Issue 
A&A
Volume 572, December 2014



Article Number  A72  
Number of page(s)  31  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201323021  
Published online  01 December 2014 
Online material
Appendix A: The minimum mass solar nebula scaling
We provide in Table A.1 the main quantities that are used in the article and present their scaling in the socalled minimum mass solar nebula formalism (Hayashi 1981; Nakagawa et al. 1986).
Expressions for the minimum mass nebula scaling used in this article.
Appendix B: Geometric collision probability with shear
We derive the expression for the geometric collision probability and show that it is identical to that derived by Ormel & Klahr (2010). From Eqs. (23a) and (23b) and accounting for both the (assumed inward) radial velocity of the gas due turbulent viscosity v_{ν} and the shear in the disk, where, following Ormel & Klahr (2010), we have used an expansion around the location of the planetesimal (Ω_{K,0} ≡ Ω_{K}(x = 0)), and x is the radial distance from the planetesimal at which the collision with the dust particle takes place. For simplicity, we assume that on average x = R_{p}/ 2. This yields the impact rate in the geometrical limit as: (B.2)where we have dropped the 0 labels. When v_{ν} = 0, this expression is equivalent to Eq. (22) of Ormel & Klahr (2010) who derive this impact rate by an analysis of the trajectory of dust particles. In most cases, we will consider ητ_{s} ≪ R_{p}/r so that shear may be neglected and : the collision rate is the product of the cross section 2R_{p} and the encounter velocity .
The 2D probability that a planetesimal will accrete a given dust grain is then given by Eq. (35): (B.3)This equation may be approximated in the different regimes (from small to large particles): (B.4)The lowest capture probability R_{p}/πr corresponds to dust particles with τ_{s} ~ 1 which drift in with a speed that is comparable to the headwind felt by the planetesimal. One can expect that these particles will be the most difficult to filter. For smaller τ_{s} values, the slower drift leads to a higher probability, limited by the gas drift. For large τ_{s} values, the drift rate also decreases and the effect of the Keplerian shear across the planetesimal becomes dominant. The capture probability then becomes a very steep function of the planetesimal size (∝ in the Epstein regime and ∝ in the Stokes regime). However, shear becomes important only for very large planetesimals ( for the MMSN). It only concerns cases for which the collisions take place in the settling or threebody regimes. Shear can thus be neglected for the geometrical regime.
Appendix C: Analytical estimates for small grains (geometric, hydro, and settling regimes)
We provide here analytical estimates for the collision probabilities and filtering efficiencies for the geometric and settling regimes. In order to provide tractable relations, we make the following simplifications:

We study only small particles such that τ_{s}< 1, corresponding generally to s_{~}^{<}1 m.

Given the above assumption, we use Δv = ηv_{K} for the dustplanetesimal encounter velocity.

We approximate χ_{α,τs} defined by Eq. (46)as follows: (C.1)We thus separate small particles with τ_{s} ≤ τ_{s,ν} ~ 0.3^{2}α which have not settled from larger particles for which h_{d}(τ_{s}) <h_{g}.

We consider cases for which the planetesimal capture radius is smaller than the dust disk scale height and thus Eq. (43)can be simplified as: (C.2)

We do not consider the Safronov and threebody regimes.
Fig. C.1
Value of the effective capture radius b_{set} in units of the Bondi radius r_{B} from Eq. (90)(colored curves) together with the approximation used in Eq. (C.3)(thick black lines). Left panel: solution when the Bondi radius is smaller than the Hill radius. Right panel: solution when the Bondi radius is larger than the Hill radius (specifically, the solution of Eq. (90)is shown for R_{B}/R_{H} = 10). 

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Given a known encounter velocity, the calculation of the collision probability only requires that of the planetesimal effective capture radius. We know that it is extremely small in the hydro regime and equal to R_{p} in the geometric regime. In the settling regime, it is given by Eq. (90). As shown in Fig. 9, the settling regime may be subdivided into a Bondi regime in which the capture radius is proportional to , and a Hill regime in which the capture radius is approximatively independent of τ_{s} and equal to the Hill radius. We thus approximate the effective capture radius of planetesimals/embryos in the different regimes as (C.3)and is defined by Eq. (88). The different regimes are defined by the following relations: (C.4)The dimensionless stopping times for the different regimes are defined by Eq. (88)for , Eq. (100)and τ_{f} = 1 for , for , and for τ^{Hill}. These and their MMSN approximations are thus respectively:
It can be seen that the value of is proportional to μ_{p} and reaches for the Hilldominated settling regime (R_{p} ~ 1000 km). Importantly, for , with
Appendix C.1: Collision probabilities
Using Eqs. (C.2), (31), (35), (80), (81), and (88), the collision probabilities for the three regimes defined in Eq. (C.3)can be written (C.11)When using the approximation from Eq. (C.1)and the MMSN scalings from Appendix A, this yields for small particles such that τ_{s}< 0.09α: (C.12)and for mediumsized particles such that : (C.13)We note that in these expressions, the scaling of the particle size (stopping time) was adjusted to be centered on the maximum collision probability in each regime.
Fig. C.2
Value of the collision probability obtained by the full theory (black contours) and by the simplified one (red contours), for the MMSN and α = 10^{4}. The two panels are for two orbital distances: 0.1 AU (left) and 5 AU (right). The yellow lines are limiting values of the stopping time, (Eq. (C.5)), (Eq. (C.6)), (Eq. (C.7)), and (Eq. (C.8)), respectively. 

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Figure C.2 shows that the simplified solutions are good approximations of the full solutions, except for large grains with τ_{s}_{~}^{>}2 and at the interface between the settling and geometric regimes (along the line). In the settling regime, the maximum collision probability is obtained for dust such that , i.e., with a stopping time equal to the Hill sphere crossing time. For a low value of the turbulence parameter α = 10^{4}, a 1 M_{⊕} embryo (corresponding to R_{p} = 11250 km for a density) would have / for particles such that , i.e., for centimetersized pebbles between 1 AU and 10 AU.
Appendix C.2: Filtering efficiency
The filtering efficiency as defined by Eq. (58)writes for the regimes considered: (C.14)The MMSN scaling then yields, for small particles such that τ_{s}< 0.09α: (C.15)and for mediumsized particles such that : (C.16)
Fig. C.3
Value of the filtering efficiency log x_{filter} obtained by the full theory (black contours) and by the simplified one (red contours), for the MMSN and α = 10^{4}. The two panels are for two orbital distances: 0.1 AU (left) and 5 AU (right). (See Fig. C.2 for a full description.) 

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Figure C.3 compares the values of x_{filter} obtained with the full theory and the simplified one, again showing good agreement except for large grains beyond meter size and near the line. For increasing planetesimal sizes, the maximum filtering efficiency is obtained along , then and finally .
Appendix D: Results for a disk with α = 10^{4}
Given the inefficient filtering obtained in the high turbulence (α = 10^{2}) case, we now consider the weak turbulence case (α = 10^{4}). This is more favorable because dust settles closer to the midplane and the mean gas flow is also slower.
Figures D.1 and D.2 show the resulting filtering efficiency, both at 1 AU and for 1 mm dust, as a function of orbital distance. At 1 AU, efficient filtering is achieved for a wider range of dust and planetesimal sizes, basically for dust of 10 μm to 1 cm and planetesimals of less than a few kilometers in radius. A small island with x_{filter}_{~}^{>}1 also appears in the settling regime, for metersized dust and planetesimals of ~1000 km in radius. Compared to Figs. 18 and 19 in the α = 10^{2} case, there is about an order of magnitude increase in the filtering efficiency.
Fig. D.1
Contours of the filtering efficiency by a MMSN planetesimal disk at 1 AU for α = 10^{4}, assuming monodisperse size distributions for planetesimals and dust grains. The contour indicating perfect filtering efficiency for log _{10}x_{filter} = 0 is shown in bold. 

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Fig. D.2
Contours of the filtering efficiency by a MMSN planetesimal disk for 1 mm dust for α = 10^{4}, assuming a monodisperse size distribution for planetesimals. The contour indicating perfect filtering efficiency for log _{10}x_{filter} = 0 is shown in bold. 

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Figure D.3, shows how a lower value of the turbulence parameter α = 10^{4} affects filtering by a distribution of planetesimals between 1 km and 1000 km. Compared to Fig. 20 for α = 10^{2}, filtering is found to be more efficient of course. For example, dust of 1 mm in size can now be efficiently captured in an MMSN disk inside about 0.03 AU while a value of x_{filter} = 1 is never reached for the high turbulence case until the gas disk has shrunk to about 1% of the MMSN value. Similarly, 1 cm grains can be captured with an efficiency close to unity inside of 0.1 AU compared to about 0.05 AU for the α = 10^{2} case. In the 0.1−1 AU range, a wide range of small grains may be captured with an efficiency between 1 and 0.1. The dependence as a function of remains relatively weak, as in the strong turbulence case.
Fig. D.3
Filtering efficiency of a swarm of planetesimals with radii between 1 km and 1000 km as a function of orbital distance and mass of the gas disk for α = 10^{4}. (See Fig. 20 for details.) 

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Appendix E: Dependence on the planetesimal scale height
All the results presented so far have assumed a rather low value of the planetesimal scale height h_{p} = 0.01h_{g}. This favors the filtering of particles able to settle to the midplane in a thin plane. In Fig. E.1 we show how the integrated filtering efficiency X_{filter} depends on the value of h_{p}. We select a low value of the turbulence parameter α = 10^{4} because this is where the differences are most important. With an infinitely thin planetesimal disk (h_{p} = 0, left panels), we obtain results that are very similar to our fiducial case (h_{p}/h_{g} = 0.01) shown in Fig. 22.
Fig. E.1
Filtering efficiency of dust from 1 micron to 100 meters by a MMSN planetesimal belt with planetesimals of 1 − 10,000 km in radius extending from 0.1 to 35 AU for a turbulent parameter α = 10^{4} and various ratios of the planetesimal to gas scale height, from an infinitely thin planetesimal disk with h_{p}/h_{g} = 0 (left panels), h_{p}/h_{g} = 0.1 (middle panels), and h_{p}/h_{g} = 1.0 (right panels). As in Figs. 21 and 22, the top panels show the contours of the diskintegrated filtering factor X_{filter} while the bottom panels show the orbital distance at which X_{filter}(r) = 1 for dust particles drifting in from beyond 35 AU. 

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When we increase the planetesimal scale height, the filtering of large particles becomes less efficient, as shown by the middle and right panels of Fig. E.1. Specifically, compared to the case when h_{p} = 0, changes occur for particles larger than about 1 cm when we consider h_{p}/h_{g} = 0.1 and for particles larger than about 0.1 cm when we consider h_{p}/h_{g} = 1.0.
We first consider the limit for which Eq. (43)becomes independent of h_{p} and the problem may be considered 2D. This occurs when the cross section of the embryos becomes so large that b> (2/. By writing β ≡ R_{H}/b, one may show that this occurs when (E.1)For the MMSN disk at 1 AU, this is equivalent to For large embryos we expect β_{~}^{>}1 with a significant dependence on τ_{s} (see Appendix C). In the regime that we considered in this work Eq. (E.1)is not satisfied, but it would be for embryos larger than a few Earth masses which may then accrete more efficiently.
The decrease of the filtering efficiency with increasing h_{p} value seen in Fig. E.1 is hence a direct consequence of the fact that when Eq. (E.1)is not satisfied, the 3D collision probability given by Eq. (43)depends on the maximum of h_{d} and h_{p}. The probability thus becomes dependent on h_{p} when h_{p}>h_{d}, that is when (E.2)For α = 10^{4}, this implies that we expect lower collision probability and filtering efficiency when compared to the h_{p} = 0 case when τ_{s} = 10^{2} for h_{p}/h_{g} = 0.1 and τ_{s} = 10^{4} for h_{p}/h_{g} = 1. According to Fig. 3 (around 1 AU), this corresponds to s ≈ 1 cm and 1 mm, in good agreement with the results of Fig. E.1.
The question whether large boulders may be efficiently filtered by planetary embryos thus depends crucially on whether these embryos effectively lie close to the midplane. In the case of a weakly turbulent disk with α = 10^{4}, this is the case when the ratio of the embryo to gas scale height h_{p}/h_{g} = 0.01 or lower, but we notice that an efficient filtering in the settling regime becomes limited to pebbles less than about a meter in size when h_{p}/h_{g} = 0.1 and that it mostly disappears for h_{p} = h_{g}. A detailed, sizedependent calculation of h_{p} is therefore critical to determine whether planetary embryos may filter large particles in the disk and grow through that mechanism.
© ESO, 2014
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