Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque S/N (1900), La Plata, Argentina IALP-CONICET, Argentina

Abstract
Aims. In this paper, we analyze the collisional and dynamical evolution of the population of Plutinos.
Methods. To do this, we test 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. This procedure allows us to estimate the size distribution of Plutinos, to study their mean collisional lifetimes, to analyze the formation of families, to obtain ejection rates of fragments from the resonance and to discuss their possible contribution to the ecliptic comet population. Our simulations are developed assuming the existence of one Pluto-sized object in the 3:2 Neptune resonance.
Results. The Plutino population larger than a few kilometers in diameter is not significantly altered by catastrophic collisions over the age of the Solar System. Thus, we infer that the break suggested by previous works at a diameter D near 40-80 km in the Plutino cumulative size distribution should be primordial and not a result of the collisional evolution. The existence of such a break is still a matter of debate. On the other hand, our analysis indicates that one large family was formed in the 3:2 Neptune resonance over the Solar System history. Concerning Plutino removal, we find that one object with a diameter D > 1 km is ejected from the 3:2 resonance with Neptune every $\sim$ 300-1200 yr. Then, we study the sensitivity of our results to the number of Pluto-sized objects in the 3:2 Neptune resonance. Our simulations suggest that the larger the number of Pluto-sized bodies, the higher the ejection rate of fragments from that resonant region and the number of families formed over 4.5 Gyr. Thus, if a maximum of 5 Pluto-sized objects are assumed to be in the 3:2 Neptune resonance, one body with a diameter D > 1 km is ejected every tens of years while $\sim$ 30 large families are formed over the Solar System history. From these estimates, we conclude that it is necessary to specify the number of Pluto-sized objects present in the 3:2 Neptune resonance to determine if this region can be considered an important source of ecliptic comets. Finally, we find that the current orbital distribution of the Plutinos does not offer a strong constraint on the dynamical origin of this population.

The 3:2 mean motion resonance with Neptune, located at $\sim$ 39.5 AU, is the most densely populated one in the Kuiper Belt. The residents of this resonant region are usually called Plutinos because of the analogy of their orbits with that of Pluto, which is its most representative member. Aside from Pluto and its largest moon Charon, the Minor Planet Center (MPC) database contains 207 Plutino candidates.

Although the Scattered Disk is assumed to be the primary source of Centaurs and Jupiter-family comets (Duncan & Levison 1997; Di Sisto & Brunini 2007), several authors have also suggested the existence of some connection between the Plutinos and comets. Duncan et al. (1995) analyzed the dynamical structure of the trans-Neptunian region and found that in the 3:2 mean motion resonance with Neptune there are currently unstable orbits that may be related to the origin of the observed Jupiter-family comets. Motivated by these results, Morbidelli (1997) studied the dynamics of the 3:2 Neptune resonance at small inclinations and showed the existence of a slow chaotic diffusion region at moderate amplitudes of libration, which should be an active source of comets at present. On the other hand, Morbidelli (1997) estimated that about 4.5 $\times$ 10⁸ comet-sized objects should presently be trapped in the 3:2 resonance to reproduce the observed flux of Jupiter-family comets in the inner Solar System. This number seems to indicate that the Plutinos should represent a collisionally evolved population.

Melita & Brunini (2000) showed that the 3:2 resonance presents a very robust stable zone primarily at low inclinations, where most of the observed Plutinos are distributed. Moreover, they suggested that the existence of Plutinos in very unstable regions can be explained by physical collisions or gravitational encounters with other Plutinos. At the same time, Nesvorný & Roig (2000) determined the most important inner resonances within the 3:2 Neptune resonance. Moreover, these authors estimated that the current number of Plutinos larger than 1-3 km in diameter is 6 $\times$ 10⁸.

Later, Dell'Oro et al. (2001) computed the values of the intrinsic collision probability and the mean impact velocity for some minor body populations, including the Plutinos.

Some years later, Bernstein et al. (2004), Elliot et al. (2005) and Petit et al. (2006) carried out several detailed surveys which allowed them to obtain results concerning the mass and size distribution parameters for the different dynamical classes of the trans-Neptunian region. Kenyon et al. (2007) summarized and discussed the main predictions derived from those surveys and analyzed the formation and collisional evolution of the Kuiper Belt objects.

In this paper, we present the first study aimed at describing the collisional evolution of the Plutinos. The main purposes of this work are to estimate the size distribution of Plutinos, to study their collisional lifetimes, to analyze the formation of families, to obtain ejection rates of fragments from the 3:2 Neptune resonance and to discuss a possible connection between such fragments and the population of ecliptic comets. To do this, we use the numerical code developed by de Elía & Brunini (2007a,b), which allows us to test different collisional parameters and to include a dynamical treatment that takes into account the stability and instability regions of the 3:2 mean motion resonance with Neptune.

In Sect. 2 we give a brief description of the main parameters of our collisional model, while the major dynamical features present in the 3:2 Neptune resonance are discussed in Sect. 3. In Sect. 4 we construct the populations of the model from recent observational surveys. In Sect. 5 we describe the full numerical model. Section 6 shows the most important results derived from the collisional and dynamical evolution of the Plutinos. Conclusions are given in the last section.

2 Collisional model

In order to describe the outcome of a collision between two bodies, we use the collisional algorithm developed by de Elía & Brunini (2007a,b) which is based on the method performed by Petit & Farinella (1993) with the corrections made by O'Brien & Greenberg (2005). Such an algorithm considers catastrophic collisions and cratering events as well as the escape and reaccumulation of the fragments resulting from those impacts.

The impact velocity V and the shattering impact specific energy $Q_{\rm S}$ determine, for a given body, if the collision must be studied in the catastrophic regime or in the cratering regime.

Finally, the intrinsic collision probability $\langle Pi_{\rm c} \rangle$ describes how frequently collisions occur, allowing us to study the evolution in time of the population.

2.1 Collision velocities and probabilities

In this work, we adopt constant values of the intrinsic collision probability $\langle Pi_{\rm c} \rangle$ and the mean impact velocity $\langle V \rangle$ for Plutinos derived by Dell'Oro et al.'s (2001) algorithm, that takes into account the resonant behavior of bodies. Based on a sample of 46 Plutinos and taking into account the libration of the critical argument $\sigma = 2 \lambda_{N} - 3 \lambda - \tilde{\omega}$ around 180 $^{\circ}$ , Dell'Oro et al. (2001) computed values of $\langle Pi_{\rm c} \rangle$ and $\langle V \rangle$ of 4.44 $\pm$ 0.04 $\times$ 10^-22 km^-2 yr^-1 and 1.44 $\pm$ 0.71 km s^-1, respectively.

2.2 Impact strength

$\begin{figure} \par\includegraphics[width=8cm,clip]{9865fig1.eps}\end{figure}$	Figure 1: Strength impact. The dashed line represents the $Q_{\rm S}$ law used in our simulations while the $Q_{\rm D}$ law from Benz & Asphaug (1999) for icy bodies at 3 km s^-1 is plotted as a solid line.
Open with DEXTER

O'Brien & Greenberg (2005) showed that the general shape of the final evolved asteroid population is determined primarily by $Q_{\rm D}$ , but variations in $Q_{\rm S}$ and $f_{\rm ke}$ can affect such a final population even if $Q_{\rm D}$ is held the same. According to these arguments, we decide to choose a combination of the parameters $Q_{\rm S}$ and $f_{\rm ke}$ that yield the $Q_{\rm D}$ law from Benz & Asphaug (1999) for icy bodies at 3 km s^-1.

In a recent paper concerning the L₄ Trojan asteroids, de Elía & Brunini (2007b) analyzed the dependence of their numerical simulations on the shattering impact specific energy $Q_{\rm S}$ . This work indicates that the smallest gaps between $Q_{\rm S}$ and $Q_{\rm D}$ curves lead to the smallest wave amplitudes in the size distribution of the final evolved population as well as to the highest ejection rates of fragments. Moreover, that study also allows us to infer that the formation of families is more effective for the simulations with a small gap between $Q_{\rm S}$ and $Q_{\rm D}$ laws. Following these arguments, in this work we decide to use an only $Q_{\rm S}$ law with a small gap with respect to the $Q_{\rm D}$ law from Benz & Asphaug (1999) for icy bodies at 3 km s^-1. This procedure allows us to obtain upper values of the ejection rate of fragments from the 3:2 Neptune resonance and to maximize the number of families that may be found in this resonant region.

The $Q_{\rm S}$ law used in our simulations is shown in Fig. 1 as a dashed line and can be represented by an expression of the form

$\begin{displaymath}% Q_{\rm S} = C_{1} D^{-\lambda_{1}}(1+(C_{2} D)^{\lambda_{2}}), \end{displaymath}$

(1)

Once the $Q_{\rm S}$ law is specified, we adjust the ineslaticity parameter $f_{\rm ke}$ to get the Benz & Asphaug (1999) $Q_{\rm D}$ law. Many authors have suggested that $f_{\rm ke}$ may vary with size (Davis et al. 1995; O'Brien & Greenberg 2005), with impact speed and probably with the material properties. Thus, according to O'Brien & Greenberg (2005), we express the parameter $f_{\rm ke}$ as

$\begin{displaymath}% f_{\rm ke} = f_{{\rm ke}_{0}} \left(\frac{D}{1000~{\rm km}}\right)^{\gamma}, \end{displaymath}$

(2)

3 Dynamical model

In order to study the orbital space occupied by the Plutino population, we develop a numerical integration of 197 Plutino candidates extracted from the Minor Planet Center database with semimajor axes between 39 and 40 AU. These objects are assumed to be massless particles subject to the gravitational field of the Sun (including the masses of the terrestrial planets) and the perturbations of the four giant planets. The simulation is performed with the simplectic code EVORB from Fernández et al. (2002) using a timestep of 0.25 years. The evolution of the test particles is followed for 10⁷ years which is a timescale greater than any secular period found in this resonance (Morbidelli 1997). From this evolution we construct 3D niches within the boundaries of the 3:2 resonance with widths of 0.02 AU, 0.0125 and 1.125 $^{\circ}$ in semimajor axis a, eccentricity e and inclination i, respectively. In each of these niches, we calculate the fraction of time each niche is occupied normalized to the total time that all the Plutinos spend within the boundaries of the resonance. This process allows us to build a map of the distribution of Plutinos in the orbital element planes (a, e) and (a, i). These maps are shown in Figs. 2a and b. Such plots indicate the regions where the Plutinos spend their time. We assume that the color zones in those maps are regions with different degrees of probability where a Plutino can be found, and they will be called ``stability niches'' in this work. On the contrary the empty zones are regions where a Plutino could not survive for a long time (remember that the neighborhoods of the resonance are rather unstable regions). These empty regions will be called ``instability niches'' in this work. They correspond to unstable zones where objects are quickly ejected from the 3:2 resonance, losing the resonance protecting mechanism and encountering Neptune on a timescale of less than some 10⁷ years (Morbidelli 1997; Nesvorný & Roig 2000).

Section 5 describes how the stability and instability niches shown in Fig. 2 are included in our numerical algorithm in order to model the dynamical treatment of the code. We discuss in Sect. 6.5 the sensitivity of our results to the way those niches are constructed as well as the dependence of our simulations on the initial orbital element distribution of the population.

4 Populations of the model

Recently, Kenyon et al. (2007) used the Minor Planet Center database and the main results from Bernstein et al. (2004), Elliot et al. (2005) and Petit et al. (2006) and found clear evidence of physical differences among the dynamical classes of the trans-Neptunian region. Particularly, their analysis indicates that the cumulative size distribution of the resonant population shows a break at a diameter D near 40-80 km, which is in agreement with Bernstein et al. (2004). Moreover, for larger resonant objects, the population seems to have a shallow size distribution with a cumulative power-law index of $\sim$ 3. On the other hand, Kenyon et al. (2007) suggested that the resonant population has $\sim$ 0.01-0.05 $M_{\oplus}$ in objects with $D \ga$ 20-40 km. Petit et al. (2006) proposed that there is no evidence of a break in the Kuiper Belt population, indicating that a single power-law luminosity function fits their data.

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{9865fig2.eps}\end{figure}$	Figure 2: Normalized distribution of Plutinos obtained from the dynamical evolution of 197 Plutino candidates extracted from the Minor Planet Center database with semimajor axes between 39 and 40 AU in the planes (a, e) a) and (a, i) b). The color scale depicts the expected density of Plutinos. Blue colors indicate where the Plutinos are statistically most likely to spend their time.
Open with DEXTER

Considering for large objects ( $D \ga$ 40-80 km) in the resonant population a cumulative power-law index of $\sim$ 3 for the size distribution and a lower limit of 0.01 $M_{\oplus}$ for the mass in objects with $D \ga$ 20-40 km, we obtain one Pluto-sized object in the resonant population. Assuming that this object is in the 3:2 mean motion resonant we have one Pluto-sized object, Pluto itself. But considering the upper limit of 0.05 $M_{\oplus}$ for the mass in resonant objects with $D \ga$ 20-40 km, we infer the existence of 5 Pluto-sized objects in the whole resonant population. If these 5 Pluto-sized objects were all in the 3:2 mean motion resonance we would have an upper limit for the large objects in this resonance. This is (though optimistic) a real possibility from the studies mentioned above, and even though only Pluto has been observed up to now, more Pluto-sized objects in the 3:2 resonance may remain to be discovered. Brown (2008), based on the completeness of the current surveys argues that two or three more large KBOs are likely awaiting discovery. We will consider the limit of 5 Pluto-sized objects in the 3:2 resonance only as an upper limit for the calculations and results addressing this subject in Sect. 6.4. In the following we assume the existence of one Pluto-sized object in the 3:2 resonance.

In order to take into account the possible existence of a break in the resonant population, the initial size distribution for $D \ga$ 60 km is assumed to follow a cumulative power-law index with a value of 3, while for $D \la$ 60 km, we assign a cumulative power-law index p in order to reproduce a given initial mass. From this, the general form of the cumulative initial population used in our model to study the collisional and dynamical evolution of the Plutinos can be written as follows

		$\displaystyle N({>}D) = C \left(\frac{1~{\rm km}}{D}\right)^{p} \qquad\qquad {\rm for} \quad D \leq 60~{\rm km},$
		$\displaystyle N({>}D) = 7.9 \times 10^{9} \left(\frac{1~{\rm km}}{D}\right)^{3} \quad {\rm for} \quad D > 60~{\rm km},$	(3)

In Sect. 6 we discuss the dependence of our simulations on the initial population. Moreover, in Sect. 6.4 we study the sensitivity of our results to the number of Pluto-sized objects in the 3:2 resonance with Neptune.

5 The full model

In order to simulate the collisional and dynamical evolution of the Plutinos, our numerical code evolves in time the number of bodies residing in a set of 135 discrete logarithmic size bins, whose central values range from D₁ = 10^-10 km to D₁₃₅ = 2816.1 km in diameter, in such a way that from one bin to the next, the mass of the bodies changes by a factor of 2 and the diameter changes by a factor of 2^1/3, adopting a density of 1 g cm^-3.

In each timestep, a characteristic orbit is generated at random for each collision between Plutinos of diameters D₁ and D₂ in the 3:2 resonance. The generated orbits are not used to compute new values of the mean impact velocity $\langle V \rangle$ and the intrinsic collision probability $\langle Pi \rangle$ (see Sect. 2.1). The main goals of this treatment are to analyze the final fate of the fragments resulting from impacts and to obtain more reliable estimates of the collisional ejection rates of Plutinos from the 3:2 Neptune resonance.

From Figs. 2a and b, we have the normalized distribution of Plutinos in each niche previously defined in Sect. 3, what is equivalent to a probability distribution function f(a,e,i). Then, we assign to our initial fictitious Plutinos orbital elements a, e, i, following the probability distribution function f(a,e,i) by von Neumann's acceptance-rejection method (Knuth 1981). This technique indicates that if a set of numbers a^*, e^* and i^* is selected randomly from a uniform distribution over the domain of the function f (namely, a^*, e^* and i^* between 39 and 40 AU, 0 and 0.5 and 0 and 45 $^{\circ}$ , respectively), and another set of numbers f^* is given uniformly at random from the range of such a function (namely, f^* between 0 and 0.0012), the condition $f^{*} \leq f(a^{*},e^{*},i^{*})$ will generate a distribution for (a^*,e^*,i^*) whose density is $f(a^{*},e^{*},i^{*}){\rm d}a^{*}{\rm d}e^{*}{\rm d}i^{*}$ . Such (a^*,e^*,i^*) values will be accepted as possible initial conditions for the semimajor axis, eccentricity and inclination of the Plutinos, in agreement with the observational data and the numerical integration developed in Sect. 3. In mean motion resonances, the evolution of a,e and i is coupled. However, here, we are treating them as uncorrelated variables. Nevertheless, a more rigorous treatment would be very difficult, and we believe that the results would be not too different to the ones found here. Finally, since we do not compute Neptune's orbit in our simulations, it is not necessary to include constraints on the resonant argument of the Plutinos. Thus, given uniformly at random the longitude of ascending node $\Omega$ , the argument of pericentre $\omega$ and the mean anomaly M between 0 and 360 $^{\circ}$ , an orbit can be assigned and from this, a position-velocity pair can be derived for each of the colliding Plutinos.

Once a typical orbit has been computed for each body participating in a given collision, the next step is to carry out the collisional treatment (including the analysis of the reaccumulation process) from the algorithm developed by de Elía & Brunini (2007a,b). In order to determine the final fate of the fragments escaping from the gravitational field of the two colliding bodies, it is necessary to calculate their orbital elements once they are ejected with a relative velocity with respect to the parent body. Immediately before the collision, the barycentric position and velocity of the fragments are assumed to be those associated with their parent body. After the collision, we consider that the barycentric position of the fragments does not change while the relative velocities with respect to their parent body are assumed to be equally partitioned between the three components. Once the barycentric position and velocity of the fragments after the collision have been obtained, it is possible to calculate their orbital elements and their final fates. To define if the fragments remain or are ejected from the resonance we use the following criterion:

6 Results

$\begin{figure} \par\includegraphics[width=7.7cm,clip]{9865fig4.eps}\end{figure}$	Figure 4: Our estimates of the Plutino cumulative size distribution after 4.5 Gyr of evolution obtained from the different initial populations proposed in Sect. 4.
Open with DEXTER

$\begin{figure} \par\includegraphics[width=7.9cm,clip]{9865fig5.eps}\end{figure}$	Figure 5: Mean collisional lifetimes of Plutinos obtained using the different initial populations defined in Sect. 4. The horizontal short-dashed line represents the age of the Solar System.
Open with DEXTER

6.1 Plutino cumulative size distribution

Figure 4a shows our estimate of the Plutino cumulative size distribution after 4.5 Gyr of evolution obtained from the Initial Population 1 (see Sect. 4) which follows a cumulative power-law index equal to 3 at all sizes. From this, we find that it is not possible to reproduce the break suggested by Bernstein et al. (2004) and Kenyon et al. (2007) at $D \sim$ 40-80 km using a single-slope power law to describe the starting population. On the contrary, the final evolved population just shows a break at a diameter D near 2 km while the larger objects are not significantly altered by collisions over the age of the Solar System. In fact, the solid line in Fig. 5 represents our results concerning the mean collisional lifetimes of Plutinos obtained from the Initial Population 1. This indicates that Plutinos with diameters greater than $\sim$ 2 km have mean lifetimes longer than the age of the Solar System, which implies that such objects have likely survived unaltered by catastrophic impacts over the Solar System history. The confirmation of the existence of a break at $D \sim$ 40-80 km in the Plutino cumulative size distribution should rule out a single-slope power law distribution as a possible initial population.

On the other hand, Fig. 4b shows our estimate of the Plutino cumulative size distribution after 4.5 Gyr, starting from the Initial Population 2 (see Sect. 4) which presents a break at a diameter D near 60 km and follows a cumulative power-law index of 2.7 for smaller sizes. Our results indicate that the final evolved population produces a break at $D \sim$ 2 km while the larger bodies do not suffer a significant collisional evolution over the Solar System age, representing a primordial population. In fact, the long-dashed line in Fig. 5 indicates that the mean collisional lifetimes of Plutinos larger than $\sim$ 2 km in diameter are longer than the age of the Solar System. In the same way, Fig. 4c presents our results concerning the Plutino cumulative size distribution after 4.5 Gyr, using the Initial Population 3 (see Sect. 4), which has a break at $D \sim$ 60 km and follows a cumulative power-law index of 2.4 for smaller sizes. In this case, the final evolved population shows a slight break at a diameter D near 1 km. Moreover, the short-dashed line in Fig. 5 allows us to infer that Plutinos with diameters greater than 1 km are primordial since their mean collisional lifetimes are longer than the age of the Solar System.

Our simulations indicate that the existence of a break at $D \sim$ 40-80 km in the Plutino cumulative size distribution should be a primordial feature of the population and not the result of the collisional evolution. This strongly agrees with the results derived by Charnoz & Morbidelli (2007) who argued that the knee at 50 km in the distribution of the Kuiper Belt objects should be primordial.

6.2 Plutino families

The existence of families in a given region represents a natural result of the collisional activity. We study the existence of families in the 3:2 resonance with Neptune formed from the breakup of parent bodies with diameters greater than 100 km that disperse fragments in such a way that the diameter of the largest one is larger than 30-40 km.

Our simulations indicate that the formation of large Plutino families does not depend strongly on the starting population. In fact, from any of the three initial populations proposed in Sect. 4, one large family is formed in the 3:2 resonance with Neptune over the age of the Solar System.

6.3 Ejection rates

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{9865fig6.eps}\end{figure}$	Figure 6: Ejection rate of fragments from the 3:2 mean motion resonance with Neptune with diameters greater than 1 km per Myr obtained from the different initial populations defined in Sect. 4.
Open with DEXTER

$\begin{figure} \par\includegraphics[width=8cm,clip]{9865fig7.eps}\end{figure}$	Figure 7: Representative values of semimajor axis, eccentricity and inclination for the fragments ejected from the 3:2 mean motion resonance with Neptune.
Open with DEXTER

Figure 6 shows the number of bodies ejected from the 3:2 Neptune resonance with diameters greater than 1 km per Myr as a function of time over the age of the Solar System, obtained from the different initial populations defined in Sect. 4. In order to study the contribution of the Plutinos to the current population of ecliptic comets, we estimate a mean ejection rate of fragments from the 3:2 resonance for each of our simulations over the last 500 Myr of evolution, where the number of bodies removed per time unit is more or less constant and the data sample is statistically significant. Our results indicate that the maximum ejection rates starting with the Initial Populations 1-3 (see Sect. 4) are of $\sim$ 3.5 $\times$ 10³, 1.7 $\times$ 10³ and 8.6 $\times$ 10² objects larger than 1 km in diameter per Myr from the 3:2 resonance, respectively. Thus, we would expect a maximum of 1 ecliptic comet with a diameter D > 1 km every $\sim$ 300-1200 yr from the 3:2 resonance with Neptune, while the estimates of Di Sisto & Brunini (2007) suggest the injection of 4 Centaurs with a radius R > 1 km each year from the Scattered Disk.

In Sect. 5 we discussed several criteria to determine the final fate of the Plutino fragments and from this to calculate the ejection rates from the 3:2 mean motion resonance with Neptune. Our simulations indicate that the ejection of fragments from the 3:2 resonance is due to variations in the a, e and i values, which are associated with some instability niches shown in Fig. 2 or exceed the boundaries assumed for the resonant region (see Sect. 3). In these cases the fragments have e< 1 (see Sect. 5). From all our numerical experiments, we find that more than 99 percent of the fragments escaping from the 3:2 resonance do it through these two mechanisms, so they have eccentricities e < 1, which rules out parabolic or hyperbolic collisional ejection as a Plutino removal source. Figure 7 shows a representative sample of the distribution of Plutinos ejected from the 3:2 resonance with respect to semimajor axis, eccentricity and inclination.

On the other hand, for all cases, most of the bodies ejected from the 3:2 mean motion resonance with Neptune with diameters greater than 1 km have diameters ranging from 1 to 5 km. In fact, our results indicate that the ejection rate of fragments with diameters D > 5 km results to be two orders of magnitude smaller than that associated to fragments with diameters D > 1 km. Moreover, it is interesting to note that the objects ejected from the resonance are fragments resulting from impacts between parent bodies with diameters ranging from 10 to 200 km.

6.4 Dependence of results on number of Pluto-sized objects

The results discussed so far have been derived assuming the existence of one Pluto-sized object in the 3:2 Neptune resonance. As discussed in Sect. 4, if we consider the upper limit of 0.05 $M_{\oplus}$ suggested by Kenyon et al. (2007) for the mass of the resonant population in objects with $D \ga$ 20-40 km, 5 Pluto-sized objects should be found in the whole resonant population according to the size distribution parameters derived by Bernstein et al. (2004) and Kenyon et al. (2007) for such population.

To test the dependence of our results on the number of Pluto-sized objects, we carry out several numerical experiments considering 5 Pluto-sized objects in the 3:2 Neptune resonance. This number is probably an overestimation (see Sect. 4 for details), but nevertheless our purpose is only to illustrate the effect of the total mass in the population on the Plutinos' collisional history.

Our results suggest that the larger the number of Pluto-sized objects in the 3:2 Neptune resonance, the higher the ejection rate of fragments from that resonant region and the number of families formed over 4.5 Gyr. In fact, if 5 Pluto-sized objects are assumed to be in the 3:2 Neptune resonance, one body with a diameter D > 1 km is ejected every tens of years while $\sim$ 30 large families are formed over the Solar System history. According to the estimates of Di Sisto & Brunini (2007), we conclude that it is necessary to specify the number of Pluto-sized objects present in the 3:2 Neptune resonance to determine if this resonant region can be considered an important source of ecliptic comets.

6.5 Robustness of results

The results shown in this paper have been obtained using the stability and instability niches defined in Sect. 3, which present widths of 0.02 AU, 0.0125 and 1.125 $^{\circ}$ in semimajor axis a, eccentricity e and inclination i, respectively. In order to test the dependence of our results on the size of those niches, we carry out several numerical experiments increasing the widths of such regions in a, e and i, which leads to a magnification of the stability region. In general terms, the larger the area of niches, the smaller the ejection rate of fragments from the 3:2 mean motion resonance with Neptune. In this work, we select small size niches in order to minimize the influence of the lowest density regions shown in Fig. 2. On the other hand, we find that the results concerning the size distribution of Plutinos, their collisional lifetimes and the formation of families are not sensitive to the size of the stability and instability regions constructed to developed our dynamical treatment.

In the same way, we also perform some numerical simulations in order to explore the sensitivity of our results to the initial mass of the population. In fact, the studies developed by Stern (1996), Stern & Colwell (1997a) and Stern & Colwell (1997b) suggest that the mass of the primordial Kuiper Belt between 30 and 50 AU was probably of the order of 10 to 50 $M_{\oplus}$ , which is $\sim$ 100 times its current value. Moreover, these authors indicate that once Neptune reached a fraction of its final mass, the disk environment became highly erosive for objects with radii smaller than $\sim$ 20-30 km, leading to a rapid mass loss on a timescale of $\sim$ 100 Myr. To take into account these results and following O'Brien & Greenberg (2005), we carry out several numerical experiments including a brief period of primordial evolution at the beginning of the simulation. To do this, each of the initial populations defined in Sect. 4 is multiplied by a factor of 100 and then its evolution followed for 100 Myr. Once this phase of primordial evolution ends, the corresponding residual population is reduced by the same factor and finally its evolution is analyzed for the rest of the 4.5 Gyr. Our results suggest that the size distribution of the Plutinos, their collisional lifetimes, the ejection rate of fragments from the 3:2 mean motion resonance with Neptune and the formation of families do not show relevant changes when this period of primordial evolution is included in the model.

$\begin{figure} \par\includegraphics[width=9cm,clip]{9865fig8.eps}\end{figure}$	Figure 8: Distribution of $D \geq$ 1 km Plutino fragments with respect to semimajor axis a, eccentricity e and inclination i. Time evolution from an initial cold population.
Open with DEXTER

The results presented in this work have been derived generating initial values of semimajor axis a, eccentricity e and inclination i from the dynamical evolution of the 197 test Plutinos shown in Fig. 2. In order to test the dependence of our results on the initial orbital distribution, we develop several simulations starting with a dynamically cold population, with eccentricities and inclinations below 0.05 and 10 $^{\circ}$ , respectively. Such e and i limit values are chosen arbitrarily. Our outcomes show that $D \geq$ 1 km Plutino fragments require timescales of the order of 100 Myr to reach the current dynamical configuration (see Fig. 8), while the smaller fragments occupy the stability niches very quickly, in only some thousands of years. In addition, we find that the results concerning the size distribution of Plutinos, their collisional lifetimes, the ejection rate of fragments from the 3:2 mean motion resonance with Neptune and the formation of families do not show a strong dependence on the initial distribution of orbital elements. From this analysis, we infer that it is possible to reach the current orbital configuration of the Plutinos starting with a dynamically very different population compared to that presently observed. Thus, we suggest that the current orbital distribution of the Plutinos does not offer a strong constraint on the dynamical origin of this population.

7 Conclusions

We have presented the first study aimed at analyzing the collisional and dynamical evolution of the Plutinos. Assuming the existence of one Pluto-sized object in the 3:2 Neptune resonance, our main results are the following:

Collisional and dynamical evolution of Plutinos

1 Introduction