mars: interior structure and excitation of free oscillations. physics of

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

Mars: interior structure and excitation of free oscillations

T.V. Gudkova∗, V.N. ZharkovSchmidt 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 volatileconstituents during the accretion of terrestrial planets, in: S.K. Attreya, J.B. Pollack, M.S. Matthews (Eds.), Origin and Evolu-tion 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 modelcomprises four submodels—a model of the outer porous layer, a model of the crust, a model of the mantle and a model of thecore. The first 10–11 km layer is considered as an averaged transition from regolith to consolidated rock. The mineral composi-tion of the crustal basaltic rock varies with depth because of the gabbro-eclogite phase transition. As a starting point for mantlemodeling 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 elasticproperties for a set of mantle minerals are used. Seismic velocities determined from new highP–T data on elastic properties are2–3% lower than the velocities calculated earlier. New high P-T measurements of the density of Fe (�-Fe), FeS and FeH enableus to refine the core model. Taking into account available chemical models and the fact that noticeable amount of hydrogen couldenter 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 theincrease of the Fe/Si ratio and decreases Fe# in the mantle due to the increase of the core radius. The higher sulfur and hydrogencontent in the core and the smaller mantle Fe#, the less likely a perovskite layer exists. The modeling shows that to obtain theFe/Si ratio up to the chondrite ratio of 1.71, more than 50 mol% of hydrogen should be incorporated into the core. In the secondpart of the paper, based on the available estimates of the Martian seismic activity and the sensitivity of current instruments, theamplitudes for different types of free oscillations have been estimated. It is found down to what depth the normal modes cansound 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 andfree oscillations of Mars. Since then more than 20years 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; Loddersand Fegley, 1997; Sohl and Spohn, 1997; Yoder andStandish, 1997; Bertka and Fei, 1997, 1998; Sanloupet al., 1999; Zharkov and Gudkova, 2000; Kavneret al., 2001). In most of these papers, to construct aMartian 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

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

Wänke, 1989; Wänke and Dreibus, 1994)is mainlyused. The DW model is based on the data on Mar-tian SNC meteorites, Earth’s materials and proper-ties of chondrite meteorites, and the assumption thatlow-volatile components must be in C1 chondrite ra-tios. Wänke and Dreibus assumed that during the for-mation of the terrestrial planets, first of all Earth andMars, the mixing of the material from different feed-ing zones of growing planets due to the influence ofJupiter took place (see the paper byZharkov (1993)devoted to this problem). It is considered that Marswas basically formed by two components: a highlyreduced componentA, existing mainly in the feedingzone of the growing Earth and an oxidized componentB, existing in the feeding zone of the asteroid belt.The protobodies consisting ofA component were freefrom all elements with volatility equal to or higherthan Na, however they contained all other elements inC1 abundance ratios. Fe and all siderophile elementswere metallic, and even Si might be partly presentin metallic form. The componentB contained all el-ements, including volatiles, with abundances like inC1 meteorites. Iron and all sidorophile and lithophileelements were present mainly as oxides.

Based on their estimates of bulk composition ofMars and Earth,Dreibus and Wänke (1989)came tothe conclusion that Mars was formed with 40% C1material (componentB), and 60% volatile depletedhighly reduced material (componentA). For the Earth,the content ratio of these components is 85:15. It wassuggested that Mars accreted homogeneously, unlikethe chemically inhomogeneous accretion of the Earth.

Wänke and Dreibus approach seems to be quite log-ical. The formation of Earth and Mars was most likelymany-component, and a two-component DW modelshould be considered as a reasonable averaged firstapproximation.

The two-component nature of this process shouldhave the most pronounced manifestations in the modelof the internal structure of Mars. This feature canbe revealed by seismic sounding and recording thetidal response of Mars. Simultaneous recording weakanomalies in the rotation of the planet could alsocontribute to the solution of the problem (Zharkovet al., 1996). First of all, to solve this problem, thesize and state of the core, and its mean density shouldbe determined. Up to now, geo- and cosmochemicalinformation as well as the data on the composition

of SNC-meteorites, Martian in their origin, have beenbasically used to judge the material composition ofthe Martian interior.

The successful completion of the Mars Pathfinder(MPF) mission provided a new value for the meandimensionless moment of inertia for Mars (Folkneret al., 1997):

I

MR2= A + B + C

3MR2= 0.3642− 0.3678, (1)

whereA andB are the values of the equatorial principalmoments of inertia,C is the polar principal momentof inertia,M andR are the mass and the mean radiusof the planet.

The estimate (1) put a rigid restriction on Martianinterior structure models. One of the main questionsconcerned 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 theinterior structure models of Mars taking into accountrestriction (1). For chondritous DW model the valueof the weight Fe/Si ratio is equal to 1.71. A number ofauthors (Sohl and Spohn, 1997; Yoder and Standish,1997; Bertka and Fei, 1997, 1998) concluded thatDW model did not satisfy current Martian models.Zharkov and Gudkova (2000)came to more carefulconclusion that there were not enough data to judgeabout the validity of the DW chemical model. Toclarify this problem in the future, it is necessary tomeasure the radius of the Martian core with a goodprecision. In interior models the Fe/Si ratio stronglydepends on the core radius. These questions will beconsidered lately in detail.

Oxygen isotopic�17O/�18O ratio is a fundamen-tal parameter for any terrestrial planet.Lodders andFegley (1997), Lodders (2000)and Sanloup et al.(1999)assume that Mars composition is a mixture ofdifferent meteorites, whose proportions are calculatedso that isotopic�17O/�18O ratio falls along a fraction-ation line of Martian SNC meteorites.Lodders andFegley (1997)andLodders (2000)considered a chem-ical Martian model corresponded to the accretion ofabout 85% H-chondritic, 11% CV-chondritic and 4%C1-chondritic material, and the model bySanloupet al. (1999)consisted of 55% ordinary chondrites Hand 45% enstatite chondrites EH.Table 1summarizesmodel bulk compositions for the mantle (a silicate

T.V. Gudkova, V.N. Zharkov / Physics of the Earth and Planetary Interiors 142 (2004) 1–22 3

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

DW Dreibus andWänke (1989)

MB Bertka andFei (1998)

LF Lodders andFegley (1997)

SJGSanloupet al. (1999)

SiO2 44.4 43.68 45.39 47.5TiO2 0.14 – 0.14 0.1Al2O3 3.02 3.13 2.89 2.5FeO 17.9 18.71 17.22 17.7MnO 0.46 – 0.37 0.4MgO 30.2 31.5 29.71 27.3CaO 2.45 2.49 2.35 2.0Na2O 0.5 0.5 0.98 1.2P2O5 0.16 – 0.17 1.2Cr2O3 0.76 – 0.68 –Mg/Si (mole) 1.01 1.075 0.976 0.7Fe/Si 1.709 1.766 1.684 1.709Mg# 75.0 75.0 75.5 72.0

Silicates (wt.%) 78.3 78.3 79.37 76.0

Core (wt.%) 21.7 21.7 20.63 24.0Fe 77.8 79.8 76.6Ni 8.0 8.05 7.2S 14.2 10.6 16.2Fe3P – 1.55 –

reservoir of the planet) and for the core. It also includesmodel MB composition (Bertka and Fei, 1998), theproperties of which have been studied experimentallyunder the Martian mantle (P, T) conditions. In the LFmodel (Lodders and Fegley, 1997) the sulfur contentin the core is lower than in the DW model, and inthe SJG model (Sanloup et al., 1999) it is somewhathigher than in the DW model. First of all, the sulfurcontent in the core and the magnesium number Mg#are varied (Mg#= atomic [Mg/(Mg + Fe2+)× 100],in order to construct interior structure models. Fromthis point of view, the compositions of new chemicalmodels are similar to the DW model.

The successful completion of the Mars Global Sur-veyor (MGS) mission provided the detailed topogra-phy mapping of the planet and the gravity field dataexpressed in spherical harmonics of degree 75. Thismade possible to apply geophysical methods for con-structing a model of the crust. Assuming a crustaldensity of 2.9 g/cm3 and a density contrast betweenthe crust and mantle of 0.6 g/cm3, Smith and Zuber(2002) obtained the mean thickness of the crust ofnearly 50 km. Noticeable zonal variation in the thick-ness of the crust is evident: the thinnest crust turnedout to be a few kilometers thick under the Isidis im-

pact basin and the thickest crust is about 90 km undersouth-central Tharsis.

Turcotte and Shcherbakov (2002)studied the corre-lation between topography, gravity, and areoid for theHellas impact basin, a major topographic feature onMars, its radius is about 1500 km. Assuming mantledensity of 3.5 g/cm3, the authors concluded that themean density of the crust is 2.96 ± 0.05 g/cm3 andthe reference (zero elevation) thickness of the cruston Mars is 90± 10 km. The same value was obtainedfor the averaged thickness of the elastic lithosphere,90± 10 km. It was noted that this is a reasonable re-sult and it can be explained by reological characteris-tics of the mantle and the crust (the crustal rocks arevery likely stronger than the mantle rocks beneath). Ashort, but clear review on the considered problem isgiven in their paper.

The spectral admittance (gravity/topography ratio)was calculated for typical Martian structures of differ-ent age and an elastic lithosphere thickness was deter-mined for them byMcGovern et al. (2001). It madepossible to estimate temperature gradients for the con-sidered features at time of their formation.

Somewhat different interpretation was appliedby McKenzie et al. (2002). To determine gravity


anomalies they used direct measurements of line ofsight velocity, rather than spherical harmonic co-efficients. Then, as usual, the functions of spectraladmittance were obtained for four large regions ofMars: Tharsis [(50N–50S)× (85–165)W], VallesMarineris [(25N–35S) × (10–100)W], Elysium[(60N–10S)× (115–175)E] and South Pole [(20–70)S× (180W–180E)]. When interpreting the data theyused a crustal thickness of 30 km and a mantle densityof 3.5 g/cm3. Such estimates of Martian lithosphericparameters as a wavelength band, the density of thecrust, 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, asimple one layer model of the crust was used, and thevalue of the mean density of the crust was determinedfrom the short wavelength behavior of the admittance.

The results described above are of great interest. Asit is assumed that the Martian crust predominantly hasa basaltic composition of SNC meteorites (McSween,2002) and their density is about 3.2–3.3 g/cm3, thenit is evident, that in the Martian crust the density pro-gressively changes with the depth. Therefore, as inthe case of the Moon (Wieczorek and Phillips, 1998),the use of two-layer or more complicated models ofthe crust can introduce some corrections in the givenabove results. Moreover, the results will be more valu-able, if a successful seismic experiment on Mars takesplace. It makes possible to sound the structure of theMartian crust by direct seismic methods in a certainplace and by such a way to obtain a benchmark.

At constructing interior structure models our ap-proach (Zharkov and Gudkova, 2000) differs fromones used earlier in a following way: (1) we speciallydevoted a paper (Babeiko and Zharkov, 2000) to thestudy of probable Martian crust structure. The Mar-tian crust comprises an outer porous (≈11 km) layer,where, by analogy with the Moon, there is a transi-tion from highly porous Martian regolith to consol-idated rocks. The bulk chemical composition of theconsolidated crust was accepted to be the averageof four basaltic SNC meteorites (Shergotty, Zagami,BETA 79001, lithologies A and B). The crust prop-erties (density distribution, mineralogical compositionand elastic wave velocities) were studied with thehelp of the petrological modeling technique (Sobolevand Babeiko, 1994); (2) according toZharkov (1996)

the fact that the Martian core can contain noticeableamount of hydrogen was taken into account. The pres-ence of hydrogen in the core decreases both the coredensity and its melting temperature.

Since the outer layers of Mars are very heteroge-neous, as in the case of the Moon (Jolliff et al., 2000), itis difficult to construct a spherically symmetric modelof its interior structure (a model of a zeroth approxi-mation), by using only a few seismometers for record-ing seismic body waves. First of all, they will allowto obtain the structure of the crust at the sites of theirlocation. That is why, in the second part of this paperthe excitation of free oscillations of the planet will beconsidered.

We used the estimates of the Martian seismicityfrom Philips and Grimm (1991), Golombek et al.(1992)andWilkins et al. (2002).

The tidal delay of Phobos was used to estimate thedissipation factor for the Martian mantle (Zharkovand Gudkova, 1993, 1997). Based on these estimates(the estimates of mantleQ for the models with a solidcore are too low) we assume that the Martian core ismost likely liquid. To the point, in order to signifi-cantly improve the Martian interior structure modeland concretize the distribution of such a fundamentalparameter as the dissipative factorQ�, it is sufficientto install one seismometer capable to record freeoscillations of Mars.

This paper is organized as follows. InSection 2the arguments in favor of the presence of hydrogen inthe core are given,Section 3is devoted to the interiorstructure models and seismic velocity profiles, then,the spectrum of free oscillations and their excitationare calculated inSection 4, at last inSection 5theobtained results are discussed.

2. An estimate for the hydrogen content in theMartian core

In the planetary interior the conditions for thedissolution of hydrogen in iron arise under high pres-sures and temperatures. The source of hydrogen isthe reaction:

Fe+ H2O → FeO+ H2. (2)

As gas pressure grows, hydrogen starts dissolving iniron:


Fe+(x

2

)H2 → FeHx, (3)

a bibliography on this point is given byFukai andSuzuki (1986)and Zharkov (1996). Particularly, themolecular ratio H/Fe≈ 0.2–0.4 is attained near themelting point at pressures as small as several tens ofkilobars, and the ratio H/Fe≈ 1 is attained at moder-ate temperatures and pressures of about 70 kbar. Thesolution of hydrogen in iron reduces the density andappreciably lowers the melting point. The idea thatthe Martian core may contain hydrogen, along withsulfur, as an admixture, is not a novelty (Fukai andSuzuki, 1986; Zharkov et al., 1991; Fukai, 1992).However, the effect of hydrogen on the model of theMartian core and, therefore, on the model of the planetas a whole, was discussed in detail only byZharkov(1996).

The oxidized component B, mentioned above, isassumed to have a composition of C1 carbonaceouschondrites. The water content attains in such bod-ies ∼7.3 wt.% (Dreibus and Wänke, 1989). Ahrenset al. (1989)found that consolidated minerals con-taining volatiles (CO2, H2O, and SO2) start losingthem at shock pressures within the range 30–50 GPa.Such shock pressures are achievable when the im-pactor has a velocity of∼2–3 km/s, colliding with atarget of the same mineral. Consequently, the dehy-dration of planetosimals that belong to component Bbegins when the radius of growing proto-Mars isr ∼0.4R and attains 75% atr = R (R being the radius ofMars).

To estimate the mass of H2 that can be buried in theinterior of growing proto-Mars, we putr ≈ 0.5R andρ ≈ 3.5 g/cm3. Then one easily obtains∼2.4×1023 gof hydrogen. The mass of the iron–nickel (Fe0.9Ni0.1)core comprises in model DW∼1.2 × 1026 g. There-fore, if all the buried hydrogen ultimately becomes aconstituent of the Martian core, we obtain the compo-sition (Fe0.9Ni0.1)0.9H0.1, where the subscripts meanmolecular fractions. The estimate can be considered asthe lower bound for the hydrogen content in the Mar-tian core. Under the assumption that component B hasthe same composition as C1 chondrites (≈7.3 wt.%H2O), we find that the maximum amount of hydrogenthat could be produced in the process of the accumu-lation of Mars is∼2 × 1024 g of H2. Thus, the upperbound for the hydrogen content in the iron core ofMars, not achievable in practice, is

NFe

NH= 1.2 × 1026

56× 2 × 1024≈ 1.2

1.12≈ 1.07 ⇒ FeH

The estimates given above were obtained with the dataon water content of about 7.3 wt.% in C1 chondritesfrom the paper byDreibus and Wänke (1989). If onetakes into account recent estimates of water content(18–22 wt.%) (Lodders and Fegley, 1998; Brearley andJones, 1998) the estimates of hydrogen content in theMartian core should be increased by a factor of 2.5–3.Then, the estimates of hydrogen in the core as H0.5and H0.7 look quite reasonable. On the other hand, thechemical models bySanloup et al. (1999)do not con-tain hydrogen at all, and maximum value of hydro-gen in the model byLodders and Fegley (1998)is 7.2times lower, than the same estimate used byDreibusand Wänke (1989). In the frame of the LF model theinfluence of hydrogen is negligible.

Zharkov (1996)found that addition of the molecularconcentrationx = 0.1 of hydrogen to the iron core ofMars reduces its density by about 0.16 g/cm3.

Hydrogen and other possible admixtures (Ni, S,C) significantly reduce the melting point of the Mar-tian 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). Onthe basis of these estimates, it is easy to calculatethe possible drop of the melting point of iron underthe conditions of the Martian core for the expectedtrial composition Fe0.9N0.1S0.2H0.1C0.03: −∆Tm =(0.1 + 3 × 0.22 + 2 × 0.1 + 3 × 0.03) × 103 K ≈1050 K. In model DW, the molecular concentrationof sulfur in the core is equal to 0.22. This estimateimplies that the Martian core is most likely liquid.The melting temperatures of core materials,�-Fe, FeSand eutectic Fe–FeS system are shown in the paperby Boehler (1996)and the melting temperature forFe–FeS is shown in paper byKavner et al. (2001).

3. The construction of Martian interiorstructure models

3.1. Crust

In the crust models ofBabeiko and Zharkov (2000),there is a transition in the outermost 10 km layerfrom highly porous Martian regolith (≈1.6 g/cm3)


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) correspondingto effective kinetic “freezing” temperatureTf of 800◦C. The figure starts at 10 km depth.

to consolidated rocks (3.2 g/cm3). The model of theouter porous layer is described in detail byBabeikoand Zharkov (2000). Because of the gabbro-eclogitephase transition, the increase of density with depth inthe consolidated crust depends strongly on the tem-perature gradient. The density and velocity profiles ofthe crust for different temperature gradients (2 (SL),6 (L) and 13.5 K/km (M)) are shown inFig. 1. Asseen fromFig. 1 the crust may be divided into severalzones according to the distribution of density and itsseismic-wave velocities. The maximum thickness ofthe crust in the models is determined by the depth, atwhich 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 thickcrust 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 man-tle, we use the experimental results ofBertka and Fei(1997, 1998).

Bertka and Fei (1997)performed high-pressuremulti-anvil experiments with an analog of theDreibusand Wänke (1985)composition to determine themodel mineralogy up to core-mantle boundary pres-sures along a model areotherm.

Following Bertka and Fei (1997)we consider theMartian mantle consisting mainly of 12 mineral as-

semblages. The mineral compositions and modelabundances of the high-pressure assemblages aregiven inBertka and Fei (1997). The sequences of thephase transitions in the Martian mantle are summa-rized 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 appearanceof �-spinel at 13.5 GPa, which then coexists with� phase, clinopyroxene and majorite. The transitionzone above 17 GPa consists of�-spinel and majorite.The lower mantle starts at 22.5 GPa and consists ofperovskite, magnesiowüstite and majorite. The weightfraction of each mineral assemblage is calculatedfrom the mass balance of the experimental productsobtained byBertka and Fei (1997).

These data are used to calculate the density of themantle as a function of pressure and temperature witha Birch–Murnaghan equation of state:

P = 3f(1 + 2f)5/2KT (1 − 32(4 − K′

T )f), (4)

where

ρ∗ = ρ0 exp

(−∫ T

T0

α(T)dT

),

f = 1

2

((ρ

ρ∗

)2/3

− 1

),

ρ0 is the STP density,ρ∗ the density atP = 0 andthe temperature corresponding to the areotherm and


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 equalto 25 unless otherwise noted in parenthesis. The abbreviationsare 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–Murnaghanequation of state,KT is the isothermal bulk modulus,K′T = ∂KT /∂P , andα(T) is the coefficient of thermal

expansion.The density of the mixture of mineral assemblages

is calculated in the approximation of additive com-ponent volumes. The database for the end-memberphases used in these calculations is given inBertkaand Fei (1998).

The calculated mantle density profile (hereinafterreferred to as B-F mantle profile) and density profilesfor different mineral assemblages are shown inFig. 2.When varying the iron content of the mantle silicates,we changed the mantle density profile. We added�ρi

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

Fe#

16 18 20 22 24 25 26 28 30

P1 14.0 13.8 13.6 13.36 13.12 13.0 12.9 12.7 12.5P2 14.45 14.35 14.25 14.15 14.05 14.0 13.94 13.82 13.7P3 17.85 17.55 17.25 16.95 16.65 16.5 16.3 15.9 15.5P4 18.35 18.05 17.75 17.45 17.15 17.0 16.8 16.4 16.0

Note: P1-� → � + �(� + �); P2-� + �(� + �) → �; P3-� → (� + �); P4-(� + �) → �.

Fig. 3. Martian mantle density profiles with different iron contentin the mantle silicates: Fe# 25 (B-F mantle profile) (solid line),Fe# 18 (dot-dashed line), and Fe# 28 (dashed line).

to the B-F mantle density profile (Fe# 25), when cal-culating the models with lower and higher iron con-tent (Fig. 3). The radial positions of high-pressurephase transitions in the Martian mantle are determinedfrom the paper byBertka and Fei (1997)and listedin Table 2. The mantle densities can be calculated ac-cording to the Fe# number of the model using empiri-cal mineral densities as a function of their iron content(Table 3). When increasing Fe# by 1�ρ is increasedby 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 forperovskite zone.

3.3. Core

The Martian core composition is considered tobe sulfur-rich, consisting of Fe with 14.2 wt.% S,7.6 wt.% Ni (Dreibus and Wänke, 1985).

New highP–T measurements of the density of Fe(�-Fe) and FeS (Kavner et al., 2001) enable us to refinethe core model byZharkov (1996).

8T.V.

Gudkova,

V.N.

Zharkov

/Physics

ofthe

Earth

andP

lanetaryInteriors

142(2004)

1–22

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

Formula (abbreviation, name) Density (g/cm3) KS (GPa) G (GPa) K′S G′ KS (GPa/K) G (GPa/K) α0 (300 K) 10−6 K−1

(Mg, Fe)2SiO4 (ol, olivine) 3.222+ 1.18XFe 129 + 9XFe 82 − 31XFe 4.2 1.4 0.017 0.014 26.6(Mg, Fe)2SiO4 (�, �-spinel) 3.472+ 1.24XFe 170 + 15XFe 114 − 41XFe 4.3 1.4 0.018 0.014 22.0(Mg, Fe)2SiO4 (�, �-spinel) 3.564+ 1.285XFe 186 + 15XFe 124 − 41XFe 4.1 1.3 0.021 0.016 21.0(Mg, Fe)SiO3 (opx, orthopyroxene) 3.204+ 0.798XFe 104 77− 24XFe 5.0 2.0 0.012 0.011 27.0(Mg, Fe)2Si2O6 (Mg-cpx, clinopyroxene) 3.188+ 0.817XFe 114 77− 24XFe 5.0 2.0 0.012 0.011 27.0Ca(Mg, Fe)Si2O6 (Ca-cpx, clinopyroxene) 3.280+ 0.376XFe 113 + 7XFe 67 − 6XFe 4.5 1.7 0.013 0.010 27.0(Mg, Fe)3Al2Si3O12 (Mg-gt, garnet) 3.566+ 0.746XFe 175 + 1XFe 90 + 8XFe 4.9 1.4 0.021 0.010 18.0Ca3Al2Si3O12 (Ca-gt, garnet) 3.6 169 104 4.9 1.6 0.016 0.015 16.0Mg4Si4O12 (Mg-mj, majorite) 3.518 175 90 4.9 1.4 0.021 0.010 20.0(Mg, Fe)SiO3 (Mg-pv, perovskite) 4.107+ 1.047XFe 266 153 3.9 2.0 0.031 0.028 17.0CaSiO3 (Ca-pv, perovskite) 4.252 227 125 3.9 1.9 0.027 0.023 17.0(Mg, Fe)O (mw, magnesiowüstite) 3.584+ 2.281XFe 163 − 8XFe 131 − 77XFe 4.2 2.5 0.016 0.024 32.510% Ni:Fe–Ni [T = 1800◦C] 7.03 105 – 4.5 – 0.025 – 75FeS [T = 800◦C] 4.94 54 – 4.0 – 0.020 – 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;K′

S, 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 coefficientof 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).


In this study the Martian core is assumed to be amixture of iron–nickel alloy, sulfur and some amountof hydrogen.

The compressibility of FeH is described by the Vinetequation:

p = 3KT0

(ρ

ρ0

)2/3[

1 −(ρ0

ρ

)1/3

× exp

{3

2(K′

T0 − 1)

[1 −

(ρ0

ρ

)1/3]}]

, (5)

in which the zero-isotherm values of the parametersareKT0 = 121± 19 GPa,K′

T0 = 5.31 ± 0.9, ρ0 =6.7 g/cm3 (Bading et al., 1992).

The mixture of�-Fe and FeS is calculated in theapproximation of additive component volumes. Theweight concentrationsXS of sulfur andXFeSof FeS arerelated by the equationXS = 0.36XFeS. In DW model,XS = 0.14; accordingly,XFeS≈ 0.4. The addition of10 mol% of hydrogen to the iron reduces its densityby 0.16 g/cm3 (Zharkov, 1996).

Experimental data of high-PT phases of�-Fe andFeS byKavner et al. (2001)were obtained for a solidstate and the temperatures of about 1300–1600 K.When the temperature is increasing from 1600 to2100 K, the density decreases by about 0.125 g/cm3.When melting core material, the density decreasesby about 0.2–0.3 g/cm3. Fig. 4 shows the density of

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 (liquidstate).

�-Fe, FeS, Fe–FeS mixtures containing 10, 14 and20 wt.% S and FeH in the pressure range of 20–40 GPaat an average temperature of 2100 K (for a liquidcore).

3.4. Seismic velocities

A seismic velocity profile in the Martian mantlemay be calculated using a method described byDuffyand Anderson (1989).

The third-order finite strain theory is used to cal-culate seismic velocities along the areotherm for a setof mantle minerals. The computation technique is de-scribed in the Appendix.

In addition to the data assembled byDuffy andAnderson (1989), some new data on the adiabatic bulkmodulusKS, shear modulusG and their pressure andtemperature derivatives for olivine,�- and �-spinelhave been used. The measured or estimated elasticproperties for a set of mantle minerals are compiledin Table 3.

Compressional and shear velocities of mineralsas a function of depth are plotted along the Martianareotherm inFig. 5.

The difference between the results, calculated byusing new data on the adiabatic bulk modulusKS,shear modulusG and their pressure derivatives forolivine, �- and�-spinel, and those of velocities, calcu-lated with the data fromDuffy and Anderson (1989),is shown inFig. 6. As seen fromFig. 6 the differenceof velocities is about 2–3%.

For a liquid core, the third-order finite strain the-ory is used to calculate the compressional velocitiesfor Fe–Ni alloy and FeS. Thermoelastic properties ofFe–Ni alloys at melting temperature were taken fromthe paper byNasch et al. (1998)and listed inTable 3.The data used for FeS are taken from the paper (Feiet al., 1995).

Then, for a solid core, the seismic velocitiesVP andVS can be calculated from:

VPS = K + 4

3

µ

ρ= 3K

ρ

(1 − σ)

(1 + σ)

VSS = µ

ρ= 3K

2ρ

(1 − 2σ)

(1 + σ)

(6)

The Poisson ratioσ is taken to be 0.3 for the Martiancore. The velocities are shown inFig. 7.


Fig. 5. Finite strain trajectories for P wave (a) and S wave (b)velocities for B-F mantle profile. The abbreviations are the sameas in Fig. 2, Mg-pv, Mg-rich perovskite. Fe# number in mineralassemblages is equal to 25 unless otherwise noted in parenthesis.

3.5. Modeling

Assuming a spherically symmetric planet we usedfundamental equations:

dM

dr= 4πr2ρ(r)

dp

dr= −ρ(r)g(r)

ρ = ρ(P, T),

(7)

Fig. 6. Finite strain trajectories for P wave (a) and S wave (b)velocities for olivine, �, � phases (for B-F mantle profile) forthe elastic properties used in this study (solid line) and the samecurves for the data used byDuffy and Anderson (1989)in earlierstudies (dashed line). The abbreviations are the same as inFig. 3.

wherer is the radial distance from the center of theplanet.

The behavior of the curveρ(P) in the crust, mantleand core is described by piece-wise polynomial func-tions of pressureρ = ∑

aipi.

For the density profiles obtained from the set ofdifferential equations (Eq. (7)), the moment of inertiais calculated:

I = 8π

3MR2

∫ R

0ρ(r)r4 dr. (8)

In our modeling we varied the following parameters:ferric number of the mantle (Fe#), sulfur content in thecore (Score) and hydrogen content in the core (Hcore).


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 with14 wt.% S (dashed line).

Core mass (Mcore), core radius (rcore), pressure at thecore-mantle boundary, crust thickness, dimensionlessvalue for the moment of inertia, calculated bulk Fecontent, weight Fe/Si ratio and the thickness of a per-ovskite layer for the calculated models are listed inTable 4. The models are gathered the following way:the model calculated based on B-F mantle profile, DWcore model and various crust thickness (10, 50, and80 km) (M0); the models with different Fe# (M1, M2,M3, M4); the models with different hydrogen con-

Table 4Parameters of the models

Models Fe#,mantle

Score

(wt.%)Hcore

(mol%)Mcore

(wt.%)rcore

(km)Pcore

(GPa)hcrust

(km)I/MR2 Bulk Fe

(wt.%)Fe/Siratio

"rpv

(km)

M0 0.25 14 0 14.3 1436 23.6 50 0.3671 23.6 1.34 830.25 14 0 14.4 1438 23.6 80 0.3669 23.7 1.34 820.25 14 0 14.2 1430 23.6 10 0.3675 23.5 1.34 86

M1 0.20 14 30 19.3 1625 21.3 50 0.3643 24.5 1.48 –M2 0.22 14 30 18.0 1590 21.7 50 0.3656 24.6 1.46 –M3 0.24 14 30 16.8 1551 22.2 50 0.3669 24.7 1.45 –M4 0.25 14 30 16.2 1532 22.4 50 0.3676 24.8 1.44 –M5 0.22 14 0 16.2 1529 22.9 50 0.3650 23.5 1.36 31M6 0.22 14 50 19.6 1662 20.8 50 0.3662 25.6 1.55 –M7 0.22 14 70 21.7 1753 19.7 50 0.3669 27.0 1.68 –M8 0.22 0 0 13.2 1327 25.4 50 0.3639 23.3 1.31 202M9 0.22 20 0 17.6 1568 22.0 50 0.3654 23.3 1.38 –M10 0.22 36 0 23.3 1816 18.9 50 0.3674 22.9 1.46 –

tent in the core (M5, M6, M7); and the models withdifferent sulfur content in the core (M8, M9, M10).For all these models the temperature gradient in thecrust was taken to be 6 K/km. The bulk iron content ofMars ranges around 23–27 wt.%. The core mass frac-tion ranges from 13 to 23% as the composition is var-ied from pure Fe to pure FeS. The mantle represents73–83% of the mass of the planet.

The temperature at the core-mantle boundaryTcm isa fundamental parameter of a planetary interior. It de-termines the temperature profile in the planet. A rangeof values ofTcm has been proposed by different au-thors: 1750–1830 K (Spohn, 1991), 1600 K (Mocquetet al., 1996), 1770–1920 K (Zohl and Spohn, 1997),1500–1900 K (Kavner et al., 2001), 1750◦C (Bertkaand Fei, 1997). A detailed discussion on the effect oftemperature is given in our previous paper (Zharkovand Gudkova, 2000). In that paper, based on availabletemperature estimates, the models with mantle temper-atures lower than the temperatures given by a modelby Bertka and Fei (1998)by 300 K were calculated.The decrease of temperature by 300 K increased thedensity by a factor of 1.01.

A trial model of Mars (M6) is shown inFig. 8.In Mars, the transformation of mantle silicates to thedense perovskite phase assemblages occurs very closeto its core-mantle boundary. In model M6 the Mar-tian mantle consists of two layers: the upper mantle(1117 km depth), the transition zone (1117–1728 kmdepth), and a perovskite-bearing lower mantle is not


Fig. 8. Distributions of densityρ, pressureP, temperatureT,compressional and shear velocities as a function of radius in theM6 trial model.

present. InTable 4there are models in which a per-ovskite layer is present. They are discussed in detaillater.

Fig. 9 shows the core radius as a function of theMartian mantle Fe# for different amount of hydrogenin the core (0–70 mol%), assuming a core composi-tion of 14 wt.% S (according to the DW model) anda 50 km thick crust. If there is no hydrogen in thecore, the Fe/Si ratio ranges from 1.34 to 1.37, and Fe#ranges from 0.26 to 0.21, respectively. The following

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 a100 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.

tendency is seen: the presence of hydrogen leads tothe 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 thecore leads to the increase of Fe/Si ratio up to aboutthe chondrite ratio.

We have calculated a series of Martian interior mod-els with core density profiles calculated for core com-positions ranging from pure Fe (0 wt.% S) to FeS(36 wt.% S).Fig. 10indicates the relation between thecore radius, the mantle Fe#, the sulfur content in thecore and the moment inertia factor. The Fe/Si ratio islower than the chondrite ratio for any of these models.

Fig. 11shows the tradeoffs between the core radius,the mantle Fe#, the sulfur and hydrogen content in thecore in determining the presence of a perovskite layer.The higher sulfur and hydrogen content in the coreand the smaller mantle Fe#, the less likely a perovskitelayer exists.

If the Martian core contains less than 20 wt.% S,and there is no hydrogen in the core, the interior willinclude a perovskite-bearing lower mantle (Fig. 11a).For 14 wt.% S in the core, its thickness ranges from0 to 150 km for Fe# 22–27. The core radius dependson its composition: at higher S abundances, the coresize increases, such that the depth of the core-mantleboundary is shallower than the depth of perovskite sta-bility. The addition of hydrogen into the core (30 and50 mol%,Fig. 11b and c, respectively) will increase


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 momentinertia factor. Fe/Si ratio is given for boundary models.

the core radius, and consequently decrease the thick-ness of the perovskite zone. For 50 mol% hydrogen inthe core, the models containing less than 9 wt.% sulfurcan include a perovskite-layer.

We have calculated a set of models assuming thecrust of 100 km thick and the temperature gradient of13 K/km (Figs. 9b and 10b). If there is no hydrogen inthe core, the Fe/Si ratio ranges from 1.34 to 1.37, asin the case of 50 km crust; but to satisfy the moment

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 thickcrust (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 perovskitestability field, the beginning of the lower mantle. Vertical lines indicate the limits of the moment inertia factor.

of inertia Fe# should be increased by about 4–5% forthe same kind of models (from 0.26 to 0.21 for 50 kmcrust (Figs. 9a and 10a) to 0.275−0.22 for 100 kmcrust (Figs. 9b and 10b)).

The phase transition zone� → � (or � → �) is ofgreat interest. If there are no any chemical changes,it is three times wider than the same zone for theEarth. In the models, for Fe# 20–25 the width of thiszone is about 55–84 km, it starts at a depth of 1082


(Fe# 25)–1140 (Fe# 20) km. The density and seismicvelocities 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 canbe found at a depth of 1374–1439 km, the width isabout 40 km. The density and seismic velocities of P-and S-waves increase by about 0.06 g/cm3, 0.22 and0.15 km/s, respectively.

4. Free oscillations

4.1. The spectrum of free oscillations

The free oscillations problem for planets was de-scribed byAlterman et al. (1959)and, as applied to thecase under consideration, byGudkova and Zharkov(1996a,b). Since the model is spherically symmet-ric, the eigenfrequencies depend only on� (the de-gree) andn, the radial number (overtone number),which is equal to the number of nodes along the ra-dius in the radial functions for torsional and spheroidaloscillations.

Torsional (Fig. 12a) and spheroidal (Fig. 12b)modes have been computed for the M6 trial model.

Consider the dependence of the periods of torsionalfundamental modes (Fig. 13a) and spheroidal funda-mental modes (Fig. 13b) on the core radius. The pe-riods of fundamental modes are shown for a numberof models for� = 2–7. It is seen that the spheroidaloscillations with� = 2–5 are quite sensitive to thecore size. The values of periods increase practicallylinearly with the increase of the planetary core radius,a change of the core radius by 1% provides a changeof the period by 1.5%. The dependence for the case ofa solid core is less sharp. If the core radius changes by1% the period only changes by 0.5%. A sharp differ-ence in periods is observed for the cases of liquid andsolid cores. The gravest normal mode is very sensitiveto the state of the core. The period of the fundamentalmode for a model with a liquid core is 23–30% largerthan periods of the same model with a solid core, withthis difference dropping to zero for� > 10.

4.2. The estimate of oscillations amplitude

The level of tectonic and geological activity on Marssuggests that it should be seismically more active than

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

the moon but less active than the Earth. Martian seis-micity thought to be of tectonic origin (Golombeket al., 1992; Wilkins et al., 2002). Tectonic featureson Mars are found primarily around the Tharsis re-gion, a large elevated volcanic plateau with associatedtectonic features. Mars, like the Moon, has one plateand is undergoing a thermoelastic cooling (Stevensonet al., 1983). The quakes are related to the cooling ofthe planet, which accumulates stresses that are thenreleased by quakes. This type of activity is the mini-mum expected activity on Mars.

Philips and Grimm (1991), and Solomon et al.(1991)considered that only the thermoelastic coolingof the lithosphere could generate marsquakes. They


Fig. 13. Period as a function of core radius (� = 2–7) for differentMartian models (seeTable 4) with liquid cores (solid lines) andwith solid cores for� = 2 (dashed line). The points denote theperiod values for the different models. The model numbers aregiven on the line for the mode� = 2.

found that more than 10 events of seismic momentgreater than 1023 dyn cm, and more than 250 events ofmagnitude greater than 1021 dyn cm, may be expectedper year. A few (2–3) per year should have a mo-ment greater than 1024 dyn cm. A 1025 dyn cm quakeis the upper bound of the estimate of the activity onMars given by Phillips and Grimm (1991). Their es-timates of seismicity are consistent with conclusionsby Golombek et al. (1992).

Golombek et al. (1992)have determined the seis-micity on Mars based on all shear faulting visible atits surface. This estimation has been calibrated witha similar calculation for the Moon, based on all ob-

served grabens and mare wrinkle ridges. They haveconcluded that Mars is seismically active today.

There have not been performed experiments onthe seismicity on Mars yet. But taking into accountthe fact that one can see giant faults on the surfaceof Mars (within Tharsis region, Tempe Terra, VallesMarineris, Olimpus region), it is not possible a priorito rule out large seismic events.

To judge whether the free oscillation method canbe used to study the Martian interiors, it is necessaryto estimate the amplitudes for different types of freeoscillations during marsquakes and to determine howthese amplitudes depend on focal depth and excitationprocesses based on the available estimates of the Mar-tian seismic activity and the sensitivity of current in-struments. The probability of detecting a seismic eventon Mars and the possibility of searching normal modefrequencies were estimated byLognonné and Mosser(1993)andLognonné et al. (1996), based on the modelby Okal and Anderson (1978).

The theory for the excitation of free oscillationswas presented byDziewonski and Woodhouse (1983).Based on their paper, it is easy to write out thecorresponding formulas for horizontal displacementcomponentsuN (northward) anduE (eastward) fortorsional oscillations anduN, uE and a vertical com-ponentuR for spheroidal oscillations. The formulasand the technique of calculations are given in thepaper byGudkova and Zharkov (2001).

Currently available broadband seismometers canmeasure accelerations (Lognonné et al., 1996):

aN,E = −ω2uN,E ≈ 10−8 cm/s2 (9)

Thus, the problem is to find the modes that satisfy con-dition (9) and to assess their diagnostic capabilities.

We have calculated the amplitudes of torsional andspheroidal oscillations for sources at different depths(0–300 km) and with different focal mechanisms forthe M6 model. The displacement componentsuN, uEanduR are proportional to the seismic momentM0 ofthe source. That is why, to estimate the values of thedisplacements for different seismic moments, they arecalculated for a unit seismic moment.

We have considered two possible locations of amarsquake: in Olympus region (135◦W, 18◦N) and inValles Marineris (80◦W, 5◦S) and located a seismome-ter at a candidate for the landing site—“Gusev” crater(14.64◦S, 175.06◦E). Gusev crater has comparable


Fig. 14. Amplitude of displacementsuN (a, b) anduE (c, d) for the modes of torsional oscillations with� = 2–20 andn = 0 vs. frequencyf = 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 andalbedo to the Viking sites and so will likely be similarto these locations, but with fewer rocks (Golombeket al., 2002).

Fig. 14shows the amplitudes of horizontal displace-ments uN and uE for the fundamental tones of tor-sional oscillations for two different focal mechanisms.We see fromFig. 14that the displacements of the tor-sional fundamental modes with� ≤ 20 lie in the rangeof 10−27 to 10−31 cm for a unit seismic moment. Fora marsquake withM0 = 1023, 1024, 1025 dyn cm, theamplitudes of oscillations lying above the correspond-ing curves are about≥10−8, t.e. they satisfy condition(9). And, consequently, the torsional modes with� ≥3 (if a marsquake withM0 = 1025 dyn cm occurs),

with � ≥ 6 (M0 = 1024 dyn cm), and with� ≥ 12(M0 = 1023 dyn cm) could be detected by currentlyavailable instruments. The torsional modes with� ≥3, 6 and 12 can sound the Martian interiors down to1600, 1100 and 700 km, respectively (Gudkova et al.,1993). The displacement amplitude for the overtonesis smaller than that for the fundamental modes.

The noise level on Mars was estimated byLognonnéand Mosser (1993). It can reach significant values, andin this case torsional modes will not be observed forseismic moments described above. Torsional modeswill be observed if a real seismic event has a largermoment or a seismometer is placed by a penetratordeeper into the ground in order to eliminate wind ef-fects.


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

The displacement for the spheroidal modes with� = 2–20 are shown inFig. 15. We see that amarsquake with a seismic moment 1025 dyn cm isrequired for them to be detected. In this case, thespheroidal modes with only� ≥ 17 could be detectedby currently available instruments. The spheroidalmodes 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 theEarth–Moon system, the impacts of large meteoriteson 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


be detected (Fig. 15). The spheroidal modes with� ≥ 6 can sound the outer layers of Mars down to2000 km (Gudkova and Zharkov, 1996a).

These results are in agreement with the resultsobtained byLognonné et al. (1996), who concludedthat normal mode detection would be clearly suc-cessful for a 1025 dyn cm seismic moment marsquakeand 10−9 ms−2 Hz−1/2 noise level and the momentmay be reduced to 1024 dyn cm for a noise level of10−10 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 theMartian interior has been constructed.

Our approach differs from those used previ-ously in the following way: a model comprises foursubmodels—a model of the outer porous layer, amodel of the crust, a model of the mantle and a modelof the core.

An actual model of the porous layer is rathercomplicated because of the presence of volatilesand regional features (Clifford, 1993;Babeiko andZharkov, 2000). In this study the first 10–11 km layeris considered as an averaged transition from regolithto consolidated rock. The mineral composition of thecrustal basaltic rock varies with depth because of thegabbro-eclogite phase transition. Mineralogical andseismic models of the Martian crust were constructedby numerical thermodynamic simulation byBabeikoand Zharkov (2000).

As a starting point for mantle modeling we haveused experimental data obtained byBertka and Fei(1997)along the areotherm, iron content of the mantlebeing varied. The effect of temperature on the man-tle density was discussed in detail earlier (Zharkovand Gudkova, 2000). A 300 K temperature decreaseresults in a density increase of about 1%. The mea-sured or estimated up to now elastic properties for aset of mantle minerals are used (Table 3). Compres-sional and shear seismic velocities in the mantle andthe core are calculated using a method described byDuffy and Anderson (1989). Seismic velocities deter-mined from new high P-T data on elastic properties

are 2–3% lower than the velocities calculated earlier(seeFig. 6).

Martian interior modeling is based on three chem-ical planetary models (Table 1). To construct interiorstructure models, first of all, the sulfur content in thecore and Fe# in the mantle are varied, and thereforethe compositions of the LF and SJC models are sim-ilar to the DW model. The DW model has a clearcosmogonical aspect. It is assumed that during theformation of the terrestrial planets, first of all Earthand Mars, due to the influence of Jupiter the mixingof the material (components A and B) from differentfeeding zones of growing planets took place. A fun-damental hypothesis is that an oxidized componentB has the same composition as C1 chondrites. Thisidea has been adopted in chemical models by famousgeochemists for a long time (Anders et al., 1971;Ringwood, 1977; Dreibus and Wänke, 1989). Zharkov(1996)has shown that during the formation of Marshydrogen could enter the core. The assumption thatthe core can contain significant amount of hydrogenis based on this hypothesis, as C1 chondrites containlarge amount of water. In DW model, the water con-tent attains in such bodies about 7.3 wt.% (Dreibus andWänke, 1989). This value was used for the estimatesof hydrogen content in the core (Zharkov (1996); andparagraph 2 in the present paper). If we take into ac-count recent estimates of water content (18–22 wt.%)(Lodders and Fegley, 1998; Brearley and Jones, 1998),the estimates of hydrogen content in the Martian coreshould be increased by a factor of 2.5–3. It indicatesthat, in principle, the core can contain significantamount of hydrogen. As concern water content, up-per mentioned chemical models are quite different.The SJC model does not contain hydrogen at all, andmaximum value of hydrogen in the LF model is 7.2times lower than the same value in the DW model.

Based on our estimates of the mean value of the dis-sipation factorQ�(r) for the mantle of Mars, by usingdata on the secular acceleration of Phobos (Zharkovand Gudkova, 1993, 1997) we assume that the Mar-tian core is liquid, because the values obtained for thedissipation factor are too low for the models with asolid core.

The DW model is presently a subject of debateconcerning its consistency with Martian interior struc-ture models constrained by the moment of inertia pro-vided by the Mars Pathfinder mission (Sohl and Spohn,


1997; Bertka and Fei, 1997, 1998), i.e. the questionis if the models can produce the bulk chondritic ratioFe/Si= 1.71.

Quantitative studying the effect of hydrogen in thecore on planetary structure is one of the main goalsof the paper. In this study we adopt the DW chemi-cal model. The parameter values, we have varied inour modeling, are ferric number of the mantle (Fe#),sulfur and hydrogen content in the core.Figs. 9–11show how these parameters influence the model Fe/Siratio. It is seen, that if there is no hydrogen in thecore, a model produces a Fe/Si ratio that is smallerthan the chondritic value of 1.71 (Sohl and Spohn,1997; Bertka and Fei, 1997, 1998). The presenceof hydrogen in the core significantly increases theFe/Si ratio up to about 1.7, and reduces the meltingtemperature of the core material. To satisfy the bulkchondritic ratio, more than 50 mol% of hydrogenmust be incorporated into the core. Then, a problemof consistency of the cosmochemical DW model withthe internal structure model of the planet is solved.It will confirm the idea that terrestrial planets wereformed from chondritic material. This is a funda-mental problem on the formation of Mars and itsevolution.

The determination of the core radius continues to beof great importance, in case of a reliable determinationof the core radius uncertainties concerning the compo-sition of Mars will be resolved. From cosmochemicalpoint of view, it is difficult to assume that the core con-tains more than 20 wt.% of sulfur. The radius of suchcore is about 1600 km (Fig. 10 and 11a). Therefore, ifthe core of Mars turns out to be larger, hydrogen couldbe such an admixture element. According to numer-ical modeling hydrogen increases the core radius anddecreases Fe# of the mantle. The decrease of Fe# leadsto the increase of seismic velocities. With decreasingFe# the phase transition zone� → � (or � → �) isgetting 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� → � zoneis about 55–84 km, it starts at a depth of 1082 (Fe#25)–1140 km (Fe# 20). The density and seismic ve-locities of P- and S-waves increase by 0.23 g/cm3, 0.6and 0.4 km/s, respectively. The transition from� to �is poorly seen for the Earth. For Mars, it can be foundat a depth of 1374–1439 km, the width is about 40 km.The density and seismic velocities of P- and S-waves

increase by about 0.06 g/cm3, 0.22 and 0.15 km/s,respectively.

Profiles of pressure, density, temperature and seis-mic velocities for a trial model M6 (the crust thicknessof 50 km) is shown inFig. 8, and the parameter valuesfor a wide set of models are summarized inTable 4.An important feature of the Martian interior is whetheror not a perovskite layer occurs. The following ten-dency is seen: the higher sulfur and hydrogen contentin the core and the smaller Fe#, the less likely a per-ovskite layer exists (Fig. 11). If the Martian core con-tains less than 20 wt.% S, and there is no hydrogen inthe core, the interior will include a perovskite-bearinglower mantle (Fig. 11a). For 14 wt.% S in the core, itsthickness ranges from 0 to 150 km for Fe# 22–27.

The models assuming the crust of 100 km thick areshown inFigs. 9b and 10b. In this case, to satisfy themoment of inertia Fe# must be increased by about4–5% for the same kind of models (from 0.26 to 0.21for 50 km thick crust to 0.275–0.22 for 100 km thickcrust).

In the future the Netlander mission will have ageodesy and seismic payload and it is of great impor-tance for studying Martian interior. The second part ofthe paper is related to the excitation of normal modesand the possibilities of detecting such modes by futureMars missions.

Since the outer layers of Mars are very heteroge-neous, as in the case of the Moon, it is difficult toconstruct a spherically symmetric model of its interiorstructure (a model of a zeroth approximation) by us-ing only few seismometers for recording seismic bodywaves. Data on seismic body waves will allow one toobtain the structure of the crust at the sites of the lo-cation of seismometers.

In this paper we would like once more to emphasizethe importance of the information on normal modefrequencies, in order to determine the very deep struc-ture of Mars. A good installation of a broadband seis-mometer is mandatory to provide this information.

It is found down to what depth the normal modescan sound the planetary interiors. A marsquake witha seismic moment of 1025 dyn cm is required forspheroidal oscillations (with� ≥ 17) to be detected.These spheroidal modes are capable sounding theouter layers of Mars down to a depth of 700–800 km.These results are in agreement with the results ob-tained byLognonné et al. (1996), who concluded that


normal mode detection would be clearly successfulfor a 1025 dyn cm seismic moment marsquake and10−9 ms−2 Hz−1/2 noise level.

Acknowledgements

This research was made possible by Grant No.03-02-16195 from the Russian Foundation for Fun-damental Research. We thank Philippe Lognonné andanonymous reviewer for their constructive and usefulcomments. Our manuscript was greatly improved dueto their careful reviews.

Appendix A

With the information available inTable 3, seismicvelocities can be computed in these minerals along atrial areotherm and corresponding pressures. The pro-cedure is first to correct for the effect of temperatureby calculating the properties at the foot of an appropri-ate adiabat. The properties are then extrapolated adia-batically to depth using finite strain theory (Duffy andAnderson, 1989).

High temperature densities are computed from:

ρ∗(T) = ρ(T0)e− ∫ TT0

α(T ′′)dT ′,

whereρ∗(T) andρ(T0) are the potential-temperatureand STP densities, respectively.

The potential temperature is a temperature at thefoot of an appropriate adiabat.

Elastic modulus at potential temperature is deter-mined by:

M(T) = M(T0)

(ρ∗(T)ρ(T0)

)(M)P

,

if parameter (M)P is assumed to be constant with tem-perature. Where the term in braces is defined as:

(M)P = 1

αM

(∂M

∂T

)P

,

whereM is an elastic modulus, andα is the coefficientof volume expansion.

The following relation for high-temperature pres-sure derivatives is available:

M ′(T) = M ′(T0)e∫ TT0

α(T ′)dT ′.

High pressure physical properties were then projectedadiabatically into the mantle using third-order finitestrain theory. The expressions for compressional andshear velocities can be cast into the form (Davies andDziewonski, 1975):

ρV 2P = (1 − 2ε)5/2(L1 + L2ε)

ρV 2S = (1 − 2ε)5/2(M1 + M2ε)

,

where the strainε is given by:

ε = 1

2

(1 − (ρ/ρ∗)2/3

),

ρ is the model density andVP andVS are the com-pressional and shear velocities, respectively. The co-efficients are:

M1 = G, L1 = K + 43G, M2 = 5G − 3KG′,

L2 = 5(K + 43G) − 3K(K′ + 4

3G′)

These parameters are determined fromTable 4, havingmade appropriate temperature corrections. In mixedphase regions, velocities were calculated by volumeaveraging using the Voigt–Reuss–Hill procedure (Wattet al., 1976):

MV =∑i

νiMi; MR =(∑

i

νM−1i

)−1

;

MVRH = MV + MR

2,

whereM represents bulk modulusKS and rigidityµ,andνi is volume proportion ofith phase.

References

Ahrens, T.J., O’Keefe, J.D., Lange, M.A., 1989. Formation ofatmospheres during accretion of the terrestrial planets. In:Attreya, S.K., Pollack, J.B., Matthews M.S. (Eds.), Origin andEvolution of Planetary and Satellite Atmospheres. Univ. ArizonaPress, pp. 328–385.

Akaogi, M., Kojitani, H., Matsuzaka, K., Suzuki, T., Ito,E., 1998. Postspinel transformation in the system MgSiO4-Fe2SiO4: element partitioning, calorimetry and thermodynamiccalculation. In: Manghnani, M.H., Yagi, T. (Eds.), Propertiesof Earth and Planetary Materials at High Pressure andTemperature. Am. Geophys. Union, Geophys. Monogr. Ser.101, 373–384.


Alterman, Z., Jarosch, H., Pekeris, C.L., 1959. Oscillations of theEarth. Proc. R. Soc. A252 (N1268), 80–95.

Anders, E.R., Ganapathy, R., Keays, R.R., et al., 1971. Volatile andsiderophile elements in lunar rocks: comparison with terrestrialand meteoritic basalts. Proc. Lunar Sci. Conf. 2, 1021–1036.

Babeiko, A.Yu., Zharkov, V.M., 2000. Martian crust: a modelingapproach. Phys. Earth Planet. Inter. 117, 421–435.

Bading, J.V., Mao, H.K., Hemley, R.J., 1992. High pressure crystalstructure and equation of state of iron hydride: implicationsfor the Earth’s core. In: Syono, Y., Manghnani, M.H. (Eds.),High-Pressure Research: Application to Earth and PlanetarySciences. Am. Geophys. Union, Tokyo, Terra, Washington, DC,pp. 363–371.

Bass J.D., 1995. Elasticity of minerals, glasses and melts. In:Ahrens, T.J. (Ed.), Mineral Physics and Crystallography: aHandbook of Physical Constants, vol. 2, pp. 45–63.

Bertka, C.M., Fei, Y., 1997. Mineralogy of the Martian interiorup to core-mantle boundary pressures. J. Geophys. Res. 102,5251–5264.

Bertka, C.M., Fei, Y., 1998. Density profile of an SNC modelMartian interior and the moment-of-inertia factor of Mars. EarthPlanet. Sci. Lett. 157, 79–88.

Boehler, R., 1996. Melting temperature of the Earth’s mantle andcore: Earth’s thermal structure. Annu. Rev. Earth Planet. Sci.24, 15–40.

Brearley, A.J., Jones, R.H., 1998. Chondritic meteorites. In: Pepike,J.J. (Ed.), Reviews in Mineralogy, vol. 36. Mineralogical Societyof America, Washington, DC.

Davies, G.F., Dziewonski, A.M., 1975. Homogeneity andconstitution of the Earth’s lower mantle and outer core. Phys.Earth. Planet. Inter. 10, 336–343.

Dreibus, G., Wänke, H., 1985. Mars, a volatile-rich planet.Meteoritics 20, 367–381.

Dreibus, G., Wänke, H., 1989. Supply and loss of volatileconstituents during the accretion of terrestrial planets. In:Attreya, S.K., Pollack, J.B., Matthews, M.S. (Eds.), Originand Evolution of Planetary and Satellite Atmospheres. Univ.Arizona Press, pp. 268–288.

Duffy, T.S., Anderson, D.L., 1989. Seismic velocities in mantleminerals and the mineralogy of the upper mantle. J. Geophys.Res. 94, 1895–1912.

Duffy, T.S., Zha, C.-S., Downs, R.T., Mao, H.-K., Hemley, R.J.,1995. Elasticity of forsterite to 16 GPa and the composition ofthe upper mantle. Nature 378, 170–173.

Dziewonski, A.M., Woodhouse, J.H., 1983. Studies of theseismic source using normal-mode theory. In: Proceedings ofthe International School of Physics Enrico Fermi, vol. 85,pp. 45–137.

Fei, Y., Mao, H.K., Shu, J., Parthasarathy, G., Bassett, W.A., 1992.Simultaneous high-P, high-T X-ray diffraction study of (�-(Mg,Fe)2SiO4 to 26 GPa and 900 K. J. Geophys. Res. 97, 4489–4495.

Fei Y., 1995. Thermal expansion. In: Ahrens, T.J. (Ed.), Handbookof Physical Constants, Am. Geophys. Union, Washington, DC,pp. 29–44.

Fei, Y., Prewitt, C.T., Mao, H.K., Bertka, C.M., 1995. Structureand density of FeS at high pressure and high temperature andthe internal structure of Mars. Science 268, 1892–1894.

Folkner, W.M., Yoder, C.F., Yuan, D.N., Standish, E.M., Preston,R.A., 1997. Interior structure and season mass redistributionof Mars from radio tracking of Mars Pathfinder. Science 278,749–1751.

Fukai, Y., Suzuki, T., 1986. The iron-water reaction under highpressure and its implication in the evolution process of theEarth. J. Geophys. Res. 91, 9229–9230.

Fukai, Y., 1992. Some properties of the Fe-H system athigh pressures and temperatures, and their implications forthe Earth’s core. In: Syono, Y., Manghnani, M.H. (Eds.),High-Pressure Research: Application to Earth and PlanetarySciences. Terra Scientific, Tokyo, pp. 373–385.

Golombek, M.P., Banerdt, W.B., Tanaka, K.L., Tralli, D.M., 1992.A prediction of Mars seismicity from surface faulting. Science258 (N 5084), 979–981.

Golombek, M.P., Grant, J., Parker, T., Schofield, T., Kass, D.,Knocke, P., Roncoli, R., Bridges, N., Anderson, S., 2002.Downselection of landing sites for the Mars exploration rovers.Abstr. Lunar Planet. Sci. Conf. XXXIII, 1245.

Gudkova, T.V., Zharkov, V.N., Lebedev, S., 1993. The theoreticalspectrum of free oscillations of Mars. Solar Syst. Res. 27, 129–148.

Gudkova, T.V., Zharkov, V.N., 1996a. The exploration of Martianinteriors using the spheroidal oscillation method. Planet. SpaceSci. 44, 1223–1230.

Gudkova, T.V., Zharkov, V.N., 1996b. On investigation of Martiancrust structure using the free oscillation method. Planet. SpaceSci. 44, 1231–1236.

Gudkova, T.V., Zharkov, V.N., 2001. On the excitation of freeiscillations on the Moon. Astronomy Lett. 27 (N10), 658–670.

Jolliff, B.L., Gillis, J.J., Haskin, L., Korotev, R.L., Wieczorek,M.A., 2000. Major lunar crustal terranes: surface expressionsand crust-mantle origins. J. Geophys. Res. 105, 4197–4216.

Kamaya, N., Ohtani, E., Kato, T., Onuma, K., 1993. High pressurephase transitions in a homogeneous model Martian mantle.Geophys. Monogr. 74, IUGG, Am. Geophys. Union 14, 19–25.

Kavner, A., Duffy, T.S., Shen, G., 2001. Phase stability and densityof FeS at high pressures and temperatures: implications for theinterior structure of Mars. Earth Planet. Sci. Lett. 185, 25–33.

Li, B., Chen, G., Gwanmesia, G.D., Liebermann, R.C., 1998.Sound velocity measurements at mantle transition zoneconditions of pressure and temperature using ultrasonicinterferometry in a multianvil apparatus. In: Manghnani, M.H.,Yagi, T. (Eds.), Properties of Earth and Planetary Materialsat High Pressure and Temperature. Am. Geophys. Union,Geophys. Monogr. Ser. 101, 41–61.

Lodders, K., Fegley, B., 1997. An oxygen isotope model for thecomposition of Mars. Icarus 126, 373–394.

Lodders, K., Fegley, B., 1998. The Planetary Scientist’sCompanion. Oxford University Press.

Lodders, K., 2000. An oxygen isotope mixing models for theaccretion and composition of rocky planets. Space Sci. Rev.92, 341–354.

Lognonné, P., Mosser, B., 1993. Planetary seismology. Surv.Geophys. 14, 239–302.

Lognonné, Ph., Beyneix, J.G., Banerdt, W.B., Cacho, S.,Karczewski, J.F., Morand, M., 1996. Ultra broad band


seismology on InterMarsNet. Planet. Space Sci. 44 (N 11),1237–1249.

McGovern, P.J., Solomon, S.C., Smith, D.E., Zuber, M.T.,Neumann, G.A., Head, J.W., Phillips, R.J., Simons, M., 2001.Gravity/topography admittances and lithospheric evolution onMars: the importance of finite-amplitude topography. LunarPlanet. Sci. Conf. 32, #1804.pdf.

McKenzie, D., Barnett, D.N., Yuan, D.-N., 2002. The relationshipbetween Martian gravity and topography. Earth Planet. Sci.Lett. 195, 1–16.

McSween, H.Y., 2002. Basalt or andesite? A critical evalution ofconstraints on the composition of the ancient Martian crust.Lunar Planet. Sci. Conf. 33, #1062.pdf.

Mocquet, A., Vacher, P., Grasset, O., Sotin, C., 1996. Theoreticalseismic models of Mars: the importance of the iron content ofthe mantle. Planet Space Sci. 44, 1251–1268.

Nasch, P.M., Manghnani, M.H., 1998. Molar volume, thermalexpansion, and bulk modulus in liquid Fe-Ni alloys at 1 bar:evidence for magnetic anomalies? In: Manghnani, M.H., Yagi,T. (Eds.), Properties of Earth and Planetary Materials at HighPressure and Temperature. Am. Geophys. Union, Geophys.Monogr. Ser. 101, 307–317.

Okal, E.A., Anderson, D.L., 1978. Theoretical models for Marsand their seismic properties. Icarus 33, 514–528.

Philips, R.J., Grimm, R.E., 1991. Martian seismicity. Abstr. LunarPlanet. Sci. Conf. XXIII, 1061.

Ringwood, A.E., 1977. Composition of the core and implicationsfor origin of the Earth. Geochim. J. 11, 11–135.

Sanloup, C., Jambon, A., Gillet, P., 1999. A simple chondriticmodel of Mars. Phys. Earth Planet. Inter. 112, 43–54.

Sinogeikin, S.V., Bass, J.D., Katsura, T., 2001. Single-crystalelasticity of �-(Mg0.91Fe0.09)2SiO4 to high pressures and tohigh temperatures. Geophys. Res. Lett. 28, 4335–4338.

Smith, D.E., Zuber, M.T., 2002. The crustal thickness ofMars. Accuracy and resolution. Lunar Planet. Sci. Conf. 33,#1893.pdf.

Sobolev, S.V., Babeiko, A.Y., 1994. Modeling of mineralogicalcomposition, density and elastic wave velocities in theanhydrous rocks. Surv. Geophys. 15, 515–544.

Sohl, F., Spohn, T., 1997. The interior structure of Mars:implications from SNC meteorites. J. Geophys. Res. 102, 1613–1635.

Solomon, S.C., Anderson, D.L., Banerdt, W.B., Butler, R.G., Davis,P.M., Duennebier, F.K., Nakamura, Y., Okal, E.A., Phillips,R.G., 1991. Scientific rationale and requirements for a globalseismic network on Mars. LPI Tech. Rpt. 91-02, Lunar andPlanetary institute, Houston, 51 pp.

Spohn, T., 1991. Mantle differentiation and thermal evolution ofMars, Mercury and Venus. Icarus 90, 222–236.

Stevenson, D.J., Spohn, T., Schubert, G., 1983. Magnetism andthermal evolution of the terrestrial planets. Icarus 54, 466–489.

Turcotte, D.L., Shcherbakov, R., 2002. Is the Martian crust alsothe Martian elastic lithosphere? Lunar Planet. Sci. Conf. 33,#1001.pdf.

Wänke, H., Dreibus, G., 1994. Chemistry and accretion history ofMars. Philos. Trans. R. Soc. London 349, 285–293.

Watt, J.P., Davies, G.F., O’Connell, R.J., 1976. The elasticproperties of composite materials. Rev. Geophys. Space Phys.14 (N 4), 541–561.

Wieczorek, M.A., Phillips, R.J., 1998. Potential anomalies ona sphere: applications to the thickness of the lunar crust. J.Geophys. Res. 103, 1715–1724.

Wilkins, S.J., Schultz, R.A., Anderson, R.C., Dohm, J.M., Dawers,N.H., 2002. Deformation rates from faulting at the TempeTerra extensional province, Mars. Geophys. Res. Lett. 29, 31-1–31-4.

Yoder, C.F., Standish, E.M., 1997. Martian precessian and rotationfrom Viking lander range data. J. Geophys. Res. 102, 4065–4080.

Zha, C.-.S., Duffy, T.S., Downs, R.T., Mao, H., Hemley, R.J.,Weidner, D.J., 1998. Single-crystal elasticity of the� and� ofMg2SiO4 polymorphs at high pressure. In: Manghnani, M.H.,Yagi, T. (Eds.), Properties of Earth and Planetary Materialsat High Pressure and Temperature. Am. Geophys. Union,Geophys. Monogr. Ser. 101, 9–16.

Zharkov, V.N., Koshlyakov, E.M., Marchenkov, K.I., 1991.Composition, structure and gravitational field of Mars. SolarSyst. Res. 25, 515–547.

Zharkov, V.N., 1993. The role of Jupiter in the formation of planets.Geophys. Monogr. 74, IUGG, Am. Geophys. Union 14, 7–17.

Zharkov, V.N., 1996. The internal structure of Mars: a key tounderstanding the origin of terrestrial planets. Solar Syst. Res.30, 456–465.

Zharkov, V.N., Molodensky, S.M., Brzezinski, A., Groten, E.,Varga, P., 1996. The Earth and its rotation. Wichmann,Heidelberg, vol. XIII, 501 pp.

Zharkov, V.N., Gudkova, T.V., 1993. Dissipative factor of theinteriors of Mars. Solar Syst. Res. 27, 3–15.

Zharkov, V.N., Gudkova, T.V., 1997. On the dissipative factor ofthe Martian interiors. Planet. Space Sci. 45, 401–407.

Zharkov, V.N., Gudkova, T.V., 2000. Interior structure models,Fe/Si ratio and parameters of figure for Mars. Phys. EarthPlanet. Inter. 117, 407–420.

mars: interior structure and excitation of free oscillations. physics of

Documents