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The Mie Representation for Mercury's Magnetic Field

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The Mie Representation for Mercury's Magnetic Field 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 magnetometer 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: s.toepfer@tu‑braunschweig.de 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 © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. Toepfer et al. Earth, Planets and Space (2021) 73:65 Page 2 of 18 (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 .
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