Chemical Diversity of Super-Earths As a Consequence of Formation
Jennifer Scora, Diana Valencia, Alessandro Morbidelli, Seth A. Jacobson
MMNRAS , 1–16 (2019) Preprint 24 February 2020 Compiled using MNRAS L A TEX style file v3.0
Chemical Diversity of Super-Earths As a Consequence ofFormation
Jennifer Scora, (cid:63) Diana Valencia, Alessandro Morbidelli and Seth Jacobson Department of Astronomy and Astrophysics, University of Toronto, Toronto, ON, Canada Centre for Planetary Sciences, University of Toronto, 1265 Military Trail, Toronto, ON, M1C 1A4, Canada Laboratoire Lagrange, Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Blvd de l’Observatoire,CS 34229, 06304 Nice Cedex 4, France Department of Earth and Environmental Sciences, Michigan State University, East Lansing, MI 48824, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Recent observations of rocky super-Earths have revealed an apparent wider distribu-tion of Fe/Mg ratios, or core to mantle ratios, than the planets in our Solar System.This study aims to understand how much of the chemical diversity in the super-Earthpopulation can arise from giant impacts during planetary formation. Planet forma-tion simulations have only recently begun to treat collisions more realistically in anattempt to replicate the planets in our Solar System. We investigate planet formationmore generally by simulating the formation of rocky super-Earths with varying initialconditions using a version of
SyMBA , a gravitational N-body code, that incorporatesrealistic collisions. We track the maximum plausible change in composition after eachimpact. The final planets span a range of Fe/Mg ratios similar to the Solar Systemplanets, but do not completely match the distribution in super-Earth data. We onlyform a few planets with minor iron-depletion, suggesting other mechanisms are atwork. The most iron-rich planets have a lower Fe/Mg ratio than Mercury, and are lessenriched than planets such as Kepler-100b. This indicates that further work on ourunderstanding of planet formation and further improvement of precision of mass andradius measurements are required to explain planets at the extremes of this Fe/Mgdistribution.
Key words: planets and satellites: terrestrial planets – planets and satellites: forma-tion
Before the discovery of exoplanets, planet formation theo-ries were limited to explaining the Solar System and thus,were unintentionally biased. Now, with thousands of extra-solar planetary systems, there is a diverse set of data to testagainst formation theories. Models so far have been con-structed to explain the masses and orbital architectures seenin the data (i.e. Hansen & Murray (2012); Ogihara et al.(2015); Izidoro et al. (2017)), where composition is just aby-product of the two. This is because composition is not auseful constraint for most classes of planets. For exo-jovians,the only possibility is a high H/He content, whereas for low-mass exoplanets the possibilities are unconstrained (Valen-cia et al. 2007a; Rogers & Seager 2010; Vazan et al. 2018).In fact, the degeneracy in the composition of mini-Neptunesarises from a trade-off between refractory material and water (cid:63)
E-mail: [email protected] (Valencia et al. 2013). On the other hand, the composition ofrocky super-Earths is a rich source of information with onlyminor degeneracies, and it has not yet been used to constrainformation scenarios. Therefore, with this study we proposeto use composition of rocky super-Earths as another axes toconstrain formation theories.The reason for these fewer compositional degeneraciesin rocky super-Earths is that the main determinant for theradius of the planet is the amount of iron (Valencia et al.2007b). In simple terms, rocky super-Earths are made of asilicate-oxide mantle overlaying an iron core, and thereforetheir relative masses can be inferred from measurements ofthe total mass and radius. Hence, we can estimate the coremass fraction (CMF) of the planet, and from this we caninfer the iron to silicon (Fe/Si) or the iron to magnesium(Fe/Mg) ratios, where silicon and magnesium are used astracers of mantle material. In reality though, the above pic-ture is more complex because: planets generally are not ex-pected (1) to have pure iron cores – Si is a candidate amongothers for the Earth’s core light alloy (Hirose et al. 2013), or © a r X i v : . [ a s t r o - ph . E P ] F e b Scora et al. (2) to be completely differentiated – some iron may remainin the mantle (e.g. Earth’s iron content in mantle minerals isabout 10% (Ringwood 1970)). However, differences in totalradius for a given mass due to these complexities are minorfor rocky super-Earths. Plotnykov and Valencia (in prep)calculate a difference in radius of 0.5% and 0.8% in the ab-sence of a light alloy in the core, and 2.6% and 2% dueto differentiation for Earth and a 5 M E -planet, respectively.Therefore, mass-radius data pairs for rocky super-Earths canprovide valuable constraints on CMF.Furthermore, observational efforts in the last decadehave now yielded hundreds of rocky super-Earths with mea-sured masses and radii, with the more precise data pairsamenable to compositional inference. The first transitingsuper-Earths with measured masses were CoRoT-7b (L´egeret al. 2009; Southworth 2011; Barros et al. 2014; Stassunet al. 2017) and Kepler-10b (Batalha et al. 2011; Dumusqueet al. 2014; Fogtmann-Schulz et al. 2014; Van Eylen et al.2016), which early on demonstrated the difficulty of collect-ing and interpreting high-quality data. The stellar activityof CoRoT-7 made it difficult to infer the mass of the planetwith high precision (Hatzes et al. 2011; Barros et al. 2014),while instead asteroseismology data on Kepler-10 allowedfor an exquisite precision of ± . R ⊕ in planetary radius(Fogtmann-Schulz et al. 2014). Kepler yielded thousands ofsuper-Earths’ with measured radii, and hundreds of themhave been followed up for mass estimates. Although, typicaldata is more precise in radius than it is in mass, we are finallyat a stage where there is enough quality mass-radius datathat we can infer the composition of rocky super-Earths asa population, and use it to constrain formation scenarios.We propose to use this chemical data to inform planetformation theories. That is, any successful theory that formssuper-Earths needs to explain their chemical diversity asseen from the Fe/Mg ratios, as well as the distribution ofmasses (5-15 M ⊕ ) and semi-major axes (0.01 - 0.2 AU) seenin rocky super-Earths from the NASA Exoplanet Archive asof July 2019. Eccentricities and inclinations need to be ac-counted for as well, although in reality these quantities arehighly unconstrained for most low-mass exoplanets.When attempting to form super-Earth and mini-Neptune planets starting from the minimum-mass solar neb-ula (MMSN), followed by standard core-accretion, fails toreproduce the exoplanet low-mass population (Raymond &Cossou 2014). Instead, efforts to explain the formation ofthese low-mass planets fall into two categories: either theyform outside their current location and migrate inwards(Ogihara et al. 2018; Terquem & Papaloizou 2007; Ida & Lin2010; Cossou et al. 2014; Izidoro et al. 2017, 2019) stoppingat the current locations due to gas-dispersal or the disk’s in-ner edge, or they form in-situ from massive disks (Hansen &Murray 2012, 2013; Chatterjee & Tan 2014). These massivedisks may have formed through inward drift of smaller mate-rial before later stages of planet formation (Hansen & Mur-ray 2012; Chiang & Laughlin 2013; Chatterjee & Tan 2014).If planets migrate inwards, they can end up with substan-tial water in their envelopes or interiors. If they form in-situ,they would have no water in either reservoir.Much of the attention has been given to explaining whymini-Neptunes exist. That is, if planets could grow enoughmass ( − M ⊕ cores) in the presence of the gas to trig-ger runaway accretion, why did they not become gas giants? The answer usually invokes a gas-poor environment due toa timing issue: either because the gas disk lifetime is short(Alibert et al. 2006; Terquem & Papaloizou 2007) or theplanets formed late when there was little gas around (Lee &Chiang 2016; Dawson et al. 2015). Alternatively, the planetsmay accrete gas more slowly than expected (Lambrechts &Lega 2017). Given that all these scenarios result in some gasenvelope accretion, then how are bare rocky super-Earthsformed? Two possibilities are (1) that rocky super-Earthsformed so slowly that by the time planets grew massiveenough to capture gas, there was no gas around (Dawsonet al. 2015), or (2) that they did acquire an envelope butlost it, either due to atmospheric photo-evaporation (Owen& Wu 2017; Lopez & Fortney 2013) or giant impact collisions(Inamdar & Schlichting 2016). The first theory does seem tobe challenged by the fast accretion timescales that close-insuper-Earths experience (Ogihara et al. 2015) (also see Sec-tion 4). No theory is without problems, especially once weconsider multi-planet systems (e.g. Kepler-36 (Carter et al.2012)) where both rocky super-Earths and mini-Neptunesexist with similar masses, and, of course, the same disk life-time and properties. However, any successful formation the-ory not only needs to explain the mass range and orbitalperiods of super-Earths as a population distinct from mini-Neptunes, but it also needs to account for their chemicaldiversity in terms of the Fe/Mg ratios seen in the data.With this in mind, we focus on the formation of rockysuper-Earths and the role of collisions in growing planetswith the observed chemical make-up. Our interest in plan-etary collisions is highly motivated by the fact that we ob-serve a spread in the Fe/Mg ratios for these planets whichseems to be beyond the spread of stellar abundances (seeSection 2), suggesting these planets do not have a primor-dial composition.Collisions may be responsible for significantly alteringthe composition of growing planets (Asphaug et al. 2006),and can vary from super-catastrophic outcomes where thetarget is heavily disrupted to perfect mergers of the two in-volved bodies and anything in between. Typical formationmodels of growing planets by gravitational interactions in-volve N-body codes that only consider perfect mergers dur-ing encounters (for a summary see Raymond et al. (2014);Bond et al. (2010)). These models are severely limited forinvestigating the chemical diversity of planets. It is only re-cently that formation simulations have begun accounting formore realistic collisions (Kokubo & Genda 2010b; Cham-bers 2013; Haghighipour & Maindl 2019). Obtaining the ac-tual outcome of a collision requires high-resolution simula-tions and is typically done with smoothed-particle hydrody-namic (SPH) codes that are too computationally expensiveto couple directly to N-body simulations. Instead, the above-quoted studies have used analytical collisional prescriptions(with varying complexity) that predict the outcomes of plan-etary encounters based on the impact parameters and veloc-ities.In this study, we use an N-body code to form planetsfrom a compact disk and the analytic equations for collisionoutcomes by Stewart & Leinhardt (2012). Our assumptionsconsider the most favorable scenarios that produce the mostchemical diversity as a limiting case. We find that impactscan explain iron enrichments similar to Mercury. However,planetary collisions among bodies that start with solar com- MNRAS000
E-mail: [email protected] (Valencia et al. 2013). On the other hand, the composition ofrocky super-Earths is a rich source of information with onlyminor degeneracies, and it has not yet been used to constrainformation scenarios. Therefore, with this study we proposeto use composition of rocky super-Earths as another axes toconstrain formation theories.The reason for these fewer compositional degeneraciesin rocky super-Earths is that the main determinant for theradius of the planet is the amount of iron (Valencia et al.2007b). In simple terms, rocky super-Earths are made of asilicate-oxide mantle overlaying an iron core, and thereforetheir relative masses can be inferred from measurements ofthe total mass and radius. Hence, we can estimate the coremass fraction (CMF) of the planet, and from this we caninfer the iron to silicon (Fe/Si) or the iron to magnesium(Fe/Mg) ratios, where silicon and magnesium are used astracers of mantle material. In reality though, the above pic-ture is more complex because: planets generally are not ex-pected (1) to have pure iron cores – Si is a candidate amongothers for the Earth’s core light alloy (Hirose et al. 2013), or © a r X i v : . [ a s t r o - ph . E P ] F e b Scora et al. (2) to be completely differentiated – some iron may remainin the mantle (e.g. Earth’s iron content in mantle minerals isabout 10% (Ringwood 1970)). However, differences in totalradius for a given mass due to these complexities are minorfor rocky super-Earths. Plotnykov and Valencia (in prep)calculate a difference in radius of 0.5% and 0.8% in the ab-sence of a light alloy in the core, and 2.6% and 2% dueto differentiation for Earth and a 5 M E -planet, respectively.Therefore, mass-radius data pairs for rocky super-Earths canprovide valuable constraints on CMF.Furthermore, observational efforts in the last decadehave now yielded hundreds of rocky super-Earths with mea-sured masses and radii, with the more precise data pairsamenable to compositional inference. The first transitingsuper-Earths with measured masses were CoRoT-7b (L´egeret al. 2009; Southworth 2011; Barros et al. 2014; Stassunet al. 2017) and Kepler-10b (Batalha et al. 2011; Dumusqueet al. 2014; Fogtmann-Schulz et al. 2014; Van Eylen et al.2016), which early on demonstrated the difficulty of collect-ing and interpreting high-quality data. The stellar activityof CoRoT-7 made it difficult to infer the mass of the planetwith high precision (Hatzes et al. 2011; Barros et al. 2014),while instead asteroseismology data on Kepler-10 allowedfor an exquisite precision of ± . R ⊕ in planetary radius(Fogtmann-Schulz et al. 2014). Kepler yielded thousands ofsuper-Earths’ with measured radii, and hundreds of themhave been followed up for mass estimates. Although, typicaldata is more precise in radius than it is in mass, we are finallyat a stage where there is enough quality mass-radius datathat we can infer the composition of rocky super-Earths asa population, and use it to constrain formation scenarios.We propose to use this chemical data to inform planetformation theories. That is, any successful theory that formssuper-Earths needs to explain their chemical diversity asseen from the Fe/Mg ratios, as well as the distribution ofmasses (5-15 M ⊕ ) and semi-major axes (0.01 - 0.2 AU) seenin rocky super-Earths from the NASA Exoplanet Archive asof July 2019. Eccentricities and inclinations need to be ac-counted for as well, although in reality these quantities arehighly unconstrained for most low-mass exoplanets.When attempting to form super-Earth and mini-Neptune planets starting from the minimum-mass solar neb-ula (MMSN), followed by standard core-accretion, fails toreproduce the exoplanet low-mass population (Raymond &Cossou 2014). Instead, efforts to explain the formation ofthese low-mass planets fall into two categories: either theyform outside their current location and migrate inwards(Ogihara et al. 2018; Terquem & Papaloizou 2007; Ida & Lin2010; Cossou et al. 2014; Izidoro et al. 2017, 2019) stoppingat the current locations due to gas-dispersal or the disk’s in-ner edge, or they form in-situ from massive disks (Hansen &Murray 2012, 2013; Chatterjee & Tan 2014). These massivedisks may have formed through inward drift of smaller mate-rial before later stages of planet formation (Hansen & Mur-ray 2012; Chiang & Laughlin 2013; Chatterjee & Tan 2014).If planets migrate inwards, they can end up with substan-tial water in their envelopes or interiors. If they form in-situ,they would have no water in either reservoir.Much of the attention has been given to explaining whymini-Neptunes exist. That is, if planets could grow enoughmass ( − M ⊕ cores) in the presence of the gas to trig-ger runaway accretion, why did they not become gas giants? The answer usually invokes a gas-poor environment due toa timing issue: either because the gas disk lifetime is short(Alibert et al. 2006; Terquem & Papaloizou 2007) or theplanets formed late when there was little gas around (Lee &Chiang 2016; Dawson et al. 2015). Alternatively, the planetsmay accrete gas more slowly than expected (Lambrechts &Lega 2017). Given that all these scenarios result in some gasenvelope accretion, then how are bare rocky super-Earthsformed? Two possibilities are (1) that rocky super-Earthsformed so slowly that by the time planets grew massiveenough to capture gas, there was no gas around (Dawsonet al. 2015), or (2) that they did acquire an envelope butlost it, either due to atmospheric photo-evaporation (Owen& Wu 2017; Lopez & Fortney 2013) or giant impact collisions(Inamdar & Schlichting 2016). The first theory does seem tobe challenged by the fast accretion timescales that close-insuper-Earths experience (Ogihara et al. 2015) (also see Sec-tion 4). No theory is without problems, especially once weconsider multi-planet systems (e.g. Kepler-36 (Carter et al.2012)) where both rocky super-Earths and mini-Neptunesexist with similar masses, and, of course, the same disk life-time and properties. However, any successful formation the-ory not only needs to explain the mass range and orbitalperiods of super-Earths as a population distinct from mini-Neptunes, but it also needs to account for their chemicaldiversity in terms of the Fe/Mg ratios seen in the data.With this in mind, we focus on the formation of rockysuper-Earths and the role of collisions in growing planetswith the observed chemical make-up. Our interest in plan-etary collisions is highly motivated by the fact that we ob-serve a spread in the Fe/Mg ratios for these planets whichseems to be beyond the spread of stellar abundances (seeSection 2), suggesting these planets do not have a primor-dial composition.Collisions may be responsible for significantly alteringthe composition of growing planets (Asphaug et al. 2006),and can vary from super-catastrophic outcomes where thetarget is heavily disrupted to perfect mergers of the two in-volved bodies and anything in between. Typical formationmodels of growing planets by gravitational interactions in-volve N-body codes that only consider perfect mergers dur-ing encounters (for a summary see Raymond et al. (2014);Bond et al. (2010)). These models are severely limited forinvestigating the chemical diversity of planets. It is only re-cently that formation simulations have begun accounting formore realistic collisions (Kokubo & Genda 2010b; Cham-bers 2013; Haghighipour & Maindl 2019). Obtaining the ac-tual outcome of a collision requires high-resolution simula-tions and is typically done with smoothed-particle hydrody-namic (SPH) codes that are too computationally expensiveto couple directly to N-body simulations. Instead, the above-quoted studies have used analytical collisional prescriptions(with varying complexity) that predict the outcomes of plan-etary encounters based on the impact parameters and veloc-ities.In this study, we use an N-body code to form planetsfrom a compact disk and the analytic equations for collisionoutcomes by Stewart & Leinhardt (2012). Our assumptionsconsider the most favorable scenarios that produce the mostchemical diversity as a limiting case. We find that impactscan explain iron enrichments similar to Mercury. However,planetary collisions among bodies that start with solar com- MNRAS000 , 1–16 (2019) hemical Diversity of Super-Earths position are insufficient to explain all the diversity seen inthe super-Earth data.Our manuscript layout is as follows: in Section 2 weexplain the chemical information that can be inferred fromthe mass-radius data, interior structure models and stellarabundances. In Section 3 we introduce our model for formingsuper-Earths and how we tracked composition during forma-tion. In Sections 4, 5 and 6 we present our results, discussthe implications, and outline our conclusions, respectively. From the 269 super-Earth planets with measured radiusand measured masses below 20 M ⊕ on the NASA exoplanetarchive as of July 2019, we chose the ones that have lowenough mass errors that a meaningful inference of compo-sition is possible. Our cut-off was to consider planets with50% mass error estimates ( ∆ M / M < ).Figure 1 shows these planets colour-coded as a functionof flux received by each planet. Highly irradiated planetsare more susceptible to atmosphere evaporation, and thusare more likely to be rocky (Lopez & Fortney 2013; Owen &Wu 2017). Since many of this sample receive high fluxes, wecan assume that most of these planets do not have significantatmospheres.We calculated the mass-radius relationships for four dif-ferent rocky compositions based on the model by Valen-cia et al. (2006, 2007b) . These compositions are: (1) Aplanet devoid of iron, (2) an Earth like composition withFe/Mg = . , (CMF = 0.326 (Stacey 2005)) (3) an iron en-riched composition (similar to Mercury) of Fe/Mg ∼ . orCMF = 0.69 (Mercury’s is estimated at 0.69-0.77 by Haucket al. (2013)), and (4) a pure iron planet. The shaded regionbetween the iron-free and pure iron planet are the bound-aries within which rocky planets exist, with iron contentincreasing from top to bottom. Any planet above the iron-free line requires an envelope massive enough to modify theradius, thus, the planet is a mini-Neptune. We term thisline the rocky-threshold-radius (RTR). Any planet belowthe RTR may be rocky, but it is not definitive, because thedegeneracy in composition means a planet can trade-off be-tween iron and H/He or H O and still have the same massand radius (Valencia et al. 2007a). Namely, a planet withthe mass and size of the Earth could also have a H/He enve-lope if it had a larger iron core and less mantle. We do notexpect to see any planets below the pure iron line, as densercompounds are not abundant enough in the solar nebula tocreate whole planets.We then investigated how planets with primordial com-positions, or those that have the same Fe/Mg ratios as theirhost stars, compare to these mass-radius relationships andthe super-Earth data. We use the Fe/Mg ratio instead of We have updated the equations of state to reflect better dataobtained by Stixrude & Lithgow-Bertelloni (2011) for the mantleand that of the core by Morrison et al. (2018). We chose theFe . Ni . equation of state for the core, but to calculate thechemical budget we include an unspecified light alloy in the corethat makes up 10% by mol and does not include magnesium, inline with the Earth (McDonough & Sun 1995), nor silicon, forsimplification the Fe/Si ratio, as silicon is somewhat volatile (McDonough& Sun 1995), and thus may not be incorporated as readilyinto planets as magnesium. We consider the stellar absoluteabundances by weight of stars with planets from the Hypa-tia Catalogue (Hinkel et al. 2014). Although there is a rangein stellar Fe/Mg, (see Fig. 2), it translates to a very nar-row spread in the relationship between mass and radius. Inother words, planets that fall outside the blue shaded regionin Figure 1 do not have a primordial composition.There are many planets that appear enriched and alsodepleted in iron with respect to the primordial compositiondistribution. The presence of significant amounts of wateror an atmosphere could inflate planet radii, causing rockyplanets to appear more iron-depleted than they are. How-ever, some of the planets that appear iron-depleted are alsohighly irradiated by their stars, and thus should be strippedof any atmosphere or volatiles. In this study we therefore as-sume that all the chosen super-Earths are completely rocky.Consequently, we consider the maximum range of refractorycompositions that these planets could possibly have. We setout to investigate if the diversity in refractory compositionseen in the data could be explained through chemical repro-cessing of planets during the giant impact phase. We consider in this work the in-situ formation scenario forsuper-Earths in the post-gas disk phase (Hansen & Murray2012). We are aware that the consistency of this scenario iscriticized (Raymond & Cossou 2014; Ogihara et al. 2015).We adopt it as a best case scenario for high-velocity colli-sions that may remove mantle material and enhance the coremass fraction of the final planets. If we do not find that thisscenario produces planets that are sufficiently iron-enriched,it is unlikely that other scenarios, including planet collisionalaccretion at larger distances and smaller relative velocitiesor the in-situ accretion of small drifting particles can explainthe large Fe/Mg ratios inferred for some exoplanets.Giant impacts occur after the oligarchic growth phaseof terrestrial planet formation, once planetary embryos haveformed. We use the basic initial conditions for in-situ planetformation developed by Kokubo & Ida (2002). Half of thedisk mass starts in embryos, or bodies that have grown largeenough to dominate gravitational interactions, and the otherhalf is in planetesimals, bodies roughly 1 km in diameter thatinteract mainly with the embryos. We distribute the embryosusing the following equations for the isolation mass, or initialmass ( M iso ), and mutual Hill radius ( r H ) for planets arounda 1 M (cid:12) star, developed by Kokubo & Ida (2002): M iso = . (cid:18) b Σ o (cid:19) α (cid:16) a AU (cid:17) α ( − α ) M ⊕ (1) r H = . × − (cid:18) b Σ o (cid:19) / (cid:16) a AU (cid:17) ( / )( − α ) AU (2)Here a is the semi-major axis of the embryo, and b is thenumber of hill radii between each embryo. Both the embryos MNRAS , 1–16 (2019)
Scora et al.
Mass ( M ) R a d i u s ( R ) observedsimulatedsimulated (short period) . . . . . . . . F l u x a t p l a n e t( l og ( W m )) EarthMercury
Figure 1.
The masses and radii of the final planets from our simulations (in purple) compared to observed super-Earths with goodmeasurements of mass and radius, which are colour-coded by the stellar flux received by the planet. Super-Earths from simulations thatstart with disks closer to the star (inner edge at 0.05 AU) are plotted as triangles, and those from the main simulation suite are plottedas circles. The errobars on these points represent the maximum and minimum CMF values that are then averaged, as per the ’mergingdisruption’ prescription described in Section 3.3. These are plotted over lines of constant composition (from top to bottom: 0 CMF,Earth (0.326), Mercury (0.69), 1 CMF). The blue shaded region is the area that would be populated by planets if they had the Fe/Mgabundances of their host stars. The different shades of blue represent the probability distribution of stellar Fe/Mg abundances, as inFigure 2, which shows the probability density of the Fe/Mg abundances of planet-hosting stars in the Hypatia Catalogue (Hinkel et al.2014).
Stellar Fe/Mg (wt%) . . . . . . . P r o b a b ili t y d e n s i t y Earth CMF (0.326) 0 . . . . . . . Figure 2.
Distribution of Fe/Mg values for stars with planets,including their error bars. The values are taken from the HypatiaCatalogue (Hinkel et al. 2014) as of July 2019. The colourbar cor-responds to the probability density values for each stellar Fe/Mgvalue, and these colours are used in Figure 1. and the planetesimals are distributed assuming some surface . . . . . . . M a ss ( M ⊕ ) α = 1 . . semi-major axis (AU) α = 2 . . α = 0 Figure 3.
Mass distribution of the planetary disk for three sur-face density slopes: α = / , / , and 0. α = / is the slope of theMMSN. The upper (purple) slope corresponds to embryo masses,which has a minimum of . M ⊕ and is capped at M ⊕ . The lower(blue) line is the planetesimals, which all share the same mass.MNRAS000
Mass distribution of the planetary disk for three sur-face density slopes: α = / , / , and 0. α = / is the slope of theMMSN. The upper (purple) slope corresponds to embryo masses,which has a minimum of . M ⊕ and is capped at M ⊕ . The lower(blue) line is the planetesimals, which all share the same mass.MNRAS000 , 1–16 (2019) hemical Diversity of Super-Earths density profile for the disk. The general equation is: Σ = Σ o (cid:16) a AU (cid:17) − α (3)where Σ o is the normalization constant, and α is the slope(Hayashi 1981). For the MMSN, Σ o = and α = . , whichgives . M ⊕ within 1 AU. Since this is too low-mass to formsuper-Earths, we base our profiles off the initial conditions ofHansen & Murray (2012), as their simulations formed super-Earths with semi-major axes between 0.05 and 1 AU, similarto those observed in the Kepler 11, HD69830, and 61 Virginissystems. In our fiducial simulations we adopt Σ o = and α = / , with embryos and planetesimals placed at a rangeof distances between 0.1 and 1 AU.Equation 1 gives initial embryo masses that can be aslarge as 5 M ⊕ , which is already a super-Earth. We place anupper limit of . M ⊕ or 1 M ⊕ on our embryos and a lowerlimit of . M ⊕ to more precisely investigate the formationprocess. In order to fit the same amount of mass between 0.1and 1 AU, we decrease the spacing of the embryos to a fewHill radii ( r H ) apart (from the ∼ r H given by Kokubo &Ida (2002)). We justify this by the fact that we are consid-ering proto-embryos before they have reached the isolationstage described by Equation 1.Our simulations do not have high enough resolution forrealistic-sized planetesimals, so instead we have larger bod-ies that represent a collection of planetesimals. Typicallyone collection of planetesimals is 3% of the mass of an em-bryo. The planetesimal locations are drawn randomly froma probability distribution calculated from the surface den-sity profile (Equation 3). Both embryos and planetesimalsare given initial random eccentricities less than 0.01 and in-clinations less than . ◦ , creating a dynamically cold disk.Figure 3 shows the initial distributions of embryos and plan-etesimals with different surface density slopes. We simulated the evolution of these initial conditions usinga modified version of the gravitational N-body code
SyMBA (Duncan et al. 1998). The code uses a multi-timestep sym-plectic integrator to efficiently resolve close encounters be-tween bodies. It tracks the movement of particles within thedisk based on gravitational interactions alone. Only embryoscan gravitationally interact with each other; planetesimalshave no gravitational effect on each other but do interactwith the embryos. To keep the simulation from stalling weset an inner limit for the bodies’ semi-major axis at . AU,and at smaller distances the body is considered to fall intothe star. We also set an outer limit at 10 AU. We run thesimulations to a maximum of 100 million years, or until theyare stable enough that there are no impacts for a few millionyears.As mentioned in Section 1, collisions beyond perfectmergers were implemented in N-body codes only recently.Kokubo & Genda (2010a) only divided their collision intotwo types: perfect mergers and hit-and-run collisions, wherethe two bodies bounce off of each other (Asphaug et al.2006). An improvement by Chambers (2013) considered eachcollision to result in fragmentation; the collision was onlyconsidered a perfect merger if the fragments were below aminimum mass. Both these simulations resulted in some mi- nor improvements over perfect merger codes, but overall theprocess and length of planet formation were similar becausemost of the fragments and bodies which escaped the imper-fect collisions remained in the same general orbits. So evenif part of the projectile was not accreted during the firstcollision, the debris eventually collided with the same em-bryo and accreted onto it at a later time. Thus, the planetsformed out of essentially the same material and in the sameplaces (Chambers 2013). The aim of these simulations wasto reproduce the Solar System, so they never tested highervelocity disks to produce a broader range of compositions.We improve upon this work in two aspects: we use amore realistic collisional outcome map, and we include theproduction and loss of debris.The version of
SyMBA we use is updated to use the an-alytic collision prescriptions of Stewart & Leinhardt (2012),Leinhardt & Stewart (2012) and Genda et al. (2012) to real-istically treat collisions between large bodies. During eachcollision, the algorithm in the Appendix of Leinhardt &Stewart (2012) is used to determine the outcome with onlysmall modifications. We sort the collision outcomes into ninetypes, as described in Table 1, which are similar to those inLeinhardt & Stewart (2012). Leinhardt & Stewart (2012) di-vide the collision outcomes into two regions, the disruptionand grazing regimes. The disruption regime is subdividedinto the super-catastrophic, erosion, and partial accretionsub-divisions. We rename the super-catastrophic regime asthe catastrophic regime, and define the super-catastrophicregime when the mass of the largest remnant is no longerdistinct from the power-law distribution of debris from thecollision, an outcome not in Leinhardt & Stewart (2012). Wealso name the divisions in the grazing regime the graze andmerge, graze, hit and run, and hit and spray regimes to keepthem distinct from the disruption regime sub-divisions. Thecriteria from Genda et al. (2012) is used to more accuratelydefine the boundary between graze-and-merge and hit-and-run. In the case of a hit-and-spray, graze-and-merge, perfectmerger, or any of the disruption regime collisions, there isdistinct from the debris a single surviving largest remnantmade mostly from target materials, which we place at thecenter of mass and momentum of the system of collidingmaterial. In the other cases, there is also a surviving secondlargest remnant made mostly from projectile material, whichwe then place a mutual Hill radii away from the largest rem-nant along the collision trajectory. The mutual velocities aredetermined by assuming that the collision is inelastic with acoefficient of restitution of 0.5 along the collision vector, butelastic in the two other directions. The debris particles areplaced in a ring orthogonal to the collision vector a Hill ra-dius away from the largest remnant and moving away fromit with a relative velocity of 5% greater than their mutual es-cape speed (similar to Chambers (2013)). For computationaltractability, we limit the number of debris particles to 38 andset a minimum mass of debris particles to . × − M ⊕ . Ifthere is not enough debris mass, the number of debris parti-cles is decreased accordingly. The total mass and momentumof the system is conserved unless some debris is removed be-cause it is considered to have been turned into dust.We also include the effects of radiation pressure on thedebris formed in the collisions. Extending the debris sizefunction developed in Stewart & Leinhardt (2012) to smaller MNRAS , 1–16 (2019)
Scora et al.
Table 1.
A description of the collision types used in our planet formation simulations. See Stewart & Leinhardt (2012) and Genda et al.(2012) for the quantitative boundaries between these collision types and the fraction of the colliding mass ejected into space.Number Collision Name Description1 Super-catastrophic disruption Only debris is left2 Catastrophic disruption Small amount of target is left, mainly debris3 Erosion The target is eroded by projectile4 Partial accretion The projectile is partially accreted by the target5 Hit and spray The projectile is eroded6 Hit and run Two bodies graze each other and produce some debris7 Graze Target and projectile bounce inelastically at large angles8 Perfect Merger Target and projectile merge9 Graze and merge Target and projectile impact and then merge
Table 2.
Parameters varied in initial conditions. For the mostpart, we perform simulations with each permutation of the firstthree parameters. Table A1 details the parameter combinationsfor each simulation set.Maximum initial Disk mass α Debris massembryo mass ( M ⊕ ) ( M ⊕ ) lost0.1 25 0 scales suggests that ∼
3% of the debris mass is small enough( < . µ m) to be pushed away by radiation pressure. Contin-ued collisional grinding, a process by which debris particlesrepeatedly collide with each other and fragment into smallerand smaller pieces, should increase this fraction significantly.As much as − of the debris could be ground downsmall enough to be blown away by radiation pressure in justa few orbits (Jackson & Wyatt 2012). We simulate this ef-fect by immediately removing a fraction of the debris masscreated in each collision. Table 2 shows the main two frac-tions of debris we remove (0% and 100%). Test simulationswere performed with 3% of the debris mass lost, howeverwe found 3% was such a small fraction it had little effect.We remove 100% (instead of 50%, for example) in order totest the maximum impact of collisional grinding. Addition-ally, comparing the two allows us to compare the impact ofdebris on the outcome of the simulations. If the removal ofdebris turned out to be the crucial ingredient to reproducethe high Fe/Mg ratio of some exoplanets, we would focus in asubsequent work on modelling more precisely the collisionalcascade within the debris population. We track the collisions that each body experiences usedto this to calculate the composition of the bodies in post-processing. All bodies are given the same initial compositionbased on the solar photosphere Fe/Mg abundance ratio, aswe assume a solar-type star for our exoplanet systems. Forconsistency, we use
Superearth (Plotnykov and Valencia,in prep) to calculate the corresponding CMF, as well as allof the final planets’ CMFs. The mantle is made of olivineand pyroxenes in the upper mantle, olivine and garnet inthe transition zone, bridgemanite and ferropericlase in thelower mantle and post-perovskite and ferropericlase in thelowermost mantle. For simplicity, and to consider the mostfavourable case for iron enrichment of planets through col- C o lli s i o n C o lli s i o n C o lli s i o n C o lli s i o n ! " %
2, 3, 4, 5 C o lli s i o n Figure 4.
Visualization of the morphology and change in core-mass fraction for all collision types. The numbers correspond tothe collision types in Table 1. Purple represents the core, and graythe mantle. The moving body is the least massive, and termed the’impactor’. The body it hits is called the ’target’. Sizes are notrepresentative of actual mass ratios. lisions, we consider iron to be only in the core and use theMg end-member for all minerals in the mantle (e.g. completedifferentiation). We find that a solar Fe/Mg abundance ra-tio of . (Palme et al. 2014) gives a CMF of ∼ . .This is larger than the Earth’s (0.326 (Stacey 2005)) as theEarth is slightly depleted in iron with respect to the Sun(McDonough & Sun 1995).We assume our embryos to start out completely differ-entiated into an iron core and mantle. Our initial embryoshave masses between 0.1 and M ⊕ , for which we can expecta large degree of differentiation. For comparison, Mars hasa mass of . M ⊕ , and Mars and the Earth are thought tohave 20% and 10% by mol of iron in the mantle, respectively(Waenke & Dreibus 1988). In addition, giant impacts deliverenergy that raises the temperature of the mantle. The tem-peratures can rise beyond the solidus to melt the mantles ofgrowing planets, which enables some of the remaining iron MNRAS000
Visualization of the morphology and change in core-mass fraction for all collision types. The numbers correspond tothe collision types in Table 1. Purple represents the core, and graythe mantle. The moving body is the least massive, and termed the’impactor’. The body it hits is called the ’target’. Sizes are notrepresentative of actual mass ratios. lisions, we consider iron to be only in the core and use theMg end-member for all minerals in the mantle (e.g. completedifferentiation). We find that a solar Fe/Mg abundance ra-tio of . (Palme et al. 2014) gives a CMF of ∼ . .This is larger than the Earth’s (0.326 (Stacey 2005)) as theEarth is slightly depleted in iron with respect to the Sun(McDonough & Sun 1995).We assume our embryos to start out completely differ-entiated into an iron core and mantle. Our initial embryoshave masses between 0.1 and M ⊕ , for which we can expecta large degree of differentiation. For comparison, Mars hasa mass of . M ⊕ , and Mars and the Earth are thought tohave 20% and 10% by mol of iron in the mantle, respectively(Waenke & Dreibus 1988). In addition, giant impacts deliverenergy that raises the temperature of the mantle. The tem-peratures can rise beyond the solidus to melt the mantles ofgrowing planets, which enables some of the remaining iron MNRAS000 , 1–16 (2019) hemical Diversity of Super-Earths in the mantle to sink to the core, enhancing differentiation(Tonks & Melosh 1992; Nakajima & Stevenson 2015). Thus,we consider our assumption of complete differentiation tobe a valid one that yields the most variation in compositionexpected.To calculate the final composition of the planets, wecreate prescriptions for how each collision type changes thecompositions of the interacting bodies. Core and mantle ma-terial are conserved throughout the collisions, but due tothe loss of material (i.e. via radiation pressure or planetsfalling into the star) there is no overall conservation of theabundance ratios in the disk. Accretion events, where smallbodies (debris or planetesimals) accrete onto an embryo, aretreated separately. The core and mantle mass of the smallbody is added to that of the embryo, and it is assumed thatthe embryo remains completely differentiated. For the col-lisions between embryos, our compositional map is split upinto two main categories: disruptive and grazing. The dis-ruptive events are those that leave one embryo and somedebris.We chose the prescription for the disruptive collisionsto follow the prescription developed in Marcus et al. (2010),where the collisions either always result in the cores of bothbodies merging, or only result in core mergers if the remain-ing body is more massive than the target body. The finalplanet CMFs are calculated assuming the first scenario, giv-ing a maximal CMF, and then with the second scenario,giving a minimal CMF for each planet. We take the averageof these two outcomes, and call this the ’merging disruption’prescription. In the case of the most disruptive events, wherethere is no embryo left (super-catastrophic), we simply as-sume that the core material is evenly distributed throughoutthe debris.On the other extreme, as disruptive events becomecloser to the hit-and-run collisions, their energy decreases,and two embryos are left instead of one. As per the simu-lations of Asphaug (2010) and Leinhardt & Stewart (2012),the projectile can strip mantle material from the target orthe projectile depending on the velocity of the impact. Thisis the only significant way that the collision can impact theCMF of the final body (or bodies) (Jackson 2020). So ourprescription still follows that of Marcus et al. (2010), wherethe largest remnant has the same core mass as the targetbody, and only the mass of the mantle changes. Similarly,the second largest remnant will keep the same core mass asthe impactor, and thus the debris from these collisions istypically mantle material. Figure 4 provides a visual repre-sentation of the collision geometry and collision types andtheir possible outcomes. For a more quantitative and de-tailed description of the compositional prescriptions we use,see Appendix B. We ran 120 simulations around 1 M (cid:12) stars that span theparameters listed in Table 2: disk surface density power lawindexes of 2.5, 1.5, and 0, maximum embryo masses of 0.5and 1 M ⊕ , total disk masses of and M ⊕ , and debrismass loss of 0 and 100%. Table A1 shows the breakdownof these simulations. The typical timescale of accretion fora planet is 3 Myr. This short timescale is a consequence of . . . . . . . . planetplanetesimaldebris . . . . . . . . . . . . . . . . . . . . . . semi-major axis (AU) . . . . . . ecce n tr i c i t y ( ◦ ) Figure 5.
The evolution of a fiducial (maximum M embr yo = . M ⊕ , M disk = M ⊕ , α = . , and no debris loss) simulationover 10 million years. The size of the circle in the plot corre-sponds to the mass of the body. The embryos are purple, and theplanetesimals and debris are blue. The mass begins concentratedinside 1 AU, and large planets quickly form close to the star.Over time they consolidate down to a few planets inside 1 AU,and most of the small debris is accreted by the planets or thrownout of the system through orbital interactions. The planetesimalsachieve such high eccentricities as there is no drag from the gasdisk to damp the perturbations from close encounters. massive disks concentrated in small annuli with fast orbitaltimescales. Figure 5 shows the evolution of a typical system,where planets have evolved to close to final mass in 1 Myr.We stop simulations at 10-50 Myr, when there are nocollisions of any type in the last ∼ M ⊕ . Theaverage total mass in planets per simulation is 22 M ⊕ , andthe total average mass of debris produced is 1.14 M ⊕ . Themajority of the planets are between 0.1 and 1 AU, thoughplanets extend out to about 4 AU.By design, and to keep computational costs reasonable,we run the majority of our simulations with disks between0.1 and 1 AU. However, we also ran a few simulations withdisks that extended from 0.05 to 1 AU to investigate shorter-period planets and see how their collisional histories woulddiffer from the planets forming further out. We ran one sim- MNRAS , 1–16 (2019)
Scora et al. − − Semi-major axis (AU) M a ss ( M ⊕ ) simulatedsimulated (short-period)observed Figure 6.
The mass and semi-major axis of observed (blue) and simulated (purple) super-Earths. Circular points are from simulationswith disks from 0.1 to 1 AU. Triangular points are from a smaller suite of simulations of super-Earths formed from disks from 0.05 to1.0 AU, where only the planets formed inside 0.2 AU are plotted. M ⊕ )0 . . . . . . . . ecce n tr i c i t y simulatedsimulated (short period)observed Figure 7.
The eccentricities of the simulated planets (purple),compared to measured eccentricities of super-Earths from theNASA Exoplanet Archive (blue). ulation for each of the parameter configurations with maxi-mum embryo masses of 0.5 and 1.0 M ⊕ (so 24 in total). Al-though the comparison size was small, we did not find anysignificant differences. Figure 6 shows the planets we form(disks starting at 0.1 AU are circles, disks starting at 0.05AU are triangles) compared to the 44 rocky super-Earths with good mass estimations (blue), which are located be-tween 0.01 and 0.3 AU.As can be seen, we can successfully grow planets to 5-15 M ⊕ between 0.1 and 1.0 AU in our simulations, and there isstrong indication from these test simulations that the sameformation scenario would form planets at shorter periods ifthe disk is extended closer to the central star.The planets have eccentricities from 0 - 0.7, with an av-erage of 0.09. Figure 7 compares the spread of eccentricitiesto those measured for super-Earths (with mass < M ⊕ , ra-dius < . R ⊕ , and around K type or larger stars) from theNASA Exoplanet Catalogue as of July 2019, demonstrat-ing that they roughly overlap. The high eccentricity spikeseen at lower masses in our simulations is likely due to thelack of damping otherwise caused by a gas disk. Includinga gas disk would reduce the excitation of eccentricities dueto close encounters, resulting in lower planet eccentricitiesoverall. This would also lead to lower impact velocities andless debris. After all the simulations are complete we analyze the col-lisional history of each planet formed and the populationas a whole. The collisions in these simulations are dividedinto two types as mentioned in Section 3.3: those betweenplanetesimals and embryos (perfect accretion), and those be-tween two embryos (giant collisions). The small collisions ac-count for ∼ of the collisions in the simulations overall,and have little effect on the composition of the final planets. MNRAS000
The eccentricities of the simulated planets (purple),compared to measured eccentricities of super-Earths from theNASA Exoplanet Archive (blue). ulation for each of the parameter configurations with maxi-mum embryo masses of 0.5 and 1.0 M ⊕ (so 24 in total). Al-though the comparison size was small, we did not find anysignificant differences. Figure 6 shows the planets we form(disks starting at 0.1 AU are circles, disks starting at 0.05AU are triangles) compared to the 44 rocky super-Earths with good mass estimations (blue), which are located be-tween 0.01 and 0.3 AU.As can be seen, we can successfully grow planets to 5-15 M ⊕ between 0.1 and 1.0 AU in our simulations, and there isstrong indication from these test simulations that the sameformation scenario would form planets at shorter periods ifthe disk is extended closer to the central star.The planets have eccentricities from 0 - 0.7, with an av-erage of 0.09. Figure 7 compares the spread of eccentricitiesto those measured for super-Earths (with mass < M ⊕ , ra-dius < . R ⊕ , and around K type or larger stars) from theNASA Exoplanet Catalogue as of July 2019, demonstrat-ing that they roughly overlap. The high eccentricity spikeseen at lower masses in our simulations is likely due to thelack of damping otherwise caused by a gas disk. Includinga gas disk would reduce the excitation of eccentricities dueto close encounters, resulting in lower planet eccentricitiesoverall. This would also lead to lower impact velocities andless debris. After all the simulations are complete we analyze the col-lisional history of each planet formed and the populationas a whole. The collisions in these simulations are dividedinto two types as mentioned in Section 3.3: those betweenplanetesimals and embryos (perfect accretion), and those be-tween two embryos (giant collisions). The small collisions ac-count for ∼ of the collisions in the simulations overall,and have little effect on the composition of the final planets. MNRAS000 , 1–16 (2019) hemical Diversity of Super-Earths Impact angle θ ( ◦ ) . . . . . . . . . v i m p / v e s c supercatastrophicdisruptioncatastrophicdisruptionerosionhit and sprayhit and rungrazepartial accretiongraze and mergeperfect merger Figure 8.
Collisional map for embryos during planet formation. The impact velocity is scaled by the escape velocity of the combinedmass of the two impacting bodies. There is a wide spread of collisions across angles, but the majority of collisions occur at low impactvelocity ratios. s u p e r c a t a s t r o p h i cc a t a s t r o p h i ce r o s i o np a r t i a l a c c r e t i o nh i t a n d s p r a y h i t a n d r u n g r a z e p e r f e c t m e r g e r g r a z e a n d m e r g e . . . . . . A v e r ag e m a ss f r a c t i o n o f d e b r i s . . . . . . N o r m a li ze d c o lli s i o n f r e q u e n c y Figure 9.
The normalized frequency of each collision type overthe simulation suite (outlined purple), combined with the averagefraction of the collision mass that becomes debris for each collisiontype (filled-in blue). The least frequent collisions (i.e. catastrophicdisruption) create the most debris.
All the planetesimals are given the same initial CMF ratio asthe embryos, so the only effect on composition occurs whenthe small body is a debris particle from a previous collisionthat may have a different composition (i.e. 0 CMF).As we are interested in how composition is affected, wefocus on the giant collisions. Figure 8 shows all the giantcollisions that our simulations produced as a function of theratio between impact and escape velocity and of the im-pact angle. This collisional map can be compared to the oneproposed by (Stewart & Leinhardt 2012). It is clear thatduring planet formation all impact angles are well-sampled during collisions, with frequency peaking around ◦ . In con-trast, reaching impact velocities exceeding more than 50%of the escape velocity of the combined mass of the collidingbodies (i.e. mutual escape velocity) is less common. This isas expected, since the impact velocities depend on the ve-locity dispersion of the disk, and the velocity dispersion ofthese disks is low. As there are no giant planets to perturbthe disk in these simulations, velocity dispersion depends onthe strength of the largest close encounters that perturb theorbits of bodies before they collide. The strength of suchencounters are set by the mutual escape velocity, and sothe relative velocities (and thus impact velocities) of em-bryos scale with the escape velocities of the embryos. Thelack of collisions at high impact to mutual escape velocityratios translates to very few erosion, catastrophic and super-catastrophic disruption collisions.We show more quantitatively the prevalence of eachtype of collision in the overall formation history of our sim-ulations, as well as the average amount of debris created byeach type of collision, in Figure 9. A strong conclusion fromour simulations is that the most common collisions duringplanet formation were those that produce little or no debris,and that the collisions with the most potential to changeplanetary composition were very infrequent.More specifically, Figure 9 shows that graze and mergecollisions were the most predominant collision type, andlike perfect mergers they do not modify planet composi-tion. Partial accretion was the next most common collisionaloutcome, followed by perfect mergers, graze, hit-and-run,and hit-and-spray. Finally, as previously mentioned, the dis-ruptive collisions such as erosion, catastrophic and super-catastrophic were rare events.By tracking debris for each collision we find that theaverage debris mass created in a collision less disruptivethan erosion is less than of the total colliding mass MNRAS , 1–16 (2019) Scora et al. . . . . . i N o r m a li ze d a bund a n ce o f p l a n e t s Figure 10.
A histogram of the CMFs of the final planets in all thesimulations. The CMFs are given as a ratio of final CMF over theinitial CMF (0.333) to show the change in CMF of each planet.Using the maximum and minimum CMF produced instead of theaverage CMF only results in a < difference in the histogrambin heights. (see Fig. 9). If all of this debris were mantle material, thiswould only produce a increase in CMF. This is con-trasted with the catastrophic disruption and erosion colli-sions, where − of the colliding mass becomes debris.This would produce a much more significant change in CMFof the remaining embryo, increasing it by a factor of 2 ormore, and could even reduce the embryo to almost a barecore. If instead we consider a fixed mass of debris, then thechange in the final embryo’s CMF will be the largest fora small embryo, and will decrease as the embryo mass in-creases. This is because a smaller mass fraction of mantle isbeing removed. So it is also the case that the larger embryoswill tend to have a smaller CMF change.In sum, the predominance of gentle collisions andscarcity of disruptive collisions translates to a difficulty inchanging the composition of the initial embryos. We showthe cumulative effects of giant impact collisions for bothcomposition prescriptions in Figure 10 by obtaining a his-togram of the core-mass enrichment or depletion after planetformation. The spread in the resulting distribution of CMFis produced by the stochasticity of giant impact collisionsduring planet formation, and the spread is nevertheless nar-row due to the dominance of collisions that produced verylittle debris. We translate the mass and composition of the planets to afinal radius using the internal structure code super-Earth (Plotnykov and Valencia, in prep), and show the results inFig. 1. The radii for the maximum and minimum CMF sce-narios are plotted as error bars on the averaged CMF. Wecompare our simulated planets to those super-Earths forwhich there are good mass estimates ( ∆ M / M < . ), as dis-cussed in Section 2.It is clear that giant impact collisions can increase thecore component up to CMFs around Mercury’s. This is onlythe case when considering the maximum CMF enrichment from the prescription, where cores always merge. The aver-age values from this prescription give a maximum CMF of0.6, which translates to an enrichment of Fe/Mg by a fac-tor of ∼
3. In comparison, the super-Earth data shows someplanets that appear to be even more enriched.For the planets for which we obtained the most iron en-richment, the collisional history shows that they tend to beformed by one or two CMF changing collisions, such as ma-jorly erosive collisions or those with core merging, with addi-tional collisions performing minor CMF enrichment. On theother end, not many iron-depleted planets were produced,since the prescription did not allow for much of a reductionin CMF. The only pathway to iron depletion is for a planetto accrete large amounts of mantle debris.Roughly 29 out of the 44 observed super-Earths fallwithin the mass-radius range of the simulated planets, whenconsidering their error bars. It is thus reasonable to concludethat giant impact collisions can explain some of the ironenrichment seen in the rocky super-Earth population. Thesimulations have difficulty achieving even the lowest CMFestimated for Mercury, so it is clear that more work needs tobe done to understand how high CMF planets form. Planetslike K2-38b (Sinukoff et al. 2016), K2-3d (Crossfield et al.2015; Almenara et al. 2015; Dai et al. 2016), and Kepler-100b (Marcy et al. 2014; Borucki et al. 2011; Batalha et al.2013) have data consistent with almost pure iron content.Forming a pure iron planet is difficult. Condensation tem-peratures of iron alloys and silicate oxides are similar, socondensation of these materials from the nebula happenswithin a few hundred Kelvin of each other. In addition, thegiant impact phase does not seem to be energetic enoughto completely remove the mantle of embryos. In fact, evenat very large impact velocities of 5 times that of mutualescape of 8 M ⊕ planets, there is always some mantle leftover (Marcus et al. 2009). One would have to invoke a se-ries of these catastrophic events to produce an iron planet.However, as seen in our calculations, these super energeticcollisions are rare. Further observational work for narrowingthe error in mass of the extremely iron-enriched planets likeKepler-100b is therefore fundamental for testing formationtheories. With current error bars, these planets could be ei-ther close to the simulated CMF range or far outside it, andthus pose a challenge to formation theories. We assume an initial Fe/Mg ratio that is constant acrossthe planetary disk. This is consistent with the assumptionthat the disk formed from the same material that formedthe star. Our simulations do not start until later in thedisk’s evolution, however, and by this time the embryos haveformed through accretion and collisions. This formation pro-cess could have already altered the CMF of the embryosfrom the primordial composition. Indeed, since some of ourembryos start at M ⊕ , it seems that these may have under-gone some collisional evolution already. Thus, instead of aconstant initial CMF, these embryos could have some initialdistribution of CMFs.A test run that began with . M ⊕ embryos confirmedthis. Once the embryos reached M ⊕ , they had a CMF rangeof 0.3-0.4. To replicate this effect without the extra simula-tion time needed to start all embryos at . M ⊕ , we use the MNRAS000
3. In comparison, the super-Earth data shows someplanets that appear to be even more enriched.For the planets for which we obtained the most iron en-richment, the collisional history shows that they tend to beformed by one or two CMF changing collisions, such as ma-jorly erosive collisions or those with core merging, with addi-tional collisions performing minor CMF enrichment. On theother end, not many iron-depleted planets were produced,since the prescription did not allow for much of a reductionin CMF. The only pathway to iron depletion is for a planetto accrete large amounts of mantle debris.Roughly 29 out of the 44 observed super-Earths fallwithin the mass-radius range of the simulated planets, whenconsidering their error bars. It is thus reasonable to concludethat giant impact collisions can explain some of the ironenrichment seen in the rocky super-Earth population. Thesimulations have difficulty achieving even the lowest CMFestimated for Mercury, so it is clear that more work needs tobe done to understand how high CMF planets form. Planetslike K2-38b (Sinukoff et al. 2016), K2-3d (Crossfield et al.2015; Almenara et al. 2015; Dai et al. 2016), and Kepler-100b (Marcy et al. 2014; Borucki et al. 2011; Batalha et al.2013) have data consistent with almost pure iron content.Forming a pure iron planet is difficult. Condensation tem-peratures of iron alloys and silicate oxides are similar, socondensation of these materials from the nebula happenswithin a few hundred Kelvin of each other. In addition, thegiant impact phase does not seem to be energetic enoughto completely remove the mantle of embryos. In fact, evenat very large impact velocities of 5 times that of mutualescape of 8 M ⊕ planets, there is always some mantle leftover (Marcus et al. 2009). One would have to invoke a se-ries of these catastrophic events to produce an iron planet.However, as seen in our calculations, these super energeticcollisions are rare. Further observational work for narrowingthe error in mass of the extremely iron-enriched planets likeKepler-100b is therefore fundamental for testing formationtheories. With current error bars, these planets could be ei-ther close to the simulated CMF range or far outside it, andthus pose a challenge to formation theories. We assume an initial Fe/Mg ratio that is constant acrossthe planetary disk. This is consistent with the assumptionthat the disk formed from the same material that formedthe star. Our simulations do not start until later in thedisk’s evolution, however, and by this time the embryos haveformed through accretion and collisions. This formation pro-cess could have already altered the CMF of the embryosfrom the primordial composition. Indeed, since some of ourembryos start at M ⊕ , it seems that these may have under-gone some collisional evolution already. Thus, instead of aconstant initial CMF, these embryos could have some initialdistribution of CMFs.A test run that began with . M ⊕ embryos confirmedthis. Once the embryos reached M ⊕ , they had a CMF rangeof 0.3-0.4. To replicate this effect without the extra simula-tion time needed to start all embryos at . M ⊕ , we use the MNRAS000 , 1–16 (2019) hemical Diversity of Super-Earths simulations that start at 1 and 0.5 M ⊕ embryos, but per-form multiple post-processing runs, one with initial CMF= 0.3 and another with CMF = 0.4. Each planet then hastwo possible CMFs, one from each run. These are consid-ered to define the range of CMFs for each planet. This ofcourse results in a wider spread of CMFs for planets in theseruns than those with initial CMF=0.333. As expected, theincrease in the spread depends on the spread of the initialCMF given to the embryos. Given our data, we can onlyplausibly expect ∆ CMF i = . , which corresponds to an in-crease in the maximum CMF of ∼ . . Simulations by Carteret al. (2015) of the collisional formation of embryos showthat their variation in CMF could reach up to a maximumof ∆ CMF i = . , corroborating our results.When given a singular initial CMF of 0.333, thefinal planets have a maximum CMF enhancement ofCMF/CMF i = 1.7 or an enrichment of Fe/Mg of . . Whengiven a spread of initial CMF, the maximum CMF enhance-ment increases to CMF/CMF i to 2, and an Fe/Mg enrich-ment of 4. This is still not enough to form the most iron richplanets, like Kepler-100b, making planets with much higherenrichment than 4 seem puzzling to form. The parameters varied in our simulations are to determinethe importance of different disk structures, embryo mass,and debris on composition. We find that, while most of theseparameters have some effect on the CMF distribution, thesize of the effect tends to be small. The overall shape of theCMF distribution as seen in Figure 10 remains roughly thesame. Since the distribution is so sharply peaked, we usethe height of the peak to identify the effect of each of thefollowing parameters.
The mass of an embryo is related to the stage of its evo-lution. Smaller embryos (0.5 M ⊕ ) are less evolved, and assuch require more collisions to form the same mass planetsas larger (1.0 M ⊕ ) embryos. This is further illustrating theeffect of initial CMF as discussed in Section 4.3. In this case,as the smaller embryos are a factor of two less massive thanthe M ⊕ embryos, there are also two times as many embryospacked into the same disk size, which increases the chances ofcollisions. Therefore, we expect smaller embryos to increasethe number of giant collisions. We see a roughly two-foldincrease in the number of giant impacts in the simulations,which increases the chance of CMF-changing collisions. Thefinal CMF distribution seen in Figure 11 is correspondinglyslightly wider for the smaller embryos. Different disk masses are meant to emulate the variation intotal exoplanet system masses. Increased disk mass trans-lates to more massive planetesimals and embryos, in addi-tion to an increase in the number of embryos inside the diskradius. As was the case with smaller embryos, there are morecollisions between embryos and thus more opportunities forCMF change. This caused M ⊕ disk simulations to have a slightly wider distribution than the M ⊕ simulations, ascan be seen from Figure 11. The M ⊕ disk also producesplanets that are M ⊕ and larger, which is beyond the max-imum mass of rocky super-Earths we observe. Disk surface density slope affects the distribution and sizeof embryos, which causes the small differences seen in theCMF distribution for each slope. Disks with slopes of α = and . have more small embryos than those with a slopeof α = . , as can be seen in Figure 3. The smaller embryosmust accrete more mass to form planets, thus there are moregiant collisions in runs with α = and 2.5. α = . runs have2 times more collisions than α = . , and α = α = . .In accordance with the above sections, more collisionstranslate to a wider spread of CMF, and thus the α = . runs have the most strongly peaked CMF distribution. Fig-ure 11 shows that the α = runs are slightly less peaked, andthe α = . runs have the widest spread of CMF. Otherwise,there seems to be little correlation between disk slope andthe CMF distribution of the final planets. Removing some (or all) of the debris mass created in a col-lision is meant to emulate the removal of very small debrisparticles via radiation pressure, as discussed in Section 3.2.We removed of the debris mass as soon as it was cre-ated in about half of the simulations. While more than ex-pected due to collisional grinding, this was done to test themaximum impact of collisional grinding, and as a test of theimportance of the debris to planet formation and composi-tion. Debris provides dynamical friction, which could changethe orbital evolution of the planetary system. It is also keyfor the composition of the planets. Debris is mostly mantlematerial, as this is easier to strip during collisions than thecore. Consequently, its removal should allow for more effi-cient mantle stripping of the bodies during the simulation,as the debris mass can no longer re-accrete onto its originalparent bodies. This is an effect that was observed in the fewsimulations of planet formation that have been performedwith debris-producing collisions (Chambers 2013).We can see some evidence that there was re-accretionof debris onto the original bodies. The CMF distribution forthe debris loss runs, seen in Figure 11, is more peakedaround the initial distribution, and the debris loss his-togram peaks at a slightly enriched CMF due to the pref-erential loss of mantle material in the form of debris. Inaddition, the debris loss histogram extends to higherCMFs, and includes almost all of the high CMF planets.Thus, it is clear that the accretion of mantle debris particlesis a major factor in determining the CMF of final planets.Without that debris, there are also no planets that haveCMF ratios lower the initial, as accretion of mantle debrisis the only way for a planet to decrease its CMF. The im-portance of debris in determining the range of final CMFspoints to the importance of understanding collisional grind-ing of debris and the subsequent loss of debris mass it causes. MNRAS , 1–16 (2019) Scora et al. M ⊕ )25500 . . . . . M ⊕ )10.5 0 . . . . . . . . . . . CMF / CMF i . . . . . . N o r m a li ze d a bund a n ce Figure 11.
Effect of parameters on final planet CMF distributions. Each panel is a normalized CMF histograms, where CMF is shown asa ratio of final planet CMF over initial CMF (0.333). Each panel shows the effect of a parameter on the histogram.
Top left:
Debris-losschanges the distribution, decreasing the peak at the initial CMF and preferentially enriching bodies.
Top right:
Increased disk masswidens the distribution slightly.
Bottom left:
Decreased embryo mass similarly decreases the peak of the distribution.
Bottom right:
Disk slopes other than 1.5 slightly decrease the main peak of the distribution.
In this section we discuss the implications of some of theassumptions we have made in our work.
Throughout the simulations, we assume that all the embryosstart with a single primordial composition determined bythe stellar composition, and we use solar values to computethe radius of the final planets (as discussed in Section 3.3).However, there is a spread in the stellar Fe/Mg abundancesfor stars that host planets (see Fig. 2). Thus, it may bethat the most iron-rich observed super-Earths (i.e. Kepler-100b) come from a star that is more iron rich than the Sun.However, one would need to invoke a substantially iron-richprimordial composition to explain the most iron-rich ob-served super-Earths, and stars that differ beyond 4 timesmore Fe/Mg content from solar are much less common. Itis beyond the scope of this project to address the statisti-cal relations between the stellar Fe/Mg distribution and thedistribution of observed super-Earth compositions, we leavethis for future work.
The simulations do not include the effect of the gas disk onthe dynamics. We have assumed, as with most terrestrialplanet formation simulations (i.e. (Chambers 2013; Kokubo& Genda 2010a)), that the oligarchic growth stage occurslate enough that the gas disk has already dissipated. How-ever, our simulations show that super-Earths grow veryquickly, so most of the action should occur within the life-time of the disk of gas (see also Ogihara et al. (2015)). Thegas disk damps the eccentricities and inclinations of theplanets, thus lowering the mutual collision speed. So, oursimulations maximize the possible CMF enhancement andtherefore provide an upper bound to the maximal CMF thatcan be achieved.Although super-Earths probably form within gaseousdisks, Izidoro et al. (2017, 2019) and Ogihara et al. (2018)showed that dynamical instabilities of super-Earth systemsare likely after the removal of the gas. This leads to highspeed giant impacts among massive planets, similar to thosesimulated in this paper. However, we pointed out in Sec-tion 4.1 that the collisions among the most massive planetstypically lead to the smallest variations in CMF. Thus, com-pared to systems with disks, we consider our simulations afavourable case to producing the largest range of CMF, orbulk compositions, from collisions.
MNRAS000
MNRAS000 , 1–16 (2019) hemical Diversity of Super-Earths . . . . . . . . . . . semi-major axis (AU) super-catastrophiccatastrophicerosionpartialaccretionhit andsprayhit and rungrazeperfect mergergraze andmerge Figure 12.
The frequency of each collision type with respect tosemi-major axis. . . . . . . semi-major axis (AU) v i m p / v e s c i m p a c t a n g l e ( ◦ ) Figure 13.
The impact velocity and angle of collisions with re-spect to semi-major axis. Collisions are shown for the main super-Earth simulations (circles), which stop at 0.1 AU. To extend thepicture inward, we add collisions interior to 0.1 AU that occur inthe short-period super-Earth simulations (triangles). The major-ity of collisions cluster at low impact velocity to escape velocityratios. The maximum velocity ratio increases only slightly withdecreasing semi-major axis.
One caveat to the above is that during the dynamicalinstability phase after gas has dispersed, it is likely thatthere will be fewer planetesimals, as planet formation hasprogressed while the gaseous disk was in place. This couldreduce the dynamical friction experienced by the planets,and actually increase their relative (and thus impact) veloc-ities, increasing CMF change. Further work will explore theeffect of the gaseous disk and the mass of planetesimals onthe CMF distribution.
For computational reasons, our fiducial simulations removeobjects inwards of 0.1 AU. Thus, we do not reproduce ex-oplanets observed within this threshold distance from thecentral star. However, we expect that planets forming within0.1 AU should not be be radically different from those wesimulated here. In fact, by investigating the type of collisions .
25 0 .
50 0 .
75 1 .
00 1 .
25 1 .
50 1 . semi-major axis (AU) . . . . . . . Surface density ( α = 1 . Figure 14.
Partial accretion collision frequency compared to typ-ical disk surface density profile. as a function of semi-major axis (illustrated by Figure 12),we find that all types of collisions happen at every location,except for a drop in super-catastrophic disruption collisionsbeyond 0.6 AU. We attribute this feature to the lack of sig-nificant material beyond the outer edge of our initial distri-bution at 1 AU. The only important feature is then that thefrequency of all collision types increases as semi-major axisdecreases, due to the increase in number density of embryosdictated by the surface density profiles chosen (see Figure 14for an example). Despite this, there was no correspondingcorrelation between CMF and semi-major axis in our sim-ulated planets between 0.1 and 1 AU. It is thus reasonableto expect that there will continue to be no correlation forplanets that form between 0.01 and 0.1 AU.Orbital speeds increase the closer in the embryo is to thestar, and thus it would seem that impact velocities shouldincrease as semi-major axis decreases. Figure 13 shows thatthe average impact to escape velocity ratio does appear toincrease slightly as semi-major axis decreases. However, asdiscussed in Section 4.1, the relative velocities of embryosactually depend on the escape velocities of those embryos,and not on the distance to the star. Thus this trend is likelydue to the tendency of more massive, fully formed embryosto be closer in. Consequently, planet formation even closerto the disk should not change drastically, as it will scalewith the escape velocity of the embryos like everywhere elsein the disk. Even a slight increase in impact velocity to es-cape velocity ratio should not drastically impact the CMFdistribution’s range or shape.To test this hypothesis we ran simulations of planetarydisks with inner edges at 0.05 AU, and we compare theseto those that terminate at 0.1 AU. We are interested in theformation of planets interior to planets formed in our mainsimulation suite, thus we only consider planets formed in-side 0.2 AU. We stopped the short period simulations assoon as planets that formed inside 0.2 AU were at least 10 r H apart at their closest approach. This satisfies general sta-bility criteria, indicating a lower probability of further col-lisions with embryos. These are the triangle planets shownin Figure 6 and Figure 13. There was no significant differ-ence in the number or type of collisions that occurred at this MNRAS , 1–16 (2019) Scora et al. smaller semi-major axis, and the planets’ CMFs largely fellwithin the range generated in the large suite of simulations.Therefore, the comparison between simulated and observedplanets in this paper is strict to planets beyond 0.1 AU, butthese preliminary runs seem to extend our conclusions toplanets inside 0.1 AU as well.
We simulate the formation of rocky super-Earths during thegiant impact stage including realistic collisions via a N-bodycode (modified
SyMBA ) that includes a parameterized mapof collisional outcomes to investigate whether the compo-sitional diversity of rocky super-Earths can be explainedfrom giant collisions during formation alone. We track thechange in composition after each impact by monitoring howthe core-mass fraction of the embryos changes over time,assuming the embryos are differentiated into an iron coreand silicate-magnesium mantle. We obtain planetary massesand compositions after planet formation which we translateto planetary radius to enable a comparison of mass-radiusdata of our simulated planets to those for which we havemeasurements. A summary of our conclusions follows:I. We find that although all types of collisions from theleast to the most energetic happen during formation, thecollisions tend to not occur at impact velocities beyond 5times the mutual escape velocity, even when occurring athigh orbital velocities close to the star. This is due to thefact that the relative velocities between embryos scale withmutual escape velocity, and thus are largely independent ofthe orbital velocity. As a result, the most prevalent type ofcollision is the low-energy graze-and-merge, which is equiva-lent to perfect merger in terms of adding the composition ofthe interacting embryos. The more disruptive collisions —catastrophic and erosion – with the highest impact velocityratios and the most potential to change the composition, oc-cur infrequently. This leaves the intermediate collision typesof partial accretion, hit and spray and hit and run, as themain means to diversify composition. A single one of thesecollisions can rarely produce a large enough amount of debristo change composition considerably. However, they are com-mon enough that through a series of collisions, cumulativeeffects can be seen in the formed planets. Thus, althoughthe most likely fate is that composition changes little from aprimordial start, the deviations from primordial are mostlyfrom multiple intermediate collisions. The outliers tend to beproduced by a collision on the more disruptive end of oneof these intermediate collisions, in combination with the re-moval of debris material that would normally re-accrete ontothe planet. This highlights the importance of understandingdebris behaviour during planet formation, and further inves-tigating the extent of collisional grinding.II. We find overall that collisions can provide a roughly2-fold increase in core-mass fraction of a planet. Disk struc-ture and embryo size do not affect this range, only the detailsin the final core-mass fraction distribution. This distributionis strongly peaked near the initial composition, and thusthere is a low probability of extreme changes in core-massfraction. Even the doubling of the initial core-mass fractionis not quite large enough to produce some of the most denseplanets discovered, such as Kepler 100b. Thus, we find that realistic collisions starting from a solar composition can ex-plain some, but not all, of the spread in composition seen inrocky super-Earths.III. Two possible scenarios may help explain the mis-match between our predicted composition distribution andthe one inferred from mass-radius super-Earth data. Eitherthe stellar hosts of the observed super-Earths have composi-tions that differ significantly from solar, and/or the planets’mass or radius is somewhat over or under-estimated. We cantest either scenario with better observational data. Thus,our work highlights the need (1) to obtain the refractoryratios of the host stars with super-Earth planets as it canhelp us compare stellar to planet chemistry for each systemand build a statistical sample overtime, and (2) to measureplanetary mass and radius at high precision to constrain thecomposition of the planets, especially the ones that seem tobe outliers.
ACKNOWLEDGEMENTS
We thank Alan Jackson for useful discussions on compo-sitional outcomes after planetary collisions, and the anony-mous reviewer for their comments. JS and DV are supportedby the Natural Sciences and Engineering Research Councilof Canada (grant RBPIN-2014-06567). The research shownhere acknowledges use of the Hypatia Catalog Database, anonline compilation of stellar abundance data as described inHinkel et al. (2014), which was supported by NASA’s Nexusfor Exoplanet System Science (NExSS) research coordina-tion network and the Vanderbilt Initiative in Data-IntensiveAstrophysics (VIDA). This research has made use of theNASA Exoplanet Archive, which is operated by the Califor-nia Institute of Technology, under contract with the NationalAeronautics and Space Administration under the ExoplanetExploration Program. We would like to acknowledge thatour work was performed on land traditionally inhabited bythe Wendat, the Anishnaabeg, Haudenosaunee, Metis andthe Mississaugas of the New Credit First Nation.
REFERENCES
Alibert Y., et al., 2006, A&A, 455, L25Almenara J. M., et al., 2015, A&A, 581, L7Asphaug E., 2010, Chemie der Erde / Geochemistry, 70, 199Asphaug E., Agnor C., Williams Q., 2006, Nature, 439, 155Barros S. C. C., et al., 2014, A&A, 569, A74Batalha N. M., et al., 2011, ApJ, 729, 27Batalha N. M., et al., 2013, ApJS, 204, 24Bond J. C., O’Brien D. P., Lauretta D. S., 2010, ApJ, 715, 1050Borucki W. J., et al., 2011, ApJ, 736, 19Carter J. A., et al., 2012, Science, 337, 556Carter P. J., Leinhardt Z. M., Elliott T., Walter M. J., StewartS. T., 2015, ApJ, 813, 72Chambers J. E., 2001, Icarus, 152, 205Chambers J. E., 2013, Icarus, 224, 43Chatterjee S., Tan J. C., 2014, ApJ, 780, 53Chiang E., Laughlin G., 2013, MNRAS, 431, 3444Cossou C., Raymond S. N., Hersant F., Pierens A., 2014, A&A,569, A56Crossfield I. J. M., et al., 2015, ApJ, 804, 10Dai F., et al., 2016, ApJ, 823, 115Dawson R. I., Chiang E., Lee E. J., 2015, MNRAS, 453, 1471MNRAS , 1–16 (2019) hemical Diversity of Super-Earths Dumusque X., et al., 2014, ApJ, 789, 154Duncan M., Levison H., Lee M., 1998, The Astronomical Journal,116, 2067Fogtmann-Schulz A., Hinrup B., Van Eylen V., Christensen-Dalsgaard J., Kjeldsen H., Silva Aguirre V., Tingley B. o.,2014, ApJ, 781, 67Genda H., Kokubo E., Ida S., 2012, ApJ, 744, 137Haghighipour N., Maindl T. I., 2019, in The Main Belt: A Gate-way to the Formation and Early Evolution of the Solar SystemVillasimius. pp 90–92Hansen B. M. S., Murray N., 2012, ApJ, 751, 158Hansen B. M. S., Murray N., 2013, ApJ, 775, 53Hatzes A. P., et al., 2011, ApJ, 743, 75Hauck S. A., et al., 2013, Journal of Geophysical Research (Plan-ets), 118, 1204Hayashi C., 1981, Progress of Theoretical Physics Supplement,70, 35Hinkel N. R., Timmes F. X., Young P. A., Pagano M. D., TurnbullM. C., 2014, AJ, 148, 54Hirose K., Labrosse S., Hernlund J., 2013, Annual Review ofEarth and Planetary Sciences, 41, 657Ida S., Lin D. N. C., 2010, ApJ, 719, 810Inamdar N. K., Schlichting H. E., 2016, ApJ, 817, L13Izidoro A., Ogihara M., Raymond S. N., Morbidelli A., Pierens A.,Bitsch B., Cossou C., Hersant F., 2017, MNRAS, 470, 1750Izidoro A., Bitsch B., Raymond S. N., Johansen A., MorbidelliA., Lambrechts M., Jacobson S. A., 2019, arXiv e-prints, p.arXiv:1902.08772Jackson A., 2020Jackson A. P., Wyatt M. C., 2012, MNRAS, 425, 657Kokubo E., Genda H., 2010a, ApJ, 714, L21Kokubo E., Genda H., 2010b, ApJ, 714, L21Kokubo E., Ida S., 2002, ApJ, 581, 666Lambrechts M., Lega E., 2017, A&A, 606, A146Lee E. J., Chiang E., 2016, ApJ, 817, 90L´eger A., et al., 2009, A&A, 506, 287Leinhardt Z. M., Stewart S. T., 2012, ApJ, 745, 79Lopez E. D., Fortney J. J., 2013, ApJ, 776, 2Marcus R. A., Stewart S. T., Sasselov D., Hernquist L., 2009,ApJ, 700, L118Marcus R. A., Sasselov D., Stewart S. T., Hernquist L., 2010,ApJ, 719, L45Marcy G. W., et al., 2014, ApJS, 210, 20McDonough W. F., Sun S. s., 1995, Chemical Geology, 120, 223Morrison R. A., Jackson J. M., Sturhahn W., Zhang D., Green-berg E., 2018, Journal of Geophysical Research (Solid Earth),123, 4647Nakajima M., Stevenson D. J., 2015, Earth and Planetary ScienceLetters, 427, 286Ogihara M., Morbidelli A., Guillot T., 2015, A&A, 578, A36Ogihara M., Kokubo E., Suzuki T. K., Morbidelli A., 2018,preprint, ( arXiv:1804.01070 )Owen J. E., Wu Y., 2017, ApJ, 847, 29Palme H., Lodders K., Jones A., 2014, Solar System Abundancesof the Elements. Elsevier, pp 15–36Raymond S., Cossou C., 2014, MNRAS, 440, L11Raymond S. N., Kokubo E., Morbidelli A., Morishima R.,Walsh K. J., 2014, in Beuther H., Klessen R. S., DullemondC. P., Henning T., eds, Protostars and Planets VI. p. 595( arXiv:1312.1689 ), doi:10.2458/azu uapress 9780816531240-ch026Ringwood A. E., 1970, Physics of the Earth and Planetary Inte-riors, 3, 109Rogers L. A., Seager S., 2010, ApJ, 712, 974Sinukoff E., et al., 2016, ApJ, 827, 78Southworth J., 2011, MNRAS, 417, 2166Stacey F. D., 2005, Reports on Progress in Physics, 68, 341Stassun K. G., Collins K. A., Gaudi B. S., 2017, AJ, 153, 136 Stewart S. T., Leinhardt Z. M., 2009, ApJ, 691, L133Stewart S. T., Leinhardt Z. M., 2012, ApJ, 751, 32Stixrude L., Lithgow-Bertelloni C., 2011, Geophysical Journal In-ternational, 184, 1180Terquem C., Papaloizou J. C. B., 2007, ApJ, 654, 1110Tonks W. B., Melosh H. J., 1992, Icarus, 100, 326Valencia D., O’Connell R. J., Sasselov D., 2006, Icarus, 181, 545Valencia D., Sasselov D. D., O’Connell R. J., 2007a, ApJ, 656,545Valencia D., Sasselov D. D., O’Connell R. J., 2007b, ApJ, 665,1413Valencia D., Guillot T., Parmentier V., Freedman R. S., 2013,ApJ, 775, 10Van Eylen V., et al., 2016, ApJ, 820, 56Vazan A., Ormel C. W., Dominik C., 2018, A&A, 610, L1Waenke H., Dreibus G., 1988, Philosophical Transactions of theRoyal Society of London Series A, 325, 545
APPENDIX A: BREAKDOWN OFSIMULATION PARAMETERS AND RESULTS
The parameter space of simulations completed is discussedin Section 3.1, and the results of these simulations are ex-plained in Section 4. Table A1 provides a more detailedbreakdown of the numbers of simulations completed for eachparameter set, and the results from each of these sets ofsimulations. The angular momentum deficit (AMD) is cal-culated as per the equations in Chambers (2001).
APPENDIX B: DETAILED COMPOSITIONALPRESCRIPTION
Section 3.3 provides a brief overview of the collision pre-scriptions we use. We choose prescriptions that maximizethe possible change in core-mass fraction (CMF). In eachcollision, there is a projectile and a target. The projectileis always the less massive body. The remaining bodies arecalled the largest remnant and/or the second largest rem-nant (if there is a second embryo), and the debris.
B1 Disruptive collisions
Disruptive collisions produce one embryo and some debris.The possible outcomes of such a collision are: 1) the coresmerge, 2) the core of the target remains unchanged, or the3) core of the target is eroded by the projectile. Remainingcore mass is distributed evenly throughout the debris par-ticles. As mentioned in Section 3.3, the more disruptive thecollision, the larger the iron fraction (Marcus et al. 2009).Marcus et al. (2009) finds that the core mass fraction in-creases following a power law:
CMF new = CMF init + . ( Q R / Q ∗ RD ) . (B1)Above, Q R is the center of mass specific energy ofthe impact and Q ∗ RD is the catastrophic disruption energythreshold, as defined in Stewart & Leinhardt (2009). Thus,the CMF increases as the energy of the collision increases.This is simply due to the fact that the more disruptive thecollision, the more mass is stripped off the target. As weare assuming differentiated bodies, the majority of the mass MNRAS , 1–16 (2019) Scora et al.
Table A1.
Breakdown of simulations done with disks from 0.1 to 1 AU by parameters. The number of runs and initial parameters arein the left-hand section. In the right hand section are the averaged outcomes of each group of simulations. he angular momentum deficit(AMD) is calculated for all planets above . M ⊕ as AMD pl , and for all remaining bodies for AMD tot .Number Maximum initial Disk mass α Percent of debris Number of Average planet AMD pl AMD tot of runs embryo mass ( M ⊕ ) ( M ⊕ ) mass lost planets mass ( M ⊕ ) (planets) (total system)1 0.1 25 1.5 eroded will be mantle material. We use the more general pre-scription, developed in Marcus et al. (2010), that providesresults that roughly follow this power law. In this prescrip-tion, we average the outcomes of two extreme prescriptions.The first produces the maximum CMF, and in this case coresalways merge. If the largest remnant is less massive than thetwo cores, the remaining core and all the mantle become de-bris. The second prescription minimizes the final CMF, andin this case the cores of the projectile and target only mergeif the largest remnant is larger than the target; essentially,if it is a partial accretion. Otherwise, the core of the largestremnant is the same mass as the target’s core, and all thatchanges is the mass of mantle material. The projectile coreand mantle become debris. B2 Grazing collisions
Grazing collisions, or larger angle collisions, such as hit-and-run, hit-and-spray, and graze collisions fall outside the dis-ruption criteria, and are therefore treated differently. Hit-and-spray collisions have a similar outcome to disruptivecollisions, with only one embryo and some debris remain-ing. The rest of these graze collisions have two remainingembryos; in hit-and-run collisions they are accompanied bydebris. The simulations of Asphaug et al. (2006) show thatthe typical outcome of a hit-and-run collision results in theerosion of the projectile, with minor erosion of the target.Since the collision angle is high, the projectile just ’grazes’the target, and simulations show that they just strip mantlefrom the target and projectile. Thus, for these types of col-lisions the core of the largest remnant is the same mass asthe target core. If there is one embryo, the projectile core issplit up in the debris. If there is another remaining embryo,it has the projectile core. The only CMF change from thesecollisions comes from the mantle stripping of the embryos.
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