FROM DUST TO PLANETESIMAL: THE SNOWBALL PHASE?

Ji-Wei Xie; Matthew J. Payne; Philippe Thébault; Ji-Lin Zhou; Jian Ge

doi:10.1088/0004-637X/724/2/1153

1. INTRODUCTION

The standard core-accretion model generally treats planet formation as occurring in two stages (Lissauer 1993; Chambers 2004; Armitage 2010, p. 294). In the first stage, mountain-sized planetesimals form from small dust grains embedded in a gas-rich protoplanetary disk (Weidenschilling 1997; Blum & Wurm 2008; Youdin 2008; Chiang & Youdin 2010) and then in the second stage these planetesimals go on to accrete one another to form planetary embryos (Kokubo & Ida 1996, 1998; Weidenschilling et al. 1997) which eventually go on to merge into full planets (Chambers & Wetherill 1998; Levison & Agnor 2003; Kokubo et al. 2006). The first stage—planetesimal formation—is of crucial importance as it sets the initial conditions upon which subsequent stages of evolution depend. However, the details of planetesimal formation are still poorly understood and remain somewhat controversial.

One model for the formation of planetesimals envisages them growing via mutual sticking collisions (Weidenschilling & Cuzzi 1993; Weidenschilling 1997). Laboratory experiments show that μm-sized dust grains can stick together efficiently through various surface forces, including the van der Waals and electrostatic forces, causing them to form mm- to cm-sized aggregates (Blum & Wurm 2008). Beyond this mm to cm range, particles become less sticky while their gravitational interactions remain weak, leading to collisional disruption rather than growth (Dominik et al. 2007). Furthermore, cm- to m-sized objects begin to decouple from the gas, so that they experience collisions with higher relative velocities and also begin to experience significant aerodynamic drag which causes them to quickly spiral into the protostar on a timescale of a few hundred orbital periods (Weidenschilling 1977)—the well-known "meter barrier."

An alternative scenario that has been suggested to try and circumvent this meter barrier envisages that the kilometer-sized planetesimals form directly via gravitational instabilities in a dense particle subdisk near the midplane of the protoplanetary disk (Safronov 1969; Goldreich & Ward 1973; Youdin & Shu 2002). However, it has been pointed out that even low levels of turbulence would prevent solids from settling down to a sufficiently dense midplane layer (Weidenschilling 1980; Cuzzi & Weidenschilling 2006).

Recently, breakthroughs have been made in both scenarios. For the collisional scenario, Teiser & Wurm (2009) found in experiments that high-velocity (tens of ms⁻¹) collisions between small dust particles (<cm) and a pre-planetesimal body (>dm) can lead to efficient growth, suggesting that some "lucky" pre-planetesimals may escape from erosive collisions and go on to form kilometer-sized planetesimals (Johansen et al. 2008).

For the instability scenario, it is found that turbulence can act to concentrate dust, thus providing a high-density, low-velocity dispersion environment in which planetesimals may form via various models of instability (Johansen et al. 2007; Cuzzi et al. 2008). The planetesimals formed in these new turbulent models can be 1–2 orders of magnitude larger than those formed in traditional models.

Planetesimal formation is thus far from being completely understood and current models are a work in progress. However, a point worth mentioning is that none of these different scenarios ensure that all planetesimals appear at the same time at a given location. In fact, any model which requires non-global density enhancements to form planetesimals, e.g., Johansen et al. (2007) and Cuzzi et al. (2008), implicitly assumes that certain regions of the disk will form planetesimals first, while the rest of the disk remains "dusty." Furthermore, two recent studies by Chambers (2010) and Cuzzi et al. (2010) argue that, for the turbulent concentration scenario, planet formation efficiency (set by the probability for turbulently formed clumps of mm-sized grains to collapse into planetesimals) could be relatively low, depending on the values of several crucial parameters such as the dust-to-gas density ratio, disk viscosity, density, and profile. This means that, at a given location in the disk, there could be a wide time gap between the moment when the first planetesimals form and the moment when most of the available solid mass has been converted into planetesimals.

There could thus be a long transition period during which individual planetesimals are embedded in a disk where most of the mass of solids is still in small grains. This issue has tended to be ignored in studies investigating the next stage of planet formation, i.e., planetesimal accretion. Most of these studies consider a system where all planetesimals are present at time t₀, possibly with a distribution of initial sizes, which then grow by pairwise accretion (Kokubo & Ida 1996, 1998; Weidenschilling et al. 1997; Kenyon & Bromley 2006; Barnes et al. 2009). Nevertheless, there are several noteworthy exceptions. Wetherill & Inaba (2000) considered that planetesimals progressively appear over a 10⁵ yr timescale, but only considered mutual planetesimal accretion as a possible growth mode, implicitly neglecting the contribution of dust. Morbidelli et al. (2009), while attempting to fit the known asteroidal size distribution, did consider in one of their simulations the possible accretion of dust by planetesimals which formed early, but did so only for one specific case: large, 100-km-sized seed planetesimals produced over 2 Myr. In addition, Leinhardt & Richardson (2005) attempted to model accretion of dust onto planetesimals as well as planetesimal–planetesimal collisions but they started with very large planetesimals. The most promising study has been recently performed by Paardekooper & Leinhardt (2010), who envisage dust accretion onto planetesimals more explicitly, but restricted to the specific case of a close-in binary, and using only a simplified two-dimensional model.

In this study, we plan to take these pioneering studies a step further and investigate in detail planetesimal growth during the transition period when isolated planetesimals and a primordial dust disk coexist. We outline a new growth mode, which dominates during this early phase, during which planetesimals experience negligible gravitational or collisional interactions with one another, and grow mainly (or solely) via the accretion of dust or ice that they sweep up—in the manner of a rolling snowball. In this paper, we refer to this dust-fueled planetesimal growth phase as the "snowball" growth phase.

The paper is organized as follows. The snowball growth rate (growth only by dust accretion) is derived, using a semi-analytical approach, in Section 2. Next, in Section 3, we show how long the snowball phase could last and to what radial size and mass fraction the planetesimals can grow via this snowball growth mode. Then, in Section 4, the implications for planet formation in both the solar system and close binary systems are discussed. Finally, we conclude in Section 5. Additionally, all variables defined in this paper are listed alphabetically in Table 1.

Table 1. List of Variables

Variables	Meaning	Definition
a_B	Semimajor axis of the orbit of binary system	First paragraph of Section 4.2.1
a_ice	Ice boundary (a semimajor axis of ice condensation)	Just behind Equation (5)
E_d	Dust accretion efficiency factor	Equation (4)
e_B	Eccentricity of the orbit of binary system	First paragraph of Section 4.2.1
e_p	Planetesimal average orbital inclination	Second paragraph of Section 4.2.1
f_d	Enhancement factor of Σ_d from a nominal disk (MMSN)	Just behind Equation (5)
f_ice	Enhancement factor of Σ_d due to ice condensation	Just behind Equation (5)
H_d	Scale height of the dust disk	Equation (4)
H_h	Scale height of the gas disk	Equation (5)
i_B	Inclination of the orbit of binary system	Second paragraph of Section 4.2.1
i_p	Planetesimal average orbital inclination	Equation (10)
M_p0	Initial individual mass of the planetesimals	Equation (1)
$\dot{M}_p$	Planetesimal mass growth rate via dust accretion	Equation (4)
M_p	Planetesimal mass	Equation (4)
N	Surface density of the planetesimals	Equation (1)
n	Planetesimal volume density	Equation (10)
PMF	Planetesimal mass fraction in the total solid mass	Equation (9)
PMF_snow	PMF at the end of snowball phase	Equation (12)
R_p0	Initial individual radial size of the planetesimals	Third paragraph of Section 2.1
R_p	Planetesimal radii	Equation (4)
$\dot{R}_p$	Planetesimal radial growth rate via dust accretion	Equation (5)
〈R_p〉	Mass-weighted average radii of the already-formed planetesimals	Equation (8)
R_snow	〈R_p〉 at the end of snowball phase	Equation (12)
t_col	Timescale for the onset of planetesimal–planetesimal collision	Equation (10)
t^b_col	t_col in binary star systems	Equation (14)
t^s_col	t_col in single-star systems	Equation (14)
t_f.ind	Formation timescale of individual planetesimals	Second paragraph of Section 2.1
t_f	Characteristic planetesimal formation timescale	Equation (2)
t_snow	Duration of snowball phase	Equation (11)
t_tran	Transition timescale from dust-dominated to planetesimal dominated	First paragraph of Section 4.1.1
v_esc	Planetesimal escape velocity	Second paragraph of Section 4.1.2
v_k	Local Keplerian velocity	Equation (6)
v_rel	Relative velocity between planetesimal and dust	Equation (4)
ΔV	Planetesimal average relative velocity	Equation (10)
_p	Planetesimal formation efficiency factor	Equation (1)
ρ_*	Planetesimal internal density	Equation (5)
Σ_d	Dust surface density	Equation (1)

Note. The table list is in the alphabetical order of the initial of the variable name. Several variables in Greek are listed at the end of the table.

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2. MODEL

2.1. Planetesimal Formation Rate

We adopt a simplified analytical model which contains only two components: dust and planetesimals. The dust in our model is assumed to be in the submillimeter to millimeter size range as (1) this is the typical size of chondrules (Scott 2007) and (2) particles of this approximate size settle down to the midplane of the disk on a relatively short timescale of ∼10³ yr, and can survive the inward drift for as long as ∼10⁶ yr (Youdin 2008).

Individual planetesimals are assumed to be quickly produced from dust, on a timescale t_f.ind, whether through collisions (Teiser & Wurm 2009) or instabilities (Johansen et al. 2007; Cuzzi et al. 2008). A reasonable reference value for t_f.ind might be ∼10⁴ yr (Lissauer 1993). However, we do not restrict our study to t_f.ind = 10⁴ yr. In fact, as we show in Section 3, the precise value of t_f.ind is unimportant to the final results as long as it is not too long (i.e., t_f.ind < t_snow, see Section 3).

We make the simplifying assumption that, once formed from the dust, all planetesimals have the same size R_p0, which we take as the initial condition for newly formed planetesimals in our model. To account for the dispersion in times at which planetesimals are created in the system, we introduce an efficiency factor epsilon _p (0 < epsilon _p ⩽ 1) for planetesimal formation at a given location in the protoplanetary disk. We then consider a given radial distance in the disk and define the local planetesimal formation rate as

$\begin{equation} \frac{{\it dN}}{{\it dt}}=\epsilon _p\times {\Sigma _{d}\over M_{p0}}\left({1\over t_{\rm{f.ind}}} \right)={\Sigma _{d}\over M_{p0}}\left({1\over t_{f}} \right), \end{equation} \tag{ 1 }$

where N and M_p0 are the surface number density and initial individual mass of the planetesimals, respectively, and Σ_d is the dust surface density. Further, following Chambers (2010), t_f is the characteristic planetesimal formation timescale defined as

$\begin{equation} t_f = \Sigma _{d} \left(M_{p0} \frac{{\it dN}}{{\it dt}} \right)^{-1}=\frac{t_{\rm{f.ind}}}{\epsilon _p}. \end{equation} \tag{ 2 }$

The surface number density of planetesimals in the system thus increases linearly according to the relation

$\begin{equation} N=\epsilon _p\times {\Sigma _{d}\over M_{p0}}\left({t\over t_{\rm{f.ind}}} \right)={\Sigma _{d}\over M_{p0}}\left({t\over t_{f}} \right), \end{equation} \tag{ 3 }$

assuming that N = 0 at t = 0.

Note that t_f does not represent the time it takes for an individual planetesimal to form (that is t_f.ind) but rather the time it takes for most of the dust to be converted into planetesimals, as parameterized by the factor epsilon _p. If epsilon _p = 1, i.e., t_f = t_f.ind, this implies that all the planetesimals are created at the same instant. This should be treated as an idealized or limiting case, and in reality, we expect epsilon _p < 1, i.e., t_f > t_f.ind. Chambers (2010) finds that t_f can easily vary over 10 orders of magnitude, from 10⁴ to 10¹⁴ yr, depending on local conditions in the protoplanetary disk.

Our model, assuming a constant rate of planetesimal creation as well as a single initial planetesimal size, is very simplified. However, it is accurate enough for the order-of-magnitude approach adopted in this present study and allows us to identify and quantify the snowball phase in a convenient manner.

2.2. Growth Rate Via Dust Sweeping

When a planetesimal is sweeping through a dust disk, the mass growth rate through dust accretion is

$\begin{equation} \dot{M}_p=E_d \pi R_p^2{\Sigma _d\over 2H_d}v_{\rm rel}, \end{equation} \tag{ 4 }$

where M_p and R_p are the mass and radii of the planetesimal, respectively, H_d is the scale height of the dust disk, and v_rel is the relative velocity between planetesimal and dust. E_d is a dimensionless coefficient accounting for the efficiency of dust accretion onto planetesimals. According to Teiser & Wurm (2009), accretion can be very efficient as long as the dust size is smaller than mm. Hence, throughout this paper, we take a medium value E_d = 0.5. From Equation (4), we can also derive the growth rate in the planetesimal radius as

$\begin{equation} \dot{R}_p={E_d\over 8}\left({\Sigma _d\over H_g\rho _*}\right)\left({H_g\over H_d}\right)v_{\rm rel}, \end{equation} \tag{ 5 }$

where ρ_* is the planetesimal density and H_g is the scale height of the gas disk. Throughout this paper, we adopt ρ_* = 3 g cm⁻³ and use a protoplanetary disk scaled by the Minimum Mass Solar Nebula (MMSN; Hayashi 1981), which gives Σ_d ∼ 10f_df_ice(a/AU)^−3/2 g cm⁻² and H_g ∼ 0.05(a/AU)^5/4 AU, where f_d is the scaling factor of the disk mass relative to the MMSN and f_ice accounts for the enhancement of solid density beyond the ice line, a_ice. We take f_ice = 4.2 if a > a_ice, and f_ice = 1 otherwise (Ida & Lin 2004). Note that in Equations (4) and (5), we do not consider the gravitational focusing effect, since dust is well coupled with the gas while the focusing effect is only well applied to two-body dynamics in which only the gravity of the two objects themselves matters. As a consequence, v_rel is basically the local differential velocity between a large planetesimal and a gas streamline. Assuming an axisymmetric and pressure-supported gas disk, this leads to

$\begin{equation} v_{\rm rel}\sim ({H_g/a})^2v_k\sim 75 \rm \,m\,s^{-1}, \end{equation} \tag{ 6 }$

where v_k is the local Keplerian velocity at a AU. Substituting this into Equation (5), we get the growth rate in the planetesimals' radii due to the direct accretion of dust:

$\begin{equation} \dot{R}_p\sim 7\times 10^{-7}f_d f_{\rm ice}\left({H_g\over H_d}\right)\left({a\over \mbox{AU}}\right)^{-11/4} \ \ \rm km\,yr^{-1}. \end{equation} \tag{ 7 }$

Taking f_d × f_ice ∼ 1–10 and H_g/H_d ∼ 1–1000, we get $\dot{R}_p\sim 10^{-6}\mbox{ to }10^{-3} \rm \,km\,yr^{-1}$ at 1 AU.

It is worth noting from Equation (7) that the planetesimal radii follow a linear growth rate. Since the planetesimal formation rate is also assumed linear as shown in Equation (3), we can derive the mass-weighted average planetesimal radius as (see the Appendix for details)

$\begin{equation} \langle R_p\rangle \sim R_{p0} +\left({1\over 4}\right)^{1/3}\dot{R}_pt, \end{equation} \tag{ 8 }$

as well as the planetesimal mass fraction (PMF) with respect to the total solid mass:

$\begin{equation} \mbox{PMF}=\epsilon _p\left({\langle R_p\rangle \over R_{p0}}\right)^3\left({t\over t_{\rm{f.ind}}}\right)=\left({\langle R_p\rangle \over R_{p0}}\right)^3\left({t\over t_{f}}\right). \end{equation} \tag{ 9 }$

Here, 〈R_p〉 and PMF are two key statistics which trace the typical size and total mass of planetesimals in the protoplanetary disk. Note that, for the growth rate we derive in Equation (7), a constant dust surface density is implicitly assumed. In reality, dust would be consumed both through the production of new planetesimals and through the growth of planetesimals which formed earlier. To account for this depletion, we use PMF as an alarm or flag: if PMF>0.5, we then understand that a large fraction of dust has been consumed and we turn off planetesimal growth via dust accretion.

2.3. Growth via Mutual (Planetesimal–Planetesimal) Accretion

As planetesimals begin to populate the disk, they can begin having mutual encounters. The timescale for the onset of mutual collisions decreases as the number density increases. Assuming equipartition between in-plane and out-of-plane motions, we get

$\begin{eqnarray} t_{\rm col} &\sim & 1/(n \Delta V\pi \langle R_p\rangle ^2) \nonumber \\ &\sim & {2\over 3\pi }\left({R_{p0}^3\rho _*\over \langle R_p\rangle ^2\Sigma _d}\right)\left({t\over t_{f}}\right)^{-1}\left({a\over \rm AU}\right)^{3/2} \ \ \rm yr, \end{eqnarray} \tag{ 10 }$

where n ∼ N/(2ai_p) and ΔV ∼ 2i_pv_k are the volume number density and average relative velocity of the planetesimals, respectively, and i_p is their average orbital inclination.

3. SNOWBALL GROWTH

We define snowball growth as beginning with the formation of the first planetesimal and ending at the point when mutual planetesimal collisions take over as the dominant growth mode, i.e., when t = t_col. If the snowball growth is efficient, i.e., 〈R_p〉 ≫ R_p0, then the duration of the snowball phase t_snow can be approximately derived as

$\begin{eqnarray} t_{\rm snow}& \sim & 10^4 \left(f_d f_{\rm ice}\right)^{-1/4}\left({t_{f}\over \rm 10^4 \,yr}\right)^{1/4}\left({R_{p0}\over \rm km}\right)^{3/4}\nonumber\\ &&\times \left({\dot{R}_p\over \rm 10^{-4}\,km\,yr^{-1}}\right)^{-1/2}\left({a\over \rm AU}\right)^{3/4} \ \ \rm yr. \end{eqnarray} \tag{ 11 }$

Given t_snow and using Equations (8) and (9), the mass-weighted size (R_snow) and the mass fraction of planetesimals (PMF_snow) at the end of the snowball growth phase can be estimated as

$\begin{eqnarray} R_{\rm snow}&\sim & R_{p0}+0.7\left(f_d f_{\rm ice}\right)^{-1/4}\left({t_{f}\over \rm 10^4 \,yr}\right)^{1/4}\left({R_{p0}\over \rm km}\right)^{3/4} \nonumber \\ && \times \left({\dot{R}_p\over \rm 10^{-4}\,km\,yr^{-1}}\right)^{1/2}\left({a\over \rm AU}\right)^{3/4} \ \rm km \end{eqnarray} \tag{ 12 }$

and

$\begin{eqnarray} \mbox{PMF}_{\rm snow} & \sim & 2.4\times 10^{-3}\left({H_g\over H_d}\right)\left({a\over \rm AU}\right)^{1/4}. \end{eqnarray} \tag{ 13 }$

We wish to emphasize that

1.
Equations (11)–(13) are approximate solutions which are valid only if 〈R_p〉 ≫ R_p0. When this approximation is invalid (〈R_p〉 ∼ R_p0), then t_snow, R_snow, and PMF_snow cannot be expressed analytically and numerical solutions are required (see Figure 2).
2.
Equations (11) and (12) depend on t_f (not t_f.ind). The coefficient of 10⁴ yr appearing in Equations (11) and (12) is not related to the value t_f.ind ∼ 10⁴ yr mentioned in Section 2.1. In fact, t_f.ind does not affect the results (Equations (11)–(13)) provided that t_f.ind < t_snow.
3.
PMF_snow is the total PMF at the end of the snowball phase, i.e., t = t_snow, it is not just the PMF contributed purely by the snowball growth phase. (See also in Sections 3.1 and 4.1.1 for details.)
4.
Since $\dot{R}_p$ is proportional to f_d and f_ice, R_snow actually increases with the disk density.

3.1. Efficiency of Snowball Growth

An efficient snowball growth phase would imply that a significant proportion of the disk's solid mass is converted into planetesimals in this phase, so one essential condition is that PMF_snow should be close to unity. Let us define this high-PMF_snow condition as corresponding to 0.1 < PMF_snow < 1. From Equation (13), we see that high PMF_snow fractions are favored by high values of H_g/H_d. At a = 1 AU from the star, for instance, 0.1 < PMF_snow requires H_g/H_d > 40. To a first order, H_g/H_d is an indicator of the degree of turbulence in the protoplanetary disk; the larger H_g/H_d, the weaker the turbulence. Therefore, snowball growth is more efficient for a disk that has weaker turbulence (higher value of H_g/H_d). This can be simply understood as being the condition that, for the same amount of dust mass, this mass is concentrated in a thinner, denser disk and is thus more easily accreted by planetesimals that always stay very close to the midplane.

However, while the above condition is necessary, it is not sufficient. This is because, according to Equation (9) (replacing R_p by R_snow and t by t_snow), PMF_snow has contributions from two sources: the snowball growth term (R_snow/R_p0)³ and the initial planetesimal creation term t_snow/t_f. Therefore, the efficiency of snowball growth should be measured using both PMF_snow and the ratio R_snow/R_p0. In this paper, we define the criteria for an efficient snowball growth phase as (1) 0.1 < PMF_snow < 1.0 (or equivalently, H_g/H_d > 40) and (2) R_snow/R_p0 ⩾ 10 (or equivalently, t_snow/t_f < 10⁻³). Note that although the efficiency of snowball growth decreases with the ratio of t_snow/t_f, the efficiency actually increases with the absolute value of t_snow, since R_snow ∝ t_snow.

3.2. Examples

As shown in Equation (13) and discussed in the preceding section, the efficiency of the snowball growth phase depends on the ratio between the gas and dust scale heights (H_g/H_d) which is an indicator of the degree of disk turbulence. Here we consider three cases with weak (H_g/H_d = 150), intermediate (H_g/H_d = 15), and strong (H_g/H_d = 1.5) disk turbulence, which lead, respectively, to $\dot{R}_p=10^{-4}, 10^{-5}$ , and 10⁻⁶ km yr⁻¹ at 1 AU in a 1 MMSN disk. For each case, we then consider two further subcases with different characteristic planetesimal formation timescales (not t_f.ind): t_f = 10⁵ yr and t_f = 10¹¹ yr and with the same initial planetesimal size of R_p0 = 1 km. Solving for t_col, 〈R_p〉 and PMF as an explicit function of time (using Equations (8)–(10)), we plot the results in Figure 1.

In Figure 1, the locations of the triangles (t_f = 10⁵ yr) and squares (t_f = 10¹¹ yr) mark the important point at which the system transits from snowball growth to the mutual collision mode. As expected, the snowball phase is much more efficient in the weakly turbulent (high H_g/H_d) case. Additionally, if the initial planetesimal creation process is inefficient (t_f = 10¹¹ yr), then the snowball phase is longer and planetesimals can grow up to R_snow ∼ 40 km purely via dust accretion, whereas when the creation process is efficient and t_f = 10⁵ yr, the planetesimals can only reach ∼3 km before mutual collisions take over. At the end of the snowball growth phase, both cases have comparable values of PMF_snow (∼35% and ∼45%, respectively), both of which are below our alarm value of PMF = 0.5.

Conversely, for the highly turbulent case, snowball growth is less efficient. Although the snowball phase lasts longer than in the low-turbulence case, planetesimals do not grow quickly enough to reach sizes bigger than ∼1.2 km (t_f = 10⁵ yr case) or ∼5 km (t_f = 10¹¹ yr case).

In summary, Figure 1 demonstrates the general trend that snowball growth is more efficient when (1) planetesimal creation rates are lower (i.e., larger t_f) and (2) when disk turbulence is weaker (i.e., larger H_g/H_d). In addition, Figure 1 provides a method (by finding the crossing point in the t–t_col plane) by which t_snow, R_snow, and PMF_snow can be found more accurately than just using the approximate expressions of Equations (11)–(13).

3.3. Mapping t_snow and R_snow

The results of Section 3.2 were obtained for a fixed initial planetesimal size of 1 km. We now extend these results to a more general case in which the initial planetesimal size, R_p0, is allowed to vary. Such a consideration is important for two reasons: first, because R_snow does not vary linearly with R_p0, and second because the initial planetesimal size is a poorly constrained parameter that strongly varies from one planetesimal formation scenario to the other.

For illustrative purposes, we focus here on the most favorable case for snowball growth, i.e., weak turbulence (H_g/H_d = 150), and vary both R_p0 and t_f as free parameters. Figure 2 shows t_snow and R_snow in the R_p0–t_f plane and was generated by numerically solving the equation t = t_col. As the values of t_snow and R_snow plotted in Figure 2 are accurately solved using this numerical method, they do not rely on the approximation 〈R_p〉 ≫ R_p0 which was assumed in Equations (11) and (12). Note that in the bottom right corner of Figure 2, the R_snow contours become vertical, while the t_snow contours become horizontal. This occurs because we suppress snowball growth whenever the PMF becomes greater than the alarm value 0.5t_snow ∼ t_tran ∼ 10⁶–10⁷.

**Figure 2.** t_snow (solid contours filled with colors) and R_snow (dashed contours) in the R_p0–*t_f* plane at 1 AU for 1 MMSN with *H_g*/*H_d* = 150 (weak turbulence) in a single-star system. The R_p0 coordinate is schematically divided into three size ranges for the initial planetesimals, forming via traditional models (Goldreich & Ward 1973; Weidenschilling & Cuzzi 1993) and via two new models which involve turbulence (Johansen et al. 2007; Cuzzi et al. 2008). Snowball growth is most efficient in the top left corner of the plot, where planetesimals of initial size 0.1 km can grow via direct dust accretion (snowball growth) to ∼100 km within 10⁶ yr, before planetesimal–planetesimal collisions commence. Note that for t_snow in the ∼10⁶–10⁷ yr range (the typical lifetime of a protoplanetary disk), we obtain 100km < R_snow < 1000 km regardless of the initial value R_p0.
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Not surprisingly, snowball growth is at its most efficient in the top part of the figure, where t_f is long (or equivalently, the planetesimal creation rate is low). In addition, from Equation (12) we see that R_snow/R_p0 ∝ R^−0.25_p0, so that snowball growth, measured in terms of the increase in planetesimal size, is more efficient for smaller initial planetesimals, i.e., on the left-hand side of the plot. This result should be expected since, for a given total mass of solids, smaller initial sizes lead to larger total collisional cross section for dust accretion.

In contrast, snowball growth gets less efficient when starting from larger initial planetesimals. However, even for R_p0 = 100 km, a factor of 10 increase in size (1000 in mass) is possible for low-turbulence disks with a low planetesimal formation rate (t_f ⩾ 10¹¹ yr). Setting this factor of 10 size increase as the criteria for an efficient snowball phase (see Section 3.1), we see that the corresponding minimum value for t_f increases from ∼10⁷ yr for R_p0 = 0.1 km to ∼10¹³ yr for R_p0 = 1000 km.

However, another crucial constraint is that t_snow should not exceed the time for disk dispersal (probably ⩽10⁷ yr). This constraint is relevant for the following considerations: if gas is dissipated, then (1) both dust and planetesimals would have Keplerian velocities, and there would thus be far less efficient dust sweeping and no snowball growth for planetesimals and (2) one would like to be able to form gas giants such as Jupiter in the solar system, and hence require that massive solid planetary cores be able to form before disk dissipation. Given this additional constraint, all the solutions in the top right part of Figure 2 are ruled out. A factor of 10 size increase in less than 10⁷ yr is thus only possible for R_p0 ⩽ 100 km.

4. DISCUSSION

4.1. Implications for Planet Formation in Single-star Systems: The Solar System and Extrasolar Systems

We emphasize that this current section of discussion is based on the results of Figure 2 for the weak turbulence case where snowball growth is most favorable.

4.1.1. Dust-to-Planetesimal Transition Timescale, t_tran

It is worth noting that in Figure 2 we always have PMF_snow ∼ 35%–50%. As a consequence, in this case t_snow can be interpreted as corresponding approximately to the typical timescale, t_tran, for which the solid disk goes from being dust-dominated to being planetesimal-dominated. This timescale is especially relevant for planet formation scenarios because it can be independently measured, through both protoplanetary disk observations (Natta et al. 2007) and meteoritic evidence in our solar system (Cuzzi & Weidenschilling 2006; Scott 2007). All these observational studies seem to agree on a transition timescale somewhere between 1 and 10 Myr, i.e., t_snow ∼ t_tran ∼ 10⁶–10⁷ yr, which is shown as the red region in Figure 2.

Further, we note that if R_snow ∼ R_p0 then

$t_{\rm tran} \sim t_{\rm snow} \sim {t_f},$

whereas if R_snow ≫ R_p0 then

$t_{\rm tran} \sim t_{\rm snow} \ll {t_f}.$

These relations occur because snowball growth implicitly includes two simultaneous processes, i.e., the formation of new planetesimals, plus the snowball growth of new planetesimals. As discussed in Section 3.1, snowball growth dominates the snowball phase only if R_snow ≫ R_p0. This requires R_p0 < 100 km for the red region of Figure 2.

In contrast to the above, if PMF_snow ≪ 0.1 then

$t_{\rm tran} \sim t_f \gg t_{\rm snow}.$

Taking into account all these considerations, one should take care not to confuse t_f with either t_snow or t_tran, as it may potentially be orders of magnitude different from either.

Irrespective of the precise scenario in question, we note that snowball growth provides an alternative channel through which dust can be converted into planetesimals. Moreover, this can be an efficient process; even if the initial planetesimal creation is very inefficient, one can still convert most of the solid mass into planetesimal within 1–10 Myr.

4.1.2. Planetesimal Birth Size

Following the previous section, we take as a reference the observationally derived constraint of t_tran ∼ 10⁶–10⁷ and the additional result that, for the weak turbulence case, t_snow ∼ t_tran. As a consequence, the most probable location for the dust-to-planetesimal transition is the red region of Figure 2. An important feature is that almost the whole of this red region corresponds to a 100km ⩽ R_snow ⩽ 1000 km range. This implies that, for the weak turbulent case, mutual planetesimal–planetesimal collisions (or planetesimal accretion) start when planetesimals are 100–1000 km in radii, regardless of their initial size R_p0.

Interestingly, this implication is consistent with the result of a recent study by Morbidelli et al. (2009), which found that the size distribution of asteroids in the solar system, in particular the knee in this distribution around 100 km, could be well reproduced if planetesimal accretion (and mutual fragmentation) starts from 100 to 1000 km-sized objects. This was interpreted by Morbidelli et al. (2009) as being an indication that asteroids were born big, i.e., they emerge from their dust-to-pebbles progenitors with a size exceeding 100 km. We argue that our results could provide an alternative explanation, i.e., that 100–1000 km is the characteristic size for the end of snowball growth and the onset of mutual collisions, a size that might be much larger than the size R_p0 of the initial seed planetesimals. In other words, the results of Morbidelli et al. (2009) could actually provide some supporting evidence for the snowball growth, since the result R_snow = 100–1000 km is robust when applying the snowball concept to parameters consistent with the solar system. However, this inference depends on whether the weak turbulence case is consistent with the solar system.

As mentioned in Section 1, a form of snowball growth was taken into account in the simulations of Morbidelli et al. (2009, see their Figure 6(b)). However, there are important differences between their work and this present study. First, dust sweeping was only considered for large (⩾100 km) seed planetesimals, implicitly neglecting its possible effect on smaller initial bodies. Another, perhaps more important difference lies in the respective models adopted for dust sweeping. Morbidelli et al. (2009) argue that it proceeds in a fast "runaway" mode—thus making it end much earlier—because the relative velocity between the planetesimals and the dust (v_vel ∼ 75 ms⁻¹) is smaller than the planetesimals' escape velocity (v_esc ∼ 100 ms⁻¹ for a 100 km size), and hence the dust is gravitationally focused onto the larger planetesimals. In this study we ignore runaway growth, even for large planetesimals, because we implicitly assume that the coupling of the dust to the gas is strong enough to prevent solid grains from being gravitationally deflected onto the planetesimals according to the usual gas-free approximation. The issue of whether or not the runaway mode is significant for snowball growth would require an investigation using high-resolution coupled N-body and hydrodynamic models that exceed the scope of this paper.

In summary, we believe that the interpretation of the present-day asteroid size distribution as a proof of their large initial sizes (Morbidelli et al. 2009) needs further investigation in light of the results presented here on the importance of dust sweeping for early planetesimal growth. At the very least, snowball growth opens up the possibility that various conditions with different combinations of R_p0 and t_f could all potentially reproduce the asteroids' size distribution, to the point that the solution may be highly degenerate. In fact, recent work by Weidenschilling (2010) has already shown that planetesimals with birth size (R_p0) of 0.1 km are also a viable initial condition for a model which can well reproduce the asteroids' size distribution.

Further work would be required, using a detailed numerical collision-and-fragmentation model (e.g., as implemented in Morbidelli et al. 2009 or Weidenschilling 2010), to understand whether the observed size distribution in the asteroid belt can at all constrain the range of R_p0 and t_f in the early solar system, i.e., it may be that certain ranges of R_p0 and t_f corresponding to subregions of the red regime of Figure 2 may give rise to size distributions that are better able to reproduce the observed size distribution in the asteroid belt, while other regions can be excluded.

However, even so, we would still be far from arriving at a final answer, as several important questions remain unaddressed, including: (1) What is the distribution of sizes with which planetesimals are initially created? (2) How are the planetesimal births distributed in time and space (linear or nonlinear)? (3) To what degree should runway growth and gravitational focusing be considered in the snowball growth model? All of these factors can change the dimension of the red regime of Figure 2, and constraining any of these factors will require further detailed studies of planetesimal formation itself. Last but not the least, in addition to studies of the formation of asteroids and planetesimals, we may also obtain constraints on planetesimal birth sizes through other observations, such as those of debris disk around A-type and G-type stars (Kenyon & Bromley 2010).

4.1.3. Planetesimal Birth Rate

The results shown in Figure 2 also allow us to place constraints on the birth rate of planetesimals or their formation efficiency for each possible formation scenario. In addition, they might help overcome some difficulties that some of these different formation scenarios encounter.

Recently, two new models (Johansen et al. 2007; Cuzzi et al. 2008) have shown that large planetesimals can form directly from the concentration of small solid particles in the turbulent disk. The numerical simulation by Johansen et al. (2007) shows that if the feedback of the solid particles within the gas is considered then the concentration of solid particles (which is most efficient for particles of ∼50 cm radial size) can grow and be maintained long enough (the so-called streaming instability; Youdin & Goodman 2005) for the formation of very large planetesimals (typically 100–1000 km). Within this framework, i.e., with such large initial planetesimals, our results (Figure 2) show that the phase of snowball growth would be inefficient. In such a scenario, the only way to convert a large amount of dust to planetesimals within 1–10 Myr is through a very effective planetesimal formation process. Nevertheless, one must be careful and note that Figure 2 ignores the runaway growth mode which might be relevant to such large planetesimals. If a runaway mode is included, then the snowball timescale, t_snow, given in Figure 2 should be considered as an upper limit. However, as discussed in Section 4.1.2, the degree to which a runaway mode should contribute must rely on future studies with high-resolution coupled N-body and hydrodynamic models. Moreover, one should also note that disk turbulence is much more intense in Johansen et al. (2007) than that in Figure 2. As in Johansen et al. (2007), the disk viscosity coefficient is adopted α ∼ 10⁻³, leading to H_g/H_d close to unity according to Equation (108) of Armitage (2010, p. 294) and thus inefficient snowball growth would result. It would be interesting to investigate the size and efficiency with which planetesimals can be formed via the mechanism of Johansen et al. (2007) but with weaker turbulence.

The other large-planetesimal model (Cuzzi et al. 2008) also relies on turbulence which can generate vortices to trap small particles (Heng & Kenyon 2010) of chondrule size (typically mm–cm). Cuzzi et al. (2008) showed that millimeter-sized particles can be sporadically concentrated by disk turbulence, forming large and gravitationally bond clumps, which can potentially shrink to solid planetesimals roughly 10–100 km in radius. However, based on the model of Cuzzi et al. (2008), Chambers (2010), and Cuzzi et al. (2010) investigated the planetesimal formation rate in details and found that it should be very low in the solar system: at 1 AU for a typical MMSN disk, the time it would take to convert most dust into planetesimals, t_f, would be over ∼10¹⁰ yr. Snowball growth might help in solving this problem by creating a second channel by which dust can be converted into planetesimals. Figure 2 shows that, for the case of weak turbulence (this turbulence level in Figure 2 is compatible with the weak turbulence adopted in Cuzzi et al. 2008), the time to convert dust into planetesimals is close to t_snow, which is much shorter than t_f. In fact, t_snow can be within the disk lifetime (1–10 Myr) even for t_f ⩾ 10¹⁰ yr. In other words, snowball growth would fully dominate in terms of transforming dust into planetesimals.

Besides the two new models described above, there are the two traditional models of planetesimal formation described in Section 1. One is the mutual sticking scenario (Weidenschilling & Cuzzi 1993; Weidenschilling 1997), in which planetesimals form by pairwise collisions of small particles. The other is the instability scenario (Safronov 1969; Goldreich & Ward 1973; Youdin & Shu 2002), which suggests that planetesimals form via gravitational instabilities in a dense particle subdisk near the midplane of the protoplanetary disk. Both scenarios remain debated.

As mentioned in the introduction, the collisional scenario is challenged by the well-known "meter barrier," while for the instability scenario it has not yet been established whether a disk of solid particles can become dense enough and dynamically cold enough to trigger instability. While these issues go well beyond the scope of this work, the present results show that, for both scenarios, snowball growth might represent an interesting way to allow planetesimal growth to happen despite these difficulties. Indeed, in both models planetesimals are thought to form with a radius of 0.1–10 km. For this size range, Figure 2 shows that snowball growth could convert most of the dust mass into planetesimals even if formation rates are very low (large t_f). This means that only a few "lucky seeds" are needed in order for planetesimal growth to proceed. Within the framework of these two scenarios, these seeds could either be a few lucky pre-planetesimals that have overcome the "meter barrier" for some reasons (Brauer et al. 2008; Johansen et al. 2008; Teiser & Wurm 2009; Wettlaufer 2010) or kilometer-sized objects that were formed in some isolated location where the critical condition for gravitational instability just happens to be satisfied (Goldreich & Ward 1973). Therefore, snowball growth lowers the requirements for those two traditional scenarios: direct planetesimal formation from dust could be very inefficient and still allow planetesimal growth to occur. This hypothesis remains to be quantified: future work should focus on deriving the planetesimal formation rates within the framework of those two traditional scenarios, to see whether they can meet these "lowered" requirements.

4.2. Implication for Planet Formation in Close Binaries

The context in which snowball growth might find its most interesting application is that of planet formation in binaries. Recent studies have shown that one of the major problems for planet formation in close binary systems, such as α Centauri, is the intermediate stage of planet formation, i.e., the mutual accretion of kilometer-sized planetesimals to form larger planetary embryos or cores (Thébault et al. 2006; Marzari et al. 2010; Haghighipour et al. 2010). The companion's gravitational perturbation, coupled to gas drag, may excite large relative velocities between the planetesimals, leading to disruptive collisions which inhibit their mutual accretion (Thébault et al. 2008, 2009; Paardekooper et al. 2008; Marzari et al. 2009; Xie et al. 2010). Recently, some mechanisms have been found to be somewhat helpful in solving this problem (Xie & Zhou 2008, 2009; Paardekooper & Leinhardt 2010), but several problems remain to be overcome.

A possible solution to these problems would be to let planetesimal collisions only begin when large (⩾50–100 km) objects are present in the system. Thébault et al. (2008) have indeed shown that the gravity of such large planetesimals is strong enough for them to survive the high-speed collisions induced by differential gas drag. One obvious way to bypass this problematic kilometer size range would be to have planetesimals that are "born big," be it by the Johansen et al. (2007) or the Cuzzi et al. (2008) scenario. However, the issue of how these planetesimal formation scenarios might work in the very specific context of binaries has not yet been addressed.

Snowball growth offers an attractive alternative way to bypass the kilometer size range, regardless of the initial size at which planetesimals appear in the disk, because it predicts that mutual planetesimal impacts will become important only by the time large objects have formed. However, it is not possible to directly apply the results of the previous sections, derived for single stars, to the close binary case. In this section, we address the issue of how snowball growth proceeds in the specific context of double stars.

4.2.1. An Example: $\dot{R}_p$ and t^b_col in α Centauri

For the sake of clarity, we consider a disk with a 1 × MMSN surface density and set a = 1 AU, and then investigate how snowball growth may occur in binary star systems, taking as a representative example the case of α Centauri AB, for which the binary separation is a_B = 23.4 AU and the eccentricity e_B∼ 0.52 (Pourbaix et al. 2002).

The main difference between the single-star case and the binary case is that in the binary case there is a much higher relative velocity between the dust and the planetesimals (v_rel). For such a highly eccentric case, we can safely neglect the small component of v_rel due to the differential Keplerian velocities between the gas and the planetesimals (because the gas is pressure-supported) and basically consider two cases, depending on the unknown angle i_B between the circumprimary disk plane and the binary's orbital plane.

1.
The coplanar binary case, where the binary orbit and the gas disk are in the same plane. In this case, to a first approximation, the average eccentricity of planetesimals (e_p) is equal to their forced eccentricity (5e_Ba/4a_B),⁴ and thus v_rel ∼ e_pv_k ∼ (5a/4a_B)e_Bv_k, leading to v_rel ∼ 840 ms⁻¹ for the α Cen case. This >v_rel value, and thus also the snowball growth rate, is roughly an order of magnitude higher than in the single-star case. We then obtain $\dot{R}_p\sim 10^{-3}\rm \,km\,yr^{-1}$ in a weakly turbulent disk with H_g/H_d = 150.Note that we here implicitly assume that the gas disk is axisymmetric (circular) in the circumprimary midplane. This is a rather crude approximation as it ignores the reaction of the gas disk to the binary perturber. Nevertheless, as a first order of approximation, we believe it is both reasonable and convenient for our semi-analytical study in this paper. In fact, if the reaction of the gas disk is considered, v_rel would probably be even larger, as shown by Paardekooper et al. (2008). Therefore, the v_rel, as well as the snowball growth rate $\dot{R}_p$ derived in our simplified axisymmetric gas disk model should be treated as a lower limit.
2.
The inclined binary case, where the binary orbital plane is tilted by an angle (i_B) relative to the gas disk plane. In such a case, as simulated by Larwood et al. (1996), if the Mach number of the gas disk is not too high, the gas disk can maintain its structure and undergo a near-rigid precession. As a planetesimal also undergoes a similar precession (regarding the binary orbital plane as the reference plane) but with a different rate, the relative angle between the gas disk plane and planetesimal orbital plane will remain approximately within the range 0–2i_B. Taking the medium value i_B as the average, we then have v_rel ∼ i_Bv_k ∼ 520(i_B/1°) ms⁻¹. Taking into account this v_rel, as well as the fraction of time that a planetesimal spends moving within the dust disk, 2H_d/πai_B, the snowball growth rate for the inclined case is $\dot{R}_p\sim 10^{-5}\rm \,km\,yr^{-1}$ , which is independent of both i_B and the turbulence factor H_g/H_d.Note that, for the inclined case, we have implicitly assumed that the vertical excursion of the planetesimals is higher than the dust disk thickness, i.e., i_B > 0.05(H_d/H_g)(a/AU)^0.25. However, even for a fully turbulent disk (H_d/H_g = 1), this condition is easily met as long as i_B > 3°. In addition, we also implicitly assume for the inclined case that v_rel is dominated by i_B rather than e_B, which requires 520(i_B/1°)>840 (i.e., comparing v_rel in the inclined and coplanar cases), i.e., i_B > 16.⁵ Given the above considerations, we shall consider that the inclined case discussed here corresponds to cases with i_B > 3°. Otherwise, it should be treated as the coplanar case.

The collision timescale, t_col, among planetesimals is also different in the binary case. The relation given in Equation (10), which assumes an unperturbed disk with equipartition between in and out-of-plane velocities, breaks down in binary systems. Here we use the empirical scaling law given by Xie et al. (2010, see Equations (6) and (7) in their paper for details) to estimate the planetesimal collisional timescale in binary star systems as

$\begin{equation} t_{\rm col}^b\sim (0.02+2i_B) t_{\rm col}^s, \end{equation} \tag{ 14 }$

where t^s_col is the collisional timescale for single-star systems as given by Equation (10).

4.2.2. An Example: The Snowball Phase in α Centauri

With the modified $\dot{R}_p$ and t^b_col, we can derive t_snow, R_snow, and PMF_snow as in the single-star case. Results are plotted in Figures 3 and 4 for the coplanar and inclined binary cases, respectively.

1.
Coplanar case. Compared to the single-star case of Figure 2, we obtain slightly smaller, but still relatively large values of PMF_snow, i.e., ∼7%–50%. R_snow is little changed, but t_snow is about an order of magnitude smaller (Figure 3). These high PMF_snow and low t_snow values imply that snowball growth can be efficient, even if the disk lifetime in such close binaries could be an order of magnitude shorter than in single-star systems (Xie & Zhou 2008; Cieza et al. 2009). Note that, as for the single-star case, results depend on the disk turbulence, which is controlled, to a first approximation, by H_g/H_d. Here, we have assumed a value of H_g/H_d = 150, corresponding to weak turbulence.
2.
Inclined binary case. PMF_snow are here higher than in the coplanar case, being close to 50% throughout the R_p0–t_f plane of Figure 4 (in fact, PMF_snow > 50%, but we stop the evolution of the system when PMF_snow reaches the alarm value 0.5 as mentioned at the end of Section 2.2). Compared to the single-star case of Figure 2, R_snow is smaller by ∼30%, but t_snow is larger by about an order of magnitude (more precisely, by a factor of ∼7). This implies that the disk lifetime has to be relatively long in order for snowball growth to be efficient. As an example, if planetesimals are born small (R_p0 < 10 km), then in order for them to reach R_snow ∼ 50–100 km, i.e., values large enough to survive the high-speed collisions induced by the companion, the disk lifetime should be no less than ∼10⁶ yr. Note that, as $\dot{R}_p$ is independent of H_g/H_d (see Section 4.2.1), the result is here independent of the level of disk turbulence.

**Figure 3.** Same as Figure 2 but for the coplanar binary case (*i_B* = 0) and *H_g*/*H_d* = 150. The binary system parameters are those of α Centauri. R_snow is close to its value for the single-star case Figure 2, but t_snow is about an order of magnitude smaller, implying that snowball growth is efficient even if the disk lifetime in close binary systems is an order of magnitude shorter than in single-star systems.
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**Figure 4.** Same as Figure 3 but for the inclined binary case with *i_B* = 10°. The binary system parameters are those of α Centauri. Note that the present results are independent of the turbulence factor *H_g*/*H_d*. Compared to the single-star case of Figure 2, R_snow is smaller by ∼30%, but t_snow is larger by about an order of magnitude. If, for example, one expects planetesimals to be born small (R_p0 < 10 km) but want them to have a large R_snow (10–100 km) in order that they can survive high-speed collisions, then the disk lifetime should be no less than ∼10⁶ yr.
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The results for both cases considered show that snowball growth might indeed be an attractive alternative to solving the planet-formation-in-binaries dilemma, i.e., the high impact velocities that prevent the mutual accretion of kilometer-sized bodies. Its main appeal is that it provides a much safer way for planetesimals to reach sizes in the 50–100 km range, after which they are large enough to be protected from destructive collisions. The only problem lies with the growth timescales, in particular for the inclined binary case for which it might take as long as 10⁶ yr to reach the safe 100 km size, by which time the gas disk might have vanished. However, this issue might not be as dramatic as it might appear because, even if 50–100 km bodies have not been formed by the time the gas disk is dispersed, the environment could be favorable to accretion anyway. Indeed, Xie & Zhou (2008) have shown that, during this gas dispersion phase, planetesimal orbits get re-phased so that, by the time the gas has dispersed, relative velocities amongst them are very low and accretion friendly. This means that, even if snowball growth is switched off (because there is no dust sweeping when grains stop to follow gas streamlines), planetesimal growth can now proceed through the classical mutual-accretion channel.

We note that the two-dimensional investigation performed by Paardekooper & Leinhardt (2010) demonstrated that planetesimals can grow to ∼50–100 km size with the help of dust accretion. Our own investigation extends this to the three-dimensional case, demonstrating that even though three-dimensional collision rates are significantly lower, dust accretion (or snowball growth) can still help planetesimals to overcome the "kilometer barrier" for planetesimal accretion (or growth) in close binary systems, such as α Centauri.

4.2.3. Limitations

One possibly problematic assumption of our snowball growth model regards the efficiency of dust accretion onto planetesimals, parameterized by E_d defined in Equation (4), for which we have implicitly assumed that it is not affected by the high v_rel values reached in the binary environment. Although recent laboratory experiments (Teiser & Wurm 2009) showed that small (millimeter-sized) particles can be efficiently accreted by large objects in a high-speed collision, their test speeds were only of a few 10 m s⁻¹, which is only relevant for the snowball growth in a single-star system with v_rel ∼ 75 ms⁻¹. In close binary systems, such as the α Centauri shown above, v_rel could be as high as 500–5000 ms⁻¹, which is beyond the limit of the test speed in any existing laboratory experiment. Therefore, the issue of whether dust can be accreted onto small planetesimal seeds (R_p0 = 0.1–1 km) for such large relative velocity remains unresolved. However, at the very least, it seems reasonable to assume that the efficiency of dust accretion onto planetesimals will be less affected by high velocities than would the efficiency of mutual planetesimal accretion.

5. CONCLUSION AND PERSPECTIVES

We have investigated the transitional phase of planet formation, dubbed "snowball growth phase," in which planetesimals are progressively emerging from the dust disk but are not yet numerous enough to enter the phase of mutual planetesimal–planetesimal collision. In this snowball phase, the planetesimals move on Keplerian orbits and grow purely through the direct accretion of subcentimeter-sized dust which is entrained with the gas in the protoplanetary disk and moves with a sub-Keplerian velocity.

Using a simplified model in which the planetesimals are progressively created from the dust disk, we find the following.

1.
In single-star systems, snowball growth can be a significant mechanism if the following conditions are met.If these conditions are met, then we find that
- (a)
  Size growth can be significant: 2 orders of magnitude in radius (6 orders in mass), before mutual planetesimal collisions take over. In addition, snowball growth can be fast enough for it to happen within the typical (1–10 Myr) lifetime of a protoplanetary gas disk (see Section 3.3).
- (b)
  Snowball growth can be the dominant mode for the transformation of dust into planetesimals. Except for highly efficient and fast planet-formation scenarios, dust is preferentially accreted onto already-formed kilometer-sized bodies rather than consumed for forming new "initial" planetesimals (see Section 4.1.1).
- (c)
  Applying the snowball growth model to a fiducial MMSN disk for the solar system and assuming weak turbulence, the turnover from snowball to mutual collisions is likely to occur when planetesimals reach a size of 100–1000 km, irrespective of their birth size (e.g., 0.1–100 km). This result, i.e., that mutual impacts become significant only when large bodies have formed, may provide an alternative explanation to the size distribution of asteroids (Morbidelli et al. 2009; see Section 4.1.2).
- (d)
  Snowball growth could help overcome some still unsolved problems inherent to some planet formation scenarios. More precisely, it lowers the requirements on the planetesimal formation mechanism itself, by allowing planetesimals to grow, and dust to be transferred onto them, even if the mechanism by which they are formed is very inefficient (see Section 4.1.3).
2.
For close binaries, snowball growth offers a way to overcome one major problem for planet formation in binaries, i.e., the high impact velocities that prevent kilometer-sized planetesimals from accreting each other. Indeed, this dynamically excited environment, which is hostile to mutual planetesimal accretion, is on the contrary favorable to growth by dust sweeping. If efficient enough, snowball growth allows planetesimals to grow large enough, 50–100 km, to be protected from mutually destructive impacts. Two main cases can be distinguished.
- (a)
  In the coplanar binary case (i_B < 3°), snowball growth is at its most efficient (a little more efficient than the single-star case, if one assumes the same level of disk turbulence) but remains sensitive to disk turbulence, i.e., weak turbulence is still required for efficient snowball growth (see Figure 3 and Section 4.2.2).
- (b)
  In the case of an inclined binary (i_B > 3°), snowball growth can still be efficient, although slightly less efficient than in the case of a single star with weak disk turbulence (H_g/H_d = 150), with the added advantage that it is independent of disk turbulence (see Figure 4 and Section 4.2.2).

Let us stress again that our model is highly simplified and that the present work, whose main objective was to identify a new and potentially significant mechanism, should be considered as a first step toward more detailed studies. In future work, the modeling of snowball growth will need to be improved in a number of aspects. The most important improvement will be to have a self-consistent model of the planetesimal size evolution under the coupled effect of snowball growth and mutual accretion, instead of the simple analytical growth law assumed here. This requires the development of a global statistical particle-in-a-box model, in the spirit of those of Weidenschilling (2010) or Morbidelli et al. (2009), spanning a wide range in sizes and incorporating all possible outcomes for dust–planetesimal and planetesimal–planetesimal collisions. Other improvements which could be made to the model include (1) the role of gravitational focusing of dust onto planetesimals, in particular to what extent it is (or is not) hampered by the action of gas and (2) the changes to E_d arising from the physics of high-velocity impacts of dust onto large bodies, especially in the specific case of close binaries where v_rel may reach values of a few 100 m s⁻¹.

This work was supported by the National Natural Science Foundation of China (Nos. 10833001, 10778603, and 10925313), the National Basic Research Program of China (No. 2007CB814800), NSF with grants AST-0705139 and 0707203, NASA with grants NNX07AP14G and NNX08AR04G, W. M. Keck Foundation, Nanjing University, and also the University of Florida. J.-W.X. was also supported by the China Scholarship Council. M.J.P. was also supported by the NASA Origins of Solar Systems grant NNX09AB35G and the University of Florida's College of Liberal Arts and Sciences.

APPENDIX: MASS-WEIGHTED AVERAGE RADII: 〈R_P〉

As we assume a linear planetesimal formation rate (see Equation (1)) and a linear growth rate (see Equation (7)), thus, at a given time t, the distribution of the planetesimals' radii will be given by R_p0, R_p0 + dr, R_p0 + 2 × dr, ..., R_p0 + i × dr, ..., R_p0 + (n − 1) × dr, where dr is the radii increase during a time interval of t_f.ind, i.e., $dr=\dot{R}_pt_{\rm{f.ind}}$ , and n denotes the total number of planetesimals. The total mass of these planetesimals is

$\begin{eqnarray} M_{\rm tot} &=& \frac{4\pi }{3}\rho _*\sum (R_{p0}+i\times dr)^3 \nonumber \\ &=&\frac{4\pi }{3}\rho _*\left(R_{p0}^3n+3R_{p0}^2dr\sum i +3R_{p0}dr^2\sum i^2\right.\nonumber\\ &&\left. +\,dr^3\sum i^3\right). \end{eqnarray} \tag{ A1 }$

We define the mass-weighted average radii, 〈R_p〉, which satisfies

$\begin{eqnarray} &&M_{\rm tot}=\frac{4\pi }{3}\rho _* \langle R_p\rangle ^3n. \end{eqnarray} \tag{ A2 }$

Comparing Equations (A1) and (A2), we have

$\begin{eqnarray} \langle R_p\rangle &= &\left(R_{p0}^3+\frac{3}{n}R_{p0}^2dr\sum i+\frac{3}{n}R_{p0}dr^2\sum i^2\right.\nonumber\\ &&\left.+\,\frac{1}{n}dr^3\sum i^3\right)^{1/3}\nonumber \\ &=&\left[R_{p0}^3+\frac{3}{2}R_{p0}^2(n-1)dr+R_{p0}(n-1)\left(n-\frac{1}{2}\right)dr^2\right.\nonumber\\ &&\left.+\,\frac{1}{4}n(n-1)^2dr^3\right]^{1/3}. \end{eqnarray} \tag{ A3 }$

Considering

$(n-1)dr\sim \left(n-\frac{1}{2}\right)dr\sim ndr\sim \dot{R}_pt,$

Equation (A3) can be rewritten as

$\begin{eqnarray} \langle R_p\rangle &\sim &\left[R_{p0}^3+\frac{3}{2}R_{p0}^2(\dot{R}_pt)+R_{p0}(\dot{R}_pt)^2+\frac{1}{4}(\dot{R}_pt)^3\right]^{1/3}\nonumber \\ &\sim & R_{p0}+\left(\frac{1}{4}\right)^{1/3}\dot{R}_pt. \end{eqnarray} \tag{ A4 }$

FROM DUST TO PLANETESIMAL: THE SNOWBALL PHASE?

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ABSTRACT

1. INTRODUCTION