Constitution of the Moon

Physics of the Earth and Planetary Interiors 107 Ž1998. 285–306

Constitution of the Moon 5. Constraints on composition, density, temperature, and radius of a core O.L. Kuskov ) , V.A. Kronrod

1

V.I. Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, 117975 Moscow, Russian Federation Received 26 March 1997; revised 15 January 1998; accepted 15 January 1998

Abstract Thermodynamic modeling of phase relations and physical properties of multicomponent mineral systems was used to develop a method for solving the inverse problem of determining the bulk composition, density and temperature distribution in the lunar mantle, and radius of a core from the totality of geophysical evidence Žseismic velocities, moment of inertia, and mass of the Moon.. A derived self-consistent model of the chemical composition and the internal structure of the Moon may be considered as a preliminary geochemical–geophysical reference model and may be used as a significant constraint for the constitution of the Moon. The geophysically admissible ranges of the chemical compositions in various zones of the lunar mantle were determined within the system CaO–FeO–MgO–Al 2 O 3 –SiO 2 . It has been found that the isochemical models fail to explain the topology of the seismic structure; the lunar mantle is chemically stratified. Compositional models of the zoned Moon are used to estimate the bulk composition of the silicate portion of the Moon Žmantleq crust., wt.%: 4 - CaO - 5%, 5.5 - Al 2 O 3 - 7%, 10 - FeO - 12%, 28 - MgO - 30%, 48 - SiO 2 - 50%. The chemical composition of the Moon bears no genetic relationship to the terrestrial material as well as to any of the known chondrites. It is suggested that the mean seismic velocities by Nakamura wNakamura, Y., 1983. Seismic velocity structure of the lunar mantle. J. Geophys. Res., 93: 6338–6352.x are too small in the middle mantle Žby about 1–2%. and almost certainly too large in the lower mantle Žby about 2–3%.. A probable law of temperature distribution in the lunar mantle is obtained. Models of the internal structure of the Moon without density inversion are constructed, and the sizes of a core are determined: 500–590 km in radius for a FeS-core and 320–390 km for a Fe-core. A comparison of an allowed ranges of Fe-core radii for the Moon and Jupiter’s satellites has been carried out. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Satellite; Lunar mantle; Seismic velocity

) Corresponding author. Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Kosygin Str. 19, 117975 Moscow, B-334, Russian Federation. Tel.: q7-095-939-7005, q7-095-245-5024; fax: q7-095-938-2054; e-mail: [email protected] 1 Also corresponding author.

0031-9201r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 1 - 9 2 0 1 Ž 9 8 . 0 0 0 8 2 - X

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O.L. KuskoÕ, V.A. Kronrodr Physics of the Earth and Planetary Interiors 107 (1998) 285–306

1. Introduction The Moon is the only extraterrestrial body for which we have information about seismic velocities as a function of depth. The mineral composition and seismic structure of the Moon depend strongly on its bulk composition and thermal regime, whereas the bulk chemical composition is controlled by the origin of a planetary body. The composition of the mantle and the bulk of the Moon, though the subject of intense geochemical interest, remains largely a mystery ŽVinogradov, 1977; Ringwood, 1979; Basaltic Volcanism Study Project, 1981; Florensky et al., 1981; Taylor, 1982; Wanke and Dreibus, 1986; Hood and Jones, 1987; Warren, ¨ 1991; Galimov, 1995.. The Moon and the Earth have similarity in oxygen and chromium isotopes and distinctly different siderophile element patterns ŽVinogradov, 1977; Taylor, 1996.. Areas of controversy include suggestions that the lunar mantle is enriched in FeO and refractory elements ŽAl, Ca.. Some investigators believe that both suggestions are likely correct ŽTaylor, 1986; Hood and Jones, 1987; Mueller et al., 1988.; others suggest that the lunar mantle is enriched in FeO by a factor 1.2–1.5, but not enriched in refractories ŽRingwood, 1979; Jones and Delano, 1989; O’Neill, 1991.. There are two approaches to obtain information on the composition and constitution of a planet. The first approach consists of a forward modeling-computer simulation of phase equilibria and thermoelastic properties at high pressures and temperatures and a comparison of the calculated seismic profiles with seismological observations. A thermodynamic approach for computation of phase diagrams ŽWood and Holloway, 1982; Jeanloz and Thompson, 1983; Kuskov and Panferov, 1991; Saxena et al., 1994; Fabrichnaya, 1995., equations of state of minerals ŽZharkov and Kalinin, 1968; Kuskov and Galimzyanov, 1986; Anderson, 1995., seismic properties, and density from known thermodynamical conditions and mineralogies ŽDuffy and Anderson, 1989; Ita and Stixrude, 1992; Bina and Helffrich, 1992; Vacher et al., 1996. applied to the planetary interiors has been extensively described. Although phase diagrams provide the basis for understanding the mineral composition of a planet, they tell us nothing about the physical properties Žthermoelastic properties, seismic velocities, density, etc... The opposite is also true; the physical properties calculated from known mineralogies are not internally consistent with databases and phase diagrams. By applying suitable thermodynamic models for the equation of state of minerals and solid solutions and the technique of free energy minimization, the profiles of the seismic properties and density can be derived entirely from an internally consistent model, with only thermodynamic data and bulk composition as input. This approach was first applied for testing the mineral and seismic structure of the Earth and Mars ŽKuskov and Panferov, 1991, 1993; Kuskov et al., 1995a.. Further development of this approach leads us to an attempt of modeling the chemical composition and internal structure of the Moon. The general aim of the series of papers under the common title ‘Constitution of the Moon’ is to combine geophysical constraints Žthe seismic data, the moment of inertia and mass of the Moon. and a strictly thermodynamic approach, and to develop, on this joint basis, a self-consistent petrological–geophysical model of the Moon, accounting for its chemical composition and internal structure. In the previous papers of this series ŽFabrichnaya and Kuskov, 1994; Kuskov and Fabrichnaya, 1994; Kuskov, 1995a, 1997., calculations of mineral composition and elastic properties from thermodynamic principles as well as an assessment of thermochemical and thermophysical properties of minerals and reliability of phase relation calculations in the FeO–MgO–CaO–Al 2 O 3 –SiO 2 ŽCFMAS. system have been described. The calculated lunar bulk silicate composition suggests that the concentrations of FeO, SiO 2 and refractory elements are significantly higher than those in the Earth’s upper mantle. It has been concluded that the lunar mantle is chemically stratified. Another approach to the solution of the problem of modeling the constitution of the Moon consists of translating the observed seismic velocity profiles into the bulk composition models ŽKuskov, 1997; Kronrod and Kuskov, 1996, 1997.. In this paper, we propose a new formulation for the solution of this problem which consists of retrieving the chemical composition of the mantle reservoirs, determining the radius of a core, and assessing the internal temperature and density distribution from the geophysical observations Žinverse modeling..

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This formulation must meet the available geophysical constraints, including the seismic data, the moment of inertia and mass of the Moon, and generate geophysically admissible bulk composition models of the zoned Moon which are internally consistent with their physical properties Želastic properties, densities and velocities.. Taking into account the recent Galileo spacecraft’s measurements, we will also focus on comparing the sizes of a core for the Moon and Jupiter’s satellites.

2. Geophysical constraints and observations 2.1. Crust The lunar crust is composed essentially of anorthosite. An Al–Ca-rich crust with the average conservative values of 3.0 g cmy3 for the density and 58 km for the crustal thickness is adopted ŽNakamura, 1983; Zuber et al., 1994.. The upper mantle density values at the mantle–crust boundary Ž r Ž58 km.. were changed in the range 3.24 - rm – cr - 3.34 g cmy3 ŽKuskov, 1997.. 2.2. Moment of inertia, density and radius For the Moon, the mass Ž M . and the mean moment of inertia Ž I . are well known ŽBills and Ferrari, 1977.: L

Ms

4

Ý is1 L

Is

Ý is1

3

pr i ) Ž R 3iq1 y R 3i .

1,6 3

Ž 1.

pr i ) Ž R 5iq1 y R 5i .

Ž 2.

where r i is the density of the i-th layer, R i is the radius of the i-th layer. A value of 0.3905 " 0.0023 for the normalized moment of inertia Ž IrMR 2 . as well as a value for the mean density of the Moon of 3.3437 " 0.003 g cm3 are adopted ŽBasaltic Volcanism Study Project, 1981; Ferrari et al., 1980.. For the case of a homogeneous planet with constant density distribution: IrMR 2 s 0.4; the observed departure of the IrMR 2 parameter is a measure of the nonhomogeneous nature of a planet. Note that the uncertainty in the mean moment of inertia parameter of 0.0023 puts significant limitations on its use as the only constraint for deep interior models ŽAnanda et al., 1977.. Distribution of pressure can be described by a formula P s P0 w1 y ŽŽ R y H .rR . 2 x, where the pressure in the center of the Moon is P0 s 47.1 kbar, R s 1738 km is the mean radius of the Moon and H is the depth. 2.3. Mantle seismic Õelocities According to the seismic data ŽGoins et al., 1981; Nakamura, 1983., the lunar mantle consists of several regions of constant seismic velocities separated by discontinuous boundaries. According to the most recent seismic velocity model of Nakamura Ž1983., the three mantle reservoirs have very different overall seismic velocities; the mean compressional Ž Vp . and shear Ž Vs . velocities can be fixed as follows: upper mantle Ž 58–270 km . :

Vs s 4.49 Ž 0.03 . km sy1 and Vp s 7.74 Ž 0.12 . km sy1 ;

middle mantle Ž 270–500 km . :

Vs s 4.25 Ž 0.1 . km sy1 and Vp s 7.46 Ž 0.25 . km sy1 ;

lower mantle Ž 500 km-core. :

y1

Vs s 4.65 Ž 0.16 . km s

y1

and Vp s 8.26 Ž 0.4 . km s

.

Ž 3.

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2.4. Thermal models Many studies are devoted to the thermal history of the Moon Že.g., McConnell and Gast, 1972; Toksoz ¨ and Solomon, 1973; Hood and Jones, 1987.. The Q values and the presence of moonquakes ŽLatham et al., 1972, Nakamura, 1983. as well as electromagnetic-sounding data are in favor of a solid state for the lunar mantle. Thus, the present-day temperature distribution in the mantle is always below solidus. As the mineral composition, density and seismic velocities are strongly dependent on the temperature, the present-day temperature profile has been derived from the geophysical constraints. The temperature distribution in the mantle has been modeled by the equation: T s To q C 1 y exp D Ž H y 58 . ,

Ž 4.

T0 is the temperature at a depth of 58 km; T0 , C and D are parameters to be determined. 2.5. The lunar core As pointed by many authors Že.g., Hood and Jones, 1987., the mass and moment of inertia of the Moon are insensitive to the presence of a very small core. The radius of a core depends on its composition, thickness of the crust, and density distribution in the mantle. Two boundary models were considered for a core of constant density: g-FeŽNi.-core and FeS-core. The densities of g-Fe Ž8.1 g cmy3 . and Fe x S Ž4.7 g cmy3 . at the core pressures and temperatures were estimated from the available experimental data ŽKuskov, 1995b..

3. Thermodynamic approach Thermodynamic approach for the calculation of phase diagrams, seismic properties, and density for an equilibrium phase assemblage has been described in earlier papers ŽFabrichnaya and Kuskov, 1994; Kuskov, 1995a,b.. For the computation of phase diagram for a given chemical composition, we have used the method of minimization of the total Gibbs free energy, i.d., the stable phase assemblage must be determined by the lowest total Gibbs free energy of the system at a chosen chemical composition, pressure and temperature Žde Capitani and Brown, 1987.. The potential method for the construction of the equation of state for solids described in detail by Kuskov and Galimzyanov Ž1986. has been used in this study. The parameters of the equation of state for minerals as well as thermodynamic data on enthalpy of formation, entropy and heat capacity of minerals are summarized in the THERMOSEISM data base Žsee Appendix A.. The assessment of thermodynamic data and their internal consistency is based on the available experimental studies involving calorimetrical data Žkey values., phase equilibria in the pure systems and solid solutions, static compression, ultrasonic and Brillouin scattering measurements. As in the work of Kuskov Ž1997., this work has been restricted to the solid solution phases in the CFMAS system such as olivine ŽOl, binary., spinel ŽSp, ŽMg, Fe.Al 2 O4 , binary., garnet ŽGar, ternary: Alm-almandine, Py-pyrope, Gross-grossular., orthopyroxene ŽOpx, En-MgSiO3 , Fs-FeSiO 3 , Di-Ca 0.5 Mg 0.5 SiO 3 , Hd-Ca 0.5 Fe 0.5 SiO 3 , Cor-Al 2 O 3 . and clinopyroxene ŽCpx, same components as in orthopyroxene. are five component solutions; other minerals such as anorthite ŽAn., kyanite ŽKy., andalusite ŽAnd. and sillimanite ŽSil. are pure phases. Standard thermodynamic properties, phase diagram and thermodynamic profiles of density Ž r ., adiabatic bulk modulus Ž K s . and seismic parameter ŽFs .

Fs s K srr s Vp2 y 4r3Vs 2

Ž 5.

for a chosen petrological model are internally consistent. At each P–T condition, the coexisting phases were mixed according to the phase diagram to form a petrological–geophysical model. In other words, profiles of the physical properties at fixed bulk composition are derived entirely from a thermodynamic model. The Hill

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average of the Voigt and Reuss bounds was used for the bulk moduli. As there is no density profile for the Moon, thermodynamic modeling allows us, strictly speaking, to compare only profiles of seismic parameter. A more reliable test of a compositional model would be the calculation of P and S wave velocities together with the density profiles. Velocities of P and S waves have been found from: Vp s  Ž K s q 4r3G . rr 4

1r2

,

Vs s  Grr 4

1r2

.

Ž 6.

At lunar P–T conditions, we assume a linear dependence on pressure and temperature for the shear modulus. For this reason, the estimation of the shear modulus at combined pressure and temperature is more uncertain than that of the bulk modulus. Typical uncertainties of the calculated density and velocity values for the lunar mantle assemblages are no more than 1% and 2%, respectively ŽKuskov, 1995a; Duffy and Anderson, 1989.. According to the seismic data ŽNakamura, 1983., the reported uncertainties in Vs and Vp are 0.7–1.5%, 2.4–3.4% and 3.4–4.8% in the upper, middle, and lower mantle, respectively.

4. Modeling the chemical composition and internal structure of the Moon It is possible to distinguish two approaches for the solution of this problem ŽKuskov, 1997; Kronrod and Kuskov, 1997.: the direct Žforward. modeling and the inverse process of modeling. The first one involves the calculation of the seismic velocities, moment of inertia and mass of the Moon starting from a proposed bulk composition, and the use of the thermodynamic properties of minerals available in the THERMOSEISM data base. We thus need a chemical model and a temperature distribution model. Since composition of the Moon is uncertain, it will take a great variety of input compositions. The second approach Žinverse problem. involves the computation of geophysically admissible bulk composition models of the zoned Moon the internal temperature distribution and radius of a core from the given seismic data, moment of inertia and mass of the Moon. The solution of inverse problem is based on the minimization of the deviations between the calculated and observed geophysical parameters. The inverse problem provides a means for retrieving or assessing a chemical composition model of the zoned Moon and the entire Moon or discriminating between several possible models. The solutions depend on the number of free parameters c i Žconcentrations of oxides. and distribution of the temperature in the mantle. The unknowns Žor independent parameters to be determined. are the temperature and concentrations of FeO, MgO, Al 2 O 3 and CaO; the content of SiO 2 is not the unknown due to normalization to 100%. Density distribution and the size of a core are not the independent parameters and are determined from the above mentioned parameters. The known values are the mean values of the P- and S-wave velocities in the upper, middle and lower mantle, the moment of inertia and mass of the entire Moon. The goal of the inversion procedure is to find the values of c i and temperature that minimize the discrepancies between the calculated and observed geophysical data. In making these calculations, we require a non-negative density gradient in the mantle Žd rrd H ) 0., assume an initial temperature distribution and a range of the Al 2 O 3 contents and assume, using the arguments on Al 2 O 3rCaO ratios in meteorites, that cŽAl 2 O 3 . s 1.25cŽCaO. ŽVinogradov, 1965; Ringwood and Essene, 1970; Dodd, 1981.. For numerical solution, we assume that the velocities and densities change linearly with a small change in free parameters. In this case we may use Taylor series expansion: 2

Z1 s Z o q Ý Ž E ZrE f . d f q O Ž d f . , Z s Ž Vp ,Vs , r . , Ž f s T , MgO, FeO, Al 2 O 3 , CaO . ,

Ž 7.

where Z0 is the value of a function at given pressure, temperature and oxide contents calculated by means of THERMOSEISM software at point f 0 , Z1 is the value of the function at the same pressure and slightly different temperature and concentrations calculated at point f 1 s f 0 q d f. Partial derivatives are computed at point f 0 . The derivatives of seismic velocities with respect to the parameters are determined from the solution of direct

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problem. Eq. Ž7. allows to calculate the velocity and density profiles using the initial values of Vp , Vs , r and their partial derivatives calculated in the limited number of depth points. For example, the calculated density can be expressed in terms of perturbations to the oxide concentrations:

r 1 s r o q Ž ErrE T . d T q Ž ErrE MgO . d MgO q Ž ErrE FeO . d FeO q Ž ErrE Al 2 O 3 . d Al 2 O 3 q Ž ErrE CaO . d CaO.

Ž 8. .2

Errors in Eq. Ž7. depend on the term O Ž d f ; the greater the second derivatives with respect to f and the greater the distance from Z0 , the higher the uncertainty in the equation is. To reduce the uncertainty, changes in Z0 and derivatives were periodically performed. According to Eq. Ž7., the dependence of the seismic velocities on the oxide contents and temperature may be expressed as Vj s Vj 0 q Ž E VjrE MgO . d MgO q Ž E VjrE FeO . d FeO q Ž E VjrE CaO . d CaO q Ž E VjrE Al 2 O 3 . d Al 2 O 3 q Ž E VjrE T . d T

Ž j s p,s . .

Ž 9.

Eq. Ž9. yields the formal possibility to determine changes in the MgO and FeO contents and their current concentrations from the seismic velocities in each calculated P–T point if the temperature and the Al 2 O 3 content Ž cŽAl 2 O 3 . s 1.25cŽCaO.. are known in this point. Thus, the independent parameters are the contents of Al 2 O 3 Žin the upper, middle and lower mantles. and the coefficients C, D and T0 in Eq. Ž4.. Note that we assume that there is no depth gradients of the Al 2 O 3 and CaO concentrations in each mantle reservoir. However, direct solution of Eq. Ž9. can lead to the density inversion and strong gradient of chemical composition in each mantle reservoir. For this reason, the contents of FeO and MgO in the mantle are determined by minimizing the F functional: 2

2

F s min a p Ž Vp y Vp0 . q a s Ž Vs y Vs 0 . ,

Ž 10 .

ErrE H ) 0,

Ž 10a. Ž 10b. Ž 10c.

E c Ž FeO . irE H ™ 0, E c Ž MgO . irE H ™ 0, E c Ž Al 2 O 3 . irE H s 0, E c Ž CaO . irE H s 0, Ž i s 1,2,3 . ,

where i s 1,2,3 denote upper, middle and lower mantle zones, respectively; EcŽFeO, MgO. irEH ™ 0 denotes minimum value of gradient of the FeO and MgO contents in the mantle zones. Minimizing Eq. Ž10., we use the relation 7. The coefficients T0 , C, D and c i ŽAl 2 O 3 . are determined by minimizing the q functional: N

qs

Ý Ý a F Ž Fio y Fi .

2

2

2

q aM Ž M o y M . q aI Ž I o y I . .

Ž 11 .

is1 F

The q function was minimized by means of the gradient-type method. In Eq. Ž11., N is the number of the calculated points throughout the mantle depth; F, M, I and F o , M o , I o are the calculated and observed parameters, Ž Fi s Vp ,Vs ., Vp , Vs , M, I are the compressional and shear velocities, the mass and moment of inertia, and a F , a M and a I are the weighing factors which adjust the solution in such a way that the relative deviations d F s <Ž F o y F .rF o < for all the F and I functions are close to each other; it is assumed that an error in the Moon’s mass is negligible. By using the weighing factors, it is possible to impose additional constraints on the behavior of a function and, for example, to require that the function sought does not fall outside the range of the values specified. For example, if d F ) d F o , where d F o is determined from the error in the experimental data F o , the a F coefficient sharply increases; therefore, the solution should be adjusted in such a way that the d F o error is reduced. In our case, for each of the function sought, we used the law of the variation of the weighing coefficients of the type a F s f F Ž d F ., where f F is a sharply increasing smooth function.

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For the case of a numerical solution, we find a point u k which satisfies the following condition: inf q Ž u . s qk ) F q Ž u k . F qk ) q m k ,

Ž 12 .

where q ) s infŽq . is the lower bound of the q function;  m k 4 is a positive sequence and

m k ™ 0 if k ™ `, u k 2 U, k s 1,2 . . . , U s Ž T0 ,C, D,c i Ž Al 2 O 3 . . , Ž i s 1,2,3 . Ž 13 . Eqs. Ž10. and Ž11. are equations to be solved simultaneously. From mathematical point of view, this procedure results in the unique solution of the inverse problem. If E rrEH - 0 or if the calculated velocities and moment of inertia fall outside the limits of the errors in Eqs. Ž2. and Ž3., the corresponding weighing factors a would increase by two orders of magnitude. This technique provides the possibility for calculation of the following parameters: T0 , C, D Žcoefficients in Eq. Ž4.., c i ŽAl 2 O 3 ., where Ž i s 1,2,3., cŽMgO., cŽFeO., Vp , Vs , r . Computational procedure on the Ž k .th step of iterative process is as follows: Ž1. T0o , C o , D o and c io ŽAl 2 O 3 . are the initial input parameters: T0o s 300–5008C, C o s 700–1500, D o s yŽ0.001–0.004. and 2 - c io ŽAl 2 O 3 . - 10 wt.%; Ž2. the values c k ŽMgO., c k ŽFeO., Vpk , Vs k , r k minimizing Eq. Ž10. are being determined from the values T0ky 1 , C ky1 , D ky1 , c iky1 ŽAl 2 O 3 ., Ž i s 1,2,3.; Ž3. the coefficients T0k , C k , D k , c ik ŽAl 2 O 3 . minimizing Eq. Ž11. are being determined on the Ž k .th step of the iterative process. The iterative procedure is terminated when k s 30–40. Minimization of Eqs. Ž10. and Ž11. means that we find ‘the best’ solution Žchemical composition, temperature and density distribution and radius of a core. which satisfies the overall agreement between the calculated parameters and the observed ones Žthe mean velocity values by Nakamura Ž1983. and the mass and moment of inertia values ŽEqs. Ž1. and Ž2... In deriving some of the thermoelastic properties of the lunar mantle and its chemical composition, we have assumed that there are no lateral changes in temperature and composition because there are no lateral variations in velocities in the Nakamura seismic model which is only an approximation of the actual velocity structure. Eqs. Ž10. and Ž11. provide strong constraints on the profiles of chemical composition and thermoelastic properties which are controlled by the seismic velocity and moment of inertia uncertainties. By combining the relations Ž10. and Eq. Ž4. for the smooth function for the temperature profile, we obtain automatically temperature derivatives in all mantle zones. Constraints on the temperature and oxide content profiles as well as the fixed values of density at the crust–mantle boundary, the condition E rrEH ) 0 and the relations Ž1,2. provide robustness of the computational procedure and possibility to find the unique solution of the inverse problem. Robustness of the computational procedure has been tested with respect to the input data. Our numerical experience has shown that initial variations in temperature in the range of 100–2008C and oxide contents in the range of 2–3 wt.% lead to the uncertainties in the final solution Žafter 30–40 iterations. in the range of 10–208C for the temperature and 0.2–0.3 wt.% for the chemical composition. The dominant error source is the uncertainty in the moment of inertia parameter. 5. Results and discussion Chemical and phase compositions of the silicate Moon Žmantleq crust., temperature and density distribution as well as radius of a core are shown in Tables 1 and 2 . It should be noted that the lower mantle thickness and core radius vary depending on the assumed composition of a core and its density. The results of determining the chemical composition and temperature substantially depend on the density jump at the crust–mantle boundary. Calculations have been carried out at two boundary upper mantle density constraints at a depth of 58 km: 3.24 - rm – cr - 3.32 g cmy3 , where rm – cr denotes the upper mantle density at a depth corresponding to the mantle–crust boundary. The calculated velocity profiles for the compositional models I Ž rm – cr s 3.24 g cmy3 . and II Ž rm – cr s 3.32 g cmy3 . are reproduced in Fig. 1. Both models are in agreement with the mean seismic velocities. However, the calculated velocity profiles Žespecially VS . are shifted to the upper-velocity bounds in the middle mantle and to the lower-velocity bounds in the lower mantle.

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Table 1 Bulk composition models Žwt.%., phase composition Žmol.%. and physical properties of the lunar mantle reservoirs Composition, properties

MgO FeO Al 2 O 3 CaO SiO 2 MGa Spinel Anorthite Quartz Orthopyroxene Clinopyroxene Olivine Garnet T Ž8C. r Žg cmy3 . Vp Žkm sy1 . Vs Žkm sy1 . RŽFeS-core. Žkm. RŽFe-core. Žkm.

Model I

Model II

Upper mantle

Middle mantle

Lower mantle

Upper mantle

Middle mantle

Lower mantle

58 km

270 km

400 km

800 km

58 km

570 km

400 km

800 km

31.0 7.9 2.5 2.2 56.4 84

31.0 8.4 2.5 2.2 55.9 84

25.4 15.1 4.2 3.3 52.0 73.5

32.9 11.4 2.0 1.6 52.1 87 1.4

34.3 11.8 2.0 1.6 50.3 87 0.0

25.9 16.6 2.6 2.1 52.8 75

2.0 0.3 93.7 4.0

0.0 0.5 95.8 3.7

78.7 5.9 14.0

72.1 3.7 24.2

96.4 3.6

435 3.239 7.674 4.493

854 3.260 7.674 4.464

466 3.319 7.748 4.511

772 3.319 7.772 4.461

922 3.375 7.526 4.294 470 310

91.9 3.5 3.8 0.8 993 3.367 7.568 4.305 590 390

34.4 10.1 6.4 5.1 44.0 86

34.5 56.5 9.0 1175 3.389 8.223 4.549

34.4 9.6 6.7 5.3 44.0 86

35.2 55.8 9.0 1254 3.376 8.202 4.532

5.1. Temperature of the lunar interior Quantitative estimation of the temperature distribution in the Moon’s interior is a key problem in lunar petrology and geophysics. The temperature profiles are shown in Fig. 2: Model I: rm – cr Ž 58 km . s 3.24 g cmy3 , T Ž 8C . s 435 q 800) 1 y exp Ž y0.0035 Ž H y 58 . .

Ž 14a.

Model II: rm – cr Ž 58 km . s 3.32 g cmy3 , T Ž 8C . s 465 q 1280 1 y exp Ž y0.0013 Ž H y 58 . .

Ž 14b.

The derived temperatures are in the range of 430–4708C at a depth of 58 km, 770–8608C at the upper–middle mantle boundary Ž270 km., and 1000–11008C at the middle–lower mantle boundary Ž500 km.. The calculated temperatures are within reasonable limits of all uncertainties involved in geophysical observations.

Table 2 Bulk composition models Žwt.%. of the silicate portion of the Moon Composition

Crust a

MgO FeO Al 2 O 3 CaO SiO 2

7.0 6.5 25.0 16.0 45.5

a

According to Taylor Ž1982..

Crustqupper mantle

Crustqupperqmiddle mantle

Crustqentire mantle

Model I

Model II

Model I

Model II

Model I

Model II

25.5 7.7 7.7 5.5 53.6

27.7 10.4 7.2 4.9 49.8

25.5 10.6 6.3 4.6 53.0

27.0 12.8 5.4 3.8 51.0

28.5 10.4 6.3 4.8 50.0

29.6 11.7 5.9 4.3 48.5

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Fig. 1. Calculated mantle velocity profiles for the compositional models I Ždashed line. and II Žsolid line. from Table 1. Seismic velocities Ždotted line. are as follows ŽNakamura, 1983.: upper mantle Ž58–270 km., Vs s 4.49 Ž0.03. km sy1 and Vp s 7.74 Ž0.12. km sy1 ; middle mantle Ž270–500 km., Vs s 4.25 Ž0.1. km sy1 and Vp s 7.46 Ž0.25. km sy1 ; lower mantle Ž500–1000 km., Vs s 4.65 Ž0.16. km sy1 and Vp s 8.26 Ž0.4. km sy1 ; error bars denote 1 SD error estimates of seismic data of Nakamura. ŽA. Compressional-wave velocity profiles; ŽB. Shear-wave velocity profiles.

Fig. 2. Temperature profiles in the lunar mantle inferred from the geophysical constraints and corresponding to Eqs. Ž14a. and Ž14b.: Model I, dashed line; Model II, solid line. The temperature gradient is estimated to be about 1.2–1.48C kmy1 at depths of 58–500 km and 0.3–0.78C kmy1 at depths of 500–1000 km.

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Fig. 3. Density profiles in the lunar mantle for the compositional models I Ždashed line. and II Žsolid line. from Table 1 inferred from the geophysical constraints and satisfying the mean seismic velocities, the moment of inertia and mass of the Moon. Density and velocity profiles have an anticorrelated behavior at the 270-km discontinuity, i.e., the negative changes in the seismic velocities Žsee Fig. 1. and the positive change in the calculated density. Opposite signs in the density and velocity jumps through the 270-km discontinuity as well as a very strong velocity jump and weak density jump through the 500-km discontinuity provide strong evidence that change in chemistry is responsible for the nature of both lunar discontinuities.

5.2. Mantle density models As there is no density profile for the Moon, the question arises as to what degree it is possible to change density models or what degree of variability of the mantle density gradient satisfying the geophysical constraints is admissible. Fig. 3 and Table 1 show the calculated density distributions that do not require density inversion and yield agreement with both the mass and moment of inertia constraints as well as with the seismic velocity profiles. Thus, a hypothesis of the density inversion in the lunar interior ŽHood and Jones, 1987. becomes redundant for explaining the physical properties of the Moon. If the density value at the top of the upper mantle ranges from 3.24 g cmy3 to 3.32 g cmy3 , a relatively broad spectrum of density models is allowed in the mantle. If the density value at the top of the upper mantle is 3.34 g cmy3 , practically, the unique solution is realized for the density distribution models in the mantle; density profile behaves monotonously even during the transition from one mantle zone to another. The lunar mantle densities are in good agreement with a model of Bills and Rubincam Ž1995.. 5.3. Composition of the lunar mantle 5.3.1. Upper mantle Table 1 shows that compositional models enriched in SiO 2 Ž50–56 wt.%., depleted in CaO and Al 2 O 3 Ž1.5–2.5%., and having 8–12% FeO are in excellent agreement with the Nakamura seismic data at depths of 58–270 km. It should be noted that the compositional models of the upper mantle are strongly dependent on the magnitude of a density jump through the crust–mantle discontinuity. The FeO and SiO 2 contents in the upper mantle is especially sensitive to this density jump; the upper mantle was found to contain only about 8 wt.% FeO at 3.24 - rm – cr - 3.29 g cmy3 . Coming from this is an important constraint on the upper mantle chemistry Žwt.%.: 1.5 - CaO, Al 2 O 3 - 2.5%, 8 - FeO - 12%, 31 - MgO - 34%, 50 - SiO 2 - 56%; MGa 84–87 Žmolar MgrŽMg q Fe..

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The calculated velocity profiles for a plagioclase-bearing pyroxenite at the top of the upper mantle and for a quartz-bearing pyroxenite at the bottom of the upper mantle ŽModel I. as well as for an olivine-bearing pyroxenite through the entire upper mantle depth interval ŽModel II. agree with seismic profiles within the limits of uncertainties in both thermodynamic and seismic data. This is a reason to believe that the upper mantle may be composed of pyroxenite with MGa 84–87 containing either free silica Ž0.3–0.5 mol.%. or olivine Ž14–24 mol.% Ol, Fo 83 – 85 ., Table 1. It follows from our modeling that orthopyroxene is the dominant phase in the upper mantle Ž72–96 mol.% Opx, En 77 – 87 .. 5.3.2. 270-km discontinuity and middle mantle Seismic velocity profiles by Nakamura Ž1983. show a sharp decrease in velocities at depths of the middle mantle. The negative jumps of velocities through the discontinuity are about: d VprVp s y3.5% and d VsrVs s y5.5%. There is no identical peculiarity for the low-velocity zones in the Earth’s mantle. This strong decrease of velocities in the sufficiently cold lunar mantle may, in principle, imply a phase transition andror dehydration, a change in chemical composition andror a very rapid increase in temperature. It has been found, however, that phase transitions in pyroxenite and peridotite are not able to explain the nature of the 270-km discontinuity ŽKuskov, 1995a, 1997.. Another potential explanation is the presence of hydrous phases which can reduce seismic properties appreciably ŽLebedev et al., 1991; Ito and Tatsumi, 1995.. Kuskov et al. Ž1995b. calculated the densities and seismic velocities for the ‘wet’ middle mantle. At depths of 270–500 km Ž13.5–23.2 kbar and 550–7508C., the inferred seismic velocities for a mantle assemblage enriched in talc Ž15% Opx q 33% Cpx q 32% Ol q 12% Gar q 8% Talc, mol.%. are Vp s 7.68 km sy1 , Vs s 4.15 km sy1 . The observed seismic velocities at these depths are 7.46 " 0.25 km sy1 and 4.25 " 0.1 km sy1 . However, unlike the Earth’s mantle, there are various problems concerning the potential presence of hydrous phases in the lunar mantle. No lunar sample returned to Earth has shown any evidence of the presence of water. The famous ‘rusty rock’ 66095 has been shown to have been hydrated in the terrestrial atmosphere ŽFlorensky et al., 1981; Taylor, 1982.. Many models derive mare basalts from the middle mantle depths but the mineralogy of mare basalts shows no evidence of having been in contact with hydrous phases. A 2008C increase in temperature through the 270-km discontinuity decreases dry silicate velocities by about 0.2 km sy1 for Vp Ž2.5%. and 0.13 km sy1 for Vs Ž3%.; however, there is no reasonable explanation for sharp increase in temperature through the upper mantle–middle mantle discontinuity. The seismic discontinuity at 270-km depth may be caused by the transition from a FeO-depleted pyroxenite ŽEn 77 – 87 . in the upper mantle to a FeO-enriched pyroxenite ŽEn 65 . in the middle mantle with a jump in the chemical composition and physical properties Žthe negative changes in the seismic velocities and the positive change in the calculated density Žwith a density jump of 1.5–3%., Table 1, Figs. 1 and 3. The calculated seismic velocities range from 7.67–7.77 km sy1 for Vp and 4.46–4.51 km sy1 for Vs in the upper mantle to 7.5–7.6 km sy1 for Vp and 4.30 km sy1 for Vs in the middle mantle with silica content varying between 50–56% in the upper mantle and ; 52 wt.% in the middle mantle, and FeO content varying between 8–12% in the upper mantle and 15–17 wt.% in the middle mantle; CaO and Al 2 O 3 contents in the upper and middle mantle are approximately similar Ž2–4%.. These velocity differences probably reflect the fact that the middle mantle is more iron-rich than the upper and lower mantle because small differences in the contents of other oxides and relative proportions of orthoŽclino.pyroxene, olivine and other minerals have a less significant influence on seismic velocities of lunar rocks ŽTable 1.. At the same P–T conditions, a 5% enrichment of a rock in FeO results in velocity decrease equal to around 0.12 km sy1 for Vs and 0.2 km sy1 for Vp . Thus, an enrichment in FeO may be responsible for the nature of the low-velocity zone at depths of 270–500 km. A FeO-depleted pyroxenite model and a FeO-enriched pyroxenite model satisfactorily account for the observed velocities in the upper and middle mantle respectively. Note that electromagnetic sounding of the Moon has suggested the existence of a layer having increased conductivity at depths of 200–500 km; this layer may be associated with the existence of a rock enriched in divalent iron oxide and having a reduced resistance ŽVan’yan et al., 1977.. Taking into account geophysical observations and these calculations, we must conclude with Ringwood and

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Essene Ž1970. and Kuskov Ž1997. that the lunar middle mantle consists essentially of pyroxenite ŽTable 1.; orthopyroxene is the dominant phase in the middle mantle Žup to 90–95 mol.% Opx.. Chemistry of the middle mantle may be expressed as follows Žwt.%.: 2 - CaO and Al 2 O 3 - 4%, 15 - FeO - 17%, ; 25% MgO and ; 52% SiO 2 ; MGa 73–75 ŽTable 1.. The 58–500-km depth interval is important in magma genesis for the interpretation of the crust composition and for understanding the early evolution and differentiation of the Moon. 5.3.3. 500-km discontinuity and lower mantle Seismic velocity profiles show a sharp increase in compressional and shear velocities through the 500-km discontinuity ŽFig. 1.. The jump in the two velocities is about 10% ŽNakamura, 1983.; unlike the strong velocity jump, there is practically no density jump across the 500-km discontinuity ŽFig. 3.. Seismic studies show that the velocity jumps at depths of 410 km and 650 km in the Earth are not greater than 2.5–5% for P wave and 3.5–7.5% for S wave. Phase changes in any model considered in the dry CFMAS system are not able to explain a 10% jump in the seismic velocities in the lunar mantle and thus the nature of the 500-km discontinuity. A concept of a ‘wet’ mantle was excluded. A very rapid decrease in temperature gradient is in conflict with our understanding of the constitution of the planetary interiors. Thus, the nature of this discontinuity can be attributed only to changes in chemical composition. The chemical and phase composition listed in Table 1 shows that a Ca–Al-rich lower mantle is not chondritic in composition and may be composed of olivine ŽFo 88 .-clinopyroxene ŽDi 46 .-garnet ŽAlm 16 Py 71Gros 13 . assemblage ŽOl:Cpx:Gars 56:35:9 mol.%.; olivine is the dominant phase. Chemistry of the lower mantle may be expressed as follows Žwt.%.: 5 - CaO, Al 2 O 3 - 7%, 9 - FeO - 10%, 34 - MgO - 35% and 43 - SiO 2 45%; MGa 86. Geophysical constraints and physicochemical modeling lend support to a chemically stratified lunar mantle. It is not clear, however, whether the lower mantle is composed of undifferentiated material of bulk lunar composition. In this case, the bulk composition of the crust q upper q middle mantle must correspond to that of the lower mantle. There is satisfactory agreement between the average concentrations of the CaO, Al 2 O 3 and FeO oxides for the upper layers Žcrust q upper q middle mantle. and for the lower mantle ŽTables 1 and 2.. This may indicate that the Moon was melted down to a depth of 500 km Ža lunar magma ocean., resulting in the formation of the crust and differentiation of material in the upper and middle mantle. Certain compositional differences for the MgO and SiO 2 contents may be attributed to the specific requirement that the calculated and observed mean seismic velocities show the best fit. If the restrictions on the seismic velocities in the lower mantle were less rigorous, the balance of major oxides would be more accurate. Thus, we suggest that the maximum thickness of the magma ocean of 500 km, rather than 800 km as assumed by Hess and Parmentier Ž1995., would result in a chemically stratified Moon and a sharp increase in velocities through the 500-km discontinuity. In order to demonstrate the influence of seismic uncertainties on composition, we have calculated the field of the seismically admissible concentrations of SiO 2 in the lunar mantle as a function of seismic velocities ŽFig. 4.. Estimates of the SiO 2 content in the mantle reservoirs have been obtained from the seismic observations, including the uncertainties of the Nakamura model Žthe mass and moment of inertia constraints are not taken into account.. Fig. 4 shows a range of the SiO 2 content from 51–56% in the upper mantle and 47–52% in the middle mantle to 44–55% in the lower mantle. The narrow range of the SiO 2 content in the upper and middle mantle Žwhere velocity resolution is higher. is due to the small velocity uncertainties Ž"0.03 km sy1 for Vs , "0.12 km sy1 for Vp in the upper mantle.. The wide range of the SiO 2 content in the lower mantle is due to the large velocity uncertainties Ž"0.16 km sy1 for Vs , "0.4 km sy1 for Vp .. It is important to emphasize that the ranges of the SiO 2 content are in agreement with the mean velocities in the upper mantle but are shifted to the upper-velocity bounds in the middle mantle and to the lower-velocity bounds in the lower mantle. It is quite possible that the mean seismic velocities by Nakamura Ž1983. are too small in the middle mantle Žby about 1–2%. and almost certainly too large in the lower mantle Žby about 2–3%..

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297

Fig. 4. The seismically admissible concentrations of SiO 2 in the lunar mantle derived from the seismic observations, including the uncertainties of the seismic model Žsolid vertical lines denote 1 SD error estimates of the seismic data of Nakamura Ž1983.; dashed lines denote the mean seismic velocities.. All three SiO 2 content–velocity regions are calculated at temperatures corresponding to Eqs. Ž14a. and Ž14b. and with oxide contents varying as follows: 2%-CaO, Al 2 O 3 -10%, 5%- FeO- 20%, 15%- MgO- 50%. In all the compositional models considered, none had the lower mantle velocities exceeding the Nakamura value for Vs s 4.65 km sy1 . An optimal fit requires that the SiO 2 content in the lower mantle is substantially lower than in the upper and middle mantle. Thus, despite the velocity uncertainties, the seismic structure of the lunar mantle provides important constraints on its chemical composition. 1, upper mantle Ž3.24 - r - 3.34 g cmy3 .; 2, middle mantle Ž3.35- r - 3.43 g cmy3 .; 3, lower mantle Ž3.35- r - 3.51 g cmy3 .. Ža. SiO 2 content vs. Vp ; Žb. SiO 2 content vs. Vs .

5.3.4. Composition of the silicate Moon The bulk composition models presented in the previous sections provide a new basis for generalization on lunar petrology and bulk chemistry Žcomposition of the silicate Moon Žcrust q mantle. is practically the same for an iron core and iron sulfide core.. Tables 2 and 3 show that the silicate Moon is significantly depleted in

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Table 3 Bulk composition models Žnormalized to 100 wt.%. of silicate portion of the Moon and Earth Žmantleqcrust. Ref.

SiO 2

FeO

MgO

CaO

Al 2 O 3

Moon Ringwood Ž1979. Taylor Ž1982. Wanke and Dreibus Ž1986. ¨ Jones and Delano Ž1989. O’Neill Ž1991. Kuskov Ž1997. This study, Model I This study, Model II

45.1 43.9 45.9 46.1 44.6 49.9 50.0 48.5

14.1 13.1 13.1 12.6 12.4 10.8 10.4 11.7

32.9 32.3 32.6 35.0 35.1 27.5 28.5 29.6

3.7 4.6 3.8 2.8 3.3 4.9 4.8 4.3

4.2 6.1 4.6 3.5 3.9 6.9 6.3 5.9

Earth Ringwood Ž1979. Wanke and Dreibus Ž1982. ¨ Taylor Ž1987. McDonough and Sun Ž1995. Kuskov et al. Ž1995a.

45.9 45.2 50.1 45.5 48.3

8.1 7.7 8.0 8.2 10.7

38.8 39.9 35.3 38.2 34.7

3.2 3.2 2.9 3.6 2.9

4.0 4.0 3.7 4.5 3.4

MgO and is enriched in FeO, SiO 2 , and refractory elements Ž4–6 wt.% CaO and Al 2 O 3 . compared to the Earth’s upper mantle and meteorites ŽVinogradov, 1965; Dodd, 1981; Taylor, 1982; McDonough and Sun, 1995.. A comparison of the calculated lunar bulk silicate compositions with a number of the available models shows that there is no agreement between the present model and some other estimates ŽTable 3.. The lunar CarSi Ž0.1. atomic ratio is greater than that of the Earth and chondrites Ž0.06–0.085.; the FerAl atomic ratio for the silicate Moon Ž1.2–1.4. is close to that Ž1.3. for the silicate Earth ŽMcDonough and Sun, 1995.. The lunar MgrSi Ž0.88. and MgrAl Ž6.0. atomic ratios are much smaller than those of the Earth and chondrites Ž1.1–1.2 and 10.8–12.5, respectively. ŽAnders and Grevesse, 1989; McDonough and Sun, 1995.. The FerSi atomic ratio is equal to 0.17–0.2 for the silicate Moon Žcrust q mantle. and 0.24–0.26 for the Moon as a whole Žcrust q mantleq core.. The lunar MGa Žin the silicate portion. of 0.82 is consistent with some recent estimates Že.g., Drake, 1986; O’Neill, 1991.; the derived lunar MGa is in conflict with that of the Earth’s upper mantle Ž; 0.89, Ringwood, 1979.. A plot of MGa in the lunar mantle vs. depth is shown in Fig. 5. As can be seen, the topological agreement between Figs. 1 and 5 is remarkably good and reflects the different influences of the FeO and MgO contents in the various mantle reservoirs on the seismic velocities by Nakamura Ž1983. and vice versa. A lower MGa for the Moon is consistent with the observations that lunar basalts are significantly more FeO-rich than terrestrial basalts ŽVinogradov, 1977; Drake, 1986; Wanke and Dreibus, 1986.. Previous and present estimates ¨ suggest that the lunar mantle contains about 10–14% of FeO. This is substantially higher than the terrestrial upper mantle Ž; 8%.. This high FeO content meets a difficulty for hypotheses that seek to form the Moon from the Earth’s mantle. For this reason, Wanke and Dreibus Ž1986. suggested that an addition of FeO was supplied ¨ by the impacting planetesimals. On the other hand, Kuskov et al. Ž1995a. estimated the bulk composition of the Earth ŽTable 3. based on a model of the fractional condensation of elements and compounds from a cooling nebula of solar composition, and found that the condensation product Žcalled ‘solar chondrite’. could be considered as corresponding to the average chemical composition of the Earth. The petrological and geophysical model of the mantle constructed from the solar chondrite composition shows that the density and velocity profiles of the bulk silicate Earth Župper q lower mantle. composed of a FeO-and SiO 2-rich assemblage Ž48% SiO 2 and 10–11% FeO. are in agreement with the global seismic models ŽKuskov and Panferov, 1991; Kuskov et al., 1995a. and does not require additional hypotheses as to the primary material of the mantle being of pyrolite type.

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299

Fig. 5. MGa Žmolar MgrMgqFe. calculated from geophysical constraints vs. depth. Model I, dashed line; model II, solid line.

Models of the Moon inferred from geophysical constraints ŽTable 3. show that the bulk compositions of the Earth and its satellite are different. Although the entire Earth’s mantle Župper q lower. and lunar mantle may or may not be similar in the FeO and SiO 2 contents, the Moon is certainly enriched in FeO and depleted in MgO compared to the Earth’s upper mantle. The Moon is also enriched in refractory elements by a factor of 1.5–2.5 compared to the terrestrial mantle; a bulk lunar Al 2 O 3 content of 6% implies a uranium abundance of 30 ppb in agreement with the heat flow estimates ŽTaylor, 1986.. None of the analyzed models of lunar formation satisfies these geochemical constraints. Such an enrichment in refractory elements and FeO and a depletion in MgO presents a paradox both from a cosmochemical point of view and from a giant impact scenario, because FeO is more volatile than MgO under vapor conditions. Moreover, it is well known that FeO content and volatile content are positively correlated in primitive nebula material ŽLewis and Prinn, 1984.. This implies that the Moon was not derived from the Earth’s mantle after the Earth’s core had segregated ŽRingwood, 1979; Wanke ¨ and Dreibus, 1986. and was not derived from the mantle of an impactor ŽNewsom and Taylor, 1989.. However, a model of the fractional condensation of material from a cooling nebula of solar composition probably accounts for this paradox ŽKuskov et al., 1995a,b.. Another explanation for the formation of a ‘nonchondritic’ Moon requires a model of the cooling vapor of nonsolar composition. The absence of genetic relationship between the compositions of the Earth and its satellite makes it possible to search new mechanisms of formation of the Moon. 5.4. The size of a core Calculations of the optimized sizes of a lunar core have been carried out at two boundary upper mantle density constraints: 3.24 g cmy3 - rm – cr Ž58 km. - 3.32 g cmy3 . The internally consistent determination of the composition of the Moon and the density distribution in the crust and mantle, combined with the mean seismic velocities as well as with the mass and moment of inertia constraints, suggests the presence of a core ŽFig. 6.: 470–590 km in radius for a FeS-core Ž2.8–5.5 wt.%. and 310–390 km for a FeŽNi.-core Ž1.2–2.7 wt.% of total mass.. The maximum and minimum values of a core radius were obtained for the upper mantle density of 3.24 and 3.32 g cmy3 , respectively. The free iron abundance in the Moon is estimated to be 2.5 " 2 wt.% from Apollo magnetometer experiments ŽDyal et al., 1977.. The current uncertainty in the moment of inertia

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Fig. 6. The geophysically admissible variations of radius of a lunar core vs. upper mantle density. Allowed ranges are calculated from the mass and moment of inertia constraints but do not include the seismic constraints. The largest core radii are estimated to be 460 km for an Fe-core Žin excellent agreement with the data of Hood and Jones, 1987. and 680 km for a FeS-core. Core sizes that are beyond the indicated ranges do not fall within the uncertainty for the moment of inertia. Values of 3.0 g cmy3 for the density and 58 km for the crustal thickness are adopted; r ŽFeS-core. s 4.7 g cmy3 ; r Žg-Fe-core. s8.1 g cmy3 . FeS-core, solid line; Fe-core, dashed line. Crosses and squares denote the optimized values of core radii obtained from the mean seismic velocities and the mass and moment of inertia constraints.

parameter does not differentiate between such different compositions as iron and iron sulfide ŽAnanda et al., 1977.. The existence of a lunar core is consistent with geochemical considerations on the extreme depletion of the mantle in siderophile elements ŽDrake, 1986; Newsom and Taylor, 1989; Galimov, 1995. and such geophysical data as attenuation of seismic velocities beginning below the deepest moonquakes at 1100 km, moment of inertia, and lunar magnetism ŽNakamura, 1983; Dyal et al., 1977; Runcorn, 1996.. The cessation of moonquakes below 1100 km Žapproximately core–mantle boundary. may have an important additional evidence for the existence of a core. Once we have constructed a model of the Moon with a core, a question which needs to be answered is the reliability of the proposed internal structure. Unfortunately, the available geophysical data are not sufficiently complete to confirm or deny the existence of a core. The model calculations indicate that the core sizes in the range 0–50 km Žcore-free models. are consistent with the seismic data within their uncertainty limits Ž"0.16 for Vs and "0.4 for Vp in the lower mantle. and with the mass and moment of inertia requirements if the following conditions are fulfilled in the mantle:

r 1 s 3.22–3.24 g cmy3 , r 2 s 3.41–3.44 g cmy3 , r 3 s 3.48–3.49 g cmy3 , Vs s 4.46–4.51 km sy1 and Vp ; 8.1 km sy1 in the lower mantle,

Ž 15 .

lower mantle composition CaO s 5%, Al 2 O 3 s 7%, FeO s 15%, MgO s 29%, SiO 2 s 44 wt.%, where r 1 , r 2 and r 3 are the densities in the upper, middle and lower mantle. Thus, the lunar model without a core differs from the model with a core mainly in showing a much higher FeO concentration Ž15%. in the lower mantle. An inspection of conditions 15 reveals that the model without a core required that the lower mantle seismic velocities would be decreased by about 2–3% from the median velocity values by Nakamura Ž1983.. Such a velocity decrease within the uncertainty limits of the inferred velocities in the lower mantle is quite possible ŽGoins et al., 1981; Nakamura, 1983.. For this reason, rigorous scientific confirmation of the existence of a lunar core can be made with some approach to precision. It is evident that new geophysical networks are required for determining the fine seismic structure of the Moon.

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Table 4 The physical properties and the sizes of iron cores of the Earth’s and Jupiter’s satellites Satellite

r Žgrcm3 .

Ir MR 2

R S Žkm.

R Fe-core , Žkm.

Mcore r MS Ž%.

Hice Žkm.

ŽFerSi.at

Moon Io Europa Ganymede

3.344 3.529 3.03 1.936

0.3905"0.0023 0.378"0.007 0.330"0.014 0.3105"0.0028

1738 1821 1569 2634

310–390 530–660 490–590 500–760

1.4–2.7 6–10 8–14 3–10

0 0 110 850

0.24–0.26 0.48–0.55 0.45–0.50 0.36–0.53

R S , MS , Hice denote the radius, mass and thickness of an ice layer for a satellite.

5.5. Comparison of the core sizes of the Earth’s and Jupiter’s satellites It is interesting to compare the core sizes of the Moon and the Galilean satellites of Jupiter. The Galileo spacecraft’s measurements of the gravitational and magnetic field yield a valuable information on the moment of inertia values of the Galilean satellites and their internal structure ŽAnderson et al., 1995, 1996, 1997.. These data as well as mean densities and radii were used to estimate the internal structure and the core sizes of Io, Ganymede, and Europa ŽKuskov et al., 1997; Kuskov and Kronrod, 1998.. We used the approach described in previous sections and considered a five-layer model of the internal structure of a satellite, including a silicate crust or an ice layer, three-layer silicate mantle in the CFMAS system, and Fe–FeS-core. The density distribution and the sizes of a core must fulfill relations 1 and 2. The results of the calculations, for the case of a Fe-core ŽTable 4., show that the ratio of a Fe-core mass to the mass of a Jupiter’s satellite is several times greater than that for the Moon at almost equal sizes and average density of the Moon, Europa, and Io. If the thick ice shell of Ganymede is excluded, this ratio also becomes higher and reaches 4.5–15%. These results show that the probability that Io, Europa and Ganymede possess a dense metallic core is much greater than the probability for the existence of a Moon’s core. The FerSi ratios and mean densities of the Galilean satellites reveal the same trend with distance from Jupiter that we see among the terrestrial planets with distance from the Sun. The Moon has the lowest FerSi ratio among the terrestrial planets, chondrites, and satellites of the outer Solar system ŽKuskov, 1997; Kuskov and Kronrod, 1998..

6. Conclusions Despite the uncertainties in the geophysical and thermodynamic data, it is believed that some firm conclusions can be drawn from this attempt to translate the geophysical constraints into the chemical, petrological and thermal models of the Moon. A derived self-consistent model of the chemical composition and the internal temperature and density distribution may be called as a preliminary geochemical–geophysical reference model and may be used as a significant constraint for the constitution of the Moon. Our analysis leads to the following conclusions. Ž1. Thermodynamic modeling of phase relations and physical properties of multicomponent mineral systems was used to develop a method for solving the inverse problem of determining the bulk composition, density and temperature distribution in the mantle and radius of a core from the totality of geophysical evidence Žseismic velocities, moment of inertia, and mass of the Moon.. The method consists of the forward and inverse processes of modeling and is based on minimizing the deviations of the calculated parameters from the observed data. By formalizing the search for the optimum solution, it is possible to take into account the full range of the factors involved, correlate a large number of the parameters optimized, and solve the problem with a precision that depends solely on the errors of the measured parameters. Ž2. The geophysically admissible ranges of the chemical composition in various zones of the lunar mantle and in the entire mantle were determined. The bulk composition of the silicate portion of the Moon

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Žmantleq crust. is estimated Žwt.%.: 4 - CaO - 5%, 5.5 - Al 2 O 3 - 7%, 10 - FeO - 12%, 28 - MgO - 30%, 48 - SiO 2 - 50%. The Earth’s mantle Župper q lower. and lunar mantle may or may not have similarity in the FeO and SiO 2 contents. The Moon is depleted in MgO and is enriched in refractory elements compared to the terrestrial mantle. The upper and middle mantles consist essentially of pyroxenite. An Al–Ca-rich model composition Žolivineq clinopyroxeneq garnet. gives the best fit to the lower mantle seismic properties. The upper, middle, and lower mantles are chemically distinct. The Moon has the lowest FerSi ratio among the terrestrial planets, chondrites, and satellites of the outer Solar system. The chemical composition of the Moon is composed of material different from that of the Earth’s mantle and bears no genetic relationship to the terrestrial material as well as to any of the known chondrites. Ž3. A probable law of temperature distribution in the lunar mantle is obtained. Models of the internal structure of the Moon without density inversion are constructed. Internally consistent determination of the composition and density distribution in the mantle, combined with the mass and moment of inertia constraints, suggests the presence of a lunar core: 470–590 km in radius for the FeS-core and 310–390 km for the Fe-core.

Acknowledgements We thank anonymous referees for their useful comments to improve the manuscript. This research was supported in part by the Russian Foundation for Basic Researches ŽGrant no. 97-05-64786..

Appendix A A.1. Comment to Appendix Comments on the thermochemical and thermophysical properties of minerals available in the THERMOSEISM data base as well as some principles of their selection were discussed by Fabrichnaya and Kuskov Ž1994., Kuskov and Fabrichnaya Ž1994. and Kuskov Ž1995a, 1997.. The potential method for construction of the equation of state for solids has been used in this study ŽKuskov and Galimzyanov, 1986.. The equation of state for a solid is written in the Mie–Gruneisen form ŽZharkov and Kalinin, 1968.: P Ž V ,T . s P0 Ž V . q Ž grV . E Ž V ,T . , where EŽ V,T . is the thermal constituent of internal energy in Debye approximation, P Ž V . is the potential constituent in Born–Mayer form, g is the Gruneisen parameter, and V is the volume. Six constants should be ¨ set for each mineral under normal conditions Ž1 bar, 298.15 K.: adiabatic bulk modulus Ž K s . and its pressure derivative Žd Krd P ., heat capacity ŽCp., Debye temperature ŽQ ., thermal expansion Ž a . and volume. Elastic moduli of minerals have been recommended if they have been determined on single-crystal and polycrystal samples with density close to crystallographic density by acoustic, static-compression and shock-wave experiments. Values of bulk Ž K s . and shear Ž G . moduli are Voigt–Reuss–Hill averages for polycrystalline aggregates; the Debye temperature is calculated from Vp and Vs values. The number in parentheses following each physical value represents the uncertainty in the last digit. New information is added for jadeite NaAlSi 2 O6 : thermoelastic constants are taken from an in situ synchrotron X-ray diffraction study ŽZhao et al., 1997.. Entropy Ž S298 s 136.5 J moly1 Ky1 . and heat capacity ŽCp s 259.08 q 0.038032T y 2518908Ty2 y 1332.57Ty1r2 y 0.0000088T 2 . are adopted from Hemingway ŽB. Hemingway, personal communication.. Enthalpy of formation of jadeite Ž D H8f ,298 s y3028.46 kJ moly1 . was calculated from the experimental P–T data of the equilibrium albite s jadeiteq quartz. The calculated Gruneisen parameter is: g s a K s VrCp s 1.21. ¨

A.2. THERMOSEISM database

Q ŽK.

G Žkbar.

ŽdG .rŽd p .

5 Ž1. 4.5 Ž20. 5 Ž2. 4.5 Ž10. 5 Ž1. 5 Ž1. 5 Ž1. 8.4 Ž19. 4.11 Ž20. 4.5 Ž18. 5.2 Ž4. 5.1 Ž3. 4 Ž2. 5 Ž1. 4.5 Ž18. 4.9 Ž10. 4 Ž2. 4.5 Ž10. 5.4 Ž20. 4.75 Ž80. 5 Ž1. 4.5 Ž20.

Cp ŽJ moly1 Ky1 . 342.6 Ž30. 204.69 Ž10. 122.6 Ž2. 212.0 Ž5. 79 Ž5. 82.0 Ž10. 87.6 Ž10. 45.4 Ž5. 79.03 Ž15. 166.63 Ž10. 131.9 Ž10. 118.6 Ž4. 92.3 Ž10. 333.2 Ž30. 175.2 Ž10. 122.0 Ž10. 99.5 Ž10. 164.1 Ž5. 121.7 Ž5. 42.05 Ž10. 79 Ž5. 166.6 Ž30.

740 Ž10. 500 Ž10. 840 Ž10. 546 Ž10. 707 Ž50. 731 Ž10. 540 Ž20. 615 Ž20. 1035 Ž5. 670 Ž10. 508 Ž5. 762 Ž5. 687 Ž20. 827 Ž5. 609 Ž10. 700 Ž20. 600 Ž20. 750 Ž20. 863 Ž25. 660 Ž10. 707 Ž50. 670 Ž20.

970 Ž10. 320 Ž15. 990 Ž13. 410 Ž10. 781 Ž100. 757 Ž10. 524 Ž40. 513 Ž20. 1630 Ž20. 670 Ž5. 505 Ž5. 818 Ž5. 785 Ž30. 1079 Ž10. 610 Ž5. 850 Ž50. 747 Ž30. 800 Ž50. 1094 Ž50. 800 Ž10. 743 Ž100. 670 Ž10.

1.4 1.5 1.5 1.5 1.7 1.7 1.7 1.3 1.8 1.7 1.5 1.8 1.7 1.6 1.7 0.5 1.7 1.7 1.5 1.64 2.0 2.0

ŽydG .rŽdT . Žkbar Ky1 . 0.10 0.10 0.13 0.10 0.10 0.10 0.10 0.11 0.19 0.10 0.13 0.13 0.17 0.125 0.10 0.10 0.17 0.13 0.13 0.14 0.11 0.11

5 Ž1. 5.0 Ž5. 4.5 Ž20.

82.0 Ž5. 87.6 Ž6. 175.2 Ž30.

731 Ž5. 540 Ž10. 609 Ž20.

757 Ž5. 524 Ž30. 610 Ž10.

2.0 2.0 2.0

0.11 0.11 0.11

4.24 Ž18. 4.5 Ž5. 6.0 Ž2. 6.8 Ž20. 4 Ž2. 4.9 Ž5. 5.1 Ž3.

37.24 Ž10. 325.5 Ž30. 44.6 Ž1. 55.05 Ž50. 121.4 Ž2. 115.9 Ž10. 43.0 Ž5.

941 Ž5. 798 Ž5. 570 Ž5. 780 Ž20. 806 Ž10. 861 Ž5. 1190 Ž25.

1310 Ž5. 920 Ž10. 445 Ž5. 1123 Ž35. 914 Ž35. 1080 Ž3. 2200 Ž30.

2.5 1.4 0.45 2.5 1.5 0.5 1.8

0.24 0.10 0.015 0.18 0.13 0.10 0.18

Žd K s rd P .

O.L. KuskoÕ, V.A. Kronrodr Physics of the Earth and Planetary Interiors 107 (1998) 285–306

Table A1. Thermoelastic properties of minerals at 1 bar and 298.15 K Mineral V a 10 6 Ks Žcm3 moly1 . ŽKy1 . Žkbar. Almandine Fe 3 Al 2 Si 3 O12 115.28 Ž6. 18 Ž2. 1780 Ž30. Analbite NaAlSi 3 O 8 100.43 Ž10. 17.5 Ž20. 677 Ž30. Andalusite Al 2 SiO5 51.46 Ž2. 25 Ž3. 1620 Ž40. Anorthite CaAl 2 Si 2 O 8 100.76 Ž10. 12 Ž2. 920 Ž15. Clinocorundum Al 2 O 3 26.2 Ž30. 25 Ž3. 1290 Ž200. Clinoenstatite, low MgSiO 3 31.65 Ž5. 24.8 Ž30. 1075 Ž20. Clinoferrosilite, low FeSiO 3 33.11 Ž10. 21 Ž5. 1010 Ž80. Coesite SiO 2 20.58 Ž3. 7 Ž1. 960 Ž30. Corundum Al 2 O 3 25.58 Ž1. 15.5 Ž5. 2525 Ž20. Diopside Ca 0.5 Mg 0.5 SiO 3 66.08 Ž8. 25.3 Ž10. 1140 Ž10. Fayalite Fe 2 SiO4 46.31 Ž3. 26 Ž1. 1278 Ž6. Forsterite Mg 2 SiO4 43.67 Ž1. 27.1 Ž5. 1288 Ž5. Geikielite MgTiO 3 30.86 Ž3. 30 Ž3. 1690 Ž90. Grossular Ca 3 Al 2 Si 3 O12 125.23 Ž11. 18.6 Ž6. 1681 Ž20. Hedenbergite Ca 0.5 Fe 0.5 SiO 3 67.84 Ž18. 20.2 Ž10. 1200 Ž10. Hercynite FeAl 2 O4 40.75 Ž10. 19 Ž3. 2100 Ž50. Ilmenite FeTiO 3 31.71 Ž5. 30 Ž3. 1725 Ž40. Jadeite NaAlSi 2 O6 60.52 Ž20. 26 Ž3. 1260 Ž40. Kyanite Al 2 SiO5 44.08 Ž2. 26.7 Ž30. 1915 Ž100. Lime CaO 16.765 Ž5. 29 Ž1. 1149 Ž20. Orthocorundum Al 2 O 3 27.4 Ž30. 25 Ž3. 1225 Ž200. Orthodiopside 67.46 Ž20. 25.5 Ž30. 1130 Ž20. Ca 0.5 Mg 0.5 SiO 3 OrthoenstatiteMgSiO3 31.37 Ž2. 25 Ž3. 1078 Ž10. OrthoferrosiliteFeSiO3 32.98 Ž1. 21 Ž3. 1010 Ž40. Orthohedenbergite 69.75 Ž18. 20.2 Ž20. 1200 Ž200. Ca 0.5 Fe 0.5 SiO 3 Periclase MgO 11.25 Ž1. 32.2 Ž8. 1628 Ž5. Pyrope Mg 3 Al 2 Si 3 O12 113.02 Ž10. 23.3 Ž10. 1728 Ž30. a-Quartz SiO 2 22.69 Ž1. 35 Ž1. 371 Ž5. Rutile TiO 2 18.8 Ž1. 22.6 Ž20. 2147 Ž50. Sillimanite Al 2 SiO5 49.84 Ž2. 14.5 Ž30. 1708 Ž40. Spinel MgAl 2 O4 39.72 Ž1. 19 Ž1. 1974 Ž5. Stishovite SiO 2 14.01 Ž2. 14 Ž2. 3100 Ž40.

303

304

A.3. THERMOSEISM database O.L. KuskoÕ, V.A. Kronrodr Physics of the Earth and Planetary Interiors 107 (1998) 285–306

Table 2. Enthalpy of formation from elements Ž D H f ,298 , kJrmol., entropy Ž S298 , Jrmol K. and heat capacity ŽCp s a q bT q cTy 2 q dTy 1r2 q eTy3 q gTy1 , Jrmol K. of minerals o D H fo,298 S298 Mineral a b c d e g Almandine Fe 3 Al 2 Si 3 O12 723.0 y0.026775 y1 992 100.0 y6043.6 y5270.26, 0.0 342.0, 0.0 Analbite NaAlSi 3 O 8 309.74 0.015278 y26 160 000.0 0.0 y3929.86, 4 109 100 000.0 221.8, 8840.0 Andalusite Al 2 SiO5 221.7 0.002581 y2 559 000.00 0.0 y2591.51, 489 900 000.0 91.420, y26 690.0 Anorthite CaAl 2 Si 2 O 8 290.90 0.0276 y34 080 000.0 0.0 y4233.15, 5 218 000 000.0 199.3, 29 625.0 Clinocorundum Al 2 O 3 157.4 0.000719 y1 896 900.0 y988.0 y1652.0, 0.0 46.0, 0.0 Clinoenstatite, low MgSiO 3 178.1 y0.0015 y298 450.0 y1592.65 y1543.19, 0.0 68.31, 0.0 Clinoferrosilite, low FeSiO 3 178.7 y0.001378 y355 550.0 y1496.3 y1192.69, 0.0 96.69, 0.0 Coesite SiO 2 74.826 0.007165 179 582.4 0.0 y907.41, 0.0 38.74, y10 011.72 Corundum Al 2 O 3 157.4 0.000719 y1 896 900.0 y988.0 y1675.73, 0.0 50.92, 0.0 Diopside Ca 0.5 Mg 0.5 SiO 3 157.3 0.0000205 y1 372 950.0 y1010.5 y1600.08, 0.0 71.35, 0.0 Faylite Fe 2 SiO4 59.9 0.07062 y5 743 700.0 2012.1 y1478.8, 0.0 151.0, 0.0 Forsterite Mg 2 SiO4 234.9 0.001069 y542 900.0 y1906.4 y2174.14, 0.0 94.1, 0.0 Grossular Ca 3 Al 2 Si 3 O12 542.6 0.01294 y3 186 000.0 0.0 y6633.65, 277 700 000.0 256.0, y56 020.0 Hedenbergite Ca 0.5 Fe 0.5 SiO 3 155.2 0.006285 y923 000.0 y1020.0 y1421.72, 0.0 87.5, 0.0 Hercynite FeAl 2 O4 238.0 0.00862 y903 100.0 y1862.7 y1951.0, 0.0 116.0, 0.0 Ilmenite FeTiO 3 y3.0 0.06505 y5 105 700.0 2426.6 y1234.5, 0.0 108.5, 0.0 Kyanite Al 2 SiO5 223.54 0.002645 y4 149 000.0 0.0 y2596.01, 760 100 000.0 82.3, y25 240.0 Lime CaO 54.25 0.001215 301 000.0 0.0 y6349.2, y59 500 000.0 38.1, y3660.0 Orthocorundum Al 2 O 3 157.4 0.000719 y1 896 900.0 y988.00 y1648.07, 0.0 55.31, 0.0 Orthodiopside Ca 0.5 Mg 0.5 SiO 3 157.3 0.0000205 y1 372 950.0 y1010.5 y1580.54, 0.0 80.0, 0.0 Orthoenstatite MgSiO 3 178.1 y0.0015 y298 450.00 y1592.65 y1546.24, 0.0 66.25, 0.0 Orthoferrosilite FeSiO 3 178.7 y0.00138 y355 550.00 y1496.3 y1193.96, 0.0 95.85, 0.0 Orthohedenbergite Ca 0.5 Fe 0.5 SiO 3 155.2 0.006285 y923 000.0 y1020.0 y1419.09, 0.0 87.235, 0.0 Periclase MgO 65.2 y0.00127 y461 900.00 y387.2 y601.6, 0.0 26.95, 0.0 Pyrope Mg 3 Al 2 Si 3 O12 545.0 0.02068 y8 331 200.0 y2283.0 y6288.53, 0.0 266.3, 0.0 a-Quartz SiO 2 71.274 0.011561 y535 661.0 0.0 y910.7, 0.0 41.46, y7270.0 Rutile TiO 2 63.1 0.011307 y986 300.00 y5.6 y944.75, 0.0 50.39, 0.0 Sillimanite Al 2 SiO5 183.87 0.01815 y123 66 000.0 0.0 y2587.51, 1 602 400 000.0 95.79, 3205.2 Spinel MgAl 2 O4 222.9 0.006127 y1 685 700.0 y1551.2 y2303.79, 0.0 82.82, 0.0 Stishovite SiO 2 74.826 0.00779 y231 960.6 0.0 y863.78, 0.0 30.03, y9383.18

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305

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