Prompt planetesimal formation beyond the snow line
aa r X i v : . [ a s t r o - ph . E P ] A ug Draft version October 15, 2018
Preprint typeset using L A TEX style AASTeX6 v. 1.0
PROMPT PLANETESIMAL FORMATION BEYOND THE SNOW LINE
Philip J. Armitage , Josh A. Eisner , and Jacob B. Simon JILA, University of Colorado and NIST, 440 UCB, Boulder, CO 80309-0440 Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder Steward Observatory, University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85721-0065 Department of Space Studies, Southwest Research Institute, Boulder, CO 80302 Sagan Fellow
ABSTRACTWe develop a simple model to predict the radial distribution of planetesimal formation. The modelis based on the observed growth of dust to mm-sized particles, which drift radially, pile-up, and formplanetesimals where the stopping time and dust-to-gas ratio intersect the allowed region for streaminginstability-induced gravitational collapse. Using an approximate analytic treatment, we first show thatdrifting particles define a track in metallicity–stopping time space whose only substantial dependenceis on the disk’s angular momentum transport efficiency. Prompt planetesimal formation is feasiblefor high particle accretion rates (relative to the gas, ˙ M p / ˙ M & × − for α = 10 − ), that couldonly be sustained for a limited period of time. If it is possible, it would lead to the deposition ofa broad and massive belt of planetesimals with a sharp outer edge. Including turbulent diffusionand vapor condensation processes numerically, we find that a modest enhancement of solids near thesnow line occurs for cm-sized particles, but that this is largely immaterial for planetesimal formation.We note that radial drift couples planetesimal formation across radii in the disk, and suggest thatconsiderations of planetesimal formation favor a model in which the initial deposition of material forgiant planet cores occurs well beyond the snow line. Keywords: accretion, accretion disks — planets and satellites: formation — protoplanetary disks —instabilities INTRODUCTIONWhere and when planetesimals form within protoplan-etary disks set the initial conditions for the gravity-dominated phase of planet formation. Meteoritic ev-idence is consistent with planetesimals forming early,with Schiller et al. (2015) suggesting that differentiatedbodies formed only 0.25 Myr after calcium-aluminum-rich inclusions, but little is known empirically about theradial profile beyond the general observation that theSolar System, and debris disks, formed planetesimals ata range of radii. One can try to back out the initialdistribution of planetesimals from the observed archi-tecture of planetary systems (as in the Minimum MassSolar Nebula, Hayashi 1981), but this is an ill-posedproblem if planets migrate (Kley & Nelson 2012). Givenrecent advances in the characterization of gas and dustin protoplanetary disks, a forward-modeling approachthat combines observations with planetesimal formationtheory may prove at least as instructive. [email protected]
Here, we develop a simplified global model for whereplanetesimals form based on known physical processes.Experiments and related modeling suggest that dustgrows rapidly up to ∼ mm sizes, defined by the onsetof bouncing (Blum & Wurm 2008; Zsom et al. 2010).We assume that this remains true beyond the snowline, although the physics of ice coagulation can differsubstantially from that of silicates (Dominik & Tielens1997; Gundlach & Blum 2015), in a direction thatallows larger particles to form (Wada et al. 2011;Kataoka et al. 2013). The small macroscopic solidsthat result from coagulation are then subject to ra-dial drift (Weidenschilling 1977) and turbulent diffusion(Clarke & Pringle 1988), which are sufficiently rapid asto lead to an approximate steady-state on small scales.As the ice-dominated solids cross the snow line theyevaporate, and the vapor diffuses outward and recon-denses forming an enhancement of the solid surfacedensity (Stevenson & Lunine 1988). At radii wherethe dust-to-gas ratio and dimensionless stopping timefall within certain ranges (determined by Carrera et al.2015), the streaming instability (Youdin & Goodman2005) results in the rapid formation of planetesimals.Planetesimals formed from the streaming instabilityare characteristically large (Johansen et al. 2007, 2012;Simon et al. 2016), and to a good approximation theywill be immune to aerodynamic drift and stay in place.The above sketch defines a model for “prompt” plan-etesimal formation that does not invoke helpful but lesswell-understood processes (such as “lucky growth” be-yond material barriers, and local concentration in zonalflows, vortices or persistent particle traps, for reviewssee Johansen et al. 2014; Armitage 2015). It is obvi-ously incomplete. Our intent is to highlight in the sim-plest possible model the critical role of the competitionbetween radial drift and planetesimal formation. A MODEL FOR PLANETESIMAL FORMATIONFROM DRIFTING SOLIDSConsider a gas disk around a star of mass M ∗ that is parameterized by the accretion rate ˙ M , an-gular momentum transport efficiency parameter α (Shakura & Sunyaev 1973), and central temperatureprofile T ( r ) ∝ r − β , with β a constant. In steady-state, ν Σ = ˙ M π , (1)where the effective viscosity is given in terms of thesound speed c s and angular velocity Ω by ν = αc s / Ω,and c s = k B T /µm H . Here k B is the Boltzmann con-stant, µ ≃ . m H is the mass of a hydrogen atom. The central density ofthe disk is ρ = Σ / √ πh , where Σ is the surface densityand h = c s / Ω is the vertical scale height.A constant α model provides a convenient link be-tween ˙ M and Σ using only a single free parameter,though it is unlikely to provide a good representationof protoplanetary disks. Simulations show that the ef-fective α due to magnetohydrodynamic processes varieswith radius (Simon et al. 2015), and that the accre-tion stress is largely divorced from the strength of tur-bulent diffusion. (In what follows we implicitly as-sume that turbulent diffusion is weak, allowing parti-cles to settle.) For particle flows, however, what mat-ters most are the profiles of surface density, tempera-ture, and (particularly) pressure. Our fiducial modelhas T = 150( r/ − . K and Σ ∝ r − . The normal-ized pressure gradient parameter,∆ ≡ − (cid:18) c s v K (cid:19) d ln P d ln r , (2)which determines in part the strength of the streaminginstability (Bai & Stone 2010), has a value (∆ ≃ . Figure 1 . Analytic tracks (dashed lines) in Z - τ s space fora disk with α = 10 − , β = 1 /
2, and various ratios of theparticle to gas accretion rate. Particles evolve from lowerright to upper left under radial drift. The red curve definesthe approximate region within which prompt planetesimalformation is possible. The green (for mm-sized particles)and blue (cm-sized) curves show tracks that include a sinkterm for planetesimal formation, for K = 10 (solid curves), K = 10 and K = 10 (dashed curves). The disk model has˙ M = 10 − M ⊙ yr − and T = 150( r/ − . K. macroscopic particles, that have grown by coagulationoutside the snow line, fall in the Epstein drag regime.For particles of material density ρ m and radius s , thedimensionless stopping time is, τ s = π ρ m Σ s. (3)The aerodynamic drift rate is (Takeuchi & Lin 2002), v r = τ − s v r , gas − ηv K τ s + τ − s , (4)where v r , gas is the radial velocity of the gas, v K is theKeplerian velocity, and η = − ( h/r ) ( β/ −
3) is a pa-rameter measuring the degree of radial pressure supportfor the gas disk. We take the limit where τ s is smalland | v r , gas | ≪ | ητ s v K | . For a steady-state radial particleflow, with accretion rate ˙ M p and surface density Σ p ,Σ p Σ = 3 α − β/
2) ˙ M p ˙ M τ − s . (5)This expression shows how the solid-phase “metallicity” Z ≡ Σ p / Σ depends upon the disk properties and ratioof solid to gas accretion rates, for given τ s .Writing the familiar physics of radial drift in the aboveform is useful because the conditions for the streaminginstability to lead to gravitational collapse and plan-etesimals can also be expressed as an allowed region in Z - τ s space. A polynomial fit to the simulation results ofCarrera et al. (2015) gives the critical metallicity as ,log Z crit = 2 . × − (log τ s ) + 0 . τ s ) +0 . τ s ) + 0 . τ s ) − . . (6)Equation (5) then defines a line in that space that ei-ther does or does not admit planetesimal formation.Whether planetesimal formation is possible depends on α and the particle flux but not on the actual physi-cal size to which solids grow. Plotting these tracksin Figure 1 for α = 10 − (at the high end of obser-vationally estimated values, Hartmann et al. 1998), wesee that if ˙ M p / ˙ M = 10 − the metallicity is too low atany τ s to allow prompt planetesimal formation. Plan-etesimal formation is only possible in the outer disk if˙ M p / ˙ M & × − , i.e. for relative accretion rates thatexceed the fiducial dust-to-gas ratio of 10 − . This im-plies that planetesimal formation would occur while theglobal average of the dust-to-gas ratio was decreasingwith time, presumably early on.A steady-state model will only be valid, even approx-imately, if particles can grow to a size set by mate-rial barriers before they drift significantly (otherwisethey will be in the regime of “drift-limited growth”,Birnstiel et al. 2012). Birnstiel et al. (2012) estimatethat a particle grows from radius s to s on a time scale, t grow ≈
1Ω ΣΣ p ln (cid:18) ss (cid:19) . (7)Adopting s = µ m, s = mm, and β = 0 .
5, we find thatthe ratio between the growth time and the drift time t drift = r/ | v r | is, t grow t drift ≈ (cid:16) α − (cid:17) − (cid:18) h/r . (cid:19) ˙ M / ˙ M p ! τ s . (8)For these parameters, the track given by equation (5)intersects the allowed streaming region at τ s ≃ . τ s growth may be slower than drift.To model the impact of planetesimal formation on thetrack in Z - τ s space, we integrate the continuity equationfor the particles, ∂ Σ p ∂t + 1 r ∂∂r ( r Σ p v r ) = − Σ p t form , (9)to find a steady-state solution. We assume that plan-etesimals form on a multiple K of the dynamical time Carrera et al. (2015) model disks with negligibly weak intrin-sic turbulence. Significant levels of turbulence would impede set-tling and planetesimal formation, especially at low τ s . Our resultsare thus most applicable to disks in which the accretion stress isprimarily laminar, with low levels of turbulence and attendantdiffusion. Figure 2 . The radial projection of the models shown in Fig-ure 1. Upper panel: the radial dependence of the particlemass accretion rate. Lower panel: the radial dependence ofplanetesimal formation. Solid curves refer to models with˙ M p / ˙ M = 1, dashed curves models with ˙ M p / ˙ M = 0 .
1. Theassumed planetesimal formation time for the mm-sized par-ticles cases increases from right to left ( K = 10 , , ).For cm-sized particles only the case with K = 10 is plotted. scale within the allowed region, t form = K Ω − . For Z < Z crit we smoothly suppress the rate by a fac-tor exp[( Z crit − Z ) / . Z crit ]. This roll-off in the plan-etesimal formation rate is imposed for numerical con-venience, though it is physically plausible that someplanetesimal formation persists at metallicities just be-low the nominal threshold. We consider icy particles( ρ m = 1 g cm − ) in a disk with ˙ M = 10 − M ⊙ yr − , α = 10 − , and T = 150( r/ − . K.Figure 1 shows solutions to this model. The key pointis that, within the allowed region, the time scale forprompt planetesimal formation scales with radius as r − / , which can be compared to the time scale forradial drift t drift = const. There is therefore a criti-cal radius within which planetesimal formation domi-nates, whereas outside radial drift leads to particle pile-up (Youdin & Chiang 2004). If the streaming instabilityleads to gravitational collapse on an essentially dynam-ical time scale ( K ∼ , as is found in simulations, e.g.Simon et al. 2016), then the derived Z - τ s tracks skirt thelower boundary of the allowed region. Slower planetesi-mal formation time scales lead to clearly defined regionswhere first radial drift and then planetesimal formationdominate. We remark that models with large values of K have an alternate interpretation in terms of stochas-tic planetesimal formation. For example, a model with˙ M p / ˙ M = 1 and K = 10 is equivalent to one with˙ M p / ˙ M = 10 − if transient local concentrations attainthe metallicity needed for prompt planetesimal forma-tion (with K = 10 ) 1% of the time.Figure 2 shows the solutions as a function of radius(notionally extended in to 1 AU, though this would beinward of the physical snow line). For ˙ M p / ˙ M & . PARTICLE AND VAPOR DIFFUSIONWe now explore how robust these conclusions areto neglected physical effects. Turbulent particle diffu-sion works against particle pile-up (Hughes & Armitage2012), further favoring large radii as sites for plan-etesimal formation, but the condensation of vapordiffusing across the snow line has the opposite ef-fect (Stevenson & Lunine 1988). We adopt a time-dependent treatment (Alexander & Armitage 2007) andsolve continuity equations that treat solids and vapor astrace species, with surface density Σ t , ∂ Σ t ∂t + 1 r ∂∂r [ r ( F diff + Σ t v r )] = 0 . (10)The diffusive term F diff = − D Σ ∂ (Σ t / Σ) /∂r (Clarke & Pringle 1988), where D is the diffusionco-efficient (here set equal to ν for both particlesand vapor), and v r is the gas radial velocity (for thevapor) or the aerodynamic drift speed (for particles).We supplement these equations with source / sinkterms appropriate for instantaneous sublimation andcondensation of water vapor, following the method andchemical constants given in Ciesla & Cuzzi (2006). Inbrief, at radii where ice is not fully sublimated, wesublimate or condense the appropriate amount of ice orvapor at each time step to maintain the actual vaporpressure of water at the equilibrium value given by theClausius-Clapeyron equation. We assume that there isno significant change to the size distribution of particlesin the vicinity of the snow line , an assumption whichminimizes its importance. In more complete models, at Figure 3 . Example steady-state surface density profiles ofmm-sized (green) and cm-sized (blue) particles, water va-por (dashed lines), and gas, for the disk model with ˙ M =10 − M ⊙ yr − . Outward diffusion and condensation of vaporleads to a modest enhancement of the solid surface densityoutside the snow line for cm-sized (and larger) particles. least some vapor condenses on to pre-existing particles,growing them to larger sizes (Ros & Johansen 2013).Figure 3 shows the resulting steady-state profiles ofvapor and solids, computed in the fiducial disk model.The solid metallicity (which is freely scalable in modelsthat ignore planetesimal formation) is set to match theanalytic results for ˙ M p / ˙ M = 1 at 50 AU. For r & D/ν >
1. Because the radial drift velocityexceeds the gas velocity, the abundance of water vaporinterior to the snow line is enhanced as long as largemasses of small solids remain present in the disk.Adopting the same “scale-free” model for local plan-etesimal formation losses as in §
2, we show in Figure 4 anillustrative example of how the results shown in Figure 1are modified by turbulent diffusion and vapor conden-sation. As before, the gas disk has ˙ M = 10 − M ⊙ yr − and T = 150( r/ − . K, which places the snowline at radii similar to those inferred for the Solar Sys-tem (Morbidelli et al. 2000). As was already obviousfrom Figure 3, for mm-sized particles the extra physicsin the numerical model makes very little difference tothe predicted radii where planetesimals could form. Forcm-sized particles there is a more significant deviation
Figure 4 . Steady-state tracks for mm-sized (green curves)and cm-sized (blue) particles in Z - τ s space, including theeffects of turbulent particle diffusion and vapor diffusion /condensation. We assume ˙ M p / ˙ M = 1, K = 10 , and thefiducial disk model. The simpler models shown in Figure 1are plotted as the dashed lines. from the analytic model results. The higher surface den-sity caused by the condensation of diffusing water vaporresults in a secondary peak of planetesimal formationjust outside the snow line, but does not alter the con-clusion that the most-favored location lies further out.This result would be reinforced at higher gas accretionrates, arguably more appropriate to an early phase ofdisk evolution, which would boost the threshold size forvapor condensation effects to matter. DISCUSSIONAt radii beyond the water snow line there is a limitedwindow to prompt planetesimal formation that invokesonly known physical processes: coagulation to a fixedsize that is of the order of mm, radial drift, and grav-itational collapse of streaming-initiated over-densities.Planetesimal formation via this route is possible early on— while radial drift is rapidly reducing the global dust-to-gas ratio — and would typically lead to the depositionof a broad and massive belt of planetesimals well outsidethe snow line. We have not attempted fine tuning of themodel, but Figure 2 makes it clear that features suchas the outer edge to the Kuiper Belt (Trujillo & Brown2001) and the large masses of primordial debris requiredin the Nice model (Tsiganis et al. 2005), are qualita-tively consistent with planetesimal formation expecta-tions, as noted by Youdin & Shu (2002).Efficient early planetesimal formation is a prerequisitefor early planet formation, which has been proposed asan explanation for the ring-like structures seen in ALMA observations of HL Tau (ALMA Partnership et al.2015). There is tension, however, between other obser-vations and any model that invokes radial drift and pile-up as key ingredients of planetesimal formation. Suchmodels predict a rapid loss of small solids via radial driftand planetesimal formation, whereas observations of COline emission toward mostly older sources favor dust-to-gas ratios in excess of 10 − (Williams & Best 2014;Eisner et al. 2016). A better understanding of gas diskmass estimates, which for now differ substantially de-pending on the technique used (e.g. Bergin et al. 2013;Manara et al. 2016), is the single most important ad-vance that would constrain theoretical models of radialdrift.Our model is deliberately simple, and is intended toprovide insight into the physical origin and robustnessof more complex approaches. It could be improved bybetter delineating the conditions needed for planetes-imal formation (e.g. by including the dependence onthe radial pressure gradient, Bai & Stone 2010), and bymodeling the coupled growth and drift of particles. Sev-eral authors have already developed such coupled mod-els. Dr¸a˙zkowska & Dullemond (2014), using a MonteCarlo dust coagulation code, argued for preferential trig-gering of the streaming instability beyond the ice line.Similarly, Krijt et al. (2016), using a detailed modelfor particle growth coupled to estimates of planetesi-mal formation thresholds similar to ours, found thatdust-rich disks with weak turbulence promote promptouter disk planetesimal formation. In the inner disk,there is general agreement that planetesimal formationis harder, and a prompt route is only possible if par-ticles grow to substantially larger sizes (limited onlyby fragmentation and radial drift, rather than bounc-ing, Drazkowska et al. 2016). Alternatively, planetesi-mal formation in the terrestrial planet region may oc-cur via a less-efficient stochastic channel (driven by lo-cal turbulent enhancements in Z , Johansen et al. 2014),and at still smaller radii at persistent traps associatedwith the inner edge of the dead zone (Lyra et al. 2009).Despite their differences all models agree that attainingthe conditions needed for planetesimal formation is rel-atively hard, and that as a result the initial distributionof planetesimals is unlikely to be a simple power-law butrather a complex function of radius.Building the cores of the giant planets requires, first,the deposition of a large mass of solids into planetesimals(Pollack et al. 1996). The preference of the streaminginstability for relatively large stopping times challengesthe common view that the favored location for this de-position is adjacent to the snow line. Even when theeffects of vapor diffusion and condensation are consid-ered, we find that the best time and place to lay downa large mass of planetesimals is early and at large radii.This suggests a model for core formation in which plan-etesimal formation and initial growth occurs well beyondthe snow line. Subsequent growth could either occur insitu via pebble accretion (Lambrechts & Johansen 2014;Levison et al. 2015), or via migration to a trap closer in.We thank the referee for a detailed and instructivereport. PJA thanks Cathie Clarke and the Instituteof Astronomy, Cambridge, for hospitality, and acknowl- edges support from NASA through grants NNX13AI58Gand NNX16AB42G, and from NSF AAG grant AST1313021. JAE acknowledges support from NSF AAGgrant 1211329. JBS’s support was provided in part un-der contract with the California Institute of Technol-ogy (Caltech) and the Jet Propulsion Laboratory (JPL)funded by NASA through the Sagan Fellowship Programexecuted by the NASA Exoplanet Science Institute.REFERENCES, Johansen et al. 2014),and at still smaller radii at persistent traps associatedwith the inner edge of the dead zone (Lyra et al. 2009).Despite their differences all models agree that attainingthe conditions needed for planetesimal formation is rel-atively hard, and that as a result the initial distributionof planetesimals is unlikely to be a simple power-law butrather a complex function of radius.Building the cores of the giant planets requires, first,the deposition of a large mass of solids into planetesimals(Pollack et al. 1996). The preference of the streaminginstability for relatively large stopping times challengesthe common view that the favored location for this de-position is adjacent to the snow line. Even when theeffects of vapor diffusion and condensation are consid-ered, we find that the best time and place to lay downa large mass of planetesimals is early and at large radii.This suggests a model for core formation in which plan-etesimal formation and initial growth occurs well beyondthe snow line. Subsequent growth could either occur insitu via pebble accretion (Lambrechts & Johansen 2014;Levison et al. 2015), or via migration to a trap closer in.We thank the referee for a detailed and instructivereport. PJA thanks Cathie Clarke and the Instituteof Astronomy, Cambridge, for hospitality, and acknowl- edges support from NASA through grants NNX13AI58Gand NNX16AB42G, and from NSF AAG grant AST1313021. JAE acknowledges support from NSF AAGgrant 1211329. JBS’s support was provided in part un-der contract with the California Institute of Technol-ogy (Caltech) and the Jet Propulsion Laboratory (JPL)funded by NASA through the Sagan Fellowship Programexecuted by the NASA Exoplanet Science Institute.REFERENCES