Collisional evolution of trans-neptunian object populations in a Nice model environment

Transcription

1 Mon. Not. R. Astron. Soc. 423, (2012) doi: /j x Collisional evolution of trans-neptunian object populations in a Nice model environment Adriano Campo Bagatin and Paula G. Benavidez Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal/Instituto Universitario de Física Aplicada a las Ciencias y la Tecnología, Universidad de Alicante. PO Box 99, Alicante, Spain Accepted 2012 March 17. Received 2012 March 16; in original form 2011 July 19 ABSTRACT Almost 20 years of observations of the trans-neptunian region have shed light on the overall dynamical structure of the trans-neptunian object (TNO) populations and absolute magnitude distributions. The TNO region can be inserted in the global frame of the dynamical evolution of the giant planets, as described by the Nice model. Any reliable collisional evolution model should account for dynamical effects and should produce results that meet the constraints imposed by current observables. With this aim, we have developed a code package [Asteroid- LIke Collisional ANd Dynamical Evolution Package (ALICANDEP)], which is a collisional evolution code that includes statistical elimination of objects by dynamical effects within the frame of a disc migrating and gradually dynamically exciting, as well as the dynamical migration of objects between regions. Moreover, we included the possibility to distinguish between dynamically cold and hot bodies in the main classical belt and to keep track of primordial bodies in the whole region. Finally, we performed a large number of numerical simulations varying physical parameters, boundary and initial conditions, in order to match the current observables and the dynamical conditions of the Nice model. Our results are in agreement with those observables and can explain the flattened size distributions in the km size range. This allows us to constrain the original mass of the belt (not less than 50 M ), which is compatible with initial shallow size distributions below 100 km. ALICANDEP also finds an extremely high probability for the existence of at least one more large (>1700 km) object yet to be discovered in the outer belt. This model supports the reliability of the Nice model, and it can be a suitable tool to statistically study many features of the trans-neptunian region. Key words: Kuiper belt: general. 1 INTRODUCTION After almost two decades of observations of the trans-neptunian region [also called the Edgeworth Kuiper belt (EKB)], we begin to understand the overall structure of the trans-neptunian object (TNOs) populations in terms of dynamical features and absolute magnitude number distributions. The trans-neptunian region is a large population of objects that orbit the Sun with semimajor axes beyond that of Neptune. Following the terminology introduced by the Canada France Ecliptic Plane Survey (CFEPS; Kavelaars et al. 2009; Petit et al. 2011), from a dynamical point of view the TNOs can be roughly classified as follows. (a) Classical, which can be divided into an inner belt (ICB), with semimajor axes smaller than the 3:2 mean motion resonances with Neptune, and a main belt acb@ua.es (MCB), in between 3:2 and 2:1 mean motion resonances with Neptune. Main belt objects can be further classified as hot, responding to high inclinations and eccentricities, and cold (both stirred and kernel populations, in Petit et al. 2011). (b) Resonant objects, trapped in the 3:2 (called plutinos) and 2:1 resonances. (c) Finally, outer belt (OB) objects, which include the so-called scattering and detached objects (Petit et al. 2011), with semimajor axes larger than 47 au. 1.1 Observables in the trans-neptunian region A number of observables are available in the trans-neptunian region; we enumerate the main ones in this section. (a) While Pluto was the only known big object (larger than 2000 km in size) for the first decade of studies of the trans-neptunian region, the second decade has provided a wider population of big objects. Large TNOs are found in all dynamical classes of the trans- Neptunian region with the exception of the cold classical population. C 2012 The Authors

2 Then, at this moment, there are at least three objects larger than some 1500 km in size, with semimajor axes below 47 au in the classical belt Pluto, in the 3:2 resonance, Makemake and Haumea, inthe MCB, and one object in the OB: Eris. All of them are classified as dwarf planets (Brown 2008). (b) The size distribution of TNOs is the outcome of accretion and subsequent collisional and dynamical evolution in the outer Solar system (Stern 1996; Davis & Farinella 1997; Stern & Colwell 1997; Kenyon & Bromley 2004; Kenyon et al. 2008). The TNOs differential size distribution for large objects (D 100 km) is estimated to be a power law of the form dn(d) D q dd,whereqis the slope of the distribution. Past measurements of the trans-neptunian region size distribution estimated an average slope between 4.5 and 5.0 (Fraser et al. 2008; Fuentes & Holman 2008; Fraser & Kavelaars 2009). Nevertheless, Fraser, Brown & Schwamb (2010) and Petit et al. (2011) find that the cold component of the main belt shows a much steeper size distribution than the hot component and the inner belt do. In fact, these authors find for the cold main belt an average value q C 7 (5.6 < q C < 7.9), while for the rest of the classical populations the average value of the slope is q H 5 (3.9 < q H < 6.6). On the contrary, little is known about the size distributions at smaller sizes. Fuentes & Holman (2008), in their Subaru archival search for faint TNOs, find a slope of around 2.5 for objects fainter than m R = 24.3, i.e. smaller than 100 km in size. Fraser & Kavelaars (2009) also found that the distribution breaks to a slope 1.9 at object size R b 60 km. (c) The change in the slope of the size distribution between 50 and 150 km to a shallower distribution is reported by many authors (Bernstein et al. 2004, 2006; Fraser & Kavelaars 2009; Fuentes, George & Holman 2009). (d) The bias-free CFEPS (L3 release, Kavelaars et al. 2009; L7 release, Petit et al. 2011) was used to implement a model generating synthetic populations of TNOs in all dynamical classes, completely consistent with its discoveries. This allows us to constrain the number of the cold and hot components of the main belt and of the inner belt, giving also estimations of the number of objects in the OB (scattering, detached), for objects larger than a given size. For the main classical trans-neptunian population, they estimated ( ) 103 objects bigger than 100 km. The cold MCB population is estimated as N C (>100 km) = They also found that the inner population lacks a cold low-inclination component, and so the population is somewhat larger than recent estimates; this analysis suggests an inner population of N(>100 km) The CFEPS confirms that a large population of objects is dwelling the 3 : 2 mean motion resonance (and to a lesser extent, the 2 : 1). An analysis of the resonant populations has also been carried out by the CFEPS team and the estimation for the number of those bodies inside the 3 : 2 mean motion resonance in the same size range is (Gladman et al. 2012). 1.2 The Nice model Some authors have proposed that perturbation of the primordial belt including resonant capture and scattering of TNOs happened during a period in which Neptune was migrating outwards (Malhotra 1995; Gomes 2003). Even after its migration ended, Neptune had continued to erode the trans-neptunian region by gravitational scattering (Holman & Wisdom 1993; Duncan & Levison 1997), sending objects inwards to become centaurs or outward to become scattered disc objects, or removing them from the Solar system. Collisional evolution of TNO populations 1255 The mass currently estimated in the EKB is insufficient to allow the formation of 500 km or larger TNOs on time-scales of 5Gyr (Stern 1996; Stern & Colwell 1997). Therefore, a much larger mass should have been present in the region originally. The TNO region can now be inserted into the global frame of the dynamical evolution of the giant planets, as described by the Nice model. This model does explain the main dynamical mechanisms that caused part of the mass depletion of the trans-neptunian region, the current orbits of the giant planets and other details of the structure of the Solar system (Gomes et al. 2005; Morbidelli et al. 2005; Tsiganis et al. 2005; Levison et al. 2008). The Nice model is based on the idea that the gas giants formed much closer together, surrounded by a disc of planetesimals stretching between 20 and 34 au. Due to interactions with the planetesimal disc, Saturn, Neptune and Uranus migrated outwards and Jupiter migrated inwards. After some Myr, Jupiter and Saturn crossed their 2:1 mean motion resonance and the system became temporarily destabilized, affecting the orbital elements of the gas giants. As Neptune moved out into the trans-neptunian region, its secular resonances dynamically excited the orbits of many of the TNOs to evolve into cometary orbits (Gomes et al. 2005). This global dynamical destabilization affected the former asteroid belt as well, which was considerably depleted. Both ejected populations of TNOs and asteroids impacted the terrestrial planets and their moons. Hence, the Nice model also explains the Late Heavy Bombardment (LHB), a period of intense bombardment in which, for instance, part of the large basins on the Moon was formed, which occurred around 3.9 billion years ago (Tera, Papanastassiou & Wasserburg 1974). This period of intense bombardment and dynamical depletion is likely to have had a significant effect on the observable properties of the debris discs of the Solar system, and in particular on the trans-neptunian region. The Nice model adds a constraint to the overall evolution of TNOs: the mass of the region, just before the onset of the LHB period, must exceed some 25 M. We do not know if the Nice model is the absolute truth on the dynamical evolution of the Solar system but we chose this dynamical frame to combine with collisional evolution because it seems to currently explain most of the known features of the outer Solar system. Nevertheless, we are aware that it may be revised in the future. For this reason, the model we present in this work has been implemented so that it can be suitably adapted to upgrades and changes in the dynamical frame. 1.3 Previous collisional evolution models A number of collisional evolution models for the TNO region have been proposed in the last 15 years. Davis & Farinella (1997) pioneered this subject by studying the collisional evolution of TNOs as an adaptation of a model developed originally to study the main asteroid belt collisional evolution. Their main results were as follows. (a) Collisional evolution is more intense in the inner part of the TNO region; its intensity decreases with increasing distance from the Sun. (b) The population of TNOs larger than 100 km in size was not significantly altered by collisions over the age of the Solar system and the size distribution in this range should resemble the original population. They also suggested that it is likely that many of these bodies have become gravitational aggregates. (c) Current TNOs smaller than km in diameter are mostly fragments from the collisional cascade.

3 1256 A. Campo Bagatin and P. G. Benavidez Kenyon & Bromley (2004) developed an analytical and numerical model of the collisional evolution of TNOs to study the formation of a break in the size distribution of TNOs, including a model of orbit stirring by Neptune. For a wide range of bulk properties, initial masses and orbital parameters, their results yield powerlaw cumulative size distributions with slopes q L 3.5 for large bodies (with D km) and q S for small bodies (with D km). The transition occurs at a break diameter D b 2 60 km. They suggest that the break size is more sensitive to the initial mass in the belt and the amount of stirring by Neptune than to the bulk properties of individual objects. However, their simulations had to assume a specific energy for disruption that is at least an order of magnitude lower than predicted by Benz & Asphaug (1999). However, the size distributions exhibited in their paper show characteristic artefacts due to the presence of a sharp cut-off at small sizes (Campo Bagatin et al. 1994). Pan & Sari (2005) assumed that the TNOs are gravity-dominated bodies and modelled analytically the collisional cascade in which all collisions were purely disruptive. They found that the break in the size distribution of TNOs is expected at 80 km, since destructive collisions are more frequent for smaller TNOs. The cumulative size distribution slope found was q = 3 for objects with sizes between about 0.2 and 80 km and could not account for the steep slope observed for large TNOs. Krivov, Sremčević & Spahn (2005) developed a kinetic model of a disc of solid particles, orbiting a primary and experiencing inelastic collisions, using mass and orbital elements as independent variables of a phase space. They found that collisional evolution is most substantial in the inner part of the EKB and that the size distribution in the EKB is not a single Dohnanyi-type power law, reflecting the size dependence of the critical specific energy in both strength and gravity regimes. Charnoz & Morbidelli (2007) proposed the first model that combines collisional evolution and dynamics to study the population of TNOs. This model is based on both a direct simulation of the dynamical evolution which is used to compute local encounter rates and a statistical erosion/fragmentation model that evolves the size distribution of bodies. They found that the mass depletion during the EKB evolution is the result of both collisional grinding and dynamical effects. They concluded that it is difficult to reconcile the collisional evolution of the EKB to the formation of the Oort cloud and the scattered disc, due to the fact that the collisional process is too efficient and avoids their formation. Consequently, these authors suggest that some other mechanism should create the three populations and that only models with little collisional activity, for D b > 20 km, may be able to do so. de Elía, Brunini & di Sisto (2008) analysed the collisional and dynamical evolution only for the population of plutinos. They consider different collisional parameters and include a dynamical treatment that takes into account the stability and instability zones of the 3:2 mean motion resonance with Neptune. They conclude that the break in size distribution suggested by previous works around km in the plutino size distribution should be primordial and not a result of the collisional evolution. Benavidez & Campo Bagatin (2009, hereafter BCB09) developed a three-zone model to study the collisional evolution of TNOs characterizing each dynamical population by their own orbital characteristics and allowing populations in different zones to partially interact collisionally. No dynamical effects due to the migration of the giant planets were taken into account. They studied the main features in the resulting size distributions exploring the parameter space of collisional physics (i.e. scaling laws, material strength and fraction of collisional energy into ejected fragments). They also estimated the ratio of gravitational aggregates in any size interval and zone. Their simulations seem to show that between 25 and 50 per cent of the initial mass is eroded in the first 100 Myr. This study also confirms that the origin for the break in the size distribution of TNO populations is due to the lack of collisional relaxation of the largest size end of the distribution, i.e. a fossil of the original one. Fraser (2009) implemented a collisional evolution model to determinate whether collisional erosion can produce the rollover at D km. He studied the collisional evolution starting from an initial size distribution similar to the one suggested by accretion models (Kenyon 2002), characterized by three ranges with different power-law size distributions, in a dynamical excited environment, as is the current EKB. His calculations show that a divot may be produced in the size distribution around D km together with a rollover from the large object accretion slope at D km. As remarked by this short review, in most cases, with the exception of Charnoz & Morbidelli (2007) and de Elía et al. (2008), no dynamical effects were taken into account in collisional modelling, leading to different mismatches with current observables. Any reliable collisional evolution model should include the main features of dynamical models of the outer Solar system, together with a detailed treatment of collisional evolution, and should produce results within the constraints imposed by current observables. With this aim, we have improved our previous collisional evolution model (BCB09) and have developed ALICANDEP, a more complex code package, as described in the following section. 2 ALICANDEP Based on our previous collisional evolution model (BCB09), we have built a numerical model of the collisional evolution of small body populations, and applied it specifically to the case of the TNO populations [Asteroid-LIke Collisional ANd Dynamical Evolution Package (ALICANDEP)]. The model includes statistical elimination of objects by dynamical effects within a frame of a disc migrating and gradually dynamically exciting, as well as the dynamical migration of objects between regions. Moreover, we implemented a feature to distinguish between dynamically cold and hot bodies in the inner and classical belt and to keep track of primordial bodies in those regions. Size distribution slopes and break sizes are also calculated. All the features included in our former model are described and explained in BCB09 and are present in ALICANDEP as well. They include (i) the computation of the number of gravitational aggregates in any size interval, (ii) the possibility to use different sets of parameters and scaling laws for shattering, and (iii) varying boundary conditions (initial overall mass, mass distribution in different regions, initial size distributions). Contrarily to most models dealing with the collisional evolution of TNOs, the fact that relative velocities of impacts are widely dispersed is suitably taken into account. We assume Maxwellian distributions around the most probable values (V p ) in each zone. The range considered for V p is between 0.1 and 4 km s 1, with the following most probable values for each zone: V p1 = 1.25 km s 1, V p2 = 0.93 km s 1 (Dell Oro et al. 2001) and V p3 = 1kms 1 (see Section 2.2 for a description of the zones). There is no previous estimation for the last value, and we figure it out from the relationship V rel e 2 + i 2,wheree and i are the average eccentricity and inclination estimated from the current TNO population. We shortly summarize here the main features of the model, emphasizing and explaining the new implemented ones.

4 Collisional evolution of TNO populations 1257 The model evolves in time the number of objects in discrete logarithmic size bins, whose central values generally span the range from 35 cm to km in diameter, in such a way that there is always a factor of 2 in mass ( 1.26 in size) between any neighbouring bins. Schematically, the model consists of two parts, as follows. 2.1 Handling of collisions The simulation of the collisional outcomes of every impact between objects belonging to any pair of size bins is performed. The outcome of a collision depends largely on the ratio of kinetic energy of the impactor to the mass of the impacted body, in the centre-of-mass reference frame, and the specific energy, commonly denoted as Q. The threshold for a shattering event is defined by Q S, which is the specific energy required to break a body so that the largest fragment produced is half the mass of the parent body. The energy of the created fragments is compared to the binding gravitational energy in order to decide what fraction of ejected mass is reaccumulated on the largest remnant of the collision. This is used to discriminate when a collision produces a gravitational aggregate, in its standard definition (the mass of reaccumulated fragments is at least half the mass of the whole gravitational aggregate body; Campo Bagatin, Petit & Farinella 2001). Such events contribute to the calculation of the number of gravitational aggregates. Bodies can be characterized by any of the scaling laws proposed in the literature. We assumed, as our nominal case, Benz & Asphaug (1999, hereafter BA99) scaling law for ice, as derived from smooth particle hydrocode numerical simulations. Those simulations provide the specific energy for dispersion Q D, which is defined as the energy per unit mass necessary to shatter and disperse half the mass of the parent body, by the following expression: ( ) a ( ) b Q D = Q Rb Rb 0 + Bρ, (1) 1cm 1cm where R b is the radius of the target, Q 0, a, b and B are values differing upon material composition and relative velocity of impact, and ρ is the density of the bodies, assumed as 1 g cm 3. For such parameters, we consider the corresponding values for ice, as given by the authors. Q 0 = erg g 1, B = 1.2 erg cm 3 g 2, a = 0.39 and b = 1.26 for impact speed of 3 km s 1. Q 0 = erg g 1, B = 2.1 erg cm 3 g 2, a = 0, 45 and b = 1.19 for impact speed of 0.5 km s 1. Suitable interpolations between 0.5 and 3 km s 1 were performed according to the relative velocity distributions. For velocities between 3 and 4 km s 1, the four parameters entering the scaling law were left constant assuming values estimated by BA99 for 3 km s 1. As the fragmentation algorithm is based on Q S, it is necessary to work it out from Q D, as explained in BCB09: [ Q S = f SH Q D GM ] T (2) f KE D f SH is the fraction of kinetic energy of the projectile delivered to the target, and is assumed to be 1/2, f KE is the fraction of kinetic energy carried by the projectile finally imparted to the created fragments; D and M T are the size and mass of the parent body, and G is the gravitational constant. More recently, Leinhardt & Stewart (2009, hereafter LS09) and Leinhardt & Stewart (2012) have derived alternative scaling laws in the case of ice, based on a different method (CTH: a shock wave physics software package) and leading to shattering critical specific energies grossly one order of magnitude smaller than BA99. Figure 1. The two scaling laws for fragmentation used alternatively by ALICANDEP. BA99 stands for Benz & Asphaug (1999) and LS09 for Leinhardt & Stewart (2009). The functional form of LS09 scaling law is the same as BA99, but their recommended values are different: Q 0 = 20 J kg 1, B = Jm 3 kg 2, a = 0.4, b = 1.3. The two scaling laws are compared in Fig. 1, and the results of assuming LS09 scaling law are reported in Section 2.3. The fragmentation algorithm is based on the fragmentation and reaccumulation model of Petit & Farinella (1993), including improvements based on recent available experimental data, numerical and theoretical studies. This part of the package computes the number of fragments produced in any possible collision between objects belonging to different size (mass) bins. Different algorithms consider shattering and cratering events. 2.2 Time evolution ALICANDEP considers three concentric toroidal zones around the Sun where the orbits of objects from different zones can cross each other and stay in common regions during a fraction or all of their periods, as a consequence of their eccentricity and semimajor axes. Therefore, the number of objects from each zone that interacts with objects from other zones depends on their mean anomaly and the range of semimajor axis that defines the corresponding common zone in each case. This is made by taking into account the interactions between zones accurately calculating the amount of time that objects belonging to any given zone spend in a region common to any other zone. (A detailed description of interactions between dynamical zones can be found in BCB09.) When collisions between objects coming from different zones happen, fragments stay in the zone of the largest body, while in the case of same size colliders they are distributed evenly in each zone. Each zone is a geometrical toroid with rectangular sections, according to the following ranges: zone 1: a 1 (1 e 1 ) < r 1 < a 2 (1 + e 1 ); zone 2: a 2 (1 e 2 ) < r 2 < a 3 (1 + e 2 ); and zone 3: a 3 (1 e 3 ) < r 3 < a 4 (1 + e 3 ).

5 1258 A. Campo Bagatin and P. G. Benavidez Table 1. Orbital elements at each dynamical phase. Semimajor axes stand for boundary values for zones 1, 2 and 3. Phase t a 1 (au) a 2 (au) a 3 (au) e 1 e 2 e 3 i 1 ( ) i 2 ( ) i 3 ( ) The values assumed by a i vary with time during migration. In Table 1, values corresponding to the beginning and the end of each dynamical phase can be found. In order to insert the collisional evolution of TNO populations into the frame described by the Nice model suitably, new features have been introduced with respect to our previous code. ALICANDEP is able to handle both migration and dynamical excitation of the disc and, at the same time, statistical dynamical depletion of mass during different periods of time and migration of bodies from one region to others. The calculation of these effects is performed at time intervals of 1 Myr by adding suitable terms - - accounting for migration and depletion to the evolutionary discrete equations corresponding to each size bin, as described in equation (3). The dynamical evolution of the planets in the Nice model requires that the trans-neptunian disc contained 25 M of material at the time when the planets became unstable, i.e. at the beginning of the LHB (Gomes et al. 2005; Tsiganis et al. 2005; Levison et al. 2008). This period of time is estimated to have lasted about 100 Myr, starting some Myr ago. Consistently with the Nice model, we consider an initially cold disc, located between 20 and 34 au. At the beginning of the evolution, the total mass, M 0, was shared between two contiguous zones, namely zone 1 and zone 2; zone 3 was initially empty. Evolution takes place in different phases, characterized by values for eccentricities, inclinations and semimajor axes as summarized in Table 1. (i) Phase 0: from 0 to 100 Myr. Little dynamical excitation. Mainly collisional evolution takes place. (ii) Phase 1: from 100 Myr to t LHB. The disc slowly starts losing mass due to dynamical interactions. (iii) Phase 2: t LHB to t LHB+100. In this period, the disc is strongly dynamically excited, migration of bodies from zone 1 to zones 2 and 3 takes place and very strong dynamical mass depletion happens in the three zones. (iv) Phse 3: t LHB+100 to the end of the evolution (4500 Myr). Dynamical excitation and migration stop, while mass loss is reduced to interactions with Neptune (mean motion resonances 3:2 and 2:1). The disc enters a quiet phase, leading to the current situation. Collisional evolution is always present in the simulations, even if its effectiveness varies depending on the evolutionary phase. As there is some uncertainty about the beginning of the LHB phase, t LHB is a parameter that is varied in the simulations, assuming values between 600 and 800 Myr (Gomes et al. 2005; Levison et al. 2008). During each phase, each zone undergoes migration and is expanded by excitation. Therefore, its volume changes, the relative velocities in every zone grow up due to dynamical excitation and collisional probabilities change according to the relationship V rel e 2 + i 2. All these values are updated at each time step when numerically solving the corresponding evolution equations. In this way, in any given zone, at any interval of time t, the number of collisions among N i objects of size D i and N j objects of size D j, corresponding to a given interval of relative velocities v l, is calculated. Time integration of the evolution equations for all size bins must be then performed taking into account the described interactions. Any size bin k, within a given zone z (z = 1, 2, 3), in a given dynamical phase a (a = 0, 1, 2, 3), must undergo time evolution according to the relationship dn(k z ) dt = n i { i m } [f ij kl p(v l ) s i,j,l ] j l [N(i) δ ij ]N(j) (1 + γ x + γ y )+ 1 + δ ij (α az + β az )N(k z ), (3) where n and m are the total number of size and velocity bins, respectively. The projectile index j ranges from 1 to i, due to the symmetrical nature of the f (i, j, k, l) matrix, which is the output of the fragmentation algorithm; p(v l ) is the probability corresponding to the lth generic discrete interval of relative velocities centred around v l,ands i,j,l is the collisional cross-section for objects of size D i and D j at speed v l. s i,j,l = 1 4 P int(d i + D j ) 2 {1 + (V esc /V l ) 2 }. Finally, 8 δ ij = 1ifi = j, and0otherwise.v esc = 3 πρgr2 is the escape velocity from the target body (ρ is its bulk density), and P int is the intrinsic collisional probability (which is a function of the relative velocity, volume and orbital elements for each zone). γ x and γ y account for collisions between bodies in the considered dynamical zone and those in any of the neighbouring zones x and y. α az is the rate of elimination due to dynamical effects. α az > 0forany dynamical phase a (a = 0, 1, 3) and for any zone z. β az is the rate of migration during the LHB phase (a = 2). β az = 0fora = 0, 1, 3. β az > 0fora = 2andz = 1, thence it is a negative contribution to dn(k z )/dt, as bodies from zone 1 migrate to zones 2 and 3. Finally, β az < 0fora = 2andz = 2, 3, representing a positive contribution to dn(k z )/dt. The set of n first-order, non-linear, differential equations described above must be solved numerically at each time step by transforming them into a set of finite difference equations. As discussed above, we assumed a Maxwellian distribution for relative impact velocities, we divided the velocity range into 20 discrete intervals, each of which has its corresponding probability, and performed the calculations according to that. As usual, in order to avoid undesired wavy effects due to an abrupt truncation of the size distribution at small sizes (Campo Bagatin et al. 1994), a number of the smallest size intervals (from 35 cm to 30 m) have been used to produce a low end of the distribution. The collisional evolution of these bins is estimated using a powerlaw extrapolation from the size distribution of the next 10 bins containing objects larger than 30 m. 2.3 Boundary conditions and model parameters The solutions of equation (3) depend on a number of initial conditions, namely (i) the initial total mass and how it was distributed between the two initially populated zones, (ii) the size distributions in each zone, (iii) the location and extension of the zones, depending on the orbital elements of the bodies. (i) and (ii) are essentially

6 unknown and must be explored in order to match current observables, while (iii) is assumed as predicted by the Nice model (see Table 1). One of the goals of ALICANDEP is to constrain (i) and (ii). It is also necessary to set boundary conditions as well. (a) At the beginning of the LHB phase, the Nice model needs some M in the region and 99 per cent of the mass is lost during that phase. (b) The migration of zones their changes in location and volumes as bodies are dynamically excited is moving geometric boundaries for the system itself. (c) Finally, current observables have to be met by the model at the end of the evolution. Other parameters that may influence the model results are those regarding the physics of collisions and those driving the migration and dynamical elimination of bodies. As for the collisional physics, the general problem of their influence on the collisional evolution of TNOs was studied in BCB09. Nevertheless, we checked the dependence of the results on scaling laws for shattering different from the nominal one. Namely, we discuss the results corresponding to the scaling law proposed by LS09 for ice. The mass depletion during each dynamical phase is handled by introducing suitable depletion rates α i (index i refers to a generic dynamical phase). A first, gross estimation of these quantities is derived from the relationship M fi = M 0i (1 α i ) t i /1Myr,where M 0i and M fi are the initial and final mass in a given depletion phase, respectively, as deduced by the Nice model, and t i is the corresponding interval of time, in Myr. A similar approach is used for the estimation of the migration rates in between zones. Fine tuning of such parameters is then performed in order to match boundary conditions. Going back to initial conditions, it is necessary to start with a total initial mass M 0, while each zone has its own initial size distribution, typically described by power laws, so that zones 1 and 2 have initial masses M 01 and M 02. The mass distribution is depending on the assumed geometric extension of each zone and on the surface density (σ D ) dependence on heliocentric distance a, normally assumed to be σ D a γ, with 1/2 <γ <1 (Weidenschilling 1977; Raymond, Quinn & Lunine 2005). M 0 itself is completely unknown. Some models that study the early accretion phase suggest that the initial mass could be between 15 and 50 M (Stern & Colwell 1997). Simulations of the Nice model suggest 35 M and larger. On this basis, we varied M 0 so that boundary conditions are met, starting at a minimum value of 30 M. Current observations and estimations constrain the slopes of the size distributions for bodies larger than some 100 km to high values that cannot be extrapolated to smaller sizes because that would lead to mass divergence. It is unknown where the rollover of the initial distribution was at the beginning of evolution; we only have loose estimations of where it currently is, depending on authors. This point is very interesting because there is no agreement on the causes of such feature. Is that a consequence of collisional evolution due to the variation with size of the critical shattering energy of bodies? Or is it rather the print of a primordial pattern in size distribution? The implications of the answers to these questions on the understanding of the processes of formation of the outer Solar Systems are remarkable indeed. We investigated that issue by varying our initial conditions over a wide range of values for the transition (break) sizes where large and small size distributions match ( km). The power-law differential distributions are represented by dn(d) D q 1 dd if D D b, dn(d) D q 2 dd if D>D b, (4) Collisional evolution of TNO populations 1259 where D b is the initial break size, q 1 and q 2 are the slopes of the size distributions for small and large sizes, respectively. It is useful to remind that q = 4.0 implies that mass is distributed evenly in each size bin, whereas for q < 4, mass is mainly in large bodies, within any given range. In the choice of our initial power-law distributions, we must distinguish between small and large size ranges. In the small size range, there is no indication of what the initial distribution was nor what the current distribution is, especially for D < 30 km. For this reason, we varied from no initial mass below the break size to equal mass bins (q 1 = 4). We started with initial size distributions with slopes close to the ones measured by the CFEPS, at the large size range. There is no point in starting with slopes smaller than current ones; in fact, dynamical effects in Nice model are not size dependent and collisional erosion tends to decrease slopes asymptotically towards q 3.5. On the other hand, starting with slopes somewhat larger than current ones is, in principle, allowed for the same reasons. Power-law indexes, q 1 and q 2, for the initial populations, are therefore varied in the following ranges. Zone 1: (q 1, q 2 ) = [no 4, 4 6]; zone 2: (q 1, q 2 ) = [no 4, 6 7.5]. We performed tens of numerical simulations varying the physical parameters and boundary conditions described above, in order to match the observables and the dynamical conditions of the Nice model. This allows us to constrain the fragmentation physics and some of the initial conditions (initial mass distribution, size distribution of objects, transition size). These issues are discussed in the following sections, together with the main conclusions of our work. 3 RESULTS The goal of this research is to set up a self-consistent collisional evolution code in the frame of the dynamical Nice model for the TNO populations. The model should be able to reproduce the available observables of that region of the Solar system. We would like to stress that the results of our simulations hold under the assumption that the Nice model describes the main dynamical mechanisms that have driven the outer Solar system to the current situation. ALICANDEP can also be adapted to other dynamical scenarios. Part of the data from the CFEPS was available and derived estimations for TNO populations were already published (L3 release, Kavelaars et al. 2009), so we initially focused our attention on choosing suitable initial populations and physical conditions in order to match such estimations and the other observables. Nevertheless, as members of the CFEPS, we had access to the new estimations derived from the full survey (L7 release, Petit et al. 2011) and we refined our boundary conditions to match L7 as well. The results matching the complete survey are reported here. Zone 1 includes the ICB and the 3:2 resonant objects ( plutinos ). Zone 2 includes the MCB, while Zone 3 contains the OB (scattering and detached objects). An interesting comparison has been made between the overall TNO evolution according to ALICANDEP (ALICANDEP-1, in Table 3, the nominal case) and the populations which resulted by taking the overall collisional evolution according to our previous model, described in BCB09, that considered no dynamical effects. In order for the two evolutions to be comparable, in the case of BCB09 we evolved the populations until 700 Myr and just scale them down artificially to get the same final total mass as in ALICANDEP-1 (Fig. 2). The two initial distributions are certainly different, with a double slope distribution in the case of ALICANDEP-1, responsible for the current observed break around 100 km, now sharper than in BCB09.

7 1260 A. Campo Bagatin and P. G. Benavidez Figure 2. BCB09 versus ALICANDEP-1: comparing the nominal case in this paper with the evolution according the BCB09 paper until t = 700 Myr, and depleted in mass to match ALICANDEP-1 final situation. The steep slope at large sizes is mostly preserved in ALICANDEP-1, as was in the case of BCB09. Instead, both final distributions at sizes smaller than the break size are very similar in shape, a consequence of the intense initial collisional evolution that drove both slopes to the corresponding steady-state values, close to Match to observables All the results of our simulations that are presented here strictly meet the requirements and constraints of the Nice model, namely having a total mass of at least 24 M at the beginning of the LHB phase and of the order of 1 per cent of it at its ending. (i) Table 2 shows that ALICANDEP results are in very good agreement with observables, including the CFEPS L7 findings (Petit et al. 2011). (ii) As far as the number of bodies larger than 100 km is concerned, we have run 20 twin runs (identical simulations) of our nominal case in order to get a statistical estimation. In other cases, we run at least eight twin runs. This is necessary because the small number statistics are handled by means of Poisson probability distributions that may provide slightly different outcomes for the number of large objects. Contrarily to BCB09, we have spread the size range up to km, assuming the initial power-law distributions described in the text, which results in some eight Mars-size objects (or larger) in the primordial populations. We find that dynamical effects are responsible for getting rid of those rogue planets during the LHB phase. Most of those objects have been probably ejected from the Solar system, but some of them may still be orbiting the Sun inside the OB. As an average value for bodies larger than 1700 km (the dwarf planets size bins) in zone 1, we get 0.6 (with a mode of 1) as the number is fluctuating around 1 in twin runs, reproducing the presence of Pluto. For the MCB, our simulations show fluctuations between one and three bodies, with an average of 1.8 and a mode of 2, in agreement with the known number of dwarf planets in the MCB. Statistics for other sets of boundary conditions that best fit observables are reported in Table 2. Due to the many surveys carried out in the last two decades to discover large bodies in the classical belt, we believe that the number of known dwarf planets up to 47 au is at completion. Our zone 3, corresponding to the OB (scattering and detached regions, beyond 47 au), allows large fluctuations in the number of bodies larger than 1700 km (from one to seven), with an average of 3.4 and mode of 3, in our nominal case (see Table 2 for other cases). The only known object past the 2:1 mean motion resonance with Neptune is Eris. On this basis, a Poisson probability of per cent is found by our simulations that at least another potential dwarf planet (larger than 1700 km) is yet to be discovered beyond 48 au (69 86 per cent of having at least two such objects). Restricting the statistics to objects larger than Pluto (and Eris), our model finds a probability of per cent of having at least one object in that size range. Finally, one Mars-size body (larger than 4500 km) might have survived in the OB with an average probability smaller than 22 per cent. It has to be kept in mind that the OB outer limit is not well defined. Even if most objects (roughly two-thirds) are listed with semimajor axes not exceeding 100 au, the CFEPS synthetic model includes bodies with semimajor axes as large as Sedna s (about 600 au) and large eccentricities. (iii) In order to check our model against further observables, we calculated the probability that a collision happened in the inner region of the belt so that the Pluto Charon system was created according to the mechanism proposed by Canup (2005). They find that an oblique impact at velocity under 900 m s 1 between a proto- Pluto and a projectile per cent the mass of the current Pluto Charon system is able to form the system. Assuming their collisional Table 2. Comparing the results of ALICANDEP s best matches to available observables. The number of objects in different populations and the corresponding slopes are compared with the model results derived from the CFEPS by Petit et al. (2011). The reported ALICANDEP estimations of the number of current dwarf planets are the averages over the best-fitting twin runs in the nominal case (ALICANDEP-1) and in the rest of best-fitting cases (ALICANDEP-2, ALICANDEP-7, ALICANDEP-12). ICB+P reports the number of objects in the ICB ( ) plus the number of objects in the 3:2 mean motion resonance with Neptune (plutinos; ). Number of objects ( 10 3 ) Slopes Dwarf planets ICB+P MCB MCB (cold) OB MCB MCB (cold) km ICB+P MCB OB Observables ± ALICANDEP ± ± ± ± ± ± ± ALICANDEP ± ± 4 97± ± ± ± ± ALICANDEP ± ± ± ± ± ± ± ALICANDEP ± ± 4 97± ± ± ± ±

8 Collisional evolution of TNO populations 1261 parameters, ALICANDEP finds a probability range of per cent of having at least one such collision before the onset of the LHB phase. A collisional probability during the LHB phase is estimated around 10 3, and to essentially nil in the last 3.9 Gyr, in agreement with the overall scenario for the formation of the Pluto Charon system. We consider that current observables of the trans-neptunian region are matched fairly well by ALICANDEP within uncertainties. 3.2 The past and present TNO region (i) From the size distributions obtained for the present populations, we estimate that the current mass of the classical belt of TNOs is close to 0.11 M, of which 0.05 M is in bodies larger than the transition size. Including the outer disc, the total mass of the classical belt of the trans-neptunian region is M,as M are predicted to be in the outer disc. ALICANDEP finds a minimum initial mass in the TNO region not less than 51 M, which is the mass necessary to get the current observables starting from an originally depleted population of bodies smaller than 100 km, an LS09 scaling law for fragmentation and considering the onset of the LHB phase after 600 Myr (ALICANDEP- 11 in Table 3). Nevertheless, this case fails to reproduce the observed number of dwarf planets in the MCB (approximately one, instead of two, predicted). Under the collisional physics described by LS09 scaling law, an LHB period starting at 700 Myr and standard break transition at 100 km (ALICANDEP-2 in Table 3), 96 M, are needed instead, which can be considered borderline among acceptable large values for initial mass. Our favourite cases are the ones found by ALICANDEP-1 (the nominal case), with 76 M, ALICANDEP-7 and ALICANDEP-12, both with 60 M. The latter two cases also happen to match the shallow distribution between 30 and 100 km. Unless the presence of the plateau below 100 km size is confirmed, the estimation of the original mass cannot be better constrained. (ii) Fig. 3 shows the evolution of mass at different phases of the collisional and dynamical evolution. This allows us to see that before the LHB phase, the collisional evolution was responsible for the loss of about two-thirds of the initial mass, with a negligible contribution by dynamical effects. At the onset of the LHB phase, in which the inclinations and eccentricities of objects were rapidly pumped up, an increase of collisional activity happens due to the increase of relative velocities. At the same time, dynamical depletion takes on, quickly damping collisional evolution and getting rid of 99 per cent of the mass still present in the disc, in 100 Myr. The situation at the end of the LHB phase is almost stationary up to present times and only modest dynamical mass loss takes place mostly through 3:2 (and 2:1) mean motion resonances with Neptune. Table 3. Dependence on boundary conditions. Zone 1 has to be compared with the ICB+P populations, while zone 2 is compared to the MCB. SL stands for scaling law. M 0 (M ) andm f (M ) are the initial and final mass (respectively) in each zone, in earth masses. D b is the break size estimated by ALICANDEP. q 1, q 2 are the average values of the slopes of the size distributions for D<D b and D>D b, respectively. Note that q 2 in zone 2 is the result of the mixture of both cold and hot populations within the MCB. Uncertainties for q 1 are typically about 0.01, and around 0.1 for q 2. Note also that q 1 stands for the slope of the distribution below 30 km in the cases with q 1 < 3.0. q 12 is the average value of the slopes of the size distribution in the km range and their uncertainties of the order of Case SL q 1 q 2 D b (km) M 0 (M ) t LHB q 1 q 2 q 12 D b (km) M f (M ) ALICANDEP-1 Zone 1 BA Zone 2 BA ALICANDEP-2 Zone 1 LS Zone 2 LS ALICANDEP-3 Zone 1 BA Zone 2 BA ALICANDEP-4 Zone 1 BA Zone 2 BA ALICANDEP-5 Zone 1 BA Zone 2 BA ALICANDEP-6 Zone 1 BA Zone 2 BA ALICANDEP-7 Zone 1 BA Zone 2 BA ALICANDEP-8 Zone 1 LS Zone 2 LS ALICANDEP-9 Zone 1 BA Zone 2 BA ALICANDEP-10 Zone 1 LS Zone 2 LS ALICANDEP-11 Zone 1 BA Zone 2 BA ALICANDEP-12 Zone 1 LS Zone 2 LS

9 1262 A. Campo Bagatin and P. G. Benavidez Figure 3. Evolution of total mass and the mass eliminated by collisions and dynamical effects. It can be noted that collisional evolution dominates mass loss until the onset of the LHB phase, in which dynamical effects quickly take over. (iii) According to our simulations, the current transition size (break) for the overall TNO populations is at km. A slightly smaller estimation results for the main belt, ranging km, varying between different zones. Estimations are given as intervals; in fact, due to collisional evolution, the transition from the high slope values for large bodies to the equilibrium slope for small ones is smooth and spreads over a few tens of km. This structural feature together with observational biases at high absolute magnitudes may be the reason for the discrepancies between different observational estimations of the transition size. We explored the effect of variations in the initial break size (from 50 to 150 km) observing that match to current observables is still possible. Nevertheless, in some case it is necessary to assume unlikely high initial masses. In the case of an initial break in size distribution at 50 km, the final break size is shifted towards 100 km due to collisional erosion in the early phases of the evolution. Table 3 reports the results for the predicted transition sizes and the slopes of the size distributions for the different sets of assumed boundary conditions. (iv) In order to match the current slopes of the size distributions suggested by the CFEPS for large bodies (larger than the break size), the initial slopes should have been close to the current ones (see Table 3). It is interesting to note that initial slopes with q 1 < 3 also match current observables in the large TNO populations (small bodies are not able to alter the size distribution and number of their larger siblings). Fig. 4 shows the time evolution of the number of objects in the ICB and MCB in the nominal case. It can be noted how the MCB initially had a steep size distribution (q 2 = 7), for bodies larger than the break size, until the beginning of the LHB phase. Immediately after that period apart from the mass depletion due to dynamical effects it can be noted that the corresponding slope of the size distribution has changed to q 2 5. This is not due to collisional evolution; the massive migration of bodies from zone 1 (ICB) that have initial size distributions with q 2 = 5 is instead responsible for that. In fact, the two populations migrating ICB Figure 4. Size distribution of (a) ICB and (b) MCB, at different phases of evolution. and indigenous MCB bodies are now mixed and show a common size distribution as a whole. The CFEPS was able to discriminate cold and hot populations in the MCB, and they show different slopes as they have different origins. (v) On the contrary, differences are found in the evolved populations of objects smaller than the break size, depending on their initial distribution. Analysing the size distributions obtained in the case of shallower power laws (q 1 from 0.5 to 2) for the initial populations of small bodies (below 100 km), it is clear that a plateau forms in the range km, followed by other typical equilibrium size distribution (Fig. 5). An average slope 2 is calculated in that range, in agreement with observational results by Fuentes & Holman (2008) and Fraser & Kavelaars (2009). This is a different outcome with respect to what is typically found when the initial distributions at small sizes are steeper than q 1 = 2. In those cases, the stationary size distribution takes on right at the