A charging model for the Rosetta spacecraft
F. L. Johansson, A. I. Eriksson, N. Gilet, P. Henri, G. Wattieaux, M. G. G. T. Taylor, C. Imhof, F. Cipriani
AAstronomy & Astrophysics manuscript no. aanda c (cid:13)
ESO 2020August 24, 2020
A charging model for the
Rosetta spacecraft
F. L. Johansson , , A. I. Eriksson , N. Gilet , P. Henri , ,G. Wattieaux , M. G. G. T. Taylor , C. Imhof , and F. Cipriani Swedish Institute of Space Physics, Uppsala, Swedene-mail: [email protected] Uppsala University, Department of Astronomy and Space Physics, Uppsala, Sweden Laboratoire de Physique et Chimie de l’Environnement et de l’Espace, CNRS, Orléans, France Laboratoire Lagrange, OCA, CNRS, UCA, Nice, France University Paul Sabatier Toulouse III, Toulouse, France ESA / ESTEC, Noordwijk, Netherlands Airbus Defence and Space GmbH, Friedrichshafen, GermanyReceived Jun 5, 2020; accepted Aug 6, 2020
ABSTRACT
Context.
The electrostatic potential of a spacecraft, V S , is important for the capabilities of in situ plasma measurements. Rosetta hasbeen found to be negatively charged during most of the comet mission and even more so in denser plasmas.
Aims.
Our goal is to investigate how the negative V S correlates with electron density and temperature and to understand the physicsof the observed correlation. Methods.
We applied full mission comparative statistics of V S , electron temperature, and electron density to establish V S dependenceon cold and warm plasma density and electron temperature. We also used Spacecraft-Plasma Interaction System (SPIS) simulationsand an analytical vacuum model to investigate if positively biased elements covering a fraction of the solar array surface can explainthe observed correlations. Results.
Here, the V S was found to depend more on electron density, particularly with regard to the cold part of the electrons, andless on electron temperature than was expected for the high flux of thermal (cometary) ionospheric electrons. This behaviour wasreproduced by an analytical model which is consistent with numerical simulations. Conclusions.
Rosetta is negatively driven mainly by positively biased elements on the borders of the front side of the solar panelsas these can e ffi ciently collect cold plasma electrons. Biased elements distributed elsewhere on the front side of the panels are lesse ffi cient at collecting electrons apart from locally produced electrons (photoelectrons). To avoid significant charging, future spacecraftmay minimise the area of exposed bias conductors or use a positive ground power system. Key words. plasmas – comets:indivdual: 67P / Churyumov-Gerasimenko – space vehicles, methods: data analysis, methods: numer-ical
1. Introduction
The European Space Agency’s (ESA) comet chaser,
Rosetta, monitored the plasma environment of comet 67P / Churyumov-Gerasimenko from August 2014 to September 2016. The sci-entific payload included the
Rosetta Plasma Consortium (RPC,Carr et al. 2007), dedicated to understanding the compositionand evolution of the comet plasma. The RPC included, amongother instruments, the
Langmuir probe (LAP, Eriksson et al.2007, 2017) and the
Mutual Impedance Probe (MIP, Trotignonet al. 2007; Henri et al. 2017). Because the instruments aremounted on
Rosetta , the RPC observations of charged particlesare influenced by the electrostatic potential of the spacecraft withrespect to its environment, V S , but several RPC measurementscan also be used to quantify this potential.All objects in space exchange charge with their surround-ings, mainly due to the collection of charged particles impactingthe object and emission of electrons via the photoelectric e ff ectand secondary emission. There are about as many negative elec-trons as positive ions in a plasma, but the electrons usually movemuch faster. In consequence, more electrons than ions tend to hitan uncharged spacecraft, giving it a negative charge unless theplasma is so tenuous that photoelectron emission dominates. An equilibrium is reached when the spacecraft becomes so negativethat most plasma electrons are repelled. When the dominatingcompensating current is photoelectron emission, the spacecraftpotential V S of a conductive spacecraft becomes (Odelstad et al.2017) V S ≈ − T e log (cid:16) C n e (cid:112) T e (cid:17) , (1)where n e is the number density the electrons, which are assumedto be a Maxwellian population of characteristic temperature T e ,given in eV, and C is a constant not depending on the plasmaproperties. The quantity in the logarithm essentially is the elec-tron flux, which, together with T e is thus expected to drive the V S in this case. If the collection of ions is a significant contributionto the current, the dependence on density becomes weaker.Predictions for the spacecraft potential of Rosetta had al-ready been produced prior to launch. In two coupled studies,Roussel & Berthelier (2004) used numerical simulations andBerthelier & Roussel (2004) investigated a spacecraft model in alaboratory plasma. Several plasma cases, including fully cooled(0.005 eV) cometary electrons were considered in the numericalsimulations. Some simulations let the solar arrays to float intotheir own equilibrium potential which was found to be beneficial
Article number, page 1 of 12 a r X i v : . [ phy s i c s . s p ace - ph ] A ug & A proofs: manuscript no. aanda
Fig. 1.
2D histogram in 80x100 bins of 88000 events of simultaneously measured spacecraft potential versus electron density ( left column ) andtemperature ( right column ) from January to September 2016 at 2 to 3.8 AU. The identified electron populations by MIP from Wattieaux et al.(2020) are separated by temperature as warm ( T ew ≈ top row ) and cold ( T ec ≈ . bottom row ). In total, 9700 outliers with either T ew >
15 eV (9400 outliers) or T ec > for reducing the otherwise often several volts positive potentialobserved in the simulations. The laboratory studies, therefore,emulated this case, with di ff erent surfaces on the model space-craft insulated from each other and thus attaining their own equi-librium. The studies, which did not include biased elements onthe solar arrays, suggested Rosetta would attain potentials be-tween a few times − k B T e / e and about +
10 V.However, the spacecraft potential was continuously mea-sured by LAP and found to be negative during most of the of
Rosetta cometary operations (Odelstad et al. 2015, 2017). Thespacecraft often reached negative potentials around and in ex-cess of -15 V, which have a severe e ff ect on in situ measure-ments of the plasma environment surrounding the spacecraft aselectrons are repelled (Eriksson et al. 2017) and positive ions areperturbed (Bergman et al. 2019, 2020). From this spacecraft po-tential result, Odelstad et al. (2017), with Eq. (1), argued that thecomponent dominating the electron flux is a thermal ≈ −
10 eVpopulation omnipresent in the parts of the comet coma visitedby
Rosetta . The existence of these warm electrons has also beenverified by direct observation, by LAP (Eriksson et al. 2017), byMIP (Wattieaux et al. 2020), as well as by the RPC Ion and Elec-tron Sensor (Broiles et al. 2016). However, there is also evidenceof a highly variable cold ( (cid:46) . Rosetta electrostatic design in Section 3 and present a series ofparticle in cell simulations of a simplified model dealing with ex-posed biased elements on the spacecraft solar array in Section 4and discuss this model’s shortcomings and merits. To improveour model, we adapt an analytical model of the vacuum poten-tial of a charged disk in Section 5 and run numerical simulations(Section 6) of a concentric disks geometry in an e ff ort to high-light spacecraft design decisions with a critical influence on thecold electron current collection. Finally, we suggest a simpli-fied model describing the Rosetta current balance by setting up asystem of equations to describe the current to the spacecraft anda positively biased surface behind a negative potential barrier inSection 7, solve it numerically, and compare it to
Rosetta results.
2. Data analysis
A reworked analysis of MIP spectra with signatures of twoelectron populations (Wattieaux et al. 2019; Gilet et al. 2019)yields an unprecedented precision in both energy and densityof the thermal ( ≈ ≈ . http://amda.cdpp.eu/ ) , give us simultaneous mea-surements of all parameters in Eq. (1) for both populations.We plot the MIP density estimates, as well as the mean of theLAP spacecraft potential estimates (typically 1 or 29 samples) Article number, page 2 of 12. L. Johansson et al.: A charging model for the
Rosetta spacecraft
Fig. 2. Top:
Example time series of LAP spacecraft potential (dia-monds), cold (circles), and warm (pluses) electron density estimatesfrom MIP in the same interval, exhibiting the strong correlation betweenspacecraft potential and cold electron density.
Middle:
Same data asshown in Figure 1, 2-D histogram of 120x150 bins of n ew vs eV S / T ew . Bottom:
As above, but with n ec on the y-axis. taken during the acquisition period of each MIP spectra (typi-cally 2 seconds) and plot them in Figure 1.In contrast to our model in Eq. (1), the temperature varia-tion in the two detected electron populations can only explainsome of the variations in the Rosetta spacecraft potential. In-stead, the cold electron density has the clearest (logarithmic) re-lation to spacecraft potential out of the four parameters inves-tigated. However, for a uniformly charged spacecraft at -10 V,these cold (0.1 eV) electrons simply cannot reach the spacecraftand meaningfully contribute to the current balance that dictatesthe spacecraft potential.The correlation between the cold electron density and thespacecraft potential is perhaps the clearest in a time-series,as plotted in Figure 2 (Top), where we also observe a ratherweak dependence on warm electron density to spacecraft po-tential. We also note that from 2016-06-12T16:00:00 to 2016-06-13T08:00:00, the average temperature of plasma electronsshould increase (up to 50 percent) as the cold electron popula- tion density decreases, which according to our relation in Eq. (1)would correspond to a more negative spacecraft potential. In-stead, the opposite is true.Normalising the spacecraft potential by e / T ew in Figure 2(middle panel) we see that an increase in warm electron den-sity (for a fixed T ew does not drive the spacecraft more nega-tive at all during the entire period from January to September2016. Instead, it seems that an increase in n ew is associated witha decrease of T ew , which is much more strongly coupled to thespacecraft potential. In general, it should not be surprising thatin denser regions of the cometary ionosphere, the denser neutralcometary gas allows for more e ffi cient cooling of all electrons(Edberg et al. 2015). What is also apparent is that the warm elec-tron density does not strongly correlate with the cold electrondensity (bottom panel, same figure), which (albeit with morescatter) still shows the same trend of linearly increased charg-ing with an exponential increase of cold electron density.In the following section, we propose a mechanism to explainthese observations.
3. Exploring what drives
Rosetta to negative
Ionospheric spacecraft have been observed to be driven negativeby exposed, positively biased conductors on solar panels. Forexample, the OGO-6 satellite was observed to reach about -20 V(Zuccaro & Holt 1982) and the International Space Station canreach as much as -140 V (Carruth et al. 2001). The reason isthat such biased elements can draw a large electron current. Toclose the circuit, the spacecraft must respond by decreasing itspotential to deflect more electrons away from it and to attractmore ions from the plasma. This phenomenon can be regularlyobserved on small spacecraft equipped with Langmuir probes,where a large positive bias on the probe can result in a smallnegative shift of V S (e.g. Ivchenko et al. 2001). On Rosetta , thesurface area of about 80 cm of each of the two LAP probes isnegligibly small compared to the total spacecraft area (includingsolar panels) around 150 m , so these cannot drive V S to thehigh negative values observed. Rosetta was not expected to be (and e ff ectively never was)exposed to the large fluxes of high energy particles often drivingspacecraft charging to dangerous levels in, for example, auro-ral zones (Eriksson & Wahlund 2006; Garrett et al. 2008). Ef-forts were taken, nonetheless, to minimise exposed dielectricsand non-grounded conductors in order to provide a stable groundfor plasma measurements. Providing a conductive and groundedouter layer was straightforward for most parts of the spacecraft.As Rosetta was to be the first spacecraft operating on only so-lar power as far from the Sun as 5.25 AU, the front side of thelarge (64 m ) solar arrays was a more complex issue, but in theend, it was decided that the solar cells would also be providedwith cover glasses with a conductive indium tin oxide surfacelayer connected to spacecraft ground. In the low-energy domi-nated environments of concern here, the charging of dielectricsis not the primary concern. More of interest are exposed conduc-tors, which can draw a significant current and, hence, influencethe spacecraft potential. Thanks to the solar cell cover glassesand the equally conductive and grounded multi-layer insulationon the spacecraft body, the dominant fraction of all exposed con-ductive surfaces (we estimate at least 95%) is at spacecraft po-tential. However, there are exceptions, particularly on the solarpanels.The Rosetta solar array (Figure 3) consists of 10 panels, eachwith 25 strings of 91 solar cells on its front side. While eachcell has a conductive and grounded cover glass, there are small
Article number, page 3 of 12 & A proofs: manuscript no. aanda
Fig. 3.
Rosetta spacecraft and one of its solar wings. The solar cell coverglass on each cell is visible as small dark glossy surfaces, with metallicreflective interconnects above and below it. There are also 25 slightlylarger reflective bus bar pairs, not to be confused with the six circu-lar Kevlar cutter / hold-down points. Adapted from Rosetta Solar pan-els on ESA’s website. Retrieved May 4, 2020, from sci.esa.int/s/w0e6nbW . Copyright 2012 ESA–A, Van der Geest. Reprinted with per-mission. exposed biased conductors (interconnects) linking the cells in astring as well as the ends of a string to the spacecraft power bus.The single largest exposed positive potentials on a panel are the25 small anodes of the bus bars at the end of each string, whichcan be seen as a sketch in Figure 4. The bus bar anode is biasedup to +
79 V from spacecraft ground (and the bus bar cathode) ona string in open-circuit condition, and +
65 V for a string operat-ing at the maximum power point . The 89 interconnects in eachstring between the anode and cathode are therefore biased to anequidistant and linearly increasing potential for each consecu-tive solar cell in the string, such that the bias voltage on the lastinterconnect before a +
79 V anode is + + + V B can easily dominate the positive current col- The string voltage on the solar array is driven by many parameters,including the temperature and degradation of the cells and the powerrequested by the spacecraft
Fig. 4.
Artist impression of a corner section of the front side of a solarpanel on
Rosetta . The black squares are individual solar cells coveredwith grounded cover glass, connected in series via (pink) interconnectsin a column that wrap around to the next column near the top edge(and bottom, not shown) of the solar panel via a longer interconnect(also pink). At the start and end of each string of 91 solar cells are busbars, marked with red (anode) and blue (cathode). The grey circle rep-resents one of six circular Kevlar cutter / hold-down points. All surfacescoloured in pink and red are exposed positively biased conductors. lection to a spacecraft as the current increases as a function of theabsolute potential of the surface for any surface except an infiniteplane (Laframboise & Parker 1973). For the simplest two-bodyproblem of a spherical, positively charged body of surface area, A , immersed in a plasma of density, n , the current collection ofelectrons is I e = Ane (cid:114) k B T e π m e (cid:32) + eUk B T e (cid:33) , (2)where U is the absolute (positive) potential of the body U = V B + V S and other constants have their usual meaning. For 0.1 eVelectrons and an exposed conductor at +
75 V as discussed in theprevious section, the current collection thus is leveraged by a fac-tor of 750. Of course, charged elements on a spacecraft is a muchmore involved circuitry with a complex geometry that needs tobe taken into account and requires numerical simulations.
4. Numerical simulations
The Spacecraft-Plasma Interaction System (SPIS) is a hybridcode package to simulate the spacecraft-plasma interaction,solving the Gauss’s Law for electric fields, pushing particles, andsimulating the spacecraft circuitry response and interaction withthe plasma (Matéo-Vélez et al. 2012; Mateo-Velez et al. 2015;Sarrailh et al. 2015). This work is a continuation of e ff orts ofmodelling the Rosetta spacecraft in a cometary plasma environ-ment by Johansson et al. (2016) and Bergman et al. (2020) tounderstand the RPC instrument measurements. The simulationparameters for a reference simulation are provided in Table 1.The cometary ion population provides little current to the space-craft system but ensures quasi-neutrality with a realistic (Sten-berg Wieser et al. 2017) but isotropic thermal velocity. To re-duce the computational time of some of the SPIS simulations,
Article number, page 4 of 12. L. Johansson et al.: A charging model for the
Rosetta spacecraft
Fig. 5.
Visualisation of electrostatic potential struture from a SPIS sim-ulation of a model with four 0 . × . +
75V biased elements on a1 . × . × .
15 m solar panel inside a spherical simulation volumeof radius 15 m. Ten equipotential surfaces (cut in the Y = Z plane) from-17 V to + Fig. 6. Top:
Spacecraft (ground) potential evolution in five SPIS sim-ulations. The reference simulation with no biased surfaces in a warm T ew =
10 eV plasma (blue line), with small charged surfaces of +
75 Vin a PIC simulation (black) or a fluid Maxwell-boltzmann simulation(yellow dot-dashed line). Also plotted, a simulation of the same plasmadensity but with 50 percent T ec = . +
75 V are strictly not valid but serve to illustrate the firstapproximation from Orbital-Motion-Limited theory.
Bottom:
Zoom-inof above, with the calculated mean (dashed black line) and a 1 σ range(dotted black lines) of spacecraft potential from the PIC simulation inthis interval. we can simulate particles also as a Maxwell-Boltzmann fluid ap-proximation instead of a full particle-in-cell (PIC) simulation.This treatment is not valid when there are positively biased el-ements present as the electron density in each simulation cellis extrapolated from the potential in that cell and, as such, wewould overestimate the electron density near positive elementsand within potential barriers. However, the reduction of compu-tational (PIC) noise from a fluid approximation is very welcomefor the purposes of demonstration. Table 1.
Table of reference SPIS simulation environment parametersused in Sections 4 & 6. n ew
100 cm − T ew
10 eV T i m i
19 amu (H3O + ) u i − j ph0 µ Am − d (cid:12) ff Number of cells 4 . × Number of PIC particles ≈ × We can also gain insights by studying simplified geome-tries; given the solar panels are the largest areas, we neglect thebody and because each solar panel is large compared to the De-bye length (which should be 30cm or less with cold electronsaround), the whole array should essentially behave as a singlesolar panel. Therefore, we approximate
Rosetta with a box ofsize 1 . × . × .
15 m, where the thickness is exaggerated forthe ease of simulation but brings in only a negligible contributionto the current balance. We also include four 0 . × . m symmet-rically placed elements on the front side of the solar panel, whichwe set to a bias potential V B =
75 V, as shown in Figure 5. Allsurfaces are simulated as indium tin oxide (ITO) for the purposeof photoemission and conductivity as it is the principal materialon all sunlit surfaces on
Rosetta and we otherwise assume this tohave a negligible e ff ect on V S in a cometary (low-energy) plasmaenvironment.In Figure 5, we show an instructive example of the potentialstructure around a -25 V solar panel with small positively biasedelements from a SPIS simulation with with the electron densityset by a Boltzmann relation with the potential, complete with athree-dimensional potential barrier of -8 V. As can be seen inFigure 6, the e ff ect of positively biased surfaces is two-fold: – The positively biased elements are collecting locally pro-duced photoelectrons from the surrounding surfaces as well,where the current magnitude as measured by SPIS corre-sponds to the photosaturation current of an area six timestheir size. E ff ectively, this turns photoemission o ff on an areasix times as large as the positively biased elements on thesolar array and drives the spacecraft potential to be morenegative. For a more realistic case, with exposed biased ele-ments that are spread over the entire (sunlit) panel, the pho-toemission of Rosetta would be heavily suppressed. This canbe part of the explanation on why
Rosetta was substantiallynegatively charged during the cometary mission and read-ily explains why
Rosetta only experienced moderate positivecharging in the solar wind. (Odelstad et al. 2017). – For standard OML, and indeed in the example SPISMaxwell-Boltzmann fluid treatments in Figures 5 and 6, thecurrent to any positively charged surface (for the barrier po-tential, U M ) is severely exaggerated as most electrons bornat a potential of 0 V at infinity cannot penetrate the barrierif their energy does not exceed eU M . The aforementioned Article number, page 5 of 12 & A proofs: manuscript no. aanda
Fig. 7.
Geometry of the concentric disk model for modelling of solarpanels with biased elements as described in Section 5.1 for two di ff erentapplications: Panel (a) represents a single exposed biased conductor onthe main area of the solar panel (Section 5.2); panel (b) exposes biasedconductors along the edge (Section 5.3). Grey areas represent the mainsolar panel at spacecraft potential, red a biased element. cold cometary electrons would contribute little to the currentto these biased elements, as has indeed been confirmed bySPIS simulations with a PIC treatment of electrons.This potential barrier e ff ect is very e ff ective in quenching thecold electron current when small positive elements are sur-rounded on all sides by grounded (negative) elements. Althoughthe cold electron density population exhibits the exact behaviourwe sought for in Section 2 in the fluid approximation simula-tions, we must look for another explanation when a realistictreatment of electrons is applied.As described in Section 3 and in Figure 4, the interconnectsare dispersed all over the solar panel surfaces, but they are (pos-sibly crucially) always present at the top and bottom border ofthe solar panel front side, as the solar cell string wraps aroundto the next column. As for all interconnects on the solar array,on average, this border is expected to be biased between + ff ect on current collection since many interconnects wouldbe repelling electrons exponentially at V B < − V S ) and can haveless restricted access to electrons as it is not surrounded by neg-atively charged surfaces, an e ff ect we investigate further in thefollowing sections.
5. Analytical model of solar panels with biasedelements
For a comparison and interpretation of the simulation results onthe formation of potential barriers, we use an analytical solutionof the Laplace equation around a thin circular disk which con-sists of two concentric parts, as illustrated by two examples inFigure 7, an inner disc of radius, a , at potential, V in , surroundedby an annulus of inner radius, a , and outer radius, b > a , at thepotential V out . At cylinder coordinates ( ρ, φ, z ) , where z = Φ ( ρ, z ) = V in π atan b √ (cid:113) r − b + (cid:112) ( r − b ) + z b ++ ( V out − V in ) √ π · (cid:90) ba (cid:115) r − s + (cid:112) ( r − s ) + z s ( r − s ) + z s · s √ s − a ds , (3) Fig. 8.
Limiting potential ratio for barrier suppression for a small biasedelement on the z axis as given by Eq. (7). where r = ρ + z . The integral can be analytically evaluated onthe z axis and in the disk plane z = Φ (0 , z ) = π V in atan bz + ( V out − V in ) z √ z + a atan (cid:115) b − a z + a , (4) Φ ( ρ, = π V in atan b (cid:112) ρ − b + ( V out − V in ) atan (cid:115) b − a ρ − b . (5)We use these expressions to model potential barriers around so-lar panels with exposed biased conductors in Sections 5.2 and5.3 below. In extending an argument used by Sherman & Parker(1971) for the z axis, the potential will have an extremum (min-imum or maximum) on exactly one of the two axes. This isbecause at large distance the two plates look like a point withcharge equal to their net charge, which must be either positiveor negative. Far away, the potential decays as ± / r , so if nega-tive at large distance, the potential must have a minimum some-where along the axis from the positively charged part, which isthe z axis if the positive part is the inner disk ρ < a and other-wise the ρ axis. Such a minimum in the potential is a maximumin electron potential energy and, hence, a potential barrier. Byconsidering the net charge of the disks, the limit for barrier for-mation is found to be (Sherman & Parker 1971): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V out V in (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:113) − (cid:16) ab (cid:17) − . (6)If the positive voltage, whether V in as in Figure 7(a) or V out as inFigure 7(b), is higher than allowed by this expression, there willbe no barrier for electron collection by the positive element. In this case, the outer annulus represents the solar panel at po-tential V out = V S < V in = V S + V B . The minimum value of V in to break Article number, page 6 of 12. L. Johansson et al.: A charging model for the
Rosetta spacecraft
Fig. 9.
Vacuum potential and electric field pattern in the ρ − z plane near the centre of a thin circular disk of radius b as sketched in Figure 7(a). Thedisk potential outside ρ = a is V in = −
10 V while the potential V out in the centre ρ < a varies as stated above each panel. In all cases, a = . b .Numerically integrated electric field lines are plotted in white. The 0 V equipotential is shown in thick red; the potential is zero also at infinity.Black curves indicate equipotentials at every integer value (in volts), with the background colour further highlighting the potential. the barrier follows from Eq. (6) as V in − V out = (cid:113) − ( ab ) − . (7)Figure 8 shows this expression evaluated for a range of the radialratio a / b . It is clear from this plot that forbiddingly large valuesof the bias ratio are needed for breaking the barrier, reinforcingthe conclusion in Section 4 that small positive elements on theinterior of a solar panel would not collect cold plasma electrons.In Figure 9, we show the vacuum electrostatic field near thecentre of the same disk as calculated from the full expressionEq. (3) for four di ff erent bias voltages V B = V in − V out . The zerovolt equipotential (red) ends at the intersection ρ = a = . b between the disks. All field lines starting on the positively biasedinner surface ends up on the main solar panel area, as is expectedsince there is a potential barrier. The radius ρ delimiting fieldlines connecting to the inner disk or to infinity can be seen toexpand from about 0 . b to 0 . b as V B increases from 30 V to75 V. If photoelectrons were massless and emitted from the solarpanel in the normal direction with zero speed, they would followthe electric field lines. Using ρ as a measure of the region fromwhich photoelectrons from the solar panel would be collectedby the biased element at the centre, we find that the area of thisregion is about 35 times bigger than the biased element itselfalready for V B =
30 V. While parameters are not perfectly com-parable, this is still significantly more than the factor of aboutsix that we found in the simulations in Section 4. This is ex-pected, as photoelectrons would follow field lines perfectly onlyif massless and emitted at zero speed, neither of which is thecase. Furthermore, our analytic model only considers a vacuum.
We now turn our attention to the positively biased elementsaround the solar panel edges and apply our analytical vacuummodel to this case. If the barrier e ff ect is as e ff ective here as wefound for biased elements on the main solar panel surface, wecould conclude that the biased elements on the solar panels can-not be responsible for the strongly negative potential of Rosetta .Here we consider whether this is indeed the case. In this situation, the general barrier limit Eq. (6) takes theform V out − V in = (cid:113) − (cid:16) ab (cid:17) − . (8)We plot this limit condition in Figure 11. For a realistic represen-tation of the Rosetta solar panels, a / b should be in the upper endof the plotted range. While the values grow large as the outer ringbecomes narrow ( a / b close to 1), they are still much more mod-est than the corresponding ratio in Figure 8, a combined e ff ect ofthe ring having much larger area then the central circle of similarwidth and of the circle being exposed to space at the edge of thesolar panels with no grounded elements surrounding it. For thevalue a / b = . , we get a limiting voltage ratio around −
4. Thismeans that the approximate maximum bias voltage V B = +
75 V(note that V B = V out − V in ) would be su ffi cient to attract coldelectrons if the spacecraft potential ( V in ) is not more negativethan −
15 V. However, this is only a vacuum model and we mayexpect the shielding provided by the plasma would lower the bar-rier height and so increase the e ffi ciency of the solar panel edgeas a driver of the spacecraft potential. In Section 6, we show thatthis actually is the case.Instead of applying the general condition of Eq. (6), we couldconsider that a barrier in the disk plane means that the potentialEq. (5) has a local maximum. By setting d Φ ( ρ, / d ρ =
0, wethen obtain the barrier position ρ from ρ = V in ba V in b + [ V out − V in ] √ b − a . (9)Requiring ρ > b ). The barrier can beseen for the two lowest bias cases, where all field lines to infinityconnect to the main solar panel area. In the two high-bias casesthe barrier is gone and, as discussed in Section 5.1, this meansthat only field lines from the solar panel edge connect to infinity,suggesting that there is an e ffi cient collection of plasma electronson this edge. Another consequence is that all field lines to themain solar panel area now originate at the biased edge, suggest-ing a strong suppression of solar panel photoemission. To find ifthis indeed is the case, we must turn back to the simulations. Article number, page 7 of 12 & A proofs: manuscript no. aanda
Fig. 10.
Vacuum potential and electric field pattern in the ρ − z plane around a thin circular disk of radius b as sketched in Figure 7(b). The diskpotential inside ρ = a is V in = −
10 V while the potential V out in the annulus a < ρ ≤ b varies as stated above each panel. In all cases, a = . b .Numerically integrated electric field lines are plotted in white. The 0 V equipotential is shown in thick red; the potential is zero also at infinity.Black curves indicate equipotentials at every integer value (in volts), with the background colour further highlighting the potential. The magentastar indicates the location of minimum electron barrier height. Fig. 11.
Minimum value of V out / ( − V in ) for full barrier suppression, asfunction of the ratio of the radii a / b , for a circular disk solar panel invacuum, where V out is positive.
6. Simulation of concentric disks
To test and extend the validity of the analytical model in Sec-tion 5.1 to incorporate a plasma (and Debye shielding), we sim-ulate two concentric disks at di ff erent bias potentials from theground (0 and +
75 V for the inner and outer disk, respectively)in SPIS. We take the models specified in Figure 7 with a discthickness of 5 mm and use the same environment and materi-als as specified in Table 1 with a PIC treatment of all particlesand simulate until the spacecraft potential converges. To improvestatistics for the electrostatic potential in the volume, we utilisethe symmetry of the problem in three dimensions (seen in Fig-ure 12). For plotting the potential along the z axis, we dividedthe axis into intervals of length d z . For each z value, we calcu-lated the median value of the potential for all points between theplanes defined by z and z + d z (as well as - z and -( z + d z )) which liewithin ρ/ a = .
2. For the plot of potential vs ρ , we did the samefor a ring-like volume containing all points between a cylinderof thickness d ρ around ρ and within 2 degrees of the z = Fig. 12. a = .
17 m , b = .
25 m) simulation with V B = +
75 V, coloured by electrostatic potential. To illustrate the po-tential in the volume, we plot the potential along the X-Z plane, as wellas 10 equipotential surfaces from -30 V to +
25 V, cut in the X-Y plane.The potential of the outer disk is -45.1 V and 29.9 V for the inner disk.
Comparing the SPIS reference simulation to the vacuum casein Figure 13, we find a potential barrier at the exact same position(as far as our SPIS simulations resolution allows) as our analyt-ical model predicts. The absolute potential of the barrier U M is,unsurprisingly, smaller, as the plasma would screen all non-zeropotentials via Debye shielding. The SPIS also allows us to runa series of simulations where we can change the Debye lengthof the plasma in the volume without changing the floating po-tential of the spacecraft (as changing the density or temperatureof the plasma would undoubtedly shift the balance of currents tothe spacecraft). These results are also shown in Figure 13. Thepotential in the volume is more e ffi ciently damped when movingaway from the spacecraft as λ D decreases and we can plot thefractional departure from the Sherman analytical model at theposition of the barrier potential vs λ D in each simulation in Fig-ure 14. Using the method of a least-squares fit to an appropriatemodel, we find that U M = U † M (cid:18) − .
78 exp − λ D . (cid:19) , (10)where U M is the SPIS barrier potential, U † M is the barrier po-tential in the vacuum solution, and λ D is the Debye length. This Article number, page 8 of 12. L. Johansson et al.: A charging model for the
Rosetta spacecraft
Fig. 13.
Electrostatic potential normalised by potential of the positiveouter ring ( +
30 V) along cylindrical axes, ρ ( left ) and z ( right ). Forthe vacuum case (blue line), the ring radii fraction, a, is identical tothe SPIS model, and the absolute potential of the disks are taken fromthe output of the reference SPIS simulation (red circles). The associ-ated analytical solution for the barrier potential, U M , is marked with adashed black line. The other SPIS simulations were simulated at identi-cal conditions except with fixed potentials on the rings, and with di ff er-ent plasma Debye lengths (modulated by changing the plasma densityor electron temperature in each simulation). Fig. 14.
Barrier potential for four SPIS PIC simulations at di ff erent De-bye lengths with the same disk potentials, divided by the analytic vac-uum model result (Sherman & Parker 1971). The fitted model is plottedin red. model has the limit U M lim λ D →∞ = U † M as expected for a vacuum.This particular model clearly does not describe the limit of veryshort Debye lengths correctly as U M here should go to zero, butit does well in describing the parameter range of our simulationsand the transition to vacuum conditions.
7. Spacecraft potential model
At distances from a negatively charged object su ffi ciently largeso that the potential has decayed by a few times, k B T e / e ,it follows from Liouville’s theorem that the electron densityequals the ambient plasma density reduced by a Boltzmann fac-tor (Laframboise & Parker 1973).As the potential is repelling from infinity up to the barrier,the plasma density at this point should be n eM = n exp (cid:32) eU M k B T e (cid:33) . (11)Olson et al. (2010) therefore proposed that the electron current, I ep , to a positive spherical probe within the negative barrier po-tential from a larger sphere some distance away is given by thestandard orbital motion limited expression (Mott-Smith & Lang-muir 1926) with the source density reduced by the Boltzmannfactor at the barrier, viz. I ep = I e0 exp (cid:32) eU M k B T e (cid:33) (cid:32) + e ( U − U M ) k B T e (cid:33) , (12)where I e0 = Ane (cid:113) k B T e π m e and it is further assumed that the currentattraction is governed not by the absolute potential of the anodewith respect to a plasma at infinity, but of the di ff erence betweenthe barrier potential and the anode. An electron passing the bar-rier may very well complete an orbit around the probe sphere andleave again, so the spherical probe form assumed by Olson et al.(2010) does not seem unrealistic. However, our case of an at-tracting ring on the edge of a repelling disk is quite di ff erent. Anelectron passing the barrier potential seen in Figure 10(ab) andFigure 12 cannot encircle the anode along the disk edge in thepoloidal direction and is e ffi ciently focused toward the edge bythe repelling field lines from the main solar panel. This is verifiedby our PIC simulations in which the current to the anode doesnot fit any spherical OML expression but is best described bythe current collection to a plate of equal area. As such, Eq. (12)is simplified to I e = I e0 exp (cid:32) eU M k B T e (cid:33) . (13) Bringing it all together, we can estimate the electron current topositive biased elements in a plasma inside a barrier potential(if any) using Equations (5), (9), (10), and (13). Along with theexpression for electron current to a negative surface in OML(Laframboise & Parker 1973), we find the general expressionfor the electron current to a charged solar panel within a barrierpotential to be I e = I e0 exp (cid:16) eU M k B T e (cid:17) for U > U M I e0 exp (cid:16) eUk B T e (cid:17) for U ≤ U M . (14)The ion current can be shown (Sagalyn et al. 1963) to be I i = − I i0 (cid:16) − eUE i (cid:17) for U < E i / e U > E i / e , (15)where E i is the ion energy, and I i0 = Ane √ E i / m i . From thelessons learned in Section 4, we also reduce the photoemission Article number, page 9 of 12 & A proofs: manuscript no. aanda
Fig. 15.
Cold electron density vs Spacecraft potential model solution ( left ) and barrier potential ( right ) for various cometary plasma parameters.
Fig. 16. Left:
Spacecraft potential model solution for a solar panel vs n ec at 2 AU, n ew = − , T ew = Right:
Barrier potential in the model vs n ec . on the spacecraft by a positive factor, γ ph ≤
1, to simulate allinterconnectors and bus bars scattered over the solar panels thatreduce the net photoelectron production for the spacecraft. Inthis way, we can simplify the photoemission current expressionin Grard (1973) for both surfaces to I ph = − π r j ph0 d γ ph , (16)where d AU is the heliocentric distance in AU, and r a is the radiusof the inner disk.We find the equilibrium (spacecraft) potential for our solarpanel when all currents to an inner disk at potential V S < U M and an outer disk at potential V S + V B , sum up to zero. Aftersome rearranging to isolate V S from Eq. 14, we find V S = k B T e e ln − I ph − I a i ( V S ) − I btot ( V S + V B , U M ) I a e0 , (17) where I btot is the sum of the currents I b e and I b i as a function ofpotential (and barrier potential), and we use a superscript to sep-arate terms for the inner and outer disks of radius a and b , re-spectively.Now we have all the tools for predicting the current to the Rosetta solar panel without the need for computationally costly3D PIC simulations. In an iterative solution of Eq. (17) of allcurrents to all surfaces on a concentric disk model of ten solarpanels and a simple conductive and photo-emitting
Rosetta
SCbox of 2x2x2.5m, where we insert also a cold (0.1 eV) electroncomponent that is otherwise computationally costly to simulate,we find the equilibrium spacecraft potential and the barrier po-tential. We plot the results in Figure 15.As we increase the cold electron component, we observea significant negative charging up until a barrier potential isformed around V S ≈ −
27 V. Beyond this potential, or beyondthe creation of a barrier potential, the electron density depen-dence on spacecraft potential tapers of rapidly as cold electrons
Article number, page 10 of 12. L. Johansson et al.: A charging model for the
Rosetta spacecraft can no longer reach the spacecraft (even when the net potentialof the biased elements is still ≈ +
35 V). When comparing thisresult to Figure 1, which prompted this study, we find an expla-nation for both the highly negative spacecraft potential and thestrong dependence on cold electron density versus spacecraft po-tential below -25 V. As we move to larger heliocentric distancesin Figure 16, we shift the curve downwards as the photoemissioncurrent decreases everywhere and find a linear trend in the sameregions of densities and potentials as in Figure 1. In reality, thepotential of the positive elements around the edge that we baseour disk model upon should be distributed on some potential be-tween + +
78 V and we see in Figure 16 that for low biaspotentials V B , the cold electron current has no coupling to thespacecraft or loses coupling even for low spacecraft charging asa potential barrier develops, indicating that moderate (absolute)positive potentials on the interconnects will have little e ff ect onthe Rosetta spacecraft current system.In reality,
Rosetta is not simply a set of solar panels and anITO-coated spacecraft box but, rather, these represent the prin-ciple surfaces for photoemission and current collection and, assuch, we believe that the current balance would only be slightlyperturbed by incorporating a more realistic
Rosetta spacecraftbody. A more accurate shape model for the solar panels wouldincrease the complexity of the barrier potential shape and thesubsequent current collection to all surfaces, but it could be ex-pected to have the same general behaviour in terms of the cre-ation of barrier potentials and current collection to the posi-tively biased surfaces. Introducing a flowing plasma and pos-sible wakes associated with this flow may also alter the detailsof the balances, but wake e ff ects should mostly be weak: thosethat are typically observed ion flow directions – between radiallyoutward from the comet and anti-sunward (Berˇciˇc et al. 2018) –in combination with the Rosetta trajectory around 67P – witha mostly terminator orbit and its solar panel length axis alongthe direction perpendicular to the nucleus and the panels them-selves normal to the sun – minimises wake e ff ects on the solarpanel front surface. We cannot be certain that all our simplifica-tions are valid when moving to a more realistic model, but Fig-ures 1 and 16 show that we have found a candidate model thatdescribes the general evolution of the spacecraft potential andcurrent collection behaviour of cold electrons to Rosetta duringits mission. Regardless of geometry, this current collection be-haviour seen on
Rosetta can only be represented by a positivelybiased conductor with su ffi cient bias to attract electrons beyond V S < −
20 V and significant surface area to both overcome bar-rier potentials from surrounding surfaces as well as yield sig-nificant cold electron current. We also note that we do not seea decoupling of cold electrons to spacecraft potential in Fig-ure 1 even at the most negative potentials, which indicates thatthe positively biased surfaces are always (at least for this set ofmeasurements taken between January to September 2016) largeor positive enough that no negative barrier potential is formed.Therefore, as shown in Figure 10, the solar array likely appearsas net positive surfaces for a charged particle far away from thespacecraft.On 12 July 2016, from 09:30 to 10:00 UTC,
Rosetta con-ducted a solar array power test composed of a 60 deg rotationof the solar array from the sun, thus reducing the available so-lar power (and photoemission on the solar array) by 50 percent,along with, perhaps, a decrease of V B on the solar array an-odes, as more strings would be at the maximum power point.At this moment, LAP registered a drop in spacecraft potential,from ≈ − . ≈ −
17 V, plotted in Figure 17, which isconsistent with a spacecraft that is photo-emitting less. As the
Fig. 17. Top:
LAP spacecraft potential (blue circles) and MIP elec-tron density (red dots) during a rotation of the Rosetta solar array of60deg from the sun, for which two dashed lines indicate the start (red)and end (blue) of the test.
Bottom:
Spacecraft potential result at var-ious gamma ph and V B from an adaptation of our solar panel model toa Rosetta spacecraft box with ten panels, for which n ew = − , T ew = plasma parameters were otherwise relatively stable (no detectedcold electron population, T ew ≈ γ ph and possibly V B .In Figure 17, we see that a decrease of 50 percent of photoemis-sion, which in our model represents a 50 percent decrease in γ ph (from before the rotation) is compatible with our measurementsfor γ ph (cid:39) . γ ph = . V B , as there are no detectable cold electrons.
8. Conclusions
In our investigation of the correlation of the LAP measured
Rosetta
Spacecraft potential to the MIP measured densities andcharacteristic temperatures of two detected cometary electronpopulations, we find the spacecraft potential to depend more onelectron density (particularly cold electron density) and muchless on electron temperature than expected in the high flux ofthermal (cometary) ionospheric electrons.To investigate the current to the positively biased borders onthe front-side panels of the
Rosetta solar array, we first apply ananalytical model to obtain the potential surrounding two concen-
Article number, page 11 of 12 & A proofs: manuscript no. aanda tric disks at di ff erent potentials in a vacuum. Comparing the re-sult to 3D PIC SPIS simulations and constructing a simple modelbridging the two, we arrive at a system of equations that canreadily explain the strong relationship between the (highly nega-tive) Rosetta spacecraft potential and an observed cold (0.1 eV)electron population that sometimes dominate the
Rosetta elec-tron environment even when barrier potential e ff ects are consid-ered. We find an explanation for the highly negative charging onthe spacecraft, the seemingly poor coupling of electron tempera-ture to spacecraft potential, and the observed log-lin relationshipof electron density to spacecraft potential that is used in our anal-ysis of LAP and MIP data to retrieve electron density estimatespublished on AMDA ( http://amda.cdpp.eu/ ) and soon onthe ESA Planetary Science Archive.To mitigate spacecraft charging on planetary plasma mis-sions in the future, especially in dense, cold environments wherelow-energy plasma particles are of particular scientific impor-tance, we suggest an inversion of polarities (i.e. setting space-craft ground as the anode) in the solar array design, which woulddrastically reduce the current drawn from the plasma. This ap-proach has been taken on ionospheric spacecraft like Atmo-spheric Explorer C and Swarm and been shown to result in astable and slightly negative potential (Samir et al. 1979; Zuccaro& Holt 1982). Increased e ff orts to insulate positively biased con-ductors, where particular attention should be directed to areasclose to edges of structures, would also help reduce spacecraftcharging and enable more sensitive plasma measurements of thecoldest plasma populations. Acknowledgements. Rosetta is an ESA mission with contributions from its mem-ber states and NASA. This work would not have been possible without the col-lective e ff orts over a quarter of a century of all involved in the project and theRPC. We are also grateful to everybody who has worked on the SPIS software, atONERA, ARTENUM and elsewhere, and to ESA for supporting this highly valu-able package. This research was funded by the Swedish National Space Agencyunder grant Dnr 168 /
15. The cross-calibration of LAP and MIP data was sup-ported by ESA as part of the Rosetta Extended Archive activities, under contract4000118957 / / ES / JD.
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