global electromagnetic induction in the moon and planets - MTNet

Physics of theEarth and Planetary Interiors, 7 (1973) 251—265
~ North-HollandPublishing Company, Amsterdam — Printed in TheNetherlands
GLOBAL ELECTROMAGNETIC INDUCTION IN THE MOON AND PLANETS
PALMER DYAL and CURTIS W. PARKIN
Space Science Division, NASA-Ames Research Center, Moffett Field, Calif (U.S.A.)
Department of Physics, University of Santa aw-a, Santa clara, Calif (U.S.A.)
Accepted for publication February 2, 1973
A summary of experiments and analyses concerning electromagnetic induction in the Moon and other extraterrestrial
bodies is presented. Magneticstep-transient measurements made on thelunar dark side show theeddy current
response to be thedominant induction mode of the Moon. Analysis of the poloidal field decay of the eddy currents
hasyielded a rangeof monotonic conductivity proilles for thelunar interior: theconductivity rises from 34(r
4 mhojm
at adepthof 170 km to 10—2 mho/m at 1000 km depth. The static magnetization field induction hasbeen measured
and thewhole-Moon relative magnetic permeability hasbeen calculated to be ~i/j~
0 = 1.01 ±0.06. The remanent magnetic
fields, measured at Apollo landing sites, range from 3 to 327 y. Simultaneous magnetometer and solar wind spectrometer
measurements show that the38-~~ 2. The solar remanent wind field confines at the theinduced Apollo 12lunar site ispoloidal compressed field; tothefield 54 7 byisa compressed solar wind
pressureincrease to the surface on of thelunar 7~10_8subsolar dyn/cmside
and extends out into a cylindrical cavity on thelunar antisolar side. This solar
wind confinement is modeled in thelaboratory by a magneticdipole enclosed in a superconducting lead cylinder;results
showthat theinduced poloidal field geometry is modified in a manner similar to that measured on the Moon. Induction
conceptsdeveloped for theMoon are extended to estimate theelectromagnetic response of other bodies in
the solar system.
1. Introduction Moon due to a fortunate combination of lunar electrical
properties and solar wind driving field properties.
In this paper we discuss experiments and analyses Similar methods may be applicable to other moonlike
that have been published on induction in the Moon bodies (e.g., Mercury) and to planets where a landing
and then present a summary of concepts concerning will be difficult or impossible (e.g., the Jovian planets).
electromagnetic induction properties of other extra- We consider first the theory of classical electromagterrestrial
bodies. The classical theory of induction netic induction in a sphere and extend this treatment
was developed by Faraday, Lenz, and Maxwell during to the case of the Moon, where poloidal eddy-current
the nineteenth century; the electromagnetic response response has been found experimentally to dominate
of a sphere has been treated by Mie (1908), Debye other induction modes. Analysis of lunar poloidal in.
(1909), Smythe (1950), and others. Induction in the duction yields lunar internal electrical conductivity
Earth and other planetary-sized bodies has been studied and temperature profiles. Two poloidal-induction anaby
investigators including Chapman (1919), Lahiri and lytical techniques will be discussed: a transient-response
Price (1939), and Rikitake (1950, 1966). The applica- method applied to time-series magnetometer data and
tion of induction methods to the case of the Earth a harmonic-analysis method applied to data numerically
withits oceans, ionosphere, and radial and azimuthal Fourier-transformed to the frequency domain, with
conductivity anomalies is difficult; however, it has emphasis on the former technique. Next we discuss
proven to be one of the more productive techniques complicating effects of the solar wind interaction with
used experimentally to probe deep into the Earth. Re- both induced poloidal fields and remanent steady fields.
suits from Apollo magnetic observatories on the lunar Then we present the static magnetization field induction
surface show that electromagnetic inductionmethods mode, from whkh we calculate bulk magnetic permeabildeveloped
to study the Earth are more suitable for the ity profiles. Finally, magnetic field measurements ob-

252 P. Dyal and C. W. Parkin, Induction in the Moon
tamed from the Moon and from fly-bys of Venus and surface; for reference we write the sum of these fields as:
Mars are examined to determine the feasibility of extending
theoretical and experimental mduction tech-
—
niques to other bodies in the solar system. Here ~A is the total magnetic field on or near the
lunar surface; BE is the total external (e.g., sold or
terrestrial) driving magnetic field; B
5 is the steady
2. Poloidal eddy-current induction in the Moon remanent field (which may be of global or local extent);
B~is the magnetization field induced in per-
In this section we discuss the electromagnetic re- meable lunar material;B~is the poloidal field caused
sponse of a particular planetary-sized body, the Moon,
treat to external in detail electromagnetic only the special driving casesfields which(Fig. have1). been We
by eddy currents induced11T in is the the lunar toroidal interior fieldbycor changing responding external to unipolar fields; electrical currents driven through
found practical in analyzing properties of the Moon;
since to be poloidaleddy-current the dominant induction response mode in has the been Moon, found
the Moon by the —,VXJ
11 electric field 8F is(V theistotal average field
velocity associated ofwith solarthe wind hydromagnetic plasma); and solar wind flow
this section will emphasize the poloidal mode and as- past the spherical body (e.g., plasma confinement or
sociated properties of the lunar interior. Two analyti- compression of the planetary permanent or induced
cal techniques wifi be discussed in reference to lunar fields, fields due to induced ionospheric currents, etc.).
induction: a transient-response (time-dependent)tech- In the case of the Moon, the external driving magnique
and a Fourier-harmonic (frequency-dependent) netic field (BE) can vary considerably with orbital
analysis technique. position (Fig. 2); therefore, it is possible to study the
Various possible induced fields and plasma-interac- various lunar properties separately through careful
tion fields will exist inthe vicinity ofthe global Moon’s selection of data timeperiods. Average magnetic field
conditions vary from relatively steady fields of magni-
/ tude —~9 ‘y (1 ~ = 10~gauss) in the geomagnetic tail
~FaLD~ ~ to mildly turbulent fields averaging 5 ‘y in the freestreaming
solar plasma region to turbulent fields aver-
\ ‘~,
~ / ~ aging 8 y in the magnetosheath. The average solar
/ / MAGNETIC wind velocity is 400 km/sec in a direction approxi-
5N~(~~—STEp TRANaEN~ mately along the Sun—Earth line.
-~ / -~ ((Z~j~MAGNETOSPHERE
~ ~~NxPLASMA FLOW’\ ~
/ I \ \ MAGNETIC -
, J ~ \ MOON FIELD
/1 ..--•~•• _____
I EXPLORER
Fig. 1. Electromagneticdriving sources. This diagram schematically
depicts external driving sources from theSun and their “ ) GEOMAGNETIC TAIL
interactionswith planetary-sized bodies. A hot plasma, con- -- SOLAR
sisting primarily of protons and electrons, streams nearly radially
outward from the Sun (at speeds averagmg 400 km/sec .-
I PLASMA
L
-
~
N
&0!
RE’
_______
MAGNETOSHEATH
_____
at Earth orbit), carrying a “frozen-in”magnetic field past the
planets. Various typesof magneticdiscontinuities and wave
phenomena act as time-varying driving functions to induce
response fields in theplanetary bodies and their ionospheres.
-~-——‘——<
Threedifferenttypes ofplanetary-solar wind interactionsare Fig. 2. Orbit of the Moon, projected onto the solar ecliptic
depicted here: one in whichtheplanetary permanent magne- plane. During a complete revolution around the Earth, the
tic field dominatesin theinteraction (in thecase of the Earth), magnetometer passes through theEarth’s bow shock, themagonewhere
the planet’s ionosphere dominates (as is probable netosheath, thegeomagnetic tail, and theinterplanetary region
inthe case of Venusand Mars), and onewhere the solid plane- dominated by solar plasma fields. The Explorer 35 satellite
tary body itself dominates (theMoon). orbit around theMoon is also shown.

P. Dyal and C. W. Parkin, Induction in the Moon 253
2.1. Time-series transient analysis
E~ORER~ 2.1.1. PoloIda! responseofa homogeneous sphere in
a vacuum
The global electromagnetic response of a planetary
body is modeled by a sphere’s response to a step-function
transient magnetic field, which canbe treated by
TI.~BULENT / e~u~is extending initially considered the theorytoof beSmythe subject (1950). to the following The sphere conis
ditions: (1) The sphere is in a vacuum, i.e., no plasma
\
\
\ \
interactions (effects of the plasma are considered in
section 3.1). (2) The sphere of radius R ishomogeneous
and has a constant permeability (it), permittivity
Fig. 3. Confinement of an induced global magnetic field by
the solar wind. Conceptuallydepicted here is the case where
(a) and conductivity (a). (3) The dimensions of the
external field transients are large compared to the
sphere, such that the sphere responds as a whole to
the transients. (4) Conduction currents dominate dis-
placement currents within the sphere. (5) There is no
a poloidal induced field Bp is confined by solar wind flow net charge on the sphere. Then, for a coordinate syspast
the Moon. The induced field is compressed on the lunar .
sunilt side and confined in thevacuum cavity region on the tem fixed m the sphere, Maxwell s equations and Ohm s
antisolardark side of the Moon. Also shown is the Explorer law in the rationalized M.K.S. system are as follows:
35 magnetometer orbit projected onto the solar ecliptic plane.
The Explorer 35 period of revolution is 11.5 h; its perilune and vx £ = — ~— (2)
apolune are approximately 0.5 and 5 lunar radii, respectively. 8t
VXB=MJ (3)
The characteristics of each of the induction fields . E = 0 4
Bp, BT and BM are dependent upon electromagnetic
properties of the Moon’s interior; therefore, these V B = 0 (5)
properties can be deduced from theoretical and ex- / = aE 6
perimental studies of the induction processes. The
toroidal response field has been found to be generally where E andIi are the electric field and the magnetic
much lower in magnitude than the poloidal field and induction and/ is the conduction current density.
perhaps not detectable by lunar magnetometers (Dyal Before we proceed to a more general transient analand
Parkin, 1972); therefore, the toroidal mode wifi ysis treatment, it is useful to consider the idealized
not be discussed in detail in this paper. The magnetiza- case of a uniformly conducting sphere in a vacuum.
tion field BM is a function of global magnetic permea- This model is particularly applicable to fields measured
biity and the iron content of the Moon; calculation on the antisolar side of the Moon, which is shielded
of the lunar permeability will be included in a later from the solar plasma. Suppose that initially there is
section. The decay characteristics of the poloidal field no magnetic field,but at t = 0 an external magnetic
~llp,the dominant lunar induction mode (Fig.3), are a field I~BEis switched on which is uniform far from
function of the global electrical conductivity distribu- the sphere. Forthe case of a homogeneous sphere in
tion, which is in turn related to the global internal tem- a vacuum Maxwell’sequations can be solved by experature
distribution. This section includes calculation tending the theory of Smythe (1950); a detailed denof
lunar conductivity and temperature profiles using vation is given by Dyal and Parkin (1971). Considering
both time-series transient analysis and Fouriór har- only the poloidal inductionmode, eq. 1 becomes
monk-analysis techniques. B 4 = BE + B1,. The solution for the vector components

254 P. Dyal and C. W. Parkin, Induction in the Moon
of the magnetic field at the surface of the sphere are northward. Qualitative representations of the solutions
listed below for the case of an extemal magnetic field
8E =
step transient of magnitude A
BEf — BEO applied
to the lunar sphere at time t = 0. BEO and BE! are miin
eq.7 and 8 are illustrated in Fig. 4A. It can be seen
that if the “decay function”F(t) can be determined em-
pirically, then the electrical conductivity can be calcutial
and fmal external fields, respectively: lated from eq.~ for the homogeneous approximation.
B = B — 31 I.~B I F(t) (7~ Over one hundred step-transient events have been
Ax Exf Ex / analyzed ~isingApollo 12 lunar surface magnetometer
B = B + ~ I ~B IF(~ ‘8’ antisolar side data and simultaneousdata from the lunar
Ay,z Ey,zf 2 Ey,z ‘‘ “ / orbiting Explorer 35 magnetometer. These data show
where: that the Moon’s response is similar to the eddy-current
F’t’ = 2 1 (—s2lr2t\ (9\ response of a conducting sphere in a vacuum; measured
‘‘ ~2 s1 ~2 exp ~ R21 “ ‘ surface magnetic field radial components have a damped
response to solar wind field step transients, while
a is the electrical conductivity and p is the magnetic components tangent to the lunar sphere have rapid repermeability;
and: sponse and overshoot initially, followed by decay to a
- steady-state value. An exampleof a step-transientevent
AB.=B.—B. z=xyz -
Ei Eif Ezo’ mApollo l2andExplorer35dataisshownmFig.4B,
The coordinate system used here has its origin on the and qualitative agreement between theory (eq. 7 and 8)
surface of the sphere withthe x..axis directed radially and data is easily seen; the deviations of the data from
outward from the sphere’s surface; they- and z-axes homogeneous-spheretheory are considered in the fol-.
are both tangential to the surface: y eastward and z lowing two sections.
TRANSIENT RESPONSE THEORY LUNAR MAGNETOMETER DATA
EXTERNAL DRIVING FIELD B -- EXPLORER 35
RA~ALXL \çIOIALSUREELD X~ O~12
TANGENTIALY B~Y
~ Bz,Y ~2468 10 12
A TIME, miii B TIME, mm
Fig. 4. Comparison of lunar magnetometer datato the theory for lunar poloidal response to a transientin thesolar wind magnetic
field. At left (A) aretheoretical solutions ofcomponents of the surfacefield (B
4) for a homogeneous sphere in a vacuum (see eq.7
and 8) which experiences step 8E changes (measured in all bycomponents lunar orbiting of external Explorerdriving 35) andfield total (BE). surfacemagnetic At right (B) are field superimposed B dataplots
of theexternal driving field
4 (measured by the
Apollo 12 magnetometer while on the antisolar side of the Moon). In orderto illustrate transients in all three componimts, these
events were selected from different times during lunar night.

P. Dyal and C. W. Parkin, Induction in theMoon 255
2.1.2. Poloidal response ofa sphere of radially varying G(r, t) = A/tiBE sin 0 and ~(r, s) be the Laplace
conductivity
To describe the response of the sphere to an arbitrary
input (Dyal et al., l972b), we define the magnetic
vectorpotential A such that V X A = B and
transform of G, eq. 10 becomes:
1 / a
2 2 ‘
_(—~ (rd) — —~)=sp
0a(r) (7
r \ar r
(15)
V A = 0. We seek the response to an input AB~b(t), for the interior. The boundary conditions are:
where b(t) = 0 fort 0, the magnetic field must be continuous at the
surface, so that A and aA/ar must always be continu-
The fmal u(r) is not unique; rather a family of a(r) is
generated with the constraint that f(t) match the exous
at r = R, the radius of the sphere. We also have the
boundary condition A(0, t) = 0 and the initial condition
perimental data.
A(r, 0; 0) = 0 inside the Moon. Outside of the Moon, 2.1.3. Magneticfield measurements and calculation of
where a = 0: electrical conductivityprofile
I.~BE Normalized lunar nighttime data from Apollo 12,
A = ~BE (~)b(t) sin 0 + f(t) sin 0 (11) giving the step-function response obtained from the
r radialsurface field component (foreleven events) are
The first term on the right is a uniform magnetic field shown in Fig. 5. Error bars are standarddeviations of
modulated by b(t); the second term is the (as yet un- the measurements. Allof these events ocurred when the
known) external transient response, which must vanish magnetometer was more than 900 km inside the optical
as r -~°° and t -÷00• Note that at r = R, where R is nor- shadow, so that plasma effects are assumed to be minimalized
to unity: mal to first order (plasma effects are considered in secrb(~\
1 tion 3.1). Effects of the lunar cavity due to solar
A = L~&BEsin 0 L~+f(t)j (12) wind diamagnetism, described by Colburn et al. (1967)
and Ness et al. (1967), are also neglected to a first apand:
proximation since the cavity field is a constant or very
aA rb(t\ 1 slowly varying function of time compared to the polo-
= t~BEsin 0 [_~.! — 2f(t)J (13) idal response time. Furthermore, recognizing the finite
length of time required for a solar-wind step transient
Therefore, at r = R I: to pass the Moon, we have in our analysis chosen a
r _~ ramp input function of 15-sec rise time, a time charac-
—2A + ~[~SB~sin 0 b(t)] (14) terizing passage of a discontinuity past the Moon. (For
a 400 km/sec solar wind, this time is 10—20 sec, de-
Since the magnetic field is continuous at r = R, this is pending on the thickness of the discontinuity and the
a boundary condition for the interior problem. Letting inclination of its normal to the solar wind velocity.)

256 P. Dyal and C. W. Parkin, Induction in the Moon
I.0
IO~
0 8 DYAL AND
~O6
DISTANCE FROM CENTER, km
0 500 000 500 740
10 ~ PARKIN(I972)~~
E10 3
b
0 I0~
204
- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 .0
- R/Rm
I I I I -‘ Fig. 6. Range of electricalconductivity profiles (Dyal et al.,
0 200 400 600 1972b) which give radialresponse time-dependentcurvesf(t)
TIME, sec that fallwithin theerror barsof Fig. 5.-A step-function input mag.
Fig. 5. Normalized transientresponse data, showing f(t)- netic field, modified by an initialramp inputof 15-sec risetime,
decay characteristics of theradial component of the total sur- is used in theanalysis. Theinformation at shallowdepths is limface
field BAX after arrival of a step transient which changes itedby theuncertainty in therisetime of the interplanetary magthe
externalmagnetic field radial componentby an amount netic field;that at largedepths is limitedby the sensitivity of the
~BEx,here normalized to one. Theshape of the curve illus- surface magnetometer. The superimposeddashedline shows
trates time characteristics of the decayof the induced poloidal the results of a three-layer Moon approximation. -
eddy-current fieldB~
Although the driving field is still approximatedby a data curve puts fairly restrictivelimits on the conducspatially
uniform field, this procedure provides a tivity in this region. Deeper than 170 km, the conducbetter
model for the very short timeresponse. tivity is seen to rise from about lO33mho/m to
The theoretical response curves corresponding to a 10—2 mho/m at R/Rm = 0.45. This is in general agreelarge
number of lunar conductivity profiles have been ment with the three-layer model of Dyal and Parkin
compared with the data of Fig. 5. It is found that a (1972). However, we note that conductivities greater
range of monotonic conductivity profiles, defining the than 10—1.5 mho/m for R/Rm

P. Dyal and C. W. Parkin, Induction in the Moon 257
-
-
-
proximationused so far, this would change the nor- the conductivity profile in Fig.6, using the expression
malization of the response curve (Fig. 5), but not its for the electrical conductivity as a function of tempershape,
and hence it does not change a(r). The effects ature given by England et al. (1968) for olivine and
of the confinement of the induced poloidal field to peridotite:
the cavity region bounded by the solar wind have a .. = 55 exp(—0.92/kT)+ 4~10~exp(—2.7/kT)
been modeled by a superconducting laboratory ex- ohvine
periment which will be discussed in section 3.1. (19)
a . . = 3.8 exp(—0.81/kT)+ l0~exp(—2.3/k~
2.1.4. Internal lunar temperature calculations pmlotite
Once a planetary electrical conductivity profile has
been determined, an internal temperature distribution
in mho/m (20)
can be inferred if the material composition is known (In the above expressions, kTis in eV.) These conductiv-
(see Rikitake, 1966). For cases where electrical con- ities are assumed to be independent of pressure below
ductivity is independentof pressure to a first approxi- 50 kbar (England et al., 1968) and independent of fremation,
the conductivity of minerals can be expressed quency below 10 Hz (Keller and Frischknecht, 1966).
in terms of temperature T as follows: For the example of a peridotite Moon, a temperature
profile that rises sharply to 850—1050°K atR/Rm
a a. exp (—E./kT)
1
(18)
~ 0.95 and then gradually to 1200— 1500°Kat
R/Rm = 0.4 is suggested by the data. At depths greater
than R/Rm = 0.4 the temperature could be higher than
where E
1 are the activation energies of impurity, intrinsic,
and ionic modes, expressed in electron volts; a1 are
1 500°K.
material-dependent constants; and k is Boltzmann’s
constant. It should be emphasized that the electrical
2.2. Fourier harmonic-analysts technique
conductivity a(a, E, T) is a strong function of the ma- The harmonic method, as applied to the Moon, is
terial composition; therefore, uncertainties in know- based on the assumption that any global induced field
ledge of the exact composition of the sphere limits the is excluded from the oncoming solar wind by currents
accuracy of the internal-temperature calculation, induced in the highly conducting solar plasma; it is as-
We calculate lunar temperature profiles (Fig. 7) from sumed that in effect the solar wind completely confines
the induced field in the lunar interior and in a
thin region above the lunar surface. By assuming this
DISTANCE FROM CENTER, km sheath region is very thin, the confinement current is
6000 500 1000 500 740 considered to be a surface current for theoretical treatment;
this provides a boundary condition of total con-
1400 finement by a spherical current around the whole
Moon, permitting solution of Maxwell’s equations be-
1200 low and on the lunar surface. This spherical confinement
case is then applied to magnetometer data meas-
T, K 000 ured on the lunar sunlit side. (Basictheoretical devel-.
opment of harmonic solutions can be found in several
eco - ~~OLI VINE references including Schubert and Schwartz, 1969;
~~PERIDOTITE Sifi and Blank, 1970.)
600 The harmonic analytical technique requires calculating
frequency-dependent “transfer functions” which
R/Rm are defined as follows:
Fig.7. Temperatureestimates for assumed lunar compositions bE,(f) + b~1(f) + b71(f)
of pure peridotite and olivine, calculated from theelectrical A,(f) = b (f) (21)
conductivity profile of Fig.6. Ez

258 P. Dyal and C. W. Parkin, Induction in theMoon
i = x, y, z, where A, are transfer functions of compo- lunar center. Other conductivity profiles have been
nents of frequency-dependent magnetic fields, ex- calculated using the Sonett et al. (1971) transfer funcpressedin
the orthogonal coordinate system with tion, showing that the spike profile is not unique but
origin on the surface of the sphere (~is radial and j) that the frontside transfer function can be fitted by
and ~ are tangent to the surface); bEj(f) is the simpler two- and three-layer models (Kuckes, 1971;
Fourier transform of the external driving magnetic Sill, 1972; Reisz et al., 1972).
field; bpj(f) andbr,(f) are Fourier transforms of
the induced global poloidal and toroidal magnetic 2.3. Comparison of transient and harmonic-analysis
fields, respectively, techniques
The harmonic data analysis approach involves
Fourier-analyzing simultaneous data from the Apollo Since transient and harmonic techniques yield dif-
12 lunar surface magnetometer, taken during lunar ferent conductivity profiles (compare Fig. 6 and 8)
daytime, and the lunar orbiting Explorer35 magneto- using data from the same magnetometers, it is useful
meter. Then ratios of the surface data to orbital data to discuss the assumptions involved in both analyses.
are used to calculate a transfer function of the form First we consider two basic assumptions of the harof
eq.21. It is assumed that the toroidal induction monic technique: (a) only the poloidal induction and
mode is neglected such that the Fourier-transformed external driving fields are measured on the daytime
surface magnetometer data represent the sum lunar side; and (b)the induced poloidal field is perbEj(f)
+ bpi(f), while the transformed orbiting mag- fectly confmed by a spherically symmetric current
netometer data represent the driving field bE,(f) layer. The first assumption considers that the total
alone, field measured by the lunar surface magnetometer is
The form of the transfer function is determined by composed of onlythe external solar wind driving field
the internal conductivity distribution in the Moon; BE and the poloidal lunar response field Bp. By reftherefore,
a “best fit” conductivity profile can be ob- erence to eq. 1, we recognize that various other fields
tamed by numerically fitting to the transfer function. may be contributing to the surface magnetometer data,
Fig. 8 shows the “best fit” radial conductivity profile e.g.,interaction fields (BF) due to such effects as cornof
Sonett et al. (1971) obtained in this manner. The pression of lunar remanent fields by the solar wind
conductivity profile is characterized by a large “spike” plasma and generation of plasma wave modes on the
of maximum conductivity about 1500 km from the sunlit side. Referring to Fig. 8, we note an asymmetry
effect which isnot predicted by simple harmonic-analysis
theory: the harmonic analysis yields two different
i02 I I I conductivity profiles, one calculated from y-axis (east—
west) data alone and the other from z-axis (north—
south) data alone. Thisy—z asymmetry can be ex-
\ plained as being in part due to the daytime compression
\ \ of the local rernanent field at the Apollo 12 magnetoio
4 meter surface site by the solar wind (Dyal et al., 1972a).
o-, mho/meter - ~ Thus, when lunar daytime magnetometer data are used
- -- -to study global lunar properties, knowledge of simultaneous
solar wind propertiesis important
a. The second assumption considers that the induced
l0”6 poloidal field is confined in the lunar sphere by the
COSOUCTIVIIY mRflI* RM solar wind plasma. Experimentally the solar wind plasma
I I I spectrometer (Snyder et al., 1970) does not measure a
4 8 2 6 20 confining plasma on the dark side lunar hemisphere, and
LUNAR RADIUS kmx 102 the transient magnetic field data show to first order a
Fig. 8. Electrical conductivity profiles from Fourier harmonic vacuum poloidal response on the lunar dark side (Dyal
analysisof lunar daytime data (from Sonett et at., 1971). and Parkin, 1971). It is questionable whether confme-

P. Dyal and C. W. Parkin, Induction in theMoon 259
ment is complete even on the sunlit side: a subsatel- resulting poloidal-induction field and any permanent
lite magnetometer has measured evidence of induced fields originating in the lunar crust. Both of these interlunar
fields at distances greater than 100 km above actions have beenmeasured by Apollo magnetometers
the lunar sunlit surface (Coleman et al., 1972), mdicating
that confinement isnot complete over dis-
and solar wind spectrometers.
tancesgreater than a proton gyroradius (‘~100 km). 3.1. Confinement of induced poloidal fieldsby the
For these reasons, the spherically symmetric plasma
confinement assumption should be modified.
solar wind plasma
In summary, it is seen that magnetic fields meas- Fig. 3 schematically portraysthe effects of the highly
ured on the lunar day side quite possibly involve a conducting solar wind on the lunar induced poloidal
very complicated set of magnetic phenomena which field; in this case the induced field geometrical con-figcan
contribute varying magnetic field amplitudes at uration is altered such that the poloidal field is cornvarying
frequencies over the entire measured spectrum, pressed onto the lunar surface on the sunlit side and is
and that these contributions cannot be adequately extended back into the cylindrical “cavity” region on
treated using the above-mentioned assumptions the dark side. This compression has been experlinen-
(a) and (b). A recent harmonic paper (Sonett et al., tally confirmed by measurements from magnetometers
1972) has considered effects such as that due to on the lunar surface. For the uncompressed (vacuum)
remanent field compression and responsevariations case the ratio of tangential components of surface
as a function of solar wind k-vector direction with total field (BAY,Z) to external driving field (BEy,z) will
respect to the Moon. As a result, the conductivity have a maximum value of 1.5 (see eq.8). Daytime
“spike”, which was emphasized earlier, is now recog- ratios, however, are considerably higher than 1.5; for
nized as only one of a larger set of possible conduc- example, at 0.01 Hz the z-component amplication has
tivity proffles, some of which are much more in agree- been measured to be 3.7 (Dyal et al., 1970). Nighttime
ment withthe transient-response three-layer models data show a smaller compressive effect in the tangential
published earlier (Dyal and Parkin, 1971, 1972). components; examples in Fig. 4B show a y-axis ratio of
Furthermore, the recent harmonic papers state that 2.0 and az-axis ratio of 1.75.
resolution of data restricts conductivity models to The confinement of the induced poloidal field by
three or four layers rather than the eight layers used the highly conducting solar wind is theoretically studied
for the “spike” model, by considering a pOint dipole field inside a supercon-~
Though of an apparently less complicated nature, ducting cylinder. Two geometrical orientations of the
there are also problems associated with the assump. point dipole are considered: along the cylinder axis
tions of the darkside transient analysis technique. and transverse to the cylinder axis. (See inserts in Fig.9
The main problems involve: (a) the finite time re- for an illustration of these orientations.) The field of
quired for a “step” transient to cross the lunar sphere; the dipole orisnted transverse to the cylinder axis has
(b) the field increase along the antisolar line due to been determinedby Parker (1962). For a dipole on
diamagnetic currents at the solar wind cavity bound- the axis of a capped superconducting cylinder (of
ary; and (c) deviations from sphere-in-a-vacuumtheory radius R), lying a distance R from the cap, the scalar
due to partial confinement of induced fields by the potential of the confined dipole is:
solar wind. These problems are considered in section 2
2.13; the partial-confinement problem (c) is considered
in more detail in the following section.
— —2m cos ~,
2 n=1
“~c— R
Np~J
1 (N~x/R)f~(r) (N2—DJ2(N
22
‘ n / 1’ n
where:
f (r) = exp (—N Jr I /R) + exp (—N (r + 2R)/R) (23)
3, Interaction of the solar wind withlunar mduced ~1 fl fl -
and remanent fields and 4 is the azimuthal angle measured around the
axis of the cylinder, from the dipole axis;m is the
The solar wind plasma transports the inducing mag- dipole moment;r is the distance along the cylinder
netic field past the Moon and also interactswith the axis; N~is the n-th root off 1 (Nn) = 0.

260 P. Dyal and C. W. Parkin, Induction in the Moon
2.5
POINT where:
/ DIPOLE
- AXIAL ~SENSORiii g~(r)= exp(—k~ri /R) — exp [_k~(~~ + 2)] (27)
o.A DATA ____________
2.0 RT~’~ andk~are roots of the zero-order Bessel function. The
A A A A TRANSXERSE L~?) axial components of the confined and unconfmed fields
3 4 are respectively —ai
TRANSVERSE’
1~iac/arand —a~,/u/ar;the ratio of
1.0 the /Bc\ radial component 1 ~ r of confined field tounconfmed
3k, 7g~(r) (28)
field along the cylinder axis is:
0 I 2 3 4 ~ n1 4(kn)
r/R
Fig.9. Confmementof apoint dipole magnetic field, shown
theoretically and experimentally. Theinserts schematically This axial confined-to-unconfmed field ratio is also
show lunar confmement by the solar wind, approximated by plottedin Fig.9.
a capped-cylinder superconductor enclosing a point dipole A laboratory experiment has been devised to exfield.
The theoretical curves showratios of confined to unconfined
dipolar field versus distance along the cylinder axis, amine the “axial” and “transverse” solutions (eq.25
Data are results of a laboratoryexperiment in which confine- and 28). The highly conducting solar wind plasma
ment of a small dipole magnet’s field by acylindrical super- cavity is modeled by a thin lead superconducting
conductor is measured experimentally, capped cylinder and the instantaneous induced polo.
idal field is modeled by a small dipolar samarium-
-
The scalar potential which describes the field due cobalt magnet placed equidistant from the closed end
to an unconfined point dipole is: and the side walls of the cylinder. The measured ratios
mx
= —i-r
(24)
of confined fields to unconfmed fields are shown in
Fig. 9 along with the theoretical value expressed in
eq.25 and 28.
The transverse component of magnetic field due to The theory and laboratory data presented in Fig. 9
either of the potentials (eq.22 or 23) canbe calculated represent to first order the effects of solar wind comas
a function of distancer along the cylinder axis, using pression on a poloidal induced lunar field, as measured
B~(r) = —ail,/ax. The ratio of the transverse component. by a lunar surface magnetometer positioned at the
of confined field to unconfmed field on the cylinder antisolar point. This compressive phenomenon, which
axis is then: has a measurable geometrical effect on the poloidal
i Bc\ 2 ~ r
field,does notsignificantly change the time dependence
3N~f trans.
3 n1
= R
1(0)f~(r) (N,~— 1)4(Nn) (25) of puted the poloidal internal electrical-conductivity eddy-current field. Therefore, profile a(r), the corn- which
is a function only of the poloidal time-dependence,
where fn(r) is expressed in eq.23; 4(0) 0.5. This will not be modified by solar wind compression to the
transverse conflned.’to-unconfined field ratio is plotted
in Fig~9.
order of approximation used here.
Now we considerthe effect of confinement field 3.2. Compression ofthe permanent lunar fieldsby the
due to a dipole lying on the cavity axis and directed
along the cavity axis (see Fig.9, insert). The potential
solar wind plasma
of the confined dipole (P. Cassen, personal communi- Remanent magnetic fields have been measured by
cation, 1972) is: Apollo magnetometers emplaced at nine sites on the
Moon. Magnitudes and locations of these steady fields
= m ~ J 0(k~x)g~(r) (26) are listed in Table I.
ac ~ n=1 J~(k~) Simultaneous solar wind plasma properties have
been measured at the Apollo 12 and 15 landing sites.
- To first order, a compression of the Apollo 12 rema-

TABLE I
Lunar surface remanent magnetic fields
P. Dyaland C. W. Parkin, Induction in theMoon 261
Site Coordinates Measured magnetic fields (‘y)
Apollo 12 3.2°S 23.4°W 387
Apollo 14 37°S 17.5°W 1037 and 43~(separated by 1.1 km)
Apollo 15 26.1°N 3.7°E 37
Apollol6 8.9’S 15.5°E 1l27,113’y,1897,232~r,and3277
(separated by 0.5 —7.1 km)
nent field in direct proportion to the solar wind pres.
sure has been measured. For one-hour averages ofmeawind
pressure increase of 7’ l0~8dyn/cm
2. The ratio
of plasma pressure to total magnetic pressure is
suTements, the increase in magnetic field pressure = 8ir nm V2/B~r= 5.9, where BST = + ~B,
~2I8,i., where L?ILR ~A — (BE +B
2 5), asshown is directly in Fig. corre! io. during stagnation maximum condition plasma ((3 ~ pressure, 1) is notindicating reaëhed and thata the local
lated The magnetic to the plasma field pressure increasesnm fromV 38 to 547 for a solar shock is probably not formed at the Apollo 12 site.
Modulation of the remanent magnetic field by time
variations in the solar wind pressure should add noise
10 to induction measurements made on the daytime side
of the Moon (Dyal et al., 1972a), as was discussed in
2 section 2.3.
b 8
N I
E I
6 4. Magnetization field induction and permeability cal-
lIo~
N
6
k ~ k ~ ~ 1’-~
culationstion
interaction immersed can
Referring
be
mode
written in B~.For modes the again steady to can times eq. be geomagnetic 1, when neglected we consider all other (e.g., tailthe induction field), formagnetiza theeq. Moon and
8A = BE + B~.For a sphere of permea- 1
bility p. immersed in a static magnetic field 8E’ the
surface components of total magnetic field are ex-
~ pressed (Jackson, 1962; Dyal and Parkin, 1972) as:
BAX=(l+2F)BEX (29)
E 4 BAYZ = (1— F) BEyz (30)
C 3 where:
E
=
-
2 (2km+1)(km_l)[1_(_~)]
I -
F=
R’3
km+1)~km+2)_2~~)(km_i)2
(31)
~30 331 332 333 334 335
YEAR 969 TIME, day
Fig. 10. Simultaneous plots of the magnetic field pressuredif- Here km is the relativepermeability p/p 0, R is the radius
ference &8~/8irand solar wind dynamic-pressureat the Apollo of the sphere, and Rc is the radius below which the
12 surface site, showing acorrelationbetween the pressures. planetary temperature is above the Curie point.

262 P. Dyal and C. W. Parkin, Induction in the Moon
5 ‘ - - urementsobtained at three locations on the lunar
surface. The increased accuracy will make possible a
T calculation of the percentage of permeable iron in the
10 ‘ . . outer layer of the lunar sphere.
5. Planets and satellites
-. /1
/‘ To consider global electromagnetic inductionin the
planets and satellites, we have divided them into three
0 ‘ main groups: (1) bodies which possess intrinsic magnetic
fields strong enough to stand off the solar wind; (2)
-5 . /
/ /.Z.~~i.Ol±0.06 bodies with weak magnetic fields but which have an
- - ionosphere; and (3) bodies that possess neitherstrong
permanent magnetic fields nor ionospheres. Table II
-10
•0”
‘ I I I I
B
I I I I I
10
I
divides the planets and satellites into these three main
groups based upon the limited number of measurements
that havebeen reported.
Ex, Y The global inductive response to the solar wind
Fig. 11. Hysteresis curve of the MOon. TheMoon is immersed
in the steady externalmagnetizing geomagnetic tail field BE
driving field will most probably be a poloidal response
for planetary bodies with either an atmosphere or a
(measured by Explorer 35); the total magnetic induction is low-conductingouter crust, either ofwhich will prevent
BA = BE + B measured by the Apollo 12 lunar surfacemagnetometer,
w~ereB is the induced lunar magnetization field the flow of umpolar currents and quench the toroidal
(thelocal remanent~heldBg at the Apollo 12 site has been mode. The probable time constant for the poloidal resubtracted
from theApollo 12 data). In this graph only the
radial (x) components are plotted. This curve is analogous to
thehysteresis B —H curve, where H AR B& andB AR B~+B~.
In this low-applied-field regime (—‘ 107) thehysteresis curve
is approximately linear;the slope of the curvecan be used to
sponse is listed for each body; estimates are calculated
from the Cowling time-constant expression
r = paR
2/ir2, where p is the global magnetic permea-
. . . . .
bility, a is the electncal conductivity, and R is the radius
calculate thelunar relativepermeability for an assumedpermeable
shell thickness ~R = Rm — R~by using eq.29 and 31.
of the body. The permeability is assumed to be that of
free space and conductivity estimates are based on extrapolations
from lunar and Earth measurements.
A plot of BAX vs BEX is in effect a plot of a B — H The properties of planetary interiors canbe studied
hysteresis curve. For cases where the ratio BAX/BEX by induction only if driving fields exist in the planetary
is a constant, i.e., for low-field BEX (the average solar environment which are of magnitude and frequency
field is approximately 5 ‘y, and the geomagnetic tail highenough to cause the body to respond electromagfield
is approximately 9 ‘y), the hysteresis curve should netically. Sources of driving magnetic fields may be
take the form of a straight line. Fig. 11 shows a plot of external to the body (e.g., solar wind magnetic field
radial’ components of Apollo 12 surface field (BAX) variations as shown in Fig. 1 and ionospheric diurnal
versus the geomagnetic tail field (BEX) measured by variations) Or internal to the body (e.g., secular varia-
Explorer35. A least-squares fit and slope calculations tions in the intrinsic permanent magnetic fields of
determine the factor F = 0.0030, which is used in Earth and Jupiter). The amplitude of the solar wind
eq. 31 to determine the relative magnetic permeability field fluctuations (at a frequency of 1 Hz)vary with
for a shell of inner radius Rc. For the bulk permeability’ distance from the Sun from approximately 107 at
of the Moon(the case Rc = 0), p/p
0 = 1.01 ±0.06. For Mercury’s orbit, to 5 y at the Earth, to approximately
a thinner permeable shell inside the Moon the permea- 0.8 7 at Jupiter (Scarf, 1970). The interplanetary field
biity is higher, as shown in eq.31, frequency spectrum varies as j_1.5 (Coleman, 1968)
A more accurate calculation oflunar permeability in the solar wind near the orbits of Earth and Mars.
will be determinedin the future from network,meas- Another driving magnetic field is that produced in the

TABLE II
Inductiveresponse times of planets and satellites
P. Dyal and C. W. Parkin, Induction in theMoon 263
2 Ionosphere Period of primary Planetary response
Group Planet or Mean radius’ Mean density
satellite (km) (g/cm3) (+ yes; — no) driving field time constant
1 Earth 6,378 5.5 + hoursto years 1 day to 10 yr3
Jupiter 70,850 1.33 + lOh
2 Venus 6,110 5.1 + lOyr
Mars 3,402 4.0 + 24 h 1 yr
Saturn 60,000 0.7 + 10 h
Uranus 25,400 1.7 + 11 h
Neptune 24,300 1.6 + 16h
Pluto 2,250 5.9 + 6 days minutes
Titan 2,425 2.4 + minutes
3 Mecury 2,430 5.4 — 10 sec days
Ganymede 2,775 2.4 — 10 sec minutes
Callisto 2,500 2.1 — 10 sec minutes
Triton 1,885 5.0 — 10 sec minutes
lo 1,750 4.0 — lOsec minutes
Moon 1,737 3.3 — 10 sec 2 mm4
Europa 1,550 3.8 — 10 sec minutes
Rhea 650 2.0 — 1 sec seconds
Tethys 600 1.2 — 1 sec < 1 sec
lapetus 575 1.9 — 1 sec < I sec
Titania 455 3.4 ‘ — 1 sec < 1 sec
Oberon 417 8.6 — 1 sec

264 P. Dyal and C.W. Parkin, Induction in theMoon
TABLE II
Apollo surfacemagnetometer characteristics
Apollo stationary Apollo portable
Parameter magnetometer magnetometer
Ranges(7) 0 —±200 0—i 256
(each sensor) 0 — ±100
0—i 50
Resolution (~‘) 0.1 1.0
Frequency response (Hz) dc — 3 dc — 0.05
Angular response Cosine of angle Cosine of angle
betweenfield between field
and sensor and sensor
Sensor geometry 3 orthogonal 3 orthogonal
sensors at ends of sensors in
100-cm booms 6-cm cube
Analog zero determination 180°flip of sensor 180°flip of sensor
Power OW) 3.5 1.5
Weight (kg) 8.9 4.6
Size(cm) 63x28X25 56x15x14
Operating temperature (‘C) —50 — +85 0 — +50
most of the solar wind magnetic driving waves with fluctuations should envelop the body and induce a
periods less than minutes and the dominant field poloidal responsewhich can be used to study the inchange
occurring over the entire globe will be due to ternal properties. The bodies possessing inductive time
the daily formation of the ionosphere whose period constants comparable to driving field periods (e.g.,
wifi be the axial rotation period of the planet. Higher- the Moon and 10) may be better explored using electrofrequency
fluctuations reaching the planetary surface
will be considerably lowerin amplitude than the diur-
magnetic induction techniques than by seismic methods.
nal variations (on Earth the daily variations are approximately
50 y whereas the shorter-period solar wind
Appendix: magnetic field measurement technique
changes are less than 10 ‘y); these diurnal ionospheric
magnetic field variations can induce a global response
Magnetic fields at the lunar surface have been meas-
uredwith magnetometers emplaced by astronauts on
in the planet’s outer layers. Since in this case the domi- the Apollo 12, 14, 15, and 16 missions. The three ornant
driving source for the planet is geometrically in thogonal vector components are measured as a function
the shape of a spherical shell, many simultaneous meas- of time and position and telemetered to Earth. Simulurements
are required to describe the total inductive taneously a magnetometer in the lunar orbiting Explorresponse
and separate it into external driving and inter- er 35 spacecraft measures the ambient solar or terresnal
response fields. This formidable task has been ac- -trial driving field and transmits this information to
complished to study the electromagneticproperties of Earth. Properties of the lunar surface magnetometers
Earth’s crust and mantle (Chapman, 1919; Lahiri and
Price, 1939; Ashour and Price, 1948; Rikitake, 1950,
1966; McDonald, 1957;Eckhardt et al., 1963). For the
are given in Table Ill.
core region of the Earth, however, seismic techniques
have provento be more useful than electromagnetic
Acknowledgements
induction techniques. We thank Richard Hartin of Philco-Ford, Ray
For the planets and satellites in group 3 of Table II Stepi~iensof the University of Santa Clara, and Jeffrey
which have neither a global permanent magnetic field Hammill and Roy Horinouchi of DeAnza College for
nor an ionosphere and which do not lie within the mag- their valuable contributions to the laboratory superconnetosphere
of a larger body, solar wind magnetic field ducting experiments.

P. Dyal and C. W. Parkin, Induction in theMoon 265
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