Field-aligned electrostatic potentials above the Martian

1

Field-aligned electrostatic potentials above the Martian exobase from MGS 1

electron reflectometry: structure and variability 2

3

Robert J. Lillis1, J. S. Halekas2, M. O. Fillingim1, A. R. Poppe1, G. Collinson3, David A. Brain4, D. L. 4

Mitchell1 5

6

1Space Sciences Laboratory, University of California Berkeley, 7 Gauss Way, Berkeley, CA 94720, 7

USA 8

2Department of Physics and Astronomy, University of Iowa, Iowa city, Iowa, USA 9

3NASA Goddard Space Flight Ctr., Greenbelt, MD, USA 10

4Laboratory for Atmospheric and Space Physics, 3665 Discovery Drive, University of Colorado, 11

Boulder, CO 80309, USA 12

13

Key points: 14

1) Field-aligned electrostatic potentials are derived from energy dependence of electron 15

loss cones measured at ~400 km altitude. 16

2) Strong potentials associated with crustal magnetic fields: positive (negative) potentials 17

for shallow (steep) magnetic elevation angles. 18

3) Potential structures also vary with solar zenith angle, solar wind pressure and EUV 19

irradiance; explanation awaits detailed modeling. 20

21

2

Abstract 22

Field-aligned electrostatic potentials in the Martian ionosphere play potentially important roles 23

in maintaining current systems, driving atmospheric escape and producing aurora. The strength 24

and polarity of the potential difference between the observation altitude and the exobase (~180 25

km) determines the energy dependence of electron pitch angle distributions (PADs) measured 26

on open magnetic field lines (i.e. those connected both to the collisional atmosphere and the 27

IMF). Here we derive and examine a data set of ~3.6 million measurements of the potential 28

between 185 km and 400 km altitude from PADs measured by the Mars Global surveyor 29

MAG/ER experiment at 2 a.m./2 p.m. local time from May 1999 until November 2006. 30

Potentials display significant variability, consistent with expected variable positive and negative 31

divergences of the convection electric field in the highly variable and dynamic Martian plasma 32

environment. However, superimposed on this variability are persistent patterns whereby 33

potential magnitudes depend positively on crustal magnetic field strength, being close to zero 34

where crustal fields are weak or non-existent. Average potentials are typically positive near the 35

center of topologically open crustal field regions where field lines are steeper and negative near 36

the edges of such regions where fields are shallower, near the boundaries with closed fields. 37

This structure is less pronounced for higher solar wind pressures and (on the dayside) higher 38

solar EUV irradiance. Its causes are uncertain at present, but may be due to differential motion 39

of electrons and ions in Mars’ substantial but (compared to Earth) weak magnetic fields. 40

3

Plain Language abstract 41

The Aurora Borealis/Australis, or northern/southern lights, occur when high-energy electrons 42

from space collide violently with Eath’s upper atmosphere. These electrons receive their energy 43

from strong electric fields that accelerate them downward along the force lines of the earth’s 44

global magnetic field. On Mars, accelerated electrons are also responsible for the aurora that 45

has been observed by both the Mars Express and MAVEN spacecraft. In this study we use the 46

observed properties of upward- and downward-traveling electrons measured by the Mars 47

Global Surveyor spacecraft to deduce the strength of these important electric fields. We find 48

that they are highly variable, as is to be expected from the dynamic interaction of the solar wind 49

(a stream of charged particles constantly ejected by the sun) with the Mars upper atmosphere. 50

However, these electric fields also show persistent patterns in regions where Mars’ crustal rocks 51

are most strongly and coherently magnetized. Here, the electric fields are predominantly 52

downward where the magnetic field lines are shallow and predominantly upward where they 53

are steep. We conjecture this pattern is due to the electric charges that build up when heavy 54

positively charged ions and light negatively charged electrons move very differently in these 55

strong magnetic fields. Detailed modeling and further observations will be required to fully 56

understand the plasma dynamics behind this phenomenon. 57

58

4

1 Introduction 59

Broadly, Mars’ induced magnetosphere is produced by the solar wind’s interaction with Mars’ 60

upper atmosphere and exosphere via mass-loading, and with the conducting obstacle of Mars’ 61

mostly photo-produced dayside ionosphere. The result is a bow shock, behind which is a 62

turbulent magnetosheath where the solar wind plasma slows, heats up and is diverted around 63

the obstacle. The interplanetary magnetic field (IMF) embedded in the solar wind piles up in 64

front of the obstacle and drapes around it, forming a double-lobed magnetotail behind Mars 65

[e.g. Brain, 2007; Nagy et al., 2004]. 66

Within this broad picture lies a rich array of electrodynamic and plasma processes, the result of 67

a) the strong and spatially inhomogeneous crustal magnetic fields rotating with the solid planet 68

and coupling with the draped IMF, b) heliospheric structures such as IMF sector boundaries, 69

stream interaction regions (SIRs) and coronal mass ejections (CMEs) convecting through the 70

system, and c) solar flares and solar energetic particle events causing episodic but sometimes 71

intense ionization of neutrals. The linkage between the collisional ionosphere and the 72

magnetosheath/tail occurs via matter and energy transport. Particles precipitate into the Mars 73

atmosphere in several forms: solar energetic ions [e.g. Leblanc et al., 2002; Lillis et al., 2016] and 74

electrons [Schneider et al., 2015], neutral hydrogen (from solar wind proton charge exchange) 75

[Halekas et al., 2015; Kallio and Barabash, 2001], solar wind protons [Diéval et al., 2013], 76

planetary ions [Hara et al., 2017; Hara et al., 2013; Leblanc et al., 2015], and solar 77

wind/magnetosheath electrons [e.g. Lillis and Brain, 2013; Xu et al., 2014]. 78

This paper focuses on the precipitation of these solar wind and sheath electrons. The manner in 79

which this precipitation occurs is influenced by a number of processes occurring in the upper 80

ionosphere and magnetosphere. These include magnetic reconnection [Eastwood et al., 2008; 81

5

Harada et al., 2015], the guiding of electrons along crustal magnetic field lines toward or away 82

from certain geographic regions [Brain et al., 2007; Lillis and Brain, 2013], the formation and 83

behavior of upward and downward current sheets [Halekas and Brain, 2010; Halekas et al., 84

2006] and the acceleration of electrons [Brain et al., 2006a; Halekas et al., 2007; Lundin et al., 85

2006]. This electron precipitation has a number of consequences, including auroral emission 86

[Bertaux et al., 2005; Leblanc et al., 2008] and ionization [Lillis et al., 2011; Lillis et al., 2009b], 87

the latter of which provides the plasma for ionospheric current systems on the nightside 88

[Fillingim et al., 2010; Riousset et al., 2014]. These current systems can close within the 89

ionosphere and/or through the magnetosphere [Dubinin et al., 2008]. 90

Ionosphere-magnetosphere currents can be enhanced by electrostatic potentials/fields parallel 91

to the magnetic field lines. In this study we explain how such parallel potentials can be detected 92

via their effects on suprathermal electron energy and pitch angle distributions measured by the 93

Mars Global surveyor (MGS) Magnetometer/Electron Reflectometer (MAG/ER) instrument in 94

Mars orbit. Section 2 describes the instrumentation and electron energy-pitch angle data set 95

used in this study. Section 3 describes the method by which the shapes of electron loss cones 96

are used to constrain both magnetic mirror and electrostatic forces on precipitating electrons 97

and hence provide an estimate of electrostatic potential, with substantially more detail provided 98

in Appendices A and B. Section 4 describes the resulting data set of the derived potentials and 99

their distribution geographically and with respect to relevant variables such as solar zenith 100

angle. Section 5 describes the results, including the average geographic structure of the derived 101

electrostatic potentials, the relationship with crustal magnetic fields, and variability with respect 102

to external variables such as solar EUV and proxies for solar wind pressure and IMF direction. 103

Section 5.3 discusses the relationship between currents and potentials and possible 104

electrodynamic scenarios to explain our observations. 105

6

2 Data used in this study 106

2.1 MGS MAG/ER 107

The Magnetometer/Electron Reflectometer (MAG/ER) experiment onboard MGS consisted of 108

two triaxial fluxgate magnetometers [Acuña et al., 2001] and a hemispherical imaging 109

electrostatic analyzer that sampled electron fluxes in sixteen 22.5 14 angular sectors, 110

spanning a 360 14 field of view. Electron fluxes in each sector were measured in 19 111

logarithmically-spaced energy channels from 10 eV to 20 keV. With knowledge of the magnetic 112

field vector measured onboard, the field of view (FOV) was mapped into pitch angle (i.e. the 113

angle between an electron’s velocity vector and the local magnetic field). During one 114

integration (2 s to 8 s), the ER measured between 8% and 100% of the 0-180 pitch angle 115

spectrum, depending on the orientation of the magnetic field with respect to the FOV plane of 116

the instrument [Mitchell et al., 2001]. In this study we use pitch angle distributions (PADs) 117

measured during the MGS mapping orbit phase, from April 1999 to November 2006 (when MGS 118

was lost) at an altitude of 370-430 km and in a sun-synchronous trajectory fixed at 2 AM/2 PM 119

local time. 120

2.2 Loss Cones in MAG/ER data 121

In a uniform magnetic field, electrons move along helical paths of constant radius and pitch 122

angle . However, if the field strength (B) increases towards the planet (as is the case for 123

regions of Mars with magnetized crust [Lillis et al., 2004]), electrons are reflected back along the 124

lines of force if reaches 90. If reflection occurs at sufficiently high altitude (before any 125

significant scattering by the collisional atmosphere), then the reflected electrons return to the 126

spacecraft after a round-trip time of ~0.1 sec (for ~200 eV) with the same energy and a pitch 127

7

angle of 180 – , according to the adiabatic condition, at least in the absence of significant 128

high-frequency waves below the spacecraft which can modify these pitch angles. Down-going 129

electrons with measured pitch angles further from 90 have lower reflection altitudes and hence 130

a higher probability of being scattered and/or absorbed by the atmosphere, so the flux of up-131

going electrons exhibits a larger attenuation for pitch angles further from 90, known as a loss 132

cone. When MGS intersects an 'open' magnetic field line (i.e. one that is connected both to the 133

collisional atmosphere and the IMF), MAG/ER observes a loss cone PAD, examples of which are 134

shown in Figure 1. The pitch angle past which the reflecting electrons encounter the collisional 135

atmosphere, as well as the shape of the loss cone, are determined by cross sections of dominant 136

electron-neutral collisions and by the profiles, along the magnetic field line to which the 137

electrons are bound, of three quantities: 1) the magnetic field magnitude, 2) the neutral density 138

and 3) the electrostatic potential, between the spacecraft altitude of 370-430 km and the 139

altitude range of significant electron absorption, as discussed in detail by Lillis et al. [2008d]. 140

MAG/ER obtains the most reliable loss cone measurements in 3 adjacent energy channels: 90-141

145 eV, 145-248 eV and 248-400 eV, for which the absorption altitude range is ~150 km - 210 142

km. This energy restriction exists due to a) unknown spacecraft potential affecting 143

measurements in the lowest energy channels, b) a ~40 times smaller instrument geometric 144

factor below 90 eV and c) insufficient counts above 400 eV to reliably measure loss cone shapes 145

in enough PADs. More than 10 million loss cone PADs were measured by MAG/ER, which we will 146

use for analysis in this paper. 147

3 Method: Measuring Field-aligned Electrostatic Potentials 148

In the simplest form of this problem, with a magnetized hard absorber such as a planetary 149

surface, nearly all electrons which do not magnetically reflect above the surface are absorbed by 150

8

that surface. In this case, electron loss cones are very close to step functions, and field-aligned 151

potential differences can be quite straightforwardly derived from the energy dependence of the 152

loss cone angle, as has been demonstrated at the Moon [Halekas et al., 2008; Halekas et al., 153

2005]. Below we describe how such potentials are derived in the complicating case of an 154

absorbing atmosphere. 155

3.1 Modeling loss cone formation 156

The mathematical details of the dependence of loss cone shapes on these quantities, as 157

well as the kinetic electron transport modeling and data analysis techniques required to 158

constrain them, using MAG/ER PADs, are explained by Lillis et al. [2008d] and are summarized 159

below. We express the probability of a downward-traveling electron of initial kinetic energy U0 160

and initial pitch angle 0, scattering off an atmospheric neutral particle as a function of the 161

distance x that it moves along the field line: 162

𝑃𝑠𝑐𝑎𝑡𝑡(𝛼0, 𝑈0, 𝑥) = 1 − exp [− ∑ ∑ ∫𝜎𝑖𝑗(𝑈(𝑥′))𝑛𝑖(𝑥′)𝑑𝑥′

√1−𝑠𝑖𝑛2𝛼(𝑥′)

𝑥

0𝑗𝑖 ] (1) 163

where the summations are over all neutral species i and all collision processes j, where ni(x) 164

is the neutral density of species i, U(x) is the electron’s energy as a function of distance along the 165

field line, ij(U(x)) is the energy-dependent cross-section for electron scattering off a neutral 166

particle i via process j and (x) is the electron’s pitch angle. The numerator inside the integral 167

accounts for scattering (i.e. it increases with both cross-section and the density of scatterers), 168

while the denominator accounts for the effective increase in path length due to the electron’s 169

gyro motion around the field line. 170

9

As it moves along the field line, the electron’s kinetic energy U(x) will increase or decrease 171

as it is accelerated or decelerated by the parallel electrostatic potential V(x). 172

𝑈(𝑥) = 𝑈0 − 𝑒∆𝑉(𝑥) (2) 173

where e is the electronic charge. The electron’s pitch angle evolution (x) is determined by 174

the adiabatic condition, and hence by the variation in both the magnetic field B(x) and the 175

potential V(x). 176

𝑠𝑖𝑛2 𝛼(𝑥) =𝐵(𝑥)

𝐵0

𝑠𝑖𝑛2 𝛼0

(1−𝑒∆𝑉(𝑥)

𝑈0) (3) 177

We express the magnetic field magnitude along the field line as the sum of a constant, 178

ambient component and a crustal component which increases as distance to crustal 179

magnetization decreases: 180

𝐵(𝑥(𝑧)) = 𝐵𝑎 + (𝐵𝑠 − 𝐵𝑎) (𝑧𝑠

𝑧)

𝑝 (4) 181

where z is the distance from the crust along the field line, Ba is the ambient (i.e. non-182

crustal) component of the field, Bs is the field measured at the MGS spacecraft and p is a power 183

law exponent for the altitude-dependence of the crustal magnetic field. The magnetic field 184

profile is therefore determined by p and the ratio Ba/Bs (convenient because it is dimensionless 185

and ranges between 0 and 1). p = 2.0 would be appropriate for an infinite line of crustal dipoles 186

and 3.0 for a single dipole. Lillis et al. [2008c] showed that loss cone shapes measured by 187

MAG/ER are not separately sensitive to both p and Ba/Bs within this reasonable range of p = 2 to 188

3 and that we can therefore fix p = 2.5 and vary Ba/Bs to reflect different strengths of crustal 189

magnetic field. 190

10

Insufficient information is contained within the loss cones shapes to constrain the specific 191

profile of the electrostatic potential difference between the spacecraft and absorption region 192

[Lillis et al., 2008d]. Therefore, in the absence of a priori information, we parameterize it by a 193

constant electric field E|| parallel to the magnetic field. 194

∆𝑉(𝑥) = 𝐸||𝑥 (5) 195

We define positive parallel electrostatic potentials/electric fields to point vertically upward 196

and negative potentials downward. Thus, positive potentials accelerate the negatively charged 197

electrons downward toward the absorbing atmosphere, reducing the fluxes that can travel back 198

up the field line to the spacecraft, thereby causing loss cones to shift nearer to 90° pitch angle. 199

In contrast, negative potentials push the electrons up and away from the atmosphere, 200

increasing the fluxes that can travel back up the field line, thereby causing loss cones to shift 201

further from 90° pitch angle. This shift in loss cone angle is greater for lower energies because 202

the electrostatic potential is a larger fraction of their total energy. This energy-dependent loss 203

cone shift is illustrated in section 5.2 of Lillis et al. [2008d] and is in contrast to changes in the 204

neutral density or magnetic field profiles, which cause loss cone shifts that are almost energy-205

independent. 206

We can use equations 1-5 to model the formation of loss cones as a function of energy 207

under any combination of neutral atmosphere profile, crustal field strength (parameterized by 208

the ratio Ba/Bs) and electric field E||. We do this in two complementary ways. The first method is 209

comprehensive, whereby we send 100,000s of electrons down into the atmosphere at 3° 210

increments of initial pitch angles, simulating every collision to correctly account for atmospheric 211

scattering. This model is called MarMCET (Mars Monte Carlo Electron Transport) and is 212

explained and applied in several publications [e.g. Lillis and Fang, 2015; Lillis et al., 2011; Lillis et 213

11

al., 2009b; Lillis et al., 2008d]. The second method ignores multiple scattering of electrons, 214

calculating only the probability that a downward-traveling electron at pitch angle will be 215

reflected (magnetically and/or electrostatically) back to the spacecraft altitude at 180 – 216

without scattering. Loss cones calculated this way are called ‘adiabatic’ because the electrons 217

that survive conserve the first adiabatic invariant of charged particle motion [e.g. Parks, 2004]. 218

This method is computationally more than 1 million times faster than the first, allowing rapid 219

calculation of loss cones for a large range of values of Ba/Bs and E|. 220

3.2 Fitting procedure to constrain field aligned potentials 221

It would be prohibitively time-consuming to use the MarMCET model to iteratively fit (i.e. 222

typically dozens of times) each of ~10 million measured loss cones for the best-fit values of Ba/Bs 223

and E||. Instead, we take advantage of a useful property of loss cone formation. Consider the 224

ratio of flux of upward-traveling electrons measured at a given pitch angle which have been 225

scattered at least once by atmospheric neutrals to the total flux of upward-traveling particles 226

(i.e. the sum of these ‘backscattered’ electrons plus electrons which have been reflected back 227

upwards solely by magnetic gradient or electrostatic forces) at the same pitch angle. Lillis et al. 228

[2008d] showed that this ratio is a known function only of the dimensionless total scattering 229

depth (akin to an optical depth, but for electrons) of atmosphere through which a non-230

scattering electron would pass. They also showed that this allows us to quickly and efficiently 231

subtract the backscattered component of the measured PAD, leaving a good approximation of 232

the loss cone shape as it would be measured if all electrons which scatter off atmospheric 233

neutrals were absorbed by the atmosphere, i.e. an adiabatic loss cone. 234

For each measured PAD, we simultaneously fit our fast adiabatic model of loss cone 235

formation to the backscatter-subtracted loss cone shapes in all three electron energy channels 236

12

(90-145, 145-248 and 248-402 eV). We employ an adaptive-step gradient search algorithm 237

called ‘amoeba’ (because it ‘slithers’ around a parameter space [Press et al., 1992] like that 238

shown in Figure 1g, h) to find the minimum misfit (reflected by the classic 2 metric [e.g. 239

Bevington and Robinson, 2003]) and hence the best-fit values of Ba/Bs and E|| for that PAD. For 240

the neutral density profile in the model, we use the same mean reference model atmosphere 241

described in Appendix A of Lillis et al. [2008c] for all cases. As was mentioned earlier and 242

explained by Lillis et al. [2008a], increasing (decreasing) atmospheric density profiles shift loss 243

cones closer to (further from) 90° in an almost-energy-independent way. Typical examples of 244

our fitting procedure are shown in figure 1, including the 2 surface and 1-sigma error ellipsoid, 245

for both positive and negative derived values of field-aligned electrostatic potentials. Note that 246

the best-fit values of Ba/Bs from the entire MGS MAG/ER data set were used to construct the 247

map of crustal magnetic field magnitude at 185 km altitude (which is the average altitude of 248

greatest sensitivity of loss cone shape to magnetic field strength) published in Lillis et al. [2008c] 249

and used in several geophysical studies subsequently [e.g. Lillis et al., 2009a; Lillis et al., 2015; 250

Lillis et al., 2008b; Lillis et al., 2010; Lillis et al., 2013a; Lillis et al., 2013b; Robbins et al., 2013]. 251

Note that the example PAD in figure 1a (where the derived field-aligned potential is 252

positive) appears to be a typical loss cone in all three energy channels, i.e. downward-traveling 253

electrons from the Martian magnetotail are precipitating into the atmosphere and then 254

scattering and magnetically reflecting back upwards in a pitch-angle dependent way, as 255

discussed in Section 2.2. 256

Figure 1 also illustrates that the uncertainties in any one determination of field-aligned 257

potential can be quite large. Therefore, we will not analyze individual orbits in this paper. 258

Instead, we will concentrate on broad trends in these potentials with respect to variables we 259

13

expect to play a role in maintaining potentials: geographic location, crustal magnetic field 260

strength, magnetic elevation angle, solar wind pressure proxy, solar zenith angle etc. 261

Place figure 1 here 262

3.3 Validation and correction of derived values of E|| 263

The examples shown in figure 1 are typical of this large dataset. However, because the 264

typical loss cone shifts are small with respect to the pitch angle bin widths and the uncertainties 265

are large, we wish to provide a statistically significant demonstration that our fitting procedure 266

(including the backscatter subtraction step) is indeed capable of estimating field-aligned 267

potentials. The interested reader is directed to Appendix A where we show a clear, intuitively 268

understandable energy-dependent pattern in the measured loss cone shapes for similar derived 269

values of Ba/Bs and E||. 270

As well as the uncertainties in the derived values of E| | discussed in Section 3.2, our ability to 271

constrain the field-aligned potential is also limited by the physics of loss cone formation. If the 272

magnetic field-aligned potential below the spacecraft is too strongly positive (upward) then the 273

negatively charged electrons are pulled downwards into the collisional atmosphere and none 274

will magnetically reflect back upwards to provide a measure of the loss cone angle and no value 275

of E| | is derived. The weaker the magnetic mirror force (i.e. the larger the value of Ba/Bs), the 276

smaller the upward potential that is required for this situation to occur. The result is that mean 277

values of E| | are biased negative for the highest values of Ba/Bs. Appendix B explains this bias in 278

more detail and describes how we correct for it. The correction is largest in regions of 279

extremely weak crustal magnetic field, such as the Tharsis volcanic province [Lillis et al., 2009a] 280

and Utopia Planitia [Lillis et al., 2008c] and is typically on the order of +0.01 V/km. This may not 281

14

seem like a large amount but it corresponds to >~2.1 V over the distance between the MGS 282

spacecraft and the collisional atmosphere and is sufficient to shift the derived values of E|| from 283

slightly negative, to approximately zero in large regions on the nightside, as discussed in the 284

next section. 285

4 Data set of Electrostatic potentials 286

The fitting algorithm results in a reliable fit for approximately 40% of measured loss cones. Poor 287

fits in the remainder result from a combination of a) insufficient fluxes to accurately determine 288

loss cone shapes in more than one energy channel, b) insufficient pitch angle coverage to 289

measure a loss cone (recall that the instrument only has a 14° x 360° field of view, so pitch angle 290

coverage depends on the magnetic field direction, varying between 14° and the full 180°) and c) 291

non-gyrotropy, i.e. substantially different fluxes measured on opposite sides of the instrument 292

anode (i.e. at the same pitch angle but different gyro phase). This leaves ~3.6 million derived 293

average values for the parallel component of electric field between ~180 km and 370-430 km at 294

2 a.m./2 p.m. solar local time. We convert these to a potential difference in volts between the 295

spacecraft and 185 km altitude by making the necessary assumption that the magnetic field 296

lines are straight over this distance, with the same elevation angle as measured at the 297

spacecraft. This leads to a small underestimation in the potential where field lines are strongly 298

curved but is a reasonable assumption for the great majority of cases, as shown in Appendix D of 299

Lillis et al. [2008c]. 300

We begin with a high-level survey of the data set. Figure 2 shows histograms of the important 301

variables defining the data set. It shows variables which are important for determining field-302

aligned potential: spacecraft altitude, magnetic elevation angle, solar zenith angle and strength 303

of the crustal magnetic field at 185 km. Magnetic elevation angle has a sharp cutoff around 30 304

15

degrees because we discard cases where the straight-line extension of magnetic field lines from 305

the spacecraft altitude does not connect to our nominal electron exobase (185 km on average 306

[Lillis et al., 2008d]). The solar zenith angle histogram reflects the fact that loss cones do not 307

form as easily in the draped magnetic topology of the dayside [Brain et al., 2007]. It also shows 308

histograms of the two variables derived directly from fitting our model to each PAD: Ba/Bs and 309

E||, as well as physical parameters derived from those quantities, i.e. the co-derived total 310

magnetic field magnitude at 185 km and the associated field-aligned potential V. Seasonal 311

coverage is approximately uniform (not shown). 312

Place Figure 2 here 313

Figure 3 shows two-dimensional histograms in planetary latitude and longitude of the data set, 314

divided up into measurements on the dayside (which we define to be SZA <70°), in the 315

terminator region (70° <SZA <110°) and on the nightside (SZA >110°). We choose 110° as the 316

(somewhat arbitrary) boundary between the terminator and nightside because the entire 317

altitude range over which we are sensitive to field-aligned potentials (~200 km-400 km) is in 318

sunlight for SZA <110°. 319

As mentioned above, the solar zenith angle distribution and geographic distribution matches the 320

locations where loss cones preferentially form, as shown by Brain et al. [2007] and Lillis et al. 321

[2008c], i.e. where magnetic field lines connect the collisional atmosphere with the draped 322

interplanetary magnetic field. Away from strong crustal magnetic fields (concentrated mostly in 323

the southern hemisphere between 120° and 240° E), this favorable magnetic topology occurs 324

more frequently on the nightside where the magnetotail is predominantly sunward-anti-325

sunward than on the dayside where the draped field is predominantly tangential to the planet. 326

We also see measurements, on both the day and night side, where crustal magnetic fields are 327

strong and vertical. We notice very few measurements where magnetic fields are 328

16

predominantly horizontal, e.g. strong crustal fields in the southern hemisphere (appearing like 329

channels of no data between long islands of measurements where the crustal field is vertical) 330

and on the dayside away from strong crustal fields. For reference, Figure 4 shows the same 331

geographic map area with contours of the radial component of the crustal magnetic fields. 332

Place Figure 3 here 333

5 Results 334

In examining this data set, we shall divide our analysis into two main areas: average 335

structures and variability. The former is concerned with the average pattern, built up 336

over tens of thousands of orbits, of field-aligned potentials with respect to Mars’ crustal 337

magnetic fields, in the three solar zenith angle bins shown in Figure 3. The latter is 338

concerned with variability, both statistical (i.e. the natural variability of the system) and 339

with respect to external conditions such as solar wind pressure, IMF direction and 340

season. 341



5.1 Average structure of field-aligned potentials 344

First we wish to examine the global geographic distribution of field-aligned electrostatic 345

potentials. We find it instructive to divide the data set into measurements taken on the 346

dayside, terminator and nightside, as discussed in the previous section. Figure 4 shows 347

the median field-aligned potential V in 1° x 1° longitude/latitude bins over the entire 348

data set, separately for dayside, terminator and nightside (similarly to Figure 3), while 349

17

Figure 5 shows the accompanying interquartile range as a function of crustal magnetic 350

field strength to give a feeling for the spread of values of V within each pixel. The radial 351

component of crustal magnetic field at 400 km altitude from the spherical harmonic 352

model of Morschhauser et al. [2014] is overlaid on both figures as contours. A few 353

broad observations can be made based on this global map. First, we notice that positive 354

(upward) potentials are generally stronger on the dayside, compared with the 355

terminator or nightside. Second, a broad correlation exists between V and crustal 356

magnetic field strength, for all SZA intervals: the strongest positive and negative values 357

of V occur in regions where the crustal field is also the strongest. Third, where crustal 358

fields are extremely weak (e.g. the large impact basins Hellas and Utopia and the Tharsis 359

volcanic region), potentials are indistinguishable from zero on the nightside and slightly 360

positive on the dayside and near the terminator. Fourth, for the nightside and 361

terminator, distinct positive and negative (i.e. blue and red) potential structures are 362

visible where the crustal magnetic fields are strong. The weaker the crustal field, the 363

less distinct and weaker these potential structures appear to be. Fifth and finally, Figure 364

5 shows that the interquartile range of values within each geographic pixel increases 365

with crustal magnetic field strength, with a slight dependence for the dayside and 366

terminator, and a more pronounced dependence for the nightside. Since the 367

uncertainty in the technique should not depend on crustal field or time of day, Figure 5 368

shows that there is substantially less natural variability in field aligned potentials in 369

weak magnetic field regions on the nightside, compared with those same regions on the 370

18

dayside/terminator or where crustal fields are stronger. Note that no significant 371

dependence of interquartile range on magnetic field elevation angle was found. 372

In order to examine these distinct positive and negative potential structures in more 373

detail, we zoom in the two strongest crustal magnetic field regions: Figure 6 shows the 374

Terra Sirenum/Cimmeria region and Figure 7 shows the Sinus Sabaeus region. Note that 375

these figures are similar to Figure 4 except a) the range of potentials (color scale) shown 376

are larger and b) the contours are not radial magnetic field but magnetic elevation 377

angle, so as to more clearly delineate magnetic geometry. Both of these figures also 378

show histograms of V for example regions of nearly vertical (~75° elevation angle) and 379

more horizontal (25°-50°) magnetic field. A few characteristics are immediately clear. 380

First, we are limited by where loss cones form based on the connectivity between 381

crustal fields and the draped IMF: Box 1 in Figure 6 has a statistically insignificant three 382

samples on the nightside, but almost 500 in the terminator region. Second, average 383

potentials are positive where magnetic elevation angles are >~50° at the dayside, 384

terminator and nightside. Note that Box 2 in both figures, near the center of an open 385

crustal magnetic field region (also called a ‘cusp’ by analogy to earth [Brain et al., 2007]) 386

is remarkably consistent at +~15-18 V, regardless of solar zenith angle. Third, where loss 387

cones do form at magnetic elevation angles <~50°, potentials are predominantly (and in 388

some places strongly) negative at night and at the terminator, but generally more 389

positive on the dayside. 390

Finally, it is useful to examine the relationship between field-aligned potential, magnetic 391

elevation angle and crustal magnetic field strength in statistical, non-geographic form, 392

19

as shown in Figure 8, where we see more clearly many of the features discernible in 393

Figure 4. The first feature we notice is the strong dependence of field-aligned potentials 394

on magnetic elevation angle. For elevation angles >45°, potentials are generally more 395

strongly positive (red) as one goes from the nightside to the terminator to the dayside; 396

indeed weak crustal field regions on the nightside show no significant potentials (< 5V) 397

at all. In contrast, for elevation angles <45° we see typically weak or negative potentials, 398

with stronger negative potentials (blue) forming for stronger crustal fields. As we go 399

from night to terminator to dayside, these negative potentials form at progressively 400

higher crustal field strengths, while negative potentials are similar across all solar zenith 401

angles for the very strongest crustal fields (>400 nT at 185 km). Lastly, the field-aligned 402

potential patterns are almost symmetric between positive and negative magnetic 403

elevation angles (with a mild asymmetry for the terminator case being likely due to the 404

sampling of the relatively smaller number of positive versus negative ‘cusps’), which 405

makes sense as similar processes should occur regardless of the direction of the 406

magnetic field (earth’s northern and southern aurora exhibit the same features). 407




5.2 Variability with external conditions 411

We now examine how our derived field-aligned electrostatic potentials and their 412

dependence on magnetic elevation angle and crustal field strength, vary with three 413

20

external drivers: solar wind pressure proxy, interplanetary field (IMF) direction proxy, 414

solar extreme ultraviolet flux (explained below). Figure 9 shows histograms of these 415

three quantities across the data set. 416


5.2.1 Variability with solar wind pressure proxy 418

Given that field-aligned electrostatic potentials are expected to be driven, at least in 419

part, by magnetosheath flow velocity and direction [Dubinin et al., 2008; Lyons, 1980], it 420

makes sense to compare our derived values with solar wind pressure. No direct 421

measurements of the upstream solar wind were made coinciding with our data set 422

(Mars Express’ Ion Mass Analyzer (2004 - present) was usually turned off more than 20 423

minutes away from periapsis [M. Holmstrom, private communication] and so did not 424

measure solar wind). However, orbit-average dayside MGS magnetometer 425

measurements of field magnitude between 50° and 60° North (i.e. away from crustal 426

magnetic fields), have been shown to be an adequate proxy for solar wind pressure 427

under the assumption that the dynamic pressure of the solar wind is converted to 428

magnetic pressure in the Mars magnetic pileup region [Brain, 2005; Crider, 2003]. This 429

proxy has been used as a way to understand the variability of Martian plasma processes 430

in numerous studies since [e.g. Edberg et al., 2008; Lillis and Brain, 2013]. 431

Figure 10 shows how V and its dependence on crustal magnetic field strength varies 432

with solar wind pressure and magnetic elevation angle. For high (i.e. steep) elevation 433

angles we see the same small positive dependence of upward potentials on crustal field 434

strength as in Figure 8, but no discernible dependence on solar wind pressure. However 435

21

for low (i.e. shallow) elevation angles and strong crustal fields, we clearly see smaller 436

negative potentials for higher solar wind pressures on both the day and night sides. 437


There are a few possible reasons for this. First, higher pressures provide more charge 439

carriers in the magnetosheath, increasing plasma conductivities and therefore 440

decreasing potential differences. Second, on the nightside at least, higher solar wind 441

pressures result in more electron impact ionization in open field regions and therefore 442

more mobile charge carriers in the ionosphere, and hence lower potentials. Third, 443

higher pressures are known to compress Mars’ crustal fields [Lillis and Brain, 2013], 444

increasing field curvature and changing field geometry, thereby modifying any plasma 445

kinetic effects that may be responsible for the potentials. 446

At the terminator, we see a similar pattern up to crustal field strengths of 100 nT but, 447

perhaps due to poor statistics, no discernible dependence on solar wind pressure for 448

stronger crustal fields. 449

450

5.2.2 Variability with IMF clock angle proxy 451

The pattern with which the IMF drapes around Mars and connects (or not) with the 452

crustal fields depends on its direction, hence we might expect this direction to affect 453

field-aligned potentials. Similarly to the solar wind pressure proxy discussed in the 454

previous section, the direction of dayside magnetic fields measured away from crustal 455

fields provides a rough proxy for the IMF clock angle, as shown by Brain et al. [2006b]. 456

They also showed that accelerated downward-travelling electron spectra (resulting from 457

22

upward potentials above the MGS spacecraft) are much more common when the clock 458

angle is less than ~240°, indicating a preferential IMF orientation for the acceleration 459

process. 460

Figure 11 is similar to Figure 10, except the colors represent two different ranges of IMF 461

clock angle proxy. We observe some marginally-significant dependences on clock angle 462

proxy only for low (shallow) elevation angles. Although these differences are small 463

compared to the interquartile ranges of the data, they are consistent as a function of 464

crustal field strength and therefore may reflect a real dependence. Here we see slightly 465

smaller positive potentials for clock angles 320-140° compared to 140-320°, for all weak 466

crustal field strengths, up to a few tens of nT. This trend is most prominent on the 467

dayside, weaker on the terminator and very slight on the nightside. Above ~50 nT there 468

is no clear pattern on the dayside or terminator, but clearly the opposite trend on the 469

nightside, where the ‘other’ IMF direction results in smaller downward field-aligned 470

potentials for shallow elevation angles. A convincing explanation for this dependence, if 471

it is real, would require detailed modeling beyond the scope of this study, as discussed 472

in Section 5.3. 473


5.2.3 Variability with solar EUV 475

Solar EUV irradiance drives ionization on the Martian dayside, its intensity linearly 476

determining ionospheric density and hence conductivity. Therefore we may also expect 477

it to affect field-aligned potentials. However, there was no EUV monitor at Mars before 478

MAVEN arrived in 2014 [Eparvier et al., 2015]. In this absence, EUV measurements 479

23

made from Earth orbit by a series of spacecraft have been used to estimate the full EUV 480

spectrum from 0.1 to 200 nm, which is then scaled and phase-shifted according to solar 481

rotation, to Mars’ position via a data assimilation model called FISM-M [Thiemann et al., 482

2017]. We convolve this spectral irradiance with the photoionization cross-section of 483

CO2 (the most common neutral source of ionospheric plasma) and integrate across 484

wavelength to get photoionization frequency, i.e. the most direct measure of the sun’s 485

ability to ionize neutrals at Mars than irradiance in any particular spectral range. The 486

calculation of photoionization frequency is explained in many textbooks, and in 487

Appendix A of Lillis et al. [2017]. 488

489

Figure 12 is similar to Figure 10 and Figure 11, except the colors represent ranges of CO2 490

photoionization frequency. On the dayside, lower solar EUV (<1.3 x 10-6) corresponds to 491

lower positive V for both steep and shallow magnetic elevation angles and weak to 492

moderate crustal field strengths, i.e. all magnetic field conditions (and essentially zero 493

V for shallow angles and weak crustal fields). At the terminator, there is no discernible 494

dependence for steep elevation angles (perhaps due to poor statistics), but a similar-to-495

dayside pattern holds for shallow angles. A simple explanation for this pattern is that 496

higher EUV leads to higher ionospheric densities, meaning higher Pedersen 497

conductivities and hence smaller field-aligned potential drops. 498

499

On the nightside where there is no photoionization, as we expect, there is no 500

dependence on EUV for steep angles, no dependence for shallow angles & weak crustal 501

24

fields and only a very slight dependence for shallow angles and strong crustal fields, 502

which may reflect the solar wind pressure dependence of Figure 10 as solar activity and 503

perihelion season both tend to see more intense EUV and solar wind together. Indeed, 504

although not shown, a similar pattern to Figure 12 is seen if the data is categorized by 505

inverse square of heliocentric distance. 506


5.3 Summary of variables controlling field-aligned potentials 508

We observe a hierarchy of the variables controlling the field-aligned potentials that we 509

infer between ~200 km and ~400 km altitude. The strongest determining factor is 510

crustal magnetic field strength. For the weakest crustal magnetic field regions (< ~20 511

nT), diurnal (day-to-night) variability appears to be the most important, with potentials 512

decreasingly positive as one moves from the dayside to the terminator to the nightside. 513

Whereas for the stronger crustal fields (> ~20 nT), magnetic elevation angle appears to 514

be the most important determinant of potentials, with shallow angles strongly negative 515

and steep angles moderately positive and this dependence being strongest for the most 516

intense crustal fields. 517

Beyond these primary correlations, a weaker correlation exists between potentials and 518

solar wind pressure proxy, with stronger negative potentials for lower solar wind 519

pressures, only for strong, shallowly-inclined crustal fields. A still weaker correlation 520

exists between potentials and solar EUV on the dayside, with consistently stronger 521

negative potentials for low solar EUV at all crustal field strengths and elevation angles. 522

Lastly, no definitive correlation can be seen with respect to IMF direction proxy. 523

25

6 Discussion 524

In attempting to explain the field-aligned potentials we have derived and analyzed in 525

this study, and their behavior with respect to time-of-day, crustal magnetic field 526

strength and geometry, and external conditions, it is instructive to look to studies of 527

field-aligned potentials in the earth’s polar cap region, where field lines connect the 528

ionosphere to the earth’s dynamic magnetosphere. Here, on a simplistic level, field-529

aligned conductivities are much higher than perpendicular conductivities in the 530

terrestrial ionosphere, and thus any potential difference between the magnetosphere 531

and the ionosphere should immediately be neutralized by the mobile electron 532

population at either end of the field-line. Thus consistently measured potential drops 533

have historically been very puzzling and, despite more than four decades of study with 534

increasingly sophisticated modeling and extensive data sets of ground magnetometers, 535

rocket flights and in-situ satellite particle and field measurements, these potential drops 536

are still an active area of research [e.g. Ergun et al., 1998; Marklund et al., 2007]. 537

Thus, given the immature state of research on this topic at Mars, we feel it wiser not to 538

speculate here too deeply as to the causes of the existence or behavior of these 539

potentials. However given what is known, an explanation likely lies in some 540

combination of the following three aspects of electrodynamics in near-Mars space: 1) 541

magnetospheric plasma flow dynamics and the currents into and out of the ionosphere 542

that are driven as a result, in rough analogy with the earth’s polar cap, 2) kinetic plasma 543

effects caused by Mars’ unique crustal magnetic fields and 3) dynamo current systems in 544

the ionosphere. Let us consider these in turn. 545

26

First, Dubinin et al. [2008] showed that plasma flow shear in the Martian 546

magnetosheath, in addition to reasonable convection (i.e. non-field-aligned) electric 547

fields of ~1 mV/m, may be capable of driving field-aligned current densities of up to ~1 548

µA/m2 (which are observed) in the case of auroral electron precipitation. They found 549

that the ‘inverted V’ structures of accelerated electrons indicative of auroral 550

acceleration at earth should occur at Mars and that upward field-aligned potentials of 551

10s to 100s of volts are required to produce those structures that have indeed been 552

observed by Mars Express and MGS [Brain et al., 2006a; Dubinin et al., 2006; Lundin et 553

al., 2006]. Plasma flow shear can be caused by the passage of solar wind structures 554

such as stream interaction regions (i.e. the interface between fast and slow solar wind 555

streams [e.g. Gosling et al., 1978]) or coronal mass ejections, or by turbulence in the 556

Martian plasma environment [e.g. Ruhunusiri et al., 2017] or by Kelvin Helmholtz 557

instabilities at the boundary between the Martian magnetosheath and magnetic pileup 558

region [Ruhunusiri et al., 2016]. In addition, so-called double layers of opposite electric 559

charge could help to maintain parallel potentials, as has been inferred from 560

observations in the Earth’s auroral zone [Mozer and Kletzing, 1998]. This broad range of 561

possible sources may explain some of the variability in our derived electrostatic 562

potentials. 563

Second, Mars’ highly inhomogeneous crustal magnetic fields make the situation 564

markedly different from the terrestrial one. The geometry of individual crustal magnetic 565

anomalies results in mini-magnetospheres with open and closed field regions, with 566

different orientations and sizes resulting from different distributions and strengths of 567

27

subsurface crustal magnetization. Not only are each of these mini magnetospheres 568

different, they are in many cases close enough to one another to substantially affect the 569

electrodynamics of each other. Further, large swaths of the planet (e.g. Tharsis) are 570

completely devoid of crustal magnetic fields, leading to a primarily Venus-like 571

interaction of the ionosphere with the draped interplanetary magnetic field [e.g. Nagy 572

et al., 2004]. Each of these different geometries is expected to lead to a different kind 573

of typical current system on the dayside, nightside and at the terminator. In addition, 574

the crustal magnetic fields themselves are hundreds to thousands of times weaker than 575

Earth’s global dipole field at ionospheric altitudes. Therefore, while electrons have 576

gyroradii of a few kilometers at most and therefore typically bound to magnetic field 577

lines at all energies where substantial electron flux exists, ion gyroradii are larger by a 578

factor of √(mi/me) where mi and me are the masses of ions and electron respectively. 579

The two most common planetary ions in near-Mars space, O+ and O2+, thus have 580

gyroradii 172 and 243 times larger than electrons respectively. Depending on magnetic 581

geometry and topology, this can lead to charge separations and resulting electrostatic 582

potentials, both within the photochemically-dominated ionosphere and the collisionless 583

region above. 584

Third and last, the current systems that link the Martian ionosphere and magnetosphere 585

must close through the ionosphere and are therefore affected by, and must be 586

consistent with, ionospheric dynamo current systems driven by thermospheric winds 587

and magnetic gradient and curvature drifts. Mittelholz et al. [2017] found a global-scale 588

pattern of external (i.e. non-crustal) current-produced magnetic fields at ~400 km quite 589

28

similar to that of predicted thermospheric winds, similar to the Sq (solar quiet) current 590

on Earth, strongly suggesting the influence of neutral winds on ionospheric currents at 591

Mars. Riousset et al. [2013] modeled these current systems at high-resolution near an 592

idealized crustal magnetic dipole and multi-dipole arcade for the simplest case of 593

uniform horizontal neutral winds. They found a rich and complex current structure, 594

even for the simplistic case of a spatially-uniform, unchanging wind velocity. 595

In reality, the vertical distribution of ionospheric currents will depend on the altitude 596

profile of Pedersen and Hall conductivity and the degree to which wind-driven currents 597

and electrojets [Fillingim et al., 2012; Opgenoorth et al., 2010] are shielded from, or 598

couple with, currents originating in a higher altitude collisionless plasma flows. 599

These three aforementioned aspects of electrodynamics in near-Mars space are likely 600

responsible, in various combinations under different conditions, 601

602

7 Summary 603

We have presented a study of electrostatic potential differences along magnetic field 604

lines between ~400 km and ~180 km above Mars, derived from the shapes of 605

suprathermal electron loss cones measured by Mars Global Surveyor at 370 - 430 km 606

altitude at 2 am and 2 pm local time. We find that potentials are stronger in regions of 607

stronger crustal magnetic fields and that an average structure exists whereby potentials 608

are generally positive near the center of topologically open crustal fields where field 609

lines are steeper and generally negative near the edges of such regions near the 610

29

boundaries with closed magnetic fields, where fields are shallower. This structure may 611

be due to charge separations resulting from magnetospheric plasma flow and kinetic 612

effects in Mars’ substantial but (compared to Earth) weak magnetic fields, whereby 613

electrons and ions are strongly and weakly magnetized respectively. This underlying 614

structure is less pronounced for the dayside compared to nightside and for higher solar 615

wind pressures and (on the dayside) higher solar EUV irradiances. 616

While this study benefited from an extremely large data set at a consistent altitude and 617

local time, the next step in understanding Mars’ extremely rich electrodynamic 618

environment is to use data from the MAVEN spacecraft [Jakosky et al., 2015], which is 619

collected over a much larger range of altitudes (150 km-6000 km) and a wide variety of 620

local times. This allows much more accurate individual measurements of electrostatic 621

potential differences due to a full suite of plasma instruments and a carefully controlled 622

and monitored spacecraft potential. In particular, this will allow us to probe the vertical 623

structure of these potentials, understand their consequences (e.g. aurora, ionization, 624

heating) and elucidate the processes responsible for their formation and variability. 625

30

8 Appendix A: validation of the fitting procedure 626

Because the uncertainties in any single derivation of E|| are typically large, we wish to 627

provide a statistically significant demonstration that our fitting procedure (including the 628

backscatter subtraction step) is indeed capable of estimating field-aligned potentials. If it is 629

working, we should see a clear, intuitively understandable energy-dependent pattern in the loss 630

cone shapes for similar derived values of Ba/Bs and E||. Therefore, let us examine raw, un-631

backscatter-subtracted ratios of upward to downward fluxes as a function of pitch angle and 632

energy, averaged over all PADs in the dataset which have a similar range of best fit values of 633

Ba/Bs and E||. 634

This is shown in Figure 13. The left column shows cases of moderate to strong crustal field, 635

where the best-fit ratio Ba/Bs of non-crustal to crustal magnetic field at the spacecraft is 636

between 0.30 and 0.50. The right column shows cases of weak-to-moderate crustal fields where 637

Ba/Bs is between 0.80 and 0.85. The top row shows cases where significant negative potentials 638

are the best fit and where, as expected, we see higher fluxes in the decreasing-flux part of the 639

loss cone for lower energies because the potential exerts a greater proportional upward force 640

on these electrons compared to higher energies, i.e. the total potential drop between the 641

spacecraft and the magnetic reflection altitude is a greater fraction of their total energy. The 642

opposite is true for the bottom row of Figure 13, which shows cases of significant positive 643

potentials, where we see lower fluxes in the loss cone for lower energies because the potential 644

exerts a greater proportional downward force on these electrons compared to higher energies. 645

The middle row of Figure 13 shows cases where the derived potential is on average zero, i.e. the 646

shapes are determined by the magnetic mirror force and by atmospheric scattering alone. Here 647

we see that, as expected, the loss cones are steeper at higher energies because the atmosphere 648

31

is a more efficient absorber due to smaller-angle scattering for elastic collisions (e.g. figure 4 of 649

Lillis and Fang [2015]) and, also as expected for a lack of field-aligned potentials, there is no 650

significant energy-dependent shift of the loss cone. Thus we see that the average loss cone 651

shapes measured by MAG/ER are intuitively consistent with the potentials derived for those loss 652

cones. 653

However it is worth mentioning a caveat here. For many of the derived positive potentials, 654

Figure 13e shows that the 245-402 eV electrons (red lines), and to a lesser extent the 148-245 655

eV electrons (blue lines), are somewhat enhanced over their downward counterparts (i.e. the 656

up/down flux ratio is > 1). The further from 90°, the larger the enhancement, reaching ~10-20% 657

just before the loss cone begins at around 50° pitch angle. However, the 95-148 eV electrons 658

(black lines) appear to be depleted, in a way that is consistent with being sucked down into the 659

atmosphere by a positive electrostatic potential or by being accelerated to higher energies 660

without being replaced by energization of lower energy electrons. However, given the typical 661

falling spectrum of sheath and tail electrons measured by MGS whereby 95-148 eV electron 662

fluxes are ~3-4 times higher than 245-402 eV fluxes [Mitchell et al., 2001], only ~20-30% of the 663

depletion of upward-traveling 95-148 eV electrons can by itself explain the increase in 148-245 664

and 245-402 eV electrons, the remainder presumably being due to positive potentials. Thus it is 665

conceivable that some mechanism acting on the precipitating electrons, perhaps resonant wave 666

heating, may at times be preferentially heating ~100 eV (but not lower energy) electrons to 667

higher energies and thus partially masking the signature of some positive electrostatic potentials 668

in situations with moderate-to-strong crustal magnetic fields such as that shown in Figure 13e. 669

Note that the loss cones in Figure 13f are more obviously shifted with respect to each other, 670

thus indicating a lower or negligible masking. In other words, our technique may at times be 671

somewhat overestimating positive potentials. 672

32


9 Appendix B: Correcting for biased values of E|| 674

9.1 Limitations imposed by loss cone formation 675

As well as the uncertainties in the derived values of E| |discussed in Section 3.2, our ability to 676

constrain the field-aligned potential is also limited by the physics of loss cone formation. If we 677

treat the atmosphere as a perfect absorber, the loss cone angle is the pitch angle furthest from 678

90° at which downward-traveling electrons can magnetically reflect back upwards before being 679

absorbed by the collisional atmosphere. In the real case of a ‘fuzzy’ atmospheric barrier, we can 680

think of the loss cone angle as the pitch angle at which the returning flux is reduced by, say, 50% 681

(e.g., 23° in figure 13c). As mentioned earlier, our fitting technique relies on being able to 682

measure how the loss cone shifts in angle for different energies, i.e. we must be able to resolve 683

the loss cone angle in at least two of our three reliable energy channels. 684

However, if the field-aligned potential below the spacecraft is too strongly positive (upward), 685

then the negatively charged electrons at all pitch angles will be pulled downwards into the 686

collisional atmosphere and none will magnetically reflect back upwards to provide a measure of 687

the loss cone angle, i.e. the loss cone will form at 90°. The weaker the magnetic mirror force, 688

the smaller the positive potential that is required for this to occur. 689

Conversely, in the rare case that the potential is both negative (i.e. downward) and more 690

than ~250 V, then electrons at all pitch angles in the 95-148 and 148-245 eV channels will be 691

pushed upwards, i.e. electrostatically reflected, before encountering any substantial 692

atmosphere and the upward flux will mirror the downward flux, with no loss cone forming at all. 693

33

This means that, for a given value of Ba/Bs, there is a maximum and minimum theoretical 694

value of E|| that can be detected using our method. So in cases where the field-aligned electric 695

field exceeds these theoretical maximum or minimum values, the loss cone cannot be measured 696

and the method does not return any derived value of E||. Figure 14 shows the same type of data 697

as in Figure 13 (i.e. up/down flux ratios as a function of pitch angle) but for a broad range of 698

derived values of Ba/Bs and E||, placed alongside loss cone shapes simulated using our adiabatic 699

model for the same values of Ba/Bs. We can clearly see that, as Ba/Bs approaches 1.0 (i.e. as the 700

magnetic mirror force weakens toward zero), a progressively narrower range of positive 701

electrostatic potentials can be derived, until at Ba/Bs = 1 (i.e. no magnetic mirror force), no 702

positive potentials may be derived at all. We show the adiabatic model results not because we 703

expect them to match the data precisely (e.g. the model does not simulate atmospheric 704

backscatter and so the loss cones appear sharper than reality), but as another check that our 705

fitting technique gives reasonable results for E||. 706

Thus when we examine mean values of E|| collected under similar sets of conditions (i.e. 707

season, geography, local time etc.), we must be aware of the bias in the data that results from 708

this limitation and how it may affect those mean values we would like to interpret. Of course, 709

we cannot know for sure the distribution of the missing values of E||, but with some reasonable 710

assumptions, we can correct for this bias insofar as possible. We now present two methods of 711

correction, along with their assumptions. The first is simple, the second more involved; the 712

results they give are quite similar to one another. 713


715

34

9.2 Simple Method of correcting values of E|| 716

Let us first examine the part of the parameter space defined by E|| and Ba/Bs where this bias 717

occurs. Figure 15a shows the maximum measurable value of E|| as a function of Ba/Bs according 718

to our adiabatic loss cone model (see Figure 14) and a quadratic fit to it, while Figure 15b 719

demonstrates how a theoretical distribution of values of E|| would truncated by this limitation. 720

In order to estimate what effect this bias has on mean values of E|| from truncated distributions, 721

we fitted Gaussian curves to the measured distributions of E|| for each 0.01 increment of Ba/Bs. 722

We used the widths of these measured distributions to create theoretical distributions for mean 723

values of E|| between -0.5 and +0.5 V/km, then truncated these theoretical distributions 724

according to the curve in Figure 15a and calculated the resulting ‘biased’ means (Figure 15c) 725

and the difference between them and the ‘real’ mean (Figure 15d). In other words, we 726

calculated the amount by which we would underestimate (i.e. bias negatively) mean values of E|| 727

from symmetric distributions due to truncation is like that shown in Figure 15b. We see clearly 728

that the bias region is triangular in shape in the upper right of the parameter space, and that the 729

bias is worse for more positive values of E|| and larger values of Ba/Bs (i.e. weaker magnetic 730

mirror forces). 731


733

Figure 15 is instructive in showing how the bias manifests itself and where the problem area lies. 734

It also provides a simplistic method of correcting individual values of E|| under the assumption 735

that the ‘real’ values are normally and symmetrically distributed in E||, i.e. for each data point, 736

take the measured value of Ba/Bs (i.e. along the x-axis in Figure 15b) and find the ‘real’ value of 737

35

E|| that corresponds to the measured value of E|| (i.e. move upward from the x-axis to the 738

contour with that measured value and read off the y-axis). 739

9.3 Kolmogorov Smirnoff correction method 740

However, we can also employ a method that uses the real distribution of values of E|| within a 741

given subset of the data for which we wish to estimate the most likely mean and the degree to 742

which the bias has affected it. Here we rely on the fact that a truncated distribution can reliably 743

be distinguished from a symmetric distribution using the classic Kolmogorov-Smirnov (KS) test, 744

i.e. for a given range of values of Ba/Bs, we can calculate the relative likelihood that our 745

measured distribution of E|| values came from various parent distributions, including truncated 746

distributions. 747

For any collection of data points for which we would like to calculate a mean value of E||, we 748

first divide up the data into separate ranges of Ba/Bs, ensuring that we have at least 30 data 749

points for each range to ensure sufficient statistics. Then for each range of Ba/Bs, we calculate 750

theoretical expected distributions of E|| for a wide range of ‘real’ values of E|| (from -1.5 V/km to 751

1.5 V/km), i.e. truncated where appropriate according to the maximum measurable E|| curve in 752

Figure 15a. Every theoretical distribution in each Ba/Bs range has a width in E|| consistent with 753

the overall data set. We then apply the KS test to calculate the probability that the measured 754

distribution of E|| values (i.e. data) came from the same parent distribution as each of our 755

theoretical distributions of E||. Since each of those distributions corresponds to a ‘real’ mean 756

value of E||, we thus calculate a probability distribution for the ‘corrected’ mean value of E||, i.e. 757

the mean we would have calculated if it were not for the aforementioned bias. We consider the 758

mean value of this distribution to be the corrected value for this range of Ba/Bs. We then 759

calculate an overall corrected mean, weighted by the number of data points in each Ba/Bs range. 760

36


Figure 16 shows an example of this method for a set of ~2000 data points collected on the 762

nightside over a small 5° x 5° box in the Tharsis region, where crustal fields are extremely weak 763

(0.74 nT mean at 185 km [Lillis et al., 2009a]), illustrating the relationship between the 764

measured and theoretical distributions for various values of the ‘real’ (i.e. corrected) values of 765

E||. Panel a) shows a range of Ba/Bs (0.944 to 0.953) where none of the measured values of E|| 766

come particularly to close to the maximum measurable value of 0.14 V/km, so none of the 767

significantly truncated theoretical distributions fit the data very well and the highest KS 768

probabilities are for values of E|| between 0 and 0.01 V/km. Panel b) shows a higher range of 769

Ba/Bs (0.979 to 0.983) where the bias problem is worse, however the measured values of E|| are 770

mostly negative, with none getting too close to the maximum measurable value of 0.045 V/km. 771

We should note that it is of course possible that we are missing a separate population of 772

substantially more positive values of E|| that fall above the maximum measurable value. 773

However, if there were a substantial population at these higher positive field values, it seems 774

unlikely that, over the whole data set, there would be no values close to the maximum. In any 775

case, it is impossible to quantify this source of uncertainty, so we shall employ Occam’s razor 776

and ignore it. 777

Figure 16c shows a case where our correction technique makes a difference. In this case, the 778

data distribution appears truncated and hence, the truncated theoretical distributions with 779

corrected values of E|| somewhat above the maximum value give the highest KS probabilities. 780

Figure 16d shows ‘corrected E||’ probability distributions for 18 ranges of Ba/Bs above 0.9 (>90% 781

of data points in this case), with 93 data points in each range. This shows that a) there appears 782

to be a real correlation between E|| and Ba/Bs for this set of data points and that b) in this case 783

37

the overall mean value of was corrected from essentially zero to 0.014 V/km, or about ~3 V 784

between the spacecraft (~400 km and the electron absorption region (~200 km). 785

9.4 Comparison of correction methods 786

The simple correction method can be applied to any data point, since we always derive a pair of 787

values of Ba/Bs and E||. The Kolmogorov-Smirnov method calculates mean values of E|| where we 788

have a sufficient number of data points (~50 or more) to make the KS probabilities meaningful. 789

Let us now compare these methods within 3° by 3° boxes over a 40° longitude by 60° latitude 790

swath of the Tharsis volcanic region where the crustal field (i.e. magnetic mirror force) is very 791

weak and hence Ba/Bs is large enough often enough for the correction to be non-negligible (52% 792

of values of Ba/Bs in the dataset are above 0.97). Figure 17 shows this comparison, for both 793

means and medians. First, we see that both the simple and KS methods shift the distribution 794

positive by ~0.01 V/km. This may not seem like a large amount, but it is ~2.2 V over the distance 795

between the spacecraft and the collisional atmosphere, a substantial fraction of the ~10 V 796

ambipolar potential seen in the ionosphere of Venus [Collinson et al., 2016]. Second, and 797

importantly, both correction methods give very similar distributions of values of E|| , especially 798

for means. Therefore, we choose to apply the simple correction to the entire data set of values 799

of E||. 800


10 Acknowledgments 802

The data used in this manuscript (magnetic field and electron pitch angle measurements 803

from Mars Global surveyor MAG/ER) may be accessed at: 804

38

https://pds.nasa.gov/ds-view/pds/viewDataset.jsp?dsid=MGS-M-ER-4-805

MAP1%2FANGULAR-FLUX-V1.0. The processed data set of derived electrostatic 806

potentials may be found, in the form of ASCII table files, in the Supplementary material. 807

11 References 808

Acuña, M. H., et al. (2001), Magnetic field of Mars: Summary of results from the 809

aerobraking and mapping orbits, Journal of Geophysical Research, 106(E10), 810

23403-23417. 811

Bertaux, J. L., F. Leblanc, O. Witasse, E. Quemerais, J. Lilensten, S. A. Stern, B. Sandel, 812

and O. Korablev (2005), Discovery of an aurora on Mars, Nature, 435(7043), 790-813

794, doi:10.1038/nature03603. 814

Bevington, P. R., and D. K. Robinson (2003), Data reduction and error analysis for the 815

physical sciences, 3rd ed., xi, 320 p. pp., McGraw-Hill, Boston. 816

Brain, D. A. (2005), Variability of the altitude of the Martian sheath, Geophysical 817

Research Letters, 32(18), doi:10.1029/2005gl023126. 818

Brain, D. A. (2007), Mars Global Surveyor Measurements of the Martian Solar Wind 819

Interaction, Space Science Reviews, 126(1-4), 77-112, doi:10.1007/s11214-006-820

9122-x. 821

Brain, D. A., J. S. Halekas, L. M. Peticolas, R. P. Lin, J. G. Luhmann, D. L. Mitchell, G. 822

T. Delory, S. W. Bougher, M. H. Acuna, and H. Reme (2006a), On the origin of 823

aurorae on Mars, Geophysical Research Letters, 33(1), doi:Artn L01201 824

Doi 10.1029/2005gl024782. 825

39

Brain, D. A., R. J. Lillis, D. L. Mitchell, J. S. Halekas, and R. P. Lin (2007), Electron 826

pitch angle distributions as indicators of magnetic field topology near Mars, Journal 827

of Geophysical Research-Space Physics, 112(A9), doi:Artn A09201 828

Doi 10.1029/2007ja012435. 829

Brain, D. A., D. L. Mitchell, and J. S. Halekas (2006b), The magnetic field draping 830

direction at Mars from April 1999 through August 2004, Icarus, 182(2), 464-473, 831

doi:10.1016/j.icarus.2005.09.023. 832

Collinson, G. A., et al. (2016), The electric wind of Venus: A global and persistent “polar 833

wind”-like ambipolar electric field sufficient for the direct escape of heavy 834

ionospheric ions, Geophysical Research Letters, 43(12), 5926-5934, 835

doi:10.1002/2016GL068327. 836

Crider, D. H. (2003), A proxy for determining solar wind dynamic pressure at Mars using 837

Mars Global Surveyor data, Journal of Geophysical Research, 108(A12), 838

doi:10.1029/2003ja009875. 839

Diéval, C., G. Stenberg, H. Nilsson, and S. Barabash (2013), A statistical study of proton 840

precipitation onto the Martian upper atmosphere: Mars Express observations, 841

Journal of Geophysical Research: Space Physics, 118(5), 1972-1983, 842

doi:10.1002/jgra.50229. 843

Dubinin, E., G. Chanteur, M. Fraenz, and J. Woch (2008), Field-aligned currents and 844

parallel electric field potential drops at Mars. Scaling from the Earth’ aurora, 845

Planetary and Space Science, 56(6), 868-872, doi:10.1016/j.pss.2007.01.019. 846

Dubinin, E., et al. (2006), Electric fields within the martian magnetosphere and ion 847

extraction: ASPERA-3 observations, Icarus, 182(2), 337-342. 848

40

Eastwood, J. P., D. A. Brain, J. S. Halekas, J. F. Drake, T. D. Phan, M. Øieroset, D. L. 849

Mitchell, R. P. Lin, and M. Acuña (2008), Evidence for collisionless magnetic 850

reconnection at Mars, Geophysical Research Letters, 35(2), 851

doi:10.1029/2007gl032289. 852

Edberg, N. J. T., M. Lester, S. W. H. Cowley, and A. I. Eriksson (2008), Statistical 853

analysis of the location of the Martian magnetic pileup boundary and bow shock and 854

the influence of crustal magnetic fields, Journal of Geophysical Research, 113(A8), 855

doi:10.1029/2008ja013096. 856

Eparvier, F., P. C. Chamberlin, and T. N. Woods (2015), The Solar Extreme Ultraviolet 857

Monitor for MAVEN, Space Sciences Reviews, 195(1-4), 293-301, 858

doi:10.1007/s11214-015-0195-2. 859

Ergun, R. E., et al. (1998), FAST satellite observations of electric field structures in the 860

auroral zone, Geophysical Research Letters, 25(12), 2025-2028, doi:Doi 861

10.1029/98gl00635. 862

Fillingim, M. O., R. J. Lillis, S. L. England, L. M. Peticolas, D. A. Brain, J. S. Halekas, 863

C. Paty, D. Lummerzheim, and S. W. Bougher (2012), On wind-driven electrojets at 864

magnetic cusps in the nightside ionosphere of Mars, Earth Planets and Space, 64(2), 865

93-103, doi:DOI 10.5047/eps.2011.04.010. 866

Fillingim, M. O., L. M. Peticolas, R. J. Lillis, D. A. Brain, J. S. Halekas, D. 867

Lummerzheim, and S. W. Bougher (2010), Localized ionization patches in the 868

nighttime ionosphere of Mars and their electrodynamic consequences, Icarus, 869

206(1), 112-119, doi:DOI 10.1016/j.icarus.2009.03.005. 870

41

Gosling, J. T., J. R. Asbridge, S. J. Bame, and W. C. Feldman (1978), Solar wind stream 871

interfaces, Journal of Geophysical Research: Space Physics, 83(A4), 1401-1412, 872

doi:10.1029/JA083iA04p01401. 873

Halekas, J. S., D. Brain, R. P. Lin, J. Luhmann, and D. L. Mitchell (2007), Distribution 874

and variability of accelerated electrons at Mars, Advances in Space Research, 41, 875

1347-1352, doi:10.1016/j.asr.2007.01.034. 876

Halekas, J. S., and D. A. Brain (2010), Global distribution, structure, and solar wind 877

control of low altitude current sheets at Mars, Icarus, 206(1), 64-73, 878

doi:10.1016/j.icarus.2008.12.032. 879

Halekas, J. S., D. A. Brain, R. J. Lillis, M. O. Fillingim, D. L. Mitchell, and R. P. Lin 880

(2006), Current sheets at low altitudes in the Martian magnetotail, Geophysical 881

Research Letters, 33(13), doi:Artn L13101, Doi 10.1029/2006gl026229. 882

Halekas, J. S., G. T. Delory, R. P. Lin, T. J. Stubbs, and W. M. Farrell (2008), Lunar 883

Prospector observations of the electrostatic potential of the lunar surface and its 884

response to incident currents, Journal of Geophysical Research-Space Physics, 885

113(A9). 886

Halekas, J. S., et al. (2015), MAVEN observations of solar wind hydrogen deposition in 887

the atmosphere of Mars, Geophysical Research Letters, 42(21), 8901-8909, 888

doi:10.1002/2015GL064693. 889

Halekas, J. S., R. P. Lin, and D. L. Mitchell (2005), Large negative lunar surface 890

potentials in sunlight and shadow, Geophysical Research Letters, 32(9). 891

Hara, T., et al. (2017), MAVEN observations on a hemispheric asymmetry of 892

precipitating ions toward the Martian upper atmosphere according to the upstream 893

42

solar wind electric field, Journal of Geophysical Research: Space Physics, 122(1), 894

1083-1101, doi:10.1002/2016JA023348. 895

Hara, T., K. Seki, Y. Futaana, M. Yamauchi, S. Barabash, A. O. Fedorov, M. Yagi, and 896

D. C. Delcourt (2013), Statistical properties of planetary heavy-ion precipitations 897

toward the Martian ionosphere obtained from Mars Express, Journal of Geophysical 898

Research: Space Physics, 118(8), 5348-5357, doi:10.1002/jgra.50494. 899

Harada, Y., et al. (2015), Magnetic reconnection in the near-Mars magnetotail: MAVEN 900

observations, Geophysical Research Letters, 42(21), 8838-8845. 901

Jakosky, B. M., et al. (2015), The Mars Atmosphere and Volatile Evolution (MAVEN) 902

Mission, Space Science Reviews, 1-46, doi:10.1007/s11214-015-0139-x. 903

Kallio, E., and S. Barabash (2001), Atmospheric effects of precipitating energetic 904

hydrogen atoms on the Martian atmosphere, Journal of Geophysical Research: 905

Space Physics, 106(A1), 165-177, doi:10.1029/2000JA002003. 906

Leblanc, F., J. Luhmann, R. E. Johnson, and E. Chassefiere (2002), Some expected 907

impacts of a solar energetic particle event at Mars, Journal of Geophysical Research, 908

107(A5), 1-10, doi:10.1029/2001JA900178. 909

Leblanc, F., et al. (2015), Mars heavy ion precipitating flux as measured by Mars 910

Atmosphere and Volatile EvolutioN, Geophysical Research Letters, 42(21), 9135-911

9141, doi:10.1002/2015GL066170. 912

Leblanc, F., et al. (2008), Observations of aurorae by SPICAM ultraviolet spectrograph 913

on board Mars Express: Simultaneous ASPERA-3 and MARSIS measurements, 914

Journal of Geophysical Research, 113(A8), doi:10.1029/2008ja013033. 915

43

Lillis, R. J., S. W. Bougher, D. L. Mitchell, D. A. Brain, R. P. Lin, and M. H. Acuna 916

(2008a), Continuous monitoring of nightside upper thermospheric mass densities in 917

the martian southern hemisphere over 4 martian years using electron reflectometry, 918

Icarus, 194(2), 562-574, doi:DOI 10.1016/j.icarus.2007.09.031. 919

Lillis, R. J., and D. A. Brain (2013), Nightside electron precipitation at Mars: Geographic 920

variability and dependence on solar wind conditions, Journal of Geophysical 921

Research-Space Physics, 118(6), 3546-3556, doi:Doi 10.1002/Jgra.50171. 922

Lillis, R. J., J. Deighan, and J. Fox (2017), Photochemical escape of oxygen from Mars: 923

First results from MAVEN in situ data, Journal of Geophysical Research: Space 924

Physics, doi:10.1002/2016JA023525. 925

Lillis, R. J., J. Dufek, J. E. Bleacher, and M. Manga (2009a), Demagnetization of crust by 926

magmatic intrusion near the Arsia Mons volcano: Magnetic and thermal implications 927

for the development of the Tharsis province, Mars, Journal of Volcanology and 928

Geothermal Research, 185(1-2), 123-138, doi:10.1016/j.jvolgeores.2008.12.007. 929

Lillis, R. J., J. Dufek, W. S. Kiefer, B. A. Black, M. Manga, J. A. Richardson, and J. E. 930

Bleacher (2015), The Syrtis Major volcano, Mars: A multidisciplinary approach to 931

interpreting its magmatic evolution and structural development, Journal of 932

Geophysical Research: Planets, 120(9), 1476-1496, doi:10.1002/2014JE004774. 933

Lillis, R. J., and X. Fang (2015), Electron impact ionization in the Martian atmosphere: 934

Interplay between scattering and crustal magnetic field effects, Journal of 935

Geophysical Research-Planets, 120(7), 1332-1345, doi:10.1002/2015je004841. 936

Lillis, R. J., M. O. Fillingim, and D. A. Brain (2011), Three-dimensional structure of the 937

Martian nightside ionosphere: Predicted rates of impact ionization from Mars Global 938

44

Surveyor magnetometer and electron reflectometer measurements of precipitating 939

electrons, Journal of Geophysical Research-Space Physics, 116, doi:Artn A12317 940

Doi 10.1029/2011ja016982. 941

Lillis, R. J., M. O. Fillingim, L. M. Peticolas, D. A. Brain, R. P. Lin, and S. W. Bougher 942

(2009b), Nightside ionosphere of Mars: Modeling the effects of crustal magnetic 943

fields and electron pitch angle distributions on electron impact ionization, Journal of 944

Geophysical Research-Planets, 114, doi:Artn E11009, Doi 10.1029/2009je003379. 945

Lillis, R. J., H. V. Frey, and M. Manga (2008b), Rapid decrease in Martian crustal 946

magnetization in the Noachian era: Implications for the dynamo and climate of early 947

Mars, Geophysical Research Letters, 35(14), doi:10.1029/2008gl034338. 948

Lillis, R. J., H. V. Frey, M. Manga, D. L. Mitchell, R. P. Lin, M. H. Acuna, and S. W. 949

Bougher (2008c), An improved crustal magnetic field map of Mars from electron 950

reflectometry: Highland volcano magmatic history and the end of the martian 951

dynamo, Icarus, 194(2), 575-596, doi:10.1016/j.icarus.2007.09.032. 952

Lillis, R. J., C. O. Lee, D. E. Larson, J. Luhmann, J. Halekas, J. Connerney, and B. 953

Jakosky (2016), Shadowing and anisotropy of solar energetic ions at Mars measured 954

by MAVEN during the March 2015 solar storm, Journal of Geophysical Research. 955

Lillis, R. J., D. L. Mitchell, R. P. Lin, and M. H. Acuna (2008d), Electron reflectometry 956

in the martian atmosphere, Icarus, 194(2), 544-561, doi:DOI 957

10.1016/j.icarus.2007.09.030. 958

Lillis, R. J., D. L. Mitchell, R. P. Lin, J. E. P. Connerney, and M. H. Acuna (2004), 959

Mapping crustal magnetic fields at Mars using electron reflectometry, Geophysical 960

Research Letters, 31(15), doi:Artn L15702, Doi 10.1029/2004gl020189. 961

45

Lillis, R. J., M. E. Purucker, J. S. Halekas, K. L. Louzada, S. T. Stewart-Mukhopadhyay, 962

M. Manga, and H. V. Frey (2010), Study of impact demagnetization at Mars using 963

Monte Carlo modeling and multiple altitude data, Journal of Geophysical Research-964

Planets, 115, doi:10.1029/2009je003556. 965

Lillis, R. J., S. Robbins, M. Manga, J. S. Halekas, and H. V. Frey (2013a), Time history 966

of the Martian dynamo from crater magnetic field analysis, Journal of Geophysical 967

Research-Planets, 118(7), 1488-1511, doi:10.1002/Jgre.20105. 968

Lillis, R. J., S. T. Stewart, and M. Manga (2013b), Demagnetization by basin-forming 969

impacts on early Mars: Contributions from shock, heat, and excavation, Journal of 970

Geophysical Research-Planets, 118(5), 1045-1062, doi:10.1002/Jgre.20085. 971

Lundin, R., et al. (2006), Plasma acceleration above martian magnetic anomalies, 972

Science, 311(5763), 980-983. 973

Lyons, L. R. (1980), Generation of large-scale regions of auroral currents, electric 974

potentials, and precipitation by the divergence of the convection electric field, 975

Journal of Geophysical Research: Space Physics, 85(A1), 17-24, 976

doi:10.1029/JA085iA01p00017. 977

Marklund, G., T. Johansson, S. Lileo, and T. Karlsson (2007), Cluster observations of an 978

auroral potential and associated field-aligned current reconfiguration during thinning 979

of the plasma sheet boundary layer, Journal of Geophysical Research: Space 980

Physics, 112(A1), n/a-n/a, doi:10.1029/2006JA011804. 981

Mitchell, D. L., R. P. Lin, C. Mazelle, H. Rème, P. A. Cloutier, J. E. P. Connerney, M. H. 982

Acuña, and N. F. Ness (2001), Probing Mars' crustal magnetic field and ionosphere 983

46

with the MGS Electron Reflectometer, Journal of Geophysical Research, 106(E10), 984

23419-23427. 985

Mittelholz, A., C. L. Johnson, and R. J. Lillis (2017), Global-scale External Magnetic 986

Fields at Mars Measured at Satellite Altitude, Journal of Geophysical Research: 987

Planets, n/a-n/a, doi:10.1002/2017JE005308. 988

Morschhauser, A., V. Lesur, and M. Grott (2014), A spherical harmonic model of the 989

lithospheric magnetic field of Mars, Journal of Geophysical Research: Planets, 990

119(6), 1162-1188, doi:10.1002/2013JE004555. 991

Mozer, F. S., and C. A. Kletzing (1998), Direct observation of large, quasi-static, parallel 992

electric fields in the auroral acceleration region, Geophysical Research Letters, 993

25(10), 1629-1632, doi:Doi 10.1029/98gl00849. 994

Nagy, A. F., et al. (2004), The plasma environment of Mars, Space Science Reviews, 995

111(1-2), 33-114, doi:Doi 10.1023/B:Spac.0000032718.47512.92. 996

Opgenoorth, H. J., R. S. Dhillon, L. Rosenqvist, M. Lester, N. J. T. Edberg, S. E. Milan, 997

P. Withers, and D. Brain (2010), Day-side ionospheric conductivities at Mars, 998

Planetary and Space Science, 58(10), 1139-1151, doi:10.1016/j.pss.2010.04.004. 999

Parks, G. K. (2004), Physics of Space Plasmas, 597 pp., Westview. 1000

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (1992), Numerical 1001

Recipes in C: the Art of Scientific Computing, Cambridge University Press. 1002

Riousset, J. A., C. S. Paty, R. J. Lillis, M. O. Fillingim, S. L. England, P. G. Withers, and 1003

J. P. M. Hale (2013), Three-dimensional multifluid modeling of atmospheric 1004

electrodynamics in Mars' dynamo region, Journal of Geophysical Research-Space 1005

Physics, 118(6), 3647-3659, doi:Doi 10.1002/Jgra.50328. 1006

47

Riousset, J. A., C. S. Paty, R. J. Lillis, M. O. Fillingim, S. L. England, P. G. Withers, and 1007

J. P. M. Hale (2014), Electrodynamics of the Martian dynamo region near magnetic 1008

cusps and loops, Geophysical Research Letters, 41(4), 1119-1125, 1009

doi:10.1002/2013gl059130. 1010

Robbins, S. J., B. M. Hynek, R. J. Lillis, and W. F. Bottke (2013), Large impact crater 1011

histories of Mars: The effect of different model crater age techniques, Icarus, 225(1), 1012

173-184, doi:DOI 10.1016/j.icarus.2013.03.019. 1013

Ruhunusiri, S., et al. (2017), Characterization of turbulence in the Mars plasma 1014

environment with MAVEN observations, Journal of Geophysical Research: Space 1015

Physics, 122(1), 656-674, doi:10.1002/2016JA023456. 1016

Ruhunusiri, S., et al. (2016), MAVEN observations of partially developed Kelvin-1017

Helmholtz vortices at Mars, Geophysical Research Letters, 43(10), 4763-4773, 1018

doi:10.1002/2016GL068926. 1019

Schneider, N. M., et al. (2015), Discovery of diffuse aurora on Mars, Science, 350(6261). 1020

Thiemann, E. M. B., P. C. Chamberlin, F. G. Eparvier, B. Templeman, T. N. Woods, S. 1021

W. Bougher, and B. M. Jakosky (2017), The MAVEN EUVM Model of Solar 1022

Spectral Irradiance Variability at Mars: Algorithms and Results, Journal of 1023

Geophysical Research: Space Physics, n/a-n/a, doi:10.1002/2016JA023512. 1024

Xu, S., M. W. Liemohn, and D. L. Mitchell (2014), Solar wind electron precipitation into 1025

the dayside Martian upper atmosphere through the cusps of strong crustal fields, 1026

Journal of Geophysical Research: Space Physics, 119(12), 10,100-110,115, 1027

doi:10.1002/2014ja020363. 1028

1029

48

1030

49

12 Figure captions 1031

Figure 1: Two examples of constraining field-aligned electrostatic potentials over the altitude 1032

range of 185-370 km using suprathermal electron pitch angle distributions (PADs). The left and 1033

right columns are examples of radially upward and downward derived field-aligned potentials 1034

respectively. Panels a) and b) show the pitch angle distributions in three adjacent energy 1035

channels 95-148 eV (black), 148-245 eV (blue) and 245-402 eV (red) with the upward-traveling 1036

and downward-traveling sides of the PAD labeled. Panels c) and d) show the ratios of upward to 1037

downward electron flux at conjugate pitch angles (e.g. flux at pitch angle divided by the flux at 1038

pitch angle 180 – or vice versa depending on the direction of the magnetic field). Panels e) and 1039

f) show adiabatic loss cones, i.e. ratios of upward to downward flux after backscattered flux has 1040

been subtracted using the technique of Lillis et al. [2008d] (section 5.3 thereof). Solid lines show 1041

the best fits to those flux ratios. Panels g) and h) show the 2 surfaces and 1-sigma error 1042

ellipsoids in the parameter space defined by Ba/Bs and E||. The best-fit values of Ba/Bs and E|| 1043

(which result in the solid lines in panels e) and f)) are shown with orange triangles. 1044

1045 Figure 2: Histograms of some key variables analyzed in this study, as follows: a) altitude of the 1046

spacecraft, b) solar zenith angle of MGS, c) magnetic elevation angle, d) crustal magnetic field 1047

magnitude at 185 km altitude from the map of Lillis et al. [2008c]. Panels e) and f) show Ba/Bs 1048

and the total (i.e. crustal plus external) value of magnetic field at 185 km that is derived 1049

therefrom [Lillis et al., 2004] respectively. Panels g) and h) show E|| in V/km and the associated 1050

field-aligned potential V between the spacecraft and 185 km altitude respectively. 1051

1052

Figure 3: The number of valid retrievals of E|| (and hence V) in each 1° x 1° bin of 1053

latitude/longitude, demonstrating the geographic distribution of data on the dayside (solar 1054

zenith angle <70°), terminator region (70° < SZA < 110°) and nightside (SZA > 110°) respectively. 1055

50

Figure 4: Maps of field-aligned potential between 185 km and MGS (370-430 km) for three 1056

ranges of solar zenith angles (SZAs): SZA < 70° (day, top panel), 70° <SZA < 110° (terminator, 1057

middle panel) and SZA > 110° (night, bottom panel). Median values of field-aligned potential 1058

derived from the complete data set (May 1999 until November 2006) are shown in 1° x 1° bins. 1059

Contours of positive (black) and negative (grey) radial crustal magnetic field at 400 km from the 1060

spherical harmonic model of Morschhauser et al. [2014] are overlaid (± 10, 20, 50 nT). 1061

1062

Figure 5: Variation of V within each 1° x 1° geographic pixel. Means and standard deviations of 1063

interquartile ranges (i.e. 75th minus 25th percentile) of V for the same pixels shown in Figure 4 1064

are plotted for each of 40 crustal field strength bins. Blue represents the nightside (SZA >110°), 1065

red represents the terminator region (70° <SZA <110°), and orange represents the dayside (SZA 1066

<70°). 1067

1068 Figure 6: The top three panels show the same data as Figure 4, zoomed into the region of Terra 1069

Sirenum and Terra Cimmeria where Mars’ strongest crustal fields exist. However, the contours 1070

are not radial magnetic field, but magnetic elevation angle (i.e. angle from the horizontal) at 400 1071

km in degrees with positive (black) and negative (gray) contours at ±25°, 50° and 75°. Also, the 1072

color scale is wider to accommodate the large negative potentials that exist at the edges of 1073

some magnetic cusps. 1074

1075

Figure 7: Identical to Figure 6 except for the Sinus Sabaeus region of Mars where also exist 1076

moderately strong crustal magnetic fields. Panels are shown only for day (SZA < 70°) and night 1077

(SZA > 110°) because MGS’ sun-locked polar orbit never samples the equatorial region at the 1078

terminator. 1079

51

1080

Figure 8: Average field-aligned potential plotted as a function of magnetic elevation angle and 1081

crustal magnetic field strength, for three SZA ranges: dayside (SZA <70°), terminator region (70° 1082

< SZA < 110°) and nightside (SZA > 110°) from top to bottom. Black and gray contours display 1083

the positive and negative 5 V contours respectively. The color scale is different from either 1084

Figure 4 or Figures 6/7. 1085

1086 Figure 9: histograms of external drivers or their proxies measured during the time of the MGS 1087

data set which may affect the formation of field-aligned a letter set of potentials. Panel a) 1088

shows solar wind pressure proxy [Crider, 2003], b) shows IMF clock angle proxy [Brain et al., 1089

2006b], c) shows CO2 photoionization frequency at Mars scaled and shifted from Earth-based 1090

assets [Thiemann et al., 2017]. 1091

1092 Figure 10: Field-aligned electrostatic potential is plotted as a function of crustal magnetic field 1093

magnitude at 185 km [Lillis et al., 2008c], for two different ranges of magnetic elevation angle: 1094

20°-40° (solid) and 60°-80° (dashed) and three different ranges of solar wind pressure proxy: 0 to 1095

30 nT (pink), 30 to 50 nT (blue) and >50 nT (red). Dayside (SZA < 70°), terminator (70° < SZA < 1096

110°) and nightside (SZA > 110°) data are shown in the top, middle and bottom panels 1097

respectively. Error bars represent the interquartile range, i.e. the difference between the 25th 1098

and 75th percentiles of all V values in each bin. 1099

1100 Figure 11: same as Figure 10 except the colors represent ranges of interplanetary magnetic field 1101

(IMF) clock angle proxy: 140° to 320° (green) and 320° to 140° (orange). 1102

1103 Figure 12: Same as Figure 10 except the colors represent different ranges of CO2 photoionization 1104

frequency (produced by solar EUV): 0.9-1.3 (pink), 1.3-1.7 (blue) and 1.7-2.3 (red) x 10-6 s-1. 1105

52

1106 Figure 13: Pitch angle distributions in 3 adjacent energy channels (95-148 eV in black, 148-245 1107

eV in blue and 245-402 eV in red) are shown as ratios of upward flux at pitch angle to 1108

downward flux at the conjugate pitch angle of 180° - . In particular, shown are the mean and 1109

standard deviations of this ratio in 3 degree pitch angle bins for all measured PADs whose best 1110

fit values for E|| and Ba/Bs fall into the same range. The top, middle and bottom rows are for 1111

significant negative (downward), approximately zero and significantly positive (upward) 1112

potentials respectively. The left column shows cases of moderate to strong crustal field, where 1113

the ratio Ba/Bs of non-crustal to total magnetic field at the spacecraft is between 0.30 and 0.50. 1114

The right column shows cases of weak-to-moderate crustal fields where Ba/Bs is between 0.80 1115

and 0.85. 1116

1117 Figure 14: The range of derivable values of E|| is limited by the physics of loss cone formation. 1118

The first and third columns show the average of all normalized measured pitch angle 1119

distributions for which the derived values of Ba/Bs and E|| are as shown by the labels on the left 1120

and on they-axes of each panel(e.g. the bottom row of colors in the top left panel shows the 1121

average ratio of the upward to downward fluxes for 90-145 eV electrons in cases where the 1122

fitting procedure results in values of Ba/Bs between 0.5 and 0.7 and values of E|| between -0.5 1123

and -0.4). The second and fourth columns show the results of the adiabatic model (i.e. 1124

excluding atmospheric backscatter) of loss cone formation discussed in section 3.1 and 3.2, for 1125

the same values of energy, Ba/Bs and E||. 1126

1127 Figure 15: bias in the retrieval technique for E||. Panel a) shows the maximum value of E|| (black 1128

diamonds) which can be derived using our method, as a function of Ba/Bs, and a quadratic fit to 1129

those values (red line). Panel b) shows a theoretical distribution of values of E|| and how it is 1130

53

truncated to worsening degrees for higher values of Ba/Bs (i.e. weaker magnetic mirror forces). 1131

Panel c) shows how measured mean values of E|| are affected by this truncation and panel d) 1132

shows the resulting error in these mean values, concentrated where E|| is positive and Ba/Bs is 1133

figure 15 > ~0.8. 1134

1135 Figure 16: Example of correction of mean values of E||. 1963 derived values of E|| were measured 1136

on the nightside of Mars in a small geographic box in the Tharsis region (very weak crustal 1137

fields): 12°-17° N, 265°-270° E. Panels a), b) and c) show three example ranges of Ba/Bs, each 1138

with 93 data points. The black histogram bars represent the derived values of E||, i.e. the data. 1139

The thin colored lines show theoretical probability distributions for different ‘real’ mean values 1140

of E||, truncated according to the curve in Figure 15a, and colored according to the color bar. 1141

The dashed pink lines show the normalized Kolmogorov-Smirnov probability that the data are 1142

drawn from the same distribution as the truncated theoretical curves, as a function of the 1143

corrected mean value of E|| corresponding to each curve. Only panel c) shows a corrected value 1144

substantially different from a straightforward mean. Panel d) shows probability distributions in 1145

corrected E|| for each of 19 ranges of Ba/Bs. 1146

1147 Figure 17: Comparison of histograms of mean (solid lines) and median (dashed lines) values of E|| 1148

in each 3° by 3° geographic box in a large swath of the Tharsis region where crustal magnetic 1149

fields are extremely weak. Three calculation methods are shown: means of values of E|| (black), 1150

means of values of E|| corrected using the simple correction method described in section 9.2 1151

(blue), and means calculated using the Kolmogorov-Smirnov method described in section 9.3 1152

(red). 1153

1154

Figure 1.

0.0 0.2 0.4 0.6 0.8 1.0-1.5-1.0

-0.5

0.0

0.5

1.01.5

0.0 0.2 0.4 0.6 0.8 1.0-1.5-1.0

-0.5

0.0

0.5

1.01.5

50

100

150

200

250

0 30 60 90 120 150 180Pitch Angle (degrees)

0.0

0.5

1.0

1.5

2.0

Rela

tive

Flux

0 15 30 45 60 75 90Pitch angle

0.0

0.5

1.0

1.5

2.0

0 15 30 45 60 75 900.0

0.5

1.0

1.5

2.0

0 30 60 90 120 150 180Pitch Angle (degrees)

0.0

0.5

1.0

1.5

2.0

Rela

tive

Flux

Downward Upward

0 15 30 45 60 75 900.0

0.5

1.0

1.5

2.0

0 15 30 45 60 75 90Pitch angle

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0-1.5-1.0

-0.5

0.0

0.5

1.01.5

0.0 0.2 0.4 0.6 0.8 1.0-1.5-1.0

-0.5

0.0

0.5

1.01.5

50

100

150

200

a)

Up/

Dow

n Fl

ux R

atio

Up/

Dow

n Fl

ux R

atio

Up/

Dow

n Fl

ux R

atio

Up/

Dow

n Fl

ux R

atio

χ2

Ba/Bs Ba/Bs

Para

llel E

field

, V/k

m

Para

llel E

field

, V/k

m

Altitude: 377.8 kmLatitude: -13.9°E. Longitude: 337.8°Bs: 14.7 nTB Elev: -80.3°

Altitude: 375.0 kmLatitude: 38.7°E. Longitude: 112.1°Bs: 37.3 nTB Elev: -67.2°

Downward Upward

180° - Pitch angle 180° - Pitch angle

Backscatter-subtracted Backscatter-subtracted Solid Lines: best FitsSolid Lines: best Fits

b)

c)

f)

d)

e)

g) h)

Fit: [low, best, high]Ba/Bs: [0.54,0.67, 0.76]E||: [0.26, 0.46, 0.76] V/km∆V: [53, 94, 154] V above 185 km

Fit: [low, best, high]Ba/Bs: [0.53, 0.69, 0.8]E||: [-0.44,-0.34, -0.19] V/km∆V: [-82, -64, -36] V above 185 km

χ2

245 - 402 eV148 - 245 eV 95 - 148 eV

245 - 402 eV148 - 245 eV 95 - 148 eV

Figure 2.

360 380 400 420 440Spacecraft altitude

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

e

0 30 60 90 120 150 180Solar Zenith Angle, degrees

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

e

-90 -60 -30 0 30 60 90Magnetic elevation angle, degrees

0.000

0.005

0.010

0.015

Nor

m. O

ccur

renc

e

0.1 1.0 10.0 100.0 1000.0|B|crust at 185 km, nT

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

e

0.0 0.2 0.4 0.6 0.8 1.0Ba/Bs

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

e

1 10 100 1000Total |B| at 185 km, nT

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

e

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8Parallel E Field, V/km

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

e

-100 -50 0 50 100 150 200Field-aligned Potential, V

0.00.2

0.4

0.6

0.81.0

Nor

m. O

ccur

renc

eg) h)

e) f )

a) b)

c) d)

Figure 3.

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

latit

ude,

deg

rees

Day

(SZA

< 7

0°)

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

1

10

100

# m

easu

rem

ents

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

latit

ude,

deg

rees

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

1

10

100

# m

easu

rem

ents

0 30 60 90 120 150 180 210 240 270 300 330 360longitude, degrees

-90

-60

-30

0

30

60

90

latit

ude,

deg

rees


-90

-60

-30

0

30

60

90

1

10

100

# m

easu

rem

ents

Term

inat

or (7

0° <

SZA

< 1

10°)

Nig

ht (S

ZA >

110°)

Figure 4.

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

latit

ude,

deg

rees

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

-30

-20

-10

0

10

20

30

�eld

-alig

ned

pote

ntia

l, V

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

latit

ude,

deg

rees

0 30 60 90 120 150 180 210 240 270 300 330 360-90

-60

-30

0

30

60

90

-30

-20

-10

0

10

20

30

�eld

-alig

ned

pote

ntia

l, V


-90

-60

-30

0

30

60

90

latit

ude,

deg

rees


-90

-60

-30

0

30

60

90

-30

-20

-10

0

10

20

30

�eld

-alig

ned

pote

ntia

l, V

Day

(SZA

< 7

0°)

Term

inat

or (7

0° <

SZA

< 1

10°)

Nig

ht (S

ZA >

110°)

Figure 5.

0.1 1.0 10.0 100.0 1000|B|crust at 185 km, nT

0

20

40

60

Pote

ntia

l di�

eren

ce

inte

rqua

rtile

rang

e, V

SZA > 110

SZA < 70

70 < SZA < 110

Figure 6.

Box 2: -4° to -2° N, 48° to 51° EBox 1: -10° to -7° N, 35° to 38° E

-150 -100 -50 0 50 100 150Field-aligned Potential, V

020406080

100120140

# oc

curr

ence

s -47.5 V13.2 VSZA < 70

SZA > 110

Day


-20-15

-10

-5

0

5

10

1520

latit

ude,

deg

rees

Day


-20-15

-10

-5

0

5

10

1520

-100-75

-50

-25

0

25

50

75100

Fiel

d-al

igne

d po

tent

ial,

V

-75

-75-75

-50

-50

-50

-50

-50-50

-25

-25

-25

-25 -25-2525

25

25

25

25

2525

50

50 50

50

5050

75

75 75

75

75

Night


-20-15

-10

-5

0

5

10

1520

latit

ude,

deg

rees

Night


-20-15

-10

-5

0

5

10

1520

-100-75

-50

-25

0

25

50

75100

Fiel

d-al

igne

d po

tent

ial,

V

-75

-75-75

-50

-50

-50

-50

-50-50

-25

-25

-25

-25 -25-2525

25

25

25

25

2525

50

50 50

50

5050

75

75 75

75

75


0

50

100

150

200

250

# oc

curr

ence

s

14.8 V17.7 VSZA < 70

SZA > 110

Figure 7.

Box 1: -56° to -55° N, 164° to 174° E


010

20

30

40

5060

# oc

curr

ence

s 19.0 V-54.2 V0.6 VSZA < 70

70 < SZA < 110SZA > 110

Box 2: -79° to -77° N, 192° to 198° E


050

100

150

200

250300

# oc

curr

ence

s

14.6 V15.0 V15.8 VSZA < 70

70 < SZA < 110SZA > 110

Day


-85

-75

-65

-55

-45

-35

-25la

titud

e, d

egre

es

Day


-85

-75

-65

-55

-45

-35

-25

-100-75

-50

-25

0

25

50

75100

Fiel

d-al

igne

d po

tent

ial,

V

-75

-75

-75

-50

-50

-50

-50

-50

-50

-50

-25-25

-25-25

-25

-25

-25-25

-25

-25

2525

25

2525

25

25

2525

50

50 50

50

5050

50 50

75

75

Terminator


-85

-75

-65

-55

-45

-35

-25

latit

ude,

deg

rees

Terminator


-85

-75

-65

-55

-45

-35

-25

-100-75

-50

-25

0

25

50

75100

Fiel

d-al

igne

d po

tent

ial,

V

-75

-75

-75

-50

-50

-50

-50

-50

-50

-50

-25-25

-25-25

-25

-25

-25-25

-25

-25

2525

25

2525

25

25

2525

50

50 50

50

5050

50 50

75

75

Night


-85

-75

-65

-55

-45

-35

-25

latit

ude,

deg

rees

Night


-85

-75

-65

-55

-45

-35

-25

-100-75

-50

-25

0

25

50

75100

Fiel

d-al

igne

d po

tent

ial,

V

-75

-75

-75

-50

-50

-50

-50

-50

-50

-50

-25-25

-25-25

-25

-25

-25-25

-25

-25

2525

25

2525

25

25

2525

50

50 50

50

5050

50 50

75

75

Figure 8.

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km

-90-60

-30

0

30

60

90

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km

-90-60

-30

0

30

60

90

-60-40

-20

0

20

40

60

Fiel

d-al

igne

d Po

tent

ial,

V

-5

-5

5 5

55

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km

-90-60

-30

0

30

60

90

Mag

netic

ele

vatio

n an

gle,

deg

rees

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km

-90-60

-30

0

30

60

90

-60-40

-20

0

20

40

60

Fiel

d-al

igne

d Po

tent

ial,

V

-5

-5

55

55

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km

-90-60

-30

0

30

60

90

Mag

netic

ele

vatio

n an

gle

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km

-90-60

-30

0

30

60

90

-60-40

-20

0

20

40

60

Fiel

d-al

igne

d Po

tent

ial,

V

-5

-5

-5

5

5

Day (SZA < 70°)

Terminator (70° < SZA < 110°)

Night (SZA > 110°)

Mag

netic

ele

vatio

n an

gle

Figure 9.

0 20 40 60 80 100Solar Wind Pressure Proxy, nT

0.0

0.2

0.4

0.6

0.8

1.0

Nor

m. O

ccur

renc

e

0 60 120 180 240 300 360IMF Azimuth Proxy, degrees

0.0

0.2

0.4

0.6

0.8

1.0

Nor

m. O

ccur

renc

e

1.0 1.5 2.0 2.5CO2 photoionization frequency, 10-6 s-1

0.0

0.2

0.4

0.6

0.8

1.0

Nor

m. O

ccur

renc

e

a)

b)

c)

Figure 10.

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

Dashed: mag. elev. 60°-80°

Solid: mag. elev. 20°-40°

0 to 300 to 30 nT

30 to 5030 to 50 nT

50 to 30050 to 300 nT

solar wind pressure proxy:

Day

(SZA

< 7

0°)

Term

inat

or (7

0° <

SZA

< 1

10°)

Nig

ht (S

ZA >

110°)

Figure 11.

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

0.9 to 1.3 1.3 to 1.7

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

Day

(SZA

< 7

0°)

Term

inat

or (7

0° <

SZA

< 1

10°)

Nig

ht (S

ZA >

110°)

1.7 to 2.3CO2 photoionization frequency, 10-6 s-1



Figure 12.

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V

320 to 140

0.1 1.0 10.0 100.0 1000.0Bcrust at 185 km, nT

-100

-50

0

50

Fiel

d al

igne

d po

tent

ial,

V



140 to 320140 to 320IMF clock angle proxy:

Day

(SZA

< 7

0°)

Term

inat

or (7

0° <

SZA

< 1

10°)

Nig

ht (S

ZA >

110°)

Figure 13.

E||: 0.25 to 0.35 V/km

0 15 30 45 60 75 90pitch angle, degrees

0.00.2

0.4

0.6

0.8

1.0

1.21.4

Up/

dow

n �u

x ra

tio


0.00.2

0.4

0.6

0.8

1.0

1.21.4

Up/

dow

n �u

x ra

tio


0.00.2

0.4

0.6

0.8

1.0

1.21.4

Up/

dow

n �u

x ra

tio


0.00.2

0.4

0.6

0.8

1.0

1.21.4

Up/

dow

n �u

x ra

tio


0.00.2

0.4

0.6

0.8

1.0

1.21.4

Up/

dow

n �u

x ra

tio


0.00.2

0.4

0.6

0.8

1.0

1.21.4

Up/

dow

n �u

x ra

tio

245 - 402 eV 148 - 245 eV 95 - 148 eV

Electron energy:

Ba/Bs = 0.30 to 0.50 Ba/Bs = 0.80 to 0.85a) b)

E||: -0.05 to 0.05 V/km

E||: -0.35 to -0.25 V/kmE||: -0.35 to -0.25 V/km

E||: -0.05 to 0.05 V/km

E||: 0.25 to 0.35 V/km

c) d)

f )e)

Figure 14.

-0.5

0.0

0.5

1.0

E, V

/km

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

0.20.4

0.6

0.8

1.01.2

Up/

dow

n �u

x ra

tio

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

0.20.4

0.6

0.8

1.01.2

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

0.20.4

0.6

0.8

1.01.2

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

0.20.4

0.6

0.8

1.01.2

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

0.20.4

0.6

0.8

1.01.2

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

-0.5

0.0

0.5

1.0

0.20.4

0.6

0.8

1.01.2


-0.5

0.0

0.5

1.0


-0.5

0.0

0.5

1.0

0 15 30 45 60 75 90Pitch angle, degrees



-0.5

0.0

0.5

1.0


-0.5

0.0

0.5

1.0



0.20.4

0.6

0.8

1.01.2

Ba/Bs = 0.50 to 0.70

Ba/Bs = 0.70 to 0.80

Ba/Bs = 0.80 to 0.85

Ba/Bs = 0.90 to 0.94

Ba/Bs = 0.94 to 0.97

Ba/Bs = 0.85 to 0.90

Ba/Bs = 0.97 to 0.98

95-148 eV 148-245 eVMGS ER

DataAdiabatic

ModelMGS ER

DataAdiabatic

Model

Decreasingmagneticmirror force

Up/

dow

n �u

x ra

tioU

p/do

wn

�ux

ratio

Up/

dow

n �u

x ra

tio

E, V

/km

E, V

/km

E, V

/km

E, V

/km

E, V

/km

E, V

/km

Figure 15.

-0.6 -0.4 -0.2 -0.0 0.2 0.4E||, V/km

0

500

1000

1500

2000

2500

# sa

mpl

es (a

rbitr

ary)

Real mean E|| = 0.050 V/kmReal mean E|| = 0.050 V/kmReal mean E|| = 0.050 V/kmReal mean E|| = 0.050 V/kmBa/Bs Mean E|| = 0.90 0.049 V/km0.93 0.043 V/km0.96 0.015 V/km0.99 -0.038 V/km

0.0 0.2 0.4 0.6 0.8 1.0-0.6-0.4

-0.2

0.0

0.2

0.4

0.6

E || rea

l, V/

km

0.0 0.2 0.4 0.6 0.8 1.0-0.6-0.4

-0.2

0.0

0.2

0.4

0.6

-0.4

-0.2

0.0

0.2

0.4

E || mea

sure

d, V

/km

-0.4

-0.2

0.0

0.2

0.4

0.0 0.2 0.4 0.6 0.8 1.0-0.6-0.4

-0.2

0.0

0.2

0.4

0.6

E || rea

l, V/

km

0.0 0.2 0.4 0.6 0.8 1.0-0.6-0.4

-0.2

0.0

0.2

0.4

0.6

0.0

0.1

0.2

0.3

0.4

Erro

r in

E || mea

sure

d, V

/km

0.0 0.2 0.4 0.6 0.8 1.0Ba/Bs

0.0

0.5

1.0

1.5

2.0

2.5

Max

Mea

sura

ble

E ||, V/

km

Quadratic �ty = 3.85 -4.71x + 0.84x2

a)

b)

c)

d)Ba/Bs

Ba/Bs

Figure 16.

Ba/Bs = 0.944 to 0.953

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2E||, V/km

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

occ

urre

nce

-0.10

-0.05

0.00

0.05

0.10

Cor

rect

ed E

||, V/

km

Ba/Bs = 0.979 to 0.983

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10E||, V/km

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

occ

urre

nce

-0.10

-0.05

0.00

0.05

Ba/Bs = 0.994 to 0.997

-0.2 -0.1 0.0 0.1E||, V/km

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

occ

urre

nce

-0.15-0.10

-0.05

-0.00

0.05

0.10

Nightside: 12° to 17° N, 265° to 270° E

Raw E|| mean: 0.001 V/kmCorrected E|| mean: 0.014 V/km

0.90 0.92 0.94 0.96 0.98 1.00Ba/Bs

-0.10-0.05

0.00

0.05

0.10

0.15

0.200.25

corr

ecte

d E |

|, V

/km

0.90 0.92 0.94 0.96 0.98 1.00Ba/Bs

-0.10-0.05

0.00

0.05

0.10

0.15

0.200.25

0.0

0.2

0.4

0.6

0.8

1.0

Prob

abili

ty

Data E||

Probability Distributionfor corrected E||

Cor

rect

ed E

||, V/

km

Cor

rect

ed E

||, V/

km

a)

b)

c)

d)

Figure 17.

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04E||, V/km

0

20

40

60

80

100Uncorrected

Simple correctionKolmogorov-Smirnov correction

Dashed: medians

Solid: means

# of

3° x

3° b

oxes

Tharsis: 0° to 60° N, 240° to 280° E

Documents

Field-aligned electrostatic potentials above the Martian