Determination of the Properties of Mercury's Magnetic Field by The

Planetary and Space Science 52 (2004) 733–746 www.elsevier.com/locate/pss

Determination ofthe properties ofMercury’s magnetic ÿeld by the MESSENGER mission

Haje Kortha;∗, Brian J. Andersona, Mario H. Acu˜nab, James A. Slavinb, Nikolai A. Tsyganenkob, Sean C. Solomonc, Ralph L. McNutt Jr.a

aApplied Physics Laboratory, The Johns Hopkins University, 11100 John Hopkins Road, Laurel, MD 20723-6099, USA bNASA Goddard Space Flight Center, Greenbelt, MD 20771, USA cCarnegie Institution of Washington, Department of Terrestrial Magnetism, 5241 Broad Ranch Road, N.W., Washington, DC 20015, USA Received 27 March 2003; received in revised form 20 November 2003; accepted 18 December 2003

Abstract

The MESSENGER mission to Mercury, to be launched in 2004, will provide an opportunity to characterize Mercury’s internal magnetic ÿeld during an orbital phase lasting one Earth year. To test the ability to determine the planetary dipole and higher-order moments from measurements by the spacecraft’s =uxgate magnetometer, we simulate the observations along the spacecraft trajectory and recover the internal ÿeld characteristics from the simulated observations. The magnetic ÿeld inside Mercury’s magnetosphere is assumed to consist of an intrinsic multipole component and an external contribution due to magnetospheric current systems described by a modiÿed Tsyganenko 96 model. Under the axis-centered-dipole approximation without correction for the external ÿeld the moment strength is overestimated by 3 ∼ 4% for a simulated dipole moment of 300 nTRM, and the error depends strongly on the magnitude ofthe simulated moment, rising as the moment decreases. Correcting for the external ÿeld contributions can reduce the error in the dipole term to a lower limit of ∼ 1–2% without a solar wind monitor. Dipole and quadrupole terms, although highly correlated, are then distinguishable at the level equivalent to an error in the position ofan oCset dipole ofa fewtens ofkilometers. Knowledge ofthe external magnetic ÿeld is thereforethe primary limiting factorin extracting reliable knowledge ofthe structure ofMercury’s magnetic ÿeld fromthe MESSENGER observations. ? 2004 Elsevier Ltd. All rights reserved.

Keywords: Mercury; MESSENGER mission; Magnetic ÿeld; Multipole inversion

1. Introduction ÿeld would presumably have decayed on timescales much faster than the timescale for thickening of Mercury’s litho- Characterizing the nature ofplanetary magnetic ÿelds is sphere (Stevenson, 1987). The latter hypothesis ofan early important for broadening our understanding of planetary dynamo being the source for thermoremanent magnetiza- origins and evolution and ofthe energetics and life-times tion at Mercury seems at ÿrst unconvincing, because ofthe ofmagnetic dynamos. A planet ofgreat interest regarding magnetostatic theorem by Runcorn (1975a,b) that prohibits its magnetic properties is Mercury, whose intrinsic mag- the existence ofa magnetic ÿeld external to the planet af- netic ÿeld was discovered during two =ybys ofthe Mariner ter dynamo activity has ceased. However, Runcorn’s theo- 10 spacecraft in 1974 and 1975. Possible sources for Mer- rem is valid only under ideal conditions and is violated ifa cury’s magnetic ÿeld are thermoremanence, an active non-uniform permeability is assumed (Stephenson, 1976), dynamo, or a combination thereof. Thermoremanent mag- cooling ofthe planetary interior occurred progressively in- netization may have been induced either by a large external ward rather than instantaneously (Srnka, 1976), or asym- (solar or nebular) magnetic ÿeld or by an internal dynamo metries persist in the thermal structure ofthe lithosphere that existed earlier in the planet’s evolution. The former (Aharonson et al., 2004). The alternative source ofMer- process seems implausible since any early solar or nebular cury’s magnetic ÿeld is a still active dynamo. Although thermal evolution models predict the solidiÿcation ofa pure iron core early in Mercury’s history (Solomon, 1976), it has ∗ Corresponding author. Fax: +1-443-778-0386. been argued that even small quantities oflight alloying el- E-mail address: [email protected] (H. Korth). ements, such as sulfur or oxygen, could have prevented the

0032-0633/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2003.12.008 734 H. Korth et al. / Planetary and Space Science 52(2004)733–746 core from freezing out (Stevenson et al., 1983). Therefore, sions perpendicular to the planet-sun line are only 5% of the hypothesis ofan active hydrodynamic ( Stevenson, 1983) those ofthe Earth, because the magnetic moment ofMer- or thermoelectric dynamo (Stevenson, 1987; Giampieri and cury is more than 1000 times smaller than the terrestrial Balogh, 2002) operating at Mercury is still viable. The moment, the solar wind density is higher, and the interplan- above source mechanisms imply distinct characteristics for etary magnetic ÿeld (IMF) is stronger at Mercury than at the structure ofthe magnetic ÿeld. In order to distinguish 1AU(Russell et al., 1988). As a consequence, the mag- among them, a detailed mapping ofthe main ÿeld, as well netic ÿelds from magnetospheric currents at Mercury are of as ofthe crustal ÿne structure, is required. the same order as the intrinsic ÿeld and vary signiÿcantly in The Mariner 10 spacecraft obtained measurements of response to the local solar wind and IMF (Luhmann et al., Mercury’s magnetic ÿeld during two ofthree Mercury =y- 1998). Careful account for the external ÿeld will therefore bys on March 29, 1974 (encounter I), and March 16, 1975 have to be made to determine the structure ofMercury’s (encounter III). On encounter I, the spacecraft made its intrinsic magnetic ÿeld. closest approach to the planet at an altitude of705 km on Extensive measurements ofthe magnetic ÿeld near the nightside ofthe planet near the equator. The maximum Mercury will be provided by the MErcury Surface, Space magnetic ÿeld strength measured on this =yby was 98 nT ENvironment, GEochemistry, and Ranging (MESSEN- (Ness et al., 1974, 1975). Encounter III was targeted closer GER) mission (Solomon et al., 2001). To be launched in to the planet, with a closest approach altitude of327 km 2004, the MESSENGER spacecraft will orbit Mercury for at 68◦N latitude. The highest ÿeld strength detected on one Earth year. The MESSENGER orbit has an initial peri- this =yby was 400 nT (Ness et al., 1976; Lepping et al., apsis altitude of200 km and an initial latitude ofperiapsis 1979). Encounter II, on September 21, 1974, was designed of60 ◦N; the orbit is inclined 80◦ to the equatorial plane of to optimize imaging observations and came only within the planet and has a 12-h period (Santo et al., 2001). In the 5 × 104 km ofthe planetary dayside surface.It showed no March 2004 launch scenario, the orbital phase lasts from evidence ofany planetary ÿeld. April 6, 2009, until April 6, 2010, after completing two Several groups have estimated the magnetic moment of =ybys ofthat planet followingtwo =ybys ofVenus. The Mercury from the observations made during the two close =ybys in July 2007 and April 2008 return signiÿcant new =ybys. Conducting a least-squared ÿt ofthe Mercury en- data early in the mission, while the orbital phase, beginning counter I =yby data to an oCset tilted dipole, Ness et al. in April 2009, enables a focused scientiÿc investigation of 3 (1974) obtained a dipole moment of227 nT RM, where RM = the planet, guided by the =yby data. 2439:7 km is the planetary radius ofMercury. Ness et al. In order to characterize Mercury’s magnetic ÿeld, MES- (1975) considered a centered dipole and an external contri- SENGER is equipped with a three-axis, ring-core =uxgate bution to the measured magnetic ÿeld and found the strength magnetometer (MAG), mounted on a 3.6-m boom in an 3 ofthe dipole to be 349 nT RM for the same data set. From anti-sunward direction (Gold et al., 2001). The MAG instru- the Mercury encounter III observations these authors deter- ment samples the magnetic ÿeld at 20 samples per second. 3 mined a dipole moment of342 nT RM (Ness et al., 1976). However, the rate ofdata transmitted will depend on the Higher-order contributions to the internal magnetic ÿeld telemetry link margin. Emphasis during data collection will were examined by Jackson and Beard (1977) (quadrupole) be on the periapsis passes, where the planetary contribution and Whang (1977) (quadrupole, octupole). Both sets ofau- to the ambient ÿeld is greatest. As the planet rotates beneath 3 ◦ thors reported 170 nTRM as the dipole contribution to Mer- the orbit ofthe spacecraft,360 -mapping ofthe magnetic cury’s intrinsic magnetic ÿeld. ÿeld is achieved in the northern hemisphere with a longitu- The large spread in the reported estimates ofthe dipole dinal track spacing of ∼ 3◦ between consecutive orbits. This term is due to the non-unique nature ofthe inversion prob- enhanced coverage ofmagnetic ÿeld observations should re- lem, caused by geometric constraints ofthe Mariner 10 tra- duce the present ambiguity in the multipole parameters de- jectories. As a consequence, not all model parameters ofthe scribed by Connerney and Ness (1988). The extended period spherical harmonic expansion can be determined indepen- ofobservations, including observations in both the magne- dently from each other. For example, there is a linear rela- tosphere and solar wind, will provide a wealth ofdata on 0 tionship between the dipole coeMcient g1 and the quadrupole the magnetospheric current systems at Mercury. 0 term g2 in the spherical harmonic expansion (Connerney and In this paper we assess how accurately the MESSENGER Ness, 1988). It is impossible to separate these terms using observations will constrain Mercury’s magnetic moment the Mariner 10 data, and detailed knowledge ofthe intrinsic including the eCects ofthe external magnetic ÿeld. This is a ÿeld will require observations from a planetary orbiter. natural extension ofthe inversion calculations of Giampieri The Mariner 10 observations showed that the modest in- and Balogh (2001), who examined multipole inversions ternal magnetic ÿeld ofMercury is suMcient to de=ect the from a simulated BepiColombo mission to Mercury. Their solar wind =ow around the planet (Siscoe and Christopher, focus was on recovering diCerent internal magnetization 1975), and the magnetic ÿeld and electron observations in- structures in the absence ofan external ÿeld. We are specif- dicate the existence ofan Earth-like magnetosphere ( Ness ically treating the eCects ofthe external ÿeld, including et al., 1974,1975; Ogilvie et al., 1975). However, its dimen- estimates for the intensity and variability that are as re- H. Korth et al. / Planetary and Space Science 52(2004)733–746 735 alistic as possible at present. This approach allows us to where the matrix A contains the derivatives ofthe spherical estimate the errors introduced by the external ÿeld, which harmonic expansion ofthe potential functioncorresponding Giampieri and Balogh (2001) did not consider. We estimate to the three magnetic ÿeld components, the data vector d m m T the external magnetic ÿeld using a modiÿed Tsyganenko 96 consists ofthe magnetic ÿeld observations, and m=(gl ;hl ) (T96) magnetospheric magnetic ÿeld model (Tsyganenko, is the unknown model vector. The least-squared solution 1995, 1996), developed for the terrestrial magnetosphere, ofa system oflinear equations ( 3) is conveniently found driven using solar wind and IMF data from the Advanced using singular value decomposition (SVD) (Lanczos, 1958), Composition Explorer (ACE), scaled to the appropriate which allows any matrix A to be decomposed into a product Mercury-orbit heliocentric distance. This scheme provides ofmatrices an estimate ofthe expected magnitude and variability in A = U V T; (4) the external ÿeld. This synthetic data set is used as input to dipole and multipole inversion procedures.We consider an where U and V are orthogonal matrices and is a diagonal internal magnetic ÿeld consisting ofdipole and quadrupole matrix containing positive or zero elements, the singular + −1 T terms, which allows us to quantify the expected error in the values. Constructing the pseudo inverse A =V U , the recovery ofthese terms and to specifythe degree to which model vector m is given by they may be independently determined. The results show m = V −1 U Td: (5) that the MESSENGER data set should allow identiÿcation The error margins ofthe model parameters can be deter- ofhigher-order terms contributing more than a fewpercent mined by analyzing the model covariance matrix (Menke, to the total surface ÿeld. We discuss the technical approach 1984) ofthe multipole inversion in Section 2. The simulation of + + T Mercury’s external magnetic ÿeld and its consideration in cov(m)=A cov(d)(A ) : (6) the inversion process are illustrated in Sections 3 and 4, Ifthe data are uncorrelated, the data covariance matrix respectively. The inversion results are then presented in cov(d) is a diagonal matrix, whose elements are the vari- 2 Section 5 and subsequently discussed in Section 6. Finally, ances d ofthe data. Assuming that all these variances are we summarize the results in Section 7. identical, the model covariance matrix becomes 2 + + T 2 −2 T cov(m)=dA (A ) = dV V : (7) 2. Multipole inversion Under the assumption that all uncertainties are instrumental, 2 the variance d can be estimated from the data (Bevington, Mercury’s intrinsic magnetic ÿeld can be determined 1969): from the magnetic ÿeld observations obtained during the 2 1 2 d ≈ (d − A m) ; (8) orbital phase ofthe MESSENGER mission by spherical Nd − Nm harmonic analysis. A technique similar to the one used here where Nd is the number ofdata points and Nm is the number has been applied to Pioneer 11 magnetic ÿeld data at Jupiter ofmodel parameters. Substituting ( 8) into (7), the statistical m m by Connerney (1981). The technique is based on the as- errors ofthe spherical harmonic coeMcients gl and hl are sumption that the data were obtained in a source-free region given by the diagonal elements ofthe model covariance ofspace, such that ∇×B=0. The magnetic ÿeld B can then matrix be expressed as the gradient ofa scalar potential function: m;j = cov(mjj): (9) B = −∇ ; (1) It is important to note that the statistical errors in (9) provide no information about systematic errors. which can be represented by a spherical harmonic expansion The oC-diagonal elements ofthe model covariance ma- series for the internal ÿeld: trix provide additional information about the dependence ∞ R l+1 oftwo model parameters on each other via the correlation = R M M r coeMcient l=1 cov(m ) r = ij ; (10) l ij m;im;j × (gm cos m + hm sin m )Pm(cos Â): (2) l l l which ranges between −1 and +1. Ideally, all model param- m=0 eters are uncorrelated, and the correlation coeMcients are m In (2) the Pl are the Schmidt-normalized associated Leg- thus zero. In reality the spatial density and distribution of m m endre polynomials, and the gl and hl are the spherical har- observations give rise to correlation between distinct model monic coeMcients describing the multipole conÿguration of parameters to a greater or lesser extent. The correlation coef- the magnetic ÿeld. For a number ofmagnetic ÿeld measure- ÿcient is a usefulmeasure ofthe independence ofthe dipole ments Eqs. (1) form a system of linear equations and quadrupole terms in the inversion. It depends only on the data kernel and the covariance ofthe data, not on the A m = d; (3) data itself, as can be seen from (7) and (9). 736 H. Korth et al. / Planetary and Space Science 52(2004)733–746

3. Magnetic ÿeld simulation 0.3 and 1:0 AU. Moreover, the present magnetic ÿeld model does not yet consider the magnetic ÿeld due to currents in- 3.1. External eld model duced in the planetary interior by variations in solar wind pressure, which may contribute up to 15% to the magnitude The basis ofour analysis is a simulated set ofmag- ofthe intrinsic magnetic ÿeld ( Glassmeier, 2000). netic ÿeld observations along the MESSENGER trajectory in orbit around Mercury. The magnetic ÿeld model 3.2. Interplanetary solar wind conditions implemented in the simulation consists ofa Mercury internal dipole source and an external contribution described by The solar wind data used to drive the model are taken a modiÿed Tsyganenko 96 model. The T96 external ÿeld from the database of the Level 2 data from the MAG (Smith is a semi-empirical magnetic ÿeld model ofthe Earth’s et al., 1998) and SWEPAM (McComas et al., 1998) instru- magnetosphere, established by ÿtting a multitude ofobser- ments on the ACE spacecraft. The ACE data set was chosen vations, parameterized by various conditions ofthe solar because it provides a full year of continuous data in a solar wind and the IMF, to the magnetic ÿelds associated with cycle phase (1998) comparable to that to be seen by MES- the most dominant terrestrial current systems, including the SENGER (2009–2010). The measurements are scaled from magnetopause current, the tail current, Birkeland currents, the ACE location at 1 AU to Mercury distances using the and the ring current. Although these sources ofthe model Parker solar wind solution (e.g., Parks, 1991): ÿeld are not speciÿc to the terrestrial magnetosphere, we U 2 dU GM dr retained only the magnetopause and tail currents in our − 1 = 2 − ; (11) 2 2 simulation ofMercury’s external ÿeld. Charged particles in Cs U Cs r r Mercury’s magnetosphere do not trace out complete drift where U is the =uid speed, Cs is the speed ofsound in a shells, so that a ring current, as it is known in the terres- gas, G is the gravitational constant, M is the mass ofthe trial magnetosphere, does not exist at Mercury, and the Sun, and r is the heliocentric distance. Assuming that Cs is associated magnetic ÿeld in the T96 model is set to zero. constant and imposing the boundary condition that U = Cs Slavin et al. (1997) attribute features in the Mariner 10 data when r = rc, the solution to the diCerential equation (11)is to ÿeld-aligned currents, so it is possible that a Birkeland U 2 U r r current system exists at Mercury and will need to be consid- − 2ln =4ln +4 c − 3 (12) 2 ered. At present however, we also omit the magnetic ÿeld of Cs Cs rc r the ÿeld-aligned currents, because we expect the conditions 2 with rc = GM=(2Cs ). Eq. (12) is used to determine the in Mercury’s magnetosphere to be quite diCerent from those =ow speed ratio between ACE and Mercury heliocentric dis- at Earth, owing to the absence ofa conducting ionosphere. tances. The solar wind speed at Mercury is then modelled as The output ofthe T96 model is a magnetic ÿeld vector, the speed measured at ACE adjusted by this ratio. At Mer- whose components depend on the position in the Earth’s cury distances between 0.31 and 0:47 AU the solution of magnetosphere, the ring current, the planetary tilt angle, as (12) may be approximated by the second-order polynomial well as the solar wind dynamic pressure and the By- and 2 Bz-components ofthe IMF. The scaling ofthe model to UM =11:06 + 201:04r − 140:11r ; (13) Mercury’s magnetosphere requires adjusting several input where C =39km=s has been assumed. parameters. Since the equatorial magnetic ÿeld strength at s Eq. (13) can be used together with the solution of( 12) Mercury B is only a fractionofthat ofthe Earth B , eq;M eq;E for the ACE location at 1 AU, U =98:67 km=s, to scale the all MESSENGER trajectory coordinates, given in units of E solar wind speed v , as measured by the ACE spacecraft, Mercury radii (R ), have to be scaled to Earth radii (R )in p;E M E to Mercury: such a way that the equatorial ÿeld strengths in both plan- U etary systems are equal. The suitable linear scaling factor v = v M ; (14) 1=3 p;M p;E between the two magnetospheres is thus (Beq;E=Beq;M) , UE with Beq;E = 30574 nT. That is, if Beq;M = 300 nT, then a where the subscripts M and E refer to Mercury and Earth, radial distance of8 RE in the T96 model corresponds to 1.7 respectively. The scaling in (14) accounts for the eCects of RM in our simulation. The T96 ring current is set to zero at variations in Cs on vp;E, but the scaling ratio, which is not a all times, corresponding to the absence ofclosed driftpaths, sensitive function of Cs, is ÿxed. as discussed above. Furthermore, we assume that Mercury’s The equivalent scaling for the solar wind density np can dipole axis is aligned with the planetary rotation axis, estab- be obtained from the continuity equation, n v r2 =constant, ◦ p p lishing a tilt angle of0 . since the number ofparticles traversing through any two It should be noted that the external ÿeld model does not spheres concentric to the Sun must be equal. Using (14) capture the magnetospheric dynamics, e.g., magnetic sub- yields storms, and the scaling ofsolar wind observations at Earth to 2 Mercury orbit does not account for the evolution of AlfvÃenic rE UE np;M = np;E : (15) =uctuations, shocks, or stream interaction regions between rM UM H. Korth et al. / Planetary and Space Science 52(2004)733–746 737

The solar wind dynamic pressure required to drive the T96 IMF 1-hour averages model can then be derived from (14) and (15): 0.31 - 0.47 AU 2 pdyn;M = mpnp;Mvp;M; (16) 6 BX where m is the proton mass. Solar wind composition ef- p 5 fects are ignored, because they amount only to about a 10% 4 correction to pdyn;M. The IMF measurements can be similarly scaled to the 3 Mercury distance range. Using the streamline equation (e.g., 2 Parks, 1991), one obtains the radial dependence ofthe radial 1 and azimuthal IMF components as 0 r 2 B (r)=B (r ) c ; (17) r r c r 10 BY

2 8 rc (r − rc) B(r)=Br(rc) ; (18) r U 6 where the Sun rotates with the angular velocity . The IMF 4 data measured by ACE are then scaled to Mercury by comparison between the two locations ofinterest: 2 Normalized Occurrence Rate 0 r 2 B (r )=B (r ) E ; (19) r M r E r M 20 Helios BZ † r − r r 2 ACE M c E 15 B(rM)=B(rE) ; (20) rE − rc rM 10 2 rM − rc rE BÂ(rM)=BÂ(rE) : (21) 5 rE − rc rM Because the streamline equation does not include any infor- 0 mation about the radial dependence ofthe polar IMF com- -80 -40 0 40 80 ponent BÂ, we adopt (20) for the BÂ-component ofthe IMF nT (Slavin and Holzer, 1979). The suitability ofthe IMF scaling can be tested by com- Fig. 1. Normalized occurrence rate ofthe three components ofthe IMF between 0.31 and 0:47 AU. The solid lines represent the HELIOS statistics paring the scaled ACE magnetic ÿeld data with measure- compiled from 6 years of data between 1975 and 1981. The dotted lines ments taken by the HELIOS spacecraft in the heliocentric show the scaled ACE data, indicated by the † symbol in the legend. distance range ofMercury’s orbit. Fig. 1 shows the occurrence rates of1-h averages ofthe three IMF components in the HELIOS and scaled ACE data sets. The HELIOS statis- the IMF environment at Mercury. The histograms ofthe tics are compiled from measurements obtained during the Bx-components show the largest diCerences. The HELIOS years 1975 through 1981, whereas the statistical ACE data distribution is more bimodal than the ACE statistics, a dif- include only the time range from April 10, 1998, to April ference attributed to a higher level of =uctuations, relative 10, 1999. The time interval for the ACE data was selected to the average absolute value, in the radial component at the to be in approximately the same phase ofthe solar cycle as Lagrange point L1 than at Mercury orbit distances. At Mer- the orbiting phase ofthe MESSENGER mission. Note that cury, Br is the dominant IMF component and AlfvÃenic =uc- the 1-h statistics ofthe ACE data presented in Fig. 1 are tuations are comparatively small, so that the Bx-distribution used only for comparison with the HELIOS observations. is more clearly bimodal than in our scaled data set. Since The time resolution employed for driving the T96 model the Mercury external ÿeld model we are using here depends is much higher. The x-direction in Fig. 1 points toward the only on By and Bz, diCerences in the Bx-distribution are not Sun, z is perpendicular to the ecliptic plane, and y com- relevant for the purpose of driving the external ÿeld model. pletes the right-handed coordinate system. The departure of The Bx-distribution is potentially ofgreat importance for the Sun’s spin axis from the ecliptic normal of about 7◦ has the actual dynamics ofMercury’s magnetosphere, however, been ignored. This coordinate system is similar to the GSE since the dominance of Bx at Mercury implies that the sub- system used in terrestrial magnetospheric physics. solar bow shock is most often quasi-parallel, i.e., the subso- The two statistical data sets are generally in good agree- lar shock normal and the magnetic ÿeld are parallel, which ment with each other, validating the scaling procedure dis- implies that the bow shock at Mercury is likely to be more cussed above for representing the statistical distribution of inherently dynamic than it is at Earth (Russell et al., 1988). 738 H. Korth et al. / Planetary and Space Science 52(2004)733–746

DiCerences among magnetosphere passes are associated 6 with diCerences in the scaled IMF and solar wind dynamic pressure. Simulated variations between orbits and during 4 an orbit range up to 100 nT or larger, consistent with the large variations observed during the Mariner 10 encounter I (Ness et al., 1974). 2

M 0

Z [R ] 4. External ÿeld correction

-2 As noted in Section 3, an accurate determination ofMer- cury’s intrinsic magnetic ÿeld requires proper correction for -4 the external ÿeld contribution. This correction can be ac- complished by modifying (5):

-6 −1 T m = V U (dobs − dext); (22) -6 -4 -2 0 2 4 6 where the vector dobs contains the magnetic ÿeld observa- X [RM ] tions and the dext are estimates ofthe external ÿeld. This Fig. 2. MESSENGER noon-midnight orbit on April 17, 2009, with peri- correction also satisÿes the condition ∇×B = 0 in an av- ◦ herm on the nightside at 260 km altitude and 60 N latitude. Along the erage sense. Clearly, for dext we do not simply subtract the orbit, the spacecraft passes through regions of magnetospheric (solid line) external ÿeld contribution, but rather we assess the solar and solar wind (dotted line) ÿeld lines. Magnetospheric ÿeld lines are wind and IMF inputs for the external ÿeld estimate directly superimposed on the plot. from the simulated data. The solar wind pressure is estimated from magnetopause crossings, and the IMF is evalu- 3.3. Simulated external eld contribution ated from the period of solar wind observations just prior to each pass through the magnetosphere. To simulate the eCects ofthe solar wind and IMF and An average dynamic pressure, to be used for the entire their variability on the MESSENGER magnetic ÿeld ob- time period ofthe inversion, is derived froma model mag- servations, we use the 1-year time series ofACE data as netopause best-ÿt to magnetopause crossings identiÿed in a time-variable input to the T96 model and evaluate the the simulated data. The detailed procedure is as follows. The net magnetic ÿeld in 1-min time steps along the MES- coordinates ofmagnetopause crossings observed in the data SENGER orbit trajectory. For this purpose, the IMF and are scaled to Earth with the zero-order estimate for Beq;M solar wind data are interpolated and scaled from 4-min av- recovered by (5). A prolate ellipsoid, erages ofthe ACE/MAG database and 64-s averages ofthe 2 2 2 ACE/SWEPAM observations, respectively. The ÿeld-line ax + bx + c + y + z =0; (23) topology (magnetospheric, solar wind) was determined by is then ÿt to the scaled magnetopause crossings, establish- tracing the ÿeld lines. Fig. 2 shows the MESSENGER tra- ing the lengths ofthe semi-major axes a, b, and c in a jectory for a noon-midnight orbit together with a set of mag- least-squared sense. Setting y = z = 0, the magnetopause netic ÿeld lines, indicating the size ofMercury’s magneto- stand-oC distance r yields sphere. The x-direction is positive toward the Sun, z is nor- s mal to Mercury’s orbit plane, and y is positive toward dusk. b b 2 c The external contribution to the magnetic ÿeld given by rs = − ± − : (24) the T96 model along a typical pass ofthe MESSENGER 2a 2a a spacecraft through Mercury’s magnetosphere is shown in the Choosing the sign ofthe square root appropriately (i.e., to bottom panel ofFig. 3. The ratio ofthe external contribution give the smaller root), the solar wind dynamic pressure can to the internal ÿeld (middle panel ofFig. 3), calculated for then be derived from the stand-oC distance as a dipole strength of300 nT R3 , indicates that the external M magnetic ÿeld strength (solid line, bottom panel) represents −6 rs a considerable contribution to the total ÿeld. The key features p = p0 (25) rs0 ofthe MESSENGER observations we seek to simulate are the signiÿcant time variations and orbit-to-orbit variability of with p0 =2:04 nPa and rs0 =11:08 RE (Sibeck et al., 1991; the external magnetic ÿeld strength that we expect due to the Tsyganenko, 1995). The resulting pressure p is used in the =uctuations in the solar wind and the IMF. An example ofthe T96 model for the entire time period to estimate dext. variability in the total external ÿeld intensity BT96 is shown in The IMF parameters driving the T96 model are deter- Fig. 4 for ÿve consecutive magnetosphere passes. mined from separate statistics for each orbit. Because the H. Korth et al. / Planetary and Space Science 52(2004)733–746 739

4 3

M 2

R [R ] 1 0 10.00 1.00

T96 Int 0.10

B/B 0.01 100

T96

B [nT]

|B| -50 BX BY BZ -100 04/17/2009 04/17/2009 04/17/2009 04/17/2009 04/17/2009 04/17/2009 21:55 22:15 22:36 22:56 23:17 23:38 Time

Fig. 3. Contribution ofthe external magnetic ÿeld along the spacecrafttrajectory during one magnetosphere pass calculated with the T96 magnetic ÿe ld model scaled to Mercury (bottom panel). The top and middle panels show the radial distance of the spacecraft from Mercury and the ratio of the external to the internal ÿeld contributions, respectively.

300 The angle in the y–z plane, called the clock angle, is measured with reference to the +z-direction, with +90◦ toward +y. The average IMF is calculated in each ofthe six IMF clock angle bins (0◦, ±60◦, ±120◦, 180◦). For every ob- 200 servation in the magnetosphere, the T96 model is evaluated using the average IMF in each clock angle bin, and the total external ÿeld contribution is the average ofthese six values,

T96 weighted by the number ofIMF measurements in each bin.

B [nT] 100 The results obtained by determining the external magnetic ÿeld contribution in this manner represent optimal conditions. Although we have evaluated the input parameters to the external ÿeld correction from the simulated observations, 0 the transfer function between the inputs and the external 0 15 30 45 60 75 90 ÿeld, the T96 model, is precisely what we used to create the Time [min] simulated data set. Our results therefore provide a best-case Fig. 4. Example oforbit-to-orbit variations ofthe T96 external magnetic benchmark for the information that one may extract from ÿeld strength BT96, shown for ÿve consecutive MESSENGER passes the MESSENGER observations. through Mercury’s magnetosphere as a function of time elapsed in each orbit after the inbound magnetopause crossing.

5. Results overall average ofthe IMF is zero whereas the average external ÿeld is not, the following procedure was used to ob- 5.1. Application to Mariner 10 tain an average external ÿeld. For a given magnetosphere pass, the magnetic ÿeld data ofthe preceding solar wind part The performance of the inversion algorithm without ex- ofthe orbit are sorted into 60 ◦-wide bins in the y–z-plane. ternal ÿeld correction is veriÿed by application to Mariner 10 740 H. Korth et al. / Planetary and Space Science 52(2004)733–746 magnetic ÿeld data from the Mercury encounters I and III. were close to the =anks ofthe magnetosphere, the subsolar Assuming that Mercury’s intrinsic magnetic ÿeld is an ax- standoC distance is poorly constrained, with the result that ial centered dipole conÿguration, a dipole moment of287 ± the correction ofthe external ÿeld as described in Section 3 13 nTRM is obtained from this data set, where the error indi- 4 is inapplicable to the Mariner 10 data. cated is the standard deviation in (9). In the approximation that the internal ÿeld can be represented as a centered dipole 4 plus a centered quadrupole, the quadrupole is 122±16 nTRM 5.2. Simulated MESSENGER observations 3 while the dipole is reduced to 210 ± 15 nTRM. However, the dipole-plus-quadrupole solution is non-unique, and the The inversion algorithm was applied to magnetic ÿeld higher-order term can also be attributed to an oC-centered data simulated along the MESSENGER trajectory by the dipole position. In order to demonstrate this equivalency, procedure described in Section 3. The intrinsic ÿeld was we relax the constraint on the location ofthe magnetic mo- simulated using a centered dipole aligned with the planet’s ment and determine dipole moment and oCset in an itera- rotation axis. Since there is a large uncertainty in Mercury’s tive approach by shifting the dipole’s position. Minimizing magnetic dipole moment, a range ofpotential planetary 3 3 the residual ÿeld vector between observations and model re- magnetic moments between 100 nTRM and 500 nTRM was 4 sults in a dipole moment of201 ± 5nTRM with an oCset considered. The diCerences between the moment and po- of x =+0:01RM, y =+0:02RM, and z =+0:20RM. Ignoring sition obtained from the inversion and the input dipole are the comparatively small oCsets in x and y, one obtains for a illustrated in Fig. 5 as functions of the input planetary mag- small displacement ofan axis-aligned dipole along its axis netic moment from the ÿrst month of the orbital mission z (Chapman and Bartels, 1940): phase. Because Mercury’s rotation period is 58.6 days, the longitude ofperiapsis ofthe MESSENGER orbit will have g0 ≈ 2Pzg0; (26) 2 1 moved by more than 180◦ over this time interval. Figs. 5a 0 0 where g1 and g2 are the dipole and quadrupole components, and b show the error in the moment as a percentage ofactual respectively, and Pz is the displacement in units ofplane- moment used in the simulation for the centered and oCset tary radii. By Eq. (26) the above oCset location of0 :2RM dipole inversions, respectively. The inversion for the oCset in +z-direction is interchangeable with a quadrupole coef- dipole is by the same technique as for the centered dipole, 0 ÿcient of g2 = 115 nT. This result agrees reasonably well except that the dipole location is a free parameter deter- 0 with the quadrupole term of g2 = 122 nT obtained from the mined by iteratively moving the position to minimize the centered dipole-plus-quadrupole conÿguration and is thus an residuals. Any deviation ofthe dipole location fromthe ori- alternative interpretation to the coeMcients obtained from gin is an error in the position, and we deÿne the norm ofthe inversion ofthe Mariner 10 observations. oCset vector as the position error (Fig. 5c). The error in the The impact ofthe external ÿeld can be demonstrated by solution at a given planetary moment is determined largely considering the minimum and maximum T96 model ÿeld by the magnitude ofthe external ÿeld strength. To illustrate as two extreme conditions, representing a very quiet and a the range ofmoment solutions that could be obtained, the greatly disturbed magnetosphere, respectively. The largest lower and upper error estimates (dashed and dotted curves in model ÿelds are obtained for high solar wind dynamic pres- Fig. 5) are inferred by adopting the extreme solar wind con- sures and a strong, southward-oriented IMF. On the other ditions used to generate the external ÿeld contributions for hand, weak solar wind pressures and a northward orienta- the Mariner 10 =ybys in Section 5.1. Driving the T96 model tion ofthe IMF result in smaller external ÿeld contributions. with the continuous stream ofscaled ACE data described The Mariner 10 solar wind data as well as the HELIOS and above gives the error estimates shown by solid lines in ACE statistics in Fig. 1 indicate that an IMF Bz = ±30 nT Fig. 5. characterizes these conditions reasonably well. Solar wind Since the contribution ofthe external magnetic ÿeld to dynamic pressures of4 :9 nPa for a quiet magnetosphere the total ÿeld decreases with increasing strength ofthe in- and 15:2 nPa for active periods were obtained as extremal trinsic ÿeld, the errors in Fig. 5 are larger for smaller plan- values by applying (16) to the statistics ofthe ACE particle etary magnetic moments. In the worst case for the centered data. The external magnetic ÿeld, evaluated by the T96 dipole (Fig. 5a) we ÿnd that the error for small planetary model driven by the solar wind parameters established moments can reach values as great as 50%, but it may be as 3 above, is then subtracted from the Mariner 10 magnetic ÿeld low as 7% for an intrinsic moment of 500 nTRM. Inversions observations prior to inversion. The resulting strength ofthe ofmagnetic ÿeld simulations using scaled ACE solar wind 3 3 dipole moment ranges between 198 nTRM and 348 nTRM, data to drive the external ÿeld show that planetary moment consistent with ÿndings from various authors, summarized can be determined within 5% ofits actual value fora dipole 3 by Lepping et al. (1979), as well as more recent modeling strength ofaround 300 nT RM. Uncertainties in the position results by Engle (1997). Therefore, we conclude that un- ofthe planetary moment increase the error in the magnitude certainties in the determination ofthe dipole strength are ofthe moment strength, as shown in Fig. 5b. For intrinsic largely due to the uncertainty ofthe external ÿeld contri- magnetic ÿelds less than 100 nT at the equator, the accu- bution. Because the Mariner 10 magnetopause crossings racy in the magnitude ofthe moment would be no better H. Korth et al. / Planetary and Space Science 52(2004)733–746 741

100 100 Centered Dipole (a) Centered Dipole (a) 80 80 [%] [%]  

60 M 60 M   40 40 Error in Error in 20 20

0 0 100 100 Offset Dipole (b) Offset Dipole (b) 80 80 [%] [%]  

60 M 60 M  

40 40 Error in Error in 20 20

0 0 0.5 0.5 (c) Offset Dipole (c) Offset Dipole

0.4 ] 0.4 M ] M

0.3 0.3

0.2 0.2 Position Error [R Position Error [R 0.1 0.1

0.0 0.0 100 200 300 400 500 100 200 300 400 500 3 3] Planetary Dipole Moment [nT RM ] Planetary Dipole Moment [nT R M

Fig. 5. Errors in dipole characteristics as functions of planetary moment Fig. 6. Errors in dipole characteristics as functions of planetary moment for the (a) centered- and (b, c) oCset-dipole models using 1 month of for the (a) centered- and (b, c) oCset-dipole models using 1 month (solid simulated MESSENGER magnetic ÿeld observations. The dashed and lines), 3 months (dashed lines), and 12 months (dotted lines) ofsimulated dotted curves depict errors obtained from the simulated data sets with MESSENGER magnetic ÿeld observations. The error in oCset ofa dipole the smallest and largest estimated external ÿeld contribution, respectively, allowed to vary in position is shown in (c). The dashed and dotted lines while the solid curve corresponds to an external ÿeld contribution acquired coincide in all three panels. by driving the T96 model with scaled ACE data. The error in oCset of a dipole allowed to vary in position is shown in (c). enhanced. The growing data set initially reduces the error, as illustrated in Fig. 6, which depicts results ofinversions than a factor of 2, and the actual position may deviate more ofsimulated magnetic ÿeld data 1 month (solid lines), 3 than 0:3RM from the inversion solution. However, if Mer- months (dashed lines), and 12 months (dotted lines) after cury’s magnetic moment is nearly consistent with previous orbit insertion. The external ÿeld model is driven by the ÿndings, we anticipate errors in the dipole strength ofabout scaled ACE solar wind data, so the solid curves in Figs. 5 10% and about 0:05RM in its position using one month of and 6 are identical. The plot format is adopted from Fig. observations. 5, showing the centered dipole evaluation in Fig. 6a and The accuracy to which Mercury’s intrinsic moment can the error estimates for the moment and position of the oC- be determined depends on the statistics ofthe observations, set dipole in Figs. 6b and c, respectively. From Fig. 6 it is on the distribution ofthe measurements, and on the knowl- evident that inversions ofsmall planetary moments beneÿt edge ofthe external magnetic ÿeld. As the orbital phase signiÿcantly from the larger data accumulation interval. The 3 ofthe mission advances, the planetary coverage ofthe ob- error estimates for the centered dipole at 100 nTRM moment servations increases while the statistics are simultaneously drop with an additional 2 months ofdata by approximately 742 H. Korth et al. / Planetary and Space Science 52(2004)733–746 a factor of 2, reaching the 10% level after three months. 10 For the oCset dipole, the error in the magnetic moment at Centered Dipole (a) 3 100 nTRM is likewise considerably reduced from 80% after 8 1 month to 33% after 3 months, and the position uncertainty [%] is lowered from 0:2RM to 0:1RM. The correlation between  6 M

the dipole and quadrupole terms, obtained from (10), ranges  from −0:84 to −0:86, depending on the number ofobserva- 4 tions included in the inversion.

The error estimates for small planetary moments improve Error in 2 noticeably during the ÿrst 3 months ofthe MESSENGER orbital phase, but only minimal improvements are achieved for 0 intrinsic moments above 300 nTR3 beyond the ÿrst month. 40 M Offset Dipole (b) This indicates that the remaining errors are systematic rather than random. As Fig. 6 illustrates, increasing the number 30

and coverage ofthe measurement points does not by itself [%]  indeÿnitely reduce the error ofthe inversion. Although the M

 20 statistical errors do decrease as the data set grows, they are ofsecondary concern because their magnitudes are smaller

than the systematical errors we obtain. The leading causes Error in 10 for systematic errors in our calculations are the orbital bias ofthe observations, being below 1000 km altitude only in 0 the northern hemisphere, and the fact that the external ÿeld 0.20 Offset Dipole is not a linear function of the solar wind dynamic pressure, (c) ]

whereas we have taken a mean magnetopause position to de- M 0.15 ÿne an average pressure. Improvements in the speciÿcation ofthe solar wind conditions may help to reduce the errors, but the results also indicate that the accuracy ofthe external 0.10 model will be critical. Since the MESSENGER data will be used to reÿne the external magnetic ÿeld model it will be 0.05 very important to use the entire 12 months ofobservations Position Error [R to make the external ÿeld model as accurate as possible. 0.00 When corrections for the external ÿeld are applied, the 100 200 300 400 500 systematic errors are reduced considerably. This result is Planetary Dipole Moment [nT R 3 ] shown in Fig. 7, where the external ÿeld correction is ap- M plied to the entire 12-month interval ofthe ACE-driven Fig. 7. Errors in dipole characteristics as functions of planetary moment MESSENGER magnetic ÿeld simulation, as described in for the (a) centered- and (b, c) oCset-dipole models using 12 months Section 4. The plot format is identical to Figs. 5 and 6, and ofsimulated MESSENGER magnetic ÿeld observations. The inversions the uncorrected curves (dotted lines in Fig. 7) are taken from represented by the solid lines have been corrected for the external ÿeld contribution, while the dotted lines represent results obtained from un- Fig. 6 for comparison. With the external ÿeld correction, corrected observations. The error in oCset ofa dipole allowed to vary in the error estimates for the planetary moment (Figs. 7a and position is shown in (c). b) are generally less than halfthat without the correction. Furthermore, the dipole location (Fig. 7c) is determined to within a few tens of kilometers after subtraction of the identiÿcation ofhigher-order terms, because only the struc- external ÿeld contribution. Similar inversions with correc- ture ofthe ÿeld can help distinguish between theories for tion for the external ÿeld were done using the 3-month sources and origins ofthe planetary ÿeld. It is crucial there- data set as well, and the systematic errors were nearly fore to consider the errors present in higher-order terms. identical to those obtained from inverting the 12-month The inversion technique described in Section 2 provides data set, consistent with the above results for the dipole the spherical harmonic coeMcients to an arbitrary degree inversion. ofsophistication, but they are subject to signiÿcant systematic errors, because the external ÿeld contribution leads to higher-order terms in the inversion that are unrelated to the 5.3. Higher-order terms intrinsic magnetic ÿeld ofthe planet. We consider the systematic error in the quadrupole term The dipole term ofthe magnetic ÿeld is fundamentalto un- and its reduction by use ofan external ÿeld correction. derstanding Mercury’s magnetosphere, but accurate knowl- Fig. 8a shows the second-order inversion solution (axial cen- 3 edge ofthe dipole is important only in so faras it allows tered dipole and quadrupole) ofthe 12-month, 300 nT RM H. Korth et al. / Planetary and Space Science 52(2004)733–746 743

-326 -305

-327 (∆Q,D) -306

-328 -307 g10 [nT] g10 [nT] -329 -308

-330 -309 δQ -331 -310 17 18 19 20 21 22 2 3 4 5 6 7 (a) g20 [nT] (b) g20 [nT]

3 Fig. 8. Magnetic moment obtained from a second-order inversion of the 12-month, 300 nTRM data set (a) without and (b) with the external ÿeld 0 0 correction applied, projected into the g1 –g2 plane. The ellipse identiÿes the 3 uncertainty level ofthe solution.

Table 1 tupole that contribute more than a few percent of the ÿeld at Multipole inversion results without and with an external ÿeld correction the planet’s surface. The large changes that result by includ- assuming a purely dipolar internal ÿeld with a moment of300 nT R3 . M ing the external ÿeld correction demonstrate that the deter- Inversions use 12 months ofsimulated data up to octupole mination ofhigher-order moments depends critically on the 0 0 0 Inversion External ÿeld g1 (nT) g2 (nT) g3 (nT) external ÿeld model. correction

Dipole (centered) No −310 — — Yes −304 — — 6. Discussion Quadrupole No −329 19.8 — Yes −307 4.3 — The results ofthe previous section provide an estimate of the uncertainties associated with the recovery ofMercury’s Octupole No −338 41.5 −16:0 Yes −308 7.1 − 2:5 magnetic ÿeld from MESSENGER observations. The key elements for a successful inversion are a large set of spatially distributed measurements and accurate correction for con- tamination by external ÿelds. The dependence ofthe com- data set without external ÿeld correction, projected into the puted multipole moments on data coverage can be assessed 0 0 g1 –g2 plane. The inversion produces an axial dipole moment from the correlation coeMcient given by Eq. (10). Corre- 0 0 of g1 = 328:5 nT and a g2 quadrupole term of19 :8 nT. The lation between model parameters is determined by the spa- 0 0 correlation coeMcient between g1 and g2 is −0:86, which tial distribution ofsamples. To illustrate this point, consider is quite high and means that errors in the dipole term cor- a spherical grid ofdata points at constant radial distance, respond directly to errors in higher-order terms. Thus, for spaced 10◦ in longitude and extending from the northern MESSENGER, the issue ofdipole moment accuracy is in- planetary pole only partway to the south pole. Fig. 9 shows extricably tied to the ability to detect higher-order terms. the correlation coeMcient between the axis-aligned dipole The width ofthe 3 error ellipse shown in Fig. 8a can and quadrupole terms as a function of the maximum colati- be interpreted as the statistical quadrupole uncertainty &Q, tude Âmax ofdata coverage. The dipole and quadrupole con- which amounts to 1.2% ofthe dipole moment. However, this tributions ofthe multipole ÿeld have zero systematic corre- error is small compared with the systematic error PQ due to lation only ifdata coverage is global. As the extent ofthe 0 the overall magnitude ofthe g2 term that was not included observations decreases, the model parameters become in- in the forward simulation. It is caused solely by the external creasingly correlated. For data only from the northern hemi- ÿeld contribution. Ifthe external magnetic ÿeld is considered sphere, the correlation coeMcient is −0:74. in the inversion, the erroneous quadrupole term is greatly MESSENGER will supply nearly global data coverage reduced, and the statistical error decreases somewhat as well between +80◦ and −70◦ latitude. However, because ofthe (Fig. 8b). The results for dipole, quadrupole, and octupole eccentricity and inclination ofthe orbit, measurements in the inversions are summarized in Table 1. southern hemisphere occur at substantially higher altitudes To be considered signiÿcant, any higher-order moment than in the northern hemisphere, meaning that signatures of must be larger in magnitude than both the systematic error higher-order terms are eCectively sampled only in the north- margin ofthe external ÿeld correction and the statistical un- ern hemisphere. The ramiÿcations ofthe altitude diCerence certainties. The external ÿeld correction used here indicates in the observations can be estimated by evaluating the cor- that it should be possible to resolve terms through the oc- relation coeMcient using the spherical grid described above, 744 H. Korth et al. / Planetary and Space Science 52(2004)733–746

0.0 spheric processes, so that it will be possible to pre-select the data to be included in the inversion, so as to reduce dynamic -0.2 eCects to a minimum. It is expected that the most reliable solutions will be aCorded by the most carefully chosen ob- -0.4 servations. The fact that the simulated inversion converges to stable results after three months of observations suggests -0.6 that there is some margin for selection of optimal data without impacting the systematic errors in the inversion. We expect that the ultimate accuracy will be determined by a

Correlation Coefficient -0.8 trade between statistical uncertainty, which grows as fewer -1.0 observations are used, and systematic error, which will be 0 45 90 135 180 reduced as the data are more carefully screened. In any case,

θ max [deg] the ultimate achievable accuracy for the dipole term may be fairly high, on the order of a few percent, and higher-order 0 terms yielding surface ÿelds comparable to this level should Fig. 9. Correlation coeMcient between the magnetic dipole g1 and 0 be reliably recovered. quadrupole g2 terms as a functionofthe maximum colatitude ofthe data coverage. Full longitudinal coverage is assumed. This study focuses on the characterization of the multipole components ofMercury’s magnetic ÿeld. Several com- but placing the data points in the northern hemisphere at plementary studies are expected to contribute to the under- 0:5RM and at 1:5RM altitude in the southern hemisphere. Un- standing ofMercury’s magnetic ÿeld. First, magnetic der these conditions, even for complete global data coverage ÿeld observations are also planned during the two MES- the correlation coeMcient between dipole and quadrupole SENGER =ybys ofMercury. These will provide the ÿrst terms is −0:72. Therefore, the dipole-quadrupole correlation new data on Mercury’s magnetic ÿeld since the Mariner of r ≈−0:85 obtained above is not surprising. The high 10 observations and will be used to update our estimates correlation implicates that extensive statistics are required ofMercury’s magnetic ÿeld prior to the orbital phase of 0 0 to separate the g1, g2, and higher-order terms. the mission. They will also be the only MESSENGER ob- The most important source ofuncertainties in obtaining servations below 1000 km at the equator (200-km =yby the planetary moments are erroneous assumptions about the closest approach). These data will likely be an important external magnetic ÿeld contribution. The need to account for point ofcomparison forthe inversion results. Simulation of external magnetic ÿelds is emphasized by the reduction of the =yby results is not included here for several reasons. the error (Fig. 7) and by misleading higher-order spherical First, in comparison with the orbital data set, the =yby harmonics (Table 1 and Section 5.3). Determining the struc- observations will still be a small data set. We conducted ture ofMercury’s intrinsic magnetic ÿeld thereforedepends inversions that included simulated data for the MESSEN- on the external ÿeld correction. The improvements obtained GER =ybys and found that their eCect on the results is not here are reasonable best-case estimates, since the same T96 statistically signiÿcant. Hence, the =yby observations need model was used for the forward simulation and inversion. to be preferentially weighted quite strongly to aCect the The use ofthe simulated data to inferan average solar wind inversions. After the data are obtained an objective justi- dynamic pressure and estimated IMF for each pass intro- ÿcation for preferential weighting may become apparent duces realistic errors. The solid curves in Fig. 7 represent and could be considered at that time. Second, one could the best achievable uncertainty when estimating upstream examine the set oftrajectories provided by the Mariner 10 conditions as done here. The true error will approach the and MESSENGER =ybys to assess the improvement in the results obtained here as the model for the external ÿeld im- statistical error (Eq. (9)) gained by adding the MESSEN- proves. Since the inversion technique can only be as accu- GER =ybys to the Mariner 10 observations. The =yby data rate as the external magnetic ÿeld model, a detailed analysis sets taken together will still be a limited set ofessentially ofthe magnetospheric current systems at Mercury will be line trajectories rather a closed surface around the planet. crucial. Both modeling calculations ofthe magnetosphere Inversions using =yby data will therefore remain subject to and analysis ofMESSENGER data formagnetospheric cur- signiÿcant systematic uncertainties due to the external ÿeld rents depend on knowledge ofthe dipole ÿeld. The factthat contribution, which will almost certainly remain larger than the dipole moment is recovered to within 10% without ap- the statistical errors, so a measure ofthe improvement in plying any corrections for the external ÿeld suggests that the statistical uncertainty would be misleading. Finally, the it should be possible to characterize accurately Mercury’s systematic errors will still be too large to identify reliably magnetosphere based on data from MESSENGER. quadrupole or higher-order terms. Although we could simu- In this paper we have assumed that all magnetic ÿeld ob- late the likely improvement in the dipole estimates aCorded servations inside Mercury’s magnetosphere are included in by the MESSENGER =ybys, the focus of this paper is on the analysis. In reality, the magnetic ÿeld data will provide the ultimate capability to extract higher-order terms in the signiÿcant clues about the occurrence ofdynamic magneto- intrinsic ÿeld. We have therefore restricted attention to data H. Korth et al. / Planetary and Space Science 52(2004)733–746 745

50 tions arising from the inversions are much smaller than the results stated above, indicating that the systematic sources 40 oferror dominate. Because ofthe elliptical MESSENGER orbit, the dipole and higher-order terms are highly corre- n [deg.] lated. This result implies that the surface ÿeld variance corre- 30 Longitude sponding to the dipole term error essentially corresponds to the minimum surface ÿeld signature that could be attributed 20 to higher-order terms. Improving the dipole accuracy therefore directly corresponds to lowering the detection level for 10 higher-order terms.

Feature Resolutio Latitude The external magnetic ÿeld is the primary factor aCect- 0 ing the inversion errors and can be signiÿcantly reduced 0 20 40 60 80 by subtracting an estimate ofthe external ÿeld contribution Latitude [deg.] prior to the inversion. The accuracy in the determinations Fig. 10. Latitudinal and longitudinal feature resolution as a function ofthe dipole and multipole components ofMercury’s mag- ofspacecraftlatitude. The dashed vertical line represents the maximum netic ÿeld is therefore tied to the accuracy of the external latitude ofthe magnetic ÿeld observations. ÿeld model. To maximize the return from the MESSEN- GER magnetic ÿeld investigation, techniques for accurately modeling Mercury’s magnetospheric ÿeld need to be devel- obtained by MESSENGER during the orbital phase ofthe oped and critically examined. The eCects ofinternal mag- mission, in as much as these data will largely supersede the netospheric dynamics and diCerences in the character ofthe =yby data sets for the purpose of characterizing the detailed solar-wind magnetosphere interaction at Mercury from that intrinsic magnetic ÿeld ofthe planet. at Earth (the absence ofan ionosphere at Mercury and the Additional analyses relate to the crustal ÿne structure increased prevalence ofsubsolar quasi-parallel shock condi- ofMercury’s magnetic ÿeld. The altitudes ofthe MES- tions) introduce signiÿcant uncertainties in our understand- SENGER orbit in the northern hemisphere are suMciently ing ofMercury’s magnetospheric ÿeld. Our ability to char- low (200-km minimum altitude) that ÿeld structures due to acterize reliably the structure ofMercury’s intrinsic mag- crustal anomalies could be directly mapped ifpresent. Large netic ÿeld is therefore determined by the extent to which crustal remanent ÿelds were found at Mars (Acu˜na et al., the contributions ofthese processes to the external ÿeld are 1998), and it is therefore reasonable to estimate the scale understood. lengths ofcrustal structures that MESSENGER could resolve. Ifwe assume that only those magnetic featureswith lateral extent larger than the spacecraft altitude can be re- Acknowledgements solved, the eCective longitudinal and latitudinal resolution is determined strictly by the spacecraft orbit. Fig. 10 illustrates We thank K.-H. Glassmeier for illuminating discussions the latitude and longitude resolution aCorded by the MES- on planetary induction currents and O. Aharonson for help- SENGER orbit. The vertical dashed line in Fig. 10 shows the ful comments on Runcorn’s theorem.We also thank the orbit inclination.We expect to be able to resolve magnetic ACE Team for use of the MAG and SWEPAM data made ◦ features with horizontal dimensions of 10 (about 400 km) available via the ACE Level 2 Database. The MESSEN- near the periapsis altitude. GER mission is supported by the NASA Discovery Program under contracts to the Carnegie Institution ofWashington (NASW-00002) and The Johns Hopkins University Applied 7. Summary Physics Laboratory (NAS5-97271).

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