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Toepfer et al. Earth, Planets and Space (2021) 73:65 https://doi.org/10.1186/s40623-021-01386-4

FULL PAPER Open Access The Mie representation for Mercury’s magnetic feld S. Toepfer1*, Y. Narita2,3, K. ‑H. Glassmeier3, D. Heyner3, P. Kolhey3, U. Motschmann1,4 and B. Langlais5

Abstract The parameterization of the magnetospheric feld contribution, generated by currents fowing in the magnetosphere is of major importance for the analysis of Mercury’s internal magnetic feld. Using a combination of the Gauss and the Mie representation (toroidal–poloidal decomposition) for the parameterization of the magnetic feld enables the anal‑ ysis of magnetic feld data measured in current carrying regions in the vicinity of Mercury. In view of the BepiColombo mission, the magnetic feld resulting from the plasma interaction of Mercury with the solar wind is simulated with a hybrid simulation code and the internal Gauss coefcients for the dipole, quadrupole and octupole feld are recon‑ structed from the data, evaluated along the prospective trajectories of the Mercury Planetary Orbiter (MPO) using Capon’s method. Especially, it turns out that a high-precision determination of Mercury’s octupole feld is expectable from the future analysis of the magnetic feld data measured by the on board MPO. Furthermore, magnetic feld data of the MESSENGER mission are analyzed and the reconstructed internal Gauss coefcients are in reasonable agreement with the results from more conventional methods such as the least-square ft. Keywords: Mie representation, Poloidal and toroidal magnetic felds, Thin shell approximation, Gauss representation, Capon’s method

Introduction contribute a signifcant amount to the total feld within Characterization of Mercury’s internal magnetic feld is Mercury’s magnetosphere (Anderson et al. 2011). one of the primary goals of the BepiColombo mission Above the planetary surface, the internal part of (Benkhof et al. 2010). Te magnetic feld in the vicinity the feld is a current-free magnetic feld which can be of Mercury is composed of internal and external parts. parameterized by the Gauss representation (Gauß 1839; Te internal feld originates in the dynamo-generated Glassmeier and Tsurutani 2014). When only data in feld, crustal remanent feld and induction feld, which source-free regions without currents are analyzed, the are mainly dominated by the dipole, quadrupole and Gauss representation also yields a parametrization for octupole feld. Te external feld originates from cur- the external parts of the feld (Olsen et al. 2010). But in rents fowing in the magnetosphere. For a clear sepa- the vicinity of Mercury signifcant currents are expected ration of the internal magnetic feld from the external (Anderson et al. 2011, eg); and therefore, the Gauss rep- feld, each part of the magnetic feld has to be mod- resentation is insufcient due to the current-generated eled properly. Especially the parameterization of the magnetic feld outside the planet. In this paper, we con- external parts is of major importance since these parts struct a parametric Mercury magnetic feld model by extending the Gauss representation to the Gauss–Mie representation. Previous studies parameterized the external parts by *Correspondence: [email protected] 1 Institut für Theoretische Physik, Technische Universität Braunschweig, making use of a global magnetospheric model, such as Mendelssohnstraße 3, 38106 Braunschweig, Germany the paraboloid model for Mercury’s magnetosphere Full list of author information is available at the end of the article

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(Alexeev et al. 2008), that has successfully been applied to coefcients are compared with the results of former the analysis of Mercury’s internal and external magnetic works by Anderson et al. (2012), Tébault et al. (2018) and feld (Alexeev et al. 2010; Johnson et al. 2021), as well as Wardinski et al. (2019). the modifed Tsyganenko model (Korth et al. 2004). As an alternative for these models the Mie representation The Mie representation is useful for decomposing the magnetic feld in current Te solenoidality of the magnetic feld B guarantees the carrying regions in the vicinity of Mercury (Backus 1986; existence of a vector potential A , so that Olsen 1997; Olsen et al. 2010). B = ∂x × A Te Mie representation, also known as toroidal-poloi- (1) dal decomposition, is based on the solenoidality of the holds, where ∂x is the spatial derivative. Using spherical magnetic feld and enables to decompose the feld into coordinates with radius r ∈ [0, ∞) , longitude ∈ [0, 2π] its toroidal and poloidal parts (Backus 1986; Backus et al. and co-latitude angle θ ∈ [0, π] the vector potential can 1996; Kazantsev and Kardakov 2019). Tis representa- be seperated into its component T r parallel to r = rer , tion has successfully been applied to several problems where er is the unit vector in radial direction, and its in space plasma physics, especially for the analysis of the components F × r perpendicular to r , yielding Earth’s magnetosphere. For example, Engels and Olsen = � + × + ∂ ϕ (1999) used the Mie representation for calculating the A T r F r x (2) magnetic efect caused by a three dimensional current with the scalar functions T and ϕ as well as a vector density model. Olsen (1997) applied the Mie representa- feld F (Jacobs 1987; Krause and Rädler 1980). Because of tion to reconstruct ionospheric F-currents in the Earth’s ∂x × ∂xϕ = 0 , the function ϕ can be chosen properly so magnetosphere from MAGSAT data. Furthermore, Bayer that ∂x × F = 0 holds, and therefore et al. (2001) introduced the wavelet Mie representation = ∂ � of the magnetic feld to calculate current densities from F x P (3) MAGSAT and CHAMP data. Tis approach has been expanded on the modeling of the Earth’s magnetic feld in with a scalar function P without changing the magnetic terms of vector kernel functions (Mayer and Maier 2006). feld (Jacobs 1987; Krause and Rädler 1980). Te vector Kosik (1984) constructed a model for the Earth’s magne- potential results in tosphere based on the Mie representation. A = �T r + ∂x�P × r (4) In this work, the magnetic feld in the vicinity of Mer- cury is parameterized by a combination of the Gauss and = � r + ∂ × (� r). the Mie representation (hereafter, Gauss–Mie representa- T x P (5) tion), based on the works of Backus (1986), Backus et al. Substituting A into Eq. (1) delivers (1996) and Olsen (1997), for analyzing Mercury’s internal magnetic feld. Te desired internal Gauss coefcients for B = ∂x × (�T r) + ∂x × ∂x × (�Pr) , (6) the dipole, quadrupole and octupole feld are estimated  with Capon’s method (Capon 1969). Te Capon method which is called the Mie representation of the magnetic serves as a powerful inversion method for linear inverse feld (Backus 1986; Backus et al. 1996). problems and was used by, e.g., Motschmann et al. (1996), Te frst term on the right hand side in Eq. (6) Glassmeier et al. (2001), Narita et al. (2006) and Narita BT = ∂x × (�T r) (2012) to evaluate spatial spectra of space plasma waves. (7) As shown by Toepfer et al. (2020a) the Capon method can is the toroidal part of the feld and the second term also be used in a generalized way to compare actual meas- urements with theoretical models. Here, we expand on BP = ∂x × ∂x × (�Pr) (8) this method. First of all, the mathematical foundations of  B the Mie representation, as derived by Backus (1986) and is the poloidal part of . B B B Backus et al. (1996) are revisited. Afterwards, the resulting From the defnition of T and P it is clear that T is B r thin shell approximation is applied to simulated magnetic perpendicular to P and also perpendicular to . Tere- feld data, that are evaluated along the future data points fore, the toroidal part of the feld does not have a radial of the Mercury Planetary Orbiter (MPO). Tis enables the component. Furthermore, poloidal magnetic felds are judgement of the expectable inversion results from the generated by toroidal currents and vice versa: data of the magnetometer (Glassmeier et al. 2010; Heyner ∂x × BT = ∂x × ∂x × (r�T ) (9) et al. 2020) on board MPO. Finally, the model is applied  to MESSENGER data and the reconstructed Gauss is a poloidal vector and Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 3 of 18

2 1 1 ∂x × B = ∂x × ∂x × ∂x × (r�P) = ∂x × −∂ �P r P x B =−∂x� =−∂r� er − ∂θ � eθ − ∂ � e ,     r r sin(θ) (10) (21) is a toroidal vector. which is known as the Gauss representation of the mag- In curl-free regions where especially the poloidal cur- netic feld (Gauß 1839; Glassmeier and Tsurutani 2014). rent density j vanishes, Ampère’s law reads as follows: P Simultanously, the Mie representation in due considera- 2 µ0j = ∂ × ∂ × (� r) =−r∂ � + ∂ ∂ (r� ) = 0 P x x T x T x r T tion that the toroidal magnetic feld vanishes in curl-free  (11) regions is given by or equivalently B = ∂x × ∂x × (�Pr) (22)  2 . ∂x∂r(r�T ) = r∂x �T (12) 1 1 2 = −∂θ (sin(θ)∂θ �P) − ∂ �P er Terefore, the gradient of ∂r(r�T ) has only a radial com- r sin(θ) sin(θ)  ponent, leaving us with (23) 1 1 ∂ �T = 0 = ∂θ �T , (13) + ∂θ ∂ (r� )e + ∂ ∂ (r� )e r r P θ r sin(θ) r P (24) so that the function T solely depends on the radial dis- tance from the center (�T = �T (r)) and thus, the toroi- Comparison of coefcients with Eq. (21) for eθ shows that dal magnetic feld vanishes 1 1 ∂r∂θ (r�P) =− ∂θ � (25) BT = ∂x × (�T r) = ∂x�T × r = ∂r�T er × r = 0. r r  (14) and analogously for e On the other hand, when the toroidal current density 1 1 vanishes ∂ ∂ (r� ) =− ∂ �. r sin(θ) r P r sin(θ) (26) 2 µ0j = ∂ × ∂ × ∂ × (� r) = ∂ × −r ∂ � e T x x x P x x P r      Consequently, when P is known, the scalar potential is (15) given by 1 1 2 2 . = ∂ϕ −r∂x �P eθ − ∂θ −r∂x �P eϕ � =−∂r(r�P) (27) r sin(θ)  r  (16) Comparison of the er-coefcients delivers = 0 (17) 1 1 2 ∂θ (sin(θ)∂θ �P) + ∂ �P = ∂r� or equivalently r sin(θ) sin(θ)  2 0 (28) ∂ϕ∂x �P = (18) or equivalently and simulanously 2 1 ∂S �P = ∂r�, (29) 2 0 r ∂θ ∂x �P = (19) 2 where the function ∂x �P solely depends on the radius. Tere- 1 1 fore, the poloidal feld ∂2 = ∂ (sin(θ)∂ ) + ∂2 S 2 sin θ θ 2 2 (30) 2 r (θ) r sin (θ) BP = ∂x × ∂x × (�Pr) =−r∂x �P + ∂x∂r(r�P) �=0  (20) is the angular part of the Laplacian. For a given , in general remains fnite in current-free regions. the function P can be found by solving Eqs. (27, 29) simultanously. Relation to the Gauss representation As a consequence, the scalar function P and the sca- When magnetic feld data in curl-free regions (where lar potential are equivalent in curl-free regions and the ∂x × B = 0 holds) are analyzed, there exists a scalar poten- Mie representation transists into the Gauss representa- tial so that the magnetic feld can be written as tion. Tus, the Gauss representation can be understood as a special case of the Mie representation. Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 4 of 18

Parameterization of the magnetic feld and Assuming that the conductivity of Mercury’s mantle e e B =−∂x� , is negligibly small (like lunar regolith (Zharkova et al. (33) 2020)), the planetary contribution to the feld outside where the scalar potentials are given by (Gauß 1839; Mercury is purely poloidal. Te currents fowing in the Glassmeier and Tsurutani 2014) magnetosphere generate poloidal and toroidal magnetic ∞ l l+1 felds that superpose with the curl-free planetary mag- i RM � = RM netic feld. To be able to separate the planetary magnetic  r  1 =0 feld out of the measured feld and to parameterize it via l= m gm cos(m ) + hm sin(m ) Pm(cos(θ)) the Gauss coefcients, a combined parametrization com- l l l (34) posed of the Mie and the Gauss representation (Gauss–   Mie representation), which is based on the works of and Backus (1986) and Olsen (1997) is necessary. ∞ l l e r Suppose that the magnetic feld in the vicinity of Mer- � = RM RM  cury is measured within a spherical shell S(a, c) with l=1 m =0 > inner radius a RM , where RM indicates the radius of qm cos(m ) + sm sin(m ) Pm(cos(θ)). b = 1 (a + c) l l l (35) Mercury, outer radius c and mean radius 2   m m as displayed in Fig. 1. Te shell can be constructed Te expansion coefcients gl and hl are the internal m m independently of the orbit’s geometry by conceptually Gauss coefcients, the coefcients ql and sl are the m covering the orbit of the . Furthermore, the external Gauss coefcients and Pl are the Schmidt nor- shell may include current carrying regions. Although malized Legendre polynomials of degree l and order m. the Mie representation enables us to analyze those cur- Since Mercury’s internal magnetic feld is dominated by rents, we focus on the analysis of Mercury’s internal the internal dipole, quadrupole and octupole felds, the magnetic feld. series expansions in Eqs. (34, 35) will be truncated at the Due to the underlying geometry, the space around degree l = 3 for the practical application. i Mercury can be decomposed into three disjoint radial It should be noted that the internal feld B is canoni- zones: cally described in a Mercury-Body-Fixed co-rotating e coordinate system (MBF), whereas the external feld B is • points in the region r < a below the shell canonically described in a Mercury-Solar-Orbital system • points in the region r > c above the shell (MSO) with the x-axis orientated towards the sun, the • points in the region a ≤ r ≤ c inside the shell layer z-axis orientated parallel to the rotation axis, i.e. antipar- allel to the internal dipole moment, and the y-axis com- Making use of the superposition principle the total pletes the right-handed systemT (Heyner et al. 2020). Let magnetic feld B measured inside the shell layer xMSO = xMSO, yMSO, zMSO defne the MSOT coordinate ( a ≤ r ≤ c ) is a composition of the feld Bj∈[a,c] gener- system and let xMBF = xMBF , yMBF, zMBF be the coor- ated by currents fowing inside the shell and the feld dinates of the co-rotating MBF system. Ten, the internal Bj∈/[a,c] generated by currents fowing outside the shell. parts of the feld are given by Again considering the underlying geometry, the second i i i i i B (xMBF) =−∂x � (xMBF) = H (xMBF) g , part can be divided into an internal part B resulting MBF (36) from currents fowing in the region r < a and an exter- Be whereas the external parts are described in the MSO sys- nal part resulting from currents fowing in the region tem, i.e. r > c , so that e e e e B (xMSO) =−∂xMSO � (xMSO) = H (xMSO) g , = i + e. (37) Bj∈/[a,c] B B (31) where the terms of the series expansion are arranged into i e i e As B and B have their sources beyond the shell they are the matrices H and H and the corresponding Gauss i e purely poloidal and especially nonrotational within the coefcients are summarized into the vectors g and g . i e shell. Tus, there exist scalar potentials and so that Te co-rotating MBF system can be transformed into the the feld can be parameterized in the shell via the Gauss MSO system via representation resulting in xMSO = A xMBF, i i (38) B =−∂x� (32) Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 5 of 18

As already mentioned in the introduction ("Introduction" section) there is no current-free shell-like region around Mercury (Olsen et al. 2010). Te currents fowing in the sh sh shell generate toroidal BT and poloidal BP magnetic i e felds which superpose with B and B . Tus, the total measured feld within the shell is composed of four parts given by i e sh sh B = Bj∈/[a,c] + Bj∈[a,c] = B + B + BT + BP (42)

i e sh sh =−∂x� − ∂x� + ∂x × ∂x × r�P + ∂x × r�T ,     (43) where each part of the feld is described either by a scalar i e sh sh potential , or a scalar function P , T . i e Te scalar potentials and are already parameter- ized by the Gauss coefcients. In the following, a proper sh sh parameterization for the scalar functions T and P is required. Because of the underlying spherical geometry it Fig. 1 Cross section of the spherical shell S(a, c) with inner radius a , outer radius c and mean radius b = (a + c)/2 . The inner sphere with is straightforward to expand the functions into spherical radius RM symbolizes Mercury harmonics ∞ l sh m m m �T = al (r) cos(m ) + bl (r) sin(m ) Pl (cos(θ)) 1 =0 where A describes a rotation matrix around the z-axis l= m   depending on the angular velocity of Mercury’s self-rota- (44) tion measured within the MSO system. and For the practical application it is convenient to describe ∞ l sh both parts of the feld in one coordinate system, for �P = RM example the MSO system. Te transformed data are l=1 m =0 given by m m m cl (r) cos(m ) + dl (r) sin(m ) Pl (cos(θ)), i e   B xMSO = A B xMBF + B xMSO (39) (45) m m m m    where al (r) , bl (r) , cl (r) and dl (r) are the expansion i i e e coefcients which in general depend on the radius r and = A H (xMBF) g + H (xMSO) g (40) m again Pl are the Schmidt normalized Legendre polyno- in the MSO system. mials. Since the toroidal and poloidal felds can be locally Since for the frst validation the model will be applied generated by currents fowing in the shell the radial to simulated magentic feld data, it is useful to match the dependences of the felds and the expansion coefcients, coordinate system of the parametrization with the coor- respectively, are unknown. dinate system of the simulation. Terefore, in the fol- Series expansion of the coefcients lowing all parts of the magnetic feld are described in a Since the exact radial dependence of the expansion coef- Mercury-Body-Fixed anti-solar orientated coordinate m m T a (r) b (r) system (MASO) with coordinates x = (x, y, z) , where fcients l and l is unknown, it is useful to expand the x-axis is orientated towards the nightside of Mercury these functions into a Taylor series in the vicinity of the (away from the sun), the z-axis is orientated antiparallel mean radius b of the shell. Within this series expansion to the internal dipole moment and the y-axis completes it is advisable not to incorporate the efect of all compo- the right-handed system, so that nents of the poloidal current density to the toroidal mag- netic feld at once. Here, we frst concentrate on the radial −100 component of the current density and consider the hori- 0 10 x =  − xMSO. (41) zontal components in higher orders of the Taylor series. 0 01 Bsh   Te toroidal magnetic feld T is generated by poloidal j currents P (cf. Eq. (11)). Ampère’s law yields Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 6 of 18

sh sh µ0j = ∂ × B = ∂ × ∂ × r� this thin shell approximation (Backus 1986; Backus et al. P x T x x T (46)   1996) as illustrated in the following section. 1 1 = −∂ sin(θ)∂ �sh − ∂2�sh e r sin θ θ T sin T r (θ)   (θ)  The thin shell approximation (47) Te thin shell approximation (Backus 1986; Backus et al. 1 sh 1 sh 1996) fnally allows the separation of the poloidal feld + ∂ ∂r r� e + ∂ ∂r r� e . r θ T θ r sin T  (θ)  into its internal and external contributions. Conferring to Eq. (6), the Mie representation for the (48) R3 Te components of the horizontal eθ - and e -direction magnetic feld in the whole space is given by sh are proportional to ∂r r�T . Terefore, the ansatz B = ∂ × (� r) + ∂ × ∂ × (� r) .  x T x x P (52) RM  am(r) = am + a′ mρ + O(ρ2) j l r l l (49) Following Ampère’s law the current density is also sole-  noidal and can as well be parameterized via the Mie rep- and resentation resulting in

m RM m ′ m O 2 µ0j = ∂x × (ŴT r) + ∂x × ∂x × (ŴPr) (53) bl (r) = bl + b l ρ + (ρ ) , (50) r   with related scalar functions ŴT and ŴP. ρ = r−b am a′ m bm b′ m where RM and l , l , l , l are constants for Since the poloidal part of the current density corre- each pair of l and m, is utilized. In the frst order of the sponds with the curl of the toroidal magnetic feld, the Taylor series expansion in the vicinity of the mean radius comparision of Eq. (53) with Eq. (9) shows that the scalar sh ∼ 1 j b, where T r , the horizontal components of P van- function T and ŴP are the same ish and only the contributions of the radial currents driv- � = Ŵ . ing the toroidal magnetic feld are considered (Olsen T P (54) 1997). Using higher orders of the Taylor series, also the j Analogously, the toroidal part of the current density cor- contributions of the horizontal components of P to the responds with the curl of the poloidal magnetic feld and toroidal magnetic feld in the vicinity of the mean radius the comparision of Eq. (53) with Eq. (10) shows that the b can be incorporated. functions P and ŴT are related via Te scalar function of the toroidal magnetic feld 2 , results in ∂x �P =−ŴT (55) ∞ l RM �sh = αm + α′ m ρ + O(ρ2) Pm(cos(θ)), so that the function P is given by the Green’s function T r l l l l=1 m =0   method ′ (51) 1 ŴT (r ) 3 ′ αm = am cos(m ) + bm sin(m ) �P(r) = d r . where l l l and 4π R3 r − r′ (56) α′ m = ′ m cos( ) + ′ m sin( ) l a l m b l m (Olsen 1997).     Tereby, each order of the Taylor series is linked with an Due to the underlying geometry the toroidal current am bm a′ m b′ m additional set of expansion coefcients l , l , l , l density j fowing in the whole space can be written as T i and so on which can be reconstructed from the data in the sum of the toroidal currents j fowing in the region e T analogy to the Gauss coefcients. r < a , the toroidal currents j fowing in the region r > c shT From a mathematical point of view the scalar func- and the toroidal currents j fowing inside the spherical sh sh T tion P of the poloidal magnetic feld BP can be para- shell in the region a ≤ r ≤ c , so that metrized analogously to the toroidal counterpart. But j = ji + je + jsh. within the reconstruction procedure the poloidal felds T T T T (57) that are generated by toroidal currents fowing inside the shell cannot be distinguished from the internally and Tereby, the Mie representation of each part is given by externally driven poloidal felds, since these felds follow ji = ∂ × Ŵi r T x T (58) the same topological structure. But when the half thick-  ness of the shell, defned by h = (c − a)/2 is smaller than the length scale on which the toroidal currents change in e e j = ∂x × ŴT r radial direction, the shell is called a thin shell and the sca- T (59) sh sh  lar function P of the poloidal feld BP vanishes within Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 7 of 18

sh jsh = ∂ × Ŵshr Substituting the Taylor series into the function T x T (60) P  delivers i e ∞ Ŵ = ŴT χ[rc] ′ n with T , T and sh 1 1 n sh (r − b) 3 ′ sh � ( ) = ∂ ′ Ŵ d . Ŵ = Ŵ χ P r r T r′=b ′ r T T [a

∞ 1 1 h n 1 �sh( ) ≤ max Ŵsh( ) d3 ′. P r T b ′ r (73) 4π n!L θ′, ′  , r − r n=0   S(b−h b+h) Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 8 of 18

R3 Te integral in Eq. (73) may be evaluated for any r ∈ density. In contrast to the toroidal magnetic feld, the but we restrict to r inside the shell as only there the mag- poloidal magnetic feld is also measurable in current- netic feld is measured. Ten, the remaining integral free region. Tus, the poloidal feld is a superposition of results in felds generated by currents fowing inside and outside 1 4π 1 d3 ′ = 3 − ( − )3 + 2π ( + )2 − 2 , ′ r r b h b h r (74) S(b−h,b+h) r − r 3 r         utilizing that the coordinate system can be chosen the shell as well as currents fowing within the shell. ′ ′ properly so that θ defnes the angle between r and r . Tis superposition is verifed in Eq. (62). Terefore, Terefore the amount of the poloidal feld generated by currents

∞ n 3 sh 1 h sh 1 2 (b − h) 1 2 2 |�P (−r)|≤ max |ŴT (b)| r − + (b + h) − r . (75) n!L θ′, ′ 3 r  2  n=0    

For all r ∈ [b − h, b + h] , the function fowing within the shell has to be compared with the amount of the internal/external contributions. Further- 1 (b − h)3 1 f (r) = r2 − + (b + h)2 − r2 more, the poloidal feld does not solely depend on the 3 2 (76) r    strength of the toroidal current density, since for the evaluation of the integrals also the volume where the is non-negative and reaches its maximum value at current density fows is vital. Tus, a small toroidal cur- r = b − h with the related function value f (b − h) = 2bh . sh rent density that fows within a large volume outside/ Terefore, an upper bound for P can fnally be esti- inside the shell can have a larger contribution to the mated as feld measured within the shell than a stronger current ∞ n sh 1 h sh fowing within the thin shell. �P (r) ≤ 2bh max ŴT (b) n!L θ′, ′ n=0   Application of the thin shell approximation ∞ 1 n+1 h sh Te thin shell approximation is applied to parameter- = 2bL max ŴT (b) . n!L θ′, ′ n=0   ize the magnetic feld in the vicinity of Mercury by the (77) Gauss–Mie representation to reconstruct Mercury’s internal magnetic feld. Te internal and external scalar When h ≪ L , the spherical shell is called a thin shell i e potentials and are expanded into spherical harmon- (Backus 1986; Backus et al. 1996) and the scalar func- sh sh ics up to third degree and order, representing the inter- tion P of the poloidal magnetic feld BP vanishes. Te sh nal and external dipole, quadrupole and octupole felds. scalar function T , however, remains fnite for all h as sh Te scalar function T of the toroidal magnetic feld shown in Appendix A. Tus, if the shell may be regarded sh BT is expanded into spherical harmonics up to second as thin, then the contribution of the toroidal currents in degree and order and additionally into a frst order Taylor this shell to the poloidal magnetic feld may be neglected. sh series for the radius. Te scalar function P of the poloi- Te poloidal magnetic feld is mainly driven by currents sh dal feld BP is neglected within the thin shell approxi- beyond the shell. Te contribution of the poloidal cur- mation. Terefore, the total feld is parameterized by 46 rents in this shell to the toroidal magnetic feld may not 0 1 expansion coefcients, i.e. Gauss internal dipole ( g1 , g1 , be neglected. 1 0 1 1 2 2 h1 ), Gauss internal quadrupole ( g2 , g2 , h2 , g2 , h2 ), Gauss From a first point of view the thin shell approxima- 0 1 1 2 2 3 3 internal octupole ( g3 , g3 , h3 , g3 , h3 , g3 , h3 ), Gauss exter- tion is not an intuitive approximation. Considering the 0 1 1 0 1 nal dipole ( q1 , q1 , s1 ), Gauss external quadrupole ( q2 , q2 , above presented nature of poloidal and toroidal mag- 1 2 2 0 1 1 2 2 3 s2 , q2 , s2 ), Gauss external octupole ( q3 , q3 , s3 , q3 , s3 , q3 , netic it can be understood as follows: 3 0 1 1 0 1 1 2 2 ′ 0 s3 ), toroidal coefcients ( a1 , a1 , b1 , a2 , a2 , b2 , a2 , b2 , a 1 , Te toroidal magnetic feld only exists within current ′ 1 ′ 1 ′ 0 ′ 1 ′ 1 ′ 2 ′ 2 a 1 , b 1 , a 2 , a 2 , b 2 , a 2 , b 2 ). Tese 46 coefcients are carrying regions (cf. Eq. 14) and thus, it is measurable estimated with Capon’s method (Capon 1969; Toepfer only within these regions. Terefore, the spatial extent et al. 2020a, b). Te Capon method and the underlying of the regions where the poloidal currents fow does not model are tested against simulated data and MESSEN- infuence the strength of the toroidal magnetic feld. It GER in situ data around Mercury. solely depends on the strength of the poloidal current Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 9 of 18

Hybrid simulation of Mercury’s magnetosphere coefcients are reconstructed from the data with good For the frst application of the thin shell approxima- precision. tion simulated magnetic feld data are analyzed. Te Concerning the BepiColombo mission the generated magnetic feld resulting from the plasma interaction circular orbits are idealized cases which are not realiz- of Mercury with the solar wind is simulated with the able in practice. To investigate the applicability of the hybrid code AIKEF (Müller et al. 2011), that has suc- thin shell approximation for elliptical orbits we analyze cessfully been applied to several problems in Mercury’s the magnetic feld data along the prospective orbits of plasma interaction, (Exner et al. 2018, 2020, e.g.). Te MPO. Te orbits are generated in analogy to the cir- g0 =−190 nT internal Gauss coefcients 1 (dipole feld), cular orbits, i.e., with the same longitudinal extend. g0 =−78 nT g0 =−20 nT 2 (quadrupole feld) and 3 (octu- Along this trajectories the distances of the data points pole feld) (Anderson et al. 2012; Tébault et al. 2018; from Mercury’s surface vary from 0.12 RM up to 0.6 RM Wardinski et al. 2019) are implemented in the simula- resulting in a shell thickness of 2h ≈ 0.48 RM . Although tion code. Te interplanetary magnetic feld with a mag- the shell is now much thicker the thin shell approxi- B = 20 nT nitude of IMF is orientated along the vector mation works successfully. Te reconstructed Gauss (x, y, z)T = (0.0, 0.43, 0.9)T in the MASO frame. Te solar coefcients are displayed in Table 2. Te optimal diag- v = 400 km/s wind velocity of sw points along the x-axis onal loading parameter for the application of Capon’s (away from the Sun) and the solar wind proton density method results in σopt. ≈ 1000 nT. n = 30 cm−3 number was chosen to sw . Te resulting Te deviation between the reconstructed and the magnitude of the magnetic feld and the corresponding implemented internal coefcients results in 4.1 nT , i.e. current density in the x-z-plane are displayed in Figs. 2, 3. 1.9% and thus, these values agree with the coefcients Te internal dipole feld dominates the geometry of reconstructed from the data evaluated along the circu- Mercury’s magnetosphere. Yet the quadrupole feld in lar orbits. Since the data are evaluated along the MPO terms of the apparently shifted dipole feld is visible. Te orbits, it is expectable that also Mercury’s internal octu- infuence of the octupole feld is not clearly noticeable on pole feld will be reconstructed with good precision from the feld lines in the fgure. Furthermore, the distribution the data of the magnetometer on board MPO. of the simulated current density shows that there exist no It should be noted that the extension of the underly- completely current-free region around Mercury. ing model by the parameterization of the external parts of the feld using the Gauss–Mie representation improves Reconstruction of the Gauss coefcients from simulated the results signifcantly. When only the internal parts data For the reconstruction of the internal Gauss coef- cients implemented in the simulation code, frst of all, magnetic feld data at a distance of 0.2 RM from Mercu- ry’s surface are evaluated. Te data are retrieved along meridional circular orbits around Mercury. Te orbital plane is rotated about the rotation axis (z-axis) from ◦ ◦ − 50 (afternoon/post-midnight sector) over 0 (noon/ ◦ midnight, x-z-plane) to 50 (morning/pre-midnight sec- tor). For this synthetically generated ideal case in terms of the thin shell approximation, the spherical shell that covers the circular orbits has a vanishing thickness h = 0 so that the application of the thin shell approxi- mation is surely valid. Te reconstructed internal Gauss coefcients are listed in Table 1. Te optimal diagonal loading parameter which determines how the data are weighted within the application of Capon’s method results in σopt. ≈ 1000 nT (Toepfer et al. 2020b). int int Te deviation g − g between the recon- C int structed internal coefcients g and the imple- int C mented internal coefcients g results in 4.0 nT , i.e. gint − gint / gint ≈ 1.9% C and thus, the implemented Fig. 2 Simulated magnitude of the magnetic feld B in the x-z-plane. The white lines represent the magnetic feld lines and the grey circle of radius 1 RM symbolizes Mercury Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 10 of 18

Table 1 Implemented and reconstructed Gauss coefcients for the dipole, quadrupole and octupole feld resulting from the simulated data along the circular orbits at a distance of 0.2 RM from Mercury’s surface Gauss coefcient Input in nT Output Capon in nT

0 g1 – 190.0 – 191.2 1 g1 0.0 – 1.9 1 h1 0.0 – 1.7 0 g2 – 78.0 – 76.0 1 g2 0.0 1.3 1 h2 0.0 – 0.5 2 g2 0.0 – 0.5 2 h2 0.0 – 0.2 0 g3 – 20.0 – 20.3 1 g3 0.0 0.5 1 Fig. 3 Simulated magnitude of the current density j in the x-z-plane. h3 0.0 0.4 2 The grey circle of radius 1 RM symbolizes Mercury g3 0.0 – 0.1 2 h3 0.0 0.9 3 g3 0.0 1.0 3 h3 0.0 – 0.2 i of the feld B are considered in the model, the devia- tion between the implemented and the reconstructed coefcients results in 29.7 nT or 14.4%, respectively (cf. Table 2). Additional parameterization of the external e e thickness h is a global quantity, the set of resulting length poloidal felds B using the external scalar potential scales is averaged over the number of points along the yields a deviation of 10.0 nT or 4.9%, respectively. Tus, orbit resulting in the mean length scale L . It should be for the analysis of Mercury’s internal magnetic feld, the noted that also the poloidal currents have horizontal Gauss–Mie representation is a suitable alternative to the components. Tus, the estimation of the length scale L application of global magnetospheric models. j with the horizontal currents H as a proxy for the toroidal currents is not exact, but it is sufcient for a qualitative Validity of the thin shell approximation discussion. Te coefcients reconstructed from the data evalu- Te deviations between the ideal coefcients imple- ated along the planned MPO trajectories are basically mented in the simulation and the reconstructed coef- in agreement with that from the data along the circular cients for varying values of h/ L are displayed in Table 3. orbits although the shell covering the MPO orbits has a When h approaches L the deviations are greater fnite thickness. than 12 nT . Tis deviation is of the same order when the To investigate the limits of the thin shell approximation parameterization is restricted to the Gauss representa- within the hybrid simulation, the thickness of the shell is tion. But for the data points along the MPO orbits, where increased incrementally. Referring to Eq. (67), the length h/�L�≈0.47 , the application of thin shell approximation scale L is estimated via radial variation of the current is justifed. distribution Since a shell of thickness 2h ≈ 0.48 RM is called thin, j the name thin shell is misleading, although this naming L = H has been adopted within the literature. Conferring to the ∂ j (78) r H global length scale of 1 RM the shell of thickness 0.48 RM is not thin, but compared with the current length scale for the horizontal currents j = j − j · er er which are H  L it is. Terefore, the term thin has to be understood in a used as a proxy for the toroidal currents at each point mathematical sense. along the orbit. Since L represents a local quantity that varies for each point along the orbit, whereas the half Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 11 of 18

Table 2 Implemented and reconstructed Gauss coefcients Table 3 Deviation between the implemented and the for the dipole, quadrupole and octupole feld resulting from the reconstructed Gauss coefcients for varying ratios of the half simulated data along the future MPO orbits. For comparision in thickness h to the mean current length scale L the last columns the reconstructed Gauss coefcients resulting Bi h in RM L in RM h/ L Devitation in nT Devitation from the sole parameterization of the internal parts (Gauss in % internal, 15 coefcients considered within the model) as well as the coefcients resulting from the parameterization of the 0.24 0.51 0.47 4.1 1.9 i e internal B and external poloidal felds B (Gauss internal external, 0.32 0.49 0.65 4.8 2.3 30 coefcients considered within the model) are presented 0.43 0.46 0.94 12.8 6.2 Gauss Input in nT Output capon in nT 0.62 0.42 1.5 13.5 7.0 coefcient 1.09 0.37 2.9 16.8 8.1 Gauss–Mie Gauss internal Gauss internal external

0 g1 – 190.0 – 190.9 – 188.3 – 190.2 1 g1 0.0 – 2.1 0.8 – 0.8 a small subset of the whole MESSENGER data set is ana- 1 h1 0.0 1.7 – 18.6 – 5.7 lyzed. In the case of a small and noisy data set the per- 0 formance of the estimator can be improved by seperating g2 – 78.0 – 76.1 – 78.5 – 76.0 1 the data set into several subsets (Meir 1994). Terefore, g2 0.0 0.7 5.6 3.3 h1 0.0 0.0 0.7 – 0.1 the Gauss coefcients are reconstructed for each pair of 2 orbits and the results are averaged over the nine pairs. g2 0.0 0.3 0.5 0.4 2 Outliers were not included within the averaging. Te h2 0.0 0.4 4.1 4.5 2 resulting mean values are listed in Table 4. Te recon- 0 – 20.0 – 19.5 – 14.3 – 20.8 g3 structed coefcients for each pair of orbits and the stand- 1 g3 0.0 0.6 – 0.5 0.4 ard deviations of the mean values are listed in Table 5 of 1 h3 0.0 – 0.1 0.9 – 0.2 Appendix C. 2 g3 0.0 – 1.9 15.2 – 2.1 Te mean values of the reconstructed internal Gauss 2 h3 0.0 0.2 0.4 – 0.2 coefcients and the external dipole coefcients are in fea- 3 g3 0.0 0.8 – 3.0 – 0.4 sible agreement with the values provided by Anderson 3 h3 0.0 – 0.4 14.4 5.1 et al. (2012), Tébault et al. (2018) and Wardinski et al. 0 (2019). Furthermore, the related standard deviations for q1 – – 34.0 – – 35.9 1 each coefcient are within the range of the variations q1 – – 8.3 – – 7.2 resulting from the time varying magnetic feld (Wardin- s1 – – 6.4 – – 9.1 1 ski et al. 2019). To classify the coefcients reconstructed from the simulated magnetic feld data (Table 2) in terms of that resulting from the MESSENGER data, the simu- lated data are furthermore evaluated along the MESSEN- Reconstruction of the Gauss coefcients from MESSENGER GER trajectories. Te reconstructed Gauss coefcients data are listed in Table 6 of Appendix C. Tese coefcients Te Gauss–Mie representation has successfully been val- are in agreement with the results presented in Table 4. idated for the simulated data. For the reconstruction of Since the reconstructed coefcients resulting from the the Gauss coefcients from the MESSENGER data only data evaluated along the MPO trajectories are in bet- data points in the northern hemisphere within a distance ter agreement with the implemented coefcients than of r ≤ 1.5 RM and x > −0.4 RM from Mercury’s surface the coefcients resulting from the data evaluated along can be considered because the orbits do not cover the the MESSENGER trajectories, the restriction of the data southern hemisphere properly. points to the northern hemisphere and the related degra- For the reconstruction of Mercury’s internal magnetic dation of the model resolution infuences the results sig- feld, the data from nine pairs of MESSENGER orbits nifcantly (Heyner et al. 2020, cf). Furthermore, it should with diferent orientations of the periapsis between 10. be noted that the analysis of the length scales of the cur- August 2012 and 14. July 2014 are analyzed. Te combi- rent densities (cf. "Validity of the thin shell approxima- nation of the orbits improves the model resolution com- tion" section) cannot be performed for the MESSENGER pared to the analysis of single orbits. A discussion of the orbits, because the current densities in the vicinity of the model resolution, as provided by Connerney (1981), can orbits are unknown. be found in the Appendix B. As a proof of concept, only Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 12 of 18

Table 4 Gauss coefcients for the internal dipole, quadrupole whereas the toroidal feld remains fnite. In the case of and octupole feld and for the external dipole feld reconstructed the planned MPO orbits, the thin shell approximation is from MESSENGER data. In the last colum the ranges of Gauss a reasonable choice. coefcients reconstructed from MESSENGER data by Anderson For the application of the thin shell approximation et al. (2012), Thébault et al. (2018) and Wardinski et al. (2019) are shown the internal Gauss coefcients for the dipole, quadru- pole and octupole feld are implemented in the simula- Gauss coefcient Output capon in nT Ranges of former tion code AIKEF and the magnetic feld data resulting works in nT from the plasma interaction of Mercury with the solar 0 g1 – 175.9 – 215.8 to – 190.0 wind are simulated in the vicinity of Mercury. Te data 1 g1 – 1.7 – 2.9 – 0.9 are evaluated along the planned MPO orbits and the 46 1 expansion coefcients, describing the internal, external h1 4.5 0.8 – 2.7 0 and the toroidal feld are reconstructed with Capon’s g2 – 82.0 – 83.2 to – 57.7 g1 0.8 – 1.5 – 3.4 method. Since the reconstructed internal Gauss coef- 2 fcients are in good agreement with that implemented h1 1.0 – 1.4 – 0.0 2 in the simulation code, the parameterization of the g2 2.9 – 7.0 to – 1.4 2 magnetic feld using the Gauss–Mie representation is a 2 1.7 – 3.3 – 0.4 h2 suitable alternative to the application of global magneto- 0 – 19.1 – 36.7 to – 15.7 g3 spheric models. Even the implemented Gauss coefcient 1 g3 1.1 1.8 – 4.1 for the octupole feld of Mercury can be reconstructed 1 h3 0.4 0.3 – 0.8 accurately and therefore, it is expectable that Mercury’s 2 g3 – 0.5 – 1.5 – 9.2 internal octupole feld will be reconstructed with high 2 h3 – 5.9 0.9 – 2.6 precision from the magnetometer data on board MPO. 3 g3 – 5.8 – 2.5 to– 1.4 Tus, it is worthwile to investigate the analysis of higher 3 multipoles, such as hexadecapole, from the data along h3 – 3.3 0.1 – 0.3 0 the MPO trajectories. q1 – 36.6 – 39.7 to – 23.2 Furthermore, the thin shell approximation is q1 – 7.4 – 0.2 – 0.7 1 applied to reconstruct Mercury’s internal magnetic s1 0.8 – 0.1 – 1.5 1 field from the data of the MESSENGER mission. Summary and outlook The results are in reasonable agreement with former In the vicinity of Mercury no completely current-free works. Since only the data points in the northern region is present. Terefore, the Gauss representation hemisphere are vital for the analysis of the MESSEN- does not yield a proper parametrization of Mercury’s GER data, the symmetrically distributed MPO orbits magnetospheric feld. Extension of the Gauss represen- will deliver a better model resolution than the MES- tation to the Gauss–Mie representation allows a more SENGER orbits. complete characterization of Mercury’s internal and Concerning the BepiColombo mission the com- magnetospheric feld. bination of the Gauss representation with the Mie For the parameterization of the magnetic feld the representation is a useful model for the analysis of orbit where the magnetic feld is measured, is conceptu- Mercury’s internal magnetic field. As the BepiCo- ally covered by a spherical shell. Due to the underlying lombo mission consits of two elements, the planetary geometry the total measured magnetic feld is a super- orbiter and the magnetospheric orbiter, measure- position of internal and external poloidal felds gener- ments of any gradients of Mercury’s magnetic field are ated by toroidal currents fowing outside the spherical possible which may lead to further improvements of shell as well as toroidal and poloidal felds generated by the methods presented here. Besides the analysis of currents fowing within the shell. Tereby, each compo- the internal magnetic field, the reconstructed coeffi- nent of the feld is either described by a scalar potential cients for the toroidal magnetic field can be used for i e sh sh calculating poloidal current systems, e.g. field aligned ( , ) or a scalar function ( P , T ). Tese potentials and functions can be expanded into spherical harmon- currents, in the vicinity of the orbit where the data are ics. When the thickness of the spherical shell is smaller evaluated. than the length scale on which the toroidal current den- sity changes in radial direction the shell is called a thin shell. Ten the poloidal feld generated by currents fow- ing inside the shell is negligible compared to the poloi- dal feld generated by currents fowing beyond the shell, Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 13 of 18

Appendix A: The thin shell approximation points. Te condition number is defned as the ratio of the for the toroidal magnetic feld largest and the smallest singular value of the shape matrix. As derived in "Te thin shell approximation" section, Terefore, some of the 46 singular values, corresponding the scalar function of the poloidal magnetic feld can be to the 46 expansion coefcients, of the shape matrix have estimated upwards as to be dropped within the numerical calculation to improve ∞ the condition number. Within the low rank approxima- 1 h n+1 sh sh tion (Eckart and Young 1936) only k singular values, where �P (r) ≤ 2bL max ŴT (b) . n!L θ′, ′ k ≤ 46 n=0   , can be considered and therefore, the shape matrix H H (79) is approximated by a shape matrix k which has a lower sh rank and a lower condition number. When h ≪ L , the function P vanishes. Now the infu- ence of the thin shell approximation for the toroidal mag- Capon’s flter matrix sh −1 netic feld BT is discussed. † † −1 † −1 sh w = H M H H M Te scalar function T of the toroidal magnetic feld (84) sh  BT can be expanded analogously into a Taylor series in the vicinity of the mean radius b which is the key parameter for calculating Capon’s ∞ estimator sh sh 1 n sh n � (r) = Ŵ (r) = ∂ Ŵ (r) (r − b) , g = w†B T P n! r P r=b (80) C (85) n=0   sh sh fulflls the distortionless constraint using that the scalar functions T and ŴP are the same † (cf. Eq. 54). w H = I, (86) If we furthermore assume that also the radial deriva- tives of the poloidal current density are bounded so where I is the identity matrix and M =�B ◦ B� describes that the data covariance matrix of the magnetic feld meas- B sh urements (Toepfer et al. 2020a, b). As a consequence n sh ŴP ∂ Ŵ ≤ , (81) of the low rank approximation Capon’s flter matrix is r P Ln modifed to sh the function �T (r) can be estimated upwards as −1 w† = H † M−1H H † M−1 k k k k (87) ∞ 1  |�sh(r)|≤ |∂nŴsh(b)|·|r − b|n T − n! r P n=0 so that ∞ (82) † 1 |Ŵsh(b)| R = w H �=I. ≤ P |r − b|n. k (88) n! Ln n=0 R = w† H Te matrix k is called the model resolution sh Using that �T (r) is solely evaluated within the spherical matrix. Because of r ∈ [b − h, b + h] r − b ≤ h shell, where and therefore gk = R g, C (89) results in ∞ n gk 1 h where C denotes the estimator resulting from the con- �sh(r) ≤ Ŵsh(b) . g T n!L P (83) sideration of k singular values and is the ideal coef- n=0 cient vector implemented in the simulation code, the sh diagonal elements of R identify the resolution of each When h ≪ L , the function T remains fnite. coefcient (Connerney 1981). When R = I each coef- cient is resolved for 100% . If the resolution is smaller than 100% Appendix B: Model resolution , there exist model parameter covariances. Te more singular values are considered within the For the analysis of Mercury’s internal magnetic feld from estimation, the better the model resolution becomes, the MESSENGER data only data points in the northern whereas the condition number of the shape matrix hemisphere can be considered. Tis limitation impairs the increases and thus, the infuence of measurement condition number κ of the shape matrix H from κ ≈ 226 8 errors increases. for the data points along the MPO orbits to κ ≈ 10 for the To achieve a compromise between the resolution and data points along a single MESSENGER orbit, where the the condition number, the coefcients are estimated shape matrix describes the spacial distribution of the data Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 14 of 18

for diferent numbers of singular values and the change the procedure is exemplarily illustrated for the coef- 0 of the coefcients resulting from k singular values to fcient g1 , which corresponds to the diagonal element the coefcients resulting from k − 1 singular values R11 by analyzing the data of one pair of MESSENGER is regarded. For the fnal computation the maximum orbits. number of singular values is chosen from which the reconstructed coefcients are almost constant. In Fig. 4 Changing the number of considered singular values incrementally from 46 to 37 the resulting coefcient changes signifcantly. For k ≤ 37 the values are almost constant and therefore, 37 singular values, correspond- 0 ing to a model resolution of 88% for the coefcient g1 are considered within the estimation.

Appendix C: Tables: Reconstructed Gauss coefcients from MESSENGER orbits

0 Fig. 4 Variation of the reconstructed coefcient g1 (red) and the related model resolution R11 (blue) with respect to the number k of singular values. For k ≤ 37 (dashed line) the reconstructed values of 0 g1 are almost constant Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 15 of 18 3.3 0.8 14.3 16.8 12.3 15.0 – 5.5 1 1 – 13.7 – 21.9 – 41.5 – 38.4 s 2.7 8.4 41.1 – 0.8 – 1.7 – 7.4 1 1 – 47.8 – 12.1 – 16.2 – 27.4 – 16.2 q 5.6 9.6 11.4 23.2 – 5.8 0 1 – 65.2 – 24.2 – 24.5 – 61.5 – 38.2 – 36.6 q 7.1 80.2 21.5 53.2 35.5 35.5 21.2 – 3.3 3 3 – 21.9 – 19.9 – 41.4 h 3.8 23.5 25.1 53.2 26.8 – 5.8 – 43.4 – 32.5 – 32.1 – 22.6 3 3 – 178.1 g 3.6 88.9 13.7 11.0 – 7.3 – 8.2 – 5.9 2 3 – 11.4 – 18.8 – 39.1 – 13.2 h 9.5 4.0 14.4 12.1 – 6.4 – 6.3 – 0.5 2 3 – 20.3 – 41.4 – 21.7 – 18.4 g 1.7 0.4 12.5 31.2 17.3 12.7 – 3.6 1 3 – 24.5 – 14.1 – 68.5 – 11.5 h 9.0 3.5 1.1 9.2 10.7 12.8 – 1.8 – 5.9 – 8.9 1 3 – 10.9 – 65.6 g 8.5 – 6.7 – 1.8 0 3 – 11.5 – 10.0 – 13.6 – 20.8 – 17.4 – 27.5 – 32.9 – 19.1 g 2.7 2.1 1.7 7.8 30.7 32.0 10.5 – 8.5 2 2 – 52.2 – 82.1 – 27.2 h 5.2 2.9 12.0 13.5 10.6 – 6.4 – 9.8 2 2 – 40.1 – 30.0 – 29.6 – 22.1 g 6.3 5.6 1.0 8.2 21.1 11.9 16.0 – 7.6 – 2.2 – 8.0 1 2 – 21.2 h 0.8 16.9 17.0 12.7 22.8 73.0 13.9 – 8.5 – 9.5 1 2 – 10.1 – 12.8 g 10.2 0 2 – 71.6 – 66.7 – 96.6 – 81.2 – 81.7 – 76.0 – 80.6 – 96.6 – 87.0 – 82.0 g 6.6 5.0 0.5 7.9 4.5 6.2 15.3 – 2.0 – 1.6 1 1 – 27.2 – 17.3 h 2.0 0.2 8.7 5.9 18.0 – 6.5 – 2.6 – 9.0 – 4.7 – 1.7 1 1 – 15.1 g 15.8 0 1 – 197.4 – 190.0 – 158.5 – 164.1 – 155.6 – 190.4 – 163.3 – 162.3 – 163.7 – 175.9 g Gauss coefcients for the dipole, quadrupole and octupole feld reconstructed from selected pairs of MESSENGER orbits quadrupole and octupole from the dipole, feld reconstructed Gauss coefcients for 2013–T245, 2014–T21 2013–T245, Orbits 5 Table 2014–T143, 2014–T195 2014–T143, 2013–T330, 2014–T107 2013–T330, 2013–T330, 2014–T195 2013–T330, 2014–T107, 2014–T143 2014–T107, 2012–T223, 2012–T311 2012–T223, 2013–T330, 2014–T21 2013–T330, 2014–T107, 2013–T245 2014–T107, 2014–T195 Mean value Standard deviation Standard Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 16 of 18 4.3 6.8 6.9 51.7 – 5.9 – 6.8 – 2.5 – 2.5 1 1 – 11.1 – 35.4 – 22.6 s 8.6 – 9.8 – 7.8 – 3.4 1 1 – 13.0 – 27.5 – 22.8 – 18.7 – 36.7 – 37.9 – 14.7 q 12.4 – 6.6 – 4.6 0 1 – 39.2 – 31.9 – 20.2 – 12.8 – 43.2 – 41.5 – 53.3 – 31.5 q 2.1 9.0 6.6 6.5 7.1 10.6 11.7 17.0 13.1 – 7.4 3 3 – 16.4 h 10.6 82.1 10.4 – 8.5 3 3 – 9.3 – 13.9 – 17.1 – 16.9 – 10.0 – 79.4 – 25.1 g 5.7 6.9 2.3 7.4 2.3 2.3 4.9 77.5 – 5.1 – 3.2 2 3 – 41.7 h 3.7 4.3 12.3 – 5.7 – 6.1 – 6.8 2 3 – 5.7 – 0.1 – 3.5 – 57.4 – 17.1 g 0.0 3.7 – 6.8 – 8.4 – 4.1 – 4.4 – 9.7 – 6.8 1 3 – 10.9 – 10.0 – 18.8 h 7.6 7.3 46.0 – 5.8 – 3.5 – 1.6 1 3 – 5.6 – 10.7 – 11.3 – 44.8 – 13.7 g 5.9 – 8.1 0 3 – 8.7 – 24.4 – 12.2 – 15.2 – 18.2 – 28.8 – 13.9 – 19.0 – 18.8 g 0.4 7.4 – 4.0 2 2 – 15.0 – 34.3 – 31.0 – 17.3 – 39.4 – 17.1 – 10.5 – 10.6 h 3.7 7.0 3.1 1.7 1.7 3.9 24.8 – 3.6 2 2 – 1.9 – 16.3 – 15.8 g 5.2 7.4 7.6 4.1 6.4 2.7 2.1 5.1 2.2 21.9 1 2 – 11.7 h 6.4 7.6 21.6 15.4 10.4 25.1 27.9 36.5 14.3 1 2 – 16.0 – 51.6 g 8.1 – 93.7 – 99.4 – 97.0 0 2 – 103.8 – 105.7 – 113.9 – 131.5 – 117.6 – 114.8 – 104.0 g 3.7 5.6 3.7 7.5 1.6 0.1 5.6 1 1 – 4.4 – 8.6 – 1.8 – 6.3 h 9.9 6.8 1.1 16.3 14.0 1 1 – 10.5 – 15.1 – 12.6 – 21.2 – 14.8 – 28.4 g 19.9 0 1 – 136.4 – 179.6 – 148.9 – 149.3 – 190.4 – 150.1 – 105.0 – 144.5 – 131.5 – 163.7 g Gauss coefcients for the dipole, quadrupole and octupole feld reconstructed from the simulated data evaluated at selected data evaluated pairs of MESSENGER orbits the simulated quadrupole and octupole from the dipole, feld reconstructed Gauss coefcients for 6 Table Orbits 2014–T21 2013–T245, 2014–T143, 2014–T195 2014–T143, 2013–T330, 2014–T107 2013–T330, 2013–T330, 2014–T195 2013–T330, 2014–T107, 2014–T143 2014–T107, 2012–T223, 2012–T311 2012–T223, 2013–T330, 2014–T21 2013–T330, 2014–T107, 2013–T245 2014–T107, 2014–T195 Mean value Standard deviation Standard Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 17 of 18

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