JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

A geophysical perspective on the bulk composition of Mars

A. Khan1, C. Liebske2, A. Rozel1, A. Rivoldini3, F. Nimmo4, J.A.D. Connolly2,

A.-C. Plesa5, D. Giardini1

Abstract. We invert the Martian tidal response and mean mass and moment of iner-tia for chemical composition, thermal state, and interior structure. The inversion com-bines phase equilibrium computations with a laboratory-based viscoelastic dissipationmodel. The rheological model, which is based on measurements of anhydrous and melt-free olivine, is both temperature and grain size sensitive and imposes strong constraintson interior structure. The bottom of the lithosphere, defined as the location where theconductive geotherm meets the mantle adiabat, occurs deep within the upper mantle (∼250–500 km depth) resulting in apparent upper mantle low-velocity zones. Assuming an Fe-FeS core, our results indicate: 1) a Mantle with a Mg# (molar Mg/Mg+Fe) of ∼0.75in agreement with earlier geochemical estimates based on analysis of Martian meteorites;2) absence of bridgmanite- and ferropericlase-dominated basal layer; 3) core composi-tions (13.5–16 wt% S), core radii (1640–1740 km), and core-mantle-boundary temper-atures (1560–1660 ◦C) that, together with the eutectic-like core compositions, suggestthe core is liquid; and 4) bulk Martian compositions that are overall chondritic with aFe/Si (wt ratio) of 1.63–1.68. We show that the inversion results can be used in tandemwith geodynamic simulations to identify plausible geodynamic scenarios and parameters.Specifically, we find that the inversion results are reproduced by stagnant lid convectionmodels for a range of initial viscosities (∼1019–1020 Pa·s) and radioactive element par-titioning between crust and mantle around 0.001. The geodynamic models predict a meansurface heat flow between 15–25 mW/m2.

1. Introduction

Knowledge of the internal constitution of the planets iscrucial to our understanding of the origin and evolution ofour solar system. Major constraints can be placed on plan-etary accretion, differentiation, and mantle evolution fromknowledge of bulk chemical composition [e.g., Taylor , 1999].By far the largest insights into the physical structure of theEarth have come from geophysical analyses, and seismologyin particular. However, the dearth of geophysical data perti-nent to the interior of other planets has made this approachless instructive and a significant part of current knowledgeon mantle and bulk composition of the terrestrial planetsderives from geochemical/cosmochemical and isotopic anal-yses of rocks and primitive solar system material [e.g., Ring-wood , 1979; Taylor , 1980; Drake and Righter , 2002; Palmeand O’Neill , 2003; Taylor et al., 2006]. In addition, geode-tic data in the form of Doppler observations obtained fromranging to orbiting and landed spacecraft (Viking, MarsPathfinder, Mars Global Surveyor, Mars Odyssey, and MarsReconnaissance Orbiter) over more than a decade, resultedin the recognition that Mars had differentiated into a silicatemantle and an Fe-rich core [e.g., Folkner et al., 1997; Yoderet al., 2003; Neumann et al., 2004; Bills et al., 2005; Laineyet al., 2007; Konopliv et al., 2006, 2011, 2016; Genova et al.,2016].

1Institute of Geophysics, ETH Zurich, Switzerland.2Institute of Geochemistry and Petrology, ETH Zurich,

Switzerland.3Royal Observatory of Belgium, Brussels, Belgium.4Department of Earth and Planetary Sciences, UC Santa

Cruz, California, USA5German Aerospace Center (DLR), Berlin, Germany.

Copyright 2017 by the American Geophysical Union.0148-0227/17/$9.00

For Mars, an increasing amount of observations, both insitu and from laboratory analyses of Martian meteorites andcosmochemical material, have become available [e.g., Nor-man, 1999; Taylor , 2013]. In addition, data and resultsfrom geophysical modeling and mantle convection studiesthat bear on interior structure [e.g., Sohl and Spohn, 1997;Yoder et al., 2003; Neumann et al., 2004; Wieczorek andZuber , 2004; Verhoeven et al., 2005; Khan and Connolly ,2008; Rivoldini et al., 2011; Baratoux et al., 2014; Hauckand Phillips, 2002; Elkins-Tanton et al., 2003; Williams andNimmo, 2004; Grott and Breuer , 2008; Kiefer and Li , 2009;Ruedas et al., 2013a; Rai and Westrenen, 2013; Plesa et al.,2015] have allowed us to refine our understanding of plane-tary processes from a Martian vantage point; yet much re-mains to be understood. Among others, how well do wereally know the composition of Mars and what is the re-lation of this to core size and state and how can currentestimates be improved? Does Mars contain the terrestrialequivalent of a lower mantle layer and what is its role inthe evolution of the core? In the broader context of planetformation, what are the implications of current bulk planetcompositions for mixing of material throughout the innersolar system during accretion of the terrestrial planets?

The aim of this study is to improve current constraintson Mars bulk composition and thermal state from inversionof currently available geophysical data (mean mass and mo-ment of inertia, global tidal dissipation, and magnitude oftidal response). To this end we will build upon our previouswork [e.g., Khan et al., 2007] to 1) invert different geophysi-cal data sets directly for compositional and thermal parame-ters, and combine this with 2) a method for computing tidaldissipation within a planet using the laboratory-based grainsize and frequency dependent viscoelastic model of Jacksonand Faul [2010], which was shown by Nimmo and Faul [2013]to be highly sensitive to mantle temperatures in Mars. Themain point is to link the dissipation model, which is basedon laboratory experiments on anhydrous melt-free polycrys-talline olivine, with thermodynamic phase equilibrium com-putations in order to self-consistently compute geophysical

1

X - 2 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

responses that can be compared directly to observations.This approach has a number of advantages: (1) It anchorstemperature, composition, dissipation, and discontinuitiesthat are in laboratory-based forward models; (2) it per-mits the simultaneous use of geophysical inverse methodsto optimize profiles of physical properties (e.g., shear mod-ulus, dissipation, density) to match geophysical data; and(3) it is capable of making quantitative predictions that canbe tested with the upcoming Mars Insight mission to belaunched in May 2018 [Banerdt et al., 2013] (InSight willemplace a seismometer, a heat flow probe, and a geode-tic experiment on Mars) and/or against results and datafrom other studies (e.g., geodynamical simulations of planetevolution, petrological analyses of Martian meteorites, andorbit-imaged surface chemistry and crustal thickness).

As a means of illustrating these points, we show thatthe inversion results can be used in combination with geo-dynamic simulations to identify plausible geodynamic sce-narios and parameters. This coupling of geodynamics andgeophysics has the advantage that it anchors geodynamicmodels in geophysically-constrained results. These simula-tions, based on the StagYY code [Tackley , 2008], explicitlyconsider grain size evolution and suggest that stagnant lidconvection is capable of explaining the various observables(crustal and lithospheric thickness and present-day grainsize and mantle temperatures) assuming reasonable initialestimates of viscosity, radioactive element partitioning, andinitial temperature field. The models are also able to pre-dict the present-day mean surface heat flow, which can becompared to surface observations to be made with InSight[e.g., Plesa et al., 2016].

In the following we discuss constraints that derive fromgeochemical and cosmochemical analyses and summarizeprevious geophysical analyses that bear on the interior ofMars (section 2); enumerate the geophysical data employedin the present analysis and detail the computation of crust,mantle, and core properties (sections 3) and numerical mod-eling aspects of solving the forward and inverse problemposited here (section 4); and, finally, describe and discussresults.

2. Background

2.1. Geochemical perspective

Our knowledge on the chemistry of the Martian mantleand core originates to a large extent from the chemical andisotopic compositions of a class of basaltic meteorites thatare believed to be fragments ejected from the Martian sur-face on meteoritic impacts and collectively known as SNCmeteorites (Shergottites, Nahklites and Chassignites) [e.g.,McSween, 1985, 1994], hereinafter referred to as Martianmeteorites. A key argument for this hypothesis is the con-centration of entrapped gases within shergottite meteoritesthat match the values measured for the Martian atmosphereby the Viking mission [Bogard et al., 2001]. The Martianmeteorites also exhibit relatively young crystallisation ages[Stolper and McSween, 1979]. Shergottites have ages in therange of 170–600 Ma [McSween and McLennan, 2014, andreferences therein], which require a sufficiently large planetas parent body because it is physically implausible, dueto thermal constraints, to maintain magmatic activity onasteroid-sized bodies so late in the solar systems history.

Different lines of arguments have been used to derive thechemical composition of the Martian mantle and core fromthe chemical compositions of the Martian meteorites. Thesecan be categorized in two general groups that consider 1) theabundance of refractory elements in the Martian meteorites[Dreibus and Wanke, 1984; Dreibus and Wanke, 1985; Hal-liday et al., 2001; Longhi et al., 1992; Taylor , 2013] and 2)oxygen isotope systematics [Lodders and Fegley , 1997; San-loup et al., 1999; Burbine and O’Brien, 2004]. Common to

both approaches is the notion that Mars is considered tohave accreted from different material which condensed fromthe solar nebular, including highly volatile-depleted and re-duced components and oxidized, volatile-rich condensates.

Exemplary of the first line of arguments was the approachof Dreibus and Wanke (hereafter DW model) [Dreibusand Wanke, 1984; Dreibus and Wanke, 1985; Dreibus andWanke, 1987]. The DW model has become the standardmodel for Mars and has served as reference in many sub-sequent studies [e.g., Bertka and Holloway , 1994; Bertkaand Fei , 1997; Matsukage et al., 2013; Collinet et al., 2015].The essence of this model lies in the assumption that Marsaccreted heterogeneously from two different cosmochemi-cal reservoirs, i.e., a highly reduced component during theearly stages, followed by the addition of oxidized, volatile-rich material during the final stages of accretion. The re-duced component is assumed to have CI-chondritic elementabundances that are more refractory than Mn, i.e., ele-ments having higher condensation temperatures than thiselement, whereas the late stage oxidized component is en-tirely CI-chondritic. Thus, a central tenet inherent of theDW model is that refractory elements follow CI-chondriticproportions. Chemical analyses of the Martian meteoritesknown at that time indicated that their Mn content is closeto CI-chrondritic; thus this element became key to the bulkchemical composition of Mars. To derive the composition ofthe mantle, fractionation trends that occur during igneousprocesses are invoked, which allow conclusions about theShergottites source region to be drawn. Recently, Taylor[2013] re-assessed the primitive mantle composition with asimilar strategy as DW using a much larger meteoritic record(∼60 versus 6). For the elements of interest here (majorand minor but no trace elements), the Taylor bulk silicateMars model is almost identical in terms of elemental con-centrations to that of DW. However, there is considerabledifference in the assumption of the S content of the planet.In the model of Taylor [2013], the core is significantly moreS-rich compared to DW.

Morgan and Anders [1979] derived a chemical composi-tion using a similar methodology as DW, i.e., to scale theconcentration of “unknown” to “known” elements (or knownelemental ratios) based on volatility trends. It should benoted that Morgan and Anders [1979] developed their modelbefore the general acceptance that Martian meteorites orig-inated from Mars and e.g., their “known” K/U ratio wastaken from the Soviet orbiter Mars 5 mission [Surkov , 1977]with several corrections applied based on various assump-tions. As pointed out by Taylor et al. [2006], the value ofK/U and K/Th as a proxy for the proportions of moder-ately volatile versus refractory elements was later corrected.This likely explains some systematic differences between theMorgan and Anders composition and other estimates.

As an illustration of the second approach to derive theMartian bulk composition, is to consider the oxygen isotopicsystematics of the Martian meteorites [e.g., Lodders and Feg-ley , 1997; Sanloup et al., 1999; Mohapatra and Murty , 2003].Oxygen is by far the most abundant element in the silicateproportions of the terrestrial planets and the Martian mete-orites form a fractionation line in a δ17O/δ18O three-isotopeplot which is distinctively different from terrestrial or othermeteoritic trends. The aforementioned two studies are basedon the assumptions that Mars is the parent body of the Mar-tian meteorites and that their isotopic compositions can bedescribed by mixing different classes of meteorites. Basedon this, the oxygen isotope composition of Mars are deducedfrom mass balancing various meteoritic end-members. Theconcentrations of all other elements are then simply definedby the chemical composition and mass fraction of the var-ious meteorite end-member components. Core and silicate

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 3

mantle compositions are finally deduced from the total oxy-gen content and redox equilibria, which leaves a proportionof Fe and the majority of Ni and S in the metallic core.

The differences in the models of Lodders and Fegley [1997]and Sanloup et al. [1999] are determined by the choices ofend-member components (meteorite classes) but generallylead to comparable results for major and minor elements.Lodders and Fegley [1997] invoked three end-members 1)mean values of reduced and volatile- depleted H- (ordinary)chondrites; 2) moderately oxidized CV-chondrites; and 3)highly oxidized and volatile-rich CI-chondrites. These end-members span a compositional triangle in the three-isotopeδ17O/δ18O diagram around the Martian meteorites. Themodel of Sanloup et al. [1999] (EH45:H55) is based on a mix-ture of two end-members, a mean value of highly reducedEH- (enstatite) chondrites (45%) and a hypothetical, but asyet un-sampled H- (ordinary) chondrite component (55%)that is located on an isotopic fractionation line along withLL-, L-, and H-chondrites. This hypothetical end-memberis required to produce a mixing line with the EH-chondritecomponent that passes through the Martian meteorite oxy-gen isotope mean value.

2.2. Summary of geochemical models

To investigate the effect of varying bulk chemical com-positions of Mars, we have selected the five possible bulkcompositions discussed above. We consider the composi-tions proposed by 1) Dreibus and Wanke (1984) and 2)Taylor (2013) as representative of the CI refractory ele-ment approach but with different core Fe/S ratios; 3) thecompositions of Lodders and Fegley (1997) and 4) Sanloup(1999) (EH45:H55) as representative of the oxygen isotopeapproach; and 5) the composition of Morgan and Anders[1979] despite the fact that some of the key elemental ra-tios in that model have been revised. Nevertheless, thismodel is distinctively different in core composition by beingsignificantly depleted in S by a factor of three to five com-pared to other models. Table 1 summarises the five modelcompositions. To facilitate treatment, some simplificationsto the proposed mantle and core compositions are made;the mantle is considered as a six component system (MgO,CaO, FeO, MgO, Al2O3, SiO2, and Na2O) to be compati-ble with the thermodynamic database (section 3.3) and thecore-system (section 3.5) is reduced to a simple Fe-S binary.

Differences in the modified bulk silicate Mars composi-tions (table 1) become apparent when considering elemen-tal ratios such as Mg/Si (molar). Mg/Si ranges from 1.07(near CI-chondritic) to 0.86 between the studies of Morganand Anders [1979] and Sanloup et al. [1999]. These latterstudies also show the widest spread in Mg-number (molarMg/Mg+Fe), which ranges from 0.77 to 0.73. Mg/Si affectsthe proportions of olivine (and its polymorphs) and pyrox-enes (and garnet), thereby influencing the properties of themantle. Very similar results in terms of major element ratiosare observed for the models of Dreibus and Wanke [1984] andLodders and Fegley [1997]. The oxygen isotope mass balancemethod predicts higher alkali concentration, but this is un-likely to influence calculated mantle phase proportions. Inregard to trace and volatile elements, differences betweenthe five models are more pronounced. From a geophysicalpoint of view, however, these differences are insignificant.We should note that the model compositions shown in ta-ble 1 are input values only, i.e., mantle and core compositionare parameters to be determined in the inversion. This willbe described in more detail below.

Finally, Borg and Draper [2003] and Agee and Draper[2004] have suggested that the Martian mantle may havea higher Mg# than the range of compositions describedabove. However, an alternative bulk Mars composition, re-porting elemental concentrations for the core and a highMg#, i.e., low-FeO, bearing mantle has, to our knowledge,

yet to be formulated. The possibility also exists that FeOis not homogeneously distributed throughout the mantle, asindicated by systematic differences in the surface distribu-tion of Fe detected by the gamma ray spectrometer onboardMars Odessey [Taylor et al., 2006]. The questions whetheralternative bulk Mars and potentially low-FeO mantle mod-els would satisfy the physical properties of Mars and whetherthe mantle is heterogeneous with regard to the distributionof major elements are beyond the scope of this paper.

2.3. Geophysical perspective

Geophysical analyses have, for the most part, relied on re-sults obtained from geochemical-cosmochemical studies withthe purpose of predicting the geophysical response of thesechemically-derived models. Geophysical and experimentalapproaches are to a large extent based on the DW modelcomposition with the goal of determining mantle mineralogyand density. Combined with equation-of-state (EOS) mod-eling this allows for determination of a model density pro-file for the purpose of making geophysical predictions thatcan be subsequently compared to observations. The studiesby Bertka and Fei [1997, 1998] represents the experimentalapproach, while numerical approaches with varying degreeof sophistication (forward/inverse modeling, number of geo-physical observations, parameterized phase diagram/phase-equilibrium computations) are embodied in the studies ofLonghi et al. [1992]; Kuskov and Panferov [1993]; Mocquetet al. [1996]; Sohl and Spohn [1997]; Sohl et al. [2005]; Ver-hoeven et al. [2005]; Zharkov and Gudkova [2005]; Khan andConnolly [2008]; Rivoldini et al. [2011]; Wang et al. [2013].Based on the available geophysical data at the time (prin-cipally the moment of inertia), Bertka and Fei [1998], forexample, concluded that the CI chondrite accretion modelfor deriving Mars is incompatible with a DW-like mantlecomposition.

According to the DW model there is evidence for chal-colphile element depletion in the Martian meteorites, whichsuggests that the otherwise Fe-Ni-rich core contains a sub-stantial amount of a sulfide component (S need not be theonly alloying element; Zharkov and Gudkova [2005], for ex-ample, considered H in addition to S in the core). Thisobservation is important because an alloying element actsto influence the physical state, while simultaneously provid-ing information on core temperature. In this context, thereis strong evidence from measurements of the deformation ofthe planet due to solar tides that Mars’ core or parts of itare currently liquid [e.g., Yoder et al., 2003; Konopliv et al.,2011; Genova et al., 2016] as had been predicted earlier [e.g.,Lognonne and Mosser , 1993; Zharkov and Gudkova, 1997].Recent experimental studies of phase relations in the Fe-Sand (Fe,Ni)-S systems at pressure and temperature condi-tions relevant for the core of Mars (1927 ◦C and 40 GPa)also point to an entirely liquid core at present Stewart et al.[2007]; Rivoldini et al. [2011]. Core size, state, and com-position are uncertain with current estimates ranging from1550–1800 km in radius, 5.9–7.5 g/cm3 in density, and 64–90 wt% FeNi and 10–36 wt% S in composition [c.f., table5 in Khan and Connolly , 2008, but see also Rivoldini et al.[2011]].

Another important issue directly related to core size is thepresence or absence of a lower mantle in Mars, i.e., a man-tle dominated by bridgmanite-structure silicates. Figure 1shows that at pressures above ∼23 GPa the major lowermantle minerals ferriperoclase and bridgmanite stabilise.However, the stability of these silicates depends stronglyon temperature and pressure conditions at the core-mantleboundary (CMB). Large cores will result in a lower mantlethat is either thin or absent altogether; whereas higher tem-peratures will tend to stabilize bridgmanite at lower pres-sures (see figure 1 and discussion below). The existence of a

X - 4 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

bridgmanite-dominated lower mantle is thus sensitive to thephysical conditions at the CMB and Fe-content of the core.Small cores tend to be Fe-rich and will favour presence ofa lower mantle whereas large cores will tend to be enrichedin S and inhibit a lower mantle. Beyond this, presence of alower mantle has implications for the dynamical evolutionof Mars. Several studies have shown that a lower mantle islikely to exert considerable control over the dynamical evolu-tion of mantle and core [e.g., Harder and Christensen, 1996;Breuer et al., 1997; van Thienen et al., 2006]. In contrast,mid-mantle phase transitions are dynamically much less im-portant and appear unlikely to prevent the entire mantlefrom convecting as a single unit Ruedas et al. [2013a].

A related question is the current thermal state of Mars’interior. While difficult to estimate directly, the areothermrepresents, on a par with mantle and core composition, themost important parameter to be determined because of thefundamental control it exerts on physical structure. This isexemplified in figure 1, which shows that phase equilibria,physical properties (here illustrate using shear-wave speed),and presence of bridgmanite-structure silicates in the lowermantle are very sensitive to the exact thermal conditionsinside the Martian mantle. Current geophysical studiestypically approach this problem by assuming examples of“cold” and “hot” mantle conditions [e.g., Longhi et al., 1992;Sohl and Spohn, 1997; Van Hoolst et al., 2003; Verhoevenet al., 2005; Sohl et al., 2005; Zharkov and Gudkova, 2005;Khan and Connolly , 2008; Rivoldini et al., 2011]. Likewise,the areotherm considered in the experimental approach ofBertka and Fei [1997] is representative of “hot” conditionsbased on the need for achieving thermodynamic equilibriumexperimentally. Thus, while the bulk chemical compositionof Mars holds the potential of constraining many aspects ofMars such as internal structure, origin and evolution, cur-rent constraints are not strong enough to reliably determinethese unequivocally.

As observed by Jackson and Faul [2010], viscoelastically-based dissipation models impose strong constraints on theareotherm. This was tested by Nimmo and Faul (2013),whose model calculations showed that Martian global tidaldissipation, based on an extended Burgers formulationof viscoelasticity (to be described in section 3.4), is notonly strongly temperature-controlled but also frequency-dependent. Here, we build upon the initial study of Nimmoand Faul (2013) and embed the viscoelastic model into ourjoint geophysical-thermodynamic framework. This extendsprevious studies of e.g., Khan and Connolly [2008] andRivoldini et al. [2011] to proper consideration of the influ-ence of dissipation on interior structure.

3. Geophysical analysis

3.1. Background

In the present analysis, we focus on the tide raised onMars by its moon Phobos and the Sun, which results in animposed potential ψ that will cause Mars to deform and giverise to an induced potential ψ′ according to

ψ′n(r) =

(R

r

)n+1

knψn(R, r∗), (1)

where R is the radius of the planet, R is a point on theplanet’s surface, r is an exterior point above the point R,while r∗ is the position of the perturbing body. The poten-tials are expanded in terms of spherical harmonics of degreen and the proportionality constants, kn, are tidal Love num-bers of degree n and determine the amplitude of the response[e.g., Efroimsky , 2012a].

The above expression (1) for the amended potential of thetidally deformed planet stays valid insofar as the planet’sresponse is purely elastic. In this approximation, the tidal

bulge raised on Mars by Phobos is aligned with the directionfrom the planet’s centre towards Phobos, with no lagging.This ensures that both the torque applied by Phobos onMars and the opposite torque with which Mars is acting onPhobos are zero. Hence, in the elastic approximation, thetides on Mars make no influence on Phobos’ semimajor axis,eccentricity, or inclination, and, as a consequence, no tidalheat is generated in Mars.

Realistic objects deviate from elasticity. So the tidalbulge acquires a complex structure and is no longer aimed atthe perturbing body. Decomposition of the bulge over thetidal Fourier modes renders harmonics, some of which lagand some advance relative to the subsatellite point. What-ever the sign of the lag, each harmonic now produces tidalheat. In this situation [following Efroimsky and Makarov ,2014], expression (1) above should be written, in the timedomain, as

ψ′n(r, t) =

(R

r

)n+1

knψn(R, r∗), (2)

where kn is a linear operator (Love operator) mapping theentire history of the perturbation (ψn(t′) over t′ ≤ t) on thevalue of ψ′ at the present time t. In the time domain, thisis a convolution:

ψ′n(r, t) =

∫ t

t′=−∞

(R

r

)n+1

kn(t− t′)ψn(R, r∗, t′)dt′, (3)

while in the frequency domain it is a product of Fouriercomponents:

ψ′n(r, ωnmpq ) =

(R

r

)n+1

kn(ωnmpq )ψn(R, r∗, ωnm

pq ) (4)

ωnmpq being the Fourier tidal modes (whose absolute values

are the physical forcing frequencies exerted in the material)and {nm

pq } integers used to number the modes. In the for-mer expression, kn denotes a time derivative, while in thelatter expression, we employ overbars to emphasize that theFourier components are complex, i.e.,

kn(ω′) = Re[kn(ω′)] + iIm[kn(ω′)] = |kn|e−εn(ω′), (5)

where we employed ω′ = ωnmpq for shorthand notation. In ex-

pression (4), ψ′n(r, ωnmpq ) is lagging behind ψn(R, r∗, ωnm

pq ) bythe phase angle εn(ωnm

pq ) which by convention is the negativeargument of the complex Love number kn(ωnm

pq ). By settingr equal to the position r∗ of the perturbing body, we obtainthe additional potential “felt” by the latter (Phobos).

From the above expression, it is also possible to calculatethe tidal torque, the radial elevation, and the tidal powerdissipated in the planet. It turns out that, at each tidalmode ωnm

pq , an appropriate Fourier contribution into each ofthese quantities is proportional to the sine of the phase lag atthat mode [e.g., Efroimsky , 2012a; Efroimsky and Makarov ,2014]. The quantity inverse to the absolute value of this sineis conventionally named the tidal quality factor and denotedwith Qn and defined as

1

Qn(ωnmpq )

= sin |εn(ωnmpq )|. (6)

The functional form of the frequency-dependence of the tidalquality factors is different for different degrees n [see e.g.,Efroimsky , 2015]. Fortunately, this difference becomes man-ifest only at extremely low frequencies. At ordinary fre-quencies, these quality factors are very close to the nom-inal (“seismic”) quality factor Q, which is usually definedthrough the relation

1

Q=

1

2πE ′

∮∂E∂t

dt, (7)

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 5

where E and E ′ refer to the energy and peak energy over onecycle and the integral is taken over the same. The expres-sion on the right-hand side of Eqn (7) turns out to be equalto the absolute value of the sine of the phase lag betweenstrain and stress [see e.g., Eqns 45–47 in Efroimsky , 2015].For ordinary (not too low) frequencies, the tidal phase lagε virtually coincides with the phase lag between the strainand the stress in the material. Accordingly, at not too lowfrequencies the tidal Qn virtually coincides with the nominal“seismic” Q.

In the following, we shall concentrate on the semidiurnaltidal mode for which {nm

pq } = {2200}; so we shall be dealing

with k2 and Q2 (henceforth labeled k2 and Q). While bothk2 and Q depend on interior properties such as density andrigidity, Q is strongly sensitive to viscosity and, thus, tem-perature and grain size. This will be described in more detailin section 3.4.

3.2. Geophysical data

There are currently few geophysical data available thatbear directly on the deep interior structure of Mars. Here,we shall focus on mean density (ρ), mean moment of inertia(I/MR2), second degree tidal Love number (k2), and globaltidal dissipation or tidal quality factor (Q). Mean densityand moment of inertia are sensitive to the density structureof the planet, whereas the sensitivity of the second degreetidal Love number and global tidal dissipation is more com-plex related to the response of a planet to tidal forcing.

The geophysical data for Mars employed here are sum-marized in table 2 and are discussed in more detail in theliterature [e.g., Van Hoolst et al., 2003; Yoder et al., 2003;Bills et al., 2005; Zharkov and Gudkova, 1997, 2005, 2009;Balmino et al., 2005; Lainey et al., 2007; Marty et al., 2009;Jacobson, 2010; Rivoldini et al., 2011; Nimmo and Faul ,2013; Konopliv et al., 2006, 2011, 2016; Genova et al., 2016].The k2 estimates determined from orbiting spacecraft data[e.g., Yoder et al., 2003; Konopliv et al., 2006, 2011; Genovaet al., 2016] are consistent with values ranging from ∼0.13–0.175 (excepting the Marty et al. [2009] determination) andtypically refer to the period of the solar (semidiurnal) tide(12h19min). Current Q estimates range from ∼70 to 110and indicate that Mars is more dissipative than the solidEarth (270±80) at the equivalent diurnal period [Ray et al.,2001].

Presently, we rely on the approach adopted by [Zharkovand Gudkova, 2005, 2009], because their estimates consis-tenly refer to the same frequency (tidal period of Phobos)and, more importantly, include consideration of the effectof anelasticity on k2 and Q to account for the proper fre-quency dependence. Based on the solar tidal k2 value of0.152±0.009 by Konopliv et al. [2006], the authors derivea value for k2 at the tidal period of Phobos (5.55 hours).The difference was found to be �1%, as a result of whichk2=0.152±0.009 will be used.

To determine the tidal dissipation factor at the same pe-riod, we follow Zharkov and Gudkova [2005], and relate themean dissipative factor Q to the tidal lag angle (ε) usingexpressions (6) and (7) above

sin ε ≈ ε =1

Q(8)

Relying on parameters relevant to the Mars-Phobos system,as summarized in Yoder [e.g., 1995], Zharkov and Gudkova[2005] find

Q

k2=

1

εk2= 559. (9)

From this expression, we obtain Q=85±5 at the tidal pe-riod of Phobos. This value agrees reasonably well with the

estimates made by Yoder et al. [2003], Bills et al. [2005],Lainey et al. [2007], Jacobson [2010], and Nimmo and Faul[2013], who determined Q’s of 92±11, ∼85, ∼80, ∼83, and88±16 respectively. In obtaining these estimates, Bills et al.[2005] used a value for k2 (0.0745) that is too low given cur-rent understanding, while Lainey et al. [2007] and Jacobson[2010] relied on the same k2 value by Konopliv et al. [2006]that we employ here. For comparison, Nimmo and Faul[2013], who used the Yoder et al. [2003] solar tide k2 value(0.149±0.017), found a 0.6% change in k2 when convertingto the synodic period of Phobos (erroneously referred to as11.1 hours) and obtained 0.148±0.017. To determine Q,Nimmo and Faul [2013] employ the estimate of Lainey et al.[2007], but correct for the influence of higher-degree terms(k3 and k4) on the orbit of Phobos (see discussion below).Assuming certain ranges for the ratios k2/k3 and k3/k4, theauthors obtain a Q estimate of 88±16. Earlier analyses ofthe orbital acceleration of Phobos estimated the tidal Q ofMars to be 100±50 [Smith and Born, 1976; Yoder , 1982;Lambeck , 1979].

Because of the proximity of Phobos to Mars (mean dis-tance 9378 km), higher-degree terms (e.g., k3, k4,. . .) appearto be significant for the orbital evolution of Phobos [Billset al., 2005; Konopliv et al., 2011]. Based on model values ofk3 and k4, Zharkov and Gudkova [2005] and Konopliv et al.[2011] considered the correction that would be introducedby including higher-order terms and found that this con-tributes less than 10% to the tidal deceleration of Phobos.In contrast, Lainey et al. [2007] considered only the degree-2term, given that degree-3 and -4 terms are not known, andargued that the tidal dissipation factor Q should be consid-ered as an effective Q that partly absorbs losses from higherharmonics. A reasonable alternative would be to compen-sate by increasing the error bars (pers. comm., V. Lainey,2017).

Additionally, tidal forcing at degree-2 will induce tidalwaves at periods other than the main tide considered here[Roosbeek , 1999]. However, the amplitudes of the tidal wavesat the other periods are much smaller. The next-largest am-plitude in the sub-diurnal spectrum, for example, is a fac-tor of ∼7 smaller than the amplitude of the main Phobos-induced tide at 5.55 hours [Van Hoolst et al., 2003]. Hence,we neglect their contribution here.

3.3. Crust and mantle model

Following our previous work [e.g., Khan and Connolly ,2008], Gibbs free-energy minimization is employed [Con-nolly , 2009] to compute stable mantle mineralogy and phys-ical properties along self-consistent mantle adiabats for eachof the five model mantle compositions listed in Table 1.We rely on the thermodynamic formulation of Stixrude andLithgow-Bertelloni [2005b] and parameters of Stixrude andLithgow-Bertelloni [2011]. In the crust and lithosphere, tem-perature is computed by a linear thermal gradient, whichfor each model is determined from surface temperature andtemperature and thickness of the lithosphere. This assump-tion implies that no crustal radioactivity is present. Al-though this is unlikely to be realistic, crustal structure haslittle influence because the data considered here are not re-ally sensitive to it. As a consequence, the exact nature ofthe crustal geotherm is not important. The assumption ofthermodynamic equilibrium is debatable at low tempera-ture [e.g., Wood and Holloway , 1984]. As a result, if for agiven model a mineralogy at a temperature below 800 ◦Cis required, equilibrium mineralogy was computed at 800◦C, whereas physical properties for the mineralogy are de-termined at the actual temperature of interest. The crustis likely to be more complex lithologically, not equilibrated,and is probably porous. The effect of porosity is taken intoaccount through a decrease in the seismic properties of thecrust. The latter will be described further in section 4.1.

X - 6 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

The fixed crustal composition employed here is also sum-marized in table 1.

Deficiency of experimental constraints on the parametersrelevant for the thermodynamic formalism and parameteri-zation of Stixrude and Lithgow-Bertelloni [2011] are the ma-jor source of uncertainty in the thermodynamic calculations.Elastic moduli and density have been estimated to be accu-rate to within ∼0.5 and ∼1–2%, respectively [Connolly andKhan, 2016].

3.4. Viscoelastic model

The dissipation model adopted here is described in detailin Jackson and Faul [2010] and relies on laboratory experi-ments of torsional forced oscillation data on melt-free poly-crystalline olivine. Broadly similar results have also beenobtained by other groups [e.g., Takei et al., 2014]. In whatfollows we base ourselves on Jackson and Faul [2010] andonly provide a summary description here.

In the Earth, Moon, and Mars, and in the absence ofmelting, dissipation (Q) has been observed to be frequency-dependent 1/Q ∼ ω−α, where ω is angular frequency and αis a constant [e.g., Efroimsky and Lainey , 2007]. that hasbeen observed to lie in the range 0.1–0.4 [e.g., Minster andAnderson, 1981; Jackson et al., 2002; Benjamin et al., 2006].Since Maxwellian viscoelasticity is unable to reproduce thisfrequency dependence, other rheological models [for a reviewsee e.g., Karato, 2008] such as the extended Burgers modelof Jackson and Faul [2010] have been proposed. Jackson andFaul [2010] argue for the extented Burgers model over otherrheological models because it describes the changeover from(anharmonic) elasticity to grain size-sensitive viscoelasticbehaviour, whereby it is able to explain the observed dis-sipation in the laboratory experiments on olivine [Jacksonand Faul , 2010].

The response of a viscoelastic material can be describedin terms of the complex frequency-dependent complianceJ(ω) = JR(ω) + iJI(ω), where i =

√−1, and subscripts R

and I denote real and complex parts, respectively. For theextended Burgers model of Jackson and Faul [2010], JR(ω)and JI(ω) can be written as

JR(ω) =1

µU

[1 + ∆

∫ τH

τL

D(τ)

1 + ω2τ2dτ

](10)

JI(ω) =1

µU

[ω∆

∫ τH

τL

τD(τ)

1 + ω2τ2dτ +

1

ωτM

], (11)

which is essentially the Laplace transform of the creep func-tion for the extended Burgers model of linear viscoelastic-ity [see e.g., Jackson and Faul , 2010]. In these expres-sions, µU represents the unrelaxed, i.e., infinite-frequency,shear modulus, τ period, τM = η/µU Maxwell viscous re-laxation time, η viscosity, ∆ strength of relaxation mecha-nism and D(τ) distribution of relaxation times. From theabove equations, local dissipation (1/Q) and shear modulus(µ) at a particular frequency can be computed from 1/Q =|JI |/

√J2I + J2

R ≈ |JI |/|JR| and µ(ω) = 1/√J2R(ω) + J2

I (ω),respectively.

The advantage with this model is that D(τ) can be usedto specify a distribution of anelastic relaxation times ac-counting for the monotonic background dissipation (DB)and the superimposed dissipation peak (DP) of elastically-accomodated grain-boundary sliding [Jackson et al., 2014],in addition to the associated modulus dispersion [Jacksonand Faul , 2010], where DB is given

DB(τ) =ατα−1

ταH − ταL, (12)

where τL and τH are the integration limits corresponding toshort and long periods, respectively. In the low-frequencylimit, JI (equation 11) reduces to Maxwellian behaviour

(JI = 1/ωη). 0 < α < 1 for τL < τ < τH and zero elsewherewith associated relaxation strength ∆B [Minster and Ander-son, 1981]. To model the dissipation peak DP that appearssuperimposed on the background a term of the followingform needs to be added

DP(ln τ) =1

τσ√

2πexp

[− ln2(τ/τP )

2σ2

], (13)

where τP indicates position of the peak with peak width σand associated relaxation strength ∆P. This peak is found tooccur at low temperature and/or short time scales and corre-sponds to elastically-accommodated grain-boundary sliding.

The timescales τL, τH , τM , and τP are all temperature(T), pressure (P), and grain-size (dgrain) dependent, whichfor each individual timescale is modeled using

τ(P,T, dgrain) = τ0

[dgrain

d0

]m

exp

[E

R

(1

T− 1

T0

)]exp

[V

R

(P

T− P0

T0

)],

(14)where τ0 is a normalized value at a particular set of referenceconditions P0 (0.2 GPa), T0 (900 ◦C), d0 (13.4 µm), and Eand V are activation energy and volume, respectively. Inaddition, different grain-size exponents m for anelastic andlong-term viscous creep processes are allowed for. All of theabove constants are tabulated in Nimmo et al. [2012] andNimmo and Faul [2013], except for α, E, and V, which areconsidered variable parameters (see section 4.1).

This model can now be directly linked with the thermo-dynamic computations in that the latter provides the unre-laxed (infinite-frequency) shear modulus µU that appears inthe above equations, in addition to P and T and any otherthermodynamic variables needed (all functions of radius).Note that the shear moduli are computed on the basis ofthermochemical models of Mars and are constrained by geo-physical data. Hence, the improvement here over previousstudies that had little control over internal structure pa-rameters required as input [e.g., Sohl and Spohn, 1997; Sohlet al., 2005; Bills et al., 2005; Nimmo et al., 2012; Nimmoand Faul , 2013]. Finally, to compute global frequency-dependent k2 and Q from the model outlined above, weemploy the viscoelastic tidal code of Roberts and Nimmo[2008]. This code assumes spherical symmetry and that alldissipation occurs in shear; for numerical reasons, we im-posed a viscosity cutoff of 1029 Pa s.

3.5. Core model

Mass-radius relations of rocky planets generally requireFe-rich metallic cores alloyed with a light element such asSi, C, or S [e.g., Birch, 1964; Poirier , 2000]. In the case ofMars, it is argued that S is the dominant light element be-cause the other candidates do not have sufficient solubilityin iron-rich liquid at the relatively low pressures that wouldhave been maintained during core formation (e.g., Stevenson2001). Evidence in support of this is the observed depletionof chalcophile elements, notably S, of the SNCs (see sec-tion 2.1). Accordingly, an Fe-S core is generally assumed ingeophysical models of Mars. To date, the most elaborate pa-rameterization of the Martian core (in the Fe-FeS system)is that of Rivoldini et al. [2011]. Here, we follow Rivol-dini et al. [2011] and assume that Mars’ core is well-mixedand convecting. To compute depth-dependent thermoelasticproperties for the core, we use equations-of-state for liquidiron and liquid iron-sulfur alloys.

Core pressure is obtained from the hydrostatic equation

dP(r)

dr= −ρ(r)g(r), (15)

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 7

where r is radius, ρ density, and g gravitational acceleration.g also obeys the Poisson equation

dg(r)

dr+

2g(r)

r= 4πGρ(r), (16)

where G is the gravitational constant. For a well-mixed andvigorously convecting core the temperature gradient is givenby

dT(r)

dr= −T(r)

γ(r)

KS(r)ρ(r)g(r), (17)

where γ is the Gruneisen parameter and KS is adiabatic bulkmodulus. These ordinary differential equations are solvednumerically [Brankin et al., 1993] subject to the followingboundary conditions

P(rcmb) = Pcmb (18)

T(rcmb) = Tcmb (19)

g(0) = 0 (20)

where rcmb, Pcmb, and Tcmb are radius, pressure, and tem-perature at the CMB, respectively. These quantities on themantle side of the CMB are those determined in section 3.3.The thermoelastic quantities ρ, KS, and γ depend on pres-sure, temperature, and core composition (S content) and arecalculated using the approach outlined in Appendix A.

4. Computational aspects

4.1. Model parameterization

In the following we briefly describe model parameteriza-tion, which is illustrated in figure 2. For present purposes,we assume our model of Mars to be spherically symmet-ric and have divided it into three layers comprising crust,mantle, and core. The crust has been further subdividedinto an additional three layers that are parameterized interms of P- and S-wave velocity, density, and Moho thick-ness. Rather than varying VP, VS, and ρ independently inthe crust, however, we introduced a variable parameter (φ)to mimick the effect of porosity and computed the aforemen-tioned physical properties using x

′i = xi · φi, where xi are

thermodynamically-computed VP, VS, and ρ (section 3.3) incrustal layer i. φi is determined from φi = φ0 +(1−φ0) ·i/Nwith φ0 being variable surface porosity and N the total num-ber of crustal layers. This parameterization ensures thatcrustal properties increase from the surface down to theMoho where porosity is expected to vanish due to pressure(i.e., 1-φi=0). The sublithospheric mantle is assumed to beuniform and modeled using the variables composition (inthe NCFMAS system) and mantle temperature. Within thelithosphere, temperature is computed by a linear areother-mal gradient, which for each model iteration is determinedfrom the variables Tsurf , Tlit, and dlit. The sublithosphericmantle adiabat is defined by the entropy of the lithologyat the temperature Tlit, i.e., at the base of the lithosphereof thickness dlit. The bottom of the lithosphere is definedas the location where the conductive lithospheric geothermintersects the mantle adiabat. Since the viscoelastic model(section 3.4) is grain size dependent, the model is addition-ally parameterized in terms of a single grain size (dgrain).Many of the parameters belonging to the viscoelastic modelare determined experimentally and therefore uncertain. Tobetter capture this, we included α, E, and V as variable pa-rameters. Uncertainty ranges are taken from Jackson andFaul [2010]. The mantle pressure profile is obtained by in-tegrating the vertical load from the surface pressure bound-ary condition. Core parameters include radius, composition(S content), and the input parameters required to computephysical properties of the core are those determined by in-tegrating the load from the surface to the CMB and the

entropy of the lithology at Tlit, which determines the tem-perature at the CMB (section 3.5). Given values of all of theabove model parameters, the forward model can be solved.

4.2. Forward problem

The forward model consists of computing geodetic data(ρ, I/MR2, k2, Q) from a knowledge of the physical struc-ture of the crust and thermochemical structure of mantleand core. To determine stable mineralogy (M), isotropicshear (µ) and bulk (κ) moduli, density (ρ), and attenua-tion structure (here designated by complex moduli µ and κ)along self-consistent mantle adiabats, we employ Gibbs free-energy minimization and a grain size dependent viscoelasticformulation. With this, the forward problem can be writtenas

{Xm,Tlit,dlit,dgrain, . . . , rcore,XS︸ ︷︷ ︸primary parameters

}

operator︷︸︸︷g1−→ M g2,g3−→ {ρ, µ, κ} g4−→ {ρ, I/MR2, k2,Q︸ ︷︷ ︸

data

}

where the primary parameters are those described abovein section 4.1 and the other parameters (M, µ, κ,. . .) repre-sent secondary parameters that are required for the purposeof computing data (section 3.2). The operators (g1, . . . , g4)represent the various forward models (e.g., Gibbs free energyminimization, extended Burgers model, viscoelastic tidal re-sponse) that are described in sections 3.3 and 3.4.

4.3. Inverse problem

To solve the inverse problem d = g(m), where d is a datavector consisting of observations and g is an operator thatmaps from the model space into the data space, we employ aBayesian approach as outlined in Mosegaard and Tarantola[e.g., 1995]

σ(m) = k · h(m)L(m), (21)

where h(m) is the prior model parameter probability dis-tribution, i.e., the information on model parameters pro-cured independently of data, L(m) is the likelihood function,which measures the misfit between observed and predicteddata, k is a normalization constant, and σ(m) is the pos-terior model parameter distribution. σ(m) represents thesolution to the inverse problem above. L(m) is determinedfrom the observed data, data uncertainties, and the mannerin which the latter are used in modeling data noise (to bedescribed in the following).

The Metropolis algorithm is employed to sample the pos-terior distribution (Eq. 21) in the model space [Mosegaardand Tarantola, 1995]. This algorithm, which is samples themodel space in a random fashion, is an importance sam-pling algorithm, i.e., it ensures that models that fit datawell and are simultaneously consistent with prior informa-tion are sampled more frequently. The Metropolis algorithmsamples the model space with a sampling density that is pro-portional to the (target) posterior probability density andthus ensures that low-probability areas are sampled less ex-cessively. This is an important feature of any algorithm thatwishes to randomly sample high-dimensional model spaceswhere the probability density over large proportions of thevolume are near-zero.

4.4. Prior information

The prior model parameter information described aboveis summarized in table 3 below. The chosen prior ranges rep-resent the information acquired from data and results fromexperimental studies supplemented with results from numer-ical studies as discussed in the foregoing sections. The priorrange on surface temperature is relatively large, but reflects

X - 8 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

the fact that the data considered here have little sensitivityon crustal and sub-crustal lithospheric thermal structure.This has little effect on computed physical properties of thecrustal and sub-crustal structure as tests have shown wheresurface temperature was fixed to that of present-day Mars.Hence, in what follows the thermal structure of the crustand sub-crustal lithosphere will not be discussed further.

4.5. Sampling the posterior

Under the assumption that data noise is Gaussian dis-tributed and that observational uncertainties and calcula-tion errors among the data sets considered are independent,the likelihood function can be written as

L(m) ∝∏

i

exp

(−|d

iobs − di

cal(m)|2

2σ2i

)(22)

where i runs over ρ, I/MR2, k2, and Q, dobs and dcal(m)denotes observed and calculated data, respectively, and σdata uncertainty.

5. Results and discussion

5.1. Datafit

Datafits for all considered compositions of table 1 aresummarized in figure 3. The various compositions are allcapable of fitting mean density and moment of inertia, inaddition to global tidal dissipation, but show scatter in k2.In particular, compositions MA and SAN are less convinc-ing than DW, LF, and TAY. The discrepancy between MAand the other models arises because MA models are morerigid (∼3–5 % higher S-wave velocity) in the depth range100–1100 km relative to the other models due to presence ofgarnet-rich phases. This coupled with smaller core sizes re-sults in the inability of MA to fit k2. Given the geochemicalpoint made earlier (section 2.1) that both MA and SAN rep-resent somewhat exotic compositions and are probably un-likely to be representative of the bulk composition of Mars,we discard MA from further analysis. While the same geo-chemical argument also apply to model SAN, we neverthe-less keep this model because it is capable of fitting data towithin uncertainties.

5.2. Mantle temperature profiles

Inverted present-day areothermal profiles are shown infigure 4. For comparison, we are also showing sampledprior areotherms, which evidently comprise a very largeparameter range. Inverted models are comparatively well-constrained over most of the mantle and core with the ex-ception of crust and lithosphere (≤400 km depth). Thisreinforces earlier analyses (e.g., Karato, 2007 and discus-sion by Efroimsky and Lainey [2007] that the constraintson the mantle temperature profile imposed by the additionof a temperature and necessarily frequency dependent visco-elastic model are strong. Relative to earlier studies that con-sidered the same problem, but without invoking a specificvisco-elastic model [e.g., Khan and Connolly , 2008; Rivoldiniet al., 2011], the advantages here are obvious. These earlierstudies had difficulty in constraining the thermal structureof the mantle.

Comparison of the obtained areotherms for the four bulkcompositions indicates relatively small differences that areunlikely to be large enough to enable us to distinguish be-tween the bulk compositions based on the available dataconsidered here. Lithospheric thickness and temperaturevariations among the four models broadly agree with es-timates in the range 250–500 km and ∼1300–1360 ◦C. Incomparison to the prior ranges (table 3), both dlit and Tlit

are relatively well-constrained. The self-consistently com-puted mantle adiabats show “structure” at depths around

1000–1100 km and 1300–1400 km, respectively, at the loca-tions where the major mantle mineral phase transitions oc-cur (olivine→wadsleyite and wadsleyite→ringwoodite). Atthe CMB, temperatures in the range 1560–1660 ◦C are ob-tained. No attempt was made to account for a thermalboundary layer between the mantle and core. While a thinthermal boundary layer with a temperature difference of∼100 ◦C is possible [e.g., Kiefer and Li , 2016], data arenot sensitive enough to “see” such a layer. This was verifiedby investigating how an increase of 100 ◦C across the CMBaffects core properties. As expected, differences in e.g., P-wave velocity and density on the core side of the CMB rela-tive to a model without a thermal boundary layer amountedto <0.5%, which from the point of view of the geophysicaldata is an insignificant change. Results are summarized intable 4.

In figure 4 we are also showing the mean temperature pro-file from a recent geodynamical study by Plesa et al. [2016].Plesa et al. [2016] modeled the thermal evolution of Mars toinvestigate the spatial variation in present-day surface heatflux. The blue and pink lines show present-day hot and coldend-member average temperature profiles from Plesa et al.[2016] assuming a core size of 1500 km. The comparisonshows that the results from the inversion are in good agree-ment with the cold end-member geodynamic model. Forreference, we also included the recent present-day dry Mar-tian solidus of Kiefer et al. [2015].

Independent direct constraints on the thermal structureof Mars are few. Petrological estimates derive principallyfrom Gamma Ray Spectrometer data (Mars Odyssey) forseveral major volcanic provinces [Baratoux et al., 2011] andanalyses of shergottite meteorites [Filiberto and Dasgupta,2015]. These estimates bracket temperatures found here,but are uncertain by several hundred ◦C for dry mantle con-ditions and increase further if water and/or other volatilesare added [e.g., Balta and McSween, 2013; Filiberto et al.,2016].

5.3. Grain size

Since the extended Burgers model is grain size depen-dent, we show sampled grain size distributions in figure 5for the four main compositions. The distributions are gener-ally in accord with grain sizes in the sub-to-millimeter rangeof 0.01–5 mm with a strong peak toward small grain sizes.Nimmo and Faul [2013] observed a trade-off between grainsize and mantle temperature (and lid thickness). This trade-off is not observed here. One reason for this is that in theinversion, models are “forced” to fit the measured k2 andQ, whereas Nimmo and Faul [2013] are simply calculatingk2 and Q based on a range of input parameters and com-paring these qualitatively to the observations. Thus, thetrade-off is not observed because the inversion forces themodel parameters to cover a more restricted range. In addi-tion, the thermodynamic parameterization coupled with thetop-down approach in which models are constructed (e.g.,dlit/Tlit determine mantle temperature) acts to break thetrade-off between various parameters that are treated as be-ing purely independent by Nimmo and Faul [2013] and re-stricts the parameter range even further. Finally, we alsoinclude ρ and I/MR2, which, relative to Nimmo and Faul[2013], exerts a strong independent control on the densitystructure.

The importance of grain size arises because of the strongcontrol it exerts on the diffusion creep rheology of the crustand mantle (viscosity ∝ dm

grain, m=2–3). As summarized byKarato [2008, Ch. 13], common estimates of grain sizes ofterrestrial rocks vary from 0.1 mm to 10 mm. Grain sizeestimates in the deeper Earth are invariably indirect withestimates ranging from small (µm) to relatively large grain

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 9

sizes (cm), with a possible increase in grain size with depth[Karato and Wu, 1993]. The latter would result from achange in deformation mechanism in Earth’s upper mantlewhere a transition from dislocation to diffusion creep oc-curs. The depth at which this transition occurs depends ona range of flow-law parameters [Hirth and Kohlstedt , 2013].

In our inverse treatment, we neglect the complexity ofincluding a depth dependent grain size. Based on the ob-servation that the geodynamical simulations (section 6) sug-gest homogeneous grain sizes throughout Mars’ mantle, thisappears to be a reasonable assumption. Generally, graingrowth is temperature controlled and on Earth the lowermantle transition is expected to limit the size of grains asmaterial is being moved across the transition and recrystal-lizes [e.g., Solomatov , 2001; Solomatov and Reese, 2008]. Incomparison, the mid-mantle phase transitions in the Mar-tian mantle at ∼13 GPa (1100 km depth) and 15 GPa(1300 km depth) (c.f., figure 1), respectively, are dynami-cally unimportant [Ruedas et al., 2013a]. As a consequence,and in view of an apparent absence of a lower mantle transi-tion in Mars (see section 5.5 below), the grain sizes for Marsfound here are not unrealistic.

For the baseline Martian model, Nimmo and Faul [2013]fixed α =0.274, whereas we generally find a slightly lower αin the range 0.2–0.22 (values for α, E, and V are summarizedin table 4). The reason for this difference is not clear, butcould possibly be related to Nimmo and Faul [2013] using11.1 hours instead of 5.55 hours as forcing period.

5.4. Attenuation

Computed shear attenuation (Qµ) profiles for the fourcompositions are also shown in figure 6 at the main pe-riod of the Phobos-induced tide (5.55 hours) and at 1 speriod (only for DW since the others are qualitatively thesame). The sampled profiles indicate that dissipation withinMars is large (small Qµ) with quasi-constant Qµ values of130–150 throughout the upper mantle (depth range 300–900 km) at seismic periods and even lower at tidal peri-ods as predicted earlier (see section 3.2). As in Nimmo andFaul [2013], smallest Qµ values are found in the depth range300–500 km, which coincides with a low-velocity zone thatis associated with relatively steep temperature gradients inthe lithosphere (this will be discussed in more detail in sec-tion 5.7 below). In spite of the near-isothermal nature ofthe areotherm in the convecting mantle of Nimmo and Faul[2013], this conclusion seems robust and serves to emphasizethat to first order dissipation appears to be controlled bythe olivine-dominated part of the mantle, while phase tran-sitions are less significant. Indeed, while variations in shearattenuation at depths of ∼1100 and ∼1300 km, respectively,are palpable, these appear to be insignificant. The observa-tion that the Martian mantle is very dissipative could havea profound effect on seismic wave propagation within Mars.In particular, significant attenuation of both short-periodbody wave and long-period surface waves is possible andmay hamper detection of distant (teleseismic) signals [seealso discussion in Lognonne et al., 1996].

5.5. Core parameters

Inverted core size, composition (S content), and densityfor the four main compositions is shown in figure 7 and sum-marized in table 4. The results reveal broadly similar coresizes and compositions for three of the compositions (DW,LF, and TAY) with core radii and S-content (XS) in therange 1640–1740 km and ∼13.5–16 wt%, respectively. Incomparison, SAN has a slightly smaller core (1590–1680 km)and lower S content (11–13.5 wt%). The agreement betweenDW and TAY is not surprising given the similarity betweenthe two compositions (see table 1 and section 2.1 for theassumptions underlying the geochemical models). The ob-served linear trends are also as expected and highlights the

notorious trade-off that exists between composition (den-sity) and core size. Relative to the original model core com-positions DW remains almost unchanged, whereas the S con-tent of LF increases by 3 wt% and that of TAY and SANdecrease by 6 and 3 wt%, respectively. Importantly, and in-spite of initial compositional differences, inverted core com-positions for models DW, LF, and TAY appear to convergeto a common composition with a mean S content around 15wt% corresponding to a mean core radius of ∼1700 km anda mean core mass fraction of 0.20-0.21, while SAN remainsa little below the lower limit of these estimates. The val-ues for DW, LF, and TAY are in good agreement with therecent estimates of Rivoldini et al. [2011], who employed asomewhat similar approach to determining interior structure(inversion of k2, M, and I/MR2) and obtained core sizes andS contents of 1794±65 km and 16±2 wt%, respectively. Thelarger ranges found here reflect the fact that Rivoldini et al.[2011] only considered two possible end-member areothermsfrom the literature (c.f., “cold” and “hot” areotherms in fig-ure 1). In addition, a slightly larger k2 (0.159±0.009) wasemployed, which, everything else being equal, will tend in-crease core size.

The obtained core sizes, however, imply that a lower man-tle transition equivalent to the “660-km” seismic discontinu-ity, is not extant inside Mars. CMB pressures attained hererange from 19.5–21.5 GPa, which is below that needed forthe ringwoodite→ferropericlase+bridgmanite transition tooccur (cf. figure 1) and no models were found to containthe equivalent of a lower mantle transition. This also holdsfor models SAN. Similar conclusions were also reached byRivoldini et al. [2011]

Absence of a lower mantle has implications for the dynam-ical evolution of the planet: large cores inhibit the existenceof a lower mantle layer as a result of which mantle convectionis expected to encompass the entire mantle. Small cores, onthe other hand, allow for a lower mantle layer that couldform a dynamically separate unit and lead to decoupling ofmantle and core [e.g., van Thienen et al., 2006; Ruedas et al.,2013a, b], although disruption of grain size growth as mate-rial passes through the phase transition would tend to lowerviscosity and hence increase convective vigour and destabi-lize any initial separation. In contrast, absence of a lowermantle transition would favour continuous grain growth andlead to larger grain sizes and thus increased viscosity at thebottom of the mantle. This argues in favour of whole-mantleconvection.

5.6. Implications for bulk composition

Inverted bulk and core compositions are summarized intable 4. The composition of bulk silicate Mars (BSM) ispresently defined by the model compositions DW, LF, TAYand SAN (table 1). These initial BSM compositions showvariability in absolute oxide concentrations and characteris-tic elemental ratios. The Mg-number, for example, rangesbetween 0.751 (LF) and 0.728 (SAN) and varies only by∼3%. The atomic Mg/Si ratio, on the other hand, rangesfrom sub-CI chondritic (0.85) for SAN to near CI-chondritic(1.04) for TAY. With the present model setup, where we al-low for a 10 wt%-variation of each mantle oxide component,the BSM models are essentially indistinguishable. A sim-plification in our phase equilibrium calculations is that the“Model mantle compositions” are directly derived from theproposed BSM models and do not account for the extractionof the crust, i.e., that bulk silicate Mars is the compositionalsum of crust and mantle. Subtracting mean crustal massesof average crustal composition (table 1) from each mantlemodel shows that the residuals are within 10 wt% of theirinitial values, except for Na. The latter component is the

X - 10 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

most incompatible element and varies by up to 24 wt%. Be-cause of its low absolute concentration, however, variationsin Na have a negligible effect on mantle phase relations. Ac-cordingly, crustal extraction is implicit in the inversion.

Core compositions are modeled in the simplified systemFe-FeS because of a lack of more detailed geophysical modelsaccounting for the effect of Ni on core properties. From ageochemical point of view, the core likely contains a signifi-cant amount of Ni, which can be estimated by assuming 1) afixed bulk Mars atomic Fe/Ni ratio of 18 (after DW) and 2)that Ni is entirely incorporated into the core. A Fe/Ni ratio∼18 is also obtained independently in the isotope analysisof Lodders and Fegley [1997], which is a consequence of theFe/Ni ratio being relatively constant across various typesof chondritic meteorite classes (ranging from 17.8 to 19.1)[Wasson and Kallemeyn, 1988]. Adjusting the core compo-sition for the presence of Ni results in very similar solutionsfor DW, LF, and TAY with a most probable core composi-tion of ∼79 wt% Fe, ∼6 wt% Ni, and ∼15 wt% S. In thecase of SAN the mean core composition would result in aslightly lower sulfur core content with ∼82 wt% Fe, ∼6 wt%Ni, and ∼12 wt% S.

The bulk planet compositions derived here result in bulkFe/Si ratios (by weight and corrected for the presence of Niin the core) that range between 1.63 and 1.68 for DW, LF,and TAY, and 1.51 for SAN. In the model of Sanloup et al.[1999], Mars consists of a mixture of 45% EH (Fe/Si∼1.74)and 55% H (Fe/Si∼1.63) chondrites with a combined Fe/Siof 1.68 ([Wasson and Kallemeyn, 1988]). However, to recon-cile the bulk silicate Mars composition originally proposedby Sanloup et al. [1999] with the geophysical data consid-ered here, current results suggest lower core mass fractionsaround 0.18–0.2. This circumstance explains the lower valuefor Fe/Si found here, relative to what had been proposed bySanloup et al. [1999], involving a larger core mass fraction(0.23) and therefore a higher Fe/Si (1.68). Ultimately, thisimplies that the compositional model of Sanloup et al. [1999]can not self-consistently explain the oxygen isotope signa-ture of Mars and simultaneously the current Martian geo-physical data. A possible explanation for this discrepancymay lie in their choice of meteoritic end-members that in-volves a hypothetical, as yet unsampled, class of meteorites.

Based on the experimental data of Stewart et al. [2007]on the system (Fe,Ni)-(Fe,Ni)S, the eutectic compositionat CMB pressures obtained here (19.5–21 GPa) is ∼17wt% S and decreases to 14 wt% S with increasing pressure(40 GPa). Corresponding eutectic temperatures are below1000◦C at CMB pressures and only increase to 1100–1200◦Cat pressures of the inner core. In comparison, inverted modeltemperatures on the core side of the CMB are >1800◦C,which together with the almost eutectic-like sulfur contentwith a mean ∼14.5 wt% (in the case of DW, LF, and TAY),suggests a fully liquid core at present in agreement withinferences made earlier [Rivoldini et al., 2011]. Based onthe mean S content determined here, Mars’ core is likelyto undergo a complex core crystallisation behaviour in thefuture. Crystallisation will start at the pressure at whichthe areotherm initially falls below the liquidus, which is ex-pected to first occur in the upper part of the core close to theCMB on the Fe-rich side of the Fe-S eutectic [e.g., Dumberryand Rivoldini , 2015], consistent with the iron-rich ”snowingcore” hypothesis of [Stewart et al., 2007]. Continued cool-ing combined with the shift in eutectic composition towardsFe with increasing depth may result in a situation wherethe areotherm also intersects the liquidus on the S-rich sideat higher pressure, causing co-crystallisation of Fe3S in theinterior of the core [e.g., Hauck et al., 2006].

5.7. Seismic profiles

Of general interest are profiles of seismic velocity and den-sity structure. These profiles are shown in figure 8 and referto seismic periods (1 s). Figure 8 also shows the frequency-dependence of shear attenuation on S-wave velocities. Atthe main tidal period of Phobos, S-wave velocities are, asexpected, lower relative to those at seismic periods. Thisoccurs for depths >200 km where shear attenuation be-comes significant in the mantle (figure 6). As already seenin the thermal profiles (figure 4), differences between thevarious models are relatively small, but point to the possi-ble presence of a lithospheric high-velocity lid followed bya low-velocity zone (LVZ). These features were noted pre-viously by Nimmo and Faul [2013] and follow directly fromthe relatively strong temperature gradients that prevail inthe conductive lithosphere. More generally, the effect oftemperature on shear modulus relative to that of pressureinside Mars is stronger and therefore results in a shear-wavevelocity reduction with depth in the lithosphere. These fea-tures on seismic wave propagation are discussed by Zhenget al. [2015]. It should be noted though that the exact con-dition that seismic waves are bent downwards in a sphericalmodel, which potentially occurs if a LVZ is present, is thattransformed velocities in a flat Mars model have a negativegradient. From a geometrical point of view, this is equiva-lent to horizontal rays being bent downwards relative to thearc at that radius. [for specific illustrations of this, see e.g.,Clinton, 2017]. As a consequence, an apparent LVZ doesnot necessarily result in a seismic shadow zone.

Models DW, LF, and TAY point to an olivine→wadsleyitetransition at around ∼1000–1050 km depth and a secondarytransition (wadsleyite→ringwoodite) at about 1250–1300km depth. For model SAN the transitions occur 50–100km shallower in the mantle. The former transition is notas sharp as on Earth (coexistence loop extends for approxi-mately 50 km), but might be detectable using seismic data inthe form of e.g., mantle triplications, PP- and SS-precursors,and receiver functions. While seismic data will be acquiredas part of the upcoming InSight mission to Mars [Banerdtet al., 2013], figure 8 indicates that it is going to be difficult,based on seismic data alone, to discriminate between thecompositional models considered here, although SAN mightpresent an exception in this regard. The same argumentsconcern determination of core size, which is otherwise de-tectable using normal modes or core-reflected and -refractedwaves. While seismic analyses in preparation for InSighthave been and are currently undertaken [e.g., Okal and An-derson, 1978; Goins and Lazarewicz , 1979; Solomon et al.,1991; Lognonne et al., 1996; Larmat et al., 2008; Teanbyand Wookey , 2011; Zheng et al., 2015; Zharkov and Gud-kova, 2014; Panning et al., 2015; Khan et al., 2016; Panninget al., 2016; Bose et al., 2017; Bozdag et al., 2017; Ceylanet al., 2017], we leave it for a future study to consider seismicpredictions of the models proposed here in more detail.

6. Geodynamic modeling

To provide an independent view on some of the geophys-ical predictions (e.g., grain size, thermal state, and litho-spheric thickness), we performed a number of mantle con-vection simulations. For this purpose, we employ the con-vection code StagYY [Tackley , 2008] in 2D spherical annulusgeometry [Hernlund and Tackley , 2008]. At this stage, wewould like to note that it is not our purpose to accuratelyreproduce all geodynamical features of Mars (such as theactivity related to the large volcanic provinces Tharsis andElysium), since this requires a larger number of simulations.Instead, the emphasis focusses on quantitatively reproduc-ing the thermal profile (internal temperature), and averagegrain size obtained in the geophysical inversion, in additionto current crustal thickness estimates [e.g., Neumann et al.,2004; Wieczorek and Zuber , 2004].

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 11

An important constraint for the geodynamic simulationsis the observation that most of Mars’ crust is older than 3.7Ga, although there is evidence for recent melting and vol-canism [Phillips et al., 2001; Neukum et al., 2004; Solomonet al., 2005; Werner , 2009; Carr and Head , 2010; Hauberet al., 2010; Grott et al., 2013]. As a consequence, “suc-cessful” models are expected to produce mafic crust mainlyin the beginning of the simulations, although localized meltproduction is possible up until the present. Interestingly,the geophysically-constrained internal temperatures are nottoo far below the solidus temperature (see figure 4). Thissuggests that “localised” features such as a large plume be-low Tharsis [Harder and Christensen, 1996; Zhong and Zu-ber , 2001; Roberts and Zhong , 2006; Zhong , 2009; Kellerand Tackley , 2009; Sramek and Zhong , 2010] would allowfor regional magmatism to occur without being widespreadas observed on e.g., Io [Breuer and Moore, 2007] and Venus[Ivanov and Head , 2013]. In the simulations performed here,we neglect the implications of a giant impact as a means offorming most of the North-South crustal dichotomy [Mari-nova et al., 2008; Nimmo et al., 2008; Reese et al., 2010;Golabek et al., 2011]. As a result, the amount of mafic crustthat forms is likely to be underestimated.

6.1. Crustal production and grain size evolution

For the simulations, we solve the equations of compress-ible thermo-chemical convection, i.e., Stokes, heat and conti-nuity equations employing the marker in cell method for theadvection of rock composition, temperature and grain sizeusing the convection code StagYY [Tackley , 2008]. Here,only a brief summary is given. Details are provided in ap-pendix B.

When temperature exceeds the mantle solidus, partialmelting ensues. The basaltic melt produced from the mixedmantle is transported towards the surface by both intru-sive (50%) and extrusive (50%) magmatism, as describedin Rozel et al. [2017]. The solid residue is harzburgitic incomposition (forming the depleted mantle) and will remainsolid as the compositionally-dependent solidus temperatureincreases [Maaløe, 2004]. Heat producing elements preferen-tially partition into the melt, implying that the mafic crustdeposited at the surface is 1/Dp times more radiogenic thanthe depleted residue where Dp is the partition coefficient forradiogenic heat producing elements. For the partitioningof heat producing elements between the mantle and crustwe use the approach described in Xie and Tackley [2004].Our simulations show that Dp is an important parameteras it controls the effective internal heat production in Mars’mantle, which directly influences the internal temperaturethrough melting and crustal production. Experimentally-determined values for Dp of radiogenic elements are of theorder of 0.001 [Hart and Dunn, 1993; Hauri et al., 1994].

In the simulations performed here, we assume a homoge-neous temperature-, pressure-, and grain size-dependent rhe-ological equation based on diffusion creep [Hirth and Kohlst-edt , 2013] over the entire computational domain

η(P,T,dgrain) = η0

(dgrain

dref

)3

exp

(E + PV

RT− E

RTref

),

(23)where η0 and dref are reference viscosity and grain size(1 mm) respectively, E and V are activation energy (375kJ/mol) and activation volume (6 cm3/mol), respectively,Tref is reference temperature (1326 ◦C), and R is the uni-versal gas constant. This expression for η is similar to Eq. 14with η = τµ, but the latter applies to timescales on the orderof Phobos’ orbital period (hours), whereas here timescalesrefer to the evolution of Mars (Gyr). To reproduce geo-dynamically the geophysically-inferred properties, we testvarious reference viscosities in the range 1018 − 1021 Pa·s.

Grain size evolution and heat producing element parti-tioning are interrelated as grain growth is strongly temper-ature dependent, in that internal heating tends to increase

grain size, while simultaneously producing mafic crust. Yet,magmatism depletes the mantle, i.e., decreases internalheating, and limits any increase in potential temperature.Consequently, it is reasonable to assume that the evolutionof Mars’ internal temperature and average grain size aregoverned by depletion in heat sources due to melting andmagmatism.

To avoid numerical diffusion, grain size evolution is com-puted on the Lagrangian advected by the flow. We use asimple grain growth equation and neglect dynamic recrys-tallisation since grain size reduction terms are negligible dueto the small stresses in the Martian mantle

D

Dtdp

grain = k0 exp

(− Ed

RT

), (24)

where k0, p, and Ed are experimentally-determined con-stants. In line with arguments made earlier (section 5.3), weuse a single grain growth law for the entire mantle. Graingrowth parameters are difficult to determine experimentallyas laboratory experiments can only be run for a limitedamount of time, far less than the scale on which geologi-cal processes occur. Following Yamazaki et al. [2005], wechose p=4.5, k0=3.9811·106 µm4.5/s and Ed=414 kJ/mol.These parameters strictly only apply to the lower mantle ofMars where ringwoodite is stable and where the tempera-ture is highest and, as a result, grain growth is fastest. Whilethe use of different growth laws in the shallower portions ofMars’ mantle would lead to a slight modification of grain sizeevolution, the important observation is that the trends foundhere would not be dramatically changed. As discussed ear-lier, mid-mantle phase transitions (olivine→wadsleyite andwadsleyite→ringwoodite) are not expected to affect grainsize [see also Solomatov and Reese, 2008].

The initial state is characterized by a potential tempera-ture of 1326 ◦C and a homogeneous grain size of 100 microns.The mantle is a mixture of 15% eclogite and 85% harzburgiteand we impose a 20 km pre-existing basaltic crust at the sur-face. Radiogenic heating of mixed mantle is initially equalto 2.3 · 10−11 W/kg, and decreases exponentially through-out time with a half-life of 2.43 Gyr [Nakagawa and Tackley ,2005]. We operate with an initial core temperature of 2226◦C and take into account self-consistent core cooling usingthe heat flux at the CMB. Core radius is fixed to 1698.3 km,which is the mean value obtained in the inversion.

6.2. Geodynamical results

Figure 9 shows final profiles of temperature, viscosity, andgrain size after 4.5 Gyr of evolution. Temperature and grainsize profiles are strongly correlated, as expected, since thefinal grain size represents an integral of the thermal his-tory; a hot mantle results in a larger grain size. Viscosity,on the other hand, does not correlate with temperature norgrain size, since these parameters compete in the rheologicalequation (Eq. 23). For comparison, geophysically-invertedthermal profiles are also shown (gray envelope). From theresults we observe the following trends: inefficient partition-ing of heat producing elements (high Dp) results in largerinternal temperatures and thinner lithospheres, whereas lowreference viscosities tend to better cool the mantle, resultingin thinner lithospheres.

To provide a more quantitative expression for thesetrends, we used the viscosity profiles to detect thelithosphere-asthenosphere boundary (LAB). The latter wasdefined as being located where the viscosity is 20% higherthan the smallest viscosity value found below the litho-sphere. From this, the following scaling laws for lithospheric

X - 12 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

thickness and temperature (generated in all simulations)were obtained

dlit(km)

500= 1.093

(lnDp

A

)0.852(η0

η′

)0.100

(25)

Tlit(K)

1600= 1.062

(lnDp

A

)−0.0740(η0

η′

)0.0333

, (26)

where dlit and Tlit are in units of km and K, respectively,and A=ln 0.01 and η′=1020 Pa·s throughout. Figure 9 alsoshows that the internal viscosity and grain size (second andthird panels) depend on both Dp and η0. At a depth of1000 km, the scaling laws for internal viscosity ηint and grainsize dgrain were found to be

ηint

1020= 3.69

(lnDp

A

)3.18(η0

η′

)0.459

(27)

dgrain

500= 1.079

(lnDp

A

)0.108(η0

η′

)0.102

, (28)

where ηint and dgrain are in units of Pa·s and m, respectively.These relations indicate that both viscosity and grain sizeincrease with increasing Dp and η0.

The present-day average surface heat flux (QHF) obtainedin our simulations ranges from 15 to 25 mW/m2. This is ingood agreement with the estimates derived by Plesa et al.[2016] where mean surface heat fluxes in the range 23–27mW/m2 were obtained. An important observation here isthat partitioning of heat producing elements and a low vis-cosity tend to slightly increase the heat flux, as summarizedby the scaling law:

QHF

25= 0.730

(lnDp

A

)−0.393(η0

η′

)−0.0158

, (29)

where the heat flux is expressed in mW/m2. The above fivescaling laws are illustrated in figure 1 in appendix B.

Figure 9 and figure 10 show that the current averagecrustal thickness, which ranges between 33 and 81 km [Wiec-zorek and Zuber , 2004; Neumann et al., 2004] (gray shadedarea in figure 10), and the geophysically-derived tempera-ture profiles obtained in this study (gray shaded area infigure 9), can be fit using the experimentally-determined Dpin the range of 0.001 and a reference viscosity in the rangeof 1019–1020 Pa·s. In particular, inefficient partitioning ofheat producing elements (Dp=0.1), irrespective of the valueof η0, can be ruled out because mantle temperatures are toohigh by several hundred degrees relative to the inverted pro-files and so is crustal production (>>200 km, not shown infigure 10). The preferred values of Dp and η0 that are consis-tent with the geophysical predictions, correspond to present-day grain sizes in the sub-mm range (0.4—0.5 mm). In spiteof the considerable overlap over the entire mantle, temper-atures in the lithosphere derived from geophysical modelsappear to be higher than those obtained in the geodynami-cal calculations including grain-size evolution. Future stud-ies should address the effects of dynamic recrystallization,which was not included in our simple grain growth equationand the effects of an initial heterogeneous grain size distri-bution as the grain size influences the viscosity and hencecooling behavior and temperature. The geodynamics results(figures 9 and 10) supporting the scaling laws (Eqs. 25–29)are summarized in table 5.

7. Summary and Conclusion

We have shown that a grain size and frequency-dependentvisco-elastic model (extended Burgers rheology) based onlaboratory deformation of melt-free polycrystalline olivine

is capable of matching a set of geophysical observations forMars. In particular, we have shown that mantle melting isnot required to reproduce the relatively dissipative Martianinterior. Starting from a set of initial geochemically-basedbulk Martian compositions, we find strong evidence for arelatively large (radius∼1640–1740 km) Fe core containing13.5–16 wt% S that appears to be entirely liquid at presentgiven current constraints, data, and modeling assumptions.The results preclude the presence of a solid inner core onaccount of the determined core S content, which is close tothe eutectic composition. The model geotherms obtainedhere remain below the present-day dry Martian solidus in-ferred by Kiefer et al. [2015] and indicate CMB tempera-tures around 1560–1660 ◦C. From the determined core sizeand CMB pressure and temperature, the presence of a lowermantle transition, equivalent of the “660-km” seismic dis-continuity in the Earth seems unlikely. The bulk Martiancompositions derived here are generally chondritic with aFe/Si (wt ratio) of 1.63–1.68. Grain size estimates rangefrom sub-mm to 5 mm, but are generally <1 mm, and wellwithin the range observed on Earth.

Following this, we performed a number of thermo-chemical evolution simulations with the purpose of fit-ting the geophysically-derived results (specifically mantlegeotherm, lithospheric thickness, and grain size). Thisserves to show that the inversion results can be used intandem with geodynamic simulations to identify plausiblegeodynamic scenarios and parameters. For a reasonable setof radioactive element partition coefficients between crustand mantle around 0.001 and reference viscosities (1019–1020Pa·s), the geodynamic models, based on 2D models andcovering 4.5 Ga of Martian history, were able to reproducecurrent areotherms, crustal and, in part, lithospheric thick-nesses, and grain sizes. The results seem to suggest thatMars could be volcanically active at present in accord withearlier suggestions based on other geodynamical studies andpresent-day surface observations. The geodynamic modelsalso predict a present-day mean surface heat flow between15–25 mW/m2, in excellent agreement with the recent heatflow estimates by [Plesa et al., 2016].

We should emphasize that the point of the geodynam-ical simulations was not to accurately reproduce the dy-namical evolution of Mars, but rather to provide 1) first-order insights on the long-term evolution (∼4.5 Gyr) interms of a few crucial parameters such as crustal produc-tion, mantle temperature, and grain size growth and 2) tosee if these can be varied within reasonable bounds, while si-multaneously fitting the present-day geophysical predictionsobtained here. In this context, we have been made awarethat more recent estimates of the second degree tidal Lovenumber have become available that are ∼5–10% larger [e.g.,Konopliv et al., 2016; Genova et al., 2016]. Everything be-ing equal, a larger k2 will result in a larger core size, but,by Eqn 9, increasing k2 also results in a larger Q, i.e., lessdissipation, which could counter the aforementioned effect.This will have to be considered in more detail in the future.

Being anhydrous and melt-free, the models constructedhere are end-member cases in that presence of e.g., waterwould lower the solidus [e.g., Pommier et al., 2012], producemelt and increase dissipation, which would have to be offsetby an overall decrease in mantle temperature. The ther-mal evolution of Mars depends crucially on mantle viscos-ity, which is also determined by conditions of temperature,pressure, and water content. Even if present at small lev-els (tens of ppm), water can lead to a reduction of effectiveviscosity and Q, thereby enhancing dissipation [Hirth andKohlstedt , 1996; Karato and Jung , 1998; Mei and Kohlst-edt , 2000; Karato, 2013]. As discussed by Filiberto et al.[2016], presence of 0.02 wt% water in the mantle source re-gion, would lower viscosity by a factor of 36 relative to a

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 13

nominally dry (≤10 ppm water) mantle with an associatedsignificant increase in convective vigor as observed in var-ious dynamical evolution studies [e.g., Hauck and Phillips,2002; Li and Kiefer , 2007; Morschhauser et al., 2011; Ogawaand Yanagisawa, 2012; Ruedas et al., 2013b]. The combinedeffect of this would be to lower the mantle geotherm belowwhat has been obtained here. Accordingly, the present man-tle geotherms represent upper bounds.

Although water in some shergottite magmas has been ob-served [McCubbin et al., 2012], the present-day water con-tent of the Martian mantle is not well constrained. It likelycontains less water than Earth with a probable value ofaround 300 ppm [Taylor , 2013]. This estimate assumes nowater loss during or after accretion due to impact heating,chemical reactions, and magma ocean crystallisaton [Wankeand Dreibus, 1994; Elkins-Tanton et al., 2005]. For now, theeffect of water needs to be better quantified and understoodbefore it can be modeled properly. In addition, current lackof thermodynamic data accounting for the influence of wa-ter on mantle phase equilibria further prohibits quantitativemodeling.

Presence of melt very much mimics the effect of water,i.e., lowers Q and increases dissipation. While there is someevidence for magmatic activity that might have occured un-til relatively recently [e.g., Neukum et al., 2004; Niles et al.,2010], Mars is unlikely to be molten on the global scale atpresent. Even if localized pockets or regions inside Mars mayremained molten up until the present, these are unlikely toaffect global dissipaton. What can be argued based on theresults here and those of Nimmo and Faul [2013] is thatlaboratory-based visco-elastic dissipation models do not re-quire melt to fit current observations. The effect of a smallmelt fraction on dissipation has been studied in the labo-ratory [e.g., Faul et al., 2004; McCarthy and Takei , 2011;Takei , 2017]. Melt acts to enhance dissipation and renderit essentially frequency-independent across the seismic fre-quency band [Faul et al., 2004]. However, modeling its in-fluence presents, on a par with water, an equally challengingproblem that will be left for a future study.

In this context, earlier studies based on viscoelastic mod-els [e.g., Bills et al., 2005; Castillo-Rogez and Banerdt , 2013],that relied on different rheologies (Maxwell and Andrade),found mantle viscosities around 1014–1016 Pa s that appearto be inconsistent with current estimates of Earth’s man-tle viscosity (∼1019–1021 Pa s). This suggested that eitherMars does not behave as Maxwell bodies or that a significantamount of partial melt is needed to lower the viscosity to theaforementioned values. The latter possibility has gained instrength in the case of the Moon, where evidence in sup-port of partial melt in the deep lunar interior has steadilyaccumulated [e.g., Williams et al., 2001; Weber et al., 2011;Efroimsky , 2012a, b; Harada et al., 2014; Khan et al., 2014;Williams and Boggs, 2015], although Nimmo et al. [2012]find that melt is not strictly required to match the obser-vations with the caveat that the frequency-dependence ofQ (i.e., α) at tidal periods is opposite in sign to what hasbeen observed through lunar laser ranging [e.g., Williamset al., 2014]. As pointed out by Efroimsky [2012a], though,the lunar laser ranging-based observation of a negative α(i.e., Q ∼ ωα instead of the common case where Q ∼ ω−α)is entirely in accordance with a low-viscosity attenuatingregion in the lunar interior. Employing a homogeneous in-compressible spherical model of the Moon Efroimsky [2012a]and Efroimsky [2012b], showed that the apparent decreaseof k2/Q with period observed through lunar laser rangingcould be explained with a function equivalent to a single re-laxation time model. Using a viscosity around (∼1015 Pa s),k2/Q was shown to peak in the tidal band in the vicinity ofthe tidal periods (see e.g., figure 1 in Efroimsky [2015]). Weleave it for a future study to consider the impact of differ-ent types of rheological models in the context of the inverseproblem posited here.

The present approach is not fully self-consistent in thatonly a single mineral (olivine) and a single composition

(Fo90) has had its viscoelastic behaviour characterized;nonetheless, Nimmo and Faul [2013] argue that the errorsintroduced by focusing exclusively on Fo90 olivine are likelyto be small, since the behaviour of olivine is believed to bebroadly representative of the viscoelastic behaviour of theupper mantle. Compostitional effects will mainly influencethe unrelaxed modulus (pers. comm., Ian Jackson, 2014).Based on a compilation of existing data, Karato [2013] sim-ilarly argues that addition of orthopyroxene to olivine haslittle effect on anelasticity. In line herewith, and becauseolivine is the dominant mineral (∼60 % by volume) through-out most of the Martian mantle, neglect of the contributionfrom other phases is expected to less likely result in signif-icant changes. In view of the different Mg#s of Earth andMars, a possible exception is the iron content that could po-tentially change the dissipative properties of olivine [Zhaoet al., 2009]. We leave it for a future treatment, based on anew and expanded experimental database, to consider theseeffects in more detail. Also, while this study is based on themost appropriate experimental constraints available, theiruse involves an extrapolation beyond the available experi-mental frequencies as well as an extrapolation in grain size.Further experiments closer to the Martian parameter spacewe explore would be valuable future work.

In spite of these caveats, the approach outlined here iscapable of making significant predictions (summarized in ta-ble 4) that provide many quantitative insights. Moreover,the study proposed here will help quantify additional fu-ture requirements – from the acquisiton of new experimentaldata to modeling aspects – that will be in need of furtherattention to extend the predictions to increasingly complexmodels.

Last, but not least, the predictions made here, will ul-timately be tested with the upcoming InSight mission thatwill perform the first in situ measurements of the interiorof Mars through the acquisition of seismic, heat flow, andgeodetic data [Banerdt et al., 2013]:

1) Analysis of the seismic data to be returned from In-Sight will rely on more “classical” global seismology tech-niques in the form of travel-time tables for various seis-mic phases, surface-wave dispersion, receiver functions, nor-mal modes, and waveforms [e.g., Okal and Anderson, 1978;Zheng et al., 2015; Panning et al., 2016]. These data willshed light on the interior physical structure of crust, man-tle, and core, including velocity gradients and dicontinuitiesassociated with mineral phase transitions and/or chemicalboundaries. Moreover, the data will help constrain Martianseismicity and tectonism and, not least, meteorite impactrates [e.g., Golombek et al., 1992; Knapmeyer et al., 2006;Teanby and Wookey , 2011].

2) On a par with the importance of acquiring seismic data,measuring heat flow with the heat flow probe (HP3) onboardInSight [Spohn et al., 2014] cannot be overestimated, be-cause it will allow for an independent means of distinguish-ing between the various bulk compositional models. Specifi-cally, by combining the heat flow measurement with numer-ical estimates of the Urey ratio [e.g., Plesa et al., 2015], itwill be possible to constrain the heat production rate andthus bulk abundance of heat producing elements in the in-terior of Mars. Camparison of the latter with the proposedcompositional models for Mars provides the necessary tool.

3) In the context of improving estimates of core structure,the RISE (Rotation and Interior Structure Experiment) in-strument onboard InSight will prove important [e.g., Folkneret al., 2012; van Hoolst et al., 2012]. Through precise radiotracking of the landed spacecraft, RISE will be able to mea-sure the precession rate and through it allow for a moreaccurate estimation of the polar moment of inertia In par-ticular, for a fluid Martian core, the nutation of the spinaxis can be resonantly amplified and allow for an indepen-dent estimation of the polar moment of inertia of the core.

X - 14 KHAN ET AL.: ON THE BULK COMPOSITION OF MARS

From precise estimation of the period of the free core nu-tation, the moment of inertia of the core can be estimated.These parameters are crucially sensitive to core size, shape,and state. RISE observations over a Martian year shouldenable a clear detection of the nutation signature and thuscore parameters.

These data sets, individually and in combination, willprove to be the Rosetta stone for unraveling the thermo-chemical structure and evolution of Mars.

Acknowledgments. We thank Walter Kiefer and Ian Jack-son for comments that helped improve the clarity of themanuscript. We also thank Michael Efroimsky for valuable com-ments on tidal deformation and additional suggestions for im-proving the manuscript. Finally, we thank Jeff Taylor and Gre-gor Golabek for helpful comments on an earlier version of thismanuscript. A.K. and D.G. would like to acknowledge supportfrom the Swiss National Science Foundation (SNF-ANR project157133 “Seismology on Mars”). Computations were performedon the ETH cluster Euler. Models can be downloaded fromhttp://jupiter.ethz.ch/~akhan/amir/Models.html.

Appendix A: Thermoelastic properties ofthe core

The thermoelastic properties of the liquid core alloy as afunction of pressure (P) and temperature (T) are calculatedfollowing Dumberry and Rivoldini [2015], with the excep-tion of eq. A12, and are contained here for completeness.In this approach, it is assumed that the properties of thealloy can be calculated from the thermoelastic properties ofliquid Fe and liquid Fe-10wt%S (hereafter referred to as Feand FeS10) and that both components mix ideally. Basedon this, the molar volume V of the solution is given by

V =

2∑i=1

χiVi, (A1)

where χ1 = 1 − χ2 is the molar fraction of Fe and χ2 themolar fraction of FeS10. The molar concentration of sulfurin FeS10 is related to the weight fraction of sulfur in thesolution, XS, by

χ2 =XS

χS10

MFe

MS(1−XS) + MFeXS(A2)

where MFe = 55.845 g/mol and MS = 32.065 g/mol are themolar masses of iron and sulfur, respectively, and χS10

=0.162137 is the atomic fraction of sulfur in FeS10.

The thermal expansivity α, isothermal bulk modulus KT,and pressure derivative KT′ of the solution can be derivedfrom Eq. A1 through the application of standard thermo-dynamic relations

α V =

2∑i=1

χiαiVi, (A3)

V

KT=

2∑i=1

χi

ViKT,i

, K′T = −1 +K2

T

V

2∑i=1

χi

ViKT,i

(1 + K′T,i).

(A4)

In order to compute the isentropic temperature gradientin the core (Eq. 17), the adiabatic bulk modulus KS andthe Gruneisen parameter γ are also required. KS is relatedto the isothermal bulk modulus KT by the following relation[e.g. Poirier , 2000]:

KS = KT (1 + αγT) . (A5)

To compute γ, we rely on the formulation of Vashchenkoand Zubarev [e.g. Poirier , 2000]:

γ =12K′T − 5

6+ 2

9PK−1

T

1− 43PK−1

T

. (A6)

To compute thermoelastic properties for each componentat P and T, we first perform an isothermal compression fromthe reference conditions (P0,T0) to (P,T0), followed by iso-baric heating to (P,T). For the isothermal compression atT0, we employ a third-order finite strain Birch-Murnaghanequation-of-state [Stixrude and Lithgow-Bertelloni , 2005a]:

P =− 3ε(1− 2ε)52

[KT,0 −

3

2KT,0(K′T,0 − 4)ε

], (A7)

KT =(1− 2ε)52

[KT,0 −KT,0

(3K′T,0 − 5

)ε+

27

2KT,0(K′T,0 − 4)ε2

],

(A8)

K′T =K′T,0 +

[3K

′2T,0 − 21K′T,0 +

143

3

]ε, (A9)

where ε = 12

(1− β−

23

)is the Eulerian strain, β = V (P)/V0,

volume, isothermal bulk modulus, and pressure derivativeat (P,T0) are denoted by V (P), KT(P) and K′T,0(P), re-spectively, and zero-subscripted quantities are evaluated atreference conditions.

Volume at (P,T) is obtained from the definition of thethermal expansivity [e.g. Poirier , 2000]:

V (P,T) = V (P) exp [α(P)(T− T0)] . (A10)

The expression for α(P) follows from the definition of theAnderson-Gruneisen parameter δT [e.g. Poirier , 2000]

α(P) = α0 exp

[−δTq

(1− βq)], (A11)

where q is a material-dependent parameter. The temper-ature dependence of the isothermal bulk modulus KT atpressure P can be derived from Eqs. A10 and A11

KT(P,T) =KT(P)

1 + (T− T0)α(P)δTβq. (A12)

Finally, we assume that K′T(P) is independent of tempera-ture. The relevant material properties for the equations ofstate for Fe and FeS10 are summarized in the table below.

Appendix B: Geodynamical simulations

Thermo-chemical compressible convection is studied us-ing the code StagYY in a 2D spherical annulus geometry[Tackley , 2008; Hernlund and Tackley , 2008]. The equations

Table 1. Equation of state parameters for core components.Fe Anderson and Ahrens [1994] and Fe − 10wt%S Balog et al.[2003]. The thermal expansivity of Fe − 10wt%S is calcu-lated from the volume and thermal expansivity of FeS Kaiuraand Toguri [1979] at 1923K (V = 12.603cm3/mol, α =16.5 × 10−5K−1) by applying Eqs. A1 and A3. ∗ Since δTis not available for Fe and Fe − 10wt%S, we made use of thefollowing approximate relation [Poirier , 2000, e.g.]: δT ≈ K′T.∗∗ is taken from Helffrich [2012].

Components T0 [K] V [cm3/mol] α [10−5K−1] KT [GPa] K′T δT qFe 1811 7.96 9.2 85.3 5.9 5.9 1.4

Fe − 10wt%S 1923 9.45 10.0 63.0 4.8 4.8∗ 1.4∗∗

KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 15

of momentum, mass and energy conservation are solved us-ing a parallel direct solver (MUMPS) available in the PETScpackage [Amestoy et al., 2000]. We use the finite differenceapproximation on a staggered grid [Harlow and Welch, 1965]with a radially varying resolution (higher in the top andbottom boundary layers). The domain is discretized by 64times 512 cells. Free slip boundary conditions are imposedon all domain boundaries. The surface temperature is fixedto 220 K and the initial core temperature is 2500 K.

Lagrangian markers are advected through the mesh us-ing second-order divergence-free spatial interpolation of thevelocities and a fourth-order Runge-Kutta scheme. Eachtracer carries several quantities such as composition, tem-perature, grain size, and radiogenic heating rate, among oth-ers. Numerical diffusion related to advection is limited bythe use of tracers. Our petrological model considers solidand molten rocks as being a linear combination of basalt-eclogite (crustal material) and harzburgite (depleted man-tle). The primordial mantle starts with an initial petrolog-ical composition of 15 % basalt (100 % pyroxene-garnet)and 85 % harzburgite (a mixture of 55 % olivine and 45 %pyroxene-garnet). For numerical efficiency the compositionis stored and transported on tracers, while the evolution ofthe melt fraction is computed on the mesh field (which gen-erates new molten tracers in each corresponding cell). Acomposition field is computed at cell centers by averaging ofthe tracers within the cell. The melt fraction is computed atevery time step comparing the pressure-temperature condi-tions to a composition-dependent solidus function, consid-ering a latent heat of 600 kJ/kg. In case melting occurs atless than ∼600 km depth, molten tracers of fully basalticcomposition are transported either to the top of the domain(eruption) or to the bottom of the crust (intrusion). Thetemperature of all erupted tracers is set to surface temper-ature, which tends to form a cold lithosphere. The temper-ature of intruded tracers (at the bottom of the crust) takesonly adiabatic decompression into account, which tends toproduce a warm lithosphere. The column of material be-tween the melt source region and the intrusion or eruptionlocation is moved downwards to conserve mass. The solidresidue left behind by the eruption-intrusion procedure ismore harzburgitic than the initial solid. For more detailsregarding implementation the reader is referred to Naka-gawa and Tackley [2004, 2012]. The density in each cellis computed as the sum of pressure, thermal and compo-sitional components, including solid-solid phase transitions.Olivine and pyroxene-garnet phases are treated separately,which enables accounting for the density increase associatedwith the basalt-eclogite phase transition.

Plastic yielding in the lithosphere is computed using adepth-dependent yield stress τy following Byerlee’s law [By-erlee, 1978]:

τy = f · P, (B1)

where P is pressure and f is a friction coefficient (we chosef=0.35 in the simulations here). The effective viscosity iscomputed from

ηeff = min(η(P,T,dgrain

),τy2ε

), (B2)

where ε is the second invariant of the strain-rate tensor. Inthe present study, viscosity depends neither on melt fractionnor on composition.

As we solve for compressible convection, adiabatic tem-perature, thermal conductivity, density, thermal expansiv-ity, and heat capacity are all pressure-dependent. We usea third-order Birch-Murnaghan equation of state which re-lates bulk modulus to pressure [see Tackley et al., 2013, fordetails]. With each mineral phase is associated individualphase transition depths and physical properties so that solid

olivine and pyroxene-garnet densities, for example, smoothlyincrease with depth, but also contain discontinuities at thephase transitions [Rozel et al., 2017]. Finally, melt densityincreases smoothly with pressure.

Figure 1. Lithosphere thickness, LAB temperature, in-ternal viscosity (at 1000km depth), internal grain size,and surface heat flux at the end of the simulations (equiv-alent to 4.5 Gyr of evolution), scaled using the approachdescribed in section 6.2.

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A. Khan, Institute of Geophysics, Swiss Federal Instituteof Technology, Sonneggstrasse 5, 8092 Zurich, Switzerland(amir.khan@erdw.ethz.ch).