JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, E10012, doi:10.1029/2012JE004165, 2012

Magnetic field direction and lunar swirl morphology:
Insights from Airy and Reiner Gamma
Doug Hemingway1 and Ian Garrick-Bethell1,2Received 21 June 2012; revised 23 August 2012; accepted 30 August 2012; published 30 October 2012.

[1] Many of the Moon’s crustal magnetic anomalies are accompanied by high albedo

features known as swirls. A leading hypothesis suggests that swirls are formed where crustal
magnetic anomalies, acting as mini magnetospheres, shield portions of the surface fromthe darkening effects of solar wind ion bombardment, thereby leaving patches that appear
bright compared with their surroundings. If this hypothesis is correct, then magnetic field
direction should influence swirl morphology. Using Lunar Prospector magnetometerdata and Clementine reflectance mosaics, we find evidence that bright regions correspondwith dominantly horizontal magnetic fields at Reiner Gamma and that vertical magnetic
fields are associated with the intraswirl dark lane at Airy. We use a genetic search algorithm
to model the distributions of magnetic source material at both anomalies, and we showthat source models constrained by the observed albedo pattern (i.e., strongly horizontal
surface fields in bright areas, vertical surface fields in dark lanes) produce fields that areconsistent with the Lunar Prospector magnetometer measurements. These findings support
the solar wind deflection hypothesis and may help to explain not only the general formof swirls, but also the finer aspects of their morphology. Our source models may also be
used to make quantitative predictions of the near surface magnetic field, which must
ultimately be tested with very low altitude spacecraft measurements. If our predictions arecorrect, our models could have implications for the structure of the underlying magnetic
material and the nature of the magnetizing field.
Citation: Hemingway, D., and I. Garrick-Bethell (2012), Magnetic field direction and lunar swirl morphology: Insights from
Airy and Reiner Gamma, J. Geophys. Res., 117, E10012, doi:10.1029/2012JE004165.

1. Introduction
1.1. Background[2] Although the Moon does not now possess a global
magnetic field [Ness et al., 1967], remanent crustal magnetization has been identified on the surface and several stableregional magnetic fields have been detected from orbit [Dyalet al., 1970; Coleman et al., 1972; Lin, 1979; Lin et al., 1998;
Hood et al., 2001]. Curiously, many, but not all of thesecrustal magnetic anomalies are accompanied by sinuous
patterns of anomalously high surface reflectance known asswirls [Hood et al., 1979; Hood and Williams, 1989;
Richmond et al., 2005; Blewett et al., 2011].
[3] Compared with their surroundings, swirls are opticallyimmature, exhibit spectrally distinct space weathering trends

1Earth and Planetary Sciences, University of California, Santa Cruz,
California, USA.
2
School of Space Research, Kyung Hee University, Yongin, South
Korea.
Corresponding author: D. Hemingway, Earth and Planetary Sciences,
University of California, 1156 High St., Santa Cruz, CA 95064, USA.
(djheming@ucsc.edu)©2012. American Geophysical Union. All Rights Reserved.
0148-0227/12/2012JE004165[Garrick-Bethell et al., 2011], and are depleted in hydroxyl
molecules [Kramer et al., 2011]. Swirls appear to overprint
local topography, having no detectable topographic or textural expression of their own [Neish et al., 2011]. So far not
identified anywhere else in the solar system, swirls are
unique natural laboratories where space weathering andcrustal magnetism intersect. As such, their study could help
address important questions in lunar science including the
Moon’s dynamo history [Garrick-Bethell et al., 2009; Dwyer
et al., 2011; Le Bars et al., 2011], the relative influences ofsolar wind and micrometeoroid bombardment on space
weathering [Hapke, 2001; Vernazza et al., 2009], and the
production and distribution of water over the lunar surface[Pieters et al., 2009; Kramer et al., 2011].
[4] Electrostatic migration of dust has been proposed as apossible mechanism for swirl formation [Garrick-Bethell
et al., 2011], as have meteoroid or comet impacts [Schultz
and Srnka, 1980; Pinet et al., 2000; Starukhina and
Shkuratov, 2004]. Another model suggests that crustal magnetic fields act as mini magnetospheres, deflecting the solar
wind and protecting portions of the surface from the opticalmaturation and darkening effects of proton bombardment[Hood and Schubert, 1980; Hood and Williams, 1989]. This
paper aims to make and test predictions based on this solar
wind deflection model for swirl formation.

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

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Figure 1. Magnetic field lines due to a single dipole located at the origin and oriented either (a) verticallyor (b) horizontally in the plane of the page. (c and d) Profiles of the horizontal component of the magnetic
field shown at the various altitudes represented by dashed lines in Figures 1a and 1b. In both cases, the dipolar source has a magnetic moment of 1012 Am2.
1.2. Predicted Influence of Magnetic Field Direction[5] The solar wind deflection model suggests that solar
wind ions are magnetically deflected due to the Lorentz
force. Neglecting the induced electric field, the cross productof particle velocity and magnetic field in the Lorentz Lawmeans the magnetic deflection force is maximized whenparticle velocity is perpendicular to the magnetic field andzero when it is parallel. Hence, if the high albedo of swirls is
the result of inhibited space weathering due to magnetic
deflection of solar wind ions, magnetic field direction should
influence swirl morphology. While the solar wind incidence
angle varies, the ion flux at the surface and thus any darkening effects will be greatest when the ion trajectories arevertical. We therefore predict that portions of the crust thatare shielded by dominantly horizontal magnetic fields should
receive maximum protection from the solar wind while portions of the crust associated with vertically oriented magnetic
fields should experience protection only at low solar windincidence angles, when darkening effects would be minimal
anyway. This suggests that the magnetic fields directly overthe bright swirls should be dominantly horizontal and thataway from swirls and in the intraswirl dark lanes, the fields
may be either closer to vertical or too weak to offer the surface any protection from solar wind darkening.
[6] Because magnetic field strength decreases rapidly with
distance from the source, swirl morphology should depend
mainly on the very low altitude structure of the magnetic
field. The structure of the field at higher altitudes, where fieldstrength is weaker, has less influence on solar wind deflection. Unfortunately, spacecraft observations are typicallylimited to higher altitudes and therefore do not directly capture the structure of the near-surface magnetic field. It istherefore important to understand the way in which the field
patterns change with observation altitude, including the rolethat is played by the direction of magnetization. Figure 1
illustrates the magnetic field due to a single dipole that iseither vertically oriented (Figures 1a and 1c) or horizontallyoriented (Figures 1b and 1d). Figures 1a and 1b illustratemagnetic field lines as seen in vertical cross section while
Figures 1c and 1d show profiles of the horizontal componentof the magnetic field as observed along the dashed lines

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

shown in Figures 1a and 1b (in both cases, the model dipole
is placed at the origin and arbitrarily assigned a magnetic
moment of 1012 Am2). Directly over the vertically oriented
dipole, field lines are vertical at any altitude (Figure 1a).
Peaks in the horizontal field strength appear to either side andare separated by a distance equal to the observation altitude(Figure 1c). Directly over the horizontally oriented dipole,field lines are horizontal at any altitude (Figure 1b). As theobserver moves along an axis parallel to the dipole direction,
horizontal field strength decreases, reaching zero (i.e., vertical field lines) at a horizontal distance of 1/√2 times theobservation altitude, before temporarily increasing againslightly (Figure 1d). Appendix A gives a quantitative treatment of the way in which magnetic fields vary with position
relative to the source.
[7] Our study uses Lunar Prospector magnetometer datacollected at altitudes of 18 km and higher. While we cannot
measure the magnetic field structure below these altitudes,
we can distinguish between vertically and horizontally oriented magnetizations (we assume that strong crustal magnetic anomalies are approximately dipolar). This means we
can select anomalies exhibiting orientations that allow us totest our predictions. Specifically, we can look for intraswirl
dark lanes at an anomaly with approximately vertical magnetization because, in that case, field lines should appear
vertical at any altitude over the dark lane (Figures 1a and 1c).
Likewise, we can look for a correlation between high albedo
and strongly horizontal fields at anomalies with approximately horizontal magnetization because field lines at suchanomalies should appear horizontal over the brightest areas
regardless of observation altitude (Figures 1b and 1d). We
find that the Airy anomaly is similar to the case illustrated in
Figures 1a and 1c while the Reiner Gamma anomaly is similar to the case illustrated in Figures 1b and 1d.

2. Data Processing
2.1. Lunar Prospector Magnetometer Data
[8] Our study uses Lunar Prospector 3-axis magnetometermeasurements (level 1B data) obtained from NASA’s Planetary Data System (ppi.pds.nasa.gov). Two distinct approaches are available for producing maps of crustal magnetic
fields from orbital measurements. One involves direct mapping using data selected from a sequence of orbits that crossthe region of interest [Hood et al., 1981, 2001], resulting inmaps of the field at the spacecraft altitude, while the other
involves developing a global spherical harmonic model andusing it to produce maps at arbitrary altitudes [Purucker andNicholas, 2010]. The spherical harmonic models are well
suited to global mapping and have the advantages of automatically accounting for varying spacecraft altitudes andguaranteeing that the mapped field is a potential field. To
date, the best available spherical harmonic models extend
to degree 170, corresponding to a horizontal wavelength of
64 km [Purucker and Nicholas, 2010]. However, because
we are interested in small-scale swirl morphology, we instead
employ a direct mapping technique similar to that of Hoodet al. [2001], allowing us to map features at wavelengths
comparable with the spacecraft altitude (as low as 18 km).
As discussed by Hood [2011], the direct mapping approach issuitable for regional mapping and has several advantages
including that it allows us to perform model fitting directly toE10012the minimally processed magnetometer measurements rather
than to a derived field model. While we focus here on direct
mapping, we did compare our maps against maps we derived
from the Purucker and Nicholas [2010] model coefficients.
We found that in the latter, crustal anomaly fields appear
morphologically similar to the same features in our maps, but
are typically broader in horizontal extent and exhibit lower
field strength.
[9] Each of our maps incorporates data from a sequence ofconsecutive orbits that cross the anomaly of interest. The
polar orbiting spacecraft obtained global coverage twice each
lunation, with consecutive orbits spaced 1 degree in longitude (30 km at the equator) and with along-track measurements recorded at 5-s intervals, corresponding to 8 km
spacing in latitude. In order to capture the undisturbed signalof the crustal magnetic anomalies, avoiding times when thefield is distorted by the impinging solar wind [Kurata et al.,2005; Purucker, 2008], we use only those magnetometermeasurements collected when the Moon was protected fromsolar wind while passing through the Earth’s magnetotail(excluding times when the field was disturbed by the plasma
sheet) or when the Moon was in the solar wind but the
spacecraft was in the lunar wake (on the dark side of theMoon and away from the terminator by at least 20 ). We
examined data from the lowest (<50 km) altitude phaseof the mission, between February and July 1999 (the final
six months of the mission). The upper portions of Tables 1
and 2 list the 13 orbit sequences we examined for each of
the Airy and Reiner Gamma anomalies, respectively. In eachcase, of the 13 orbit sequences we examined, 5 took placein the lunar wake and 2 in the Earth’s magnetotail.
[10] We examined orbit segments spanning 15 of latitude
approximately centered on each anomaly. Having eliminatedmeasurements taken outside of wake and tail times, we
assume that any remaining external fields are steady overthese 15 of latitude and that fields from isolated crustal
sources span no more than a few degrees. After subtracting
the mean background field, we assume the remaining fields
are due to crustal sources. We then convert the spacecraft
position into selenographic spherical coordinates, taking the
Lunar Orbiter Laser Altimeter reference radius of 1737.4 km[Smith et al., 2010] as zero altitude, and transform the magnetometer measurements into local east, north, and radial
components. We next combine data from consecutive orbits
to produce magnetic field maps by fitting regular square
meshes to the data using Delauny triangulation. In order tocapture the structure of the signal, the grid cells must be nolarger than half the spacecraft altitude, which can be as low as
18 km. We therefore use 0.25  0.25 (about 7.6 km 
7.6 km at the equator) as the grid spacing when fitting to the
Lunar Prospector magnetometer (LP MAG) data. This grid
resolution is also finer than the spacing between the observations (8 km in latitude, up to 30 km in longitude)
meaning that no observed signal variations are lost in thegridding process. Using still finer resolution has no effect on
the resulting linearly interpolated surface and is undesired
as it increases computation time unnecessarily. Conversely,
using a larger (more coarse) grid spacing leads to undesired
smoothing in latitude, and as cell size approaches 1 , undesired smoothing occurs in longitude as well. Despite ourefforts to remove external fields and avoid times withsignificant field distortion, portions of some of the orbit

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Table 1. Lunar Prospector Data Used for AiryaDay(s) of 1999

Solar Wind
Exposure/Protection

Minimum Solar Zenith Angle(deg)

Mean Altitude(km)

Number of
Measurements

Orbit
Segments

35,36,37
49,50
63,64
76,77,78
90,91
104,105
117,118
131,132
145,146
158,159
172,173
186,187
199,200
131,132
91,11849,50,77,104

solar wind
wakesolar wind
wake
tail
wake
tail
wake
solar wind
wakesolar windsolar windsolar wind
wake
tail
wake

56
131
29
150
9
151
26
134
52
111
77
88
102
136
15
134

27.5
18.7
28.3
18.5
28.3
18
28.3
34.4
38.5
34.3
38.6
35.9
37
34.4
28.4
18.5

734
772
759
615
610
678
776
782
838
627
824
660
785
48
82
130

15
15
14
12
11
13
14
14
15
12
15
12
14
4
7
12

a
The upper part of the table lists the 13 orbit sequences (each consisting of segments of between 11 and 15 consecutive orbits)
we examined in a 15  15 window centered on the anomaly. The last three rows of the table show the 260 measurements
retained after discarding data collected outside of wake and tail times, after combining like-altitude orbit sequences, and after
cropping to a 3.25  4 study area.

sequences still appear to be contaminated by transient signals
and are therefore discarded. In order to obtain maps withimproved spatial coverage, we combine measurements fromdifferent orbit sequences if the observation altitudes are sufficiently similar (differing by less than 1 km). Since thespacecraft altitude varies only slightly (<1 km) over the scaleof our maps (>120 km), we do not perform upward or
downward continuation of the signal to some fixed altitude. Instead, our maps represent the magnetic field at theslowly varying spacecraft altitude. Finally, we define study
areas within the boundaries of the processed data: the Airy
study area is illustrated in Figures 2a–2c while the Reiner
Gamma study area is illustrated in Figures 2d–2f. As
summarized in the lower portions of Tables 1 and 2, thisresults in Airy study area maps at altitudes of 18 km,
28 km, and 34 km, using a total of 260 LP MAG
measurements, and Reiner Gamma study area maps ataltitudes of 18 km, 34 km, and 40 km, using a totalof 245 LP MAG measurements (only the lowest altitude
maps are presented here).
2.2. Clementine Reflectance Mosaics[11] Our study uses Version 2 Clementine 750 nm reflectance mosaics produced by the USGS (United States Geological Survey) map-a-planet service (www.mapaplanet.
org). Compared with the Version 1 (V1) maps, the Version 2(V2) maps use a newer geodetic control network [Archinal
et al., 2006], refining the horizontal registration by as muchas 10 km in some locations. We have manually adjusted
the V2 reflectance values to match those of the V1 maps,
which have been more carefully controlled to represent true
reflectance.

3. Observations[12] We focus our discussion here on two specific examples: Airy and Reiner Gamma. The Airy anomaly exhibits a

Table 2. Lunar Prospector Data Used for Reiner Gammaa
Day(s) of 1999

Solar Wind
Exposure/Protection

Minimum Solar Zenith Angle(deg)

Mean Altitude(km)

Number of
Measurements

Orbit
Segments

40,41
54,55
67,68
81,82
95,96
108,109
122,123
136,137
149,150
163,164
177,178
190,191
204,205150
136,163,164
54,81,82,109

solar wind
wakesolar wind
wakesolar wind
wakesolar wind
wake
tail
solar wind + wakesolar wind + tailsolar windsolar wind
tail
wake
wake

51
139
24
162
3
158
29
135
56
109
82
83
107
57
110
141

27.7
19
28.5
18.6
28.6
18.3
28.5
34.8
39.3
34.5
39.4
36.3
37.7
39.7
34.3
18.3

820
769
832
795
834
634
833
703
817
798
842
838
769
48
79
118

15
14
15
15
16
12
15
13
15
15
15
15
14
4
7
10

a
The upper part of the table lists the 13 orbit sequences (each consisting of segments of between 12 and 16 consecutive orbits)
we examined in a 15  15 window centered on the anomaly. The last three rows of the table show the 245 measurements
retained after discarding data collected outside of wake and tail times, after combining like-altitude orbit sequences, and after
cropping to a 3.25  4 study area.

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

Figure 2. Magnetic field maps derived from Lunar Prospector magnetometer data over Clementine
750 nm reflectance maps at the (a–c) Airy and (d–f) Reiner Gamma anomalies. Figures 2a and 2d show
the direction of field lines at the spacecraft altitude (18 km in both cases), with arrow lengths showing
relative horizontal field strength. Figures 2b and 2e show contours of the total magnetic field strength,and Figures 2c and 2f show contours of the horizontal component alone. The Airy maps shown here arederived from LP MAG data collected on days 49–50, 77, and 104 of 1999 while the Reiner Gamma maps
are derived from data collected on days 54, 81–82, and 109 of 1999.

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012field strength peaks is expected at lower altitudes, potentially allowing for better alignment with the swirl’s brightlobes.

Figure 3. East-west component of magnetic field as measured by Lunar Prospector over Clementine 750 nm reflectance at Airy. Contour line colors and labels indicate
magnitude of eastward field strength, with warm colors indicating east-pointing fields and cool colors indicating westpointing fields. The dashed white line indicates zero fieldstrength in the east-west direction. The map is derived fromLP MAG data collected on days 49–50, 77, and 104 of 1999.
magnetic field resembling the case illustrated in Figure 1awhile the Reiner Gamma anomaly field resembles the caseillustrated in Figure 1b. The distinct orientations of these two
anomalies reveal different aspects of the relationship between
magnetic field direction and swirl morphology.
3.1. Airy[13] First described by Blewett et al. [2007], the Airy swirl
is found near Airy crater in the lunar nearside highlands. At
the Lunar Prospector spacecraft altitude, the magnetic fieldlines tend to point inward toward the middle of the anomaly,
becoming increasingly vertical and downward pointing at thecenter (Figure 2a), consistent with a source magnetizationthat is pointed mainly downward. This inferred magnetization direction is based on examination of the vector components and is supported by the models described in section 4.
Here, contour maps of the total magnetic field strength
illustrate only that the albedo anomaly is approximately
centered on the magnetic anomaly (Figure 2b). However,maps of the horizontal field alone (Figure 2c) reveal structurethat is more closely related to albedo morphology. For
example, the dark lane through the center of the albedo
anomaly forms an approximate plane of symmetry in theobserved horizontal magnetic field map. Even more striking
is the alignment between the dark lane in the center of theanomaly and the line representing zero east-west magnetic
field strength (dashed white line in Figure 3). The brightest
parts of the swirl are organized into two roughly parallel
lobes on either side of the dark lane. The center-to-center
horizontal distance between the lobes is approximately 8–
10 km. Peaks in the horizontal field strength are also organized into two lobes on either side of the dark lane but witha peak-to-peak separation of roughly 30 km (Figure 2c).
As illustrated in Figure 1c, a closer spacing of horizontal

3.2. Reiner Gamma[14] Reiner Gamma, the type example for lunar swirls,
is found in Oceanus Procellarum near the western limb of
the lunar nearside. At the Lunar Prospector spacecraft altitude, the magnetic field lines are south pointing (Figure 2d),
consistent with a source magnetization that is mainly horizontal and north pointing. This inferred magnetization direction is based on examination of the vector components, issupported by the models described in section 4, and is inagreement with Kurata et al. [2005]. Contour maps of totalmagnetic field strength, seen here (Figure 2e) and elsewhere[e.g., Hood et al., 2001], illustrate only that the albedo
anomaly is approximately co-located with the magnetic
anomaly; there is no clear relationship between total fieldstrength and the morphology of the albedo anomaly. However, contour maps of the horizontal component alone(Figure 2f) show that regions of high horizontal field strength
correspond well with the bright swirl. Based on SELENE
magnetometer data, Shibuya et al. [2010] have reported
similar findings for Reiner Gamma and other anomalies.
Patches of high albedo tend not to occur where fields are
weak, or where fields are strong but lack a large horizontal
component. This observation is consistent with the hypothesis that darkening due to solar wind ion bombardment will
be minimized where magnetic fields are strongly horizontal.
[15] We tested the degree to which magnetic field direction
is related to reflectance by measuring the correlation between
reflectance and the angle the field makes with the vertical.
Figure 4 plots reflectance versus magnetic field direction
using data points gathered from all six retained sets of LP

Figure 4. Correlation between magnetic field direction andreflectance at Reiner Gamma. ‘Angle from vertical’ is 90
where field lines are horizontal. Data points are color-coded
by total field strength with cool colors representing weak
fields and warm colors representing strong fields. Blue andred regression lines are fit to the weakest and strongest thirdsof the data points, respectively, demonstrating that magnetic
field direction becomes increasingly important with increasing field strength.

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012the data points and illustrates that, for stronger fields,reflectance values are low where field lines are vertical andhigh where field lines are horizontal. Dashed lines are used toshow 95% confidence intervals on the estimated slopes,
indicating that the observed differences in slope are statistically significant. These trends are precisely what we would
expect if the albedo anomalies owe their brightness to magnetic deflection of solar wind.

Figure 5. Locations of the 245 LP MAG data points (green
circles) used to generate Figure 4. Background is a Clementine 750 nm reflectance mosaic with resolution reduced tobe comparable with the LP MAG data resolution.
MAG measurements. Since the field strength varies withaltitude, values were normalized to the maximum total fieldstrength for each map before being combined into the set of
245 data points used to generate the scatterplot. Figure 5
illustrates the locations of the data points and correspondingreflectance values used to make the scatterplot. The reflectance map in Figure 5 has been smoothed by a 0.25  0.25
window moving average filter in order to reduce its effective
resolution to be comparable to the LP MAG data resolution.
This removes the high frequency component of the albedo
signal that could not possibly be captured in the lower frequency magnetic field data. The data points in Figure 4 arecolor-coded according to normalized total field strength.
Cool colors represent weak fields and warm colors represent
strong fields. A blue regression line is fit to the weakest thirdof the data points and illustrates that, as expected for weak
fields, reflectance values are low regardless of magnetic field
direction. A red regression line is fit to the strongest third of

3.3. Discussion[16] The examples of Airy and Reiner Gamma illustratedistinct aspects of the solar wind deflection phenomenon: theAiry case shows that dark lanes may be associated with vertical magnetic fields (Figures 2a–2c and Figure 3), while theReiner Gamma example demonstrates that bright areas may be
associated with strongly horizontal fields (Figures 2d–2f andFigure 4). In principle, however, both of these effects should
be present at both locations. That is, we should expect to seealignment between the bright lobes of Airy and peaks in horizontal magnetic field strength and we should expect to see
vertical field lines over the dark lanes of Reiner Gamma. But
as discussed in section 1.2, we expect the near-surface field
patterns to dominate solar wind deflection and we do not yet
have observations at sufficiently low altitudes to map the nearsurface field directly. However, we can use source models todetermine whether the near surface field pattern we predict(horizontal over bright areas and vertical over dark lanes) is
consistent with the observational constraints we do have.

4. Source Modeling[17] Using a combination of techniques, including a
genetic search algorithm, we developed subsurface magnetization source models to gain insight into the possible distribution of source material at the Airy and Reiner Gamma
anomalies. We use these models to support our interpretationof the observations discussed above and to allow us to further
test the plausibility of our hypothesis—that areas of highalbedo should coincide with dominantly horizontal nearsurface magnetic fields.

Figure 6. The Airy source model obtained via the genetic search algorithm described in section 4.1.1.
(left) Each square represents a single dipole covering 0.25  0.25 (roughly 5.5  107 m2) with the color
indicating each dipole’s total magnetic moment (typical values in the center are 1.25  1011 Am2 per
dipole). (right) The same information as contours over Clementine albedo, suggesting approximate agreement
between the source structure’s longitudinal axis and that of the swirl’s dark lane.
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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

Figure 7. Comparison of (left) the observed field at 18 km
over Airy (derived from LP MAG data collected on days
49–50, 77, and 104 of 1999) and (right) the model field(obtained by fitting a grid of dipoles to the LP MAG datausing the algorithm described in Appendix B).
4.1. Lunar Prospector Data Fitting[18] As a first step, we use the Lunar Prospector magnetometer (LP MAG) measurements to determine the mostprobable characteristics of the magnetic source (the approximate distribution of magnetic material as well as the magnetic moment magnitude and orientation). The results are
spatially coarse as the data are insensitive to variations onscales smaller than the observation altitudes. The results arealso inherently non-unique as trades can be made between
depth and magnetic moment magnitude, for example. Previous studies have modeled the source of the central part of
the Reiner Gamma anomaly as a single dipole [Kurata et al.,2005] or as a grid of dipoles [Nicholas et al., 2007] using
inversion techniques [Von Frese et al., 1981; Purucker et al.,E10012

1996; Dyment and Arkani-Hamed, 1998; Parker, 2003] toobtain models that best fit the LP MAG data in a least squares
sense. Here, we employ an alternative approach involving
iterative forward modeling to identify the characteristicsof the best fitting solutions. Our approach demonstrates thata range of different solutions can deliver similarly goodresults. Throughout our modeling, we make the simplifying
assumption that the source material is coherently magnetizedin a single direction; much larger remanent magnetizations
would be required to produce the observed fields if the source
materials were not unidirectionally magnetized, and theinferred values are already large (as discussed in section 5.1).
4.1.1. Airy Model[19] We begin with a single dipole model to find themagnetic moment direction that best fits, in a least squares
sense, the 260 LP MAG measurements of the Airy study
area. We define the ‘effective error’ at each data point as the
maximum of the three vector component residuals. The leastsquares solution thus minimizes residuals in all three vector
components simultaneously. We vary the burial depth from0 to 20 km in 1-km increments and the magnetic moment
magnitude from 0 to 8  1012 Am2 in increments of 5 1011 Am2. We vary the inclination (angle positive downward
from the horizontal) from 90 to +90 and the declination(angle positive clockwise from north) through 360 , eachwith 1 resolution. We find the measurements are wellaccommodated by a dipole buried between 10 and 20 km
below the surface, with magnetic moment magnitude between
4  1012 Am2 and 7  1012 Am2, inclination between 79
and 81 and declination between 14 and +29 (i.e., pointed
steeply downward and slightly to the north). Since thismethod delivers equally good solutions over a wide range of
depths and magnetic moment magnitudes, we do not attempt
to constrain depth and magnitude at this stage. In any case,because we are assuming a single dipole, the model fit
overestimates the true source depth.
[20] Next, we expand the parameter space by replacing thesingle dipole with a grid of dipoles, separated by 0.25 inboth latitude and longitude (a distance comparable withhalf the minimum spacecraft altitude), resulting in a total of
208 dipoles. We allow each of the 208 dipoles’ magnitudes tovary independently. We allow the burial depth of the grid tovary as a whole but burial depth does not vary between
dipoles and we do not allow the dipoles to be above the lunar
surface. Likewise, we allow the magnetization direction tovary as a whole but direction does not vary between dipoles.
This opens a parameter space that is too large to explore
completely, even with coarse resolution. Instead, we employ
a genetic search algorithm that iteratively adjusts each of
the 211 independent parameters (208 independent dipole
moment magnitudes plus depth of the grid, inclination anddeclination), gradually progressing toward improving leastsquares solutions (Appendix B gives a thorough descriptionof the algorithm). Based on the single dipole model fit
described above, we initiate the genetic search algorithmwith a downward pointing magnetization (80 inclination,
8 declination) and an arbitrary burial depth of 10 km. Again,
the algorithm attempts to find a solution that minimizes thesum of squares over the ‘effective’ error at each of the 260 LP
MAG data points. The dipoles in the final model grid have
inclination 79 , declination 20 and a total magnetic momentof 4.7  1012 Am2 distributed as shown in Figure 6. The final

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Figure 8. The Reiner Gamma source model obtained via the genetic search algorithm described in
section 4.1.2. (left) Each square represents a single dipole covering 0.25  0.25 (roughly 5.7 
107 m2) with the color indicating each dipole’s total magnetic moment (typical values in the center are3.4  1011 Am2 per dipole). (right) The same information as contours over Clementine albedo.
grid has a burial depth of 650 m, but again, depth is not wellconstrained by the LP MAG data as it can be traded against
magnetic moment magnitude and the lateral extent of thesource; an equally good fit could be obtained with a sourcethat is deeper, stronger and more horizontally concentrated.
Although the model is spatially coarse, the distribution ofmagnetic moments appears to be consistent with a source
structure that is centered on the anomaly and elongated in anorth-south direction. Figure 7 shows that the resulting
model field compares very well with the observed field. The
magnetization direction is close to that of the hypotheticalfield illustrated in Figure 1a, indicating that the intraswirl
dark lane may well coincide with vertical field lines at thesurface.
4.1.2. Reiner Gamma Model[21] Again, we begin with a single dipole oriented so that it
best fits, in a least squares sense, the 245 LP MAG measurements at the Reiner Gamma study area. Using the‘effective’ error metric described for the Airy model above,
we search for the least squares best fit by varying the burial
depth from 0 to 20 km in 1-km increments, the magnetic
moment magnitude from 0 to 2  1013 Am2 in increments of1012 Am2, the inclination from 90 to +90 and the declination through 360 , with 1 resolution for both inclinationand declination. We find that the measurements are wellaccommodated by a dipole that is buried between 8 and
15 km below the surface with magnetic moment between 9 1012 Am2 and 16  1012 Am2, inclination between 6
and +5 and declination between 18 and 3 (i.e., lying
approximately in the plane of the lunar surface and pointing
slightly west of north). For comparison, Kurata et al. [2005]
modeled this part of the anomaly as a single dipole buried
11.1 km below the surface with magnetic moment 11.3 1012 Am2, inclination +1.3 and declination 11 . Onceagain, depth and magnitude are not well constrained by thedata and are overestimated by this single dipole model.
[22] As with the Airy model, we next expand the parameter
space by replacing the single dipole with a grid of 208 dipoles
and use the genetic algorithm described in Appendix B tofind a solution that best fits the 245 LP MAG data points. The

final model dipole grid is magnetized with inclination +4
and declination 12 at a depth of 1.6 km (but again, depth is
not well constrained by the LP MAG data), and has totalmagnetic moment 1.4  1013 Am2 distributed as shown in
Figure 8. The source model is spatially coarse but is consistent with a source structure that is elongated in the east-west
direction and is most intense in the brightest part of thealbedo anomaly. Figure 9 shows that the resulting model
field compares well with the observed field. For comparison,
Nicholas et al. [2007] modeled the Reiner Gamma source asa grid of dipoles separated by 0.1 and placed at the surface,coincident with the albedo anomaly. Those authors assumed
a northward magnetization and solved for the magnitude andsign at each of the dipoles, obtaining the distribution ofmagnetization illustrated in Figure 10. In spite of the very
different methods employed, the two results resemble one
another, predicting strong magnetization in the brightest partsof the anomaly.
4.2. Albedo Pattern Matching[23] The results of the genetic search algorithm suggest that the distribution of the underlying source material
coincides roughly with the shape of the albedo anomaly:
a north-south distribution for Airy and an east-west distribution for Reiner Gamma. Here, we refine our source models
by applying the constraint that the near-surface field be
structured according to our hypothesis: strongly horizontal
over the brightest parts of swirls and vertical in the intraswirl
dark lanes. This allows for greatly improved spatial resolution since Clementine reflectance mosaics are available
at 256 pixels/degree whereas LP MAG data are limited to1– 4 pixels/degree. If our hypothesis is correct, source
models constrained by the albedo pattern should produce
fields that are consistent with the LP MAG observations
made at higher altitudes. Below, we show that even simple
source models are sufficient to accomplish this, suggestingthat our models are highly plausible.
4.2.1. Airy Model[24] Given the downward-pointing magnetization at Airy(section 4.1.1) and the associated field pattern (Figure 1c),

9 of 19

E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012

with peaks in the surface horizontal magnetic field profile,then the depth of this line source must be 7.8 km. Because
we assume an infinitely narrow source, this represents the
maximum possible source depth consistent with the hypothesis that the bright lobes align with peaks in the horizontal
magnetic field profile. The LP MAG data are too spatiallycoarse to constrain the distribution of magnetic moments
within the source structure so we assign equal strength toeach of the 32 dipoles in the array. With the depth fixed andthe dipole moment magnitudes set to be equal, the parameter
space is reduced to three dimensions (total magnitude, inclination, and declination), allowing for a complete search(varying magnitude in increments of 1011 Am2 and inclination and declination in 1 increments) to find the solution thatbest fits, in a least squares sense, the LP MAG data. The
resulting best fit solution has the dipoles pointed steeply
downward (inclination 75 , declination 20 ) with totalmagnetic moment 3.8  1012 Am2 (Table 3). This source
model produces a magnetic field pattern at the surface thatmimics the morphology of the albedo anomaly, with the field
being strongly horizontal in high albedo areas and vertical in
the intraswirl dark lane (Figure 11). Remarkably, this same
simple source model produces a field pattern at the spacecraft
altitude that substantially resembles the LP MAG observations (Figure 12).
4.2.2. Reiner Gamma Model[25] The Airy model (section 4.2.1) suggests that thedark lane is aligned with a cusp between two peaks in thehorizontal magnetic field profile. Based on the Airy example,
we may view the two dark lanes at Reiner Gamma assuggestive of cusps in the near-surface horizontal magnetic
field profile (see Harnett and Winglee [2003] for a related
discussion). As illustrated in Figure 1d, a horizontallymagnetized source will produce strongly horizontal fields
directly above the source while producing cusps (where the

Figure 9. Comparison of (left) the observed field at
18 km over Reiner Gamma (derived from LP MAG datacollected on days 54, 81–82, and 109 of 1999) and (right)
the model field (obtained by fitting a grid of dipoles to the
LP MAG data using the algorithm described in Appendix B).
the morphology of the Airy swirl suggests a roughly linear
source structure following the dark lane. In order to be consistent with our hypothesis, the vertically magnetized source
material cannot substantially underlie the bright lobes since
this would mean vertical field lines in bright areas. We
therefore replace the grid of dipoles described in section 4.1.1
with an array of 32 dipoles arranged as a two-segmented
linear feature following the dark lane (Figure 11, left).
In cross-section, the field pattern from such a line source
resembles the pattern from a single dipole (illustrated in
Figures 1a and 1c). However, for a linear source, the peaks in
the horizontal field profile are shifted outward slightly suchthat the peak-to-peak separation is 2/√3 times the altitude
above the source rather than being equal to the altitude above
the source (details are given in Appendix A). If we assume
the two bright lobes at Airy (separated by 9 km) are aligned

Figure 10. Distribution of source magnetization at Reiner
Gamma as estimated by Nicholas et al. [2007] assuming
the source layer is 40 km thick. The contour interval is
0.1 A/m with warm colors positive, cool colors negative, andthe dashed white contour representing zero magnetization.
The contours are superimposed over Clementine V1 750 nm
reflectance. Note that our study instead uses Clementine V2
mosaics, which differ in horizontal registration by 7 km in
the vicinity of Reiner Gamma.

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

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Figure 11. (left) Airy study area showing Clementine albedo map with red circles indicating the locationsof the source model’s 32 dipoles. (right) Resulting horizontal magnetic field strength predicted at the surface. White arrows indicate where field lines become vertical in the intraswirl dark lane. Figure 12 compares the model field predicted at the spacecraft altitude with the LP MAG observations.
horizontal field strength drops to zero) on either side of thesource. This suggests that the horizontally magnetized source
material underlies the brightest parts of the Reiner Gamma
albedo feature and that the dark lanes may be aligned with thecusps. The appearance of the two dark lanes and the relatively bright material between them may be explained by asuperposition of two sources (see Figure 13 and comparewith Figure 1).
[26] As with the case illustrated in Figures 1b and 1d, thefield illustrated in Figure 13 is strongly horizontal at any
altitude over the two horizontally magnetized sources.
However, in this case there is an additional region of elevatedhorizontal field strength where the side lobes interfere constructively between the two sources. Taking this pattern as acue, we replace the grid of dipoles described in section 4.1.2
with an array of 55 dipoles arranged as curvilinear structures
beneath the three brightest parts of the swirl (Figure 14, left).
In cross-section, the field pattern from the two approximately
linear sources adjacent to the dark lanes resembles the patternfrom the two-dipole case illustrated in Figure 13, but thecusps in the horizontal magnetic field profile are shifted
outward slightly such that they are displaced laterally (in thiscase to the north and south) from the center of the sources
by a distance equal to the altitude above the source rather
than 1/√2 times the altitude above the source (details are
given in Appendix A). If we assume the dark lanes at Reiner
Gamma (which are displaced 5 km from the centers of thebright lobes) are aligned with the cusps in the surface horizontal magnetic field profile, then the depth of the sourcemust be 5 km. Because we assume infinitely narrow source
structures, this represents the maximum possible source depth
consistent with the hypothesis that the dark lanes align withcusps in the horizontal magnetic field profile. The LP MAG
data are too spatially coarse to constrain the distribution ofmagnetic moments within the source structure so we assign
equal strength to each of the 55 dipoles in the array. As with
the Airy case, we perform a complete search to find the solution that best fits, in a least squares sense, the LP MAG data.
The resulting best fit solution has the dipoles pointed withinclination +2 and declination 8 (i.e., pointed nearly horizontally and slightly west of north) and total magnetic moment

1.0  1013 Am2 (Table 4). This source model produces amagnetic field pattern at the surface that mimics the morphology of the albedo anomaly, with the field being strongly
horizontal in high albedo areas and vertical in the intraswirl
Table 3. Final Dipole Array Model for Airy as Described in
Section 4.2.1a
Latitude( N)

Longitude
( E)

Depth
(km)

Magnetic
Moment(Am2)

Inclination(deg)

Declination(deg)

17.48917.512
17.534
17.557
17.579
17.602
17.624
17.647
17.669
17.692
17.714
17.737
17.75917.782
17.815
17.860
17.906
17.951
17.99618.041
18.087
18.132
18.177
18.223
18.268
18.313
18.359
18.404
18.449
18.49418.540
18.585

2.826
2.862
2.899
2.935
2.972
3.008
3.044
3.081
3.117
3.153
3.190
3.226
3.263
3.299
3.305
3.313
3.322
3.330
3.338
3.347
3.355
3.363
3.372
3.380
3.389
3.397
3.405
3.414
3.422
3.430
3.439
3.447

7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8
7.8

1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11
1.19E+11

75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75
75

20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20

a
See Figure 11. The model consists of 32 dipoles at the indicated latitudes
and longitudes, all buried 7.8 km below the surface. The total magnetic
moment is 3.8  1012 Am2. Inclination is measured positive downward fromthe horizontal and declination is measured positive clockwise from north.

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012

typical magnetizations at the Reiner Gamma anomaly are on
the order of 1 A/m. If the magnetized layer is only 1 km thick,the implied magnetization approaches 10 A/m at Reiner
Gamma. For comparison, Nicholas et al. [2007] predict a
minimum magnetization of 1 A/m for a layer 1 km thick at
Reiner Gamma and Wieczorek et al. [2012] calculate 2 A/m
for the same layer thickness assuming an anomaly that produces a 10 nT field at 30 km altitude. If we suppose thesource structures are further horizontally concentrated, as
the albedo-pattern-constrained models of section 4.2 suggest(Figures 11 and 14), the magnetizations would have to beeven greater.
5.2. Magnetizing Field[28] The strong magnetizations at the Airy and Reiner
Gamma anomalies could be the result of the source material
having cooled in a long lasting global magnetic field, perhaps
generated by a core dynamo [Garrick-Bethell et al., 2009;

Figure 12. Comparison of (left) the observed field at
18 km over Airy (derived from LP MAG data collectedon days 49–50, 77, and 104 of 1999) and (right) the model
field (obtained as described in section 4.2.1). Figure 11
shows the horizontal component of the model field predicted at the surface.
dark lanes (Figure 14). Remarkably, this same source model
produces a field pattern at the spacecraft altitude that agrees
with the Lunar Prospector observations (Figure 15).

5. Discussion
5.1. Magnetization[27] The models described in section 4.1 suggest that thesource material is concentrated under the central parts of thealbedo anomalies. Based on the source material distributions
illustrated in Figures 6 and 8, we can compute the impliedmagnetizations at Airy and Reiner Gamma for various
assumed layer thicknesses. Figure 16 illustrates that evenwhen the magnetized layer is assumed to be 10 km thick,Figure 13. (top) Magnetic field lines due to a pair of dipoles
separated by 15 km and oriented horizontally in the plane ofthe page. (bottom) Profiles of the horizontal component of
the magnetic field shown at the various altitudes represented
by dashed lines in Figure 13 (top). Each of the dipoles has amagnetic moment of 1012 Am2.

12 of 19

E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012

Figure 14. (left) Reiner Gamma study area showing Clementine albedo map with red circles indicating
the locations of the source model’s 55 dipoles. (right) Resulting horizontal magnetic field strength predicted
at the surface. White arrows indicate where field lines become vertical in one of the intraswirl dark lanes.
Figure 15 compares the model field predicted at the spacecraft altitude with the LP MAG observations.
Dwyer et al., 2011; Le Bars et al., 2011; Shea et al., 2012].
As we have shown here, however, the Airy anomaly (locatedat approximately 17 S, 3 E) is magnetized with a steep
downward inclination while the Reiner Gamma anomaly(located at approximately 7 N, 59 W) is magnetized withalmost zero inclination and points approximately toward thenorth. If these two anomalies acquired their magnetizations
by cooling in a dipolar dynamo field, they could not haveformed contemporaneously. Instead, the two anomalies may
have formed during different global field orientation epochs.
Alternatively, the Moon’s dynamo field may have had substantial higher order components (i.e., beyond dipolar).
[29] Another possibility that avoids the difficulties associated with inconsistent magnetization directions is that, rather
than being the thermal remanent magnetization signaturesof an extinct core dynamo, the Moon’s crustal magnetic
anomalies may instead be the result of shock remanent
magnetization that occurs following basin-forming impact
events when magnetohydrodynamic shock waves converge
near the basin antipode producing strong transient fields andpotentially magnetizing iron-rich ejecta materials [Hoodet al., 2001; Halekas et al., 2001; Richmond et al., 2005;
Hood and Artemieva, 2008].

6. Conclusions[30] Our examination of swirls at Airy and Reiner Gamma,
two magnetic anomalies with dissimilar orientation, suggeststhat magnetic field direction and swirl morphology are
related in the way we predict based on the solar wind
deflection hypothesis: the Reiner Gamma case delivers evidence that swirls are brightest where magnetic field lines are
dominantly horizontal and the Airy case demonstrates aconnection between dark lanes and vertically oriented fieldlines. These findings support the solar wind deflection model
for swirl formation, implying that differential solar winddarkening is largely responsible for creating the albedo
anomalies. Although our source models do not represent
unique solutions, they agree with observational constraints
while plausibly accounting for the alternating bright and darkbands at both Airy and Reiner Gamma. Our model results

suggest that swirl morphology could potentially be used toinfer small-scale structure in the near-surface magnetic field
as well as the layout and burial depth of the magnetic source
material. For Airy and Reiner Gamma, we infer maximum
source burial depths of 8 km and 5 km, respectively, andhorizontally concentrated sources with strong magnetizations(10 A/m or greater for a layer 1 km thick). Examination of
additional swirls may help to further our understanding of
how magnetic field direction relates to swirl morphology, butultimately, very near-surface magnetic field and solar wind
flux measurements (i.e., from altitudes of hundreds of metersor less) will be required to confirm our predictions.

Appendix A: Magnetic Field Profile Calculations[31] Here, we calculate the locations of the maxima andminima in the horizontal magnetic field profiles due to bothvertically and horizontally magnetized sources. We firstconsider the single dipole sources illustrated in Figure 1 andthen repeat the calculations for the line sources discussed in
section 4.
A1.

Single Dipole Source[32] If we take the horizontal and vertical positions inFigures 1a and 1b to be the x and z coordinates, respectively,
and if the magnetic dipoles are each located at the origin withmagnitude and orientation represented by a vector, m, then^the resulting magnetic field, B, at a position, r = (x^{ + y^| + zk),
is given by:
B¼

1
m0 
3ðm ⋅ rÞr  mr2 5
r
4p

ðA1Þ

Equation (A1) is adapted from Blakely [1995, p. 75]. Here,we employ SI units with m0 being the magnetic permeabilityof free space. For points in the x-z plane, the horizontal
component of the magnetic field (Bh) is identical to themagnitude of the x-component (Bx) since, by symmetry,magnetic field lines cannot cross the x-z plane (i.e., cannot
have a y-component). Considering the case of the vertically

13 of 19

E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

Table 4. Final Dipole Array Model for Reiner Gamma as
Described in Section 4.2.2a
Latitude( N)

Longitude
( E)

Depth
(km)

Magnetic
Moment(Am2)

Inclination(deg)

Declination
(deg)

7.649
7.673
7.693
7.710
7.723
7.731
7.735
7.735
7.731
7.723
7.710
7.693
7.673
7.649
7.622
7.592
7.559
7.525
7.488
7.451
7.315
7.278
7.241
7.207
7.174
7.144
7.117
7.093
7.073
7.056
7.043
7.035
7.031
7.031
7.035
7.043
7.056
7.073
7.093
7.117
7.542
7.538
7.543
7.555
7.573
7.593
7.612
7.628
7.636
7.637
7.629
7.615
7.596
7.576
7.557

301.460
301.418
301.371301.320301.264301.205301.144301.080301.015300.950
300.885
300.821300.759
300.700
300.645300.593300.546300.505
300.469
300.439301.525301.495
301.459301.418
301.371301.319301.264301.205301.143301.079
301.014300.949300.884
300.820300.759
300.700
300.644300.593300.546300.504
301.774
301.819301.878301.943302.001
302.043302.062302.056302.024301.973301.910301.847301.794301.759301.749

5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5

1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11
1.82E+11

2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2

8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8

E10012Equation (A2) demonstrates that field lines directly over a
vertically oriented dipole are vertical at any altitude because
Bh = 0 when x = 0 (this is also evident in Figure 1a).
Differentiating equation (A2) with respect to x and setting
the result to zero yields the positions of the peaks in thehorizontal field strength profile:1
xBh max ¼  z
2

ðA3ÞThis means that the separation between peaks (2x) is equal
to the altitude above the source (z), which is evident in
Figure 1c.
[33] Considering the case of the horizontally oriented
dipole (m = [m 0 0]T ), and still being restricted to points ina
See Figure 14. The model consists of 55 dipoles at the indicated latitudes
and longitudes, all buried 5 km below the surface. The total magnetic
moment is 1.0  1013 Am2. Inclination is measured positive downward fromthe horizontal and declination is measured positive clockwise from north.

oriented dipole (m = [0 0 m]T ), the horizontal componentof the magnetic field becomes:
Bh ¼ jBx j ¼

(
)
m0
3mzx
4p ðx2 þ z2 Þ52

ðA2Þ

Figure 15. Comparison of (left) the observed field at
18 km over Reiner Gamma (derived from LP MAG datacollected on days 54, 81–82, and 109 of 1999) and (right)
the model field (obtained as described in section 4.2.2).
Figure 14 shows the horizontal component of the model field
predicted at the surface.

14 of 19

E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012the y axis, and the magnetization per unit length is m′, thenthe magnetic field is given by:
B¼

1
m0 
4ðm′ ⋅ rÞr  2m′r2 4
4p
r

ðA7Þ

Equation (A7) is adapted from Blakely [1995, p. 96]. Again,
considering only points in the x-z plane, the horizontal
component of the magnetic field due to the vertically magnetized line source becomes:
Bh ¼ jBx j ¼

Figure 16. Rock magnetization implied by source models
illustrated in Figures 6 and 8 versus assumed layer thickness. Solid lines are based on the maximum individual dipole
moments found in those models (2.1  1011 Am2 for Airy,
6.5  1011 Am2 for Reiner Gamma), and the dashed lines
are based on characteristic values of 1.25  1011 Am2 for
Airy and 3.4  1011 Am2 for Reiner Gamma.

the x-z plane, the vertical (z) component of the magnetic field
becomes:
(
)
m0
3mxz
Bv ¼ jBz j ¼
4p ðx2 þ z2 Þ52

ðA4Þ

Equation (A4) demonstrates that field lines directly over a
horizontally oriented dipole are horizontal at any altitude
because Bz = 0 when x = 0 (Figure 1b). Again, starting fromequation (A1), and considering only points in the x-z plane,
the horizontal component of the magnetic field due to ahorizontally oriented dipole that is aligned with the x axis
becomes:()
m0 mð2x2  z2 ÞB h ¼ jB x j ¼
4p ðx2 þ z2 Þ52

ðA5Þ

Setting equation (A5) to zero shows that the horizontal fieldstrength drops to zero when:1xBh ¼0 ¼  pﬃﬃﬃ z
2

(
)
m0
4m′zx
4p ðx2 þ z2 Þ2

ðA8ÞAs with the dipole case, field lines directly over a verticallymagnetized line source are vertical at any altitude. Differentiating equation (A8) with respect to x and setting the result tozero yields the positions of the peaks in the horizontal field
profile:1xBh max ¼  pﬃﬃﬃ z3

ðA9Þ

Considering the case of a horizontally magnetized line source,
the vertical (z) component of the magnetic field becomes:
(
)
m0
4m′xz
Bv ¼ jBz j ¼
4p ðx2 þ z2 Þ2

ðA10ÞHence, field lines directly over a horizontally magnetizedlinear source are horizontal at any altitude. Again, startingfrom equation (A7), and considering only points in the x-z
plane, the horizontal component of the magnetic field due to alinear source coincident with the y axis and magnetization
parallel to the x axis becomes:()
m0 2m′ðx2  z2 Þ
Bh ¼ jBx j ¼
4p ðx2 þ z2 Þ2

ðA11Þ

Setting equation (A11) to zero shows that the horizontal fieldstrength drops to zero when:xBh ¼0 ¼ z

ðA12ÞHence, the cusps in the horizontal magnetic field profile arelaterally displaced from the line source by a distance equal tothe altitude above the source.

Appendix B: Genetic Search Algorithm

ðA6Þ[35] The magnetic source models discussed in section 4.1
were obtained using a heuristic search technique known asHence, the cusps in the horizontal magnetic field profile
are laterally displaced from the source by a distance of1/√2 times the altitude above the source (this is evident in
Figure 1d).

Table B1. The 211 ‘Genes’ That Define an Individual Source
Model

A2.

Gene Number

Controls

1
2
3
4
…
211

Dipole grid burial depth (km)Magnetic moment inclination (deg)Magnetic moment declination (deg)Magnitude of dipole #1 (Am2)…
Magnitude of dipole #208 (Am2)Linear Source[34] Imagine that instead of a single dipole, the source is alinear structure such as a long, uniformly magnetized cylinder. If this linear source is infinitely long and coincident with
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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

Figure B1. Snapshots at generations 1, 100, 300, and 600 during the evolution of the dipole grid source
models for (left) Airy and (right) Reiner Gamma. Each panel shows the distribution of magnetic momentsfrom the best-performing model of the specified generation.
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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

E10012

Figure B2. Evolution of dipole grid source model error over 600 generations for (left) Airy and(right) Reiner Gamma. The error shown is the sum of squared ‘effective’ errors (as described in
section 4.1.1) over all LP MAG data points for the best-performing model in each generation.

a genetic algorithm. Genetic algorithms have a diverse range
of applications [Holland, 1992; Goldberg, 1989] including
large-scale optimization problems such as ours: to find asource model (defined by 211 parameters) that produces
minimal error between predicted and observed magnetic
fields. Genetic algorithms employ concepts borrowed fromgene-centered biological evolution [Dawkins, 1976] in orderto iteratively (i.e., over many generations) progress toward
solutions with increasing degrees of ‘fitness’. In our case, this‘fitness’ is a measure of how well the magnetic field produced by the source model matches the Lunar Prospector
magnetometer (LP MAG) observations.
[36] Our algorithm begins by generating a population ofindividual source models with randomly distributed characteristics. The characteristics of each individual source
model are defined by its ‘genes’, with each individual having
the 211 distinct genes listed in Table B1. At each iteration, or
generation, the individual members of the population areevaluated according to how well they predict the magnetic
field at each of the LP MAG observation points. We computethe error between the prediction and the observation using the‘effective error’ metric described in section 4.1.1. We thenrank the individuals from lowest to highest sum of squared
effective errors. The individuals with the lowest errors arethen selected as the ‘parents’ of the next generation. The next
iteration begins by generating a new population to replace theprevious generation. Each individual in the new generation
is formed by setting each of its 211 genes equal to thecorresponding gene from one of its parents in a process
known as crossover (analogous to chromosomal crossover).
For example, an individual’s gene for inclination will matchthe inclination gene of one of its parents (chosen at random
from among the parents). Because only the best-performing
individuals contribute their genes to the next generation,individuals of each new generation tend to inherit the best
characteristics of the previous generation. We then apply
random mutations (i.e., randomly adjust several isolated

genes). This step ensures variation in the population andallows for the possibility of introducing advantageous genes
not possessed by the parents. Finally, members of the newgeneration are evaluated and their best performers are
selected as the parents of the next generation. With eachgeneration, genes that result in large model errors tend to bediscarded while genes that result in smaller model errors tend
to be retained. Inheritance and selection lead the gene pool to
be increasingly rich in genes that form good source models
while crossover and mutations ensure variation, allowing
for innovations that can lead offspring to outperform theirparents. The result is that the population progresses gradually
toward optimality in terms of minimum model error.
[37] While this explanation captures the essence of thealgorithm, our implementation includes additional details
such as gene mutation rate (the probability that any given
gene will mutate) and how “random” mutations are distributed (i.e., it is not useful to have sudden changes in burial
depth on the order of 1000 km, for example, so mutations
must be limited to reasonable adjustments according to some
distribution). We experimented with various population sizes
and numbers of parents selected at each generation (there
must be at least two parents but more than two is also
allowed). For computational efficiency, a small population
size is preferred. The algorithm was found to operate effectively with a population of just 10 individuals in which the
top-performing 3 contribute their genes to the next generation. We allow mutations to occur with a probability of
0.1 and when they do occur, the gene is adjusted from its
current value according to a normal distribution with somepre-defined standard deviation (which depends on whether
the gene has units of kilometers, degrees or Am2). We also
apply smaller mutations with a higher probability in order toensure a measurable degree of variation in the population.
The initial population is generated based on a seed model;
each individual in the initial population is formed by applying normally distributed adjustments to each of the genes in

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E10012HEMINGWAY AND GARRICK-BETHELL: MAGNETIC FIELD DIRECTION AT LUNAR SWIRLS

the seed model. While the choice of seed model and parameter settings influences the efficiency of the algorithm (i.e.,
how quickly it reaches a good solution), the end results tend
to be very similar across a wide range of parameters andinitial conditions. For the dipole grid source models presented in section 4.1, the algorithm was allowed to iterate
through 600 generations (Figure B1). Based on the trends inerror evolution (Figure B2), we would not expect additional
iterations to yield substantial improvements.
[38] Acknowledgments. This research was performed as part of theWorld Class University Project at Kyung Hee University and sponsoredby the Korean Ministry of Education, Science and Technology. Additional
support was provided by the NASA Ames Research Center/University ofCalifornia Santa Cruz University Affiliated Research Center. We would liketo thank Lon Hood, Robert Lin, Jasper Halekas, Greg Delory, Robert Lillis,
and Michael Purucker for their comments and suggestions and Joseph
Nicholas for the data used to generate Figure 10.

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