A quantification of hydrodynamical effects on protoplanetary dust growth
AAstronomy & Astrophysics manuscript no. sellentinetal_v8.1 c (cid:13)
ESO 2018September 15, 2018
A quantification of hydrodynamical effects onprotoplanetary dust growth
E. Sellentin, J. P. Ramsey, F. Windmark (cid:63) , C. P. Dullemond
Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Überle-Str. 2, D-69120 Heidelberg,Germany e-mail: [email protected]
September 15, 2018
ABSTRACT
Context.
The growth process of dust particles in protoplanetary disks can be modeled via numerical dust coagulation codes. In thisapproach, physical e ff ects that dominate the dust growth process often must be implemented in a parameterized form. Due to a lackof these parameterizations, existing studies of dust coagulation have ignored the e ff ects a hydrodynamical gas flow can have on graingrowth, even though it is often argued that the flow could significantly contribute either positively or negatively to the growth process. Aims.
We intend to qualitatively describe the factors a ff ecting small particle sweep-up under hydrodynamical e ff ects, followed bya quantification of these e ff ects on the growth of dust particles, such that they can be parameterized and implemented in a dustcoagulation code. Methods.
Using a simple model for the flow, we numerically integrate the trajectories of small dust particles in disk gas around aproto-planetesimal, sampling a large parameter space in proto-planetesimal radii, headwind velocities, and dust stopping times.
Results.
The gas flow deflects most particles away from the proto-planetesimal, such that its e ff ective collisional cross section, andtherefore the mass accretion rate, is reduced. The gas flow however also reduces the impact velocity of small dust particles ontoa proto-planetesimal. This can be beneficial for its growth, since large impact velocities are known to lead to erosion. We alsodemonstrate why such a gas flow does not return collisional debris to the surface of a proto-planetesimal. Conclusions.
We predict that a laminar hydrodynamical flow around a proto-planetesimal will have a significant e ff ect on its growth.However, we cannot easily predict which result, the reduction of the impact velocity or the sweep-up cross section, will be moreimportant. Therefore, we provide parameterizations ready for implementation into a dust coagulation code. Key words. accretion, accretion disks – protoplanetary disks – stars: circumstellar matter – planets and satellites: formation
In the classical incremental growth scenario, planetesimals areformed by the coagulation of dust across several orders of mag-nitude in mass. During this growth phase, many di ff erent phys-ical processes are important, but not all are beneficial. Indeed,several barriers have been found that stop dust particles fromgrowing to sizes where gravity can aid in coagulation. For exam-ple, the charge barrier (Okuzumi et al. 2009), the radial drift andfragmentation barriers (Nakagawa et al. 1986; Weidenschillinget al. 1997; Brauer et al. 2008; Birnstiel et al. 2010), or thebouncing barrier (Zsom et al. 2010; Windmark et al. 2012b).In order to overcome these barriers, it is important to exam-ine what e ff ects neglected physical processes might have. Onesuch e ff ect is the hydrodynamical flow of disk gas around largerdust particles such as pebbles or small planetesimals, to whichwe will collectively refer to as proto-planetesimals. In this paper,we focus on the question of whether such a flow pattern is bene-ficial for the growth of protoplanetary dust. Although this ques-tion has already been partially addressed in Sekiya & Takeda(2003, hereafter ST03), a quantification of these e ff ects over alarge parameter space such that the results can be implementedin a numerical dust growth code still lacks in the literature.Whether colliding dust particles stick to each other and grow,bounce o ff each other, or disrupt one or both of the collision part-ners depends on a series of parameters, out of which the impactvelocity is one of the most important. Experimental work has (cid:63) Member of IMPRS for Astronomy & Cosmic Physics at the Uni-versity of Heidelberg shown (for a summary, see Blum & Wurm 2008; Güttler et al.2010) that particles preferentially stick to each other if they col-lide at low velocities (Wurm & Blum 1998), and bounce o ff ordisrupt at high collision velocities (e.g. Wurm et al. 2005; Kotheet al. 2013). It was also found that larger dust aggregates areeroded due to high velocity impacts of dust monomers, a pro-cess that has become known as monomeric erosion (Schräpler& Blum 2011). Many of these experimental findings are in-cluded in the numerical simulations that model dust growth inprotoplanetary disks (e.g. Weidenschilling 1997; Ormel & Cuzzi2007; Brauer et al. 2008; Birnstiel et al. 2010; Zsom et al. 2010;Okuzumi et al. 2012). Numerical studies also show that thegrowth of dust does not always proceed by a hierarchical coagu-lation of equal sized particles, but instead that collisions betweenparticles of large size ratios can lead to the formation of planetes-imals (Xie et al. 2010; Windmark et al. 2012a). In this scenario,proto-planetesimals can grow by sweeping up a secondary pop-ulation of particles kept small by the collisional growth barriers.To date, collision velocities in dust growth codes have beencalculated from Brownian motion, radial and azimuthal drift,vertical settling, and turbulence induced motions of the dust par-ticles. The collisional cross section of the particles is further-more assumed to be equal to the geometrical cross section. Thisneglects potential e ff ects of hydrodynamical gas flow past dustparticles on the growth process. It is however expected that thesee ff ects can be quite important for collisions between particles ofa large size ratio, i.e., in a sweep-up growth scenario. Geomet- Article number, page 1 of 9 a r X i v : . [ a s t r o - ph . E P ] N ov ical cross sections are only applicable if the trajectories of thecolliding particles follow straight lines. This is not necessar-ily the case for a small dust particle passing near a large proto-planetesimal surrounded by a hydrodynamical flow pattern: Thedrag of the gas flow around the proto-planetesimal can deflect thesmall dust particle, resulting in a curved trajectory. In the mostextreme case, this deflection can prevent a collision, preventingthe sweep-up of the dust particle. The collisional cross sectionmust therefore be modified to take into account this deflection.The drag force of the gas flow on a small particle can also af-fect its speed. This then modifies the collision velocity betweenthe dust particle and proto-planetesimal, which is decisive fordetermining whether a collision leads to sticking, bouncing, orerosion.In protoplanetary disks, we expect hydrodynamical flow pat-terns to form around dust particles much larger than the meanfree path of the gas because the gas orbits at sub-Keplerian speedwhile the dust orbits at Keplerian speed. This gives rise to a rel-ative motion between disk gas and dust, appearing in the restframe of the dust as a headwind. In this work, we aim to usenumerical simulations to study the e ff ect that such a flow patternwill have on the dust coagulation.We assume that our proto-planetesimals are small enoughso that gravity is negligible, and constraints on this assump-tion are presented below. For studies which include the grav-ity of larger planetesimals, see, for e.g., Lambrechts & Johansen(2012); Morbidelli & Nesvorny (2012); Ormel (2013).This paper is organized as follows: In Sect. 1, we present ourassumptions and general properties of the flow pattern around aproto-planetesimal. In Sect. 2, we describe the changes to thecollisional cross section and the impact velocity which we ob-serve from integrating trajectories of small dust particles in theflow around a proto-planetesimal. In Sect. 3, we turn to the ques-tion of whether gas flow around proto-planetesimals can returndebris from a disruptive collision to the proto-planetesimal sur-face. In Sect. 4, we summarize our results.
1. Validity range of the model
In the limit of high viscosity (i.e. small Reynolds numbers), thesteady-state ( ∂/∂ t =
0) velocity field v g of a gas flow around asphere of radius R in spherical polar coordinates can be derivedfrom the Navier-Stokes equations (Greiner & Stock 1991; origi-nally due to Stokes): v g ( r s , θ ) = v ∞ ( θ ) (cid:32) R r + R r s − (cid:33) + R r ( v ∞ ( θ ) · r s ) r s (cid:32) − R r (cid:33) , (1)where v ∞ ( θ ) = v ∞ (cid:16) cos θ ˆ r − sin θ ˆ θ (cid:17) is the upstream velocity, v ∞ is the headwind velocity and constant, and r s denotes the radialdistance measured from the center of the sphere. As r s → ∞ , v g reduces to − v ∞ . Eq. (1) describes the flow around a sphericalproto-planetesimal if a) the proto-planetesimal radius R is muchlarger than the mean free path λ in the disk, b) the flow is in-compressible, c) laminar and unseparated, d) the gravity of theproto-planetesimal is negligible, and e) the proto-planetesimaldoes not rotate.In the following, we assume (e) to be true and calculatewhen (a) – (d) are valid for a protoplanetary disk in both theminimum mass solar nebula model (MMSN) (Weidenschilling1977b; Hayashi et al. 1985) and the model of Desch (2007). Al-though the two models provide similar constraints, we quote theresults for both. Assumption a)
Hydrodynamics is suitable to describe themacroscopic properties of a flow around a proto-planetesimal if R (cid:38) λ . The mean free path λ is related to the number density n of gas molecules by λ = / ( n σ g ), where σ g is the collisionalcross section of the gas molecules. For n , we use the numberdensity n at the mid-plane of a protoplanetary disk, which isrelated to the surface density Σ ( r ) by: n ( r ) = Σ ( r ) √ π h µ m p , (2)with the disk scale height h = c s / Ω K , and average molecularmass µ m p , where m p is the mass of a proton, c s = (cid:112) k B T ( r ) /µ m p is the isothermal sound speed, k B is the Boltzmann constant, and Ω K is the Keplerian angular frequency.For the disk model, we assume µ = . σ g = · − m (ST03) for the collisional cross section of Hydrogen, and a stellarmass of 1 M (cid:12) . We adopt the disk temperature profile of Alexan-der et al. (2004), which is consistent with observations: T ( r ) =
100 ( r / − / K , (3)where r is the heliocentric distance, expressed in astronomicalunits. The gas surface densities for the Desch (2007) and MMSN(Hayashi et al. 1985) models are, respectively: Σ Desch ( r ) = · ( r / − . kg m − ; (4) Σ MMSN ( r ) = . · ( r / − / kg m − . (5)Substituting the surface densities into Eq. (2), the minimal radius R min = λ that a proto-planetesimal must have for assumption(a) to hold is then: R min , Desch ( r ) = · − ( r / . m; (6) R min , MMSN ( r ) = · − ( r / . m , (7)which increases with heliocentric distance for both disk models. Assumption b)
Incompressibility approximately holds true forsmall Mach numbers, M (cid:46) .
1. From the temperature profile(3), the isothermal sound speed can be written as: c s ( r ) =
600 ( r / − / m s − . (8)The headwind velocity v ∞ = v K − v g originates from the sub-Keplerian rotation of the disk gas, and can be written as v ∞ = ηv K with (ST03): η = − r Ω µ m p n ∂ P ∂ r , (9)where P is the gas pressure as given by the ideal gas law.Substituting the surface density profiles Eqs. (4) & (5), andthe temperature profile Eq. (3) into the expression for η , we findthe headwind velocity for the Desch and MMSN models to be v ∞ , Desch ≈
24 m s − ; (10) v ∞ , MMSN ≈
20 m s − , (11)and independent of heliocentric distance.From Eq. (8), the Mach number of a gas flow around a proto-planetesimal is thus on the order of 10 − to 10 − in both modelsfor reasonable values of r , and assumption (b) is valid. Article number, page 2 of 9ellentin et al.: Hydrodynamical e ff ects on dust growth Assumption c)
To determine whether the flow around a proto-planetesimal can be described as laminar, we first estimate themolecular viscosity of gas in a protoplanetary disk, which canbe determined from: ν =
12 ¯ u th λ. (12)Using Eq. (8), the mean thermal speed of the disk gas is given by¯ u th ( r ) = √ /π c s ( r ). The mean free path can be calculated fromEq. (2), and thus we have, ν Desch ( r ) = .
13 ( r / . m s − ; (13) ν MMSN ( r ) = . r / . m s − . (14)Based on experimental results, viscous hydrodynamic flow pasta sphere will remain perfectly laminar and unseparated if theReynolds number of the flow is Re crit (cid:46)
22 (e.g. Taneda 1956),where the Reynolds number of flow around a spherical proto-planetesimal is given by:Re = R v ∞ ν . (15)Other than the proto-planetesimal radius, all variables in Eq. (15)are now fixed by the choice of disk model. The statement that aflow will remain perfectly laminar below Re crit ≈
22 can then beused to solve for the maximal radius of a proto-planetesimal: R crit , Desch = .
059 ( r / . m; (16) R crit , MMSN = .
98 ( r / . m . (17)Laminarity therefore holds if the proto-planetesimal radius doesnot exceed R crit ( r ). Assumption d)
We assume that the gravity of the proto-planetesimal is negligible if the drag force due to the frictionof the fluid, F D , is at least F D > F G , where F G is the gravita-tional force of the proto-planetesimal.The drag force can be written as F D = m v/ t s where m , v ,and t s are a dust particle’s mass, speed, and stopping time. Thelargest stopping time that leads to a deflection in the flow patternis approximately equal to the time t p that a particle needs to beadvected by the gas flow through the perturbed region aroundthe proto-planetesimal. This timescale is thus t p ≈ L /v ∞ , where L is roughly the extent of the perturbed region. At a distance of r s = R , the flow pattern Eq. (1) virtually equals the upstreamvelocity v ∞ . We therefore set the extent of the perturbed regionto L = R and thus F D = m v ∞ / R .We evaluate the gravitational force at the surface of theproto-planetesimal by assuming early proto-planetesimals havea volume filling factor of 0 . · kg m − (Carry 2012). From this, we estimate the density of the proto-planetesimal to be ρ p = . × · kg m − .The statement F D > F G then becomes: m v ∞ R > Gm π R ρ p R , (18)where G is the gravitational constant. The maximally allowedproto-planetesimal radius before the gravitational force on thedust particles becomes comparable to the drag force is then R grav ≈
250 m . (19) Note also that the gravity of a proto-planetesimal with R (cid:46) R grav does not a ff ect the disk gas because the escape velocityfrom its surface is much smaller than the local thermal speed ofthe gas molecules.In assumption (a), we found that the minimally required ra-dius of a proto-planetesimal which induces a hydrodynamicalflow grows with heliocentric distance. There must consequentlybe a distance when the radius exceeds the above specified R grav .Setting R min = R grav in Eqs. (6) & (7), this occurs at r grav , Desch =
14 AU; (20) r grav , MMSN = , (21)for the Desch and MMSN models, respectively. Beyond theseheliocentric distances, a proto-planetesimal cannot fulfill bothprerequisites of being large enough to induce a hydrodynami-cal flow and small enough for gravity to be negligible, and ourmodel is no longer valid.
2. Hydrodynamical effects on collisions betweenproto-planetesimals and dust grains
In order to investigate the change of the e ff ective sweep-up crosssection σ e ff and the impact velocity v imp due to the drag in a hy-drodynamical flow pattern, we calculate the trajectories of dustparticles from the equation of motion,˙ v = − ( v − v g ) t s , (22)where the dot denotes a time derivative, v is the dust particle’svelocity, v g is the velocity of the gas flow (Eq. 1), t s = | p | / F D is the stopping time of the dust particle, and | p | = m | v | is thedust particle momentum. For example, in the Epstein regime,when the particle radius a < λ/
4, the stopping time is givenby t s , Ep = ρ p a /ρ g ¯ u th . This quantity can be regarded as a parti-cle property parameterizing the e ff ects which contribute to thedrag of this particle in a specified medium. Hence, the follow-ing work can be applied to real protoplanetary disks if stoppingtimes of dust particles are known. Theoretical stopping timesfor particles in di ff erent drag regimes are summarized in Wei-denschilling (1977a). Far upstream and downstream of a spherical proto-planetesimal,the streamlines of the chosen flow pattern Eq. (1) are parallel.Meanwhile, near the proto-planetesimal, the stream lines divergeupstream and converge again downstream, creating a flow pat-tern that is rotationally symmetric about the flow axis. If dustparticles approach the proto-planetesimal in this flow patternfrom all directions – which could be possible if they are stirredup by nearby turbulent eddies and acquire randomly orientatedvelocities, or if the headwind velocity is modified by a large-scale turbulent eddy – then they should experience the followingscenarios, all of which are depicted in Fig. 1:arrow a) Dust particles that approach the proto-planetesimalfrom upstream and parallel to the flow axis will be de-flected away from the proto-planetesimal due to thediverging flow pattern. In particular, particles witha large impact parameter that would collide with theproto-planetesimal in the absence of the flow pattern
Article number, page 3 of 9 an be deflected such that they pass over the rim of theproto-planetesimal and miss it. The collisional crosssection of the proto-planetesimal is then reduced rela-tive to its geometrical cross section. Furthermore, themagnitude of the deflection will depend on the dustparticle stopping time: The larger the stopping time,the smaller the deflection. Lastly, due to the rotationalsymmetry of the flow pattern, the collisional cross sec-tion will be circular.arrow b) Particles that approach the proto-planetesimal fromdownstream and parallel to the flow axis do so undera converging gas flow. Even particles that lie outsidethe geometrical cross section of the proto-planetesimalcan then be pushed onto a collision course by the gasflow. A collision, however, only occurs if the particleshave su ffi cient inertia such that the gas flow cannot re-verse their direction of motion (i.e. a large stoppingtime). For particles that approach from downstreamand collide with the proto-planetesimal, the collisionalcross section must then be larger than the geometricalcross section, but will remain circular due to rotationalsymmetry.arrow c) Particles that approach the proto-planetesimal at anangle θ to the flow axis experience the flow patternas a cross wind. Particles not initially on a collisioncourse can be deflected such that they now collidewith the proto-planetesimal. In this scenario, the crosswind introduces an additional dependence on | sin θ | tothe collisional cross section. For a spherical proto-planetesimal, this causes a small stretching of the col-lisional cross section in the upstream direction, result-ing in a top-down symmetric oval. The slight breakingof left-right symmetry occurs because the flow patternalso depends on θ (Eq. 1).arrow d) In contrast, particles that approach the proto-planetesimal under an angle to the flow axis, but thatwere originally on a collision course, can be deflectedsuch that they now miss the proto-planetesimal.To test for the above qualitative features of the dust trajectories,we ran a set of example simulations (see Sect. 2.2), and foundthat all cases occur as described. The reduction of the proto-planetesimal collisional cross section for particles that approachfrom upstream (arrow (a) in Fig. 1) was found to be the mostsignificant, while the enhancement (or distortion) of the crosssection for particles that approach from downstream (or underan angle) is e ff ectively negligible.We expect particles that approach from upstream to be themost frequently occurring situation in a sweep-up scenario, andthus in the following we focus only on the case of upstream par-ticles travelling parallel to the flow axis. In total, we performed 16000 simulations that numerically inte-grate Eq. (22) for the trajectories of dust particles in the flow pat-tern around a proto-planetesimal using a 4 th -order Runge-Kuttabased, backwards Euler method with adaptive stepping. Eachsimulation uses a di ff erent combination of proto-planetesimalradii, headwind velocities, and particle stopping times, con-strained in Sect. 1 and summarized in Table 1.The calculation of the headwind velocity v ∞ in Sect. 1 onlyaccounts for the sub-Keplerian gas velocity as the origin of arelative motion between disk gas and proto-planetesimals. The acb d Fig. 1.
Trajectories of dust particles resulting from their interactionwith the gas flow are indicated by the solid arrows. The gray linesrepresents stream lines of the gas flow around the proto-planetesimal,as calculated from Eq. (1). The feathered arrows indicate the directionof gas flow. See the text for a discussion of scenarios (a) – (d). total headwind velocity should also include, for example, diskturbulence, and for the sake of generality we vary the headwindvelocities between v ∞ ∈ [15 ,
60] m s − (c.f. the headwind veloc-ities calculated in Weidenschilling 1977a).In our simulations of particle trajectories, we assume the par-ticles are initially at rest with respect to the gas, and thus we setthe initial velocity of the dust particles equal to the headwindvelocity. Although dust particles that react significantly to thedisturbed gas flow around the proto-planetesimal consequentlymust have a relatively small stopping time, and are thus essen-tially comoving with the gas, only dust particles with t s = H · R upstream fromthe proto-planetesimal, where the parameter 1 < H ≤
150 is amultiple of the proto-planetesimal radius. The larger the value of H , the closer the gas is to a uniform flow pattern, and the longerthe particles will remain at rest with respect to the gas.To investigate which impact parameters lead to collisions,we vary the initial impact parameter p in steps of 10 − R betweenzero and 0 . R , and then by steps of 0 . R up to the planetesimalradius R . Fig. 2 shows a representative example of trajectoriesfor dust particles with di ff erent impact parameters.While particles with small impact parameters collide withthe proto-planetesimal, beyond a certain value of p = p max ( < R )the particle will be deflected around the proto-planetesimal, andimpacts cease to occur. The maximum impact parameter, p max ,that leads to a collision is related to the e ff ective sweep-up crosssection σ e ff by: σ e ff = π p . (23) Article number, page 4 of 9ellentin et al.: Hydrodynamical e ff ects on dust growth Table 1.
Simulation parameter space. The range of values for each parameter is determined with regards to Sect. 1.
Parameter Definition Range As restricted by R proto-planetesimal radius 1 −
150 m gas mean free path & neglect of gravity v ∞ headwind velocity 15 −
40 m s − protoplanetary disk model t s dust particle stopping time 10 − − s size of perturbed gas region -15-10-505101520 -15 -10 -5 0 5 10 15 20 y [ m ] x [m] Fig. 2.
The numerically calculated trajectories of dust particles withdi ff erent impact parameters in the viscous laminar flow around a spheri-cal proto-planetesimal are indicated by solid lines. In this representativeexample, the proto-planetesimal radius is 10 m, the headwind velocityof the gas is 20 m s − , and the particle stopping time is 1 s. The dustparticles enter the plot from the right, and the axes have units of meters. For the entire set of 16000 simulations, we find the outcomedepends only on the dimensionless parameter: x : = R v ∞ t s . (24)This parameter x is the ratio of the hydrodynamical time scalefor flow past the proto-planetesimal, t p = R /v ∞ , and the parti-cle stopping time, t s . Note that x is equal to the reciprocal ofthe dimensionless stopping time (c.f. ST03). We prefer to use x here because it characterises the strength of the deflection bythe flow for a particle with a given t s . As x increases, so does theparticle deflection, eventually reaching a point where the particleno longer collides with the proto-planetesimal and the e ff ectivesweep-up cross section goes to zero. Conversely, as x →
0, theinfluence of the gas flow on the particle becomes insignificant,and the resulting e ff ective cross section reduces to the geometri-cal cross section.When x > .
8, we observe that σ e ff /σ geom < − , and wetherefore consider it to be zero. As such, for x > .
8, the deflec-tion of dust particles is substantial enough to entirely preventimpacts onto the proto-planetesimal, and the particles are in-stead advected around it. That collisions cease for x < ff ective sweep-up cross sectionvaries with the parameter x . The open points are taken from oursimulations, while the dashed line plots the function which best-fits the data points. This function is given by σ e ff σ geom = exp ( − D σ ( H ) x ) , (25)where σ geom = π R is the geometrical cross section. The func-tion D σ ( H ) accounts for the dependence of the e ff ective crosssection (i.e. the amount of deflection) on the starting distance H from the surface of the proto-planetesimal.Although Eq. (25) typically fits the simulation results towithin 10% di ff erence, it does not reproduce the e ff ect that par-ticles no longer impact the proto-planetesimal for x > . = x cuto ff . We therefore suggest the following form for the best-fitfunction: σ e ff σ geom = (cid:40) exp ( − D σ ( H ) x ) x < x cuto ff ;0 x ≥ x cuto ff . (26) σ e ff / σ geo m x Fig. 3.
Decreasing e ff ective sweep-up cross section as a function ofthe parameter x for H =
10. The open circles represent simulation data,and the dashed line corresponds to the best-fitting function Eq. (25).Above x > x cuto ff , the simulations show that particles no longer impactthe proto-planetesimal. In this regime, the best-fitting function overes-timates the sweep-up cross section. If a dust particle starts very close to the proto-planetesimalsurface, i.e. H (cid:38)
1, the flow does not have a significant opportu-nity to deflect the particle and the sweep-up cross section is thenvery close to the geometrical cross section. Comparing simula-tions with di ff erent values of H , we find: D σ ( H ) = . − . · exp (cid:18) − H . (cid:19) . (27) Article number, page 5 of 9 he maximum deviation of Eq. (27) from the simulation resultsis 10%, although the typical deviation is only ∼ H > D σ ( H ) (cid:39)
4. This behaviour isphysical: In Sect. 1 we calculated that the viscous, laminar flowpattern reaches a gas velocity of 98% the upstream velocity v ∞ at a distance of 100 R . That the function D σ ( H ) is constant be-yond H =
100 therefore originates from the gas velocity beinge ff ectively constant in this regime.In a protoplanetary disk, it is reasonable to assume that thecollision timescale for small dust is long enough that the parti-cles have spent several stopping times in the gas flow, and theyare at rest with respect to the gas as they approach the proto-planetesimal. Thus, one can safely choose the asymptotic value D σ ( H ) = H and D σ ( H ) <
4. However, this means the particle is not ini-tially at rest with respect to the gas flow, nor travelling parallel tothe flow axis, and this is beyond the scope of the current study.
Here, we summarize the e ff ects that we observe for the flow pat-tern on the measured impact velocity v imp = | v imp | of dust par-ticles onto the proto-planetesimal. From Eq. (1), the gas veloc-ity in the perturbed region steadily decreases as a particle ap-proaches the proto-planetesimal surface. Directly at the surface,the gas velocity is zero. The reduced gas velocity exerts a dragforce on approaching dust particles, thereby reducing their im-pact velocity.In our simulations, we find the impact velocity to be virtuallyindependent of the impact parameter – under the condition thatthe chosen impact parameter leads to a collision at all. In thefollowing, we therefore take the average of the impact velocities,weighted by the impact parameter p , over all impact parametersthat lead to collisions and refer to it as the impact velocity v imp . V i m p / V ∞ x Fig. 4.
Decreasing impact velocity of dust particles onto a proto-planetesimal as a function of the parameter x . The open circles repre-sent simulation data, while the solid line corresponds to the best-fittingfunction Eq. (28). Fig. (4) demonstrates how the impact velocity of dust parti-cles varies with x . The open circles are taken from our simula- tions and the solid line depicts the best fitting function: v imp v ∞ = exp ( − D v x ) ; D v = . . (28)Eq. (28) describes that particles with small x will impact theproto-planetesimal with velocity equal to v ∞ . The impact ve-locity decreases with increasing x because the drag that brakesthe particles also increases with x .Eq. (28) fits the simulation results to better than 10% and, aswith Eq. (25), Eq. (28) is only applicable for x < x cuto ff becausean impact velocity is ill-defined for particles that do not impactthe proto-planetesimal.Schräpler & Blum (2011) find that large dust particles areprone to erosion by the impact of smaller particles in a processthat has become known as monomeric erosion. More specifi-cally, they find the mass loss of the target scales linearly with theimpact velocity of the monomers, and that the process can resultin losses of up to ∼ × the mass of the impactor.We find that a flow around proto-planetesimals reduces theimpact velocity of small dust particles (i.e. large x , all else beingequal), and thus will have the consequence of reducing the e ffi -ciency of monomeric erosion. In the extreme limit of x > x cuto ff ,the flow entirely prevents impacts and thereby also monomericerosion. Thus, if a proto-planetesimal grows large enough thata hydrodynamical gas flow develops around it, it can becomepartially shielded against erosive high velocity impacts. From Eqs. (13), (14), and the range of values for R and v ∞ inTable 1, one can calculate the minimal and maximal values ofthe Reynolds numbers for our parameter space:Re max = R max v ∞ , max /ν ( r );Re min = R min v ∞ , min /ν ( r ) . The resulting range of values, constrained by the assumptionsof Sect. 1, are shown in Fig. 5. For example, at r = − (20 m s − ), requiring Re ≤
22 restricts our results to proto-planetesimal radii R ≤ . ≤ . (cid:28)
1, and is therefore strictly valid onlyin this regime. Meanwhile, experimental results a ffi rm that theflow streamlines upwind of a sphere remain remarkably similarto Eq. (1) for Reynolds numbers Re (cid:46)
22 (e.g. Taneda 1956; VanDyke 1982 and references therein). At larger Reynolds numbershowever, the flow pattern downstream of the sphere qualitativelychanges as the flow separates and vortices begin to form.As we are only concerned with the sweep-up of dust particleson the upstream side of the proto-planetesimal, the application ofEq. (1) is thus expected to accurately describe the deflection ofparticles even when Re (cid:54)(cid:28) ff ers from the issue that the ve-locity at the surface of the sphere depends on a term O (Re). Incontrast, Eq. (1) correctly predicts zero velocity at the surface ofthe sphere.The second, which we refer to here as “Proudman and Pear-son’s two-term approximation” (Proudman & Pearson 1957; Article number, page 6 of 9ellentin et al.: Hydrodynamical e ff ects on dust growth r [AU]10 − − R e Desch Re min Re max r [AU]10 − − MMSN Re min Re max Fig. 5.
Reynolds number as a function of heliocentric distance r for the Desch and MMSN disk models. The Re min (dashed) line corresponds to aproto-planetesimal radius and headwind velocity of ( R , v ∞ ) = (1 m ,
15 m s − ), while the Re max (dotted) line corresponds to (150 m ,
40 m s − ). Thesolid horizontal lines denote Reynolds numbers of 2 and 22, while the solid vertical line indicates the critical gravitational heliocentric distancefor each disk model (Eqs. 20 & 21). The successively darker shading distinguishes the regions of our parameter space that fall below Reynoldsnumbers of 22 and 2, respectively. “PP’s approximation” for short), is derived with the region nearthe sphere in mind, and preserves a zero velocity at the surface.Indeed, in the immediate vicinity of the sphere, this approxi-mation does a remarkable job of matching experiment, even forRe >
22 (Van Dyke 1964, p. 160). The trade-o ff , however, is thatthe velocity far from the sphere can significantly exceed v ∞ whenRe > v ∞ , impacting the proto-planetesimal with v imp /v ∞ > >
1. Furthermore, asour results strongly depend on the conditions in the immedi-ate vicinity of the proto-planetesimal, and Oseen’s approxima-tion does not recover a zero velocity at the surface of the proto-planetesimal, we choose to henceforth discuss only PP’s approx-imation.Relative to PP’s approximation, we expect simulations ap-plying Eq. (1) to progressively overestimate the deflection ofdust particles as Re increases. Indeed, additional simulationsrun using PP’s approximation demonstrate that x cuto ff increaseslinearly with Re. In other words, as the relative importance of in-ertial forces increase, all else being equal, a successively smallerstopping time is required for a particle to be deflected around theproto-planetesimal.Similarly, the additional simulations show that D σ ( H ) and D v decrease with increasing Re before asymptoting to small butnon-zero [ O (10 − )] values for Re (cid:38)
15. This manifests as anupward shift of the curves in Figs. 3 & 4 and, as with x cuto ff , thisillustrates that the amount of particle deflection decreases withincreasing Re.In regards to the accuracy of using Eq. (1) when Re (cid:54)(cid:28)
1, weobserve that x cuto ff , D σ ( H ), and D v match the results from PP’sapproximation for Re (cid:46) .
1. Moreover, these variables agreewith PP’s approximation to within 50% for Re ≤
1, and withina factor of two for Re ≤
2. Beyond Re =
2, notwithstanding thevariation of x cuto ff , D σ ( H ), and D v , the results applying Eq. (1) remain qualitatively correct with respect to PP’s approximation.Thus, Fig. 5 depicts where in our parameter space the results ofStokes’ and PP’s approximations agree to within a factor of 2(dark grey), and where these approximations still qualitativelyagree (light gray).That said, keep in mind that the additional approximationsconsidered here are derived under the assumption of small Re,and thus the above trends are at best estimates. It should also benoted that the e ff ects of Re on x cuto ff , D σ ( H ), and D v can at leastbe partially explained by the locally enhanced particle velocities(relative to v ∞ ) observed when using PP’s approximation. Directnumerical simulations of subsonic, laminar flow past a proto-planetesimal are needed to verify these results, and this is left tofuture work. .
3. Hydrodynamical effects on the reaccretion ofcollisional debris
If a dust aggregate collides with a proto-planetesimal and thecollision velocity is large enough, the aggregate will be dis-rupted into fragments. Debris of the particle can then re-enterthe gas flow around the proto-planetesimal. In the literature, thequestion has been raised whether the gas flow can return suchfragments to the surface of the proto-planetesimal, and therebyincrease the e ffi ciency of sweep-up growth (Wurm et al. 2001,2004; Sekiya & Takeda 2003, 2005).In this section, we attempt to address the escape of collisionaldebris from an impact site by launching dust particles directlyfrom the surface of the proto-planetesimal into the flow patternwith random directions. Sekiya & Takeda (2003) find that the gas can return collisionaldebris to the proto-planetesimal surface in the free molecularflow. The term free molecular flow describes the situation where For direct numerical simulations of particle accretion in fully turbu-lent flow see, for e.g., Mitra et al. 2013. Article number, page 7 of 9 he proto-planetesimal radius is comparable to or smaller thanthe mean free path of the gas. The possibility of averaging overthe random thermal motion of the gas particles is then no longeravailable, and the fluid has instead to be described by the motionof individual particles.Even in free molecular flow, the transformation into therest frame of the proto-planetesimal induces an apparent macro-scopic motion of the gas, namely the headwind. ST03 assumethe gas motion can then be described by straight, parallel stream-lines that intersect with the proto-planetesimal.However, the thermal motion of the gas will be superimposedon the ordered headwind. From the temperature profile Eq. (3),the mean thermal speed of gas molecules in the disk is then:¯ u th ( r ) (cid:39)
960 ( r / − / m s − . (29)At 1 AU, the random thermal motion outweighs the headwindvelocity of v ∞ ≈
24 m s − (20 m s − ) by a factor of 40 (48), andby a factor of (cid:38)
20 (28) within the inner 14 AU (8 AU) of the diskfor the Desch (MMSN) model. Clearly, the ordered macroscopicmotion of the headwind will disappear under the random thermalmotion of the free molecular flow, and thus parallel stream linesare not physically applicable in this context.However, if we launch dust particles from the surface of theproto-planetesimal into a flow pattern of parallel stream lines,then we do reproduce the result of ST03, namely that parallelstreamlines lead to reaccretion of collisional debris on the up-stream side of the proto-planetesimal.
We also launched collisional debris from the surface of the proto-planetesimal into the laminar flow pattern of Eq. (1) with multi-ple ejection velocities and angles in the range 0 ≤ α ≤ π .Independent of the ratio between headwind velocity and theejection velocity of the collisional debris, we find virtually noreaccretion, in agreement with the findings of ST03. In our simu-lations however, we observe a maximal fraction of ∼ − of par-ticles return to the proto-planetesimal surface. These re-impactsonly occur for angles nearly tangential to the proto-planetesimalsurface, and it is our opinion that they are the result of limitednumerical precision and thus, physically, no reaccretion occurs.Fig. 6 shows a representative example of the possible trajec-tories for collisional debris. Plotted are the trajectories of fourdust particles that leave the proto-planetesimal surface with thesame initial location, direction α , and ejection velocity (2 m s − ),but with di ff erent stopping times.Particles with small stopping times only escape into the fluidlayers just above the proto-planetesimal surface before being de-flected (trajectories 1 and 2). Since the gas flows around theproto-planetesimal, and the particles are dragged with it, theyare not returned to the surface.Particles with a larger stopping time that are ejected againstthe headwind can reach the region upstream of the proto-planetesimal where the gas velocity points towards the proto-planetesimal. In this region, the particle direction is reversed bythe gas flow and it then approaches the planetesimal (trajectory3). However, because the direction has been reversed, the parti-cle stopping time must consequently still be relatively small. Inour simulations, stopping times that lead to a reversal of direc-tion in front of the planetesimal also lead to a complete deflectionof the particle and thus no reaccretion.For particles with even larger stopping times (trajectory 4),the dust particles leave the collisional cross section of the proto-planetesimal before their motion is reversed. In all four cases above, the ejected particles do not collidewith the proto-planetesimal, and reaccretion does not occur. -30-20-100102030-30 -20 -10 0 10 20 30
431 2
Fig. 6.
Trajectories of collisional debris in a laminar flow. In thisrepresentative example, the headwind velocity is 20 m s − , and the axeshave units of meters. All dust particles begin with the same location,direction α , and velocity (20 m s − ), indicated by a dot on the proto-planetesimal surface. The stopping times of the dust particles are, fortrajectories 1 – 4, t s = .
1, 1, 8, and 40 s, respectively. See the text for adiscussion of the di ff erent outcomes.
4. Discussion & conclusions
We have examined a number of issues related to the e ff ects of hy-drodynamical gas flow around proto-planetesimals, with an in-terest in the consequences for coagulation e ffi ciency, expandingupon the work of ST03. By numerically integrating the trajec-tories of dust particles in the gas flow described by Eq. (1), wehave quantified how these particles are deflected and their impactvelocities a ff ected. We have also studied whether reaccretion ofcollisional debris remains possible in the presence of a laminarhydrodynamical flow.We have found that small particles that would impact theproto-planetesimal in the absence of a flow pattern can insteadbe deflected by the streaming gas and pass over the rim of theproto-planetesimal, avoiding a collision. Even if an impact oc-curs, the gas flow can decrease the relative velocity between thesmall particle and the proto-planetesimal, leading to generallyless disruptive collisions. These e ff ects are mostly important ina sweep-up scenario, occurring in the regime between small dustaggregates and larger proto-planetesimals, as studied by, e.g.,Xie et al. (2010); Windmark et al. (2012a).Our model is valid for spherical, non-rotating proto-planetesimals with radii (cid:46)
250 m, and within 8 AU of the centralstar in a MMSN disk, 14 AU in a Desch (2007) type disk.The results of our model are summarized as follows: – Sect. 2: The amount of deflection or deceleration experi-enced by a dust particle as a result of the flow pattern around
Article number, page 8 of 9ellentin et al.: Hydrodynamical e ff ects on dust growth a proto-planetesimal is purely a function of the parameter x = R / ( v ∞ t s ), where R is the radius of the proto-planetesimal, t s is the dust particle stopping time, and v ∞ is the headwindvelocity of the disk gas. – Sect. 2.1–2.3: The flow of disk gas around a proto-planetesimal reduces the e ff ective cross section with whichit can sweep up dust particles from the surrounding gas. Fora spherical proto-planetesimal, the e ff ective cross section iseasily parameterized (Eq. 25). When x > x cuto ff = . .
25, in rough agreement with ST03.For example, a proto-planetesimal with a radius R =
100 m at5 AU would, in a MMSN (Desch) type disk with a headwindvelocity of v ∞ =
20 m s − (24 m s − ), have its e ff ective crosssection reduced by half (i.e. x (cid:39) .
17 when D ( H ) =
4) forcollisions with particles of stopping time t s ∼
30 s (25 s),corresponding to a particle size of 0 . µ m (4 µ m). Particlesof stopping time t s (cid:46) . . x ≥ x cuto ff ), corresponding toparticle sizes of (cid:46) . µ m (0 . µ m). – Sect. 2.4: The flow pattern reduces the impact velocity ofsmall dust particles onto the proto-planetesimal, and this ef-fect can be parameterized with Eq. 28. The reduced impactvelocity decreases the erosion e ffi ciency, and could result inenhanced sticking. – Sect. 3.1: A flow pattern of parallel stream lines in freemolecular flow leads to enhanced reaccretion, in agreementwith ST03. However, because the random thermal motion ofgas molecules in the disk will dominate over the headwind,parallel streamlines are not physically justifiable for a proto-planetesimal with radius comparable to the mean free pathof the gas. – Sect. 3.2: A laminar flow pattern around a spherical proto-planetesimal does not result in an enhanced reaccretion rate,in agreement with ST03. In the cases studied here, col-lisional debris of di ff erent stopping times is either su ffi -ciently deflected by the flow to be advected past the proto-planetesimal, or moves beyond the e ff ective cross section be-fore it can be reaccreted.The adopted flow pattern, Eq. 1, is strictly valid only forRe (cid:28)
1, but should provide accurate quantitative results for thesweep-up of dust particles by a spherical proto-planetesimal ifRe (cid:46) ff ects oflaminar hydrodynamic flow on the accretion of dust particles. Ofcourse, the dependence of the flow pattern on the Reynolds num-ber influences the results, and while we have explored three an-alytical approximations (Stokes, Oseen, and Proudman & Pear-son), only full hydrodynamical simulations of the Navier-Stokesequations over the broad parameter space presented here candefinitively determine the e ff ects. These simulations will be thesubject of future work.Additional and obvious refinements to our model would en-compass surface irregularities, velocity shear of the disk gas,and a non-spherical or rotating proto-planetesimal. Althougha non-spherical, or rotating proto-planetesimal, or a shear ve-locity could produce significant di ff erences in the hydrodynami-cal flow pattern (e.g. Kurose & Komori 1999; Ormel 2013), weexpect that surface irregularities of planetesimals will introduceonly minor di ff erences.For proto-planetesimals of size R ∼
100 m at 5 AU, ourresults predict that hydrodynamical deflection and deceleration will considerably reduce the e ff ect of monomeric erosion (Schrä-pler & Blum 2011). Meanwhile, the detailed consequences ofincluding the reduced sweep-up cross section and impact veloc-ities into a dust growth code are not easily predicted. The rela-tively low Reynolds numbers ( ≤
22) used in this study preventus from generalising to larger proto-planetesimal radii at low he-liocentric distances, and therefore to larger dust particle stoppingtimes. Even when considering the results of ST03 (Re =
50 andcut-o ff point x ∼ r in theDesch & MMSN disk models ( > ffi cultto predict what e ff ect a ∼
100 m size body might have on thedeflection and deceleration of dust particles at r ∼ ff ectthe ability of a proto-planetesimal to sweep-up smaller particles.To further quantify these e ff ects, Eq. (25) for the e ff ective sweep-up cross section and Eq. (28) for the impact velocities must beimplemented into a dust coagulation code which treats sweep-up growth. In this regard, the appropriate value of D ( H ) to usein coagulation models is the asymptotic limit of 4 (i.e. the dustparticles are already well-coupled to the gas flow). Acknowledgements.
We thank the anonymous referee for a careful and thoroughreview which improved the manuscript. JPR is supported by DFG grant DU414 / References