtained from NRLMSISE-00 [29] and IRI-2012 [30], whilst their velocity is assumed to be equivalent to that of the orbital velocity at 400km.The coefficient of thermal accommodation (eq.3) is calculated for O + whereas the rest of the ambient species have a thermal accommodation coefficient set to unity.This is a valid assumption due to most of the drag in a Low-Earth-Orbit is induced by O + , and therefore modelling the atomic oxygen surface interaction is the main priority.Furthermore, there is insufficient available data on the behaviour of the other present ambient species when they interact with surfaces in Low-Earth-Orbit.The propellant particles are set to diffuse reflectively with a complete thermal accommodation due to the current uncertainty in literature about their behaviour.
A series of runs varying the surface temperature of the CubeSat and solar panels are performed in order to study the effect the temperature of the surfaces has on the incoming particles.The variation in temperature is performed for 253K, 288K and 323K.This accounts for the lowest, median and highest surface temperature the CubeSat will experience at an altitude of 400km [31].The results for the temperature variation study for both Xenon and Iodine is shown throughout figure 7, for both CubeSat solar panel configurations.Figure 7 shows that the the flux of incoming particles on the solar panel increases as the temperature decreases for the solar sailing configuration.This is due to the particles that are diffuse reflecting and re-emitting with a velocity corresponding to the surface temperature.This lower velocity increases the number density near the solar panel region and can cause further collisions to occur with the ambient species or other ionised propellant particles and thus, causing a buildup.Xenon has an overall lower flux than Iodine by ≈ 5% for the solar sailing configuration, however, for the solar panel mounted in the CubeSat surface, the difference is insignificant.The temperature to flux relation is also maintained for the other solar panel configuration on the CubeSat itself.However, for this set-up, the solar panel particle flux is quasi-constant across the entire solar panel, unlike for the solar sailing configuration.There is a greater flux of particles, particularly for Iodine, at the junction of the solar panel with the CubeSat for the solar sailing configuration, which will lead over time to a build up of charge and the formation of a sheath.The propellant velocity is also plotted throughout figure 7 illustrating a greater expansion of the ionised Iodine, carrying a similar velocity to that of Xenon, wrapping around the the solar sailing panels.This suggests that the solar panels would suffer greater pitting due to the increase in flux of the incoming ionised Iodine particles and the ambient species, and thus lower their efficiency.
Overall, even though Xenon and Iodine have similar propulsive performance, their physical behaviour due to the plume expansion and collisions are better suited to specific solar panel configurations.When the solar panels are mounted on the CubeSat surface, the surfaces experience approximately the same amount of exposure to both propellants and ambient species.However, for the solar sailing configuration, Xenon is marginally superior, leading to fewer interactions with the surface of the satellite, which would cause less pitting.Additionally, Xenon is inert, and would not cause any further chemical reaction when it comes into contact with the surface of the satellite.

Conclusion
This study compares the exposure different configurations of solar panels have on a 3U CubeSat to two propellants, Xenon and Iodine in a Low-Earth-Orbit.In addition, a specific thermal accommodation coefficient for atomic oxygen (being the major contributor to drag on satellites in Low-Earth-Orbit) is calculated.In recent years, Iodine has been seen as a potential alternative to the most used propellant, Xenon.This study offers further evaluation of alternative propellants by including collisions between the neutral and charged particles in the plume.The results have shown that, for a standard solar panel configuration, the Iodine and Xenon fluxes are similar.However, the same cannot be said for a solar sailing configuration, where Iodine has an overall ≈ 5% higher flux than Xenon.Therefore, it is concluded that Xenon is more appropriate for certain solar panel configurations such as solar sailing due to both the reduced flux, and its inert nature making chemical interactions with the spacecraft surfaces or ambient species less likely.It is also established that there is an inverse relationship between flux and surface temperature.These findings highlight the impact of the plume expansion of electric propulsion deviceson the spacecraft system.It may be necessary to research protective coating layers on satellite surfaces for certain propellants such as Iodine.

Figure 4 :
Figure 4: Xe + number density # m −3 for the solar panel placement on the CubeSat surface (top half) and the solar sailing configuration (bottom half).

Figure 5 :
Figure 5: Xe + velocity ms −1 for the solar panel placement on the CubeSat surface (top half) and the solar sailing configuration (bottom half).

Figure 6 :
Figure 6: Pressure (P a) contour (black and white scale) and Xenon flux #/m 2 /s (blue to yellow scale) for the solar panel placement on the CubeSat surface (left) and the solar sailing configuration (right).

Figure 7 :
Figure 7: Velocity of Xenon (top half) and Iodine (bottom half) ms −1 (black and white scale) and the flux of incoming particles #/m 2 /s (blue to yellow scale) comparison between the solar panels set on the CubeSat surface (left column) and a solar sailing configuration (right column) with a variation in surface temperature of 253K, 288K and 323K.

Table 2 :
Allowed particle interactions with the corresponding numerical algorithms that model these collisions.

Table 3 :
Xenon and Iodine thruster configuration used based on the BIT-3 thruster.