global electromagnetic induction in the moon and planets - MTNet

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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.)
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