Parker Solar Probe FIELDS Instrument Charging in the Near Sun Environment: Part 1: Computational Model

The Spacecraft Interaction Plasma Software package (SPIS), a three‐dimension particle in cell (PIC) code, was used to model the Parker Solar Probe (PSP) spacecraft and FIELDS instrument and their interactions with the Solar wind. Our SPIS modeling relied on material properties of new spacecraft materials that we had obtained in previous work. The model was used to find the floating potentials of the spacecraft and FIELDS antennas at different distances from the Sun (from 1AU to 0.046AU). We find the following results: At greater distances from the Sun, the shadowed spacecraft charges negatively while the illuminated Thermal Protection System (TPS) charges positive due to the high resistance of the TPS Alumina shield at low temperatures. As the spacecraft approaches the Sun, the temperature of the TPS increases, the resistance between it and the spacecraft drops, and its photoemission increases, driving the spacecraft more positive. At the same time, an electrostatic barrier forms near the illuminated surface of the TPS and reflects the photoelectrons back leading to negative charging of some surfaces. The FIELDS antennas and shield also see this barrier forming but on a smaller scale. The FIELDS antennas charge positively at all distances modeled when no current bias is applied. Current biasing of the antennas affects their floating potential.

shows a rendering of the PSP spacecraft and elements of the FIELDS instrument. Previous PSP spacecraft surface charging models involved the PSP spacecraft, including the Thermal Protection System (TPS), the spacecraft radiators, and the bus. However, they did not include the FIELDS antennas (Donegan et al., 2010(Donegan et al., , 2014Ergun et al., 2010;Guillemant et al., 2012) because the necessary information to model the antennas, such as probe surface properties were not available, and the final geometry of these thin (0.0031 m diameter) 2 m long probes were unknown at the time of their publishing.
Throughout the entire mission, PSP slowly decreases its perihelion, getting closer to the Sun with each Venus encounter (Fox et al., 2016). The maximum illuminated surface temperatures rise with each pass, annealing and "baking out" those surfaces, leading to subtle changes in the spacecraft and antenna charging behavior. In addition, the solar photon flux changes as ∼1/R 2 over each orbit, where R is the distance from the Sun (Ergun et al., 2010], leading to significant changes in photoemission between aphelion and perihelion, and over the course of the mission. In order to survive and operate at the high temperatures expected at perihelion, the PSP FIELDS antennas utilize for the first time a refractory Niobium alloy, Nb-C103. A testing campaign (Diaz-Aguado et al., 2019, 2020 quantitatively characterized the physical process involved and determined the charging properties of this new material, along with other refractory materials that were used on other parts of the FIELDS antennas. This test campaign obtained the material properties for photoemission, secondary DIAZ-AGUADO ET AL. A self-consistent modeling of plasma interactions with the spacecraft, taking into consideration the actual spacecraft configuration, is needed in order to predict the FIELDS antenna charging correctly. Many spacecraft-plasma interaction software models exist, including: EMSES, iPic3D, LASP, PTetra, Multiutility Spacecraft Charging Analysis Tool (MUSCAT), NASA Charging Analyzer Program (NASCAP; Mandell et al., 2005) and Spacecraft Plasma Interaction Software package SPIS (Marchand et al., 2014;Roussel et al., 2008). This study utilized SPIS because it's an open source design allows for ready inclusion of the novel material properties and surface geometries of the PSP FIELDS antennas and spacecraft.
The main purpose of this paper is to evaluate the charging environment of the PSP FIELDS antennas and quantify its effect on measurements of the SW plasma structure and dynamics by presenting the results of the SPIS model at close encounters and by comparing the model results with 1AU. First, an overview of the spacecraft charging theory is described, including photoemission, SE emission, BSE emission, and thermionic emission. Electron and ion current theories are also included. Second, an outline of how SPIS functions is provided, followed by environments explored and materials used specific to FIELDS and PSP. Finally, the model results are discussed.

Spacecraft Charging Overview
In steady state the spacecraft's antenna floating potential is determined by the balance of various charging currents to and from the spacecraft or probe. The floating potential of a spacecraft or probe relative to the ambient plasma is determined by the current balance condition, that is the net current to any exposed surface must sum to zero (Garrett, 1981;Grard, 1973;Hastings & Garrett, 1996;Mullen et al., 1986;Whipple, 1965Whipple, , 1981. For PSP these currents are determined by the SW plasma environment, solar photon flux, spacecraft orientation and material properties, which depend on the floating potential. PSP is designed to be electrically conductive between all surfaces during perihelia to obtain similar potentials throughout the spacecraft, except for specific instrumentation, including the FIELDS antennas and shields that remain isolated to make electric field measurements possible. The current balance condition for PSP and FIELDS is I  e   I  I  I  I  I  I  I  I  (1) where I ph is the photoelectron current from photoelectron emission, I I and I e are the ion current and electron current from the plasma environment, respectively, I se secondary electron current, and I bse backscattered secondary electron current resulting from the electrons leaving a surface due to the plasma interaction with surfaces, I therrm thermionic electron current from electrons emitted from a hot body, and I other could be other currents such as sensor bias currents. Each of the currents varies with the spacecraft or probe potential (Φ) relative to the plasma potential, also known as the floating potential of each surface.
It is important to note that PSP is in a mesothermal plasma environment with plasma ion thermal velocities lower than the SW (∼300 km/s) and spacecraft speeds (up to 197 km/s), and plasma electron thermal velocities that remain greater than the spacecraft and solar wind velocities. A spacecraft in a mesothermal plasma forms a wake behind it (Ergun et al., 2010;Wang & Hastings, 1992;Wang & Hu, 2018). In addition to negative potential wake, for PSP an electrostatic barrier forms in the front of the spacecraft, as the ambient electrons penetrate the barrier while the photoelectrons and SE cannot (Ergun et al., 2010). These potential barriers can cause the spacecraft to charge negatively in the Sun-exposed surfaces, as shown in the results below.
Photoemission, emitted electrons that escape from a surface where an energetic photon has impacted, is material, solar flux and angle of incidence dependent (Feuerbacher et al., 1972;Garrett, 1981;Hastings & Garrett, 1996;Whipple, 1981). Photocurrent is the highest current PSP experiences due to 1/R 2 increase from aphelion to perihelion, and the expected densities and temperatures of the ambient plasma.
Past measured J ph0 have ranged from 20 to 60 μA/m 2 at 1 AU, which is often several times the flux observed in ground experiments (Ergun et al., 2010). The FIELDS antennas and shields, composed of Nb-C103, have a tested range of J ph0 between 49 to 139 μA/m 2 (Diaz-Aguado et al., 2019) depending on the material state (annealed vs. unannealed) and solar cycle and activity (Diaz-Aguado et al., 2019;Sternovsky et al., 2008). Current flight estimated Nb-C103 J ph0 is closer to 240 μA/m 2 .
As shown in the results, the photocurrent is the highest induced current source on PSP, though followed by SE currents, at one order of magnitude smaller for FIELDS and two orders of magnitude smaller for the spacecraft, reducing the influence of the material properties of SE. SE are defined as emitted electrons due to kinetic impacts with primary incident electrons or ions. For the FIELDS antennas, the ambient electron temperature of the SW is increasing from a few eV to tens of eV, in the range where the SE yield becomes greater than one, causing the departing current of SE to be greater than the arriving environment electron current. Secondary electron (ion induced) yield results in similar induced currents as the ion current due to high impact efficiencies, I sei ∼ I i (Ergun et al., 2010). SE can be approximated by an isotropic Maxwellian distribution, with a characteristic energy of T se ∼2 eV (Hachenberg & Brauer,1959;Lai, 2012Lai, , 2013. Estimates of the SE yield due to electrons have been difficult to predict theoretically, but by knowing that the yield is strictly dependent on the incoming energy and angle, we can look at J e /n e and material dependent J se /n e (T e ) figures to predict at what ambient electron temperatures the SE currents are greater than the electron current (Diaz-Aguado et al., 2020).
Note that past studies of the SE yield of conductive materials at temperatures greater than 600 K, have shown a decrease in yield on the order of 0.05%/K (Michizono et al., 2004;Sternglass, 1954;Warnecke, 1936). For the first perihelia the predicted antenna surface temperatures are ∼885 K, and the yield reduced by 44% relative to results at 270-300 K; for the closest perihelia, the expected decrease in SE yield is even greater, on the order of 79% (Diaz-Aguado et al., 2020). These temperature-dependent effects were not modeled in this analysis as photocurrent dominated the current balance by an order of magnitude.
SW ambient electron currents are in the same order of magnitude as SE currents for PSP and are considered Maxwellian. If the surfaces are charging positively, not only are the ambient electrons attracted to the surface, but photoelectrons, SE and BSE are also attracted, creating currents.
In contrast with the ambient electrons, the ions can be considered as a cold, nearly monoenergetic beam. The velocity of the SW varies depending on the distance from the Sun and solar activity, but it is in the range of 300 km/s, corresponding to a proton kinetic energy of 1-2 keV in the spacecraft frame. This means that the SW can penetrate barriers as high as 1-2 kV (Ergun et al., 2010), and as shown by other modeling studies, narrow negative potential wake forms behind a cylindrical body (Engwall et al., 2006;Guillemant, 2014]. The BSE current is in the same order of magnitude as the ion current for PSP. BSE occur when electrons impact and enter the surface, but through collisions they eventually reverse direction to leave the material. The currents can be similarly calculated as the SE currents but using J bse /n e (T e ) figures. In this study, the BSE are defined as those SE backscattered with energies above 50 eV (Katz et al., 1977). This definition is the most commonly used in the spacecraft charging community, and necessary to define material properties for NASCAP and SPIS.
As PSP approaches the Sun with each orbit with a smaller perihelion, the TPS and instruments could experience sufficiently hot temperatures to undergo thermionic emission during the closest perihelion passes. The FIELDS instrument thermal predictions have temperatures reaching 1600 K, and thermionic emission becomes a primary current with similar orders of magnitude to photoemission. This thermionic current can be calculated using the Richardson-Dushman Law (Modinos, 1982;Richardson, 2013], and assuming similar orbit limited behavior: and where A TS is the total surface area, G therm is a shape factor and 0 therm J is defined as: where A is the material specific Richardson constant, k is the Boltzmann constant, T is the surface temperature (in eV),  is the work function of the metal, and G therm is the fraction of the thermionic electrons that escape as a function of surface potential, surface radius, and sheath radius. The temperature of the thermionic electron leaving the surface is assumed to be 2 eV, similar to SE and photocurrent electron emission temperatures.
For FIELDS, we only have the workfunction of Nb C103 (Diaz-Aguado et al., 2019, 2020) and not the Richardson Constant, A. The analysis presented uses the value of A for pure Nb to obtain the thermionic current of the antenna.
The Richardson Constant A varies significantly between materials and within the same material depending on measurement method. For example, for pure Nb, two different research measurements gave values of A between 32.7 and 57 (A/cm 2 /K 2 ; Fomenko, 1956). Figure 2, a., shows the predicted thermionic electron current emitted by the antenna during the mission as the spacecraft perihelion decreases in altitude. Figure 2, b. shows the thermionic and photoemission current density of the antenna versus the distance to the Sun. While the thermionic current from the antenna was not included in this study because of issues with the current SPIS version (6.0.0), one can see that at heliocentric distances of 30-40 Rs, the thermionic current is at least four orders of magnitude less than the photoelectron current, and so does not have significant effect on the model results shown below.

SPIS Software and Numerical Simulations
The PSP charging models shown below were implemented using the SPIS package (Thiebault, 2013). New materials and material properties were added to the SPIS database. SPIS is an electrostatic unstructured three dimensional mesh, particle in cell (PIC), plasma modeling software, that uses JAVA, making it highly modular. SPIS provides a modeling framework to build up the antenna and spacecraft geometry and materials, the plasma environment and the interaction between them. It then uses the Vlasov-Poisson equations to self-consistently solve for the potential distribution, Φ, including the potentials on the FIELDS antenna, FIELDS shield and spacecraft surfaces. For example, the time independent Vlasov equation is shown in the following equation:  Where v is the velocity of the particle, f is the distribution function, E is the electric field, B is the magnetic field, q is the particle charge, m is the particle mass. To obtain the potential of the spacecraft or probe, we use Poisson's equation: where ε 0 is the permittivity and the current is the summation of different particle densities, where n is the number density given by the following distribution function equation: where x is the position of the particle, v its velocity, t is the time elapsed.
The software uses a particle-in-cell, or PIC approach, in which it is important to specify a sufficiently small simulation time step Δt. This Δt should be selected to ensure that the fastest particles in the simulation move less than one simulation cell in a single time step. It should also be smaller than the plasma characteristic timescale, or the plasma period T p = 1/ω pe , where ω pe is the plasma frequency, and can be calculated using the following equation: For PIC simulations, the plasma Δt should be less than 0.2T p to ensure proper modeling of the electric fields and avoid modeling erroneous electron oscillations.
SPIS offers the use of various velocity distribution functions, f, for the particles. For ambient electron an isotropic, non-drifting Maxwellian distribution function was used: where v th is the average thermal velocity.
Past solar wind observations have found the Kappa function could be a better fit than Maxwellian function (Gloeckler et al., 2006;Guillemant, 2014;Halekas et al., 2008;Leitner et al., 2009;Maksimovic et al., 2005;Pierrard & Lazar, 2010), including non-Maxwellian supra-thermal tails. Kappa distribution function can be defined as: where θ is the Kappa average thermal velocity and is defined as: where m e is the electron mass.
As the SC nears the Sun the Kappa values increase and become more Maxwellian. The exact Kappa values were unknown at the time of this publication, but past research (Ko et al., 1996;Maksimovic et al., 2005) suggest that it is κ > 7 (at 0.25AU), and closer to κ = 10 for 9.8 Rs. This research found that Kappa function does not heavily influence the SC potential results (few volts), since the current balance equation is dominated by the photoelectron current, as shown in the results section.
While past observational results have found that the ambient electron distributions are better described by heavy-tailed Kappa distributions rather than a Maxwellian (Gloeckler et al., 2006;Guillemant, 2014 have utilized Maxwellian electrons, and so our study will also in order to better facilitate comparison with those prior modeling results.
From this modeling framework and results, various case studies in different operational regimes are used to provide a full set of predictions for antenna, spacecraft, and antenna-spacecraft plasma interactions on PSP FIELDS. Figures 3 and 4, the PSP spacecraft and FIELDS antennas have unique shapes, driven by the requirement to protect the thermally sensitive portions of the instruments and spacecraft from the radiant heat of the Sun. PSP's sun facing side consists of a TPS, or sun and heat shield that protects the rest of the spacecraft. The TPS is attached to a Ti frame that holds the spacecraft radiators and four of the FIELDS antennas. As shown in Figure 3, the FIELDS antennas and shields were first stowed along the spacecraft body in order to fit them within the launch fairing and secure them against launch loads and vibration. They were later deployed and are exposed to the solar flux and SW.

As shown in
The FIELDS antennas had their own small Sun shields near the spacecraft to be able to reduce the heat flux going from the antennas to the instrument electronics. We modeled and analyzed the deployed state of the antennas as follows.
As shown in Figure 4, a simplified geometry was used in SPIS for the spacecraft body, TPS, FIELDS shield and antenna. A 1 m diameter and 1 meter long, cylinder with similar outside surfaces was used for the spacecraft (in actuality it is hexagon prism). The solar panels were not modeled, as this paper was mostly concerned about the FIELDS antennas. The solar arrays were not modeled, as the arrays were mostly in the shade at 35 Rs, and the dominating current of the spacecraft was the photoelectron emission of the TPS.
10.1029/2020JA028688 7 of 22  Because of meshing convergence difficulties due to geometries of the model, only one shield and one antenna were modeled, and several cases run with different orientations of the ram velocity, 90° and 180°, at the first perihelion (35 Rs), to reveal any differences in charging. Figure 4b shows the simulation unstructured mesh, with a size of 1 m at the sphere, 10 cm at the spacecraft, 8 cm at the TPS, 1 cm at the shield and 3 cm at the antenna. The spacecraft model is centered within a 16-m radius simulation volume, at least twice the Debye length of the plasma environment, and tens of times the effective Debye lengths of the photoelectron and SE populations. Figure 4 also shows the x,y,z axis, with the z axis aligned with the spacecraft away from the Sun, y axis aligned with the antenna, and x forming a right-handed triad with y and z. A finer mesh, as seen in Table 5, was used for 9.5 Rs to ensure that the Debye length was greater than the mesh cell near the spacecraft, ensuring that the grid was less than half the Debye length in the sheath.  Table 2 for Al 2 O 3 conductive properties as they are thermally dependent, and therefore dependent on distance to the Sun.

Table 1
Material Properties Used in Surface Charging Calculations (Diaz-Aguado et al., 2020;Donegan et al., 2014;Thiebault, 2013) The ion SE yield properties of Nb-C103 were not known at the time of publishing, instead the properties of Aluminum were used. Given that the ion current is small, this assumption has no significant effect on the modeling results. The average photoelectron yield values between solar maximum and solar minimum were used for both the Nb-C103 annealed and unannealed. The photocurrents are predicted to vary by up to ± 17% for the unannealed Nb-C103 and ± 15% for the annealed Nb-C103, depending on solar activity (Diaz-Aguado et al., 2019). Table 2 shows the variable conductivity of Al 2 O 3 due to temperature. As the spacecraft nears the Sun, the temperature of the Alumina increases, and its conductivity increases, improving the electrical connection between the illuminated and shadowed portions of the TPS and spacecraft.
As shown in Figure 5, the charging model consists of four different groups of surfaces, or nodes: Spacecraft, Radiators and TPS-foam were node 0, TPS-Sun is node 1, FIELDS Shield is node 2, FIELDS antenna is node 3. As shown in Table 1, the spacecraft, Radiators and TPS foam are considered all to be Node 0. Node 1 is the TPS shield and is connected to Node 0 through a variable resistor, which is dependent on the electrical properties of alumina, as shown in Table 2. Node 2 is the antenna and Node 3 is the antenna shield.
The models were run in two configurations: First, Node 2 and Node 3 were free floating for all environmental cases; second, Node 0 and Node 3 and Node 0 and Node 2 were connected with a variable differential voltage for the first perihelion pass environments only to model the conditions during the inflight I-V bias current sweeps, shown in Part II. The authors would like to note that the current future closest approach will be at 9.8 Rs, compared to the modeling results shown at 9.5 Rs. The modeling results have a 6% higher radiant flux and photoelectrons flux, which increases the number of electrons leaving compared to current estimated closest approach, influencing the photocurrents but with a small impact on the potential trends. PSP is exposed to the SW plasma environment near the Sun's equatorial plane. Table 3 summarizes the predicted parameters of that plasma environment during various phases of the PSP mission. The response of the ambient electrons, PE, and SE to the potential structures around the spacecraft and antennas was modeled using PIC. The SW ions were also modeled using PIC.
To account for the great changes in PSP orbital velocity from aphelion to perihelion, the SW ion velocity in the spacecraft frame was modeled differently depending on the distance from the Sun. As the spacecraft nears perihelion, the velocity increases, as shown in Table 3. Near the Earth and Venus, the velocity vector of the SW had a 45° angle in the z-y plane. At science operations (0.25AU and closer) the SW was more radial, in the z direction, and the velocity of the spacecraft was perpendicular to it in the y or ram direction.
It is worthy to note that the ambient electron Debye length decreases as the distance from the Sun decreases and is within the same dimensions of the spacecraft and FIELDS antennas during perihelion passes. The photoelectron and SE Debye lengths also decrease as the distance from the Sun decreases but are smaller than the dimensions of the spacecraft. These Debye's lengths are shown in the results.
SPIS uses super-particles injected in each cell to represent dynamics of individual groups of particles. The smaller number of super-particles are easier to track and make the PIC processing less intensive. Each of them represents a group of physical particles ruled by physics. For this analysis, super-particle numbers per cell ranged between 10 and 15 for electrons and ions, 5 for photoelectrons, 3-2 for SE, 1-2 for BSE and ion induced SE, totaling 9.3 million super-particles for 35 Rs, and 16.4 million super-particles for 9.5 Rs. As shown in Table 4, different ram directions (0°, 90° and 180°) in the x-y plane were studied at 35 Rs, in order to discern any aspect-dependent charging effects on the antennas or spacecraft. For reference, the Sun is at -z direction. Table 5 shows the major numerical inputs from the SPIS runs and typical mesh size. The timestep (Dt) was held smaller than usual because the mesh had tetrahedron angles smaller than 60°. The capacitance of the spacecraft was held at 2 × 10 −10 F, but was varied to slow/speed up modeling results. Table 3 also shows the electron plasma frequency, Debye lengths and electron gyrofrequencies.

Numerical Results Unbiased FIELDS Antenna and Shield
Tables 6 and 7 show the floating potentials (i.e., potential of surface relative to the potential of the outer simulation boundary fixed at 0 V) of the SC, Radiators, TPS, FIELDS antenna and FIELDS shield, for heliocentric distances and predicted plasma conditions from 1AU to 0.0495AU (9.5 Rs), and for both unannealed and annealed photoelectron and SE yields. All models included a magnetic field, but it had little effect on the spacecraft and FIELDS instrument potentials from models run with no magnetic field. Studies by (Guillemant et al., 2017) showed how ∼30 times the magnetic field expected for Solar Orbiter (0.25AU) changed the potential of the Radio and Plasma Wave (RPW) antennas by only a few volts, and so the relative insensitivity of the model to B-field effects is consistent with past results.
In these runs, one can see that the floating potential of the spacecraft and TPS were highly dependent on the conductance of the Alumina. At closer distances to the Sun, the TPS and spacecraft floated to similar potentials as shown in Tables 6 and 7. At Earth and Venus where the Alumina temperature is predicted to be markedly lower and significantly more resistive, this significant isolation resistance allows the shadowed spacecraft to charge negative as the current from the ambient electrons and SE is higher than the ambient ions, while the illuminated TPS charges positive due to high photoelectron currents. Table 6 also shows the effect of changes in the ram direction of the SW as described in Table 4, and of reductions of SE yield due to temperature increases of the surface.
The proximity of the ion wake negative potential to the antenna at Ram 0° had minor effects on the antenna, a few tenths of a volt of change. The direction of the ion flow had very little influence on the floating potentials of any of the surfaces. This isn't surprising given the three orders of magnitude difference be-DIAZ-AGUADO ET AL.    tween the dominant photoelectron, SE, and ambient electron currents and the ion currents.
Spacecraft floating potential does not depend significantly upon the FIELDS antenna and shield characteristics and their floating potentials as can be seen by comparing Table 5 with Table 6. Spacecraft floating potentials were similar to predictions, except for 9.5 Rs which was more negative (Donegan et al., 2014) modeled the slow and fast SW at different heliocentric distances, predicting spacecraft floating potentials at 0.25AU between −0.2 and 9.2 V, at 35 Rs between 1.0 and 8.0 V, and at 9.5 Rs between −3.3 V and −8.8 V. For the 35 Rs, the spacecraft potential predictions are within past models, but for the 9.5 Rs case, the potential of the spacecraft is more negative (Donegan et al., 2014). also modeled an extreme, post shock-case, with a spacecraft floating potential prediction of −31 V.
The annealing effects in Table 6 show the floating potential of the shield and antenna lower than the unannealed cases as expected given the reduced PE yield of the annealed materials. The spacecraft material properties and dimensions stayed constant, while the PE yield of the antenna and shield materials were reduced by annealing, decreasing their floating potentials and that of the spacecraft as well; as fewer electrons leave the shield and antenna surfaces, the potential becomes less positive to maintain current balance. Table 8 shows the current source comparison between 1AU (215 Rs) and 0.16AU (35 Rs) for unannealed Nb-C103. At 219 Rs the total currents are two order of magnitude smaller than at 35 Rs. The photocurrent is dominating in both cases, by two orders of magnitude at 219 Rs and by one order of magnitude at 35 Rs. The SE current is two orders of magnitude smaller at 219 Rs compared to 35 Rs, which is only one order of magnitude smaller for the entire spacecraft, including the antennas. The ion current is two orders of magnitude smaller than the photocurrent at 1AU, while it is up to three orders of magnitude smaller at 0.16AU. The SE due to ions are in the same order of magnitude as the ion current. The BSE currents are also small, several orders of magnitude smaller than the photoemission and not an important factor in the current balance in both environments.
As explained in the materials section, the high temperature on the antennas could have effects on their potentials by reducing the SE current. The potential differences are shown in Tables 5-6, while the currents are shown in Tables 5-8 The SE currents are reduced on the FIELDS antenna by 44% at 35 Rs because of the SE yield reduction due to temperature, but with similar potential results. The spacecraft potential is negative at 219 Rs as it is isolated from the TPS shield, compared to the positive charging of the TPS and FIELDS instrument. At 35 Rs, the TPS shield becomes more conductive, making the spacecraft dependent on the photocurrent of the TPS and charge positive.   Table 4 Graphical Representation of Different Ram Direction Cases relative to outer simulation boundary, respectively). Negative potential wells with a depth of −3.25 V form in front of the TPS and in the ion wake of the spacecraft, not as deep as those previously found at 9.5 Rs by (Donegan et al.. 2014;Ergun et al., 2010;Guillemant et al., 2012). These wells do differ in depth and dimension from those observed in previous studies because previous studies focused on 9.5 Rs, while this research focused on 35 Rs. At 35 Rs, the photoemission fluxes are lower as the solar flux is lower, the electron and ion number density and temperature, are lower, which translate also to lower SE fluxes. That said (Donegan et al.. 2014;Ergun et al., 2010;Guillemant et al., 2012), found the negative potential well in front of the TPS was less deep than the wake potential well. Figure 8 shows these differences in a comparison between our runs at 35 and 9.5 Rs. Note that the color scales for the potential are different between the two cases in order to show the extent and depth of the potential wells. In these cases (Donegan et al.. 2014), predicted negative wake potential wells charging from −20 V to −36 V, compared to −23.9 V in Figures 5 and 6, but their plasma parameters varied from the ones in this study with their densities ranging from 1.2 × 10 3 cm −3 to 4.1 × 10 4 cm −3 , electron temperatures ranging from 48.6 and 59.7 eV, and ion temperatures ranging from 40.5 to 223.1 eV.
The plasma potential around the shield (a.) and the antenna (b.) are shown in detail in Figure 9, for the antenna at 90° RAM. The antenna and shield wakes at 0° RAM case join the wake of the spacecraft and therefore are not discernable and more negative. Similarly to the TPS in Figure 7, a negative potential well forms in front of the shield and the antenna. Note that the well in front of the antenna is not as negative as the shield and TPS. The wake from the significant proton flow (solar wind plus PSP orbital velocity) can also be seen.  Figure 11 shows the plasma potential of PSP and the antenna in Volts. The negative potential well in front of the TPS and antennas are not seen at 1AU compared to 0.16AU. The wake potentials are also different due to a different angle of attack of the ions, and lower density of the ions at 1AU. The TPS shield and the spacecraft are isolated from each other at 1AU and charging at different potentials. At 35 Rs the TPS shield and spacecraft charge to similar potentials. The FIELDS antennas decrease their potential. All potentials are shown in Table 6. Figure 11 shows the Log electron plasma charge density. The electron density increases as the spacecraft approaches the Sun. The electron number density figure at 1AU (left) is smoother due to a larger scale, compared to the 35 Rs figure (right) which has a much smaller scale. Figure 12 shows the ion number density. The wake is seen in both 1AU and 0.16AU, but as the velocity of the spacecraft increases, the wake has a larger ram component. The ion number density is shown with a linear scale to better capture the wake DIAZ-AGUADO ET AL.   structure. A low-density ion region forms opposite the impinging ions from the ram and solar wind. Higher electron mobility leads to negative space charge filling the wake, which forms a negative potential area, as seen in previous simulations (Ergun et al., 2010;Guillemant et al., 2012). The negative potential well in front of the TPS supported by PE and SE populations, as well as the one in the wake supported by ambient, PE, and SE electrons, repel ambient electrons, leading to reduced ambient electron densities in those locations, as seen in Figure 12b. Figure 13 shows the photoelectron number density. The photoelectron density is much higher for 0.16AU as expected, as it is closer to the Sun. The photoelectrons produced fill the wells in front of the TPS and to a lesser degree, the antenna well. Figures 5-14 shows the SE charge density. The SE number density is one order of magnitude smaller than photoelectron number density at 0.16AU.
It is important to note that near the antenna and TPS, the photoelectron number density is one order of magnitude greater than the SE number density, and two orders of magnitude greater than the ambient electrons and ions. Compared to the spacecraft, the antenna and its shield are exposed to the Sun and are photoelectron current dominated. The photoemission electron number density (∼1e10 1/m 3 ) is the highest of all particle densities by at least an order of magnitude within a region several meters away from the spacecraft and antennas. Figure 13 shows how the photoelectrons at 35 Rs occupy the environment near the spacecraft, compared to that at 1AU, where it concentrates mainly on the TPS and the antenna. Similarly, Figure 14 shows the electrons occupying the near spacecraft environment at 35 Rs, in contrast to 1AU, where a singular structure forms around it. The SE concentrate in front of the TPS shield, near the antenna, close to the side of the spacecraft and in the wake. It is significant to note that the Debye lengths are shorter at 35 Rs than at 1AU. Figure 15 shows the BSE charge density, at two orders of magnitude smaller than photoelectron density. The BSE are attracted to the TPS shield and antennas which are charging positive at both 1AU and 0.16AU. Figure 16 shows the ion SE charge density, also at two orders of magnitude smaller than the photoelectron charge density. At 1AU the ions hit the left corner of the spacecraft, causing a small ion SE source. At 35 Rs, the ions impact a larger surface, showing a larger ion SE source on the spacecraft. This difference is due to the greater ram velocity at 35 Rs.    Figure 17 shows the particle densities in front of the TPS as a function of distance for 1AU, 0.16AU and 0.045AU. The TPS shield is located at −0.2 m. It also includes the potential as a function of distance. For 0.16AU and 0.045 AU the potential in front of the spacecraft has virtual cathodes (negative potential wells as shown in Figures 7, 8 and 9). Space-charge-limited currents which cause this virtual cathode are determined by the PSP spacecraft sheath. They have been studied extensively by many authors, including Langmuir (Langmuir, 1929). Bohm (Guthrie, 1949), Crawford and Cannara (Crawford & Cannara, 1965), Prewett and Allen (Prewett & Allen, 1976), Marese et al. (Ketsdever & Micci, 2000), Wang and Lai (Wang & Lai, 1997). Figure 18 show that the minimum of these wells occurs when the number density of the photoelectrons (plus other negative charge densities) become larger than the ion charge density, creating an    In addition, the modeling results show the net current density of the antenna, where the tip has positive current density while the area near the shield has negative current density. This negative density is mostly due to photoelectrons and SE both attracted from the shield. To reduce the influence of this current from the shield on the antenna's floating potential, a voltage bias is imposed between the shield and antenna, with the results shown in Part II of this paper.

Sensitivity Analysis
Various sensitivity analyses were conducted on the PSP FIELDS model to find both the main current source contributor and verify the model. A variation on electron and ion density, ion velocity, photocurrent and magnetic field were introduced in the modeling parameters at both 1AU and 35 Rs. The previous section showed the differences in currents between the photoelectron yield of the FIELDS Nb-C103 annealed versus unannealed. The potentials clearly showed a dependence on the photocurrent yield, but there was also a change in the SE yield due to annealing, even though this change was less prominent. Multiple runs were performed at 1AU and 0.16 AU (35 Rs) with a variation of environmental inputs.
10.1029/2020JA028688 16 of 22  The models were run with a variation of the distance from the Sun to vary photon flux, and hence photoemission, while keeping the electron and ion densities and velocities the same. The 1AU model (nonconductive TPS) was run at 0.7AU, double the photon flux, and hence the Sun exposed surface photoemission, and at 1.4AU, half the photon flux. At 0.7AU the photoemission current and hence the total currents doubled. The TPS shield and FIELDS antenna and shield remained positive and attracted the electrons. Their potential though decreased by a few volts (∼3-5 V). The spacecraft potential remained equal as the electron and ion environment were not changed and the TPS and the spacecraft were isolated from each other. At 1.4AU, there was half the photon flux than at 1AU. Photocurrent also halves, decreasing the number of electrons emitted which reduces the potential of the surface. The photoelectron density decreases with respect to other electrons at 1AU. Photon flux variation was also run at 35 Rs with different results, simulating the photon flux at 53 and 27 Rs while maintaining plasma densities constant. At 35 Rs, the potential of the surfaces was greater than at 53 Rs and continued to increase at 27 Rs. The photoelectrons increased, which increased the potential.
The model was run with no SE yield to confirm that the SE current was not a predominant influence on the potential during close encounters. At 35 Rs it was found that the SE yield did not influence the potential charging greatly (minus a few millivolts on the antenna and minus one volt on the spacecraft). Like the antenna, the shield and the spacecraft were charging positive, and the emitted electrons were attracted by the positive potential surfaces. At 1AU, the SE had a greater influence on the surface potentials, but just by a few volts (∼2 V). If the SE yield is removed, the spacecraft potential decreases by a few volts (∼3 V), the TPS shield decreases by half a volt and the antenna reduces its potential by five volts.
The models were also run with a variation of electron and ion density while keeping the photoemission constant. At 1AU, the model was run with higher ambient plasma density (by one order of magnitude). The higher density of the plasma causes the potential of the antenna, the TPS shield and the spacecraft to DIAZ-AGUADO ET AL.  decrease, showing a dependency on the ambient thermal electron density. The model was also run with a smaller order of magnitude of ambient plasma density. In comparison with the higher density, the potential increased by tens of volts for the spacecraft, TPS shield, and antenna. At 35 Rs (0.16AU) similar electron and ion density variation was performed. When the electron and ion densities were doubled, the potentials decreased by few volts on the spacecraft, TPS and the antenna. The model was then run with the density halved. The potentials of the spacecraft and the antennas increased by a few volts. At 35 Rs the TPS shield is conductive, making the TPS shield and spacecraft float at the same potential.
The magnetic field was also modified from expected values to see no changes at either 1AU, 35 Rs nor 9.5 Rs. The gyroradius of the electrons remains much larger than the spacecraft and FIELDS antenna dimensions. Changes in the magnetic field do not affect PSP and FIELDS charging. Finally, the model was run at 35 Rs with the finer mesh of 9.5 Rs in order to do a comparison run. The currents and potentials were compared. The model found that for the spacecraft the difference in the total collected and total emitted current to be 0%, while for the FIELDS antenna the total collected current and emitted current difference was 0.3%. The maximum error on the spacecraft was of 3% for the collected SE, while for the FIELDS antenna it was 31% of the collected SE ion. This contrast was probably due to the reduction of the super particles which was done in order to be able to run the models in a shorter period of time. These errors had little effect on the total current and final potential of the antenna during the I-V curve models, and with the lower number of super-particles the models ran faster with the author's limited computing resources. The plots shown in the previous section were created using runs with a larger number of super particles to decrease the maximum current errors down to 7%. The average potential errors were low, with an overall average of 2.9%, and a maximum error of 8.9% on the spacecraft.
This sensitivity study of varying the photoemission yield, SE yield, electron and ion densities reinforced the importance of knowing the material properties of the PSP and FIELDS shield and antenna. Material properties of the TPS shield at high temperatures and photoemission and SE yields must be known to predict the plasma environment near the spacecraft and instrument.

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18 of 22  Further sensitivity studies beyond the scope of this paper, should include an investigation into the development and response of trapped particle population in a non-monotonic potential structure to initial conditions and perturbations of equilibrium. A possible example of this would be a "triple root jump" floating potential behavior predicted for and observed on terrestrial geosynchronous satellites in energetic electron environments and eclipse conditions (Hastings & Garrett, 1996;Lai, 2012).

Conclusions
The theoretical spacecraft charging overview was shown, introducing the use of thermionic emission as a primary current. The PSP spacecraft and FIELDS antennas were modeled using SPIS software to predict their potential and current interactions with the environment. Results predict the FIELDS antennas charging positive for all cases. The plasma potentials show the ion wake and negative potentials in front of the DIAZ-AGUADO ET AL.

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19 of 22 TPS, as previously predicted. The plasma potential also shows the ion wake and negative wells in front the FIELDS shields and antennas. The antenna had a shallower negative potential well surrounding the cylinder, but similar negative potential well due to ion wake. It is important to note, the spacecraft potential predictions were within past estimates, except at 9.5 Rs, which was closer to an extreme SW event. Finally, a sensitivity analysis was performed which reinforced the importance of knowing the material properties of the FIELDS antenna and the rest of the spacecraft to predict the potential charging of the antenna.

Data Availability Statement
Flight and environmental data used in this paper are publicly available at http://fields.ssl.berkeley.edu/data. Model data are available at Zenodo, http://doi.org/10.5281/zenodo.4035319.