Implications for the Collisional Strength of Jupiter Trojans from the Eurybates Family

First, Figure 1 shows the cumulative SFD of the two Trojan swarms using the WISE data (Grav et al. 2011). We often find that SFDs follow a power law where the number of objects larger than a certain diameter can be written as

$$\begin{eqnarray}&&N(\gt D)={C}_{0}{D}^{-q};\quad {C}_{0}={N}_{0}{D}_{0}^{q};\end{eqnarray} \tag{ 1 }$$

where q is the cumulative power-law exponent or slope, and C₀ is the normalization constant such that for a reference diameter, D₀, the number of objects larger than D₀ is N₀. The differential power-law exponent, p, differs with respect to the cumulative power-law exponent by one: p = q + 1.

The Jupiter Trojan SFD has a well-defined cumulative slope of q = 2.1 between 10 and 100 km for both the L₄ and L₅ swarms (Figure 1). At larger sizes the slope becomes steeper (similar to the KBO SFD) but is poorly defined because of the limited number of objects. Note that 100 km planetesimals are thought to be the preferred size (e.g., Morbidelli et al. 2009a; Klahr & Schreiber 2020, 2021) for the formation of planetesimals by streaming instability (Youdin & Goodman2005). On the other end of the size range, the population of small Trojans has been explored for L₄ (Jewitt et al. 2000; Yoshida & Nakamura 2008; Wong & Brown 2015; Yoshida & Terai 2017). The latter three studies find that the slope of the SFD continues below 10 km, below which there are some indications that the SFD might become shallower. Unfortunately, the papers do not agree where (or if) this turnover takes place. The latest study (Yoshida & Terai 2017), which uses the powerful Hyper Suprime-Cam attached to the Subaru Telescope, finds that a single power law is a better overall fit than a broken power law. It remains uncertain whether a bend in the Jupiter Trojan SFD has actually been detected. If such a turnover exists, it remains an open question how it relates to the observed turnover in the Edgeworth-Kuiper Belt, which occurs at roughly 1 km (Singer et al. 2019; Morbidelli et al. 2021).

Second, we discuss the Eurybates family. To identify the family, we use the proper elements computed by Mira Brož (Holt et al. 2020) using the hierarchical clustering method (HCM, Zappala et al. 1990; Nesvorný et al. 2015) with a cutoff of 40 m/s. We have varied the cutoff to probe the change in family definition. Increasing the cutoff leads to clustering in the wider neighborhood, while a smaller cutoff leaves out many peripheral members. In total the HCM identified 400 family members, which roughly doubles the family members previously found by Nesvorný et al. (2015). This change reflects the increased number of known JTs since the time of that paper. The full list of family members with relevant properties presented in this work can be found in Tables 3 and 4.

The Eurybates family is well defined with respect to the proper elements (Figure 2) and is roughly located in a_prop ∈ [5.27 au, 5.32 au], e_prop ∈ [0.032, 0.066], and i_prop ∈ [6.99°, 7.86°].

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The family is located close to the edge of the stable orbit region (Figure 2, and Robutel & Gabern 2006). In inclination, the family is extremely well defined (within half a degree for most members; Brož & Rozehnal 2011) and thus stands out in the right panel of Figure 2.

Figure 3 shows the SFD of the Eurybates family as retrieved by the HCM (red line in the left panel). This raw SFD has a shallow slope between 20 and 60 km of q ∼ 2.1, similar to the overall JT slope (Figure 1). At small sizes the SFD steepens up to a slope of q ∼ 3.7, a value that is steeper than that of the L₄ Trojans. Slopes steeper than those of the background population are a common property of collisional families (Durda et al. 2007).

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Before going any further, we need to establish if some of the identified members could be interlopers. This is particularly important for the largest family member because the shape of the SFD at those sizes can be diagnostic of the family-forming collision itself (e.g., Benavidez et al. 2012).

To validate the possible detection of interlopers we first assess the likelihood of the background L₄ population contaminating the Eurybates family region. We have used the following dynamical constraint to determine the background population. First, we defined an ellipsoid in proper (a, e, i)-space to encircle the family. The center of the ellipsoid was placed at (a, e, i) = (5.30 au, 0.049, 7.4°) and the respective radii set to (R_a , R_e , R_i ) = (0.069 au, 0.040, 1.04°). These values were chosen to maximize the number of family members identified by HCM and to minimize the number of false defections. All JTs within this ellipsoid were considered as a simplified definition of the family members and only slightly differed from the actual family members determined by HCM. We then increased the volume of the ellipsoid by a factor of eight by increasing the radii by a factor of two in each dimension. The Trojan density within this larger volume minus the family volume was used to estimate the number of interlopers we can expect in the family. For objects with magnitude H < 12 we expect on average 1.4 interlopers. We also determined the probability of having N interlopers. There is a 40% chance that N ≥ 2, and a 20% chance that N ≥ 3. Consequently there is a non-negligible probability that there could be at least three large interlopers in the family. Over all sizes about 8% of the family members might be interlopers.

To investigate which of the largest members might be interlopers, we inspected their proper orbital elements as well as color data (Figure 4).

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We find that the second brightest/largest object identified by HCM, (5258) Rhoeo, is clearly displaced in proper inclination and eccentricity with respect to the center of the family. This would imply, that a very large fragment from the family-forming event would have been ejected at a high speed. That is not to be expected. Collisional families typically have their largest members in the center of the orbital element distribution (e.g., Milani et al. 2017). That is a consequence of the velocity distribution being a function of 1/D. We hence strongly suspect that (5258) Rhoeo is an interloper. This conclusion is strengthened by its anomalous color compared to the other family members.

We collected colors from the Sloan Digital Sky Survey (SDSS; Ivezić et al. 2001), spectral slopes between 0.3 to 0.9 μm (Fornasier et al. 2007), and colors from the Zwicky Transient Facility (ZTF) observations (Schemel & Brown 2021). The SDSS g–i color (top panel in Figure 4) and the spectral slopes from Fornasier et al. 2007 (middle panel in Figure 4) are connected, and thus their color scales have been chosen such that they correspond to each other. The color scale of the ZTF data cannot be directly compared to the other two.

Eurybates and the bulk of the family have ZTF g–r of ∼0.5 whereas (5258) Rhoeo has 0.6, which is significantly redder. This together with the orbital displacement of (5258) Rhoeo previously discussed provides strong evidence that it is indeed an interloper.

The Eurybates family is significantly bluer than the overall L₄ population (Figure 5). With a g–i color of 0.96 ± 0.09 (313024) 2000 AV210 is on the very red end of the color distribution and thus another likely interloper. It has a diameter of 10.805 ± 0.599 and will thus not affect the family SFD and the work described here.

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There are two other family members that might be interlopers. (8060) Anius has a somewhat larger eccentricity (middle panel in Figure 4) but its ZTF color is close to the average color of the family. It is therefore unclear if (8060) Anius is indeed an interloper. We leave this as an open question. Future survey data might be able to provide additional insights. Similarly, (9818) Eurymachos has a slightly low semimajor axis for its size, comparable to (5258) Rhoeo (see the top panel in Figure 4). Other than that, the body is unremarkable with respect to the colors or orbital elements of the other family members. For this reason, we do not consider (9818) Eurymachos an interloper in this work.

The other larger family members do not have unusual colors or orbital elements. Consequently, for our work here we consider (5258) Rhoeo an interloper and exclude it from what we consider to be the nominal family. The family SFD excluding (5258) Rhoeo is shown in Figure 3. We can use these data to explore what the SFD tells us about the family-forming event.

A substantial amount of work has been done to simulate the outcome of asteroid impacts (e.g., Benz & Asphaug 1999; Durda et al. 2007; Benavidez et al. 2012; Jutzi 2015; Benavidez et al. 2018; Jutzi et al. 2019) using primarily a smoothed particle hydrodynamics (SPH) approach. We have access to the simulation results from Durda et al. 2007, who used 100 km diameter monolithic target bodies, and Benavidez et al. (2012), covering collisions of 100 km rubble pile target bodies. Given that these two works cover a large part of parameter space, we consider this to be a good first-order reference for possible outcomes.

We used three criteria to determine whether a simulation outcome matched the observed family SFD: (1) the mass ratio of the two largest family members shall be within 25%, (2) the slope between the second and tenth largest object shall be within ±0.2, and (3) the slope between the tenth and 30th largest body shall be within ±0.2. These criteria were chosen because they fully characterize the observed family SFD and should so be fairly diagnostic of the family-forming collision. For the 387 SPH runs available to us, the mass ratio varied between 3 × 10⁻⁵ and 1 (median of 0.03), the slope between the second and tenth largest object varied between −0.6 and −15 (median of −3.7), and the slope between the tenth and 30th largest body varied between −1.6 and −10 (median of −4.4).

We found that only one model outcome out of 387 satisfied all three criteria (see right panel of Figure 3). The model SFD is remarkably close to the actual family SFD. The simulation corresponded to a collision between a 100 km rubble pile target ⁴ (Benavidez et al. 2012) and a ∼35 km impactor ($\mathrm{log}({M}_{\mathrm{tar}}/{M}_{\mathrm{imp}})=1.4$) at 6 km s⁻¹ and an impact angle of 75° from vertical. Though this match is unlikely to be a unique solution, it is an important piece of information to be evaluated. This is the first indication in our analysis that Jupiter Trojans might be collisionally weak. We will examine this issue below.

A strong constraint for our modeling would be the estimated age of the family, but unfortunately, it is rather poorly constrained. Brož & Rozehnal (2011) and Rozehnal et al. (2016) estimated the family age between 1 and 4 Gy. Milani et al. (2017) were unable to retrieve an age for the Eurybates family from their dynamical methodology. Finally, using the escape probability of family members as derived from dynamical considerations, Holt et al. (2020) estimated the minimum age at ≥1.0 ± 0.4 Gy. We are hence left with a weak constraint for the family age between 1 Gy and the age of the solar system.

Our last and arguably most important constraint comes from Eurybates itself. Noll et al. (2020) reported the discovery of a 1.2 ± 0.4 km satellite around (3548) Eurybates, which has since been named Queta. Eurybates is only the fourth Trojan with a known satellite (Noll et al. 2020).

Since its discovery, Queta's orbit has been further constrained by Hubble Space Telescope observations (Brown et al. 2021). It has a semimajor axis of 2350 ± 11 km, an orbital period of 82.46 ± 0.06 days, and a small eccentricity of 0.125 ± 0.009 (Brown et al. 2021). We find the low eccentricity to be particularly noteworthy because impact simulations (e.g., Durda et al. 2007) typically generate companions around the target body that have highly eccentric orbits. In dynamics, "particles always return to the scene of the perturbation" unless some mechanism has changed their orbit. In Questa's case this can be due to the Kozai effect, which lets Queta's orbit go through oscillations in eccentricity (Brown et al. 2021).

The existence of Queta as a satellite of Eurybates, the most sizeable member of the largest collisional family within the Jupiter Trojans, immediately poses the question as to the long-term survival of Queta. To date, the dynamical stability of Queta has been studied by Brown et al. (2021). Here we will examine whether Queta can survive the expected collisional environment within L₄ . We will summarize the dynamical findings from Brown et al. (2021) in Section 3.

In Section 3, we mentioned the input parameters that are needed for this work. First, we have described that the Boulder code requires population-specific parameters. These are the initial SFD, intrinsic impact probability, P_i , and the average impact speeds, v_i , of the population. In the case of the JTs we can treat the two swarms separately because they do not overlap dynamically and thus do not contribute to the collisional evolution of each other. For the SFD we, therefore, assume a slightly larger population than L₄. The largest object in our SFD has a diameter of 140 km. Between our largest size and 100 km we have a steep slope leading to 15 Trojans larger than 100 km. Below 100 km we impose a cumulative slope of 2.1 down to our smallest size of 2 m. This initial SFD is shown in later figures (e.g., Figure 10) and has a total mass of 4 × 10⁻⁶ M_⊕. This mass is consistent with the estimate by Nesvorný et al. (2013). The collisional environment within each swarm is defined by P_i = 7 × 10⁻¹⁸ km⁻² yr⁻¹ and v_i = 4.6 km s⁻¹ (Davis et al. 2002; Nesvorný et al. 2018). The intrinsic impact probability takes into account the resonant dynamics of the Trojans.

The second set of input parameters relates to the material properties of the JTs through the ${Q}_{D}^{* }$. As described above we have tested two types of disruption laws, one with a canonical shape and a minimum in ${Q}_{D}^{* }$ at D = 250 m and one with a minimum at D = 20 m. The former ${Q}_{D}^{* }$ form a set with the weakest material labeled Q₀. Scaled versions of Q₀ have been tested up to 100 × Q₀ (see Figure 6). The parameters ${Q}_{D}^{* }$ (Equation (2)) used in this work are listed in Table 1.

Table 1. Parameters used for ${Q}_{D}^{* }$ in This Work, and Resulting Minimum of ${Q}_{D}^{* }$, D_min

Cs	ss	Cg	sg	ρ	Dmin	Description
(erg g−1)		(erg cm3 g−2)		(g cm−3)	(m)
1.50 × 106	−0.40	0.05	1.30	1	250	Q0 case
7.65 × 105	−0.36	1.40	1.36	1	20	${Q}_{D}^{* }$ with minimum at 20 m

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In a first step we assume that the SFD of the JT population has not evolved over the age of the solar system. To do this we switch off the collisional evolution of the SFD in Boulder. This retains only the part of the code that monitors impacts on the Eurybates-Queta system and as a consequence allows us to separate the effects from the evolving Trojan SFD (see Section 4.2) and the inherent survival probability of Queta given a certain Trojan population. Though this is a simplification it does allow us to retrieve a first estimate for the timescales. As described above we assume two different initial SFDs.

First, we assume that the SFD below 10 km continues with the same slope as at larger sizes (2.1, see orange line in Figure 10). In this case the probability that Queta survives is a simple exponential decay over time and depends only on ${Q}_{D}^{* }$ (left panel of Figure 7). At a certain age in the family the probability that Queta would have survived to this day becomes so small that we should not consider these cases. For our purpose we use a 10% survival probability as a discriminator between a scenario that we still consider plausible and scenarios where it would simply be too unlikely for Queta to have survived. We find that a large part of parameter space can be excluded (right panel of Figure 7).

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Therefore, Figure 7 can answer the question of the age of the Eurybates family as a function of collisional strength. Any cases below the 10% contour line in the right panel of Figure 7 should be considered statistically improbable. Given this static SFD with a slope of 2.1 between 2 m and 100 km, we find that ${Q}_{D}^{* }$ needs to be strictly larger than 4.6Q₀. Otherwise Queta would not survive for more than 1 Gy, the lower limit for the family age. Interestingly, 4.6Q₀ corresponds to the ${Q}_{D}^{* }$ found by Benavidez et al. (2018) for their rubble pile asteroids. As we have seen above, one of their simulations is a good fit for the SFD of the Eurybates family. If the parent body of the Eurybates family was indeed this weak (4.6Q₀), and this disruption law is representative of Jupiter Trojans, and the present-day SFD continues with the same slope down to small sizes, then this result would suggest that the Eurybates family is a "young" family (1 Gy).

We also observe that the age of the family and the collisional strength are directly linked. As the collisional strength increases so does the age of the family. Strictly speaking these are all upper limits for the age of the family. For example, our strongest material (100Q₀) never falls below 10% and hence the family age could be up to the age of the solar system. On the other hand the 46Q₀ case, which in fact corresponds roughly to the ${Q}_{D}^{* }$ of basaltic material in Benz & Asphaug (1999; see blue line in Figure 6) gives us an upper limit of ∼3.7 Gy for the family age.

We can also examine what contributes to the destruction of the Eurybates-Queta system. Figure 8 shows our results for different ${Q}_{D}^{* }$ and a family age of 1 Gy (left) and 4.5 Gy (right). We find that in the overwhelming fraction of cases, an impact occurs on Queta that catastrophically disrupts it (pink in Figure 8). There are no cases where an impact on Queta perturbs its orbit enough for the binary to be dissolved (red in Figure 8). In a minor fraction of cases (up to 10% after 4.5 Gy for 100Q₀), an impact on Eurybates leads to the ejection of Queta. It is interesting to note that the relative fraction of cases where an impact on Eurybates destroys Eurybates (light blue in Figure 8) decreases with respect to when such an impact dissolves the binary (dark blue in Figure 8). This result makes sense because as the material strength increases it becomes harder to catastrophically disrupt Eurybates but "easier" to nudge Eurybates such that the binary orbit is dissolved and Queta is lost from the system. The fact that the survival of the Euybates-Queta system is dominated by impacts on Queta is noteworthy because it is different than what has been observed for other binaries, in particular the large Trojan binary pair of Patroclus and Menoetius (P-M binary; Nesvorný et al. 2018), the final fly-by target of the Lucy mission. The P-M system is an almost equal size binary with diameters of 113 km and 104 km, respectively. The survival of the P-M binary is limited by impacts on one of the components that alter the orbit such that the binary is dissolved. The binaries survival is thus sensitive to the semimajor axis of the binary (Nesvorný et al. 2018) because binaries with the larger semimajor axis are dissolved more easily than tight binaries. Nesvorný et al. (2018) estimated that the specific energy needed to dissolve the P-M binary through a noncatastrophic impact is much smaller than the respective ${Q}_{D}^{* }$. In our case, it is much easier to catastrophically disrupt the much smaller Queta (∼1 km). Therefore, in contrast to the P-M binary, the Eurybates-Queta system's survival is not sensitive to the semimajor axis.

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Up to now we have assumed that the Jupiter Trojan SFD simply continues below 10 km with respect to the slope above 10 km. Because there are indications that the SFD bends/breaks at or somewhat below 1 km (Yoshida & Nakamura 2008; Wong & Brown 2015; Yoshida & Terai 2017), we now assume that the Trojan SFD has a break at 10 km and continues at small sizes with a slope of q = 1 (significantly flatter than above 10 km). In this case, we find a very different picture from Figure 9. We find that the age and collisional strength become unconstrained. Queta can survive 4.5 Gy in all cases of ${Q}_{D}^{* }$. In the worst case scenario with extremely weak material (Q₀) there is still a 50/50 chance of Queta surviving. In most cases (${Q}_{D}^{* }\geqslant 4.6{Q}_{0}$) the survival probability of Queta over 4.5 Gy is ∼80%.

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This behavior is readily explained. Due to the very flat SFD below 1 km, the impactor population that could destroy Queta is simply not very large. Consequently the probability for Queta to be catastrophically disrupted (pink in Figures 8 and 9) shrinks to the single percent digits. In this situation, the probability of the Eurybates-Queta system being disrupted is dominated by impacts on Eurybates. This happens either by direct disruption of Eurybates in the case of weak material or dissolution of the binary for strong material. Further, we observe a transition from disruptions of Eurybates dominating the survival probability at low collisional strength to impacts on Eurybates dissolving the binary at large collisional strengths. At low strength, Eurybates is easily disrupted and thus becomes the primary mode limiting the retention of Eurybates-Queta system. As the strength increases Eurybates becomes hard to disrupt but can be nudged strongly enough to dissolve the orbit of Queta.

As argued above, the assumption of a static Jupiter Trojan SFD over the age of the solar system is likely to be an unrealistic simplification. Accordingly, we now consider an initial Trojan SFD with a slope of 2.1 below 10 km (see the orange line in Figure 10) and let it evolve over the age of the solar system.

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The results of 300 such simulations for a ${Q}_{D}^{* }$ of Q₀ and 100Q₀ are shown in Figure 10. We show 300 simulations to illustrate the stochastic nature of the collisional evolution of the Trojan population. The sets of simulations differ only in the random seed given to the code. Each simulation, therefore, signifies a possible future of the same initial state of the system.

We find that the slope between ∼6 km and 100 km remains unaltered though the population is depleted slightly in both cases. In both cases shown in Figure 10 the SFD below ∼6 km is significantly affected by collisional grinding. In particular, the population is depleted between 10 m and 3 km. This results in a turnover between 2 and 6 km. The location of this bend depends on the ${Q}_{D}^{* }$. It is likely that this is the cause of the bend in the SFD seen by Yoshida & Nakamura (2008) and Wong & Brown (2015). But we should also point out that Yoshida & Terai (2017) reported no detection of this bend, and thus its existence and location require further observations. From a theoretical standpoint we expect there to be a bend in the Trojan SFD (Figure 10), therefore it is likely "merely" a matter of getting more observations to identify it.

The depletion of the JT population between 10 m and 1 km directly affects the survival probability of Queta because this is the impactor population responsible for its destruction (see above). We find that the survival probability of Queta significantly increases in all cases (see bottom row of Figure 11). Compared to the static SFD (Figure 8), all of the ${Q}_{D}^{* }$ allow Queta to plausibly survive with a family age of more than 1 Gy. But for these cases, we also need a material strength of at least 10Q₀ for Queta to have a 10% chance of surviving the age of the solar system.

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Figures 12 shows the survival probability as a function of the family age. Compared to the static SFD (Figure 7) we observe that the decay is no longer exponential but becomes almost linear in time for the strongest material. Further, the part of parameter space consistent with Queta's existence (cases above the 10% contour) is greatly expanded compared to the static SFD (right panel of Figure 7). This result is not a surprise because there are fewer projectiles in the population capable of destroying Queta. The collisional strength derived from the Benavidez et al. (2012) rubble piles with ${Q}_{D}^{* }=4.6{Q}_{0}$ now puts an upper limit on the family age. The family is no older than 3.7 Gy with a probability of 90% for this size distribution and disruption law.

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Finally, we will examine what happens to our results if we employ a more unconventional ${Q}_{D}^{* }$. As described in Section 4.2 (see also Figure 6) the minimum value of ${Q}_{D}^{* }$ in literature for rocky and icy targets is typically around a diameter of ${D}_{\min }=200$ m. There are indications from cratering results obtained by the New Horizons mission to the Pluto system and to the cold classical KBO Arrokoth (Singer et al. 2019; Morbidelli et al. 2021); however, this might not hold for icy bodies of the outer solar system. The critical clue is that the cumulative power-law slope of the Kuiper Belt SFD is approximately q = 1 over the size range between 1 km and tens of meters in diameter Morbidelli et al. (2021).

As discussed in Section 4.2, the location where the intermediate slope steepens back up to the slope in the strength regime is indicative of the minimum in ${Q}_{D}^{* }$. For the asteroid belt, this change in slope is observed at 200 m (e.g., Bottke et al. 2020). For the Kuiper Belt, we have yet to identify the size of where this putative change in slope would take place. All we can say is that if it exists, it has to occur at sizes smaller than tens of meters. Accordingly, we infer that ${Q}_{D}^{* }$ for Kuiper Belt objects, and presumably Trojans, need to have a minimum value smaller than a few tens of meters. We therefore assume an unconventional ${Q}_{D}^{* }$ (purple line in the right panel of Figure 6) which follows the Benz & Asphaug (1999) basaltic material in the gravity regime but then continues down to include continuously weaker material strength all the way down to ${D}_{\min }=20$ m before transitioning to the strength regime. This minimum value is consistent with the arguments made by Morbidelli et al. (2021), namely that an upturn near that point would provide the means to explain the quantity of dust observed in the Edgeworth-Kuiper Belt by New Horizons.

Figure 13 shows what the SFD evolved to after 4.5 Gy. The SFD is more complex than the ones previously shown (Figure 10). We interpret it using the local slopes shown in the right panel. Here we have defined the local slope as the power-law index connecting the two adjacent size bins within the Boulder output results. The typical distance between size bins is 1.25D.

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O'Brien & Greenberg (2003) showed for a population in collisional equilibrium that the differential power-law exponent, p, is connected to the slope, s, of the gravity or strength regime respectively via:

$$\begin{eqnarray}&&p=\displaystyle \frac{7+\tfrac{s}{3}}{2+\tfrac{s}{3}}.\end{eqnarray} \tag{ 3 }$$

If ${Q}_{D}^{* }$ is independent of size (i.e., s = 0) we retrieve the classical Dohnanyi steady-state solution of p = 3.5 (Dohnanyi 1969). In our case the gravity regime of ${Q}_{D}^{* }$ has a slope of s = 1.3 resulting in p = 3.05, which corresponds to a cumulative slope of 2.05. This is close to the observed slope between 10 and 100 km, which could indicate that this part of the population was in collisional equilibrium when the Trojans were captured. The slope at small sizes should correspond to the respective slope predicted for the strength regime. In between the two steep slopes of very small and large sizes we find a shallow slope in the diameter range from roughly 30 m to 1 km. Note that the location where the SFD transitions to the slope in the strength regime is precisely the location of the minimum in ${Q}_{D}^{* }$ at roughly ${D}_{\min }\,=100-200$ m of the traditionally shaped ${Q}_{D}^{* }$ (Figure 10), and ${D}_{\min }=20$ m for our unconventional ${Q}_{D}^{* }$ (Figure 13). This makes sense as objects smaller than D_min are stronger than their larger companions. As a consequence they can more easily break larger objects but not be easily broken up themselves.

Starting at large JTs (10 km < D < 100 km) in Figure 13, the SFD is largely unchanged from the initial SFD, and retains q ∼ 2. Few of these objects have been disrupted. At 10 km the slope begins to steepen slightly, reaching a peak at around 2.5 km before beginning to flatten. This indicates a steady-state "bump" is being formed by collisional evolution; it is analogous to the bump seen in the asteroid belt near 2–3 km, and we will discuss its origin below. The flattest part of the SFD occurs at ∼200 m. From there the slope steepens up again to reach the predicted slope in the strength regime for a collisionally-evolved Dohnyani-like SFD (O'Brien & Greenberg 2003). This final slope is reached as expected around ${D}_{\min }=20$ m. As with the asteroid belt, the limited number of projectiles near this minimum means there should be an excess of bodies at larger sizes corresponding to the targets that would be disrupted by 20 m projectiles. That explains the change of the aforementioned "bump" at around 1 km. Similarly, the slight "dip" in the population around 10 km is produced by the excess number of 1 km bodies in the bump. Intriguingly, there may be some limited evidence of a "dip" near 10 km Trojans. The tentative shallowing of the slope at small sizes (∼5 km Yoshida & Nakamura 2008; Wong & Brown 2015) is associated with a slightly steeper slope than the nominal 2.1 just before the apparent turnover. While this is not definitive proof, and details need to be worked out in future work and observations, we argue that there is sufficient support here to warrant seriously entertaining the possibility of an unorthodox ${Q}_{D}^{* }$ with a D_min at ∼20 m.

It is also worth mentioning that the bump at ∼1 km can cause the present-day population to exceed the initial population at that size (left panel of Figure 13). This is not observed in any of the cases that use a traditionally shaped ${Q}_{D}^{* }$. The reason is that traditionally shaped ${Q}_{D}^{* }$ produce a turnover at larger sizes (3–7 km) than for our unconventional ${Q}_{D}^{* }$ (below 1 km). This leads to a smaller 1 km impactor population for traditional ${Q}_{D}^{* }$ functions that is not large enough to disrupt many Trojans of the order of 10 km, i.e., it cannot produce a dip at those sizes. These larger breakups act as a source population for further 1 km Trojans and, therefore, allow a "bump" in the unconventional ${Q}_{D}^{* }$ case.

This model not only has the advantage of fitting into the data returned by the New Horizons mission (Singer et al. 2019; Morbidelli et al. 2021) and qualitatively matching the flattening of the Jupiter Trojan SFD at small sizes (Yoshida & Nakamura 2008; Wong & Brown 2015) but it also liberates the constraint on the collisional lifetime of Queta and thus the age of the Eurybates family.

In fact, this unconventional ${Q}_{D}^{* }$ would allow Queta to survive under any of our assumptions for at least the age of the solar system (Figures 8 and 11). The reason is that the projectile population capable of disrupting Queta is low and stays that way for 4.5 Gyr.

This unconventional ${Q}_{D}^{* }$, though, would not match the strength of Proto-Euryabes derived from SPH simulations in Section 2.2. Note also that we have tested smaller values for D_min, but they produce SFDs that are less consistent with New Horizons crater constraints because the shallow slope in the SFD continues to even smaller sizes than shown in Figure 13. This would make it difficult to reconcile the New Horizons crater record with the Kuiper Belt dust observations (see the argument in Morbidelli et al. 2021).