OBSERVED BINARY FRACTION SETS LIMITS ON THE EXTENT OF COLLISIONAL GRINDING IN THE KUIPER BELT

Our collisional modeling simulations employ Boulder, a new code capable of simulating the collisional fragmentation of multiple planetesimal populations using a statistical particle-in-the-box approach. It was constructed along the lines of other published codes (Weidenschilling et al. 1997; Kenyon & Bromley 2001). A full description of the Boulder code, how it was tested, and its application to both accretion and collisional evolution of the early asteroid belt, can be found in Morbidelli et al. (2009). Examples of its previous use for the asteroid belt, Hildas, Trojans, irregular satellites, and primordial trans-planetary disk are described in Levison et al. (2009) and Bottke et al. (2010).

The code's procedure for modeling impacts is as follows. For a given impact between a projectile and a target object, the code computes the specific impact energy Q, defined as the kinetic energy of the projectile divided by the target mass, and the critical impact energy Q*_D, defined as the energy per unit target mass needed to disrupt and disperse 50% of the target (e.g., Davis et al. 2002). For reference, Q < Q*_D values correspond to cratering events, Q ≈ Q*_D correspond to barely catastrophic disruption events, and Q > Q*_D correspond to super-catastrophic disruption events.

For each collision identified by the code, the mass of the largest remnant is computed from the scaling laws found in hydrodynamic simulations of impacts (Benz & Asphaug 1999; Leinhardt & Stewart 2009; Stewart & Leinhardt 2009). The mass of the largest fragment and the slope of the power-law SFD produced by each collision is set as a function of Q/Q*_D by empirical fits to the hydrocode results of Durda et al. (2004, 2007) and Nesvorný et al. (2006). See Bottke et al. (2010) for the explicit definition of these fits. These results apply to monolithic target bodies. We also tested approximate scaling laws for impacts on pre-fragmented and rubble-pile targets. Again, these scaling relations were drawn from the fits to hydrocode impact simulations (Benavidez et al. 2011).

The Q*_D function was assumed to split the difference between the impact experiments of Benz & Asphaug (1999), who used a strong formulation for ice, and those of Leinhardt & Stewart (2009), who used the finite volume shock physics code to perform simulations into what they describe as weak ice. To do this, we divide the Benz & Asphaug (1999) strong-ice Q*_D function by a factor, f_Q. We typically used f_Q = 3, 5, and 10. Note that because we sampled a broad section of parameter space, we chose not to include still more complicating factors (e.g., Q*_D may vary with impact velocity, etc.). The bulk density was set to ρ = 1 g cm⁻³.

The main input parameters for the Boulder code are the (1) initial SFD of the simulated populations (see Section 3); (2) intrinsic collision probability, P_i, defined as the probability that a single member of the impacting population will hit a unit area of a body in the target population over a unit of time; and (3) mean impact speed, v_i. For collisions between present-day CKBOs, we used P_i = 4 × 10⁻²² km⁻² yr⁻¹ and v_i = 1 km s⁻¹ (Davis & Farinella 1997; Dell'Oro et al. 2001).⁵ See Section 3 for our specific choices of P_i and v_i for other populations.

Petit & Mousis (2004) identified three main processes that can dissolve KBO binaries: (1) one of the components is hit by a small impactor. While the component survives essentially intact, except for a new crater on its surface, the linear momentum transferred from the impactor imparts a "kick" on the velocity vector of the binary orbit. If the kick is large enough, the component becomes unbound from its companion. (2) One of the components can be shattered by a large impactor. The fragments produced the object's breakup are expected to escape on unbound trajectories because the ejection speeds (∼100 m s⁻¹) largely exceed that of the binary orbit (∼1 m s⁻¹). (3) The binary system has a close gravitational encounter with another KBO. The tidal gravity of that object can unbind the binary provided that the object is massive enough (e.g., Stern et al. 2003).

Petit & Mousis (2004) found that mechanism (1) is by far the most efficient way to dissolve binaries in the present KB. Thus, we focus on modeling (1) in this work. Mechanism (2) is also considered, but we find that its effects are negligible compared to (1), except if extremely weak (and probably unrealistic) disruption laws are adopted. This result stems from the fact that it is generally easier to dissolve a wide KBO binary orbit by an impact than to physically disrupt a 100 km object. Mechanism (3) could have been important during the early phases of the KB evolution provided that the cold CKB overlapped with a population of numerous, very massive KBOs. We do not model (3) in this work because we do not yet have an adequate understanding of these early stages.

We will assume in the following that small impacts with Q ≪ Q*_D can be treated as inelastic collisions. Thus, we will ignore any linear momentum that can be potentially carried away by ejected fragments. In this approximation and assuming that m_i ≪ m, where m_i and m are the impactor and binary component masses, the velocity vector of the binary orbit, v, will change by

$$\begin{equation} \delta {\bf v} \simeq \frac{m_i}{m}\,{\bf v}_i. \end{equation} \tag{ 1 }$$

See Dell'Oro & Cellino (2007) for a discussion of the linear momentum transfer for different impact angles and in the case where the escaping ejecta affect the linear momentum budget.

The binary components are assumed to have equal mass, which should be a good approximation for the real binaries in the cold CKB (see Section 1). This assumption is conservative in the sense that it is generally harder to disrupt a binary with equal-size components than the one in which the secondary is smaller than the primary (if other parameters are the same). By using equal-size binaries we may thus slightly underestimate the real decay rate.

The radial, tangential, and normal components of δv will be denoted by δv_R, δv_T, and δv_Z, respectively, in the following. With this notation, the semimajor axis a, eccentricity e, and inclination I of the binary orbit will change by

$$\begin{eqnarray} \delta a &=& \frac{2}{n\eta } \left[\delta v_T \left(1+e\cos f\right) + \delta v_R\, e \sin f\right], \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \delta e &=& \frac{\eta }{na} \left[\delta v_R \sin f + \delta v_T \left(\cos f+\frac{e+\cos f}{1+e\cos f}\right)\right], \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \delta (\cos I) &=& \frac{\eta }{na}\, \delta v_Z\, \frac{\sin (\omega +f)}{1+e\cos f}, \end{eqnarray} \tag{ 4 }$$

where ω is the argument of pericenter, f is the true anomaly at the time of impact, and $\eta =\sqrt{1-e^2}$ (e.g., Bertotti et al. 2003).

Our statistical code does not deal with the detailed geometry of each individual impact. Instead, it follows the mean quadratic changes of orbital elements produced by the average effect of impacts with different orientations relative to the binary orbit and times corresponding to different orbital phases. Specifically, we assume that the impact direction is isotropic and impacts are not correlated with the binary orbit phase, which should be the case for the well-mixed system that we deal with. The algorithm is as follows.

Assuming that v_i = v_i n, where v_i is the characteristic impact speed and n is the unit vector with isotropic orientation, we have

$$\begin{equation} \langle \left(\delta v_R\right)^2 \rangle = \langle \left(\delta v_T\right)^2 \rangle = \langle \left(\delta v_Z\right)^2 \rangle = \frac{1}{3}\,\frac{m^{\prime 2}}{m^2}\,v_i^2 \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \langle \delta v_R\,\delta v_T \rangle = \langle \delta v_R\,\delta v_Z \rangle = \langle \delta v_T\,\delta v_Z \rangle = 0, \end{equation} \tag{ 6 }$$

where the average was taken over n's orientations. Speed v_i can be determined as the rms value of impact speeds with the distribution that is characteristic to the studied population (e.g., Davis & Farinella 1997).

In the next step, we average over the phase of the binary orbit at which the impact occurs. We obtain

$$\begin{equation} \frac{\langle\left(\delta a\right)^2\rangle}{a^2} = \frac{4}{3}\,\left(\frac{m_i v_i}{m v} \right)^2, \end{equation} \tag{ 7 }$$

$$\begin{equation} \langle\left(\delta e\right)^2\rangle = \frac{5}{6}\,\eta ^2\,\left(\frac{m_iv_i}{m v} \right)^2, \end{equation} \tag{ 8 }$$

and

$$\begin{equation} \langle\left(\delta \cos I\right)^2\rangle = \frac{2+3e^2}{12\eta ^2}\, \left(\frac{m_iv_i}{m v}\right)^2. \end{equation} \tag{ 9 }$$

Since the orbital speed of a KBO binary, v = na, where n is mean motion, is typically ∼1 m s⁻¹, while v_i ∼ 1 km s⁻¹, the binary orbit can be dissolved by a single impact generating δa/a, δe, or δ(cos I) of the order of unity, if m_i/m ≳ v/v_i ∼ 10⁻³.

In addition, the cumulative effect of impactors with m_i < 10⁻³m can also be important as it leads to a random walk in orbital elements. It can be shown that

$$\begin{equation} \langle\delta a\, \delta e\rangle = \langle\delta a\, \delta \left(\cos I\right) \rangle = \langle\delta e\, \delta \left(\cos I\right)\rangle = 0. \end{equation} \tag{ 10 }$$

Thus, the changes in different orbital elements due to small impacts are uncorrelated and can be treated separately.

The binary system is assumed to become dissolved either if e > 1 or if the semimajor axis exceeds some critical value, a_crit. We set a_crit = 0.5R_Hill as a rough limit dictated by the Hill stability criterion (e.g., Donnison 2010) and numerical integrations (e.g., Nesvorný et al. 2003). By experimenting with the code we find that the main channel of binary disruption is reaching a > a_crit in one or a small number of collisions. Accordingly, the results are insensitive to the initial distribution of e.

Note that the inclination changes cannot result in binary splitting directly, but, if coupled to effects arising from the Kozai dynamics (Kozai 1962), they can also be important. We do not consider the coupling of collisional effects with dynamics of inclined binary orbits in this work. Therefore, the initial direction of the binary angular momentum vector does not need to be specified.

The binary module was inserted in the Boulder code. This was done by attaching additional data structures to each size bin. These data structures describe the initial distributions of a, e, and i of binary orbits and track how these distributions change over time. As the population of impactors evolves with time due to collisional fragmentation, the code calculates the time-dependent rate of change of binary orbits and evolves them according to Equations (7)–(9).

To start with, we illustrate the case studied by PS05. In this case, the KB is assumed to evolve in isolation over τ = 4.5 Gyr. The initial population is given a two-slope SFD with q_< = 3, q_> = 5, and R* between 1 km and 50 km. The total number of objects with R > 50 km is normalized to N₀ = 50,000, which is roughly the estimated number of objects in the present KB. We also consider cases with N = f_NN₀, where f_N is the scaling parameter. For example, assuming that p_V = 0.2, ρ = 1 g cm⁻³, q_< = 3, q_> = 5, and R* = 37 km, f_N ∼ 0.4 gives the total mass ≈0.01M_E, where M_E is the Earth mass, which is what the observations seem to indicate for the cold CKB (Fuentes & Holman 2008), although this value is still quite uncertain. On the other hand, f_N = 13 corresponds to the population used in PS05 (see discussion in Section 1).

Figure 2 shows the result of collisional grinding for f_N = 1 and two different values of f_Q: f_Q = 3, roughly corresponding to the PC05's strong disruption law, and f_Q = 500, roughly corresponding to the PC05's weak disruption law (see Figure 1). We used P_i = 4 × 10⁻²² km⁻² yr⁻¹ and v_i = 1 km s⁻¹. The initial distribution was set so that q_< = 3 for R < 1 km and q_> = 5 for R > 1 km. For ρ = 1 g cm⁻³ this gives the total initial mass of 0.84 M_E. After having evolved the population with the Boulder code over τ = 4.5 Gyr, the remaining masses were 0.056 M_E for f_Q = 3 (Figure 2(a)) and 0.027 M_E for f_Q = 500 (Figure 2(b)).

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For f_Q = 3, the SFD slope just below ∼50 km becomes slightly shallower over time, mainly due to the effects of large cratering impacts and reaches q ∼ 4 at τ = 4.5 Gyr. This final slope is significantly steeper than the present slope of cold CKBOs at these radii. With f_Q = 500, a sharp rollover to a very shallow slope develops at R ∼ 10 km, because most objects with R < 10 km suffer catastrophic disruptions. Thus, even with the unrealistically weak disruption law, the SFD break is still significantly below the actual rollover radius in the present cold CKB. This result is insensitive to the specific choice of model parameters related to the generation of fragments, resolution, and plausible changes of P_i and/or v_i.

We therefore find that the observed break at R* = 25–50 km cannot be produced by collisional grinding, except if the number of objects with R > R* was much larger in the past and was depleted by dynamical processes, or if the cold CKBO overlapped with a much larger population of impactors in the past. We consider these possibilities in the following text.

Levison et al. (2008a) proposed that most of the complex orbital structure seen in the KB region today (see, e.g., Gladman et al. 2008) can be explained if bodies native to 15–35 AU were scattered to >35 AU by eccentric Neptune (Tsiganis et al. 2005). If these outer solar system events coincided in time with the Late Heavy Bombardment (LHB) in the inner solar system (Gomes et al. 2005), binaries populating the original planetesimal disk at 15–35 AU would have to withstand ∼600 Myr before being scattered into the KB. Even though their survival during this epoch is difficult to evaluate, due to major uncertainties in the disk's mass, SFD, and radial profile, the near-absence of wide and equal-sized binaries among 100 km sized hot classical KBOs (Noll et al. 2008a, 2008b) seems to indicate that the unbinding collisions and scattering events must have been rather damaging.

The contrasting characteristics of cold CKBOs discussed in Section 1 may indicate that the cold CKBOs formed in a relatively quiescent environment at >40 AU rather than having been scattered to their current orbits from <35 AU, because more resemblance between different trans-Neptunian populations would be expected in the latter model. The in situ formation of cold CKBOs is also supported by the results of Parker & Kavelaars (2010) who showed that some of the widest binaries observed in the cold CKB today would probably not survive scattering encounters with Neptune.

To understand the collisional history of the cold CKB, we consider both the pre-LHB and post-LHB epochs. For the pre-LHB epoch, we assume that the cold CKBOs evolved in relative isolation at 40–50 AU, where most collisions occurred between cold CKBOs themselves. The results of these pre-LHB simulations should also apply, with minor modifications, if the cold CKBOs formed closer in, as far as the source population of cold CKBOs can be considered as a closed collisional system. The coupling of CKBOs to the scattered trans-planetary disk during the LHB is not considered here because Benavidez & Campo Bagatin (2009) showed that collisional fragmentation during this stage was unlikely to produce substantial changes in the SFD.

We start by discussing the collisional evolution of the KB after LHB. We consider collisions between two CKBOs, one CKBO and one scattered disk object, and two scattered disk objects. The scattered disk is massive initially and becomes dynamically depleted over time, with an estimated depletion factor of 100–250 over 4 Gyr (Levison & Duncan 1997; Dones et al. 2004; Tsiganis et al. 2005). This dynamical depletion is taken into account in Boulder by gradually decreasing the number of scattered disk objects in all size bins.

The collisional probabilities, impact speeds, initial SFDs, and dynamical decay rates were taken from Levison et al. (2008b). Specifically, we used P_i = 4 × 10⁻²² km⁻² yr⁻¹ and v_i = 1 km s⁻¹ for CKB collisions, P_i = 2 × 10⁻²² km⁻² yr⁻¹ and v_i = 1.5 km s⁻¹ for CKB–scattered-disk collisions, and P_i = 10⁻²² km⁻² yr⁻¹ and v_i = 3 km s⁻¹ for collisions between scattered disk objects (see also Brown et al. 2007). We varied these (and other) parameters in test simulations to determine the sensitivity of results to different assumptions.

The SFD of the scattered disk was normalized as in Levison et al. (2008b). We assumed that there are presently 50,000 scattered disk objects with R > 50 km and that this population decayed by a factor of 250 times since LHB (Tsiganis et al. 2005). We fixed f_N = 1, because the dynamical depletion of CKBOs should be relatively minor during this stage. For R* = 37 km, this gives the initial mass ∼0.2 M_E. Also, to promote collisional grinding, we used the weak disruption law with f_Q = 10.

Figure 3 shows the SFD of CKBOs and scattered disk objects that were obtained for two different assumptions on the initial SFD of cold CKB. In Figure 3(a), we assumed that the initial SFD was steep down to R = 1 km. In Figure 3(b), we used the present SFD shape of cold CKB with R* = 37 km.

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In Figure 3(a), the rollover in the final SFD of CKBOs occurs at R ≈ 5 km, which is significantly below the observed R* value. We experimented with a range of f_N, f_Q, v_i, and P_i values and various initial SFDs. These tests showed that the results illustrated in Figure 3(a) are representative. Specifically, if we start with the initial break at R < 10 km, the final break ends up being at R < 10 km as well. These results therefore suggest that the observed SFD rollover of cold CKBOs at R* = 25–50 km cannot be produced by the collisional grinding after LHB.

On the other hand, if we start with the break at R* = 37 km (Figure 3(b)), the SFD of CKBOs remains nearly unchanged for R ≳ 10 km. The total mass loss due to the collisional grinding of small objects is only ∼15% in the CKB and ∼5% in the scattered disk. These results show that, while the SFD may have been shaped by collisional grinding in epochs prior to the LHB, it remained essentially constant since then. The binary fraction does not provide any interesting constraints on the collisional evolution of CKBOs after LHB because the vast majority of binaries with R ≳ 10 km survive.

Fraser (2009) and Benavidez & Campo Bagatin (2009) suggested that the observed SFD rollover in the cold CKB was produced by collisional grinding during the ∼600 Myr before the LHB. The collisional modeling of the pre-LHB phase is complicated by the fact that the state of the KB before LHB is poorly understood. What we seem to infer from observations (see discussion in Sections 1 and 3.2) is that the cold CKB probably formed in situ at 40–50 AU. The LHB modeling would then indicate that various dynamical processes probably removed ∼90% of its mass during LHB (Morbidelli et al. 2008). Using these results as a guideline, we find that there is indeed a potential for the observed SFD rollover being a fossil remnant of the collisional grinding of massive population before LHB.

The impact speeds in the pre-LHB CKB depend on the dynamical state of the disk. In its present state, the impact speeds are relatively high (∼1 km s⁻¹) because orbits have relatively high eccentricities (e ∼ 0–0.2) and inclinations (i ∼ 0°–30°). On the other hand, KBOs can only have formed if e and i were much lower (e.g., Kenyon et al. 2008, and the references therein). This implies that some dynamical processes must have excited e and i to their present values. Two main possibilities exist. Either (1) the cold CKB was dynamically excited by some primordial process that dates back to KBO formation or (2) the excitation was produced by the LHB itself (see Morbidelli et al. 2008 for a discussion). Following Fraser (2009) we first study (1), in which case v_i ∼ 1 km s⁻¹. Low v_i values implied by (2) will be discussed in Section 3.7.

Fraser (2009) considered the initial SFDs that were steep (q₁ = 5) down to R_a, then became nearly flat (q₂ = 0–2) down to R_b, with the collisional equilibrium below R_b (q₃ = 3.5). These initial SFDs were motivated by the results of published simulations of the coagulation growth in KB, which tend to produce such distributions (Kenyon et al. 2008). Since it is not clear, however, whether KBOs formed by two-body coagulation in the first place (see Chiang & Youdin 2010 for a review), it is not guaranteed that these initial conditions actually apply. Nevertheless, we use Fraser's initial SFDs as a starting point and consider other options in Section 3.6.

Figure 4 illustrates the Fraser's case with R_a = 2 km, R_b = 0.5 km, q₁ = 5, q₂ = 1, and q₃ = 3.5. As in Fraser (2009), we used v_i = 1 km s⁻¹, f_Q = 3 (corresponding to Fraser's weak disruption law) and evolved the populations with Boulder over τ = 600 Myr. Factor f_N was varied to obtain different initial masses and thus different collisional histories. Each of these cases would imply a different dynamical depletion factor during LHB.

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The best results were obtained with f_N ∼20–50, implying the initial mass between 7 and 15 M_E (Figure 4(a)). With f_N < 20 (<7 M_E) and f_N > 50 (>15 M_E), the SFD rollover occurred at radii that were either too small or too large, respectively, compared to observations. These values are only soft limits, however, because the rollover radius is also sensitive to the assumed R_a value. For example, smaller initial mass values would still be plausible if R_a = 3–10 km. In addition, slightly larger rollover radii can be produced with f_Q = 10. We therefore find, in agreement with Fraser (2009), that a reasonably conservative lower limit on the initial disk's mass is ∼1 M_E.

The model SFDs discussed here share a common trait. While they bend to a shallow slope at R*, the slope below R* is never shallower than q ≈ 3 and, even in the best cases such as the one shown in Figure 4(a), just barely matches the observational constraint. Moreover, the shallow SFD segment below R* generally only extends down to R = 10–20 km and steepens back to q ≈ 5 for R ≲ 10 km (Figure 4(a)). This does not contradict the existing observations because very little is known about KBOs with R ≲ 10 km. We were unable to obtain a case where the final SFD would be uniformly shallow below R* down to R < 10 km, except for very large and probably implausible initial masses. This is therefore clearly not the idealized case considered in PS05, where it was assumed that the slope below R* can be approximated by q ≈ 3 to some very small (indefinite) R.

Our simulations show that the disk mass is reduced by a factor of ∼10 by collisional grinding. Thus, starting with ∼1 M_E, the remnant mass just before LHB would be ∼0.1 M_E. This is plausible because the LHB modeling in Morbidelli et al. (2008) showed that the population at 40–50 AU becomes dynamically depleted by a factor of ∼10. The expected final mass of cold CKBOs from these order-of-magnitude considerations is therefore ∼0.01 M_E, which is in the right ballpark when compared to observations (e.g., Fuentes & Holman 2008; Brucker et al. 2009).

The initial masses smaller than ∼1 M_E would imply R_a > 10 km and indicate that the observed break at R* ∼ 37 km was essentially in place before fragmentation processes had begun. The initial masses larger than ∼10 M_E are probably implausible, because these large masses would imply ≳1 M_E mass before LHB and would require a dynamical depletion factor at LHB in excess of ∼100. Such a large depletion factor is difficult to explain by LHB processes if the cold CKBOs formed in situ at 40–50 AU (Morbidelli et al. 2008).

We now consider the binary survival. Figure 4(b) shows the fraction of binaries surviving the pre-LHB epoch for f_N = 30 and Fraser's initial SFD shape. There is a clear trend with the physical size of binary components. Specifically, more than 50% of binaries with radii R > 50 km survive, while the survival rate for R = 10–20 km is only ∼0.5%. This trend is easy to understand because, according to Equations (7)–(9), smaller binary mass m implies larger orbital change.

The trend is reversed for R < 10 km because of the lack of small impactors with R ∼ 0.5 km that could unbind binaries with R ∼ 5 km (see Figure 4(a)). The behavior of the surviving binary fraction for R < 10 km is sensitive to the initial SFD. For example, if R_a = R_b = 2 km, in which case the Dohnanyi's slope is directly attached to the steep SFD slope at large sizes, the survival rate of R < 10 km binaries more monotonically drops with decreasing R (see Section 3.6). We will concentrate on binaries with R > 10 km in the following discussion, because that is where things can be constrained by the existing observational data.

Figure 5(a) shows the physical parameters of known binaries in the cold CKB (Noll et al. 2008a; Grundy et al. 2009). The radii of primary components range from ∼30 to ∼100 km. The separations are between ∼2 × 10⁻³ and ∼0.2 Hill radii. From Equation (1), the survival rate should decrease with separation because $\delta v/v \propto v_i \sqrt{a}$, which is larger for larger a. Given the spread of separations in Figure 5(a), we therefore consider cases with a = 0.001, 0.01, and 0.1 R_H. These initial values should cover the interesting range of separations. Considering these cases separately, rather than using some continuous initial distribution of a, is the right thing to do, because the real distribution of a produced by the formation process is not well understood.

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Figure 5(b) shows the surviving binary fraction for these semimajor axis values. As expected, the wider binaries have lower survival rates than the tighter ones. For example, R = 30 km binaries with a = 0.1 R_H would be reduced by a factor of ∼100 in the Fraser's pre-LHB collisional model illustrated in Figure 4(a), while R = 30 km binaries with a = 0.01 R_H would be reduced by a factor of ∼10. These considerations provide an interesting test on the level of collisional grinding in the cold CKB, because the binary fraction could have been strongly reduced by collisions, especially for R ≲ 50 km. To pass this test, any plausible collisional scenario needs to match not only the SFD of cold CKBOs, including the rollover at R* = 25–50 km, but also explain how the large fraction of cold CKBOs binaries survived, as indicated by observations (see Section 4).

Before we compare our results to observational data, we test the sensitivity of the binary survival to different assumptions. Figures 6 and 7 illustrate the dependence on the initial SFD. In Figure 6, we choose to extend the steep slope (q = 5) from large sizes down to R = 2 km and assume that there are no bodies initially with R < 2 km (case A). In Figure 7, we impose a break from q = 5 to q = 3 at R = 2 km (case B). In each case, f_N, or equivalently the total initial mass, is set so that the collisional grinding leads to the SFD rollover at R = 25–50 km. The population is evolved over τ = 600 Myr.

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The best results were obtained with f_N = 14 (4 M_E initial mass) for case A and f_N = 30 (26 M_E) for case B. These initial masses grind down to 0.6 and 1 M_E, respectively. While the final distribution obtained in case A matches constraints reasonably well (Figure 6(a), but see discussion in Section 3.4), the distribution for R < 30 km in case B is always steeper than q = 3. Interestingly, it is difficult to produce a shallower slope in case B unless we use the initial masses in excess of 100 M_E, which is clearly implausible, or f_Q ≫ 10, which would conflict with the published results of impact simulations (see Section 3.1).

It thus appears that a rather abrupt change in the initial SFD slope is needed to produce a shallow slope with q = 2–3 below R*. This can be easily understood. The objects with R near the initial slope change are long-lived, because they see a small number of impactors, if the transition is sharp. Thus, they can break larger bodies and create a sharp SFD rollover at about 10 times their radius. We find that it is easier to fit constraints if the initial break is placed at R ∼ 5 km, rather than at R ≲ 2 km, because R ∼ 5 km objects are long-lived in that case and can disrupt KBOs near R* with our disruption laws (f_Q = 1–10).

Additional dependencies exist on the assumed SFDs of fragments produced by the cratering and catastrophic impacts. We find that catastrophic impacts tend to produce a sharp transition from steep to shallow slope near the largest object in the population that can be disrupted by them. The cratering impacts, on the other hand, tend to smooth this transition and produce more gentle waves in the SFD. This may explain some of the subtle differences between our simulations, which tend to produce gentle SFD waves, and those of Fraser (2009), which show stronger variations of slope for R ∼ R* (e.g., q < 2 below the break).

The fraction of binaries surviving in case A is within a factor of ∼2 to that we obtained with the Fraser's initial distribution. Interestingly, the binary survival rate is slightly larger in case B, which has a larger mass than case A, and should thus lead to more perturbations on binary orbits. The larger binary survival rate in case B is related to the shorter lifetime of intermediate-size impactors (R ∼ 5 km) in case B, which are disrupted by smaller impactors from the Dohnanyi's tail. These small impactors are nearly absent in case A.

So far we described tests with v_i ∼ 1 km s⁻¹. It is uncertain whether these assumed impact speeds should apply to the pre-LHB collisions, because the dynamical state of the KB region at 40–50 AU could have been very different. Specifically, the dynamical effects during LHB, such as, e.g., passing resonances, could have excited the orbits of cold CKBOs. If so, the cold CKBOs could have had smaller eccentricities and inclinations than they have now, implying lower collision speeds before the LHB. Here, we test cases with v_i < 1 km s⁻¹. The results need to be considered with caution, because it is not clear whether our disruption laws (Section 2.1) are applicable with low impact speeds.

Figure 8 illustrates the case with v_i = 300 m s⁻¹. To create the rollover at R* with this low collisional speed, we needed to assume f_N = 300, giving the initial mass of 88 M_E for the case A initial SFD. The population grinds down to 6.7 M_E at τ = 600 Myr, which would imply a dynamical depletion factor of >200. Collision speeds v_i < 300 m s⁻¹ would require even larger initial masses and dynamical depletion factors that are clearly implausible. We therefore find that it is difficult to produce the SFD rollover by collisional grinding with low collisional speeds (v_i ≲ 300 m s⁻¹).

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The surviving binary fraction for v_i = 300 m s⁻¹ (Figure 8(b)) shows the usual trend with R in that the binaries with physically smaller components survive at lower rate than the larger ones. Below R = 10–20 km, the surviving binary fraction drops to <10⁻³. The results for R > 20 km are similar to those obtained with v_i ∼ 1 km s⁻¹ indicating that the survival of larger binaries is not overly sensitive to the assumption on v_i.

Additional tests show that it might be more plausible to create the rollover at R* with low v_i, if r₁ ∼ 5 km (initial 20 M_E grinds to 5 M_E) and/or for f_Q = 10 (20 M_E grinds to 2 M_E). The results for binary survival are similar in these two cases. While r₁ = 5 km generates a slightly stronger gradient with R and minimum survivability for R < 20 km, f_Q = 10 produces a slightly softer gradient that is similar to that in Figure 6(b).

So far we discussed the binary survival in different collisional scenarios. Here, we describe the distribution of binary orbits of the surviving binaries. Figure 9(a) shows the semimajor axis distribution of binary orbits produced in simulations with initial a = 0.01 R_H and R = 20, 50, and 100 km. Figure 9(b) shows the same case for N(a)da∝da/a for a < 0.1 R_H and N(a)da = 0 for a > 0.1 R_H. These results were obtained for the Fraser's case (see Figure 4).

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While the binaries with R ≳ 100 km tend to retain the shape of their original distribution, the binary orbits for R ≲ 50 km become significantly modified. For example, Figure 9(a) shows that binaries with R = 50 km and a = 0.01 R_H become tighter or looser, producing a wide range of separations with a tail extending to a = 0.1 R_H. The number of very wide binaries produced by collisions, however, is not expected to be large, because the tail of the distribution at a ∼ 0.1 R_H in Figure 9(a) only represents a very small fraction.

Figure 9(b) shows how the initial gradient of binary fraction with a can be modified by collisions. For example, the initial gradient ∝a⁻¹ for R = 20 km binaries, becomes ∝a^−3.5. Thus, if the collisional effects on these binary systems were important, it could be difficult to try to infer their primordial semimajor distribution from present observations. On the other hand, the semimajor axis distribution of binaries with R ≳ 100 km should not have changed much since their formation.