Electrical conductivity structure determined for the lunar lower mantle from Lunar Prospector and Kaguya-Selene magnetic data

The determination of induced magnetic fields in the Moon, produced by the electrical currents that flow within the Moon as a result of the time-variable external magnetic field, have formed the basis of successful electrical-conductivity structure studies of the lunar crust and mantle (e.g., Dyal and Parkin, 1971;Kuckes, 1971Kuckes, , 1973;;Sonett et al., 1971aSonett et al., , 1971b;;Leavy and Madden, 1974;Sonett and Duha, 1975;Hood et al., 1982;Khan et al., 2006;Mittelholz et al., 2021). Also, the lunar core size has been studied considering that it is a highly conducting metallic sphere overlain by an insulating mantle (e.g., Shimizu et al., 2013). In all these studies, the measures of the lunar magnetic fields (at the surface, in orbit, or in passing the Moon) have been provided by several missions (e.g., Apollo, Lunar Prospector and Kaguya Selene). However, none of the aforementioned studies have achieved a precise determination of the electrical conductivity structure for the lunar lower mantle.Consequently, the goal of this study is this precise determination applying the methodology described in the sections 2 (filtering) and 3 (modelling), to the magnetometer data provided by the Lunar Prospector (1998-1999) and Kaguya Selene (2007-2009) satellite missions. This methodology is needed to reduce noise and other undesirable perturbations (Hwang and Mitchell, 1986), increasing the signal/noise ratio to improve the computation of the transfer functions, and the modelling performed in this study has been successfully applied to other previous studies (Corchete 2026a and2026b). Thus, the above-mentioned studies of the lunar crust, upper and midmantle will be completed with the present characterization (in terms of electrical conductivity) of the lunar lower mantle structure.Both, Lunar Prospector (LP; Hood et al., 2001) and Kaguya Selene (KS; Takahashi et al., 2009) missions have provided the magnetic data used in this study. For each dataset (LP and KS magnetic data), the records of each magnetic field component (B x (t), B y (t), B z (t)), with the time t give in seconds and (XYZ) in the Cartesian coordinate system (with origin in the lunar center), are converted to magnetic field components (B r (t), B  (t), B  (t)) measured in a spherical coordinate system (with the same origin). This observed magnetic field B(r, t)  (B r (t), B  (t), B  (t)) is assumed as a potential field, i.e., B(r, t)  grad (V(r, t)), where r  (r, , ) is the position vector of a given location, grad () is the gradient operator, and V(r, t)  V e (r, t) + V i (r, t) is a scalar potential given by (Mittelholz et al., Gauss coefficients of degree n and order m, respectively. The potential field assumption given by (1)implies that the satellites (LP and KS) are in vacuum, but really, they are moving in a low-density plasma (Baumjohann and Treumann, 1996). However, Hood and Schubert (1978) concluded that the magnetohydrodynamic effects for induction sounding are negligible at low frequencies (< 10 3 Hz).Then, the field representation given by ( 1) is justified. On the other hand, for Earth and at periods longer than one day, the structure of the time-varying external field is mostly governed by the magnetospheric ring current, and it can be described by a zonal spherical harmonic of degree one (Price, 1967;Banks, 1969;Olsen, 1999;Kuvshinov et al., 2021). In this study, the external (inducing)source is also assumed of magnetospheric origin (Olsen, 1999), with a spatial structure described by the spherical harmonic n  1 and m  0, i.e., a potential V(r, t) given the coefficients (□□ 1 0 , □□ 1 0 ).Following this assumption, the external (□□ 1 0 ) and internal (□□ 1 0 ) parts can be computed from B(r, t)  grad (V(r, t)) with V(r, t) given by (1) as□□ 1 0 (t) = 1 3 (2 B θ (t) sin θ - B r (t) cos θ ) □□ 1 0 (t) = 1 3 ( □□ □□ ) 3 ( B θ (t) sin θ + B r (t) cos θ ) (2)where (B r (t), B  (t)) are the above-mentioned time series. The coefficients (□□ 1 0 (t), □□ 1 0 (t)) given by ( 2) are independent of latitude and radius (Olsen, 1999). Then, it is possible to average them. The nearzero denominators in (2) has been avoided, considering only the values of sin  and cos  greater than 0.65, because small values of these terms introduce big scatter in the coefficients. Supplement 1 shows these coefficients averaged each 24 hours (red line) for LP (a) and KS (b) data. A low-pass filter at 10 6 Hz is applied to them to smooth these time series (Supplement 1, blue line; Bath, 1974;Brigham, 1988.). The transfer function S ob (f)  G(f)/Q(f), where (Q(f), G(f)) are the Fourier spectra of the smoothed time series (Supplement 1, blue line), is calculated by means of a useful technique used in other fields of Geophysics (as Seismology) called the Wiener deconvolution (Hwang and Mitchell, 1986), in which an spectral ratio as G(f)/Q(f) is calculated byG(f) Q(f) = G(f)Q * (f) Q(f)Q * (f) = C(f) A(f) (3)where (*) denotes conjugate complex, and C(f )The electrical conductivity () of the lunar lower mantle is determined fitting the observed data S ob (f) (Figure 1a, circles), with the theoretical data S th (f) (Figure 1a, black line) given by an electrical conductivity model. This theoretical data can be calculated byS(f) = 1 -T(f) 1 + T(f) (4)where T(f) is defined by Schmucker (1970) and calculated from a conductivity model as described by Corchete (2026a and2026b). The difference between observed (S ob ) and theoretical data (S th ) is calculated by□□ = √ ∑ [S ob -S th ] 2 M i=1 M (5)where M is the number of observations and S th (f) is given by (4) with T(f) determined from a conductivity model. This model must achieve a minimum difference between observed and theoretical data, i.e. it must be minimum  given by ( 5). In this study, the model considered for the depth range from 300 to 900 km (Supplement 4), has been defined through the mean of the conductivity profiles determined by Mittelholz et al. (2021). This model is completed for two depth ranges: from 0 km to 300 km and from 900 km to 1410 km-depth (mantle-core boundary), which have been unsatisfactorily resolved by Mittelholz et al. (2021). In the first depth range (0300 kmdepth), two layers: crust (045 km-depth) and uppermost mantle (45300 km-depth) have been considered. The location of crust-mantle boundary (45 km-depth) is defined as determined by Matsumoto et al. (2015). The conductivity values for this depth range (0300 km-depth), has been defined through a mean of the values determined in previous studies (Dyal and Parkin, 1971;Kuckes, 1971;Sonett and Duha, 1971;Leavy and Madden, 1974;Hood et al., 1982;Khan et al., 2006). For the second depth range (the lower mantle, 9001410 km-depth), the present model has also two layers: 9001170 km-depth and 11701410 km-depth. The first layer is defined from the middlelower mantle boundary (900 km-depth; Matsumoto et al., 2015;Briaud et al., 2023) to the uppermost limit of the transition zone (1170 km-depth; Shimizu et al., 2013;Matsumoto et al., 2015). The second layer is defined by the thickness of this transition zone (240 km; Shimizu et al., 2013;Matsumoto et al., 2015). The conductivity values for the two lower-mantle layers (9001410 km-depth) has been determined with the above-described methodology, considering fixed the conductivity values defined from 0 to 900 km-depth. The value of the wave number k (used in the computations) is determined considering n 1 (Corchete, 2026a). The semi-infinite medium is considered the lunar core, which is characterized by a high conductivity (10  S/m; Shimizu et al., 2013). The results of this computational procedure are shown in Figure 1b. In this figure, the results of other models considered for comparison are also shown. Supplement 5 shows the errors  given by ( 5), for each model considered, and the present model (Figure 1b, black line) shows the best fit of the observed data (0.0324).Figure 1b (and Supplement 5) shows a clearly better fit of the observed data for the present model, which shows a clearly increase in conductivity for the lower mantle ( 0.8 S/m, from 900 to 1170 km-depth), with a notably jump in conductivity for the transition zone ( 8 S/m, from 1170 to 1410 km-depth). This increase of conductivity is not surprising, because it has been also determined for the lower mantle of Mars (Mocquet and Menvielle, 2000;Vacher and Verhoeven, 2007), and confirmed in recent conductivity studies (Civet andTarits, 2013, 2014). The olivine/wadsleyite phase transition may be the cause of the rapid conductivity increase determined for the Martian lower mantle, according to mineralogy models (Bertka and Fei 1997). The cause of the conductivity increase in the lunar lower-mantle could be also related to a similar phase transition, or may be related to changes in the lower-mantle mineralogical composition (e.g., Xu et al., 1998). This increase is especially high in the transition zone ( 8 S/m), which is considered by Shimizu et al. (2013) as a possible melt layer. The justification of this hypothesis is given in their study through interesting references that support their conclusions and results, and it is also justified in other studies (Khan et al., 2014;Walterová et al., 2023;Goossens et al., 2024). A higher conductivity value (10 S/m or greater) is difficult to observe in an olivine-basalt melt system (Yoshino et al., 2010). Then, the high conductivity value (8 S/m) determined in this study is a possible value for partial melt layer as affirmed by Shimizu et al. (2013), which is also a possible value because it is less than the limit value (10 S/m), proposed by Yoshino et al. (2010) for a olivine-basalt melt layer. On the other hand, the conductivity value of 10 S/m is also an upper limit in the study performed by Shimizu et al. (2013), because a higher value can produce the overestimation of the lunar core size by some methods employed for this analysis. These methods are described in their study, in which specific references to justify their affirmation. Fortunately, the conductivity value determined in this study ( 8 S/m) confirms that this overestimation cannot be produced. The precise estimation of the lunar core size is a very important issue, because a lunar core dynamo probably could not be activated, in the early Moon, with a small core (Takahashi and Tsunakawa, 2009).The LP and KS datasets have provided the observed data for the determination of the electrical conductivity structure of lunar lower mantle, which has been calculated with well-known techniques.The results of this computational procedure are compared with those determined for other previous studies, showing very different values of error. The present model gives the best fit of the observed data (0.0324), and shows a clearly increase in conductivity for the lower mantle, with a notably jump in conductivity for the transition zone, which is a possible partial melt layer. The cause of this conductivity increase could be related to phase transitions, or could be related to changes in the lowermantle mineralogical composition. The high increase in conductivity of the transition zone is very important, because it can produce the overestimation of the lunar core size, and this estimation is very important to know whether or not a lunar core dynamo could be activated in the early Moon. On the other hand, the methodology described in the sections 3 and 4, can be used also for shallow depth determinations, e.g., it can be useful to identify buried ice layers in the polar region or deep root