- Email: [email protected]
Mercury's thermoelectric dynamo model revisited
Planetary and Space Science 50 (2002) 757 – 762
www.elsevier.com/locate/pss
Mercury’s thermoelectric dynamo model revisited Giacomo Giampieri ∗ , Andr$e Balogh Space and Atmospheric Physics Group, Imperial College, The Blackett Laboratory, Prince Consort Road, London SW7 2BW, UK Received 19 October 2001; received in revised form 20 March 2002; accepted 25 April 2002
Abstract We reconsider the thermoelectric dynamo model for the generation of the magnetic /eld of Mercury, as proposed by Stevenson (Earth Planet. Sci. Lett. 82 (1987) 114). We brie4y review the status of our knowledge about the nature of Mercury’s planetary /eld, and describe the main features of the /eld generated through a thermoelectric dynamo, including its multipole expansion. We then discuss the possibility of testing this model, by correlating future measurements of Mercury’s magnetic and gravitational /elds. We conclude that the test is, in principle, very robust, and discuss possible limits on its interpretation. ? 2002 Elsevier Science Ltd. All rights reserved. PACS: 96.30.Dz; 96.35; 95.55.Pe; 91.10.Qm; 91.25.Cw Keywords: Mercury; Planetary characteristics and properties; Planetary probes; Gravity /eld harmonics; Dynamo theories
1. Introduction After Mariner 10’s unexpected discovery of Mercury’s planetary scale, mostly dipolar magnetic /eld (Ness et al., 1975), models of the planet’s thermal evolution and internal structure needed revisiting, in order to /nd a plausible explanation for the generation of the magnetic /eld, without contradicting what was already known about the terrestrial case. None of the models has proved completely convincing, although none could be completely discounted (see, e.g., Schubert et al., 1988). In particular, the hydromagnetic dynamo, assuming the possibility that Mercury’s large iron core has retained to the present epoch a molten outer layer, remains probably the most likely explanation, although initial di?culties with it have not been completely overcome. One problem concerned the quantity of lighter elements, speci/cally sulphur, that was needed in models of thermal evolution to ensure a molten outer core. In this respect, more recent models of thermal evolution have shown that only small amounts of sulphur, of order 1–2% of the total mass, would su?ce to keep a molten layer in the core (Stevenson et al., 1983; Spohn et al., 2001). Another point of concern regarding the hydromagnetic dynamo is the strength of the magnetic /eld that would be generated in such a core. In fact, energetic considerations, and earth-like assumptions ∗
Corresponding author. E-mail addresses: [email protected] (G. Giampieri), [email protected] (A. Balogh).
about the ratio of toroidal to poloidal magnetic /eld in the dynamo, had originally led to estimates of Mercury’s magnetic /eld up to three orders of magnitudes larger than that observed by Mariner 10 (which was equivalent to a dipole moment of 200 –300 nT R3 , where R is the planetary radius, Ness et al. (1975)). It is di?cult to identify the qualitative and quantitative diDerences, between the Earth’s and Mercury’s cases, that would explain why the conversion of toroidal to poloidal /eld should be much less e?cient in Mercury than in the Earth. More recent parameterized estimates of the magnetic /eld (Spohn et al., 2001) have apparently reduced the discrepancy between the estimated and measured /eld to about one order of magnitude, but the basic di?culty of modelling a realistic magnetic /eld, based on the terrestrial example, remains. The recent discovery of Mars’ crustal magnetisation (Acu˜na et al., 1998) shows the complexity and variety of planetary magnetic /elds in terrestrial planets. Although crystal magnetisation at Mercury is unlikely to be at the origin of the Mariner 10’s observations (see, e.g., Schubert et al., 1988), only future measurements can rule it out completely. In order to widen the range of possible explanations for Mercury’s magnetic /eld, an alternative dynamo mechanism has been proposed by Stevenson (1987, hereafter S87), capable of providing an “intermediate” value between the small (and unlikely) permanent magnetism, and the large hydromagnetic dynamo. This model is based on the thermoelectric eDect and, with an appropriate choice of some internal
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 2 ) 0 0 0 2 0 - X
758
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
parameters, would be able to reproduce more closely the observed values. The thermoelectric dynamo is discussed in detail in the next section. Despite the presence of adjustable parameters, this model can be tested on future Mercury orbiter missions, which will allow, in particular, a determination of the internal magnetic /eld up to multipoles of order 10 or more. In the following, after reviewing the main features of the thermoelectric dynamo model, and its prediction in terms of multipoles coe?cients, we will propose a test of the model, based on the combination of the expected results from the magnetic and gravitational mappings of the planet. 2. Thermoelectric dynamo We will give a schematic representation of the mechanism which generates the external magnetic /eld, according to the thermoelectric dynamo model. Convective motion in the mantle creates long-wavelength undulations at the core–mantle boundary, of amplitude Lh. The corresponding temperature variations at the core–mantle interface are LT ∼ ∇ad T · Lh, where ∇ad T is the adiabatic temperature gradient. Because of the diDerent composition of the core and the mantle, these temperature 4uctuations produce a thermoelectric e.m.f. ∼ QLT , where Q is the relative thermopower between the liquid iron and the solid silicate. The resulting current generates, at the core–mantle interface, a toroidal /eld of amplitude BT ∼ 0 m QLT , where m is the conductivity of the mantle (a much poorer conductor than the core, i.e., m c ), and 0 = 4 × 10−7 (SI units) is the magnetic permeability. The /eld lines of the toroidal /eld run parallel to the core–mantle boundary. As in most conventional dynamo models, convective motion in the liquid shell transforms, through the -eDect, part of the toroidal /eld into the observed poloidal /eld, of amplitude B ∼ RM BT , where RM is the magnetic Reynolds number. In S87, the following values for the model’s parameters are adopted: Lh ∼ 1 km, ∇ad T ∼ −1 K km−1 , Q ∼ 10−3 V K −1 , m ∼ 103 A V−1 m−1 , and RM ∼ 10 − 100. The amplitude of the external /eld at the planet’s surface turns out to be B ∼ 104 (RM =10) nT. Note that this rough estimate is between one and two orders of magnitude too large, when compared with the observed /eld’s amplitude at the surface, of order a few hundreds nT. As mentioned in Section 1, this is exactly the same problem faced by the hydromagnetic dynamo model, and that originally motivated Stevenson’s investigation. For this reason, the appeal of the thermoelectric dynamo model could be greatly reduced. However, as we will see in the next paragraph, the exact value of the generated /eld can be substantially less than presently estimated, due to ine?ciencies in the source mechanism, namely, due to the fact that both the mantle and the liquid outer core are relatively narrow. Also, the parameters concerning the composition and physical properties of the mantle, as given above, are subject to a great deal of uncertainty, as discussed in S87. In order to determine the range of acceptable val-
ues, we need to proceed with a more detailed analysis of the thermoelectric dynamo model. An encouraging aspect of the model is its ability to predict the speci/c multipole structure of the /eld. Following S87, we consider a three-layer model, with a solid inner core of radius Ri , a liquid outer core of radius Rc , and a silicate mantle up to the planetary radius R. The temperature 4uctuations at the core–mantle junction can be expressed as lm Ylm ( ; ’); (1) LT ( ; ’) = lm
where is the co-latitude, ’ is the longitude, the Ylm are spherical harmonics polynomials, and the lm are the corresponding harmonic coe?cients. Taking into account that the magnetic /eld is rotor-free everywhere, except in the convective liquid shell, we can solve Maxwell equations and match the solutions in the three regions (inside the core, in the liquid shell, and in the mantle=outer space). We eventually /nd that the magnetic scalar potential in the outer region can be expressed as (see S87, and Appendix A) (˜r) ≡ lm (Rc =r)l+1 Ylm ( ; ’) lm
= − 0 m QRM Rc
l (Rc =r)l+1 lm Ylm ( ; ’);
(2)
lm
where l(l + 2+2l − (1 + l)3+2l )(1 − 2l+1 ) (3) l ≡ (1 + l)2 (1 + 2l)2 (1 + l2l+1 =(l + 1)) and ≡ Ri =Rc ¡ 1; ≡ R=Rc ¿ 1. Although Eq. (2) is in agreement with the order of magnitude estimate obtained in the previous section, we point out that the actual /eld strength is signi/cantly reduced by the numerical factor l . In particular, for reasonable values of the outer core radius Rc and inner core radius Ri , Eq. (3) gives |1 | . 0:01, thus reducing the order of magnitude estimate for B to values closer to the measured ones. The coe?cients l , normalized to the dipole term 1 , are shown in Fig. 1, assuming Rc =1850 km, and for two values of the liquid core thickness. Note that the actual amplitude of the magnetic multipoles depends also on the unknown quantities lm . For this reason, we conclude that mapping the magnetic /eld around the planet, by itself, will not provide a test of the model, unless one assumes a particular angular distribution of the temperature 4uctuations at the core–mantle boundary. In the next section we will propose a possible test based on the simultaneous measurements of the magnetic and gravitational /elds of the planet, which will not rely on any a priori knowledge about the lm . 3. Correlation between gravitational and magnetic elds Thanks to the availability of future Mercury orbiters, both the gravitational and internal magnetic /elds of the planet will be accurately mapped (see, e.g., Grard and Balogh, 2001). In particular, it is estimated that the magnetometer
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
759
1.6 d=50
1.4
Normalized δl
1.2
d=
50
0
1
0.8 d=50
0.6
0.4
d=50
0
0.2 1
2
3
4
5
6
7
8
l
Fig. 1. Coe?cients l for the thermoelectric model, normalized to the l = 1 value. When multiplied by lm , these coe?cients determine the multipole terms of the magnetic /eld potential. We have assumed a core radius of Rc = 1850 km, and two values for the shell thickness, d = 50 km (top), and d = 500 km (bottom). The dotted curves are the same, using the l from Eq. (15) of S87 (see Appendix).
onboard the BepiColombo planetary orbiter will allow measuring the magnetic multipole coe?cients lm , which appear in Eq. (2), up to order l ≈ 10, although the actual /gure will depend on our ability of measuring and eliminating the magnetospheric /elds (Giampieri and Balogh, 2001). In analogy with Eq. (2), the Newtonian potential, resulting from the internal density distribution of the planet, can be expanded in spherical harmonics as V (˜r) = − lm (R=r)l+1 Ylm ( ; ’): (4) lm
The gravitational coe?cients lm are ordinarily estimated from accurate radio tracking of an orbiting spacecraft. With the BepiColombo orbiter, we can obtain the multipoles lm up to order l ≈ 20 (Milani et al., 2001). We now claim that the two sets of coe?cients, lm and lm , when correlated, can provide stringent constraints on the source of the planetary magnetic /eld. In fact, the topography at the core–mantle boundary contributes to the gravitational /eld multipoles an amount ˜lm =
4 G 0 Rl+2 c lm ; l+1 (2l + 1)R ∇ad T
(5)
where 0 is the density of the outer core, G is the gravitational constant, and the adiabatic gradient ∇ad T is assumed to be constant in the shell. Obviously, we cannot disentangle a possible contribution to these coe?cients from density anomalies internal or external to the core–mantle boundary. In particular, high-order multipoles (l ¿ 2) of the gravitational potential (4) are likely to contain information about the latitudinal and longitudinal variation of the crust thickness, a fact which is often exploited in planetary modelling.
In our case, this constitutes a limitation, since we want to be able to isolate the particular contribution from the core– mantle boundary. Spohn et al. (2001) have considered this issue, and found that the undulations at the core–mantle boundary are hidden by those in the crust, for wavelengths shorter than approximately 1700 km. Thus, the contribution to the gravitational multipoles from the core–mantle topography dominates over the crust only at orders l 6 5. Note, however, that this argument relies on the assumption that the undulations at the core–mantle and at the mantle– crust boundaries have the same amplitude. In the presence of diDerential isostatic compensation in the crust and in the mantle, this would not be a valid conjecture. For the moment, let us assume that the core–mantle boundary dominates the gravity /eld only in the range 3 6 l 6 5, and neglect other possible contributions to these coe?cients, due to the matter distribution in other parts of the planet. We return later to the consequences of this assumption. The correlation parameter between the potentials and V at r = R is given by lm lm lm ; (6) k= [ lm ( lm )2 ]1=2 [ lm (lm )2 ]1=2 where l goes from 3 to 5 (by choice), and m from −l to +l. Note that the sums over m involve only the quantity 2 lm , and we can assume without much loss of generality 2 that m lm ≡ l2 ≈ const, at least over the small range of values of l that we are considering. It is easy to verify that the correlation is very strong (k & 0:99), for every reasonable value of the parameters Rc ; Ri ; and l , thanks to the fact that the temperature 4uctuations lm determine both the
760
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
1 0.9 0.8 0.7 p=0.001 |k|
0.6 p=0.01
0.5 0.4
p=0.1 0.3 0.2 0.1 0
0.1
0.2
0.3 0.4 Fractional error in multipoles
0.5
0.6
0.7
Fig. 2. Monte-Carlo analysis of the measurement of the correlation parameter k. On the x-axis are the fractional errors in the magnetic and gravitational multipoles (assumed to be the same for both /elds, for simplicity). The error bars are 1 error estimates, for normally distributed and uncorrelated measurement errors. The horizontal dotted lines give the signi/cance levels for p = 0:001; 0:01; 0:1, respectively.
magnetic /eld potential (through the thermoelectric eDect) and the gravitational /eld potential (through the density perturbations). 4. Discussion The high correlation between the gravitational and magnetic /elds is a very clear prediction of the thermoelectric model under consideration. However, a crucial assumption, so far, is that the gravitational potential at the wavelengths considered (between ∼ 1500 and 2500 km) is dominated by the core–mantle topography. The relationship between topography and gravity varies considerably among the terrestrial planets, because of diDerent isostatic mechanisms for the gravity anomalies. In the Earth case, long-wavelength topography is isostatically compensated and is not correlated with the gravity /eld, whereas, for example, Venus’ topography correlates well with gravity anomalies (Kiefer et al., 1986). If the topography is isostatically compensated, or if residual terms of the same degree, resulting from density contrasts in other parts of the planet’s interior, turn out to be more signi/cant, then the correlation between the magnetic and gravitational /elds will obviously decrease. Similarly, if the magnetic /eld multipoles are not correctly determined, either because of errors in the magnetospheric model or in the actual measurements, the correlation factor will not be as high as predicted from Eq. (6). In order to determine the robustness of the test, in Fig. 2 we show the results of a Monte-Carlo simulation of the measurement of the correlation factor k, assuming the presence of a thermoelectric dynamo. We have included, in our simulation, normally dis-
tributed random errors in the multipoles of both the gravitational and magnetic /elds, assumed to be uncorrelated with each other. We conclude from this simple analysis that the proposed test of the thermoelectric dynamo hypothesis is very robust. For example, a 30% fractional error in lm and lm would still allow a test with & 99% con/dence. Note that in this analysis we have not considered the possible effect of correlated errors within the multipoles of a speci/c /eld; for instance, an underestimate of lm , for a particular degree and order (l; m), is expected to produce an ‘aliasing’ eDect in the adjacent terms. Another important caveat in the present analysis is that we have assumed that the -eDect is spatially constant. Adopting a more realistic, latitude or longitude dependent eDect will change the relationship between the harmonics of the temperature 4uctuations and the corresponding ones in the expansion of the magnetic /eld potential, as already pointed out in S87. This, in turn, will reduce the correlation coe?cient (6). Thus, we can conclude that a statistically signi/cant correlation, if found in the real data, should be seen as a strong support for Stevenson’s suggestion, since it would be di?cult to explain otherwise. However, the converse is not necessarily true: if a signi/cant correlation between the two /elds is not found, we would not be able to immediately discard the thermoelectric dynamo model. In fact, one could always hope to /nd a particular model able to account for the low correlation. This can be obtained, for example, by assuming a particular spatial dependency of the -eDect or a speci/c isostatic model, capable of enhancing or decreasing those coe?cients of the magnetic and gravitational /elds, which are responsible for the low correlation.
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
Note that a strong correlation (k & 0:8) was found years ago between the Earth’s magnetic and gravitational /elds (Hide and Malin, 1970). In that case, however, the two /elds showed a signi/cant correlation only when allowing ◦ for a 160 relative longitude displacement of the two /elds. This was explained as due to the interaction of the 4uid motions with the undulations of the liquid layer boundary, which in turns produce a distortion in the geomagnetic /eld. Some authors argued that allowing for an arbitrary longitude displacement makes the measurement statistically insigni/cant (Khan, 1971; Lowes, 1971; Hide and Malin, 1971). Note, however, that the nature of the thermoelectric dynamo does not allow for this additional degree of freedom. In conclusion, the combination of gravitational and magnetic mappings of the planet can be exploited to provide what appears to be a very promising test of the thermoelectric dynamo model. We have discussed the parametric dependence of the correlation on the errors, for example due to the anisotropic distribution of the matter density outside the region of interest. As far as the magnetic /eld measurement is concerned, the most important likely source, additional to the assumed thermoelectric dynamo (or, in fact, any dynamo) is not so much internal but rather external to the planet. Mercury’s magnetospheric currents, resulting from the interaction of the solar wind with the planet’s magnetic /eld, are likely to be highly variable and to contribute to the magnetic /eld measurements. However, these currents are expected to vary greatly in time and will be, on that basis, relatively easy to distinguish from the more permanent inner terms. Given the robustness of the test, even considering that other sources may aDect the high degree of correlation calculated above, conclusions concerning the source of Mercury’s magnetic /eld can be drawn from future observations. However, it is clear that the real tests of the method and its limitations will only be fully apparent once the data have been collected.
Also, we introduce the su?x − to indicate quantities in the inner core, the su?x 0 for the same quantities in the outer core, and the su?x + for the mantle and the exterior of the planet. The electric potential outside the liquid core satis/es Laplace equation, and can be written as $+ = [A+ (r=Rc )l + B+ (Rc =r)l+1 ]Ylm : (A.1) By imposing the boundary conditions $|r=Rc = and @$=@r|r=Rc = 0 we can solve for A+ and B+ 1 ; (A.2) A+ = Q lm 1 + (l=l + 1)2l+1 B+ = Q lm
1 ; 1 + (l + 1=l)−(2l+1)
From the continuity of the radial component at r = Rc we /nd 1 − 2l+1 : (A.5) Clm = 0 m Q lm 1 + (l=l + 1)2l+1 We can now proceed to calculate the magnetic /eld vector in the three regions. Since the /eld is rotor-free everywhere, except in the liquid shell, we can write −∇ − r 6 Ri ; ˜B = −∇ 0 + ˜B0 Ri 6 r 6 Rc ; (A.6) + −∇ r ¿ Rc ; where ˜B0 is the solution of 0 RM ˜ J Rc
∇2 ˜B0 = −
We thank R. Hide, T. Spohn, and D. Stevenson for helpful comments and discussions. G.G. contributed to this work while holding an ESA Research Fellowship at the Imperial College of Science, Technology and Medicine.
and the are Laplace potentials, i.e. − =
− (r=Ri )l Ylm ;
In this appendix we present the detailed calculations that lead to Eq. (2), following S87. We start from the spherical harmonics expansion of the temperature 4uctuations, Eq. (1). The analogous expansion for the thermoelectric e.m.f. follows directly from Eq. (1), multiplied by the thermopower Q. To simplify notations, we omit the indexes lm in the sum.
(A.3)
where ≡ R=Rc ¿ 1. The current in the conductive, outer ˜ core is generated by the electric potential via ˜J = −m ∇$, and can be expanded in spherical harmonics as ˜ ˜J = − 1 ∇ Clm (r=Rc )l Ylm : (A.4) 0
Acknowledgements
Appendix A. The multipoles of the thermoelectric dynamo model
761
0
=
+
=
[ 0 (r=Ri )l + 0 (Rc =r)l+1 ]Ylm ;
+ (Rc =r)l+1 Ylm :
(A.7)
(A.8) (A.9) (A.10)
We need to solve for − ; 0 ; 0 ; + by imposing the continuity of the radial components of ˜B and its derivative at r = Ri and Rc . First, however, we need to solve for ˜B0 in Eq. (A.7), after replacing ˜J with its spherical harmonics expansion, given by Eq. (A.4). The radial component of ˜B0 is
762
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
thus found to be l RM Clm Rc Rlc 2l + 1 2l+2 − Ri2l+2 r l × + r (R − r) Ylm : c (2l + 2)r l+1
and Table 1 of the same paper. In any case, this does not aDect the conclusions of this work.
Br0 (˜r) = −
References (A.11)
The four continuity equations are equivalent to the system
−l l l+1
0
1 −1 l(l−1) l 0 − (l+1)(l+2) 1
−1
−1 l+1 −(l+1) l
−1 −(l+1) − (l+1)(l+2) l(l−1) 2l+2
l(1− ) − 2(2l+1)(l+1) 2
l+1 (1−) (2l+1) = − l(1−2l+2 ) 2(2l+1)(l+1)(l+2) ll+1 (1−) (l−1)(2l+1)
−
0 0 0 (A.12) 1 + 0 1
× Clm RM Rc ;
(A.13)
where ≡ Ri =Rc ¡ 1. Since we are interested in the external /eld, we only need to solve for + . We get + = −
l(l + 2+2l − (1 + l)3+2l ) Clm RM Rc (1 + l)2 (1 + 2l)2
(A.14)
which can be inserted in Eq. (A.10), along with Eq. (A.5), to obtain the external potential (2), with l given by Eq. (3). As evident from Fig. 1, our expression for l is somewhat diDerent from the one given in S87, Eq. (15). We can only attribute this discrepancy to an algebraic error in either our or his calculation. Note, however, that there is also a signi/cant disagreement between the expression in Eq. (15) of S87,
Acu˜na, M.H., Connerney, J.E.P., Wasilewski, P., Lin, R.P., Anderson, K.A., Carlson, C.W., McFadden, J., Curtis, D.W., Mitchell, D., Reme, H., Mazelle, C., Sauvaud, J.A., D’Uston, C., Cros, A., Medale, J.L., Bauer, S.J., Cloutier, P., Mayhew, M., Winterhalter, D., Ness, N.F., 1998. Magnetic /eld and plasma observations at Mars: initial results of the Mars global surveyor mission Science 279, 1676. Giampieri, G., Balogh, A., 2001. Modelling of magnetic /eld measurements at Mercury. Planet. Space Sci. 49, 1637. Grard, R., Balogh, A., 2001. Return to Mercury: science and mission objectives Planet. Space Sci. 49, 1395. Hide, R., Malin, S.R.C., 1970. Novel correlations between global features of the Earth’s gravitational and magnetic /elds. Nature 225, 605. Hide, R., Malin, S.R.C., 1971. Novel correlations between global features of the Earth’s gravitational and magnetic /elds: further statistical considerations Nature 230, 63. Khan, M.A., 1971. Correlations between the Earth’s gravitational and magnetic /elds. Nature 230, 57. Kiefer, W.S., Richards, M.A., Hager, B.H., Bills, B.G., 1986. A Dynamic model of Venus’s gravity /eld. Geophys. Res. Lett. 13, 14. Lowes, F.J., 1971. Signi/cance of the correlation between spherical harmonics /elds. Nature 230, 61. Milani, A., Rossi, A., Vokrouhlicky, D., Villani, D., Bonanno, C., 2001. Gravity /eld and rotation state of Mercury from the BepiColombo Radio Science experiments. Planet. Space Sci. 49, 1579. Ness, N.F., Behannon, K.W., Lepping, R.P., Whang, Y.C., 1975. Magnetic /eld of Mercury con/rmed. Nature 255, 204. Schubert, G., Ross, N.M., Stevenson, D.J., Spohn, T., 1988. Mercury’s thermal history and the generation of its magnetic /eld. In: Vilas, F., Chapman, C.R., Matthews, M.S (Eds.), Mercury. The University of Arizona Press, Tucson, pp. 429. Spohn, T., Sohl, F., Wieczerkowski, K., Conzelmann, V., 2001. The interior structure of Mercury: what we know, what we expect from BepiColombo Planet. Space Sci. 49, 1561. Stevenson, D.J., 1987. Mercury’s magnetic /eld: a thermoelectric dynamo? Earth Planet. Sci. Lett. 82, 114. Stevenson, D.J., Spohn, T., Schubert, G., 1983. Magnetism and thermal evolution of the terrestrial planets. Icarus 54, 466.
www.elsevier.com/locate/pss
Mercury’s thermoelectric dynamo model revisited Giacomo Giampieri ∗ , Andr$e Balogh Space and Atmospheric Physics Group, Imperial College, The Blackett Laboratory, Prince Consort Road, London SW7 2BW, UK Received 19 October 2001; received in revised form 20 March 2002; accepted 25 April 2002
Abstract We reconsider the thermoelectric dynamo model for the generation of the magnetic /eld of Mercury, as proposed by Stevenson (Earth Planet. Sci. Lett. 82 (1987) 114). We brie4y review the status of our knowledge about the nature of Mercury’s planetary /eld, and describe the main features of the /eld generated through a thermoelectric dynamo, including its multipole expansion. We then discuss the possibility of testing this model, by correlating future measurements of Mercury’s magnetic and gravitational /elds. We conclude that the test is, in principle, very robust, and discuss possible limits on its interpretation. ? 2002 Elsevier Science Ltd. All rights reserved. PACS: 96.30.Dz; 96.35; 95.55.Pe; 91.10.Qm; 91.25.Cw Keywords: Mercury; Planetary characteristics and properties; Planetary probes; Gravity /eld harmonics; Dynamo theories
1. Introduction After Mariner 10’s unexpected discovery of Mercury’s planetary scale, mostly dipolar magnetic /eld (Ness et al., 1975), models of the planet’s thermal evolution and internal structure needed revisiting, in order to /nd a plausible explanation for the generation of the magnetic /eld, without contradicting what was already known about the terrestrial case. None of the models has proved completely convincing, although none could be completely discounted (see, e.g., Schubert et al., 1988). In particular, the hydromagnetic dynamo, assuming the possibility that Mercury’s large iron core has retained to the present epoch a molten outer layer, remains probably the most likely explanation, although initial di?culties with it have not been completely overcome. One problem concerned the quantity of lighter elements, speci/cally sulphur, that was needed in models of thermal evolution to ensure a molten outer core. In this respect, more recent models of thermal evolution have shown that only small amounts of sulphur, of order 1–2% of the total mass, would su?ce to keep a molten layer in the core (Stevenson et al., 1983; Spohn et al., 2001). Another point of concern regarding the hydromagnetic dynamo is the strength of the magnetic /eld that would be generated in such a core. In fact, energetic considerations, and earth-like assumptions ∗
Corresponding author. E-mail addresses: [email protected] (G. Giampieri), [email protected] (A. Balogh).
about the ratio of toroidal to poloidal magnetic /eld in the dynamo, had originally led to estimates of Mercury’s magnetic /eld up to three orders of magnitudes larger than that observed by Mariner 10 (which was equivalent to a dipole moment of 200 –300 nT R3 , where R is the planetary radius, Ness et al. (1975)). It is di?cult to identify the qualitative and quantitative diDerences, between the Earth’s and Mercury’s cases, that would explain why the conversion of toroidal to poloidal /eld should be much less e?cient in Mercury than in the Earth. More recent parameterized estimates of the magnetic /eld (Spohn et al., 2001) have apparently reduced the discrepancy between the estimated and measured /eld to about one order of magnitude, but the basic di?culty of modelling a realistic magnetic /eld, based on the terrestrial example, remains. The recent discovery of Mars’ crustal magnetisation (Acu˜na et al., 1998) shows the complexity and variety of planetary magnetic /elds in terrestrial planets. Although crystal magnetisation at Mercury is unlikely to be at the origin of the Mariner 10’s observations (see, e.g., Schubert et al., 1988), only future measurements can rule it out completely. In order to widen the range of possible explanations for Mercury’s magnetic /eld, an alternative dynamo mechanism has been proposed by Stevenson (1987, hereafter S87), capable of providing an “intermediate” value between the small (and unlikely) permanent magnetism, and the large hydromagnetic dynamo. This model is based on the thermoelectric eDect and, with an appropriate choice of some internal
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 2 ) 0 0 0 2 0 - X
758
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
parameters, would be able to reproduce more closely the observed values. The thermoelectric dynamo is discussed in detail in the next section. Despite the presence of adjustable parameters, this model can be tested on future Mercury orbiter missions, which will allow, in particular, a determination of the internal magnetic /eld up to multipoles of order 10 or more. In the following, after reviewing the main features of the thermoelectric dynamo model, and its prediction in terms of multipoles coe?cients, we will propose a test of the model, based on the combination of the expected results from the magnetic and gravitational mappings of the planet. 2. Thermoelectric dynamo We will give a schematic representation of the mechanism which generates the external magnetic /eld, according to the thermoelectric dynamo model. Convective motion in the mantle creates long-wavelength undulations at the core–mantle boundary, of amplitude Lh. The corresponding temperature variations at the core–mantle interface are LT ∼ ∇ad T · Lh, where ∇ad T is the adiabatic temperature gradient. Because of the diDerent composition of the core and the mantle, these temperature 4uctuations produce a thermoelectric e.m.f. ∼ QLT , where Q is the relative thermopower between the liquid iron and the solid silicate. The resulting current generates, at the core–mantle interface, a toroidal /eld of amplitude BT ∼ 0 m QLT , where m is the conductivity of the mantle (a much poorer conductor than the core, i.e., m c ), and 0 = 4 × 10−7 (SI units) is the magnetic permeability. The /eld lines of the toroidal /eld run parallel to the core–mantle boundary. As in most conventional dynamo models, convective motion in the liquid shell transforms, through the -eDect, part of the toroidal /eld into the observed poloidal /eld, of amplitude B ∼ RM BT , where RM is the magnetic Reynolds number. In S87, the following values for the model’s parameters are adopted: Lh ∼ 1 km, ∇ad T ∼ −1 K km−1 , Q ∼ 10−3 V K −1 , m ∼ 103 A V−1 m−1 , and RM ∼ 10 − 100. The amplitude of the external /eld at the planet’s surface turns out to be B ∼ 104 (RM =10) nT. Note that this rough estimate is between one and two orders of magnitude too large, when compared with the observed /eld’s amplitude at the surface, of order a few hundreds nT. As mentioned in Section 1, this is exactly the same problem faced by the hydromagnetic dynamo model, and that originally motivated Stevenson’s investigation. For this reason, the appeal of the thermoelectric dynamo model could be greatly reduced. However, as we will see in the next paragraph, the exact value of the generated /eld can be substantially less than presently estimated, due to ine?ciencies in the source mechanism, namely, due to the fact that both the mantle and the liquid outer core are relatively narrow. Also, the parameters concerning the composition and physical properties of the mantle, as given above, are subject to a great deal of uncertainty, as discussed in S87. In order to determine the range of acceptable val-
ues, we need to proceed with a more detailed analysis of the thermoelectric dynamo model. An encouraging aspect of the model is its ability to predict the speci/c multipole structure of the /eld. Following S87, we consider a three-layer model, with a solid inner core of radius Ri , a liquid outer core of radius Rc , and a silicate mantle up to the planetary radius R. The temperature 4uctuations at the core–mantle junction can be expressed as lm Ylm ( ; ’); (1) LT ( ; ’) = lm
where is the co-latitude, ’ is the longitude, the Ylm are spherical harmonics polynomials, and the lm are the corresponding harmonic coe?cients. Taking into account that the magnetic /eld is rotor-free everywhere, except in the convective liquid shell, we can solve Maxwell equations and match the solutions in the three regions (inside the core, in the liquid shell, and in the mantle=outer space). We eventually /nd that the magnetic scalar potential in the outer region can be expressed as (see S87, and Appendix A) (˜r) ≡ lm (Rc =r)l+1 Ylm ( ; ’) lm
= − 0 m QRM Rc
l (Rc =r)l+1 lm Ylm ( ; ’);
(2)
lm
where l(l + 2+2l − (1 + l)3+2l )(1 − 2l+1 ) (3) l ≡ (1 + l)2 (1 + 2l)2 (1 + l2l+1 =(l + 1)) and ≡ Ri =Rc ¡ 1; ≡ R=Rc ¿ 1. Although Eq. (2) is in agreement with the order of magnitude estimate obtained in the previous section, we point out that the actual /eld strength is signi/cantly reduced by the numerical factor l . In particular, for reasonable values of the outer core radius Rc and inner core radius Ri , Eq. (3) gives |1 | . 0:01, thus reducing the order of magnitude estimate for B to values closer to the measured ones. The coe?cients l , normalized to the dipole term 1 , are shown in Fig. 1, assuming Rc =1850 km, and for two values of the liquid core thickness. Note that the actual amplitude of the magnetic multipoles depends also on the unknown quantities lm . For this reason, we conclude that mapping the magnetic /eld around the planet, by itself, will not provide a test of the model, unless one assumes a particular angular distribution of the temperature 4uctuations at the core–mantle boundary. In the next section we will propose a possible test based on the simultaneous measurements of the magnetic and gravitational /elds of the planet, which will not rely on any a priori knowledge about the lm . 3. Correlation between gravitational and magnetic elds Thanks to the availability of future Mercury orbiters, both the gravitational and internal magnetic /elds of the planet will be accurately mapped (see, e.g., Grard and Balogh, 2001). In particular, it is estimated that the magnetometer
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
759
1.6 d=50
1.4
Normalized δl
1.2
d=
50
0
1
0.8 d=50
0.6
0.4
d=50
0
0.2 1
2
3
4
5
6
7
8
l
Fig. 1. Coe?cients l for the thermoelectric model, normalized to the l = 1 value. When multiplied by lm , these coe?cients determine the multipole terms of the magnetic /eld potential. We have assumed a core radius of Rc = 1850 km, and two values for the shell thickness, d = 50 km (top), and d = 500 km (bottom). The dotted curves are the same, using the l from Eq. (15) of S87 (see Appendix).
onboard the BepiColombo planetary orbiter will allow measuring the magnetic multipole coe?cients lm , which appear in Eq. (2), up to order l ≈ 10, although the actual /gure will depend on our ability of measuring and eliminating the magnetospheric /elds (Giampieri and Balogh, 2001). In analogy with Eq. (2), the Newtonian potential, resulting from the internal density distribution of the planet, can be expanded in spherical harmonics as V (˜r) = − lm (R=r)l+1 Ylm ( ; ’): (4) lm
The gravitational coe?cients lm are ordinarily estimated from accurate radio tracking of an orbiting spacecraft. With the BepiColombo orbiter, we can obtain the multipoles lm up to order l ≈ 20 (Milani et al., 2001). We now claim that the two sets of coe?cients, lm and lm , when correlated, can provide stringent constraints on the source of the planetary magnetic /eld. In fact, the topography at the core–mantle boundary contributes to the gravitational /eld multipoles an amount ˜lm =
4 G 0 Rl+2 c lm ; l+1 (2l + 1)R ∇ad T
(5)
where 0 is the density of the outer core, G is the gravitational constant, and the adiabatic gradient ∇ad T is assumed to be constant in the shell. Obviously, we cannot disentangle a possible contribution to these coe?cients from density anomalies internal or external to the core–mantle boundary. In particular, high-order multipoles (l ¿ 2) of the gravitational potential (4) are likely to contain information about the latitudinal and longitudinal variation of the crust thickness, a fact which is often exploited in planetary modelling.
In our case, this constitutes a limitation, since we want to be able to isolate the particular contribution from the core– mantle boundary. Spohn et al. (2001) have considered this issue, and found that the undulations at the core–mantle boundary are hidden by those in the crust, for wavelengths shorter than approximately 1700 km. Thus, the contribution to the gravitational multipoles from the core–mantle topography dominates over the crust only at orders l 6 5. Note, however, that this argument relies on the assumption that the undulations at the core–mantle and at the mantle– crust boundaries have the same amplitude. In the presence of diDerential isostatic compensation in the crust and in the mantle, this would not be a valid conjecture. For the moment, let us assume that the core–mantle boundary dominates the gravity /eld only in the range 3 6 l 6 5, and neglect other possible contributions to these coe?cients, due to the matter distribution in other parts of the planet. We return later to the consequences of this assumption. The correlation parameter between the potentials and V at r = R is given by lm lm lm ; (6) k= [ lm ( lm )2 ]1=2 [ lm (lm )2 ]1=2 where l goes from 3 to 5 (by choice), and m from −l to +l. Note that the sums over m involve only the quantity 2 lm , and we can assume without much loss of generality 2 that m lm ≡ l2 ≈ const, at least over the small range of values of l that we are considering. It is easy to verify that the correlation is very strong (k & 0:99), for every reasonable value of the parameters Rc ; Ri ; and l , thanks to the fact that the temperature 4uctuations lm determine both the
760
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
1 0.9 0.8 0.7 p=0.001 |k|
0.6 p=0.01
0.5 0.4
p=0.1 0.3 0.2 0.1 0
0.1
0.2
0.3 0.4 Fractional error in multipoles
0.5
0.6
0.7
Fig. 2. Monte-Carlo analysis of the measurement of the correlation parameter k. On the x-axis are the fractional errors in the magnetic and gravitational multipoles (assumed to be the same for both /elds, for simplicity). The error bars are 1 error estimates, for normally distributed and uncorrelated measurement errors. The horizontal dotted lines give the signi/cance levels for p = 0:001; 0:01; 0:1, respectively.
magnetic /eld potential (through the thermoelectric eDect) and the gravitational /eld potential (through the density perturbations). 4. Discussion The high correlation between the gravitational and magnetic /elds is a very clear prediction of the thermoelectric model under consideration. However, a crucial assumption, so far, is that the gravitational potential at the wavelengths considered (between ∼ 1500 and 2500 km) is dominated by the core–mantle topography. The relationship between topography and gravity varies considerably among the terrestrial planets, because of diDerent isostatic mechanisms for the gravity anomalies. In the Earth case, long-wavelength topography is isostatically compensated and is not correlated with the gravity /eld, whereas, for example, Venus’ topography correlates well with gravity anomalies (Kiefer et al., 1986). If the topography is isostatically compensated, or if residual terms of the same degree, resulting from density contrasts in other parts of the planet’s interior, turn out to be more signi/cant, then the correlation between the magnetic and gravitational /elds will obviously decrease. Similarly, if the magnetic /eld multipoles are not correctly determined, either because of errors in the magnetospheric model or in the actual measurements, the correlation factor will not be as high as predicted from Eq. (6). In order to determine the robustness of the test, in Fig. 2 we show the results of a Monte-Carlo simulation of the measurement of the correlation factor k, assuming the presence of a thermoelectric dynamo. We have included, in our simulation, normally dis-
tributed random errors in the multipoles of both the gravitational and magnetic /elds, assumed to be uncorrelated with each other. We conclude from this simple analysis that the proposed test of the thermoelectric dynamo hypothesis is very robust. For example, a 30% fractional error in lm and lm would still allow a test with & 99% con/dence. Note that in this analysis we have not considered the possible effect of correlated errors within the multipoles of a speci/c /eld; for instance, an underestimate of lm , for a particular degree and order (l; m), is expected to produce an ‘aliasing’ eDect in the adjacent terms. Another important caveat in the present analysis is that we have assumed that the -eDect is spatially constant. Adopting a more realistic, latitude or longitude dependent eDect will change the relationship between the harmonics of the temperature 4uctuations and the corresponding ones in the expansion of the magnetic /eld potential, as already pointed out in S87. This, in turn, will reduce the correlation coe?cient (6). Thus, we can conclude that a statistically signi/cant correlation, if found in the real data, should be seen as a strong support for Stevenson’s suggestion, since it would be di?cult to explain otherwise. However, the converse is not necessarily true: if a signi/cant correlation between the two /elds is not found, we would not be able to immediately discard the thermoelectric dynamo model. In fact, one could always hope to /nd a particular model able to account for the low correlation. This can be obtained, for example, by assuming a particular spatial dependency of the -eDect or a speci/c isostatic model, capable of enhancing or decreasing those coe?cients of the magnetic and gravitational /elds, which are responsible for the low correlation.
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
Note that a strong correlation (k & 0:8) was found years ago between the Earth’s magnetic and gravitational /elds (Hide and Malin, 1970). In that case, however, the two /elds showed a signi/cant correlation only when allowing ◦ for a 160 relative longitude displacement of the two /elds. This was explained as due to the interaction of the 4uid motions with the undulations of the liquid layer boundary, which in turns produce a distortion in the geomagnetic /eld. Some authors argued that allowing for an arbitrary longitude displacement makes the measurement statistically insigni/cant (Khan, 1971; Lowes, 1971; Hide and Malin, 1971). Note, however, that the nature of the thermoelectric dynamo does not allow for this additional degree of freedom. In conclusion, the combination of gravitational and magnetic mappings of the planet can be exploited to provide what appears to be a very promising test of the thermoelectric dynamo model. We have discussed the parametric dependence of the correlation on the errors, for example due to the anisotropic distribution of the matter density outside the region of interest. As far as the magnetic /eld measurement is concerned, the most important likely source, additional to the assumed thermoelectric dynamo (or, in fact, any dynamo) is not so much internal but rather external to the planet. Mercury’s magnetospheric currents, resulting from the interaction of the solar wind with the planet’s magnetic /eld, are likely to be highly variable and to contribute to the magnetic /eld measurements. However, these currents are expected to vary greatly in time and will be, on that basis, relatively easy to distinguish from the more permanent inner terms. Given the robustness of the test, even considering that other sources may aDect the high degree of correlation calculated above, conclusions concerning the source of Mercury’s magnetic /eld can be drawn from future observations. However, it is clear that the real tests of the method and its limitations will only be fully apparent once the data have been collected.
Also, we introduce the su?x − to indicate quantities in the inner core, the su?x 0 for the same quantities in the outer core, and the su?x + for the mantle and the exterior of the planet. The electric potential outside the liquid core satis/es Laplace equation, and can be written as $+ = [A+ (r=Rc )l + B+ (Rc =r)l+1 ]Ylm : (A.1) By imposing the boundary conditions $|r=Rc = and @$=@r|r=Rc = 0 we can solve for A+ and B+ 1 ; (A.2) A+ = Q lm 1 + (l=l + 1)2l+1 B+ = Q lm
1 ; 1 + (l + 1=l)−(2l+1)
From the continuity of the radial component at r = Rc we /nd 1 − 2l+1 : (A.5) Clm = 0 m Q lm 1 + (l=l + 1)2l+1 We can now proceed to calculate the magnetic /eld vector in the three regions. Since the /eld is rotor-free everywhere, except in the liquid shell, we can write −∇ − r 6 Ri ; ˜B = −∇ 0 + ˜B0 Ri 6 r 6 Rc ; (A.6) + −∇ r ¿ Rc ; where ˜B0 is the solution of 0 RM ˜ J Rc
∇2 ˜B0 = −
We thank R. Hide, T. Spohn, and D. Stevenson for helpful comments and discussions. G.G. contributed to this work while holding an ESA Research Fellowship at the Imperial College of Science, Technology and Medicine.
and the are Laplace potentials, i.e. − =
− (r=Ri )l Ylm ;
In this appendix we present the detailed calculations that lead to Eq. (2), following S87. We start from the spherical harmonics expansion of the temperature 4uctuations, Eq. (1). The analogous expansion for the thermoelectric e.m.f. follows directly from Eq. (1), multiplied by the thermopower Q. To simplify notations, we omit the indexes lm in the sum.
(A.3)
where ≡ R=Rc ¿ 1. The current in the conductive, outer ˜ core is generated by the electric potential via ˜J = −m ∇$, and can be expanded in spherical harmonics as ˜ ˜J = − 1 ∇ Clm (r=Rc )l Ylm : (A.4) 0
Acknowledgements
Appendix A. The multipoles of the thermoelectric dynamo model
761
0
=
+
=
[ 0 (r=Ri )l + 0 (Rc =r)l+1 ]Ylm ;
+ (Rc =r)l+1 Ylm :
(A.7)
(A.8) (A.9) (A.10)
We need to solve for − ; 0 ; 0 ; + by imposing the continuity of the radial components of ˜B and its derivative at r = Ri and Rc . First, however, we need to solve for ˜B0 in Eq. (A.7), after replacing ˜J with its spherical harmonics expansion, given by Eq. (A.4). The radial component of ˜B0 is
762
G. Giampieri, A. Balogh / Planetary and Space Science 50 (2002) 757–762
thus found to be l RM Clm Rc Rlc 2l + 1 2l+2 − Ri2l+2 r l × + r (R − r) Ylm : c (2l + 2)r l+1
and Table 1 of the same paper. In any case, this does not aDect the conclusions of this work.
Br0 (˜r) = −
References (A.11)
The four continuity equations are equivalent to the system
−l l l+1
0
1 −1 l(l−1) l 0 − (l+1)(l+2) 1
−1
−1 l+1 −(l+1) l
−1 −(l+1) − (l+1)(l+2) l(l−1) 2l+2
l(1− ) − 2(2l+1)(l+1) 2
l+1 (1−) (2l+1) = − l(1−2l+2 ) 2(2l+1)(l+1)(l+2) ll+1 (1−) (l−1)(2l+1)
−
0 0 0 (A.12) 1 + 0 1
× Clm RM Rc ;
(A.13)
where ≡ Ri =Rc ¡ 1. Since we are interested in the external /eld, we only need to solve for + . We get + = −
l(l + 2+2l − (1 + l)3+2l ) Clm RM Rc (1 + l)2 (1 + 2l)2
(A.14)
which can be inserted in Eq. (A.10), along with Eq. (A.5), to obtain the external potential (2), with l given by Eq. (3). As evident from Fig. 1, our expression for l is somewhat diDerent from the one given in S87, Eq. (15). We can only attribute this discrepancy to an algebraic error in either our or his calculation. Note, however, that there is also a signi/cant disagreement between the expression in Eq. (15) of S87,
Acu˜na, M.H., Connerney, J.E.P., Wasilewski, P., Lin, R.P., Anderson, K.A., Carlson, C.W., McFadden, J., Curtis, D.W., Mitchell, D., Reme, H., Mazelle, C., Sauvaud, J.A., D’Uston, C., Cros, A., Medale, J.L., Bauer, S.J., Cloutier, P., Mayhew, M., Winterhalter, D., Ness, N.F., 1998. Magnetic /eld and plasma observations at Mars: initial results of the Mars global surveyor mission Science 279, 1676. Giampieri, G., Balogh, A., 2001. Modelling of magnetic /eld measurements at Mercury. Planet. Space Sci. 49, 1637. Grard, R., Balogh, A., 2001. Return to Mercury: science and mission objectives Planet. Space Sci. 49, 1395. Hide, R., Malin, S.R.C., 1970. Novel correlations between global features of the Earth’s gravitational and magnetic /elds. Nature 225, 605. Hide, R., Malin, S.R.C., 1971. Novel correlations between global features of the Earth’s gravitational and magnetic /elds: further statistical considerations Nature 230, 63. Khan, M.A., 1971. Correlations between the Earth’s gravitational and magnetic /elds. Nature 230, 57. Kiefer, W.S., Richards, M.A., Hager, B.H., Bills, B.G., 1986. A Dynamic model of Venus’s gravity /eld. Geophys. Res. Lett. 13, 14. Lowes, F.J., 1971. Signi/cance of the correlation between spherical harmonics /elds. Nature 230, 61. Milani, A., Rossi, A., Vokrouhlicky, D., Villani, D., Bonanno, C., 2001. Gravity /eld and rotation state of Mercury from the BepiColombo Radio Science experiments. Planet. Space Sci. 49, 1579. Ness, N.F., Behannon, K.W., Lepping, R.P., Whang, Y.C., 1975. Magnetic /eld of Mercury con/rmed. Nature 255, 204. Schubert, G., Ross, N.M., Stevenson, D.J., Spohn, T., 1988. Mercury’s thermal history and the generation of its magnetic /eld. In: Vilas, F., Chapman, C.R., Matthews, M.S (Eds.), Mercury. The University of Arizona Press, Tucson, pp. 429. Spohn, T., Sohl, F., Wieczerkowski, K., Conzelmann, V., 2001. The interior structure of Mercury: what we know, what we expect from BepiColombo Planet. Space Sci. 49, 1561. Stevenson, D.J., 1987. Mercury’s magnetic /eld: a thermoelectric dynamo? Earth Planet. Sci. Lett. 82, 114. Stevenson, D.J., Spohn, T., Schubert, G., 1983. Magnetism and thermal evolution of the terrestrial planets. Icarus 54, 466.