Electromagnetic induction in a stratified Moon with a trailing cavity is ... the Moon. The magnetic field due to these currents does not penetrate into the highly.
ELECTROMAGNETIC
INDUCTION
IN THE MOON*
L. L. VANYAN and I. V. EGOROV
Institute of Cosmic Physics, Academy of Sciences, Moscow, U.S.S.R.
(Received 18 December 1974) Abstract. Electromagnetic induction in a stratified Moon with a trailing cavity is discussed. The influence of the Moon wake is studied by using a two-layer lunar model with a perfectly conductive core. The magnetic field is shown to be independent of the wake length when that quantity is greater than 3 lunar radii. Regions on the sunlit and dark sides where the magnetic field may be described in terms of its first spatial harmonic have been distinguished, together with the corresponding errors admitted. It is in these regions that the electrical conductivity of the Moon can be found with very high accuracy, by simultaneous observations on the lunar surface and in the undisturbed solar wind. Results of these observations can be conveniently related to values of the apparent resistivity. At the basis of numerical methods of investigation of the electromagnetic induction in the M o o n it is necessary to take into account the asymmetric lunar wake. The region. where the main contribution to the interplanetary spectrum of the magnetic variation is in the first spatial harmonic is selected. In such regions the distribution of electroconductivity may be determined either by the increase of the variation of the interplanetary field by day or its diminution by night, or by the apparent resistivity in the harmonic and transient regimes. It is well known (Blank and Sill, 1969; Sonett et al., 1971; Sonett et al., 1972) that the solar wind has an immediate interaction with the illuminated side of the Moon, which then forms behind it a significant lowering of the concentration of the solar wind plasma over a length of several lunar radii. Inhomogeneities of the magnetic field, 'frozen' into the solar wind induce electric currents in the conductive regions of the Moon. The magnetic field due to these currents does not penetrate into the highly conductive solar stream, but gives rise to a considerable compression. Propagation of the induced field into the conducting body of the M o o n is accompanied by heavy attenuation. I f we are interested in measuring depths to hundreds of kilometres it is necessary to observe oscillations with periods delayed by tens of seconds. We shall investigate such slow processes neglecting the flight time of inhomogeneities past the M o o n which take only a few seconds. This circumstance in combination with the large (in comparison to the M o o n ) scale of inhomogeneities of the interplanetary magnetic field allows us to consider it to be spatially uniform and varying only in time. At higher frequencies, taking into account the finite flight time of the inhomogeneities leads to a number of important effects. In all studies devoted to the detailed analysis of magnetic field induction (Blank and Sill, 1969; Schubert and Schwartz, 1969), asymmetries of the lunar wake were neglected. This leads to a considerable simplification of the mathematical exposition, * Translated by Miss Eva Vokfilov~i of the Astronomical Institute, Charles University, Prague, Czechoslovakia. The Moon 112(1975) 277-298. All Rights Reserved Copyright © 1975 by D. Reidel Publishing Company, Dordrecht-Holland
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L.L. VANYAN AND I. V. EGOROV
but at the same time the question arises as to the degree of correspondence of the simplified model with reality.':" In this paper we shall consider the influence of the extent of the lunar wake on the character of the electromagnetic induction. It is evident that the greater the compression of the magnetic field the more noticeable should be the effects of asymmetry. Sonett et al. (1971) investigated the relationship of the intensification of the interplanetary field strength/~ on the oscillation period. The increase is a m a x i m u m for oscillation periods T < 100 sec. The field may then attain a value o f / ~ 3 to 4, but with an increase in T of approximately one second. We shall consider the influence of the lunar cavity at relatively short periods where the induced field is greatest. It is clear that the effect will decrease in other cases. According to Sonett et al. (1971), Vanyan et al. (1971), in the M o o n we may distinguish between the upper envelope, where the conductivity is as small as 10-6 to 10 -8 ohm -1, and the much more conductive interior regions ( ~ 10 -3 o h m - l ) . At fairly short periods this internal region behaves like a perfect conductor in comparison to the envelope.
Y Fig. 1. Model of the induction field around the Moon with wake.
1. Numerical Solution On the basis of what has been said, we may estimate the role of asymmetry in the case of a simple two-layer lunar model consisting of a perfectly conductive core of radius R, surrounded by a dielectric medium with outer radius a. This model is shown in Figure 1. The surface P1 for Z ~~0 the surface is defined * Since completing this paper the authors have received preprints (Schubert, Sonett, Schwartz and Lee, 1972) in which an infinitely extended lunar wake has been considered. This represents a limiting case of our model.
279
ELECTROMAGNETIC I NDUCTI ON IN THE MOON
by an ellipsoid of revolution whose equation assumes the form Z = L ~ / I - (x 2 +
y2)/a2,
where L is the wake length. At low frequencies the current displacement may be neglected. Then the upper nonconductive layer in this model of the Moon is effectively devoid of plasma, as is the cavity behind the Moon and magnetic field fluctuations are quasi-static. They satisfy the condition r o t H = 0 although they do also vary in time. If the magnetic field due to currents induced in the Moon does not spread out through the solar wind, then outside the lunar cavity there is only the interplanetary field He. In particular, on the outside of the boundary of the surface P1, the normal component of the magnetic field is equal to the corresponding projection of the interplanetary magnetic field Hen. As a consequence of the continuity of 14,,this value will be carried through to the inside of the surface P1. Concerning the surface of the interior core P2, as a result of the assumed perfect conductivity the normal component of the magnetic field must vanish here. Hence, we have the internal boundary problem consisting of the equations rot/4 = 0, div/7 = 0, = (He.),1.
(H.),2 = o.
The required magnetic field may be expressed in terms of a scalar magnetic potential ( H = - grad U), as a solution of the internal Neumann problem AU=0, ~U ~n
fO, atP2 ( f ( ~ , 2), a t P 1 .
(1.1)
The f u n c t i o n f (~9, 2) may be expressed in terms of/4e over the boundary PI, so that f (~, 2) must satisfy the condition
f
f'(~, 2) ds = O.
(1.2)
P~
We choose the centre of the Moon as the origin of a spherical system of co-ordinates with polar axis coinciding with the axis of the cavity (see Figure 1). The solution of the Neumann problem may then be expressed in the usual form 1
( r t + a p . - t - i ) p~, (cos ~) ("tin COSm2 +
firmsin m2),
(1.3)
1=0 m=O
where the constant coefficients a~, cq,,, film, are determined from the boundary conditions.
280
L.L. VANYAN AND I.V. EGOROV
Pz
Over the surface
we have
P2
=
r=R
(1.4)
~, [IR ;-1 - (1 + 1) atR -t-2] x t=l m=0
P~' (cos ¢)
(aimcos m2 + fit,. sin m2)
= O,
from which l at
=
_ _
l+ 1
(1.5)
RZt + 1
Substituting Equation (1.5) into (1.3) we obtain U=
~ ~ (
IRzt+a -t-l)
rz+ - -
t=l m=O
r
P~' (cos ~/) x
/+1
x (e;m cos m2 + fi,,. sin m2).
(1.6)
We may determine f (O, 2) on the assumption that the outer field is homogeneous, with components Hex, Hez. In spherical co-ordinates we may then write
/-/~={gexsinOcos2+HezcosO, HexCOSOCOs2-HezsinO, -Hexsin,~}, (1.7) from which
-f=-
>3600 sec,/~,~ 1. 2.4. INTENSIFICATIONDUE TO A TRANSIENTEVENT Using the Fourier transform and a complex frequency characteristic, it is not difficult
ELECTROMAGNETIC
INDUCTION
IN THE MOON
293
to obtain the increase of the horizontal magnetic field due to aperiodic excitation of the Moon by a sudden variation of the interplanetary magnetic field. For example, for a square-wave pulse, 1
H~(t)
e
= 2re j -
H o -- - dicoc o ,
(2.41)
oo
from which we obtain the increase in the static regime oo
h~(t) = ~
J h ~ ( c o ) - -ico - dco.
(2.42)
-oo
Transferring to a Fourier sine-transform, we may express/~= (t) as oo
2f
h~(t) = ~ Reh~
(co) sin cot dco.
co
(2.43)
0
It is well known that the initial stage of the magnetic pulsation may be represented by high frequency harmonics in the spectrum. Thus for any finite electroconductivity of the upper layer as t ~ 0 we find that oo
t - 1i~ d c o = a x / ~ t . h, (t) ~ ~1 fKlae_iO, 2
(2.44)
0
However, in the presence of a poorly conductive, fairly thin envelope the increase of magnetic field in the initial stage approximates to the familiar asymptotic value of h~ (t) = a/2di. As for the final stage of the transient process/~ (t) -+ 1, which reflects the delay of induced currents. An example of a transient characteristic is shown in Figure 5, evaluated by Phelan's method (Hemming, 1968) applied to Equation (2.43). 2.5. APPARENTRESISTANCE Let us turn our attention to the monotonic character of both [/~,(co)[ and/~ (t), which is independent of the variation of electroconductivity with depth. It would be desirable to have the results of magnetic field measurements in such a form as would be more obviously related to the distribution of electroconductivity in the Moon. In theory electromagnetic sond-techniques provide functions which satisfy the imposed requirements. This identifies a resistivity value which may be introduced in a suitable region of the illuminated hemisphere in the following way. From Equation (2.44) we find that, as co -+ 0% co/2a2 01-4 (,o)?" If we make use of this equation, over a wide band of frequencies and not just in the
294
L . L , V A N Y A N A N D I. V. E G O R O V
high frequency range, we obtain the a p p a r e n t resistivity as 6 X 10 6
costa 2
le, I - 4 ifi~(co)l z - Tlfi~(co)l 2 • This is the result of averaging the true resistivity over a thin surface layer of the M o o n at a given frequency. Hence Qa is a function of co. In a similar way, we m a y determine the a p p a r e n t resistivity in the transient regime /~a 2
3 x l0 s
~Oa= 4~th~ (t) -
t/~z (t)
The apparent resistivity (Figure 6) increases with T or t at relatively small values of these variables, after which a characteristic bend occurs, due to the presence of the conductive layer. F o r large T and t b o t h [Q,(co)[ and O,(t) a p p r o x i m a t e to an asymptotic value which corresponds to the considerable increase h~ (co) = 1;~( 0 = 1. In this region the induced magnetic field declines very little in c o m p a r i s o n to the first situation, and, correspondingly, provides little information a b o u t the internal electroconductivity.
V-T, V-t, sec 1/2 2
z,
6 8 10
i
i
i
i
i
20 i
40 6 0 80100 i
i
i
i
104
103
~j
Wga,ohm.m Fig. 6. Curves of the apparent resistivity.
ELECTROMAGNETIC INDUCTION
IN THE MOON
295
3.1. THE DARK HEMISPHERE
It may be seen from Figures 2 and 3 that the magnetic field on the unilluminated hemisphere essentially depends on the length of the cavity. However, like the sunlit side, this dependence practically disappears when L >~3a. Let us consider the tangential component of the magnetic field along the principal meridian with an external field directed at right angles to the cavity axis (Figure 3). With increasing separation from the antisolar point along the midday-midnight meridian up to about + 72 ° latitude, the value of H~ hardly varies (H~,= 1.7 H0). This is an intermediate value between H~ = 3.67 Ho, which corresponds to a Moon symmetrically scorched by the solar wind, and H~ = 1.32 H o which corresponds to a Moon immersed in a perfect vacuum. Hence the current contribution at the cavity boundary to the tangential component of the night-side magnetic field is rather significant. If we take into account that the tangential component at the surface of the vacuum-immersed globe includes an original magnetic field contribution, which is more than double the induced field, then we may obtain results which are relatively insensitive to the internal electroconductivity distribution. The radial component is very much more sensitive. 3.2. THE RADIAL COMPONENT OF THE MAGNETIC FIELD OVER THE DARK SIDE OF THE MOON
If the external field is perpendicular to the cavity axis then the radial component for latitudes in the range - 4 5 ° to 45 ° does not exceed 10% of Ho. Significantly larger values of Hr are found in the case of longitudinal polarization which is the basis of study of this component. The dark side of the Moon, unlike the illuminated side, is covered not by the solar wind plasma, but by a gas of much greater cross-section. In the zero-order approximation this may be taken to be a vacuum. For this reason the normal component of the induced field does not vanish at the lunar surface. It annihilates the normal component of the external field at the surface of the conductive core and significantly decreases that field over the dark side of the adopted model. From calculations it follows that for L>~ 3a the net radial field (initial plus induced) amounts to 0.45 H o at the antisolar point. At the same time, without taking into account the influence of the cavity boundary the reduced field/~n = Hr/Ho would be given by /~,= 1 - (R/a) 3 =0.36. Consequently, the contribution of currents induced TABLE I
R/1500 1 0.95 0.9 0.8 0.7
L
L
M,~/h-~
(sym.)
(asym.)
(%)
0.36 0.445 0.535 0.67 0.78
0.45 0.50 0.62 0.75 0.85
25 12 15 12 9
296
L . L . VANYAN AND I. V. EGOROV
over the cavity boundary amounts to 25% of the total effect in the case considered. Results of calculations carried out for different sizes of conductive nucleus are given in Table I. In the second row the reduced field is given without taking account of the cavity 'wall', and below there are given the relative contributions of the boundary currents. Evidently for/~, < 0.6 it is necessary to take into account compression of the magnetic field for the dark side. With increasing distance from the antisolar point the influence of the cavity wall diminishes. It is possible to select a region suitable for sonding, within which the contribution of boundary currents on the radial field is less then 25%. The angular extent of this region (centred on the antisolar point) is given in the lowest row of Table II. TABLE II
L
.4L/#.
si e of
(%)
suitable region
0.36 0.54 0.67 0.78
25 15 12 9
18° 30° 37° 45°
With decrease of the cavity length the induced magnetic field also decreases so that the net field becomes less related to the internal conductivity distribution and approximates to H o (see Table III). Since the radial magnetic field on the dark side responds to the lunar model, this may be used in order to determine the lunar electroconductivity. If, with the aid of Table II, we may take account of the cavity wall, then the problem TABLE III L/a
HdHo
3 1.4 1.2 1.1
0.45 0.5 0.6 0.7
1.0
1.0
reduces to that of a layered sphere immersed in a vacuum. It is well known that an external uniformly varying magnetic field will induce currents in a sphere, whose field extends into the external medium (as at the sphere's surface) and is equivalent to that of a magnetic dipole, whose moment we may represent by M. The components of this field at the surface of the globe then take the form Ho = - Ho (1 + ½M) sin 0,
Hr = Ho (1 -- M) cos 0. It is interesting to note the ratio Ho/Hr c o t 0 = (1 + { M ) / ( M - 1 )
depends only on the
ELECTROMAGNETIC I NDUCTI ON IN THE MOON
297
electroconductivity distribution within the Moon and not on the properties of the external medium. Hence, we may compare its value o f / ~ , from which M = (/~ + 1)/ (/~-½). Using the expression for ]~, we find that M = 1 - ( 3 / K t a ) (R 1 - 1/K~a), and the radial component of interest to us then assumes the form Hr=Ho~
R1-
cos0.
Noting that the radial component of the original field Hro = Ho cos 0, we find the weakened magnetic field over the dark side of the Moon becomes ,, h" (o))
-
3(
U,o
1)
-
-
It is not difficult to obtain a simple relation between/~, (co) and h~(co): i.e., /~ (co) = 3/[1 + 2/~ (e))]. If by d a y / ~ (o))= 4, then the radial component decreases by a factor of 3 at night, although in the low frequency domain as Ih=(o))l--, 1, Ih.(o))l also tends to unity. For very large values of IKta I the normal component tends to zero as 3/K~a. However in a single important case the modulus of reduced magnetic field has a finite high-frequency asymptote. This case involves the poorly conductive layer which surrounds the relatively highly conductive nucleus. If we may neglect the fly-by time of inhomogeneities in comparison with the oscillation period, we find that /~. (ca) = 1 -
~
a
.
(3.21)
The qualitative example of Figure 5 shows an amplitude-frequency characteristic for the reduced normal component of the magnetic field on the dark side. The electroconductivity distribution is the same as that previously used in the determination of the day time intensification. 3.3. DIMINUTIONDUE TO A TRANSIENTEVENT If the external field does not vary sinusoidally the normal component of the nocturnal magnetic field may be easily found using the Fourier transform
H n (t) = 2~
#ln (co) Ho (03) e -i°~t do).
- oo In particular, for a perpendicular step we have
U.(t)
=
H° i -oo
e-lOt
;'" (o)) -
do).
298
L . L . VANYAN AND I. V. EGOROV
Consequently the intensification in the pulse regime assumes the form 1 (" h, ( t ) = 2(-c I
e -i`°t 2 - do = - io9 n
OO
sin ogt R e h , (o9) - o9
do9.
- o o
The diminution in the radial magnetic field due to excitation from square-wave pulsations was calculated with the aid of this last equation and results are shown in Figure 5. Similarly, it is easy to determine the asymptotic behaviour of the magnetic field on the dark side of the Moon. Evidently, as t--+ 0% h , ( t ) ~ 1. If t ~ 0 , Ho ~ 3 e -i'~t do9. H n(t) ~ ~ k ~ a - i~o
J
Hence, for any finite electroconductivity of the upper layer the disturbed radial magnetic field grows in the initial stages in proportion to t. If the upper layer is perfectly insulating, Hn (t) at small time values is practically independent of t and is determined by the same Equation (3.21) as the high frequency asymptotic ~, (co). In Figure 5 the region of weak dependence on time begins at t~0.1 sec. 4. C o n c l u s i o n s
Numerical calculations of electromagnetic induction in the Moon, taking into account asymmetry of the lunar cavity have been carried out on the assumption that the fly-by time of solar wind inhomogeneities past the M o o n is small in comparison to the field oscillation period. The results show the necessity to take into account the influence of the cavity boundary both on the daytime tangential component and the nocturnal radial component. Regions suitable for electromagnetic sonding are situated around the subsolar and antisolar points. The intensification of the interplanetary magnetic field variation by day and its diminution at night (both in the harmonic and transient regimes), which may be determined from synchronous observations on the M o o n and in the solar wind, provide information about the electroconductivity distribution inside the Moon. The apparent resistivity appears in a convenient form in presenting the results of such observations. References
Blank, J. L. and Sill, W. R. : 1969, J. Geophys. Res. 74, 736-743. Hemming, R. V. : 1968, Numerical Methods, Izd, Nauka, Moscow. Schubert, G. and Schwartz, K. : 1969, The Moon 1, 106-117. Schubert, G., Sonett, C. P., Schwartz, K., and Lee, H. J. : 1972, 'The Induced Magnetosphere of the Moon', preprint, NASA TM X-62, 227. Sochelnikov, V. V. : 1968, Physika Zernli, No. 7, 65-71. Sonett, C. P. et al. : 1971, Nature 230, 359-362. Sonett, C. P. et al. : 1972, Nature 238, 145-146. Vanyan, L. L. et al." 1967, Depth Electromagnetic Soundings, New York. Vanyan, L. L., Berdichevskii, M. N., Egorov, I. V., Krass, M. S., Okulesskii, B. A., and Fadeev, E. V. : 1971, 'Depth Electromagnetic Sounding of the Moon'. Communication to the XV General Assembly of MAGA, 1971.
