Planetary Core Formation with Collisional Fragmentation and Atmosphere to Form Gas Giant Planets
aa r X i v : . [ a s t r o - ph . E P ] J un Planetary Core Formation with Collisional Fragmentation andAtmosphere to Form Gas Giant Planets
Hiroshi Kobayashi , Hidekazu Tanaka , Alexander V. Krivov Astrophysical Institute and University Observatory, Friedrich Schiller University, Schillergaesschen 2-3,07745, Jena, GERMANY Institute of Low Temperature Science, Hokkaido University, Kita-Ku Kita 19 Nishi 8, Sapporo 060-0819,JAPAN [email protected]
ABSTRACT
Massive planetary cores ( ∼
10 Earth masses) trigger rapid gas accretion to form gas giantplanets such as Jupiter and Saturn. We investigate the core growth and the possibilities for coresto reach such a critical core mass. At the late stage, planetary cores grow through collisions withsmall planetesimals. Collisional fragmentation of planetesimals, which is induced by gravitationalinteraction with planetary cores, reduces the amount of planetesimals surrounding them, andthus the final core masses. Starting from small planetesimals that the fragmentation rapidlyremoves, less massive cores are formed. However, planetary cores acquire atmospheres thatenlarge their collisional cross section before rapid gas accretion. Once planetary cores exceedabout Mars mass, atmospheres significantly accelerate the growth of cores. We show that, takinginto account the effects of fragmentation and atmosphere, initially large planetesimals enableformation of sufficiently massive cores. On the other hand, because the growth of cores is slowfor large planetesimals, a massive disk is necessary for cores to grow enough within a disk lifetime.If the disk with 100 km-sized initial planetesimals is 10 times as massive as the minimum masssolar nebula, planetary cores can exceed 10 Earth masses in the Jovian planet region ( > Subject headings: planet and satellites:formation
1. Introduction
Gas giant planets such as Jupiter and Sat-urn form in gaseous disks. In the core-accretionmodel, the accretion of planetesimals producescores of giant planets. Once a core reaches acritical mass ∼
10 Earth masses, it can rapidlyaccrete gas to form a gas giant planet (Mizuno1980; Bodenheimer & Pollack 1986; Ikoma et al.2000). Gas giants must form within the lifetimeof gaseous disks ( .
10 Myr).For km-sized or larger planetesimals, gravita-tional focusing enhances their collisional cross sec-tions, resulting in a high collision probability forlow relative velocities. Relative velocities of largebodies are kept lower than those of small ones dueto dynamical friction. A combination of gravita- tional focusing and dynamical friction brings rapidgrowth of large bodies, which is referred to as run-away growth (Wetherill & Stewart 1989). Eventu-ally, the runaway growth generates a small popu-lation of large bodies called planetary embryos.Planetary embryos keep their orbital separationsand hence grow through collisions with surround-ing remnant planetesimals more slowly than in therunaway mode (Kokubo & Ida 1998). This regimeis called oligarchic growth.Kobayashi et al. (2010) pointed out that theoligarchic growth halts due to fragmentation ofplanetesimals. In the oligarchic growth, the rel-ative velocities of planetesimals are controlled bythe viscous stirring of embryos and gas drag. Asembryos grow, the velocities of remnant planetes-imals are increased so greatly that collisions be-1ween planetesimals become destructive. Suchcollisions eject numerous fragments, which col-lide with each other to produce further smallerbodies. Planetesimals are therefore ground downthrough such successive collisions (collision cas-cade). The random velocities of small bodies arestrongly damped by gas drag and thereby the colli-sional cascade no longer occurs for fragments withradii . . .
100 mand damps the relative velocities to halt collisioncascade at 1–10 m as mentioned above. Therefore,only a small amount of coupled bodies are pro-duced and hence they hardly contribute to embryogrowth (Kobayashi et al. 2010).Many authors have investigated embryo growthwith N -body, statistical, and hybrid simula-tions (Kokubo & Ida 1996, 1998, 2000, 2002;Inaba et al. 1999, 2001, 2003; Weidenschilling et al.1997; Weidenschilling 2005, 2008; Kenyon & Bromley2004, 2008; Chambers 2006, 2008; Kobayashi et al.2010). Although providing most accurate dynam-ical results, N -body simulations have difficultyin producing numerous fragments and followingtheir fate. The fragmentation effect on embryogrowth has thus not been treated in detail in spiteof its importance. Recently, Levison et al. (2010)included fragment production in their N -body simulation. However, it is still difficult to treatfragment–fragment collisions. Such successive col-lisions are essential in the collision cascade (e.g.,Kobayashi & Tanaka 2010). Therefore, statisti-cal simulations are a better method to accuratelyinvestigate planet formation with fragmentation.In the statistical simulation, the collisionalmass evolution of bodies is calculated within a“particle-in-a-box” approximation. Bodies havehorizontal and vertical components of random ve-locity relative to a circular orbit that are deter-mined by their eccentricities and inclinations, re-spectively. These velocities are changed by gravi-tational interactions between the bodies and henceaffected by their mass spectrum, while the colli-sion rates between the bodies depend on the ve-locities. Therefore, the coupled mass and velocityevolution needs to be solved (Wetherill & Stewart1993). While the statistical method has ad-vantages, its weak point is the inability totrack the individual positions of planetesimals.However, progress in planetary dynamic theory(Greenzweig & Lissauer 1992; Ida & Nakazawa1989; Ohtsuki 1999; Stewart & Ida 2000; Ohtsuki et al.2002) has helped to overcome this problem.For example, Greenzweig & Lissauer (1992) andIda & Nakazawa (1989) provided detailed ex-pressions for the probability of collisions be-tween planetesimals orbiting a central star, whileStewart & Ida (2000) and Ohtsuki et al. (2002)derived improved equations for calculating theevolution of random planetesimal velocities causedby gravitational interactions. Finally, it hasbeen shown that the recently developed statis-tical codes can describe some aspects of the plan-etary accumulation processes with the same ac-curacy as N -body simulations (Inaba et al. 2001;Kobayashi et al. 2010).Since the timescale of collision cascade stronglyaffects the final mass of planetary embryos (Kobayashi et al.2010), fragmentation outcome models are es-sential for embryo growth. Collisional frag-mentation includes several uncertain parame-ters. Kobayashi & Tanaka (2010) constructeda simple fragmentation model which is consis-tent with laboratory experiments (Fujiwara et al.1977; Takagi et al. 1984; Holsapple 1993) andhydrodynamical simulations (Benz & Asphaug1999) and analytically clarified which parame-ters are essential. They found that the mass de-2letion due to collision cascades is sensitive tothe total ejecta mass yielded by a single colli-sion, while it is almost independent of the massof the largest ejecta fragment and the size dis-tribution of ejecta over a realistic parameter re-gion. Furthermore, fragmenting collisions are sub-divided into two types, catastrophic disruptionand cratering (erosive collision). Although somestudies neglected or underestimated the effect ofcratering (Dohnanyi 1969; Williams & Wetherill1994; Wetherill & Stewart 1993; Inaba et al. 2003;Bottke et al. 2005), Kobayashi & Tanaka showedthat cratering collisions make a dominant contri-bution to the collision cascade.A planetary embryo larger than ∼ − Earthmasses acquires a tenuous atmosphere of gas fromthe disk. Fragments are captured by the atmo-sphere even if they do not collide directly withthe embryo, implying that the collisional crosssection of the embryo is enhanced. This ef-fect advances the growth of Mars-mass or largerembryos (Inaba & Ikoma 2003). Embryos withthe atmospheres accrete fragments prior to theirdrift inward and can acquire more than 10 Earthmasses starting from 10km-sized planetesimals(Inaba et al. 2003). However, erosive collisionsand initial planetesimal sizes strongly affect finalembryo masses (Kobayashi et al. 2010).This paper investigates the embryo growth tak-ing into account erosive collisions and embryo’satmosphere. Although growing embryos may fallinto a central star due to the type I migration (e.g.,Tanaka et al. 2002), we neglect the migration here.We perform both analytical studies and statisticsimulations, which extend those of Kobayashi etal. (2010) by including atmospheric enhancementof embryo growth. The goal is to find out what de-termines embryo growth and whether an embryocan reach the critical core mass. We introducethe theoretical model in Section 2 and derive finalembryo masses taking into account atmosphere inSection 3. In Section 4, we check solutions for fi-nal masses against the statistical simulations. Sec-tions 5 and 6 contain a discussion and a summaryof our findings.
2. THEORETICAL MODEL2.1. Disk Model
We introduce a power-law disk model for theinitial surface mass density of solids Σ s , and gasΣ g , such thatΣ s , = f ice Σ (cid:16) a (cid:17) − q g cm − , (1)Σ gas , = f gas Σ (cid:16) a (cid:17) − q g cm − , (2)where a is a distance from a central star, Σ isthe reference surface density at 1 AU, and q is thepower-law index of the radial distribution. Thegas-dust ratio f gas = 240 (Hayashi 1981). The fac-tor f ice that represents the increase of solid densityby ice condensation beyond the snow line a ice isgiven by f ice = 1 ( a < a ice ) and 4.2 ( a ≥ a ice ). Inthe MMSN model, Σ = 7 . − and q = 3 / a ice = 2 . (cid:18) L ∗ L ⊙ (cid:19) / AU , (3)where L ∗ and L ⊙ are the luminosities of the cen-tral star and the sun, respectively. In reality, disksmay be optically thick even after planetesimal for-mation. However, we assume Equation (3) for sim-plicity. Kobayashi & Tanaka (2010) showed erosive col-lisions to dominate the collision cascade. Weshould take into account such collisions prop-erly. We assume that fragmentation outcomes arescaled by the impact energy and hence the totalejecta mass m e produced by a single collision be-tween m and m is given by a function of the di-mensionless impact energy φ = m m v / m + m ) Q ∗ D , where v is the collisional velocity be-tween m and m and Q ∗ D is the specific en-ergy needed for m e = ( m + m ) /
2. FollowingKobayashi & Tanaka (2010) and Kobayashi et al.(2010), we model m e m + m = φ φ . (4)Inaba et al. (2003) derived m e from the frag-ment model developed by Wetherill & Stewart31993) with a value of Q ∗ D found by Benz & Asphaug(1999) for ice. Fig. 1 shows their model and Equa-tion (4). As discussed in Kobayashi & Tanaka(2010), most of the laboratory experiments andthe hydrodynamic numerical simulations of colli-sional disruption showed m e not to have a discon-tinuity at φ = 1 (Housen et al. 1991; Takagi et al.1984; Benz & Asphaug 1999). Therefore, Equa-tion (4) includes erosive collisions ( φ <
1) moreaccurately. φ m e / ( m + m ) Fig. 1.— The total ejecta mass m e produced bya single collision with m and m , as a functionof the dimensionless energy φ = m m v / m + m ) Q ∗ D . The solid line indicates Equation (4).For reference, the dotted lines are shown for thefragment model of Inaba et al. (2003) with m =10 m = 4 . × g.The critical energy Q ∗ D is given by Q ∗ D = Q r ! β s + Q ρ p r ! β g + C gg Gmr , (5)where r and m are the radius and mass of a body, ρ p is its density, and G is the gravitational con-stant. The first term on the right-hand side ofEquation (5) is dominant for r . –10 cm,the second term describes Q ∗ D of r . cm,and the third term controls Q ∗ D for the largerbodies. Benz & Asphaug (1999) performed thehydrodynamical simulations of collisional disper-sion for r = 1–10 cm and provided the values of Q , β s , Q , and β g . For r & cm, Q ∗ D is purely determined by the gravitational binding energy,being independent of material properties. Thecollisional simulation for gravitational aggregatesyields C gg ∼
10 (Stewart & Leinhardt 2009).
Once a planetary embryo has grown larger thanthe Moon, it acquires an atmosphere. It helps theaccretion of planetesimals or fragments onto anembryo; small bodies are captured by the atmo-sphere of the embryo.Inaba & Ikoma (2003) provided an analyticalmodel for a density profile of the atmosphere. Weconsider the atmosphere at a distance R e froman embryo center. We assume that R e is muchsmaller than that at the outer boundary of the at-mosphere and that its temperature is much higherthan that at the boundary. The atmospheric den-sity ρ a is then proportional to R − (Mizuno 1980;Stevenson 1982). Applying the temperature T neb ,pressure P neb , and density ρ neb of the nebula in thedisk midplane as those at the outer boundary ofatmosphere, the density profile of the atmospherearound an embryo with mass M is given by ρ a ( R e ) ρ neb = 16 πσ SB GM T κL e P neb (cid:18) GM ρ neb P neb R e (cid:19) , (6)where κ is the opacity of the atmosphere and σ SB is the Stephan-Boltzmann constant. The plane-tary luminosity L e mainly comes from the accre-tion of bodies. We approximate L e = GMR dMdt , (7)where R is the embryo radius. To validate the as-sumption of ρ a ∝ R − , we will apply the completemodel by Inaba & Ikoma (2003) to our statisticsimulation and compare our analytical solutionswith the statistical simulations in Section 4.When a body passes by a planetary embryowith an atmosphere, the embryo can accrete thebody without direct collision due to the atmo-sphere. The relative velocity between the bodyand the embryo at infinity is determined by theeccentricity e of the small body; it is given by ev k with the Keplerian velocity v k = p GM ∗ /a and M ∗ being the mass of a central star. The rela-tive velocity is typically smaller than the surface4scape velocity of the embryo during the embryogrowth. If the orbital energy of a body is suffi-ciently reduced by the atmospheric gas drag, thebody is captured by the embryo. The maximumradius r of bodies captured at distance R e is givenby (Inaba & Ikoma 2003) r = 9 ah M e ρ a ( R e ) ρ p , (8)where h M = ( M/ M ∗ ) / is the reduced Hillradius of the embryo and ˜ e = e/h M . Equa-tion (8) is derived under the two-body approxima-tion. Tanigawa & Ohtsuki (2010) confirmed thatEquation (8) is valid in the case where the three-body effects are included.Equation (8) means that R e is the effective col-lisional radius of an embryo for bodies with radius r . The enhanced radius of the embryo with atmo-sphere is thus derived from Eqs. (6)–(8) as R e R = F M / m / ˙ M / , (9)where F = (cid:20) π aσ SB T ρ G (˜ e + 6)(3 M ∗ ) / κP (cid:21) / . (10)The enhancement factor R e /R given by Equa-tion (9) is shown in Fig. 2, where the power-law density profile given by Equation (6) iscompared with a more realistic profile given byInaba & Ikoma (2003). As we discuss later, plan-etary embryos mainly grow through collisions withplanetesimals of the initial size or with fragmentsof radius r ∼
10 m. The enhancement factor cal-culated with Equation (9) reproduces well themore realistic one for km-sized or larger planetesi-mals, but Equation (9) significantly overestimates R e /R for fragments. However, since the accre-tion rate due to collision with such fragments hasa weak dependence on the enhancement factor( ∝ ( R e /R ) / ; see Equation (30)), this discrep-ancy produces insignificant errors.
3. FINAL EMBRYO MASS3.1. Isolation Mass
Planetary embryos can grow until they haveaccreted all planetesimals within their feeding r [cm] R e / R κ = . c m / g κ = c m / g f = . f = f = 0.0001 Fig. 2.— The ratio of the enhanced radius of plan-etary embryo to its physical radius with M = M ⊕ , ρ p = 1g cm − , and ˙ M = 1 × − M ⊕ / yr for˜ e = 4 in the MMSN disk around the star withmass M ⊙ . The ratios are calculated by the formu-lae of Inaba & Ikoma (2003) for the opacity ob-tained from Equation (39) with the grain deple-tion factor f = 10 − –1 (solid lines) and by Equa-tion (9) for the constant opacity κ = 0 .
01 cm g − and 1 cm g − (dotted lines).zones. The width of a feeding zone is givenby the orbital separation of neighboring em-bryos, ˜ b (2 M/ M ∗ ) / a , where ˜ b ≃
10 is theseparation measured in their mutual Hill radii(Kokubo & Ida 2000, 2002). The maximum massor “isolation mass” is M iso = 2 πa (2 M iso / M ∗ ) / ˜ b Σ s , .It can be expressed as M iso = 2 . ˜ b ! / Σ s , . − ! / × a ! M ∗ M ⊙ ! − / M ⊕ , (11)where M ⊕ is the Earth mass and M ⊙ is thesolar mass. The planetary embryo mass ap-proaches the isolation mass if fragmentation isignored (Kokubo & Ida 2000, 2002). However, iffragmentation is included, the embryo mass canreach only about Mars mass for a MMSN disk(Kobayashi et al. 2010).5 .2. Planetesimal Accretion As shown by Kobayashi et al. (2010), a plan-etary embryo accretes planetesimals with massescomparable to original ones or fragments resultingfrom collisional grinding of planetesimals. In theformer case, a final embryo mass is determined bythe equilibrium between the accretion of planetes-imals and their removal due to collisional grind-ing. In the latter case, an embryo can grow un-til fragments are depleted by the gas drag. Fol-lowing Kobayashi et al. (2010), we want here toderive final masses determined by the accretionof planetesimals in the case with atmospheric en-hancement, while we treat the fragment accretionin Section 3.3.At the oligarchic stage, embryos mainly growthrough collisions with planetesimals that domi-nate the surface density. The growth rate of anembryo with mass M is given by dMdt = C acc Σ s a h M h P col i Ω k , (12)where Ω K is the Keplerian frequency and C acc isthe correction factor on the order of unity. Thedimensionless collision rate h P col i is formulated asa function of the eccentricities e and inclinations i of bodies accreted onto the embryo. We assume e = 2 i in this analysis.Embryos have a constant ratio of their sepa-rations to their Hill radii (Kokubo & Ida 1998).When the ratio decreases as embryos grow, rela-tively smaller embryos are culled and thereby re-maining embryos keep the ratio constant. Sup-posing the cull occurs instantaneously, the growthrate of embryos due to the cull is estimated to be ahalf of that from planetesimals. Therefore, we set C acc = 1 . e are controlled by the em-bryo stirring and gas drag. The stirring rateis written as de /dt = n M a h M h P VS i Ω k , where n M is the surface number density of embryos andthe dimensionless stirring rate h P VS i is given by h P VS i = C VS h M ln(Λ + 1) /e with C VS = 40 andΛ = 5˜ e /
96 for e ≫ h M (Ohtsuki et al. 2002).Although ln(Λ + 1) in h P VS i is weakly dependenton e , we adopt, in this analysis, ln(Λ + 1) ≃ e = 3–10. The gas-drag time τ is characterized as (Adachi et al. 1976) τ = 2 mπr C D ρ neb v k , (13)where the dimensionless gas drag coefficient C D =0 . τ is the stopping time due to gasdrag only when the relative velocity u betweengas and a body is equal to the Keplerian ve-locity; hence τ is almost always much shorterthan the stopping time for realistic relative ve-locities. The e -damping rate due to gas drag isgiven by de /dt = − C gas e /τ with C gas = 2 . n M = (2 πaδa ) − withthe orbital separation of neighboring embryos of δa = 2 / h M a ˜ b (Kokubo & Ida 2000) and equat-ing the stirring and damping rates result in theequilibrium eccentricity: ˜ e = " C VS ln(Λ + 1)Ω k τ / π ˜ bC gas / . (14)Since we roughly estimate ˜ e ∼ ( τ Ω k ) / fromEquation (14), the eccentricities of the kilometer-sized and larger bodies are larger than h M , ac-cording to the assumption e ≫ h M .Taking into account the enhancement dueto the atmosphere, the dimensionless colli-sional probability for e ≫ h M is given by(Greenzweig & Lissauer 1992; Inaba et al. 2001;Inaba & Ikoma 2003) h P col i = C col ˜ R ˜ e R e R , (15)where C col = 36 and ˜ R = R/ah M = (9 M ∗ / πρ p ) / /a .Inserting Eqs. (9) and (15) to Equation (12), weobtain ˙ M as dMdt = A ca M / Σ / , (16)where A ca = " C acc a C col ˜ RF Ω k (3 M ∗ ) / ˜ e m / / . (17) Ida & Makino (1993) and Thommes et al. (2003) pre-sented a similar equation from the stirring timescale de-rived by Ida & Makino (1993). We apply the formula ofOhtsuki et al. (2002), which weakly depends on e throughln(Λ + 1). However, since we adopt a constant value forln(Λ +1) in this analysis, there is no substantial differencebetween their and our treatment, except for the definitionof the coefficient for the viscous stirring.
6s embryos grow, destructive collisions be-tween planetesimals are induced by the stirringof embryos and generate a lot of small fragments,which produce further small bodies through mu-tual collisions. Since very small bodies re-sulting from successive collisions are rapidly re-moved by the gas drag, the collision cascade re-duces the surface density of solids. In the colli-sion cascade, collisional fragmentation dominatesthe mass flux along the mass coordinate. Sincethe mass flux is independent of mass in a steadystate, the mass distribution of fragments follows apower law and the power-law exponent α is givenby α = (11 + 3 p ) / (6 + 3 p ) for e /Q ∗ D ∝ m − p (Kobayashi & Tanaka 2010). The steady-statemass flux determines the surface density reductionas (Kobayashi & Tanaka 2010; Kobayashi et al.2010) d Σ s dt = − B ca Σ M α − / . (18) B ca = (2 − α ) Ω k s ( α ) m / × (cid:18) ˜ e v M ∗ ) / Q ∗ D (cid:19) α − , (19)where s ( α ) = Z ∞ (cid:20) φ − b − φ ln ǫφ (1 + φ ) + ln(1 + φ ) (cid:21) × φ − α φ dφ, (20)and h = 1 . ρ − / . For the derivation of Equa-tion (18), we apply the fragmentation outcomemodel of Kobayashi & Tanaka (2010); ejectayielded by a single collision between m and m are characterised by their total mass m e and theirpower-law mass spectrum with an exponent b be-low the mass m L = ǫ ( m + m ) φ/ (1 + φ ) , where ǫ < s reduction rate is in-sensitive to ǫ and b (Kobayashi & Tanaka 2010).We set b = 5 / ǫ = 0 . M and the surface density Σ s :64 α − h M (4 α − / − M (4 α − / i = 4(Σ − / − Σ − / , ) A ca B ca , (21) where M is the initial embryo mass. Note thatthe derivation of Equation (21) assumed that theplanetesimal density reduction is caused by col-lisional grinding, but the planetesimal accretiononto embryos significantly contributes to the Σ s -reduction when the surface density of planetesi-mals, Σ s , is much smaller than that of embryos, M n M . When an embryo reaches a final mass M ca , Σ s may be described as C Σ s M ca n M with aconstant C Σ s ≪
1; henceΣ s Σ s , = C Σ s (cid:18) M ca M iso (cid:19) / . (22)For C Σ s ∼ .
1, a final mass is consistent with sim-ulations (Kobayashi et al. 2010). We thus set C Σ s = 0 . M ca = " α − A ca C − / s Σ − / , M / B ca / α − . (23)Here, we assume M ca ≫ M .For kilometer-sized or larger planetesimals, Q ∗ D = Q ρ p r β g with constants Q and β g . Weapply Q = 2 . g − and β g = 1 .
19 for ice(Benz & Asphaug 1999) and ˜ e ≫
6, and Equa-tion (23) can then be re-written as M ca = 1 . × − (cid:16) a (cid:17) . (cid:18) m × g (cid:19) . × (cid:18) Q . g − (cid:19) . (cid:18) κ .
01 g cm − (cid:19) − . × (cid:18) f gas Σ . × g cm − (cid:19) . M ⊕ . (24)Since planetesimals grow before planetesimals’fragmentation starts, planetesimal mass m isslightly larger than initial planetesimal mass m .Kobayashi et al. (2010) showed that planetesimalsmainly accreting onto embryos have m = 100 m .For m & g ( r & × km), final embryomasses exceed 10 M ⊕ at 5 AU in a MMSN disk,but embryos cannot reach it within a disk life-time due to their slow growth. The final mass M ca is independent of Σ s , , while high Σ g , in-creases M ca because gas drag highly damps ˜ e . ForΣ = 71 g cm − (10 × MMSN), initial planetesi-mals with r &
50 km can produce an embryowith 10 M ⊕ at 5 AU.7or comparison, we also show the final mass M c in the same situation but neglecting the atmo-sphere (Kobayashi et al. 2010): M c = 0 . (cid:16) a (cid:17) . (cid:18) m × g (cid:19) . × (cid:18) ln(Σ s , / Σ s )4 . (cid:19) . (cid:18) Q . g − (cid:19) . × (cid:18) f gas Σ . × g cm − (cid:19) . M ⊕ , (25)where ln(Σ s , / Σ s ) ≃ . C Σ s = 0 . M = 0 . M ⊕ in theMMSN model. The collisional enhancement dueto the atmosphere is inefficient for m = 4 × g; M ca < M a . If m & × g, the atmospherecontributes to embryo growth. As described above, planetesimals are grounddown by collision cascade and resulting small frag-ments spiral into the central star by gas drag.In the steady state of collision cascade, the sur-face density of planetesimals is much larger thanthat of fragments. However, when the grinding ofplanetesimals is much quicker than the removal ofsmall fragments by gas drag, fragments accumu-late at the low-mass end of collision cascade anddetermine the total mass of bodies. Embryos thengrow through the accretion of such fragments.The specific impact energy between equal-sizedbodies, e v /
8, should be much smaller than Q ∗ D at the low-mass end; thus the typical fragments atthe low-mass end have e v = C L Q ∗ D , (26)where C L ∼ C L = 1 to determine the typi-cal fragment mass, we apply C L = 0 . e . Suchsmall fragments feel strong gas drag in Stokesregime; C D = 5 . cl g /ur , where c is the soundvelocity and l g = l g , /ρ g is the mean free pathof gas molecules with l g , = 1 . × − g cm − (Adachi et al. 1976). The eccentricities of frag-ments at the low-mass end are much smaller than h M and η , where η = ( v k − v gas ) /v k is the de-viation of the gas rotation velocity v gas from the Keplerian velocity. The dimensionless viscous stir-ring rate is given by h P VS i = h P VS , low i = 73 for e ≪ h M (Ohtsuki et al. 2002) and the dampingrate is expressed as de /dt = − ηe /τ for e ≪ η (Adachi et al. 1976). The equilibrium eccentricitybetween stirring by embryos and damping by gasdrag is obtained as (Kobayashi et al. 2010) e = h M h P vs , low i τ Ω K / π ˜ bη , (27)Using Eqs. (13), (26), and (27) under the Stokesregime, we have the fragment mass m f at the low-mass end of collision cascade: m f = m f0 M − / , (28)where m f0 = " M ∗ ˜ bC L Q ∗ D h P VS , low i a Ω cl g , (cid:18) πρ p (cid:19) / / . (29)For e ≪ h M , Ida & Nakazawa (1989) foundthat the dimensionless collision rate for e ≪ h M is given by h P col , low i = 11 . p ˜ R , where the coeffi-cient is determined by Inaba et al. (2001). Sincethe atmosphere effectively enhances an embryo ra-dius for the accretion of bodies, the collision rateis modified to be (Inaba & Ikoma 2003) h P col i = h P col , low i r R e R . (30)We obtain the accretion rate of fragments by anembryo, ˙ M , from Eqs. (12) and (30) as dMdt = A fa M / Σ / , (31) A fa = " F / h P col , low i C acc a Ω k m / (3 M ∗ ) / / . (32)Fragments with m f at the low-mass end ofcollision cascade that dominate the surface den-sity of solids Σ s are no longer disrupted by col-lisions and drift inward by gas drag. The driftvelocity is given by 2 η a/τ and then the Σ s -reduction rate due to the radial drift is expressedas d Σ s /dt = − / − q ) η Σ s /τ with the assump-tion of Σ s ∝ a − q . Since τ of fragments with m f isdetermined by Equations (26) and (27), we have8Kobayashi et al. 2010) d Σ s dt = − B fa Σ s M, (33) B fa = (cid:18) − q (cid:19) h P vs , low i Ω k ηv / πM ∗ C L ˜ bQ ∗ D . (34)Since fragments are later produced by embryogrowth in an outer disk, the radial distributiondepends on time in contrast to the assumption ofΣ s ∝ a − q . Nevertheless, the effect is negligible forembryo growth unless the atmosphere is consid-ered (Kobayashi et al. 2010). We discuss this ef-fect with the atmospheric enhancement in § § M fa from ˙ M and ˙Σ s for fragment accretion, similar tothe case of planetesimal accretion. Integration ofEquation (31) divided by Equation (18) results in M fa = (cid:18) A fa B fa (cid:19) / Σ / , , (35)where we assume M fa ≫ M and Σ s , ≫ Σ s . For q = 3 /
2, we have M fa = 0 . (cid:16) a (cid:17) / (cid:18) κ .
01 g cm − (cid:19) / × (cid:18) f ice Σ
30 g cm − (cid:19) / × (cid:18) Q ∗ D . × erg g − (cid:19) / M ⊕ . (36)Here, we adopted f ice and Σ for the minimummass solar nebula model. The weak dependenceof M fa on κ implies that the overestimate of R e /R due to the power-law radial profile is insignificant,as discussed in Section 2.3.For the case without an atmosphere, Kobayashi et al.(2010) derived a final mass for the fragment ac-cretion, M f = 0 . (cid:16) a (cid:17) / (cid:18) f ice Σ
30 g cm − (cid:19) / × (cid:18) Q ∗ D . × erg g − (cid:19) / M ⊕ . (37)Eqs. (36) and (37) imply that the final massesincrease due to the atmosphere, but the enhance-ment is insignificant; M fa /M f ≃ . × MMSN. If we neglect the collisional enhancement due toatmosphere, the final mass M na is determined bythe larger of M c and M f (Kobayashi et al. 2010).In the case with atmosphere, a final mass M a isalso given by the larger of M ca and M fa . The finalmass M a is shown in Figs. 3–5. For the initialplanetesimal radius r = 10 km, M a is dominatedby M fa inside the point where the line of M a bendsin Fig. 3 and by M ca outside. The final mass M a is determined only by M fa for r = 1 km (Fig. 4)and by M ca for r = 100 km (Fig. 5) in the rangeof interest. M na M a M iso M na M a M iso M na M a M iso Distance [AU] E m b r yo M a ss [ M ] Fig. 3.— Embryo masses with (circles) andwithout (squares) atmosphere after 10 years for m = 4 . × g ( r = 10 km), as a func-tion of distance form the central star. We setΣ = 71 g cm − (top), Σ = 21 g cm − (middle),and Σ = 7 . − (bottom). Solid lines in-dicate M a which is the larger of M ca and M fa for κ = 0 . g − . Dotted lines represent M na which is the larger of M c and M f . Thin lines show M iso .9 M na M a M iso M na M a M iso M na M a M iso Distance [AU] E m b r yo M a ss [ M ] Fig. 4.— Same as Fig. 3, but for m = 4 . × g(radii of 1 km).
4. NUMERICAL SIMULATION
Regarding the method of numerical simulation,we basically follow Kobayashi et al. (2010). Themethod of Kobayashi et al. (2010) is briefly ex-plained here. In the calculation, a disk is dividedinto concentric annuli and each annulus containsa set of mass batches. We set the mass ratiobetween the adjacent batches to 1.2, which canreproduce the collisional growth of bodies result-ing from N -body simulation without fragmenta-tion (Kobayashi et al. 2010) and the analytical so-lution of mass depletion due to collisional grinding(Kobayashi & Tanaka 2010). The mass and veloc-ity evolution of bodies and their radial transportare calculated as follows.- The mass distribution of bodies evolvesthrough their mutual collisions that pro-duce mergers and fragments. The total massof fragments ejected by a single collision is M na M a M iso M na M a M iso M na M a M iso Distance [AU] E m b r yo M a ss [ M ] Fig. 5.— Same as Fig. 3, but for m = 4 . × g(radii of 100 km).given by Equation (4) and the remnant be-comes a merger. The collision rates betweenthe bodies are calculated from the formulaeof Inaba et al. (2001).- The random velocities given by e and i of thebodies simultaneously evolve through theirmutual gravitational interactions, gas drag,and collisional damping. The formulae ofOhtsuki et al. (2002) are applied to describethe changing rates of e and i . The gas-drag damping rates of e and i are describedas functions of e , i , η , and τ according toInaba et al. (2001). To determine τ , we takeinto account Stokes and Epstein drag as wellas a drag law with a quadratic dependenceon velocity. For the collisional damping,both fragments and a merger resulting froma single collision have the velocity dispersionat the gravity center of colliding bodies.10 In each annulus there is a loss and gainof bodies due to their inward drift. Thenumber loss rate from an annulus is givenby R ( N ( m ) v drift / ∆ a ) dm , where v drift is thedrift velocity of bodies, N ( m ) dm is the num-ber of bodies with mass ranging from m to m + dm in the annulus, and ∆ a is thewidth of the annulus. The bodies lost fromeach annulus are added to the next innerannulus. The drift velocity is given by(Kobayashi et al. 2010) v drift = 2 aητ ˜ τ τ (cid:20) (2 E + K ) π e + 4 π i + η (cid:21) / , (38)where E = 2 . K = 1 .
211 and the dimen-sionless stopping time ˜ τ stop = Ω k τ / ( e + i + η )is adopted.In this paper, we add a collisional enhance-ment due to the atmosphere. Although the sim-ple power-law radial density profile of the atmo-sphere (Equation (6)) is used for the derivation offinal masses ( M ca , M fa ), the simulation incorpo-rates a more realistic profile provided by the for-mulae of Inaba & Ikoma (2003). The opacity ofthe embryo’s atmosphere in their model is givenby κ = κ gas + f κ gr , where κ gas is the gas opacity, κ gr is the opacity of grains having an interstellarsize distribution, and f is the grain depletion fac-tor. Following Inaba & Ikoma, we adopt κ = .
01 + 4 f cm g − for T ≤
170 K , .
01 + 2 f cm g − for 170 K < T ≤ , .
01 cm g − for T > . (39)The enhancement factor R e /R due to the atmo-sphere is shown in Fig. 2.We perform the simulations for embryo for-mation starting from a monodisperse mass pop-ulation of planetesimals of mass m and ra-dius r with e = 2 i = (2 m /M ∗ ) / and ρ p =1 g cm − around the central star of mass M ⊙ with a set of eight concentric annuli at 3.2,4.5, 6.4, 9.0, 13, 18, 25, and 35 AU contain-ing Σ gas and Σ s for q = 3 /
2. To compute Q ∗ D , we use Equation (5) with Q = 7 . × erg g − , β s = − . Q = 2 . g − , β g = 1 .
19, and C gg = 9 (Benz & Asphaug 1999;Stewart & Leinhardt 2009). We artificially ap-ply the gas surface density evolution in the form Σ gas = Σ gas , exp( − t/T gas , dep ), where T gas , dep is the gas depletion timescale, which we setto T gas , dep = 10 years. Assuming a constantΣ gas gives almost the same results for final em-bryo masses, because we consider time spans t ≤ T gas , dep .Fig. 6 shows the embryo-mass evolution at6.4 AU for f = 0 .
01. Runaway growth initially oc-curs; embryo mass exponentially grows with timeduring the stage. The runaway-growth timescaleis proportional to r / Σ s , (Ormel et al. 2010a,b).When the embryo masses exceed 0.001-0.01 M ⊕ ,oligarchic growth starts. Since massive embryosdynamically excite planetesimals, the reductionof planetesimals due to collisional fragmentationstalls the embryo growth (Kobayashi et al. 2010).For Σ = 7 . − (MMSN), the fragmenta-tion limits the final mass to about Mars mass ( ∼ . M ⊕ ) and the atmosphere is insignificant. Onceembryo masses exceed the Mars mass, atmospheresubstantially accelerates the embryo growth. ForΣ ≥
21 g cm − (3 × MMSN), the atmosphere leadsto further embryo growth. Nevertheless, embryosfinally attain asymptotic masses.Results for these simulations are summarised inFig. 3, where the embryo masses after 10 years arecompared to analytical formulae for final embryomasses. Embryo masses finally reach M a inside5 AU (Σ = 7 . − ), 10 AU (Σ = 21 g cm − ),and 20 AU (Σ = 71 g cm − ). However, embryosexceed M a inside 5 AU for Σ = 71 g cm − . Thisexcess comes from the embryo growth through col-lisional accretion with bodies drifting from out-side, which effect we did not consider in the anal-ysis described in Section 3. To confirm the contri-bution from drifting bodies, we show the surfacedensity evolution in Fig. 7. For Σ = 71 g cm − ,the surface density of solids increases after 2 × years. Since the drift timescale shortens in-ward, bodies from outside cannot raise the surfacedensity unless embryos accrete them. Therefore,the increase in the surface density implies that em-bryo grows through the accretion of such bodies.The initial mass m of planetesimals in the sim-ulations depends on their formation process, whichis not well understood yet. We perform the em-bryo growth starting from different m (Figs. 4and 5). Small planetesimals are relatively easilyfragmented due to low Q ∗ D and quickly grounddown to the low-mass end of collision cascade.11 .00010.0010.010.11100.00010.0010.010.111010 Time [yr] E m b r y o M a ss [ M ] Fig. 6.— Evolution of embryo mass at 6.4 AUwith m = 4 . × g ( r = 10 km) forΣ = 71 g cm − (10 × MMSN; top), 21 g cm − (3 × MMSN; middle), and 7 . − (MMSN; bot-tom). Solid lines show the case with atmosphereand dotted lines represent the result without at-mosphere.The resulting fragments with low e actively ac-crete onto embryos. For m = 4 . × g( r = 1 km), embryos can reach a final mass M a in a relatively wide region inside 10 AU (MMSN),20 AU (3 × MMSN), and 30 AU (10 × MMSN). Onthe other hand, large initial planetesimals de-lay the runaway growth of embryos (Ormel et al.2010a,b) and the following oligarchic growth isalso slower than that for small planetesimals be-cause embryos mainly accrete original planetes-imals rather than fragments with low e . For r = 100 km, embryos attain the final massesonly inside 4 AU for 3 × MMSN and inside 6 AUfor 10 × MMSN, and embryos cannot reach finalmasses beyond 2.7 AU in the MMSN disk. In ad-dition, small bodies drifting from outside are ef-fectively captured by embryos and thereby em- Σ = 71 g/cm Σ = 21 g/cm Σ = 7.1 g/cm Time [yr] Σ s [ g / c m ] Fig. 7.— The solid surface density evolution at3.2 AU.bryos exceed final masses M a inside 4 AU for10 × MMSN.In the case without an atmosphere, initiallylarger planetesimals can form massive embryos.Since large planetesimals delay embryo growth,embryos made from 100 km-sized initial planetesi-mals can reach 10 M ⊕ but the location is only in-side 3–4 AU even for 10 × MMSN (Kobayashi et al.2010). The case with the atmosphere shows asimilar dependence of the final embryo masseson initial planetesimal mass. However, since theatmosphere accelerates embryo growth, embryoslarger than 10 M ⊕ are produced inside 8–9 AU ofa 10 × MMSN disk with 100 km-sized initial plan-etesimals.While the final masses of embryos exceed 10 M ⊕ for large initial planetesimals of r &
100 km, em-bryos must reach the critical core mass within thedisk lifetime T gas , dep to form gas giant planets.The growth timescale is estimated to be M/ ˙ M ,where ˙ M is given by Equation (16). The crit-ical distance a c inside which embryos can reach10 M ⊕ is approximately obtained from the condi-tion M/ ˙ M < T gas , dep with M = 10 M ⊕ , a c = 9 . (cid:18) T dep years (cid:19) / (cid:18) Σ
71 g cm − (cid:19) / × (cid:16) r
100 km (cid:17) − / AU , (40)12here we adopt m = 100 m and q = 3 / r = 100 km, the massive disk with Σ &
70 g cm − can form such large embryos around10 AU. In addition, we estimate a c ∼ × MMSN disk with r = 10 km. Indeed, the simulation with Σ =71 g cm − and r = 100 km shows embryos cannotreach 10 M ⊕ beyond 5 AU (see Fig. 8). Therefore,the condition of 10 × MMSN with r ∼
100 km isnecessary to form gas giants around 10 AU. M na M iso Distance [AU] E m b r yo M a ss [ M ] Fig. 8.— Same as Fig. 3, but for Σ = 71 g cm − with m = 4 . × g ( r = 1000 km). The finalmass M a with atmosphere is estimated to be largerthan 200 M ⊕ .We also give a constraint on f . For f . .
01, afinal mass is almost independent of f (see Fig. 9).This is because the gas opacity dominates over thegrain opacity (see Equation (39)). For f = 1, em-bryos at 3–4 AU become larger due to the captureof bodies drifting from outside, while final embryomasses in the outer disk are similar to the casewithout atmosphere. The condition of f . . f is acceptable;the depletion factor f should be much smallerthan unity after planetesimal formation. In ad-dition, a low-opacity atmosphere reduces the crit-ical core mass (Mizuno 1980; Ikoma et al. 2000;Hori & Ikoma 2010). M na M a M iso M na M a M iso M na M a M iso Distance [AU] E m b r yo M a ss [ M ] f = 1 f = 0.01 f = 0.0001 Fig. 9.— The final embryo masses for f = 0 . × MMSN with m = 4 . × g ( r = 100 km).Lines and symbols are the same as in Fig. 3, butwe apply κ = 1cm g − to derive M a for f = 1.
5. DISCUSSION
We derived final embryo masses analyticallyand numerically. They agree with each other quitewell in the inner disk where the embryo forma-tion timescale is shorter than the nebula lifetime( ∼ years). The analytical formula for finalmasses M a implies that initial planetesimal radiishould be larger than about 3 × km to form em-bryos with 10 M ⊕ at 5 AU in a MMSN disk. How-ever, the critical distance a c inside which embryosreach 10 M ⊕ within 10 years (Equation (40)) isestimated to be much smaller than 5 AU; a massivedisk is likely to form gas giant planets. Embryosinside 5 AU of a ∼ × MMSN disk exceed finalembryo masses M a due to the accretion of smallbodies drifting from outside. In spite of such fur-ther growth, embryos starting from small planetes-13mals cannot reach the critical core mass ∼ M ⊕ .In addition, further growth is insignificant beyond5 AU. The formulae for M a and a c suggest thatinitial planetesimals with r ≃ M ⊕ at 5 AU in the10 × MMSN disk.Inaba et al. (2003) performed similar simula-tions incorporating collisional fragmentation andenhancement due to the embryo’s atmosphereand showed a planetary core with
M > M ⊕ could be produced around 5 AU with m =4 . × g ( r = 10 km) for 10 × MMSN. Inour simulation, embryos cannot reach 10 M ⊕ un-der this condition and larger planetesimals arenecessary to form such massive embryos beyond5 AU. As Kobayashi & Tanaka (2010) discussed,Williams & Wetherill (1994) underestimated thetotal ejecta mass produced by a single collision forcratering; Inaba et al. adopted the fragmentationmodel similar to theirs that Wetherill & Stewart(1993) developed (see Fig. 1). Erosive collisionsshorten the depletion time of 10km-sized plan-etesimals in collision cascade by a factor of 4–5 (Kobayashi & Tanaka 2010) and hence reducefinal embryo masses. As seen from Eqs. (24)and (36), final embryo masses M ca , M fa in-crease with Q ∗ D ; the results of Inaba et al. cor-respond to embryo masses for higher Q ∗ D . Al-though we and Inaba et al. applied Q ∗ D pro-vided by Benz & Asphaug (1999), porous bodieswith r .
10 km may have much lower Q ∗ D (e.g.,Stewart & Leinhardt 2009; Machii & Nakamura2011). For initial planetesimals with radii &
100 km, Q ∗ D of slightly larger bodies determinesfinal embryo masses and is almost entirely deter-mined by the gravitational binding energy; theuncertainty from their structure would be minor.Therefore, such large planetesimals are possible toproduce cores for gas giant planets.The mechanisms of planetesimal formation arehighly debated but, despite intensive effort, re-main fairly unknown. The formation through col-lisional coagulation in which dust smoothly growsto planetesimals with r ∼ M ca is large enough to start core ac-cretion, while embryo growth is slow. If the ra-dial slope of surface density q = 3 / × MMSNis necessary for embryos to reach the final massaround 10 AU. However, observations of proto-planetary disks infer their relatively flatter ra-dial distributions over several hundred AU (e.g.,Kitamura et al. 2002). In such a disk, dust grainsaccumulate in an inner disk due to radial drift dur-ing their growth, which increases the solid surfacedensity in the inner disk (Brauer et al. 2008). Theenhancement of solid surface density acceleratesembryo growth and hence embryos may achievethe critical core mass in less massive disks.To form gas giants via core accretion, rapidgas accretion onto a core with ∼
10 Earth massesmust occur prior to gas depletion. However, thesecores migrate inward due to their exchange of an-gular momentum with the surrounding gas (TypeI). From linear analysis, the characteristic orbitaldecay time of Earth-mass cores at several AU inthe MMSN model is about 1 Myr (Tanaka et al.2002). Several processes to delay the timescale ofType I migration have been pointed out, for exam-ple, disk surface density transitions (Masset et al.2006b), intrinsic turbulence (Nelson & Papaloizou2004), and hydrodynamic feedback (Masset et al.2006a). There is still uncertainty about this es-timate of the migration time. Indeed, the dis-tribution consistent with observations of exoplan-ets can be reproduced only if the timescale of thetype I migration is at least an order of magnitudelonger than that derived from the linear analysis(Ida & Lin 2008). We should also investigate the14trength of such migration for the survival of coresof gas giant planets in our future work.
6. SUMMARY
In this paper, we investigate the growth of plan-etary embryos by taking into account, among oth-ers, two effects that are of major importance. Oneof them is collisional fragmentation of planetesi-mals, which is induced by their gravitational inter-action with planetary cores. Another effect is anenhancement of collisional cross section of a grow-ing embryo by a tenuous atmosphere of nebulargas, which becomes substantial when an embryohas reached about a Mars mass.The main results are summarized as follows.1. If the atmosphere is not taken into account,collisional fragmentation suppresses plane-tary embryo growth substantially. As a re-sult, embryos cannot reach the critical coremass of ∼ M ⊕ needed to trigger rapidgas accretion to form gas giants. The fi-nal masses are about Mars mass in a MMSNdisk (Kobayashi et al. 2010). Embryo’s at-mosphere accelerates the embryo growth andmay increase the final embryo mass by up toa factor of ten.2. Planetary embryos attain their final massesasymptotically. We have derived the finalmass analytically. The final mass of an em-bryo is predicted to be the larger of M ca and M fa , which are given by Eqs. (24) and (36),respectively. These final masses are in goodagreement with the results of statistical sim-ulations.3. Our solution indicates that an initial plan-etesimal radius r & × km is necessaryto form a planetary core with 10 M ⊕ at 5 AUin a MMSN disk. However, such initiallylarge planetesimals delay embryo growth; amassive disk is required to produce massivecores within a disk lifetime. The analyticalsolution for the final mass and the embryoformation time show that planetesimals withan initial radius of r ≃ × MMSN. 4. The embryo growth depends on the diskmass, initial planetesimal sizes, and theopacity of atmosphere. We have performedstatistical simulations to calculate the finalembryo masses over a broad range of param-eters. We took the surface density of solidsat 1 AU in the range of Σ = 7 . − (1–10 × MMSN), initial planetesimal radius r = 1–1000 km, and the grain depletionfactor f in planetary atmosphere between f = 10 − –1. We found that planetary em-bryos can exceed 10 M ⊕ within 8-9 AU for10 × MMSN, r = 100 km, and f ≤ . M ⊕ for r = 10 kmonly inside 4 AU. Therefore, we concludethat a massive disk ( ∼ × MMSN) with r ∼
100 km and f . .
01 is necessary toform gas giant planets around 5–10 AU. Thiscondition for large embryo formation is in-dependent of the material strength and/orstructure of bodies, because Q ∗ D of 100km-sized or larger bodies is largely determinedby their self-gravity.We thank Chris Ormel for helpful discussionsand the reviewer, John Chambers, for useful com-ments on the manuscript. REFERENCES
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