© ESO, 2014

1. Introduction

Fragmenting collisions are important in a range of astrophysical systems. While the slope of the fragment size distribution and the size of the largest fragment are well characterized and can be used confidently in models, the smallest fragment size is less well understood and is usually assumed to be constant for all collisions. We provide a framework for self-consistently calculating the smallest fragment size as a function of material and collision parameters (Sect. 2), and discuss its implications for modelling debris disks (Sect. 3).

Numerous experimental studies have looked at the fragment size distribution of destructive collisions, focussing on the slope of the power law(s), and on the size of the largest fragment (Davis & Ryan 1990; Ryan et al. 1991; Nakamura & Fujiwara 1991; Ryan 2000). The smaller end of the size distribution has received considerably less attention; the smallest fragments are hard to count experimentally, and require a very high resolution to be captured in numerical simulations. Fragment distributions are therefore incomplete below sizes of 100 μm, or masses below 10^-3 gr (Fujiwara et al. 1977; Takagi et al. 1984). Molecular dynamics (e.g. Dominik & Tielens 1997) or smooth particle hydrodynamics (Geretshauser et al. 2010) simulations have limited resolution and tend to focus on the fragmentation threshold velocity rather than the smallest fragments.

2. Minimum fragment size in a single collision

We consider collisions below the hypervelocity regime, i.e. the relative velocity of the colliders is much smaller than their internal sound speed, generally implying v_rel ≲ 1 km s^-1. Based on experiments, we adopt the standard fragment size distribution (1)with 3 <α< 4, and C a coefficient we express below. While the mass is dominated by the largest particles, the surface area and thus the surface energy is dominated by the smallest fragments. As the creation of infinitely small fragments would require an infinite amount of energy, while the amount of kinetic energy available in a collision is finite, the power law must stop or flatten at some small fragment size. To the best of our knowledge, however, the regime of fragment sizes relevant for the analysis below has not yet been probed by available experimental data nor described theoretically in an astrophysical context.

Assuming spherical fragments with sizes between s_min and s_max( ≫s_min), the total fragment mass and surface area are (2)For a collision between two bodies of size s₀ and mass , mass conservation implies M_frag = 2M₀, and thus (3)The pre-collision kinetic energy is simply , where μ = M₀/2 denotes the reduced mass. The difference in surface energy before and after the collision equals , where γ equals the surface energy per unit surface of the material. Assuming that only a fraction η of the kinetic energy is used for creating new surface, we can combine Eqs. (2) and (3) to obtain a lower limit for the smallest fragment size. For the specific case of α = 3.5, this reduces to (4)and gives the size of the smallest fragments created in a collision at v_rel, assuming α = 3.5 and s_max ≫ s_min.

Instead of forming a fragment distribution, we imagine the limiting case in which the kinetic energy just suffices to split both colliders in half¹, i.e. . Solving for s₀, we obtain (5)which is the smallest particle that can be split in half. The smallest fragment is slightly smaller, but does not have a rigorously defined radius because we assume spherical particles. Equation (5) is similar to the result of Biermann & Harwit (1980) if η = 1.

The same limit can be explored using Eq. (4), by forcing s_min ~ s_max ~ 2^{−1 /3}s₀. This results in (6)which is very similar to Eq. (5). To summarise, in an energetic collision in which many fragments are created, the size of the smallest fragment is given by Eq. (4). When the relative velocity is decreased, the fragment distribution becomes more and more discrete, until we reach the limit described by Eq. (6), in which particles can only just be split into two.

Fig. 1 Top: minimum fragment size smin in a 20 m/s collision (Eq. (4)) for three materials, as a function of collider size s0. We have assumed η = 10-2 and smax = 2−1 / 3s0. The grey shaded areas are excluded because of mass conservation (left), and self-gravity (right). The dotted lines indicate the limit of Eq. (6). Bottom: critical energies versus size for equal-sized collisions. The energy needed to split a particle (e.g. Eq. (6)) is shown for η = 10-3 (dotted) and η = 1 (dashed). The solid curves correspond to catastrophic fragmentation of aggregates (Beitz et al. 2011), and ice and basalt (Benz & Asphaug 1999), showing both the strength-dominated (small sizes) and self-gravity dominated regimes (large sizes).

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Figure 1 shows the minimum size from Eq. (4) as a function of collider size, assuming v_rel = 20 m/s, η = 10^-2, and a maximum fragment that carries half of the initial collider mass. Gravity is important for bodies larger than 100 m (see below). Smaller bodies are weaker, and can produce fragments down to the s_min indicated by the solid curve. For example, SiO₂ fragments smaller than a micron can, at this velocity, only be formed by collisions of bodies larger than a few centimetres. The shaded region, top left, is forbidden, as there M_frag> 2M₀ and mass is not conserved. Close to s_min ~ s₀, the solid curves are non-linear as the pre- and post-collision surface areas become comparable.

It is interesting to compare Eq. (6) with the traditional form of the catastrophic fragmentation threshold velocity in equal-sized collisions: . The critical energy, Q^∗, has units of erg/g, and varies with particle mass. For small bodies, the strength is dominated by cohesion, and for large ones by gravity (Benz & Asphaug 1999). For solid bodies, this transition occurs around 100 m in size. Values of Q^∗ ~ 10⁷ erg/g are often taken as typical for asteroids, and experimentally obtained values for small grains (mm to cm sizes) can be several orders of magnitude smaller (Blum & Münch 1993; Beitz et al. 2011). Figure 1 shows the critical energy for splitting predicted by Eq. (6) as a function of size for the materials in Table 1 assuming η = 10^-3 and η = 1. The solid lines indicate critical fragmentation energies for basalt and ice (Benz & Asphaug 1999) and silicate aggregates (Beitz et al. 2011). The critical fragmentation energies exceed the splitting energy, indicating that substantially more energy is required to destroy – rather than split – colliders. The values plotted for ice and basalt were obtained at collision velocities of 3 km s^-1, substantially higher than the velocities considered here, and Q^∗ is known to depend on velocity (Leinhardt & Stewart 2012). While such a velocity dependence appears absent in the splitting energy, it might be implicitly included in η. In fact, η is expected to vary with material and impact energy. We adopt a constant value of η = 10^-2.

Appendix B investigates similar limits for colliders with different mass ratios, and shows that collisions with a mass ratio close to unity are the most effective at creating small fragments.

3. Application to debris disks

Debris disks are leftovers of planet formation, and are usually described by a birth-ring of km-sized asteroid-like particles orbiting their parent star, together with a population of smaller bodies formed in a collisional cascade (for a recent review, see Matthews et al. 2014). A steady-state and scale-independent population of bodies will follow a size distribution given by a power law with α = 3.5 (Dohnanyi 1969). Some variation in α has been found in different simulations. Pan & Schlichting (2012) find up to α = 4 for cohesion-dominated collisional particles, and up to α = 3.26 for gravity-dominated ones.

Models of debris disks most often assume a smallest fragment size equal to the blow-out size, s_blow (Wyatt & Dent 2002; Wyatt et al. 2010), or some constant, but arbitrary, s_min<s_blow for all collisions (Thébault et al. 2003; Krivov et al. 2008). The blow-out size corresponds to particles with β = 1/2, where β = 1.15Q_pr(L_⋆/L_⊙)(M_⋆/M_⊙)^-1(ρ/ g cm^-3)^-1(s/μm)^-1 is the ratio of the radiation and gravitational force. Particles with β > 1/2 are removed from the system by radiation pressure. Alternatively, Gáspár et al. (2012) calculate a collision-dependent s_min from mass conservation, but do not study the surface energy.

If, however, for any relevant collision Eq. (4) predicts s_min>s_blow, extrapolating the fragment size distribution down to these sizes is not justified. For example, starting from Eq. (2) of Krivov et al. (2008), cm-sized bodies have Q^∗ ≃ 5 × 10⁶ erg/g. In the Krivov et al. framework, a collision between two such bodies at 70 m/s will then result in fragmentation, as the kinetic energy (≃12 J, assuming ρ = 2.35 g/cm³) slightly exceeds the critical energy (=2mQ^∗ ≃ 10 J), and fragments will be created from a size comparable to the impactor (Eq. (21) of Krivov et al. 2006) down to the blow-out size. For this particular collision, Eq. (4) yields s_min ≪ s_blow for η = 0.1 and γ = 0.1 J / m², but s_min ≃ 6 μm for η = 10^-3. Thus, the difference between our results and the fragment sizes of Krivov et al. may be substantial, depending on the true value of η. We stress that our theory is valid for 3 > α > 4, and does not apply to models that use shallower power laws, for example Sect. 4.2 of Krivov et al. (2013).

The importance of the limit given by Eq. (4) depends on the parameters, and can vary per individual collision, depending on the collision velocity and choice for s_max. In the rest of this section, we explore in which cases this limit is most relevant.

In a debris disk, a particle of size s₀ is most likely formed in a collision between only slightly larger particles. In addition, we focus on collisions between equal-sized particles, as these are most efficient at forming small fragments (Appendix B). Therefore, we use Eq. (6) as an indication for the lower limit of the particle size distribution. Quantitative comparisons require relative collision velocities, which for the largest bodies are often written in terms of the Keplerian orbital velocity at the corresponding distance from the central object. For bodies on orbits with identical semi-major axes, the relative velocity can then be written in terms of orbital eccentricity and inclination as f ≡ v_rel/v_K = (1.25e² + i²)^{1 /2}, with v_K = (GM_⋆/a)^{1 /2} the Keplerian orbital velocity (Wyatt & Dent 2002). In a debris disk, a range of eccentricities and inclinations will be present. For a rough comparison, we use average quantities ⟨ e ⟩ and ⟨ i ⟩ to obtain typical collision velocities. In reality, ⟨ e ⟩ and ⟨ i ⟩ are poorly constrained. Estimates range from ⟨ e ⟩ ~ ⟨ i ⟩ ~ 10^-3−10^-1, depending on the level of stirring (Matthews et al. 2014).

	Fig. 2 Predicted smin/sblow in debris disks, as a function of disk radius. Arrows indicate how the ratio changes with stellar luminosity, surface energy, f, and η. Coloured regions indicate observational constraints on smin/sblow for various systems (see text), and the diagonal solid lines give our calculations for each system. We fix γ = 0.1 J m-2, f = 10-2, and η = 10-2 for this comparison.
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The ratio between the smallest grain size from Eq. (6) and the blow-out size then becomes (7)where both the stellar mass and the material density drop out, and we assumed Q_pr = 1. Figure 2 compares this ratio with observations of debris disks at large radii, where we predict the most pronounced effect. We have used a fixed γ = 0.1 J m^-2, f = 10^-2, and η = 10^-2, and the arrows indicate the dependence of s_min/s_blow on various parameters. For the main dust belt around Fomalhaut, Min et al. (2010) found the scattering properties to be consistent with predominantly ~100 μm silicate grains (s_blow = 13 μm Acke et al. 2012). The relative velocities in Fomalhaut are typically taken a factor of 10 higher (Wyatt & Dent 2002). For HD 105, Donaldson et al. (2012) derived s_min = 8.9 μm (s_blow = 0.5 μm) at orbital distances above ~50 AU. Notably, very large grain sizes of ~100 μm (s_blow ≲ 1 μm) are inferred for the recently discovered “Herschel cold debris disks” (Krivov et al. 2013), which are seen around F, G, and K type stars. Krivov et al. were not able to model these systems with a collisional cascade reaching down to s_min = 3 μm, and proposed that the large grains in these systems are primordial, unstirred material. Our calculations suggest that they can also be explained as the outcome of a collisional cascade. However, the model is highly degenerate, as material properties (η and γ) and belt properties (f) are usually poorly known, and all have a large impact on s_min.

In Fig. 2, we assume constant and equal relative velocities for all particles. In reality, radiation pressure will also increase the eccentricities of small particles with β ≲ 0.5. The enhanced eccentricity can be written as e_β = β/(1−β). The relative velocity of such a radiation-influenced particle scales with its size as v_rel ∝ β ∝ s^-1, while Eq. (6) predicts the fragmentation velocity scales as . Hence, the relative velocity between the smallest particles increases faster than the velocity needed for fragmentation. As a result, particles can reach arbitrarily small sizes in this regime. Particles smaller than s_blow are then removed on a short timescale. For a more detailed estimation, β should be evaluated for each particular case, considering the optical properties of the material and the shape of the stellar spectrum.

A dearth of small grains in weakly stirred disks is also predicted by Thébault & Wu (2008), but the cause is not a limit on s_min. In their scenario, s_min is fixed and the production rate of the smallest grains decreases with weaker stirring, while the destruction rate is determined by radiation forces and is unaffected by stirring. While the smallest grains present are always of blow-out size, their abundance is set by the balance between their creation and destruction (their Fig. 7).

While the theory developed in this work predicts that the smallest particles that can fragment further can be quite large and sizes just below this will be depleted, some smaller particles will still be created as a result of erosive collisions, collisions between larger bodies, and collisions that occur above the average collision velocity. Detailed debris disk models implementing the surface energy constraint are needed to determine the resulting size distribution.

4. Discussion

Of the fundamental parameters in our model, the largest uncertainty affects η, the fraction of kinetic energy used for the creation of new surface. While information may be available about the kinetic and surface energy of the largest fragments, it is hard to quantify whether the remainder of the available energy went into surface creation, heat generation, or kinetic energy of the smallest fragments. Experimentally, studying η is challenging, since it requires sensitive and complete measurements down to very small sizes. Once the functional form of η is quantified by laboratory and numerical experiments, observations of s_min in a system of interest may constrain f and thus the local relative velocities.

During the preparation of this manuscript, we discovered that a similarly defined s_min to the one we present has been explored in a more abstract framework, and without elaborating on applications, by Bashkirov & Vityazev (1996). We note that the lack of data on the size distribution of small collision fragments, as well as the fraction of kinetic and internal energy in the fragments already noted by Bashkirov & Vityazev, still prevails and we encourage further experiments to quantify these important parameters.

Thus far, we have focussed on equal-sized collisions. While collisions with a larger mass ratio might not lead to catastrophic fragmentation, cratering and erosion may still be important, and might be able to form small particles (Appendix B). Assuming a fixed relative velocity, we focus on a particle of size s₁. We define a mass loss rate ṁ(s) for the larger particle, dependent on collider size s. Assuming a collision with a particle of size s<s₁ erodes a mass ∝s³, and noting the collision timescale is proportional to the particle density and collision cross-section, we obtain for the total mass loss rate Ṁ = ṁ(s) ds ∝ s^-3.5s³(s + s₁)² ds. If the collisional cross-section is dominated by s₁, we find , and thus the mass loss is dominated by the larger bodies. When s ~ s₁, we obtain Ṁ ∝ s^{5 /2}.

We have adopted a constant value of 50% of the collider mass for the largest fragment. However, experiments show s_max can be substantially smaller as a function of material and impact velocity (e.g. Davis & Ryan 1990; Ryan et al. 1991). Such results can easily be implemented in Eq. (4) (and Eqs. B.2 and B.3) as necessary. Since , smaller sizes for the largest remnant will make the production of small particles even more difficult.

Other collisional systems where the proposed fragment size limit operates include planetary rings. Our calculations are consistent with the observed dominant grain sizes in the rings of Saturn, Jupiter, and Uranus. Because of additional relevant physics, such as tidal and electromagnetic effects, consistency does not directly imply the dominant grain size in all these rings is fragmentation-dominated.

The full implications of an energy-limited s_min on systems such as debris disks and planetary rings can only be assessed with models tracking the full particle population with all relevant processes included. For example, if small particles cannot be destroyed in collisions, Poynting-Robertson (PR) drag will influence their orbits, and cause particles to drift towards the star on timescales of Gyr in the outer parts of disks (Wyatt 2005; van Lieshout et al. 2014). Such modelling is outside the scope of the present paper.

5. Conclusions

We investigated the energetic constraints on the lower size limit in a distribution of collision fragments. A quantification of the lower limit of such size distributions is relevant for the modelling

of debris disks and other astrophysical systems where collisional fragmentation is important.

• 1.

  Based on surface energy constraints, we derive a parameterised recipe for the smallest fragment size in individual grain-grain collisions.

• 2.

  The smallest size in a distribution of fragments from a two-particle collision, constrained by the collision energy, is given by Eq. (4), and illustrated in Fig. 1. For example, at 20 m/s, submicron silicate particles can only be effectively produced by centimeter-sized colliders.

• 3.

  In the limit where the colliding bodies are split in half, the fragmentation threshold velocity is given by Eq. (6).

• 4.

  While dedicated models are needed to reveal the full implications of the fragment size limit, Fig. 2 offers an indication of where the size distribution is expected to be influenced.

• 5.

  In systems where the collision velocities are low, our theory may offer a natural explanation for a paucity of small grains in debris at large orbital distances, such as observed in Fomalhaut and the Herschel cold debris disks (Fig. 2).

Acknowledgments

The authors wish to thank Carsten Dominik and Xander Tielens for comments and discussions. Dust studies at Leiden Observatory are supported through the Spinoza Premie of the Dutch science agency, NWO. Astrochemistry in Leiden is supported by NOVA, KNAW and EU A-ERC grant 291141 CHEMPLAN.

References

Online material

Appendix A: Applicability to aggregates

For porous aggregates, the basic principles explored here are expected to hold, but some material properties have to be altered. First, aggregates have an internal filling factor φ = ρ_agg/ρ that is <1, and might be as low as 10^-4 in some extreme cases (Okuzumi et al. 2012; Kataoka et al. 2013). Second, the ’effective’ surface energy γ_agg will be smaller, since there is only limited contact between the aggregate’s constituents to begin with. Assuming the parent bodies are built up of spherical monomers, the effective surface energy can be estimated as γ_agg ~ (a/R)²φ^2/3γ, where a and R denote the radius of the contact area shared by monomers, and the radius of the monomers themselves. The fraction (a/R) depends on the size of the monomers and the material properties, but for 0.1-micron-sized monomers, (a/R) ~ 0.1 is reasonable.

For aggregates, N-body simulations have been performed with particles containing up to 10⁶ monomers, and values of η range from close to unity (Dominik & Tielens 1997; Wada et al. 2009), to several orders of magnitude less (Ringl et al. 2012), and depend on the employed contact model (Seizinger et al. 2013).

Appendix B: Collisions with different mass ratios

Here we extend the theory to collisions between “targets” and “projectiles” of arbitrary sizes s_t>s_p. Assuming the fragment distribution can be described as before, we can still use Eq. (2), while the pre-collision kinetic energy now equals (B.1)For a given collision velocity, we might then think of two cases, complete fragmentation when s_p ~ s_t, and erosion/cratering when s_t ≫ s_p.

Appendix B.1: Catastrophic fragmentation

Since both particles are destroyed completely we may set M_frag = m_p + m_t. For simplicity we will assume in this section that the change in surface energy is dominated by the fragments ΔU_S = γA_frag. Using the same definition for η as before and focussing on the α = 3.5 case, we obtain (B.2)

Appendix B.2: Erosion

When the mass ratio becomes very large, it is no longer realistic to assume the target is completely disrupted. Rather, such collisions result in erosion, and the eroded mass is typically of the order of the mass of the projectile (Schräpler & Blum 2011). Thus, we write M_frag = κm_p, with κ of the order of unity. For large mass ratios μ → m_p. Furthermore, assuming that the change in surface energy is dominated by the new fragments, ΔU_S = γA_frag, we obtain (B.3)Consider now a particle with a size s₀ close to the smaller end of the size distribution, colliding with particle of sizes s_x, ranging from slightly smaller to much larger than s₀. Figure B.1 shows the minimum size of the fragments produced as a function of collider size s_x. The y-axis is normalized to the value obtained in equal-sized collisions (i.e. between two s₀ particles). For mass ratios below unity, s₀ acts as the target, and the mass of the largest fragment is assumed to equal m₀/2. For large mass ratios, s₀ acts as a projectile instead, and the largest fragment mass is set to m_x/ 2. Collisions at mass ratios above 10² are assumed to be erosive (Seizinger et al. 2013), with the largest fragment equalling m₀/2. Since both the excavated mass, and the largest fragment mass, depend on the projectile, the curves in the erosive regime do not depend on the mass ratio directly. However, the size of the target does set an upper limit on κ.

	Fig. B.1 Minimum fragment size resulting from destructive unequal collisions, assuming smin ≪ smax, and relating the largest fragment size to the heavier collider. The y-axis has been normalized with the value for smin in equal-mass collisions.
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All Tables

All Figures

	Fig. 1 Top: minimum fragment size smin in a 20 m/s collision (Eq. (4)) for three materials, as a function of collider size s0. We have assumed η = 10-2 and smax = 2−1 / 3s0. The grey shaded areas are excluded because of mass conservation (left), and self-gravity (right). The dotted lines indicate the limit of Eq. (6). Bottom: critical energies versus size for equal-sized collisions. The energy needed to split a particle (e.g. Eq. (6)) is shown for η = 10-3 (dotted) and η = 1 (dashed). The solid curves correspond to catastrophic fragmentation of aggregates (Beitz et al. 2011), and ice and basalt (Benz & Asphaug 1999), showing both the strength-dominated (small sizes) and self-gravity dominated regimes (large sizes).
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In the text

	Fig. 2 Predicted smin/sblow in debris disks, as a function of disk radius. Arrows indicate how the ratio changes with stellar luminosity, surface energy, f, and η. Coloured regions indicate observational constraints on smin/sblow for various systems (see text), and the diagonal solid lines give our calculations for each system. We fix γ = 0.1 J m-2, f = 10-2, and η = 10-2 for this comparison.
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In the text

	Fig. B.1 Minimum fragment size resulting from destructive unequal collisions, assuming smin ≪ smax, and relating the largest fragment size to the heavier collider. The y-axis has been normalized with the value for smin in equal-mass collisions.
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In the text