Collisional process on Comet 9/P Tempel 1: Mass loss of its dust and ice by impacts of asteroidal objects and its collisional history

Abstract

We have studied the collisional process on the nucleus of Comet 9/P Tempel 1 (T1) and estimated the mass loss of surface materials due to the impacts of asteroidal objects. The mass loss rate is around ∼2×10⁵ kg yr^-1, which is smaller than that of water sublimation of ∼6×10⁹ kg yr^-1. We then estimated the number density of craters formed by asteroid impacts to infer the collisional history of T1. We found that the time necessary to accumulate the crater population on T1 is as long as ∼ a few 10⁴ years, suggesting that T1 crossed the asteroid belt region more than ∼ a few 10⁴ years ago, though this estimated time may be reduced to several thousands of years by taking into account the influence of the sublimation process on the crater size. Alternatively, the total period of the recent orbit, in which the perihelion distance is small enough to cause significant sublimation by erasing a large number of small craters, may be less than ∼2×10³ years.

1. Introduction

Short-period comets were originally trans-Neptunian objects formed from dust and ice in the outer cold region of the primordial solar nebula. When such a comet leaves the trans-Neptunian region and approaches the Sun, volatile components of its nucleus begin to sublime and form a dust mantle on the surface. Vapor pressure of the sublimating components preferentially carries the dust having a high surface-to-mass ratio off the surface. The sublimation process also leads to chemical differentiation of a nucleus owing to preferential sublimation losses of highly volatile materials in the deeper parts of the nucleus. In addition, trapped primordial volatile gasses are released when pristine amorphous water ices just below the dust mantle crystallize to form a crystalline ice crust. Heating caused by radioactive decay might have altered even the central region of the nucleus. Therefore, primordial dust and ice would remain only local regions well below the surface and well above the center of the nucleus (e.g., Jewitt, 1992; Meech, 2000; Prialnik et al., 2004, 2008). This scenario, which is hereafter called the standard model, has turned out to describe well the nucleus of Comet 9P/Tempel 1 (T1), the target of NASA’s Deep Impact (DI) mission: (1) a low thermal inertia of its surface estimated from thermal maps of the nucleus indicates the presence of a dust mantle (A’Hearn et al., 2005); (2) the existence of large grains in a dust trail along the trajectory of the comet would result from the preferential elimination of small-sized dust during the formation of a dust mantle (Sykes and Walker, 1992; Reach et al., 2007); (3) the optical and infrared properties of dust excavated by the DI event are explained well by the existence of a dust mantle composed of compact aggregates, below which fluffy aggregates are also embedded with icy volatiles (Yamamoto et al., 2008); (4) near-infrared spectroscopic observations of volatiles excavated by the DI event revealed thermal processing of icy layers below a dust mantle (Mumma et al., 2005); (5) the sublimation rate of 6 x 10²⁷ water molecules s^-1 (∼180 kg s^-1) from the T1 nucleus before the DI event implies that the top surface of several meters could have been lost simply by the sublimation process (e.g., Schleicher et al., 2006). In addition, the standard model has been supported by numerical studies on thermal evolution of the T1 nucleus: the formation of a dust mantle with chemical differentiations of the T1 nucleus (De Sanctis et al., 2007); a crystalline ice crust with a thickness of 40-240 m below a dust mantle (Bar-Nun et al., 1989).

In the above scenario, sublimation is an important process by which a comet nucleus can evolve. In addition to the sublimation process, however, impacts by asteroidal bodies may play a vital role in the evolution of the nucleus (e.g., Fern’andez, 1981). This is the most likely case for T1 because the present orbit of T1 crosses the asteroid belt region. Indeed, many crater-like features have been found on the T1 surface (Thomas et al., 2007). Thus, a study on the degree of the impact process is the key to understanding the evolution of the T1 nucleus. Furthermore, an application of crater chronology to the T1 nucleus may allow us to infer the collisional history on T1 and consequently provide us with clues about the orbital evolution of T1 in the past.

There are several studies on the impacts of asteroids onto the T1 nucleus, the results of which strongly depend on the model of the asteroidal population (e.g., Hawkes and Eaton, 2004; Gronkowski, 2007; Yamamoto et al., 2008). For example, Hawkes and Eaton (2004) estimated the mass flux of asteroids into the T1 nucleus to range from ∼10² to 10⁷ kg yr^-1, depending on the model they used. While the previous models are based on the data of asteroids with sizes larger than ∼ 100 m, small asteroids with sizes of less than 100 m play an important role in the collisional history of the T1 nucleus. The asteroidal model by O’Brien and Greenberg (2005) or O’Brien et al. (2006) (hereafter OGR model) is a self-consistent numerical model for the main-belt and near-Earth Asteroid (NEA) populations of small asteroids. These researchers developed the model based on a wide range of observational constraints, including small crater population on NEAs (see OGR model). Michel et al. (2009) reported that the asteroid population model by Bottke et al. (2005) (hereafter BTK model) may actually be more realistic than the OGR model to explain the crater population on NEA (25143) Itokawa. We are, therefore, convinced that either the OGR model or the BTK model is the best available model for the population of small asteroids colliding with comet nuclei. On the basis of these models, we have studied collisional and sublimation processes on the T1 nucleus to assess the mass loss rate of surface materials by impacts of asteroids and to infer the collisional history of the T1 nucleus.

2. Mass Loss of Surface Materials from T1 due to Impacts of Asteroidal Objects

We first estimate the mass loss by impacts of asteroids by assuming that both T1 and asteroids are homogeneous spheres. The rate of the excavated mass M_col by impacts of asteroids into a comet with radius R_T is given as:

((1))

where P_i is the intrinsic collision probability of T1, n(r)dr is the number of asteroids with radii ranging from r, r+dr, r_min and r_max are the minimum and maximum radii of asteroids impacting into T1, respectively, and m_e(r) is the total mass of impact ejecta escaping from the T1 surface by an impact of an asteroid with radius r.

The description of n(r) in the OGR model was not given explicitly in O’Brien and Greenberg (2005) nor O’Brien et al. (2006). Thus, we determine n(r) for the OGR model using the cumulative number N_as (d) of asteroids with larger than a given diameter d, as shown in figure 3 of Michel et al. (2009). From this figure we fit N_as(d) with an assumption of a power-law size distribution as:

((2))

Considering d = 2r, we obtain

((3))

The m_e(r) can be obtained from Eq. (A.7) in Appendix as:

((4))

where we assume the impact velocity υ_i, the escape velocity υ_esc, the density of impactor ρ_i, and the density of comet nucleus ρ_t as listed in Table 1, respectively (we refer to Table 1 for the values of parameters that will appear in the text hereafter).

Table 1.

Parameters used in this study.

Substituting Eqs. (3) and (4) into Eq. (1), we obtain

((5))

where we assume r_min ≪ r_max ≪ R_T. Equation (5) indicates that the mass loss rate by impacts of asteroidal objects is around ∼2×10⁵ kg yr^-1 for asteroids of r_max = 1 m, which would produce the largest crater with a radius of R_cr ∼ 200 m observed on T1 (cf. Eq. (11)). Alternatively, the mass loss rate of water sublimation near the perihelion is ∼6×10⁹ kg yr^-1, which is derived from the observational value of 6 × 10²⁷ molecules s^-1 (Schleicher et al., 2006). Although the sublimation rate depends on the solar distance, the sublimation fluxes of volatiles for a large part of the T1 current orbit are higher than the value of 2 × 10⁵ kg yr^-1 estimated to be due to impacts of asteroids (see De Sanctis et al., 2007). In addition, the mass ejection rate of solid dust due to the sublimation is measured to be ∼4×10⁸ kg yr^-1 (Reach et al., 2007), which is also larger than that by the impact. Therefore, the mass loss of the T1 nucleus is not primarily controlled by the collisional processes but by sublimation processes. We suggest that the mass loss by impacts would make only a minor contribution to a decrease in the thickness of a dust mantle. Nevertheless, this does not necessarily mean that the impact is not important for the evolution of the T1 nucleus. The collision may play a role to trigger off the formation of active regions on the surface. Britt et al. (2004) and Basilevsky and Keller (2006) proposed that the formation of depression feature, which can be formed by impacts, is an important process on the surface of T1 to sublime the material below a dust mantle.

We also examine the mass loss rate by using the asteroid population of the BTK model. Similarly, we determine n(r) for the BTK model using N_as(d) as shown in figure 3 in Michel et al. (2009). From the figure, we fit N_as(d) with an assumption of a power-law size distribution as:

((6))

Considering d = 2r, we obtain

((7))

By using Eq. (7) with the assumptions used above, we obtain

((8))

The mass loss rate by impacts of asteroidal objects around ∼3×10⁴ kg yr^-1 for r_max = 1 m is much lower than that of water sublimation. Thus, even for the BTK model, the sublimation process dominates over the mass loss by impact.

3. Collisional History of T1 Nucleus

3.1 Number density of craters formed by impacts of asteroids

We then estimate the collision number into the T1 nucleus to infer the collisional history from a comparison with the number of craters observed on T1 as follows. The total number N_col of impacts by asteroids with radii ≥ r into a comet with radius of R_T is given as:

((9))

Substituting Eq. (3) into Eq. (10), we obtain

(_)

where we assume r ≪ rmax ≪ R_T.

The rim radius R_cr of a crater formed by an asteroid with radius r can be estimated from Eq. (A.3) as:

((11))

Using the crater rim diameter D_cr = 2R_cr, we can rewrite Eq. (11) as

((12))

Substituting Eq. (12) into Eq. (10),

((13))

From Eq. (13), we estimate that a crater with D_cr = 100 m is formed at each ∼70 yr, which is lower than the previous estimate (Yamamoto et al., 2008). This is because these authors used the maximum asteroidal population model by Hawkes and Eaton (2004), which was based on an extrapolation from the population of large observed asteroids (>∼100 km) with a single power-law distribution. However, the population of small asteroids with r < ∼ 100 m does not follow the extrapolation from the population of large asteroids with r > ∼ 100 km (see OGR and BTK models).

To compare our results with observational data on T1, we use the R-plot defined by equation (10.2.4) in Melosh (1989):

((14))

where T is the exposure age and A is the surface area of the nucleus. The R plot is a conventional method to see the ratio between the actual cumulative number distribution of craters and a cumulative distribution with a slope of -2, which is thought to be a typical distribution of a saturated crater population on planetary bodies (see e.g., Melosh, 1989). Substituting Eq. (13) into Eq. (14), we obtain

((15))

In Fig. 1, we plot R(D_cr) against D_cr for various T. Also plotted are the observational data of likely impact crater remnants on T1 by Thomas et al. (2007). From this figure, we estimate T, although the observational data intersects several lines with various T. We first consider that T must be longer than 300 years. While the R value for the largest craters (D_cr = 400 m) shows an older age of T ∼ 10⁵ years, the data for the largest few craters may have a stochastic problem. Namely, there are too few the largest craters to give us an exact age. If we exclude the data with D_cr = 400 m, the R values for craters with D_cr > ∼150 m seem to follow the line with T = 10⁴ years. In contrary, the R value for smaller craters with D_cr < 100 m shows T < 10⁴ years, but this may be due to the sublimation process, as discussed later. Therefore, it is likely that the time necessary to accumulate the crater population on the T1 nucleus exceeds ∼10⁴ years (up to 10⁵ years).

Fig. 1.

R-plot against crater diameter for various exposure age T . Solid circles indicate the data observed on T1 (Thomas et al., 2007). Dash-dotted line indicates the empirical saturation line (R ∼ 0.2). Dashed-line indicates the R value for our sublimation erase model in Eq. (22).

Numerical calculations suggest that before T1 had close approaches to Jupiter ∼300–400 years ago, the perihelion of T1 may have been further than the asteroid belt region (e.g., Yeoman et al., 2005). However, this does not necessarily mean that T1 has never crossed the asteroid belt region prior to ∼400 years ago. Levison and Duncan (1997) reported that typical Jupiter-Family Comets (JFCs) change their perihelion and aphelion distances frequently after becoming JFCs. Thus, it is likely that T1 may have crossed the asteroid belt region many times in the past, forming craters corresponding to the R-plot for at least 10⁴ years. Note that the actual duration time may be longer (or shorter) than 10⁴ years, because we used the values of P_i and v_i for the present orbit of T1 (see Table 1), which might have been smaller (or larger) than the present values.

We also examine the case for the BTK model. Using Eq. (7), we also derived

((16))

In Fig. 2, we plot R(D_cr) against D_cr for various T. Also plotted are the observational data on T1 regarded as impact crater remnants by Thomas et al. (2007). We can see that the R values for craters with D_cr ∼ 100–300 m seem to follow the line with T = 10⁴ to ∼3× 10⁴ years, and the R value for the largest craters (D_cr = 400 m) shows an older age of T ∼ 105 years. Thus, even for the BTK model, the time necessary to accumulate the crater population on the T1 nucleus exceeds ∼10⁴ years (up to 10⁵ years), which is almost the same result as that for the OGR model.

Fig. 2.

The same as Fig. 1, but for the BTK model. Dash-dotted line indicates the empirical saturation line (R ∼ 0.2). Dashed line indicates the R value fo sublimation erase model as: which is derived from Eqs. (16) and (21).

3.2 Effects of sublimation process on crater population

As we see in Section 2, the mass loss by the sublimation process is more significant than that by impacts of asteroids. In this case, we may consider the possibility that the sublimation process erases or changes the crater population. Our question is: how would the sublimation process change the discussion in Section 3.1? We thus investigate the effect of sublimation on the crater population of the T1 surface hereafter.

We first estimate the thickness H_s of the T1 surface lost by sublimation as follows. The sublimation rate of 6 × 10²⁷ water molecules s^-1 is much higher than that expected from the water spot regions on the T1 nucleus (a few % in area) (see e.g., Schleicher et al., 2006; De Sanctis et al., 2007). We thus assume that dust particles are ejected not only from the active regions but also from the dust mantle by the sublimation of water vapor that passes through the dust mantle during the sublimation process. Assuming homogeneous sublimation from the whole surface of the T1 nucleus, we estimate the total mass lost by sublimation during T_s as

((17))

where Ṁ_s is the mass-loss rate of surface materials due to sublimation. From Eq. (17), the thickness H_s can be written as

((18))

((19))

The mass loss rate of large dust particles due to sublimation is reported to be Ṁ_s ∼ 14kg s^-1 (Reach et al., 2007). In this case, we estimate H_s ∼ 2.6 m for T = 300 years. If we use Ṁ_s ∼ 180 kg s^-1 for the water sublimation rate (e.g., Schleicher et al., 2006), we estimate H_s ∼ 34 m for T = 300 years. Moreover, Lisse et al. (2005) reported a twofold decrease in the water sublimation rate from 1983 to 1994, suggesting that the rate of water sublimation in the past was higher than the present rate. Therefore, the T1 primordial top layer of at least several meters could have been lost owing to the sublimation process. Namely, any primordial materials on the surface layers of T1 (e.g., residue created in trans-Neptunian region) cannot remain on the surface.

We next estimate to what degree craters are erased by the sublimation process. Consider that a crater with depth h_cr and diameter D_cr is completely erased when H_s = h_cr. Namely,

((20))

where ƒ is a depth-diameter ratio of a crater. Hereafter, we call this sublimation-driven process of erasing craters “sublimation erase model”. Substituting Eq. (19) into Eq. (20), we obtain the time T_er needed to erase a crater with diameter D_cr as

((21))

where we assume Ṁ_s = 14 kg s^-1 (Reach et al., 2007) and adopt ƒ = 0.2, which was measured for the crater population on 81P/Wild 2 nucleus (Basilevsky and Keller, 2006). Thus, when T_s > T_er, the number of craters with diameter of D_cr remaining on the T1 surface is suppressed to Ncol(D_cr) × T_er. Substituting Eq. (21) into Eq. (15), we obtain the R-D_cr relation for the sublimation erase model as

((22))

Equation (22) is also plotted in Fig. 1 (dashed-line). We see that the observational data with D_cr > ∼100 m are located above the line of Eq. (22). This indicates that the total duration of sublimation is T_s < ∼2 × 10³ years or that Ṁ_s in the past was lower than the value used here. In any case, this may suggest that the duration of the present significant sublimation is less than ∼2×10³ years.

We see that the observational data with D_cr ∼ 50 m is less than that by the sublimation erase model in Fig. 1. This may be explained as follows. First, water sublimation under the dust mantle may also contribute to erasing such small craters. The dust mantle layer would subside when the water ices below the dust mantle are lost by the sublimation process. During this subsidence, some of the small craters might have collapsed. Second, involved with the escape of water vapor through the dust mantle, the shape of small craters may be deformed too much to be identified as crater features. Third, large-sized dust fallen back to the surface may cover small craters. Fourth, the redistribution of re-golith by seismic-shaking induced by large impacts proposed by Richardson et al. (2005) may erase small craters. The decrease in the R value for small craters is a typical feature for the small crater population with D < ∼100 m of near-Earth asteroids, such as Eros or Itokawa, which can be explained by a smoothing effect from the redistribution of regolith by the seismic-shaking (e.g., Richardson et al., 2005; Michel et al., 2009). However, the report by Thomas et al. (2007) that T1 has higher surface slopes than other small bodies suggests a lack of the smoothing effect from the redistribution of regolith. In any case, although it is difficult to uniquely attribute the erasure of craters to one of the above-mentioned processes, some erase processes may be needed to explain the small crater population on the T1 nucleus.

3.3 Possible scenario for the evolution of T1

From the above discussion, we expect the following scenario for T1. T1 might have become JFC ∼ 10⁴–10⁵ years ago. T1 subsequently changed the perihelion distance as well as the aphelion distance many times, which was predicted by numerical calculations for JFCs (Levison and Duncan, 1994, 1997). During this period, T1 crossed the asteroid-belt frequently to accumulate the observed crater population on the surface. Recent close approaches to Jupiter about ∼400 years ago changed the T1 perihelion to be ∼1.5 AU (Yeomans et al., 2005). During this period, the smaller crater population decreased owing to significant sublimation. We cannot rule out the possibility that T1 suffered from significant sublimation before 400 years ago, but the total duration time of such significant sublimation is most likely less than ∼2×10³ years.

This scenario is based on the assumption that crater size does not change during the sublimation process. On the other hand, crater sizes may increase with time, because the sublimation process may erode the inside wall of craters, as suggested by Britt et al. (2004). From Eq. (19) with the water sublimation of Ṁ_s = 180 kg s^-1, the retreat rate by sublimation may be ∼0.11 m yr^-1. Thus, we estimate that the increase in D_cr by sublimation is ∼88 m for 400 years. In this case, the original crater sizes of D_cr = 200–300 m (with T ∼ 10⁴ yr) might have been D_cr ∼ 100–200 m, which corresponds to T ∼ 2000–5000 years estimated from Eq. (15), i.e., T ∝ D¹⁴⁶_cr. Thus, the age that T1 crossed the asteroid belt region may be as short as several thousands of years. Nevertheless, there is currently no clear correlation between the active regions and the impact craters on T1 (see Thomas et al., 2007). If the sublimation process eroded the inside wall of craters, we would observe many active regions inside the craters. This is, however, not the case. We expect that craters formed by asteroid impacts can be eroded by the sublimation process and become bigger, but the sublimation process would be quenched immediately owing to the formation of a dust mantle on the crater wall. The increase in the crater size by sublimation may not change significantly the above scenario for T1.

We also assume that there are no ancient craters formed by impacts with the other trans-Neptunian objects before T1 entered the inner Solar System. The reasoning for this is as follows. If the observed craters were ancient craters formed in the trans-Neptunian region, a large number of impacts with trans-Neptunian objects may be expected (e.g., Stern, 1996), and the R value for T1 would be around the empirical saturation line with R ∼ 0.2. 81P/Wild 2 is an example showing that the most of crater population follow the empirical saturation line (Basilevsky and Keller, 2006). It has been suggested that the surface of 81P/Wild 2 nuclei remains very old ancient terrain, presumably dating back to the comet’s residence in the trans-Neptunian region (e.g., Brownlee et al., 2004). Furthermore, while 81P/Wild 2 has many large craters with D_cr > 400 m, there are no large craters with D_cr > 400 m, suggesting that the T1 surface is not old enough to accumulate such large craters. We conclude that resurface events over the T1 surface in the past might have erased most of ancient craters on T1.

4. Summary

We have studied the mass-loss rate of surface materials by impacts of asteroidal objects. The mass-loss rate by impacts of asteroidal objects is around ∼2×10⁵ kg yr^-1, which is smaller than that of the water sublimation of ∼6×10⁹ kg yr^-1. This result suggests that the mass loss of a comet nucleus is mainly due to sublimation processes, and such a significant sublimation process has lost most, if not all, of the primordial materials on the surface layers of T1. However, this does not necessarily mean that the impact is not important for the evolution of the T1 nucleus because the collision process may play a role in triggering the formation of active regions on the surface. We have then estimated the number density of craters by asteroid impacts and compared this with the observational data. We found that the time necessary to accumulate the crater population on T1 is longer than 300 years, probably ∼ a few 10⁴ years, suggesting that T1 crossed the asteroid belt region during at least ∼10⁴ years in the past. If the crater size increases with time owing to sublimation, this age may be as short as several thousands of years. The present sublimation rate for T1 is very significant because of the recent orbit close to the Sun. A large number of small craters would be erased by such significant sublimation processes. Comparing the number density of craters with the sublimation erase model, we estimate that the timing of T1’s entry into the recent orbit may be later than ∼2×10³ years ago.

Appendix A. Scaling Formulae for the Crater Rim Radius and the Total Mass of Impact Ejecta Escaping from a Target Body

In this appendix, we derive the formulae for a crater rim radius R_cr, which is defined by the distance from the center to the rim of the crater, and for the total mass of ejecta m_e escaping from a target body, provided that an object of radius r and density ρ_i (then the mass m_i = 4πρ_ir³/3) impacts with velocity ν_i onto the comet surface of density ρ_t and gravity g. We assume that the cratering process is in the gravity regime, namely independent of the material strength (see Holsapple, 1993).

The apparent crater radius R_at, which is measured along a pre-impact surface level, is expressed in the gravity regime (e.g., Housen et al., 1983) as:

((A.1))

where K₁ and ε_x are constants. When we use the values of ε_x = 0.167 and K₁ = 0.85 for craters formed in Ottawa sand (Schmidt, 1980), we obtain

((A.2))

Assuming the rim diameter of a crater is 1.3 times larger than the apparent diameter (Housen et al., 1983; Holsapple, 1993), the rim radius is given by

((A.3))

Housen et al. (1983) formulated the volume V_t(>ν_e) of impact ejecta with velocity higher than ν_e in the gravity regime as:

((A.4))

where K₂ and ε_v are constants. According to Housen et al. (1983), ε_v and ε_x are related each other by the formula ∈_v=6∈_x/1-∈_x). Using Eq. (A.4), the total mass of the escaping ejecta m_e is expressed as

((A.5))

((A.6))

Substituting Eq. (A.2) into Eq. (A.6), we obtain

((A.7))

where we use the values of K₂ = 0.32 and ε_v = 6ε_x(1 - ε_x) = 1.20 for cratering processes in the gravity regime (Housen et al., 1983).