RECONCILING THE ORBITAL AND PHYSICAL PROPERTIES OF THE MARTIAN MOONS

During the 1970s and 1980s, dynamicists demonstrated that the present low eccentricity, low inclinations, and prograde orbits of Phobos and Deimos are very unlikely to have been produced following capture (Burns 1978; Pollack et al. 1979), thus favoring a formation of the moons around Mars (Goldreich 1963; Cazenave et al. 1980; Szeto 1983). Despite such early robust evidence against a capture scenario, the fact that the moons share similar physical properties (low albedo, red and featureless VNIR reflectance, low density) with outer main belt D-type asteroids has kept the capture scenario alive (Fraeman et al. 2012, 2014; Pajola et al. 2013).

Whereas the present orbits of the moons are hardly compatible with a capture scenario, they correspond to the expected outcome of an in situ formation scenario as the result of either co-accretion or a large impact. Co-accretion with Mars appears unlikely because Phobos and Deimos would consist of the same building block materials from which Mars once accreted. Those building blocks would most likely comprise water-poor chondritic meteorites (enstatite chondrites, ordinary chondrites) and/or achondrites (e.g., angrites), which are all suspected to have formed in the inner (≤2.5 au) solar system, namely, interior to the snowline. This assumption is supported by the fact that the bulk composition of Mars can be well reproduced assuming ordinary chondrites (OCs), enstatie chondrites, and/or angrites as the main building blocks (Sanloup et al. 1999; Burbine & O'Brien 2004; Fitoussi et al. 2016). Yet OCs, as well as the remaining candidate building blocks (enstatite chondrites, angrites), are spectrally incompatible with the Martian moons, even if space weathering effects are taken into account (see panel (b) in Figure 1).

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It thus appears from above that accretion from an impact-generated accretion disk remains as the only plausible mechanism at the origin of the Martian moons. As a matter of fact, the large impact theory has received growing attention in recent years (Craddock 2011; Rosenblatt & Charnoz 2012; Canup & Salmon 2014; Citron et al. 2015). This hypothesis is attractive because it naturally explains the orbital parameters of the satellites, as well as some features observed on Mars, such as (i) its excess of prograde angular momentum possibly caused by a large impact (Craddock 2011) and (ii) the existence of a large population of oblique impact craters at its surface that may record the slow orbital decay of ancient moonlets formed from the impact-generated accretion disk (Schultz & Lutz-Garihan 1982). Along these lines, Citron et al. (2015) have recently shown that a large impact (impactor with 0.01–0.02 Mars masses) would generate a circum-Mars debris disk comprising ∼1%–4% of the impactor mass, thus containing enough mass to form both Phobos and Deimos. Although the impact scenario has become really attractive, it has not yet been demonstrated that it can explain the physical properties and spectral characteristics of the Martian moons.

Here we investigate the mineralogical composition and texture of the dust that would have crystallized in an impact-generated accretion disk. Since there are no firm constraints regarding the thermodynamic properties of the disk, we perform our investigation for various thermodynamic conditions and impactor compositions. We show that under a specific disk's pressure and temperature conditions, Phobos's and Deimos's physical and orbital properties can be finally reconciled.

Because of the absence of constraints regarding the composition (Mars-dominated or impactor-dominated) and the thermodynamic conditions of the impact-generated disk, several configurations must be investigated in order to understand the formation conditions of Phobos and Deimos within such a scenario. As a first step, we considered the protolunar disk as a reference case because it is so far the most studied impact-generated accretion disk. Its structure has been investigated by Thompson & Stevenson (1988) and subsequently by Ward (2012, 2014). These studies have shown that the disk's midplane consists of a liquid phase surrounded by a vapor atmosphere. Beyond the Roche limit, gravitational instabilities developed and large clumps formed directly from the magma (e.g., Kokubo et al. 2000; Salmon & Canup 2012). Those clumps then agglomerated to form the Moon. In this case, because of internal evolutionary processes (differentiation, convection, etc.), the mineralogical composition and thus the spectral properties of the lunar mantle and crust will differ from the clump ones. In the Martian case, the situation is different in the sense that the clumps possess right away a mass/size comparable to that of Phobos and Deimos (Rosenblatt & Charnoz 2012). This implies that the final composition and spectral properties of the Martian moons would directly reflect those of the minerals that crystallized from the magma disk.

2.1. Methods

2.2. Results

A different formation mechanism is thus required to explain both the current orbits of the moons and their spectral characteristics. Of great interest, Rosenblatt & Charnoz (2012) suggested that the moons could have formed in an extended gaseous disk farther from Mars than in the two-phase disk case. Such a disk should be initially hot so that it would thermally expand under pressure gradients and be gravitationally stable beyond the Roche limit. As the disk would thermally expand, it would cool down rapidly. The extended disk would also have a lower pressure and a larger radiative surface, allowing, again, a faster cooling than a compact disk residing inside the Roche limit.

In this scenario, the conditions under which Phobos and Deimos formed could have been similar to those that occurred in the protosolar nebula in the sense that small solid grains could have condensed directly from the gas without passing through a liquid (magma) phase (see Section A.2.). In this case, Phobos and Deimos would consist of material that has the same texture—not necessarily the same composition—as the one that has been incorporated into comets and D-type asteroids, namely, fine-grained dust (grain size ≤2 μm; see A2; Vernazza et al. 2015). It would therefore not be surprising that these objects share similar physical properties, including (i) a low albedo (pv ∼ 0.06) and (ii) featureless and red spectral properties in the visible and near-infrared range (see Figure 1, panel (c)—right; Vernazza et al. 2015). It is important to stress here that there are currently no available laboratory reflectance spectra in the visible and near-infrared range for submicron-sized particles. The spectral behavior in this wavelength range can be reproduced via the Mie theory as implemented by Vernazza et al. (2015). These authors showed that a space weathered mixture of submicron-sized olivine and pyroxene grains would possess spectral properties similar to those of P- and D-type asteroids. Future laboratory measurements will be necessary in order to characterize the reflectance properties of all kinds of minerals (silicates, phyllosilicates, iron oxides, etc.) in order to provide more accurate constraints on the composition of the moons. Note that a further comparison of the Phobos and Deimos spectra with those of fine-grained lunar mare soils (grain size ≤ 10 μm) reinforces the idea that the moons may effectively be aggregates of submicron-sized grains. Indeed, although the average grain size of the lunar soils is small (≤10 μm), absorption bands at 1 μm are still visible, suggesting that the grains at the surface of the Martian moons must be even smaller than these already fine-grained lunar samples.

Finally, accretion from such poorly consolidated submicron-sized material would also naturally explain the low densities (∼1.86 g cm⁻³ for Phobos and ∼1.48 g cm⁻³ for Deimos) and high internal porosities (∼40%–50% assuming an anhydrous silicate composition) of the moons (Andert et al. 2010; Rosenblatt 2011; Willner et al. 2014). Importantly, such high porosity is not observed in the case of S-type asteroids with diameters in the 10–30 km size range (a ∼20%–30% porosity is observed for these objects; Carry 2012), reinforcing the idea that the building blocks of the Martian moons must be drastically different in texture from those of S-type asteroids (i.e., OCs).

The fact that the building blocks of the moons should avoid a magma phase provides interesting constraints on the thermodynamic conditions that prevailed inside the disk. Whereas a magma layer inside the Roche limit is not inconsistent with our findings (the moon or moons that would have formed from this layer would have impacted Mars a long time ago), future modeling of the disk should account for an extended disk where the pressure and the temperature allow for a direct condensation of the vapor into solid grains.

In order to provide constraints for future models of the Martian disk, we determined the pressure–temperature ranges where the gas would directly condense into solid grains assuming a BSM composition of the gas. We restricted our analysis to the condensation of olivine only. Gail (1998) found that olivine would first condense as nearly pure forsterite and iron might be included later on in the solution. We consequently considered the condensation of pure forsterite. The pressure up to which solid or liquid forsterite is stable for a given temperature can be calculated with the following formula:

$$\begin{eqnarray}&&P{(T)}^{3}=\displaystyle \frac{1}{{x}_{\mathrm{MgO}}^{2}{x}_{{\mathrm{SiO}}_{2}}{e}^{-{\rm{\Delta }}{G}_{s,l}/{RT}}},\end{eqnarray} \tag{ 1 }$$

where ΔG_s,l is the Gibbs free energy of formation of solid or liquid forsterite from gaseous MgO and SiO₂ (it can be calculated with the JANAF online tables), and x_MgO and ${x}_{{\mathrm{SiO}}_{2}}$ are the molar fractions of the gases. An extensive chemical model would thus be required in order to infer these molar fractions. As such detailed modeling is beyond the scope of the present work, we simply assumed that different fractions (f) of Mg and Si were bound into MgO and SiO₂ molecules in the gas phase and thus ${x}_{\mathrm{MgO},{\mathrm{SiO}}_{2}}=f{\epsilon }_{\mathrm{Mg},\mathrm{Si}}$, being the fraction of the element.

The stability curves are shown in Figure 2 for f = 1, 0.1, and 0.01. The last two cases are more realistic given that Mg is usually found as a free atom whereas Si is mainly found in SiO molecules. We find that the solid phase of forsterite becomes more stable than the liquid one below a temperature of ∼2200 K. At this temperature, the condensation of forsterite occurs at a pressure lower than 10⁻⁶ to 10⁻³ bar depending on the partial pressures of MgO and SiO₂.

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Here we have opened the possibility that gas-to-solid condensation in the external part of an extended gaseous disk is a likely formation mechanism for the Martian moons' building blocks, as it would lead to the formation of small (≤2 μm) dust particles. Accretion from such tiny grains would naturally explain the similarity in spectral properties between D-type asteroids (or comets) and the Martian moons, as well as their low densities. It therefore appears that accretion in the external part of an impact-generated gaseous disk is a likely formation mechanism for the Martian moons that would allow reconciling their orbital and physical properties. Future work should attempt modeling the mineralogy resulting from the gas-to-solid condensation sequence and verify that a space weathered version of the derived composition sieved to submicron-sized grains is compatible with the spectral properties of the moons. Such investigation would greatly benefit from detailed numerical models that would constrain the thermodynamic properties of a circum-Mars impact-generated disk as a function of radial distance.

Finally, note that our proposed scenario is not incompatible with the presence of weak hydration features in the spectra of Phobos and Deimos (Giuranna et al. 2011; Fraeman et al. 2012, 2014). Water has been delivered to the surfaces of most if not all bodies of the inner solar system, including the moon (Sunshine et al. 2009), Mercury (Lawrence et al. 2013), and Vesta (Scully et al. 2015). It has thus become clear recently that space weathering processes operating at the surfaces of atmosphere-less inner solar system bodies not only comprise the impact of solar wind ions and micrometeorites, which tend to redden and darken the spectra of silicate-rich surfaces, but also comprise contamination and mixing with foreign materials, including water-rich ones (Pieters et al. 2014).

We warmly thank the referee for his very constructive review. P.V. and O.M. acknowledge support from CNES. This work has been partly carried out thanks to the support of the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR).

A.1. Grain Size Resulting from Magma Solidification

The grain size of a crystal resulting from magma solidification has been extensively studied and appears closely related to its cooling history. Rapid cooling (≳10² K hr⁻¹) will lead to the formation of smaller grains (∼10⁻² mm), whereas slow cooling (≲1 K hr⁻¹) will lead to the formation of larger grains (∼1 mm), as is observed in experimental studies and Earth's samples (Flemings et al. 1976; Ichikawa et al. 1985; Cashman & Marsh 1988; Grove 1990; Cashman 1993). This is well illustrated in the case of Earth's rocks via basalts and gabbros. These rocks possess the same composition but a very different texture. Basalts are extrusive igneous rocks that experienced rapid cooling within either Earth's atmosphere or oceans and possess a fine-grained structure. On the other end, gabbros are intrusive igneous rocks that crystallized below Earth's surface on much longer timescales and thus exhibit coarse grains.

In the present case, assuming that the temperature of the magma layer is regulated by the disk radiative cooling only, an order of magnitude of the disk cooling rate is

$$\begin{eqnarray}&&\displaystyle \frac{{dT}}{{dt}}=-\displaystyle \frac{2\pi {R}_{\mathrm{disk}}^{2}{\sigma }_{\mathrm{SB}}{T}_{{ph}}^{4}}{{M}_{\mathrm{disk}}{C}_{p}}\sim -2\,{\rm{K}}\,{\mathrm{hr}}^{-1},\end{eqnarray} \tag{ 2 }$$

where ${R}_{\mathrm{disk}}=4{R}_{\mathrm{Mars}}$, which is approximately the Roche limit, σ_SB = 5.67 × 10⁻⁸ J m⁻² K⁻¹ is the Stefan–Boltzmann constant, T_ph = 2000 K is the temperature of the disk at the photosphere that is maintained at ∼2000 K throughout its lifetime (Ward 2012), M_disk = 5 × 10²⁰ kg following the results of Citron et al. (2015), and C_p = 4 × 10³ J kg⁻¹ K⁻¹ is the heat capacity of the vapor.

One may also consider that the clumps formed still molten at the Roche limit. In this case, assuming a density ρ = 3300 kg m⁻³ for the melt and clumps with typical sizes in the 1–10 km range, an order of magnitude of the clump cooling rate would be

$$\begin{eqnarray}&&\displaystyle \frac{{dT}}{{dt}}=-\displaystyle \frac{3\pi {\sigma }_{\mathrm{SB}}{T}_{s}^{4}}{{R}_{\mathrm{clump}}\rho {C}_{p}}\sim -0.2\mbox{--}2\,{\rm{K}}\,{\mathrm{hr}}^{-1},\end{eqnarray} \tag{ 3 }$$

where the largest clumps would possess the slowest cooling rates and vice versa. In either case, the cooling timescales are comparable to those derived from laboratory experiments, and the rocks that would have crystallized from the magma should typically exhibit the same grain sizes as those found in magmatic rocks on Earth.

In a different register, it is interesting to note that chondrules, which formed as molten or partially molten droplets in space before being accreted to their parent asteroids, have typical sizes in the 0.1–1 mm range (Hutchison 2004). In summary, magma condensates appear always coarse grained (grain size in the 0.1–1 mm range), regardless of their formation mechanism.

A.2. Grain Size Resulting from Gas to Solid Condensation