Mars: interior structure and excitation of free oscillations

Physics of the Earth and Planetary Interiors 142 (2004) 1–22

Mars: interior structure and excitation of free oscillations T.V. Gudkova∗ , V.N. Zharkov Schmidt Joint Institute of Physics of the Earth, Russian Academy of Sciences, B. Gruzinskaya, 10, Moscow 123995, Russia Received 14 November 2002; received in revised form 7 May 2003; accepted 10 October 2003

Abstract Based on available chemical models of the planet [Philos. Trans. R. Soc. London 349 (1994) 285; Supply and loss of volatile constituents during the accretion of terrestrial planets, in: S.K. Attreya, J.B. Pollack, M.S. Matthews (Eds.), Origin and Evolution of Planetary and Satellite Atmospheres, Univ. Arizona Press, pp. 268–288; Icarus 126 (1997) 373; Phys. Earth Planet. Inter. 112 (1999) 43; Space Sci. Rev. 92 (2000) 34], a new set of global models of the Martian interior has been constructed. A model comprises four submodels—a model of the outer porous layer, a model of the crust, a model of the mantle and a model of the core. The first 10–11 km layer is considered as an averaged transition from regolith to consolidated rock. The mineral composition of the crustal basaltic rock varies with depth because of the gabbro-eclogite phase transition. As a starting point for mantle modeling the experimental data obtained by Bertka and Fei [J. Geophys. Res. 102 (1997) 525; Earth Planet. Sci. Lett. 157 (1998) 79] along the areotherm have been used, iron content of the mantle being varied. The measured or estimated up to now elastic properties for a set of mantle minerals are used. Seismic velocities determined from new high P–T data on elastic properties are 2–3% lower than the velocities calculated earlier. New high P-T measurements of the density of Fe (␥-Fe), FeS and FeH enable us to refine the core model. Taking into account available chemical models and the fact that noticeable amount of hydrogen could enter the Martian core during its formation [Solar Syst. Res., 30 (1996) 456], such parameters as a ferric number of the mantle (Fe#), sulfur and hydrogen content in the core are varied. The following tendency is seen: the presence of hydrogen leads to the increase of the Fe/Si ratio and decreases Fe# in the mantle due to the increase of the core radius. The higher sulfur and hydrogen content in the core and the smaller mantle Fe#, the less likely a perovskite layer exists. The modeling shows that to obtain the Fe/Si ratio up to the chondrite ratio of 1.71, more than 50 mol% of hydrogen should be incorporated into the core. In the second part of the paper, based on the available estimates of the Martian seismic activity and the sensitivity of current instruments, the amplitudes for different types of free oscillations have been estimated. It is found down to what depth the normal modes can sound the planetary interiors. A marsquake with a seismic moment of 1025 dyn cm is required for spheroidal oscillations (with
≥ 17) to be detected. These spheroidal modes are capable sounding the outer layers of Mars down to a depth of 700–800 km. © 2004 Elsevier B.V. All rights reserved. Keywords: Mars; Interior structure; Free oscillations

1. Introduction Okal and Anderson (1978) studied the structure and free oscillations of Mars. Since then more than 20 years have passed and this problem has been further ∗ Corresponding author. Fax: +7-95-255-60-40. E-mail address: [email protected] (T.V. Gudkova).

developed in a number of publications (Kamaya et al., 1993; Zharkov, 1996; Mocquet et al., 1996; Lodders and Fegley, 1997; Sohl and Spohn, 1997; Yoder and Standish, 1997; Bertka and Fei, 1997, 1998; Sanloup et al., 1999; Zharkov and Gudkova, 2000; Kavner et al., 2001). In most of these papers, to construct a Martian interior structure model, the chemical model (DW) proposed by Wänke and Dreibus (Dreibus and

0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.pepi.2003.10.004

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T.V. Gudkova, V.N. Zharkov / Physics of the Earth and Planetary Interiors 142 (2004) 1–22

Wänke, 1989; Wänke and Dreibus, 1994) is mainly used. The DW model is based on the data on Martian SNC meteorites, Earth’s materials and properties of chondrite meteorites, and the assumption that low-volatile components must be in C1 chondrite ratios. Wänke and Dreibus assumed that during the formation of the terrestrial planets, first of all Earth and Mars, the mixing of the material from different feeding zones of growing planets due to the influence of Jupiter took place (see the paper by Zharkov (1993) devoted to this problem). It is considered that Mars was basically formed by two components: a highly reduced component A, existing mainly in the feeding zone of the growing Earth and an oxidized component B, existing in the feeding zone of the asteroid belt. The protobodies consisting of A component were free from all elements with volatility equal to or higher than Na, however they contained all other elements in C1 abundance ratios. Fe and all siderophile elements were metallic, and even Si might be partly present in metallic form. The component B contained all elements, including volatiles, with abundances like in C1 meteorites. Iron and all sidorophile and lithophile elements were present mainly as oxides. Based on their estimates of bulk composition of Mars and Earth, Dreibus and Wänke (1989) came to the conclusion that Mars was formed with 40% C1 material (component B), and 60% volatile depleted highly reduced material (component A). For the Earth, the content ratio of these components is 85:15. It was suggested that Mars accreted homogeneously, unlike the chemically inhomogeneous accretion of the Earth. Wänke and Dreibus approach seems to be quite logical. The formation of Earth and Mars was most likely many-component, and a two-component DW model should be considered as a reasonable averaged first approximation. The two-component nature of this process should have the most pronounced manifestations in the model of the internal structure of Mars. This feature can be revealed by seismic sounding and recording the tidal response of Mars. Simultaneous recording weak anomalies in the rotation of the planet could also contribute to the solution of the problem (Zharkov et al., 1996). First of all, to solve this problem, the size and state of the core, and its mean density should be determined. Up to now, geo- and cosmochemical information as well as the data on the composition

of SNC-meteorites, Martian in their origin, have been basically used to judge the material composition of the Martian interior. The successful completion of the Mars Pathfinder (MPF) mission provided a new value for the mean dimensionless moment of inertia for Mars (Folkner et al., 1997): I A+B+C = = 0.3642 − 0.3678, MR2 3MR2

(1)

where A and B are the values of the equatorial principal moments of inertia, C is the polar principal moment of inertia, M and R are the mass and the mean radius of the planet. The estimate (1) put a rigid restriction on Martian interior structure models. One of the main questions concerned in a number of previous papers (Zharkov, 1996; Sohl and Spohn, 1997; Yoder and Standish, 1997; Bertka and Fei, 1997, 1998; Kavner et al., 2001) was to find out if the DW model satisfied the interior structure models of Mars taking into account restriction (1). For chondritous DW model the value of the weight Fe/Si ratio is equal to 1.71. A number of authors (Sohl and Spohn, 1997; Yoder and Standish, 1997; Bertka and Fei, 1997, 1998) concluded that DW model did not satisfy current Martian models. Zharkov and Gudkova (2000) came to more careful conclusion that there were not enough data to judge about the validity of the DW chemical model. To clarify this problem in the future, it is necessary to measure the radius of the Martian core with a good precision. In interior models the Fe/Si ratio strongly depends on the core radius. These questions will be considered lately in detail. Oxygen isotopic ␦17 O/␦18 O ratio is a fundamental parameter for any terrestrial planet. Lodders and Fegley (1997), Lodders (2000) and Sanloup et al. (1999) assume that Mars composition is a mixture of different meteorites, whose proportions are calculated so that isotopic ␦17 O/␦18 O ratio falls along a fractionation line of Martian SNC meteorites. Lodders and Fegley (1997) and Lodders (2000) considered a chemical Martian model corresponded to the accretion of about 85% H-chondritic, 11% CV-chondritic and 4% C1-chondritic material, and the model by Sanloup et al. (1999) consisted of 55% ordinary chondrites H and 45% enstatite chondrites EH. Table 1 summarizes model bulk compositions for the mantle (a silicate

T.V. Gudkova, V.N. Zharkov / Physics of the Earth and Planetary Interiors 142 (2004) 1–22

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Table 1 Bulk model compositions of Mars (values are in wt.%, except for Mg#, Mg# = atomic [Mg/(Mg + Fe2+ ) × 100]) DW Dreibus and Wänke (1989)

MB Bertka and Fei (1998)

LF Lodders and Fegley (1997)

SJG Sanloup et al. (1999)

SiO2 TiO2 Al2 O3 FeO MnO MgO CaO Na2 O P 2 O5 Cr2 O3 Mg/Si (mole) Fe/Si Mg#

44.4 0.14 3.02 17.9 0.46 30.2 2.45 0.5 0.16 0.76 1.01 1.709 75.0

43.68 – 3.13 18.71 – 31.5 2.49 0.5 – – 1.075 1.766 75.0

45.39 0.14 2.89 17.22 0.37 29.71 2.35 0.98 0.17 0.68 0.976 1.684 75.5

47.5 0.1 2.5 17.7 0.4 27.3 2.0 1.2 1.2 – 0.7 1.709 72.0

Silicates (wt.%)

78.3

78.3

79.37

76.0

Core (wt.%) Fe Ni S Fe3 P

21.7 77.8 8.0 14.2 –

21.7

20.63 79.8 8.05 10.6 1.55

24.0 76.6 7.2 16.2 –

reservoir of the planet) and for the core. It also includes model MB composition (Bertka and Fei, 1998), the properties of which have been studied experimentally under the Martian mantle (P, T) conditions. In the LF model (Lodders and Fegley, 1997) the sulfur content in the core is lower than in the DW model, and in the SJG model (Sanloup et al., 1999) it is somewhat higher than in the DW model. First of all, the sulfur content in the core and the magnesium number Mg# are varied (Mg# = atomic [Mg/(Mg + Fe2+ ) × 100], in order to construct interior structure models. From this point of view, the compositions of new chemical models are similar to the DW model. The successful completion of the Mars Global Surveyor (MGS) mission provided the detailed topography mapping of the planet and the gravity field data expressed in spherical harmonics of degree 75. This made possible to apply geophysical methods for constructing a model of the crust. Assuming a crustal density of 2.9 g/cm3 and a density contrast between the crust and mantle of 0.6 g/cm3 , Smith and Zuber (2002) obtained the mean thickness of the crust of nearly 50 km. Noticeable zonal variation in the thickness of the crust is evident: the thinnest crust turned out to be a few kilometers thick under the Isidis im-

pact basin and the thickest crust is about 90 km under south-central Tharsis. Turcotte and Shcherbakov (2002) studied the correlation between topography, gravity, and areoid for the Hellas impact basin, a major topographic feature on Mars, its radius is about 1500 km. Assuming mantle density of 3.5 g/cm3 , the authors concluded that the mean density of the crust is 2.96 ± 0.05 g/cm3 and the reference (zero elevation) thickness of the crust on Mars is 90 ± 10 km. The same value was obtained for the averaged thickness of the elastic lithosphere, 90 ± 10 km. It was noted that this is a reasonable result and it can be explained by reological characteristics of the mantle and the crust (the crustal rocks are very likely stronger than the mantle rocks beneath). A short, but clear review on the considered problem is given in their paper. The spectral admittance (gravity/topography ratio) was calculated for typical Martian structures of different age and an elastic lithosphere thickness was determined for them by McGovern et al. (2001). It made possible to estimate temperature gradients for the considered features at time of their formation. Somewhat different interpretation was applied by McKenzie et al. (2002). To determine gravity

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anomalies they used direct measurements of line of sight velocity, rather than spherical harmonic coefficients. Then, as usual, the functions of spectral admittance were obtained for four large regions of Mars: Tharsis [(50N–50S) × (85–165)W], Valles Marineris [(25N–35S) × (10–100)W], Elysium [(60N–10S) × (115–175)E] and South Pole [(20–70)S × (180W–180E)]. When interpreting the data they used a crustal thickness of 30 km and a mantle density of 3.5 g/cm3 . Such estimates of Martian lithospheric parameters as a wavelength band, the density of the crust, the thickness of the elastic lithosphere, the age, the temperature at the bottom of the elastic lithosphere, and the thermal flux were obtained for these regions. Both, in this paper and in the above cited papers, a simple one layer model of the crust was used, and the value of the mean density of the crust was determined from the short wavelength behavior of the admittance. The results described above are of great interest. As it is assumed that the Martian crust predominantly has a basaltic composition of SNC meteorites (McSween, 2002) and their density is about 3.2–3.3 g/cm3 , then it is evident, that in the Martian crust the density progressively changes with the depth. Therefore, as in the case of the Moon (Wieczorek and Phillips, 1998), the use of two-layer or more complicated models of the crust can introduce some corrections in the given above results. Moreover, the results will be more valuable, if a successful seismic experiment on Mars takes place. It makes possible to sound the structure of the Martian crust by direct seismic methods in a certain place and by such a way to obtain a benchmark. At constructing interior structure models our approach (Zharkov and Gudkova, 2000) differs from ones used earlier in a following way: (1) we specially devoted a paper (Babeiko and Zharkov, 2000) to the study of probable Martian crust structure. The Martian crust comprises an outer porous (≈11 km) layer, where, by analogy with the Moon, there is a transition from highly porous Martian regolith to consolidated rocks. The bulk chemical composition of the consolidated crust was accepted to be the average of four basaltic SNC meteorites (Shergotty, Zagami, BETA 79001, lithologies A and B). The crust properties (density distribution, mineralogical composition and elastic wave velocities) were studied with the help of the petrological modeling technique (Sobolev and Babeiko, 1994); (2) according to Zharkov (1996)

the fact that the Martian core can contain noticeable amount of hydrogen was taken into account. The presence of hydrogen in the core decreases both the core density and its melting temperature. Since the outer layers of Mars are very heterogeneous, as in the case of the Moon (Jolliff et al., 2000), it is difficult to construct a spherically symmetric model of its interior structure (a model of a zeroth approximation), by using only a few seismometers for recording seismic body waves. First of all, they will allow to obtain the structure of the crust at the sites of their location. That is why, in the second part of this paper the excitation of free oscillations of the planet will be considered. We used the estimates of the Martian seismicity from Philips and Grimm (1991), Golombek et al. (1992) and Wilkins et al. (2002). The tidal delay of Phobos was used to estimate the dissipation factor for the Martian mantle (Zharkov and Gudkova, 1993, 1997). Based on these estimates (the estimates of mantle Q for the models with a solid core are too low) we assume that the Martian core is most likely liquid. To the point, in order to significantly improve the Martian interior structure model and concretize the distribution of such a fundamental parameter as the dissipative factor Q␮ , it is sufficient to install one seismometer capable to record free oscillations of Mars. This paper is organized as follows. In Section 2 the arguments in favor of the presence of hydrogen in the core are given, Section 3 is devoted to the interior structure models and seismic velocity profiles, then, the spectrum of free oscillations and their excitation are calculated in Section 4, at last in Section 5 the obtained results are discussed. 2. An estimate for the hydrogen content in the Martian core In the planetary interior the conditions for the dissolution of hydrogen in iron arise under high pressures and temperatures. The source of hydrogen is the reaction: Fe + H2 O → FeO + H2 .

(2)

As gas pressure grows, hydrogen starts dissolving in iron:

T.V. Gudkova, V.N. Zharkov / Physics of the Earth and Planetary Interiors 142 (2004) 1–22

Fe +


x 2

H2 → FeHx ,

(3)

a bibliography on this point is given by Fukai and Suzuki (1986) and Zharkov (1996). Particularly, the molecular ratio H/Fe ≈ 0.2–0.4 is attained near the melting point at pressures as small as several tens of kilobars, and the ratio H/Fe ≈ 1 is attained at moderate temperatures and pressures of about 70 kbar. The solution of hydrogen in iron reduces the density and appreciably lowers the melting point. The idea that the Martian core may contain hydrogen, along with sulfur, as an admixture, is not a novelty (Fukai and Suzuki, 1986; Zharkov et al., 1991; Fukai, 1992). However, the effect of hydrogen on the model of the Martian core and, therefore, on the model of the planet as a whole, was discussed in detail only by Zharkov (1996). The oxidized component B, mentioned above, is assumed to have a composition of C1 carbonaceous chondrites. The water content attains in such bodies ∼7.3 wt.% (Dreibus and Wänke, 1989). Ahrens et al. (1989) found that consolidated minerals containing volatiles (CO2 , H2 O, and SO2 ) start losing them at shock pressures within the range 30–50 GPa. Such shock pressures are achievable when the impactor has a velocity of ∼2–3 km/s, colliding with a target of the same mineral. Consequently, the dehydration of planetosimals that belong to component B begins when the radius of growing proto-Mars is r ∼ 0.4R and attains 75% at r = R (R being the radius of Mars). To estimate the mass of H2 that can be buried in the interior of growing proto-Mars, we put r ≈ 0.5R and ρ ≈ 3.5 g/cm3 . Then one easily obtains ∼2.4 × 1023 g of hydrogen. The mass of the iron–nickel (Fe0.9 Ni0.1 ) core comprises in model DW ∼1.2 × 1026 g. Therefore, if all the buried hydrogen ultimately becomes a constituent of the Martian core, we obtain the composition (Fe0.9 Ni0.1 )0.9 H0.1 , where the subscripts mean molecular fractions. The estimate can be considered as the lower bound for the hydrogen content in the Martian core. Under the assumption that component B has the same composition as C1 chondrites (≈7.3 wt.% H2 O), we find that the maximum amount of hydrogen that could be produced in the process of the accumulation of Mars is ∼2 × 1024 g of H2 . Thus, the upper bound for the hydrogen content in the iron core of Mars, not achievable in practice, is

5

1.2 × 1026 1.2 NFe = ≈ ≈ 1.07 ⇒ FeH NH 1.12 56 × 2 × 1024 The estimates given above were obtained with the data on water content of about 7.3 wt.% in C1 chondrites from the paper by Dreibus and Wänke (1989). If one takes into account recent estimates of water content (18–22 wt.%) (Lodders and Fegley, 1998; Brearley and Jones, 1998) the estimates of hydrogen content in the Martian core should be increased by a factor of 2.5–3. Then, the estimates of hydrogen in the core as H0.5 and H0.7 look quite reasonable. On the other hand, the chemical models by Sanloup et al. (1999) do not contain hydrogen at all, and maximum value of hydrogen in the model by Lodders and Fegley (1998) is 7.2 times lower, than the same estimate used by Dreibus and Wänke (1989). In the frame of the LF model the influence of hydrogen is negligible. Zharkov (1996) found that addition of the molecular concentration x = 0.1 of hydrogen to the iron core of Mars reduces its density by about 0.16 g/cm3 . Hydrogen and other possible admixtures (Ni, S, C) significantly reduce the melting point of the Martian iron core. The rate of melting-point reduction −dTm /dx × (103 K) due to solute atoms (Ni, H, C, S) in Fe is 1, 2, 3, 3, respectively (Fukai, 1992). On the basis of these estimates, it is easy to calculate the possible drop of the melting point of iron under the conditions of the Martian core for the expected trial composition Fe0.9 N0.1 S0.2 H0.1 C0.03 : −∆Tm = (0.1 + 3 × 0.22 + 2 × 0.1 + 3 × 0.03) × 103 K ≈ 1050 K. In model DW, the molecular concentration of sulfur in the core is equal to 0.22. This estimate implies that the Martian core is most likely liquid. The melting temperatures of core materials, ␥-Fe, FeS and eutectic Fe–FeS system are shown in the paper by Boehler (1996) and the melting temperature for Fe–FeS is shown in paper by Kavner et al. (2001).

3. The construction of Martian interior structure models 3.1. Crust In the crust models of Babeiko and Zharkov (2000), there is a transition in the outermost 10 km layer from highly porous Martian regolith (≈1.6 g/cm3 )

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Fig. 1. Density and velocity profiles of the crust for three different marsotherms: M (13.5 K/km), L (6 K/km) and SL (2 K/km) corresponding to effective kinetic “freezing” temperature Tf of 800 ◦ C. The figure starts at 10 km depth.

to consolidated rocks (3.2 g/cm3 ). The model of the outer porous layer is described in detail by Babeiko and Zharkov (2000). Because of the gabbro-eclogite phase transition, the increase of density with depth in the consolidated crust depends strongly on the temperature gradient. The density and velocity profiles of the crust for different temperature gradients (2 (SL), 6 (L) and 13.5 K/km (M)) are shown in Fig. 1. As seen from Fig. 1 the crust may be divided into several zones according to the distribution of density and its seismic-wave velocities. The maximum thickness of the crust in the models is determined by the depth, at which the crust density is equal to the mantle density. Otherwise, the crust would be dynamically unstable. For the mantle with Mg# = 0.75 we have 50 km thick crust for L and SL temperature gradients in the crust, for M temperature gradient the crust can be thicker. 3.2. Mantle For the modeling of the density profiles in the mantle, we use the experimental results of Bertka and Fei (1997, 1998). Bertka and Fei (1997) performed high-pressure multi-anvil experiments with an analog of the Dreibus and Wänke (1985) composition to determine the model mineralogy up to core-mantle boundary pressures along a model areotherm. Following Bertka and Fei (1997) we consider the Martian mantle consisting mainly of 12 mineral as-

semblages. The mineral compositions and model abundances of the high-pressure assemblages are given in Bertka and Fei (1997). The sequences of the phase transitions in the Martian mantle are summarized as follows: an upper mantle consists of olivine, clinopyroxene, orthopyroxene and garnet up to 9 GPa, above 9 GPa orthopyroxene is no longer present. The transition zone is marked by the appearance of ␥-spinel at 13.5 GPa, which then coexists with ␤ phase, clinopyroxene and majorite. The transition zone above 17 GPa consists of ␥-spinel and majorite. The lower mantle starts at 22.5 GPa and consists of perovskite, magnesiowüstite and majorite. The weight fraction of each mineral assemblage is calculated from the mass balance of the experimental products obtained by Bertka and Fei (1997). These data are used to calculate the density of the mantle as a function of pressure and temperature with a Birch–Murnaghan equation of state: P = 3f(1 + 2f)5/2 KT (1 − 23 (4 − KT )f),

(4)

where

  T  ρ∗ = ρ0 exp − α(T) dT , T   0  2/3 1 ρ f = −1 , 2 ρ∗ ρ0 is the STP density, ρ∗ the density at P = 0 and the temperature corresponding to the areotherm and

T.V. Gudkova, V.N. Zharkov / Physics of the Earth and Planetary Interiors 142 (2004) 1–22

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Fig. 3. Martian mantle density profiles with different iron content in the mantle silicates: Fe# 25 (B-F mantle profile) (solid line), Fe# 18 (dot-dashed line), and Fe# 28 (dashed line). Fig. 2. The model mantle density (B-F mantle profile) (solid line) and density profiles for various mineral assemblages (dashed line) as a function of depth. Fe# number of mineral assemblages is equal to 25 unless otherwise noted in parenthesis. The abbreviations are ol, olivine; opx, orthopyroxene; cpx, clinopyroxene; Mg-cpx, Mg-rich clinopyroxene; Ca-cpx, Ca-rich clinopyroxene; Mg-gt, Mg-rich garnet; Ca-gt, Ca-rich garnet; Mg-mj, Mg-rich majorite; ␤, ␤-spinel; ␥, ␥-spinel; mw, magnesiowüstite.

ρ is the density in the isothermal Birch–Murnaghan equation of state, KT is the isothermal bulk modulus, KT = ∂KT /∂P, and α(T) is the coefficient of thermal expansion. The density of the mixture of mineral assemblages is calculated in the approximation of additive component volumes. The database for the end-member phases used in these calculations is given in Bertka and Fei (1998). The calculated mantle density profile (hereinafter referred to as B-F mantle profile) and density profiles for different mineral assemblages are shown in Fig. 2. When varying the iron content of the mantle silicates, we changed the mantle density profile. We added ␦ρi

to the B-F mantle density profile (Fe# 25), when calculating the models with lower and higher iron content (Fig. 3). The radial positions of high-pressure phase transitions in the Martian mantle are determined from the paper by Bertka and Fei (1997) and listed in Table 2. The mantle densities can be calculated according to the Fe# number of the model using empirical mineral densities as a function of their iron content (Table 3). When increasing Fe# by 1 ␦ρ is increased by about 0.01 g/cm3 for olivine zone, 0.0083 g/cm3 for ␤-zone, 0.011 g/cm3 for ␥-zone and 0.0125 g/cm3 for perovskite zone. 3.3. Core The Martian core composition is considered to be sulfur-rich, consisting of Fe with 14.2 wt.% S, 7.6 wt.% Ni (Dreibus and Wänke, 1985). New high P–T measurements of the density of Fe (␥-Fe) and FeS (Kavner et al., 2001) enable us to refine the core model by Zharkov (1996).

Table 2 Phase boundaries of pressure in the Martian mantle as function of Fe# (Bertka and Fei, 1997) Fe#

P1 P2 P3 P4

16

18

20

22

24

25

26

28

30

14.0 14.45 17.85 18.35

13.8 14.35 17.55 18.05

13.6 14.25 17.25 17.75

13.36 14.15 16.95 17.45

13.12 14.05 16.65 17.15

13.0 14.0 16.5 17.0

12.9 13.94 16.3 16.8

12.7 13.82 15.9 16.4

12.5 13.7 15.5 16.0

Note: P1 -␣ → ␣ + ␤(␣ + ␥); P2 -␣ + ␤(␣ + ␥) → ␤; P3 -␤ → (␤ + ␥); P4 -(␤ + ␥) → ␥.

8

Formula (abbreviation, name)

Density (g/cm3 )

(Mg, Fe)2 SiO4 (ol, olivine) (Mg, Fe)2 SiO4 (␤, ␤-spinel) (Mg, Fe)2 SiO4 (␥, ␥-spinel) (Mg, Fe)SiO3 (opx, orthopyroxene) (Mg, Fe)2 Si2 O6 (Mg-cpx, clinopyroxene) Ca(Mg, Fe)Si2 O6 (Ca-cpx, clinopyroxene) (Mg, Fe)3 Al2 Si3 O12 (Mg-gt, garnet) Ca3 Al2 Si3 O12 (Ca-gt, garnet) Mg4 Si4 O12 (Mg-mj, majorite) (Mg, Fe)SiO3 (Mg-pv, perovskite) CaSiO3 (Ca-pv, perovskite) (Mg, Fe)O (mw, magnesiowüstite) 10% Ni:Fe–Ni [T = 1800 ◦ C] FeS [T = 800 ◦ C]

3.222 3.472 3.564 3.204 3.188 3.280 3.566 3.6 3.518 4.107 4.252 3.584 7.03 4.94

+ 1.18XFe + 1.24XFe + 1.285XFe + 0.798XFe + 0.817XFe + 0.376XFe + 0.746XFe + 1.047XFe + 2.281XFe

KS (GPa) 129 170 186 104 114 113 175 169 175 266 227 163 105 54

+ 9XFe + 15XFe + 15XFe + 7XFe + 1XFe

− 8XFe

G (GPa)

KS

G

˙ S (GPa/K) K

˙ (GPa/K) G

α0 (300 K) 10−6 K−1

82 − 31XFe 114 − 41XFe 124 − 41XFe 77 − 24XFe 77 − 24XFe 67 − 6XFe 90 + 8XFe 104 90 153 125 131 − 77XFe – –

4.2 4.3 4.1 5.0 5.0 4.5 4.9 4.9 4.9 3.9 3.9 4.2 4.5 4.0

1.4 1.4 1.3 2.0 2.0 1.7 1.4 1.6 1.4 2.0 1.9 2.5 – –

0.017 0.018 0.021 0.012 0.012 0.013 0.021 0.016 0.021 0.031 0.027 0.016 0.025 0.020

0.014 0.014 0.016 0.011 0.011 0.010 0.010 0.015 0.010 0.028 0.023 0.024 – –

26.6 22.0 21.0 27.0 27.0 27.0 18.0 16.0 20.0 17.0 17.0 32.5 75 68.52

All the values are for zero pressure and 300 K temperature unless otherwise noted. In all cases, XFe denotes the molar proportion of the iron end-member component (0 ≤ XFe ≤ 1). KS , adiabatic bulk modulus; G, shear bulk modulus; KS , pressure derivative of the adiabatic bulk modulus; G , pressure derivative of the shear modulus; KS , absolute value of the temperature derivative of the adiabatic bulk modulus; G, absolute value of the temperature derivative of the shear modulus; ␣0 , the volume coefficient of thermal expansion at 300 K. Data sources: Duffy and Anderson (1989), Bass (1995), Duffy et al. (1995), Fei et al. (1992), Fei (1995), Fei et al. (1995), Akaogi et al. (1998), Bertka and Fei (1998), Li et al. (1998), Nasch et al. (1998), Zha et al. (1998), Sinogeikin et al. (2001).

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Table 3 Elastic properties of mantle minerals and core materials used for velocity calculations

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In this study the Martian core is assumed to be a mixture of iron–nickel alloy, sulfur and some amount of hydrogen. The compressibility of FeH is described by the Vinet equation:  1/3  2/3  ρ0 ρ p = 3KT 0 1− ρ0 ρ   1/3 

3 ρ0 × exp (KT 0 − 1) 1 − , (5) 2 ρ in which the zero-isotherm values of the parameters are KT 0 = 121 ± 19 GPa, KT 0 = 5.31 ± 0.9, ρ0 = 6.7 g/cm3 (Bading et al., 1992). The mixture of ␥-Fe and FeS is calculated in the approximation of additive component volumes. The weight concentrations XS of sulfur and XFeS of FeS are related by the equation XS = 0.36XFeS . In DW model, XS = 0.14; accordingly, XFeS ≈ 0.4. The addition of 10 mol% of hydrogen to the iron reduces its density by 0.16 g/cm3 (Zharkov, 1996). Experimental data of high-PT phases of ␥-Fe and FeS by Kavner et al. (2001) were obtained for a solid state and the temperatures of about 1300–1600 K. When the temperature is increasing from 1600 to 2100 K, the density decreases by about 0.125 g/cm3 . When melting core material, the density decreases by about 0.2–0.3 g/cm3 . Fig. 4 shows the density of

␥-Fe, FeS, Fe–FeS mixtures containing 10, 14 and 20 wt.% S and FeH in the pressure range of 20–40 GPa at an average temperature of 2100 K (for a liquid core). 3.4. Seismic velocities A seismic velocity profile in the Martian mantle may be calculated using a method described by Duffy and Anderson (1989). The third-order finite strain theory is used to calculate seismic velocities along the areotherm for a set of mantle minerals. The computation technique is described in the Appendix. In addition to the data assembled by Duffy and Anderson (1989), some new data on the adiabatic bulk modulus KS , shear modulus G and their pressure and temperature derivatives for olivine, ␤- and ␥-spinel have been used. The measured or estimated elastic properties for a set of mantle minerals are compiled in Table 3. Compressional and shear velocities of minerals as a function of depth are plotted along the Martian areotherm in Fig. 5. The difference between the results, calculated by using new data on the adiabatic bulk modulus KS , shear modulus G and their pressure derivatives for olivine, ␤- and ␥-spinel, and those of velocities, calculated with the data from Duffy and Anderson (1989), is shown in Fig. 6. As seen from Fig. 6 the difference of velocities is about 2–3%. For a liquid core, the third-order finite strain theory is used to calculate the compressional velocities for Fe–Ni alloy and FeS. Thermoelastic properties of Fe–Ni alloys at melting temperature were taken from the paper by Nasch et al. (1998) and listed in Table 3. The data used for FeS are taken from the paper (Fei et al., 1995). Then, for a solid core, the seismic velocities VP and VS can be calculated from: VPS = K + VSS

Fig. 4. Density of FeS, Fe, their mixtures, labeled in wt.% S (10, 14 and 20), and FeH as a function of pressure at 2100 K (liquid state).

9

4µ 3K (1 − σ) = 3ρ ρ (1 + σ)

µ 3K (1 − 2σ) = = ρ 2ρ (1 + σ)

(6)

The Poisson ratio σ is taken to be 0.3 for the Martian core. The velocities are shown in Fig. 7.

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Fig. 6. Finite strain trajectories for P wave (a) and S wave (b) velocities for olivine, ␤, ␥ phases (for B-F mantle profile) for the elastic properties used in this study (solid line) and the same curves for the data used by Duffy and Anderson (1989) in earlier studies (dashed line). The abbreviations are the same as in Fig. 3. Fig. 5. Finite strain trajectories for P wave (a) and S wave (b) velocities for B-F mantle profile. The abbreviations are the same as in Fig. 2, Mg-pv, Mg-rich perovskite. Fe# number in mineral assemblages is equal to 25 unless otherwise noted in parenthesis.

3.5. Modeling Assuming a spherically symmetric planet we used fundamental equations: dM = 4πr2 ρ(r) dr dp = −ρ(r)g(r) dr ρ = ρ(P, T),

(7)

where r is the radial distance from the center of the planet. The behavior of the curve ρ(P) in the crust, mantle and core is described by piece-wise polynomial functions of pressure ρ = ai pi . For the density profiles obtained from the set of differential equations (Eq. (7)), the moment of inertia is calculated:  R 8π I= ρ(r)r 4 dr. (8) 3MR2 0 In our modeling we varied the following parameters: ferric number of the mantle (Fe#), sulfur content in the core (Score ) and hydrogen content in the core (Hcore ).

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Fig. 7. Finite strain trajectories for P wave (liquid (
) and solid (s) state of the core) and S wave velocities for Fe (solid line), FeS (dot-dashed line) and an ideal mixture of Fe and FeS with 14 wt.% S (dashed line).

Core mass (Mcore ), core radius (rcore ), pressure at the core-mantle boundary, crust thickness, dimensionless value for the moment of inertia, calculated bulk Fe content, weight Fe/Si ratio and the thickness of a perovskite layer for the calculated models are listed in Table 4. The models are gathered the following way: the model calculated based on B-F mantle profile, DW core model and various crust thickness (10, 50, and 80 km) (M0); the models with different Fe# (M1, M2, M3, M4); the models with different hydrogen con-

11

tent in the core (M5, M6, M7); and the models with different sulfur content in the core (M8, M9, M10). For all these models the temperature gradient in the crust was taken to be 6 K/km. The bulk iron content of Mars ranges around 23–27 wt.%. The core mass fraction ranges from 13 to 23% as the composition is varied from pure Fe to pure FeS. The mantle represents 73–83% of the mass of the planet. The temperature at the core-mantle boundary Tcm is a fundamental parameter of a planetary interior. It determines the temperature profile in the planet. A range of values of Tcm has been proposed by different authors: 1750–1830 K (Spohn, 1991), 1600 K (Mocquet et al., 1996), 1770–1920 K (Zohl and Spohn, 1997), 1500–1900 K (Kavner et al., 2001), 1750 ◦ C (Bertka and Fei, 1997). A detailed discussion on the effect of temperature is given in our previous paper (Zharkov and Gudkova, 2000). In that paper, based on available temperature estimates, the models with mantle temperatures lower than the temperatures given by a model by Bertka and Fei (1998) by 300 K were calculated. The decrease of temperature by 300 K increased the density by a factor of 1.01. A trial model of Mars (M6) is shown in Fig. 8. In Mars, the transformation of mantle silicates to the dense perovskite phase assemblages occurs very close to its core-mantle boundary. In model M6 the Martian mantle consists of two layers: the upper mantle (1117 km depth), the transition zone (1117–1728 km depth), and a perovskite-bearing lower mantle is not

Table 4 Parameters of the models Models

Fe#, mantle

Score (wt.%)

M0

0.25 0.25 0.25

14 14 14

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10

0.20 0.22 0.24 0.25 0.22 0.22 0.22 0.22 0.22 0.22

14 14 14 14 14 14 14 0 20 36

Mcore (wt.%)

rcore (km)

Pcore (GPa)

hcrust (km)

I/MR2

Bulk Fe (wt.%)

Fe/Si ratio

"rpv (km)

0 0 0

14.3 14.4 14.2

1436 1438 1430

23.6 23.6 23.6

50 80 10

0.3671 0.3669 0.3675

23.6 23.7 23.5

1.34 1.34 1.34

83 82 86

30 30 30 30 0 50 70 0 0 0

19.3 18.0 16.8 16.2 16.2 19.6 21.7 13.2 17.6 23.3

1625 1590 1551 1532 1529 1662 1753 1327 1568 1816

21.3 21.7 22.2 22.4 22.9 20.8 19.7 25.4 22.0 18.9

50 50 50 50 50 50 50 50 50 50

0.3643 0.3656 0.3669 0.3676 0.3650 0.3662 0.3669 0.3639 0.3654 0.3674

24.5 24.6 24.7 24.8 23.5 25.6 27.0 23.3 23.3 22.9

1.48 1.46 1.45 1.44 1.36 1.55 1.68 1.31 1.38 1.46

– – – – 31 – – 202 – –

Hcore (mol%)

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Fig. 8. Distributions of density ρ, pressure P, temperature T, compressional and shear velocities as a function of radius in the M6 trial model.

present. In Table 4 there are models in which a perovskite layer is present. They are discussed in detail later. Fig. 9 shows the core radius as a function of the Martian mantle Fe# for different amount of hydrogen in the core (0–70 mol%), assuming a core composition of 14 wt.% S (according to the DW model) and a 50 km thick crust. If there is no hydrogen in the core, the Fe/Si ratio ranges from 1.34 to 1.37, and Fe# ranges from 0.26 to 0.21, respectively. The following

tendency is seen: the presence of hydrogen leads to the increase of the Fe/Si ratio and the decrease of Fe# in the mantle due to the increase of the core radius. The incorporation of 50 mol% of hydrogen into the core leads to the increase of Fe/Si ratio up to about the chondrite ratio. We have calculated a series of Martian interior models with core density profiles calculated for core compositions ranging from pure Fe (0 wt.% S) to FeS (36 wt.% S). Fig. 10 indicates the relation between the core radius, the mantle Fe#, the sulfur content in the core and the moment inertia factor. The Fe/Si ratio is lower than the chondrite ratio for any of these models. Fig. 11 shows the tradeoffs between the core radius, the mantle Fe#, the sulfur and hydrogen content in the core in determining the presence of a perovskite layer. The higher sulfur and hydrogen content in the core and the smaller mantle Fe#, the less likely a perovskite layer exists. If the Martian core contains less than 20 wt.% S, and there is no hydrogen in the core, the interior will include a perovskite-bearing lower mantle (Fig. 11a). For 14 wt.% S in the core, its thickness ranges from 0 to 150 km for Fe# 22–27. The core radius depends on its composition: at higher S abundances, the core size increases, such that the depth of the core-mantle boundary is shallower than the depth of perovskite stability. The addition of hydrogen into the core (30 and 50 mol%, Fig. 11b and c, respectively) will increase

Fig. 9. Core radius as a function of Martian mantle Fe#, assuming a core composition of 14 wt.% S and a 50 km thick crust (a), and a 100 km thick crust (b) for different amount of hydrogen in the core (0–70 mol%). Dashed lines show the lower (left) and upper (right) limits of the moment inertia factor. Fe/Si ratio is given for boundary models.

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Fig. 10. Core radius as a function of Martian mantle Fe# for a core composition ranging from 0 wt.% S (Fe-core) to 36 wt.% S (FeS core) assuming a 50 km thick crust (a) and a 100 km thick crust (b). Dashed lines show the lower (left) and upper (right) limits of the moment inertia factor. Fe/Si ratio is given for boundary models.

the core radius, and consequently decrease the thickness of the perovskite zone. For 50 mol% hydrogen in the core, the models containing less than 9 wt.% sulfur can include a perovskite-layer. We have calculated a set of models assuming the crust of 100 km thick and the temperature gradient of 13 K/km (Figs. 9b and 10b). If there is no hydrogen in the core, the Fe/Si ratio ranges from 1.34 to 1.37, as in the case of 50 km crust; but to satisfy the moment

of inertia Fe# should be increased by about 4–5% for the same kind of models (from 0.26 to 0.21 for 50 km crust (Figs. 9a and 10a) to 0.275−0.22 for 100 km crust (Figs. 9b and 10b)). The phase transition zone ␣ → ␤ (or ␣ → ␥) is of great interest. If there are no any chemical changes, it is three times wider than the same zone for the Earth. In the models, for Fe# 20–25 the width of this zone is about 55–84 km, it starts at a depth of 1082

Fig. 11. Core radius as a function of S wt.% in the core for different Martian mantle Fe# (18, 20, 22, 24, 25, 26) assuming a 50 km thick crust (a, no hydrogen in the core; b, 30 mol%; c, 50 mol% of hydrogen in the core). The horizontal lines mark the depth to the perovskite stability field, the beginning of the lower mantle. Vertical lines indicate the limits of the moment inertia factor.

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(Fe# 25)–1140 (Fe# 20) km. The density and seismic velocities of P- and S-waves increase by 0.23 g/cm3 , 0.6 and 0.4 km/s, respectively. The transition from ␤ to ␥ is poorly seen for the Earth. For Mars, it can be found at a depth of 1374–1439 km, the width is about 40 km. The density and seismic velocities of Pand S-waves increase by about 0.06 g/cm3 , 0.22 and 0.15 km/s, respectively. 4. Free oscillations 4.1. The spectrum of free oscillations The free oscillations problem for planets was described by Alterman et al. (1959) and, as applied to the case under consideration, by Gudkova and Zharkov (1996a,b). Since the model is spherically symmetric, the eigenfrequencies depend only on
(the degree) and n, the radial number (overtone number), which is equal to the number of nodes along the radius in the radial functions for torsional and spheroidal oscillations. Torsional (Fig. 12a) and spheroidal (Fig. 12b) modes have been computed for the M6 trial model. Consider the dependence of the periods of torsional fundamental modes (Fig. 13a) and spheroidal fundamental modes (Fig. 13b) on the core radius. The periods of fundamental modes are shown for a number of models for
= 2–7. It is seen that the spheroidal oscillations with
= 2–5 are quite sensitive to the core size. The values of periods increase practically linearly with the increase of the planetary core radius, a change of the core radius by 1% provides a change of the period by 1.5%. The dependence for the case of a solid core is less sharp. If the core radius changes by 1% the period only changes by 0.5%. A sharp difference in periods is observed for the cases of liquid and solid cores. The gravest normal mode is very sensitive to the state of the core. The period of the fundamental mode for a model with a liquid core is 23–30% larger than periods of the same model with a solid core, with this difference dropping to zero for
> 10. 4.2. The estimate of oscillations amplitude The level of tectonic and geological activity on Mars suggests that it should be seismically more active than

Fig. 12. Torsional (a) and spheroidal (b) modes of the Martian model M6.

the moon but less active than the Earth. Martian seismicity thought to be of tectonic origin (Golombek et al., 1992; Wilkins et al., 2002). Tectonic features on Mars are found primarily around the Tharsis region, a large elevated volcanic plateau with associated tectonic features. Mars, like the Moon, has one plate and is undergoing a thermoelastic cooling (Stevenson et al., 1983). The quakes are related to the cooling of the planet, which accumulates stresses that are then released by quakes. This type of activity is the minimum expected activity on Mars. Philips and Grimm (1991), and Solomon et al. (1991) considered that only the thermoelastic cooling of the lithosphere could generate marsquakes. They

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Fig. 13. Period as a function of core radius (
= 2–7) for different Martian models (see Table 4) with liquid cores (solid lines) and with solid cores for
= 2 (dashed line). The points denote the period values for the different models. The model numbers are given on the line for the mode
= 2.

found that more than 10 events of seismic moment greater than 1023 dyn cm, and more than 250 events of magnitude greater than 1021 dyn cm, may be expected per year. A few (2–3) per year should have a moment greater than 1024 dyn cm. A 1025 dyn cm quake is the upper bound of the estimate of the activity on Mars given by Phillips and Grimm (1991). Their estimates of seismicity are consistent with conclusions by Golombek et al. (1992). Golombek et al. (1992) have determined the seismicity on Mars based on all shear faulting visible at its surface. This estimation has been calibrated with a similar calculation for the Moon, based on all ob-

15

served grabens and mare wrinkle ridges. They have concluded that Mars is seismically active today. There have not been performed experiments on the seismicity on Mars yet. But taking into account the fact that one can see giant faults on the surface of Mars (within Tharsis region, Tempe Terra, Valles Marineris, Olimpus region), it is not possible a priori to rule out large seismic events. To judge whether the free oscillation method can be used to study the Martian interiors, it is necessary to estimate the amplitudes for different types of free oscillations during marsquakes and to determine how these amplitudes depend on focal depth and excitation processes based on the available estimates of the Martian seismic activity and the sensitivity of current instruments. The probability of detecting a seismic event on Mars and the possibility of searching normal mode frequencies were estimated by Lognonné and Mosser (1993) and Lognonné et al. (1996), based on the model by Okal and Anderson (1978). The theory for the excitation of free oscillations was presented by Dziewonski and Woodhouse (1983). Based on their paper, it is easy to write out the corresponding formulas for horizontal displacement components uN (northward) and uE (eastward) for torsional oscillations and uN , uE and a vertical component uR for spheroidal oscillations. The formulas and the technique of calculations are given in the paper by Gudkova and Zharkov (2001). Currently available broadband seismometers can measure accelerations (Lognonné et al., 1996): aN,E = −ω2 uN,E ≈ 10−8 cm/s2

(9)

Thus, the problem is to find the modes that satisfy condition (9) and to assess their diagnostic capabilities. We have calculated the amplitudes of torsional and spheroidal oscillations for sources at different depths (0–300 km) and with different focal mechanisms for the M6 model. The displacement components uN , uE and uR are proportional to the seismic moment M0 of the source. That is why, to estimate the values of the displacements for different seismic moments, they are calculated for a unit seismic moment. We have considered two possible locations of a marsquake: in Olympus region (135◦ W, 18◦ N) and in Valles Marineris (80◦ W, 5◦ S) and located a seismometer at a candidate for the landing site—“Gusev” crater (14.64◦ S, 175.06◦ E). Gusev crater has comparable

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Fig. 14. Amplitude of displacements uN (a, b) and uE (c, d) for the modes of torsional oscillations with
= 2–20 and n = 0 vs. frequency f = 1/T and degree of oscillation
for two focal mechanisms. M0 is equal to 1. The focal mechanisms are 45◦ , 45◦ , 45◦ (a, c) and 90◦ , 90◦ , 90◦ (b, d) for the dip, strike and slip angles. The seismometer coordinates are 15◦ S, 185◦ W. The epicenter coordinates are 18◦ N, 135◦ W (Olympus), the epicentral distance is 59.3◦ (open circles: a focal depth of 0.3 km and open squares: a focal depth of 300 km) and 5◦ S, 80◦ W (Valles Marineris), the epicentral distance is 103.1◦ (filled circles: a focal depth of 0.3 km and filled squares: a focal depth of 300 km).

thermal inertia, fine component thermal inertia and albedo to the Viking sites and so will likely be similar to these locations, but with fewer rocks (Golombek et al., 2002). Fig. 14 shows the amplitudes of horizontal displacements uN and uE for the fundamental tones of torsional oscillations for two different focal mechanisms. We see from Fig. 14 that the displacements of the torsional fundamental modes with
≤ 20 lie in the range of 10−27 to 10−31 cm for a unit seismic moment. For a marsquake with M0 = 1023 , 1024 , 1025 dyn cm, the amplitudes of oscillations lying above the corresponding curves are about ≥10−8 , t.e. they satisfy condition (9). And, consequently, the torsional modes with
≥ 3 (if a marsquake with M0 = 1025 dyn cm occurs),

with
≥ 6 (M0 = 1024 dyn cm), and with
≥ 12 (M0 = 1023 dyn cm) could be detected by currently available instruments. The torsional modes with
≥ 3, 6 and 12 can sound the Martian interiors down to 1600, 1100 and 700 km, respectively (Gudkova et al., 1993). The displacement amplitude for the overtones is smaller than that for the fundamental modes. The noise level on Mars was estimated by Lognonné and Mosser (1993). It can reach significant values, and in this case torsional modes will not be observed for seismic moments described above. Torsional modes will be observed if a real seismic event has a larger moment or a seismometer is placed by a penetrator deeper into the ground in order to eliminate wind effects.

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Fig. 15. Amplitude of displacements uN (a, b), uE (c, d) and uR (e, f) for the modes of spheroidal oscillations with
= 2–20 and n = 0 vs. frequency f = 1/T and degree of oscillation
for two focal mechanisms. M0 is equal to 1. See also the caption to Fig. 14.

The displacement for the spheroidal modes with
= 2–20 are shown in Fig. 15. We see that a marsquake with a seismic moment 1025 dyn cm is required for them to be detected. In this case, the spheroidal modes with only
≥ 17 could be detected by currently available instruments. The spheroidal modes with
≥ 17 can sound the outer layers of

Mars down to 700–800 km (Gudkova and Zharkov, 1996a). As Mars is closer to the asteroid belt than the Earth–Moon system, the impacts of large meteorites on the surface of Mars are more likely expected. For a marsquake with a higher seismic moment (1026 dyn cm) the spheroidal modes with
≥ 6 could

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be detected (Fig. 15). The spheroidal modes with
≥ 6 can sound the outer layers of Mars down to 2000 km (Gudkova and Zharkov, 1996a). These results are in agreement with the results obtained by Lognonné et al. (1996), who concluded that normal mode detection would be clearly successful for a 1025 dyn cm seismic moment marsquake and 10−9 ms−2 Hz−1/2 noise level and the moment may be reduced to 1024 dyn cm for a noise level of 10−10 ms−2 Hz−1/2 .

5. Discussion and conclusion Based on available chemical models of the planet (Wänke and Dreibus, 1994; Dreibus and Wänke, 1989; Lodders and Fegley, 1997; Sanloup et al., 1999; Lodders, 2000), a new set of global models of the Martian interior has been constructed. Our approach differs from those used previously in the following way: a model comprises four submodels—a model of the outer porous layer, a model of the crust, a model of the mantle and a model of the core. An actual model of the porous layer is rather complicated because of the presence of volatiles and regional features (Clifford, 1993; Babeiko and Zharkov, 2000). In this study the first 10–11 km layer is considered as an averaged transition from regolith to consolidated rock. The mineral composition of the crustal basaltic rock varies with depth because of the gabbro-eclogite phase transition. Mineralogical and seismic models of the Martian crust were constructed by numerical thermodynamic simulation by Babeiko and Zharkov (2000). As a starting point for mantle modeling we have used experimental data obtained by Bertka and Fei (1997) along the areotherm, iron content of the mantle being varied. The effect of temperature on the mantle density was discussed in detail earlier (Zharkov and Gudkova, 2000). A 300 K temperature decrease results in a density increase of about 1%. The measured or estimated up to now elastic properties for a set of mantle minerals are used (Table 3). Compressional and shear seismic velocities in the mantle and the core are calculated using a method described by Duffy and Anderson (1989). Seismic velocities determined from new high P-T data on elastic properties

are 2–3% lower than the velocities calculated earlier (see Fig. 6). Martian interior modeling is based on three chemical planetary models (Table 1). To construct interior structure models, first of all, the sulfur content in the core and Fe# in the mantle are varied, and therefore the compositions of the LF and SJC models are similar to the DW model. The DW model has a clear cosmogonical aspect. It is assumed that during the formation of the terrestrial planets, first of all Earth and Mars, due to the influence of Jupiter the mixing of the material (components A and B) from different feeding zones of growing planets took place. A fundamental hypothesis is that an oxidized component B has the same composition as C1 chondrites. This idea has been adopted in chemical models by famous geochemists for a long time (Anders et al., 1971; Ringwood, 1977; Dreibus and Wänke, 1989). Zharkov (1996) has shown that during the formation of Mars hydrogen could enter the core. The assumption that the core can contain significant amount of hydrogen is based on this hypothesis, as C1 chondrites contain large amount of water. In DW model, the water content attains in such bodies about 7.3 wt.% (Dreibus and Wänke, 1989). This value was used for the estimates of hydrogen content in the core (Zharkov (1996); and paragraph 2 in the present paper). If we take into account recent estimates of water content (18–22 wt.%) (Lodders and Fegley, 1998; Brearley and Jones, 1998), the estimates of hydrogen content in the Martian core should be increased by a factor of 2.5–3. It indicates that, in principle, the core can contain significant amount of hydrogen. As concern water content, upper mentioned chemical models are quite different. The SJC model does not contain hydrogen at all, and maximum value of hydrogen in the LF model is 7.2 times lower than the same value in the DW model. Based on our estimates of the mean value of the dissipation factor Q␮ (r) for the mantle of Mars, by using data on the secular acceleration of Phobos (Zharkov and Gudkova, 1993, 1997) we assume that the Martian core is liquid, because the values obtained for the dissipation factor are too low for the models with a solid core. The DW model is presently a subject of debate concerning its consistency with Martian interior structure models constrained by the moment of inertia provided by the Mars Pathfinder mission (Sohl and Spohn,

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1997; Bertka and Fei, 1997, 1998), i.e. the question is if the models can produce the bulk chondritic ratio Fe/Si = 1.71. Quantitative studying the effect of hydrogen in the core on planetary structure is one of the main goals of the paper. In this study we adopt the DW chemical model. The parameter values, we have varied in our modeling, are ferric number of the mantle (Fe#), sulfur and hydrogen content in the core. Figs. 9–11 show how these parameters influence the model Fe/Si ratio. It is seen, that if there is no hydrogen in the core, a model produces a Fe/Si ratio that is smaller than the chondritic value of 1.71 (Sohl and Spohn, 1997; Bertka and Fei, 1997, 1998). The presence of hydrogen in the core significantly increases the Fe/Si ratio up to about 1.7, and reduces the melting temperature of the core material. To satisfy the bulk chondritic ratio, more than 50 mol% of hydrogen must be incorporated into the core. Then, a problem of consistency of the cosmochemical DW model with the internal structure model of the planet is solved. It will confirm the idea that terrestrial planets were formed from chondritic material. This is a fundamental problem on the formation of Mars and its evolution. The determination of the core radius continues to be of great importance, in case of a reliable determination of the core radius uncertainties concerning the composition of Mars will be resolved. From cosmochemical point of view, it is difficult to assume that the core contains more than 20 wt.% of sulfur. The radius of such core is about 1600 km (Fig. 10 and 11a). Therefore, if the core of Mars turns out to be larger, hydrogen could be such an admixture element. According to numerical modeling hydrogen increases the core radius and decreases Fe# of the mantle. The decrease of Fe# leads to the increase of seismic velocities. With decreasing Fe# the phase transition zone ␣ → ␤ (or ␣ → ␥) is getting narrower. If there are no any chemical changes, it is three times wider than the same zone for the Earth. In the models, for Fe# 20–25 the width of ␣ → ␤ zone is about 55–84 km, it starts at a depth of 1082 (Fe# 25)–1140 km (Fe# 20). The density and seismic velocities of P- and S-waves increase by 0.23 g/cm3 , 0.6 and 0.4 km/s, respectively. The transition from ␤ to ␥ is poorly seen for the Earth. For Mars, it can be found at a depth of 1374–1439 km, the width is about 40 km. The density and seismic velocities of P- and S-waves

19

increase by about 0.06 g/cm3 , 0.22 and 0.15 km/s, respectively. Profiles of pressure, density, temperature and seismic velocities for a trial model M6 (the crust thickness of 50 km) is shown in Fig. 8, and the parameter values for a wide set of models are summarized in Table 4. An important feature of the Martian interior is whether or not a perovskite layer occurs. The following tendency is seen: the higher sulfur and hydrogen content in the core and the smaller Fe#, the less likely a perovskite layer exists (Fig. 11). If the Martian core contains less than 20 wt.% S, and there is no hydrogen in the core, the interior will include a perovskite-bearing lower mantle (Fig. 11a). For 14 wt.% S in the core, its thickness ranges from 0 to 150 km for Fe# 22–27. The models assuming the crust of 100 km thick are shown in Figs. 9b and 10b. In this case, to satisfy the moment of inertia Fe# must be increased by about 4–5% for the same kind of models (from 0.26 to 0.21 for 50 km thick crust to 0.275–0.22 for 100 km thick crust). In the future the Netlander mission will have a geodesy and seismic payload and it is of great importance for studying Martian interior. The second part of the paper is related to the excitation of normal modes and the possibilities of detecting such modes by future Mars missions. Since the outer layers of Mars are very heterogeneous, as in the case of the Moon, it is difficult to construct a spherically symmetric model of its interior structure (a model of a zeroth approximation) by using only few seismometers for recording seismic body waves. Data on seismic body waves will allow one to obtain the structure of the crust at the sites of the location of seismometers. In this paper we would like once more to emphasize the importance of the information on normal mode frequencies, in order to determine the very deep structure of Mars. A good installation of a broadband seismometer is mandatory to provide this information. It is found down to what depth the normal modes can sound the planetary interiors. A marsquake with a seismic moment of 1025 dyn cm is required for spheroidal oscillations (with
≥ 17) to be detected. These spheroidal modes are capable sounding the outer layers of Mars down to a depth of 700–800 km. These results are in agreement with the results obtained by Lognonné et al. (1996), who concluded that

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T.V. Gudkova, V.N. Zharkov / Physics of the Earth and Planetary Interiors 142 (2004) 1–22

normal mode detection would be clearly successful for a 1025 dyn cm seismic moment marsquake and 10−9 ms−2 Hz−1/2 noise level.

Acknowledgements This research was made possible by Grant No. 03-02-16195 from the Russian Foundation for Fundamental Research. We thank Philippe Lognonné and anonymous reviewer for their constructive and useful comments. Our manuscript was greatly improved due to their careful reviews.

Appendix A With the information available in Table 3, seismic velocities can be computed in these minerals along a trial areotherm and corresponding pressures. The procedure is first to correct for the effect of temperature by calculating the properties at the foot of an appropriate adiabat. The properties are then extrapolated adiabatically to depth using finite strain theory (Duffy and Anderson, 1989). High temperature densities are computed from: ρ∗ (T) = ρ(T0 )e

−

T

T0

α(T

)dT

,

where ρ∗ (T) and ρ(T0 ) are the potential-temperature and STP densities, respectively. The potential temperature is a temperature at the foot of an appropriate adiabat. Elastic modulus at potential temperature is determined by:   ρ∗ (T) (M)P M(T) = M(T0 ) , ρ(T0 ) if parameter (M)P is assumed to be constant with temperature. Where the term in braces is defined as:   ∂M 1 (M)P = M , α ∂T P where M is an elastic modulus, and α is the coefficient of volume expansion. The following relation for high-temperature pressure derivatives is available:

M (T) = M (T0 )e

T

T0

α(T ) dT

.

High pressure physical properties were then projected adiabatically into the mantle using third-order finite strain theory. The expressions for compressional and shear velocities can be cast into the form (Davies and Dziewonski, 1975): ρVP2 = (1 − 2ε)5/2 (L1 + L2 ε) ρVS2 = (1 − 2ε)5/2 (M1 + M2 ε)

,

where the strain ) is given by:  1
ε= 1 − (ρ/ρ∗ )2/3 , 2 ρ is the model density and VP and VS are the compressional and shear velocities, respectively. The coefficients are: M1 = G,

L1 = K + 43 G,

M2 = 5G − 3KG ,

L2 = 5(K + 43 G) − 3K(K + 43 G ) These parameters are determined from Table 4, having made appropriate temperature corrections. In mixed phase regions, velocities were calculated by volume averaging using the Voigt–Reuss–Hill procedure (Watt et al., 1976):  −1   −1 MV = νi M i ; MR = νMi ; i

MVRH

MV + MR , = 2

i

where M represents bulk modulus KS and rigidity µ, and νi is volume proportion of ith phase. References Ahrens, T.J., O’Keefe, J.D., Lange, M.A., 1989. Formation of atmospheres during accretion of the terrestrial planets. In: Attreya, S.K., Pollack, J.B., Matthews M.S. (Eds.), Origin and Evolution of Planetary and Satellite Atmospheres. Univ. Arizona Press, pp. 328–385. Akaogi, M., Kojitani, H., Matsuzaka, K., Suzuki, T., Ito, E., 1998. Postspinel transformation in the system MgSiO4 Fe2 SiO4 : element partitioning, calorimetry and thermodynamic calculation. In: Manghnani, M.H., Yagi, T. (Eds.), Properties of Earth and Planetary Materials at High Pressure and Temperature. Am. Geophys. Union, Geophys. Monogr. Ser. 101, 373–384.

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