Numerical Simulation of the Interactions Between Solar Arrays and the Surrounding Plasma Environment DRAFT IEPC-2005-214 Presented at the 29th International Electric Propulsion Conference, Princeton University October 31 – November 4, 2005 Ren´e Sedmik∗, Carsten Scharlemann†, and Martin Tajmar‡ ARC Seibersdorf research Gmbh., Seibersdorf, A-2444, Austria J. Gonz´alez del Amo§, A. Hilgers¶, and D. Estublierk ESA-ESTEC, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands G. Noci∗∗ and M. Capacci†† LABEN, Proel, Firenze, Italy and C. Koppel‡‡ Snecma Monteurs DMS DPES, Site de Villaroche Nord, BP No 93, 77552, Moissy Cramayel Cedex., France
SMART–1 is the first European satellite utilizing an electric thruster as main propulsion system. In the exhaust plume of the PPS-1350 Hall collisions lead to concentrations of low energy charge exchange ions. These low energy particles are easily redirected towards the spacecraft where they cause sputtering and electrical charging of the satellite, both leading for example to degradation and malfunction of solar cells and electronics. Analysis of flight-data has shown a cyclic variation of the spacecraft floating potential with the orbit. ARC Seibersdorf research is investigating the plasma environment and interactions with the solar arrays using the numeric code SmartPIC. A direct correlation to the rotation angle of the two solar array wings could be demonstrated. Previous versions of this tool have already been able to predict floating potential variations Review of plasma physics shows that the quasineutrality assumption used in most plasma simulations does not hold for regions acting as source for backflow currents on SMART–1. This paper describes the enhancements in model as well as in the numerical methods. A a fast multigrid solver working on a modified multigrid in conjunction with domain subdivisioning allows for instant solution of Poisson’s equation at high accuracy in arbitrary distributed high resolution domains. A new kind of electron model enables self-consistent solution of the electron density distribution throughout the large sized domain. ∗ Junior
Scientist, Aerospace and Advanced Technologies,
[email protected] Scientist, Aerospace and Advanced Technologies,
[email protected] ‡ Head of Department, Aerospace and Advanced Technologies,
[email protected] § Head of the Electric Propulsion Section, ESTEC,
[email protected] ¶ Head of the SMART–1 Workinggroup, ESTEC,
[email protected] k Senior Electric Propulsion Engineer, ESTEC,
[email protected] ∗∗
[email protected] ††
[email protected] ‡‡
[email protected] † Senior
1 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
Nomenclature ne , ni me , mi kB e ε0
= = = = =
electron, and ion number densities, m−3 masses of electrons, and ions, 9.11 × 10−31 kg, 2.2 × 10−25 kg Boltzmann constant, 1.381 × 10−23 J/K elementary charge, 1.602 × 10−19 C electric permittivity of vacuum, F/m
I.
Introduction
mart–1, the first mission of ESA’s “Small Missions for Advanced Research in Technology” program S is the premiere European satellite utilizing an electric thruster as main propulsion system. The main objective of this mission was the demonstration of this new technology. Several instruments have been compiled in the Electric Plasma Diagnostic Package (EPDP) and the Spacecraft Potential Electron and Dust Experiment (SPEDE) allowing for determination of plasma parameters such as its potential, temperature, or electron density. The main body is roughly cubic with approximate dimensions 1 m × 1 m × 1 m. The PPS-1350 Hall thruster built by SNECMA is situated on the +Z-panel. On each of the ±Y panels of the main body an extension arm of 90 cm length is mounted pivotable around the Y -axis attaching the two solar array wings to the satellite. The thruster exhausts primary beam ions at energies of approximately 350 eV. The hollow cathode neutralizer causes concentrations of neutral particles at thermal energies around 1, 000 K. Charge exchange (CEX) collisions of these neutrals with primary beam ions create a source for slow CEX particles. These are easily redirected by electric fields in the plume and in the vicinity of surfaces. Since the spacecraft can be seen as a floating body immersed in ambient plasma its potential is expected to be negative. Hence the satellite attracts positive CEX ions. The resulting backflow current is responsible for spacecraft charging, surface sputtering, and material deposition. These effects are known to cause degradation of solar arrays, differential charging especially on payload devices, and even sparking. Damage of sensible electronics and short circuits are possible. Hence investigation of the interactions of CEX plasma with the satellite is of vital importance for future spacecraft design Analysis of the data provided by plasma diagnostic instruments has shown a cyclic variation of the spacecraft floating potential with the orbit. The relative changes exhibit direct dependence on the rotation angle of the solar array wings following the sun. ARC Seibersdorf research is investigating the plasma environment and interactions with the solar arrays numerically with a code called SmartPIC. However, review of plasma physics for the special configuration on SMART–1 shows that the quasineutrality assumption used for backflow calculations is violated in regions of low CEX density above the solar array. Detailed sheath modeling is necessary to include these effects and acquire correct backflow data. This is the motivation for the development of a new kind of electron fluid model allowing for a self consistent solution of electron densities throughout the computational domain. Such a model requires major changes to the simulation in both, physical modeling, and computational methods.
II.
Floating Potentials on SMART–1
As has been pointed out previously4,11 the spacecraft floating potential ΦSC on SMART–1 correlates exactly with the rotation of the solar array which is conducted in discrete steps of 5 ◦ . These stepwise changes have been the decisive indicator to rule out other effects like photoemission or dependence on the orbit. Measurements of the cathode reference potential Ucrp are representative for the spacecraft floating potential. Exemplary data is depicted in Figure 1.
III. A.
Model utilized in Previous Versions
Physical Model
The electric propulsion exhaust and plasma interactions simulation SmartPIC is being developed since the year 2000. It is a hybrid PIC code, meaning that ions and neutral particles are treated as particles while electrons are assumed to form an ambient fluid of equal density than the ions. Hence for calculation
2 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
Figure 1. Cathode reference potential and solar array rotation angle. Data by ESA from SMART-1
of plasma potentials ΦP a simplified version of the momentum equation, the Boltzmann relation, is used. µ ¶ kB Te ne Φ= ln + Φ0 (1) e ne,0 where the index 0 indicates a known reference state. The assumption of quasineutrality gives ne,0 ≈ ni . Electron temperatures are obtained by an adapted adiabatic model kB Te n−γ+1 = C1 e
(2)
where γ = 1.28 and C1 = 8.76 × 10−5 are derived from experimental results (see Ref.9 and references therein). Static potentials on the satellite are solved in an initial step by a successive over-relaxation (SOR) algorithm with TBD acceleration. Solutions of potentials emerging from the spacecraft, and plasma potentials are superimposed to a total solution Φtot from which the electric field E can be computed by E = ∇Φtot (3) Collisions are modelled by a statistical Monte Carlo method assuming Markov processes. The probability for a collision, thus, does not depend on the particles history but solely on the current timesteps’ configuration. The types of collisions included in the current simulation are collected in Table 1. Probabilities P for collisions are computed using the simple model P (∆v) = 1 − e−∆vσαβ nβ ∆t
(4)
where ∆v is the relative velocity of the two collision partners α, and β, σαβ is the cross section for the specific collision, and ∆t is the time step. CEX cross sections are calculated using the model by Miller10 already introduced in Ref.9 . Elastic collisions utilize the model by Oh12 . B.
Computational Model
Exploiting the symmetry of SMART-1 the simulation includes one half of the satellite’s main body, one extension arm and one solar array. The computational domain is sized in a way to provide sufficient space around the spacecraft for CEX plasma flows. The computational grid is of cubic type with a constant size of ∆x = 2.5 cm. Ions and neutrals are represented by so called “super-particles” representing 108 −1011 physical particles. Ions are introduced at the thruster exit, neutrals enter the simulation at the thruster, and the hollow cathode position. Background neutrals and electrons are considered as a fluid represented by densities nn and ne ≈ ni respectively. Particles move freely through the domain while fields, densities, and 3 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
Table 1. Types of collisions implemented in the current version
Type of collision
Model
Xe + Xe
elastic
−→
Xe + Xe
σel =
Xe2+ + Xe
elastic
−→
Xe2+ + Xe
σel =
Xe+ + Xe
CEX
Xe + Xe+
+
+
−→
a = 1.03 × 10−18 b = 7.70 × 10−20
Xe+ + Xe2+
σCEX (∆v) = a − b ln ∆v
a = 4.32 × 10−19 b = 1.18 × 10−19
Xe+ + Xe+
σCEX (∆v) = a − b ln ∆v √ σre = k Te
a = 1.03 × 10−18 b = 2.8 × 10−20
Xe2+ + Xe+
CEX
−→
Xe2+ Xe
CEX
Xe2+ e−
recomb.
−→
−→
k = 1.28 × 10−15
σCEX (∆v) = a − b ln ∆v
Xe + Xe
+ Xe
k = 6.42 × 10−16 a = 1.71 × 10−18 b = 1.18 × 10−19
−→
Xe
Parameters
σCEX (∆v) = a − b ln ∆v
CEX
2+
k ∆v k ∆v
Xe+
2+
k = 2.0 × 10−18
Figure 2. The computational domain. Outer dimensions are (DDSx × DDSy × DDSz ) = (2.2 m × 7.5 m × 2.2 m. The offset in Z-direction is given by SDOz which is varied according to the solar arrays rotation angle.
potentials are calculated on grid points. According to the PIC scheme the electric forces are projected onto the particle positions by a first-order tri-linear interpolation scheme. Movement is conducted in two half-steps. In the first particle energy and thrust are calculated while in the second the positions are updated. This method is usually referred to as the “leap frog scheme”. The simulation runs are done on Dell workstations with 3.0GHz and 1GB of memory under Windows XP. Equilibrium is defined by a stationary plasma flow to the solar array and is reached after approximately 1 ms. Computation takes approximately two days per run. C.
Floating Potentials
In general the spacecraft floating potential ΦSC is calculated by balancing ion currents against electron currents. SmartPIC implements a model introduced by Samanta Roy14 . The surface potential will vary until the net charge flux to the spacecraft vanishes. The total ion flux Ii is composed of several components: Static thermal flux IT H , static ram component IRAM for the spacecraft moving through ambient plasma at speed vSC exhibiting an effective ram area ARAM , thruster ion exhaust current IEX , and CEX ion backflow IB . Hence Ii
= =
IT H + IR AM + IEX + IB r eni,0 ASC 8kB Te,0 + ene,0 vSC ARAM + IB + IEX 4 πmi
(5)
Accordingly this current has to be met by a total electron current obtained by the equation # " r r ZZ −ΦP ) ene,0 ASC 8kB Te,0 e(ΦSC enCEX 8kB TCEX e(ΦkSCT−ΦCEX ) kB Te e e B CEX − IN − IINT (6) Ie = − − df 4 πme 4 πme ASC The components of this current (ordered by appearence of the summation terms) are: Thermal current IeT H , electron backflow current IeB , neutralizer exhaust current IeN , and the backflow current addition4 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
ally collected by interconnector structures on the solar arrays IeIN T . The CEX ion current IB is calculated by counting ions colliding with surfaces. For reasons of analysis SmartPIC distinguishes currents to different parts of the spacecraft.
IV.
Analysis and Review of Plasma Physics
SmartPIC has already led to good results regarding the calculation of backflow currents18 , prediction of plasma parameters in the plume17,9 , and floating potentials4 . The physical models the simulation is based on have been successful and sufficient for previous analysis. Nonetheless for higher accuracy and quantitative correctness the simplifying assumptions leading to quasineutrality and the Boltzmann relation have to be rethought critically. The theory of plasma sheaths in the vicinity of metallic surfaces is well known. Potentials Φs emerging from surfaces into the plasma are damped until they reach a value of Φ0 = 1e Φs at the sheath boundary q ε0 kB Te which is defined by a distance of approximately one Debye length λD = ne e2 . Potentials Φ ≤ kB T /e propagate nearly unhindered deeper into the plasma. In general at the sheath boundary the quasineutrality assumption ne ≈ ni is valid. For calculation of the spacecraft floating potential the surface integral in Equation (6) is computed by summation over grid points adjacent to surfaces which defines a constant sheath length of 2.5 cm. This is not correct in terms of the model introduced by Samanta Roy14 where nCEX , TCEX , and ΦCEX (in Equation (6)) have to be evaluated at the sheath boundary. However determination of the correct values for these three variables is not a simple issue.
Z
2
1
Ion Density: 1E+08 1E+09 1E+10 1E+11 1E+12 1E+13 1E+14 1E+15 1E+16 1E+17 0
1
2
3
4
5
6
0
7
I
Y
(a) Ion density distribution and approximate sheath boundary (black line)
(b) Theoretical potential distribution in the sheath
Figure 3. Debye length and Sheath
The Debye length λD in this simulation ranges from 3 × 10−5 m close to the thruster exit and up to 1 m in the far regions above the solar array. One main stability criteria for PIC simulations is that the grid size fulfills ∆x ≤ λD . Hence a grid of higher spatial resolution is needed. On the other hand a basic requirement for SmartPIC is to run on a standard workstation PC which has limited memory resources. Filling the complete domain depicted in Figure 2 with a grid featuring ∆x = 3 × 10−5 m would take 1.7×1015 grid points which is not feasible. The solution is definitely to use adaptive grid sizes.
V. A.
Enhancements
Semi-adaptive Multigrid
A special grid of rectangular type has been developed allowing for local high resolution at a minimum of total grid points. A modified domain subdivision scheme is used to distribute nested sub-domains of increasing spatial resolution throughout the computational domain. A total of 6 levels provides grid sizes of 8.6 × 10−3 m to 0.275 m. Domains are distributed to adapt to the expected density distribution and according Debye lengths. The total number of grid points does not exceed 1.2 × 106 . 8.6 × 10−3 m still doesn’t meet the Debye length in the regions of density ne = 1014 m−3 but since microscopic particle movement is not subjected in SmartPIC this is not seen to be a problem. Special treatment is needed for the solar array which is pivotable and features extremely fine interconnector structures and high electric fields. An automatism in the grid generation distributes high resolution domains stepwise on the array to adapt its rotation angle and resolve fine structures. This 5 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
2
Z 1
0 0
1
2
3
4
5
6
7
Y
(a) YZ-Slice of the entire domain
Z
X
Y
Z
Y
X
(c) Adaption to solar array at 45 ◦
(b) Domains above the thruster
Figure 4. Exemplary views of the sub-domain distribution
advanced technique allows further enhancements in modeling of the solar array structures, mainly the interconnectors. These metallic structures have been found4 to be the main electron current drain to the spacecraft and, hence play a major role in the calculation of floating potentials. Exact implementation of geometries is an essential requirement for accurate results. The solar array model implemented in the simulation is depicted in Figure 6. Secondary electrons tend to be attracted by the solar array’s positive potentials. On dielectric surfaces they build up space charges that shield out the underlying potentials In order to simulate this effect a “shielding factor” ηsh has been introduced. η = 100 % means
7
4
5
6
01
1
2
3
X 0
-1
Z
2
1
0
1 Z
2
3
5
6
4Y
X Y
7
Figure 5. Distribution of sub-domains throughout the computational domain
6 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
Figure 6. Modeling of the solar array structures and potentials. The two solar array wings are each divided into three panels of equal size. Each panel has 6 sections (in y-direction) which consist of three strings(in x-direction). The strings again are placed such that the voltage increases meander-shaped from 0 V to a maximum of 50 V along along 5 substrings. These are connected at their ends via “interconnectors” of 8 mm width. Supporting structures have metallic surfaces and are biased to spacecraft ground. Interconnectors are bare metallic and biased to the potential of adjacent solar cells. Solar cells are covered by glass of 150 µm thickness and are not conductive.
total shielding which corresponds to a potential of 0 V. According to Hastings7 space charge limitation is not applicable to typical solar array structures in low density plasma. Hence (in contrast to previous versions4 ) only glass surfaces are shielded, interconnectors are not. B.
Multigrid Poisson Solver
The method to obtain a solution for the electric potential in most current PIC simulations is based on quasineutrality (for example see Refs.1,14,18 ). Principally the Poisson equation ∆Φ =
e (ne − ni ) ε0
(7)
is solved where the electron density ne is obtained by the Boltzmann relation (1). This is a simple and widely used method to include sheath effects at biased surfaces. In fact it is inaccurate for special configurations. The point is that in applying the Boltzmann relation one states the assumption of a completely uniform density ni,0 = ne,0 at the sheath boundaries, and a homogeneous undisturbed plasma of indefinite extension behind. These requirements are necessary to sustain ion and electron currents building up space charges in the sheath. According to Chapman5 , and Chen6 sheaths take only the known form described by the Boltzmann relation if the currents drawn from the plasma do not represent a serious drain. This can be quantified by the well known sheath criterion by Bohm. On SMART–1 the creation of CEX ions can be estimated to be approximately 100 mA. Backflow currents are 20 mA-30 mA, ion currents are one order of magnitude below. According to the sheath theory by Langmuir this is definitely a serious drain. The CEX plasma cloud cannot sustain such high currents to the sheath which results in invalidation of the linear theory. The multigrid solver is inherently bound to the usage of a multi-grid. The domain subdivision technique requires to adapt the standard scheme found in the literature20 . Basically an iterative Gauss-Seidel scheme is utilized to propagate the potential according to Equation (7) on multiple layers gaining a complete solution throughout the computational domain in approx. 5 s. Generally the multigrid scheme
7 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
is one of the fastest known numeric algorithms for problems of this type.
C.
Virtual Sensors
SmartPIC implements various virtual sensors for comparison to real data. Two of these have been utilized for the current investigation: RPA sensors and Langmuir probes. 1.
Retarding Potential Analyzer
RPAs measure the energy of incident ions and, thus, provide information about the flow. Positions are defined relative to the thruster exit (see Figure 7(a)). Positioning data for RPAs utilized in ground tests as well as on SMART–1 are collected in Table 2.
(a) Geometry definition of RPA positions
(b) Position of EPDP and SPEDE
Figure 7. Positioning of sensors
Table 2. Positions of RPA sensors used for data comparison
No. 1 2 3 4 5 6
2.
d [mm] 670 670 670 670 670 670
α [◦ ] -10.0 0.0 5.0 10.0 15.0 20.0
β [◦ ] 0.0 0.0 0.0 0.0 0.0 0.0
data from LABEN LABEN LABEN LABEN LABEN LABEN
No. 7 8 9 10 11 12
d [mm] 1,092 1,035 1,016 800 215 570
α [◦ ] 21.5 11.2 0.0 60.0 88.0 85.0
β [◦ ] 0.0 0.0 0.0 0.0 61.0 25.0
data from LABEN LABEN LABEN LABEN LABEN SMART–1
SPEDE Probes
The two SPEDE probes have already been implemented by Scharlemann4 et al. Investigation has been conducted without new multigrid techniques. SPEDE consists of two cylindrical shaped metallic probes mounted on 60 cm long booms extending from the ±X planes of the main body. SPEDE provides three operating modi. First it can be used as an electric field mode to measure plasma waves. In the second mode the sensor operates as a Langmuir probe biased at constant or variable voltage to obtain either relative changes in the electron density or electron densities and temperatures.
D.
Verification
For verification of the new grid and the multigrid solver several crosschecks of current results with analytical predictions, ground data, in-flight data, and previous results by Scharlemann4 et al. have been conducted.
8 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
1.
Poisson solver
Verification of the multigrid solver has been done by comparison to analytical solution. Test objects are point charges immersed in plasma of different density. The potential was measured across several grid domains of different spatial resolution to check for possible instabilities induced at the grid boarders. 2.
RPA verification
The ground test campaign by LABEN originally intended to give reference measurements for the STENTOR satellite that was lost in the 2000 Ariane–5 failure. Tests have been conducted with an SPT-100 that was able to simulate the expected PPS-1350 environment by increased discharge voltage. Explicit documentation of the test campaign and detailed implementation in SmartPIC can be found in an earlier work9 . The original report by LABEN is unpublished. The analysis included a sweep through the beam from α = −10 ◦ to α = +20 ◦ . In general an offset of 25 V has to be added to experimental results due to the potential configuration in the experiment. In low angle measurements (α < 42 ◦ ) which are taken in the plume SmartPIC’s results in general agree very well with the measurements. Full Widths at Half Maximum (FWHM) in the simulation are less due to a uniform kinetic exit velocity of the ions according to the discharge voltage. Peak values in contrast fit very well. Far angle RPAs are representative for the CEX environment. Again the experimental results are met.
Stentor measurement
1.0
1.0
SmartPIC 0.9
1.0
0.9
SmartPIC (1035mm)
0.7
0.8
I E
SmartPIC
(670mm)
Relative intensity, normalized d /d
I E
0.9
of SmartPIC data
0.8
Stentor measurement
Relative intensity, normalized d /d
8 point FFT smoothing
0.8
0.7
0.6
0.5
0.4
0.3
0.6 0.7 0.5
0.4
0.6
0.3 0.5
0.2
0.1 0.4 0.0 0
0.3
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30
0.2
0.2 0.1
0.1 0.0 0
0.0 0
50
100
150
200
250
Ion Energy
300
350
400
450
50
100
150
200
Ion Energy
500
250
300
E [eV]
E [eV]
(b) Comparison of RPA 10 data at α = 60 ◦
(a) Comparison of RPAs 4 and 8 with simulation data at α ≈ 10 ◦
400
360
Peak Ion Enery E
max
/eV
380
340
320 E
(shifted +25eV)
E
(simulation)
E
(raw data)
max
300
max
max
280 -10
-5
0
5 Beam Angle
10
15
20
/°
(c) RPA angle sweep Figure 8. Comparison of LABEN ground test data to simulation results
9 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
350
400
3.
Reproduction of Previous Results
The floating potential calculation has been conducted for different rotation angles of the solar array. The investigation was done in the range of 0 ◦ to 180 ◦ in steps of 45 ◦ . The main differences appear
1 0.01
]
Main body (prev.)
Ext.
arm (curr.)
Ext.
arm (prev.)
0.1
2
Solar array (curr.)
I Ion currents
Mean collective surface [m
Solar array (prev.)
i
[A]
Main body (curr.)
1E-3
1E-4
1E-5
0.01
Main body (curr.)
1E-3
Main body (prev.)
Solar
array (prev.)
Solar
array (curr.)
Interconnectors (prev.) Interconnectors (curr.)
1E-4
1E-6 0
30
60
90
120
Solar array rotation angle
150
180
0
30
60 Solar
[°]
(a) Ion backflow currents
90
120
array rotation angle
150
180
[°]
(b) Collective areas
0.1 0.01 1E-3
Absolute electron currents
I
i
[A]
1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 1E-11 Interconnectors (prev.)
1E-12
Interconnectors (curr.) Main body (curr.)
1E-27
Solar array (curr.)
1E-29
Main body (prev.) Solar array (prev.)
1E-31 0
30
60
90
120
Solar array rotation angle
150
180
[°]
(c) Electron backflow currents
Figure 9. Comparison of the most important parameters for floating potential calculation. (a) depicts ion currents on different parts of the spacecraft. (b) compares the mean surface areas collecting electrons. c) shows the calculated electron currents based on Equation (6). Abbreviations: prev. for previous version, and curr. for current version results.
in effective collecting surfaces of the interconnectors. Due to not shielding these structures the collected current is lower by several orders in magnitude. For low absolute currents the floating potential calculated by Equations (5) and (6) reacts more sensitively on relative changes. This explains the amplitude of the variation ∆ΦSC = 30 V compared to ∆ΦSC = 7 V of previous SmartPIC versions and ∆ΦSC = 14 V of SMART-1 data in in Figure 10. However absolute values of the floating potential varied significantly throughout the mission.
10 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
0
-5
Spacecraft floating potential
SC
[V]
-10
-15
-20 SmartPIC (current version) SmartPIC (previous results) SMART-1 derived data
-25
-30
-35
-40 0
15
30
45
60
75
90
105
120
Solar array rotation angle
135
150
165
180
[°]
Figure 10. Comparison of the floating potential dependence achieved with previous and current versions of SmartPIC, and flight data from November 2004.
E. 1.
Results Current Measurements by SPEDE probes
For implementation of SPEDE data in SmartPIC the domain size had to be changed to (DDSx ×DDSy × DDSz ) = (3 m×5 m×2.2 m) which resulted in approximately 2×106 grid points. Accommodation of the probes in X-dimension required widening of the domain. In order to keep the total memory consumption the Y-dimension was cut from 7.5 m to 5 m. For comparison to flight data the virtual SPEDE probes were set to biasing voltages −3 V, and 3 V. Measurements on SMART–1 have shown an asymmetry between the two probes. Currents on the -X panel appear to be higher than on the opposite side. Outcomes of SmartPIC confirm the asymmetry (Figure 11(b)). Currents on the +X side which is opposite to the neutralizer appear definitely lower compared to the -X side. The difference remains roughly constant at 170 nA to 200 nA, independent of the biasing voltage. This emphasizes the assumption that the neutralizer affects the local CEX flow field. SPEDE 2003-10-10 -3,110-6
5
-3,210-6
1.2x10
-6
-X
Side
+X Side
0
-3,510-6
-3,610-6
-3,710-6 5:33:20
6:56:40
cu rre n t -x
b ia s -x
cu rre n t + x
b ia s + x
8:20:00
9:43:20
11:06:40
1.0x10
8.0x10 Current [A]
Current [A]
-3,410-6
Bias Voltage [V]
-3,310-6
6.0x10
4.0x10
2.0x10
-5 12:30:00
Time
(a) SMART–1 SPEDE data from Oktober 10th , 2003
-6
-7
-7
-7
-7
0.0 -3
-2
-1
0
1
2
3
Biasing Voltage [V]
(b) Simulation of SPEDE probes for biasing voltages −3 V, and 3 V
Figure 11. SPEDE data from SMART–1 compared to simulation outputs
2.
SMART–1 RPA Flight-Data
In addition to the floating potentials, and SPEDE data comparisons mentioned earlier the EPDP RPA data has been reproduced. Since measured ion energy is relative to the electric ground of EPDP the 11 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
according floating potential Φf,EP DP for the sensor has to be taken into account. Unfortunately there is no time-correlated data for the EPDP floating potential and RPA measurements. Analysis of available data for Φf,EP DP has shown that the variation, and hence the shift of the RPA curve, is between −11 V and −18.5 V. The resulting upper and lower boundary curves are depicted in dark and light grey in Figure 12 respectively. Unfortunately the actual tilting angle β of the EPDP RPA is unknown. The 3.5x10
-7
SMART-1 EPDP RPA
3.0x10
SMART-1
-7
20pt
FFT smoothed
Maximum shift boundary Minimum shift boundary
Relative Intensity d
IRPA/dE [A/eV]
2.5x10
-7
Simulation
4pt
2.0x10
1.5x10
1.0x10
5.0x10
Simulation
-7
4pt
β=0°
FFT smoothed
0°
β=25°
FFT smoothed
25°
-7
-7
-8
0.0
-5.0x10
-1.0x10
-8
-7
0
10
20
30
40
50
Ion Energy
60
E [eV]
70
80
90
100
Figure 12. Comparison of simulation data with SMART-1 RPA data.
figure shows the results for two runs at different angles. In fact there is a strong dependence on this setting. For β = 15 ◦ the best fitting between simulation and experiment is obtained.
VI.
Conclusion
Revision of the physical and computational model utilized in SmartPIC has led to several enhancements in accuracy. A modified multigrid technique with domain subdivisioning has been implemented to allow higher spatial resolution in regions of interest. The grid size is variable between 8.6 × 10−3 m and 0.275 m. A special multigrid solver for obtaining potential solutions in shortest times and at high accuracy has been implemented and tested . The resulting plasma flow field has been verified by extensive comparison to experimental ground test data. RPA data acquired on SMART–1 has been predicted correctly by the simulation. Floating potential calculations obtained with previous versions of SmartPIC have been emphasized.
VII.
Outlook
Completing the restructuring campaign for SmartPIC a new kind of electron model has been developed. It is dedicated to gain a complete solution for the electron density distribution without relying on the Boltzmann relation. This guarantees conservation of charge and currents. Furthermore the spacecraft potential will be calculated by a new model based on the electric capacity of the satellite immersed in the plasma. This will form the physically most complete and accurate model for the calculation of spacecraft charging A.
Kinetic Density Fluid Model
This model is based on the idea to create a kinetic flow field without the use of discrete particles. This circumvents the problem of the tremendous rise in computational times which scales with O(N 2 ) in the total number of particles for some parts of the simulation. Kinetic aspects are necessary in order to permit electrons to escape the CEX flow and follow the potential fields emerging from the solar array to account for electron backflow currents. The basic idea is to calculate flow speeds that apply to densities located at the grid points. In every time step the density is moved virtually by its flow vector ve ∆te and is then interpolated to the surrounding
12 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
grid points by a weighting function. In order to conserve inertia and energy the weighting has to include the mass and velocity of the density fractions. Starting at the equation of motion one obtains me ne
∂ve ene j + me ne (ve · ∇)ve + ene E + ∇p = ∂t σei
(8)
Using a standard forward discretization scheme one can directly convert this equality to a discrete expression for the change in velocity ∆ve based on the current set of data (ne (x, t), ve (x, t), E(x, t), T (x, t)) · ¸ e kB ∆ve = ∆t ve2 ni σ − (ve · ∇)ve − E− ∇(ne Te ) (9) me ne me Now that the flow speeds are known the move can be performed. In order to respect the continuity
Figure 13. (a) For each grid point (x, y) the electron density ne is moved in direction of the velocity vector ve to a position (x, y) + ve ∆t. (b) In the second step the density is assigned to the closest grid points by trilinear interpolation. Each point acquires a density fraction ∆ne = fi ne where i indicates the target grid point. e equation ∂n ∂t − ∇(ve ne ) = 0 which guaranties charge and mass conservation a method is necessary which can be represented by an operator normalized to 1. Such an operator is the well known trilinear interpolation. In order to include Maxwellian velocity distributions explicitly in the model the weighting can also be done by a Maxwell-Boltzmann distribution function around the interpolation point. In either case a displacement vector ∆x = (ve + ∆ve )∆t is calculated which gives a virtual position xv = x + ∆x. Based on this position the density originated at x can be interpolated to the surrounding grid points p. Finally the weighting has to include conservation of momentum which is guaranteed by
vp =
ne (p)ve (p) + ne (x)fi (xv ) (ve (x) + ∆ve (x)) ne (p) + ne (x)fi (xv )
Hence charge, momentum, and mass are inherently conserved by the algorithm. Regarding the electron temperature the standard adiabatic model utilized in previous versions of SmartPIC and other simulations (for example Refs.2,1 ) is used. Following Boyd2 and Taccogna16 the adiabatic exponent γ is set to 5/3. Using such a simple model might be subject to criticism but the model presented is at an early stage of development. Further refinements in the electron temperature calculation are planned. In order to save computing time an artificial mass ratio15 mi /me = 100 has been chosen which allows accordingly to use a timestep ratio ∆t(ion)/∆t(e− ) = 10. This erases microscopic movement of the particles like gyration or drifts. Since SmartPIC is a pure electrostatic simulation this is not seen to be a drawback. First tests with the new electron model put on some issues of stability. The usage of superparticles according to the PIC scheme rises stability problems. Since one such particle represents 109 physical particles one can estimate for densities of approximately 1017 m−3 there are only 5-10 particles per grid cell on the highest resolution. Hence there is high noise in the local charge distribution that accounts for appropriate electric fields. A method to cut down the electric field strength in a physically correct way is being investigated at the moment. Some aspects of this model might give rise to criticism but it has to be pointed out that in fact no existing model is capable of simulating electron fluids throughout such a huge domain; this one is intended to be. 13 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
B.
New model for floating potential calculation
Although there are many interesting approaches in the literature8,13 existing models are still based on assumptions. The availability of independent fluxes of electrons and ions in the new electron model will enable to calculate the floating potential in a different way. Returning to the very rudimental physics of spacecraft charging the basic assumption common to all usual models is conservation of currents. This, undoubtable, is a reasonable and valid basis for calculations. But conservation of currents is not the most basic principle in charging. It is the collection of discrete charges, causing electric potentials and, thus, react on the amount of newly collected particles which define a current. Hence the most accurate calculation of a floating potential should include the influence of collected particles (which define a backflow current). From the theory of electromagnetic fields it is known that for a configuration of N insulated conductors the following relations hold N X Qi = Cij Φj (10) j=1
where i and j are indices of the conductors, Φi depicts the electric potential, and Cij are the electric capacity coefficients depending solely on the geometry of the configuration. If one adapts this picture for a satellite surrounded by plasma in fact the system can be seen as a badly insulated capacitor. The total space charge in the plasma is very low and only effective in the sheaths near surfaces. Hence for a first approximation one can assume the spacecraft to be insulated and floating in vacuum. For such a single conductor Equation (10) simplifies to Q = CΦSC . The difficulty now is the determination of the exact capacity C of the spacecraft. It strongly depends on the geometry and cannot be obtained analytically. A possible solution is provided by the multigrid solver. If one assumes the spacecraft as just a vacuum filled conductive hull of equal geometry and places a known uniform charge density on the surface one is able to calculate the internal potential. Faraday states that ∇Φ = E = 0 internally. Hence the potential is constant and should equal the spacecraft floating potential for the given charge. Furthermore the result should depend linear on the total charge applied. The coefficient of this linearity is the capacity sought. This scheme has already been implemented and results for Smart–1 in a capacity of 1.7368 × 10−10 F against vacuum. Bearing in mind that the expected total current from ambient plasma obtained by Equations (5), and (6) for a density 108 m−3 and uniform temperature of 0.6 eV initially is of the order −10−5 A. Assuming a transient of 0.1 ms one obtains for the spacecraft potential ΦSC = −5.8 V. This correlates excellent with theory and measurements for floating potentials on spacecraft.
Acknowledgments This work is being funded under ASAP contract no. W 45100 27 132-1. Ren´e Sedmik would like to thank Mr. Densis Estublier from ESTEC for great support with SMART–1 flight data Additionally we would like to thank all participants from the SMART-1 working group at ESTEC for the continued discussions on the topic, in particular the contributions from Alain Hilgers on the floating potential calculations are greatly appreciated.
References 1 Boyd, I.D., “Review of Hall Thruster Plume Modeling”, Journal of Spacecraft and Rockets, Vol. 38, No. 3, pp. 381, May-June, 2001 2 Boyd, I.D., “Modeling of the near field plume of a Hall thruster”, Journal of Applied Physics, Vol. 95, No. 9, pp. 4575, 2004
Figure 14. Convergence of total charge density. Slices are printed at every 2nd electron timestep.
14 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
3 Boyd, I.D., “Numerical Simulation of Hall Thruster Plasma Plumes”, Paper presented at the 9th Spacecraft Charging Technology Conference, Tsukuba, Japan, April 2005 4 Scharlemann, C., Tajmar, M., Gonz´ alez, J., Estublier, D., Noci, G., and Carpacci, M., “Influence of the Solar Arrays on the Floating Potential of SMART-1: Numerical Simulations”, 41th Joint Propulsion Conference, Arizona, July 2005, AIAA-2005-3859, 2005 5 Chapman, B., Glow Discharge Processes, John Wiley & Sons, New York, 1980 6 Chen, F.F., Introduction to Plasma Physics and Controlled Fusion, 2nd ed., Vol. 1:Plasma Physics, Plenum Press, London and New York, 1984 7 Hastings, D.E., and Chang, P., “The physics of positively biased conductors surrounded by dielectrics in contact with a plasma”, Journal of Physics of Fluids B, Vol. 1, No. 5, pp. 1123, May 1989 8 Hilgers, A., Thi´ ebault, Estublier, D., Gengembre, E., Gonz´ alez, J., Roussel, J.F., Tajmar, M., Markelov, G., and Forest, J., “Modelling of SMART-1 Interaction with the Electric Thruster Plume”, Proc. 9th Spacecraft Charging Conference , Tsukuba, Japan, April 2005 9 Tajmar, M., Meissl, W., Gonz´ alez del Amo, J., Foing, B., Laasko, H., Noci, G., Capacci, M., M”alkki, A., Schmidt, W., and Darnon, F., “Charge-Exchange Plasma Cotamination on SMART-1: First Measurements and Model Verification”, 40th Joint Propulsion Conference, Ft. Lauderdale, FL, July 2004, AIAA-2005-3437, 2004 10 Miller, J.S., Pullins, S.H., Levandier, D.J., Chiu, Y., and Dressler, R.A., “Xenon charge exchange cross sections for electrostatic thruster models”, Journal of Applied Physics, Vol. 91, No. 3, pp. 984 11 Milligan, D., Camino, O. “SMART1 EP floating potential and SA rotation correlation”, ESA technical note, S1ESC-TN-5540, December 17th , 2004, unpublished 12 Oh, D., “Computational Modeling of Expanding Plasma Plumes in Space using a DSMC Algorithm”, Ph.D. Dissertation, Masschusetts Institute of Technology, Boston, MA, 1997 13 Scudder,J.D., Cao, X., and Mozer, F.S., “Photoemission current-spacecraft voltage relation: Key to routine, quantitative low-energy plasma measurements”, Journal of Geophysical Research, Vol. 105, No. A9, pp. 21,281, September 2000 14 Samanta Roy, R.I., “Numerical Simulation of Ion Thruster Plume Backflow for Spacecraft Contamination Assessment”, Ph.D. Dissertation, Massachusetts Institute of Technology, Boston, MA, 1995 15 Szabo (Jr.), J.J., 2001, “Fully Kinetic Numerical Modeling of a Plasma Thruster”, Ph.D. Dissertation, Massachusetts Institute of Technology, , Boston, MA, 2001 16 Taccogna, F., Longo, S., and Capitelli, M., “Particle-in-Cell with Monte Carlo Simulation of SPT-100 Exhaust Plumes”, Journal of Spacecraft and Rockets, Vol. 39, No. 3, pp. 409, 2002 17 Tajmar, M., Gon´ zalez, J., Estublier, D., and Saccoccia, G., “Modelling and Experimental Verification of Hall and ION Thrusters at ESTEC”, Proc. 3rd International Conference on Spacecraft Propulsion, Cannes, October 2000, ESA SP-465, December 2000 18 Tajmar, M., Gonz´ alez, J., Foing, B., Marini, A., Noci, G., and Laasko, H., “Modelling of the Electric Propulsion Induced Plasma Environment on SMART-1”, Proc. 7th Spacecraft Charging Technology Conference, Noordwijk, The Netherlands, April 2001, ESA SP-476, November 2001 19 Tajmar, M., “Electric Propulsion Plasma Simulations and Influence on Spacecraft Charging”, Journal of Spacecraft and Rockets, Vol. 39, No. 6, November-December 2002 20 Tannehill, J.C., Anderson, D.A., and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, Second Edition, Taylor & Francis, 1997
15 The 29th International Electric Propulsion Conference, Princeton University, October 31 – November 4, 2005
