Are exoplanetesimals differentiated?
Amy Bonsor, Philip J. Carter, Mark Hollands, Boris T. Gaensicke, Zoe Leinhardt, John H. D. Harrison
aa r X i v : . [ a s t r o - ph . E P ] J a n MNRAS , 1–18 (2015) Preprint 15 January 2020 Compiled using MNRAS L A TEX style file v3.0
Are exoplanetesimals differentiated?
Amy Bonsor ⋆ , Philip J. Carter , Mark Hollands , Boris T. G¨ansicke , Zo¨e Leinhardt and John H. D. Harrison Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK Department of Earth and Planetary Sciences, University of California Davis, One Shields Avenue, Davis, CA 95616, USA Department of Physics, University of Warwick, Coventry CV4 7AL, UK School of Physics, University of Bristol, HH Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK.
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Metals observed in the atmospheres of white dwarfs suggest that many have re-cently accreted planetary bodies. In some cases, the compositions observed suggestthe accretion of material dominantly from the core (or the mantle) of a differentiatedplanetary body. Collisions between differentiated exoplanetesimals produce such frag-ments. In this work, we take advantage of the large numbers of white dwarfs where atleast one siderophile (core-loving) and one lithophile (rock-loving) species have beendetected to assess how commonly exoplanetesimals differentiate. We utilise N-bodysimulations that track the fate of core and mantle material during the collisional evo-lution of planetary systems to show that most remnants of differentiated planetesimalsretain core fractions similar to their parents, whilst some are extremely core-rich ormantle-rich. Comparison with the white dwarf data for calcium and iron indicatesthat the data are consistent with a model in which 66 +4 − % have accreted the remnantsof differentiated planetesimals, whilst 31 +5 − % have Ca/Fe abundances altered by theeffects of heating (although the former can be as high as 100%, if heating is ignored).These conclusions assume pollution by a single body and that collisional evolutionretains similar features across diverse planetary systems. These results imply thatboth collisions and differentiation are key processes in exoplanetary systems. We high-light the need for a larger sample of polluted white dwarfs with precisely determinedmetal abundances to better understand the process of differentiation in exoplanetarysystems. Key words: planets and satellites: general < Planetary Systems, (stars:) circum-stellar matter < Stars, (stars:) planetary systems < Stars, (stars:) white dwarfs < Stars
Elements heavier than helium sink below the observable at-mospheres of white dwarfs on timescales of days (young,hydrogen-rich or DA white dwarfs) to millions of years (old,helium-rich or DB white dwarfs) (Koester 2009). The pres-ence of heavy elements in the atmospheres of >
30% ofwhite dwarfs (Zuckerman et al. 2003, 2010; Koester et al.2014) can only be explained by their recent or on-goingaccretion. The consensus in the literature is that we areobserving the accretion of planetary bodies that have sur-vived the star’s evolution in an outer planetary systemorbiting the white dwarf ( e.g.
Jura 2003; Farihi 2016; ⋆ E-mail: [email protected]
Veras 2016). Dynamical instabilities following stellar massloss can scatter planetary bodies onto star-grazing orbits(Debes & Sigurdsson 2002; Bonsor et al. 2011; Veras et al.2013; Debes et al. 2012) where they are disrupted by strongtidal forces (Veras et al. 2014). Fragments of disrupted bod-ies are accreted onto the star, with observations of gas anddust tracing this accretion in action (see Farihi (2016) for arecent review).The abundances observed in the atmospheres of whitedwarfs provide unique insights regarding the compositionof exoplanetary building blocks; the planetesimals accretedby the white dwarfs. Whilst most white dwarf pollutantsexhibit abundances that are broadly similar to rocky plan-ets (Jura & Young 2014), a handful show the presence ofvolatiles, including oxygen and nitrogen ( e.g.
Raddi et al. c (cid:13) A. Bonsor et al. i.e. iron or lithophile (rock-loving) species, such as calcium,magnesium or silicon, have led to the suggestion that thesepolluted white dwarfs have accreted a fragment of a largerbody that differentiated into a core and mantle. For exam-ple, for SDSS J0845+2257 (Wilson et al. 2015), the high ironabundance could be explained by the accretion of a planetes-imal stripped of its mantle. In order to explain the observedabundances, not only must the planetary bodies differenti-ate, but collisions must be sufficiently catastrophic that atleast some fragments have extreme compositions e.g. core-rich.Both collisions and differentiation are common fea-tures of our asteroid belt. Samples of differentiated bod-ies arrive to Earth as meteorites, most famously the ironmeteorites. The range of different spectral classificationsfor asteroids could be explained in part by their differ-entiation and collisional evolution (Burbine et al. 2002).The budget of short-lived radioactive nuclides in the So-lar System, including Al, is sufficient to differentiate bod-ies larger than >
10 km (Urey 1955; Ghosh & McSween1998), and there is a growing suite of evidence that differ-entiation occurred early (Kleine et al. 2005; Scherst´en et al.2006; Kruijer et al. 2014). We note here that there is suf-ficient potential energy imparted during formation aloneto differentiate bodies larger than around 1,000 km, with-out the need for short-lived radioactive nuclides ( e.g.
Davison et al. 2010; Elkins-Tanton et al. 2011). In fact, thedifferentiation of planetary building blocks influences thecomposition of the terrestrial planets, most notably thebudgets of highly siderophile elements (Rubie et al. 2011,2015; Fischer et al. 2017), but also potentially the bulkcomposition (Bonsor et al. 2015; Carter et al. 2015). How-ever, the budget of short-lived radioactive nuclides in ex-oplanetary systems is greatly debated ( e.g.
Young 2014;Lichtenberg et al. 2016; Gounelle 2015; Boss & Keiser 2010;Gritschneder et al. 2012) and it is not clear how widespreadthe differentiation of small exoplanetary bodies is. In fact,Jura et al. (2013) previously used the white dwarf observa-tions to suggest that the Solar System’s abundance of Al was not so unusual.Collisions between differentiated bodies can lead to frag-ments with a diverse range of compositions. Simulationsof disruptive collisions produce fragments with a range ofcompositions, including those dominated by core materialor mantle material (Marcus et al. 2010; Bonsor et al. 2015;Carter et al. 2015). Mercury could be a collision fragmentdominated by core material; stripped of its mantle in a highvelocity collision (Benz et al. 1988). The merging of plane-tary cores and the stripping of mantles are common featuresof high velocity collisions (Benz et al. 1988; Marcus et al.2009; Landeau et al. 2016). Similar processes have been hy-pothesised to occur in exoplanetary systems (Marcus et al.2009). In this work we consider how the differentiation andcollisional evolution of planetesimal belts may influence thecompositions of planetary bodies accreted by white dwarfs.With the growing number of polluted white dwarfs where both lithophile ( e.g. calcium) and siderophile ( e.g. iron) elements have been detected, it is now possible toassess the population of exoplanetary systems as a whole.We hypothesise that all polluted white dwarfs have accretedplanetesimals from outer planetesimal belts that have sur-vived the star’s evolution. These outer planetesimal beltsare commonly observed around main-sequence stars andknown to be collisionally active due to the large quantitiesof small dust continuously replenished in collisions betweenlarger bodies (Wyatt 2008; Hughes et al. 2018). Collisionsbetween planetesimals that have differentiated to form a coreand mantle should lead to a spread in total abundances ofsiderophile ( e.g. iron) and lithophile ( e.g. calcium) species.If polluted white dwarfs sample this distribution, it will bereflected by the spread in their observed calcium and ironabundances.The aim of this work is to collate as large a sample aspossible of polluted white dwarfs where both calcium andiron are detected and use these, compared to models forthe collisional evolution of differentiated planetesimals, toinfer the prevalence of differentiation in exoplanetesimals.Do most planetary systems have planetesimals that differ-entiate, or is the differentiation of planetesimals a uniquefeature of the Solar System?We start by summarising the aims and approach of thepaper in §
2, followed by the observational data sample in § §
4, we firstly consider the possibility that the distributionof abundances could be explained by a range of initial abun-dances for the planet forming material. In § § § § In this work we aim to investigate what the observed pop-ulation of polluted white dwarfs can tell us about how fre-quently exoplanetesimals are differentiated. We do this bycomparing the observed population to a model population,developed from the results of simulations. The N-body simu-lations follow the collisional evolution of planetary systems,tracing the fate of core-like and mantle-like material. We usecollision simulations to predict the population of fragmentsthat accrete onto white dwarfs and compare to the observedpopulation of white dwarf pollutants.Calcium and iron are both commonly observed in pol-luted white dwarfs, whilst being a pair of lithophile andsiderophile elements that behave differently during differ-entiation. In addition to which both elements sink at rel-atively similar timescales through the white dwarf atmo-sphere (Koester 2009), which means that the observed abun-dances should match those of the material accreted onto thestar and do not need to be adjusted to take into accountdifferential sinking. The ratio of Ca to Fe should remain un-changed even if the sinking timescales change, for example inhot DA white dwarfs due the onset of thermohaline (finger-
MNRAS , 1–18 (2015) ing) convection sinking timescale may be significantly longer(Deal et al. 2013; Zemskova et al. 2014; Wachlin et al. 2017;Bauer & Bildsten 2018, 2019) or accreted material is mixedmuch deeper into the white dwarf (Cunningham et al. 2019).Ca and Fe are, therefore, a very useful pair of speciesfor diagnosing the levels of differentiation in exoplanetesi-mals. We collate as large an observational sample as possi-ble of white dwarfs where both calcium and iron have beendetected and compare these to the model predictions. Weutilise the cumulative distribution of Ca/Fe ratios, k , tocompare the model to the observations. For the observations,this essentially equates to a list of observed Ca/Fe values inascending order, each with an associated error, σ Ca / Fe . We,therefore, define X obsCa / Fe ( k ) as the value of Ca/Fe for whicha fraction k of the observed sample have a lower Ca/Fe mea-surement, or in other words a fraction k of the sample havean observed Ca/Fe ratio lower than X obsCa / Fe ( k ). In a simi-lar manner, a fraction k of the model population are pre-dicted to have a lower Ca/Fe value than X modelCa / Fe ( k ). We can,thus, assess the quality of the model fit using a reduced chi-squared of: χ = 1N WD X k =0 X modelCa / Fe ( k ) − X obsCa / Fe ( k )2 σ Ca / Fe ( k ) ! , (1)where N WD is the number of white dwarfs in the sample.In order to determine the most likely values of the modelparameters, we use a Bayesian framework. The posteriorprobability distribution ( p ( θ | M i , D )) of the model param-eters, θ , given the model, M i and the data, D , is propor-tional to the likelihood of the data, given the model andparameters, L ( D | θ, M i ) and the prior on the model param-eters p ( θ | M i ) (see § § θ ) is the fraction of exoplanetesimalsthat are differentiated, f diff . The Markov Chain Monte Carlo(MCMC) fitting routine (Foreman-Mackey et al. 2013) isused to maxmimise this likelihood function in order to findposterior distribution for each model parameter, assuming alikelihood of the form : L ( X obsCa / Fe | θ, M i ) = (2) Q k =0 1 √ πσ Ca / Fe ( k ) exp − (cid:18) X modelCa / Fe ( k ) − X obsCa / Fe ( k )2 σ Ca / Fe ( k ) (cid:19) ! ln L ( X obsCa / Fe | θ, M i ) = − X k =0 ln( √ π σ Ca / Fe ( k )) + (cid:18) X obsCa / Fe ( k ) − X modelCa / Fe ( k )2 σ Ca / Fe ( k ) (cid:19) ! . The observational sample are collated from the literature,some are the most highly polluted white dwarfs where multi-ple species have been detected (Jura et al. 2012; Farihi et al.2013; Dufour et al. 2012; G¨ansicke et al. 2012; Klein et al.2011; Xu et al. 2013; Zuckerman et al. 2011; Klein et al.2011; Raddi et al. 2015; Farihi et al. 2016; Wilson et al.2015; Hollands et al. 2017; Kawka & Vennes 2012, 2016;Swan et al. 2019), whilst most are cool ( T ∗ < , Figure 1.
The calcium and iron abundances measured in the 179white dwarfs plotted in the sample. Lines of constant Ca/Fe ratioare over-plotted as dotted lines. ( σ Ca is the error on 10 Ca / H(e) and σ Fe on 10 Fe / H(e) ). Wenote that quoted errors are often conservative, with an at-tempt to fold in uncertainities on the atomic data, as wellas uncertainities derived from models of the white dwarfatmospheres. Standard error propagation is used to findthe error on the Ca/Fe ratio, assuming that the errorson the Ca and Fe abundances are independent: σ Ca / Fe = Ca / H(e) Fe / H(e) ln(10)( σ + σ ) / . This may not be valid asabundances in Ca and Fe may be correlated. For theHollands et al. (2017) sample, errors are estimated assumingthat they are a sum of systematic errors plus the statisticaluncertainty on the abundances, where the systematic errorsfor Ca and Fe are taken to be σ Ca , sys = σ Fe , sys = 0 .
05 dexand σ = σ , sys + (cid:16) . S/N ) (cid:17) , and an equivalent equation for σ , where S/N is the mean spectral signal-to-noise ratio be-tween 4500–5500˚A, taken from (Hollands private communi-cation) and the scaling factor 1 .
27 is a conservative estimate,based on the weakest lines in the noisiest spectra match-ing the maximum error on an individual element detection(Hollands et al. 2017). For those white dwarfs with low sig-nal to noise observations, particularly from Hollands et al.(2017), the Ca and Fe abundances are poorly known and donot provide information regarding differentiation, we, there-fore, focus on a sample of white dwarfs where
S/N >
In the scenario that no exoplanetesimals differentiate, a nar-row range in the Ca/Fe ratios of exoplanetesimals is ex-pected resulting from the narrow range of Ca/Fe ratios foundin the initial conditions of the material from which theseplanetesimals formed. If we consider that stars and planetarysystems form out of the same material, then we can assumethat the range of compositions of nearby stars will be fairly
MNRAS , 1–18 (2015)
A. Bonsor et al.
Figure 2.
The cumulative distribution of Ca/Fe ratios observedin the white dwarf pollutants (magenta solid line, plus grey shadedregion indicating 1 − σ errors), compared to the cumulative distri-bution of Ca/Fe abundances predicted for the scenario in whichno exoplanetesimals differentiate (stars: black dashed line). similar to the range of initial compositions present in theirplanetary systems. We can easily determine the composi-tions of nearby stars, whereas determining the initial compo-sitions of their planetary systems is challenging. We, there-fore, consider the range of Ca/Fe ratios found in a sampleof nearby stars to be a good proxy for the potential spreadexpected for undifferentiated, pristine exoplanetesimals. Wecompare the cumulative distribution of Ca/Fe ratios foundin a sample of nearby FGK stars taken from Brewer et al.(2016) to the population of polluted white dwarfs. FGK starsare used as they are more likely to have formed at similartimes, thus, with potentially similar compositions, to theprogenitors of the white dwarfs considered. Fig. 2 shows thecumulative distribution of Ca/Fe ratios observed in pollutedwhite dwarfs (magenta line), along with a corresponding er-ror range shown in grey. This is calculated by considering thecumulative distribution that would occur if all white dwarfsin the sample were measured to have their measured Ca/Fe(de)increased by one sigma. This is compared to the dis-tribution of Ca/Fe ratios seen in nearby stars (black line).The polluted white dwarfs show a much broader range ofCa/Fe ratios. We find a reduced χ = 3 .
5, indicating arelatively poor fit. If we consider those 35 white dwarfs with
S/N > χ = 12 . Planetesimal belts evolve over many billions of years to thewhite dwarf phase. Debris discs around main-sequence starsprovide evidence that planetesimal belts are collisionally ac-tive (Wyatt 2008). If the planetesimals have differentiatedto form a core and a mantle, collisions between these dif-ferentiated planetesimals lead to fragments with a range ofcompositions and, of particular relevance here, a range ofCa/Fe ratios (Marcus et al. 2009, 2010; Bonsor et al. 2015;Carter et al. 2015). The scattering and accretion of planetes-imals from planetary systems that have survived the star’s main-sequence evolution to the white dwarf phase is thoughtto be a dynamical process (Veras 2016) and thus, broadly in-dependent of a body’s collisional or geological history. Thus,we anticipate that if the exoplanetesimals accreted by whitedwarfs originate from a system where both collisions anddifferentiation have occurred, they should randomly samplethis distribution of Ca/Fe ratios.The aim here is to produce a model population whichpredicts the distribution of Ca/Fe ratios following collisionalevolution and random sampling of the collision fragments bythe white dwarfs. The exact distribution of Ca/Fe ratios willdepend upon the precise architecture, in particular orbitallocation of material and evolution timescales, of individualsystems. Given that it is not computationally feasible toconsider the full range of system architectures, we use a sin-gle system as a proxy for all systems. We base our resultson simulations of our own Solar System, early in its evolu-tion, namely considering a planetesimal belt in the regionaround Earth’s orbit. Whilst this system goes on to formsome proto-planets, it leaves behind a population of colli-sion remnants, produced from a range of destructive andconstructive collisions. We use these remnants are a proxyfor the bodies accreting onto the white dwarfs.We hypothesise that if the distribution of Ca/Fe ratiosin collision fragments is broadly similar across all planetarysystems, the result of the polluted white dwarfs samplinga range of systems will be similar to our model in whicha single system is sampled. Crucially, we note that even ifthe distribution of Ca/Fe ratios exhibits stark differencesbetween individual systems, these will be very difficult todisentangle from the general population and therefore, as afirst approach, we can derive significant insights from ourmodel.In particular, we anticipate that similar evolution willoccur on longer timescales further from the star. We sim-ulate a massive belt (see below), close to the star ( ∼ r as r / , such that the evolution occurring in a beltat 1au in 10Myr, occurs at 5au in approximately 10Gyr.Thus, we anticipate that the distribution from a close-insystem on short timescales can be used as a proxy for thedistribution from a system further out on longer timescales,potentially more relevant to planetary systems around whitedwarfs. We note here that most collisional evolution will oc-cur whilst the planetary systems are on the main-sequence,whilst some evolution may continue during the white dwarfphase, at which point the system will have expanded in or-bital radii, potentially by a factor of ∼ § MNRAS , 1–18 (2015) to predict the distribution of Ca/Fe ratios in the populationof planetesimals polluting white dwarfs. This assumes thatthe generic form of the predicted distribution of Ca/Fe ra-tios can be applied across a wide range of systems and we,therefore, try not to pin our conclusions down to the specificdetails of the distribution predicted here.
The N-body simulations were performed using a state-of-the-art N-body code, pkdgrav . The collisional evolution ofa planetesimal belt is followed, taking into account bothdestructive and accretional collisions. The fate of colli-sion fragments are tracked using the EDACM collisionmodel (Leinhardt et al. 2015). Every planetesimal is as-sumed to start with a given size and initial core mass frac-tion. The fate of core and mantle material during collisionsis tracked separately using prescriptions based on simula-tions of (Marcus et al. 2009, 2010). Full details of the codeand simulations can be found in Leinhardt et al. (2015);Bonsor et al. (2015); Carter et al. (2015). The simulationswere originally designed to focus on terrestrial planet forma-tion in the inner regions of planetary systems and, therefore,lead to the formation of several proto-planets. For the pur-poses of this work we focus on the collisionally evolved pop-ulation of fragments that remain at the end of the simula-tions, only considering bodies with a mass less than 0 . M ⊕ .This population will have undergone similar collisional evo-lution to that which might occur in a planetesimal belt. Theformation of proto-planets can stir the smaller fragments,inciting further collisional evolution, in a similar manneras is thought to occur in outer debris discs either due toself-stirring or planet stirring (Wyatt 2008; Mustill & Wyatt2009; Kennedy & Wyatt 2010). For a planetesimal belt sig-nificantly further from the star, or significantly less dense,the evolution seen in these simulations is representativeof the evolution that would occur on significantly longertimescales, timescales that would be computationally infea-sible to simulate. Each simulation takes about one month torun. Two types of collision simulations are considered, thefirst considers Earth formation in a Calm scenario and wasrepeated 5 times, whilst the second considers a more spe-cialised scenario involving Earth’s formation, but this timeincluding Jupiter’s Grand Tack. Full details of both scenar-ios can be found in Carter et al. (2015): • Calm:
The evolution of 100,000 planetesimals with aninitial radius of between 196 km and 1530 km in a belt of2 . M ⊕ between 0.5 and 1.5 au is followed for 20 Myr . Allplanetesimals start with an initial core mass fraction of C f (0) = 0 .
35. These are equivalent to Simulations 8–11(
Calm This is the effective time elapsed, which differs from the timefor which the simulation was run by a correction to take intoaccount the expanded radii particles used in order to speed thecomputation by a factor of 6. See Carter et al. (2015) for moredetails • Grand Tack ( GT ): The evolution of 10,000 planetes-imals with an initial radius of between 528 km and 2250 kmin a belt of 4 . M ⊕ between 0.5 and 3.0 au is followed for20 Myr . The migration of a Jupiter mass planet in a so-called Grand Tack is included, where Jupiter starts at 3.5 au,migrates inwards to 1.5 au and then back outwards to finishat 5.2 au. Gas drag is included with a surface density pro-file based on hydrodynamical simulations of giant planetsembedded in a disc (Morbidelli & Crida 2007). The effect ofJupiter’s grand tack is essentially to increase collision veloci-ties, in a similar manner as might occur later in the evolutionof a planetary system due to other effects, including stirringby a giant planet. (Simulation 28 in Carter et al. (2015).In both simulations, a range of collision fragments are cre-ated following each (partially) destructive collision. Frag-ments that fall below the minimum resolution limit (196 kmfor the Calm simulations) are added to the mass in unre-solved dust. This debris is distributed in a series of circularannuli each 0.1 au wide, with the debris placed in the annuluscorresponding to the location of the collision that producedit. This material is assumed to have circular Keplerian or-bits. The unresolved debris is accreted onto all planetesimalsat each time-step, and the fraction of core and mantle ma-terial in the unresolved debris is tracked. Given that mantlestripping is common for small fragments, this unresolved de-bris tends to be dominated by mantle material and leads toall fragments accreting extra mantle-like material.
In this work we are interested in the collisionally evolvedfragments and their range of potential abundances. We,therefore, focus on the distribution of core mass fractionsof planetesimals, with masses less than 0 . M ⊕ , that surviveto the end of the simulations. Fig. 3 shows these distribu-tions. All the simulations display two key features that werealso seen in other similar simulations with different initialparameters. Firstly, the distribution of core mass fractionsis peaked close to the initial core mass fraction of the plan-etesimals. Secondly, the simulations produce a wide range offragments, spread between core-rich and mantle-rich frag-ments, with the potential for some fragments that are ex-tremely core or mantle-rich.In these simulations the peak in the distribution is al-ways shifted slightly towards mantle-rich fragments. Thiscan be attributed to two effects. Firstly, the prescriptionused in Carter et al. (2015) favours the accretion of cores bythe largest remnants following a collision. This leads to thecore material being predominantly found in the larger, lessnumerous remnants, such that most remnants tend towardsbeing more mantle-rich. Secondly, most bodies have grownby accreting the predominantly mantle-rich, unresolved frag-ments, which fill the orbital parameter space around theoriginal bodies. Whether this is a realistic effect, or a resultof the limited resolving power of the simulations that canonly track planetesimals down to around 200 km in size (see § MNRAS , 1–18 (2015)
A. Bonsor et al.
Figure 3.
The distribution of core mass fractions for fragmentswith masses less than 0 . M ⊕ left at the end of the N-body sim-ulations (see § Calm simulations are averaged for clar-ity. The dotted line indicates the initial core mass fraction of C f (0) = 0 .
35 given to planetesimals. produce more debris. As collisions are more disruptive thisdebris contains more core material and thus, the accretionof this debris leads to a population of fragments with coremass fractions very close to the average values.In order to compare the simulation results to the whitedwarf observations, we convert the distribution of core massfractions to a distribution of Ca/Fe ratios in the simplestmanner possible, by assuming that the mass fraction of Caand Fe in the planetesimal’s core and mantle are the sameas in bulk Earth, respectively. This means that: (cid:18) C a F e (cid:19) sims = (cid:18) A Fe A Ca (cid:19) (cid:18) (1 − C f )Ca ⊕ mantle C f Fe ⊕ core + (1 − C f )Fe ⊕ mantle (cid:19) , (3)where C f is the core mass fraction of the planetesimal, A Fe and A Ca are the atomic weights for iron and calcium,Ca ⊕ mantle = 26 . × − the mass fraction of Ca in Earth’smantle, Fe ⊕ core = 0 .
85 the mass fraction of iron in thecore, and Fe ⊕ mantle = 0 .
063 (all values from Palme & O’Neill(2003)). The simulations predict the existence of some al-most pure core fragments. These will have so little calciumthat they are unlikely to be detected and we, therefore, re-move from the model population all fragments with coremasses higher than C f > .
826 or Ca/Fe < . × − , whereCa/Fe= 8 . × − is the lowest observed Ca/Fe ratio in anypolluted white dwarf in the sample.We consider that the distribution of core mass fractionsproduced by the collisional evolution seen in these simula-tions will be fairly typical of collisional evolution in mostplanetesimal belts, although the exact details will clearlyvary depending on the exact collisional and dynamical his-tory of the individual planetary system. Not all bodies that accrete onto white dwarfs will be frag-ments of differentiated planetesimals. In some planetary sys-tems, small exoplanetesimals may not differentiate and thesemay be the planetesimals that accrete onto white dwarfs. In
Figure 4.
The same as Fig. 2 now including the cumulative distri-bution of Ca/Fe abundances predicted from the simulations, theblue solid lines show the
Calm simulations and the green solidline the GT simulations. Figure 5.
The posterior distribution for the fraction of exoplan-etesimals that are differentiated, f diff , considering only observa-tions with low Ca/Fe (Ca/Fe < . our Solar System, some bodies that formed early in the innersystem show clear evidence for differentiation ( i.e. iron mete-orites), whilst other asteroids remain unaltered, such as thechondritic meteorites. In fact, the exact frequency of differ-entiated bodies within our asteroid belt remains a subject ofdebate (DeMeo et al. 2015). In exoplanetary systems, theremay not have been a sufficient heat budget to lead to differ-entiation ( e.g. short-lived radioactive nuclides), or planetes-imals may have formed too late to take advantage of this.We, therefore, introduce a parameter f diff , which representsthe fraction of exoplanetesimals accreted by white dwarfsthat are fragments of differentiated bodies. We consider thisto be a good proxy for the fraction of exoplanetary systemsin which planetesimals are differentiate. The model is nowdefined as: k diff ( X Ca / Fe ) = f diff k sims ( X Ca / Fe )+(1 − f diff ) k stars ( X Ca / Fe ) , (4) MNRAS , 1–18 (2015) where k diff ( X Ca / Fe ) is the cumulative distribution of Ca/Feratios predicted by the model, which is found as a sum of thecumulative distribution of Ca/Fe ratios predicted from thesimulations, k sims ( X Ca / Fe ) and the cumulative distributionof Ca/Fe ratios predicted for the initial conditions at thestart of planet formation, k stars ( X Ca / Fe ), which are both afunction of the Ca/Fe ratio, X Ca / Fe . A 1-D interpolation isused to convert between k diff ( X Ca / Fe ) and X diffCa / Fe ( k ). TheMCMC fitting routine (Foreman-Mackey et al. 2013) is usedto maximise the likelihood function (Eq. 2) in order to findthe posterior distribution of f diff , assuming a uniform priorin which 0 < f diff <
1. The best fit value is very close to1, with f diff = 99 +0 . − . % averaged over all Calm simulations.In other words, the data is consistent with almost all whitedwarf pollutants being the fragments of differentiated plan-etesimals.Fig. 4 shows the cumulative distribution of Ca/Fe ratiospredicted by the model with differentiation, compared tothe observed population. The uncertainties in the observedabundances are such that this model is consistent with theobservations ( χ < χ maybe indicative that the errors on the abundances are in somecases overestimated. In most cases, quoted errors are veryconservative, taking into account both potential errors onthe atomic data and white dwarf atmosphere models.By comparing the likelihoods (Table 1), we can see thata model in which all planetesimals are differentiated is sig-nificantly more likely as an explanation for the data than amodel in which the range of Ca/Fe ratios is explained onlyby the small variation in the initial conditions for planetformation (no exoplanetesimals are differentiated).However, whilst the model is a reasonable fit to the ob-servations at low Ca/Fe ratios, it is less successful at highCa/Fe ratios. The reasons for this will be discussed in detailin the next section, however, we consider those observationswith low Ca/Fe ratios (Ca/Fe < .
07) as a useful samplefor better constraining the fraction of planetesimals that aredifferentiated, f diff . Ca/Fe= 0 .
07 is the median of the distri-bution of Ca/Fe in nearby stars, whilst also being equivalentto a core mass fraction of C f = 0 .
35, assuming typical pa-rameters for bulk Earth (Eq. 3). Maximising the likelihood(Eq. 2) considering only low Ca/Fe ratios (Ca/Fe < . f diff as shown in Fig. 5.Whilst all exoplanetesimals being differentiated ( f diff = 1 . There are more polluted white dwarfs with high Ca/Fe ra-tios in the observed population than in the model popula-tion in which all exoplanetesimals are differentiated. Thiscan be seen on Fig. 4, where the third quartile (84%) occursat Ca/Fe= 0 . . Figure 6.
The overabundance of white dwarfs with high Ca/Feratios, as discussed in §
6, is apparent when only the Ca/Fe ratiois used to predict the core mass fractions of the bodies accretedby the white dwarf sample (Eq. 3), shown in the figure comparedto the distribution of core mass fractions from the simulations(see § § , we discuss alternative explanations to explain theoverabundance of apparently core-rich fragments, derived fromthe overabundance of polluted white dwarfs with high Ca/Fe. the planetesimals accreted by the white dwarfs, using theinverse of Eq. 3. This figure clearly shows that the Ca/Feratios observed would represent a significant overabundanceof mantle-rich fragments compared to the simulated popu-lation. Here, we discuss other potential reasons for this di-vergence.The observed population is by no means selected in anunbiased manner. The white dwarfs used here have beencollated from the literature, where in general the objectswith the best determined abundances are published. Eventhe large number of white dwarfs in Hollands et al. (2017)have not been selected in a uniform manner. Rather, theywere identified in SDSS DR 12 due to the presence of suf-ficient metal lines in the spectra to alter their broad-bandmagnitudes, such that they have redder u-g colours than themain-sequence in a u-g vs. g-r colour-colour diagram, ratherthan bluer like most white dwarfs. In addition to which, onlywhite dwarfs where all three major elements Ca, Fe and Mgwere detected are included.We can, however, consider the impact of requiring thatboth Ca and Fe are detected might have on the model pop-ulation. Calcium is easier to detect than iron, yet, presentin smaller quantities in planetary material. No calcium willbe present in purely core fragments, which have, thereforebeen removed from the model population as undetectable.Fig. 1 shows that the fragments with low Fe abundanceshave high Ca/Fe ratios and thus, the non-detection of thesefragments leads the observational sample to be biased to-wards low Ca/Fe ratios. This is the opposite trend to thatseen in the observations. We, therefore, consider that we donot find strong evidence that observational biases are re-sponsible for the overabundance of high Ca/Fe ratios in theobserved population.One tendency of the simulations, as discussed in § MNRAS , 1–18 (2015)
A. Bonsor et al. to resolve these small fragments and therefore, they are as-sumed to reaccrete onto all fragments uniformly. This leadsto many fragments with core fractions slightly below aver-age. However, if these fragments could be followed in detail,or if collisions were more violent, such as may occur in a sys-tem stirred by giant planets, it is plausible that the distribu-tion of core mass fractions may be skewed to contain morevery mantle-rich fragments. Such fragments would producea model population that tends to have more mantle-richfragments and be more similar to that predicted from theobservations, however, there will still be many more moder-ately mantle-rich fragments than extremely mantle-rich frag-ments. Fig. 6 indicates how extreme this overabundance ofhigh Ca/Fe ratios in the observed population is comparedto the simulations, which makes it hard to explain withthis model. The observations were sensitive enough to de-tect white dwarfs with low Ca/Fe ratios, as demonstratedby a handful of extreme examples, but these were found tobe rare in Hollands et al. (2017).The simulations assume a single initial Ca/Fe ratio ofCa/Fe= 0 .
07 ( C f = 0 .
35) for all planetesimals. A rangeof initial Ca/Fe may be more appropriate across diverseexoplanetary systems, and indeed a range of Ca/Fe ratiosare seen across chondritic meteorites in our Solar System(0.04-0.11 Wasson & Kallemeyn (1988), although some ofthe spread may be due to the effects of heating). We do notdeem a range in the initial Ca/Fe as a likely explanation forthe observed overabundance, as a large spread towards highCa/Fe would be required.The collision model did not include crustal differentia-tion. This has the potential to increase the number of frag-ments with high calcium abundances (Carter et al. 2018).Crustal stripping is particularly efficient at producing smallfragments with high Ca/Fe ratios. Such fragments wouldalso have altered abundances of other species. In additionto this, crustal material is generally a significantly smallerproportion of a planetary body’s overall mass budget ( e.g. < In this section, we investigate whether a model in whichsome planetesimals accreted by white dwarfs have higherCa/Fe ratios due to the effects of heat processing can ex-plain the white dwarf observations. The increased Ca/Fe ra-tio could have occurred due to incomplete condensation ofiron-rich minerals during planet formation, or evaporationduring formation or subsequent evolution, for example onthe giant branch. The exact distribution of Ca/Fe is, thus,unknown. There is an additional complication that heatingand differentiation can occur in the same planetesimals. We,therefore, decide not to focus on producing a detailed modelof heating, but make a broad, all encompassing model thatcan tell us whether this explanation has the potential to beconsistent with the population. We, therefore, parametrisethe distribution of Ca/Fe ratios due to heating as a nor-mal distribution centred at d Ca / Fe , of width σ W , where both d Ca / Fe and σ W are model parameters. Clearly, such a modelis sufficiently flexible to fit the observations, given appro-priate values of d Ca / Fe and σ W , however, it can provide anindication of the fraction of the population for which heatingis important.We create a model in which a fraction, f hot , of the sam-ple have undergone heating which leads to a cumulative dis-tribution of Ca/Fe abundances, k hot ( X Ca / Fe ), given by thecumulative distribution function of a Gaussian centred on d Ca / Fe and of width σ w . At the same time we consider thata fraction, f diff , of the sample are fragments of differentiatedplanetesimals, with a distribution of abundances that followthose of the simulations, k sims ( X Ca / Fe ). Those white dwarfpollutants that are not differentiated, nor have experiencedheating (1 − f diff − f hot ), have abundances that originatefrom the distribution of potential initial abundances, i.e. from nearby stars, k stars ( X Ca / Fe ). This leads to a model withfour free parameters, f diff , f hot , d Ca / Fe and σ w . We use thisto calculate the model population, where k model ( X Ca / Fe ) isthe cumulative distribution of Ca/Fe ratios in the modelpopulation, where: k model ( X Ca / Fe ) = f diff k sims ( X Ca / Fe ) + (5)(1 − f diff − f hot ) k stars ( X Ca / Fe ) + f hot k hot ( X Ca / Fe ) . We find the best-fit values of f diff , f hot , d Ca / Fe and σ w by maximise the posterior distribution (Eq. 2) using theMCMC fitting routine of (Foreman-Mackey et al. 2013) andassuming uniform priors of 0 < f diff <
1, 0 < f hot < . < d Ca / Fe < . < σ W < .
3. The mid-pointof the Gaussian is fixed to occur at high Ca/Fe ratios, i.e. above Ca/Fe= 0 .
07, equivalent to C f = 0 .
35, otherwise thepriors are designed to be non-informative and encompass thefull range of potential values for the parameters.The results of the fitting procedure are listed in Table 1,whilst the posterior probability distributions of the four pa-rameters are shown in Fig. 7. Fig. 7 plots all 50 walkers atsteps 1,000 to 2,000, in the 4
Calm simulations, with con-tours overlaid at (0 . , . , . , . , . σ . The distributionsshow the existence of a clear best-fit solution.Fig. 7 shows there is a correlation between f diff and f hot .Those models in which more exoplanetesimals are differen-tiated ( f diff is higher) show less effects of heating ( f hot islower). This naturally makes sense as both parameters help MNRAS , 1–18 (2015) to explain the excess of polluted white dwarfs with highCa/Fe ratios. However, solutions for f diff and f hot all liewithin defined ranges, with best-fit values of 66 +4 − % for f diff and f hot = 31 +5 − % , averaged across all Calm simulations.The fraction of planetesimals that are differentiated is in-line with the model that focussed only on low Ca/Fe ratios( § d Ca / Fe =0 . +0 . − . , σ w = 0 . +0 . − . %, averaged across all Calm sim-ulations. In terms of the mid-point ( d Ca / Fe ) and width ( σ w )of the Gaussian, the majority of solutions indicate the addi-tion of planetesimals with Ca/Fe ratios between 0.1-0.3 dueto heating. These are very plausible values for objects wheresome iron rich minerals have been removed and the Ca/Feratio was fixed prior to differentiation. Such high Ca/Fe ra-tios are less likely to occur following differentiation, as theywould require the removal of more iron than generally foundin the body’s mantle. In this work we present a model in which the population ofcalcium and iron abundances observed in a sample of 179polluted white dwarfs can be explained by the differentia-tion and collisional processing of a substantial fraction ofthe accreted planetesimals and the effects of processing attemperatures higher than 1,350K. We aim to constrain howfrequently planetesimals accreted by white dwarfs are thecollision fragments of differentiated bodies. In this sectionwe discuss how robustly we can come to a conclusion.Our null hypothesis was that the range of abundancesobserved in the planetary bodies accreted by white dwarfsresulted from a range of initial abundances present in thematerial out of which the planetary bodies formed. Us-ing nearby stars as a proxy for this range of composi-tions, we show that this model is unlikely to explain thedata. We deem it likely that differentiation, rather thanother unknown processes, are responsible for the abun-dances in at least some white dwarfs due to the correla-tions observed in multiple siderophile species (core) or multi-ple lithophile species (mantle) which point towards segrega-tion due to melting and differentiation (Harrison et al. 2018;Jura & Young 2014). In fact, those polluted white dwarfsin this sample with the lowest Ca/Fe ratios also have thelowest Mg/Fe ratios, pointing towards a common origin tothe depletion of both lithophiles (Hollands et al. 2018). In asimilar manner correlations between the abundances of mul-tiple species linked by similar condensation temperatures,provides evidence for heat processing in individual objects(Harrison et al. 2018).A model in which all white dwarf pollutants are frag-ments of differentiated bodies is consistent with the data,given the uncertainties. However, an excess of high Ca/Fe ra-tios in the observed sample compared to the model remains.We hypothesise that this excess can be explained by plan-etesimals that have suffered the effects of heat processing attemperatures between 1,000K to 2,000K. At such temper- atures calcium-rich and iron-rich minerals exhibit differentbehaviours, with iron-rich minerals tending to be removedpreferentially from the solid phase compared to calcium-richminerals, which can lead to high Ca/Fe ratios in planetarybodies. We know from the Solar System that the effectsof the depletion of moderately volatile elements, includingtrends featuring Ca and Fe are seen in meteoritic samples( e.g.
Palme & O’Neill 2003). It, therefore, seems probablethat such processes would have occurred in the planetesi-mals accreted by white dwarfs, long prior to their accretiononto the white dwarfs.The key question regards the fraction of exoplanetes-imals that are differentiated. Based on the model analysis,we consider that between 60% and 100% of exoplanetesimalsare differentiated, with a most likely value of 66 +4 − %. Thebest way to improve upon these conclusions would be to in-crease the sample size, particularly the sample size of whitedwarfs with precisely determined abundances of at least Caand Fe. For example, there are less than 10 white dwarfs withlow Ca/Fe < .
07 and high
S/N ( S/N >
10) from which themost can be learnt about the fraction of planetesimals thatare differentiated. This is not yet large enough to smoothout the effects of small number statistics.
The current model suffers from many limitations, which willbe discussed here. However, we note that the limitations ofthe model must be considered in the context the size of thedata sample with sufficiently precise abundance determina-tions and our lack of knowledge regarding any biases in itsselection. The most significant weakness of this work is thatwe focus solely on Ca and Fe. Whilst this allows us to pro-cess a larger data sample in the same manner, we are nottaking advantage of the full information available for eachwhite dwarf. Ca and Fe are a very useful pair of elementsas the effects of differential sinking in the white dwarf at-mosphere can be ignored. Conclusions regarding individualobjects will be found in Harrison et al, in prep.The model is relatively simplistic. We create a modelof collision fragments based on the collisional history of asingle planetary system. Whilst the white dwarf pollutantsnecessarily originate from a wide range of systems, it is notfeasible to simulate even a small range of the potential archi-tectures. We do, however, predict that the range of core massfractions produced will vary in width and height, rather thansignificantly in form. We hypothesise that the generic form,whereby collisional evolution produces produces a centrallypeaked distribution of Ca/Fe ratios, or indeed core massfractions, with fewer fragments possessing extreme values,will be consistent across a wide range of system architec-tures.The main conclusions of this work are based on theexistence of many more bodies with Ca/Fe (core mass frac-tions) close to a central (original) value than extreme val-ues. In fact, the fraction of exoplanetesimals that must bedifferentiated would only increase, if the distribution weremore centrally peaked, a plausible consequence of more dra-matic collisional evolution, as seen in the GT simulation (seeFig. 3). If the distribution of Ca/Fe ratios were spread morebroadly in systems with particular collision histories, it isplausible that the fraction of exoplanetesimals that are dif-
MNRAS , 1–18 (2015) A. Bonsor et al.
Figure 7.
The posterior probability distributions of each parameter in the empirical fit to observations, calculated by maximising thelikelihood (Eq. 6), assuming that a fraction f diff of planetesimals are differentiated, whilst f hot are subject to heating. Results are includedfor 4 high resolution Calm simulations. Plotted are individual walkers, with density contours overplotted at (0 . , . , . , . , . σ , createdusing Foreman-Mackey (2016). Model χ ln L f diff f hot d Ca / Fe σ w Stars 3.5 -180Diff: GT 0 .
83 230 98 . +1 − %Diff: Calm1 0 .
46 300 98 . +1 − %Diff: Calm2 0 .
26 330 98 . +1 − %Diff: Calm3 0 .
43 300 98 . +1 − %Diff: Calm4 0 .
42 305 98 . − %Hot: GT 0 .
57 330 69 +1 − % 30 +1 − % 0 . ± .
01 0 . ± . .
13 370 62 +5 − % 34 +5 − % 0 . ± .
02 0 . ± . .
08 374 66 +8 − % 29 +6 − % 0 . ± .
03 0 . ± . .
16 367 67 +3 − % 31 +3 − % 0 . ± .
02 0 . ± . .
09 373 66 +5 − % 31 +4 − % 0 . ± .
02 0 . ± . Table 1.
A table to show the results of comparing the model populations to the white dwarf observations. Best-fit parameters are shown,alongside reduced chi-squared (Eq. 1) and likelihoods (Eq. 2). ferentiated has been overestimated. However, the number ofwhite dwarf pollutants with extreme Ca/Fe ratios limits anypotential reduction.We sample the Ca/Fe ratios produced in a single plan-etary system at a single epoch. We can consider this to bea reasonable approximation to the collisional evolution atlarger orbital radii on longer timescales, as discussed in § MNRAS , 1–18 (2015) Figure 8.
The same as Fig. 2 now including the cumulative distri-bution of Ca/Fe abundances predicted from the model with bothheating and differentiation (red dot-dashed line), compared to theblue dotted line which shows the
Calm 1 simulation. f hot = 31%, f diff = 66%, d Ca / Fe = 0 .
24 and σ W = 0 . We assume that each white dwarf has accreted a sin-gle fragment. If instead, multiple fragments were impor-tant, this could potentially act to smooth the distribution ofCa/Fe ratios (Turner & Wyatt 2019) and we would expectto see fewer examples of white dwarf pollutants with ex-treme abundances. This would make it harder to concludethat any white dwarf pollutants are not the fragments ofdifferentiated exoplanetesimals. It is, however, potentiallypossible that white dwarfs accrete differentiated bodies in abiased manner, for example accreting first material from themantle, followed by material from the core.We ignore crustal fragments, which do have the poten-tial to alter the Ca/Fe ratio. This would likely mean thatsome pollutants with high Ca/Fe may result from crustalfragments, rather than heating, thus reducing f hot , whilstleaving f diff unaffected. This is an important avenue for fu-ture investigations.The heating model is very simplistic and presentedmerely to show that heating is a plausible explanation forthe overabundance of white dwarfs with high Ca/Fe ratios,rather than as a precise distribution of the Ca/Fe ratioslikely to exist in planetesimals following the effects of tem-peratures higher than 1,000K. We refer the interested readerto (Harrison et al. 2018) for a more detailed description ofthe Ca/Fe ratios that may result from the effects of heatingon exoplanetesimals. We have shown that many white dwarf pollutants are likelyto be the collision fragments of planetesimals that differen-tiated to form a core and a mantle. The implications of thisconclusion depends on the size of planetesimals that are ac-creting onto white dwarfs, which remains an open question.Whilst we can measure the mass of material currently inthe atmosphere of the white dwarf, which in some cases isgreater than the mass of Ceres (Raddi et al. 2015), exactlyhow this relates to the size of the body accreted, or the sizeof the parent body, if the pollutant is a collision fragment,remain unclear. If the white dwarf pollutants are the collision frag-ments of parent bodies larger than around > ,
000 km indiameter, their differentiation can be explained by the heat-ing imparted by impacts occurring during planet formation(Davison et al. 2010). This would imply that a large pro-portion of exoplanetary systems include collisionally evolvedpopulations of Pluto-sized bodies, something which does notseem to be the case within our own Solar System, for exam-ple where
D >
100 km bodies in the asteroid belt are mostlyprimordial ( e.g.
Bottke et al. 2015). However, white dwarfplanetary systems have the potential for dynamical instabil-ities to have occurred due to mass loss post-main sequence( e.g.
Veras et al. 2013; Mustill et al. 2014).On the other hand, the planetesimals accreted by whitedwarfs could in general be collisional fragments of bodiessmaller than 1 ,
000 km. In this case it is harder to under-stand what powers their differentiation unless there is a sig-nificant source of heating present from short-lived radioac-tive nuclides in most exoplanetesimals, as suggested by ( e.g.
Jura et al. 2013; Young 2014). Such a theory would supporta model in which the presence of these nuclides leads totriggered star formation, such that all planetary systems ini-tially have a large budget of such nuclides ( e.g.
Boss et al.2008; Li et al. 2014). In addition to which it would implythat most white dwarf pollutants originate from sufficientlyclose to the star that planetesimal growth had made it tolarge enough sizes to differentiate prior to the decay of anyshort-lived radioactive nuclides present in the planet formingmaterial e.g. Al with its half life of 0.7Myr.The requirement for the collisional evolution of largeplanetesimals, whether they are of the order of ∼
10 km insize, or much larger, in most white dwarf planetary systemsimplies that most of these systems possess debris belts sig-nificantly more massive than our own Solar System’s aster-oid or Kuiper belt. This agrees with observations of main-sequence debris discs, which find that, for example, ∼ Spitzer (Su et al. 2006; Wyatt 2008), whilst our asteroidand Kuiper belt lie orders of magnitude below the detec-tion limit. Another possibility is that the dynamical insta-bilities that lead to white dwarf pollution ( e.g.
Bonsor et al.2011; Debes & Sigurdsson 2002; Debes et al. 2012) lead tocollisional processing in large bodies. The most likely sce-nario is that in some systems, the white dwarf pollutantsresult from the collisional processing of very large bodies( > M ⊙ star can reach aluminosity of 16 , L ⊙ at the tip of the AGB (Hurley et al.2000), which would imply an equilibrium temperature of > , MNRAS , 1–18 (2015) A. Bonsor et al. tained volatiles such as water originated from planetesimalbelts outside of the ice line (Farihi et al. 2011; Raddi et al.2015; Malamud & Perets 2016). Thus, in combination withthe need for heating, white dwarf abundances imply that alarge spread in the original radii of white dwarf pollutantsmust exist.
Abundances of calcium and iron in the atmospheres of whitedwarfs can be used to study the differentiation of exoplan-etesimals. We use a sample of 179 white dwarfs collated fromthe literature, where both Ca and Fe are detected, to showthat the distribution of Ca/Fe ratios are unlikely to occur asa result of a distribution in the initial compositions (Ca/Fe)at the start of planet formation. We hypothesise that, in-stead, a fraction of the white dwarfs have accreted the frag-ments of differentiated exoplanetesimals.If exoplanetesimals differentiate, collisions can lead tofragments with a range of core mass fractions (Ca/Fe). Wepresent results from a set of N-body simulations in which thefate of core and mantle material during collisions is tracedseparately. These simulations show that the distribution ofcore mass fractions in the remnant planetesimals is alwaysdominated by values close to the core mass fraction of theparent bodies, with a few extremely core-rich or mantle-richfragments. This means that whilst it is easy to conclude thatwhite dwarf pollutants with extreme Ca/Fe ratios are likelyto be core-rich or mantle-rich fragments, for every extremeobservation, we anticipate many more white dwarf pollu-tants with less extreme Ca/Fe ratios.Using the simulation results, we created a model pop-ulation of planetesimals that could accrete onto pollutedwhite dwarfs. We use this model to show that the observedrange of Ca/Fe ratios in polluted white dwarfs is consistentwith all polluted white dwarfs having accreted the collisionfragment of a differentiated planetesimal, however, there isan overabundance of polluted white dwarfs observed withhigh Ca/Fe ratios. We suggest that this is unlikely to be anobservational bias and more likely a result of processing attemperatures between 1,000K and 2,000K during the forma-tion or subsequent evolution of the accreted planetesimals.In this case, our best-fit model finds that 31 +5 − % of planetes-imals accreted by white dwarfs have increased Ca/Fe due tothe effects of heating, whilst 66 +4 − % are the fragments ofdifferentiated planetesimals.These results imply that the collisional evolution oflarge planetesimals (at least larger than tens of kilometres)is a typical feature of exoplanetary systems, in line with ob-servations of debris discs around main-sequence stars. Thepopulation of white dwarf pollutants suggest that differenti-ation occurs commonly in exoplanetesimals and that eithershort-lived radioactive nuclides are present in many exoplan-etary systems, or white dwarf pollutants are typically thecollision remnants of planetary bodies sufficiently large thatimpacts during planet formation can lead to differentiation.We highlight the need for a larger sample of white dwarfswith precisely determined abundances to investigate furtherthe geological process of differentiation in exoplanetary sys-tems. ACKNOWLEDGEMENTS
AB acknowledges funding from a Royal Society DorothyHodgkin Fellowship. JHDH acknowledges an STFC stu-dentship. BTG was supported by the UK STFC grantST/P000495. PJC acknowledges support from UC Office ofthe President grant LFR-17-449059. The research leading tothese results has received funding from the European Re-search Council under the European Union’s Horizon 2020research and innovation programme no. 677706 (WD3D)
APPENDIX
MNRAS , 1–18 (2015) Name T eff Ca/H(e) σ Ca Fe/H(e) σ F e
Ca/Fe σ Ca / Fe K dex dex dex dexWD0122-227 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ α β γ ǫη ǫ z η θ ι κλ λ λ λ ζ ρ ζ † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † MNRAS , 1–18 (2015) A. Bonsor et al.
SDSS J0818+1247 † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † MNRAS , 1–18 (2015) SDSS J1257-0310 † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † † φ MNRAS , 1–18 (2015) A. Bonsor et al.
NLTT 6390 φ ξ α Jura et al. (2012) β Farihi et al.(2013) γ Dufour et al. (2012) ǫ Xu et al. (2013) ζ Zuckerman et al.(2011) η Klein et al. (2011) θ Raddi et al. (2015) ι Farihi et al. (2016) κ Wilson et al. (2015) λ G¨ansicke et al. (2012) † Hollands et al. (2017) φ Kawka & Vennes (2012) ξ Kawka & Vennes (2016) ∗ Swan et al. (2019) z Jura et al. (2012); Zuckerman et al. (2010)
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