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The Mie representation for Mercury’s magnetic field
Earth, Planets and Space volume 73, Article number: 65 (2021)
Abstract
The parameterization of the magnetospheric field contribution, generated by currents flowing in the magnetosphere is of major importance for the analysis of Mercury’s internal magnetic field. Using a combination of the Gauss and the Mie representation (toroidal–poloidal decomposition) for the parameterization of the magnetic field enables the analysis of magnetic field data measured in current carrying regions in the vicinity of Mercury. In view of the BepiColombo mission, the magnetic field resulting from the plasma interaction of Mercury with the solar wind is simulated with a hybrid simulation code and the internal Gauss coefficients for the dipole, quadrupole and octupole field are reconstructed from the data, evaluated along the prospective trajectories of the Mercury Planetary Orbiter (MPO) using Capon’s method. Especially, it turns out that a highprecision determination of Mercury’s octupole field is expectable from the future analysis of the magnetic field data measured by the magnetometer on board MPO. Furthermore, magnetic field data of the MESSENGER mission are analyzed and the reconstructed internal Gauss coefficients are in reasonable agreement with the results from more conventional methods such as the leastsquare fit.
Introduction
Characterization of Mercury’s internal magnetic field is one of the primary goals of the BepiColombo mission (Benkhoff et al. 2010). The magnetic field in the vicinity of Mercury is composed of internal and external parts. The internal field originates in the dynamogenerated field, crustal remanent field and induction field, which are mainly dominated by the dipole, quadrupole and octupole field. The external field originates from currents flowing in the magnetosphere. For a clear separation of the internal magnetic field from the external field, each part of the magnetic field has to be modeled properly. Especially the parameterization of the external parts is of major importance since these parts contribute a significant amount to the total field within Mercury’s magnetosphere (Anderson et al. 2011).
Above the planetary surface, the internal part of the field is a currentfree magnetic field which can be parameterized by the Gauss representation (Gauß 1839; Glassmeier and Tsurutani 2014). When only data in sourcefree regions without currents are analyzed, the Gauss representation also yields a parametrization for the external parts of the field (Olsen et al. 2010). But in the vicinity of Mercury significant currents are expected (Anderson et al. 2011, eg); and therefore, the Gauss representation is insufficient due to the currentgenerated magnetic field outside the planet. In this paper, we construct a parametric Mercury magnetic field model by extending the Gauss representation to the Gauss–Mie representation.
Previous studies parameterized the external parts by making use of a global magnetospheric model, such as the paraboloid model for Mercury’s magnetosphere (Alexeev et al. 2008), that has successfully been applied to the analysis of Mercury’s internal and external magnetic field (Alexeev et al. 2010; Johnson et al. 2021), as well as the modified Tsyganenko model (Korth et al. 2004). As an alternative for these models the Mie representation is useful for decomposing the magnetic field in current carrying regions in the vicinity of Mercury (Backus 1986; Olsen 1997; Olsen et al. 2010).
The Mie representation, also known as toroidalpoloidal decomposition, is based on the solenoidality of the magnetic field and enables to decompose the field into its toroidal and poloidal parts (Backus 1986; Backus et al. 1996; Kazantsev and Kardakov 2019). This representation has successfully been applied to several problems in space plasma physics, especially for the analysis of the Earth’s magnetosphere. For example, Engels and Olsen (1999) used the Mie representation for calculating the magnetic effect caused by a three dimensional current density model. Olsen (1997) applied the Mie representation to reconstruct ionospheric Fcurrents in the Earth’s magnetosphere from MAGSAT data. Furthermore, Bayer et al. (2001) introduced the wavelet Mie representation of the magnetic field to calculate current densities from MAGSAT and CHAMP data. This approach has been expanded on the modeling of the Earth’s magnetic field in terms of vector kernel functions (Mayer and Maier 2006). Kosik (1984) constructed a model for the Earth’s magnetosphere based on the Mie representation.
In this work, the magnetic field in the vicinity of Mercury is parameterized by a combination of the Gauss and the Mie representation (hereafter, Gauss–Mie representation), based on the works of Backus (1986), Backus et al. (1996) and Olsen (1997), for analyzing Mercury’s internal magnetic field. The desired internal Gauss coefficients for the dipole, quadrupole and octupole field are estimated with Capon’s method (Capon 1969). The Capon method serves as a powerful inversion method for linear inverse problems and was used by, e.g., Motschmann et al. (1996), Glassmeier et al. (2001), Narita et al. (2006) and Narita (2012) to evaluate spatial spectra of space plasma waves. As shown by Toepfer et al. (2020a) the Capon method can also be used in a generalized way to compare actual measurements with theoretical models. Here, we expand on this method. First of all, the mathematical foundations of the Mie representation, as derived by Backus (1986) and Backus et al. (1996) are revisited. Afterwards, the resulting thin shell approximation is applied to simulated magnetic field data, that are evaluated along the future data points of the Mercury Planetary Orbiter (MPO). This enables the judgement of the expectable inversion results from the data of the magnetometer (Glassmeier et al. 2010; Heyner et al. 2020) on board MPO. Finally, the model is applied to MESSENGER data and the reconstructed Gauss coefficients are compared with the results of former works by Anderson et al. (2012), Thébault et al. (2018) and Wardinski et al. (2019).
The Mie representation
The solenoidality of the magnetic field \(\underline{B}\) guarantees the existence of a vector potential \(\underline{A}\), so that
holds, where \(\partial _{\underline{x}}\) is the spatial derivative. Using spherical coordinates with radius \(r\in \left[ 0,\infty \right)\), longitude \(\lambda \in \left[ 0,2\pi \right]\) and colatitude angle \(\theta \in \left[ 0,\pi \right]\) the vector potential can be seperated into its component \(\Psi _T\underline{r}\) parallel to \(\underline{r}=r\,\underline{e}_r\), where \(\underline{e}_r\) is the unit vector in radial direction, and its components \(\underline{F}\times \underline{r}\) perpendicular to \(\underline{r}\), yielding
with the scalar functions \(\Psi _T\) and \(\varphi\) as well as a vector field \(\underline{F}\) (Jacobs 1987; Krause and Rädler 1980). Because of \(\partial _{\underline{x}}\times \partial _{\underline{x}}\varphi =0\), the function \(\varphi\) can be chosen properly so that \(\partial _{\underline{x}}\times \underline{F}=0\) holds, and therefore
with a scalar function \(\Psi _P\) without changing the magnetic field (Jacobs 1987; Krause and Rädler 1980). The vector potential results in
Substituting \(\underline{A}\) into Eq. (1) delivers
which is called the Mie representation of the magnetic field (Backus 1986; Backus et al. 1996).
The first term on the right hand side in Eq. (6)
is the toroidal part of the field and the second term
is the poloidal part of \(\underline{B}\).
From the definition of \(\underline{B}_T\) and \(\underline{B}_P\) it is clear that \(\underline{B}_T\) is perpendicular to \(\underline{B}_P\) and also perpendicular to \(\underline{r}\). Therefore, the toroidal part of the field does not have a radial component. Furthermore, poloidal magnetic fields are generated by toroidal currents and vice versa:
is a poloidal vector and
is a toroidal vector.
In curlfree regions where especially the poloidal current density \(\underline{j}_P\) vanishes, Ampère’s law reads as follows:
or equivalently
Therefore, the gradient of \(\partial _r\left( r\Psi _T \right)\) has only a radial component, leaving us with
so that the function \(\Psi _T\) solely depends on the radial distance from the center \(\left( \Psi _T=\Psi _T(r)\right)\) and thus, the toroidal magnetic field vanishes
On the other hand, when the toroidal current density vanishes
or equivalently
and simulanously
the function \(\partial _{\underline{x}}^2\Psi _P\) solely depends on the radius. Therefore, the poloidal field
in general remains finite in currentfree regions.
Relation to the Gauss representation
When magnetic field data in curlfree regions (where \(\partial _{\underline{x}}\times \underline{B}=0\) holds) are analyzed, there exists a scalar potential \(\Phi\) so that the magnetic field can be written as
which is known as the Gauss representation of the magnetic field (Gauß 1839; Glassmeier and Tsurutani 2014). Simultanously, the Mie representation in due consideration that the toroidal magnetic field vanishes in curlfree regions is given by
Comparison of coefficients with Eq. (21) for \(\underline{e}_\theta\) shows that
and analogously for \(\underline{e}_\lambda\)
Consequently, when \(\Psi _P\) is known, the scalar potential is given by
Comparison of the \(\underline{e}_r\)coefficients delivers
or equivalently
where
is the angular part of the Laplacian. For a given \(\Phi\), the function \(\Psi _P\) can be found by solving Eqs. (27, 29) simultanously.
As a consequence, the scalar function \(\Psi _P\) and the scalar potential \(\Phi\) are equivalent in curlfree regions and the Mie representation transists into the Gauss representation. Thus, the Gauss representation can be understood as a special case of the Mie representation.
Parameterization of the magnetic field
Assuming that the conductivity of Mercury’s mantle is negligibly small (like lunar regolith (Zharkova et al. 2020)), the planetary contribution to the field outside Mercury is purely poloidal. The currents flowing in the magnetosphere generate poloidal and toroidal magnetic fields that superpose with the curlfree planetary magnetic field. To be able to separate the planetary magnetic field out of the measured field and to parameterize it via the Gauss coefficients, a combined parametrization composed of the Mie and the Gauss representation (Gauss–Mie representation), which is based on the works of Backus (1986) and Olsen (1997) is necessary.
Suppose that the magnetic field in the vicinity of Mercury is measured within a spherical shell \(S(a,c)\) with inner radius \(a>R_\text {M}\), where \(R_\text {M}\) indicates the radius of Mercury, outer radius c and mean radius \(b=\frac{1}{2}\left( a+c \right)\) as displayed in Fig. 1. The shell can be constructed independently of the orbit’s geometry by conceptually covering the orbit of the spacecraft. Furthermore, the shell may include current carrying regions. Although the Mie representation enables us to analyze those currents, we focus on the analysis of Mercury’s internal magnetic field.
Due to the underlying geometry, the space around Mercury can be decomposed into three disjoint radial zones:

points in the region \(r<a\) below the shell

points in the region \(r>c\) above the shell

points in the region \(a \le r \le c\) inside the shell layer
Making use of the superposition principle the total magnetic field \(\underline{B}\) measured inside the shell layer (\(a \le r \le c\)) is a composition of the field \(\underline{B}_{\underline{j}\in \left[ a,c \right] }\) generated by currents flowing inside the shell and the field \(\underline{B}_{\underline{j}\notin \left[ a,c \right] }\) generated by currents flowing outside the shell. Again considering the underlying geometry, the second part can be divided into an internal part \(\underline{B}^i\) resulting from currents flowing in the region \(r<a\) and an external part \(\underline{B}^e\) resulting from currents flowing in the region \(r>c\), so that
As \(\underline{B}^i\) and \(\underline{B}^e\) have their sources beyond the shell they are purely poloidal and especially nonrotational within the shell. Thus, there exist scalar potentials \(\Phi ^i\) and \(\Phi ^e\) so that the field can be parameterized in the shell via the Gauss representation resulting in
and
where the scalar potentials are given by (Gauß 1839; Glassmeier and Tsurutani 2014)
and
The expansion coefficients \(g_l^m\) and \(h_l^m\) are the internal Gauss coefficients, the coefficients \(q_l^m\) and \(s_l^m\) are the external Gauss coefficients and \(P_l^m\) are the Schmidt normalized Legendre polynomials of degree l and order m. Since Mercury’s internal magnetic field is dominated by the internal dipole, quadrupole and octupole fields, the series expansions in Eqs. (34, 35) will be truncated at the degree \(l=3\) for the practical application.
It should be noted that the internal field \(\underline{B}^i\) is canonically described in a MercuryBodyFixed corotating coordinate system (MBF), whereas the external field \(\underline{B}^e\) is canonically described in a MercurySolarOrbital system (MSO) with the xaxis orientated towards the sun, the zaxis orientated parallel to the rotation axis, i.e. antiparallel to the internal dipole moment, and the yaxis completes the righthanded system (Heyner et al. 2020). Let \(\underline{x}_{\text {MSO}}=\left( x_{\text {MSO}},y_{\text {MSO}},z_{\text {MSO}}\right) ^T\) define the MSO coordinate system and let \(\underline{x}_{\text {MBF}}=\left( x_{\text {MBF}},y_{\text {MBF}},z_{\text {MBF}}\right) ^T\) be the coordinates of the corotating MBF system. Then, the internal parts of the field are given by
whereas the external parts are described in the MSO system, i.e.
where the terms of the series expansion are arranged into the matrices \(\underline{\underline{H}}^i\) and \(\underline{\underline{H}}^e\) and the corresponding Gauss coefficients are summarized into the vectors \(\underline{g}^i\) and \(\underline{g}^e\).
The corotating MBF system can be transformed into the MSO system via
where \(\underline{\underline{A}}\) describes a rotation matrix around the zaxis depending on the angular velocity of Mercury’s selfrotation measured within the MSO system.
For the practical application it is convenient to describe both parts of the field in one coordinate system, for example the MSO system. The transformed data are given by
in the MSO system.
Since for the first validation the model will be applied to simulated magentic field data, it is useful to match the coordinate system of the parametrization with the coordinate system of the simulation. Therefore, in the following all parts of the magnetic field are described in a MercuryBodyFixed antisolar orientated coordinate system (MASO) with coordinates \(\underline{x}=(x,y,z)^T\), where the xaxis is orientated towards the nightside of Mercury (away from the sun), the zaxis is orientated antiparallel to the internal dipole moment and the yaxis completes the righthanded system, so that
As already mentioned in the introduction ("Introduction" section) there is no currentfree shelllike region around Mercury (Olsen et al. 2010). The currents flowing in the shell generate toroidal \(\underline{B}_T^{sh}\) and poloidal \(\underline{B}_P^{sh}\) magnetic fields which superpose with \(\underline{B}^i\) and \(\underline{B}^e\). Thus, the total measured field within the shell is composed of four parts given by
where each part of the field is described either by a scalar potential \(\Phi ^i\), \(\Phi ^e\) or a scalar function \(\Psi _P^{sh}\), \(\Psi _T^{sh}\).
The scalar potentials \(\Phi ^i\) and \(\Phi ^e\) are already parameterized by the Gauss coefficients. In the following, a proper parameterization for the scalar functions \(\Psi _T^{sh}\) and \(\Psi _P^{sh}\) is required. Because of the underlying spherical geometry it is straightforward to expand the functions into spherical harmonics
and
where \(a_l^m(r)\), \(b_l^m(r)\), \(c_l^m(r)\) and \(d_l^m(r)\) are the expansion coefficients which in general depend on the radius r and again \(P_l^m\) are the Schmidt normalized Legendre polynomials. Since the toroidal and poloidal fields can be locally generated by currents flowing in the shell the radial dependences of the fields and the expansion coefficients, respectively, are unknown.
Series expansion of the coefficients
Since the exact radial dependence of the expansion coefficients \(a_l^m(r)\) and \(b_l^m(r)\) is unknown, it is useful to expand these functions into a Taylor series in the vicinity of the mean radius b of the shell. Within this series expansion it is advisable not to incorporate the effect of all components of the poloidal current density to the toroidal magnetic field at once. Here, we first concentrate on the radial component of the current density and consider the horizontal components in higher orders of the Taylor series.
The toroidal magnetic field \(\underline{B}_T^{sh}\) is generated by poloidal currents \(\underline{j}_P\) (cf. Eq. (11)). Ampère’s law yields
The components of the horizontal \(\underline{e}_\theta\) and \(\underline{e}_\lambda\)direction are proportional to \(\partial _r\left( r\Psi _T^{sh} \right)\). Therefore, the ansatz
and
where \(\rho =\frac{ rb }{R_\text {M}}\) and \(a_l^m\), \(a_{\,l}^{\prime \,m}\), \(b_l^m\), \(b_{\,l}^{\prime \,m}\) are constants for each pair of l and m, is utilized. In the first order of the Taylor series expansion in the vicinity of the mean radius b, where \(\Psi _T^{sh}\sim \frac{1}{r}\), the horizontal components of \(\underline{j}_P\) vanish and only the contributions of the radial currents driving the toroidal magnetic field are considered (Olsen 1997). Using higher orders of the Taylor series, also the contributions of the horizontal components of \(\underline{j}_P\) to the toroidal magnetic field in the vicinity of the mean radius b can be incorporated.
The scalar function of the toroidal magnetic field results in
where \(\alpha _l^m=a_l^m\,\cos (m \lambda )+ b_l^m\,\sin (m \lambda )\) and \(\alpha _l^{\prime \,m}=a_{\,l}^{\prime \,m}\,\cos (m \lambda )+ b_{\,l}^{\prime \,m}\,\sin (m \lambda )\) (Olsen 1997). Thereby, each order of the Taylor series is linked with an additional set of expansion coefficients \(a_l^m\), \(b_l^m\), \(a_{\,l}^{\prime \,m}\), \(b_{\,l}^{\prime \,m}\) and so on which can be reconstructed from the data in analogy to the Gauss coefficients.
From a mathematical point of view the scalar function \(\Psi _P^{sh}\) of the poloidal magnetic field \(\underline{B}_P^{sh}\) can be parametrized analogously to the toroidal counterpart. But within the reconstruction procedure the poloidal fields that are generated by toroidal currents flowing inside the shell cannot be distinguished from the internally and externally driven poloidal fields, since these fields follow the same topological structure. But when the half thickness of the shell, defined by \(h=(ca)/2\) is smaller than the length scale on which the toroidal currents change in radial direction, the shell is called a thin shell and the scalar function \(\Psi _P^{sh}\) of the poloidal field \(\underline{B}_P^{sh}\) vanishes within this thin shell approximation (Backus 1986; Backus et al. 1996) as illustrated in the following section.
The thin shell approximation
The thin shell approximation (Backus 1986; Backus et al. 1996) finally allows the separation of the poloidal field into its internal and external contributions.
Conferring to Eq. (6), the Mie representation for the magnetic field in the whole space \({\mathbb {R}}^3\) is given by
Following Ampère’s law the current density \(\underline{j}\) is also solenoidal and can as well be parameterized via the Mie representation resulting in
with related scalar functions \(\Gamma _T\) and \(\Gamma _P\).
Since the poloidal part of the current density corresponds with the curl of the toroidal magnetic field, the comparision of Eq. (53) with Eq. (9) shows that the scalar function \(\Psi _T\) and \(\Gamma _P\) are the same
Analogously, the toroidal part of the current density corresponds with the curl of the poloidal magnetic field and the comparision of Eq. (53) with Eq. (10) shows that the functions \(\Psi _P\) and \(\Gamma _T\) are related via
so that the function \(\Psi _P\) is given by the Green’s function method
Due to the underlying geometry the toroidal current density \(\underline{j}_T\) flowing in the whole space can be written as the sum of the toroidal currents \(\underline{j}_T^i\) flowing in the region \(r<a\), the toroidal currents \(\underline{j}_T^e\) flowing in the region \(r>c\) and the toroidal currents \(\underline{j}_T^{sh}\) flowing inside the spherical shell in the region \(a\le r \le c\), so that
Thereby, the Mie representation of each part is given by
with \(\Gamma _T^i=\Gamma _T\,\chi _{\left[ r<a \right] }\), \(\Gamma _T^e=\Gamma _T\,\chi _{\left[ r>c \right] }\) and \(\Gamma _T^{sh}=\Gamma _T\,\chi _{\left[ a<r<c \right] }\), where
is the indicator function of the interval I. Using this segmentation the scalar function of the poloidal field can be rewritten as
Thus, the part of the scalar function that corresponds to the poloidal magnetic field which is generated inside the shell is given by
Since \(2b=a+c\) and \(2h=ca\) the bounds of integration can be rewritten as \(a=bh\) and \(c=b+h\), so that
Analogously to the scalar function \(\Psi _T\), the function \(\Gamma _T\) can be expanded into a Taylor series in the vicinity of the mean radius b, resulting in
Substituting the Taylor series into the function \(\Psi _P^{sh}\) delivers
For the further evaluation of the integral it is assumed that the derivatives of the toroidal currents with respect to r are bounded, i.e., there exists a constant \(L>0\), so that
for \(n\in {\mathbb {N}}\). Thus, L represents the length scale on which the toroidal currents change in radial direction. From
and therefore
it follows that
or equivalently
Since \(r^\prime \in \left[ bh,b+h \right]\), each summand within the Taylor series can be estimated upwards via
delivering for Eq. (66)
The integral in Eq. (73) may be evaluated for any \(\underline{r}\in {\mathbb {R}}^3\) but we restrict to \(\underline{r}\) inside the shell as only there the magnetic field is measured. Then, the remaining integral results in
utilizing that the coordinate system can be chosen properly so that \(\theta ^\prime\) defines the angle between \(\underline{r}\) and \(\underline{r}^\prime\). Therefore
For all \(r\in \left[ bh,b+h \right]\), the function
is nonnegative and reaches its maximum value at \(r=bh\) with the related function value \(f(bh)=2bh\). Therefore, an upper bound for \(\Psi _P^{sh}\) can finally be estimated as
When \(h\ll L\), the spherical shell is called a thin shell (Backus 1986; Backus et al. 1996) and the scalar function \(\Psi _P^{sh}\) of the poloidal magnetic field \(\underline{B}_P^{sh}\) vanishes. The scalar function \(\Psi _T^{sh}\), however, remains finite for all h as shown in Appendix A. Thus, if the shell may be regarded as thin, then the contribution of the toroidal currents in this shell to the poloidal magnetic field may be neglected. The poloidal magnetic field is mainly driven by currents beyond the shell. The contribution of the poloidal currents in this shell to the toroidal magnetic field may not be neglected.
From a first point of view the thin shell approximation is not an intuitive approximation. Considering the above presented nature of poloidal and toroidal magnetic fields it can be understood as follows:
The toroidal magnetic field only exists within current carrying regions (cf. Eq. 14) and thus, it is measurable only within these regions. Therefore, the spatial extent of the regions where the poloidal currents flow does not influence the strength of the toroidal magnetic field. It solely depends on the strength of the poloidal current density. In contrast to the toroidal magnetic field, the poloidal magnetic field is also measurable in currentfree region. Thus, the poloidal field is a superposition of fields generated by currents flowing inside and outside the shell as well as currents flowing within the shell. This superposition is verified in Eq. (62). Therefore, the amount of the poloidal field generated by currents flowing within the shell has to be compared with the amount of the internal/external contributions. Furthermore, the poloidal field does not solely depend on the strength of the toroidal current density, since for the evaluation of the integrals also the volume where the current density flows is vital. Thus, a small toroidal current density that flows within a large volume outside/inside the shell can have a larger contribution to the field measured within the shell than a stronger current flowing within the thin shell.
Application of the thin shell approximation
The thin shell approximation is applied to parameterize the magnetic field in the vicinity of Mercury by the Gauss–Mie representation to reconstruct Mercury’s internal magnetic field. The internal and external scalar potentials \(\Phi ^i\) and \(\Phi ^e\) are expanded into spherical harmonics up to third degree and order, representing the internal and external dipole, quadrupole and octupole fields. The scalar function \(\Psi _T^{sh}\) of the toroidal magnetic field \(\underline{B}_T^{sh}\) is expanded into spherical harmonics up to second degree and order and additionally into a first order Taylor series for the radius. The scalar function \(\Psi _P^{sh}\) of the poloidal field \(\underline{B}_P^{sh}\) is neglected within the thin shell approximation. Therefore, the total field is parameterized by 46 expansion coefficients, i.e. Gauss internal dipole (\(g_1^0\), \(g_1^1\), \(h_1^1\)), Gauss internal quadrupole (\(g_2^0\), \(g_2^1\), \(h_2^1\), \(g_2^2\), \(h_2^2\)), Gauss internal octupole (\(g_3^0\), \(g_3^1\), \(h_3^1\), \(g_3^2\), \(h_3^2\), \(g_3^3\), \(h_3^3\)), Gauss external dipole (\(q_1^0\), \(q_1^1\), \(s_1^1\)), Gauss external quadrupole (\(q_2^0\), \(q_2^1\), \(s_2^1\), \(q_2^2\), \(s_2^2\)), Gauss external octupole (\(q_3^0\), \(q_3^1\), \(s_3^1\), \(q_3^2\), \(s_3^2\), \(q_3^3\), \(s_3^3\)), toroidal coefficients (\(a_1^0\), \(a_1^1\), \(b_1^1\), \(a_2^0\), \(a_2^1\), \(b_2^1\), \(a_2^2\), \(b_2^2\), \(a_{\,1}^{\prime \,0}\), \(a_{\,1}^{\prime \,1}\), \(b_{\,1}^{\prime \,1}\), \(a_{\,2}^{\prime \,0}\), \(a_{\,2}^{\prime \,1}\), \(b_{\,2}^{\prime \,1}\), \(a_{\,2}^{\prime \,2}\), \(b_{\,2}^{\prime \,2}\)). These 46 coefficients are estimated with Capon’s method (Capon 1969; Toepfer et al. 2020a, b). The Capon method and the underlying model are tested against simulated data and MESSENGER in situ data around Mercury.
Hybrid simulation of Mercury’s magnetosphere
For the first application of the thin shell approximation simulated magnetic field data are analyzed. The magnetic field resulting from the plasma interaction of Mercury with the solar wind is simulated with the hybrid code AIKEF (Müller et al. 2011), that has successfully been applied to several problems in Mercury’s plasma interaction, (Exner et al. 2018, 2020, e.g.). The internal Gauss coefficients \(g_1^0=190\,\,\text {nT}\) (dipole field), \(g_2^0=78\,\,\text {nT}\) (quadrupole field) and \(g_3^0=20\,\,\text {nT}\) (octupole field) (Anderson et al. 2012; Thébault et al. 2018; Wardinski et al. 2019) are implemented in the simulation code. The interplanetary magnetic field with a magnitude of \(B_{\text {IMF}}=20\,\,\text {nT}\) is orientated along the vector \((x,y,z)^T=(0.0 , 0.43 , 0.9)^T\) in the MASO frame. The solar wind velocity of \(v_{\text {sw}}=400\,\,\text {km}/\text {s}\) points along the xaxis (away from the Sun) and the solar wind proton density number was chosen to \(n_{\text {sw}}=30\,\,\text {cm}^{3}\). The resulting magnitude of the magnetic field and the corresponding current density in the xzplane are displayed in Figs. 2, 3.
The internal dipole field dominates the geometry of Mercury’s magnetosphere. Yet the quadrupole field in terms of the apparently shifted dipole field is visible. The influence of the octupole field is not clearly noticeable on the field lines in the figure. Furthermore, the distribution of the simulated current density shows that there exist no completely currentfree region around Mercury.
Reconstruction of the Gauss coefficients from simulated data
For the reconstruction of the internal Gauss coefficients implemented in the simulation code, first of all, magnetic field data at a distance of \(0.2\,R_\text {M}\) from Mercury’s surface are evaluated. The data are retrieved along meridional circular orbits around Mercury. The orbital plane is rotated about the rotation axis (zaxis) from \( \;50^{^\circ }\) (afternoon/postmidnight sector) over \(0^\circ\) (noon/midnight, xzplane) to \(50^\circ\) (morning/premidnight sector). 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 approximation is surely valid. The reconstructed internal Gauss coefficients are listed in Table 1. The optimal diagonal loading parameter which determines how the data are weighted within the application of Capon’s method results in \(\sigma _{opt.}\approx 1000\,\text {nT}\) (Toepfer et al. 2020b).
The deviation \(\big \underline{g}^{int}\underline{g}_C^{int} \big \) between the reconstructed internal coefficients \(\underline{g}_C^{int}\) and the implemented internal coefficients \(\underline{g}^{int}\) results in \(4.0\,\,\text {nT}\), i.e. \(\big \underline{g}^{int}\underline{g}_C^{int} \big /\big \underline{g}^{int} \big \approx 1.9\%\) and thus, the implemented coefficients are reconstructed from the data with good precision.
Concerning the BepiColombo mission the generated circular orbits are idealized cases which are not realizable in practice. To investigate the applicability of the thin shell approximation for elliptical orbits we analyze the magnetic field data along the prospective orbits of MPO. The orbits are generated in analogy to the circular orbits, i.e., with the same longitudinal extend. Along this trajectories the distances of the data points from Mercury’s surface vary from \(0.12\,R_\text {M}\) up to \(0.6\,R_\text {M}\) resulting in a shell thickness of \(2h\approx 0.48\,R_\text {M}\). Although the shell is now much thicker the thin shell approximation works successfully. The reconstructed Gauss coefficients are displayed in Table 2. The optimal diagonal loading parameter for the application of Capon’s method results in \(\sigma _{opt.}\approx 1000\,\,\text {nT}\).
The deviation between the reconstructed and the implemented internal coefficients results in \(4.1\,\,\text {nT}\), i.e. \(1.9\%\) and thus, these values agree with the coefficients reconstructed from the data evaluated along the circular orbits. Since the data are evaluated along the MPO orbits, it is expectable that also Mercury’s internal octupole field will be reconstructed with good precision from the data of the magnetometer on board MPO.
It should be noted that the extension of the underlying model by the parameterization of the external parts of the field using the Gauss–Mie representation improves the results significantly. When only the internal parts of the field \(\underline{B}^i\) are considered in the model, the deviation between the implemented and the reconstructed coefficients results in \(29.7\,\,\text {nT}\) or 14.4%, respectively (cf. Table 2). Additional parameterization of the external poloidal fields \(\underline{B}^e\) using the external scalar potential \(\Phi ^e\) yields a deviation of \(10.0\,\,\text {nT}\) or 4.9%, respectively. Thus, for the analysis of Mercury’s internal magnetic field, the Gauss–Mie representation is a suitable alternative to the application of global magnetospheric models.
Validity of the thin shell approximation
The coefficients reconstructed from the data evaluated along the planned MPO trajectories are basically in agreement with that from the data along the circular orbits although the shell covering the MPO orbits has a finite thickness.
To investigate the limits of the thin shell approximation within the hybrid simulation, the thickness of the shell is increased incrementally. Referring to Eq. (67), the length scale L is estimated via radial variation of the current distribution
for the horizontal currents \(\underline{j}_H=\underline{j}\left( \underline{j}\cdot \underline{e}_r \right) \underline{e}_r\) which are used as a proxy for the toroidal currents at each point along the orbit. Since L represents a local quantity that varies for each point along the orbit, whereas the half thickness h is a global quantity, the set of resulting length scales is averaged over the number of points along the orbit resulting in the mean length scale \(\langle L \rangle\). It should be noted that also the poloidal currents have horizontal components. Thus, the estimation of the length scale L with the horizontal currents \(\underline{j}_H\) as a proxy for the toroidal currents is not exact, but it is sufficient for a qualitative discussion.
The deviations between the ideal coefficients implemented in the simulation and the reconstructed coefficients for varying values of \(h/\langle L \rangle\) are displayed in Table 3.
When h approaches \(\langle L \rangle\) the deviations are greater than \(12\,\text {nT}\). This deviation is of the same order when the parameterization is restricted to the Gauss representation. But for the data points along the MPO orbits, where \(h/\langle L \rangle \approx 0.47\), the application of thin shell approximation is justified.
Since a shell of thickness \(2h\approx 0.48\,R_\text {M}\) is called thin, the name thin shell is misleading, although this naming has been adopted within the literature. Conferring to the global length scale of \(1\,R_\text {M}\) the shell of thickness \(0.48\,R_\text {M}\) is not thin, but compared with the current length scale L it is. Therefore, the term thin has to be understood in a mathematical sense.
Reconstruction of the Gauss coefficients from MESSENGER data
The Gauss–Mie representation has successfully been validated for the simulated data. For the reconstruction of the Gauss coefficients from the MESSENGER data only data points in the northern hemisphere within a distance of \(r\le 1.5\,R_\text {M}\) and \(x>  0.4\,R_\text {M}\) from Mercury’s surface can be considered because the orbits do not cover the southern hemisphere properly.
For the reconstruction of Mercury’s internal magnetic field, the data from nine pairs of MESSENGER orbits with different orientations of the periapsis between 10. August 2012 and 14. July 2014 are analyzed. The combination of the orbits improves the model resolution compared to the analysis of single orbits. A discussion of the model resolution, as provided by Connerney (1981), can be found in the Appendix B. As a proof of concept, only a small subset of the whole MESSENGER data set is analyzed. In the case of a small and noisy data set the performance of the estimator can be improved by seperating the data set into several subsets (Meir 1994). Therefore, the Gauss coefficients are reconstructed for each pair of orbits and the results are averaged over the nine pairs. Outliers were not included within the averaging. The resulting mean values are listed in Table 4. The reconstructed coefficients for each pair of orbits and the standard deviations of the mean values are listed in Table 5 of Appendix C.
The mean values of the reconstructed internal Gauss coefficients and the external dipole coefficients are in feasible agreement with the values provided by Anderson et al. (2012), Thébault et al. (2018) and Wardinski et al. (2019). Furthermore, the related standard deviations for each coefficient are within the range of the variations resulting from the time varying magnetic field (Wardinski et al. 2019). To classify the coefficients reconstructed from the simulated magnetic field data (Table 2) in terms of that resulting from the MESSENGER data, the simulated data are furthermore evaluated along the MESSENGER trajectories. The reconstructed Gauss coefficients are listed in Table 6 of Appendix C. These coefficients are in agreement with the results presented in Table 4. Since the reconstructed coefficients resulting from the data evaluated along the MPO trajectories are in better agreement with the implemented coefficients than the coefficients resulting from the data evaluated along the MESSENGER trajectories, the restriction of the data points to the northern hemisphere and the related degradation of the model resolution influences the results significantly (Heyner et al. 2020, cf). Furthermore, it should be noted that the analysis of the length scales of the current densities (cf. "Validity of the thin shell approximation" section) cannot be performed for the MESSENGER orbits, because the current densities in the vicinity of the orbits are unknown.
Summary and outlook
In the vicinity of Mercury no completely currentfree region is present. Therefore, the Gauss representation does not yield a proper parametrization of Mercury’s magnetospheric field. Extension of the Gauss representation to the Gauss–Mie representation allows a more complete characterization of Mercury’s internal and magnetospheric field.
For the parameterization of the magnetic field the orbit where the magnetic field is measured, is conceptually covered by a spherical shell. Due to the underlying geometry the total measured magnetic field is a superposition of internal and external poloidal fields generated by toroidal currents flowing outside the spherical shell as well as toroidal and poloidal fields generated by currents flowing within the shell. Thereby, each component of the field is either described by a scalar potential (\(\Phi ^i\), \(\Phi ^e\)) or a scalar function (\(\Psi _P^{sh}\), \(\Psi _T^{sh}\)). These potentials and functions can be expanded into spherical harmonics. When the thickness of the spherical shell is smaller than the length scale on which the toroidal current density changes in radial direction the shell is called a thin shell. Then the poloidal field generated by currents flowing inside the shell is negligible compared to the poloidal field generated by currents flowing beyond the shell, whereas the toroidal field remains finite. In the case of the planned MPO orbits, the thin shell approximation is a reasonable choice.
For the application of the thin shell approximation the internal Gauss coefficients for the dipole, quadrupole and octupole field are implemented in the simulation code AIKEF and the magnetic field data resulting from the plasma interaction of Mercury with the solar wind are simulated in the vicinity of Mercury. The data are evaluated along the planned MPO orbits and the 46 expansion coefficients, describing the internal, external and the toroidal field are reconstructed with Capon’s method. Since the reconstructed internal Gauss coefficients are in good agreement with that implemented in the simulation code, the parameterization of the magnetic field using the Gauss–Mie representation is a suitable alternative to the application of global magnetospheric models. Even the implemented Gauss coefficient for the octupole field of Mercury can be reconstructed accurately and therefore, it is expectable that Mercury’s internal octupole field will be reconstructed with high precision from the magnetometer data on board MPO. Thus, it is worthwile to investigate the analysis of higher multipoles, such as hexadecapole, from the data along the MPO trajectories.
Furthermore, the thin shell approximation is applied to reconstruct Mercury’s internal magnetic field from the data of the MESSENGER mission. The results are in reasonable agreement with former works. Since only the data points in the northern hemisphere are vital for the analysis of the MESSENGER data, the symmetrically distributed MPO orbits will deliver a better model resolution than the MESSENGER orbits.
Concerning the BepiColombo mission the combination of the Gauss representation with the Mie representation is a useful model for the analysis of Mercury’s internal magnetic field. As the BepiColombo mission consits of two elements, the planetary orbiter and the magnetospheric orbiter, measurements of any gradients of Mercury’s magnetic field are possible which may lead to further improvements of the methods presented here. Besides the analysis of the internal magnetic field, the reconstructed coefficients for the toroidal magnetic field can be used for calculating poloidal current systems, e.g. field aligned currents, in the vicinity of the orbit where the data are evaluated.
Availability of data and materials
Simulation data can be provided upon request.
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Acknowledgements
The authors are grateful for stimulating discussions and helpful suggestions by Alexander Schwenke and the technial support by Willi Exner.
Funding
Open Access funding enabled and organized by Projekt DEAL. We acknowledge support by the German Research Foundation and the Open Access Publication Funds of the Technische Universität Braunschweig. The work by Y. Narita is supported by the Austrian Space Applications Programme at the Austrian Research Promotion Agency under contract 865967. D. Heyner and K.H. Glassmeier were supported by the German Ministerium für Wirtschaft und Energie and the German Zentrum für Luft und Raumfahrt under contract 50 QW1501.
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Appendices
Appendix A: The thin shell approximation for the toroidal magnetic field
As derived in "The thin shell approximation" section, the scalar function of the poloidal magnetic field can be estimated upwards as
When \(h\ll L\), the function \(\Psi _P^{sh}\) vanishes. Now the influence of the thin shell approximation for the toroidal magnetic field \(\underline{B}_T^{sh}\) is discussed.
The scalar function \(\Psi _T^{sh}\) of the toroidal magnetic field \(\underline{B}_T^{sh}\) can be expanded analogously into a Taylor series in the vicinity of the mean radius b
using that the scalar functions \(\Psi _T^{sh}\) and \(\Gamma _P^{sh}\) are the same (cf. Eq. 54).
If we furthermore assume that also the radial derivatives of the poloidal current density are bounded so that
the function \(\Psi _T^{sh}(\underline{r})\) can be estimated upwards as
Using that \(\Psi _T^{sh}(\underline{r})\) is solely evaluated within the spherical shell, where \(r\in \left[ bh,b+h \right]\) and therefore \(\big  rb\big \le h\) results in
When \(h\ll L\), the function \(\Psi _T^{sh}\) remains finite.
Appendix B: Model resolution
For the analysis of Mercury’s internal magnetic field from the MESSENGER data only data points in the northern hemisphere can be considered. This limitation impairs the condition number \(\kappa\) of the shape matrix \(\underline{\underline{H}}\) from \(\kappa \approx 226\) for the data points along the MPO orbits to \(\kappa \approx 10^8\) for the data points along a single MESSENGER orbit, where the shape matrix describes the spacial distribution of the data points. The condition number is defined as the ratio of the largest and the smallest singular value of the shape matrix. Therefore, some of the 46 singular values, corresponding to the 46 expansion coefficients, of the shape matrix have to be dropped within the numerical calculation to improve the condition number. Within the low rank approximation (Eckart and Young 1936) only k singular values, where \(k\le 46\), can be considered and therefore, the shape matrix \(\underline{\underline{H}}\) is approximated by a shape matrix \(\underline{\underline{H}}_k\) which has a lower rank and a lower condition number.
Capon’s filter matrix
which is the key parameter for calculating Capon’s estimator
fulfills the distortionless constraint
where \(\underline{\underline{I}}\) is the identity matrix and \(\underline{\underline{M}}=\langle \underline{B}\circ \underline{B} \rangle\) describes the data covariance matrix of the magnetic field measurements \(\underline{B}\) (Toepfer et al. 2020a, b). As a consequence of the low rank approximation Capon’s filter matrix is modified to
so that
The matrix \(\underline{\underline{R}}=\underline{\underline{w}}^{\dagger }_k \underline{\underline{H}}\) is called the model resolution matrix. Because of
where \(\underline{g}_C^k\) denotes the estimator resulting from the consideration of k singular values and \(\underline{g}\) is the ideal coefficient vector implemented in the simulation code, the diagonal elements of \(\underline{\underline{R}}\) identify the resolution of each coefficient (Connerney 1981). When \(\underline{\underline{R}}=\underline{\underline{I}}\) each coefficient is resolved for \(100\%\). If the resolution is smaller than \(100\%\), there exist model parameter covariances.
The more singular values are considered within the estimation, the better the model resolution becomes, whereas the condition number of the shape matrix increases and thus, the influence of measurement errors increases.
To achieve a compromise between the resolution and the condition number, the coefficients are estimated for different numbers of singular values and the change of the coefficients resulting from k singular values to the coefficients resulting from \(k1\) singular values is regarded. For the final computation the maximum number of singular values is chosen from which the reconstructed coefficients are almost constant. In Fig. 4 the procedure is exemplarily illustrated for the coefficient \(g_1^0\), which corresponds to the diagonal element \(R_{11}\) by analyzing the data of one pair of MESSENGER orbits.
Changing the number of considered singular values incrementally from 46 to 37 the resulting coefficient changes significantly. For \(k\le 37\) the values are almost constant and therefore, 37 singular values, corresponding to a model resolution of 88% for the coefficient \(g_1^0\) are considered within the estimation.
Appendix C: Tables: Reconstructed Gauss coefficients from MESSENGER orbits
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Toepfer, S., Narita, Y., Glassmeier, K.H. et al. The Mie representation for Mercury’s magnetic field. Earth Planets Space 73, 65 (2021). https://doi.org/10.1186/s40623021013864
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DOI: https://doi.org/10.1186/s40623021013864
Keywords
 Mie representation
 Poloidal and toroidal magnetic fields
 Thin shell approximation
 Gauss representation
 Capon’s method