Transient response of moon to variations in interplanetary magnetic fiel

Schubert, G.

Schwartz, K.

English

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https://ntrs.nasa.gov/search.jsp?R=19690014252 2020-03-12T03:47:26+00:00Z
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Transient Electromagnetic Interaction of
the Moon with the Solar Wind
Kenneth Schwartz
American Nucleonics Corporation
1007 Ai rway
Glendale, California 91201
Gerald Schubert
Department of Planetary and Space Science
University of California
Los Angeles, California 90024
N69-236-Q
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ABSTRACT
The response of a homogeneous conducting moon to varia-
tions and irregularities in the interplanetary magnetic
field is determined. The significant lunar response is forced
by fluctuatio,is in the motional electric field. 	 In the limit
of a highly conducting plasma surrounding the moon and for
length scales of the interplanetary magnetic field irregularities
much larger than the moon's radius, the magnetic field fluctua-
tions in the lunar interior are the sum of variations in the
driving interplanetary magnetic field plus field fluctuations
driven by the motional electric field. The magnitude of the
motionally induced time-varying magnetic field inside the moon
is proportional to the product of the magnitude of the inter-
planetary magnetic field fluctuations and the lunar magnetic
Reynolds number. This induce,, field is toroidal about an axis
in the direction of the forcing motional electric f'r-eld. The
motionally induced magnetic field in the plasma is also toroidal
and is confined to a thin layer near the lunar surface. The
electric field is everywhere equal to the forcing motional
electric field.
INTRODUCTION
The steady state interaction of the moon and the solar
wind plasma has been described by SONETT and COLBURN [1967,
1968; in terms of a unipolar generator. HOLLWEG [1968] has
applied this induction generator model to the steady inter-
action of a two-layered moon with the solar wind. The motional
electric field measured by an observer moving with the moon
generates currents in the lunar interior which, in the steady
state, close in the highly conducting interplanetary plasma.
In this paper we present a theory for the time-dependent inter-
action of the moon and the solar wind. Time variations in the
driving or incident interplanetary magnetic field appear as
time-dependent magnetic field fluctuations in the lunar co-
ordinate system. In addition, the relative motion of the moon
with respect to spatial irregularities in the incident inter-
planetary magnetic field produces time-dependent magnetic
field fluctuations in the reference frame moving with the moon.
In fact, the power density spectrum of the interplanetary
magnetic field [COLEMAN,. 19681 shows that a significant fraction
of the power is at the frequencies associated with field
reversals caused by the corotation of the field's sector
structure with the sun. Also, the time scale of the sector
structure field reversals is in the range of possible Cowling
diffusion times for the moon.
In addition to forcing a direct time-varying response in
-2-
the lunar interior, variations and irregularities in the
interplanetary magnetic field generate a'time-dependent
motional electric field in the reference frame moving with
the moon which, as in the steady state unipolar generator
model, forces an important time-dependent lunar response.
BLANK and SILL [1969] in their ad hoc model of the transient
interaction problem, have failed to consider this significant
phenomenon. We solve for the fluctuating electromagnetic
fields in the moon and in the plasma which are driven by the
superposition of both a time-varying magnetic field and a
time-varying motional electric field. The magnetic field
fluctuations in the lunar interior are the sum of the fluctua-
tions in the incident field and those induced by the incident
motional electric field. The motional electric field also
induces magnetic field fluctuations in the solar wind which
are confined to a thin layer surrounding the moon. For fre-
quencies of the order of 10 -5 sec-1 and a plasma conductivity
of 104 mhos/m the length scale of this layer is approximately
4 km. The theory presented in this paper is necessary for the
correct interpretation of data from lunar surface magnetometer
experiments which can provide information on the electrical
conductivity and consequently the temperature of the lunar
interior.
f
s
THEORETICAL. MODEL AND SOLUTION
Consider the.case of a homogeneous moon with electrical
conductivity a, and magnetic permeability p immersed in a
plasma of conductivity 
a2 and permeability p	 In a coordinate
system moving with the moon let the driving electromagnetic
fields be of the form a y
 HO exp[ i (2rrz/X-wt )] and a x pvH 0 exp
[ i (2?Tz/),- wt )] , where H O
 is the amplitude of the magnetic field
oscillation, p.vH 0 = Em
 is the amplitude of the motional electric
field fluctuation associated with the relative velocity v of
the moon and the spatially periodic magnetic field of length
scale X , the frequency err = T , and ax and a y are unit vectors
in the x and y-directions, respectively. This form for the
forcing electromagnetic field models the relative motion of
the moon and the rotating sector structure of the interplanetary
magnetic field.
In the lunar int-rior the forcing fields produce the
response E t exp(-i(ut), H t exp(-iu)t) and in the plasma, the
electromagnetic field is the sum of the forcing field and the
reflected field E r exp(-iwt), H r exp(-iwt).	 Since the driving
fields are in mutually perpendicular directions, the transmitted
and reflected fields possess no simple symmetry properties.
The reflected magnetic field satisfies the vector Helmholtz
equation
- It -
v 2 H r + k2H r = 0	 ( 1 )
where v2 is the Laplacian operator,
2	 2
+ iak	
(2)
2 = pew 	 ,
and c is the permittivity of the plasma.	 Similarly, the
magnetic.field inside the moon is a solution of the vector
Helmhoitz equation v2 H t + k^ H t = 0, where k^ = uc^tU2 +
io l uw	 The reflected electric field and the electric field
in the lunar interior are determined from the appropriate
forms of Ampere's law. The boundary conditions are the continuity
of the tangential components of the electric and magnetic fields
at the moon's surface r=a( (r,8,c^) are spherical polar coordi-
nates).
T!,e solution of this problem is readily obtained with the
aid of STRATTON [19 1,17. The spherical harmonic expansions of
the forcing fields are
	
- i to t
	
+
H = - HOe	
^n{Mn,i + i Nni )	 ,	 (3)
n=1
E - Eme- iu t C	 S n (Mn i - i N- i )	 (4)
n=1
where
i n ( 2n+1 )
S n = n ( -n+]-) —	 { 5 )
1-J-
(	 }	 P I (cos©)	 dPI(cos©)
	
Cs	 0Mn, i = ^n\^/^ " n s" rn0	 sin C ' 	-	 n	 cos CP }	 (6)
Nn, i - l^J n(n+l) j n (2^-) Pn(cos) cos	 t +
d	 P1( cosA )
	
2n rl l f d	 i	 s i n
	 n	 cosCam) C l r^ n`,—JJ Lai p 	 cos 0 8 + s r n ?^  sin CP
(7)
The spherical Besse] functions j n
 (and later hn l) ) and the
associated Legendre functions Pn(cos8) are defined as in
STRATTON [1941 -1. The vectors
	 6, cp are the orthogonal unit
vectors of the spherical coordinate system.
The electric and magnetic fields inside the moon are
kl	 ( t	 t + 1	 ( 8)H t - - vH0 W t,
	 ^n ` b n M^,t+ i y n Nn,t)	
n=1
CO
I t = Em 	 S (y t M+	 - i b t N- )	 ,	 (9)
n=1	 n	 n n, t	 n, t
where Mn t and N t are given by replacing 2TT with kl
in equations (6) and (7), respectively. The reflected fields
are
W
H r =- vH0 W 	 Sn\ n Mn, r + r Y n Nn, r)	 ,	 (10)
n=1
r
-6-
CO
r	 r	 1
Er	 E m ^^	 S n Y n Mn, r - i S n Nn , r/
n=1
where Mn^ r are given by replacing ^ with k2 and j n with
h(l) in equations (6) and (7), respectively.
	 Continuity of
the tangential components of the E and H fields at r = a
leads to
Y t	 r-a h(1)(k a) 1 -r h(i)(k a) d	 Ia j ^ a /f jd-1,(12)n 	 ^`	 3a t	 n	 2 1	 n	 2 3a l	 n	 Y
Yr	
D n (ka) d la j ^2na11 j 	 r 	 dajk a)dn 	1	 2a L	 n A /^	 n( 2Xa)  ^a	 n 1 ^J Y
bt = ^k2^A h(1)(k a) d	 a j 	 _ 2TT j 2na1 d ah(1){k a) 1d - 1n --7n n	 2 -da
	
n
( 2TT a)j
 A 	 k^ nl^l cT—a n	 2 J b
r	 rklA	 d	 (2ri a ` _ 2n	 2na d j	 1	 -1
n CT,- ^n (k l a) c?a { a ^n\^/) Tc l1 i n\^^ 3alain(kla)db
(15)
where
d Y = hnl)(k2a) da {a j n (k l a)f- 	 in(kla) a ja hn l) (k2a) f	 (t6)
d S =	 hnl)(k2a) ii s i n(k l a)f - ^ jn(kia) a { ah(l) ( k 2 a) f •(17)
1	 2
DISCUSSION OF THE SOLUTION
The formulas of the preceding section can be simplified
I
	 considerably for the values of the parameters of interest
here. Since we are mainly concerned with the transient re-
spouse of the moon to the rotating sector structure of the
interplanetary magnetic field  we will
 1 l take u) = 0(10 -5 sec- 1) .
The electrical conductivity of the solar wind plasma lies in
the range 10 2
 to 10 14 mhos/m (Sonett and Col burn 1 96$) and the
absence of a
	
lunar bow shock indicates
	 that a representative
lunar conductivity would be smaller	 than	 10-5 mhos/m.	 The
actual value of o f
 depends on the thermal state of the lunar
interior. Thus k
^, 2 	iµwa1,2 and lk 2a) is,--60   for 6 2 ki
10 2
 mhos/m. The parameter 27a/X = wa/v ^ 10 4 for v of
0(10 2 km/sec). Thus we will be interested in the limits
27a/%— O and k2 8- 0 .
The parameter k^a 2 is equal to the product of i 27a/X
and the magnetic Reynolds number o l µva	 The magnetic Reynolds
number for the moon is < 1 so that in the limit 2na /% —O we
will assume a l µva i.s 0(1) and k 1 a is 0( (21Ta/X) 2 ).	 This in-
sures that the lunar transient response to the forcing motional
field fluctuations is consistent with the absence of a lunar
bole shock. From equations ( 12) - ( 17) we find that in the
limit ` a 0
n	 n_ 1
Y n' ti n' b n' b n	0 2aa	 { 1$)
-8-
We also note that as {? ^^-0.
n
+	 f	 2TTan'7,' 0P1n
	M^	 Mn	 = OC--^- -) 19 }^ i	 ,	 t	 , r
' 	 i
+	 f	 +	 _
 O(ITT)	 l 172 - 1	 0'
N n 	 ^n,t	 Nn,r	 0 ^- (20),i
The dimensionless
	 transmitted magnetic field H t/HO	 is
thus the sum of terms of order (2 1 ra/X) n-l .	 The	 terms	 involv-
ing bn and vn are of	 the same order.	 The dimensionless	 lunar
e l ectric	 y ield E t /Em is	 the	 sum of	 terms of order (2TTa/X)n
(associated wi ih
	 ,t )	 and	 terms of order (2TTa/X) n-1
	(associated
rith	 b').	 Similarly,	 %.he nth	 term	 in	 the	 summation for Hr/HO
is of	 order
	 (2T?^.,'; 1'''' 1	 and	 the	 nth	 term
	 in	 the	 expression	 for
E r "Cm is of ardor (2TT^1 /? ) n (bn and yn terms are of	 the same
order).	 The folln-aing are formulas for H t /H O ,	 H r /H0 ,	 Et/Em
and E r/Em
 correct to zeroth order
	 in the small
	 parameter 2TTa/X
(the	 simplification
	 k 2 a---	 has also been employed):
H t /H 0 = a y -	 (a I pva)(	 )( s  n:p0	 + coscPcos8 P) (21)
E t /Em _ ax (22)
E r/Em = 0	 , ( 23 )
ik2 ( r-a)
	
..1H r /H O = -	 a(7 l Eiva)	 e	 (sing	 +cos: cos9 cp) (24)I	 r
w-
-9-
To zeroth order, the electric field both in the solar
wind and in the lunar interior is the incident motional
4
electric field. To this same order, the magnetic field in
'	 the lunar interior is the sum of the incident interplanetary
magnetic field and a field induced by the motional electric
field fluctuation. The magnitude of this motionally induced
magneti, field fluctuation is proportional to H O times the
lunar magnetic Reynolds number and is toroidal with respect
to the x-axis, the direction of the incident motional electric
field. It.is the time-dependent analog of the steady state
unipolar generator solution for the induced magnetic field
inside the moon [SONETT and COLBURN, 1968]. The disturbance
magnetic field in the solar wind is also toroidal about the
x-axis and is confined to a chin layer with length scale (o 2P:u) 2
adjacent to the lunar surface.
From the theory presented here [see equation (21)x], data
from lunar surface magnetometer experiments can be used to
evaluate a, and consequently obtain knc ! vledge of the thermal
state of the lunar interior. The normal component of the magnetic
field fluctuation at the lunar surface will equal the normal
component of the incident interplanetary magnetic field. After
subtracting the tangential components of the forcing inter-
p l anetary field  from the lunar surface field,  the net fluctua-
tions  •rill give a direct measure of the lunar magnetic Reynolds
number. In a future publication we will consider the application
of this theory to a two-layered lunar conductivity model.
-10-
Acknowledgments
The authors Would like to thank Dr. C.P. Sonett
for encouraging us to consider this problem. We also
thank Dr. Sonett, Dr. D.S. Colburn and Dr. E.J. Putzer
for helpful discussions. This paper is based on NASA
CR 7-
	
prepared for Ames Research Center under NASA
Contract NAS 2-5153.
REFERENCES
Blank, J.L. and W.R. Sill, Response of the Moon to the time-
varyin g interplanetary magnetic field, J. Geophys. Res.,
74,736, 1969.
Coleman, P.J., Jr.,Turbulence, viscosity and dissipation in
the solar wind plasma, Astrophys. J., 153, 371, 1968.
Hollwcg, J.V., Interaction of the solar wind with the moon and
formation of a lunar limb shock wave, J. Geophys. Res.,
73, 7269, 1968.	 —^
Sonett, C.P. and D.S. Colburn, Establishment of a lunar uni-
polar generator and associated shock and wake by the solar
wind, Nature, 216, 340, 1967.
Sonett, C.P. and D.S. Colburn, The principle of solar wind
induced planetary dynamos, Phys. Earth Planet. Interiors,
1, 326, 1968.
Stratton, J.A., Electromagnetic Theory, McGraw-Hill, New York,
1941. 


Transient electromagnetic interaction of the moon with the solar wind

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