Adv. SpaceRes. Vol. 10, No. 1. (1)55—(1)64. 1990. 0273—1177/90$0.00+ .50Printedin Great Britain. All tightsreserved. Copyright© 1989 COSPAR

PLANETARY MAGNETIC FIELDS:A COMPARATIVE VIEW

Michael Schulz and George A. Paulikas

The AerospaceCorporation, El Segundo,CA 90245, U.S.A.

ABSTRACT

The sums of the squares of the non—axial Cm ~ 0) spherical—harmonic expansion coefficients(g~,hm) yield a readily interpretable exponential spectrum when normalized so as to reflectthe g~obal energy content per degree of freedom and plotted against n for each magnetizedplanet. Indeed, the resulting spectra for Earth, Jupiter, and Uranus suggest an equiparti—tion of magnetic energy among the available degrees of freedom and yield reasonable valuesfor the nominal core radii r~ (= 0.432 RE, 0.756 R~, 0.464 RU), which can be interpreted asthe mean values of r within the respective fluid cores. Results for Saturn are indeterminatebecause the most reliable field models are axisymmetric. Results for Mercury are ambiguousbecause spatial coverage during the Mariner—lO encounters was too limited. Results for Nep-tune await encounter by Voyager 2. The implications of maguetic—energy equipartition for dy-namo theory are unclear but presumably significant, since equipartition seems to suggest theimportance of the a—effect (turbulent—eddy formation) for the symmetry—breaking required byCowling’s Theorem in the context of an a—~~dynamo. It seems significant, moreover, that theequatorial projection of the dipole moment belongs to the same exponential spectrum as theanalogous projections of the quadrupole and higher moments. If (as seems probable) the spec-trum persists during a magnetic reversal (characterized by the condition g~ 0), then thedipole moment at the time of polarity transition would be perpendicular to the planetary ro-tation axis and about 20% as strong as the time—averaged moment, which is inclined ._100 tothe rot~tion axis. Observation of a 58.60 inclination for Uranus’ magnetic axis suggeststhat Ig~I is abnormally small at the present epoch (i.e., that Uranus is in the process ofundergoing a magnetic reversal). A more usual (_.100) inclination typical of Earth and Jupi-ter at times not near polarity transitions would correspond to a dipole moment ~ 1 G—R~andan offset 0.07 RIT for Uranus and (by im~lication, according to Blackett’s Law and similarconsiderations) a dipole moment i~ — 1 G—RNfor Neptune.

BACKGROUND

The usual opening of a paper on this topic includes a plot (see Figure 1) of magnetic moment(i~) against angular momentum (L) for various objects in the solar system, in order to deter-mine whether Blackett’s Law (which asserts that ~ ~ L) is satisfied. It is not clear howseriously to take the superficial agreement of Figure 1 with Blackett’s Law, however, sinceit is known that magnetic moments vary with time, and it is not clear whether Blackett’s Lawshould apply to ~ itself or in

9ead to the component of ~ in the direction of the rotationaxis ~2 (i.e., to the quantity a where a is the planetary radius and g~is the spherical—harmonic expansion coefficient corresponding to the axial part of the dipole moment.

HARNONICSPECTRA

The real purpose of this paper is to consider the construction of “spectra” from spherical—harmonic coefficients (g~,h~) other than ~ so as to learn what an optimally constructed‘spectrum” may reveal about the physical principles underlying magnetic dynamos, the dimen-sions of the dynamo region for each magnetized planet, and the satisfaction of Cowling’s The-orem (which requires a self—maintained dynamo field to be other than axisymmetric). For thispurpose it is convenient to define the quantities

5 ~ [(gm)2 + (hrn)2] (is)

and

~ [(gm)2 + (hrn)21 = Sn — (g~)2, (ib)

(1)55

(1)56 M. Schulzand G. A. Paulikas

8 —

—2 - —

I I I I I I I—4 —2 0 2 4 6 8

Iog10(L/L~)

Fig. 1. Empirical test /1,2/ of Blackett’s Law (proportionality of magnetic momentto angular momentum L) for the Sun (0) and various planets (denoted by initial

letters of names, except for Mars, ci), with revised values /3/ of L for the giantplanets. Note: (E) denotes Earth—Moon system /2/; (N) represents Blackett’s—Lawprediction for Neptune; V and (V) correspond to different interpretations of Venera—4 and Mariner—5 data /4/, with (V) representing a proposed upper bound.

lO~ I I I I I+

-

IGRF 1985.0

-z •.. -

10 ~

- N.,~

-4 — ,‘Z.~,, -10 ~N..

- ~

~ io6 - + + -

U ~+ (n+1)S~

U) -

(n+1)S,,2n(2n+1)

10 — (n+1)5~ —

0 2 —

—10 I I I I I I I I

10 0 2 4 6 8 10

n, degree

Fig. 2. Harmonic spectra of geomagnetic field /9/ normalized as by Loves /5/ so asto reflect contribution (+) to <32>and alternatively so as to reflect global mag-netic energy content per degree of freedom (d.f.), including (•) and excluding (0)m 0.

PlanetaryMagnetic Fields (1)57

using Schmidt normalization for the spherical—harmonic expansion coefficients (gm,h~) corre-sponding to degree n and order m. The quantity usually plotted against n (e.g. Y5,6/) is(n+l)S~. since it is easy to show /7,8/ that the mean—square magnitude of B (averaged overthe planetary surface) is given by

<82> = (n + 1)S. (2)

The results are shown in Figure 2 for a typical geomagnetic field model /9/. The plot issemi—logarithmic, and the crosses (+) denote the values of (n+1)S for 1 ~ n ~ 10. The re-sulting “spectrum” seems to be fitted fairly well for 3 ~ n ~ 10 ~y the dashed straight linewhose equation is (n+1)S~ — 0.325(0.487) n G

2. As usual, however, the data show an excessin the dipolar (n = 1) term and a deficit in the quadrupolar (n = 2) term, compared to this“best—fitting” straight line (cf. /6/). The usual interpretation of the straight line isthat the various terms in (2) would contribute equally (for 3 ~ n 5 10) to the value of <B2>on a sphere of radius r 0.487 RE if the spherical—harmonic expansion of B — VV could beextrapolated that far inward from the Earth’s surface. Such an extrapolation is not liter-ally permissible, of course, since the fluid core (which presumably carries dynamo currents)extends from r = 0.191 RE to r = 0.547 RE /10/.

Moreover, it seems to us that (n+1)Sn is not the optimal quantity to be plotting against n.From a fundamental (thermodynamic) standpoint, a more significant quantity to plot against nwould be a quantity indicative of the global energy content per degree of fr~edom (d.f.).It is shown below that quantities such as ((n+i)/2n(2n+1)]S~ and (n+1)(S~j/4n ) have this de-sirable property. These are plotted by means of filled and open cir~les, respectively~ inFigure 2. Both seem to satisfy the same spectrum, viz., (n+i)(S~1~/4n ) 0.0091(0.432) ~ G2/d.f., for n ~ 2. Indeed, several of the filled circles in Figure 2 obscure the correspond-ing open circles. There is no longer a deficit in the quadrupolar (n 2) term relative tothe “best—fitting” straight line. Moreover, even the non—axial (m = 1) part of the dipolemoment (open circle, n = 1) seems to satisfy the same spectrum.

The rationale for having plotted quantities such as [(n+i)/2n(2rt+1)]S~ and (n+1)(S~/4n2) inFigure 2 follows from an admittedly oversimplified description and model of the geomagneticdynamo field. Specifically, the fluid core is idealized as a geocentric spherical shell ofradius r~ (to be determined). The model B field is derived everywhere else from the scalarpotentials

V(r,8,~) a~ ~ (a/r)~l[gm cos m~+ h~sin m~]P~(0), r > r (3a)

n1 m0

and (to make ~B continuous at r = r~)

V(r,8,~) = - a ~ [~±~_](a/r~)2~+1(r/a)m(gm cos m~+ h~sin m~]Pm(O), r < r, (3b)

n”i m0

since it is assumed that currents are restricted to the spherical shell r = rc. The parame-ter a in (3) denotes the planetary radius, and the associated Legendre functions P~(e) areSchmidt—normalized. The global energy content corresponding to (3) can be derived by brut~force, but it is more convenient to invoke Gauss’ divergence theorem. The result (since V V= 0 for a spherical—harmonic expansion) is

2ir +1 m

u .g.L j J f (V V)2 r2 dr d(cos e) d~, r ~ rc (4a)

2Tr +1 m

= .~J_J J j v~(vvV) r2 dr d(cos 8) d~, r ~ rc (4b)

2 2ir +1

= —~ J J [V(r,0,~) - V(r~,8,~)](dV/8r) d(cos 8) d~ (4c)8ir 0 —i C

Na 2 2n+3 I (0+1)2 n+1~

= ~ r (air) Ln2n-+-U + ~4tj S~ (4d)

n= 1

(1)58 M. SchulzandG. A. Paulikas

— a~ ~ [ii] (a/r~)2~~~ ((gtfl)2 + (h~)2]. (4e)

The number of degrees of freedom (d.f.) is 2n+1 if the m — 0 term i~ (4e) is included and 2nif the axial (m — 0) term is excluded.

The filled circles in Figure 2 denote the product of (n+1)/2n with S0/(2n+1). The open cir-

cles denote the product of (n+1)/2n with S~j/2n. Both are indicative of energy content perdegree of freedom in the sense that the contribution to U per degree of freedom is equal toa

3(a/rc)2h1+~ tim!s the quantity plotted. Since the data seem to satisfy a spectrum of theform (n+1)(S~/4n ) —

0~0091(0•432)2nG2/d.f., it follows that a value of r~ 0.432 RE would

correspond to an equipartition of global energy content among the non—axial (in # 0) degreesof freedom for 1 s n s 10. Indeed, it has been mentioned above that the values of ((n+1)/2n(2n+1)]S

0 satisfy the same spectrum for 2 ~ n s 10. This suggests that the equipartitionextends even to the axial (in — 0) degrees of freedom for a ~ 2.

Strictly speaking, the expression for U derived in (4) includes only the poloidal degrees offreedom, whereas the dynamo field must involve toroidal components as well. The toroidalfield would be restricted (in the present idealization) to the surface of the sphere r =

and so (unless it were infinite there) it would not contribute to the global energy content.More realistically, the toroidal field would extend throughout the fluid core (0.191 RE ~ r5 0.547 RE) and probably into the metallic solid core (r s 0.191 RE) as well. It seems like-ly from the thermodynamic perspective that each (unobserved) toroidal degree of freedom wouldcontribute as much to the global energy content of the B field as each poloidal degree offreedom has been observed to contribute. In other words, the principle of equipartitionshould apply equally to toroidal and poloidal degrees of freedom, in which case the actualvalue of U should be twice that estimated via (4).

The thermodynamic interpretation of the solid line in Figure 2 thus suggests the formation ofmutually coupled turbulent eddies (the so—called a—effect) in the fluid core as a means ofachieving the symmetry—breaking required by Cowling’s Theorem. This, together with differen-tial rotation of the fluid, seems capable of producing an a—udyn~ipo (cf. /11/). It is note-worthy in this context that the global energy content (m 2.90X 10 erg) of the axial—dipolefield, i.e., of the term (n,m~4= (1,0) in (4), is greater than the global energy content perdegree of freedom (m 5.50X10 erg/d.f.) form s~ 0, but only by a factor—SO. The number ofnon—axial degrees of freedom for n s N in (3) is N(N+1), which amounts to 110 for N = 10 (cf.Figure 2). The spectrum compiled by Langel and Estes /6/ from MACSATdata suggests thatequipartition remains valid at least to N 12 (156 non—axial d.f.), but that crustal anoma-lies dominate (and thus obscure) the core contributions for n ~ 14. It follows in any casethat the aggregate global energy content of the non—axisymmetric degrees of freedom signifi-cantly exceeds that of the axial dipole.

It turns out that the value of r~ (= 0.432 RE) deduced from Figure 2 (solid line) is remark-ably close to the mean value of r within a more realistic terrestrial fluid core, viz.,

r r 4 4(23 (2 2 3(r2—r1)

— r dr + r dr = ~ 0.422 ~ (5)J .1 4(r2—r1)

where r1 0.191 RE and r2 = 0.547 RE /10/. A discrepancy of only 0.01 RE between <r> in (5)and rc in (4) strikes us as an encouraging result that lends support to our thermodynamic in-terpretation of the spherical—harmonic spectra shown in Figure 2. Other investigators, how-ever, might regard this result as too good to be true.

Spherical—harmonic spectra constructed by the present method for Jupiter and Uranus are shown(for comparison with Earth) in Figure 3. Filled and open squares (circles) correspond toEarth (Uranus) at values <0.04 G

2/d.f.; the terrestrial results are identical (of course) tothose shown in Figure 2. The Uranian results are based on the Q.~ model of Connerney et al./12/, who caution that the octupolar (n = 3) coefficients determined by least—squares fromthe Voyager—2 data are “not resolved” in the s~nseof generalized—inversetheory. The expo-nential spectrum (n+1)(S~/4nL) =

0•0881(0•464)L0 G2/d.f. represented by a straight line in

Figure 3 has accordingly been determined solely from the non—axial (m ~ 0) dipolar (n 1)and quadrupolar (n 2) terms. Although characterized as “not resolved,” the best—fittingUranian octupolar (n 3) coefficients deduced from the Voyager—2 data seem to satisfy thesame spectrum. The significance of this finding is debatable, but it seems to support athermodynamic interpretation of the Uranian spectrum of spherical—harmonic expansion coeff i—dents (g~,h~), just as Figure 2 seems to support a thermodynamic interpretation of the ter-restrial spectrum.

The Jovian spectrum (n+1)(S~/4n2) = 0.452(0.756) 2n G2/d.f. in Figure 3 is based on various

PlanetaryMagneticFields (1)59

100 - I I —

JUPITER -

lO~

- U EARTH + (g~)2 -

- URANUS ° (n+1)S1

1j -

i02 __

EARTH ‘~ ~‘, -

I~(IGRF~~\

n, degree

Fig. 3. Harmonic spectra of planetary magnetic fields, normalized so as to reflectglobal magnetic—energycontent per degree of freedom (d.f.), including (filled sym-bols) and excluding (open symbols) in = 0. Crosses(+) correspond to (n,m) = (1,0)term only.

Pioneer—il and Voyager—i models /13—15/. Filled plotting symbols include the in = 0 term (ifavailable), and open plotting symbols exclude it. The filled diamond (~)obscuresthefilled circle (•) in the quadrupole (a = 2) column and the open circle (0) in the octupolecolut~n. T~eopen square (0) in the octupole (a = 3) column denotes (1/6)((g~)2 + (h~)2 +

(g~) + (h3)

2], which is the mean contrib8tio~ to (nt1)(S~j/2n) or to (n+1)(S~72n) per re-

solved degree of freedom; no valu~s for g3, g3, or h3 were specified by Connerney et al./15/. The Jovian values for (g~) and are off—scale by 1.2—1.3 and 0.7—0.8 decades, re-spectively, since the angle 8~Letween the dipole axis and the rotation axis is only about10 . The axial part of the dipole field thus contains about 70 times as much global energyas the “typical” (n,m) ,~(1,0) degree of freedom. The corresponding terrestrial ratio (seeabove) is —50.

It follows f~om (4e~ t1~at the global energy content per “typical” ~m ~ 0) degree of freedomis U~/2n = a~(a/rc) ~ ~n+1)(S14j/4n

2) and from Figure 3 that (a/rc) ~‘(n+1)(S~/4n2) is a plan-etary constant (0.452 GL/d.f. for Jupiter, 0.0881 G2/d.f. for Uranus, 0.009? G2/d.f. for theEarth). Figure 3 further im~lies that rc/a = 0.756 for Jupiter, 0.464 for Uranus, and 0.432for Earth). The values of a for these three planets are related as 1403:63:1. Figure 4tests the hypothesis that the global magnetic—energy content per “typical” degree of freedom

JASS 10/1—5

(1)60 M. Schulz andG. A. Paulikas

I I (U1t

10~— E~ —

10-2/I I I I I I -

102 io0 102 1o~ 106

(LQ/v) 0=1 LQ

Fig. 4. Empirical test of proportionality of global magnetic—energycontent per de-gree of freedom (d.f.) to rotational kinetic energy for three planets. Parenthesizedsymbols test the same relationship per unit planetary volume.

(m5.50X1024 erg/d.f. for Earth) might be proportional to the rotational kinetic energy (LW

2) of the planet (m3.38X1035 erg for Earth), where L is the angular momentum and ~2 is therotation rate (angular velocity). This is a new hypothesis not to be regarded as a corollaryof Blackett’s Law. The non—parenthesized data points in Figure 4 constitute a plot of U1/2nagainst L~2, relative to the terrestrial values of these parameters, and agreement with t~ediagonal straight line (indicating direct proportionality) seems fairly good. It might beargued, of course, that the factor a3 (which is not a magnetic property of any planet) ac-counts for most of the interplaneta~y variability of U~/2n in Figure 4, whereas the moment ofinertia (which is proportional to a for a given density profile) accounts for most of theinterplanetary variability of L~there. Perhaps a better test of any hypothesized relation-ship between planetary magnetic and kin~matical properties could be achieved through divisionof each by the planetary volume V = 4ira~/3. The parenthesized data points in Figure 4 re-flect this latter test (viz., of the proportionality of U~/2nV to LWV). The correlationpersists, of course, but the argument for direct proportionality seems less compelling whenthe agreement spans only two decades instead of five. Uranus nevertheless occupies a posi-tion intermediate between Earth and Jupiter in Figure 4, just as it does in Figure 3.

Two other obviously magnetized planets (Mercury and Saturn) produce indeterminate spectra inthe sense that the contribution to (n+1)(S1/2n) per resolved degree of freedom cannot be ob-tained from an axisymmetric field model, whereas reliable values of g~and hin with in ,~ 0 seemnot to be resolvable for these planets from the Mariner—lO and Voyager data ?16,17/. Relia-ble values of S~j are likewise unavailable for Mars, Venus, and the various planetary satel-lites in the solar system. Preliminary analyses /18/ of the Sun’s spherical—harmonic spec-trum for various Carrington rotations suggest that rc — 0.8 R

5, which is in fact a represen-tative value of r within the convective zone.

INTERPRETATION

It seems from Figures 3 and 4 that Uranus’ magnetic field is anomalous only in the sense that(g~)

2 is anomalously small (—0.4 ~t)• The Uranian spectrum is otherwise qualitatively simi-lar to the terrestrial and Jovian spectra, and (as might have been anticipated) the Uranianspectral parameters are intermediate between the terrestrial and the Jovian. It seems natu-ral /19—21/ to interpret the anomalously small value of (g

1)2/Sf for Uranus as the signature

of a planet undergoing magnetic reversal (polarity transition). Indeed, Ness et al. /22/ hadacknowledged this as one possible interpretation of their preliminary data.

PlanetaryMagneticFields (1)61

T~emagnetic—reversal scenario contemplated here /23—25/ entails the continuous transition of

g1(t) through zero /19—211 from (for example) positive to negative values while Sf, defined

by (ib), remains nonvanishing. This is a scenario which has remained popular among space re-searchers (e.g. /26/) but which has been out—of—favor for at least a decade in the geomagne—tism/paleomagnetism community (e.g. /27—30/), mainly because it has not been supported by VGP(virtual geomagnetic pole) analyses of paleomagnetic data from different sites during thesame polarity transition (e.g. /31—35/). Such evidence against the present scenario and infavor of a purely or predominantly non—dipolar transition field is not to be dismissed light-ly, but neither does it seem entirely compelling. The geographical coverage available forpaleomagnetic studies is necessarily limited to a few sites, whereas observational coverageof the Sun (which tends to support the present scenario) is practically global (although theavailable 27—day time—resolution leaves much to be desired). A major discrepancy between theVGP paths inferred from the paleomagnetic data at two widely separated sites does, of course,rule out a purely dipolar transition field, but the present scenario entails a transitionfield for which

in ( 3+(r/a)2

2Sf + n(2n+i)(r /a)20_2 = c 2 3 — i~ sf (6)

n=2 c ([1 — (na) I )

at the Earth’s surface when g~ 0. Since this amounts to 4.9218 Sf for Earth (rc = O.432a),it follows that <B

2> consists of a 40.6% dipolar (n = 1) contribution, a 37.9% quadrupolar(n = 2~contribution, and a 21.5% contribution from higher (a ~ 3) moments at the instantwhen g

1 = 0. The corresponding percentages for Uranus (r = O.464a) at reversal would be35.4% dipolar (n = 1), 38.1% quadrupolar (a = 2), and 26,g% from n ~ 3. The Q3 model of Con—nerney et al. /12/ yields 2(g~)

2 — 0.0283 C2, 2St = 0.0760 G2, 3S2 = 0.0765 C

4, and (with thereservation that the octupolar coefficients are not r~solved” in the sense of generalized—inverse theory) 4S

3 = 0.0390 C2 as contributions to <BL> at the surface of Uranu~ during the

Voyager—2 encounter. In other words, the quadrupolar (a = 2) contribution to <B ) at r = RUis indeed about the same as the non—axial dipolar (a = n = 1) contribution.

The transition field described by (6) is, of course, incompatible with axisymmetnic (zonal—harmonic) models (e.g. /36/) of geomagnetic reversal. Indeed, the contribution of axisymmet—nc (in = 0) terms to (6) is only 0.5115 sf (i.e., about 10.4%) for Earth (rc = 0.432a) andonly 0.6240 sf (i.e., about 11.0%) for Uranus (rc = 0.464a), for which (6) yields <B2>5.6543 Sf at transition (i.e., for g~= 0). This is not necessarily an argument against thepresent model, however, since there is by now considerable evidence /30/ for non—axisymmetrictransition fields during at least some geomagnetic reversals.

To encounter a planet in the process of undergoing a magnetic reversal is not probable aon, but neither is it to be regarded as extremely improbable. The Sun, for example, rever-ses its magnetic polarity about every 11 years, and the a priori probability of finding theSun’s dipole moment tilted by >600 relative to its rotation axis is estimated to be —6%fromvarious models for the temporal variation of g~. The a priori probability of finding Earth’sdipole moment tilted by >60° relative to its rotation axis is thus—’O.O6s (i.e.,—0.3%),where � (— 0.05) is the probability /37/ that a magnetic excursion will develop into a mag-netic reversal (polarity transition). Hathaway and Dessler /38/ have argued (on the basis ofdynamo theory) that = 1 for Jupiter and Saturn, and Stevenson /39/ has proposed an atmos-pheric instability mechanism that would make = 1 for Uranus.

If Uranus is indeed undergoing a polarity transition at the present time, then Uranus’ pres-ent value of ~ (m 0.228 G—R~)must be regarded as anomalously small for the planet, just asthe present 58.6° tilt and O.3—RU offset of the best—fitting dipole (in uranographic coordi-nates) must be regarded as anomalously large. If, for example, the typical value of g~j forUranus were—8 times the value (m 0.119 C) inferred from the Voyager—2 data /12/, then Ura-nus’ typical value of ~ (— 1 G—R~)would place its data point in Figure 1 near the parenthe-sized N on the straight diagonal line (indicating conformity with Blackett’s Law relative toEarth). Under this same hypothesis about g~, moreover, the typical angle 00 between Uranus’dipole moment and rotation axis would be ~~120, and the offset r

0 of the best—fitting pointdipole from the center of Uranus would be—0.07 RU (under the assumption that sf and S2 wouldretain their present values). Uranus’ “typical” magnetic configuration would then not differsubstantially from Earth’s or Jupiter’s.

Figure 5 is a plot (normalized to Earth) of ~/V against L/V (i.e., of magnetic moment perunit volume against angular momentum per unit volume) for various objects in the solar sys-tem. This is essentially a test of Blackett’s Law (as is Figure 1), but one which removesthe correlation between a

3 and a5 (cf. Figure 4) as a source of correlation between the vari-ables plotted. Deviations from the diagonal straight line are (of course) the same here asin Figure 1, but they seem more significant in Figure 5 because the quantities plotted herespan fewer decades (orders of magnitude).

Whether or not Figure 5 is taken as a confirmation of Blackett’s Law, the structural and kin—

(1)62 M. Schulzand G. A. Paulikas

:: I I

(N)

0 F/I F) -

(V) —2 —~1 +I~ +2

log10(L VE/LEV)

Fig. 5. Empirical test of Blackett’s Law per unit planetary volume, employing thesame symbols as in Figure 1. Vertical dashed line denotes angular momentum of Nep-tune (relative to Earth) per unit planetary volume and intersects solid line (repre-senting Blackett’s Law, normalized to Earth) at (N), which corresponds to ~ — 1 G—R~.

ematical similarities that exist between Uranus and Neptune suggest that their “typical” mag-netic properties should be very similar to each other. Since Uranus and Neptune have (towithin a few percent) the same angular momentum per unit volume, Blackett’s ~aw would suggestthat ~ — 1 G—R~for Neptune (which would correspond in Figure 5 to ~i — 1 C—Ru for Uranus).There was no reason to expect a priori that Uranus would be encountered in the course of amagnetic reversal (i.e., that ~ would be as small as 0.228 G—R~at the time of the Voyager—2encounter). Similarly, there is no reason to expect that Neptune will be undergoing a mag-netic polarity transition in August 1989. The a priori expectation for Neptune, based on thearguments presented above, is a magnetic moment — 1 G—R~,which would correspond to an in-clination 00 — 100 of the dipole axis relative to the rotation axis and an offset r0 — 0.07RN of the best—fitting point dipole relative to the center of the planet. These numbers areuncertain by perhaps ±40%, in view of anticipated “se~ular” variations comparable to thoseexperienced by the Earth’s B field during the past 10 yr /37/, but deviations of andfrom their nominal values should be positively correlated with each other and anti—correlatedwith deviations of j.i from its nominal value—i C—Rd. This, therefore, is our prediction forNeptune. Voyager 2 will reveal the truth soon enough.

ACKNOWLEDGMENTS

This work was supported by the AerospaceSponsoredResearch (ASR) program of The AerospaceCorporation.

The authors are very grateful to J.E.P. Connerney, N.H. Acui~a, and N.F. Ness for having sup-plied (and for having granted permission to use) a table of values for the Uranian expansioncoefficients g~and h~prior to their publication /12/ of these Voyager—2 results, and to J.E.P. Connerney for having suggested that (n+1)(S~/4n

2) be plotted along with our originalquantity [(n+1)/2n(2n+1)]Sn for all values of n ~ 1.

The authors are pleased to thank Neil Divine /3/ for having privately communicated the fol—8

lowing values of planetary rotational angular momentum: 5.84 X1033 J—s for Earth, 4.26X io~

PlanetaryMagneticFields (1)63

J—s for Jupiter, 7.46 X1037 J—s for Saturn, 1.33X 1036 J—s for Uranus, and 1.17X 1036 J—s for

Neptune. These are the values being used by the Voyager Project in planning the August 1989encounter with Neptune.

Finally, it is a pleasure to thank J.C. Cain, D.J. Corney, G.L. Siscoe, R.C. Elphic, Cather-ine Constable, C.T. Russell, D.J. Stevenson, K.A. Hoffman, Scott W. Bogue, L.J. Lanzerotti,Gerald Schubert, and the late Allan Cox for various helpful and encouraging comments. We arelikewise grateful for the help of several colleagues who have commented anonymously on vari-ous aspects of the present work.

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phys. Res. Lett. 3, 125—128 (1976)

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the core, Geophys. J. Royal Astrom. Soc. 36, 717—730 (1974)

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