Model description. This work follows the models developed by Love and Brownlee (34) (hereafter LB) and Genge (14) (hereafter MG). The LB model describes the entry and evaporation of silicate micrometeorites on modern Earth. The MG model expands the model of LB to include Fe-rich micrometeorites and their oxidation during atmospheric entry. Below, we describe our model, which is an implementation of the MG model but for oxidation of Fe by CO 2 .

Following MG and LB, our model describes the motion, heating, evaporation, and oxidation of iron micrometeorites. We assume an initial velocity, v (m s−1), an initial mass, m (kg), and an initial entry angle from zenith θ with θ = 90∘ being tangential to Earth’s surface and θ = 0∘ indicating the particle is moving directly toward Earth’s surface. The motion of the particle in two dimensions, accounting for atmospheric drag, can be calculated via dv dt = g − 3 ρ a v 2 4 ρ r v ̂ (1)where g is gravity (m s−2), v is velocity (m s−1), t is time (s), ρ a is the atmospheric density (in kg m−3), ρ is the density of the micrometeorite (kg m−3), and r is the particle radius (m). The altitude, a, of the particle was assumed to start at 190 km above Earth’s surface, following LB, and tracked throughout the model. For a given altitude, g is easily calculated from g = GM ⊕ / r alt 2 for gravitational constant G = 6.67 × 10−11 N m2 kg−2 and Earth mass M ⊕ = 5.97 × 1024 kg; here, r alt is the radial distance from the center of Earth to altitude, a. The MSIS-E-90 Atmosphere Model (available at https://ccmc.gsfc.nasa.gov/modelweb/models/msis_vitmo.php) was used to generate an atmospheric density profile for modern Earth, which we used for both the modern and Archean atmospheres following (35). Even if the atmospheric pressure in the Archean were lower than modern (22), the density profile of the upper atmosphere generated by the MSIS-E-90 model would likely remain similar and produce similar model results, as noted by (10).

We used the MSIS-E-90 model to generate atmospheric densities, ρ a , and total atmospheric oxygen densities (both O and O 2 ) at 1-km intervals from Earth’s surface to 190 km (see data file S1 atmosphere_data.txt for MSIS-E-90 input parameters and resulting data). We linearly interpolated between each data point to find the atmospheric density for a given altitude. When calculating atmospheric CO 2 abundance in the model, we specify a CO 2 wt %, and then multiply the wt % by ρ a . This was done to make conversions from the MSIS-E-90 density data to CO 2 abundances a simple conversion. For example, if the model is run with 30 wt % CO 2 , then the atmospheric CO 2 density will be found via 0.3ρ a , with ρ a coming directly from the MSIS-E-90 data. When modeling CO 2 atmospheres, we assume the remainder of the atmosphere is N 2 .

With the velocity of the micrometeorite known, the heat flux of the particle, dq/dt, in watts can be described by dq dt = π r 2 ρ a v 3 2 − L v dm evap dt − 4 π r 2 σ T 4 + Δ H ox dm ox dt (2)following equations 3 and 14 from MG. The first term on the right hand side of Eq. 2 describes the heat flux due to collisions with air, which incorporates the ram pressure, ρ a v2, experienced by the micrometeorite. The second term describes the heat flux due to evaporative mass loss with the latent heat of vaporization for both FeO and Fe given by L v = 6 × 106 J kg−1 and the mass loss given by dm evap /dt (in kg s−1). The third term accounts for radiative heat loss with the Stefan-Boltzmann constant σ = 5.67 × 10−8 W m−2 K−4 and micrometeorite temperature T (in K). We assume a blackbody emissivity of unity for the radiative term, following MG. The final term describes the heat of oxidation of an Fe particle with dm ox /dt being the mass growth of the oxide layer (in kg s−1). From MG, Fe oxidation by oxygen is exothermic, with an oxidation enthalpy of ΔH ox = 3.716 × 106 J kg−1. Oxidation by CO 2 is endothermic and has an oxidation enthalpy of ΔH ox = − 4.65 × 105 J kg−1.

Following MG, the ΔH ox for CO 2 is approximated from the standard enthalpies of formation at standard temperature and pressure. It is estimated for the reactants and products in Fe + CO 2 → FeO + CO (3)where the energies for Fe, CO 2 , FeO, and CO are 0, −393.5 (36), −249.5 (37), and −110.5 kJ mol−1 (37), respectively. Putting these values into Eq. 3, we see that 33.5 kJ mol−1 is consumed in the reaction, or 4.65 × 105 J kg−1 of FeO. The heat of oxidation has only minor impact on the model results, so we neglect the temperature dependence of ΔH ox following MG. The ΔH ox for CO 2 is an order of magnitude smaller than for O 2 in absolute value, so this assumption is especially reasonable for the CO 2 -rich atmosphere modeled here. It has been argued that the reaction described in Eq. 3 is the only plausible pathway of oxidation of Fe by CO 2 under the conditions considered in this work, so we do not consider other Fe + CO 2 products (28).

The heat flux can be related to the specific heat capacity, and to temperature via dq dt = mc sp and dT dt = dq dt ⋅ dT dq (4)for mass m and specific heat of wüstite c sp = 400 (J kg−1 K−1), which coats the molten Fe micrometeorite upon entry (38). As shown in MG, Eqs. 2 and 4 give an equation for rate of temperature change dT dt = 1 rc sp ρ ( 3 ρ a v 3 8 − 3 L v 4 π r 2 dm evap dt − 3 σ T 4 + 3 Δ H ox 4 π r 2 dm ox dt ) (5)

Equation 5 is the same as equation 6 of MG, but with the heat of oxidation term included. We note that MG is missing a 3 in the 3σT4 term, likely due to a typesetting error.

In our model, we assume that Fe is only oxidized to FeO and do not consider further oxidation, following MG, as the process of oxidation past FeO is uncertain. As such, we only consider the ratio of unoxidized Fe to oxidized Fe in micrometeorites that are not fully oxidized in our model results. Following MG, we assume that any liquid oxide that forms during melting is immiscible with the molten Fe core and coats the exterior of the micrometeorite. Thus, we can calculate the evaporative mass loss rate, dm evap /dt, by considering the rate of FeO (or Fe) evaporation using the Langmuir approximation, which is given by dm evap dt = − 4 π r 2 p v M 2 π R gas T (6)where R gas = 8.314 J mol−1 K−1 is the ideal gas constant, M is the molar mass (0.0718 kg mol−1 for FeO or 0.0558 kg mol−1 for Fe), and p v is the vapor pressure of the evaporating FeO or Fe (in Pa). The vapor pressure was determined experimentally by Wang et al. (39) and from MG is given by log ( p v ) = 10.3 − 20126 / T (7)for FeO (note that Eq. 7 is the same as MG’s equation 13 but for units of Pa rather than dynes cm−2).

In addition to evaporation of the liquid oxide layer, our model allows the unoxidized Fe to evaporate as well. This is necessary because we consider low-CO 2 atmospheres where the formation of a liquid oxide layer surrounding the micrometeorite can be slower than the rate of evaporation. To handle this in our model, at each time step we calculate the oxide mass loss rate via Eq. 6, and if it exceeds the total oxide mass remaining in the particle, we evaporate liquid Fe for the remainder of the time step. The liquid Fe evaporation is calculated from Eq. 6 as well, but p v is defined by log ( p v ) = 11.51 − 1963 / T (8)from the data in (39). Thus, the total evaporation can be given by dm evap dt = dm evap _ m dt + dm evap _ ox dt (9)where dm evap_m /dt is the metallic Fe evaporated and dm evap_ox /dt is the Fe oxide evaporated via Eq. 6.

The final step is to track the mass of Fe metal and FeO oxide in the micrometeorite. In an oxygen-rich atmosphere, we assume the total oxygen accumulated by the micrometeorite is given by dO dt = γ ρ O π r 2 v (10)where ρ O is the total density of oxygen (both O and O 2 ) encountered (in kg m−3), following MG. The γ term is a dimensionless factor between 0 and 1 that determines what fraction of the encountered oxidant is used to oxidize Fe (γ = 1 in this work, see MG for a discussion of γ and O 2 ).

For oxidation by CO 2 , we calculate the reaction rate of Fe and CO 2 from r CO 2 = k [ Fe ] [ CO 2 ] (11)for rate constant k = 2.9 × 108 exp(− 15155/T) m3 mol−1 s−1 from (28) with r CO 2 in mol m−3 s−1. The Fe concentration is given by [ Fe ] = m Fe V (12)where m Fe is the mass of Fe in the micrometeorite (in mol), and V is the volume of the micrometeorite (in m3). The CO 2 concentration per unit volume is given by the total CO 2 encountered per second multiplied by the time step, Δt, i.e. [ CO 2 ] = γ ρ CO 2 π r 2 v V Δ t (13)where ρ CO 2 is in mol m−3. We compare the rate in Eq. 11 to the total CO 2 encountered per unit volume per time step and take the lesser of the two as the amount of oxygen accumulated by the Fe. We assume γ = 1 in Eq. 13, so the oxidation from CO 2 calculated with this model should be considered an upper bound. From Eq. 3, each CO 2 that reacts with the micrometeorite will add one O to the particle as FeO so d O dt = V · min ( [ CO 2 ] Δ t , r CO 2 ) (14)

When calculating dO/dt for CO 2 , we first calculate in mol s−1 and then convert to kg s−1 for ease of use in the model.

It is important to note that the reaction rate for Fe oxidation via CO 2 used in this model was derived from laboratory measurements of gas-phase interactions of Fe and CO 2 (28). This likely represents an upper bound on Fe oxidation via CO 2 and may overestimate the oxidation of liquid Fe in a CO 2 -rich atmosphere, where diffusion of the oxidant through the liquid Fe oxide could be the rate-limiting step. However, the kinetics of the reaction for the pressures and temperatures considered in this model are uncertain (noting that temperatures often exceed ~2000 K). As such, the model presented here should be considered an upper bound on the oxidation rate by CO 2 . Future laboratory measurements are desirable to constrain the reaction rate described by Eq. 11.

Following MG, our model only allows oxidation, while unoxidized Fe remains in the micrometeorites. Thus, one Fe atom will be removed from the metallic Fe mass for each O atom accumulated. The total metallic Fe in the particle at each time step is then calculated by the amount Fe converted to Fe oxide, minus evaporated Fe, giving an equation for the mass of metallic Fe in the particle dm m dt = − M Fe M O d O dt − dm evap _ m dt (15)for Fe molar mass M Fe = 0.0558 kg mol−1 and atomic O molar mass M O = 0.0160 kg mol−1. Similarly, the mass of oxide will grow for each O atom encountered, minus the evaporated FeO, which can be described by dm ox dt = M FeO M O d O dt − dm evap _ ox dt (16)

With initial altitude, velocity, and mass known, Eqs. 1, 5, 9, 15, and 16 can be solved numerically to simulate the entry of an Fe micrometeorite. We assume all particles start as pure Fe and use a simple Euler approximation to numerically integrate the equations. We set the maximum time step to 0.01 s for our integration but allow the time step to adjust dynamically such that the maximum change in temperature of the particle never exceeds 0.1%. Following MG, we assume that no oxidation occurs until the micrometeorite melts at 1809 K for Fe and oxidation shuts off when the micrometeorite solidifies at 1720 K (the FeO melting temperature). We assume the liquid FeO has a density of 4400 kg m−3 and Fe has a density of 7000 kg m−3, from MG. In addition, for Fe, we use a specific heat of 400 J K−1 kg−1. The Python script containing our model is available in the Supplementary Materials. Figure 1 shows an example model run for a single 50-μm particle entering a 50 wt% CO 2 , 50 wt% N 2 atmosphere at 12 km s−1, and an entry angle of 45° from the Zenith.

Of interest in our model is the fractional area of unoxidized Fe compared with the total cross-sectional area of the micrometeorite (i.e., unoxidized Fe plus oxidized Fe) after they solidify. For a given micrometeorite, we assume that the unoxidized Fe forms a spherical bead at the center of the particle and is evenly surrounded by any produced oxide (FeO). We then “sectioned” the simulated micrometeorites at the midpoint and compared the total surface area of exposed metallic Fe to the total area of the sectioned micrometeorite. This quantity can be easily compared to measurements like those in figure 4 of (15), which reports Fe-phase abundance in sectioned micrometeorites. Our assumption that the metallic Fe is centered in the micrometeorite means we assume an upper bound on the sectioned area, as an uncentered bead may not measure Fe at the widest point. Despite this assumption, our model is able to accurately reproduce the data reported in figure 4 of (15), which shows the ratio of metallic Fe to oxidized Fe for 34 modern micrometeorites collected from Antarctica (we consider both FeO and Fe 3 O 4 as oxides here and do not differentiate between them). Note that in (15), the captions of figures 4 and 5 are switched so the interested reader should look at the data presented in figure 4, but apply the caption of figure 5 to avoid confusion. Figure 4 in this paper shows our model prediction of Fe fractional area compared with the data inferred from figure 4 of (15). The agreement between our simulated data (blue) and the modern micrometeorite collected data (orange) is shown in the figure.

Following MG, we do not consider magnetite formation in our model because the process by which magnetite forms during entry is uncertain. The liquid Fe oxide that forms while the micrometeorite is molten could crystalize as Fe 3 O 4 if enough oxygen is accumulated while molten. However, magnetite may also form after solidification due to decomposition of FeO to Fe 3 O 4 or possibly from further oxidation at low temperatures. In addition, the central Fe bead could separate from the molten micrometeorite during entry, leading to a highly oxidized melt as a remainder, which could solidify as magnetite [see (14) for a discussion of Fe 3 O 4 formation]. Thus, we only consider micrometeorites that still retain unoxidized Fe in both our model results and collected data and do not differentiate between the phases of oxidized Fe.

When calculating our simulated micrometeorite areas, we impose several conditions on the final particle. First, we do not consider simulated particles that are smaller than 2 μm in final radius. Despite the abundance of such small particles, they are not easily found when extracting micrometeorites from sedimentary rocks given their small size, so we do not consider them to better represent collected data. The smallest micrometeorite found by Tomkins et al. was ~4 μm in radius. Second, we do not consider fully oxidized or unoxidized micrometeorites (i.e., pure FeO or pure Fe). This is done as pure Fe micrometeorites only exist in our model because they enter the atmosphere slow enough, or with a shallow enough angle that they do not reach the Fe melting temperature. These unmelted micrometeorites represent a fixed feature in the model data that cannot inform of atmospheric composition. Pure FeO micrometeorites represent the limit of our model calculations, so aside from showing their production is expected from the model (dashed blue curve in Fig. 3), they are not considered when calculating the mean Fe fractional area (black/orange lines in Figs. 2 and 3). Neglecting these edge cases does not hinder our ability to reproduce modern micrometeorite data (Fig. 4) since we exclude fully oxidized micrometeorites from the collected data as well, ensuring our data sets are comparable.