Formation of dustrich planetesimals from sublimated pebbles inside of the snow line
^{1} EarthLife Science Institute, Tokyo
Institute of Technology, Meguroku, Tokyo 1528550, Japan
email: ida@elsi.jp
^{2} Laboratoire J.L. Lagrange,
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
06304
Nice,
France
Received:
10
September
2016
Accepted:
2
November
2016
Context. For up to a few millions of years, pebbles must provide a quasisteady inflow of solids from the outer parts of protoplanetary disks to their inner regions.
Aims. We wish to understand how a significant fraction of the pebbles grows into planetesimals instead of being lost to the host star.
Methods. We examined analytically how the inward flow of pebbles is affected by the snow line and under which conditions dustrich (rocky) planetesimals form. When calculating the inward drift of solids that is due to gas drag, we included the backreaction of the gas to the motion of the solids.
Results. We show that in lowviscosity protoplanetary disks (with a monotonous surface density similar to that of the minimummass solar nebula), the flow of pebbles does not usually reach the required surface density to form planetesimals by streaming instability. We show, however, that if the pebbletogasmass flux exceeds a critical value, no steady solution can be found for the solidtogas ratio. This is particularly important for lowviscosity disks (α< 10^{3}) where we show that inside of the snow line, silicatedust grains ejected from sublimating pebbles can accumulate, eventually leading to the formation of dustrich planetesimals directly by gravitational instability.
Conclusions. This formation of dustrich planetesimals may occur for extended periods of time, while the snow line sweeps from several au to inside of 1 au. The rocktoice ratio may thus be globally significantly higher in planetesimals and planets than in the central star.
Key words: planets and satellites: formation / planetdisk interactions / accretion, accretion disks
© ESO 2016
1. Introduction
Determining the fate of solids in protoplanetary disks is key for understanding the birth and growth of planets and planetary systems. While small grains are coupled to the disk gas, large particles drift inward as a consequence of angular momentum loss by aerodynamical gas drag. For meter sizes (assuming compact grains), the inward drift velocity is ~10^{2} au/yr (e.g., Weidenschilling 1980; Nakagawa et al. 1981). For small dust grains, growth through pairwise collisions is faster than drift so that they grow in situ until they reach 1 to 100 cm, at which point drift starts to dominate (e.g., Okuzumi et al. 2012; Lambrechts & Johansen 2014). These socalled pebbles then drift rapidly with limited growth, implying that without a mechanism to suppress the drift, they would be lost to the central star.
Planetesimals would form directly by gravitational instability (GI) in the dust disk if it is sufficiently thin and dense (e.g., Goldreich & Ward 1973). However, even in lowturbulence disks, the KelvinHelmholtz (KH) instability generated by the vertical shear between the dust subdisk and the gas prevents the development of a disk that is thin enough (e.g., Weidenschilling 1995; Sekiya 1998). Youdin & Shu (2002) proposed that migrating dust (or pebbles) would pile up in the inner disk to become gravitationally unstable (see also Laibe et al. 2012), but they neglected grain growth, which was then shown to prevent this pileup (see Krijt et al. 2016).
Another possibility to form planetesimals is to invoke streaming instabilities (SI) in the drifting pebble flow: when their density is high enough, clumps can form, and because they undergo relatively less gas drag, they accrete individual pebbles to rapidly form planetesimals of 100 to 1000 km (Youdin & Goodman 2005; Johansen et al. 2007). However, this mechanism requires a high solidtogas ratio (Z) and has also been shown to be difficult to achieve in realistic disks (Krijt et al. 2016).
With analytical calculations, we examine here the formation of planetesimals from drifting pebbles in smooth disks (i.e., without pressure bumps, gaps, or vortexes) through these two mechanisms. We highlight the importance of sublimation across the snow line. Instead of examining the consequence of ice deposition beyond the snow line in turbulent disks (e.g., Stevenson & Lunine 1988; Ros & Johansen 2013; Armitage et al. 2016), following Saito & Sirono (2011), we concentrate on the region inside the snow line where dust grains ejected from sublimated pebbles are present.
2. Pebbletogas surface density ratio
We consider a protoplanetary disk characterized by a steady gas accretion rate Ṁ_{∗} in which the solids are in the form of pebbles migrating inward at a mass flux Ṁ_{peb}. The surface density of the migrating pebbles Σ_{p} and the disk gas Σ_{g} are given by (1)where ν is the turbulent viscosity of the gas disk, h_{g} the gas scale height, Ω_{K} the Keplerian frequency, and v_{r} is the pebble migration speed. We use the αprescription, that is, .
The inward radial drift speed of solids was calculated in the limit of a static disk by Nakagawa et al. (1986) and in the limit of a low solidtogas ratio by Guillot et al. (2014). Combining the two yields (2)where the backreaction of the gas to the motion of solids has been included through Λ ≡ ρ_{g}/ (ρ_{g} + ρ_{p}), and ρ_{g} and ρ_{p} are the midplane densities of gas and solids, respectively. We included the Λdependence of u_{ν} as well for later purposes. In Eq. (2), the size of the solids is defined through their Stokes number τ_{s}, which is the ratio of their stopping time due to gas drag (t_{stop}) to the Kepler frequency as (3)u_{ν} is the radial velocity of the accreting disk gas, which in the inner regions of a vertically uniform disk may be approximated by (4)and η(≪ 1) is the deviation fraction of the gas orbital angular velocity (Ω) relative to the Keplerian angular velocity (Ω_{K}) that is due to the radial pressure gradient in the disk, (5)From Eqs. (2), (4), and (5), (6)Using Eq. (1), we then obtain the solidtogas ratio as (7)
Now, the parameter Λ may be estimated in the limit of a vertically isothermal disk as (8)This thus leads to the following secondorder equation in Z: (9)where we have defined a few quantities, (10)and we adopt a_{0} ≈ 1.75 as a_{0} is estimated to be ≈1.70−1.85 for radially smooth disks with both viscous heating and stellar irradiation (see Ida et al. 2016), β is the ratio of the gastodust pressure scale heights (Dubrulle et al. 1995; Youdin & Lithwick 2007), (11)and ξ is the ratio of the solid mass flux to the gas mass flux: (12)The solutions to Eq. (9)are(13)
Figure 1 shows the solutions to Eq. (13) obtained for Z as a function of τ_{s} for different values of the pebbletogasmass flux ratio ξ. For pebbles, we expect τ_{s} ~ 0.1 (Sato et al. 2016; Ida et al. 2016), so that τ_{s} ≫ α and hence A/a_{0} ≃ β^{2} ≫ 1, yielding the approximate solution(14)which fits the lowerZ solutions in Fig. 1 for τ_{s} ≳ 10^{2}. This approximate solution can be easily derived from Eq. (7) with τ_{s} ≫ α and Λ ≃ 1. The dependence on α appears because, as shown by Eq. (1), Σ_{g} ∝ 1 /α and Σ_{p} is independent of α for τ_{s} ~ 0.1. The fast drift of pebbles is responsible for the small Z_{peb}.
Fig. 1 Steadystate solutions for the solidtogas mixing ratio Z as a function of the Stokes number of solid particles τ_{s} for different values of the solidtodustmass flux ratio ξ (as labeled), assuming a value of the turbulent viscosity α = 10^{3}. The values of τ_{s} corresponding to expected pebble sizes are highlighted with larger symbols. The two solutions provided by Eq. (13)are indicated by filled and open symbols, respectively. The gray area highlights the region in which planetesimals should form by a streaming instability (Carrera et al. 2015). 

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We expect planetesimal formation to occur either by GI in the dust disk when ρ_{p} ≳ ρ_{R}, where ρ_{R} ~ M_{∗}/r^{3} is the Roche density, independently of the dust size, or by SI with Z as low as ~0.02 but a limited range of τ_{s} values (Dra¸żkowska & Dullemond 2014; Carrera et al. 2015, also see Fig. 1). In general, the condition for GI, ρ_{p} ≳ ρ_{R}, is difficult to reach because of the vertical shear that is due to KH instabilities and thus requires very high values of Z.
We see in Fig. 1 that the formation of pebbles by SI is possible but requires high values of ξ. The criterion for planetesimals to form directly from τ_{s} ~ 0.1 pebbles is Z_{pebble}_{~}^{>}0.02. From Eq. (14), this implies (15)where α_{3} = α/ 10^{3}. We show below that this condition, which requires the mass flux of pebbles to be equivalent to the mass flux of gas, is difficult to reach (see also Krijt et al. 2016). One possibility is to advocate high values of α (see Armitage et al. 2016), but this is generally not favored by the latest magnetohydrodynamical simulations of protoplanetary disks (e.g., Bai 2015). Other possibilities exist that require local perturbations in the disk to modify the pressure gradient term (e.g., Johansen et al. 2014, and references therein) or favorable conditions in terms of fragmentation threshold velocity and disk properties (Laibe 2014; Drazkowska et al. 2016). We show below that the conditions necessary to form planetesimals can be reached at lower ξ values and for small αdisks next to a snow line. Before we examine this possibility, it is worth noting that for small particles (with τ_{s} ≲ 10^{3}), no solution is found for ξ ≳ 1, meaning that no steadystate exists: if there were a way to have small particles drift in at a very high rate or to deplete disk gas preferentially, the particles would pile up and accumulate, eventually forming planetesimals by direct gravitational instability.
3. Solidtogas density ratio inside of the snow line
When the inwarddrifting pebbles cross the snow line, they progressively sublimate until only small refractory (silicate dust) seeds remain (e.g., Saito & Sirono 2011; Morbidelli et al. 2016). Observations of the interstellar medium and of interplanetary dust particles indicate that these dust seeds should be of submicron size, corresponding to τ_{s} ~ 10^{7}−10^{5}. Even for the much larger millimetersized chondrules, we expect τ_{s} to be between 10^{4} and 5 × 10^{2} at most in a rarefied disk. The presence of a snow line is thus a way of transforming a highmass flux of fastdrifting pebbles into a flux of small, slowdrifting dust particles.
Two additional factors need to be considered. First, the sublimation of the ice decreases the amount of solid material by a factor ζ_{0} ~ 1/3 corresponding to the ratio of the mass of dust (silicate components) to the total mass of condensates (dust+ice) (Lodders 2003). By assuming for simplicity that the pebbles instantaneously form small dust particles, the flux of material to inside of the snow line that is to be considered in Eq. (13)is now therefore ξ → ζ_{0}ξ_{peb}, where ξ_{peb} corresponds to the ratio of the nonsublimated pebble mass flux to the gas mass flux.
Second, we expect dust grains to retain a memory of the vertical scale height of the pebbles. Their vertical mixing timescale can be estimated to be (16)where is the estimated vertical mean free path and T_{K} is Kepler period. Comparing a sublimation timescale with a migration timescale for pebbles, we can derive the radial width for completion of the sublimation as Δr ~ 10^{2}(R/ 10 cm)^{1/2}r (see also Ciesla & Cuzzi 2006). With Eq. (1), the timescale for the pebble flux to establish Z ≳ 1 in the sublimation region is estimated as . Although t_{Z} for R = 10 cm is 10−100 times longer than t_{mix}, the effective R for sublimation would be much smaller and t_{Z} would be much shorter for more realistic fluffy pebbles (e.g., Kataoka et al. 2013). We can thus assume that the dust seeds released by the sublimating pebbles have the same vertical thickness as the pebbles themselves. This is done in Eq. (13)by replacing β by the value set by the pebble subdisk β → β_{0} ~ (1 + τ_{s,peb}/α)^{1/2}.
Fig. 2 Steadystate solutions for the solidtogas mixing ratio Z as a function of the solidtogasmass flux ratio ξ for different values of the Stokes number of solid particles τ_{s} (from 10^{5} to 10^{2}, as labeled), assuming two values of the turbulent viscosity α = 10^{4} (in blue) and α = 10^{3} (in red). Equation (13) is numerically solved. In contrast to Fig. 1, we now consider that initially icy pebbles with τ_{s,peb} ~ 0.1 and containing a mass fraction ζ_{0} = 1/3 in dust sublimate inside of the snow line. The thicker lines corresponds to the preferred value for the dust particles, τ_{s} = 10^{4}. 

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As shown in Fig. 2, the situation is now much more favorable toward the development of GI of the layer of sublimated pebbles. As pointed out by Saito & Sirono (2011) and Birnstiel et al. (2012), the release of small slowdrifting dust by fastdrifting pebbles naturally yields values of Z that are much higher than for pebbles. It first increases linearly with ξ_{peb}. However, when the pebble flux is high enough to yield ρ_{p}/ρ_{g} ≳ 1, the dust migration velocity decreases as a result of the inertia of the dust, which further enhances ρ_{p}/ρ_{g}. The value of Z then increases more than linearly with ξ_{peb} until the positive feedback becomes so strong that no steadystate solution can be found. At this point, we expect ρ_{p} to rapidly exceed ρ_{R}, and planetesimals are formed by GI.
It is useful at this point to rederive Eq. (13) for these small grains, that is, in the limit of A ≪ 1 and τ_{s} ≪ 1: (17)For these small grains, the critical value ξ_{crit} above which no steadystate solution exists can be easily derived by calculating when the denominator of Eqs. (13) or (17) becomes zero, that is, (18)The critical value ξ_{crit} is thus independent of the local pressure gradient. This is because the small dust grains ejected from sublimating pebbles are coupled to the disk gas motion and migrate with disk gas accretion and not by gas drag.
As shown by Fig. 2, higher values of ξ_{crit} are possible for τ_{s} ≫ α, a possibility that we do not consider here because we expect seed grains to be such that τ_{s} ≲ 10^{5}. Typically, we thus obtain ξ_{crit} ≃ 0.3 for α = 10^{3}, ζ_{0} = 1/3 and τ_{s,peb} = 0.1. The fact that ξ_{crit} scales with α^{1/2} is governed by the height of the dust and pebble subdisk. The formation of dustrich planetesimals inside of the snow line is thus favored in weakly turbulent disks.
4. Pebble flux and planetesimal formation
The pebble mass flux is calculated by the mass in dust swept by the pebble formation front at r ≃ r_{peb} per unit time (Lambrechts & Johansen 2014; Ida et al. 2016), (19)where Z_{0} is the solidtogas ratio in the pebble formation region. The growth time of pebbles from μm sized dust grains is given by (Takeuchi & Lin 2005; Okuzumi et al. 2012; Ida et al. 2016) (20)where Z_{02} = Z_{0}/ 0.01. From this equation, the pebble formation front radius is given by (21)where t_{6} = t/ 10^{6} yr. Because ice needs to condense, r_{peb} ≳ 1 au, implying that pebbles may start forming when t ≳ 200 yr. Substituting this relation into Eq. (19) with ṙ_{peb}/r_{peb} = (2/3t_{grow}), we obtain (22)where L_{∗ 0} = L_{∗}/L_{⊙}, M_{∗ 0} = L_{∗}/M_{⊙}, α_{3} = α/ 10^{3}, and we considered the irradiationdominated regime with , for the pebble formation region in the outer disk. This corresponds to the disk aspect ratio, (23)(The expressions are slightly simplified compared to Ida et al. 2016.) The pebble mass flux is given by ξ_{peb,pf}Ṁ_{∗}, implying (24)where Ṁ_{∗ 8} = Ṁ_{∗}/ (10^{8}M_{⊙}/ yr). This mass flux is higher than obtained by Lambrechts & Johansen (2014) because they assumed a lower initial surface density in the gas disk (Σ_{g} = 500 g/cm^{2} at 1 au, not tied to the mass flux in the disk) and a 1/2 coagulation probability.
When r_{peb} exceeds the disk size r_{out}, that is, when t ≳ 2 × 10^{5}(r_{out}/ 100 au)^{3/2} yr, we would expect Ṁ_{peb} to decay more rapidly than Ṁ_{∗} because most of the solid material has been made into pebbles and drifted in (Sato et al. 2016). This may be inconsistent with the observational data that show that mm or cm sized particles survive in the disks for several million years (Brauer et al. 2007). However, we expect strong turbulence due to GI of the gas disk (not GI of the dust subdisk) to limit this fast spread of the pebble front, which could explain the presence of mm or cm sized particles for relatively long times.
Disks with Σ_{g} given by Eq. (1) are gravitationally unstable in their outer parts unless the disk is compact. In the unstable parts, their surface density should evolve to become marginally unstable so that 1 ~ Q = c_{s}Ω /πGΣ_{g,GI}, or equivalently (25)where Q is a factor on the order of unity. In these regions, the turbulence generated by gravitational waves (assumed to lead to α_{GI} ~ 0.1) may be high enough that collisions between icy grains result in fragmentation rather than coalescence. Typical collision velocities are estimated to be (Sato et al. 2016). Assuming a sound speed c_{s} ~ 270(r/ 100 au)^{− 3/14} m/s implies v_{col} ~ 50(α_{GI}/ 0.1)^{1/2}(τ_{s}/ 0.1)^{1/2}(r/ 100 au)^{− 3/14} m/s. The threshold velocity for fragmentation of icy particles is predicted to be around 20−100 m/s (e.g., Blum & Wurm 2000; Zsom et al. 2011; Wada et al. 2011). We therefore assume that pebbles may form only in the stable parts of the disks, that is, for r<r_{GI}, implying that the location of the pebble formation front is given by min(r_{peb},r_{GI}).
Since Σ_{g} given by Eq. (1) is equal to Σ_{g,GI} at r = r_{GI}, we obtain Q(r_{GI}/h_{g})^{3} ≃ 3α(M_{∗}/Ṁ_{∗})Ω, where α is the turbulence parameter for the inner regions. With Eq. (23), (26)As Ṁ_{∗} decreases with time, r_{GI} increases. From (dr_{GI}/ dt) /r_{GI} ~ −(14/9)(dṀ_{∗}/ dt) /Ṁ_{∗} and Eq. (25), the pebble mass flux that is due to the outward spread of r_{GI} is (27)We now use the relation between accretion rate and age suggested by the observations of young clusters, (Hartmann et al. 1998). This yields (28)and (29)Equivalently, (30)
Figure 3 shows the evolution of the pebble formation front, the pebble mass flux Ṁ_{per} , and ξ_{peb}. The mass flux ratio ξ_{peb} reaches a maximum when r_{GI} = r_{peb}. This maximum value is proportional to α^{− 29/45}, whereas ξ_{crit} ∝ α^{1/2}. For the α = 10^{3} case, we see that ξ_{peb} becomes equal to ξ_{crit} obtained from Eq. (18)only for a very short time. The α = 10^{4} case, in contrast, leads to a prolonged period in which planetesimals can form inside the snow line. Thus, direct formation of planetesimals is possible in lowturbulence disks. Conversely, in highturbulence disks, the relatively low maximum values of ξ_{peb} obtained imply that other mechanisms have to be sought so that planetesimals can form. This may involve disk photoevaporation, disk winds, or growth of pebbles by ice condensation.
Fig. 3 Time evolution of a) radius of the pebble formation front, b) pebble accretion rate (Ṁ_{peb}), and c) ξ_{peb} = Ṁ_{peb}/Ṁ_{∗} for two values of α, 10^{4} (red) and 10^{3} (blue). The lines labeled “grow” (dotted) and “GI” (dashed) represent the pebble growth and disk GI limits, respectively. The thick solid lines express the actual values obtained by the minima of the two limits. Here we assumed τ_{s,peb} = 0.1 and Z_{0} = 0.01. In panel c), the small squares represent the points with ξ_{peb}>ξ_{crit}, see Eq. (18). 

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When formed by GI, the mass of a clump is , where ρ_{R} ~ M_{∗}/r^{3} is the Roche density. If the clump shrinks into a planetesimal with physical radius R and bulk density ρ_{s} ( ~1 g cm^{3}), then for the solar case, (31)Although a more detailed analysis would be required, it appears that the planetesimals formed through this mechanism are as large as those formed by SI (e.g., Johansen et al. 2014).
5. Discussion and conclusion
With a simple model of pebble growth, drift, and sublimation at the snow line, we have examined the conditions for the formation of planetesimals in protoplanetary disks.
We first showed that forming planetesimals from streaming instability in the flow of icy pebbles requires both a high level of turbulence (α ≳ 0.01) and an unrealistically high pebble flux. We note that for these high turbulence levels, water vapor diffusion can decrease the requirement on the pebble mass flux (e.g., Ros & Johansen 2013; Armitage et al. 2016).
By including the often neglected massloading factor in the equations for the drift of solids, we have shown that the pileup of solids inside of the snow line leads to the formation of dustrich planetesimals directly by gravitational instability in the dust subdisk. This instability exists for relatively high values of ξ, the pebbletogasmass flux ratio, and for relatively low values of α.
With a simple model of the formation of pebbles, we have demonstrated that supercritical values of the pebbletogas mass flux ξ ≥ ξ_{crit} are reached in disks with α ≲ 10^{3}. This condition may be reached more easily, that is, for higher values of α or lower values of ξ, by taking into account disk gas depletion mechanisms other than viscous disk accretion such as photoevaporation or disk winds. The planetesimals that are formed in this way are expected to be large, probably larger than 100 km in radius.
During that time, the H_{2}O snow line could move from several au to inside of 1au (e.g., Oka et al. 2011). These planetesimals are expected to be composed of a high fraction of dust (silicates), which may explain why the rocktoice fractions inferred in minor planets and moons in the outer solar system (e.g., Schubert et al. 2010) or the dusttoice ratio in comets (e.g., Rotundi et al. 2015; Lorek et al. 2016) are often significantly higher than the expected 1/2 to 1/3 value obtained from purely solar composition (e.g., Lodders 2003).
Our model requires a fast breakup of pebbles, however, so that the dust particles are released over a small annulus. Given that we expect these pebbles to be porous (e.g., Kataoka et al. 2013), this should be verified. We note that the possibility that disks have flow in the midplane that are directed outward (Takeuchi & Lin 2005) or are stochastic (Suzuki & Inutsuka 2014) will favor the mechanism that we propose.
Finally, this process may apply to other sublimation lines (e.g., Drozdovskaya et al. 2016) if these lead to the breakup of fluffy pebbles into much smaller grains. Indeed, sintering has been shown to have this effect and thus might explain the rings that were recently observed in young disks (Okuzumi et al. 2016). The pileup and planetesimal formation mechanism that we propose may thus naturally explain the formation of rings of planetesimals in lowturbulence disks.
Acknowledgments
We thank Chris Ormel, Anders Johansen, Satoshi Okuzumi, and an anonymous referee for helpful comments. S. I. thanks for the hospitality he experienced during his visit to the Observatoire de la Côte d’Azur, which was made possible thanks to support from OCA BQR. S.I. is also supported by MEXT Kakenhi grant 15H02065. We acknowledge support by the French ANR, project number ANR1313BS05 000301 projet MOJO (Modeling the Origin of JOvian planets).
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All Figures
Fig. 1 Steadystate solutions for the solidtogas mixing ratio Z as a function of the Stokes number of solid particles τ_{s} for different values of the solidtodustmass flux ratio ξ (as labeled), assuming a value of the turbulent viscosity α = 10^{3}. The values of τ_{s} corresponding to expected pebble sizes are highlighted with larger symbols. The two solutions provided by Eq. (13)are indicated by filled and open symbols, respectively. The gray area highlights the region in which planetesimals should form by a streaming instability (Carrera et al. 2015). 

Open with DEXTER  
In the text 
Fig. 2 Steadystate solutions for the solidtogas mixing ratio Z as a function of the solidtogasmass flux ratio ξ for different values of the Stokes number of solid particles τ_{s} (from 10^{5} to 10^{2}, as labeled), assuming two values of the turbulent viscosity α = 10^{4} (in blue) and α = 10^{3} (in red). Equation (13) is numerically solved. In contrast to Fig. 1, we now consider that initially icy pebbles with τ_{s,peb} ~ 0.1 and containing a mass fraction ζ_{0} = 1/3 in dust sublimate inside of the snow line. The thicker lines corresponds to the preferred value for the dust particles, τ_{s} = 10^{4}. 

Open with DEXTER  
In the text 
Fig. 3 Time evolution of a) radius of the pebble formation front, b) pebble accretion rate (Ṁ_{peb}), and c) ξ_{peb} = Ṁ_{peb}/Ṁ_{∗} for two values of α, 10^{4} (red) and 10^{3} (blue). The lines labeled “grow” (dotted) and “GI” (dashed) represent the pebble growth and disk GI limits, respectively. The thick solid lines express the actual values obtained by the minima of the two limits. Here we assumed τ_{s,peb} = 0.1 and Z_{0} = 0.01. In panel c), the small squares represent the points with ξ_{peb}>ξ_{crit}, see Eq. (18). 

Open with DEXTER  
In the text 