KBO binaries: how numerous were they?

Transcription

1 Icarus 168 (2004) KBO binaries: how numerous were they? J.-M. Petit and O. Mousis Observatoire de Besançon, B.P. 1615, Besançon Cedex, France Received 9 May 2003; revised 9 December 2003 Abstract Given the large orbital separation and high satellite-to-primary mass ratio of all known Kuiper Belt Object (KBO) binaries, it is important to reassess their stability as bound pairs with respect to several disruptive mechanisms. Beside the classical shattering and dispersing of the secondary due to a high-velocity impact, we consider the possibility that the secondary is kicked off its orbit by a direct collision of a small impactor, or that it is gravitationally perturbed due to the close approach of a somewhat larger TNO. Depending on the values for the size/mass/separation of the binaries that we used, 2 or 3 of the 9 pairs can be dispersed in a timescale shorter than the age of the Solar System in the current rarefied environment. A contemporary formation scenario could explain why we still observe these binaries, but no convincing mechanism has been proposed to date. The primordial formation scenarios, which seem to be the only viable ones, must be revised to increase the formation efficiency in order to account for this high dispersal rate. For the reference current KBO population, objects like the large-separation KBO binaries 1998 WW 31 or 2001 QW 322 must have been initially an order of magnitude more numerous. If the KBO binaries are indeed primordial, then we show that the mass depletion of the Kuiper belt cannot result from collisional grinding, but must rather be due to dynamical ejection Elsevier Inc. All rights reserved. Keywords: Kuiper belt; Trans-neptunian objects; Binaries 1. Introduction Over the past decade, the Edgeworth Kuiper belt has changed status, from a theoretically predicted entity to a collection of more than 800 comets orbiting beyond Neptune. At first, those (not so) small icy bodies were thought to be lonely wanderers, except for the pair Pluto Charon. At the end of 2000, Veillet et al. (2002) found the first Kuiper Belt Object (KBO) satellite. Since then, this discovery was followed by eight others, representing more than 1% of the total known KBO population. Five observing programs using HST have been conducted to discover binary KBOs (Noll, 2003). During these programs, a total of 155 KBOs have been observed, 7 binaries were detected, of which 6 were new. So the frequency of binaries is more likely 4.5% ± 2%. This represents a lower limit since these programs may have missed very small separation binaries and very small secondaries. The main characteristics of the KBO binaries, when compared with the asteroid binaries, are large separations ( 5000 to 130,000 km, or 20 to almost 2000 times the * Corresponding author. address: petit@obs-besancon.fr (J.-M. Petit). primary radius of order a few to 10 for asteroids) and high satellite-to-primary mass ratio of 0.1 to 1 ( 10 4 to 10 3 for asteroids). The set of known KBO binaries suffers from a very strong observational bias. KBO binaries with a small separation are impossible, or at least very difficult to detect as binaries because of their large distance to Earth. Their angular separation is smaller than the typical seeing, and still smaller than the diffraction limit (achievable with Adaptive Optics) if the separation is comparable to that of the asteroid binaries. Likewise, KBO binaries with low satellite-to-primary mass ratios cannot be recognized as binaries, because the secondary falls beyond the limiting magnitude of most observations. However, the very existence of the known binaries is a great novelty with respect to what is known in the asteroid belt or in the NEO population. This has prompted several authors to study their formation mechanisms (Goldreich et al., 2002; Stern, 2002; Weidenschilling, 2002). Goldreich and co-workers and Weidenschilling concluded that collisions in the current Edgeworth Kuiper cannot account for the large number of binaries found, nor for their large separation and high satellite-to-primary mass ratios. They proposed various mechanisms that must have occurred in the late stage of the /$ see front matter 2004 Elsevier Inc. All rights reserved. doi: /j.icarus

2 410 J.-M. Petit, O. Mousis / Icarus 168 (2004) formation of the Solar System, at the end of the accretion phase. According to Goldreich and co-workers and Weidenschilling, the binaries would be primordial. Although contemplating similar primordial scenarios, Stern favors more contemporary collisional formation mechanisms, and reconciles the number of required impactors with the actual number of bodies by assuming a surface albedo of the binaries to be 15%, 2 to 4 times larger than usually assumed. Once formed, a binary object can disappear either because one of the components (usually the secondary) is destroyed (shattered and dispersed) through a high velocity impact, or the pair gains enough orbital energy to become unbound, due to the close approach or direct collision of another object. For asteroids, the major mechanism to eliminate a binary is the destruction of the secondary through high-velocity impacts. Since it seems well established that all known KBO binaries cannot be efficiently collisionally destroyed in less than 4 Gyr, all work to date have assumed that the KBO binaries would be stable over the age of the Solar System, except for Weidenschilling (2002), who mentioned, without any development, the possibility of disrupting the most loosely bound binaries. We show that long term stability is not guaranteed, and some of the KBO binaries may very well have lifetimes of order 1 2 Gyr. In the present work, we estimate the stability of these binaries with respect to several dispersal mechanisms. The data describing the known binaries and their dynamical and collisional environments are listed in Section 2. Beside the classical shattering and dispersing of the secondary through a direct collision, we also consider the possibility that the secondary is knocked off its orbit by a direct collision of a rather small impactor, or is gravitationally perturbed by the close approach of a somewhat larger TNO. All these mechanisms are described in Section 3. In Section 4, we present the lifetimes of the KBO binaries with respect to all three disruption mechanisms in the current rarefied environment. The case of the dense initial environment is considered in Section 5. After presenting the possible changes in the model parameters in Section 6, we discuss the implications of these results on the formation scenarios in Section 7. Finally, a summary of our findings is given in Section The facts To address our goal, we first need to know the parameters defining the binaries, and then the population of potential impactors. The binary parameters we use here (Table 1) are from two different compilations for the first seven of them, the first one by Merline et al. (2003), the second one by Stern (2002), yielding different sizes, masses, and separations. The secondary to primary size ratios have been checked for consistency with the latest values given by Noll (2003). The only noticeable discrepancy occurs for 1997 CQ 29 which has a magnitude difference between primary and secondary of 0.2. For 1998 WW 31, the sizes given by Stern is obviously outdated since we now have an accurate knowledge of the orbit and hence the total mass of the system (Noll, 2003). The sizes given in Table 1 are obtained from this mass, a magnitude difference of 0.4 and Stern s assumed density of ρ = 2gcm QC 298 was discovered in October, 2002, and reported by Noll et al. (2002) RZ 253 was discovered in April, 2003, and reported by Noll et al. (2003). Very little information is given in the discovery announcements. From the published magnitudes and distance, we estimated the equivalent radius of the pair R eq = RP 2 + R2 S to be 212 km for 2001 QC 298 and 202 km for 1999 RZ 253,assuming an albedo of 0.04, the usual default value for KBOs. The separation projected on the sky is estimated to be 5000 ± 2000 km (2001 QC 298 ) and 6300 ± 600 km (1999 RZ 253 ). The differences in magnitude between primary and secondary are given by Noll (2003): 0.5 for 2001 QC 298 and 0 for 1999 RZ 253. The resulting parameters are displayed in the last two lines of Table 1. The number of objects in the Edgeworth Kuiper belt is not yet very well known. For the sake of simplicity and to allow comparison with previous work, we use the same differential size distribution as proposed by Weissman and Levison (1997), and Durda and Stern (2000), i.e., a twocomponent power law of the form N(r i ) r b i dr i, where b = 3forr<r 0 and b = 4.5 forr>r 0, with r 0 = 5 km. The differential size distribution is assumed to be continuous at r = r 0. Although, we are more concerned with the cumulative size distribution, we give the differential one because it is simpler to write, and because this is what is generally published. Following Durda and Stern (2000), the normalization constant should be at least 70,000 objects with radius larger than 50 km, and perhaps twice that many. So we use 10 5 objects larger than 50 km in radius. This constitutes our reference size distribution. In Section 6, we investigate the range of possible changes to this size distribution and their effect on the lifetime of the KBO binaries. The final piece we need to estimate the number of collisions on a given target from a given set of impactors is the intrinsic collision probability. This number depends on the actual orbital distribution of the TNOs, and is, therefore, not well determined. It also depends on the location of the target in the belt. Here, we use the average value proposed by Farinella et al. (2000) P i = km 2 yr 1. Note that this is an old determination obtained from a poor representation of the dynamical structure of the Kuiper belt. A better determination is needed, but this would need an unbiased knowledge of the orbital structure of the belt which we do not have as yet (Petit and Gladman, 2003). In Section 6, we present other estimates of this probability. (1) (2)

3 KBO binaries: how numerous were they? 411 Table 1 Characteristics of the binaries, according to Merline et al. (2003) (columns 2 to 4), and Stern (2002) (columns 5 to 7). R P is the radius of the primary, R S the radius of the secondary, the heliocentric distance, and columns 4 and 7 give the distance between the 2 components of the binary. The assumed albedo is 0.04 in all cases, except for 1998 WW 31, for which Merline et al. assumed an albedo of Stern used a density of 2 g cm 3, while Merline et al. assumed a more conventional density of 1 g cm 3. Values for 2001 QC 298 are derived from Noll et al. (2002) and Noll (2003). Values for 1999 RZ 253 are derived from Noll et al. (2003) and Noll (2003). Here also the assumed albedo is 0.04 and no indication of density is given Object name Merline et al. (2003) Stern (2002) (AU) R P (km) R S (km) Separ. (km) R P (km) R S (km) Separ. (km) 1998 WW , , QT , , QW , , TC , SM CQ CF , , Noll (2003) 2001 QC RZ Disruption mechanisms In this work, we consider three different ways (Fig. 1) of eliminating a KBO binary. The first one is the shattering of the secondary by a collision, followed by the dispersing of the resultant fragments (Fig. 1a). This possibility has been studied at length in previous works, in particular, in the framework of the asteroid belt. Davis and Farinella (1997) show that bodies of radius larger than 50 km cannot be shattered and dispersed in the current dynamical and collisional environment. Since all binaries considered here have a secondary larger than 50 km in radius, it is clear that this process cannot be an efficient mechanism for eliminating the known KBO binaries. However, we consider this case as a reference, and as a mean of comparison with the other mechanisms. We use the value of Q D for ice (minimal energy per unit mass of target to shatter and disperse the target) given by Benz and Asphaug (1999) to compute the required minimum impactor size ( ) α ( ) β Q D = Q Rpb Rpb 0 + Bρ. (3) 1cm 1cm R pb is the radius of the parent body to shatter and disperse (expressed in cm) and ρ is the density of the parent body (in g cm 3 ). α, β, B, andq 0 are constants determined by a fit of results of numerical experiments, for impact velocity of 500 and 3000 m s 1. Since we will use impact velocities of 500 and 1500 m s 1 (see below), the values of the parameters for the latter case are derived by linear interpolation from those given by Benz and Asphaug. The second mechanism is the collision of a small projectile, not big enough to shatter the secondary, but that gives enough momentum to unbind the secondary from the primary (Fig. 1b). For all known KBO binaries, it is easier to unbind the secondary than to send it colliding with the primary, i.e., e 1. When an impactor of mass M i hits the secondary of mass M S M i, the secondary undergoes a change in velocity of V = M i V i /M S, where V i is the impactor s relative velocity. At this point, it is convenient to introduce the total mas of the binary, M = M P + M S,whereM P is the mass of the primary, and the reduced mass µ = M P M S /(M P + M S ). Before the kick, we assume the secondary to be on a circular orbit around the primary, with speed V S = GM/r, whereg is the gravitational constant, and r the separation between the primary and the secondary. The velocity after the kick is V S = V S + V. We look for a value of that velocity such that the total energy of the system vanishes, that is 1 (4) 2 µ(v S )2 = GM PM S = µ(v S ) 2 r (circular initial orbit). The square modulus of the velocity is given by (V S )2 = (V S ) 2 + ( V ) 2 + 2V S V = (V S ) 2 + ( V ) 2 + 2V S V cos θ, (5) where θ is the angle between the impactor s and the secondary s velocities. Averaging over all impact directions, we obtain (V S ) 2 = (V S ) 2 + ( V ) 2 + V S V. (6) Combining with Eq. (4) we solve for the change in velocity 5 1 GM V =V S (7) 2 r So, finally, the average impactor s mass necessary to dislodge the secondary from its orbit by direct collision is M i = 0.62 M S GM. (8) V i r

4 412 J.-M. Petit, O. Mousis / Icarus 168 (2004) Fig. 1. (a) The secondary is shattered and its fragments are dispersed due to a high-velocity impact. (b) The secondary is kicked off its orbit around the primary due to a direct collision by another TNO. (c) The secondary is dislodged from its orbit around the primary due to the gravitational perturbation from a passing TNO. The last possibility is gravitational perturbation from an encounter with a third body, that will transfer enough energy to the binary to unbind it (Fig. 1c). We have performed numerical integrations of the 3-body problem to determine the unbinding gravitational cross-section for a perturber of mass M i =10 19,10 20,10 21, and kg, with velocity V i. For each value of the mass and incoming velocity, we have selected a set of impact parameters, from 150 to 660,000 km, with 1.5 ratio increments. For each impact parameter, we ran 10,000 simulations with all other parameters taken at random, to evenly sample the space of possible orientation. Integrations were performed using the well-known general purpose, self-adaptive Bulirsch Stoer integrator (Stoer and Bulirsch, 1980) with relative precision of Fromthis we determined the probability of unbinding of the binary as a function of the impact parameter. Figure 2 shows this probability for 4 different masses of the projectile (10 19,10 20,10 21, and kg) arriving at 500 m s 1 on 2001 QW 322. This case has been chosen as being representative of all cases, with no particular behavior. The probability of unbinding P(h) for impact parameter h determines the gravitational unbinding cross-section σ = 0 2πhP(h)dh, (9) Fig. 2. Probability of disruption of 2001 QW 322 due to a projectile arriving at 500 m s 1 with mass of (solid line), (dashed line), (dash-dotted line), and kg (dotted line). The dash-triple dot line indicates the orbital separation of 2001 QW 322. from which one can derive the frequency of occurrence of such unbinding, and, finally, define the equivalent radius R g = σ/π. Note that on Fig. 2, the distance between curves decreases between the last two on the right. This results in a maximum efficiency (min-

5 KBO binaries: how numerous were they? 413 imum lifetime) for the gravitational unbinding mechanism somewhere in the range of mass studied. 4. Lifetimes In order to determine the frequency of disruption events, or conversely the expected lifetime with respect to disruption, one need to know the number of projectiles, the disruption cross-section, and the intrinsic encounter probability. Given the size of the projectile, one can easily determine the number of such projectiles using the size distribution given by Eq. (1). For the first two disruption mechanisms (direct collision), the disruption cross-section is simply the collisional cross-section, that is the physical cross-section π(r S + R i ) 2 times the gravitational focusing (1 + Vesc 2 /V i 2), where V esc is the escape velocity of the pair (secondary, impactor). However, the π factor is already included in the definition of P i. Hence we only need to compute (R S + R i ) 2 times the gravitational focusing. For the third disruption mechanism, we compute σ/π from Eq. (9). Note that the gravitational focusing correction is almost negligible in all our cases, since the escape velocity of the KBO binaries turns out to be less than 1 m s 1. For each set of binary parameters of Merline et al. (2003) and Stern (2002), we have estimated the impactor size and/or the disruption cross-section for the three mechanisms, assuming encounter velocities of 500 and 1500 m s 1 which roughly bracket the actual encounter velocities in the present day Edgeworth Kuiper belt. In Table 2, we report the shortest lifetime and the corresponding impactor size for each mechanism for the seven KBO binaries listed by Merline et al. (2003) and Stern (2002). For collisional shattering and collisional unbinding, this corresponds to the highest velocity, 1500 m s 1, while for gravitational unbinding, this corresponds to 500 m s 1. The cases of 2001 QC 298 and 1999 RZ 253 are presented in Table 3. Here we have considered the binary parameters given in the two last lines of Table 1, and a density of either 1 or 2 g cm 3. For each of these sets, we have run simulations for the same encounter speeds as before, and we report the shortest lifetime for each parameter set. The first obvious trend is that ejection of the secondary due to a direct collision is the most efficient way to eliminate a KBO binary. As was already well known, we find that collisional shattering and dispersing of the secondary is not efficient here. Gravitational disruption is also inoperative here because of the large size needed for the projectile, and the steep slope of the size distribution at large sizes. Interestingly, thanks to the high encounter speed, we never saw an exchange between the projectile and one of the components of the binaries in any of our integrations. As can be seen from the tables, 2 or 3 of the 9 known KBO binaries have mean unbinding lifetimes shorter than the age of the Solar System, even if all secondaries would survive shattering disruption over that time span QW 322 cannot survive in its current state for more than 1 to 2 Gyr CF 105 would most certainly have been destroyed if it was primordial WW 31 is potentially the less stable against collisional unbinding with a lifetime of Table 2 Minimum size of impactor (even columns) and corresponding lifetime (odd columns) for the KBO binaries, for each of the three disruption mechanisms: shattering, hitting, gravity perturbation. Binary parameters correspond to Merline et al. (2003) for columns 2 to 7, and to Stern (2002) for columns 8 to 13 Object name Shattering Collisional unbinding Gravity Shattering Collisional unbinding Gravity R shat T shat R h T h R g T g R shat T shat R h T h R g T g (km) (Gyr) (km) (Gyr) (km) (Gyr) (km) (Gyr) (km) (Gyr) (km) (Gyr) WW QT QW TC SM CQ CF Table 3 Same as Table 2, but for the various parameters for 2001 QC 298 and 1999 RZ 253 Object name Shattering Collisional unbinding Gravity R shat (km) T shat (Gyr) R h (km) T h (Gyr) R g (km) T g (Gyr) ρ = 1gcm QC RZ ρ = 2gcm QC RZ

6 414 J.-M. Petit, O. Mousis / Icarus 168 (2004) order 1 Gyr. However, this may be slightly increased, but not drastically, if one account for the actual eccentricity of its orbit. Up to now, we have solved the disruption equation for a single encounter. Since the number of small impactors is larger than the number of large impactors, we must also consider the effect of multiple collisions by small impactors on the secondary. In this case, the secondary will experience a random walk. The total change in velocity will grow like V = (δv ) 2, (10) where δv is the change of velocity due to each collision from a small impactor of mass m i. As before, δv m i, and the number of collisions, in a fix timespan, is proportional to the number of impactors of mass m i, n(m i ).Inthe following, we only consider a single power law size distribution, meaning that we will only be able to compare lifetimes or efficiency for masses on the same side of r 0.Fromthe differential size distribution of Eq. (1), the differential mass distribution is n(m i ) m (b 2)/3 i dm i. (11) So the effect of collisions from impactors of mass m i varies like V m (b+4)/6 i. (12) Hence, for b> 4, the largest impactors have the dominant effect, while for b< 4, the cumulative effect of small impactors overcomes the effect of a single collision by a big impactor. It is important to note that the effect on the velocity of the secondary is a continuous function of the impactor s mass, while the collisional erosion rate exhibits a large discontinuity for masses smaller than the critical mass for shattering and dispersing the secondary. The size distribution we have used so-far has b = 4.5 in the range of sizes of the disruptive impactors (Tables 2 and 3). Hence impactors of size r 0 would be collectively more efficient at disrupting the binaries. Since for r i <r 0, b = 3, smaller projectiles would be less efficient at disrupting the binaries. Equaling Eqs. (7) and (12), and noting that the change in velocity is proportional to the square root of the timespan, we relate the disruption lifetime T s due to multiple collisions from bodies of size r 0 to the one (T l ) computed earlier for a single collision ( ) b+4 ml 3 T s = T l, (13) m s where m l is the mass of the large impactor and m s the mass of the small impactors. Here, we have used the fact that the impactors are always small compared to the secondary, and then only the mass of the secondary governs the gravitational focusing. This reduces the lifetimes given in Tables 2 and 3, although not in a way that changes our previous conclusions. The same 3 binaries are unbound, maybe a little faster, and the other ones can still survive for the age of the Solar System. A word of caution is in order here. Dynamical friction from a swarm of small bodies has been said to cause a hardening of the binaries. It is not clear that bodies of radius r 0 = 5 km actually participate in the dynamical friction, hence hardening the binaries instead of unbinding them. But the single unbinding collisions still occur on the time scales given in Tables 2 and 3, which then set an upper limit for the lifetimes. 5. Primordial grinding In the previous calculations, we have considered a population of projectiles corresponding to today s Edgeworth Kuiper belt. However, it seems most likely that the primordial belt had to be much more massive, as much as 100 times more massive, in order to grow bodies as large as those observed today (Stern, 1996). The increase in mass can be achieved by simply multiplying the number of objects of each size by a constant factor of order 100, retaining the same size distribution, or by keeping the same number of large bodies, and increasing the mass in small bodies. Some authors (Stern and Colwell, 1997; Davis and Farinella, 1997) have argued that the mass loss of the Kuiper belt is due to collisional erosion. From our previous estimates, we can see that a long lasting intense collisional activity can have profound effects on the KBO binaries. We now investigate these effects on the direct collision ejection mechanism. We now suppose that the mass loss of the belt is actually due to collisional grinding. In this case, both Davis and Farinella (1997) and Stern and Colwell (1997) concluded that all primordial bodies of radius 50 km or less would have been destroyed, the ones we see now being fragments due to the shattering of bigger parent bodies. For each density and impact velocity assumed so-far, we can estimate the minimum size of an impactor capable of shattering and dispersing a 50 km radius body from Eq. (3). We compare this size and the corresponding collisional cross-section of a body of 50 km radius to the size and collisional cross-section of an impactor large enough to push the secondary out of its orbit, as in our second disruption mechanism. The occurrence frequency of these two types of events is the product of the collisional cross-section time the number of impactors time the intrinsic collision probability. The intrinsic collision probability has changed over the age of the Solar System, and cannot be given by Eq. (2) at all times, but it is the same for both types of events at any times. So we do not need to know its value to compare the frequencies. We just need to compare the cross-sections and numbers of impactors. Assuming a power-law size distribution like in Eq. (1), we can derive a condition on the slope b so that a KBO binary would be unbound by a direct impact more often than a 50 km radius body would be shattered and dispersed. This corresponds to the open region in Fig. 3, while the hashed regions corre-

7 KBO binaries: how numerous were they? 415 Fig. 3. The hashed regions correspond to slopes of the differential size distribution for which collisional unbinding of a binary occurs less frequently than shattering and dispersing of a 50 km body in the massive primordial environment. The 4 slope (dashed line) is the limit below, which multiple collisions of small impactors are more efficient than single collisions of larger impactors. The dash-dotted line corresponds to the large-end size distribution exponent in Eq. (1), the dotted line corresponding to the small-end. spond to slopes for which a KBO binary would be unbound by direct impact less frequently. The current slope for large bodies, 4.45 ± 0.3 (Gladman et al., 2001), is a relic of the accretion phase. Later collisional evolution tend to push the slope toward 3.5 oreven 3, starting with the small bodies. So clearly KBO binaries like 1998 WW 31, 2000 CF 105, and 2001 QW 322 (large orbital separation) cannot survive an intense initial collisional activity. Even a large fraction of objects like 1998 SM 165 would be unbound. They would resist only if the slope is 4.5 or steeper. The five remaining binaries could, in some cases, resist disruption even with a size distribution shallower than b = 4. This would happen only if all collisions occur at high speed ( 1500 m s 1 ). Large speeds favor shattering and dispersing over ejection since the former depends on the square of the velocity, while the latter depends on the velocity CQ 29, 1999 TC 36, 2001 QC 298, and 1999 RZ 253 are all very close binaries, with a large secondary, increasing their stability QT 297 is a rather well separated binary (20,000 km), but has the largest of all secondaries. One can also consider that all the present widely-separated binaries were more closely-bound originally, and have evolved to their current state through collisions. But, thus, either all binaries were much closer bound originally, or there was a much larger fraction of them. This point will be further discussed in Section What if... In this section, we discuss the major assumptions used so far. We present the plausible range of variation of some of the parameters, in particular, the size distribution, and investigate the effect of changing these parameters. As noted above, the collisional unbinding is the most efficient mechanism for eliminating a binary and we will concentrate our attention on it. However, we will also see whether and why another mechanism could become more efficient. The first assumption we consider is that of perfectly inelastic collision for the collisional unbinding mechanism. In the computation, we suppose that the momentum of the incoming projectile is entirely and exactly given to the secondary. To be more rigorous, we should consider that part of the projectile mass will stick to the secondary, and hence the departing mass may be (very) slightly larger than the initial secondary mass. However, this may not even be the case as all laboratory experiments and numerical models have shown that collisions at speed of order 1 km s 1 on targets a few tens to a few hundred kilometers tend to erode the target. So the departing mass is certainly smaller than the initial mass. In addition, in all experiments and modeling, the mass ejected from the crater tend to go backward with respect to the incoming projectile. As a consequence of the conservation of the total momentum, the change of momentum of the secondary is larger potentially much larger than the one assumed here and the projectile size is (much) smaller, and so is the unbinding lifetime. We stick to the present model because we know very little on this subject and do not want to introduce too much modeling which would give a false impression of precision when it will only increase the uncertainty. The number of 50 km radius objects is derived from a p = 0.04 albedo. As hypothesized in (Stern, 2002), one could consider that the albedo is larger (4 times larger for Stern), then the size of the objects is smaller. Thus, the size of the KBO binary component is smaller. But in the same time, the size of the background impacting objects is also smaller. Assuming the factor of 4 change prescribed by Stern this means that there are 10 5 objects of radius 25 km instead of radius 50 km. Since for the collisional unbinding the only relevant parameter is the ratio of the secondary mass to the impactor mass, the number of efficient projectile is unchanged. The only negligible change is due to gravitational focusing. The target being smaller, the gravitational focusing is less effective. But as mentioned earlier, this focusing is already negligible. However, decreasing the size of the secondary will help the collisional shattering. The value of Q D decreases with the size of the target. So the mass ratio of the secondary to the impactor increases, and the number of effective projectiles increases. However, this is a slight effect which is unlikely to make collisional shattering the most efficient mechanism. Considering only the collisional unbinding is probably good enough, and at worst gives an upper limit to the binary lifetime. The size distribution of KBOs is a very important factor in the lifetimes computed before. This distribution is correctly described by a two-component power-law defined by two exponents or slopes and a normalization factor, here the number of bodies with radius larger than 50 km. Durda and Stern

8 416 J.-M. Petit, O. Mousis / Icarus 168 (2004) (2000) claim that this number is somewhere between 70,000 and 150,000. The lower limit is in agreement with estimates by Trujillo et al. (2000, 2001) who consider that there are 31,000 Scattered Disk Objects (SDOs) and 38,000 KBOs of this size. The number of KBOs seems rather well established. The number of SDOs results from surveys where not all objects were followed. As explained in Petit and Gladman (2003), failing to track on the long term all objects in a survey introduces strong biases. In the survey presented by Petit and Gladman, where all objects were tracked to accurate orbit determination, the fraction of SDOs is twice as large as that of Trujillo et al. It follows that the number of SDOs is most likely 62,000. Hence, the total number of objects to consider is 100,000, which is the value used in the previous section. In addition to these, one must also consider the newly discovered Extended Scattered Disk (ESD) (Gladman et al., 2002), which is likely 10 to 100 times more populated than the Classical Kuiper belt and the Scattered Disk. This population has no interaction with bodies that are inside 40 AU from the Sun, as this is the minimum perihelion distance of the ESD Objects. But KBO binaries further out are more and more prone to collision from an ESDO. However, without a good description of the ESD, we prefer to ignore it, keeping in mind that doing so decreases the collision frequency and increases the binary lifetime. The normalization constant appears as a multiplicative factor in the collision frequency. According to the minimum values of the populations from Trujillo et al. (2000, 2001) and applying the Petit and Gladman (2003) correction factor, the normalization constant is at least 60,000 object larger than 50 km. Hence, the binary lifetimes must at most be multipliedby5/3. The other quantity that may be changed is the exponent of the power-law. For the large end of the differential size distribution the slope given in the literature varies from 4.15 (Trujillo et al., 2001) to 4.5 (Gladman et al., 2001) to 4.6 (Kinoshita et al., 2003). For the small size end, the only observational estimate of the slope is given by Kinoshita et al. (2003) to be The power-law exponent of a collisionally evolved population is 3.5 (Dohnanyi, 1969). To bracket the entire range of possible size distributions, we now consider a large population case with b = 4.6 for r>r 0 and b = 3.5 forr<r 0, and a low population cases with b = 4.1 forr>r 0 and b = 3forr<r 0.In both cases, we keep the reference value of the normalization constant. As expected, the lifetime of all binaries decreases when assuming the large population case, but only slightly, by a factor of As a result the unstable KBO binaries are still the same as in the reference case. With the low population size distribution, the lifetimes of the three unstable KBO binaries are multiplied by about 2. Hence they remain unstable, at least marginally in the case of 2000 CF 105. As noted in Section 2, the value of P i may be inaccurate. More recent estimates (Stern et al., 2003; Bottke, personal communication) give a value 3 to 5 times smaller. The collision probability is directly proportional to P i and, therefore, the binary lifetime is inversely proportional. If P i really turns out to be as small as proposed by Stern et al. (2003), then all KBO binaries would at least be marginally stable over the age of the Solar System in its current dynamical structure. Note, however, that these low values have been derived from a strongly biased orbital database and may very well be as inaccurate as that of Farinella et al. (2000). Nevertheless, the argument presented in Section 5 remains valid. In this argument, we do not consider the details of the collision probabilities. We only compare the respective efficiencies of collisional shattering and collisional unbinding and show that for a given range of values of the size distribution slope, collisionally eroding bodies as large as 50 km in radius implies that most or all widely separated KBO binaries would disappear. Stern et al. (2003) also claim that the mean encounter velocity of KBOs with Pluto and Charon is 2.1 km s 1.Using this value for the other binaries would reduce the critical impactor mass by a factor of 1.4, increasing the number of impactors by 1.4 (slope b = 3) up to 1.7 (slope b = 4.5). This would partly compensate for the decrease of the intrinsic collision probability. Here again we only intend to show a trend. A more accurate modeling of the momentum distribution (as opposed to separate mass and velocity distribution) would be misleading as long as the real orbital distribution is not better constrained. As we were doing some final revisions to this work, we became aware of a new paper by Bernstein et al. (2004) who predict that the number of objects of radius 6 and 10 km is 45 and 12 times less than the reference case described before. If this estimate is confirmed, then all known binaries would have no problem surviving for the age of the Solar System. This would also imply that recent formation of these binaries is much more difficult than estimated by Stern (2002) (see next section). However, this result is based on detecting fewer objects than expected in a very difficult deep small-scale search. In the same time, the luminosity function determined by Bernstein et al. predict far less objects than have been actually detected in the work of Gladman et al. (2001) and Larsen et al. (2001). So caution is in order when using Bernstein et al. results. Here again, this only concerns the current situation, and has no implication on the primordial evolution. 7. Discussion The existence of objects like 1998 WW 31 and 2001 QW 322, with lifetimes of 1 to 2 Gyr in the current rarefied environment means than there were, at least, between 7 and 50 times more similar KBO binaries 4 Gyr ago. The lifetime given above is the mean duration of a binary t = 0 tp(t)dt, wherep(t) is the probability that a binary still exists at time t. The probability of eliminating a KBO binary during a given interval of time being proportional to the current number of binaries, P(t)= 1/τ exp ( t/τ) and

9 KBO binaries: how numerous were they? 417 t =τ. The number of binaries left at time t is N(t) = N 0 exp ( t/τ).fort = 4Gyr,t/τ is 2 4, so N(t) = N 0. The number of binaries have been divided by a factor of 7 to 50. As expected, the most largely separated are the easiest to disrupt. In other words, the KBO binaries easiest to disrupt are also the ones that are the most difficult to create in Goldreich et al. (2002; L 2 s for two large bodies and a sea of small bodies) and Stern (2002; lcl 2 for late collision of two large bodies) scenarios. Weidenschilling (2002; cl 2 L for collision of two large bodies in the vicinity of a third large body) scenario, on the other hand, tends to create more large separation binaries. Since large separation binaries are more prone to disruption, this could reconcile cl 2 L with the observations, showing more binaries with small separations (5 with a distance km, 3 with a distance 20,000 23,000 km and only 1 with a distance > 100,000 km) than with large separations. L 2 s tend to form enough large separation binaries, if considering only their current number. However, since one must assume that there were at least an order of magnitude more large separation binaries 4 Gyr ago, this implies that at least the same increase in formation frequencies occurred for small separation binaries. This means that binaries like 1998 SM 165 or 1999 TC 36 should be at least 10 times more numerous. One possible explanation as to why this is not reflected in the current sample is that such binaries already suffer from a strong observational bias. The HST survey for KBO binaries will bring important information to try and answer this question. Another possibility is that both L 2 s and cl 2 L have been active to form KBO binaries, L 2 s forming most of the small separation ones, while cl 2 L produced a large number of large separation binaries. Finally, the short lifetime of the latter would explain the current distribution of KBO binaries. Due of typos introduced at print time, the main equation of Stern (2002) scenario (left column of p in that paper) is dimensionally wrong. The correct equation should read m min = 2M totv 2 orbit f KE v 2 i 2GM pm sat R tot f KE vi 2, (14) where m min is the minimum impactor mass to create a skimming satellite, v i its impact speed, f KE the fraction of its kinetic energy imparted into fragment ejection velocity, M p, M sat,andm tot are the masses of the primary, satellite, and sum of both, respectively, R tot is the sum of the radii of the primary and the satellite, and v orbit is the orbital velocity of the satellite. However, the derivation of the last term of Eq. (14) is only an approximation and a more rigorous one is given in Appendix A. The conclusion of Stern is that contemporary formation of KBO binaries is impossible with the assumed collisional environment. But instead of accepting a primordial formation scenario, Stern favors a recent formation of the KBO binaries, arguing that the binary albedo could be as much as 4 times larger than usually assumed. In this case, the size of the binary components would be divided by 2, and, therefore, the required impactor mass would decrease by almost an order of magnitude. This in turn would increase drastically their number. As explained in Section 6, this last part of reasoning is true only if the albedo of all KBOs is kept at the usual value of 0.04, except for the binaries. This is quite difficult to justify. If, as seems more reasonable, all albedos have to be increased by a factor of 4, then the size distribution of KBOs would keep its shape, but shifted to sizes half the usual ones. Finally, the number of potential impactors with the required mass would be essentially the same as with the classical computation. In such a scenario, we only gain because the critical specific energy Q D is an increasing function of mass. We have seen that if the much denser initial environment lasts long enough to allow the elimination of most of the mass of the Kuiper belt through collisional grinding (Davis and Farinella, 1997; Stern and Colwell, 1997), then all the large separation KBO binaries would be unbound. Simultaneously, many of the small separation binaries would also be unbound. Since it seems impossible to create the large separation binaries in the current environment, this implies that the dense initial environment did not last long enough to allow for collisional grinding. Furthermore, collisional grinding can be effective at removing mass only for very steep slopes (b 4.5) down to very small sizes, which is steeper than that predicted by accretion models (Davis et al., 1999; Kenyon and Luu, 1999), at least in the 1 10 km range. Hence, the mass reduction of the Kuiper belt by a factor of 100 must result from a dynamical mechanism, as proposed by Gomes (2003) and Levison and Morbidelli (2003). This explanation is actually supported by the current inclination of the KBOs which is much larger than their eccentricity on average (Gladman et al., 2001). Note that this remains true even if we must change the parameters describing the current collisional environment of the Kuiper belt. This also allows to address the main criticism of the Weidenschilling scenario by Goldreich et al. that Weidenschilling did not propose a mechanism for disposing of the surplus of large bodies needed at the beginning. If it is the case that dynamics rather than collision erosion is responsible for eliminating most the Kuiper belt mass, then the belt erosion is independent of the size of the objects. In such a case, one can easily dispose of 99% of the belt mass, and, in particular, of numerous large bodies. Both primordial formation scenarios (L 2 s and cl 2 L) form widely-separated binaries with the latter forming then preferentially. Moreover, it seems unlikely that one mechanism could have been acting while the other was not. So if erosion of the belt was collisional, even assuming that present day widely-separated binaries are formerly close bound pairs, there is still a noticeable fraction of them that would have been unbound, while the components themselves would survive. So the fraction of binaries should have

10 418 J.-M. Petit, O. Mousis / Icarus 168 (2004) been larger than currently, which is not easy to explain with current formation models. 8. Summary In this paper, we have shown that the stability of the KBO binaries with respect to perturbations other than the usual shattering and dispersing of fragments had to be investigated. Ejection of the secondary from its orbit around the primary by a direct collision of a rather small impactor turns out to be an efficient way to eliminate KBO binaries. Assuming the current estimates of the KBO population and impact velocities, the lifetime of 1998 WW 31 and 2001 QW 322 is 1 to 2 Gyr, and hence there must have been at least 10 times more similarly widely separated binaries formed at the beginning. Therefore, both Goldreich et al. (2002) and Weidenschilling (2002) scenarios of primordial formation must have been acting, Goldreich et al. mechanism forming most of the close binaries, Weidenschilling s forming most of the large separation binaries. However, this statement strongly rely on the assumed intrinsic collisional probability (Farinella et al., 2000) which may have to be lowered by a factor of 3 to 5 (Stern et al., 2003; Bottke, personal communication). In this case, all KBO binaries may be marginally stable over the age of the Solar System. The binary lifetime could also be lowered slightly since the velocity distribution seems to have larger values than what we used in our reference distribution. Unless one can find a viable mechanism for forming numerous KBO binaries in the current dynamical and collisional environment, we also showed that the erosion of the Kuiper belt cannot be due to collisional grinding. It has to be the result of some dynamical effect that occurred on a time scale shorter than that necessary to collisionally erode most of the belt. Acknowledgments We thank E. Howell and M. Nolan for discussions at ACM2002 that initiated this work. We are also grateful to A. Morbidelli and B. Gladman who made interesting suggestions that helped us improve our first manuscript. Many thanks to the referees D. Davis and W. Bottke whose criticism have forced us to investigate in more details the range of possible parameters, providing us with new results and strengthening our manuscript. Appendix A To derive Eq. (14) one must first find the kinetic energy of the incoming impactor M i v 2 i /2, multiply it by f KE to obtain the energy that will be imparted to the fragments, and then equate this quantity to the energy required to extract the satellite material from the parent body and send it into skimming orbit. Derivation by Stern of this last term is a crude approximation of the minimum energy needed to place the secondary in orbit around the primary. To compute this quantity, we must first define the initial and final states A and B, and determine their energy. Following the common practice, we define the zero-energy state as the one with no matter, or when each individual piece of matter is infinitely separated from any other. With this definition, the internal energy of a spherical body of mass M and radius R is E = 3G M 2 (A.1) 5 R. Our initial state A is a single parent body of mass M tot and radius R 1 = (Rp 3 + R3 sat )1/3. So the energy of this state is E A = 3G Mtot 2. (A.2) 5 R 1 The final state B consist of 2 components orbiting each other. Its energy must account for the internal energy of each component and their gravitational interaction energy (the orbital energy) E B = 3G Mp 2 3G 5 R p 5 M 2 sat GM pm sat R sat 2a, (A.3) where a is the semi-major axis of the orbit. So the energy required to create such a binary from a parent body is E required = 3G 5 [ M 2 tot R 1 ] M2 p M2 sat R p R sat GM pm sat. 2a (A.4) This is minimum for the smallest possible value of a, namely, a = R p + R sat,so E min = 3G [ M 2 ] tot M2 p M2 sat GM pm sat (A.5) 5 R 1 R p R sat 2(R p + R sat ). This could be of the same order than (GM p M sat )/R tot,but not necessarily. References Benz, W., Asphaug, E., Catastrophic disruptions revisited. Icarus 142, Bernstein, G.M., Triling, D.E., Allen, R.L., Brown, M.E., Holman, M., Malhotra, R., The size distribution of trans-neptunian bodies. Astron. J. Submitted for publication. Davis, D.R., Farinella, P., Collisional evolution of Edgeworth Kuiper belt objects. Icarus 125, Davis, D.R., Farinella, P., Weidenschilling, S.J., Accretion of a massive Edgeworth Kuiper belt. In: Proc. Lunar Planet. Sci. Conf. 30th. Abstract Dohnanyi, J.S., Collisional model of asteroids and their debris. J. Geophys. Res. 74, Durda, D.D., Stern, S.A., Collision Rates in the present-day Kuiper belt and Centaur regions: applications to surface activation and modification on comets, Kuiper Belt Objects, Centaurs, and Pluto Charon. Icarus 145,

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