Dust Size Distributions in Coagulation/Fragmentation Equilibrium: Numerical Solutions and Analytical ﬁts

A&A 525, A11 (2011) Astronomy DOI: 10.1051/0004-6361/201015228 & c ESO 2010 Astrophysics

Dust size distributions in coagulation/fragmentation equilibrium: numerical solutions and analytical ﬁts

T. Birnstiel, C. W. Ormel, and C. P. Dullemond

Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany e-mail: [email protected] Received 17 June 2010 / Accepted 9 September 2010

ABSTRACT

Context. Grains in circumstellar disks are believed to grow by mutual collisions and subsequent sticking due to surface forces. Results of many fields of research involving circumstellar disks, such as radiative transfer calculations, disk chemistry, magneto-hydrodynamic simulations largely depend on the unknown grain size distribution. Aims. As detailed calculations of grain growth and fragmentation are both numerically challenging and computationally expensive, we aim to find simple recipes and analytical solutions for the grain size distribution in circumstellar disks for a scenario in which grain growth is limited by fragmentation and radial drift can be neglected. Methods. We generalize previous analytical work on self-similar steady-state grain distributions. Numerical simulations are carried out to identify under which conditions the grain size distributions can be understood in terms of a combination of power-law distributions. A physically motivated fitting formula for grain size distributions is derived using our analytical predictions and numerical simulations. Results. We find good agreement between analytical results and numerical solutions of the Smoluchowski equation for simple shapes of the kernel function. The results for more complicated and realistic cases can be fitted with a physically motivated “black box” recipe presented in this paper. Our results show that the shape of the dust distribution is mostly dominated by the gas surface density (not the dust-to-gas ratio), the turbulence strength and the temperature and does not obey an MRN type distribution. Key words. accretion, accretion disks – stars: pre-main-sequence – circumstellar matter – planets and satellites: formation – protoplanetary disks

1. Introduction that this result is exactly independent of the adopted collision model and that the resulting slope α is only determined by the Dust size distributions are a fundamental ingredient for many mass-dependence of the collisional cross-section if the model of astrophysical models in the context of circumstellar disks and collisional outcome is self-similar (in the context of fluid dynam- planet formation: whenever dust is present, it dominates the ics, the same result was independently obtained by Hunt 1982; opacity of the disk, thereby influencing the temperature and con- 11 and Pushkin & Aref 2002). The value of 6 agrees well with the sequently also the vertical structure of the disk (e.g., Dullemond size distributions of asteroids (see Dohnanyi 1969) and of grains ff & Dominik 2004). Small grains e ectively sweep up electrons in the interstellar medium (MRN distribution, see Mathis et al. ff and therefore strongly a ect the chemistry and the ionization 1977; Pollack et al. 1985) and is thus widely applied, even at the fraction (also via grain surface reactions, see Vasyunin et al., gas-rich stage of circumstellar disks. in prep.) and thereby also the angular momentum transfer of the disk (e.g., Wardle & Ng 1999; Sano et al. 2000). However, in protoplanetary disks gas drag damps the mo- Today, it is well established that the dust distributions in as- tions of particles. Very small particles are tied to the gas and, teroid belts and debris disks are governed by a so-called “col- as a result, have a relative velocity low enough to make stick- lision cascade” (see Williams & Wetherill 1994): larger bod- ing feasible. The size distribution therefore deviates from the ies in a gas free environment exhibit high velocity collisions MRN power-law. Theoretical models of grain growth indicate − μ (∼>km s 1), far beyond their critical fragmentation threshold, that particles can grow to sizes much larger than a few m which lead to cratering or even complete shattering of these (see Nakagawa et al. 1981; Weidenschilling 1980, 1984, 1997; objects. The resulting fragments in turn suffer the same fate, Dullemond & Dominik 2005; Tanaka et al. 2005; Brauer et al. thus producing ever smaller grains down to sizes below about 2008a; Birnstiel et al. 2009). Indeed, the observational evidence a few micrometers where Poynting-Robertson drag removes the suggests that growth up to cm-sizes is possible (e.g., Testietal. dust particles (e.g., Wyatt et al. 1999). The grain number density 2003; Natta et al. 2004; Rodmann et al. 2006; Ricci et al. 2010). distribution in the case of such a fragmentation cascade has been However, as particles grow, they become more loosely cou- derived by Dohnanyi (1969)andWilliams & Wetherill (1994) pled to the gas. This results in an increase in the relative veloc- and was found to follow a power-law number density distribu- ity between the particles, a common feature of most sources of ∝ −α α = 11 tion n(m) m with index 6 (which is equivalent to particles’ relative velocity (i.e., turbulence, radial drift, and set- n(a) ∝ a−3.5), with very weak dependence on the mechanical tling motions). Therefore, we expect that the assumption of per- parameters of the fragmentation process. Tanaka et al. (1996), fect sticking will break down at some point and other collisional Makino et al. (1998)andKobayashi & Tanaka (2010)showed outcomes (bouncing, erosion, catastrophic disruption) become Article published by EDP Sciences A11, page 1 of 16 A&A 525, A11 (2011) possible (see Blum & Wurm 2008). It is expected, then, that at collision always result in coagulation. As explained above, the a certain point growth will cease for the largest particles in the physical motivation for this choice is that relative velocities in- distribution. Collisions involving these particles result in frag- crease with mass. This also means that in our theoretical model mentation, thus replenishing the small grains. On the other hand we neglect collision outcomes like bouncing or erosion (erosion mutual collisions among small particles still result in coagula- is included in the simulations). Even though, small particles will tion. As a result, a steady-state emerges. In this paper, this situa- in reality cause cratering/growth instead of complete fragmentation is referred to as a fragmentation-coagulation equilibrium. tion of the larger particle, we find that this assumption is often The situation in protoplanetary disks differs, therefore, from justified because fragmenting similar-sized collisions prevent that in debris disks. In the latter only fragmentation operates. any growth beyond mf. Thus, collisions with much smaller par- The mass distribution still proceeds towards a steady state but, ticles (m mf ) do not have an important influence on the max- ultimately, mass is removed from the system due to, radiation imum size, however, they can significantly change the amount pressure or Poynting-Robertson drag. of small particles in the stationary distribution (see Sect. 5). We In this paper, we consider the situation that the total mass further assume a constant porosity of the particles, which relates budget in the system is conserved. For simplicity, we will ig- the mass and size according to nore motions due to radial drift in this study. This mechanism π ff 4 3 e ectively removes dust particles from the disk as well as pro- m = ρs a , (1) viding particles with a large relative motion. However, the de- 3 rived presence of mm-size dust particles in protoplanetary disks where ρs is the internal density of a dust aggregate. is somewhat at odds with the usual (laminar) prescriptions for In our analysis, we will identify three different regimes, the radial drift rate (Weidenschilling 1977). Recently, it was which are symbolically illustrated in Fig. 1: shown that mm observations of protoplanetary disks can be explained by steady-state size distributions if radial drift is ineffi- – Regime A represents a case where grains grow sequentially cient (Birnstiel et al. 2010b). On the other hand, if radial drift (i.e. hierarchically) by collisions with similar sized grains would operate as effectively as the laminar theory predicts, then until they reach an upper size limit and fragment back to the the observed populations of mm-sized particles at large disk radii smallest sizes. The emerging power-law slope of the size dis- cannot be sustained (Brauer et al. 2007). Possible solutions to tribution depends only on the shape of the collisional kernel. reduce the drift rate include bumps in the radial pressure profile – Regime B is similar to regime A, however in this case the (see Kretke & Lin 2007; Brauer et al. 2008b; Cossins et al. 2009) fragmented mass is redistributed over a range of sizes and, or zonal flows (see Johansen et al. 2009). thus, influencing the out-coming distribution. In this work we analytically derive steady-state distributions – In Regime C, the upper end of the distribution dominates of grains in the presence of both coagulation and fragmentation. grain growth at all sizes. Smaller particles are swept up by The analytical predictions are compared to numerical simula- the upper end of the distribution and are replenished mostly tions and applied to grain size distributions in turbulent circum- by redistributed fragments of the largest particles. The result- stellar disks. Both the theoretical and numerical results presented ing distribution function depends strongly on how the frag- in this work are used to derive a fitting formula for steady-state mented mass is distributed after a disruptive collision. grain size distributions in circumstellar disks. The paper is outlined as follows: in Sect. 2,webrieflysum- For each of these regimes, we will derive the parameter ranges marize and then generalize previous results by Tanaka et al. for which they apply and the slopes of the resulting grain size (1996)andMakino et al. (1998). In Sect. 3, we test our theoret- distribution. ical predictions and their limitations by a state-of-the-art grain evolution code (see Birnstiel et al. 2010a). Grain size distribu- 2.1. The growth cascade tions in circumstellar disks are discussed in Sect. 4. In Sect. 5, we present a fitting recipe for these distributions that can eas- The fundamental quantity that governs the time-evolution of the , ily be used in models where grain properties are important. Our dust size distribution is the collision kernel Cm1 m2 .Itisdefined findings are summarized in Sect. 6. such that · · Cm1,m2 n(m1) n(m2)dm1dm2 (2) 2. Power-law solutions for a dust gives the number of collisions per unit time per unit volume be- coagulation-fragmentation equilibrium tween particles in mass interval [m1, m1 + dm1]and[m2, m2 + dm2], where n(m) is the number density distribution. Once spec- In this section, we begin by summarizing some of the previ- ified it determines the collisional evolution of the system. In the ous work on analytical self-similar grain size distributions on case that the number density n(m) does not depend on posi- which our subsequent analysis is based. We will then extend tion, Cm1,m2 is simply the product of the collision cross section this to include both coagulation and fragmentation processes and the relative velocity of two particles with the masses m in a common framework. Under the assumption that the rel- 1 and m2. evant quantities, i.e., the collisional probability between parti- We use the same Ansatz as Tanaka et al. (1996), assuming cles, the distribution of fragments, and the size distribution, be- that the collision kernel is given in the self-similar form have like power-laws, we will solve for the size distribution in coagulation-fragmentation equilibrium. For simplicity, we con- ν m2 C , = m h · (3) sider a single monomer size only of mass m0. This is therefore m1 m2 1 m the smallest mass in the distribution. 1 Another key assumption of our analytical model is that Here, h is any function which depends only on the masses / we assume the existence of a sharp threshold mass mf , above through the ratio of m2 m1. By definition, the kernel Cm1,m2 has which collisions always result in fragmentation and below which to be symmetric, therefore, Eq. (3) implies that h(m1, m2) is not

A11, page 2 of 16 T. Birnstiel et al.: Size distributions of grains in coagulation/fragmentation equilibrium

log(mass) Substituting the deﬁnitions of above and using the dimen- sionless variables x1 = m1/m, x2 = m2/m, x0 = m0/m,and sequential growth xf = mf /m one obtains Case A) 1 xf x m m = ν−2α+3 ν+1−α −α 2 , 0 fragmentation to monomers f F(m) m dx1 dx2x1 x2 h (7) − x x0 1 x1 1 :=K

where K approaches a constant value in the limit of m m0 and m m . sequential growth f Postulation of a steady state (i.e. setting the time derivative Case B) in Eq. (5) to zero), leads to the condition m0 mf F(m) ∝ mν−2α+3 = const., (8)

fragment distribution fragmentation barrier from which it follows that the slope of the distribution is sweep-up growth ν + 3 α = · (9) 2 Case C) m0 mf This result was already derived for the case of fragmentation by Tanaka et al. (1996)andDohnanyi (1969) and for the coagula- fragment distribution tion by Klett (1975)andCamacho (2001). The physical interpre- tation of this is a “reversed” fragmentation cascade: instead of a Fig. 1. Illustration of the three different regimes for which analytical resupply of large particles which produces ever smaller bodies, solutions have been derived. Case A represents the growth cascade this represents a constant resupply of monomers which produce discussed in Sect. 2.1, case B the intermediate regime (see Sect. 2.2) ever larger grains (cf. case A in Fig. 1). and case C the fragmentation dominated regime which is discussed in Sect. 2.3. Particles are shattered once they reach the fragmentation barrier mf since collision velocities for particles >mf exceed the fragmen- 2.2. Coagulation fragmentation equilibrium tation threshold velocity. As Tanaka et al. (1996)andMakino et al. (1998) pointed out, the previous result is independent of the model of collisional symmetric (see Eq. (21)). ν is called the index of the kernel outcomes as long as this model is self-similar (Eq. (3)). However or the degree of homogeneity. As we will see in the follow- this is no longer the case if we consider both coagulation and ing, ν is one of the most important parameters determining the fragmentation processes happening at the same time. resulting size distribution. Different physical environments are We will now consider the case with a constant resupply of represented by different values of ν. Examples include the con- matter, due to the particles that fragment above mf. We assume stant kernel (i.e., ν = 0, mass independent), the geometrical these fragments obey a power-law mass distribution and are pro- kernel (i.e., ν = 2/3, velocity independent) or the linear kernel duced at a rate (i.e., ν = 1, as for grains suspended in turbulent gas). = · −ξ, It is further assumed that the number density distribution of n˙f(m) N m (10) dust particles follows a power-law, where ξ reflects the shape of the fragment distribution and N is n(m) = A · m−α. (4) a constant. If there is a constant flux of particles F(m) as defined above, The time evolution of the mass distribution obeys the equation then the flux of fragmenting particles (i.e., the flux produced by of mass conservation, particles that are growing over the fragmentation threshold) is ∂ ∂ given by mn(m) + F(m) = , 0 (5) = · ν−2α+3 ∂t ∂m F(mf) K mf (11) where the flux F(m) does not represent a flux in a typical where K is the integral defined in Eq. (7). continuous way since coagulation is non-local in mass space The resulting (downward) flux of fragments Ff(m) can then (each mass can interact with each other mass) but rather an in- be derived by inserting Eq. (10) into the equation of mass con- tegration of all growth processes which produce a particle with servation (Eq. (5)), mass greater than m out of a particle that was smaller than m ∂ (i.e. collisions of m1 < m with any other mass m2 such that Ff(m) = −m · n˙f. (12) m1 + m2 > m). The flux in the case of pure coagulation was ∂m derived by Tanaka et al. (1996) Integration from monomer size m0 to m yields m mf = , 1 −ξ 2−ξ F(m) dm1 dm2 m1 Cm1,m2 n(m1) n(m2) (6) = − · · 2 − . − Ff(m) N m m0 (13) m0 m m1 2 − ξ where we changed the lower bound of the integration over m1 to The normalization factor N can be determined from the equilib- start from the mass of monomers m0 (instead of 0) and the upper rium condition that the net flux vanishes, bound of the integral over m2 to a finite upper end mf (instead of infinity), compared to the definition used by Tanaka et al. (1996). Ff(mf) = −F(mf), (14)

A11, page 3 of 16 A&A 525, A11 (2011) and was found to be Physically, this means that the total number of collisions of any grain is determined by the largest grains in the upper end mν−2α+3 N = (2 − ξ) K f · (15) of the distribution. Hence, smaller sized particles are predomi- 2−ξ − 2−ξ nantly refilled by fragmentation events of larger bodies instead m m0 f of coagulation events from smaller bodies and they are predom- In Eq. (13), we can distinguish two cases: inantly removed by coagulation events with big particles (near the threshold m ) instead of similar-sized particles. This corre- ξ> f – If 2, the contribution of m0 dominates the term in brack- sponds to case C in Fig. 1. ets. This means that most of the fragment mass is redis- To determine the resulting power-law distribution, we again tributed to monomer sizes and the situation is the same as in focus on Eq. (6). The double integral of the flux F(m) can now the pure coagulation case (cf. case A in Fig. 1). The steady- be split into three separate integrals, + = state condition F(m) Ff(m) 0 (i.e., the net flux is zero) m m yields that 2 f = F(m) dm1 dm2 m1 Cm1,m2 n(m1) n(m2) m m−m · ν−2α+3 = − · 1 · 2−ξ, 0 1 K m N m0 (16) m m1 2 − ξ + dm1 dm2 m1 Cm1,m2 n(m1) n(m2) m m−m is constant, which leads to the same result as Eq. (9). 2 1 m mf Intuitively, this is clear since the majority of the redistributed + dm1 dm2 m1 Cm1,m2 n(m1) n(m2) (22) mass ends up at m ∼ m . m 0 m1 – If ξ<2, the m-dependence dominates the term in brackets 2 in Eq. (13) and postulation of a steady-state, according to whether m2 is larger or smaller than m1. It can be derived (see Appendix A) that if the condition above 1 K · mν−2α+3 N · · m2−ξ, (17) (Eq. (19)) is violated and if mf m, then the first and the third 2 − ξ integral in Eq. (22) dominate the flux due to the integration un- 2+γ−α til mf. In this cases, the flux F(m) is proportional to m . leads to an exponent A stationary state in the presence of fragmentation (Eq. (13)) ν + ξ + 1 is reached if the fluxes cancel out, which leads to the condition α = , (18) 2 α = ξ + γ. (23) less than Eq. (9). In this case, the slope of the fragment dis- This is the sweep-up regime where small particles are cleaned tribution matters. This scenario is represented as case B in out by big ones (cf. case C in Fig. 1). Fig. 1.

2.4. Summary of the regimes 2.3. Fragment dominated regime Summarizing these findings, we find that the resulting distribu- The result obtained in the previous section may seem to be quite ξ tion is described by three scenarios (depicted in Fig. 1), depend- general. However, it does not hold for low -valuesaswewill ing on the slope of the fragment distribution: show in the following. In our case, the integrals do not diverge due to the finite in- ν + 3 Case A (growth cascade): ξ>2 α = tegration bounds. However Makino et al. (1998)used0and∞ 2 as lower and upper bounds for the integration and thus needed ν+ξ+1 (24) Case B (intermediate regime): ν−2γ+1<ξ<2 α = to investigate the convergence of the integral. They derived the 2 following conditions for convergence: Case C (fragment dominated): ξ<ν−2γ+1 α = ξ + γ. ν − γ − α + 1 < 0 (19) 3. Simulation results interpreted γ − α + 2 > 0, (20) In this section, we will test the analytical predictions of the previ- γ / where gives the dependence of m2 m1 in the h-function of ous section by a coagulation/fragmentation code (Birnstiel et al. Eq. (21)(seeMakino et al. 1998): 2010a; see also Brauer et al. 2008a). The code solves for the ⎧ γ time evolution of the grain size distribution using an implicit in- ⎪ m2 m2 ⎪ m for m 1 tegration scheme. This enables us to find the steady-state distri- m2 ⎨ 1 1 h = h0 · ⎪ (21) bution by using large time steps. In this way, the time evolution m ⎪ ν−γ 1 ⎩ m2 m2 . is not resolved, but the steady-state distribution is reliably and m for m 1 1 1 very quickly derived. The first condition (Eq. (19)) considers the divergence towards We start out with the simplest case of a constant kernel and the upper masses, whereas the second condition pertains the then – step by step – approach a more realistic scenario (in the lower masses. We assume that Eq. (20) is satisfied and consider context of a protoplanetary disk). In Sect. 4, we will then con- the case of decreasing ξ for α givenbyEq.(18) which results in a sider a kernel taking into account relative velocities of Brownian steeper size distribution (where the mass is concentrated close to motion and turbulent velocities and also a fragmentation proba- the upper end of the distribution). We see that for ξ<1 + ν − 2γ, bility which depends on the masses and the relative velocities of Eq. (19) is no longer fulfilled. The behavior of the flux integral the colliding particles. changes qualitatively: growth is no longer hierarchical, but it be- The following results are only steady-state solutions, comes dominated by contributions by the upper end of the inte- whether or not this state is reached depends on several condi- gration bounds. tions. Firstly, particles need to fragment. If there is no upper

A11, page 4 of 16 T. Birnstiel et al.: Size distributions of grains in coagulation/fragmentation equilibrium

2.2 ξ = 0.5 simulation result 0 ξ growth cascade (Case A) 10 = 1.0 2 ξ = 1.5 intermediate regime (Case B) ξ = 2.0 1.8 fragment dominated (Case C) ] 3 ξ − −2 = 2.5 10 ξ = 3.0 1.6

[g cm 1.4 a α · −4 m 10 1.2 · ) a

( 1 n

−6 0.8 10 0.6

−8 0.4 10 0.5 1 1.5 2 2.5 3 −4 −2 0 −ξ 10 10 10 ξ wheren ˙ f(m) ∝ m grain size [cm] Fig. 3. Exponent of grain size distributions for a constant kernel (i.e. ν = Fig. 2. Grain size distributions2 for a constant kernel (i.e., ν = 0) and 0) and diﬀerent distributions of fragments (solid line) and the analytic diﬀerent distributions of fragments. The peak towards the upper end solution for the growth cascade (case A, dotted line), the intermedi- of the distribution is due to the fragmentation barrier (explanation in ate regime (case B, dashed line), and the fragment dominated regime the text). The slope of the mass distribution corresponds to 6 − 3α. (case C, dash-dotted line).

boundary for growth, it will proceed unlimited and a steady state truncated at the upper end (defined as amax, see also Eq. (43)), will never be reached. Secondly, radial motion needs to be slow particles close to the upper end lack larger collision partners, enough to allow for a steady state. If quantities like the surface the growth rate at these sizes would decrease if the distribu- density or the temperature vary smoothly between neighbouring tion keeps its power-law nature. This, in turn means that the regions in the disk, the steady state solutions will also be simi- flux could not be constant below amax. To keep a steady state, lar and radial transport will happen on top of a steady-state grain the number of particles at that point has to increase in order to distribution. However if radial drift is acting strongly on particles replace the missing collision partners at larger sizes. of the distribution, the steady state will not be reached. Thirdly, a Figure 3 shows how the slope of the resulting distribution distribution of initially sub-μm sized grains will need some time depends on the fragmentation slope ξ, where the three previously to get into an equilibrium state. This time scale can be as small discussed regimes can be identified: as <1000 years inside of a few AU, while it can be of the order – Case A, growth cascade: as predicted, this scenario holds of a million years at 100 AU. We provide a rough estimate of ξ > this time scale in Appendix C. for values of ∼ 2 where most of the fragmenting mass is redistributed to fragments. This case corresponds to a “reversed” collisional cascade (just the direction in mass space 3.1. Constant kernel is reversed as collisions mostly lead to growth instead of shattering). In the following section, we consider the case of a constant ker- – Case B, the intermediate regime: when the fragment mass is nel and will include fragmentation above particle sizes of 1 mm more or less equally distributed over all sizes, mass gain by because this represents an instructive test case. redistribution of fragments and the mass loss due to coagu- We iteratively solve for the steady-state size distribution be- lation have to cancel each other. tween coagulation and fragmentation. The outcome of these sim- ν = – Case C: most of the fragmented mass remains at large sizes. ulations for a constant kernel (i.e., 0) are power-law dis- Therefore the mass distribution is dominated by the largest tributions where the slope of the distribution depends on the particles. In other words, growth is not hierarchical anymore. fragmentation law (the slope ξ,seeEq.(10)). Figure 2 shows γ = ff In this test case 0, and from Eq. (19) it follows that the the corresponding size distributions for some of the di erent transition between the intermediate and fragment-dominated fragment distributions: the steepest distribution corresponds to regime lies at ξ = 1, which can be seen in Fig. 3. the case of ξ = 0.5. For larger ξ-values, the slope of the mass distribution flattens. In all cases, a bump develops towards the The measured slopes of the size distribution for m mf are in upper end of the distributions. The reason for this “pile-up” is excellent agreement with the model outlined in Sect. 2. the following: grains typically grow mostly through collisions with similar-sized or larger particles. Since the distribution is 3.2. Including settling effects 2 It should be noted that in this paper we will plot the distributions · · The simplest addition to this model is settling: as grains become typically in terms of n(a) m a which is proportional to the distribution larger, they start to settle towards the mid-plane. However, tur- of mass. The advantage of plotting it this way instead of plotting n(a) is the following: when n(m) follows a power-law m−α, then the grain bulent mixing counteracts this systematic motion. The vertical size distribution n(a) describes a power-law with exponent 2 − 3α and distribution of dust in a settling-mixing equilibrium can (close the mass distribution exponent is 6 − 3α. For typical values of α,this to the mid-plane) be estimated by a Gaussian distribution with a is less steep and differences between a predicted and a real distribution size-dependent dust scale height Hd. Smaller particles are well are more prominent. enough coupled to the gas to have the same scale height as the

A11, page 5 of 16 A&A 525, A11 (2011)

0 10 Similar to Eq. (2), we can write the vertically integrated number simulation of collisions as −1 fit functions 10 ˜ · · , Cm1,m2 N(m1) N(m2)dm1 dm2 (29)

] −2 2 10 which gives the total number of collisions that take place over − the entire column of the disks. −3 The dependence of the collisional probability on scale [g cm 10 a height, is now reﬂected in the modiﬁed kernel C˜m ,m (see · 1 2

m −4 Birnstiel et al. 2010a, Appendix A for derivation):

· 10 )

a , ( Cm1 m2 C˜ , = · (30) N −5 m1 m2 10 π 2 + 2 2 H1 H2

−6 sett 10 a The point to realize here is that, due to the symmetry between Eqs. (2)and(29), the analysis in Sect. 2 holds also for disk-like −7 10 configuration, if the kernel is now replaced by C˜m ,m . The result- −4 −3 −2 −1 0 1 2 10 10 10 10 10 ing exponent α then concerns the column density dependence grain size [cm] (N(m) ∝ m−α). >α Fig. 4. Simulation result (solid line) and a “two power-law fit” to the If we consider the case that St t and substitute Eqs. (25) vertically integrated dust distribution for a constant kernel with settling and (26)into(30), we find that included. The dotted, vertical line denotes the grain size above which C , grains are affected by settling. ˜ = m1 m2 Cm1,m2 π 2 + 2 2 H1 H2 gas, H , while larger particles decouple and their scale height is ⎛ ⎞− 1 g ⎜ H2 ⎟ 2 −1 ⎜ 2 ⎟ decreasing with grain size, = Cm ,m · H · ⎝1 + ⎠ 1 2 1 H2 1 Hd αt = , for St >α (25) 1/6 m2 t = C , · m · h , Hg St m1 m2 1 (31) m1 (see e.g., Dubrulle et al. 1995; Schräpler & Henning 2004; has an index ν = 1/6 for grain sizes larger than asett and Youdin & Lithwick 2007) where the Stokes number St is a di- ν = 0 otherwise (it should be noted that H and H are the dust mensionless quantity which describes the dynamic properties of 1 2 scale heights whereas h(m2/m1) represents the function defined a suspended particle. Very small particles have a small Stokes in Eq. (3)). number and are therefore well coupled to the gas. Particles which ff The theory described in Sect. 2 is strictly speaking only valid have di erent properties (e.g., size or porosity) but the same St for a constant ν-index, but if this index is constant over a signif- behave aerodynamically the same. In our prescription of tur- icant range of masses, then the local slope of the distribution bulence, St is defined as the product of the particles stopping τ Ω will still adapt to this index. In the case of settling, we can there- time st and the orbital frequency k. We focus on the Epstein fore describe the distribution with two power-laws as can be seen regime where the Stokes number can be approximated by in Fig. 4. ρ π The fact that the distribution will locally follow a power-law =Ω · τ a s St k st (26) is an important requirement for being able to construct fitting Σg 2 formulas which reproduce the simulated grain size distributions. with Σg being the gas surface density and ρs being the inter- It allows us in some cases to explain the simulation outcomes nal density of the particles which relates mass and size via with the local kernel index (although coagulation and fragmenta- 3 m = 4π/3ρs a , tion are non-local processes in mass space, since each mass may Settling starts to play a role as soon as the Stokes number interact with each other mass). A physically motivated recipe to becomes larger than the turbulence parameter αt which can be fit the numerically derived distribution functions for the special related to a certain size case of ξ = 11/6 is presented in Sect. 5. 2α Σ = t g · asett (27) 3.3. Non-constant kernels πρs Equation (27) only holds within about one gas pressure scale We performed the same tests as in Sect. 3.1 also for non-constant ν> height because the dust vertical structure higher up in the disk kernels, i.e. kernels with 0. The measured slopes for the ν = / ,γ= deviates from the a Gaussian profile (see also Dubrulle et al. cases of ( 5 6 0) (corresponding to the second turbulent ν = / ,γ = / ν = / ,γ = 1995; Schräpler & Henning 2004; Dullemond & Dominik 2004). regime, see Sect. 4.1), ( 1 3 1 3), and ( 1 6 − / The mass-dependent dust scale height causes the number 1 2) (i.e., Brownian motion, see Sect. 4.1)areshowninFig.5. density distribution n(m) to depend on the vertical height, z.In Similar to Fig. 3, the distribution always follows the minimum ff − disk-like configurations, it is therefore customary to consider the index of the three di erent regimes (cases A C). It should be ξ column density, noted that for all indices of the fragmentation law, all processes – coagulation, fragmentation, and re-distribution of frag- ∞ ments – take place, but the relative importance of them is what N(m) = n(m, z)dz. (28) −∞ determines the resulting slope. A11, page 6 of 16 T. Birnstiel et al.: Size distributions of grains in coagulation/fragmentation equilibrium

In Fig. 5, it can be seen that case B (defined in Sect. 2.2) vanishes for the kernel in the upper panel, while it is present for ν = 5/6, γ =0 a large range of ξ for the kernel in the middle panel. This can 2 be explained by the definitions of the three regimes which were summarized in Eq. (24): with (ν = 5/6,γ = 0) (cf. upper panel in Fig. 5), case B is confined between ξ = 11/6 and 2. The grain 1.5 size distribution, therefore, switches almost immediately from being growth dominated (case A) to fragmentation dominated α (case C). In the case of a kernel with ν = 1/3andγ = 1/3, this 1 range spans from 2/3 to 2, as can be seen in the central panel of Fig. 5. 0.5 simulation result The lower panel shows the distribution for a Brownian mo- growth cascade (Case A) tion kernel (i.e., ν = 1/6andγ = −1/2). The grey shaded area intermediate regime (Case B) highlights the range of ξ values where our predictions do not ap- fragment dominated (Case C) 0 ply, that is, Eq. (20) is no longer fulfilled and thus, the resulting steady-state distributions are no longer power-law distributions. ν = 1/3, γ = 1/3 2

4. Grain size distributions in circumstellar disks 1.5 In this section, we will leave the previous “clean” kernels and focus on the grain size distribution in circumstellar disks includ- α ing relative velocities due to Brownian and turbulent motion of 1 the particles, settling effects, and a fragmentation probability as function of particle mass and impact velocity. Combining all these effects makes it impossible to find 0.5 simple analytical solutions in the case of a coagulation- fragmentation equilibrium. We will therefore use a coagulation fragmentation code to find the steady-state solutions and to show 0 how the steady-state distributions in circumstellar disks depend ν = 1/6, γ = -1/2 on our input parameters. 2 We will first discuss the model “ingredients”, i.e. the relative velocities and the prescription for fragmentation/sticking. Afterwards, we will define characteristic sizes at which the shape 1.5 of the size distribution changes due to the underlying physics. In the last subsection, we will then show the simulation results α and discuss the influence of different parameters. 1

4.1. Relative velocities 0.5 We will now include the effects of relative velocities due to Brownian motion, and due to turbulent mixing. The Brownian 0 motion relative velocities are given by 0.5 1 1.5 2 2.5 3 ∝ − ξ ξ wheren ˙ f(m) m 8kb T (m1 + m2) Fig. 5. Exponents of the grain size distributions for three different ker- ΔuBM = (32) π m1 m2 nels as function of the fragment distribution index. The plot shows the simulation results (solid lines), the analytic solution for the growth cas- where kb is the Boltzmann constant and T the mid-plane tem- cade (case A, dotted lines), the intermediate regime (case B, dashed perature of the disk. Ormel & Cuzzi (2007) have derived closed lines), and the fragment dominated regime (case C, dash-dotted lines). form expressions for particle collision velocities induced by tur- The upper panel was calculated with a (ν = 5/6,γ = 0)-kernel bulence. They also provided easy-to-use approximations for the (i.e. turbulent relative velocities, see Sect. 4.1), the middle panel with a ff (ν = 1/3,γ = 1/3)-kernel and the lower panel with a (ν = 1/6,γ = di erent particle size regimes which we will use in the follow- − / ing. Small particles (i.e. stopping time of particle eddy cross- 1 2)-kernel (i.e. Brownian motion relative velocities). In the grey shaded area, Eq. (20) is not fulfilled, and the size distribution is not ing time) belong to the first regime of Ormel & Cuzzi (2007) apower-law. with velocities proportional to

Δu ∝|St − St | . (33) I 1 2 ratio of the Stokes numbers which we will neglect in the follow- The relative velocities of particles in the second turbulent regime ing discussion. σ = π + of Ormel & Cuzzi (2007)aregivenby Together with the geometrical cross section geo (a1 2 a2) , it is straight-forward to estimate the indices of the ker- ν γ ΔuII ∝ Stmax, (34) nel, and , as deﬁned in Eqs. (3)and(21) for all these regimes, without settling (assuming that only Brownian motion or turbu- where Stmax is the larger of the particles Stokes numbers. lent motion dominates the relative velocities). The indices for Velocities in this regime show also a weak dependence on the these three sources of relative particle motion are summarized in

A11, page 7 of 16 A&A 525, A11 (2011)

Table 1. Kernel indices for the diﬀerent regimes without settling. Simulations of silicate grain growth around 1 AU show that also bouncing (i.e. collisions without sticking or fragmentation) can Regime νγUpper end play an important role (see Weidling et al. 2009; Zsom et al. 2010). However, changes in material composition such as or- Brownian motion regime 1 − 1 a 6 2 BT ganic or ice mantles or the monomer size are expected to change Turbulent regime I 1 0 a12 this picture. As there is still a large parameter space to be ex- 5 Turbulent regime II 6 0 amax plored, we continue with the rather simple recipe of sticking, fragmentation and cratering outlined above.

Table 1 (for a derivation, see Appendix B) . If settling is to be 4.3. Regime boundaries included, then the ν index for particle sizes above asett has to be 1 γ ν increased by 6 (see Sect. 3.2) while remains the same. The From Eq. (24), we can calculate the slope of the distribution in and γ indices for all three regimes can be found in Table 1. the different regimes if we assume that the slope of the distribution at a given grain size always follows from the kernel index ν. 4.2. Fragmentation and cratering To construct a whole distribution consisting of several power- laws for each regime, we need to know where each of the differ- We introduce fragmentation and cratering according to the ent relative velocity regime applies. recipe of Birnstiel et al. (2010a): whether a collision leads to It is important to note that relative velocities due to Brownian sticking or to fragmentation/cratering is determined by the frag- motion decrease with particle size whereas the relative velocities mentation probability induced by turbulent motion increase with particle size (up to ⎧ St = 1). Therefore, Brownian motion dominates the relative ve- ⎪ 0ifΔu < uf − δu ⎪ locities for small particles, while for larger particles, turbulence ⎨⎪ dominates. From numerical simulations, we found that at those = Δu≥u pf ⎪ 1iff (35) sizes where the highest turbulent relative velocities (i.e. col- ⎪ ⎩⎪ −Δ lisions with the smallest grains) start to exceed the smallest − uf u 1 δu else Brownian motion relative velocities (i.e. collisions with similar sized grains), the slope of the distribution starts to be determined which means that all impacts with velocities above the critical by the turbulent kernel slope. By equating the approximate rela- break-up threshold u lead to fragmentation or cratering while f tive velocity of Ormel & Cuzzi (2007)andEq.(32), the accord- impacts with velocities below u − δu lead to sticking. The width f ing grain size can be estimated to be of the linear transition region δu is chosen to be 0.2 uf,since laboratory experiments suggest that there is no sharp fragmenta- ⎡ ⎤ 2 − 1 5 ⎢ Σ μ π 2 ⎥ tion velocity (Blum & Muench 1993; Güttler 2010, priv. comm.). ⎢8 g − 1 mp 4 ⎥ aBT ≈ ⎣⎢ · Re 4 · · ρs ⎦⎥ , (37) Our simulation results do not show a strong dependence on this πρs 3παt 3 value, which was tested by varying δu by a factor of 2. The mass ratio of the two particles determines whether the where we approximate the Reynolds number (i.e., the ratio of impact completely fragments the larger body (masses within one turbulent viscosity νt = αcs Hp over molecular viscosity) near order of magnitude) or if the smaller particle excavates mass the disk mid-plane by from the larger body (masses differ by more than one order of α Σ σ magnitude). This distinction between an erosion and a shatter- t g H2 Re ≈ · (38) ing regime follows the numerical studies of Paszun & Dominik 2 μ mp (2009)andOrmeletal.(2009) and experimental studies of Güttler et al. (2010). These works do not precisely constrain the Here, σH is the cross section of molecular hydrogen (taken to 2− mass ratio which distinguishes between both regimes, however be 2 × 10 15 cm2)andμ = 2.3 is the mean molecular weight in our simulation results do only weakly depend on it. proton masses mp. In the case of fragmentation, the whole mass of both colli- Turbulent relative velocities strongly increase for grains with sion partners is redistributed to all masses smaller than the larger a stopping time that is larger or about the turn-over time of the body according to Eq. (10). smallest eddies. More specifically, the Stokes number of the par- In the case of cratering, it is assumed that the smaller particle ticles at this change in the relative velocity is (with mass mimp) excavates its own mass from the larger body. 1 tη 1 − 1 The mass of the impactor as well as the excavated mass is then 2 St12 = = Re , (39) redistributed to masses smaller than the impactor mass according ya tL ya to Eq. (10). Thus, the total redistributed mass equals − 1 where tL = 1/Ωk and tη = tL · Re 2 are the turn-over times of the mimp Ω · = largest and the smallest eddies, respectively, and k is the Kepler n(m) m dm 2 mimp (36) y m0 frequency. Ormel & Cuzzi (2007) approximated the factor a to be about 1.6. The corresponding grain size in the Epstein regime and the mass of the larger body is reduced by 2 mimp. is therefore given by Most parameters such as the fragmentation velocity or the Σ amount of excavated material during cratering are not yet well 1 2 g − 1 = · 2 . enough constrained (for the most recent experimental work, see a12 Re (40) ya πρs Blum & Wurm 2008; Güttler et al. 2010, and references therein). Experiments suggest fragmentation velocities of a few m s−1 and As mentioned above, the Brownian motion relative velocities of fragment distributions with ξ between 1 and 2 (Güttler et al. small grains decrease with their size. For larger sizes, the rela- 2010 find ξ values between 1.07 and 1.37 for SiO2 grains). tive velocities due to turbulent motion are gaining importance,

A11, page 8 of 16 T. Birnstiel et al.: Size distributions of grains in coagulation/fragmentation equilibrium which are increasing with size until a Stokes number of unity. to smaller sizes. Consequently, the mass distribution at smaller For typical values of the sound speed sizes increases relative to the values of smaller ξ values. The strong influence of the fragmentation threshold velocity u can be seen in the upper right panel in Fig. 6: according to = kb T f cs (41) Eq. (43), an order of magnitude higher u leads to a 100 times μmp f larger maximum grain size. and the turbulence parameter α , the largest turbulent relative The grain size distributions for different levels of turbulence √ t ff velocity Δumax ≈ αt cs exceed the critical collision velocity are shown in the middle left panel of Fig. 6.Thee ects are two- − α of the grains (which is of the order of a few m s 1) and therefore fold: firstly, an increased t leads to increased turbulent relative leads to fragmentation of the dust particles. In the case of very velocities, thus, moving the fragmentation barrier amax to smaller α quiescent environments and/or larger critical collision velocities, sizes (cf. Eq. (43)). Secondly, a larger t shifts the transition particles do not experience this fragmentation barrier and can within the turbulent regime, a12, to smaller sizes. Consequently, α continue to grow. Hence, a steady state is never reached. The the second turbulent regime gains importance as t is increased work presented here focuses on the former case where since its upper end lower boundary extend ever further. The middle right panel in Fig. 6 displays the influence of an Δumax > uf, (42) decreased gas surface density Σg (assuming a fixed dust-to-gas ratio). It can be seen that not only the total mass is decreased and grain growth is always limited by fragmentation. due to the fixed dust-to-gas ratio but also the upper size of the As relative turbulent velocities are (in our case) increasing distribution amax decreases. This is due to the coupling of the with grain size, we can relate the maximum turbulent relative ve- dust to the gas: with larger gas surface density, the dust is bet- locity and the critical collision velocity to derive the approximate ter coupled to the gas. This is described by a decreased Stokes maximum grain size which particles can reach (see Birnstiel number (see Eq. (26)) which, in turn, leads to smaller relative et al. 2009) velocities and hence a larger amax. Interestingly, the shape of the 2 grain size distribution does not depend on the total dust mass, 2Σg u a · f · (43) but on the total gas mass, as long as gas is dynamically dominat- max πα ρ 2 t s cs ing (i.e., Σg Σd). If more dust were to be present, grains would collide more often, thus, a steady-state would be reached faster 4.4. Resulting steady-state distributions and the new size distribution would be a scaled-up version of the former one. However the velocities at which grains collide are The parameter space is too large to even nearly discuss all pos- determined by the properties of the underlying gas disk. In this sible outcomes of steady state grain size distributions. We will way, the dust grain size distribution is not only a measure of the therefore focus on a few examples and rather explain the basic dust properties, but also a measure of the gas disk physics like features and the most general results only. For this purpose, we the gas density and the amount of turbulence. will adopt a fiducial model and consider the influences of several The shape of the size distribution for different grain volume ξ α Σ parameters on the resulting grain size distribution: , uf, t, g, densities ρs does not change significantly. However most regime ρ s,andT (see Fig. 6). boundaries (asett, a12,andamax) are inversely proportional to ρs The fiducial model (see the solid black line in Fig. 6)shows (because of the coupling to the gas, described by the Stokes num- the following features: steep increase from the smaller sizes un- ber). A decrease (increase) in ρs therefore shifts the whole dis- til a few tenth of a micrometer. This relates to the regime domi- tribution to larger (smaller) sizes as can be seen in the lower left nated by Brownian motion relative velocities. The upper end of panel in Fig. 6. this regime can be approximated by Eq. (37). The flatter part The upper end of the distribution, amax, is inversely propor- of the distribution is caused by a different kernel index ν in the tional to the mid-plane temperature T (as in the case of the tur- parts of the distribution which are dominated by turbulent rela- bulence parameter) whereas the transition between the different μ tive velocities. The dip at about 60 m(cf.Eq.(40)) is due to the turbulent regimes a12 does not. Therefore, increasing the tem- jump in relative velocities as the stopping time of particles above perature decreases amax in the same way as decreasing Σg does. this size exceeds the shortest eddy turn-over time (see Ormel & However a12 is not influenced by temperature changes, there- Cuzzi 2007). fore the shape of the size distribution changes in a different way The upper end of the distribution is approximately at amax. than in the case of changing Σg as can be seen by comparing the The increased slope of the distribution and the bump close to the middle right and the lower right panels of Fig. 6. upper end are caused by two processes. Firstly, a boundary effect: grains mostly grow by collisions with similar or larger sized particles. Grains near the upper end of the distribution lack larger 5. Fitting formula for steady-state distributions sized collision partners and therefore the number density needs to increase in order to keep the flux constant with mass (i.e. in In this section, we will describe a simple recipe which allows us order to keep a steady-state). Secondly, the bump is caused by to construct vertically integrated grain size distributions which cratering: impacts of small grains onto the largest grains do not fit reasonably well to the simulation results presented in the pre- cause growth or complete destruction of the larger bodies, in- vious section. stead they erode them. Growth of these larger bodies is therefore The recipe does not directly depend on the radial distance slowed down and, similar to the former case, the mass distribu- to the star or on the stellar mass. A radial dependence only en- tion needs to increase in order to fulfill the steady-state criterion ters via radial changes of the input parameters listed in Table 4. (“pile-up effect”). This recipe has been tested for a large grid of parameter values3 The upper left panel in Fig. 6 shows the influence of the (shown in Table 2), however there are some restrictions. distribution of fragments after a collision event: larger values of ξ mean that more of the fragmented mass is redistributed 3 See also http://www.mpia.de/distribution-fits

A11, page 9 of 16 A&A 525, A11 (2011)

ξ =1.1 u f = 1 m/s ξ =1.5 u f = 3 m/s ξ =1.8 u f = 10 m/s

] 0 2 10 − [g cm a ·

m −2 · 10 ) a ( N

−4 10

−4 −2 α t =10 Σg =10gcm −3 −2 α t =10 Σ =100gcm − g α =10 2 −2 t Σg =500gcm

] 0 2 10 − [g cm a ·

m −2 · 10 ) a ( N

−4 10

−3 ρS =0.1gcm T =10K −3 T =50K ρS =1.6gcm −3 T = 500 K ρS =3.0gcm

] 0 2 10 − [g cm a ·

m −2 · 10 ) a ( N

−4 10

−5 −4 −3 −2 −1 0 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 10 10 10 10 10 grain size [cm] grain size [cm]

Fig. 6. Fiducial model and variations of the most important parameters: fragmentation power-law index ξ, threshold fragmentation velocity uf , turbulence parameters αt, surface density Σg, particle internal density ρs, and mid-plane temperature T. The shape of the vertically integrated size distributions does not depend on the stellar mass or the distance to the star (only via the radial dependence of the parameters above).

5.1. Limitations Table 2. Parameter values for which the recipe presented in Sect. 5 has been compared to simulation results. These fits strictly apply only for the case of ξ = 11/6. In this case, the slopes of the distribution agree well with the predic- Parameter Unit Values tions of the intermediate regime (case B, defined in Sect. 2.2). α −4 −3 −2 Forsmallervaluesofξ, the slopes do not strictly follow the an- t 10 10 10 ––– T [K] 10 100 500 103 –– alytical predictions. This is due to the fact that we include cra- −2 3 4 Σg [g cm ] 0.1 1 10 100 10 10 tering, which is not covered by our theory. Erosion is therefore u [m s−1]1310––– an important mode of fragmentation: it dominates over complete f disruption through the high number of small particles (see also Notes. αt is the turbulence parameter, T is the mid-plane temperature, Kobayashi & Tanaka 2010) and it is able to redistribute signifi- Σg is the gas surface density, and uf is the critical collision velocity. cant amounts of mass to the smallest particle sizes. One important restriction for this recipe is the upper size of the particles amax: it needs to obey the condition of Eq. (42), high-velocity impacts and a steady state will never be reached since otherwise, particles will not experience the fragmenting since particles can grow unhindered over the meter-size barrier.

A11, page 10 of 16 T. Birnstiel et al.: Size distributions of grains in coagulation/fragmentation equilibrium

3 There are also restrictions to a very small amax:ifamax is 10 closetoorevensmallerthana12, then the fit will not represent Δu mon the true simulation outcome very well. In this case, the upper Δu eq end of the distribution, and in more extreme cases the whole fit: Δu mon distribution will look much different. Thus, the sizes should obey 2 fit: Δu eq the condition 10 ] 1 − 5 μm < a12 < amax, (44) [cm s

where each inequality should be within a factor of a few. u Δ 1 10 5.2. Recipe The following recipe calculates the vertically integrated mass distribution of dust grains in a turbulent circumstellar disk within the the above mentioned limitations. The recipe should be ap- 0 10 plied on a logarithmic grain size grid a with a lower size limit −3 −2 −1 0 i < 10 10 10 10 of 0.025 μm and a fine enough size resolution (ai+1/ai ∼ 1.12). grain size [cm] For convenience, all variables are summarized in Table 4.The steps to be performed are as follows: Fig. 7. Comparison of the turbulent relative velocities of Ormel & Cuzzi (2007) to the fitting formula used in the recipe (see Eqs. (48)and(45)) 1. Calculate the grain sizes which represent the regime bound- for grain size distributions. The largest error in the resulting upper grain size a derived from the fitting formula is about 25%. aries aBT, a12, asett whicharegivenbyEqs.(37), (40), P and (27). 2. Calculate the turbulent relative velocities for each grain size. Table 3. Power-law exponents of the distribution n(m) · m · a. For this, we approximate the equations which are given by Ormel & Cuzzi (2007). Collision velocities with monomers δ Regime i are given by ai < asett ai ≥ asett Δ mon = 3 5 ui Brownian motion regime ⎧ 2 4 ⎪ 1 ⎪Re 4 ·(St −St )fora . where μ = 2.3, σH = 2 × 10 cm and ya = 1.6. 3 ui for ai a12 2 5. The power-law indices of the mass distribution δi for each A comparison between these approximations and the formu- interval between the regime boundaries (aBT, a12, asett, las of Ormel & Cuzzi (2007) is shown in Fig. 7. aP) are calculated according to the intermediate regime 3. Using Eqs. (45)and(48) for the relative velocities, and the (cf. Eq. (18)). The slopes have to be chosen according to the transition width δu = 0.2uf, find the grain sizes which corre- kernel regime (Brownian motion, turbulent regime 1 or 2) spond to the following conditions: and according to whether the regime is influenced by settling – aL: particles above this size experience impacts with or not (i.e., if a is larger or smaller than asett). The resulting Δ mon ≥ − δ slopes of the mass distribution are given in Table 3.Thefirst monomers with velocities of ui uf u (i.e., cratering starts to become important). version of the fit f (ai)(whereai denotes the numerical grid – a : particles above this size experience impacts with point of the particles size array) is now constructed by using P δ Δ eq ≥ − δ ∝ i equal sized grains with velocities of ui uf u power-laws ( ai ) in between each of the regimes, up to aP. (i.e., fragmentation becomes important). The fit should be continuous at all regime boundaries except

A11, page 11 of 16 A&A 525, A11 (2011)

Table 4. Deﬁnition of the variables used in this paper.

Variable Deﬁnition Unit

αt Turbulence strength parameter – −1 uf Fragmentation threshold velocity cm s −2 Σg Gas surface density gcm −2 Σd Dust surface density gcm ξ Power-law index of the mass distribution of fragments, see Eq. (10)– T Mid-plane temperature K Input variables −3 ρs Volume density of a dust particle g cm 1/3 a Grain size, a = (3 m/(4πρs)) cm α Slope of the number density distribution n(m) ∝ m−α,seeEq.(4)– aBT Eq. (37) cm a12 Eq. (40) cm amax Eq. (43) cm aL Left boundary of the bump function b(a), see Sect. 5, paragraph 3 cm aP Peak size of the bump function b(a), see Sect. 5, paragraph 3 cm aR Right boundary of the bump function b(a), see Sect. 5, paragraph 3 cm asett Eq. (27) cm ainc Eq. (52) cm b(a) Bump function, see Eq. (53) – 3 −1 Cm1,m2 Collision kernel, see Eq. (3) cm s ˜ ff 2 −1 Cm1,m2 Collision kernel including settling e ects, see Eq. (3)cms −1 cs Sound speed, see Eq. (41) cm s δi Slopes of the fit-function, see Table 3 – −1 δu Width of the transition between sticking and fragmentation, taken to be 0.2 uf cm s Interpolation parameter, see Eq. (47)– γ Power-law index of the function h(m2/m1) for large ratios of m2/m1,asdefinedinEq.(21)– Other variables J Empirical parametrization of the discontinuity, see Eq. (50)– K Integral defined in Eq. (7) – −1 kb Boltzmann constant erg K 3 m Mass of the particle, m = 4π/3 ρs a g m0 Monomer mass g m1 Mass above which particles fragment g μ Mean molecular weight in proton masses, taken to be 2.3 – mp Proton mass g n(m) Number density distribution gcm−3 N(m) Vertically integrated number density distribution g cm−2 ν Degree of homogeneity of the kernel as defined in Eq. (3)– Re Reynolds number, see Eq. (38) – −3 ρs Density of a dust grain gcm St Mid-plane Stokes number in the Epstein regime, see Eq. (26)– σ 2 H2 Cross-section of molecular hydrogen cm −1 ugas Mean square turbulent gas velocity, see Eq. (46)cms Δ mon −1 ui Relative velocities between monomers and grains of size ai,seeEq.(45)cms Δ eq −1 ui Relative velocities between grains of size ai,seeEq.(48)cms V Parameter used in the calculation of J,seeEq.(49)cms−1 ya Equals 1.6, parameter from Ormel & Cuzzi (2007)–

Notes. The variables grouped as “input variables” are the parameters of the ﬁtting recipe in Sect. 5. “Other variables” summarizes the deﬁnitions of all other variables used in this recipe.

adropof1/J at the transition at a12. An example of this first 7. The bump due to cratering is mimicked by a Gaussian, version is shown in the top left panel of Fig. 8. 6. Mimic the cut-off effects which cause an increase in the dis- (a − a )2 b(a ) = 2 · f (a ) · exp − i P , (53) tribution function close to the upper end: linearly increase i L σ2 the distribution function for all sizes ainc < a < aP: a − a where σ is defined as f (a ) → f (a ) · 2 − i P , (51) i i a − a inc P min (|a − a | , |a − a |) σ = R √ P L P , (54) with ln(2) = . . ainc 0 3 aP (52) but should be limited to be at least The resulting fit after this step is shown in the upper right panel of Fig. 8. σ>0.1 aP. (55)

A11, page 12 of 16 T. Birnstiel et al.: Size distributions of grains in coagulation/fragmentation equilibrium

−1 10 power−laws boundary effects

−2 ]

2 10 −

[g cm −3