Dust production in debris discs: constraints on the smallest grains
LESIAObservatoire de Paris, UPMC Univ. Paris 06, Univ.
ParisDiderot,
92195
Meudon
France
email:
philippe.thebault@obspm.fr
Received: 23 October 2015
Accepted: 4 January 2016
Context. The surface energy constraint puts a limit on the smallest fragment s_{surf} that can be produced after a collision. Based on analytical considerations, this mechanism has been recently identified as being having the potential to prevent the production of small dust grains in debris discs and to cut off their size distribution at sizes larger than the blowout size.
Aims. We numerically investigate the importance of this effect to find out under which conditions it can leave a signature in the smallsize end of a disc’s particle size distribution (PSD). An important part of this work is to map out, in a disc at steadystate, what is the most likely collisional origin for μmsized dust grains, in terms of the sizes of their collisional progenitors.
Methods. For the first time, we implement the surface energy constraint into a collisional evolution code. We consider a typical debris disc extending from 50 to 100 au and two different stellar types: sunlike and A star. We also consider two levels of stirring in the disc: dynamically “hot” (⟨e⟩ = 0.075) and “cold” (⟨e⟩ = 0.01). In all cases, we derive s_{surf} maps as a function of target and projectile sizes, s_{t} and s_{p}, and compare them to equivalent maps for the dustproduction rate. We then compute discintegrated profiles of the PSD and estimate the imprint of the surface energy constraint.
Results. We find that the (s_{p},s_{t}) regions of high s_{surf} values do not coincide with those of high dust production rates. As a consequence, the surface energy constraint generally has a weak effect on the system’s PSD. The maximum s_{surf}induced depletion of μmsized grains is ~30% and is obtained for a sunlike star and a dynamically “hot” case. For the e = 0.01 cases, the surface energy effect is negligible compared to the massive small grain depletion that is induced by another mechanism: the “natural” imbalance between dust production and destruction rates in lowstirring discs.
Key words: planetary systems / circumstellar matter
© ESO, 2016
1. Introduction
Circumstellar debris discs have been detected around mainsequence stars because of the excess luminosity produced by μm to mmsized dust. Because of its relatively short lifetime, this dust cannot be primordial and is thought to be steadily replenished by a chain of erosive collisions starting from a reservoir of large, planetesimallike, parent bodies (e.g. Wyatt 2008; Krivov 2010). In an idealized case, the size distribution of bodies within such a collisional cascade should settle toward a power law of the form dN ∝ s^{q}ds, where the index q is close to −3.5 (Dohnanyi 1969). Such a power law has the interesting characteristics that, while most of the disc’s mass is contained in the biggest objects of the cascade, most of the geometrical cross section should be contained in the smallest grains of size s_{min}. This means that, at all wavelengths λ ≲ 2πs_{min}, the disc’s luminosity should be dominated by these smallest grains.
Determining s_{min} is consequently of crucial importance. Luckily, for most stars, stellar radiation pressure imposes a “natural” minimum cutoff size that is, in principle, easy to estimate. Indeed, because this pressure is both ∝1 /s and, like stellar gravity, ∝1 /r^{2} (r being the radial distance to the star), there is a minimum size s_{blow} below which radiation pressure overcomes gravity and grains are quickly blown out from the system. Because small grains are probably produced from parent bodies on Keplerian orbits, the criteria for estimating s_{blow} is given by β(s_{blow}) = F_{Rad.Press.}/F_{grav.} = 0.5 (for parent bodies on circular orbits). For typical astrosilicates (Draine 2003), values for β range from ~0.5 μm for solartype stars to 2−10 μm for latetype A stars (Krivov 2007)^{1}. As a consequence, in scattered light, disc luminosities should always be dominated by grains close to s_{blow}, and, for A stars, s_{blow} grains should also dominate the thermal flux up to midIR wavelengths.
However, observations have sometimes challenged this basic view. Some discs have, for example, been found to contain important quantities of submicron grains below s_{blow} (Ardila et al. 2004; Fitzgerald et al. 2007; Johnson et al. 2012). While there does not seem to be one single straightforward scenario for the presence of such tiny particles, some possible explanations have been suggested. One of them is that, at very small sizes, the β(s) curve starts to decrease with decreasing s before reaching a constant value that can be <0.5 for K, G or F stars, thus placing submicron grains on bound orbits (Johnson et al. 2012). Another possibility is that we are witnessing a powerful collisional chainreaction, called an “avalanche”, of highβ particles that are passing through a dense ring of larger grains (Grigorieva et al. 2007).
1.1. Discs with “too large” minimum dust sizes
Here we will focus on the opposite problem, i.e., the systems for which the observationally derived s_{min} has been found to be larger than s_{blow}. The existence of such systems was first unambiguously determined by Pawellek et al. (2014), who considered a sample of 34 resolved discs^{2} and found that, while discs around Astar have s_{min} ~ s_{blow}, the s_{min}/s_{blow} ratio increased towards lowmass stars and could reach ~10 for solartype stars. These results were later confirmed by Pawellek & Krivov (2015), who showed that the s_{smin}/s_{blow} trend is robust and does not depend on material compositions or grain porosity.
These results seem to be in contradiction with the aforementioned behaviour expected in discs where a size distribution in dN ∝ s^{q}ds holds down to the blowout size s_{blow}, for which luminosities should be dominated by grains close to the blowout size. Interestingly, numerical investigations of the collisional evolutions of debris discs have shown that, at least in the smallsize domain, size distributions can in fact significantly depart from an idealized dN ∝ s^{q}ds powerlaw. The simulations of Thebault et al. (2003), Krivov et al. (2006), Thebault & Augereau (2007) have indeed shown that sizedistributions can exhibit pronounced “wavy” patterns in the ≲100s_{blow} domain, triggered by the absence of potential destructive projectiles below s_{blow}. However, this waviness cannot deprive the system from grains close to s_{blow}, and, generally, it even has just the opposite effect, inducing a density peak of grains in the 1.5−2s_{blow} region (Thebault & Augereau 2007).
A possible cause of smallgrain depletion could be the wellknown PoyntingRobertson drag, which causes small particles to slowly spiral starwards. However, this effect is likely to be only noticeable for tenuous discs for which the collisional timescale of small grains becomes larger than t_{PR} (Wyatt 2005). The first plausible mechanism for smallgrain depletion that could also work for bright discs was proposed by Thebault & Wu (2008), who showed that, in discs with a low dynamical excitation (low orbital eccentricities e and inclinations i), there is an imbalance between the production and destruction rates of small grains. Indeed, while their production rate is controlled by the erosion of larger particles and is thus low because of these particles’ low ⟨e⟩, their destruction rate, which is controlled by impacts involving the small grains themselves, is much higher because these grains are placed on highe orbits by radiation pressure regardless of the dynamical excitation in the rest of the system. As a result, depending on the value for ⟨e⟩, there could be a strong depletion of grains up to submm sizes. The main issue with this scenario is that it requires values of ⟨e⟩ that could possibly be unrealistically low (see Sect. 4).
1.2. Maximum surface energy and minimum fragment size
Another, potentially more generic scenario has recently been proposed by Krijt & Kama (2014). It is based on an energyconservation criteria, related to the physics of collisions themselves, which, remarkably enough, had never before been invoked in the context of debris discs. The argument is that a destructive collision cannot produce a size distribution of fragments that reaches an infinitely small value, as this would require an infinite amount of energy. This is because creating new fragments means increasing the amount of exposed surface, and this requires energy. There is thus a minimum size s_{surf} in the fragment distribution, which is given by the requirement that the total surface energy of all fragments cannot exceed the kinetic energy of the impact. For the specific case of a fully destructive impact between two equalsized bodies and for a q = −3.5 size distribution, Krijt & Kama (2014) derived a relation linking s_{surf} to s_{lfr}, the size of the largest fragment, and the impactor’s size s_{0}: (1)where v_{rel} is the impact velocity, γ is the surface energy per unit surface, and η is the fraction of the kinetic energy that is used to create a new surface. Interestingly, this formula gives the counterintuitive result that, for a given v_{rel} and a fixed s_{lfr}/s_{0} ratio (i.e., for selfsimilar impacts), the size of the smallest fragment increases with decreasing s_{0} sizes. For lowvelocity impacts, low values of η and for material with high γ (like ice), s_{surf} can be significantly higher than s_{blow} for s_{0} ≲ 1 m (see Fig. 1 of Krijt & Kama 2014).
However, while deriving estimates of s_{surf} is analytically possible for one given collision, giving global estimates for a whole debris disc is a much more problematic task, since the values of s_{surf} will vary greatly depending on the absolute and relative sizes of the impacting objects. In their pioneering study, Krijt & Kama (2014) attempt to derive a discintegrated s_{surf} by only considering impacts between equalsized s_{0} objects and by restricting themselves to “barely catastrophic” impacts where each impactor is split into two identical fragments. This simplifying choice relied on two main assumptions: 1) for a given target size, and if s_{lfr} is fixed, it is with a projectile of the same size that the smallest “smallest fragments” should be obtained, hence the choice of equalsized impacts as the most constraining ones, and 2) in a collisional cascade, small grains should preferentially originate from collisions that involve particles that are barely larger than themselves; so that even if collisions amongst large objets have very small s_{surf} values (because if s_{lfr}/s_{0} is fixed) the amount of small dust they produce cannot compensate for the amount of small dust “not produced” (because of their larger s_{surf}) by collisions amongst small grains, hence the focus on the smallest “barely catastrophic” impacts . Under these simplifying hypotheses, the discintegrated is then the s_{surf} obtained for the smallest s_{0} object that can be split in two by an impact at v_{rel}(2)where f = (1.25e^{2} + i^{2})^{0.5}.
1.3. Need for a numerical approach
As rightfully stated by Krijt & Kama (2014) themselves, Eq. (2) should only be taken as an order of magnitude estimate, as it only takes into account a very limited range of collision types. But even this role as an orderofmagnitude indication should be taken with caution, because the simplifying assumptions that justify the predominant role of equalsized barelycatastrophic impacts might not hold in a realistic disc. The first problem is that this predominant role has been inferred by assuming that s_{lfr} is fixed, which is far from being the case in realistic conditions, where s_{lfr} strongly depends on the two impactors’ mass ratio and on v_{rel} (e.g. Leinhardt & Stewart 2012). Another problem is that it was derived assuming that a q = −3.5 power law holds for the whole size distribution, which several debris disc studies have shown to be erroneous, especially in the small grainsize domain. Last but not least, it neglects the contribution of cratering impacts, which might have a dominant role in the global dust production and destruction balance (Thebault & Augereau 2007; Kobayashi & Tanaka 2010)^{3}.
Pawellek & Krivov (2015) improved on Eq. (2) by taking into account the fact that velocities of impacts involving small grains might have higher values because radiation pressure places small grains on highly eccentric or even parabolic orbits. This significantly reduces the values of with respect to Eq. (2). Interestingly, this did not, however, improve the fit to the observationallyderived , for which the original Krijt & Kama (2014) formula seems to provide a better match (see Fig. 14 of Pawellek & Krivov 2015). However, even the improved formula used by Pawellek & Krivov (2015) is still a discaveraged analytical estimate that relies on the same simplifying assumptions mentioned earlier.
We propose here to take these studies a step further by incorporating, for the first time, the surfaceenergy constraint within a numerical collisional evolution code, estimating s_{surf}(i,j) for all pairs of impacting bodies of sizes s_{i} and s_{j} and taking these values into account in the collisionoutcome prescription of our code. We describe our numerical model and the setups for the different cases that we explore in Sect. 2. Results are presented in Sect. 3, where we first compare the s_{surf}(i,j) maps to equivalent (s_{i}, s_{j}) maps for the level of dust production in the disc. We then display the steady state particle size distributions obtained for all considered cases, and investigate the signature of the surfaceenergy constraint by comparing these particle size distributions (PSDs) to those obtained for control runs without the s_{surf} criteria. In Sect. 4, we then discuss the importance of the surfaceenergy constraint, in particular with respect to the concurrent “lowstirring” dustdepletion mechanism identified by Thebault & Wu (2008).
2. Model
We use the statistical collisional model developed by Thebault et al. (2003) and later upgraded by Thebault & Augereau (2007). This has a “particleinabox” structure, where particles are sorted into logarithmic size bins separated by a factor 2 in mass. It also has a 1D spatial resolution, being divided into radially concentric annuli. Collision rates between all size bins are computed using an estimate of the average orbital eccentricity and inclinations in each size bin. Crucially, the code takes into account the increased eccentricities, and thus impact velocities of small grains whose orbits are affected by stellar radiation pressure, as well as the fact that these grains are able to cross several concentric annuli (see Appendix of Thebault & Augereau 2007). Collision outcomes are then divided into two categories, cratering and fragmentation, depending on the ratio between the specific impact kinetic energy and the specific shattering energy Q ∗, which depends on object sizes and composition. In both regimes, the size of the largest fragment and the size distributions of the other debris are derived through the detailed energyscaling prescriptions that are presented in Thebault et al. (2003) and Thebault & Augereau (2007).
The main upgrade for the present runs is that we implement a prescription for s_{surf}. We derive s_{surf}(i,j) for all possible colliding pairs of sizes s_{i} and s_{j} and implement this parameter into our collisionoutcome prescription: if, for a given collision between the bins “i” and “j”, s_{surf}(i,j) >s_{min−num} (where s_{min−num} is the smallest size bin considered in the code), we cut the postimpact fragment distribution at s_{surf}(i,j) instead of s_{min−num}.
We estimate the values of s_{surf}(i,j) for both fragmenting and cratering impacts, using the equations presented in the Appendix of Krijt & Kama (2014): where we assume that i is the target and j the projectile (s_{i} ≥ s_{j}). For the cratering case, κ is the ratio m_{crat}/m_{j}, where m_{crat} is the total mass excavated from the target i. The crucial point here is that, instead of having to assume values for the largest fragment’s size s_{lfr} and the ratio κ, we can retrieve both quantities in a selfconsistent way from our collisionoutcome prescription.
2.1. Setup
Setup for the numerical simulations.
We do not consider the full range of possible L^{∗}/L_{⊙} explored by Pawellek & Krivov (2015), but restrict ourselves to the two illustrative cases of a sunlike L^{∗} = L_{⊙} star and a βPiclike A5V star with L^{∗} = 9 L_{⊙}. We consider a reference debris disc extending from r_{min} = 50 au to r_{max} = 100 au. The disc’s total initial mass is M_{disc} = 0.5 M_{⊕}, distributed between s_{max−num} = 50 m and s_{min−num} = s_{blow}, which corresponds to ~0.01 M_{⊕} of <1 mm dust^{4}. The value of s_{blow} is equal to 0.5 μm for the sunlike case (with 80 size bins from s_{max} to s_{blow}) and 4 μm for the A star one (71 size bins). As for the dynamical state of the disc, we consider one dynamically “hot” case with ⟨e⟩ = 0.075 and one dynamically “cold” case with ⟨e⟩ = 0.01. For each L^{∗} and ⟨e⟩ case, we run both a simulation taking into account the s_{surf} prescription and a reference case with no s_{surf}. We let the runs evolve for 10^{7} yr, which is enough to reach a collisional steadystate in the dust size domain (<1 cm).
As for the free parameters of the s_{surf} prescriptions (Eqs. (3) and (4)), we adopt a conservative approach and chose, amongst the η and γ values considered by Krijt & Kama (2014), the ones that are in principle the most favourable to yielding large s_{surf} values. We thus take η = 0.01 (i.e., 1% of the kinetic energy is used for creating new surface) and γ = 0.74 J m^{2}, which is the value for water ice^{5}.
All main parameters for the setup are summarized in Table 1.
Main results for all four collisional runs.
3. Results
3.1. Dust progenitors and s_{surf} maps
Fig. 1 Discintegrated mean values of the s_{surf}/s_{blow} ratio for all possible targetprojectile pairs of sizes s_{i} and s_{j}. Upper left panel: Astar primary and ⟨e⟩ = 0.075. Upper right: Astar primary and ⟨e⟩ = 0.01. Bottom left: solartype primary and ⟨e⟩ = 0.075. Bottom right: solartype primary and ⟨e⟩ = 0.01. The dashed white line delineates the s_{i} = s_{j} diagonal of equalsized impactors. 

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Fig. 2 Discintegrated and normalized production rates of s ≤ 10 μm dust as a function of target and projectile sizes. These maps are estimated, at collisional steadystate, for the “control runs” where the s_{surf} criteria is turned off. This allows for an easy comparison with Fig. 1 by directly showing which high s_{surf}(i,j) regions will have a significant damping effect on the smalldust production rate. Upper left panel: Astar primary and ⟨e⟩ = 0.075. Upper right: Astar primary and ⟨e⟩ = 0.01. Bottom left: solartype primary and ⟨e⟩ = 0.075. Bottom right: solartype primary and ⟨e⟩ = 0.01. 

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For each of the considered setups, Fig. 1 presents the s_{surf}(i,j) value for every pair of impacting target and projectile of sizes s_{i} and s_{j}. We also show, as a useful comparison tool, the equivalent (s_{i}, s_{j}) maps of the amount of μmsized dust produced as a function of target and projectile sizes (Fig. 2).
3.1.1. Astar
For the dynamically “hot” (⟨e⟩ = 0.075) Astar case, Eq. (2) predicts a low of only ~0.4, and our s_{surf}(i,j) map is in good agreement with this prediction, since almost all (i,j) impacts result in s_{surf}/s_{blow}< 1 (Fig. 1a) . There is, admittedly, a very narrow range of impacts for which s_{surf}/s_{blow} ≥ 1, but even there the ratio does not exceed 2. Moreover, this limited family of s_{surf}>s_{blow} collisions corresponds to a region that has only a very limited contribution to the total dust production. The dust production is instead dominated by impacts that involve s ~ s_{blow} = 5 μm projectiles and larger 5s_{blow} ≲ s ≲ 20s_{blow} targets (see Fig. 2a). This is because s ~ s_{blow} grains are placed on highe orbits by radiation pressure and impact all other grains at very high velocities. They are thus very efficient at fragmenting targets and producing dusty debris over a wide range of target sizes.
The situation is slightly different for the dynamically “cold” Astar case, for which the fraction of s_{surf}>s_{blow} impacts is more extended and the maximum value for s_{surf}(i,j) reaches ~5 (Fig. 1b). This value is, however, well below the analytical prediction of Eq. (2). We note that these highs_{surf}(i,j) impacts are located close the s_{i} = s_{j} diagonal, which could appear to contradict Fig. B.1 of Krijt & Kama (2014), which shows instead a minimum of the s_{surf} curve for equalsized impactors. But this is because this analyticallyderived curve is only valid for fragmenting impacts, whereas here, with ⟨e⟩ = 0.01, the highcollisionalenergy requirement for fragmentation can only be met when projectiles have sizes comparable to the target. All other cases result in cratering, for which s_{surf} are, in general, much smaller. As was the case for the ⟨e⟩ = 0.075 run, the (i,j) region of high s_{surf} values does not match the region of highest dust production, which corresponds, here again, to fragmenting impacts by s ~ s_{blow} projectiles on 5s_{blow} ≲ s ≲ 20s_{blow} targets (Fig. 2b). Nevertheless, the contribution of the high s_{surf} region is not fully negligible, so that we expect the maximum surface energy criteria to have at least some effect on the system’s evolution.
3.1.2. Solartype star
For a Sunlike star, orbital velocities are lower and collisions less energetic than for an Astar for equivalent orbital parameters. The s_{surf}/s_{blow} values are thus logically higher, as is clearly seen in Figs. 1c and d. The s_{surf}/s_{blow}> 1 domain is much more extended than for an A star, with peak values reaching ~25 for the ⟨e⟩ = 0.075 case and even ≥100 for the dynamically cold case. As was already the case for the Astar runs, these peak s_{surf} values are located, in the (s_{i}, s_{j}) map, on a line corresponding to the transition from cratering to fragmenting impacts^{6}. For the ⟨e⟩ = 0.075 run, this transition always occurs for s_{j} ≤ 0.5s_{i}, with a s_{j}/s_{i} ratio that is decreasing towards larger sizes following an approximate law in s_{2}/s_{1} ~ 0.4−0.1log (s_{2}/ 10 μm)^{7}. For ⟨e⟩ = 0.01, the peak s_{surf} line still corresponds to the fragmentation/cratering transition, but is now much closer to the s_{i} = s_{j} diagonal, which reflects the fact that, for these lower impact velocities, it takes a larger projectile to fragment a given target.
We also note that, for both the ⟨e⟩ = 0.075 and ⟨e⟩ = 0.01 cases, there are no high s_{surf}/s_{blow} values for impactors ≲10 μm. This is because such small grains have more energetic impacts (able to produce smaller fragments) because of their radiationpressureaffected orbits. Incidentally, this s ≲ 10 μm region is the one where the μmdust production is the highest (Figs. 2c and d). This means that a large fraction of the dustgenerating impacts are, in fact, able to produce fragments down to the blowout size. The regions of high s_{surf}/s_{blow}> 5 only contribute to approximately 20% of the ≤10 μm dust production for both the ⟨e⟩ = 0.075 and ⟨e⟩ = 0.01 cases.
3.2. Size distribution
Fig. 3 Astar case. Geometrical crosssection as a function of particle size, at t = 10^{7} yr, integrated over the whole 50−100 au disc, for both the dynamically “hot” (⟨e⟩ = 0.075) and “cold” (⟨e⟩ = 0.01) cases. For each case, a reference run with the s_{surf} constraint switchedoff is also presented. The two vertical dotted lines represent the analytical values given by Eq. (2). 

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Fig. 4 Same as for Fig. 3, but for a solartype star. 

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Figures 3 and 4 present the PSD, in terms of the differential distribution of geometric cross section dΣ / ds, at collisional steadystate^{8}, for all four of the explored setups, as well as for the control runs where the s_{surf} constraint is ignored. Visualizing the dΣ / ds distribution allows us to immediately identify which particle sizes dominate the system’s optical depth, and thus its luminosity.
3.2.1. Generic features
Before assessing the additional effect of the s_{surf} constraint, let us first emphasise some characteristics of the obtained size distributions that are well known generic features of steadystate collisional discs, and that also affect the smallsize domain of the PSDs.
A first “classic” result is that, for all cases, the size distribution displays a clear wavy structure in the ≲100 μm domain. This waviness is more pronounced than in Pawellek & Krivov (2015) because we consider a wider disc, so that small grains, placed on highe orbits par radiation pressure, will be able to impact target particles at a higher impact angle and thus higher v_{rel}. This will reinforce their shattering power and thus the amplitude of the size distribution wave (see Thebault & Augereau 2007, for a detailed discussion on “waveenhancing” factors).
Another crucial result is the strong depletion of small particles for all ⟨e⟩ = 0.01 cases. This is the signature of the mechanism identified by Thebault & Wu (2008), i.e., the strong imbalance between the smalldust production and destruction rates for dynamically cold systems (see Sect. 1.1). As underlined by Pawellek & Krivov (2015), this effect is more pronounced for solartype stars, for which we find a clear drop of the dΣ / ds curve at s ≲ 50s_{blow}, than for A stars, where the PSD is roughly flat in the same s ≲ 50s_{blow} domain. Nevertheless, for both cases the depletion of small grains is very strong, exceeding one order of magnitude as compared to a dynamically “hot” system.
3.2.2. Effect of the s_{surf} constraint
As expected from the s_{surf}(i,j) and dustproduction maps (see Sect. 3.1), the effect of the s_{surf} constraint is very weak for the Astar runs. For the ⟨e⟩ = 0.075 case, the PSD is almost indistinguishable from the control run with s_{surf} switched off (Fig. 3). Some differences between the with and withouts_{surf} runs are visible for the ⟨e⟩ = 0.01 case, but they remain relatively limited. Interestingly, they are not so much visible as a dearth of grains in the s ≤ 10−20 μm domain where the few s_{surf} ≥ s_{blow} values lie (see Fig. 1b), but rather as an excess of larger particles in the ~30−50 μm range. This is a direct consequence of the specific dynamics, and thus destructive efficiency, of grains close to s_{blow}. In the “s_{surf}free” case, it is indeed this high destructive efficiency that is responsible for the strong depletion of 30−50 μm particles in the PSD (the first dip in the “wave”)^{9}. So removing a small number of small destructive impactors will necessarily increase the number of 30−50 μm grains. And given that we start from a strongly depleted population of 30−50 μm particles in the s_{surf}free case, their relative increase in the s_{surf} runs will be higher than the relative decrease of s ≤ 10−20 μm grains. And this is exactly what we observe (see Table 2): a ~5% depletion of s ~ s_{blow} grains that causes a ~20% excess of s ~ 30 μm particles, which is the size of the biggest objects that can be destroyed by projectiles close to s_{blow}. However, for this ⟨e⟩ = 0.01 case, the s_{surf}induced depletions and excesses remain marginal when compared to the much more significant global depletion of s ≲ 50s_{blow} grains caused by the low dynamical excitation of the system (see Discussion).
The imprint of the s_{surf} limit is, logically, much more visible around a Sunlike star (Fig. 4). Interestingly, this is especially true of the dynamically hot disc, for which the depletion of s ≤ 2s_{blow} grains is close to 30%, and the related excess of 4s_{blow} ≤ s ≤ 20s_{blow} particles exceeds 40% (Table 2). For the ⟨e⟩ = 0.01 run, these excesses and depletions are only ~12% and ~14%, respectively. This is a rather counterintuitive result, as Eq. (2) predicts a ratio that is 50 times higher for ⟨e⟩ = 0.01 than for ⟨e⟩ = 0.075, and even the more accurate Figs. 1c and d show peak s_{surf}(i,j) values that are still 6−7 times higher for the dynamically cold case. However, as discussed in Sect. 3.1.2, the crucial point is that these (s_{i}, s_{j}) regions of high s_{surf} values do not match those of high dust production. Most of the dust is indeed created by collisions on s ≤ 10 μm (≲20s_{blow}) targets (Fig. 2), for which s_{surf} rarely exceeds s_{blow} (Fig. 1). Another reason for which the s_{surf} constraint only leaves a weak signature on the ⟨e⟩ = 0.01 system is that, in the s ≤ 100 μm range, the PSD is already massively depleted because of the dust production imbalance inherent to dynamically “cold” discs. This effect is even stronger than for the Astar case, with a depletion that reaches almost two orders of magnitude for s ~ s_{blow} grains (Fig. 4).
4. Discussion and conclusion
Figures 3 and 4 seems to indicate that the two dynamically “hot” cases bear some similarities with the prediction of Eq. (2) regarding the imprint of the s_{surf} constraint on the PSD. In the Astar case, this similarity is, of course, simply that the surface energy constraint has no effect on the PSD, but in the ⟨e⟩ = 0.01 case, we do indeed find a depletion of grains in the s ≤ 2s_{blow} domain, which is roughly consistent with the value given by Eq. (2). However, this quantitative agreement is probably largely a coincidence, firstly because the depletion of s ≤ s_{surf} grains does not come from the “smallestbarelycatastrophicequalsizeimpactors” collisions considered in deriving Eq. (2) (see Figs. 1c and 2c), and secondly because this depletion stops at s ~ 2s_{blow} largely because of the “natural” waviness of the PSD, combined with the excess of ≥3s_{blow} particles. We also note that this depletion is limited to ~30% and does not create a sharp cutoff in the size distribution. This limited amplitude, combined to the aforementioned excess of particles in the ~3s_{blow} to ~20−30s_{blow} range, renders the PSD plateaulike at sizes smaller than ~10s_{blow}. It is thus difficult to define a proper s_{min} for the size distribution, but, at least from a qualitative point of view, we confirm that, for this case, the maximum energy criteria does have a visible effect on the lowerend of the PSD.
The situation is radically different for the dynamically cold cases, for which the simulated size distributions are much less affected by the s_{surf} constraint than would be expected from simple analytical estimates, which predict values in excess of 20, or even 150 (see Table 2). Even though we do find some (s_{i}, s_{j}) targetprojectile configurations that result in large s_{surf} comparable to these values (Fig. 1), the decisive point is that these collisions only have a marginal contribution to the disc’s total smalldust production. As a result, the differences with the reference s_{surf}free cases’ PSDs remain very limited: for neither the Astar nor the Sunlike cases do we obtain a depletion that exceeds ~10% in the small grain domain close to s_{blow}. Crucially, the s_{surf} constraint is never strong enough to change the size of grains that dominate the system’s geometrical crosssection, and thus its luminosity.
More importantly, we confirm that the most efficient way of depleting a collisional debris disc from its small grains is, by far, to reduce its dynamical excitation. The production/destruction imbalance mechanism identified by Thebault & Wu (2008) for lowstirring discs has an effect that exceeds, by more than one order of magnitude, that of the s_{surf} limit. This is true both for the amplitude of the smallgrain depletion and for the sizerange that is affected. For low ⟨e⟩ values, the size s_{cold} below which the PSD is depleted is, to a first order, given by the relation (5)which translates into (6)leading to s_{cold} ~ 50s_{blow} for our ⟨e⟩ = 0.01 case, a value that roughly agrees with the one obtained in the simulations (Figs. 3 and 4). It should be noted that this dominance of the lowstirring imbalance effect over the surface energy constraint is probably even stronger than that witnessed in Figs. 3 and 4, since we have taken, for our s_{surf} prescription, the parameters that were the most likely to lead to high s_{surf} values (see Sect. 2.1). On a related note, choosing disc configurations that enhance the amplitude of the surfaceenergy constraint, by, for instance, decreasing ⟨e⟩ or increasing [r_{min},r_{max}], would not change this dominance either, because these configurations would also enhance the dustproductionimbalance effect by further decreasing the level of stirring in the disc.
We emphasise that we did not attempt to fit observed s_{min}/s_{blow} trends with an exhaustive parameter exploration (⟨e⟩, M_{disc}, [ r_{min},r_{max} ], etc.) in the spirit of the Pawellek & Krivov (2015) study. Our goal was here to quantify, in a selfconsistent way, the relative effects of the two potential dustdepletion mechanisms that are the surface energy constraint and the lowstirring dustproduction imbalance. For the sake of clarity, and to clearly identify the mechanisms at play, we restricted our study to a reference wide disc and to the two illustrative reference cases of a sunlike and an Atype star. Test runs with a narrower, ringlike disc, have, however, been performed. They gave relatively similar results as to the relative imprint of the s_{surf} and lowstirring effects on the PSD, although with a less pronounced waviness in the PSD’s shape, which was to be expected, since small grains close to s_{blow} will impact larger targets within a narrow ring at a lower velocity than they would have in an extended disc.
Despite this limited parameter exploration, we note however that our conclusions seem to agree with the numerical investigations of Pawellek & Krivov (2015), who were able to find a reasonable fit to the s_{min}/s_{blow} trend observed on their 34stars sample with the lowstirringinduced mechanism alone. The main assumption for this fit to work is that, while the dynamical stirring of A stars should be of the order of e ~ 0.1, it should decrease towards lower mass stars and be as low as ~0.01 for solartype objects. Such values might appear unrealistically low if we assume the classical view that debris discs are stirred by large lunartoMarssized large planetesimals (e.g., Thebault 2009). However, the level and the cause of stirring is still an open issue in present debris discs studies (see discussion in Pawellek & Krivov 2015). Moreover, we know of at least two discs, around the G0Vstar HD 207129 (Löhne et al. 2012), and the K2Vstar HIP 17429 (Schüppler et al. 2014), for which collisional modelling predicts ⟨e⟩ barely larger than 0.01.
5. Summary
Based on analytical considerations, the pioneering study of Krijt & Kama (2014) finds that the surface energy constraint, which limits the size of the smallest fragment produced after destructive collisions, could potentially affect the evolution of debris discs by limiting the smallest sizes of observable grains to a value s_{surf} larger than the blowout size s_{blow}.
Here we numerically quantify the importance of this mechanism by incorporating, for the first time, the surface energy constraint into a statistical code that follows the collisional evolution of a debris disc. Instead of relying on an analytical systemaveraged value, we compute s_{surf}(i,j) for all impacting targetprojectile pairs of sizes s_{i} and s_{j} in the disc. We consider two stellar types, sunlike and AV5, and two levels of stirring for the disc, ⟨e⟩ = 0.075 and ⟨e⟩ = 0.01. Our main results and conclusions are as follows:

We confirm that, for all considered setups, there is a fraction of the (s_{i}, s_{j}) space for which s_{surf} ≥ s_{blow}, especially for low ⟨e⟩ and/or lowmass stars.

However, the (s_{i}, s_{j}) regions of high s_{surf} do not coincide with the regions of high dust production. Indeed, most of the μmsized dust is produced by impacts involving ~3−20s_{blow} targets and small ≲2−3s_{blow} projectiles, for which s_{surf} is, in general, smaller than the blowout size. This is mainly due to the fact that impacts involving small grains are very energetic, because these particles are placed on highe orbits by radiation pressure.

Because of this discrepancy between highs_{surf} and dustproducing collisions, the global effect of the surface energy constraint is generally relatively limited. The only setup for which it has a visible signature on the PSD is for a dynamically “hot” disc around a solartype star. But even there, the depletion of small dust does not exceed 30%.

For the low ⟨e⟩ = 0.01 cases, which should in principle be the most favourable to strong surfaceenergy constraints, the depletion of small grains never exceeds 12% and is never able to change the sizes of particles that dominate the system’s geometrical crosssection.

At such lowstirring levels, the system’s PSD in the small size domain is, instead, totally dominated by another mechanism: the imbalance between small dust production and destruction rates identified by Thebault & Wu (2008) for lowe discs. This imbalance creates a depletion of small grains that is at least one order of magnitude more pronounced than that caused by the surfaceenergy constraint, and it affects grains over a much wider size range.
Even if its effect is not as decisive as could be analytically expected, we do, however, recommend implementing the surfaceenergy constraint in collisionalevolution codes, as it might leave visible signatures in the lows end of PSDs for some stardisc configurations.
For subsolar stars, radiation pressure is too weak to overcome gravity, but stellar wind could play a similar role for active M stars like AU Mic (Augereau & Beust 2006; Schüppler et al. 2015).
There is an inherent degeneracy between grain size and disc radius that can only be broken for resolved systems (see discussion in Pawellek & Krivov 2015).
To their credit, Krijt & Kama (2014) briefly investigate the role of cratering impacts in their Appendix, but again assuming a fixed s_{lfr}/s_{target} ratio (see Fig. B.1. of that paper).
This relatively sharp transition at the cratering/fragmentation boundary is due to the discontinuity in the s_{surf} prescriptions of Krijt & Kama (2014; Eqs. (3) and (4)) at this boundary (s_{lfr} = 2^{−1/3}s_{1}). More refined and selfconsistent s_{surf} prescriptions should probably be derived in the future, but we chose to stick to the laws given by Krijt & Kama (2014) for this exploratory work.
This negative slope reflects the fact that, in the strength regime, the specific shattering energy Q ∗ decreases with increasing sizes (Benz & Asphaug 1999).
Or, more exactly, when steadystate is at least reached where it matters here, that is in the s< 1 cm domain. In the largesize domain, there is a difference of PSD slopes between the dynamically hot and cold discs that is due to the ⟨e⟩ = 0.01 systems not yet having had time to reach collisional steadystate for these larger objects yet. Collisional steadystate does indeed work its way up the PSD with time, at a rate that strongly increases with decreasing dynamical stirring (e.g., Löhne et al. 2008).
Acknowledgments
The author would like to thank M. Kama, S. Krijt and A. Krivov for fruitful discussions.
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All Tables
All Figures
Fig. 1 Discintegrated mean values of the s_{surf}/s_{blow} ratio for all possible targetprojectile pairs of sizes s_{i} and s_{j}. Upper left panel: Astar primary and ⟨e⟩ = 0.075. Upper right: Astar primary and ⟨e⟩ = 0.01. Bottom left: solartype primary and ⟨e⟩ = 0.075. Bottom right: solartype primary and ⟨e⟩ = 0.01. The dashed white line delineates the s_{i} = s_{j} diagonal of equalsized impactors. 

Open with DEXTER  
In the text 
Fig. 2 Discintegrated and normalized production rates of s ≤ 10 μm dust as a function of target and projectile sizes. These maps are estimated, at collisional steadystate, for the “control runs” where the s_{surf} criteria is turned off. This allows for an easy comparison with Fig. 1 by directly showing which high s_{surf}(i,j) regions will have a significant damping effect on the smalldust production rate. Upper left panel: Astar primary and ⟨e⟩ = 0.075. Upper right: Astar primary and ⟨e⟩ = 0.01. Bottom left: solartype primary and ⟨e⟩ = 0.075. Bottom right: solartype primary and ⟨e⟩ = 0.01. 

Open with DEXTER  
In the text 
Fig. 3 Astar case. Geometrical crosssection as a function of particle size, at t = 10^{7} yr, integrated over the whole 50−100 au disc, for both the dynamically “hot” (⟨e⟩ = 0.075) and “cold” (⟨e⟩ = 0.01) cases. For each case, a reference run with the s_{surf} constraint switchedoff is also presented. The two vertical dotted lines represent the analytical values given by Eq. (2). 

Open with DEXTER  
In the text 
Fig. 4 Same as for Fig. 3, but for a solartype star. 

Open with DEXTER  
In the text 