A Geophysical Perspective on the Bulk Composition of Mars A

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 inertia for chemical composition, thermal state, and interior structure. The inversion com- bines phase equilibrium computations with a laboratory-based viscoelastic dissipation model. The rheological model, which is based on measurements of anhydrous and melt- free olivine, is both temperature and grain size sensitive and imposes strong constraints on interior structure. The bottom of the lithosphere, defined as the location where the conductive 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 ∼0.75 in agreement with earlier geochemical estimates based on analysis of Martian meteorites; 2) absence of bridgmanite- and ferropericlase-dominated basal layer; 3) core compositions (13.5–16 wt% S), core radii (1640–1740 km), and core-mantle-boundary temperatures (1560–1660 ◦C) that, together with the eutectic-like core compositions, suggest the core is liquid; and 4) bulk Martian compositions that are overall chondritic with a Fe/Si (wt ratio) of 1.63–1.68. We show that the inversion results can be used in tandem with geodynamic simulations to identify plausible geodynamic scenarios and parameters. Specifically, we find that the inversion results are reproduced by stagnant lid convection models for a range of initial viscosities (∼1019–1020 Pa·s) and radioactive element partitioning between crust and mantle around 0.001. The geodynamic models predict a mean surface heat flow between 15–25 mW/m2.

1. Introduction For Mars, an increasing amount of observations, both in situ and from laboratory analyses of Martian meteorites and Knowledge of the internal constitution of the planets is cosmochemical material, have become available [e.g., Nor- crucial to our understanding of the origin and evolution of man, 1999; Taylor, 2013]. In addition, data and results our solar system. Major constraints can be placed on plan- from geophysical modeling and mantle convection studies etary accretion, differentiation, and mantle evolution from that bear on interior structure [e.g., Sohl and Spohn, 1997; knowledge of bulk chemical composition [e.g., Taylor, 1999]. Yoder et al., 2003; Neumann et al., 2004; Wieczorek and By far the largest insights into the physical structure of the Zuber, 2004; Verhoeven et al., 2005; Khan and Connolly, Earth have come from geophysical analyses, and seismology 2008; Rivoldini et al., 2011; Baratoux et al., 2014; Hauck in particular. However, the dearth of geophysical data perti- and Phillips, 2002; Elkins-Tanton et al., 2003; Williams and nent to the interior of other planets has made this approach Nimmo, 2004; Grott and Breuer, 2008; Kiefer and Li, 2009; less instructive and a significant part of current knowledge Ruedas et al., 2013a; Rai and Westrenen, 2013; Plesa et al., on mantle and bulk composition of the terrestrial planets 2015] have allowed us to refine our understanding of plane- derives from geochemical/cosmochemical and isotopic anal- tary processes from a Martian vantage point; yet much re- yses of rocks and primitive solar system material [e.g., Ring- mains to be understood. Among others, how well do we wood, 1979; Taylor, 1980; Drake and Righter, 2002; Palme really know the composition of Mars and what is the re- and O’Neill, 2003; Taylor et al., 2006]. In addition, geode- lation of this to core size and state and how can current tic data in the form of Doppler observations obtained from estimates be improved? Does Mars contain the terrestrial ranging to orbiting and landed spacecraft (Viking, Mars equivalent of a lower mantle layer and what is its role in Pathfinder, Mars Global Surveyor, Mars Odyssey, and Mars the evolution of the core? In the broader context of planet Reconnaissance Orbiter) over more than a decade, resulted formation, what are the implications of current bulk planet in the recognition that Mars had differentiated into a silicate compositions for mixing of material throughout the inner mantle and an Fe-rich core [e.g., Folkner et al., 1997; Yoder solar system during accretion of the terrestrial planets? et al., 2003; Neumann et al., 2004; Bills et al., 2005; Lainey The aim of this study is to improve current constraints et al., 2007; Konopliv et al., 2006, 2011, 2016; Genova et al., on Mars bulk composition and thermal state from inversion 2016]. of currently available geophysical data (mean mass and moment of inertia, global tidal dissipation, and magnitude of tidal response). To this end we will build upon our previous 1Institute of Geophysics, ETH Zürich, Switzerland. work [e.g., Khan et al., 2007] to 1) invert different geophysi- 2Institute of Geochemistry and Petrology, ETH Zürich, cal data sets directly for compositional and thermal parame- Switzerland. ters, and combine this with 2) a method for computing tidal 3Royal Observatory of Belgium, Brussels, Belgium. dissipation within a planet using the laboratory-based grain 4Department of Earth and Planetary Sciences, UC Santa size and frequency dependent viscoelastic model of Jackson Cruz, California, USA and Faul [2010], which was shown by Nimmo and Faul [2013] 5German Aerospace Center (DLR), Berlin, Germany. to be highly sensitive to mantle temperatures in Mars. The main point is to link the dissipation model, which is based on laboratory experiments on anhydrous melt-free polycrys- Copyright 2017 by the American Geophysical Union. talline olivine, with thermodynamic phase equilibrium com- 0148-0227/17/$9.00 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. both approaches is the notion that Mars is considered to This approach has a number of advantages: (1) It anchors have accreted from different material which condensed from temperature, composition, dissipation, and discontinuities the solar nebular, including highly volatile-depleted and re- that are in laboratory-based forward models; (2) it per- duced components and oxidized, volatile-rich condensates. mits the simultaneous use of geophysical inverse methods Exemplary of the first line of arguments was the approach to optimize profiles of physical properties (e.g., shear mod- of Dreibus and Wänke (hereafter DW model) [Dreibus ulus, dissipation, density) to match geophysical data; and and Wänke, 1984; Dreibus and Wänke, 1985; Dreibus and (3) it is capable of making quantitative predictions that can Wanke, 1987]. The DW model has become the standard be tested with the upcoming Mars Insight mission to be model for Mars and has served as reference in many sub- launched in May 2018 [Banerdt et al., 2013] (InSight will sequent studies [e.g., Bertka and Holloway, 1994; Bertka emplace a seismometer, a heat flow probe, and a geode- and Fei, 1997; Matsukage et al., 2013; Collinet et al., 2015]. tic experiment on Mars) and/or against results and data The essence of this model lies in the assumption that Mars from other studies (e.g., geodynamical simulations of planet accreted heterogeneously from two different cosmochemi- evolution, petrological analyses of Martian meteorites, and orbit-imaged surface chemistry and crustal thickness). cal reservoirs, i.e., a highly reduced component during the As a means of illustrating these points, we show that early stages, followed by the addition of oxidized, volatile- the inversion results can be used in combination with geo- rich material during the final stages of accretion. The re- dynamic simulations to identify plausible geodynamic sce- duced component is assumed to have CI-chondritic element narios and parameters. This coupling of geodynamics and abundances that are more refractory than Mn, i.e., ele- geophysics has the advantage that it anchors geodynamic ments having higher condensation temperatures than this models in geophysically-constrained results. These simula- element, whereas the late stage oxidized component is en- tions, based on the StagYY code [Tackley, 2008], explicitly tirely CI-chondritic. Thus, a central tenet inherent of the consider grain size evolution and suggest that stagnant lid DW model is that refractory elements follow CI-chondritic convection is capable of explaining the various observables proportions. Chemical analyses of the Martian meteorites (crustal and lithospheric thickness and present-day grain known at that time indicated that their Mn content is close size and mantle temperatures) assuming reasonable initial to CI-chrondritic; thus this element became key to the bulk estimates of viscosity, radioactive element partitioning, and chemical composition of Mars. To derive the composition of initial temperature field. The models are also able to pre- the mantle, fractionation trends that occur during igneous dict the present-day mean surface heat flow, which can be processes are invoked, which allow conclusions about the compared to surface observations to be made with InSight Shergottites source region to be drawn. Recently, Taylor [e.g., Plesa et al., 2016]. [2013] re-assessed the primitive mantle composition with a In the following we discuss constraints that derive from similar strategy as DW using a much larger meteoritic record geochemical and cosmochemical analyses and summarize (∼60 versus 6). For the elements of interest here (major previous geophysical analyses that bear on the interior of and minor but no trace elements), the Taylor bulk silicate Mars (section 2); enumerate the geophysical data employed Mars model is almost identical in terms of elemental con- in the present analysis and detail the computation of crust, centrations to that of DW. However, there is considerable mantle, and core properties (sections 3) and numerical mod- difference in the assumption of the S content of the planet. eling aspects of solving the forward and inverse problem posited here (section 4); and, finally, describe and discuss In the model of Taylor [2013], the core is significantly more results. S-rich compared to DW. Morgan and Anders [1979] derived a chemical composition using a similar methodology as DW, i.e., to scale the 2. Background concentration of “unknown” to “known” elements (or known elemental ratios) based on volatility trends. It should be 2.1. Geochemical perspective noted that Morgan and Anders [1979] developed their model Our knowledge on the chemistry of the Martian mantle before the general acceptance that Martian meteorites orig- and core originates to a large extent from the chemical and inated from Mars and e.g., their “known” K/U ratio was isotopic compositions of a class of basaltic meteorites that taken from the Soviet orbiter Mars 5 mission [Surkov, 1977] are believed to be fragments ejected from the Martian sur- with several corrections applied based on various assump- face on meteoritic impacts and collectively known as SNC tions. As pointed out by Taylor et al. [2006], the value of meteorites (Shergottites, Nahklites and Chassignites) [e.g., K/U and K/Th as a proxy for the proportions of moder- McSween, 1985, 1994], hereinafter referred to as Martian ately volatile versus refractory elements was later corrected. meteorites. A key argument for this hypothesis is the con- This likely explains some systematic differences between the centration of entrapped gases within shergottite meteorites Morgan and Anders composition and other estimates. that match the values measured for the Martian atmosphere As an illustration of the second approach to derive the by the Viking mission [Bogard et al., 2001]. The Martian Martian bulk composition, is to consider the oxygen isotopic meteorites also exhibit relatively young crystallisation ages systematics of the Martian meteorites [e.g., Lodders and Feg- [Stolper and McSween, 1979]. Shergottites have ages in the ley, 1997; Sanloup et al., 1999; Mohapatra and Murty, 2003]. range of 170–600 Ma [McSween and McLennan, 2014, and Oxygen is by far the most abundant element in the silicate references therein], which require a sufficiently large planet proportions of the terrestrial planets and the Martian mete- as parent body because it is physically implausible, due 17 18 to thermal constraints, to maintain magmatic activity on orites form a fractionation line in a δ O/δ O three-isotope asteroid-sized bodies so late in the solar systems history. plot which is distinctively different from terrestrial or other Different lines of arguments have been used to derive the meteoritic trends. The aforementioned two studies are based chemical composition of the Martian mantle and core from on the assumptions that Mars is the parent body of the Mar- the chemical compositions of the Martian meteorites. These tian meteorites and that their isotopic compositions can be can be categorized in two general groups that consider 1) the described by mixing different classes of meteorites. Based abundance of refractory elements in the Martian meteorites on this, the oxygen isotope composition of Mars are deduced [Dreibus and Wänke, 1984; Dreibus and Wänke, 1985; Hal- from mass balancing various meteoritic end-members. The liday et al., 2001; Longhi et al., 1992; Taylor, 2013] and 2) concentrations of all other elements are then simply defined oxygen isotope systematics [Lodders and Fegley, 1997; San- by the chemical composition and mass fraction of the var- loup et al., 1999; Burbine and O’Brien, 2004]. Common to 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- yet to be formulated. The possibility also exists that FeO gen content and redox equilibria, which leaves a proportion is not homogeneously distributed throughout the mantle, as of Fe and the majority of Ni and S in the metallic core. indicated by systematic differences in the surface distribu- The differences in the models of Lodders and Fegley [1997] tion of Fe detected by the gamma ray spectrometer onboard and Sanloup et al. [1999] are determined by the choices of Mars Odessey [Taylor et al., 2006]. The questions whether end-member components (meteorite classes) but generally alternative bulk Mars and potentially low-FeO mantle mod- lead to comparable results for major and minor elements. els would satisfy the physical properties of Mars and whether Lodders and Fegley [1997] invoked three end-members 1) the mantle is heterogeneous with regard to the distribution mean values of reduced and volatile- depleted H- (ordinary) of major elements are beyond the scope of this paper. chondrites; 2) moderately oxidized CV-chondrites; and 3) highly oxidized and volatile-rich CI-chondrites. These end- 2.3. Geophysical perspective members span a compositional triangle in the three-isotope δ17O/δ18O diagram around the Martian meteorites. The Geophysical analyses have, for the most part, relied on re- model of Sanloup et al. [1999] (EH45:H55) is based on a mix- sults obtained from geochemical-cosmochemical studies with ture of two end-members, a mean value of highly reduced the purpose of predicting the geophysical response of these EH- (enstatite) chondrites (45%) and a hypothetical, but as chemically-derived models. Geophysical and experimental yet un-sampled H- (ordinary) chondrite component (55%) approaches are to a large extent based on the DW model that is located on an isotopic fractionation line along with composition with the goal of determining mantle mineralogy LL-, L-, and H-chondrites. This hypothetical end-member and density. Combined with equation-of-state (EOS) mod- is required to produce a mixing line with the EH-chondrite eling this allows for determination of a model density pro- component that passes through the Martian meteorite oxy- file for the purpose of making geophysical predictions that gen isotope mean value. can be subsequently compared to observations. The studies by Bertka and Fei [1997, 1998] represents the experimental 2.2. Summary of geochemical models approach, while numerical approaches with varying degree of sophistication (forward/inverse modeling, number of geo- To investigate the effect of varying bulk chemical com- physical observations, parameterized phase diagram/phase- positions of Mars, we have selected the five possible bulk equilibrium computations) are embodied in the studies of compositions discussed above. We consider the composi- Longhi et al. [1992]; Kuskov and Panferov [1993]; Mocquet tions proposed by 1) Dreibus and Wänke (1984) and 2) et al. [1996]; Sohl and Spohn [1997]; Sohl et al. [2005]; Ver- Taylor (2013) as representative of the CI refractory ele- hoeven et al. [2005]; Zharkov and Gudkova [2005]; Khan and ment approach but with different core Fe/S ratios; 3) the Connolly [2008]; Rivoldini et al. [2011]; Wang et al. [2013]. compositions of Lodders and Fegley (1997) and 4) Sanloup Based on the available geophysical data at the time (prin- (1999) (EH45:H55) as representative of the oxygen isotope cipally the moment of inertia), Bertka and Fei [1998], for approach; and 5) the composition of Morgan and Anders example, concluded that the CI chondrite accretion model [1979] despite the fact that some of the key elemental ra- for deriving Mars is incompatible with a DW-like mantle tios in that model have been revised. Nevertheless, this composition. model is distinctively different in core composition by being According to the DW model there is evidence for chal- significantly depleted in S by a factor of three to five com- colphile element depletion in the Martian meteorites, which pared to other models. Table 1 summarises the five model suggests that the otherwise Fe-Ni-rich core contains a sub- compositions. To facilitate treatment, some simplifications to the proposed mantle and core compositions are made; stantial amount of a sulfide component (S need not be the the mantle is considered as a six component system (MgO, only alloying element; Zharkov and Gudkova [2005], for example, considered H in addition to S in the core). This CaO, FeO, MgO, Al2O3, SiO2, and Na2O) to be compati- ble with the thermodynamic database (section 3.3) and the observation is important because an alloying element acts core-system (section 3.5) is reduced to a simple Fe-S binary. to influence the physical state, while simultaneously provid- Differences in the modified bulk silicate Mars composi- ing information on core temperature. In this context, there tions (table 1) become apparent when considering elemen- is strong evidence from measurements of the deformation of tal ratios such as Mg/Si (molar). Mg/Si ranges from 1.07 the planet due to solar tides that Mars’ core or parts of it (near CI-chondritic) to 0.86 between the studies of Morgan are currently liquid [e.g., Yoder et al., 2003; Konopliv et al., and Anders [1979] and Sanloup et al. [1999]. These latter 2011; Genova et al., 2016] as had been predicted earlier [e.g., studies also show the widest spread in Mg-number (molar Lognonnéand Mosser, 1993; Zharkov and Gudkova, 1997]. Mg/Mg+Fe), which ranges from 0.77 to 0.73. Mg/Si affects Recent experimental studies of phase relations in the Fe-S and (Fe,Ni)-S systems at pressure and temperature condi- the proportions of olivine (and its polymorphs) and pyrox- ◦ enes (and garnet), thereby influencing the properties of the tions relevant for the core of Mars (1927 C and 40 GPa) mantle. Very similar results in terms of major element ratios also point to an entirely liquid core at present Stewart et al. are observed for the models of Dreibus and Wänke [1984] and [2007]; Rivoldini et al. [2011]. Core size, state, and com- Lodders and Fegley [1997]. The oxygen isotope mass balance position are uncertain with current estimates ranging from 3 method predicts higher alkali concentration, but this is un- 1550–1800 km in radius, 5.9–7.5 g/cm in density, and 64– likely to influence calculated mantle phase proportions. In 90 wt% FeNi and 10–36 wt% S in composition [c.f., table regard to trace and volatile elements, differences between 5 in Khan and Connolly, 2008, but see also Rivoldini et al. the five models are more pronounced. From a geophysical [2011]]. point of view, however, these differences are insignificant. Another important issue directly related to core size is the We should note that the model compositions shown in ta- presence or absence of a lower mantle in Mars, i.e., a man- ble 1 are input values only, i.e., mantle and core composition tle dominated by bridgmanite-structure silicates. Figure 1 are parameters to be determined in the inversion. This will shows that at pressures above ∼23 GPa the major lower be described in more detail below. mantle minerals ferriperoclase and bridgmanite stabilise. Finally, Borg and Draper [2003] and Agee and Draper However, the stability of these silicates depends strongly [2004] have suggested that the Martian mantle may have on temperature and pressure conditions at the core-mantle a higher Mg# than the range of compositions described boundary (CMB). Large cores will result in a lower mantle above. However, an alternative bulk Mars composition, re- that is either thin or absent altogether; whereas higher tem- porting elemental concentrations for the core and a high peratures will tend to stabilize bridgmanite at lower pres- Mg#, i.e., low-FeO, bearing mantle has, to our knowledge, 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 the bulge raised on Mars by Phobos is aligned with the direction physical conditions at the CMB and Fe-content of the core. from the planet’s centre towards Phobos, with no lagging. Small cores tend to be Fe-rich and will favour presence of This ensures that both the torque applied by Phobos on a lower mantle whereas large cores will tend to be enriched Mars and the opposite torque with which Mars is acting on in S and inhibit a lower mantle. Beyond this, presence of a Phobos are zero. Hence, in the elastic approximation, the lower mantle has implications for the dynamical evolution tides on Mars make no influence on Phobos’ semimajor axis, of Mars. Several studies have shown that a lower mantle is eccentricity, or inclination, and, as a consequence, no tidal likely to exert considerable control over the dynamical evolu- heat is generated in Mars. tion of mantle and core [e.g., Harder and Christensen, 1996; Realistic objects deviate from elasticity. So the tidal Breuer et al., 1997; van Thienen et al., 2006]. In contrast, bulge acquires a complex structure and is no longer aimed at the perturbing body. Decomposition of the bulge over the mid-mantle phase transitions are dynamically much less im- tidal Fourier modes renders harmonics, some of which lag portant and appear unlikely to prevent the entire mantle and some advance relative to the subsatellite point. What- from convecting as a single unit Ruedas et al. [2013a]. ever the sign of the lag, each harmonic now produces tidal A related question is the current thermal state of Mars’ heat. In this situation [following Efroimsky and Makarov, interior. While difficult to estimate directly, the areotherm 2014], expression (1) above should be written, in the time represents, on a par with mantle and core composition, the domain, as most important parameter to be determined because of the fundamental control it exerts on physical structure. This is R n+1 ψ0 (r, t) = kˆ ψ (R, r∗), (2) exemplified in figure 1, which shows that phase equilibria, n r n n physical properties (here illustrate using shear-wave speed), where kˆn is a linear operator (Love operator) mapping the and presence of bridgmanite-structure silicates in the lower 0 0 entire history of the perturbation (ψn(t ) over t ≤ t) on the mantle are very sensitive to the exact thermal conditions 0 inside the Martian mantle. Current geophysical studies value of ψ at the present time t. In the time domain, this is a convolution: typically 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; Verhoeven Z t n+1 0 R ˙ 0 ∗ 0 0 et al., 2005; Sohl et al., 2005; Zharkov and Gudkova, 2005; ψn(r, t) = kn(t − t )ψn(R, r , t )dt , (3) Khan and Connolly, 2008; Rivoldini et al., 2011]. Likewise, t0=−∞ r the areotherm considered in the experimental approach of Bertka and Fei [1997] is representative of “hot” conditions while in the frequency domain it is a product of Fourier based on the need for achieving thermodynamic equilibrium components: experimentally. Thus, while the bulk chemical composition n+1 of Mars holds the potential of constraining many aspects of 0 nm R nm ∗ nm ψ¯ (r, ω ) = k¯n(ω )ψ¯n(R, r , ω ) (4) Mars such as internal structure, origin and evolution, cur- n pq r pq pq rent constraints are not strong enough to reliably determine nm these unequivocally. ωpq being the Fourier tidal modes (whose absolute values As observed by Jackson and Faul [2010], viscoelastically- are the physical forcing frequencies exerted in the material) nm based dissipation models impose strong constraints on the and {pq } integers used to number the modes. In the for- areotherm. This was tested by Nimmo and Faul (2013), mer expression, k˙ n denotes a time derivative, while in the whose model calculations showed that Martian global tidal latter expression, we employ overbars to emphasize that the dissipation, based on an extended Burgers formulation Fourier components are complex, i.e., of viscoelasticity (to be described in section 3.4), is not 0 0 0 0 −n(ω ) only strongly temperature-controlled but also frequency- k¯n(ω ) = Re[k¯n(ω )] + iIm[k¯n(ω )] = |k¯n|e , (5) dependent. Here, we build upon the initial study of Nimmo 0 nm and Faul (2013) and embed the viscoelastic model into our where we employed ω = ωpq for shorthand notation. In ex- ¯0 nm ¯ ∗ nm joint geophysical-thermodynamic framework. This extends pression (4), ψn(r, ωpq ) is lagging behind ψn(R, r , ωpq ) by nm previous studies of e.g., Khan and Connolly [2008] and the phase angle n(ωpq ) which by convention is the negative Rivoldini et al. [2011] to proper consideration of the influ- ¯ nm argument of the complex Love number kn(ωpq ). By setting ence of dissipation on interior structure. r equal to the position r∗ of the perturbing body, we obtain the additional potential “felt” by the latter (Phobos). From the above expression, it is also possible to calculate 3. Geophysical analysis the tidal torque, the radial elevation, and the tidal power dissipated in the planet. It turns out that, at each tidal 3.1. Background nm mode ωpq , an appropriate Fourier contribution into each of In the present analysis, we focus on the tide raised on these quantities is proportional to the sine of the phase lag at Mars by its moon Phobos and the Sun, which results in an that mode [e.g., Efroimsky, 2012a; Efroimsky and Makarov, imposed potential ψ that will cause Mars to deform and give 2014]. The quantity inverse to the absolute value of this sine rise to an induced potential ψ0 according to is conventionally named the tidal quality factor and denoted with Qn and defined as n+1 0 R ∗ 1 ψn(r) = knψn(R, r ), (1) nm r nm = sin |n(ωpq )|. (6) Qn(ωpq ) where R is the radius of the planet, R is a point on the The functional form of the frequency-dependence of the tidal planet’s surface, r is an exterior point above the point R, ∗ quality factors is different for different degrees n [see e.g., while r is the position of the perturbing body. The poten- Efroimsky, 2015]. Fortunately, this difference becomes man- tials are expanded in terms of spherical harmonics of degree ifest only at extremely low frequencies. At ordinary fre- n and the proportionality constants, kn, are tidal Love num- quencies, these quality factors are very close to the nom- bers of degree n and determine the amplitude of the response inal (“seismic”) quality factor Q, which is usually defined [e.g., Efroimsky, 2012a]. through the relation The above expression (1) for the amended potential of the tidally deformed planet stays valid insofar as the planet’s 1 1 I ∂E = dt, (7) response is purely elastic. In this approximation, the tidal Q 2πE 0 ∂t KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 5 where E and E 0 refer to the energy and peak energy over one estimates made by Yoder et al. [2003], Bills et al. [2005], cycle and the integral is taken over the same. The expres- Lainey et al. [2007], Jacobson [2010], and Nimmo and Faul sion on the right-hand side of Eqn (7) turns out to be equal [2013], who determined Q’s of 92±11, ∼85, ∼80, ∼83, and to the absolute value of the sine of the phase lag between 88±16 respectively. In obtaining these estimates, Bills et al. strain and stress [see e.g., Eqns 45–47 in Efroimsky, 2015]. [2005] used a value for k2 (0.0745) that is too low given cur- For ordinary (not too low) frequencies, the tidal phase lag rent understanding, while Lainey et al. [2007] and Jacobson virtually coincides with the phase lag between the strain [2010] relied on the same k2 value by Konopliv et al. [2006] and the stress in the material. Accordingly, at not too low that we employ here. For comparison, Nimmo and Faul frequencies the tidal Qn virtually coincides with the nominal [2013], who used the Yoder et al. [2003] solar tide k2 value “seismic” Q. (0.149±0.017), found a 0.6% change in k2 when converting In the following, we shall concentrate on the semidiurnal to the synodic period of Phobos (erroneously referred to as nm 22 11.1 hours) and obtained 0.148±0.017. To determine Q, tidal mode for which {pq } = {00}; so we shall be dealing with k and Q (henceforth labeled k and Q). While both Nimmo and Faul [2013] employ the estimate of Lainey et al. 2 2 2 [2007], but correct for the influence of higher-degree terms k2 and Q depend on interior properties such as density and rigidity, Q is strongly sensitive to viscosity and, thus, tem- (k3 and k4) on the orbit of Phobos (see discussion below). perature and grain size. This will be described in more detail Assuming certain ranges for the ratios k2/k3 and k3/k4, the in section 3.4. authors obtain a Q estimate of 88±16. Earlier analyses of the orbital acceleration of Phobos estimated the tidal Q of Mars to be 100±50 [Smith and Born, 1976; Yoder, 1982; 3.2. Geophysical data Lambeck, 1979]. There are currently few geophysical data available that Because of the proximity of Phobos to Mars (mean dis- bear directly on the deep interior structure of Mars. Here, tance 9378 km), higher-degree terms (e.g., k3, k4,...) appear we shall focus on mean density (ˆρ), mean moment of inertia to be significant for the orbital evolution of Phobos [Bills 2 (I/MR ), second degree tidal Love number (k2), and global et al., 2005; Konopliv et al., 2011]. Based on model values of tidal dissipation or tidal quality factor (Q). Mean density k3 and k4, Zharkov and Gudkova [2005] and Konopliv et al. and moment of inertia are sensitive to the density structure [2011] considered the correction that would be introduced of the planet, whereas the sensitivity of the second degree by including higher-order terms and found that this con- tidal Love number and global tidal dissipation is more com- tributes less than 10% to the tidal deceleration of Phobos. plex related to the response of a planet to tidal forcing. In contrast, Lainey et al. [2007] considered only the degree-2 The geophysical data for Mars employed here are sum- term, given that degree-3 and -4 terms are not known, and marized in table 2 and are discussed in more detail in the argued that the tidal dissipation factor Q should be considered as an effective Q that partly absorbs losses from higher literature [e.g., Van Hoolst et al., 2003; Yoder et al., 2003; harmonics. A reasonable alternative would be to compen- Bills et al., 2005; Zharkov and Gudkova, 1997, 2005, 2009; sate by increasing the error bars (pers. comm., V. Lainey, Balmino et al., 2005; Lainey et al., 2007; Marty et al., 2009; 2017). Jacobson, 2010; Rivoldini et al., 2011; Nimmo and Faul, Additionally, tidal forcing at degree-2 will induce tidal 2013; Konopliv et al., 2006, 2011, 2016; Genova et al., 2016]. waves at periods other than the main tide considered here The k2 estimates determined from orbiting spacecraft data [Roosbeek, 1999]. However, the amplitudes of the tidal waves [e.g., Yoder et al., 2003; Konopliv et al., 2006, 2011; Genova at the other periods are much smaller. The next-largest am- et al., 2016] are consistent with values ranging from ∼0.13– plitude in the sub-diurnal spectrum, for example, is a fac- 0.175 (excepting the Marty et al. [2009] determination) and tor of ∼7 smaller than the amplitude of the main Phobos- typically refer to the period of the solar (semidiurnal) tide induced tide at 5.55 hours [Van Hoolst et al., 2003]. Hence, (12h19min). Current Q estimates range from ∼70 to 110 we neglect their contribution here. and indicate that Mars is more dissipative than the solid Earth (270±80) at the equivalent diurnal period [Ray et al., 3.3. Crust and mantle model 2001]. Presently, we rely on the approach adopted by [Zharkov Following our previous work [e.g., Khan and Connolly, and Gudkova, 2005, 2009], because their estimates consis- 2008], Gibbs free-energy minimization is employed [Con- tenly refer to the same frequency (tidal period of Phobos) nolly, 2009] to compute stable mantle mineralogy and phys- and, more importantly, include consideration of the effect ical properties along self-consistent mantle adiabats for each of anelasticity on k and Q to account for the proper fre- of the five model mantle compositions listed in Table 1. 2 We rely on the thermodynamic formulation of Stixrude and quency dependence. Based on the solar tidal k value of 2 Lithgow-Bertelloni [2005b] and parameters of Stixrude and 0.152±0.009 by Konopliv et al. [2006], the authors derive Lithgow-Bertelloni [2011]. In the crust and lithosphere, tem- a value for k2 at the tidal period of Phobos (5.55 hours). perature is computed by a linear thermal gradient, which The difference was found to be 1%, as a result of which for each model is determined from surface temperature and k2=0.152±0.009 will be used. temperature and thickness of the lithosphere. This assump- To determine the tidal dissipation factor at the same pe- tion implies that no crustal radioactivity is present. Al- riod, we follow Zharkov and Gudkova [2005], and relate the though this is unlikely to be realistic, crustal structure has mean dissipative factor Q to the tidal lag angle () using little influence because the data considered here are not re- expressions (6) and (7) above ally sensitive to it. As a consequence, the exact nature of the crustal geotherm is not important. The assumption of 1 sin ≈ = (8) thermodynamic equilibrium is debatable at low tempera- Q ture [e.g., Wood and Holloway, 1984]. As a result, if for a given model a mineralogy at a temperature below 800 ◦C Relying on parameters relevant to the Mars-Phobos system, is required, equilibrium mineralogy was computed at 800 as summarized in Yoder [e.g., 1995], Zharkov and Gudkova ◦C, whereas physical properties for the mineralogy are de- [2005] find termined at the actual temperature of interest. The crust Q 1 = = 559. (9) is likely to be more complex lithologically, not equilibrated, k2 k2 and is probably porous. The effect of porosity is taken into From this expression, we obtain Q=85±5 at the tidal pe- account through a decrease in the seismic properties of the riod of Phobos. This value agrees reasonably well with the crust. 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- (JI = 1/ωη). 0 < α < 1 for τL < τ < τH and zero elsewhere marized in table 1. with associated relaxation strength ∆B [Minster and Ander- Deficiency of experimental constraints on the parameters son, 1981]. To model the dissipation peak DP that appears relevant for the thermodynamic formalism and parameteri- superimposed on the background a term of the following zation of Stixrude and Lithgow-Bertelloni [2011] are the ma- form needs to be added jor source of uncertainty in the thermodynamic calculations. Elastic moduli and density have been estimated to be accu- 1 ln2(τ/τ ) D (ln τ) = √ exp − P , (13) rate to within ∼0.5 and ∼1–2%, respectively [Connolly and P 2σ2 Khan, 2016]. τσ 2π where τ indicates position of the peak with peak width σ 3.4. Viscoelastic model P and associated relaxation strength ∆P. This peak is found to The dissipation model adopted here is described in detail occur at low temperature and/or short time scales and corre- in Jackson and Faul [2010] and relies on laboratory experi- sponds to elastically-accommodated grain-boundary sliding. ments of torsional forced oscillation data on melt-free poly- The timescales τL, τH , τM , and τP are all temperature crystalline olivine. Broadly similar results have also been (T), pressure (P), and grain-size (dgrain) dependent, which obtained by other groups [e.g., Takei et al., 2014]. In what for each individual timescale is modeled using follows we base ourselves on Jackson and Faul [2010] and only provide a summary description here. m dgrain E 1 1 V P P0 In the Earth, Moon, and Mars, and in the absence of τ(P, T, dgrain) = τ0 exp − exp − , melting, dissipation (Q) has been observed to be frequency- d0 R T T0 R T T0 dependent 1/Q ∼ ω−α, where ω is angular frequency and α (14) is a constant [e.g., Efroimsky and Lainey, 2007]. that has where τ0 is a normalized value at a particular set of reference ◦ been observed to lie in the range 0.1–0.4 [e.g., Minster and conditions P0 (0.2 GPa), T0 (900 C), d0 (13.4 µm), and E Anderson, 1981; Jackson et al., 2002; Benjamin et al., 2006]. and V are activation energy and volume, respectively. In Since Maxwellian viscoelasticity is unable to reproduce this addition, different grain-size exponents m for anelastic and frequency dependence, other rheological models [for a review long-term viscous creep processes are allowed for. All of the see e.g., Karato, 2008] such as the extended Burgers model above constants are tabulated in Nimmo et al. [2012] and of Jackson and Faul [2010] have been proposed. Jackson and Nimmo and Faul [2013], except for α, E, and V, which are Faul [2010] argue for the extented Burgers model over other considered variable parameters (see section 4.1). rheological models because it describes the changeover from This model can now be directly linked with the thermo- (anharmonic) elasticity to grain size-sensitive viscoelastic dynamic computations in that the latter provides the unre- behaviour, whereby it is able to explain the observed dissipation in the laboratory experiments on olivine [Jackson laxed (infinite-frequency) shear modulus µU that appears in and Faul, 2010]. the above equations, in addition to P and T and any other The response of a viscoelastic material can be described thermodynamic variables needed (all functions of radius). Note that the shear moduli are computed on the basis of in terms of the complex frequency-dependent√ compliance Jˆ(ω) = JR(ω) + iJI (ω), where i = −1, and subscripts R thermochemical models of Mars and are constrained by geo- and I denote real and complex parts, respectively. For the physical data. Hence, the improvement here over previous extended Burgers model of Jackson and Faul [2010], JR(ω) studies that had little control over internal structure pa- and JI (ω) can be written as rameters required as input [e.g., Sohl and Spohn, 1997; Sohl et al., 2005; Bills et al., 2005; Nimmo et al., 2012; Nimmo 1 Z τH D(τ) J (ω) = 1 + ∆ dτ (10) and Faul, 2013]. Finally, to compute global frequency- R µ 1 + ω2τ 2 U τL dependent k2 and Q from the model outlined above, we 1 Z τH τD(τ) 1 employ the viscoelastic tidal code of Roberts and Nimmo JI (ω) = ω∆ 2 2 dτ + , (11) [2008]. This code assumes spherical symmetry and that all µU 1 + ω τ ωτM τL dissipation occurs in shear; for numerical reasons, we im- 29 which is essentially the Laplace transform of the creep func- posed a viscosity cutoff of 10 Pa s. tion for the extended Burgers model of linear viscoelasticity [see e.g., Jackson and Faul, 2010]. In these expres- 3.5. Core model sions, µU represents the unrelaxed, i.e., infinite-frequency, Mass-radius relations of rocky planets generally require shear modulus, τ period, τM = η/µU Maxwell viscous re- Fe-rich metallic cores alloyed with a light element such as laxation time, η viscosity, ∆ strength of relaxation mecha- Si, C, or S [e.g., Birch, 1964; Poirier, 2000]. In the case of nism and D(τ) distribution of relaxation times. From the Mars, it is argued that S is the dominant light element be- above equations, local dissipation (1/Q) and shear modulus (µ) at a particular frequency can be computed from 1/Q = cause the other candidates do not have sufficient solubility p 2 2 p 2 2 in iron-rich liquid at the relatively low pressures that would |JI |/ JI + JR ≈ |JI |/|JR| and µ(ω) = 1/ JR(ω) + JI (ω), respectively. have been maintained during core formation (e.g., Stevenson The advantage with this model is that D(τ) can be used 2001). Evidence in support of this is the observed depletion to specify a distribution of anelastic relaxation times ac- of chalcophile elements, notably S, of the SNCs (see sec- counting for the monotonic background dissipation (DB) tion 2.1). Accordingly, an Fe-S core is generally assumed in and the superimposed dissipation peak (DP) of elastically- geophysical models of Mars. To date, the most elaborate pa- accomodated grain-boundary sliding [Jackson et al., 2014], rameterization of the Martian core (in the Fe-FeS system) in addition to the associated modulus dispersion [Jackson is that of Rivoldini et al. [2011]. Here, we follow Rivol- and Faul, 2010], where DB is given dini et al. [2011] and assume that Mars’ core is well-mixed and convecting. To compute depth-dependent thermoelastic α−1 ατ properties for the core, we use equations-of-state for liquid DB(τ) = α α , (12) τH − τL iron and liquid iron-sulfur alloys. Core pressure is obtained from the hydrostatic equation where τL and τH are the integration limits corresponding to short and long periods, respectively. In the low-frequency dP(r) = −ρ(r)g(r), (15) limit, JI (equation 11) reduces to Maxwellian behaviour dr KHAN ET AL.: ON THE BULK COMPOSITION OF MARS X - 7

where r is radius, ρ density, and g gravitational acceleration. entropy of the lithology at Tlit, which determines the tem- g also obeys the Poisson equation perature at the CMB (section 3.5). Given values of all of the above model parameters, the forward model can be solved. dg(r) 2g(r) + = 4πGρ(r), (16) dr r 4.2. Forward problem where G is the gravitational constant. For a well-mixed and The forward model consists of computing geodetic data 2 vigorously convecting core the temperature gradient is given (¯ρ, I/MR , k2, Q) from a knowledge of the physical struc- by ture of the crust and thermochemical structure of mantle dT(r) γ(r) = −T(r) ρ(r)g(r), (17) and core. To determine stable mineralogy (M), isotropic dr KS(r) shear (µ) and bulk (κ) moduli, density (ρ), and attenua- where γ is the Gr¨uneisenparameter and KS is adiabatic bulk tion structure (here designated by complex moduliµ ˆ andκ ˆ) modulus. These ordinary diﬀerential equations are solved along self-consistent mantle adiabats, we employ Gibbs free- numerically [Brankin et al., 1993] subject to the following energy minimization and a grain size dependent viscoelastic boundary conditions formulation. With this, the forward problem can be written as P(rcmb) = Pcmb (18) operator

T(rcmb) = Tcmb (19) z}|{g g ,g g {X , T , d , d ,..., r , X } −→1 M −→2 3 {ρ, µ,ˆ κˆ} −→4 {ρ,¯ I/MR2, k , Q} g(0) = 0 (20) m lit lit grain core S 2 | {z } | {z } primary parameters data where rcmb,Pcmb, and Tcmb are radius, pressure, and temperature at the CMB, respectively. These quantities on the where the primary parameters are those described above mantle side of the CMB are those determined in section 3.3. in section 4.1 and the other parameters (M, µ,ˆ κˆ,...) repre- The thermoelastic quantities ρ,KS, and γ depend on pres- sent secondary parameters that are required for the purpose sure, temperature, and core composition (S content) and are of computing data (section 3.2). The operators (g1,..., g4) calculated using the approach outlined in Appendix A. represent the various forward models (e.g., Gibbs free energy minimization, extended Burgers model, viscoelastic tidal re- 4. Computational aspects sponse) that are described in sections 3.3 and 3.4. 4.1. Model parameterization 4.3. Inverse problem In the following we briefly describe model parameteriza- To solve the inverse problem d = g(m), where d is a data tion, which is illustrated in figure 2. For present purposes, vector consisting of observations and g is an operator that we assume our model of Mars to be spherically symmet- maps from the model space into the data space, we employ a ric and have divided it into three layers comprising crust, Bayesian approach as outlined in Mosegaard and Tarantola mantle, and core. The crust has been further subdivided [e.g., 1995] into an additional three layers that are parameterized in σ(m) = k · h(m)L(m), (21) terms of P- and S-wave velocity, density, and Moho thick- where h(m) is the prior model parameter probability dis- ness. Rather than varying VP,VS, and ρ independently in the crust, however, we introduced a variable parameter (φ) tribution, i.e., the information on model parameters pro- to mimick the effect of porosity and computed the aforemen- cured independently of data, L(m) is the likelihood function, 0 which measures the misfit between observed and predicted tioned physical properties using x = x · φ , where x are i i i i data, k is a normalization constant, and σ(m) is the pos- thermodynamically-computed VP,VS, and ρ (section 3.3) in crustal layer i. φ is determined from φ = φ +(1−φ )·i/N terior model parameter distribution. σ(m) represents the i i 0 0 solution to the inverse problem above. L(m) is determined with φ0 being variable surface porosity and N the total number of crustal layers. This parameterization ensures that from the observed data, data uncertainties, and the manner crustal properties increase from the surface down to the in which the latter are used in modeling data noise (to be Moho where porosity is expected to vanish due to pressure described in the following). (i.e., 1-φi=0). The sublithospheric mantle is assumed to be The Metropolis algorithm is employed to sample the pos- uniform and modeled using the variables composition (in terior distribution (Eq. 21) in the model space [Mosegaard the NCFMAS system) and mantle temperature. Within the and Tarantola, 1995]. This algorithm, which is samples the lithosphere, temperature is computed by a linear areother- model space in a random fashion, is an importance sam- mal gradient, which for each model iteration is determined pling algorithm, i.e., it ensures that models that fit data from the variables Tsurf ,Tlit, and dlit. The sublithospheric well and are simultaneously consistent with prior informa- mantle adiabat is defined by the entropy of the lithology tion are sampled more frequently. The Metropolis algorithm at the temperature Tlit, i.e., at the base of the lithosphere samples the model space with a sampling density that is pro- of thickness dlit. The bottom of the lithosphere is defined portional to the (target) posterior probability density and as the location where the conductive lithospheric geotherm thus ensures that low-probability areas are sampled less ex- intersects the mantle adiabat. Since the viscoelastic model cessively. This is an important feature of any algorithm that (section 3.4) is grain size dependent, the model is addition- wishes to randomly sample high-dimensional model spaces ally parameterized in terms of a single grain size (dgrain). where the probability density over large proportions of the Many of the parameters belonging to the viscoelastic model volume are near-zero. are determined experimentally and therefore uncertain. To better capture this, we included α, E, and V as variable pa- 4.4. Prior information rameters. Uncertainty ranges are taken from Jackson and Faul [2010]. The mantle pressure profile is obtained by in- The prior model parameter information described above tegrating the vertical load from the surface pressure bound- is summarized in table 3 below. The chosen prior ranges rep- ary condition. Core parameters include radius, composition resent the information acquired from data and results from (S content), and the input parameters required to compute experimental studies supplemented with results from numer- physical properties of the core are those determined by in- ical studies as discussed in the foregoing sections. The prior tegrating the load from the surface to the CMB and the range 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 sensitivity 1000–1100 km and 1300–1400 km, respectively, at the loca- on crustal and sub-crustal lithospheric thermal structure. tions where the major mantle mineral phase transitions oc- This has little effect on computed physical properties of the cur (olivine→wadsleyite and wadsleyite→ringwoodite). At crustal and sub-crustal structure as tests have shown where the CMB, temperatures in the range 1560–1660 ◦C are ob- surface temperature was fixed to that of present-day Mars. tained. No attempt was made to account for a thermal Hence, in what follows the thermal structure of the crust boundary layer between the mantle and core. While a thin and sub-crustal lithosphere will not be discussed further. thermal boundary layer with a temperature difference of ∼100 ◦C is possible [e.g., Kiefer and Li, 2016], data are 4.5. Sampling the posterior not sensitive enough to “see” such a layer. This was verified ◦ Under the assumption that data noise is Gaussian dis- by investigating how an increase of 100 C across the CMB tributed and that observational uncertainties and calcula- affects core properties. As expected, differences in e.g., P- tion errors among the data sets considered are independent, wave velocity and density on the core side of the CMB rela- the likelihood function can be written as tive to a model without a thermal boundary layer amounted to <0.5%, which from the point of view of the geophysical Y |di − di (m)|2 data is an insignificant change. Results are summarized in L(m) ∝ exp − obs cal (22) 2 table 4. 2σi i In figure 4 we are also showing the mean temperature pro- 2 file from a recent geodynamical study by Plesa et al. [2016]. where i runs overρ ¯, I/MR , k2, and Q, dobs and dcal(m) denotes observed and calculated data, respectively, and σ Plesa et al. [2016] modeled the thermal evolution of Mars to data uncertainty. investigate the spatial variation in present-day surface heat flux. The blue and pink lines show present-day hot and cold end-member average temperature profiles from Plesa et al. 5. Results and discussion [2016] assuming a core size of 1500 km. The comparison 5.1. Datafit shows that the results from the inversion are in good agreement with the cold end-member geodynamic model. For Datafits for all considered compositions of table 1 are reference, we also included the recent present-day dry Mar- summarized in figure 3. The various compositions are all tian solidus of Kiefer et al. [2015]. capable of fitting mean density and moment of inertia, in Independent direct constraints on the thermal structure addition to global tidal dissipation, but show scatter in k2. of Mars are few. Petrological estimates derive principally In particular, compositions MA and SAN are less convinc- from Gamma Ray Spectrometer data (Mars Odyssey) for ing than DW, LF, and TAY. The discrepancy between MA and the other models arises because MA models are more several major volcanic provinces [Baratoux et al., 2011] and rigid (∼3–5 % higher S-wave velocity) in the depth range analyses of shergottite meteorites [Filiberto and Dasgupta, 2015]. These estimates bracket temperatures found here, 100–1100 km relative to the other models due to presence of ◦ garnet-rich phases. This coupled with smaller core sizes re- but are uncertain by several hundred C for dry mantle conditions and increase further if water and/or other volatiles sults in the inability of MA to fit k2. Given the geochemical point made earlier (section 2.1) that both MA and SAN rep- are added [e.g., Balta and McSween, 2013; Filiberto et al., resent somewhat exotic compositions and are probably un- 2016]. likely to be representative of the bulk composition of Mars, we discard MA from further analysis. While the same geo- 5.3. Grain size chemical argument also apply to model SAN, we neverthe- Since the extended Burgers model is grain size depen- less keep this model because it is capable of fitting data to dent, we show sampled grain size distributions in figure 5 within uncertainties. for the four main compositions. The distributions are generally in accord with grain sizes in the sub-to-millimeter range 5.2. Mantle temperature profiles of 0.01–5 mm with a strong peak toward small grain sizes. Inverted present-day areothermal profiles are shown in Nimmo and Faul [2013] observed a trade-off between grain figure 4. For comparison, we are also showing sampled size and mantle temperature (and lid thickness). This trade- prior areotherms, which evidently comprise a very large off is not observed here. One reason for this is that in the parameter range. Inverted models are comparatively well- inversion, models are “forced” to fit the measured k2 and constrained over most of the mantle and core with the ex- Q, whereas Nimmo and Faul [2013] are simply calculating ception of crust and lithosphere (≤400 km depth). This k2 and Q based on a range of input parameters and com- reinforces earlier analyses (e.g., Karato, 2007 and discus- paring these qualitatively to the observations. Thus, the sion by Efroimsky and Lainey [2007] that the constraints trade-off is not observed because the inversion forces the on the mantle temperature profile imposed by the addition model parameters to cover a more restricted range. In addi- of a temperature and necessarily frequency dependent visco- tion, the thermodynamic parameterization coupled with the elastic model are strong. Relative to earlier studies that con- top-down approach in which models are constructed (e.g., sidered the same problem, but without invoking a specific d /T determine mantle temperature) acts to break the visco-elastic model [e.g., Khan and Connolly, 2008; Rivoldini lit lit trade-off between various parameters that are treated as be- et al., 2011], the advantages here are obvious. These earlier ing purely independent by Nimmo and Faul [2013] and re- studies had difficulty in constraining the thermal structure stricts the parameter range even further. Finally, we also of the mantle. 2 Comparison of the obtained areotherms for the four bulk includeρ ¯ and I/MR , which, relative to Nimmo and Faul compositions indicates relatively small differences that are [2013], exerts a strong independent control on the density unlikely to be large enough to enable us to distinguish be- structure. tween the bulk compositions based on the available data The importance of grain size arises because of the strong considered here. Lithospheric thickness and temperature control it exerts on the diffusion creep rheology of the crust m variations among the four models broadly agree with es- and mantle (viscosity ∝ dgrain, m=2–3). As summarized by timates in the range 250–500 km and ∼1300–1360 ◦C. In Karato [2008, Ch. 13], common estimates of grain sizes of comparison to the prior ranges (table 3), both dlit and Tlit terrestrial rocks vary from 0.1 mm to 10 mm. Grain size are relatively well-constrained. The self-consistently com- estimates in the deeper Earth are invariably indirect with puted mantle adiabats show “structure” at depths around estimates 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 notorious trade-off that exists between composition (den- [Karato and Wu, 1993]. The latter would result from a sity) and core size. Relative to the original model core com- change in deformation mechanism in Earth’s upper mantle positions DW remains almost unchanged, whereas the S con- where a transition from dislocation to diffusion creep oc- tent of LF increases by 3 wt% and that of TAY and SAN curs. The depth at which this transition occurs depends on decrease by 6 and 3 wt%, respectively. Importantly, and in- a range of flow-law parameters [Hirth and Kohlstedt, 2013]. spite of initial compositional differences, inverted core com- In our inverse treatment, we neglect the complexity of positions for models DW, LF, and TAY appear to converge including a depth dependent grain size. Based on the ob- to a common composition with a mean S content around 15 servation that the geodynamical simulations (section 6) sug- wt% corresponding to a mean core radius of ∼1700 km and gest homogeneous grain sizes throughout Mars’ mantle, this a mean core mass fraction of 0.20-0.21, while SAN remains appears to be a reasonable assumption. Generally, grain growth is temperature controlled and on Earth the lower a little below the lower limit of these estimates. The val- mantle transition is expected to limit the size of grains as ues for DW, LF, and TAY are in good agreement with the material is being moved across the transition and recrystal- recent estimates of Rivoldini et al. [2011], who employed a somewhat similar approach to determining interior structure lizes [e.g., Solomatov, 2001; Solomatov and Reese, 2008]. In 2 comparison, the mid-mantle phase transitions in the Mar- (inversion of k2, M, and I/MR ) and obtained core sizes and tian mantle at ∼13 GPa (1100 km depth) and 15 GPa S contents of 1794±65 km and 16±2 wt%, respectively. The (1300 km depth) (c.f., figure 1), respectively, are dynami- larger ranges found here reflect the fact that Rivoldini et al. cally unimportant [Ruedas et al., 2013a]. As a consequence, [2011] only considered two possible end-member areotherms and in view of an apparent absence of a lower mantle transi- from the literature (c.f., “cold” and “hot” areotherms in fig- tion in Mars (see section 5.5 below), the grain sizes for Mars ure 1). In addition, a slightly larger k2 (0.159±0.009) was found here are not unrealistic. employed, which, everything else being equal, will tend in- For the baseline Martian model, Nimmo and Faul [2013] crease core size. fixed α =0.274, whereas we generally find a slightly lower α The obtained core sizes, however, imply that a lower man- in the range 0.2–0.22 (values for α, E, and V are summarized tle transition equivalent to the “660-km” seismic discontinu- in table 4). The reason for this difference is not clear, but ity, is not extant inside Mars. CMB pressures attained here could possibly be related to Nimmo and Faul [2013] using range from 19.5–21.5 GPa, which is below that needed for 11.1 hours instead of 5.55 hours as forcing period. the ringwoodite→ferropericlase+bridgmanite transition to occur (cf. figure 1) and no models were found to contain 5.4. Attenuation the equivalent of a lower mantle transition. This also holds Computed shear attenuation (Qµ) profiles for the four for models SAN. Similar conclusions were also reached by compositions are also shown in figure 6 at the main pe- Rivoldini et al. [2011] riod of the Phobos-induced tide (5.55 hours) and at 1 s Absence of a lower mantle has implications for the dynam- period (only for DW since the others are qualitatively the ical evolution of the planet: large cores inhibit the existence same). The sampled profiles indicate that dissipation within of a lower mantle layer as a result of which mantle convection Mars is large (small Qµ) with quasi-constant Qµ values of is expected to encompass the entire mantle. Small cores, on 130–150 throughout the upper mantle (depth range 300– the other hand, allow for a lower mantle layer that could 900 km) at seismic periods and even lower at tidal peri- form a dynamically separate unit and lead to decoupling of ods as predicted earlier (see section 3.2). As in Nimmo and mantle and core [e.g., van Thienen et al., 2006; Ruedas et al., Faul [2013], smallest Qµ values are found in the depth range 300–500 km, which coincides with a low-velocity zone that 2013a, b], although disruption of grain size growth as mate- is associated with relatively steep temperature gradients in rial passes through the phase transition would tend to lower the lithosphere (this will be discussed in more detail in sec- viscosity and hence increase convective vigour and destabi- tion 5.7 below). In spite of the near-isothermal nature of lize any initial separation. In contrast, absence of a lower the areotherm in the convecting mantle of Nimmo and Faul mantle transition would favour continuous grain growth and [2013], this conclusion seems robust and serves to emphasize lead to larger grain sizes and thus increased viscosity at the that to first order dissipation appears to be controlled by bottom of the mantle. This argues in favour of whole-mantle the olivine-dominated part of the mantle, while phase tran- convection. sitions are less significant. Indeed, while variations in shear attenuation at depths of ∼1100 and ∼1300 km, respectively, 5.6. Implications for bulk composition are palpable, these appear to be insignificant. The observation that the Martian mantle is very dissipative could have Inverted bulk and core compositions are summarized in a profound effect on seismic wave propagation within Mars. table 4. The composition of bulk silicate Mars (BSM) is In particular, significant attenuation of both short-period presently defined by the model compositions DW, LF, TAY body wave and long-period surface waves is possible and and SAN (table 1). These initial BSM compositions show may hamper detection of distant (teleseismic) signals [see variability in absolute oxide concentrations and characteris- also discussion in Lognonnéet al., 1996]. tic elemental ratios. The Mg-number, for example, ranges between 0.751 (LF) and 0.728 (SAN) and varies only by 5.5. Core parameters ∼3%. The atomic Mg/Si ratio, on the other hand, ranges from sub-CI chondritic (0.85) for SAN to near CI-chondritic Inverted core size, composition (S content), and density (1.04) for TAY. With the present model setup, where we al- for the four main compositions is shown in figure 7 and sum- low for a 10 wt%-variation of each mantle oxide component, marized in table 4. The results reveal broadly similar core sizes and compositions for three of the compositions (DW, the BSM models are essentially indistinguishable. A sim- plification in our phase equilibrium calculations is that the LF, and TAY) with core radii and S-content (XS) in the range 1640–1740 km and ∼13.5–16 wt%, respectively. In “Model mantle compositions” are directly derived from the comparison, SAN has a slightly smaller core (1590–1680 km) proposed BSM models and do not account for the extraction and lower S content (11–13.5 wt%). The agreement between of the crust, i.e., that bulk silicate Mars is the compositional DW and TAY is not surprising given the similarity between sum of crust and mantle. Subtracting mean crustal masses the two compositions (see table 1 and section 2.1 for the of average crustal composition (table 1) from each mantle assumptions underlying the geochemical models). The ob- model shows that the residuals are within 10 wt% of their served linear trends are also as expected and highlights the initial 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- Of general interest are profiles of seismic velocity and den- cause of its low absolute concentration, however, variations sity structure. These profiles are shown in figure 8 and refer in Na have a negligible effect on mantle phase relations. Ac- to seismic periods (1 s). Figure 8 also shows the frequency- cordingly, crustal extraction is implicit in the inversion. dependence of shear attenuation on S-wave velocities. At Core compositions are modeled in the simplified system the main tidal period of Phobos, S-wave velocities are, as Fe-FeS because of a lack of more detailed geophysical models expected, lower relative to those at seismic periods. This accounting for the effect of Ni on core properties. From a occurs for depths >200 km where shear attenuation be- geochemical point of view, the core likely contains a signifi- comes significant in the mantle (figure 6). As already seen cant amount of Ni, which can be estimated by assuming 1) a in the thermal profiles (figure 4), differences between the fixed bulk Mars atomic Fe/Ni ratio of 18 (after DW) and 2) various models are relatively small, but point to the possi- that Ni is entirely incorporated into the core. A Fe/Ni ratio ble presence of a lithospheric high-velocity lid followed by ∼18 is also obtained independently in the isotope analysis a low-velocity zone (LVZ). These features were noted pre- of Lodders and Fegley [1997], which is a consequence of the viously by Nimmo and Faul [2013] and follow directly from Fe/Ni ratio being relatively constant across various types the relatively strong temperature gradients that prevail in of chondritic meteorite classes (ranging from 17.8 to 19.1) the conductive lithosphere. More generally, the effect of [Wasson and Kallemeyn, 1988]. Adjusting the core compo- temperature on shear modulus relative to that of pressure inside Mars is stronger and therefore results in a shear-wave sition for the presence of Ni results in very similar solutions velocity reduction with depth in the lithosphere. These fea- for DW, LF, and TAY with a most probable core composi- tures on seismic wave propagation are discussed by Zheng tion of ∼79 wt% Fe, ∼6 wt% Ni, and ∼15 wt% S. In the et al. [2015]. It should be noted though that the exact con- case of SAN the mean core composition would result in a dition that seismic waves are bent downwards in a spherical slightly lower sulfur core content with ∼82 wt% Fe, ∼6 wt% model, which potentially occurs if a LVZ is present, is that Ni, and ∼12 wt% S. transformed velocities in a flat Mars model have a negative The bulk planet compositions derived here result in bulk gradient. From a geometrical point of view, this is equiva- Fe/Si ratios (by weight and corrected for the presence of Ni lent to horizontal rays being bent downwards relative to the in the core) that range between 1.63 and 1.68 for DW, LF, arc at that radius. [for specific illustrations of this, see e.g., and TAY, and 1.51 for SAN. In the model of Sanloup et al. Clinton, 2017]. As a consequence, an apparent LVZ does [1999], Mars consists of a mixture of 45% EH (Fe/Si∼1.74) not necessarily result in a seismic shadow zone. and 55% H (Fe/Si∼1.63) chondrites with a combined Fe/Si Models DW, LF, and TAY point to an olivine→wadsleyite of 1.68 ([Wasson and Kallemeyn, 1988]). However, to recon- transition at around ∼1000–1050 km depth and a secondary cile the bulk silicate Mars composition originally proposed transition (wadsleyite→ringwoodite) at about 1250–1300 by Sanloup et al. [1999] with the geophysical data consid- km depth. For model SAN the transitions occur 50–100 ered here, current results suggest lower core mass fractions km shallower in the mantle. The former transition is not around 0.18–0.2. This circumstance explains the lower value as sharp as on Earth (coexistence loop extends for approxi- for Fe/Si found here, relative to what had been proposed by mately 50 km), but might be detectable using seismic data in Sanloup et al. [1999], involving a larger core mass fraction the form of e.g., mantle triplications, PP- and SS-precursors, (0.23) and therefore a higher Fe/Si (1.68). Ultimately, this and receiver functions. While seismic data will be acquired implies that the compositional model of Sanloup et al. [1999] as part of the upcoming InSight mission to Mars [Banerdt can not self-consistently explain the oxygen isotope signa- et al., 2013], figure 8 indicates that it is going to be difficult, ture of Mars and simultaneously the current Martian geo- based on seismic data alone, to discriminate between the physical data. A possible explanation for this discrepancy compositional models considered here, although SAN might may lie in their choice of meteoritic end-members that in- present an exception in this regard. The same arguments volves a hypothetical, as yet unsampled, class of meteorites. concern determination of core size, which is otherwise de- Based on the experimental data of Stewart et al. [2007] tectable using normal modes or core-reflected and -refracted on the system (Fe,Ni)-(Fe,Ni)S, the eutectic composition waves. While seismic analyses in preparation for InSight at CMB pressures obtained here (19.5–21 GPa) is ∼17 have been and are currently undertaken [e.g., Okal and An- wt% S and decreases to 14 wt% S with increasing pressure derson, 1978; Goins and Lazarewicz, 1979; Solomon et al., (40 GPa). Corresponding eutectic temperatures are below 1991; Lognonnéet al., 1996; Larmat et al., 2008; Teanby 1000◦C at CMB pressures and only increase to 1100–1200◦C and Wookey, 2011; Zheng et al., 2015; Zharkov and Gud- at pressures of the inner core. In comparison, inverted model kova, 2014; Panning et al., 2015; Khan et al., 2016; Panning et al., 2016; Böseet al., 2017; Bozda˘get al., 2017; Ceylan temperatures on the core side of the CMB are >1800◦C, et al., 2017], we leave it for a future study to consider seismic which together with the almost eutectic-like sulfur content predictions of the models proposed here in more detail. with a mean ∼14.5 wt% (in the case of DW, LF, and TAY), suggests a fully liquid core at present in agreement with inferences made earlier [Rivoldini et al., 2011]. Based on 6. Geodynamic modeling the mean S content determined here, Mars’ core is likely to undergo a complex core crystallisation behaviour in the To provide an independent view on some of the geophys- future. Crystallisation will start at the pressure at which ical predictions (e.g., grain size, thermal state, and litho- the areotherm initially falls below the liquidus, which is ex- spheric thickness), we performed a number of mantle con- pected to first occur in the upper part of the core close to the vection simulations. For this purpose, we employ the con- CMB on the Fe-rich side of the Fe-S eutectic [e.g., Dumberry vection code StagYY [Tackley, 2008] in 2D spherical annulus and Rivoldini, 2015], consistent with the iron-rich ”snowing geometry [Hernlund and Tackley, 2008]. At this stage, we core” hypothesis of [Stewart et al., 2007]. Continued cool- would like to note that it is not our purpose to accurately ing combined with the shift in eutectic composition towards reproduce all geodynamical features of Mars (such as the Fe with increasing depth may result in a situation where activity related to the large volcanic provinces Tharsis and the areotherm also intersects the liquidus on the S-rich side Elysium), since this requires a larger number of simulations. Instead, the emphasis focusses on quantitatively reproduc- at higher pressure, causing co-crystallisation of Fe3S in the interior of the core [e.g., Hauck et al., 2006]. ing the thermal profile (internal temperature), and average grain size obtained in the geophysical inversion, in addition 5.7. Seismic profiles to 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 simulations grain size, while simultaneously producing mafic crust. Yet, is the observation that most of Mars’ crust is older than 3.7 magmatism depletes the mantle, i.e., decreases internal Ga, although there is evidence for recent melting and vol- heating, and limits any increase in potential temperature. canism [Phillips et al., 2001; Neukum et al., 2004; Solomon Consequently, it is reasonable to assume that the evolution et al., 2005; Werner, 2009; Carr and Head, 2010; Hauber of Mars’ internal temperature and average grain size are et al., 2010; Grott et al., 2013]. As a consequence, “suc- cessful” models are expected to produce mafic crust mainly governed by depletion in heat sources due to melting and in the beginning of the simulations, although localized melt magmatism. production is possible up until the present. Interestingly, To avoid numerical diffusion, grain size evolution is com- the geophysically-constrained internal temperatures are not puted on the Lagrangian advected by the flow. We use a too far below the solidus temperature (see figure 4). This simple grain growth equation and neglect dynamic recrys- suggests that “localised” features such as a large plume be- tallisation since grain size reduction terms are negligible due low Tharsis [Harder and Christensen, 1996; Zhong and Zu- to the small stresses in the Martian mantle ber, 2001; Roberts and Zhong, 2006; Zhong, 2009; Keller and Tackley, 2009; Srámekandˇ Zhong, 2010] would allow D E dp = k exp − d , (24) for regional magmatism to occur without being widespread Dt grain 0 RT as observed on e.g., Io [Breuer and Moore, 2007] and Venus [Ivanov and Head, 2013]. In the simulations performed here, where k0, p, and Ed are experimentally-determined con- we neglect the implications of a giant impact as a means of stants. In line with arguments made earlier (section 5.3), we forming most of the North-South crustal dichotomy [Mari- use a single grain growth law for the entire mantle. Grain nova et al., 2008; Nimmo et al., 2008; Reese et al., 2010; Golabek et al., 2011]. As a result, the amount of mafic crust growth parameters are difficult to determine experimentally that forms is likely to be underestimated. as laboratory experiments can only be run for a limited amount of time, far less than the scale on which geologi- 6.1. Crustal production and grain size evolution cal processes occur. Following Yamazaki et al. [2005], we 6 4.5 chose p=4.5, k0=3.9811·10 µm /s and Ed=414 kJ/mol. For the simulations, we solve the equations of compress- These parameters strictly only apply to the lower mantle of ible thermo-chemical convection, i.e., Stokes, heat and continuity equations employing the marker in cell method for the Mars where ringwoodite is stable and where the tempera- advection of rock composition, temperature and grain size ture is highest and, as a result, grain growth is fastest. While using the convection code StagYY [Tackley, 2008]. Here, the use of different growth laws in the shallower portions of only a brief summary is given. Details are provided in ap- Mars’ mantle would lead to a slight modification of grain size pendix B. evolution, the important observation is that the trends found When temperature exceeds the mantle solidus, partial here would not be dramatically changed. As discussed ear- melting ensues. The basaltic melt produced from the mixed lier, mid-mantle phase transitions (olivine→wadsleyite and mantle is transported towards the surface by both intru- wadsleyite→ringwoodite) are not expected to affect grain sive (50%) and extrusive (50%) magmatism, as described size [see also Solomatov and Reese, 2008]. in Rozel et al. [2017]. The solid residue is harzburgitic in The initial state is characterized by a potential tempera- composition (forming the depleted mantle) and will remain ◦ solid as the compositionally-dependent solidus temperature ture of 1326 C and a homogeneous grain size of 100 microns. increases [Maaløe, 2004]. Heat producing elements preferen- The mantle is a mixture of 15% eclogite and 85% harzburgite tially partition into the melt, implying that the mafic crust and we impose a 20 km pre-existing basaltic crust at the sur- deposited at the surface is 1/Dp times more radiogenic than face. Radiogenic heating of mixed mantle is initially equal −11 the depleted residue where Dp is the partition coefficient for to 2.3 · 10 W/kg, and decreases exponentially through- radiogenic heat producing elements. For the partitioning out time with a half-life of 2.43 Gyr [Nakagawa and Tackley, of heat producing elements between the mantle and crust 2005]. We operate with an initial core temperature of 2226 we use the approach described in Xie and Tackley [2004]. ◦C and take into account self-consistent core cooling using Our simulations show that Dp is an important parameter the heat flux at the CMB. Core radius is fixed to 1698.3 km, as it controls the effective internal heat production in Mars’ mantle, which directly influences the internal temperature which is the mean value obtained in the inversion. through melting and crustal production. Experimentally- determined values for Dp of radiogenic elements are of the 6.2. Geodynamical results order of 0.001 [Hart and Dunn, 1993; Hauri et al., 1994]. Figure 9 shows final profiles of temperature, viscosity, and In the simulations performed here, we assume a homogeneous temperature-, pressure-, and grain size-dependent rhe- grain size after 4.5 Gyr of evolution. Temperature and grain ological equation based on diffusion creep [Hirth and Kohlst- size profiles are strongly correlated, as expected, since the edt, 2013] over the entire computational domain final grain size represents an integral of the thermal history; a hot mantle results in a larger grain size. Viscosity, 3 dgrain E + PV E on the other hand, does not correlate with temperature nor η(P, T, dgrain) = η0 exp − , grain size, since these parameters compete in the rheological dref RT RTref (23) equation (Eq. 23). For comparison, geophysically-inverted where η0 and dref are reference viscosity and grain size thermal profiles are also shown (gray envelope). From the (1 mm) respectively, E and V are activation energy (375 results we observe the following trends: inefficient partition- 3 kJ/mol) and activation volume (6 cm /mol), respectively, ing of heat producing elements (high Dp) results in larger ◦ Tref is reference temperature (1326 C), and R is the uni- internal temperatures and thinner lithospheres, whereas low versal gas constant. This expression for η is similar to Eq. 14 reference viscosities tend to better cool the mantle, resulting with η = τµ, but the latter applies to timescales on the order in thinner lithospheres. of Phobos’ orbital period (hours), whereas here timescales To provide a more quantitative expression for these refer to the evolution of Mars (Gyr). To reproduce geo- dynamically the geophysically-inferred properties, we test trends, we used the viscosity profiles to detect the various reference viscosities in the range 1018 − 1021 Pa·s. lithosphere-asthenosphere boundary (LAB). The latter was Grain size evolution and heat producing element parti- defined as being located where the viscosity is 20% higher tioning are interrelated as grain growth is strongly temper- than the smallest viscosity value found below the litho- ature dependent, in that internal heating tends to increase 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) is capable of matching a set of geophysical observations for were obtained Mars. In particular, we have shown that mantle melting is not required to reproduce the relatively dissipative Martian d (km) ln D 0.852 η 0.100 lit = 1.093 p 0 (25) interior. Starting from a set of initial geochemically-based 500 A η0 bulk Martian compositions, we find strong evidence for a −0.0740 0.0333 relatively large (radius∼1640–1740 km) Fe core containing T (K) ln D η lit p 0 13.5–16 wt% S that appears to be entirely liquid at present = 1.062 0 , (26) 1600 A η given current constraints, data, and modeling assumptions. The results preclude the presence of a solid inner core on where dlit and Tlit are in units of km and K, respectively, account of the determined core S content, which is close to and A=ln 0.01 and η0=1020 Pa·s throughout. Figure 9 also the eutectic composition. The model geotherms obtained shows that the internal viscosity and grain size (second and here remain below the present-day dry Martian solidus in- third panels) depend on both D and η . At a depth of p 0 ferred by Kiefer et al. [2015] and indicate CMB tempera- 1000 km, the scaling laws for internal viscosity ηint and grain ◦ size d were found to be tures around 1560–1660 C. From the determined core size grain and CMB pressure and temperature, the presence of a lower 3.18 0.459 mantle transition, equivalent of the “660-km” seismic dis- η ln D η int = 3.69 p 0 (27) continuity in the Earth seems unlikely. The bulk Martian 1020 A η0 compositions derived here are generally chondritic with a d ln D 0.108 η 0.102 Fe/Si (wt ratio) of 1.63–1.68. Grain size estimates range grain = 1.079 p 0 , (28) 500 A η0 from sub-mm to 5 mm, but are generally <1 mm, and well within the range observed on Earth. where ηint and dgrain are in units of Pa·s and m, respectively. Following this, we performed a number of thermo- These relations indicate that both viscosity and grain size chemical evolution simulations with the purpose of fit- increase with increasing Dp and η0. ting the geophysically-derived results (specifically mantle The present-day average surface heat flux (QHF) obtained geotherm, lithospheric thickness, and grain size). This in our simulations ranges from 15 to 25 mW/m2. This is in serves to show that the inversion results can be used in good agreement with the estimates derived by Plesa et al. tandem with geodynamic simulations to identify plausible [2016] where mean surface heat fluxes in the range 23–27 geodynamic scenarios and parameters. For a reasonable set mW/m2 were obtained. An important observation here is of radioactive element partition coefficients between crust that partitioning of heat producing elements and a low vis- and mantle around 0.001 and reference viscosities (1019– cosity tend to slightly increase the heat flux, as summarized 1020Pa·s), the geodynamic models, based on 2D models and by the scaling law: covering 4.5 Ga of Martian history, were able to reproduce current areotherms, crustal and, in part, lithospheric thick- Q ln D −0.393 η −0.0158 nesses, and grain sizes. The results seem to suggest that HF = 0.730 p 0 , (29) 25 A η0 Mars could be volcanically active at present in accord with earlier suggestions based on other geodynamical studies and where the heat flux is expressed in mW/m2. The above five present-day surface observations. The geodynamic models scaling laws are illustrated in figure 1 in appendix B. also predict a present-day mean surface heat flow between 2 Figure 9 and figure 10 show that the current average 15–25 mW/m , in excellent agreement with the recent heat crustal thickness, which ranges between 33 and 81 km [Wiec- flow estimates by [Plesa et al., 2016]. zorek and Zuber, 2004; Neumann et al., 2004] (gray shaded We should emphasize that the point of the geodynam- area in figure 10), and the geophysically-derived tempera- ical simulations was not to accurately reproduce the dy- ture profiles obtained in this study (gray shaded area in namical evolution of Mars, but rather to provide 1) first- figure 9), can be fit using the experimentally-determined Dp order insights on the long-term evolution (∼4.5 Gyr) in in the range of 0.001 and a reference viscosity in the range terms of a few crucial parameters such as crustal produc- of 1019–1020 Pa·s. In particular, inefficient partitioning of tion, mantle temperature, and grain size growth and 2) to heat producing elements (Dp=0.1), irrespective of the value see if these can be varied within reasonable bounds, while si- of η0, can be ruled out because mantle temperatures are too multaneously fitting the present-day geophysical predictions high by several hundred degrees relative to the inverted pro- obtained here. In this context, we have been made aware files and so is crustal production (>>200 km, not shown in that more recent estimates of the second degree tidal Love figure 10). The preferred values of Dp and η0 that are consis- number have become available that are ∼5–10% larger [e.g., tent with the geophysical predictions, correspond to present- Konopliv et al., 2016; Genova et al., 2016]. Everything be- day grain sizes in the sub-mm range (0.4—0.5 mm). In spite ing equal, a larger k2 will result in a larger core size, but, of the considerable overlap over the entire mantle, temper- by Eqn 9, increasing k2 also results in a larger Q, i.e., less atures in the lithosphere derived from geophysical models dissipation, which could counter the aforementioned effect. appear to be higher than those obtained in the geodynami- This will have to be considered in more detail in the future. cal calculations including grain-size evolution. Future stud- Being anhydrous and melt-free, the models constructed ies should address the effects of dynamic recrystallization, here are end-member cases in that presence of e.g., water which was not included in our simple grain growth equation would lower the solidus [e.g., Pommier et al., 2012], produce and the effects of an initial heterogeneous grain size distri- melt and increase dissipation, which would have to be offset bution as the grain size influences the viscosity and hence by an overall decrease in mantle temperature. The ther- cooling behavior and temperature. The geodynamics results mal evolution of Mars depends crucially on mantle viscos- (figures 9 and 10) supporting the scaling laws (Eqs. 25–29) ity, which is also determined by conditions of temperature, are summarized in table 5. pressure, and water content. Even if present at small lev- els (tens of ppm), water can lead to a reduction of effective 7. Summary and Conclusion viscosity and Q, thereby enhancing dissipation [Hirth and Kohlstedt, 1996; Karato and Jung, 1998; Mei and Kohlst- We have shown that a grain size and frequency-dependent edt, 2000; Karato, 2013]. As discussed by Filiberto et al. visco-elastic model (extended Burgers rheology) based on [2016], presence of 0.02 wt% water in the mantle source re- laboratory deformation of melt-free polycrystalline olivine 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 associated (Fo90) has had its viscoelastic behaviour characterized; significant increase in convective vigor as observed in var- nonetheless, Nimmo and Faul [2013] argue that the errors ious dynamical evolution studies [e.g., Hauck and Phillips, introduced by focusing exclusively on Fo90 olivine are likely 2002; Li and Kiefer, 2007; Morschhauser et al., 2011; Ogawa to be small, since the behaviour of olivine is believed to be and Yanagisawa, 2012; Ruedas et al., 2013b]. The combined broadly representative of the viscoelastic behaviour of the effect of this would be to lower the mantle geotherm below upper mantle. Compostitional effects will mainly influence what has been obtained here. Accordingly, the present man- the unrelaxed modulus (pers. comm., Ian Jackson, 2014). tle geotherms represent upper bounds. Based on a compilation of existing data, Karato [2013] sim- Although water in some shergottite magmas has been ob- ilarly argues that addition of orthopyroxene to olivine has served [McCubbin et al., 2012], the present-day water con- little effect on anelasticity. In line herewith, and because tent of the Martian mantle is not well constrained. It likely olivine is the dominant mineral (∼60 % by volume) through- contains less water than Earth with a probable value of out most of the Martian mantle, neglect of the contribution around 300 ppm [Taylor, 2013]. This estimate assumes no from other phases is expected to less likely result in signif- water loss during or after accretion due to impact heating, chemical reactions, and magma ocean crystallisaton [Wänke icant changes. In view of the different Mg#s of Earth and and Dreibus, 1994; Elkins-Tanton et al., 2005]. For now, the Mars, a possible exception is the iron content that could po- effect of water needs to be better quantified and understood tentially change the dissipative properties of olivine [Zhao before it can be modeled properly. In addition, current lack et al., 2009]. We leave it for a future treatment, based on a of thermodynamic data accounting for the influence of wa- new and expanded experimental database, to consider these ter on mantle phase equilibria further prohibits quantitative effects in more detail. Also, while this study is based on the modeling. most appropriate experimental constraints available, their Presence of melt very much mimics the effect of water, use involves an extrapolation beyond the available experi- i.e., lowers Q and increases dissipation. While there is some mental frequencies as well as an extrapolation in grain size. evidence for magmatic activity that might have occured un- Further experiments closer to the Martian parameter space til relatively recently [e.g., Neukum et al., 2004; Niles et al., we explore would be valuable future work. 2010], Mars is unlikely to be molten on the global scale at In spite of these caveats, the approach outlined here is present. Even if localized pockets or regions inside Mars may capable of making significant predictions (summarized in ta- remained molten up until the present, these are unlikely to ble 4) that provide many quantitative insights. Moreover, affect global dissipaton. What can be argued based on the the study proposed here will help quantify additional fu- results here and those of Nimmo and Faul [2013] is that ture requirements – from the acquisiton of new experimental laboratory-based visco-elastic dissipation models do not re- data to modeling aspects – that will be in need of further quire melt to fit current observations. The effect of a small attention to extend the predictions to increasingly complex melt fraction on dissipation has been studied in the labo- models. ratory [e.g., Faul et al., 2004; McCarthy and Takei, 2011; Last, but not least, the predictions made here, will ul- Takei, 2017]. Melt acts to enhance dissipation and render timately be tested with the upcoming InSight mission that it essentially frequency-independent across the seismic fre- will perform the first in situ measurements of the interior quency band [Faul et al., 2004]. However, modeling its in- of Mars through the acquisition of seismic, heat flow, and fluence presents, on a par with water, an equally challenging problem that will be left for a future study. geodetic data [Banerdt et al., 2013]: In this context, earlier studies based on viscoelastic mod- 1) Analysis of the seismic data to be returned from In- els [e.g., Bills et al., 2005; Castillo-Rogez and Banerdt, 2013], Sight will rely on more “classical” global seismology tech- that relied on different rheologies (Maxwell and Andrade), niques in the form of travel-time tables for various seis- found mantle viscosities around 1014–1016 Pa s that appear mic phases, surface-wave dispersion, receiver functions, nor- to be inconsistent with current estimates of Earth’s man- mal modes, and waveforms [e.g., Okal and Anderson, 1978; tle viscosity (∼1019–1021 Pa s). This suggested that either Zheng et al., 2015; Panning et al., 2016]. These data will Mars does not behave as Maxwell bodies or that a significant shed light on the interior physical structure of crust, man- amount of partial melt is needed to lower the viscosity to the tle, and core, including velocity gradients and dicontinuities aforementioned values. The latter possibility has gained in associated with mineral phase transitions and/or chemical strength in the case of the Moon, where evidence in sup- boundaries. Moreover, the data will help constrain Martian port of partial melt in the deep lunar interior has steadily seismicity and tectonism and, not least, meteorite impact accumulated [e.g., Williams et al., 2001; Weber et al., 2011; rates [e.g., Golombek et al., 1992; Knapmeyer et al., 2006; Efroimsky, 2012a, b; Harada et al., 2014; Khan et al., 2014; Teanby and Wookey, 2011]. Williams and Boggs, 2015], although Nimmo et al. [2012] 2) On a par with the importance of acquiring seismic data, find that melt is not strictly required to match the obser- measuring heat flow with the heat flow probe (HP3) onboard vations with the caveat that the frequency-dependence of InSight [Spohn et al., 2014] cannot be overestimated, be- Q (i.e., α) at tidal periods is opposite in sign to what has cause it will allow for an independent means of distinguish- been observed through lunar laser ranging [e.g., Williams ing between the various bulk compositional models. Specifi- et al., 2014]. As pointed out by Efroimsky [2012a], though, cally, by combining the heat flow measurement with numer- the lunar laser ranging-based observation of a negative α α −α ical estimates of the Urey ratio [e.g., Plesa et al., 2015], it (i.e., Q ∼ ω instead of the common case where Q ∼ ω ) will be possible to constrain the heat production rate and is entirely in accordance with a low-viscosity attenuating thus bulk abundance of heat producing elements in the in- region in the lunar interior. Employing a homogeneous incompressible spherical model of the Moon Efroimsky [2012a] terior of Mars. Camparison of the latter with the proposed and Efroimsky [2012b], showed that the apparent decrease compositional models for Mars provides the necessary tool. 3) In the context of improving estimates of core structure, of k2/Q with period observed through lunar laser ranging could be explained with a function equivalent to a single re- the RISE (Rotation and Interior Structure Experiment) in- laxation time model. Using a viscosity around (∼1015 Pa s), strument onboard InSight will prove important [e.g., Folkner k2/Q was shown to peak in the tidal band in the vicinity of et al., 2012; van Hoolst et al., 2012]. Through precise radio the tidal periods (see e.g., figure 1 in Efroimsky [2015]). We tracking of the landed spacecraft, RISE will be able to mea- leave it for a future study to consider the impact of differ- sure the precession rate and through it allow for a more ent types of rheological models in the context of the inverse accurate estimation of the polar moment of inertia In par- problem posited here. ticular, for a fluid Martian core, the nutation of the spin The present approach is not fully self-consistent in that axis can be resonantly amplified and allow for an indepen- only a single mineral (olivine) and a single composition 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- To compute γ, we rely on the formulation of Vashchenko tation, the moment of inertia of the core can be estimated. and Zubarev [e.g. Poirier, 2000]: These parameters are crucially sensitive to core size, shape, and state. RISE observations over a Martian year should 1 K0 − 5 + 2 PK−1 γ = 2 T 6 9 T . (A6) enable a clear detection of the nutation signature and thus 1 − 4 PK−1 core parameters. 3 T These data sets, individually and in combination, will To compute thermoelastic properties for each component prove to be the Rosetta stone for unraveling the thermo- at P and T, we first perform an isothermal compression from chemical structure and evolution of Mars. the reference conditions (P0,T0) to (P,T0), followed by iso- baric heating to (P,T). For the isothermal compression at Acknowledgments. We thank Walter Kiefer and Ian Jack- T , we employ a third-order finite strain Birch-Murnaghan son for comments that helped improve the clarity of the 0 equation-of-state [Stixrude and Lithgow-Bertelloni, 2005a]: manuscript. We also thank Michael Efroimsky for valuable comments on tidal deformation and additional suggestions for im- 5 3 proving the manuscript. Finally, we thank Jeff Taylor and Gre- 2 0 P = − 3ε(1 − 2ε) KT,0 − KT,0(KT,0 − 4)ε , (A7) gor Golabek for helpful comments on an earlier version of this 2 manuscript. A.K. and D.G. would like to acknowledge support 5 27 2 0 0 2 from the Swiss National Science Foundation (SNF-ANR project KT =(1 − 2ε) KT,0 − KT,0 3KT,0 − 5 ε + KT,0(KT,0 − 4)ε , 157133 “Seismology on Mars”). Computations were performed 2 on the ETH cluster Euler. Models can be downloaded from (A8) http://jupiter.ethz.ch/~akhan/amir/Models.html. 0 143 K0 =K0 + 3K 2 − 21K0 + ε, (A9) T T,0 T,0 T,0 3 Appendix A: Thermoelastic properties of 2 1 − 3 the core where ε = 2 1 − β is the Eulerian strain, β = V (P)/V0, volume, isothermal bulk modulus, and pressure derivative The thermoelastic properties of the liquid core alloy as a 0 at (P,T0) are denoted by V (P), KT(P) and KT,0(P), re- function of pressure (P) and temperature (T) are calculated spectively, and zero-subscripted quantities are evaluated at following Dumberry and Rivoldini [2015], with the excep- reference conditions. tion of eq. A12, and are contained here for completeness. Volume at (P,T) is obtained from the definition of the In this approach, it is assumed that the properties of the thermal expansivity [e.g. Poirier, 2000]: alloy can be calculated from the thermoelastic properties of liquid Fe and liquid Fe-10wt%S (hereafter referred to as Fe V (P, T) = V (P) exp [α(P)(T − T0)] . (A10) and FeS10) and that both components mix ideally. Based on this, the molar volume V of the solution is given by The expression for α(P) follows from the definition of the Anderson-Grüneisenparameter δT [e.g. Poirier, 2000] 2 X V = χiVi, (A1) δT q α(P) = α0 exp − (1 − β ) , (A11) i=1 q where χ1 = 1 − χ2 is the molar fraction of Fe and χ2 the where q is a material-dependent parameter. The temper- molar fraction of FeS . The molar concentration of sulfur 10 ature dependence of the isothermal bulk modulus KT at in FeS10 is related to the weight fraction of sulfur in the pressure P can be derived from Eqs. A10 and A11 solution, XS, by KT(P) XS MFe KT(P, T) = q . (A12) χ2 = (A2) 1 + (T − T0)α(P)δTβ χ MS(1 − XS) + MFeXS S10 0 Finally, we assume that KT(P) is independent of tempera- where MFe = 55.845 g/mol and MS = 32.065 g/mol are the ture. The relevant material properties for the equations of molar masses of iron and sulfur, respectively, and χ = S10 state for Fe and FeS10 are summarized in the table below. 0.162137 is the atomic fraction of sulfur in FeS10. The thermal expansivity α, isothermal bulk modulus KT, Appendix B: Geodynamical simulations and pressure derivative KT0 of the solution can be derived from Eq. A1 through the application of standard thermo- Thermo-chemical compressible convection is studied us- dynamic relations ing the code StagYY in a 2D spherical annulus geometry 2 [Tackley, 2008; Hernlund and Tackley, 2008]. The equations X α V = χi αiVi, (A3) i=1 2 2 2 Table 1. Equation of state parameters for core components. V X Vi 0 KT X Vi 0 Fe Anderson and Ahrens [1994] and Fe − 10wt%S Balog et al. = χi , KT = −1 + χi (1 + KT,i). KT KT,i V KT,i [2003]. The thermal expansivity of Fe − 10wt%S is calcu- i=1 i=1 (A4) lated from the volume and thermal expansivity of FeS Kaiura and Toguri [1979] at 1923K (V = 12.603cm3/mol, α = −5 −1 ∗ 16.5 × 10 K ) by applying Eqs. A1 and A3. Since δT In order to compute the isentropic temperature gradient is not available for Fe and Fe − 10wt%S, we made use of the in the core (Eq. 17), the adiabatic bulk modulus KS and 0 following approximate relation [Poirier, 2000, e.g.]: δT ≈ KT. ∗∗ the Grüneisenparameter γ are also required. KS is related is taken from Helffrich [2012]. to the isothermal bulk modulus KT by the following relation [e.g. Poirier, 2000]: 3 −5 −1 0 Components T0 [K] V [cm /mol] α [10 K ] KT [GPa] KT δT q Fe 1811 7.96 9.2 85.3 5.9 5.9 1.4 ∗ ∗∗ KS = KT (1 + αγT) . (A5) 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- olivine and pyroxene-garnet densities, for example, smoothly ing a parallel direct solver (MUMPS) available in the PETSc increase with depth, but also contain discontinuities at the package [Amestoy et al., 2000]. We use the finite difference phase transitions [Rozel et al., 2017]. Finally, melt density approximation on a staggered grid [Harlow and Welch, 1965] increases smoothly with pressure. with a radially varying resolution (higher in the top and bottom boundary layers). The domain is discretized by 64 times 512 cells. Free slip boundary conditions are imposed on all domain boundaries. The surface temperature is fixed to 220 K and the initial core temperature is 2500 K. Lagrangian markers are advected through the mesh using second-order divergence-free spatial interpolation of the velocities and a fourth-order Runge-Kutta scheme. Each tracer carries several quantities such as composition, temperature, grain size, and radiogenic heating rate, among others. Numerical diffusion related to advection is limited by the use of tracers. Our petrological model considers solid and molten rocks as being a linear combination of basalt- eclogite (crustal material) and harzburgite (depleted mantle). The primordial mantle starts with an initial petrological composition of 15 % basalt (100 % pyroxene-garnet) and 85 % harzburgite (a mixture of 55 % olivine and 45 % pyroxene-garnet). For numerical efficiency the composition is stored and transported on tracers, while the evolution of the melt fraction is computed on the mesh field (which gen- erates new molten tracers in each corresponding cell). A composition field is computed at cell centers by averaging of the tracers within the cell. The melt fraction is computed at every time step comparing the pressure-temperature conditions to a composition-dependent solidus function, considering a latent heat of 600 kJ/kg. In case melting occurs at less than ∼600 km depth, molten tracers of fully basaltic composition are transported either to the top of the domain (eruption) or to the bottom of the crust (intrusion). The temperature of all erupted tracers is set to surface temperature, which tends to form a cold lithosphere. The temperature of intruded tracers (at the bottom of the crust) takes only adiabatic decompression into account, which tends to produce a warm lithosphere. The column of material between the melt source region and the intrusion or eruption location is moved downwards to conserve mass. The solid residue left behind by the eruption-intrusion procedure is Figure 1. Lithosphere thickness, LAB temperature, in- more harzburgitic than the initial solid. For more details ternal viscosity (at 1000km depth), internal grain size, regarding implementation the reader is referred to Naka- and surface heat flux at the end of the simulations (equiv- gawa and Tackley [2004, 2012]. The density in each cell alent to 4.5 Gyr of evolution), scaled using the approach is computed as the sum of pressure, thermal and compo- described in section 6.2. sitional components, including solid-solid phase transitions. Olivine and pyroxene-garnet phases are treated separately, which enables accounting for the density increase associated with the basalt-eclogite phase transition. Plastic yielding in the lithosphere is computed using a depth-dependent yield stress τy following Byerlee’s law [By- References erlee, 1978]: Agee, C. B., and D. S. 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