PLANETESIMAL COLLISIONS AS A CHONDRULE FORMING EVENT

Figure 1 shows snapshots of a collision between the same sized planetesimals (10 km in diameter). Each panel denotes temperatures on the right-side and velocities on the left-side at each position at a time of 2.0 s and 4.0 s after the impact. We find that narrow flow (so-called jet) emerges from the collisional surface between the planetesimals. Some fraction of such ejecta could eventually become chondrules because both of their temperatures and velocities exceed T_m and v_imp, respectively. As time goes on, the temperature of ejecta decreases (see Figure 1(b)). This indicates that materials ejected at a later time are very unlikely to become chondrules since they cannot reach T_m (see below and Figure 3).

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Figure 2 also shows the same physical quantities, but the original positions of ejects are identified in order to specify from which regions of planetesimals the ejected materials emerge. As seen from Figure 2, progenitor of chondrules originates from the surface region of the impactor and the target. In addition, our results show that the temperature and velocity of materials located in the central region of the planetesimals are not affected by the collision very much. In other words, ejecta from such a region cannot contribute to chondrule formation. We will confirm below that these are the general trend for various kinds of collisions (see Section 3.2 and Figure 5).

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It is important to quantitatively examine how much mass of chondrules are formed by single collisions as a function of time. Figure 3 shows the time evolution of F_ch (see the green solid line, also see Equation (3)). We find that the value of F_ch initially increases with time. It then becomes constant at a time of ∼2.0 s or later after the impact. Our simulations show that the total mass of ejecta increases after 2.0 s. These ejecta, however, do not end up with chondrules. This is simply because ejecta which emerge at a later time do not experience melting at that time (see Figure 1). Thus, F_ch takes a similar value between 2.0 s and 4.0 s. We also confirm that the resultant value of F_ch does not change after 4.0 s. Furthermore, our results show that the total mass of chondrules' progenitor (F_ch) both from an impactor and from a target, are similar (see Section 3.2 and Figure 5). In the following sections, we will use the value of F_ch at 2.0 s after the impact.

We now check the convergency of our results by changing the value of cell per projectile radius (CPPR), that is, the cell size. Figure 3 shows the value of F_ch for different CPPR with the same impact velocity (2.5 km s⁻¹). Each line represents different CPPR. Our results show that there is no significant difference between CPPR = 400 and CPPR = 800. Thus, our specific choice of CPPR = 400 is assured for the estimate of F_ch, and will be adopted in all the following simulations in this paper. Note that we also perform a simulation of a collision with a flat target and confirm that our results (F_ch) are in good agreement with the previous work (Johnson et al. 2015) (not shown in this paper).

We examine how F_ch varies as a function of the impact velocity for collisions between the same sized planetesimals.

Figure 4 shows the resultant behavior for collisions with the 10 km sized target planetesimals (see the solid line). We find that as v_imp increases F_ch also increases. This occurs simply because a high impact velocity generates large impact energy, which creates more chondrules. Interestingly, even if v_imp = 2.0 km s⁻¹, which is lower than the threshold velocity adopted in the previous work (v_c = 2.5 km s⁻¹), chondrules could form for the same sized planetesimal collisions. Since the impact velocity of v_imp = 2.0 km s⁻¹ can be achieved at the early stage of planetary accretion, our results suggest that when planetesimal–planetesimal collisions are considered, impact jetting can start forming chondrules at an earlier time than the previous studies indicate (Johnson et al. 2015; Hasegawa et al. 2016a, 2016b). This finding is important because the meteoritic data suggest that the formation time of the oldest chondrules overlaps with that of CAIs (Connelly et al. 2012). In other words, impact jetting may be able to produce such chondrules via planetesimal–planetesimal collisions, if planetesimals formed contemporary with CAIs (see Section 4.2 for further discussion). Note that the value of F_ch is small (0.1%) for this case, so that the mass budget of chondrules may not be affected by the collisions significantly. We also find that when the impact velocity is slower than 1.5 km s⁻¹, the resultant chondrule formation would be negligible since F_ch is much less than 10⁻³%.

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Figure 5 shows the cumulative mass fraction of chondrules (f_ch(r)) as a function of depth (r) from the surface of an impactor and a target (see Equation (2)). The contribution from the impactor is denoted by the dashed line, and the total ejected mass both from the impactor and from the target is denoted by the solid line. In other words, the contribution only from the target is represented by the difference between the solid and the dashed lines. It clearly shows that progenitor of chondrules can be ejected at deeper regions from the surface when collisions occur with higher impact velocities. Our results also show that the value of f_ch(r) increases at the same depth for higher impact velocities. These explain the results shown in Figure 4: F_ch is an increasing function of the impact velocity. Furthermore, we find that the deepest region that can eject progenitor of chondrules is located only at about 400 m from the surface even for the case of v_imp = 4.0 km s⁻¹. It is important that half of the chondrule mass (F_ch) arises from a very thin surface layer, which is within 100 m of the surface. In addition, our results show that the chondrule masses formed by impact jetting from the impactor and the target are almost the same. Planetesimals with diameter of 10 km hardly experience differentiation, especially at their surface. Therefore, produced ejecta from such collisions can surely become chondrules which can be compatible with the ones found in chondrites.

We explore how the diameter of target planetesimals affects impact jetting and the resultant chondrule formation. In this section, we adopt the impact velocity of 4.0 km s⁻¹ for all the runs, and change only the diameter of the target. In Section 3.4, we will discuss the results in which both the impact velocity and the target's size vary.

Figure 4 shows that the total mass of chondrules becomes largest when a 10 km sized planetesimal serves as a target. As the diameter of target planetesimals increases, the produced mass of chondrule becomes smaller. Our simulations nonetheless show that the resultant values of F_ch are comparable for a wide range of the target size. We can therefore conclude that the results obtained from flat targets can be used as a reference value for both planetesimal–planetesimal and protoplanet–planetesimal collisions. More specifically, the chondrule mass estimated by the previous studies would still hold even for planetesimal–planetesimal collisions (Johnson et al. 2015; Hasegawa et al. 2016a, 2016b).

We also plot the cumulative chondrule mass (f_ch(r)) from the target and the impactor as a function of depth (r) from their surface in Figure 6 (as done in Figure 5). For the case of collisions with a 10 km sized target, progenitor of chondrules is ejected from the deepest region ($r\simeq 400$ m). Equivalently, the largest amount of chondrules is produced. We find that f_ch(r) decreases at all depths with increasing size of targets. This indicates that the original locations of chondrules are the key to evaluating f_ch(r) (and hence F_ch). It is interesting that progenitor of chondrules emerges from a deeper region of targets than that of impactors when targets are bigger than impactors. Thus, our results suggest that the composition of targets is more important for determining that of chondrules more accurately when there is a size difference between targets and impactors.

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We are now in a position to change both the impact velocity and the target size to examine how these two parameters affect chondrule formation.

Figure 4 shows the corresponding results. We find that the behaviors of F_ch are very complicated. For instance, we consider collisions with larger target planetesimals (≥40 km in diameter, see the green triangles and the open squares). When the impact velocity is slower than 3.5 km s⁻¹, the value of F_ch increases monotonically. Once the impact velocity exceeds 3.5 km s⁻¹, however, the value becomes smaller. It should be pointed out that such complicated behaviors are also observed in the previous work (Johnson et al. 2014), while our numerical setups are not entirely the same as the work.

Based on this parameter study, we may be able to draw the following two conclusions. The first one is that when the impact velocity is smaller than $\lt 2.0$ km s⁻¹, the resultant value of F_ch is around 0.1%. Consequently, such collisions may be important to regulate the formation timing of chondrules, but not their total abundance. The second one is that while the dependence of F_ch on impact velocities and the target sizes is not so simple, the condition adopted by the previous work can be applicable for various kinds of collisions that can occur during planetary accretion: chondrule formation can take place at planetesimal collisions with the impact velocity of ${v}_{\mathrm{imp}}\simeq 2.5$ km s⁻¹ or higher, and the resultant abundance of chondrules formed by a collision is ${F}_{\mathrm{ch}}{m}_{\mathrm{pl}}$ with F_ch of 0.01.

We here discuss the assumptions that are used in our numerical simulations.

First, we assume that the bulk compositions of an impactor and a target are the same in our simulations; porous dunite. It is well known that ordinary chondrites have a porosity of $\lt 10 \%$ (Consolmagno et al. 2008). The value of porosity adopted in our simulations, which is 1%, is in such a range. The bulk composition of planetesimals used in this paper, dunite, is similar to that of ordinary chondrites (Svetsov & Shuvalov 2015). Thus, our assumption that colliding, undifferentiated planetesimals would be represented by porous dunite may be reasonable, and hence our results would be useful for producing chondrules found in ordinary chondrites. Note that the properties of melt ejecta that can finally be transformed to chondrules would be determined both by the porosity and by the composition of impactors and/or targets (Wünnemann et al. 2008).

Second, we fix the initial temperature of planetesimals at 300 K in all the simulations. If this value would be decreased down to ∼150 K, (which is more appropriate for the gas temperature at 2–3 au in the solar nebula), the resultant mass of melting ejecta would be smaller. On the contrary, if the surface of planetesimals has a much higher temperature (>600 K) following the formation and growth of planetesimals, then the produced chondrule mass might increase (e.g., Sanders & Scott 2012). A more comprehensive parameter study would be needed to fully develop a quantitative argument.

Third, only head-on collisions are examined in this paper. This arises because we have made use of the iSALE-2D code. Such collisions would probably produce a lower amount of chondrules, compared with other types of collisions. Thus, our results are very likely to provide a conservative estimate in the resultant abundance of chondrules. In our future work, we will use the iSALE-3D code to explore the three-dimensional effect on planetesimal collisions and compute the amount of chondrules formed by the collisions.

We discuss a possibility about the occurrence of differentiation in planetesimals and its effect on chondrule formation via impact jetting.

Differentiation is indeed an important process for the impact jetting scenario to produce chondrules. This is because once planetesimals experience differentiation, the composition of colliding planetesimals may not be entirely the same as that of the currently sampled chondrules. As a result, it may be difficult to generate chondrules from ejected materials. In addition, while our results suggest that chondrules formed via planetesimal–planetesimal collisions might be able to generate the oldest chondrules (see Section 3.2), this suggestion would be valid only if colliding planetesimals did not experience differentiation at that time. Thus, differentiation can affect both the composition of chondrules and their formation timing.

Based on thermal modelling of planetesimals, the onset of differentiation in planetesimals is regulated predominantly by their formation timing (e.g., Gail et al. 2014, p. 571; Wakita et al. 2014). When planetesimals formed at a few Myr after CAI formation, which occurred 4567 Myr ago (Connelly et al. 2012), most planetesimals are hard to differentiate. Differentiation might occur for planetesimals that are larger than 20 km in diameter, when they formed shortly after the formation of CAIs.

In our model, the formation of planetesimals has to occur prior to the formation of chondrules, since planetesimal collisions are the mechanism to generate chondrules. In other words, the impact jetting scenario can become a crucial process for chondrule formation only if the formation of planetesimals took place in the solar nebula well after the CAI formation and these planetesimals underwent impact jetting. Given that it is still poorly known how to form planetesimals (Johansen et al. 2014, p. 547), it is unlikely that a definitive statement can be made here. It may nonetheless be possible to suggest that if planetesimal formation occurred continuously in the solar nebula, impact jetting can contribute to chondrule formation for a long time. In fact, chondrule formation continued 3–5 Myr after CAI formation started (Connelly et al. 2012; Bollard et al. 2015). Furthermore, in order for impact jetting to generate the oldest chondrules, colliding planetesimals should have been small enough (∼10 km in diameter) to avoid differentiation. Taking into account that at least about 10⁵ years are needed to form protoplanets once a large number of planetesimals are present (Johnson et al. 2015; Hasegawa et al. 2016b), planetesimal–planetesimal collisions would have more chance to generate the oldest chondrules via impact jetting (than protoplanet–planetesimal collisions).

It should be noted that the above argument has been made based on the assumption that when colliding bodies are fully differentiated, they cannot yield jetted materials of an undifferentiated composition. As pointed out by Johnson et al. (2015), however, differentiated bodies may be able to have surface layers that contain yet-to-be differentiated materials as long as the bodies are still growing (e.g., Weiss & Elkins-Tanton 2013). Since our results show that the progenitor of chondrules arises from thin (∼100 m) surface layers (see Figures 5 and 6), there may still be a possibility that largely differentiated, growing bodies (such as massive planetesimals and protoplanets) can generate chondrules via impact jetting. A more comprehensive work would be needed to fully address these issues, wherein thermal modeling of massive planetesimals (and/or protoplanets) and the impact jetting process are examined simultaneously.

While we have so far computed the mass fraction (F_ch) of ejecta that experience a high temperature (T_m) and have a high ejection velocity (v_imp), we have not discussed the fate of such ejecta yet. Here, we attempt to explore how the ejecta can eventually become chondrules and will be accreted by parent bodies of (ordinary) chondrites.

As suggested by Johnson et al. (2015), chondrules can be formed from the resultant melts via impact jetting. For this case, the typical size of chondrules could be estimated from the surface tension and the velocity of ejecta (Melosh & Vickery 1991; Johnson & Melosh 2014; Johnson et al. 2015). We here apply a formula (see Equation (6) in Johnson & Melosh 2014) to our results under the assumption that the surface tension of dunite is the same as that of silica. We find that the diameter of chondrules generated from the impact jetting would be around 0.4 mm. The high impact velocity results in the high velocity of ejecta, which leads to the production of smaller chondrules. Our results nonetheless show that the diameter of chondrules shrinks down to only 0.2 mm for the highest impact velocity case. Based on the above simulations, we have confirmed that the velocity of ejecta does not depend on the size of target planetesimals very much. As a result, the diameter of chondrules is not so sensitive to the impact velocities and the size of targets.

Could the resultant droplets of chondrules return or be accreted onto planetesimals that can serve as parent bodies of chondrites? Hasegawa et al. (2016a) discussed this based on the idea of pebble accretion in a magnetized disk. Small particles including chondrules can be accreted onto massive planetesimals via pebble accretion (e.g., Johansen et al. 2015). They utilize the estimate of the magnetic field strength that is derived from chondrules in Semarkona ordinary chondrite (Fu et al. 2014). They found that in order for planetesimals to accrete chondrules and to become parent bodies of chondrites, the solar nebula should be $\lt 5$ times more massive than the minimum mass solar nebula (Hayashi 1981) and the planetesimals are $\lt 500$ km in diameter ($\sim {10}^{21}$ kg). Matsumoto et al. (2016) undertook a follow-up study in which pebble accretion of chondrules in a gas disk is investigated carefully. They consider a situation that a protoplanet, planetesimals and chondrules all coexist. They have found that although about half of produced chondrules are accreted onto the protoplanet, ten percent of chondrules are accreted onto planetesimals and can make a thin chondrule-rich layer near their surface. If such planetesimals can act as parent bodies of chondrites, it may be possible to create chondrites with the high abundance of chondrules; chondrites would be fragments originating from the surface region of the planetesimals.

In addition to the ejecta that can eventually become chondrules, it would also be important to discuss the fate of unmelted ejecta that cannot be transformed to chondrules. This is because planetesimal–planetesimal collisions can potentially generate a considerable number of such unmelted broken rocks (see Figures 1 and 2).⁵ The presence of such broken rocks may become problematic in reproducing the high abundance of chondrules in ordinary chondrites; a good mixture of chondrules and broken rocks may end up with difficulty generating purely chondritic meteorites. One of the possible processes to filter out the broken rocks from a swarm of chondrules may arise from a size difference between chondrules and the broken rocks. As discussed above, small-sized particles like chondrules will be accreted onto massive bodies via pebble accretion. If the broken rocks would be much larger than or much smaller than chondrules, then the accretion timescale of such rocks onto massive planetesimals/protoplanets would be different from that of chondrules. As a result, it may be possible to selectively remove unmelted broken rocks from a collection of chondrules, and hence to create purely chondritic meteorites. An intriguing conjecture can be developed from this argument: since the particle size is the main quantity to trigger this filtering process, some chondrites may possess both chondrules and unmelted rocks with a size of ∼1 mm. It is interesting that some (breccia) chondrites indeed contain the unmelted and lightly shocked, chondrule-sized materials (e.g., Stoeffler et al. 1991). The detail will remain to be explored for future work.

Thus, the coupling of impact jetting with pebble accretion of chondrules is promising to obtain new insights about the formation of chondrules, chondrites, and up to planets in the solar system.

It is interesting to compare our results with the actually found chondrules.

As discussed above, our results suggest that chondrules are about 0.4 mm in diameter. On the other hand, the diameter of chondrules in ordinary chondrites is estimated as 0.3–0.6 mm (Scott & Krot 2005; Scott 2007; Friedrich et al. 2015). Thus, our results are in good agreement with the actual measurements of chondrules in ordinary chondrites.

While we have focused mainly on chondrules in ordinary chondrites in above sections, chondrules are also found in carbonaceous chondrites. Since most chondrules are composed mainly of silicate, our results may be applicable for chondrules in CB chondrites (one type of carbonaceous chondrites) as well. In fact, it is suggested that chondrules in CB chondrites would form via collisional processes (Krot et al. 2005; Bollard et al. 2015).

Note that the bulk composition of CB chondrites is quite different from that of ordinary chondrites, so that parent bodies of these two chondrites should be largely different with each other. The diameter of chondrules in CBb chondrites (a subgroup of CB chondrites) is ∼0.5 mm (Scott & Krot 2005; Scott 2007), which is comparable to the size of our results. Thus, planetesimal collisions and the subsequent impact jetting may be a crucial process to account for the properties of chondrules found in chondrites.

It should be noted that chondrites contain various sizes of chondrules and their texture also varies (Scott & Krot 2005; Scott 2007). For instance, the diameter of chondrules in CBa chondrites (another subgroup of CB chondrites) is measured as ∼5 mm (Scott & Krot 2005; Scott 2007; Bollard et al. 2015), which is much larger than what our results predict. A more detailed analysis would be needed to fully understand the properties of currently sampled chondrules such as their size distribution, their mineralogy and their texture.