Planetary and Space Science 73 (2012) 145–150

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Planetary and Space Science

0032-06

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n Corr

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journal homepage: www.elsevier.com/locate/pss

Earth magnetic field effects on Swarm electric field instrument

S. Rehman a,n, J. Burchill b, A. Eriksson c, R. Marchand a

a Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2E1b Department of Physics and Astronomy, Calgary, Alberta, Canada T2N 1N4c Swedish Institute of Space Physics, POB 537, SE-75121, Uppsala, Sweden

a r t i c l e i n f o

Article history:

Received 6 April 2012

Received in revised form

1 October 2012

Accepted 1 October 2012Available online 13 October 2012

Keywords:

Thermal ion imager

Swarm satellites

Electric field instruments

Low earth orbit spacecraft

Earth magnetic field

Space plasma

33/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.pss.2012.10.004

esponding author. Tel.: þ1 780 729 1640.

ail address: surehman@ualberta.ca (S. Rehma

a b s t r a c t

Earth magnetic field effects on the particle sensors carried by the Swarm satellites are investigated

using particle in cell (PIC) and test-particle modelling. In the reference frame of the spacecraft in which

plasma flows at relative velocity v!

, Earth magnetic field leads to an ambient electric field

E!¼� v!� B!

, which affects the shape of particle distribution functions at the particle sensors. This

in turn impacts the distribution of particle fluxes on the microchannel plate (MCP) in the ram face

mounted thermal ion imagers (TIIs). Shifts in the centroid of these distributions depend on the direction

and magnitude of the local magnetic field and, as such, are expected to vary periodically along the

spacecraft orbit. The magnitude of these shifts is estimated quantitatively, and the effect of their

variation on the calibration and interpretation of the electric field instrument (EFI) are also discussed.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Spacecraft interaction with ionospheric plasma is known toaffect measurements made with particle sensors. The primarycause of perturbations in these measurements associated withspace environment results from the electric sheath surroundingthe spacecraft and its instruments. These effects have beenstudied by a number of authors, using various theoretical andcomputational approaches. For example, Robinson and Coakley(1992) considered the basic physics of spacecraft charging anddischarging as a part of general study of dielectric and plasmainteraction. A comprehensive review of plasma interaction withspacecraft in low Earth orbit (LEO) is given by Hastings (1995).The subjects considered in that paper included plasma environ-ments in equatorial and polar orbits, the theory of plasma wakestructure behind spacecraft. It was explained, in particular thatthe main source of charging of equatorial LEO spacecraft are theambient ion and electron current densities. More recently Olsonet al. (2010) used two and three-dimensional PIC simulations tostudy the evolution of the potential structure around the Cassinispacecraft near the orbit of Enceladus. They found good agree-ment between simulation results and measurements. In particu-lar they predict similar spacecraft floating potential, surroundingpotential sheath, density and temperature electron profiles. They

ll rights reserved.

n).

also noted that an increased drift speed or a reduction in plasmatemperature tends to extend the wake region and make thespacecraft potential less negative.

In parallel with in situ observations and laboratory measure-ments, a number of advanced simulation codes have been devel-oped and used for modelling the interaction of spacecraft withspace environment. The main models in use today are NASCAP-2k, SPIS and MUSCAT, driven respectively by NASA and the U.S.Air force, the European Space Agency (ESA), and the Japan Aero-space Exploration Agency (JAXA). Detailed descriptions of thesemodels and of the numerical approaches that they use can befound in the literature (Mandell et al., 2006, 2008; Roussel et al.,2008, 2012; Muranaka et al., 2008; Hatta et al., 2009). Theapplication of these models to specific spacecraft–plasma inter-action problems has also been reported in several recent articles.For example spacecraft interaction with tenuous plasma ingeosynchronous Earth orbit, solar wind environments was mod-elled with NASCAP-2k (Mandell et al., 2005). In another study byDonegan et al. (2010) the effects of extreme environments inspace (near to sun) on pyrolytic born nitride, barium zirconiumphosphate and Al2O3 materials were made using NASCAP-2k. Itwas found that absolute and differential surface charging arefunctions of temperature and radiation fluxes. This study revealedthat among the materials considered, Al2O3 coatings minimiseboth absolute and differential spacecraft charging. SPIS simula-tions were used to investigate wake effects on the sheath regionaround DEMETER satellite (Yang et al., 2010). It was found thatfast moving electrons make the wake region more negative and

Fig. 1. Simplified Swarm satellite geometry (left). The assemblage of the face

plate, the cylindrical TII sensors (shown on right), and Langmuir probes constitute

the EFI. Each sensor entrance is flanked with two gold strips deposited on the

shells. Note that while the position of the two Langmuir probes is illustrated here

for reference, these probes are not included in the satellite geometry used in the

simulations that follow.

S. Rehman et al. / Planetary and Space Science 73 (2012) 145–150146

that the sheath can extend up to 2.5 m in the wake region. Theauthors noted that the sheath around the spacecraft looked ‘like apeach’.

In a recent paper Marchand et al. (2010) used a combination ofPIC and test-particle techniques to simulate the electrostaticsheath on the ram face of the Swarm satellites to study its effecton ion velocity measurements from the Thermal Ion Imager (TII).Their analysis made use of a newly developed computer code,PTetra, based on an unstructured adaptive tetrahedral mesh,capable of describing complex objects, and accounting for manyphysical processes such as secondary and photo-electron emis-sion, and satellite component differential biasing. A detaileddescription of the methodology used in this model can be foundin a recent paper by Marchand (2012). This study led to thefinding that sheath effects could cause aberrations in the inferredplasma flow velocities of order 37 m/s or less. The study, however,was carried out without accounting for ambient magnetic fieldeffects. In this paper we use a more recent version of PTetra tomake the first assessment of magnetic field effects on the Swarmelectric field instrument (EFI). The calculation proceeds in threesteps. First, the structure of the electrostatic sheath surroundingthe spacecraft and the TII is calculated with PTetra. Test-particlemodelling with particle backtracking is then used to compute iondistribution functions at the TII apertures. Finally, these com-puted distribution functions are used to track ions into thesensors, down to the microchannel plate (MCP), from whichfluxes are calculated on the 64�64 pixel array. These in turncorrespond directly to what will be measured with the TII. Theremainder of this paper is organised as follows. The numericalapproach used in this analysis is summarised in Section 2. Thephysical problem, and simulation results from PIC and test-particle simulations are presented in Section 3. Finally, a sum-mary and a discussion are given in Section 4.

2. Swarm EFI modelling

Swarm, a European Space Agency Earth Observation missionthat will begin in 2013, consists of three identical satellites, eachcarrying many instruments and electrical components on a pay-load exceeding 9 m in length. The Electric Field Instrument ofinterest here, is mounted on the ram face of each of the threeSwarm satellites. Each EFI consists of two Thermal Ion Imagersmounted on a face plate on the ram face of each satellite, and twoLangmuir probes ‘below’ the TIIs, pointing away from the satellite.The TIIs will be used to infer the ambient electric fields from therelation E

!¼� v!� B!

, where v!

is the plasma flow velocity in thesatellite rest frame and B

!is the local magnetic field. Describing

the full geometry of the spacecraft with a detailed representationof its instruments would be prohibitive in terms of requiredcomputing resources, as well as unnecessary for the purpose ofstudying sheath and magnetic field effects on the ram face. Forthat reason, the geometry of a satellite is simplified considerablyby (a) truncating it to only a fraction of the ram section and(b) limiting a detailed description of the geometry to theimmediate vicinity of the front plate and the TII. The geometryconsidered in the simulations that follow is illustrated in Fig. 1.The assumption made here is that this truncated and geometri-cally simplified representation of the spacecraft is sufficient todescribe the main effect of the electric sheath and ambientmagnetic fields on EFI. This assumption was verified with simula-tions that accounted for a more complete description of thespacecraft over the � 9 m of its length. The main effect was adecrease in the floating potential (an increase in absolute value)by � 28%. The structure of the sheath near EFI and its effect on

particle distributions and fluxes described below were otherwiseunchanged.

The basic computational approach used in PIC simulations hasbeen described in detail by Birdsall and Langdon (1985), and byHockney and Eastwood (1981). Several authors have also dis-cussed and used the test-particle approach including, for exampleSpeiser (1965), Delcourt (2002), Takeuchi (2005), Mendillo et al.(1997), Moore et al. (2005, 2007), Delcourt et al. (2007), Richardet al. (1994) and Marchand et al. (2008). In the following, we usetwo versions of the test-particle approach referred to as ‘‘Back-tracking Liouville’’ and ‘‘Forward Liouville’’ in Marchand (2010)and Voitcu and Echim (2012).

A description of the computational approach used in PTetrawas recently given by Marchand (2012). It is summarised here forcompleteness. The model uses a combination of particle in cell(PIC) and test-particle simulations. PIC simulations are used tocalculate the satellite floating potential and sheath electric fieldnear EFI. PTetra uses an adaptive unstructured tetrahedral gridcapable of representing boundaries with complex or irregularshapes. It also uses this grid to solve for the electrostatic potentialand associated electric fields from a finite element discretisationof Poisson’s equation. At present the code is electrostatic; but itdoes account for a time-independent, uniform magnetic field.Photo-electron emission from satellite surfaces can be taken intoaccount by PTetra. It may account for an arbitrary number ofspecies with different densities, masses, charges, temperatures,drift velocities. It describes all particle species fully kineticallywith physical mass ratios. Test-particle simulations, both back-ward and forward in time, are used to calculate ion distributionfunctions at the sensors’ apertures and track them into thesensors, down to the microchannel plate and detector arrays.The underlying assumption in both formulations is that theplasma is well described by the Vlasov equation and that thesingle particle distribution function is constant along particletrajectory. The advantage of using test-particle simulations,compared to a direct calculation of particle distribution functionsfrom PIC simulation results, is that distribution functions can beobtained with essentially no statistical errors (Marchand, 2010).Using particle backtracking, or the ‘backward Liouville’ approach,ion distribution functions are computed around the entranceapertures of both TII sensors. In practice, this is achieved byintegrating ion trajectories at these points backward in time untilthey reach the outer boundary where the distribution function isa known drifting Maxwellian. The initial velocities used in thisintegration are parametrised on an block adaptive mesh invelocity space. Making use of Liouville’s theorem the distributionfunction is then obtained at the initial position. If, on the otherhand, a backward integrated trajectory intersects a component ofthe satellite before reaching the outer boundary, the correspond-ing distribution function is set to zero. The Forward Liouvilleapproach is similar except that particle trajectories are now

S. Rehman et al. / Planetary and Space Science 73 (2012) 145–150 147

integrated forward in time. Here again, making use of the fact thatthe numerical value of a particle distribution function remainsconstant along a particle trajectory, ion fluxes are computed onthe MCP detector array, from the particle distribution functionscomputed at each sensor aperture.

We recall that fields and particle trajectories are not calculatedfully self-consistently in the test-particle approach. The assump-tion there is that the fields used to integrate particle trajectoriesare a good approximation of the actual fields that would beobtained self-consistently. This should be satisfied when applyingbacktracking with fields obtained from PIC simulations. It shouldalso be valid with the Forward Liouville approach used to trackparticles in the EFI sensors, where space charge effects areexpected to be negligible.

3. Case studies

In this section we present results from two case studies obtainedwith Earth magnetic field and plasma parameters that are repre-sentative of those expected along Swarm orbits. For reference andcomparison purposes, one case is also carried out without magneticfield. Specifically, in all cases the background plasma assumed in thesimulations consists of singly ionised oxygen and hydrogen ionswith densities nOþ ¼ 2:25� 1010 m�3 and nHþ ¼ 0:25� 1010 m�3.The electron density is determined from quasi-neutrality to bene ¼ 2:5� 1010 m�3. All species temperatures far from the space-craft are assumed to be equal to T¼0.2 eV. Also all species areassumed to drift in the spacecraft reference frame, with the negativesatellite orbital velocity; that is with velocity v

!d ¼ 7587 m=sx.

Referring to Fig. 1, the two cases considered with magnetic fieldeffects assume B

!¼�40 mTz and B

!¼ 40 mTz expected near the

geographic North and South poles respectively. By comparing resultsobtained in the three cases it will be possible to assess the effect ofthe magnetic field both qualitatively and quantitatively.

3.1. PIC simulations of the floating potential and the sheath

Spacecraft floating potential and the surrounding electrostaticsheath are computed with PTetra for the three cases mentionedabove. In these simulations all species are treated fully kineticallyand the code is run in time-dependent mode until a steady state isreached. In the three cases the spacecraft floating potential, given inTable 1, is approximately �3kT=e, with the least negative valuesfound when a magnetic field is taken into account. In the presenceof a magnetic field transverse to the plasma flow, the spacecraftpotential depends on its position in the simulation domain(Marchand, 2012) because there should be a potential gradientassociated with the uniform E

!¼� v!� B!

field even in the absenceof any field perturbation due to a spacecraft. The physically mean-ingful ‘floating potential’ in this case is therefore not the value of thepotential computed at steady state, when no net current is collected,

Table 1Spacecraft floating potential and potentials calculated at the tips of the Langmuir

probes (mV) from three different simulations at steady state. The x and y positions

of the centroids of the Oþ fluxes, defined in pixel units (see Eq. (1)), are also given

in each case. The uncertainty in the potentials is estimated to be approximately

70.5 mV.

Magnetic field (10�5 T) B¼�4 B¼0 B¼4

Floating potential �653 �657 �653

Left Langmuir probe �9.2 �53.8 �88.8

Right Langmuir probe �89.1 �54.5 �8.1

x 46.43 46.46 46.40

y �0.36 0.004 0.38

but rather the difference between this potential and the onecomputed when the spacecraft carries no charge and surroundingplasma is charge neutral. It is this difference that is reported in theTable. The reduction in the absolute value of the negative floatingpotential with a magnetic field, is consistent with the fact that, forthe parameters considered, the electron thermal gyro-radius(re � 3 cm) is smaller than the size of the spacecraft, while theion gyro-radii (rHþ � 1 m, rOþ � 4:5 m) are larger. As a result,electrons are magnetised and constrained to move to the spacecrafteffectively in one dimension along the field line. Ions, on the otherhand, are effectively unmagnetised and can reach the spacecraftfrom any direction. The result is that the floating potential has to beless negative in order to repel a smaller incoming electron flux andensure zero net collected current at steady state (Marchand, 2012).While the geometry considered in these simulations did not accountfor the Langmuir probes shown in Fig. 1, it is of interest to considerthe plasma potential at their locations. The solution to Poisson’sequation at steady state is used to determine the potential at theprobes’ tips in the three cases considered. The resulting values aregiven in Table 1. Considering the separation of 30 cm of the probesin the y direction, our calculations indicate that a reading of thepotential from these positions would lead to an estimate of theelectric field in the y direction EyC�0:27 V=m for B¼�4� 10�5 T,and EyC0:27 V=m for B¼ 4� 10�5 T. These are close to the valuesexpected from E

!¼� v!� B!

C80:30 V=my corresponding toB!¼ 84� 10�5 Tz. The discrepancy here is largely due to the

proximity of the probe tips to the spacecraft body. Accounting forthe actual probe geometry and relative biases between their variouscomponents would naturally change these potentials. A study ofprobe characteristics accounting for their detailed geometry isbeyond the scope of the present paper. The effect of Bz is bestunderstood by looking at equipotentials in the x�y plane. Three setsof equipotentials, corresponding to the three cases considered, areshown in Fig. 2. In this figure, the cross-section chosen for the

Fig. 2. Equipotential contours for B¼�4� 10�5 T (top), B¼0 T (middle), B¼ 4�

10�5 T (bottom).

Fig. 4. Cross-section of TII showing an example particle trajectory penetrating

through the tunnel, being deflected by the radial field generated between two

concentric hemispheres and precipitating on the MCP. The potential difference

between the two hemispheres is 60 V.

S. Rehman et al. / Planetary and Space Science 73 (2012) 145–150148

equipotentials is such that it goes through the aperture of thehorizontal TII sensor. In the absence of a magnetic and associatedelectric field, the equipotentials surrounding the spacecraft displayan approximate mirror symmetry in the y¼0 plane. With amagnetic field, however, the associated � v

!� B!

electric field isseen to cause a significant distortion of the equipotentials in the y

coordinates. With B¼�4� 10�5 Tz, the ambient E!

is directedalong �y, and equipotentials are seen to wrap around the satelliteand ‘pile up’ on the y40 side. This in turn, leads to a significantlystronger electric field on that side of the spacecraft than on theopposite (yo0) side. The opposite is observed for B¼ 4� 10�5 Tz.

3.2. Test-particle calculation of distribution functions

Given the potential field surrounding the spacecraft and the TIIaperture, particle backtracking is used to compute the velocitydistribution functions for both ion species at 31 points uniformlydistributed around each TII aperture. Each distribution function f

is discretised on an unstructured block adaptive mesh in velocityspace, in which more mesh points are created where f issignificant and varies appreciably. The effect of the induced � v

!�

B!

electric field on the sheath is clearly visible in the cross-sections of f shown for Hþ ions in Fig. 3. In the reference case(B¼0) the distribution function is seen to have approximatemirror symmetry about the vy¼0 plane. The other two cases, onthe other hand, show up-down asymmetries in vy that areconsistent with the shape of the equipotentials seen in Fig. 2.For example, ‘wrapping’ of equipotentials around the spacecraftfrom left to right (negative to positive y), leads to an electric field�rV with a negative y component at the horizontal aperturecentre. The induced field combines with the three-dimensionalelectric field in the sheath to cause this distortion. In the absenceof the spacecraft, the background � v

!� B!

electric field wouldlead to a bulk E

!� B!

plasma drift and the distribution functionwould simply be a Maxwellian drifting at the negative ramvelocity.

The presence of the spacecraft, the fact that it is an equipo-tential and the resulting contouring of the equipotentials notedabove, cause a strong asymmetry in the sheath in the y direction.This in turn leads to a shift of the Hþ ion distribution towardnegative y velocities, a feature clearly visible in the left panel ofFig. 3. The same qualitative features are seen with Bz ¼ 4� 10�5 T,except that the asymmetry induced in the sheath by the � v

!� B!

field now results in a shift in the distribution function in thepositive vy direction. We note that the shifted distributioncomputed in the vx�vy cross-section, with Bz ¼�40 mT is almosta mirror image (with respect to the vx¼0 axis) of the distributionfunction computed with Bz ¼ 40 mT. There are in fact smalldeviations from exact mirror symmetry due in part to the factthat the horizontal sensor is not centred on the ram face. It isslightly to the left (toward positive y) of the centre and, as a

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5 6 7

Fig. 3. Hydrogen ion distribution function at the central point of the horizontal TII

Numbers along the axes represent the velocities of hydrogen ions, normalised by the t

result, when equipotentials contour the ram face from left to rightas with Bz ¼�4� 10�5 T, the sheath electric field at the centre ofthe horizontal aperture is slightly stronger than when contouringis from right to left. Another cause of the lack of symmetry is theproximity of the horizontal sensor aperture to the other (vertical)sensor to the right in Fig. 1. The results shown here are forhydrogen ions which, owing to their low mass, are most affectedby sheath effects. Similar qualitative effects are found with Oþ

ions, but these are considerably smaller quantitatively due tothese ions’ larger mass.

3.3. Ion fluxes on the microchannel plate (MCP)

Given the parametrisation of the particle distribution func-tions around the apertures, it is now straightforward to do aMonte Carlo simulation of ions injected into the sensors andcompute fluxes on each pixel of the detector array. Here again, inorder to minimise statistical errors, use is made of the one-particle Liouville theorem. Specifically, for each particle reachinga pixel a contribution to the flux of vnf is added, where f is theinterpolated value of the distribution function computed for theinjected particle and vn is the component of the velocity particlecrossing the pixel perpendicular to the MCP plane. The geometryof the TII is illustrated in Fig. 4 for reference. More detailsconcerning this instrument can be found in Knudsen et al.(2003). Normalised fluxes computed on a 32�64 array of pixelsof the horizontal sensor are shown in Fig. 5 for the three casesconsidered. For a z directed magnetic field considered here, the� v!� B!

electric field is in the y direction, and it only has asignificant effect in the ‘horizontal’ (x�y) direction. For thatreason, we limit the following discussion to the horizontal sensoronly. Similar effects would be found for the vertical sensor if theambient magnetic field were in the y direction (corresponding to

4 5 6 7-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5 6 7

sensor aperture for B¼�4� 10�5 T (left), B¼0 T (centre), B¼ 4� 10�5 T (right).

hermal velocityffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikT=mHþ

p.

Fig. 5. Normalised ion fluxes computed on the 32�64 array of horizontal MCP sensor for B¼�4� 10�5 T (left), B¼0 T (middle), B¼ 4� 10�5 T (right). Tick marks along

the axes represent pixel indices on the MCP. The largest flux is from the majority oxygen ions, whereas the lower flux is from the minority (10%) hydrogen ions. The

horizontal white line in the figures indicates the boundary between rows 32 and 33. It is the centre of the pixel array in the y direction.

S. Rehman et al. / Planetary and Space Science 73 (2012) 145–150 149

� v!� B!

in the z direction). Here, the centre of the pixel arrays isbetween rows 32 and 33 in y and only columns 33–64 in x areused for measuring particle fluxes. In the absence of sheatheffects, or of ambient electric fields, ion fluxes on the MCP shouldbe located halfway between rows 32 and 33 in y (we shall refer tothis as row 32.5). That is, they should peak at row index 32.5,and they should display mirror symmetry in y with respectto that position. To good approximation this is what is observedin the B

!¼ 0 reference case. The other two cases, with

B!¼ 84� 10�5 Tz, on the other hand, show up-down asymme-

tries in the flux profiles. In particular there is a clear shift in theflux maximum toward lower row indices (lower values of y) whenB!¼�4� 10�5 Tz. This shift is consistent with the shift computed

in the Hþ distribution function when the ambient electric fieldpoints in the negative y direction, as shown in Fig. 3. The oppositeis seen when B

!¼ 4� 10�5 Tz and, again, it is consistent with the

shift in the velocity distribution function found in this case.In closing, it is instructive to quantify the shift introduced in

the ion fluxes to the MCP by computing moments of the pixelindices for the three cases considered. We recall that the ion flowvelocity is to be determined with EFI from these moments. MonteCarlo simulations have been made to establish a nearly linearrelation between the displacement of the centroid of the Oþ fluxfrom the centre of the pixel array, and incoming plasma flowvelocity. The relation between the two is such that a shift by onepixel corresponds to a plasma velocity of approximately 600 m/s.Furthermore, laboratory testing of the EFI instruments demon-strates a velocity sensitivity of 5 m/s, corresponding to a sensi-tivity of 0.01 pixel (D. Knudsen, personal communication).

Taking the centre of the pixel array at row 32.5, we calculatethe x, y positions of the centroid of the oxygen flux distribution onthe MCP as

ðx,yÞ ¼Xi2

ix ¼ i1

X64

iy ¼ 1

ðix,iy�32:5ÞFðix,iyÞ

, Xi2

ix ¼ i1

X64

iy ¼ 1

Fðix,iyÞ, ð1Þ

where Fðix,iyÞ is the flux in pixel ix, iy, and indices i1 and i2 arechosen so as to limit the calculation of the moment to that of theOþ ion flux only. Thus, for the cases considered above, we wouldchoose i1 � 42 and i2 � 52. Operationally, the EFI instrumentdetects the Oþ signal for bulk flow analysis since Oþ typicallydominates other ion species at Swarm altitudes (� 500 km), andbecause Oþ ions, on account of their larger mass, are less affectedby the sheath than lighter ions. Accordingly, here we calculatemoments of the Oþ peak to determine what impact the magne-tised sheath may have on EFI ion velocity measurements. Thepixel coordinates of the centroid ðx,yÞ obtained in the three casesconsidered are given in Table 1. Comparing these cases, it followsthat magnetic fields are expected to lead to systematic andperiodic (along the orbits) shifts in the pixel coordinates ofapproximately 70:37 in y. Owing to the proportionality betweenpixel position and velocity mentioned above, if magnetic fieldeffects were not accounted for in the interpretation of the fluxes,

there would result systematic aberrations of C7200 m=s in y

component of the plasma flow velocity.The effect in the x component of the plasma velocity is seen to

be much smaller. Comparing with the reference B¼0 case, the x

coordinates of the Oþ flux centroids calculated with Bz ¼�4, andþ4� 10�5 Tz, are seen to vary by �0.03 and �0.06 respectively.This would correspond to aberrations in the x component of thevelocity of t36 m=s.

4. Summary and conclusion

A first study of Earth magnetic field effects on the Swarm’sElectric Field Instrument (EFI) is presented. The study was madewith a combination of PIC and test-particle simulations. Fullykinetic PIC simulations were carried out with a newly developedcode PTetra, which uses an adaptive unstructured tetrahedralmesh capable of representing complex geometries. PTetra wasused to determine the spacecraft floating potential as well as thestructure of the electrostatic sheath near EFI. This potential wasinterpolated at the positions of the two Langmuir probes that arepart of EFI, and inferred electric fields were calculated, that are inreasonable agreement with expected values. Also, using thepotential field obtained with PTetra, particle backtracking andthe one-particle Liouville theorem were used to compute Hþ andOþ ion distribution functions around the aperture of the particlesensors. These distribution functions were then used to initiate aMonte Carlo simulation of ions entering the sensors, down to themicrochannel plate (MCP), where detailed fluxes were computedon a 32�64 pixel array. The plasma parameters considered in thesimulations correspond to those expected to be representativealong Swarm orbits. Three cases were considered, including areference case without magnetic field, and two with magneticfields B

!¼ 84� 10�5 Tz expected near the North and South

geographic poles respectively. The local magnetic field and theassociated � v

!� B!

electric field, where v!

is the plasma flowvelocity in the spacecraft reference frame, are found to lead to up-down asymmetries in the electric field at the aperture of theThermal Ion Imagers (TIIs) and in the distribution function of ions.This in turn leads to shifts in the centroid of ion flux profiles onthe MCP, with hydrogen being significantly more affected thanthe heavier oxygen ions. Our estimates indicate that the shifts inthe Oþ flux centroids associated magnetic field effects could leadto aberrations in the inferred lateral velocity by as much asC7200 m=s. These aberrations would be systematic and peri-odic, as the magnitude and orientation of the magnetic fieldwould vary along Swarm orbits. We therefore conclude that anoptimal interpretation of EFI measurements will require thatshifts induced by magnetic fields and associated v

!� B!

electricfields be modelled and accounted for in the calibration of theinstruments. In addition to Swarm, these terrestrial magnetic fieldeffects will be important for low-Earth orbit spacecraft measure-ments of ion velocity and electric field in general.

S. Rehman et al. / Planetary and Space Science 73 (2012) 145–150150

Acknowledgments

The authors are grateful to Professor D. Knudsen for usefuldiscussions and input. This work was supported by the NaturalSciences and Engineering Research Council of Canada and theCanadian Space Agency. Our collaboration was supported by theInternational Space Science Institute in Bern, Switzerland. We alsothank COM DEV Ltd and the European Space Agency for providinginformation on the physical characteristics of the EFI instruments.The calculations done in this work made use of the WestGridcomputing infrastructure.

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