A&A 487, 363-367 (2008)
DOI: 10.1051/0004-6361:20079236

Collisional activation of asteroids in cometary orbits[*]

C. G. Díaz2 - R. Gil-Hutton1,2


1 - Complejo Astronómico El Leoncito - CONICET - Av. España 1512 sur, J5402DSP San Juan, Argentina
2 - Universidad Nacional de San Juan - Av. J. I. de la Roza 590 oeste, 5400 Rivadavia, San Juan, Argentina

Received 12 December 2007 / Accepted 27 April 2008

Abstract
Aims. We study the time an asteroid in a cometary orbit must wait to receive a collision producing a crater depth enough to expose subsurface volatiles, aiming to analyze the possibility of collisional reactivation of these objects if they are dormant comets.
Methods. We perform a numerical integration of the asteroids in cometary orbits and a population of projectiles to find the mean intrinsic collision probabilities and mean impact velocities of the targets. The projectile population was obtained as a sample with the same distribution of orbital elements as observed for main belt asteroids, and we also take into account that its size distribution changes for different size ranges. Only 206 asteroids in cometary orbits, that are not members of other asteroid groups, with a Tisserand parameter $2\le T_{\rm J} \le 2.9$ and perihelion distance q>1.3 AU were considered.
Results. A large fraction of the objects in the sample receive at least 1 collision energetic enough to break the comet crust and allow a dormant comet to reach an active state in a period shorter than a Jupiter Family Comet dynamical lifetime. A large fraction of the objects in the sample with $r_{\rm t}\ge 8$-9 km receive several collisions and could be active for more than $3\times 10^4$ yr. We found an excess in the number of dormant comet candidates from the expected values which is indicative of the presence in the ACOs population of objects that are not comets in a dormant state. These objects could be asteroids with $T_{\rm J}<3$ that reach their present orbits by some dynamical mechanism that perturbs the original asteroidal orbit changing its Tisserand invariant.

Key words: minor planets, asteroids - comets: general - solar system: general

1 Introduction

The different formation regions of asteroids and comets in the Solar System produced differences in their physical properties. Asteroids are planetesimals formed during the early ages of the Solar System in the region between Mars and Jupiter while comets were formed in a region that extends from the giant planets to the outer limits of the pre-solar nebula. As a consequence, there is a significant difference in the content of volatile material in both populations. This fact has provided the most apparent distinction between members of these two populations: comet nuclei, when close to the Sun, are usually surrounded by a coma produced by the out-gassing of volatiles caused by solar heating, while asteroids are not. This simple distinction has some complications due to the discovery of icy objects that rarely develop a coma due to their distance from the Sun, the discovery of asteroids with dynamical properties similar to those of comets, and the discovery of objects in typical asteroidal orbits that show temporary comet-like activity.

At the end of their active life some comets might develop an asteroidal appearance when sublimation stops and reaches dormancy or extinction due to the depletion of volatile material or by a crust built up on their surfaces (Rickman et al. 1990; Benkhoff & Huebner 1996; Kührt & Keller 1994; Jewitt 2002). The gas activity of a cometary nucleus coming close to the Sun can form a layer of dust grains that are too heavy to be blown off by the gas outflow. This crust could eventually become so thick that subsurface volatiles cannot be warmed up to sublimation temperature, ceasing any cometary activity, and the nucleus appears observationally identical to an asteroid. Since this crust does not completely stop the vapor production in the comet interior, it could be possible that a large amount of ice is present below it even though the comet reaches an inactive state (Prialnik & Mekler 1991).

One way to reactivate dormant comets is by means of impacts with interplanetary bodies. Fernández (1990) and Matese & Whitman (1994) tried to explain the outburst activity of comets by impacts with small asteroids or meteoroids, and some authors have the opinion that this is the case for comets 41P/Tuttle-Giacobini-Kresák (Fernández 1981; Kresák 1974), 72P/Denning-Fujikawara (Beech 2001), and 133P/Elst-Pizarro (Toth 2000), while Toth (2001) suggested that the splitting of the comet C/1994 S4 (LINEAR) was a result of a collision of the comet with asteroid debris. Collisions of interplanetary boulders with a dormant comet could result in craters that partially destroy the crust on its surface, triggering reactivation by allowing the fresh material buried below the crust to begin sublimation in the next perihelion passage.

The dynamical criterion used to define the sample of objects that are candidate dormant comets is related to the Tisserand parameter (Kresák 1979), which is defined by the relation $ T_{\rm J}=a_{\rm J}/a+2\cos I \sqrt{(a/a_{\rm J})(1-e^2)}$, where a and $a_{\rm J}$ are the semimajor axis of the orbits of the object and Jupiter, respectively, while e and I are the eccentricity and inclination relative to the orbital plane of Jupiter of the object's orbit. By this criterion, cometary orbits are defined as those having $T_{\rm J}<3$, while asteroidal orbits are those with $T_{\rm J}>3$. Therefore, all the objects with $T_{\rm J}<3$ that do not present any signature of cometary activity are defined as an asteroid in a cometary orbit (ACO). Objects with $T_{\rm J}<2$ have been called Damocloids by Jewitt (2005) and are asteroids in Halley-type cometary orbits, while those with $2\le T_{\rm J}\le 3$ have orbits similar to the Jupiter family comets (JFCs). Therefore, ACOs are good candidates to be extinct or dormant comets.

Licandro et al. (2005) found differences in the spectroscopic properties of two sub-samples of ACOs, the near Earth objects (NEOs) with perihelion distance $q\le 1.3$ AU and the non-NEOs, these last objects being spectroscopically similar to cometary nuclei. These authors also found that ACOs with featured spectra typical of the main belt have $T_{\rm J}\ge2.9$ while those with $T_{\rm J}<2.9$ shown comet-like spectra, suggesting that the sub-sample of ACOs with $2.9\le T_{\rm J}\le 3.0$ could be contaminated by a large fraction of interlopers from the inner part of the belt. On the other hand, Alvarez-Candal & Licandro (2006) found that the sub-sample of ACOs with q>1.3 AU has a size distribution similar to that of the Jupiter family comets and can be composed of a significant fraction of dormant comets, while a large fraction of ACOs with q<1.3 AU could be scattered objects from the outer main belt.

The purpose of this paper is to analyze the possibility of reactivation of dormant comets by collisions with interplanetary boulders. Since the physical properties of JFCs are better known than for other comets and taking into account the above considerations, we only consider as dormant comet candidates ACOs with $2\le T_{\rm J} \le 2.9$ and q>1.3 AU. In Sect. 2 we describe the computational method used and in Sects. 3 and 4 we present and discuss our results. In Sect. 5 we present our conclusions.

2 Computational method

In order to break the cometary crust and allow the dormant comet to reactivate in the next perihelion passage, it is necessary to break the target surface with impact craters deep enough to reach the buried ice. The crater diameter $D_{\rm c}$ produced by a collision with a projectile of radius $r_{\rm p}$ and impact velocity $v_{\rm p}$ is found using the expression proposed by Zahnle et al. (1998):

\begin{displaymath}D_{\rm c}=1.70r_{\rm p}^{0.78}g^{-0.22}v_{\rm p}^{0.44}\left ...
...rho_{\rm p}}{\rho_{\rm t}}\right)^{0.333}(\cos \theta)^{0.44},
\end{displaymath} (1)

which is essentially the expression recommended by Schmidt & Housen (1987) but considering that only the normal component of the impact velocity contributes to cratering. In this equation $\rho_{\rm t}$ and $\rho_{\rm p}$ are the densities for the comet and projectile, respectively, g is the surface gravity on the comet, $\theta$ is the incidence angle measured from the zenith, and the equation must be evaluated in cgs units. The mean value of $\theta$ for the normal component of isotropic velocities is $45\hbox{$^\circ$ }$ and we always assume densities of $\rho_{\rm t}=0.5$ g cm-3 and $\rho_{\rm p}=2.5$ g cm-3 for ACOs and projectiles, respectively. The exposed area produced by the crater is:

\begin{displaymath}A_{\rm c}= \pi \left(\frac{D_{\rm c}^2}{4}+ h_{\rm c}^2\right),
\end{displaymath} (2)

where $h_{\rm c}$ is the crater depth. If the crust thickness is h, to reach the ice below the crust we need $h_{\rm c} \ge h$ and using the crater depth/diameter ratio, which is almost constant for simple craters and equal to 0.18-0.20 for the Moon and icy Galilean satellites (Schenk et al. 2004), it is possible to find the radius $r_{\rm p}$ of the projectile needed to form such crater.

On the other hand, the mean number of impacts received by the comet with projectiles of radius larger than $r_{\rm p}$ in a time $\Delta t$ is:

            $\displaystyle \langle N_{\rm col}(>r_{\rm p})\rangle$ = $\displaystyle \langle P_{\rm i} \rangle(r_{\rm t}+r_{\rm p})^2\Delta t N_{\rm pro}(>r_{\rm p})$  
  $\textstyle \quad\approx$ $\displaystyle \langle P_{\rm i} \rangle r_{\rm t}^2\Delta t N_{\rm pro}(>r_{\rm p}),$ (3)

where $r_{\rm t}$ is the comet radius, $\langle P_{\rm i} \rangle$ is the mean intrinsic collision probability between the comet and the projectile population, $N_{\rm pro}(>r_{\rm p})$ is the number of projectiles in the population with radius larger than $r_{\rm p}$, and $r_{\rm p}\ll r_{\rm t}$. The comet radius has been computed from the absolute magnitude of the object, H, by:

\begin{displaymath}\log (p_v \pi r_{\rm t}^2)=16.85+0.4(m_{\odot}-H),
\end{displaymath} (4)

where the result is in kilometers, $m_{\odot}=-26.77$ is the apparent visual magnitude of the Sun, and a standard albedo pv=0.04 was assumed.

Mean intrinsic collision probabilities and mean impact velocities can be inferred from statistical studies of the occurrence of orbital encounters between the comet and the projectile population, so we decided to estimate these parameters using the numerical method developed by Marzari et al. (1996). In this method the target and a projectile population were numerically integrated over a time span $T_{\rm int}$ and the encounter distance and encounter velocity between the target and any projectile were recorded. Since Marzari et al. showed that the distribution of the cumulative number of encounters for an encounter distance less than $d_{\rm enc}$ is proportional to $d_{\rm enc}^2$, a distribution of the form:

\begin{displaymath}N_{\rm enc}(<d_{\rm enc})=P_1\times d_{\rm enc}^2
\end{displaymath} (5)

was assumed. P1 is found by a fit to the data, taken as the standard deviation for each point $\sqrt{N_{\rm enc}}$. Then, the mean intrinsic collision probability is obtained from:

\begin{displaymath}\langle P_{\rm i} \rangle=\frac{P_1}{n_{\rm pair}T_{\rm int}},
\end{displaymath} (6)

where $n_{\rm pair}$ is the number of different pairs of objects that can be formed within the interacting population.

Since the asteroid belt is the main source of projectiles, it is enough to use as the interacting population a sample of particles with the same orbital element distribution as that observed for the objects in the asteroid belt. This sample was obtained as follows: first, the objects that form the complete known asteroid population, i.e., those asteroids with mean apparent opposition V magnitudes V(a,0)<15.75 (Tedesco et al. 2005), and with semimajor axis a>2 AU were taken from the ASTORB database (ftp://ftp.lowell.edu/pub/elgb/astorb.html) to get a first sample of 4549 objects. Then, a six-dimensional distribution of the orbital elements was calculated and a final sample of 350 particles was obtained at random from it. The hybrid integrator EVORB (Fernández et al. 2002) was used for the numerical integration of the targets and the particles of the interacting population, under the gravitational influence of the Sun and the planets from Mercury to Neptune. The integration was performed over a time span of $T_{\rm int}=10^5$ yr, and a encounter was recorded every time the mutual distance between the target and a particle was less than 0.05 AU.

To find a value for $N_{\rm pro}(>r_{\rm p})$ it is necessary to know the cumulative size distribution of the real projectile population, i.e. the size distribution of the main asteroid belt. We assume an exponential size distribution of the form ${\rm d} N_{\rm pro}(>r)\propto r^{-b}{\rm d}r$, where b is a characteristic exponent. Taking into account that the main belt size distribution changes for different size ranges, we have:

$\displaystyle N_{\rm pro}(>r_{\rm p})=N_{\rm pro}(>500~{\rm m})
+ K_0\int_{r_{\...
...^{500~{\rm m}}r^{-b_0}{\rm d}r+K_1\int_{r_{\rm p}}^{r_{\rm c}}r^{-b_1}{\rm d}r,$     (7)

where $N_{\rm pro}(>500~{\rm m})=1.36~\times~~10^6$ is the number of objects in the asteroid belt with radius larger than 500 m (Farinella & Davis 1992; Tedesco & Desert 2002; Bottke et al. 2005; Morbidelli & Vokrouhlický 2003). The size distribution of very small main belt asteroids is not well known, so it is not easy to choose values for the parameters $r_{\rm c}$, K0, b0, K1, and b1. We decide to use two size distributions with $r_{\rm c}=200$ m and 100 m, respectively, and to assume the size distribution proposed by Yoshida & Nakamura (2007) in the size range $r_{\rm c}<r<500$ m (b0=2.29), and a Dohnanyi (1969) size distribution for objects smaller than $r_{\rm c}$ (b1=3.5). Using this combined size distribution the total number of objects with a radius larger than 0.5 m is $1.42\times 10^{13}$ in the first case and $6.11\times 10^{12}$ in the second.

3 Results

The minimum ice exposed area needed to consider the comet as active could be obtained from the known JFC population. Almost all the studied JFCs have a fraction of active surface area below $\sim$10-20%, with a large fraction of active area for the smaller comets and a very small one for the few observed comets with $r_{\rm t}>3$ km (Tancredi et al. 2006). This could be indicative of a condition of the minimum effective exposed area to consider a comet as active, which could be proportional to $r_{\rm t}^{-2}$. Thus, we assume that a comet could reach an active state during the next perihelion passage when its exposed area is at least $10\%$ of the total surface area of an object with radius 1 km (1.257 km2). With this definition an object must have a radius larger than 0.316 km to have the possibility of being active, because for smaller objects its total surface area is always less than the required exposed one.

There are different values for the mantle thickness in the literature, ranging from a few millimeters to several centimeters. Since we must be sure that the crater excavates the surface enough to reach the ice below it, only craters with a minimum depth of h=1 m are considered to calculate the exposed area.

A sample of 206 dormant comets candidates was obtained from the ASTORB database, considering only ACOs with $2\le T_{\rm J} \le 2.9$ and q>1.3 AU, and excluding objects with $3.03~{\rm AU}<a<3.70~{\rm AU}$, e<0.4 and $i<25\hbox{$^\circ$ }$ (possibly main belt or Cybele asteroids), with $3.70~{\rm AU}<a<4.20~{\rm AU}$, e<0.4 and $i<20\hbox{$^\circ$ }$ (Hildas), and with $5.00~{\rm AU}<a<5.40~{\rm AU}$, and e<0.3 (Trojans). The absolute magnitude and radius of the ACO, mean intrinsic collisional probability, mean collision velocity and its error, the radius of the projectile that produces a crater with the minimum exposed area, and the times needed to receive such a collision for the two projectile size distributions ( $r_{\rm c}=200$ m and 100 m, respectively) are listed in Table 1. The errors of the mean intrinsic collision probabilities are always less than 10-22 km-2 yr-1 for all the objects. Twenty seven ACOs (5164, 30512, 32511, 37117, 96177, 1983 JZ1, $1995\; WL_3$, $2000~{\rm AU}_{242}$, $2000~{\rm QD}_{181}$, $2000\; WT_{168}$, $2000\; XO_8$, $2001\; QG_{288}$, $2003\; BM_1$, $2003\; UR_{267}$, $2004\; BT_1$, $2004\; KZ_7$, $2004\; RW_{141}$, $2005\; EB_{127}$, $2005\; JE_{173}$, $2005\; NX_{43}$, $2005\; XK_{57}$, 2005; YW24, $2006\; BV_7$, $2006\; HS$, $2006\; HP_{131}$, $2006\; JO_{65}$, and $2006\; SO_{134}$) had one or more close encounters with the planets and escaped before the integration ended. Then, their mean intrinsic collisional probability and mean collision velocity were calculated over a shorter time span, as indicated in Table 1.

4 Discussion

If the ACOs in our sample are dormant comets from the Jupiter family, their dynamics are also dominated by close encounters with this planet producing strong perturbations in their orbits. Fernández et al. (2002) found that the dynamical lifetime of Jupiter family comets is $\sim$ $ 2\times 10^5$ yr, which are relatively short compared to other populations. Assuming that this dynamical lifetime is also valid for the ACOs in our sample, the results presented here show that 53% and 32% of the objects in the sample receive at least 1 collision energetic enough to break the comet crust and allow a dormant comet to reach an active state in a period shorter than a JFC dynamical lifetime, 37% and 16% receive 2 or more collisions, 26% and 11% receive at least 3, and 16% and 7% receive not fewer than 4, for projectile distributions with $r_{\rm c}=200$ m and 100 m, respectively (Fig. 1). Then, many objects in the ACOs sample received 1 or more collisions during their dynamical lifetimes and, if they are dormant comets, they could be reactivated.


  \begin{figure}
\par\resizebox{8.8cm}{!}{\includegraphics[angle=270]{9236fig1.ps}}\end{figure} Figure 1: Fraction of ACO sample reaching the minimum exposed area to be considered as active comets in a certain time, for a projectile size distribution with a) $r_{\rm c}=200$ m, and b) $r_{\rm c}=100$ m. Only times shorter than the dynamical lifetime of Jupiter family comets are considered.
Open with DEXTER

On the other hand, in spite of their short dynamical lifetimes, comets become inactive mainly by physical causes due to a steady mass loss by sublimation of volatiles or by formation of a dust mantle. Based on the mean rate of secular brightness decrease, Kresák & Kresáková (1990) have estimated a mean active lifetime of $\sim$ $ 6\times 10^3$ yr for JFCs with a perihelion distance less than 1.5 AU, while Fernández (1985) found a lifetime of $\sim$ $ 2\times 10^4$ yr for these objects. Taking $1\times 10^4$ yr as a working value for the mean active lifetime of JFCs before they become dormant, and assuming that each time a dormant comet receives a collision energetic enough to expose fresh ices it becomes active for at least half the initial active period ( $5\times 10^3$ yr), it is expected to see these objects in an active state during a significant fraction of their dynamical lifetimes. As shown in Fig. 2 and Table 1, a large fraction of the largest objects in the ACO sample ( $r_{\rm t}\ge 8$-9 km) receives more than 4 collisions during this period and could be active during more than $3\times 10^4$ yr.


  \begin{figure}
\par\resizebox{8.8cm}{!}{\includegraphics[angle=270]{9236fig2.ps}}\end{figure} Figure 2: Time needed to receive a collision energetic enough to reactivate a dormant comet in function of its radius. Only objects with collision time shorter than $2.5\times 10^5$ yr are shown. The results for the projectile distributions with $r_{\rm c}=200$ m and 100 m are indicated with squares and circles, respectively.
Open with DEXTER

Using this result we can compare the present number of objects in the active comet and ACO sample to test our initial assumption that they are dormant comets. Since the population of dormant comet candidates must be seriously affected by an observational bias, which is more serious for the smaller and fainter objects, to allow a direct comparison only the largest objects in both populations are taken into account to minimize the bias effect. Then, using the radii of JFCs obtained by Tancredi et al. (2006), we found for the active population only 1 comet with radius larger than 9 km, 2 objects with radius larger than 5 km, and 5 objects with radius larger than 4 km. Taking into account these values and the period the comets are in an active state found previously, we must expect $\sim$6, $\sim$12 and $\sim$32 dormant comets in each radius range, respectively. In the ACO sample we found 15, 46 and 59 objects, respectively, that are in excess from the expected values and are indicative of the presence in the ACO population of objects that are not comets in a dormant state. These objects could be asteroids (possibly Hildas) with $T_{\rm J}<3$ obtaining their present orbits by some dynamical mechanism that perturbed the original asteroidal orbit changing its Tisserand invariant (Di Sisto et al. 2005).

The objects (1373) Cincinatti, (2938) Hopi, and (20898) Fountainhills receive collisions energetic enough to reactivate a dormant comet at a rate of >1- $2\times 10^{-4}$ yr-1, depending on the projectile size distribution. With these collision rates, they are the best candidates in the ACO sample to be observed during a reactivation if they are dormant comets.

While all the assumptions made here are reasonable, it is necessary to consider two of them more deeply. First, if the projectile size distribution used for the calculations is not accurate, the results presented here could be highly modified. Since the size distribution changes for different size ranges and for very small objects is not well known, it is not easy to choose values for their parameters in the smaller size end of the distribution. As a first guess we used the size distribution proposed by Yoshida & Nakamura (2007) for objects with a radius between 500 m and 200 or 100 m, in spite that these authors proposed it for objects larger than 250 m. It is possible that this size distribution could be still valid for smaller objects (for example, for $r\la 100$ m) producing a shallow size distribution and a shortage of small projectiles, making the ACO collisional reactivation process very improbable. Nevertheless, this shallow size distribution extended to very small sizes is difficult to reconcile with the cratering records of (243) Ida, (253) Mathilde, and (951) Gaspra (Chapman 2002).

Second, the choice of the minimum exposed area on the cometary surface to be considered active during its next perihelion passage is rather arbitrary. Taking the double of the original value (20% of the total surface area of an object with radius 1 km), the radius of the projectile needed to make a crater with that dimension is 1.56 times the original radius and the time needed to receive such a collision is 2-3 times longer, depending on the size distribution used and if the radius of the projectile is larger or smaller than $r_{\rm c}$. In this case, 37% and 16% of the objects in the sample receive at least 1 collision energetic enough to break the comet crust and allow a reactivation, and $16\%$ and $7\%$ receive more than 2 collisions for the projectile distributions with $r_{\rm c}=200$ m and 100 m, respectively. Then, the objects in the ACO sample with $r_{\rm t}\ge 8$-9 km receive more than 2 collisions during this period and could be active during >10% of their dynamical lifetimes, and we must expect $\sim$9, $\sim$18 and $\sim$45 dormant comets for a comet radius of >9 km, >5 km, and >4 km, respectively, which are also indicative of the presence in the ACO population of objects that are not comets in a dormant state.

5 Conclusions

We compute the mean intrinsic collision probability, the mean collision velocity, and the time needed to be reactivated by a collision for a sample of 206 ACOs that are dormant comet candidates from the Jupiter family.

The results presented here show that a large fraction of the objects in the sample receive at least 1 collision energetic enough to break the comet crust and allow a dormant comet to reach an active state in a period shorter than a JFC dynamical lifetime. A large fraction of the objects in the ACO sample with $r_{\rm t}\ge 8$-9 km receive several collisions and could be active during more than $3\times 10^4$ yr.

Three objects, (1373) Cincinatti, (2938) Hopi, and (20898) Fountainhills, receive collisions energetic enough to reactivate a dormant comet at a high rate, being the best candidates in the ACO sample of being observed during a reactivation if they are dormant comets.

Comparing only the largest objects in the active JFC population and ACO sample, we found an excess in the number of dormant comet candidates from the expected values which is indicative of the presence in the ACO population of objects that are not comets in a dormant state. These objects could be asteroids with $T_{\rm J}<3$ obtaining their present orbits by a dynamical mechanism that perturbs the original asteroidal orbit, changing its Tisserand invariant.

Acknowledgements
The authors wish to thanks J. A. Fernández for his comments and suggestions. The authors acknowledge the partial financial support by CICITCA, Universidad Nacional de San Juan, through a research grant.

References

 

  
Online Material

Table 1: Physical and collisional parameters for ACOs in the dormant comet candidate sample.



Copyright ESO 2008