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Mars' formation can constrain the primordial orbits of the gas giants

Recent high precision meteoritic data infers that Mars finished its accretion rapidly within 10 Myr of the beginning of the Solar system and had an accretion zone that did not entirely overlap with the Earth's. Here we present a detailed study of the accretion zone of planetary embryos from high resolution simulations of planetesimals in a disc. We found that all simulations with Jupiter and Saturn on their current eccentric orbits (EJS) result in a similar accretion zone between fast-forming Mars and Earth region embryos. Assuming more circular orbits for Jupiter and Saturn (CJS), on the other hand, has a significantly higher chance of forming Mars with an accretion zone not entirely dominated by Earth and Venus region embryos, however CJS in general forms Mars slower than in EJS. By further quantifying the degree of overlap between accretion zones of embryos in different regions with the average overlap coefficient (OVL), we found that the OVL of CJS shows a better match with the OVL from a chondritic isotopic mixing model of Earth and Mars, which indicates that the giant planets are likely to have resided on more circular orbits than today during gas disc dissipation, matching their suggested pre-instability orbits. More samples, including those from Mercury and Venus, could potentially confirm this hypothesis.


Introduction
Recently improved isotopic measurements of martian meteorites, mainly from the most abundant shergottite-nakhlite-chassignite (SNC) meteorites, allows us to constrain the bulk composition of Mars. Early oxygen isotopic mixing models suggested that Mars is composed of more than 80% ordinary chondrite (OC) like material (Lodders and Fegley 1997). Recent improvements of laboratory apparatus allows high precision isotopic measurements on lower-concentration nucleosynthetic isotopes, such as 48 Ca, 54 Cr, 50 Ti, etc. Chondritic isotope mixing models involving these nucleosynthetic isotopes provide a more complete comparison between Earth's and Mars' bulk composition. Assuming chondritic material as the building blocks of the terrestrial planets, recent models suggest that Mars accreted about half of its mass from OC-like material, and another half from enstatite chondrite (EC) like material (Sanloup et al. 1999;Tang & Dauphas 2014;Brasser et al. 2018), whereas Earth accreted more than 70% EC-like material and the remaining 30% mostly from OC-like material (e.g Javoy,

2010; Dauphas 2017). To account for the isotopic differences between Earth and
Mars, Mars is suggested to have formed in a different region from that of the Earth, potentially even in the asteroid belt (Brasser et al. 2017).
Isotopic measurements of martian meteorites not only constrain Mars' bulk composition, but also its formation timescale. Among all the radioactive decay systems, the 182 Hf-182 W system is the most widely adopted for dating the coreformation time of a body because of its ideally short half-life (8.9 Myr) and the nature of the parent lithophile isotope decaying to its daughter siderophile isotope. By applying the Hf-W chronology, Dauphas & Pourmond (2011) showed that Mars accreted very rapidly; approximately 50% of its mass after 2 Myr and finished most of its accretion between 5 to 10 Myr after the formation of the Solar System. This result is supported by another radioactive decay system 60 Fe-60 Ni (Tang & Dauphas 2014). On the other hand, Earth's accretion took much longer. Yin et al. (2002) and Kleine et al. (2009) (Levison et al. 2015), these cores and their fully formed giant planets may not be in very circular orbits (e < 0.01). Furthermore, if gas giants formed from gravitational instability of gas fragments in the protosolar disc, which could be an alternative path for giant planet formation (e.g. Boss, 1997;Vorobyov and Elbakyan, 2018), gas giants' initial eccentricities are not well determined. Hence, we study both EJS and CJS scenarios.

Method
Here we briefly summarize the simulation set up from Woo et al. (2021). In Appendix A we outline the initial conditions used for simulations of both the classical (Chambers 2001) as well as the depleted disc model (Izidoro et al. 2014 (2021)). The effect of decaying gas disc gravitational potential, aerodynamic gas drag (Adachi et al., 1976) and Type-I migration (Tanaka & Ward 2004)

Results
We found that most Mars embryos do not grow fast enough in our simulations (Woo et al. 2021) when compared to the timescale suggested by Dauphas & Pourmand (2011). However, this timescale from Dauphas and Pourmand (2011) is model dependent, as they extrapolates their measurement in the context of Mars accretion following the simplistic oligarchic growth curve of Chambers (2006).
On the other hand, the formation of embryos in our simulations is directly To conclude, the classical model has a much higher probability (~80% for EJS) of yielding fast-forming Mars embryos than the depleted disc model (<10% for EJS), because of the higher solid surface density beyond the Mars region.
For the classical cases, EJS results in a higher probability of rapidly forming Mars region embryos than for the CJS simulations. Nevertheless, we still have ~57% of CJS simulations yielding at least one fast-forming Mars embryo, which is not low. However, the final system architecture may not match the sizes and separations of the terrestrial planets. We will study the system architecture in a future paper.

Accretion zone analysis
The sweeping secular resonance, however, also induces severe mixing within the terrestrial planet region, causing the accretion zones of embryos to be similar to each other. In the following, we present the accretion zone of the fast-forming Mars embryos in comparison with the accretion zone of embryos in other regions in each simulation (Appendix C gives the precise definition of the accretion zone which we used). We define the Mercury, Venus, Earth and Mars regions as 0.27 AU < a < 0.5 AU, 0.5 AU < a < 0.85 AU, 0.85 AU < a < 1.25 AU and 1.25 AU < a < 1.75 AU.  (Dauphas et al. 2017;Brasser et al. 2017). This is not the case in Figure 1: we observe that the fast-forming Mars embryos all have accretion zones covered entirely by those of the Earth region embryos. This is also true for the comparison between Venus region embryos and Mars region embryos. Since the subsequent giant impact phase (Agnor et al. 1999;Chambers, 2001) involves scattering between embryos and the viscous stirring caused by the embryos, which dynamically heats up the disc even further, we expect the final planetary system would retain this highly mixed state until the assembly of the terrestrial planets is completed.  is the dominant effect for the extensive mixing in EJS simulations, since Type-I migration is less effective in a gas disc with a shorter decay timescale. On the other hand, the mixing effect of sweeping secular resonances remains similar between long lived and short lived discs (comparing Figures 1 and 2).   Table 1 shows the average OVL for each embryo pair in different simulation sets. The larger the OVL the higher degree of overlap between the accretion zones of two

Conclusions
We performed over 40 state-of-the-art N-body simulations to study the formation of embryos directly from a disc of only planetesimals. We focus on whether embryos in the Mars region can form rapidly (reaching ~0.8 MMars within 10 Myr) while simultaneously having an accretion zone different from Earth region embryos. We found that the CJS simulations produce fast-forming Mars embryos with less Earth-overlapping accretion zones ( Figure 3) and results in an overlap coefficient (OVL) closely matching the chondritic mixing model's result (Table 1). This indicates that even though the EJS model has higher probability in forming Mars fast (~80 %, whereas in CJS ~57 %) and clearing the mass in the Mars region and the asteroid belt due to sweeping secular resonances, its severe mixing effect could violate the isotopic differences between Earth and Mars (e.g. Sanloup et al. 1999) and the inferred heliocentric isotopic gradient in the inner Solar System (Yamakawa et al. 2010;Mezger et al. 2020).
The CJS case, on the other hand, matches predictions for the giant planets' orbital state during gas disc dispersal (e.g. Tsiganis et al. 2005) and also the isotopic data from various meteorites including the martian meteorites that we showed in this study. Hence, we postulate that the giant planets were likely to possess more circular orbits than today during gas disc dispersal. As suggested in our previous paper (Woo et al. 2021), due to the differences of the accretion zones of embryos in the Mercury and Venus region between EJS and CJS simulations, samples from Mercury and Venus could settle this debate.
The remaining challenge is to solve the problems of CJS, which produces a Mars and an asteroid belt that are both too massive (Chambers 2001;O'Brien et al 2006;Raymond et al. 2009), as well as producing too many low mass planets in the terrestrial planet region (Clement et al. 2020). One of the proposed solutions is to have the giant planets undergo an early instability phase in < 100 Myr after gas disc dispersal (e.g. Clement et al. 2018;Ribeiro et al. 2020). Radiogenic reset ages from the asteroid belt support an instability before 80 Myr (Mojzsis et al., 2019). Giant planet migration and subsequent instability excites the planetesimals and subsequently clears them around the Mars and asteroid belt region (e.g. Clement et al. 2018Clement et al. , 2019. Clement et al. (2020) suggested this instability may also excite the terrestrial planets region and help overcome the problem of having too few giant impacts during the embryo collision phase, which results in forming too many planets in the terrestrial planet region in their simulations. However, the giant planet's instability may also induce mixing within the disc, which could possibly erase the composition gradient developed in our CJS simulations ( Figure 3). Future simulations need to be carried out in order to verify this claim.
There are factors that we did not consider in our simulations. Fragmentation, which has been suggested to speed up the initial growth of the embryos during the gas disc phase (Kobayashi & Dauphas 2013), should be implemented in our future simulation. Inducing a slight excitement of the dynamical state to the planetesimals by gas disc turbulence (Laughlin et al. 2004;Ogihara et al. 2007) may also help in speeding up the growth of embryos in the Mars region, so as to more closely match with the timescale inferred by Hf-W dating. A comprehensive set of future high-resolution simulations including various dynamical effects could help solve the puzzles of the growth rate, composition and architecture of the terrestrial planets and constrain the early evolution of the giant planets.
We run simulations for both the classical model (e.g. Chambers, 2001;O'Brien et al., 2006;Raymond et al. , 2009) and the depleted disc model (Izidoro et al., 2014;Mah and Brasser, 2021). In the classical model, the solid surface density profile Σs = 7 gcm -2 (a/ 1 AU) -3/2 , which follows the minimum mass Solar nebula (MMSN, Hayashi, 1981). In the depleted disc model, the solid surface density profile is given as the following: where adep = 1, 1.25 or 1.5 AU is the depletion location, β = 50% or 75% is the depletion factor and χ = 1, 1.5 or 2 is the scaling factor of MMSN. The parameters are chosen from simulations in Mah and Brasser (2021) that achieved the highest probability of forming Mars with the correct mass. Table A1 shows the initial conditions of our simulations for this study. at 10 Myr as the "fast-forming Mars embryo". Table A2 shows the percentage of simulations that yield at least one fast-forming Mars embryo. We have discussed the detailed statistics in Section 3.1.  (Brasser et al., 2017;Fischer et al., 2018;Kaib and Cowan, 2015;Woo et al., 2018), where mi and ai are the initial mass and semi-major axis of the accreted planetesimals and Nacc is the number of planetesimals accreted by the embryo.
Its mass weighted standard deviation is written as weight = √ ∑ ( − weight ) 2 acc (( acc −1)/ acc ) ∑ acc (A3) (Brasser et al., 2017;Kaib and Cowan, 2015;Mah and Brasser, 2021;Woo et al., 2018). The accretion zone for each embryo in Figure 1, Figure 2 and In order to calculate the average OVL of each set of simulations in Table   3, we proceed as follows: 1) Obtain the initial semi-major axes of the accreted planetesimals for the embryos in different regions.
2) Combine the initial semi-major axes of the accreted planetesimals for embryos in the same region into one file.
3) Bin the data with an interval of 0.1 AU and calculate the mass fraction of planetesimals originating from each bin for each embryo region. The mass fraction of planetesimals is relative to the total mass of embryos in that region.
4) Comparing the mass fraction of planetesimals originated from each bin between embryos from two different regions, take the number with a smaller mass fraction and subsequently summarize this mass fraction for each bin.
For example, if we are comparing the OVL between Venus region embryos and Earth region embryos in a particular simulation, the OVL can be calculated as: OVL = ∑ min[ Venus , Earth 1 ], (A4) where fVenus and fEarth is the fraction of mass that originated from a particular bin (e.g. 0.5-0.6 AU, 0.6-0.7 AU, …, 2.9 to 3 AU) for embryos in Venus region and the Earth region, respectively, and n is the total number of bins.  Table 1.