Because the Embedded Image varies in a narrow range of 10 to 31 for the relevant P-T-fO2 conditions and the S content in the alloy, we used a fixed mean value of 19. At 3 GPa, the Embedded Image decreases by a factor of 2 with an increase in T from 1600° to 1800°C at low S and intermediate S contents in the alloy; however, at high S contents, the Embedded Image at 1600° and 1800°C (15 and 13, respectively) are almost identical (Fig. 2A). As our model calculations for a volatile-rich impactor are based on an equilibrating alloy containing >15% S, it is reasonable to use a fixed mean value for the Embedded Image based on the relevant experimental data. For core formation in a small planetary embryo, the range of the Embedded Image for the relevant fO2 conditions and S content in the alloy (>10 wt %) lies between 73 and 165, with a mean value of 119 (fig. S9). Assuming a shallow MO during alloy-silicate equilibration of the volatile-bearing impactor, we used a peridotite liquidus temperature of T = 2200 K at P = 5 GPa (67).

The Embedded Image, in conjunction with the alloy/silicate ratio, would determine the S content in the core of the impactor, which, in turn, would determine the C solubility and the Embedded Image. In addition, we also performed a sensitivity analysis for the entire range of applicable Embedded Image and Embedded Image values. The variation of the Embedded Image between 5 and 35 (Fig. 2A) does not show any significant effect on the model results. For the lower end, with a Embedded Image = 73, the most probable S content in the core of the impactor lies in a lower range (21 to 26 wt %) (fig. S10A), which, in turn, would mean that a higher bulk C (0.17 to 0.73 wt %) is required to attain the C enrichment in the impactor’s mantle (fig. S10B) via C expulsion from the core, resulting in a relatively smaller size (Embedded Image MEarth) of the pertinent impactor (fig. S10C). For the higher end, with a Embedded Image = 165, the most probable S content in the core of the impactor lies in a higher range (26 to 31 wt %) (fig. S10D), indicating that a lower bulk C (0.09 to 0.37 wt %) is needed to attain the C enrichment in the impactor’s mantle (fig. S10F) via C expulsion from the core, resulting in a relatively larger size (Embedded Image MEarth) of the corresponding impactor (fig. S10F). Variation of the Embedded Image over the entire applicable range shows that an increase in the Embedded Image from 73 to 165 decreases the most probable bulk C content of the impactor from 0.5 to 0.2 wt % and increases the most probable mass of the impactor from 0.06 to 0.11 MEarth (fig. S11). For the mean value of Embedded Image = 119, the most probable bulk C content of the impactor was calculated to be 0.3 wt %, and the most probable mass ratio of the impactor was calculated to be 0.085 MEarth, i.e., similar to a Mars-sized body (fig. S11).

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