Journal of Geophysical Research: Planets
RESEARCH ARTICLE
10.1029/2018JE005604
Key Points:• Lunar swirl morphology suggests
ﬁnely structured near-surface
magnetic ﬁelds, requiring narrow,shallow magnetic source bodies• The source bodies underlying
swirls must be much more strongly
magnetized than is typical of lunar
mare basalts• The inferred source geometry andmagnetization could be explained
both by magmatic intrusions andassociated thermochemical alteration

Correspondence to:
D. J. Hemingway,
djheming@berkeley.edu

Citation:
Hemingway, D. J., & Tikoo, S. M. (2018).
Lunar swirl morphology constrains thegeometry, magnetization, and origins
of lunar magnetic anomalies. Journalof Geophysical Research: Planets, 123.
https://doi.org/10.1029/2018JE005604

Received 3 MAR 2018
Accepted 18 JUL 2018
Accepted article online 26 JUL 2018

Lunar Swirl Morphology Constrains the Geometry,
Magnetization, and Origins of Lunar
Magnetic Anomalies
Douglas J. Hemingway1

and Sonia M. Tikoo2

1 Department of Earth and Planetary Science, University of California, Berkeley, Berkeley, CA, USA, 2 Department of Earthand Planetary Sciences, State University of New Jersey, Rutgers, Piscataway, NJ, USA

Abstract Lunar swirls are collections of ﬁnely structured bright and dark surface markings, alternatingover length scales of typically 1–5 km. If swirls are the result of plasma interactions with crustal magnetic
anomalies or electrostatic or magnetic sorting of ﬁne materials, the magnetic ﬁeld orientation must varyover similar length scales. This requires that the associated source bodies be both shallow and narrow inhorizontal extent. The correspondingly restricted volume of the source bodies in turn implies strong rockmagnetization. Here we show that if ∼300-nT surface ﬁelds are necessary to produce observable swirl
markings, the required rock magnetization must be > 0.5 A/m, even for very shallow sources and likely
closer to ∼ 2 A/m or more. This strong source rock magnetization, together with the geometric constraints
that favor magmatic structures such as dikes or lava tubes, requires a mechanism to enhance the magnetic
carrying capacity of the rocks. We propose that heating associated with magmatic activity could
thermochemically alter host rocks and impart them with magnetizations an order of magnitude stronger
than is typical of lunar mare basalts. Our results both place constraints on the geometry and magnetizationof the source bodies and provide clues about the possible origins of the Moon’s crustal magnetic anomalies.
Plain Language Summary Lunar swirls are collections of ﬁnely structured, bright and dark
surface markings, alternating over length scales of typically 1–5 km. Swirls are thought to form where local
magnetic ﬁelds shield parts of the lunar surface from exposure to the solar wind or where those magnetic
ﬁelds lead to sorting of some of the ﬁnest lunar soils. In either case, the length scales of swirls are eﬀectively
telling us about the structure of near-surface magnetic ﬁelds on scales that are ﬁner than what can bemeasured from lunar orbit. Here we use this length scale information, along with estimates of the strengthof those magnetic ﬁelds, to obtain new constraints on the underlying magnetized rocks, showing that theymust be shallow, narrow, and strongly magnetized. This result helps us to better understand the origin of
these magnetized rocks and the history of lunar magnetism more generally. In particular, we suggest thatthese rocks were likely injected into the crust in the form of dikes or subsurface channels of ﬂowing lava andthat they cooled slowly, leading to enhancement of their metal content and enabling the rocks to capture astable record of the Moon’s ancient global magnetic ﬁeld.

1. Introduction
1.1. Lunar Magnetism Background
Although the Moon does not have an internally generated global magnetic ﬁeld today, paleomagnetic analysis of Apollo samples shows that many lunar rocks exhibit a thermoremanent magnetization (TRM), implyingthat a long-lived global ﬁeld was present early (before ∼1–2.5 Ga) in the Moon’s history (e.g., Tikoo et al., 2017;
Weiss & Tikoo, 2014); during the period ∼3.85–3.56 Ga, lunar surface ﬁelds are inferred to have reachedintensities of 78 ± 43 μT (comparable to the Earth’s modern ﬁeld).

©2018. American Geophysical Union.
All Rights Reserved.

HEMINGWAY & TIKOO

In situ Apollo magnetometer observations found surface ﬁelds as large as 327 nT at the Apollo 16 landingsite (Dyal et al., 1974). Magnetometer and electron reﬂectometer experiments aboard orbiting spacecraft (theApollo 15/16 subsatellites, Lunar Prospector, and Kaguya), however, have revealed that other portions of thelunar crust are even more strongly magnetized (e.g., Mitchell et al., 2008; Purucker & Nicholas, 2010; Richmond& Hood, 2008; Tsunakawa et al., 2015). These orbit-based measurements necessarily miss the ﬁnest structuresin the near-surface magnetic ﬁelds and therefore provide only lower bounds on the magnitude of surface
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ﬁelds (Halekas et al., 2010). For instance, using downward continuation techniques, Tsunakawa et al. (2015)
estimate a surface ﬁeld of ∼ 20 nT at the Apollo 16 landing site—more than an order of magnitude lessthan what was measured in situ. Meanwhile, the same technique estimates surface ﬁelds of > 200 nT at the
Descartes magnetic anomaly (∼ 50 km away) and more than 500 nT in a number of other locations, such as
at Reiner Gamma. Given that even these estimates may be an order of magnitude too small, it is not unreasonable to expect strong crustal magnetic anomalies to exhibit surface ﬁelds of thousands of nanoteslas(Halekas et al., 2010; Hemingway & Garrick-Bethell, 2012). In principle, the technique of electron reﬂectometry(Halekas et al., 2001; Mitchell et al., 2008) can be used to determine the magnetic ﬁeld strength at the surface, but the technique has limited sensitivity to surface ﬁelds that vary over short length scales (<∼ 10 km;
Halekas et al., 2010). Accordingly, surface ﬁeld estimates from electron reﬂectometry (Mitchell et al., 2008) aresimilar to the magnetometer-based results of Tsunakawa et al. (2015), with both eﬀectively missing the ﬁner
structures where surface ﬁelds may be strongest (Halekas et al., 2010).
Several ideas have been proposed to explain how the Moon might have generated and sustained an
intrinsic dynamo ﬁeld that could have magnetized parts of the crust (e.g., Dwyer et al., 2011; Evans et al.,2014; Laneuville et al., 2014; Le Bars et al., 2011; Scheinberg et al., 2015, 2018; Stegman et al., 2003),
but it remains a challenge to explain how even the weakly magnetized Apollo samples acquired their magnetization from such a ﬁeld because most lunar thermal evolution models predict dynamo-driven surface
ﬁelds well below the inferred ∼ 100 μT (Weiss & Tikoo, 2014). The origins of the much stronger, localized
crustal magnetic anomalies are even more puzzling. Some studies suggest that very strong transient ﬁelds
may have magnetized parts of the crust during basin-forming impact events (e.g., Hood & Artemieva, 2008;
Mitchell et al., 2008), while others suggest that the localized crustal anomalies are related to the emplacementof especially iron-rich exogenic materials (Wieczorek et al., 2012) or intrusions (Purucker et al., 2012). Distinguishing between the possible formation mechanisms has been diﬃcult due in part to our limited knowledge
about the near-surface ﬁeld structure and the geometry of the underlying magnetic source bodies. Putting
constraints on these aspects of crustal magnetic sources could provide valuable insights into their possible
origins and, in turn, the thermal history and evolution of the Moon.
1.2. Exploiting Lunar Swirls
Another intriguing aspect of the Moon’s crustal magnetic anomalies is that many of them are colocated withpeculiar clusters of optical anomalies, comprising alternating bright and dark markings, known as lunar swirls(Figure 1; e.g., Blewett et al., 2011; Denevi et al., 2016; El-Baz, 1972). One possible explanation for the formation of swirls is that they are the result of spatially variable space weathering that occurs due to magnetic
deﬂection of the incoming solar wind (e.g., Hood & Schubert, 1980; Hood & Williams, 1989), which is known
to aﬀect the optical properties of exposed surfaces over time (e.g., Hapke, 2001; Noble et al., 2001). Roughly
speaking, portions of the surface with open magnetic ﬁeld lines may be expected to experience normal oreven accelerated solar wind-related space weathering, leading to darkening, whereas portions of the surface beneath closed magnetic ﬁeld lines should experience greater protection from solar wind weathering,
thus remaining relatively bright (Hemingway & Garrick-Bethell, 2012). Indeed, magnetometer-based studies
of magnetic ﬁeld structure (Hemingway & Garrick-Bethell, 2012; Shibuya et al., 2010; Tsunakawa et al., 2015),
as well as hybrid and kinetic plasma simulations (e.g., Bamford et al., 2012, 2016; Deca et al., 2015, 2016;
Fatemi et al., 2015; Giacalone & Hood, 2015; Jarvinen et al., 2014; Poppe et al., 2012, 2016; Zimmerman et al.,2015), indicate that swirl morphology may be dictated by magnetic ﬁeld topology in precisely this way. It
has also been proposed that swirls may be the result of electrostatic (Garrick-Bethell et al., 2011) or magnetic(Pieters et al., 2014) sorting of ﬁne-grained materials, rather than of deﬂection of solar wind. In this case, thenear-surface magnetic ﬁeld structure dictates the spatial variation in the forces responsible for the sorting andshould thus be tightly linked to the structure of the resulting optical anomaly. The connection between swirl
morphology and ﬁeld topology is less obvious under the hypothesis that swirls are the result of scouring due
to cometary impacts (see Schultz & Srnka, 1980; Syal & Schultz, 2015), though even in that model, it remains
possible that the near-surface ﬁeld structure would play a role in determining the way the impacting material interacts with and aﬀects the optical properties of the surface. In any case, we will proceed here on theassumption that there is a close connection between the near-surface magnetic ﬁeld structure and the morphology of the swirls. That is, following Hemingway and Garrick-Bethell (2012), we will use swirl morphology
as a proxy for the structure of the near-surface magnetic ﬁelds.
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Figure 1. The archetypal lunar swirl complex—Reiner Gamma—consisting of numerous alternating bands of brightlobes and dark lanes. The largest features have bright-to-dark transition length scales of up to ∼5 km, but much moreﬁnely structured features are discernible as well. Image is from the Lunar Reconnaissance Orbiter Camera Wide Angle
Camera (Mahanti et al., 2016; Robinson et al., 2010; Speyerer et al., 2012).

1.3. Outline
In section 2, we develop a model and combine it with observations to place constraints on the geometries and magnetizations of the source bodies associated with lunar swirls. We discuss the implications ofthese results for the distribution of swirls and the possible origins of these crustal magnetic anomalies,
showing that unusually intense rock magnetization is required. In section 3, we explore mechanisms by whichthese intense rock magnetizations could be realized through magnetic enhancement associated with heating due to the emplacement of dikes or lava tubes in the reducing environment of the shallow subsurface.
Finally, we provide concluding remarks and discuss future work in section 4.

2. Constraining Source Geometry and Magnetization
2.1. Model Description
Here we wish to relate the near-surface magnetic ﬁeld structure, parameterized by what we call the transition
length scale (L, see below), and the estimated surface ﬁeld strength (B) to the source body’s magnetization(or magnetic moment per unit volume, M) and the geometry of the underlying magnetized source body. Following Hemingway and Garrick-Bethell (2012), we will assume that the bright-to-dark transition length scalefound at lunar swirls can be used as a proxy for the length scale over which the near-surface magnetic ﬁeld
direction transitions from vertical (perpendicular to the surface) to horizontal (parallel to the surface). Moreprecisely, we will deﬁne this bright-to-dark transition length scale as the distance between the brightest partof a bright lobe and the darkest part of the adjacent dark lane as measured by albedo proﬁles taken perpendicular to the long axis of the feature (Figure 2). Although large swirl complexes, such as the archetypal Reiner
Gamma, can span several tens of kilometers, they are invariably ensembles of smaller, elongate structures,with the transition from bright to dark taking place over distances of typically 1–5 km (Figure 1). We call thistransition length scale L, and we use the term interchangeably to describe both the optical bright-to-dark
transition and the magnetic ﬁeld’s vertical-to-horizontal transition (Figure 2).
Given the elongate nature of the ﬁnest structures in swirl complexes, we model the underlying sources as elongate structures. Our model sources are buried rectangular prisms that, for the sake of simplicity, are inﬁnitely
long. This simpliﬁes the expressions, reducing the problem to two dimensions, and is a reasonable approximation when examining areas far from the ends of the source. We focus on a cross section through the surface
and the underlying source body, which we take to have height h, width w, and whose top is at a depth d
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beneath the surface (Figure 3). In principle, we can model source bodies with arbitrarily more complicated cross sections by superimposing a series of such prisms. Likewise, it is possible to model collections of separate but nearby sources, which may produce complicatedsurface ﬁelds. For clarity and simplicity, however, we will focus for
now on the case of a single rectangular prism. If the source is not
unidirectionally magnetized, the ﬁeld structure may be highly complex near the source but, due to partial mutual cancelation of distinctly oriented adjacent magnetic moments, will reduce to weaker andsmoother ﬁelds with increasing distance from the source. From the perspective of constraining source rock magnetization, the assumptionof unidirectional magnetization is thus conservative in that it demands theleast in terms of source rock magnetization. Although any magnetization
direction is possible, we will focus on the two end member cases of horizontal (parallel to the surface) and vertical (perpendicular to the surface)magnetization. We also restrict the magnetization direction to be perpendicular to the long axis of the source. That is, the magnetization direction
is assumed to lie within the plane of Figure 3. This is broadly consistent
with previous inversions for isolated, swirl-bearing magnetic anomalies
such as Airy and Reiner Gamma (e.g., Hemingway & Garrick-Bethell, 2012;
Kurata et al., 2005; Oliveira & Wieczorek, 2017). Moreover, in the limit of
an inﬁnitely long source, magnetization components parallel to the longaxis (i.e., out of the plane of Figure 3) do not contribute to the external
magnetic ﬁeld and hence have no eﬀect on the geometric constraints wederive below.
Under these assumptions, it can be shown (Appendix A) that the magnitude of the surface ﬁeld strength per unit magnetization, measured
directly above the source, is given by
(
(
)
(
))
𝜇0
d+h
d
B
=
arctan− arctan
(1)
M
𝜋
w∕2
w∕2

Figure 2. Illustration of the transition length scale, L. (a) Close-up of a lunar
swirl centered on its dark lane (this particular feature is located at 17.4∘ N,
87.6∘ E). (b) Reﬂectance proﬁle across the bright and dark parts of the swirl,
based on the pixels in the blue box illustrated in (a), with reﬂectancenormalized to the brightest part of the swirl. The transition length (L) fromthe brightest part of the bright lobe to the center of the adjacent dark lane
is indicated. (c) Vertical cross section illustrating notional magnetic ﬁeld
vectors, with the near-surface ﬁeld direction transitioning from vertical tohorizontal over a distance L. Image in (a) is from the Lunar Reconnaissance
Orbiter Camera Narrow Angle Camera (Humm et al., 2016; Robinson et al.,2010; Speyerer et al., 2012).

where the depth to width ratio (d∕w) is constrained by the transition
length scale (L) according to the geometric relationship√L = (d + h) d + (w∕2)2(2)(see Appendix A for details). Equation (1) is obtained by integrating the
dipole ﬁeld equation (A1) over the volume of the prism illustrated in
Figure 3. In the case of horizontal magnetization, we obtain the transition
length scale by solving (A4) for the position where the ﬁeld becomes vertical at the surface. Similarly, in the case of vertical magnetization, we obtainthe transition length scale by solving (A13) for the position where the ﬁeld
becomes horizontal at the surface. Owing to symmetries in the establishedproblem geometry, (1) and (2) hold regardless of whether the magnetization direction is horizontal or vertical. Rearrangement and substitution of(2) into (1) allows the problem to be fully described in terms of the surface ﬁeld strength per unit magnetization (B∕M) and the dimensionless quantities of the prism’s aspect ratio (h∕w) and its relative burial depth(d∕L).

2.2. Model Results
It follows from (2) that the source depth can never be greater than the transition length scale (d < L) andthat the width of the source can never exceed twice the transition length scale (w < 2L). Hence, the sourcesmust be both shallow and narrow in order to produce the transition length scales observed at lunar swirls(L ≈ 1 –5 km). Similarly, it follows that, in order to preserve the transition length scale at the surface (L), as
the depth to the top of the source increases, the source must become increasingly compact (Figure 4a), withHEMINGWAY & TIKOO

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Figure 3. Model cross section of buried magnetized rectangular prism illustrating the relationship between its width(w), height (h), burial depth (d), and the length scale (L) over which the surface ﬁeld orientation transitions between
vertical and horizontal. The surface ﬁeld strength directly above the source (B) and the source magnetization (M) arealso indicated. Field vectors are shown over a regular grid with the surface ﬁeld vectors exaggerated for emphasis. The
two panels illustrate the cases of horizontal (a) and vertical (b) magnetization.

the precise relationship depending in part on the prism’s aspect ratio. The implied rock magnetization thusincreases with increasing burial depth and, equivalently, with decreasing source width (Figure 4b).
Considering a range of possible aspect ratios and relative burial depths, we can compute from (1) the magnitude of the surface ﬁeld per unit magnetization in the source body (Figure 5a). Conversely, we can solve (1)for the source magnetization required to produce a given surface ﬁeld (Figure 5b) across the same parameter space. The rock magnetization required to account for the observed surface ﬁelds increases with relative
burial depth and becomes impossibly large as the burial depth approaches the transition length scale. But
even in the limit of an inﬁnitely tall prism, with zero burial depth, the required magnetization is still
(A/m)
M
2
=
≈ 0.159B
𝜇0(100nT)(3)

Given that the surface ﬁeld estimates of Tsunakawa et al. (2015) provide only a lower bound and given thetheoretical arguments given by Halekas et al. (2010) and Hemingway and Garrick-Bethell (2012), we will take
300 nT to be a conservative estimate of the surface ﬁelds present where swirls are found. Given this assumption, equation (3) and Figure 5 demonstrate that rock magnetization at swirl-forming magnetic anomalies

Figure 4. Relationship between relative burial depth (d∕L), relative source width (w∕L), and relative requiredmagnetization (M∕B) for various aspect ratios (h∕w). (a) The maximum burial depth (d) is equal to the transition lengthscale (L) and requires an inﬁnitely compact source (w → 0). The maximum source width (w) is twice the transition lengthscale (L) and requires the top of the source to be at the surface (d = 0). (b) In the shallow source limit of d = 0, therequired magnetization is determined by equation (1), is entirely a function of aspect ratio, and reaches its minimum
value of ≈ 0.159 (A/m)/(100 nT) for an inﬁnitely tall source (h∕w → ∞). As relative burial depth (d∕L) increases, therelative required magnetization (M∕B) increases rapidly, becoming inﬁnite as the burial depth approaches its maximum
value (d∕L → 1) and the source becomes inﬁnitely compact (w → 0).

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Figure 5. Surface ﬁeld strength and required magnetization as a function of source geometry. (a) Surface ﬁeld strength
drops rapidly as relative burial depth (d∕L) increases and as the prism’s aspect ratio (h∕w) becomes very small.
(b) Required source magnetization per 100-nT surface ﬁeld. The required magnetization increases rapidly as relative
burial depth (d∕L) increases and as the prism’s aspect ratio (h∕w) becomes very small.

must be at least ∼ 0.5 A/m, even for zero depth sources, and ∼ 2 A/m for sources that are buried by a depthof more than half the transition length scale (i.e., buried more than 0.5–2.5 km beneath the surface). For surface ﬁelds of > 1, 000 nT, the corresponding values are more than 1.5 and 6 A/m, respectively. If the source ismore deeply buried (i.e., larger d∕L), or has a larger aspect ratio (h∕w), as expected for dikes, the implied rockmagnetization could easily exceed several amperes per meter (Figure 5b).
2.3. Implications for Swirl Distribution
A long-standing question has been: why is it that swirls are found at many, but not all, of the localized crustal
magnetic anomalies? Our analysis suggests that part of the answer may have to do with the burial depth ofthe magnetized source bodies. It is intuitive, and Figure 5 makes it clear that, for a given rock magnetization,
the surface ﬁeld strength drops oﬀ rapidly with burial depth. Hence, if the formation of swirls requires verystrong surface ﬁelds, then deep sources will be unable to produce swirls.
What is perhaps less obvious is that the geometric relationship expressed in equation (2) also implies that any
magnetic source capable of producing swirls, with their characteristic bright-to-dark transition taking placeover no more than a few kilometers, must be both narrow (less than a few kilometers wide) and shallow (withina few kilometers of the surface). Deeper sources produce surface ﬁelds that are not only weaker, for a given
source magnetization, but also whose orientations vary only gradually across the surface. Without the strong
and spatially rapid changes in surface ﬁeld direction that arise from narrow, shallow sources, the formation ofswirls may be impossible.
Indeed, a comparison between the locations of swirls, as mapped by Denevi et al. (2016), and an estimate ofthe depth to the top of all the crustal magnetic sources globally (Wieczorek, 2018) conﬁrms that swirls tend
to be found only in places where the magnetic sources are closest to the surface (Figure 6a). The Crisium
basin, for example, features two strong magnetic anomalies, but, because the depth to the top of the source
is more than ∼ 15 km, the surface ﬁelds must be relatively weak and their orientation must vary only gradually across the surface, precluding the formation of swirls. More generally, whereas the global distribution ofsource depths is broad and centered on a depth of ∼ 10 km, swirl regions are strongly biased toward shallow
sources (Figure 6b). Not surprisingly, swirls also tend to be found in regions where the magnetic ﬁeld strength
is relatively high (Figure 6c).
2.4. Implications for Origin Scenarios
There are several forms of remanence that could account for the natural remanent magnetization (NRM) foundin parts of the lunar crust, including TRM, whereby the initially hot source materials cool through the ferromagnetic grains’ Curie temperature in the presence of an ambient magnetic ﬁeld; thermochemical remanent
magnetization (TCRM), where new magnetic carriers are formed or existing carriers are altered due to exposure to high temperatures; shock remanent magnetization (SRM), involving remagnetization of grains due
to transient high pressures accompanying impact shock waves; and/or isothermal remanent magnetization
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Figure 6. Lunar swirls and magnetic source depths. (a) Map showing the depth to the top of the magnetic source material as estimated by Wieczorek (2018), withblack overlay indicating the locations of lunar swirls as mapped by Denevi et al. (2016). Gray contours are derived from LOLA topography and are included for
geographic context. (b) Area-weighted histograms of source depth showing that swirls (black) tend to correspond to regions with source depths that are shallow
compared to the overall distribution (blue). (c) Area-weighted histograms of ﬁeld strength showing that, whereas ﬁeld strength is low over the vast majority ofthe surface (blue), swirls (black) tend to be found where ﬁeld strength is relatively high. Recall that the ﬁeld strength estimates of Tsunakawa et al. (2015) may bean order of magnitude too small at strong crustal magnetic anomalies and that this eﬀect is compounded by the map’s limited resolution of 5 pixels per degree.

(IRM), which involves exposure to transient magnetic ﬁelds with intensities far in excess of any likely long-lived
ambient ﬁelds. In order to understand the origins of the magnetic anomalies accompanying swirls in particular, the challenge is to identify processes that can account for not only the inferred intensity of magnetization
but also the geometric constraints.
Depending on lithology, endogenic lunar rocks in the sample record typically have saturation remanent magnetizations (Mrs ) ranging between 0.05 A/m (feldspathic highland rocks) and 30 A/m (maﬁc impact melt
breccias; Wieczorek et al., 2012). Assuming a TRM acquired during primary cooling, we can estimate theexpected NRM (Mnr ) intensity according to
BMrs
(4)
a
where B is the paleoﬁeld intensity and a is a calibration constant with a value of ∼ 3000 μT (though this value
is uncertain by a factor of 2–3; Gattacceca & Rochette, 2004; Weiss & Tikoo, 2014). Paleomagnetic studies
suggest that surface ﬁelds on the Moon were likely <∼ 100 μT in intensity for nearly all of lunar history (Weiss& Tikoo, 2014). This implies that, assuming the rocks carry a TRM, the NRMs of known lunar materials and
Mnr =HEMINGWAY & TIKOO

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melt breccias are unlikely to exceed ∼ 1 A/m, even when utilizing upper limit values for Mrs (Figure 7). For
comparison, the NRMs of most lunar breccia samples are ∼ 0.1 A/m and the NRMs of most mare basalt samples
range between ∼ 0.001 and ∼ 0.01 A/m (Weiss & Tikoo, 2014). However, given that the Apollo missions only
sampled rocks at the lunar surface, and only in a limited number of locations, it is possible that unsampledendogenic lunar materials have higher NRMs and saturation remanent magnetizations.
Impacts can produce strong, transient plasma ﬁelds that could have magnetized target rocks via either TRM
or SRM (Figure 8). As discussed above, however, endogenic lunar materials may not have suﬃciently high ironcontents to produce TRMs with the inferred intensity of 0.5–2 A/m. SRM may be even less likely given thatthe intensity may be more than an order of magnitude weaker than that of TRM produced in an equivalentﬁeld (Tikoo et al., 2015). On the other hand, basin-forming impact events could lead to extremely intensetransient ﬁelds (perhaps ∼ 1 mT) near the basin antipode where the most distal ejecta is converging (e.g.,
Hood & Artemieva, 2008; Srnka, 1977), leading to acquisition of a strong SRM within the ejecta deposits. In
general, however, impact-generated ﬁelds may be complex as well as time varying, making it diﬃcult for themto impart a coherent remanent magnetization.
Large impactors could also have delivered unusually metal-rich exogenic materials, with the hot ejecta acquiring a TRM that could be much more intense than any produced from endogenic materials (Wieczorek et al.,2012). While such impact ejecta or impact melt scenarios could be consistent with the inferred shallowness ofthe sources, however, the broad, quasi-circular shapes of impact basins or melt sheets are not consistent with
the narrow, elongate source structures we infer for swirls (Figure 8). Ejecta could explain elongate sources,
although one might expect rays to be locally parallel and unlike the morphologically complex structures
found at swirls (Figure 1).
It has also been suggested that compression of a cometary coma colliding with the Moon could produce transient magnetic ﬁelds as intense as 1–100 mT (Schultz & Srnka, 1980; Syal & Schultz, 2015). For ferromagnetic
grains within lunar rocks, magnetic coercivity is typically tens of milliteslas (see Table 2 by Rochette et al.,2010), meaning that it may be possible for such strong transient ﬁelds to magnetize mare basalts to several
amperes per meter. However, it is unclear whether these ﬁelds could magnetize a suﬃcient volume of rock to
generate a substantial magnetic moment or whether these ﬁelds can have the complex geometries required
to make the narrow, elongate structures inferred for swirl-bearing magnetic anomalies. On the other hand,under the cometary impact hypothesis for swirl formation, the source geometry need not closely match swirl
morphology (a key assumption in our analysis) since this model proposes that swirls are the result of scouring
during the impact event rather than being the result of space weathering or magnetic or electrostatic sorting.
Another possibility is that the sources formed due to magmatic processes, either intrusive, like dikes or lava
tubes, or extrusive, such as mare basalt ﬂows. Like impact basins and melt sheets, surface lava ﬂows satisfy therequirement of being shallow sources but are too broad to be consistent with the ﬁnely structured surface
ﬁelds necessary to give rise to swirls (Figure 8). Dike complexes, on the other hand, typically produce linear
structures that trend roughly parallel to one another, and can sometimes exhibit curved geometries, as in
the case of ladder dikes (e.g., Hodge et al., 2012). It is possible that undiscovered dikes could exist within theupper portions of the lunar crust, especially given that they may not be wide enough to produce density
anomalies strong enough to be observable in gravity observations. Subsurface lava tubes, which are likely
present within mare plains, should have the shallow depths and narrow, elongate structures expected forthe sources underlying lunar swirls. The parent magmas of such dikes and lava tubes would likely be similar
in composition to those of mare basalts. The main challenge associated with such source bodies is that, inorder to produce the observed strong remanent crustal ﬁelds, they would need to contain far more metallic
iron than what has so far been reported for mare basalts. In section 3, we explore scenarios in which theemplacement of dikes or lava tubes can lead to enhancement of magnetic carriers (raising the mare basalt
NRM values to beyond those in Figure 7) through reduction reactions and thermochemical alteration.

3. Magnetic Enhancement of Lunar Rocks
In section 2.4, we argued that the morphology of swirls is suggestive of magnetic source structures resemblingdikes or lava tubes but that the required magnetizations are at least an order of magnitude larger than typicalmare basalts in the sample record. Localized magmatism could produce such strong magnetizations within
the lunar crust in multiple ways. First, maﬁc dikes or lava tubes might be sourced from unusually iron-rich
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magmas, thereby producing more metal-rich rocks. Given that the Apollo
missions sampled only a limited range of rock types and locations, it is conceivable that more metal-rich endogenic lunar rocks exist in other places
that have not yet been explored. Alternatively, the emplacement of dikes
and lava tubes may reset the magnetization within immediately adjacentrocks via a baked contact. In this scenario, host rocks next to a baked contact are exposed to very high temperatures that may be suﬃcient to impartthem with full or partial TRM (pTRM) as they cool below their Curie temperatures (Everitt & Clegg, 1962). Heating may also alter the remanent
magnetization within rocks via the modiﬁcation of preexisting ferromagnetic grains or the creation of new ones (e.g., Buchan, 2007; Coe et al.,2014). In the terrestrial environment, emplacement of magmatic products
may alter primary titanomagnetite within nearby host rocks, leading to theformation of more oxidized phases such as titanomaghemite. Such subsolidus alteration has been observed in mid-ocean ridge basalts (e.g., Shauet al., 2004). On Earth, water-rock interactions may also lead to hydrotherFigure 7. Natural Remanent Magnetization, assuming thermoremanent
mal alteration of basaltic materials within the seaﬂoor. This process may
magnetization, as a function of paleoﬁeld and rock type (characterized heretransform titanomagnetite into more Ti-poor magnetite and ilmenite (e.g.,
by saturation remanent magnetization, Mrs ).
Shau et al., 2000). Alteration of mid-ocean ridge basalt occurs almost
immediately after emplacement (Kent & Gee, 1996), and the resulting
TCRM carried by more oxidized phases typically has a weaker magnetization intensity than the original TRM
acquired during primary cooling (Xu et al., 1997). In the case of the Moon, heating may produce subsolidus
reduction of primary nonferromagnetic minerals such as ilmenite that would lead to the formation of new
metallic iron. This process would impart new or altered ferromagnetic grains with TCRM, likely enhancing theoriginal magnetization within the rock.
3.1. Formation of Ferromagnetic Grains in Lunar Rocks
Following Oliveira et al. (2017), and assuming TRM, we determined that the > 0.5-A/m magnetization intensities we infer for lunar swirl magnetic sources require rocks to have abundances of >0.3 wt.% Fe and >31 wt.%
Fe, assuming magnetizing ﬁeld strengths of 100 and 1 μT, respectively. These Fe abundances are subject toa cumulative factor of ∼4 uncertainty that is related to (factor of ∼2) variations in the magnetic hysteresis
properties of mare basalts and the (factor of ∼2) uncertainty in the relationship between magnetizing ﬁeld
intensities and Mrs values of Fe-bearing rocks (see Oliveira et al., 2017, section 2.2). For our calculations, weFigure 8. Possible origins of the crustal magnetic anomalies underlying lunar swirls. Listed are several possible source
origin scenarios and magnetization processes (TRM is thermoremanent magnetization, TCRM is thermochemical
remanent magnetization, SRM is shock remanent magnetization, and IRM is isothermal remanent magnetization) alongwith the ability of each to satisfy the inferred geometric and intensity constraints (check marks indicate whereconstraints are satisﬁed easily; cross marks indicate where constraints are unlikely to be satisﬁed; question marks indicate
where constraints may be met under the right circumstances or where there is no clear conclusion). Image credits: NASA.

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used a hysteresis loop squareness of 0.0064 and we assumed that the source rocks had mare basalt-like densities of 3,300 kg/m3 . In the high ﬁeld (100 μT) regime, the required Fe contents are similar to those inferredby Oliveira et al. (2017) for basin impact melt sheets, as well as measured metal contents in melt-bearing lunar
impact breccias (Gose & Butler, 1975). While mare basalts typically have lower metal contents of ∼ 0.08 wt.%
Fe (Gose & Butler, 1975), pervasive reduction of magmatic bodies of similar initial composition may well becapable of generating suﬃcient Fe to produce magnetizations of >0.5 A/m (see below).
Experimental petrology studies indicate that mare basalts generally formed in a reducing environment withoxygen fugacities (f O2 ) approximately one log unit below the iron-wüstite buﬀer (i.e., IW-1; Sato et al.,1973). The reduction of mare materials was likely achieved when their parent magmas ascended to depths<∼ 8.5 km (Nicholis & Rutherford, 2009). At these shallow depths (and accompanying low pressures), carbon
becomes oxidized to carbon monoxide. The liberation of carbon monoxide causes subsolidus reduction ofoxides and silicates and the associated creation of the metallic ferromagnetic minerals observed in lunar rocks(Sato et al., 1973). A number of reduction reactions that are capable of forming metallic iron from diﬀerent
primary mineral phases may take place during deuteric cooling of initially molten material. Some example
reaction pathways for diﬀerent minerals (from Haggerty, 1971) include the following:
1
O
2 2
1
Chromite ∶ Fe2 CrO4 = Cr2 O3 + Fe + O2
2

Ilmenite ∶ FeTiO3 = TiO2 + Fe +

(5)
(6)

The amount of metallic Fe produced by subsolidus reduction during primary cooling will depend on factors such as the initial abundances of precursor minerals (e.g., ilmenite), as well as the cooling rate. Intrusive
bodies have longer cooling timescales than mare ﬂows, so it is likely that they would experience more subsolidus reduction than basalts at the lunar surface. Mare basalts on the Moon are often classiﬁed into two
categories based on Ti content: low Ti and high Ti. The low-Ti basalts tend to have modal mineralogies with<∼ 8 vol.% ilmenite (Dowty et al., 1973; Neal et al., 1994), whereas high-Ti mare basalts have been found withup to ∼ 22 vol.% ilmenite (e.g., Longhi et al., 1974). Low-Ti mare basalts are also less dense (𝜌 ≈ 3, 270 kg/m3 )than high-Ti mare basalts (𝜌 ≈ 3, 460 kg/m3 ; Kiefer et al., 2012). In the extreme scenario wherein all ilmenite ina rock with high-Ti mare basalt composition (e.g., 22 vol.% ilmenite) is reduced and converted to Fe and TiO2 ,the rock could be enriched with an additional ∼ 11 wt.% Fe. Such a concentration of Fe would permit rockmagnetizations of ∼ 17.5 A/m if magnetized in the high ﬁeld (100 μT) regime (Oliveira et al., 2017). Even in arock with low-Ti basalt composition (e.g., 2 vol.% ilmenite), full breakdown of ilmenite would result in ∼ 1 wt.%
neoformed Fe (corresponding to M ≈ 1.6 A/m, assuming the high ﬁeld regime). These Fe contents are at least
an order of magnitude higher than those reported for Apollo mare basalt samples (typically ∼ 0.08 wt.%)(Gose & Butler, 1975). While it is unlikely that all ilmenite would actually experience such a pervasive breakdown in a natural setting, the preceding calculations nevertheless demonstrate the potential for subsolidus
reduction to create substantial amounts of metallic Fe, especially in slowly cooled magmatic bodies.
Laboratory heating of mare basalts and synthetic analogs in a controlled oxygen fugacity atmosphere to temperatures up to 800 ∘ C has produced subsolidus reduction and the associated growth of metal grains in several
paleomagnetism studies. Pearce et al. (1976) conducted a Thellier-Thellier paleointensity experiment (Thellier,1959) on a synthetic sample intended to approximate mare basalt 10017. During the experiment, the sample was heated in a ∼20% H2 and ∼80% N2 mixture that the authors measured (using a yttria-doped zirconia
sensor) to have an f O2 of approximately IW-1 at 800 ∘ C. As part of the paleointensity experiment, the sample
was heated to increasing temperatures and imparted with a pTRM at each temperature step. At low temperatures, the amount of pTRM gained at each heating step increased steadily. However, between the 700 and
800 ∘ C heating steps, the authors observed a >1 order of magnitude increase in the intensity of pTRM gained(Figure 9). The majority of this increase was attributed to an experimentally determined factor of 5 increasein the amount of metallic iron in the sample. This increase in metal content was caused by rapid subsolidus
reduction at the rims of ilmenite grains. The neoformed iron would have acquired a TCRM as it cooled. Petrographic analyses of altered ilmenite grains revealed that heating the sample at 800 ∘ C for 30 min produced
approximately twice as much reduction along the rims of ilmenite grains as a 20-min heating experiment. This
indicates that the amount of reduction increases with the duration of heating, although in both cases reduction was limited to the outermost 1–2 μm of ilmenite grains. However, due to the lack of long-term heating
experiments on lunar rocks, it is unclear what the rate of transformation of ilmenite is or how long it takes for
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this alteration to reach its maximum extent. Pearce et al. (1976) also
reported that the reduction of troilite in their experiments produced amodest (<10%) increase in the abundance of metallic iron.
More recently, Tikoo et al. (2014) conducted several heating experiments
on ilmenite basalt 12022 using the controlled f O2 (IW-1) H2 -CO2 atmosphere oven system described by Suavet et al. (2014). This system has beenused successfully to retrieve paleointensities from mare basalt analogs(Suavet et al., 2014) as well as impact melt rock portions of a lunar regolith
breccia (Tikoo et al., 2017), with limited sample alteration as constrainedby pTRM checks (after Tauxe & Staudigel, 2004). The experiments by
Suavet et al. (2014) produced Fe from ilmenite, with the Fe grains ranging from submicron to a few microns in size—large compared to thesingle domain to pseudosingle domain size transition of ∼ 25 nm for
cubic Fe grains—suggesting that the Fe grains produced by the reduction mechanisms discussed here are probably multidomain grains. Someof the heating experiments on 12022 (not all of which were presented by
Tikoo et al., 2014, but are included here) involved imparting subsamplesof the rock with laboratory IRM acquired in a near-saturating 886.5-mT
ﬁeld before or after heating the subsamples for 20 min to temperaturesof 575–600 ∘ C. The mean intensities of the laboratory IRMs acquired after
heating were a factor of ∼3 higher than those acquired by unheatedsubsamples (Figure 10). These experiments suggest that heating-related
reduction of ilmenite and troilite led to the formation of additional metal
within subsamples of 12022. In summary, experimental evidence indicatesthat heating lunar rocks to temperatures of at least 575 ∘ C in a reducing
environment facilitates the creation of new ferromagnetic grains, providing a mechanism for enhancing rock magnetization within and nearintrusive bodies.
Figure 9. Magnetic enhancement through heating experiments on asynthetic lunar sample. Black squares represent the amount of partial
thermoremanent magnetization (pTRM) gained by the sample after heating
to diﬀerent temperatures. Gray circles represent the inferred metallic Fe
content in the sample following each heating step. Modiﬁed from Pearce
et al. (1976).

3.2. Heating by Dikes and Lava Tubes
Many of the Moon’s crustal magnetic anomalies, including the Reiner
Gamma anomaly, are located within, or in proximity to, mare plains withmaximum thicknesses of a few kilometers. These mare plains were likely
fed via intrusions and lava tubes that contained material that originatedfrom mantle sources and were emplaced shallowly. The dikes associatedwith mare volcanism are generally inferred to have lengths ranging from tens to hundreds of kilometers andwidths of tens of meters to ∼ 250 m (Head & Wilson, 1993; Wilson & Head, 1981). Much wider dikes (tensof kilometers wide) may also exist, such as those inferred to exist on the lunar nearside in the substructureof the Procellarum region (Andrews-Hanna et al., 2014). Collapsed lunar lava tubes have been observed tohave widths of up to ∼ 3 km, although the vast majority have widths <1.5 km (e.g., Hadley Rille is ∼ 1.2 km
wide; Howard et al., 1972). Dikes form relatively rapidly, having typical emplacement timescales of minutes todays (Montgomery-Brown et al., 2010; Spence & Turcotte, 1985). In contrast, lava tubes may serve as magma
conduits for months (Calvari & Pinkerton, 1998, 1999). The emplacement of these dikes and lava tubes could
account for the strong magnetization of swirl-bearing magnetic anomalies provided that either the bodies themselves acquire a strong magnetization or that suﬃcient volumes of the adjacent rocks are raised to
suﬃciently high temperatures.
Here we use a heat conduction model to determine the extent of the region that is heated to at least 600 ∘ C following the emplacement of a dike or ﬂowing lava tube. We assume a lunar surface temperature of Ts = −20 ∘ C
and apply a geothermal gradient of dT= 13.5 ∘ C/km (approximated by averaging the initial and currentdz
model mantle temperature proﬁles from Laneuville et al., 2013) to determine preintrusion temperatures ofhost rocks, given by

Thost (x, z) = Ts +HEMINGWAY & TIKOO

dT
z
dz

LUNAR MAGNETIC SOURCES CONSTRAINED BY SWIRLS(7)
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Figure 10. Magnetization intensities for Apollo 12022 subsamples that were imparted with 886.5-mT isothermal
remanent magnetization before and/or after heating to 575–600 ∘ C. Dashed lines illustrate mean values before and after
heating. The right scale shows the conversion to units of amperes per meter assuming a rock density of 3,300 kg/m3 .

where x is the horizontal position with respect to the center of the intrusion and z is the depth. Assuming
mainly horizontal heat ﬂow away from the intrusion, we compute the temperature ﬁeld at time t according
to (e.g., Delaney, 1987)
))
)
( (b
(b
+x
−x
(
)1
2
2
+ erf √T (x, z, t) = Thost (x, z) + T0 − Thost (x, z)
erf √
(8)
2
4𝜅t
4𝜅t
where 𝜅 is the thermal diﬀusivity, which we take to be 10−6 m2 /s for the intrusion and the host rock alike, b
is the width of the hot magmatic body, and T0 is its initial temperature. As end member scenarios we assume
initial temperatures of either 1200 or 1500 ∘ C. These temperatures lie within the 1100–1500 ∘ C range of liquidus temperatures determined for lunar mare basalts (Wieczorek et al., 2001). High temperatures (∼1500∘ C)
may also be achieved by emplacement of ultramaﬁc komatiites, which may have formed on the Moon in rare
circumstances (Green, 1972; Ringwood et al., 1987).
Intrusions are treated here as instantaneous heat pulses, with heat then diﬀusing into the surrounding hostrock as the body cools. For various depths and widths, we determine peak temperatures experienced in andaround the magmatic body following its initial emplacement by evaluating (8) at a series of times in steps ofat most 100 years (peak temperatures are reached within a few thousand years). Peak rock temperature is astrong function of the initial temperature and width of the intrusive body (i.e., the amount of thermal energy
deposited) as well as distance away from the body (Figure 11a). The width of the heated region is also a weak
function of depth (Figure 11b). With an initial temperature of 1200 ∘ C, host rock temperatures do not exceed
600 ∘ C except within a few meters of the intrusion. With initial temperatures of 1500 ∘ C, however, the region
heated to at least 600 ∘ C extends some 20% beyond the body’s own width.
Small-scale swirls (e.g., L = 1 km or less) are readily accounted for by shallow dikes or lava tubes, provided
those dikes or lava tubes are not too wide (see Figure 4a). Conversely, for the largest-scale swirls, the corresponding magmatic bodies must not be too narrow either. This is because very narrow sources can give riseto large transition length scales only if they are deeply buried (Figure 4a), which in turn requires strong rockmagnetization (Figure 4b). For example, with L = 5 km, a 100-m-wide dike with an aspect ratio h∕w = 10, hasw∕L = 0.02 and d∕L ≈ 0.9 and requires the rock magnetization to exceed 300 A/m assuming a surface ﬁeldof 300 nT. On the other hand, the same result is achieved with < 1 A/m given a 1500 ∘ C, 2,000-m-wide dike,which heats a region ∼ 2.5 km wide, yielding w∕L = 0.5 and d∕L ≈ 0.18. Lava tubes could also account forsome of the larger-scale swirl features if they are suﬃciently large and/or deeply buried. For example, a 3-km
diameter lava tube that is initially ﬂowing with T0 = 1500 ∘ C heats a region roughly 3.5 km × 3.5 km. If the topof this region is 3 km beneath the surface, then the result (neglecting the cross-sectional diﬀerences between
lava tubes and the prism represented in Figure 3) is L ≈ 4.8 km (w∕L ≈ 0.74, d∕L ≈ 0.63). To produce a 300-nT
ﬁeld at the surface, the corresponding rock magnetization would have to be ∼ 2.8 A/m.
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Figure 11. (a) Peak temperatures reached in the vicinity of a cooling magmatic body as a function of the body’s width,
shown here for the example case of an initial temperature of 1500 ∘ C and measured at a depth of 3 km. The dashed line
marks the region heated to at least 600 ∘ C. (b) Width of the heated region (the intrusion plus the part of the surrounding
host rock that is heated to at least 600 ∘ C) as a function of depth and shown for a series of diﬀerent intrusion widths(from 100 to 2,000 m) and shown for two diﬀerent initial temperatures: 1200 ∘ C (solid lines) and 1500 ∘ C (dashed lines).

The preceding analysis assumed an instantaneous heat pulse into an otherwise cold host rock, which may be
a good assumption for dikes that are isolated in space and time. Although lava tubes may remain active formonths, an instantaneous heat pulse may nevertheless be an appropriate assumption in the case of deeply
buried lava tubes, for which the thermal diﬀusion timescale could be relatively long (e.g., for a depth of 1 km,
the diﬀusion timescale is > 104 years). On the other hand, the assumption of an initially cold host rock isconservative in the sense that it neglects the possibility of residual heat from previous episodes of intrusive
activity. That is, the volume of rock heated to at least 600 ∘ C could be considerably larger in regions wherepreviously used magmatic pathways are periodically reactivated.
As discussed in section 3.1, temperatures of 600 ∘ C are suﬃcient to thermochemically alter host rocks
and form new ferromagnetic grains. Because temperatures of several hundred degrees Celsius may persist
for > 103 years (far longer than laboratory heating timescales), there is ample opportunity to pervasively
reduce minerals such as ilmenite and troilite in mare basalts, as well as thermally remagnetize host rocks(Garrick-Bethell & Weiss, 2010). If such thermochemical alteration and cooling occur in the presence of a stable dynamo ﬁeld, preexisting and neoformed ferromagnetic grains may acquire a mixture of TRM and TCRM
that could be orders of magnitude more intense than the primary magnetizations of unaltered rocks. Sucha process would result in a substantial magnetization contrast between altered rocks and their unalteredsurroundings, producing a magnetic anomaly.
If dikes and lava tubes are indeed the origin of these swirl bearing magnetic anomalies, the natural limits on
their depths and widths may eﬀectively limit the scale of swirls, potentially explaining why their bright-to-dark
transition length scales rarely exceed ∼ 5 km. Given that most known lunar dikes have widths <250 m andgiven that lava tube depths are not likely to exceed ∼ 2 km, the conditions necessary for making the largest
swirl-bearing magnetic anomalies may be present only in rare circumstances. This may also help account for
why it is that lunar swirls and strong crustal magnetic anomalies are not found everywhere across the mareplains, in spite of the fact that lunar magmatism and the dynamo persisted for over two billion years after
the formation of the Moon. Similarly, modest variations in oxygen fugacity between magma sources (stemming from lunar mantle heterogeneity) may cause rocks forming in some locations to be more reduced andmetal rich than rocks forming in other locations. Finally, the intensity of magnetization in a host rock depends
strongly on the intensity of the ambient magnetic ﬁeld in which its ferromagnetic grains were magnetized.
Rocks magnetized during the lunar high ﬁeld period (between at least 3.85 and 3.56 Ga) are more likely tocarry strong magnetizations than older rocks that formed before the high ﬁeld period or younger rocks thatformed after the lunar dynamo declined.

4. Conclusions
Following from several magnetometer-based studies of magnetic ﬁeld structure (Hemingway &Garrick-Bethell, 2012; Shibuya et al., 2010; Tsunakawa et al., 2015), together with a series of hybrid and
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kinetic plasma simulations (e.g., Bamford et al., 2012, 2016; Deca et al., 2016; Poppe et al., 2012, 2016;
Zimmerman et al., 2015), we use lunar swirl morphology as a proxy for the structure of near-surface magnetic
ﬁelds. This allows us to place strong constraints on the geometry of the underlying magnetic source bodies.
Incorporating conservative estimates of near-surface ﬁeld strength (Tsunakawa et al., 2015), we additionallyplace constraints on the intensity of magnetization in these source bodies (Figure 5). We conclude that, atleast where lunar swirls are present, the underlying magnetic source bodies must be narrow, elongate structures, whose upper surfaces must be shallow (less than 0.5–2.5 km below the surface), with magnetization
intensities of at least 0.5 A/m and perhaps closer to ∼ 2 A/m or even more.
Although their ﬁne scale and complexity has made it diﬃcult to characterize swirl-bearing magnetic anomalies comprehensively, magnetometer-based inversions of isolated anomalies, such as Airy and Reiner Gamma,
support our suggestion of elongate sources (with widths less than the spacecraft altitude), even when the
optical features are ignored (e.g., Hemingway & Garrick-Bethell, 2012; Oliveira & Wieczorek, 2017). The requirement of narrow, shallow sources may account for why it is that not all crustal magnetic anomalies areaccompanied by swirls. If they are not suﬃciently shallow and narrow, even strong crustal magnetic anomalies (e.g., the Mare Crisium anomalies) will not produce the very strong near surface ﬁelds and spatially rapidly
varying ﬁeld orientations required to produce swirls (section 2.3 and Figure 6).
Our result also sheds light on the possible origins of the magnetic anomalies accompanying lunar swirls(section 2.4 and Figure 8). In particular, we argue that magnetization of mare basalt ﬂows, impact melt sheets,or rocks that experienced cometary collisions, is unlikely to produce the narrow, elongate magnetic source
bodies required to account for the near-surface magnetic ﬁeld structure that gives rise to swirls. In termsof magnetization intensity, impact ejecta deposits, especially if they are enriched in metals, may be goodcandidates for the sources underlying swirls, although it is not obvious how well they can account for swirl
morphology. Dikes and lava tubes, on the other hand, readily account for the narrow and often sinuous morphology of lunar swirl complexes. The natural limits on their depths and widths may also account for why thebright-to-dark transition length scales of swirls rarely exceed ∼ 5 km.
The main challenge with the dike and lava tube hypotheses is that they require a process that could enhance
the metal content to substantially beyond what is currently represented in the sample record. In section 3, we
showed that subsolidus reduction of minerals such as ilmenite and chromite during deuteric cooling may leadto elevated metal contents and, in turn, strong remanent magnetization within intruding magmatic bodies.
We also showed that the emplacement of dikes or lava tubes could heat the surrounding rocks enough to
impart them with additional TRM and/or TCRM that could be an order of magnitude more intense than the
primary remanence found in lunar basalts. The largest swirls, however, require large and/or high-temperature(∼1500 ∘ C) magmatic bodies, which may be rare. Further studies regarding the range of conditions that could
allow for the formation of strong magnetic carriers could be an important part of accounting for the presence
and distribution of lunar crustal magnetic anomalies.

Appendix A: Source Geometry and Surface Fields]T[
The vector magnetic ﬁeld arising from a magnetic dipole moment, m = mx my mz , is given by (e.g.,
Blakely, 1995)) 1
𝜇 (B (r) = 0 3 (m ⋅ r) r − mr2 5
(A1)
4𝜋
r
]T
[
where r = rx ry rz is the vector position of the evaluation point with respect to the location of themagnetic dipole.

We will use a coordinate system whose x -y plane coincides with the surface and whose z axis points upward.
Consider a buried magnetized rectangular prism that is inﬁnitely long in the y direction and symmetric about
the y-z plane. If the prism has width w (in the x direction), height h (in the z direction), and is buried a depth
d below the surface (Figure 3), then the magnetic ﬁeld can be obtained by integrating equation (A1) over theprism’s volume. We will focus on the two end member cases of horizontal (parallel to the x axis) and vertical(parallel to the z axis) magnetization, always in the plane of Figure 3. Since we assume that the source has
inﬁnite extent parallel to the y axis, magnetization components parallel to the y axis do not contribute to theexternal magnetic ﬁeld.
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A1. Horizontal Magnetization
Supposing that the magnetization (magnetic moment per unit volume) is purely in the +x direction, that is,[]T
M = M 0 0 , where M is the magnitude of the magnetization (A/m). In this case, the contribution fromeach inﬁnitesimal volume element in the source is

B (r) =

⎡ rx ⎤ ()−5∕2 ⎡ M ⎤ ()−3∕2 ⎞𝜇0 ⎛⎜⎟ drx dry drz
3Mrx ⎢ ry ⎥ rx2 + ry2 + rz2− ⎢ 0 ⎥ rx2 + ry2 + rz2
⎢ ⎥
⎢ ⎥
⎟
4𝜋 ⎜
⎣0⎦
⎣ rz ⎦⎝⎠(A2)

Considering the x component of the vector ﬁeld at a position (x, z), we haveBx (x, z) = M

𝜇0
4𝜋 ∫ ∫ ∫(()−5∕2 ()−3∕2 )
− rx2 + ry2 + rz2
drx dry drz
3rx2 rx2 + ry2 + rz2(A3)

where if the volume element is located at x ′ , y′ , and z′ , then rx = x − x ′ , ry = y − y′ , and rz = z − z′ , so thatdrx = −dx ′ , dry = −dy′ , and drz = −dz′ . The integration in y′ is from −∞ to ∞, the integration in x ′ is from one
face of the prism (x ′ = −w∕2) to the other (x ′ = w∕2), and the integration in z′ is from the top of the prism(z′ = −d) to the bottom (z′ = − (d + h)). After integrating, we obtain
Bx (x, z) = M

𝜇0
2𝜋

(
(
)
(
)
z+d
z+d+h
arctan− arctan···x + w∕2
x + w∕2(− arctanz+dx − w∕2)(+ arctanz+d+hx − w∕2

)

(

(A4)

))

At the origin, this simpliﬁes toBx (0, 0) = M

𝜇0
𝜋

(

(
arctandw∕2− arctan

d+h
w∕2

))

(A5)

By symmetry, the vector ﬁeld has no y component (By (x, z) = 0).
Considering the z component of the vector ﬁeld at a position (x, z), we haveBz (x, z) = M

𝜇0
4𝜋 ∫ ∫ ∫(()−5∕2 )
drx dry drz
3rx rz rx2 + ry2 + rz2(A6)

where the integration limits are as above. After integrating, we obtain
Bz (x, z) = M

𝜇0
ln
4𝜋

[(

(x + w∕2)2 + (z + d + h)2)((x + w∕2)2 + (z + d)2(x − w∕2)2 + (z + d)2)](x − w∕2)2 + (z + d + h)2(A7)

At the origin, Bz (0, 0) = 0, meaning the ﬁeld is purely horizontal. Using (A7), it can be shown that Bz = 0
whenever x = 0 as well as whenever z = − (d + h∕2). Using (A4), it can be shown that the ﬁeld is vertical, thatis, Bx = 0, whenever(A8)(x + w∕2) (x − w∕2) = (z + d) (z + d + h)
A2. Vertical Magnetization[]T
Supposing that the magnetization is instead in the +z direction, that is, M = 0 0 M , where M is themagnitude of the magnetization (A/m). In this case, we haveB (r) =

⎡ rx ⎤ ()−5∕2 ⎡ 0 ⎤ ()−3∕2 ⎞
𝜇0 ⎛⎜⎟ dr dr dr
3Mrz ⎢ ry ⎥ rx2 + ry2 + rz2− ⎢ 0 ⎥ rx2 + ry2 + rz2⎢ ⎥⎢ ⎥⎟ x y z4𝜋 ⎜⎣M⎦⎣ rz ⎦⎝⎠(A9)

The x component of the ﬁeld is given by
Bx (x, z) = M

HEMINGWAY & TIKOO

𝜇0
4𝜋 ∫ ∫ ∫(()−5∕2 )
drx dry drz
3rz rx rx2 + ry2 + rz2

LUNAR MAGNETIC SOURCES CONSTRAINED BY SWIRLS

(A10)

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10.1029/2018JE005604

After integrating over the volume of the source body, we obtain
Bx (x, z) = M

𝜇0
ln
4𝜋

[(

(x + w∕2)2 + (z + d + h)2)((x + w∕2)2 + (z + d)2(x − w∕2)2 + (z + d)2)](x − w∕2)2 + (z + d + h)2(A11)

which is identical to (A7).
The z component of the ﬁeld is given by

Bz (x, z) = M

𝜇0
4𝜋 ∫ ∫ ∫(()−5∕2 ()−3∕2 )
− rx2 + ry2 + rz2
drx dry drz
3rz2 rx2 + ry2 + rz2(A12)

After integrating over the volume of the source body, we obtain
Bz (x, z) = M

𝜇0
2𝜋

(

(
arctanx + w∕2z+d()− arctanx − w∕2z+d)(− arctanx + w∕2
z+d+h

(

))

+ arctanx − w∕2
z+d+h

)
···

(A13)

which diﬀers from (A4) only( )because the integration limits are diﬀerent in the x and z directions. However,using the identity arctan 𝛼1 = 𝜋2 − arctan (𝛼), this can also be written as
Bz (x, z) = −M

𝜇0
2𝜋

(

− arctan(arctanz+dx + w∕2()

z+dx − w∕2)(− arctanz+d+h
x + w∕2())+ arctanz+d+hx − w∕2)

which diﬀers from (A4) only in its sign. At the origin, this again simpliﬁes to
(
(
)
(
))
𝜇
d
d+h
− arctanBz (0, 0) = −M 0 arctan
𝜋
w∕2
w∕2

···

(A14)

(A15)

Using (A11), it can be shown that the ﬁeld is vertical, that is, Bx = 0, whenever x = 0 as well as wheneverz = − (d + h∕2). Using (A13), it can be shown that the ﬁeld is horizontal, that is, Bz = 0, whenever(x + w∕2) (x − w∕2) = (z + d) (z + d + h) ,(A16)which is identical to (A8).
A3. Transition Length Scale
When the magnetization is horizontal, the ﬁeld direction transitions from horizontal at (x, z) = (0, 0) to verticalat (x, z) = (L, 0), where L is obtained by setting (A4) to zero and solving. Similarly, when the magnetization isvertical, the ﬁeld direction transitions from vertical at (x, z) = (0, 0) to horizontal at (x, z) = (L, 0), where L is
obtained by setting (A13) to zero and solving. In either case, we obtain L by solving (A8) for x with z = 0. The
result is√
L = (d + h) d + (w∕2)2(A17)
regardless of whether the magnetization direction is horizontal or vertical.
It follows from (A17) that d < L for all positive h and w and that w < 2L for all d > 0. It can be shown that thewidth to depth ratio (w∕d) can be written in terms of the dimensionless quantities of prism aspect ratio (h∕w)
and relative burial depth (d∕L) as
√
w
=2
d

HEMINGWAY & TIKOO

(

h
w

)2

(
( )2 ) ( )
L
h
− 1−
−
d
w

LUNAR MAGNETIC SOURCES CONSTRAINED BY SWIRLS

(A18)

16

Journal of Geophysical Research: Planets

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Substitution of (A18) into (A5) helpfully allows all the important relationships to be captured in terms of
dimensionless quantities. The resulting expression for the magnitude of the magnetic ﬁeld at the origin,relative to the source magnetization, is
⎡
⎞
⎛
⎢
⎜
)⎟
(
𝜇 ⎢
⎟
⎜
(d∕L)
h
B
= 0 ⎢arctan ⎜ √
+
···
(
)
M
𝜋 ⎢
w∕2 ⎟⎟( )2( )( )⎜ ( d )2 ( h )2dhd
+ 1− L
− w
⎢
⎟
⎜
L
w
L
⎣
⎠
⎝
⎞⎤
⎛
⎟⎥
⎜
⎟⎥
⎜
(d∕L)
− arctan ⎜ √
⎟⎥
(
)
( )2
( ) ( ) ⎟⎥⎜ ( d )2 ( h )2dhd
+ 1− L
− w
⎟⎥
⎜
L
w
L ⎠⎦
⎝
which, again, holds regardless of whether the magnetization direction is horizontal or vertical.
Acknowledgments
This paper beneﬁtted from insightful
and constructive reviews by Catherine
Johnson and an anonymous reviewer.
We also thank Michael Manga, Ian
Garrick-Bethell, Tushar Mittal, Andrew
Poppe, Claude Herzberg, and Juliane
Gross for helpful comments andsuggestions. We thank Brett Denevi andMark Wieczorek for supplying their swirl
and source depth maps, respectively.
We thank Benjamin Weiss for access torock magnetic data from Apollo sample
12022 that was previously acquired by
the MIT Paleomagnetism Laboratory.
All other data and models are freely
available in the published literature.
D. J. H. was supported by the Miller
Institute for Basic Research in Science at
the University of California Berkeley.
S. M. T. was supported by the Rutgers
University School of Arts and Sciences.

HEMINGWAY & TIKOO(A19)

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HEMINGWAY & TIKOO

LUNAR MAGNETIC SOURCES CONSTRAINED BY SWIRLS

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