The Formation of Jupiter's Diluted Core by a Giant Impact
Shang-Fei Liu, Yasunori Hori, Simon Müller, Xiaochen Zheng, Ravit Helled, Doug Lin, Andrea Isella
TThe Formation of Jupiter’s Diluted Core by a Giant Impact
Shang-Fei Liu , (cid:63) , Yasunori Hori , , Simon M¨uller , Xiaochen Zheng ,Ravit Helled , Doug Lin , & Andrea Isella School of Physics and Astronomy, Sun Yat-sen University Zhuhai Campus, 2 Daxue Road,Tangjia, Zhuhai 519082, Guangdong Province, P.R. China; Department of Physics and Astronomy, Rice University, 6100 Main St., MS-108, Houston, TX77005, USA; Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan; National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan; Institute for Computational Science, Center for Theoretical Astrophysics and Cosmology, Uni-versity of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland; Department of Physics and Center for Astrophysics, Tsinghua University, Beijing 10084, P.R.China; Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95060,USA; Institute for Advanced Study, Tsinghua University, Beijing 100084, P.R. China. (cid:63)
To whom correspondence should be addressed; Email: [email protected]
The
Juno mission is designed to measure Jupiter’s gravitational field with an extraordinaryprecision . Structure models of Jupiter that fit Juno gravity data suggest that Jupiter couldhave a diluted core and a total heavy-element mass M Z ranging from ten to two dozens ofEarth masses ( ∼ − M ⊕ ). In that case the heavy elements are distributed within an ex-tended region with a size of nearly half of Jupiter’s radius R J3, 4 . Planet formation modelsindicate that most of the heavy elements are accreted onto a compact core , and that almostno solids are accreted during runaway gas accretion (mainly hydrogen and helium, hereafterH-He), regardless to whether the accreted solids are planetesimals or pebbles . Therefore,the inferred heavy-element mass in the planet cannot significantly exceeds the core’s mass.The fact that Jupiter’s core could be diluted, and yet, the estimated total heavy-element massin the planet is relatively large challenges planet formation theory. A possible explanationis erosion of the compact heavy-element core. Its efficiency, however, is uncertain and de-pends on both the immiscibility of heavy materials in metallic hydrogen and the efficiency a r X i v : . [ a s t r o - ph . E P ] J u l f convective mixing as the planet evolves
11, 12 . Neither can planetesimal enrichment andvaporization produce such a large diluted core. Here we show that sufficiently energetichead-on collisions between additional planetary embryos and the newly emerged Jupiter canshatter its primordial compact core and mix the heavy elements with the outer envelope. Thisleads to an internal structure consistent with the diluted core scenario which is also found topersist over billions of years. A similar event may have also occurred for Saturn. We suggestthat different mass, speed and impact angle of the intruding embryo may have contributedto the structural dichotomy between Jupiter and Saturn . Giant impacts
19, 20 are likely to occur shortly after runaway gas accretion when a gas giantplanet’s gravitational perturbation significantly intensifies (about thirty-time increases in a fractionof a million years) and therefore destabilizes the orbits of nearby protoplanetary embryos. Thistransition follows oligarchic growth and the emergence of multiple embryos with isolation massin excess of a few M ⊕ . Some of these massive embryos may collide with the gas giant duringtheir orbit crossing
23, 24 . Through tens of thousands of gravitational N -body simulations with dif-ferent initial conditions such as Jupiter’s growth model, orbital configuration, etc. (see Methods),we find that an emerging Jupiter has a strong influence on nearby planetary embryos. As a result,a significant fraction of these embryos could collide with Jupiter within a few million years, i.e.,within the Solar nebula lifetime. Among those catastrophic events, head-on collisions are morecommon than grazing ones due to Jupiter’s gravitational focusing effects.In order to investigate the influence of such impacts on the internal structure of the youngJupiter we use the hydrodynamics code FLASH with the relevant equation of state (EOS). Detailsof the computational setup and the simulations are presented in the Methods section. Generally,the disintegration of the intruding embryo leads to the disruption of the planet’s original core.However, to establish a large diluted-core structure as inferred from recent Jupiter structure modelsbased on Juno ’s measurements, the core and embryos’ fragments need to efficiently mix with thesurrounding convective envelope, which requires a large embryo to strike the young Jupiter almosthead-on. Massive embryos are available at the advanced stage of Jupiter’s formation and our N -body simulations also suggest that head-on collisions are common (see Methods).In Figure 1 we show the consequence of a head-on collision between an embryo and Jupiterwith an initial M core = 10 M ⊕ silicate/ice core, a H-He envelope, approximately present-day total2ass and radius (The young Jupiter may have a size up to twice of its current-day value, however, toavoid introducing additional free parameters, we consider models more similar to current Jupiter).In fact, the post-impact core-envelope structure mainly depends on the mass of the initial coreand envelope as well as the impactor’s mass and impact velocity V imp . We adopt an impact speed V imp ∼ km s − which is close to the free-fall speed onto Jupiter’s surface (see Methods) andassume the impactor is comprised of an 8 M ⊕ silicate-ice core and a 2 M ⊕ H-He envelope. Thetotal mass of proto-Jupiter’s and embryo’s core M Z,total is chosen to be compatible with the derivedmass of heavy elements in Jupiter’s diluted core models . Note that at Jupiter’s distance of 5.2AU from the sun, the impactor’s speed relative to the gas giants is limited by the planets’ surfaceescape speed. The acquisition of protoplanetary embryos would not lead to any major changes inthe spin angular momentum and orientation of the targeted planet. The total energy injected intothe young Jupiter by the intruding embryo is only a few percent of its original values so that thereis little change in Jupiter’s mean density and mass.The impact results in little mass loss (see Table 1), while Jupiter’s initial core is completelydisrupted. During the impactor’s plunge and collision with the primordial core, a large amount ofkinetic energy is dissipated. The heat release near the center increases the local temperature T ,offsets the pressure P balance, and induces oscillations (see the full video in Supplementary Infor-mation). The steep negative entropy gradient near the core overturns the local negative molecularweight gradient µ and leads to convection in the inner part of the envelope. Vigorous turbulencestirs up efficient mixing between the heavy elements and H-He envelope. After a few dynamicaltime-scales (a characteristic time scale to measure expansion or contraction of a planet; Jupiter’sdynamical time scale is roughly a third of an hour), the initial silicate/ice core is thoroughly ho-mogenized with the surrounding H-He and their mass fraction Z ≤ . interior to ∼ . R J . Within ∼ dynamical time-scales, Jupiter’s interior settles into a quasi-hydrodynamic equilibrium witha diluted core extending to R ∼ . − . R J (see Table 1 and panel a of Figure 2). In the outerhalf of the envelope, the gas density is slightly elevated and a small trace of the dredge-up heavyelements ( Z ) leads to the formation of a composition gradient.The post-impact heavy-element distribution leads to a composition gradient that could evolveand become similar to an internal structure with a diluted-core. However, the hydrodynamic simu-lation is terminated ten hours after the impact. In order to explore under what conditions a diluted-core-like structure persists after the 4.56 Gyrs of Jupiter’s evolution, we compute the thermal-3volution shortly after the impact until today. The hydro-simulation sets the initial heavy-elementgradient as shown in panel a of Figure 2. Since the post-impact temperature profile is unknown ∗ ,we consider various temperature profiles with different central temperatures. Furthermore, we con-sider an initial thermal structure that accounts for the accretion shock during runaway gas accretionas suggested by a recent Jupiter formation model (see Methods for details). We find that for thehead-on collision, a post-impact central temperature of ∼ ∼ ∼ . A gradual accretion of planetesimals along with therunaway gas accretion may also produce a diluted core
15, 28 . A relevant issue to be investigateelsewhere is whether the steep compositional gradient needed to preserve the diluted core can alsobe established after a series of planetesimal-accretion events rather than a single embryo’s giantimpact. Finally, extra-solar gas giant planets could also suffer giant impacts which could explain ∗ the exact temperature profile depends on the formation process
26, 27 , the energetics of the impact, etc. . Acknowledgements
We thank S.M. Wahl and Y. Miguel for sharing their results with us. Wethank J.J. Fortney, P. Garaud and H. Rein for helpful conversations. S.-F.L. thanks the support andhospitality from Aspen Center for Physics during the early stage of this work. D.L. thanks Institutefor Advanced Study, Princeton, Institute of Astronomy and DAMTP Cambridge University forsupport and hospitality when this work was being completed. R.H. acknowledges support fromSNSF grant 200021 169054. A.I. acknowledges support from the National Aeronautics and SpaceAdministration under award No. 80NSSC18K0828 and from the National Science Foundationunder grant No. AST-1715719.
Author Contributions
D.L. had the impact idea. S.-F.L. and A.I. examined its feasibility. S.-F.L. coordinated this study. S.-F.L. and Y.H. designed and analysed the hydrodynamic simulations.X.Z. and S.-F.L. performed and analysed the N -body simulations. S.M. and R.H. designed thelong-term thermal evolution study. All authors contributed to discussions, as well as editing andrevising the manuscript. Author Information
The authors declare that they have no competing financial interests.Correspondence and requests for materials should be addressed to S.-F.L. (email: [email protected]).5 ensity( g / cm )18.013.59.04.510 -4 Befrore impact During impact After impact a b c
Figure 1:
3D cutaway snapshots of density distributions during a merger event between aproto-Jupiter with a 10 M ⊕ rock/ice core and a 10 M ⊕ impactor. a, just before the contact. b, the moment of core-impactor contact. c,
10 hours after the merger. Due to impact-inducedturbulent mixing, density of Jupiter’s core decreases by a factor of three after the merger, resultingin an extended diluted core. A 2D presentation of density slices of the same event is shown inExtended Data Figure 3. 6 b pre-impactpost-impactH-4.5H-4.5-rockH-radenvH-4.5H-4.5-rockH-radenvWahl+ 17Normalized Radius0.0 0.2 0.4 0.6 0.8 1.0012345 D en s i t y [ g / c m ] H ea vy - e l e m en t F r a c t i on Figure 2:
Post-impact thermal evolution models. a,
Heavy-element distribution vs. normalizedradius before (dotted) and after (dashed) the giant impact. The solid lines show the compositionafter 4.56 Gyrs of evolution for the three best-fit models that result in a diluted core, see theMethods section and Table 3 for more details. b, Density vs. normalized radius after 4.56 Gyrsof evolution (solid) and from the diluted-core interior structure model of Wahl et al. 2017 (dash-dotted). 7able 1: Initial conditions and final outcomes of the head-on giant impact simulation. M T and M I are the total mass of the proto-Jupiter and the impactor, respectively. M core is the mass ofheavy elements in the proto-Jupiter’s core. M Z, I and M Z, total are the total mass of heavy elementscontained in the impactor and the system, respectively. After the merger, values of total mass ofJupiter M T and heavy elements M Z, total are measured within 1 and 2 R J , respectively. Those valuesreveal that the majority of Jupiter’s mass still resides in its original size, albeit a hot extended low-density envelope mostly made of H-He forms after the merger (see also Extended Data Figure 3).The size of a diluted core was defined as the central region enclosed by a sphere with Z > . .The last three rows list values for the best-fit evolution models to the interior structure model ofJupiter with a diluted core . R core /R J is the radius of the proto-Jupiter’s core scaled to the Jupiter’scurrent radius. All mass quantities are in unit of M ⊕ . M T M core M I M Z, I M Z, total R core /R J Before merger 306.714 9.962 9.967 7.975 17.937 0.15 ∼
10 hrs after merger 304.946 / 313.360 17.693 – – 17.901 / 17.925 0.423H-4.5: After 4.56 Gyrs 313.36 10.61 – – 17.925 0.30H-radenv: After 4.56 Gyrs 313.36 17.24 – – 17.925 0.39H-4.5-rock: After 4.56 Gyrs 313.36 15.92 – – 17.925 0.458 ensity ( g cm -3 )10 -3 -1 -4 -2 a bc de f Figure 3:
2D snapshots of an off-center collision between the proto-Jupiter with a 10 M ⊕ solidcore and a 10 M ⊕ impactor. a, Density contours in the orbital plane before the impact; b-e, theimpactor being disrupted and accreted; e, at ∼
30 hours after the impact. See Methods for detaileddiscussion. 9
MethodsA statistical N -body study of embryo collisions: We investigate the statistics of collisions be-tween an emerging Jupiter and planetary embryos with the open-source N -body code REBOUND version 3.6.2. To simulate the evolution of a planetary system we choose the built-in hybrid HER-MES integrator † , which uses the WHFast integrator for the long-term dynamics and switches tothe IAS15 integrator when close encounters (such as scattering and collisions) happen.Our N -body simulations start from a coplanar configuration in which five ten-Earth-massplanetary embryos ( M p = 10 M ⊕ ) orbit the Sun ( M ∗ = 1 M (cid:12) ∼ . × M ⊕ ) on circular progradeorbits. The embryo at 5.2 astronomical units (a.u.) from the Sun grows into a Jupiter-mass planet atthe end of the simulation. Initially, two embryos are placed interior to Jupiter’s orbit and the othertwo embryos are placed exterior to Jupiter’s orbit. The orbital separation between two adjacentembryos i and i + 1 is determined by a dimensionless number k = 2 a i r H (cid:18) a i +1 − a i a i +1 + a i (cid:19) , (1)where a i and a i +1 are the semi-major axes of each embryo, and r H = a i (cid:16) M p M ∗ (cid:17) / is the Hill radiusof embryo i . It is convenient to express Equation 1 in terms of q = a i +1 /a i , the ratio of semi-majoraxis between embryos i and i + 1 k = 2 (cid:18) µ (cid:19) / (cid:18) q − q + 1 (cid:19) , (2)where µ = M p /M ∗ (cid:39) × − is the mass ratio between the embryo and the Sun. A larger k willgive rise to a wider separation, i.e. a more dynamically stable configuration. Extended Data Table1 summarizes the locations of all embryos for a given parameter k in our N -body simulation suite.In addition, we also consider a configuration, in which all four embryos are beyond Jupiter’s orbit.At the onset of the simulation, the runaway gas accretion of Jupiter’s core starts. The massaccretion rate is an exponential decay function characterized by an exponential time parameter t grow ranging from 0.1 Myr to 0.5 Myr in this study. At a given time t , the mass of an emergingJupiter is determined by M ( t ) = M J − ( M J − M p ) e − t/t grow , (3) † In recent updates of REBOUND, the HERMES integrator has been replaced by the MERCURIUS integrator,which offers a similar capability in a single scheme. M J = 317 . M ⊕ is one Jovian mass. In this model, Jupiter quickly acquires more than 90%of its mass within 3 t grow and steadily gains another a few percent of its mass until t = 10 t grow .For simplicity, we assume that all other four embryos do not grow during the whole time, since atypical hydrostatic growth stage of an embryo before it entering the runaway gas accretion phaseis around a few Myr and the embryo mass barely increases.Size is another crucial factor as a larger cross-section can boost the probability of collisions.We adopt the mean density of the Earth for embryos, so their sizes R p (cid:39) . R ⊕ , where R ⊕ isEarth’s mean radius. For the emerging Jupiter, its mean density could be as low as half of itscurrent-day value. We use the parameter f to describe the degree of inflation.Thus, we design a simple classification for our N -body simulation suite with three free pa-rameters k , t grow and f . For each combination set of ( k , t grow , f ), we run thousands of simulationswith other orbital parameters (e.g., true anomaly, argument of periapsis) randomly chosen between0 and π .At the end of an N -body simulation ( t = 10 t grow ), a planetary embryo may remain boundto the Sun with considerable changes in its orbit, or coalesce with Jupiter and other embryos orescape from the system after a close encounter. The statistics of final outcomes of four planetaryembryos under the influence of an emerging Jupiter is shown in Extended Data Figure 1. Theresults are grouped by different parameters to compare their impacts. In all subsets of our N -body simulations, we observe an efficient pathway to deliver planetary embryos to collide with anemerging Jupiter.Because embryos are equally distributed on both sides of Jupiter’s orbit (except for the lastgroup with all embryos in the ”Outward” state to begin with), the results suggest that embryosboth interior or exterior to Jupiter could collide with Jupiter within the simulation time. Whileembryos beyond Jupiter may have a slightly larger chance to strike Jupiter as there are less embryosasymptotically result in an ”Outward” destiny. Among the three key parameters, orbital tightnesscharacterized by k plays the most substantial role in affecting the collision probability. For the sameorbital configuration, Jupiter inflation factor f can slightly change the collision rate. However,Jupiter’s accretion history determined by t grow has the least influence on the results.11e analyze the distribution of collision angle using our N -body simulation suite. And thehistograms of collision angles are plotted in Extended Data Figure 2. Each histogram represents adetailed breakdown of ”Merger” events of a simulation set presented in Extended Data Figure 1.Unlike collisions between similar-sized planetary bodies, in which 45 ◦ collisions are common ,the statistical results suggest that half of the merger events have collision angles less than ∼ ◦ inall cases we investigated. We suggest that low-angle impacts are very common because of Jupiter’sstrong gravitational focusing effect.It is often useful to define a two-body escape velocity as V esc = (cid:18) G ( M J + M p ) R R + R p (cid:19) / , (4)which is around 51 km s − for the proto-Jupiter and the 10 M ⊕ impactor studied in the hydrody-namic simulation. In general, an embryo’s impact velocity V imp is related to V esc as well as the localKeplerian velocity V kep . Gravitational perturbation during close encounters can produce an impactvelocity with a magnitude up to the escape velocity . On the other hand, the Keplerian orbitalvelocity gives rise to the random velocity dispersion during impacts. At Jupiter’s current location, V kep ∼
13 km s − is much smaller than V esc , so the impact velocity V imp is approximately at theescape velocity V esc . Indeed, we find the impact velocity is quantitatively similar to V esc ratherthan V kep , although V imp is always slightly smaller than V esc in the N -body simulation suite, be-cause initial separations between Jupiter and embryos are finite (a two-body system has a negativegravitational potential energy).This simple statistical model can be improved in the future to compare with other formationmodels of the outer Solar system. For example, because Jupiter’s inward migration is much slowerthan those planetary embryos, the presence of Jupiter in the Solar nebula acts like a barrier forinward migrating planetary embryos formed exterior to Jupiter . Consequently, collisions amongthose planetary embryos may become frequent and some of those events may eventually formUranus and Neptune . Hydrodynamic simulations:
Our 3D hydrodynamic simulation of giant impacts between a proto-Jupiter and a protoplanetary embryo is based on the framework of the Eulerian FLASH code which utilizes the adaptive-mesh refinement. The setup of giant impact simulations has been laidout in our previous study . Here, we briefly describe the model of the planetary interior. The12rimordial Jupiter is modeled with a three-layer structure: a silicate core, an icy mantle, and a H-Heenvelope. We calculate two thermodynamic (density and internal energy) properties of silicate andice material and their velocities with the governing continuity, momentum, and energy equation.For computational efficiency, these quantities are converted into pressure and temperature with theTillotson EOS . The mass fraction between ice to silicate is assumed to be 2.7 according to thatof protosun (2–3). In addition, the H-He EOS is modeled with an n = 1 , γ = 2 polytropic relation,where n and γ are the polytropic and adiabatic indexes. Although this idealized treatment ignoreseffects such as the H-He phase transition and separation, it reasonably matches the density profileof Jupiter’s envelope calculated with ab-initio EOS , and is good enough for dynamic processesthat happen in a few hours (see detailed discussion below). Collisions between a proto-Jupiter with a 10 M ⊕ core and a 10 M ⊕ embryo: From N -bodysimulations we learn that most collisions have collision angles less than 30 degrees, so we firststudy the head-on collision as one of the representative cases in the main text and the consequenceis shown in Figure 1. Here we also plot its 2D counterpart in Extended Data Figure 3. The generalbehavior of head-on collisions has been studied extensively in previous works , . To recapitulate,the solid material of the impactor can penetrate Jupiter’s gaseous envelope and smash into its coreas a whole. As a result, Jupiter’s core gets completely destroyed after the impact. The releaseof a large amount energy inside the proto-Jupiter drives large scale turbulence and the primordialcompact core is homogenized subsequently. We compare the enclosed internal energy of Jupiteras a function of radius before and after the impact. The results are shown in Extended Data Figure4. Although Jupiter gains internal energy through the release of kinetic and gravitational energyof the impactor as well as impactor’s own internal energy, the core region gets barely heated. Infact, there is even a little decrease of internal energy inside the core region right after the impactpossibly due to mixing with H-He. The analysis suggests that the impactor dumps most of itsenergy outside the original core region.Our simplified EoS for H/He causes less efficient dissipation of the impactors kinetic energywithin the H/He envelope. As a vigorous mixing between H/He and core material, however, isdriven by a merger between the core of a photo-Jupiter and an impactor, we can expect formationof a dilute core to occur regardless of EoS models. In addition, a temperature profile inside a coreis not strongly affected by the choice of a H/He EoS model because the impact causes only a smallchange in internal energy inside the core. 13o illustrate the effects of off-center collisions, we run the same setup of simulation exceptthat the collision angle is at 45 degrees. The consequence is shown in Figure 3. Because the initialimpact velocity is at the escape velocity, the impactor misses Jupiter’s core and overshoots untilJupiter’s gravitational force pulls it back. During its course, the impactor gradually loses angularmomentum and gets torn apart. The remnant is gently accreted by Jupiter’s solid core later on. Asa result, the impact has little influence on Jupiter’s core-envelope structure. A head-on collision between a proto-Jupiter with a massive core and a small impactor:
Inaddition, we perform a head-on collision between a proto-Jupiter with a massive primordial coreof 17 M ⊕ and a 1 M ⊕ impactor, which is composed of pure silicate, at the same impact velocity.The total amount of heavy elements is the same as that in previous head-on and off-center models(hereafter, case-1 and case-2). Unlike case-1, the impactor disintegrates in the proto-Jupiter’senvelope before making contact with the core. A strong shock wave induced by the entry of theimpactor propagates throughout the entire planet and deforms the core (see panel c of ExtendedData Figure 5). The heavy elements are well mixed with a small fraction of H-He (only ∼ Post-impact thermal evolution:
We simulate Jupiter’s long-term evolution after the giant impactin order to identify the evolutionary paths that lead to a diluted core structure at present-day. The14lanetary evolution is modelled using the 1D stellar evolution code Modules for Experiments inStellar Astrophysics (MESA), where the planet is assumed to be spherically symmetric and in hy-drostatic equilibrium . The evolution is modeled with a modification to the equation of state ,where the H-He EOS is based on SCVH with an extension to lower pressures and temperatures,and the heavy-element (H O/SiO ) EOS is QEOS
46, 47 . Conductive opacities are from Cassisi et al.(2007) , and the molecular opacity is from Freedman et al. (2007) .The planetary evolution is governed by the energy transport in the interior, which can oc-cur via radiation, conduction, or convection. We use the standard Ledoux criterion to determinewhether a region with composition gradients is stable against convection, i.e., ∇ T < ∇ ad + B ,where ∇ T = d log T /d log P , with ∇ ad and B being the adiabatic temperature and compositiongradient, respectively. If the composition gradient is such that the mean molecular weight in-creases towards the planetary center, then B > and the composition gradient could inhibit con-vection. For a homogeneous planet, B = 0 and the Ledoux criterion reduces to the Schwarzschildcriterion ∇ T < ∇ ad . A region that is Ledoux stable but Schwarzschild unstable could developsemi-convection. In that case, double-diffusive processes can lead to additional mixing .In the planet evolution code, convective mixing is treated via the mixing length theory (MLT),which provides a recipe to calculate ∇ T and the diffusion coefficient, fully determining the con-vective flux. The MLT requires the knowledge of a mixing length l m = α mlt H P , where H P is thepressure scale-height and α mlt is a dimensionless parameter. The expected value of α mlt for planetsis poorly constrained. Following previous work on Jupiter’s evolution with convective mixing we use α mlt = 0 . as our baseline. It is found that the mixing is relatively insensitive to the choiceof the mixing length within about an order of magnitude. This is because its value does not di-rectly determine when mixing occurs, but the mixing efficiency. To investigate the sensitivity ofthe results on this parameter we also included a model with α mlt = 10 − . While our conclusionson the diluted core are robust, a detailed and rigorous investigation on mixing in giant planets isclearly desirable, and will be presented in future work .The case of semi-convection is treated as a diffusive processes which requires the calcu-lation of the temperature gradient and diffusion coefficient in the semi-convective region. Therecipe includes a free parameter that can be interpreted as the layer-height of the double-diffusiveregion
54, 55 , which is unknown and could range over a few orders of magnitude. In the case where15e include semi-convection, we set the value to − pressure scale heights, which is an interme-diate value in the range given in the literature .The hydro-simulation of the giant impact sets the post-impact composition profile to be usedby the evolution model. The initial temperature profile is crucial for determining the energy trans-port for the subsequent evolution. Since proto-Jupiter’s thermal state at the time of impact is un-known, we consider various initial temperature profiles and explore how the mixing is affected bythis choice. Giant planet formation calculations estimate the central temperature of proto-Jupiterto be ∼ K . The exact temperature, however, is unknown and can change by a factor of a few.For determining the convective mixing efficiency such factors can lead to large differences in thelong-term evolution and the final internal structure. Also, recent work has shown that accountingfor the accretion shock during the runaway gas accretion phase can lead to a radiative envelope anda non-monotonic temperature profile in the deep interior
26, 27 . We include this possibility in one ofour models (H-radenv). Our nominal models use α mlt = 0 . , no semi-convection with the heavyelements being represented by water. A summary of the model parameters is given in ExtendedData Table 2.In Extended Data Figure 6 we present the starting models that are evolved to Jupiter’s age.The solid and dashed lines correspond to the head-on and oblique (at an angle of 45 degrees) col-lisions, respectively. The temperatures are increasing towards the interior for all models exceptH-radenv, as explained above. Here, a temperature inversion occurs in the deep interior, corre-sponding to the location of the accretion shock during early runaway gas accretion. Note that inthis model the location of the temperature-inversion occurs within the same region of the com-position gradient, which supports the stability of the region against convection. While the exactlocation of the temperature jump is not well determined, it can be estimated due to the requirementof reaching the so called cross-over mass to enter the runaway phase . As the heavy-elementfraction increases, the interior becomes hotter due to the change in opacity and the increase indensity. If the collision is head-on, the composition gradient is shallower and extends farther intothe envelope.Extended Data Figures 7 & 8 show the density profiles after 4.56 Gyrs of evolution for thehead-on and oblique collision, respectively. The crucial influence of the initial thermal profile onthe mixing is clear: For the log T c [ K ] = 4 . head-on collision case (H-4.7), the end-result is a fully16omogeneous Jupiter without a core. For the oblique impact, even the very steep compositiongradient, with the highest temperatures, is insufficient to inhibit substantial mixing of the deepinterior. The intermediate temperature profiles lead to varying degrees of mixing. In general, thehead-on collision results in an extended core that is highly enriched in H-He, while for the obliqueimpact the core is more compact and less diluted. Despite a substantial fraction of proto-Jupiterbeing very hot in the model H-radenv, there is not enough mixing to erase the composition gradient.In this case, the envelope is radiative at early times when mixing would be most efficient. If a lowermixing length is chosen (H-4.5-low α ), the composition gradient is less eroded and extends fartherinto the envelope. Because the energy transport is also affected by the chosen mixing length,Jupiter’s interior is hotter and denser compared to H-4.5.In H-4.5-semiconv, we consider the same model as H-4.5 but allow semi-convective mixing.with a layer height of − pressure scale-heights. In this case, semi-convection is insufficient toovercome the stabilizing composition gradient. While some additional mixing occurs, particularlyat early times, there are no semi-convective regions towards the end of the evolution. In otherwords, the final interior structure is such that the radiative regions are Schwarzschild and Ledouxstable. This demonstrates that also when semi-convection is included we infer a Jupiter with adiluted core.In order to completely erase the composition gradient created by the giant impact the im-pact must be head-on with a very hot interior ( ∼
44, 52 . If the core is defined as the region that is heavy-elementrich in comparison to the envelope, then most of our models imply that Jupiter has a diluted andextended core extending to ∼ − of the planet’s radius. All the oblique collisions lead toa relatively compact core since the initial composition gradient is very steep.Figure 2 shows the models that best match the diluted-core density profile from Wahl etal. (2017) (H-4.5-rock, H-4.5, H-radenv). We find that for the head-on collision, a post-impact17entral temperature of ∼ ), the diluted core extends fartherinto the envelope and is thus more consistent with a Jupiter structure with a diluted core. Anotherpathway to the diluted core is when Jupiter’s deep interior is radiative due to the accretion shockas predicted by recent giant planet formation models (H-radenv). Videos that demonstrate theplanetary evolution for three selected cases can be found in the Supplementary Information. Data availability.
The datasets generated and analysed during the current study are availablefrom the corresponding authors upon reasonable request.
Code availability.
The FLASH code is publicly available for download at http://flash.uchicago.edu/site/flashcode.The implementation of giant impact simulations in the framework of FLASH is available upon re-quest. The REBOUND code is publicly available for download at https://github.com/hannorein/rebound.The MESA code is an open source stellar evolution code and is publicly available at http://mesa.sourceforge.net.The modified version of the MESA code is not yet ready for public release - it will be presented infuture work . Gnuplot, Jupyter Notebook, Mathematica, VisIt and yt python package have beenused for data reduction and presentation in this study.18 I n w a r d M e r ge O u t w a r d E sc ape I n w a r d M e r ge O u t w a r d E sc ape I n w a r d M e r ge O u t w a r d E sc ape I n w a r d M e r ge O u t w a r d E sc ape F r a c ( % ) t grow = 0.2 Myr k = 10.0 t grow = 0.1 Myr t grow = 0.1 Myr k = 10.0 f = 2.0 f = 2.0Group 1: Group 2: Group 3: Group 4: I n w a r d M e r ge O u t w a r d E sc ape I n w a r d M e r ge O u t w a r d E sc ape I n w a r d M e r ge O u t w a r d E sc ape I n w a r d M e r ge O u t w a r d E sc ape F r a c ( % ) f = 1.0f = 1.5f = 2.0 k = 5.0k = 6.0k = 8.0 t grow = 0.2 Myrt grow = 0.3 Myrt grow = 0.5 Myr f = 1.0, k = 10.0f = 2.0, k = 10.0f = 2.0, k = 6.0t = 0 t = 10 t grow ab Figure 4:
Extended Data Figure 1 — Statistics of outcomes of four planetary embryos underthe influence of an emerging Jupiter. a, the initial configurations of four planetary embryos di-vided into four groups based on fixed parameters shown under the group numbers. In group 1-3,half of the embryos are placed inside Jupiter’s orbit (labeled as ”Inward”), the other half are out-side Jupiter’s orbit (labeled as ”Outward”). In group 4, all embryos are outside Jupiter’s orbit. Theexact location of every embryo is shown in Table 1 in the Supplementary Information. b, the statis-tical outcomes of the dynamic evolution after 10 t grow . Because Jupiter’s growth can substantiallymodify orbits of those embryos. Some embryos collided with Jupiter (labeled as ”Merger”), andsome have been ejected from the Solar system (labeled as ”Escape”). Other embryos are labeledeither ”Inward” or ”Outward” depending on their orbital locations inside or outside Jupiter’s orbit.Colours indicate different choices of the free parameter displayed in legend in each group.19
10 20 30 40 50 60 70 80 900255075100 02468Collision angle N t grow = 0.1 Myr, k = 10.0 P r obab ili t y ( % ) f = 1.0 0 10 20 30 40 50 60 70 80 900255075100125 02468Collision angle N P r obab ili t y ( % ) f = 1.5 0 10 20 30 40 50 60 70 80 900255075100125 02468Collision angle N P r obab ili t y ( % ) f = 2.00 10 20 30 40 50 60 70 80 900255075100125150175 02468Collision angle N t grow = 0.1 Myr, f = 2.0 P r obab ili t y ( % ) k = 5.0 0 10 20 30 40 50 60 70 80 900255075100125150175 02468Collision angle N P r obab ili t y ( % ) k = 6.0 0 10 20 30 40 50 60 70 80 900255075100125150 02468Collision angle N P r obab ili t y ( % ) k = 8.00 10 20 30 40 50 60 70 80 900255075100125 02468Collision angle N k = 10, f = 2.0 P r obab ili t y ( % ) t grow = 0.2 Myr 0 10 20 30 40 50 60 70 80 900255075100125 02468Collision angle N P r obab ili t y ( % ) t grow = 0.3 Myr 0 10 20 30 40 50 60 70 80 900255075100125 02468Collision angle N P r obab ili t y ( % ) t grow = 0.5 Myr0 10 20 30 40 50 60 70 80 900255075100 02468Collision angle N t grow = 0.2 Myr P r obab ili t y ( % ) f = 1.0, k = 10.0 0 10 20 30 40 50 60 70 80 900255075100125 02468Collision angle N P r obab ili t y ( % ) f = 2.0, k =10.0 0 10 20 30 40 50 60 70 80 900255075100125150 02468Collision angle N P r obab ili t y ( % ) f = 2.0, k= 6.0 a Group 1: b Group 2: c Group 3: d Group 4:
Figure 5:
Extended Data Figure 2 — Histograms of collision angles of each data set presentedin Extended Data Figure 1. a, group 1. b, group 2. c, group 3. d, group 4. The bin size is 5 ◦ ,and there are 18 bins in each plot. The red dashed line indicate the median value in each case. Theresults suggest head-on collisions are more common than grazing ones.20 ensity ( g cm -3 )10 -3 -1 -4 -2 a bc d Figure 6:
Extended Data Figure 3 — 2D snapshots of a merger between the proto-Jupiterwith a 10 M ⊕ solid core and a 10 M ⊕ impactor. a, Density contours in the orbital plane beforethe impact; b, before the impactor arriving at the core; c, after the destruction of the core; d, at ∼
10 hours after the impact. 21 .0 0.2 0.4 0.6 0.8 1.0- 1 ×10
01 ×10 Radius [R J ] Δ E i n t e r na l [ e r g s ] PreimpactPostimpact0.0 0.2 0.4 0.6 0.8 1.002×10 Radius [R J ] E i n t e r na l [ e r g s ] ab Figure 7:
Extended Data Figure 4 — The change of internal energy caused by the mergerin case-1. a,
The enclosed internal energy of Jupiter before and after the impact as a function ofradius. b, The net change of enclosed internal energy of Jupiter as a function of radius.22 ensity ( g cm -3 )10 -3 -1 -4 -2 bdac Figure 8:
Extended Data Figure 5 — 2D snapshots of a merger between the primordialJupiter with a 17 M ⊕ core and a 1 M ⊕ impactor. a, Density contours in the orbital plane beforethe impact; b, before the impactor arriving at the core; c, after the merger with the core; d, at ∼ b Figure 9:
Extended Data Figure 6 — Initial conditions for post-impact evolution. a, the initialpost-impact heavy-element profile and b, temperature profiles of the models that are used for thethermal evolution. The heavy-element distribution is taken from the hydro simulation ten hoursafter the giant impact. Solid lines correspond to a head-on collision, while dashed-dotted linesshow the result of an oblique collision at a 45 degree angle. The colours depict models withdifferent initial thermal states. See text and Extended Data Table 2 for further details.24 .0 0.2 0.4 0.6 0.8 1.0 Normalised Radius D e n s i t y [ g / c m ] H-4.3H-4.5H-4.7H-radenvH-4.5-lowH-4.5-semiconvH-4.5-rock
Figure 10:
Extended Data Figure 7 — Density vs. normalized radius for the head-on collisionafter 4.56 Gyrs of evolution.
The colors correspond to distinct model assumptions: H-4.3, H-4.5, H-4.7 correspond to initial thermal profiles with different central temperatures at the time ofthe impact, while H-radenv assumes a proto-Jupiter with a radiative envelope. H-4.5-low α uses ashorter mixing length, H-4.5-semiconv allows for semi-convective mixing, and in H-4.5-rock theheavy elements are represented by rock instead of water for EOS purposes. See text and ExtendedData Table 2 for further details. 25 .0 0.2 0.4 0.6 0.8 1.0 Normalised Radius D e n s i t y [ g / c m ] O-4.3O-4.5O-4.7O-4.5-lowO-4.5-semiconvO-4.5-rock
Figure 11:
Extended Data Figure 8 — Density vs. normalized radius for the oblique collisionafter 4.56 Gyrs of evolution.
The colors correspond to distinct model assumptions: O-4.3, O-4.5, O-4.7 correspond to initial thermal profiles with different central temperatures at the time ofthe impact, while O-radenv assumes a proto-Jupiter with a radiative envelope. O-4.5-low α uses ashorter mixing length, O-4.5-semiconv allows for semi-convective mixing, and in O-4.5-rock theheavy elements are represented by rock instead of water for EOS purposes. See text and ExtendedData Table 2 for further details. 26igure 12: Extended Data Table 1 — List of initial orbital semi-major axis of each embryo ofour N -body simulation suite. The location of the embryo that grows into a Jupiter in each case isin bold face.Figure 13:
Extended Data Table 2 — Description of the evolutionary models that are dis-cussed throughout this work.
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