Composition of the lunar mantle for lower mantle high-velocity seismic model

We investigated models of the internal structure of initially homogeneous Moon differentiated as a result of partial melting, using data on seismic velocities according to the seismic models assume the zonal structure of the lunar mantle is a model of the Moon which was obtained with using the array processing methods of high velocities in the lower mantle. As a result of inversion of gravity (mass, moment of inertia), seismic (P- and S-waves velocities) and petrological (balance ratios) data, the Monte Carlo method was used to reconstruct the chemical composition and internal structure of the Moon. The phase composition and physical properties of the mantle were obtained with Gibbs free energy minimization method and equations of state in the five-component system CaO-FeO-MgO-Al2O3-SiO2. For all models, possible values of seismic velocities and concentrations of the main oxides in three zones of the mantle were obtained, satisfying the geochemical and geophysical constraints and the possible sizes of the Fe-10%S core were determined. It was found that the lunar mantle chemical composition (concentration of FeO, Al2O3 and CaO) differs depending on the mantle zone. Constraints on the values of seismic velocities in the lower mantle and the most probable size of the lunar core were determined: VP ≤ 8.45 km/s; Fe-10%S core radius is ∼360 km.


Introduction
Seismic data is one of the most important data in determining the internal structure of the Moon. From 1969 to 1977, the Apollo seismic experiment was carried out, during which four seismic stations on the lunar surface recorded seismic events. The processing of the data obtained by Apollo made it possible to obtain information about the structure of the lunar interior. Seismic models of the Moon were proposed in [1,2,3,4,5]. Most seismic models assume the zonal structure of the lunar mantle [3,4,6] Weber et al. (2011) ( [6]) proposed a model of the Moon obtained using the array processing methods. The model [6] at depth less than 740 km is similar to earlier models proposed by groups of French scientists [3,4]; at greater depths, V P velocity is significantly higher compared to previous models (up to 8.5 km/s, while in the model [4] V P = 8.15 ± 0.23 km/s).
In this study, the problem of determining the chemical composition and internal structure of the Moon is solved by Monte Carlo method using gravity (mass and moment of inertia), seismic (P-and S-wave velocities), and petrological data [7]. The internal structure of the Moon and the composition of the silicate mantle were obtained using the data of a high-velocity in the lower mantle seismic

Calculation method
The approach used in this work allows the density and isotropic velocities VP and VS to be calculated for phase association, which depend on the chemical and phase composition of the lunar material. THERMOSEISM software package was used to calculate the equilibrium phase associations and physical properties (seismic velocities and density) in the lunar mantle [17,19]. The THERMOSEISM database includes thermodynamic parameters (enthalpy, entropy, heat capacity, Grüneisen parameter, thermal expansion, bulk modulus and shear modulus for minerals) as well as mixing parameters of solid solutions. Using the Gibbs free energy minimization method, the chemical composition of the individual phases and the ratio between the phases were determined. The solution of the equation of state (EOS) of minerals is carried out in the quasi-harmonic Mie-Grüneisen-Debye approximation [20].
When constructing models of the internal structure of the Moon, based on seismic data, we assume a model of the Moon, which consists of five spherical shells: crust, three-layer (upper, middle and lower) mantle and iron-sulfide core. The core size is calculated during the inversion process. The thickness of the lunar crust is assumed to be 50 km, the upper mantle is located at depths from 50 to 250 km, the middle mantle is from 250 to 625 km, the lower mantle extends from 625 km to the mantle-core boundary. It is assumed that within each mantle layer, the composition (concentrations of MgO, FeO, Al2O3, CaO, SiO2) and density are almost constant. It is also assumed that there is no density inversion with depth. The composition of the outer mantle layers of the mantle region is determined from the condition of maintaining a balance in the system of oxides CaO-FeO-MgO-Al2O3-SiO2, taking into account geophysical constraints [21].
A detailed description of the inverse problem of determining the chemical and mineral composition and radius of the lunar core from geophysical constraints can be found in [15,19,22]. This paper uses the Monte Carlo method for a uniform distribution [23]. The applied approach makes it possible to determine the probable distributions of the concentration of rock-forming oxides and seismic

Results
The values of the parameters calculated as a result of solving the problem (bulk composition, concentration of rock-forming oxides, core size) depend on the constraints that were imposed on the model of the Moon. The models of the Moon considered are shown below. The models under consideration differ in the constraints imposed on seismic velocities and oxide concentrations in the lunar mantle. The main input parameters of the considered models are given in Table 1 and the constraints on the model parameters are given in Table 2.   [4], the lower ones are according to [3].
1-3. The seismic velocities and associated uncertainties in these models represent mean values and standard deviations (assuming normal distribution).
4. V P in the lower mantle is 4.5±0.05; there are no restrictions for V S 5. V P in the lower mantle is 4.45±0.05; there are no restrictions for V S Notes: 1. Model MI -constraints on the mass and moment of inertia of the Moon. 2. MIS model -restrictions are imposed on mass, moment of inertia and seismic velocities in the upper and middle mantle. The highest values for seismic velocities were taken according to the model [4], the lowest ones are according to [3].
1, 2 -The seismic velocities and associated uncertainties in these models represent mean values and standard deviations (assuming normal distribution).

Conclusions
Models of the internal structure of differentiated as a result of partial melting of the initially homogeneous Moon for a high-velocity seismic model [6] were investigated by the method of numerical simulation. Reconstruction of the chemical composition and physical properties in the lunar mantle was carried out using the Monte Carlo method with gravity and seismic constraints. Possible