# Satellite Formation around the Largest Asteroids

Kevin J Walsh, Ronald-Louis Ballouz, Harrison F Agrusa, Josef Hanuš, Martin Jutzi, Patrick Michel

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Kevin J Walsh, Ronald-Louis Ballouz, Harrison F Agrusa, Josef Hanuš, Martin Jutzi, et al. Satellite Formation around the Largest Asteroids. The Astrophysical Journal Letters, 2025, 986, 10.3847/2041-8213/adc562. hal-05403786

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# Satellite Formation around the Largest Asteroids

Kevin J. Walsh<fnref n="1" />, Ronald-Louis Ballouz<fnref n="2" />, Harrison F. Agrusa<fnref n="3" />, Josef Hanuš<fnref n="4" />, Martin Jutzi<fnref n="5" />, and Patrick Michel<fnref n="3" />,<fnref n="6" />

*Received 2024 October 14; revised 2025 March 17; accepted 2025 March 26; published 2025 June 5*

## Abstract

Satellites around large asteroids are preferentially found among those with the most rapid rotation and elongated shape. The taxonomic statistics are similarly skewed; in total, 13 asteroids larger than 100 km are known to have satellites, but none have been discovered among S-type asteroids. Previous modeling suggests that satellites could be generated by impacts, but spin and shape have never been tracked in models to relate collisional circumstances with those two observed properties concerning the primary. Here we show, by combining simulations of impacts into porous low-density asteroids, their subsequent disruption, reaccumulation, and long-term satellite stability, a direct pathway for the formation of satellites. The immediate distortion and elongation of a rotating target body provide a launching point for some debris distinct from simple ballistic ejecta trajectories. The debris that are found to originate from the distorted long axis is sourced primarily from 10 to 20 km below the surface and can be placed directly onto eccentric orbits with sufficiently large pericenter distances that avoid rapid reimpact. The specific energy and resultant total mass loss in satellite-forming collisions are not constraining, which explains the observed lack of correlation between asteroids with satellites and those that are part of large asteroid families.

*Unified Astronomy Thesaurus concepts: Asteroid satellites (2207); Main belt asteroids (2036)*

## 1. Introduction

The known population of asteroids with satellites is constantly growing with new discoveries. They have been found widely across the various small-body populations in the solar system, are being better characterized with new technologies (P. Vernazza et al. 2021), and have been the subject of in situ study from recent space missions (R. T. Daly et al. 2023). There are current space missions that will encounter more in the near future (M. E. Brown et al. 2021; K. S. Noll et al. 2023), while current and future observatories are likely to dramatically increase the list of candidate systems (L. Liberato et al. 2024).

Asteroid satellites provide important information regarding the collisional history of different populations, by way of their existence and properties. Asteroid mass can be measured from satellites' orbits, and when combined with sizes and shapes, which can be constrained with ground-based telescopes, densities can be estimated. Changes in orbital properties of satellites over time can also provide insight into the internal properties of the primary (e.g., M. Brož et al. 2022a). Finally, any differing binary properties between asteroid taxonomic classes may provide insight into the satellite formation process (e.g., K. Minker & B. Carry 2023), the internal properties of different types of large asteroids, or both.

Detection and characterization of asteroid satellites have been done in numerous ways: in situ detection with spacecraft (C. R. Chapman et al. 1995; R. T. Daly et al. 2023), via

occultation/eclipses in light-curve amplitude data (P. Pravec & G. Hahn 1997; S. Mottola & F. Lahulla 2000), direct imaging from ground- or space-based platforms (W. J. Merline et al. 1999), or via radar (J. L. Margot et al. 2002). The properties of asteroid satellites have a strong dependence on the primary's size, where satellites among asteroids with diameter larger than 100 km are relatively small (<0.1 times the diameter of the primary) and slightly more distant than those found among asteroids smaller than 10 km (see J. L. Margot et al. 2015; K. J. Walsh & S. A. Jacobson 2015). There are few known satellites in between, with primaries between 10 and 100 km, where discoveries may be frustrated by observing biases (see K. S. Noll et al. 2023), although astrometric studies have identified new candidates in this size range yet to be confirmed with other techniques (L. Liberato et al. 2024).

The known satellites around the large asteroids (hereafter "large asteroids" are those with diameter larger than ~100 km) are all on orbits between 3 and 14 primary radii (Figure 1), which occupy between 0.5% and 4% of the primary body's Hill sphere. Where known, their orbital eccentricities are low (below ~0.1). Five of the 13 known systems have multiple satellites, with (130) Elektra having three (A. Berdeu et al. 2022). The satellites are relatively small compared to their primary, where all of the satellites have radii smaller than ~0.2 primary radii and most are below 0.1 primary radii. Thus, it is expected that more than 99% of a system's mass is typically contained in the primary of a multiasteroid system.

The primaries of large main-belt asteroid systems are also among the most elongated among their peers, as evident both from analysis of light curves for a huge number of bodies (B. D. Warner et al. 2021) and via direct imaging of ~30 of the largest asteroids (P. Vernazza et al. 2021). Light-curve

<figure><img src="image_3.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Southwest Research Institute, 1301 Walnut St., Suite 400, Boulder, CO 80305, USA</footnote>

<footnote n="2">²Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA</footnote>

<footnote n="3">³Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, France</footnote>

<footnote n="4">⁴Charles University, Faculty of Mathematics and Physics, Institute of Astronomy, V Holesovíkách 2, 18000 Prague 8, Czech Republic</footnote>

<footnote n="5">⁵Space Research and Planetary Sciences, Physikalisches Institut, University of Bern, Switzerland</footnote>

<footnote n="6">⁶The University of Tokyo, Department of Systems Innovation, Japan</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1.** The population of observed large asteroids with satellites. (a) The size ratio of satellites to primary body as a function of their orbit semimajor axis scaled by the primary radii for the observed population of satellites among asteroids with diameter greater than 100 km. Multiple systems are connected with lines, and the size of each point gives a relative scale of the size of the primary, where *D* = 286 km (87) Sylvia is indicated for scale. (b) Rotation periods of asteroids larger than 50 km as a function of diameter derived from light-curve observations, where S-complex and C-complex objects are in red and blue, respectively, and those with observed satellites are aqua. (c) Asteroid light-curve amplitudes in magnitude plotted as a function of the asteroid's rotation period in hours, where all asteroids are in purple and those with satellites are in aqua. The light-curve amplitudes are from the same light-curve database and can be roughly translated into axis ratios via $\Delta m = 2.5 \log(a/b)$, where $\Delta m$ is the light-curve amplitude and *a* and *b* are the long and intermediate principal axes, respectively.</figcaption></figure>

<table><caption>**Table 1**

Data on Asteroids with Diameters Larger Than 100 km That Have Confirmed Satellites</caption><thead><tr><th rowspan="2">Primary</th><th rowspan="2">Diameter<br>(km)</th><th colspan="2">Light-curve Data</th><th rowspan="2">p<sub>v</sub></th><th rowspan="2">Tax</th><th rowspan="2">Fam</th><th rowspan="2">Dimensions<br>(km)</th><th rowspan="2">Reference</th></tr><tr><th>Per Hour</th><th>Amplitude</th></tr></thead><tbody><tr><td>(22) Kalliope</td><td>150</td><td>4.14820</td><td>0.53</td><td>0.17</td><td>M/x</td><td>Y</td><td>207,145,122</td><td>M. Ferrais et al. (2022)</td></tr><tr><td>(31) Euphrosyne</td><td>268</td><td>5.529595</td><td>0.11</td><td>0.05</td><td>C</td><td>Y</td><td>286,274,247</td><td>B. Yang et al. (2020)</td></tr><tr><td>(41) Daphne</td><td>187</td><td>5.98798</td><td>0.34</td><td>0.08</td><td>Ch</td><td>N</td><td>235,183,153</td><td>B. Carry et al. (2019)</td></tr><tr><td>(45) Eugenia</td><td>188</td><td>5.699152</td><td>0.41</td><td>0.04</td><td>C</td><td>N</td><td>252,191,138</td><td>P. Vernazza et al. (2021)</td></tr><tr><td>(87) Sylvia</td><td>274</td><td>5.183641</td><td>0.62</td><td>0.05</td><td>P/X</td><td>Y</td><td>363,249,191</td><td>P. Vernazza et al. (2021)</td></tr><tr><td>(93) Minerva</td><td>154</td><td>5.981768</td><td>0.20</td><td>0.07</td><td>C</td><td>N</td><td>...</td><td>J. Hanuš et al. (2017)</td></tr><tr><td>(107) Camilla</td><td>254</td><td>4.843928</td><td>0.53</td><td>0.06</td><td>P/X</td><td>N</td><td>...</td><td>J. Hanuš et al. (2017)</td></tr><tr><td>(121) Hermione</td><td>187</td><td>5.550877</td><td>0.70</td><td>0.08</td><td>Ch</td><td>N</td><td>...</td><td>M. Viikinkoski et al. (2017)</td></tr><tr><td>(130) Elektra</td><td>201</td><td>5.224663</td><td>0.58</td><td>0.09</td><td>Ch</td><td>N</td><td>267,202,151</td><td>M. Fuksa et al. (2023)</td></tr><tr><td>(216) Kleopatra</td><td>118</td><td>5.385280</td><td>1.22</td><td>0.15</td><td>M</td><td>N</td><td>267,61,48</td><td>F. Marchis et al. (2021)</td></tr><tr><td>(283) Emma</td><td>142</td><td>6.89523</td><td>0.57</td><td>0.03</td><td>P</td><td>N</td><td>...</td><td>M. Viikinkoski et al. (2017)</td></tr><tr><td>(702) Alauda</td><td>202</td><td>8.354</td><td>0.12</td><td>0.06</td><td>C</td><td>Y</td><td>...</td><td>...</td></tr><tr><td>(762) Pulcova</td><td>142</td><td>5.839</td><td>0.42</td><td>0.04</td><td>F/C</td><td>N</td><td>...</td><td>...</td></tr></tbody></table>

**Note.** The geometric visible albedo values *p*<sub>v</sub> and taxonomies are taken from the MP3C database. The sizes and rotation periods are based on published shape models derived from disk-resolved images (references in the last column). The shape model is unavailable for Alauda and Pulcova, so these parameters are taken from the MP3C database. The “Fam” column indicates whether the asteroid is associated with a collisional family (“Y” for Yes, “N” for No) in which it is assumed to be the largest remnant of the family-forming event. Not listed is (90) Antiope, a system of two nearly 90 km components on a close orbit whose combined brightness often leads to it being listed as a body larger than 100 km.

observations have been published for 222 objects larger than *D* > 100 km (with a quality of *U* = 3 that ensures unambiguous period measurements), and they can immediately detect a lower limit for *a/b* axis ratio (long and intermediate, respectively), but

they require favorable observing geometry over numerous apparitions to constrain the short axis *c* (B. D. Warner et al. 2021). Direct imaging can provide more detailed shape information, including concavities (P. Vernazza et al. 2021).
The primaries of known large main-belt asteroids are of the taxonomic types C, M, and P, with a notable lack of S-type primaries (Table 1). There is also a lack of satellites among the very largest asteroids ($D > 300$ km), where Ceres and Vesta were both visited by spacecraft with instrumentation capable of finding even very tiny orbiting debris (L. A. McFadden et al. 2018). Nor have satellites been found around Hygiea or Pallas, the third- and fourth-largest asteroids, despite being observed with modern ground-based adaptive optics imaging (M. Marsset et al. 2020; P. Vernazza et al. 2021). Meanwhile, asteroids with diameters between approximately 10 and 100 km have a lower frequency of known satellites and less strict correlation between satellite presence and primary shape and spin (Figure 1). Notably, with decreasing primary diameter, satellites with semimajor axes of ~3–14 primary radii would be increasingly hard to spatially resolve with direct imaging techniques (K. S. Noll et al. 2008). Therefore, the scarcity of asteroid satellites around these bodies may simply be due to an observational bias, rather than the product of a physical mechanism that would prevent their formation.

It is generally thought that impacts are responsible for creating satellites around large asteroids (P. Michel et al. 2003; D. D. Durda et al. 2004). Impacts are an inescapable event for large asteroids; furthermore, they are known to happen, via in situ observations of large impact basins on asteroids such as Vesta (P. Schenk et al. 2012), the existence of many tens of families of asteroids (D. Nesvorný et al. 2015), and collisional probabilities calculated for the well-measured orbital and size distribution of bodies inhabiting the main asteroid belt (W. F. Bottke et al. 2005). Numerical models of collision and reaccumulation have found satellite formation to be possible (P. Michel et al. 2003; D. D. Durda et al. 2007; P. G. Benavidez et al. 2012; M. Brož et al. 2022b).

Modeling of impact probabilities, the size distribution of asteroids, and the energy required to disrupt and disperse half of a target's mass finds that roughly four asteroids larger than 100 km should be disrupted every 1 Gyr (W. F. Bottke et al. 2005). Analysis of the main asteroid belt population finds that roughly 10–20 known asteroid families resulted from the disruption of targets larger than 100 km (P. Tanga et al. 1999; W. F. Bottke et al. 2005; M. Brož et al. 2013). However, the largest remnants of very large families do not have observed satellites. Rather, only half of the observed large asteroids with satellites are the largest members of families, all of which are modest or small in size (see D. Nesvorný et al. 2015). The lack of correlation between observed satellites and membership in a very large family is suggestive of subcatastrophic impacts being more important than catastrophic collisions in the formation of satellites.

Numerical simulations of the entire satellite formation process are challenging owing to the different timescales and size scales involved in the different phases of the whole collisional process. Modeling the impact shock propagation through the target requires simulation times that scale with the asteroid size and sound speed, on the order of tens to hundreds of seconds for 100 km objects. The subsequent ejection and reaccumulation of the debris back onto the largest remnant and into a family of smaller remnants—some of which may end up as bound satellites—requires simulation times on the order of tens of hours to days for 100 km objects. Orbital periods of temporary satellites can be many days, and their orbits may evolve over many years. Previous large simulation efforts have

also attempted to cover a wide range of possible collision scenarios, varying impact speeds, angles, impactor size (D. D. Durda et al. 2007), and the parent body's internal properties (e.g., P. Michel et al. 2003; M. Jutzi et al. 2010). The size of the parameter space and the limitation of the computational tools required simplifications such as neglecting the spin and shape of the remnants by performing perfect mergers for colliding particles. These simulations found satellite formation to be more prevalent for disruptions where more than half of the target mass was lost, which occurs when the specific impact energy exceeds the value called the catastrophic disruption threshold, $Q_D^*$ (D. D. Durda et al. 2007). Highly oblique impacts with impact angles of 75° resulted in no satellites, likely due to impacts that were not very disruptive, with more than 90% of the mass of the target remaining in the largest remnant in each case tested.

## 2. Results

We present impact models of asteroid collisions typical of the main asteroid belt with a parameter space designed to explore direct pathways to generate satellites. Specifically, the parameters are not chosen to cover all parameter space, but rather to ensure impact outcomes with a range of final shapes and spin states to examine the correlation of satellite production with these post-impact properties. A hydrodynamic model of each impact is performed, and the outcome is handed off to *N*-body gravitational models. The *N*-body modeling is done with pkdgrav, which is capable of modeling granular mechanical interactions that allow for the evolving shape of the various remnants to be tracked throughout the reaccumulation process. The final shape and spin of the largest remnant are then used in a longer-term dynamical simulation of the initial temporary satellites to determine which, if any, are stable. We simulate head-on and oblique impacts into spherical targets that have a range of preimpact rotation rates.

### 2.1. Simulation Framework for Hypervelocity Impacts into Asteroid Parent Bodies and Long-term Dynamical Evolution of Ejecta

The smoothed particle hydrodynamics (SPH) model is specific to primitive (low-albedo, low-density) asteroid classes with compaction as a key physical process (M. Jutzi et al. 2019). The SPH simulations used an initial porosity of 50% and bulk densities around $1.3 \text{ g cm}^{-3}$, which are typical of dark C-complex asteroids (B. Carry 2012). The SPH simulations used nonrotating targets and had a resolution of 400,000 particles.

To retain the post-collision shape and spin of reaccumulated remnants, we utilize a handoff routine between an SPH code (M. Jutzi et al. 2008; M. Jutzi 2015) that models the impact and shock propagation and an *N*-body code pkdgrav (D. C. Richardson et al. 2000) that handles the subsequent reaccumulation due to self-gravity and the granular mechanics that govern the final shapes and spins of reaccumulated fragments (S. R. Schwartz et al. 2012). The shape of the largest remnant is captured during the handoff by “wrapping” the SPH particles in a 3D $\alpha$-shape model. This wrapped shape model is the outcome of a geometric algorithm that constructs concave shapes around a set of points in Euclidian space (R.-L. Ballouz et al. 2019). This effectively captures the irregular shape of the largest body at this point in time. This is then used
to “carve” the same shape from discrete *N*-body particles (R.-L. Ballouz et al. 2019). The mass and velocity distribution are then transferred to the *N*-body shape, ensuring linear and angular momentum conservation. The *N*-body shape is then integrated forward with a soft-sphere discrete element (SSDEM) version of pakdgrav (S. R. Schwartz et al. 2012; Y. Zhang et al. 2017). This process is applied only to the largest remnant, which is the primary body for orbiting debris. It has previously been shown that escaping ejecta can be bound to other escaping ejecta (D. D. Durda et al. 2004; “EEBS”), but these are unlikely to include large enough debris to be in the size ranges considered here. The debris is analyzed to determine whether they are bound to the largest remnant and their instantaneous orbital properties. A more detailed analysis of the evolution of their orbits that considers the final shape and spin of the largest remnant is described below.

To reduce the computational burden, the targets in the SPH models are not rotating, so an additional preprocessing step is completed during the handoff from SPH to *N*-body models to simulate the effect of a rotating target. This preprocessing step is completed to expand the parameter space of our study with little added computational cost compared to the computational expense of running multiple cases in SPH with varying preimpact target spin. At the time of handoff, 400 s after the time of impact, an additional change in linear and angular momentum is added to reflect a modeled preimpact rotation of the primary (see the Appendix). This allows an assessment of preimpact rotation on the outcome of the impact and reaccumulation process. More importantly, this produces a wider range of post-impact shape and spin characteristics for the largest remnant, allowing a comparison between satellite formation processes and post-impact system properties.

The SSDEM models have various parameters (static and rolling friction, etc.) that control the outcome of contact and interpenetration of particles. Here we use a few simple friction settings previously used in other studies that bracket expectations for asteroid behavior—high friction, low friction, and zero friction (M. Marsset et al. 2020). The reaccumulation stage is modeled for 38 hr, during which the largest remnant converges toward its final shape and spin state and the population of initial orbiting debris stabilizes. The final shape is characterized by dynamically equivalent equal-volume ellipsoids (DEEVEs; see H. F. Agrusa et al. 2024), where the long semiaxis ($a_{pri}$) is typically used to compare with satellites’ close approach distances and is distinct from the volume-equivalent radius ($R_{pri}$).

The SPH simulation resolution of 400,000 means that for a target of diameter 100 km a single simulation particle would have a diameter of 1.4 km, which is similar in size to the smallest known satellite in this population (S/2014 (130) 2 orbiting Elektra is $1.6 \pm 0.4$ km; A. Berdeu et al. 2022). At the end of the 38 hr reaccumulation, nearly all simulated cases have liberated material that is bound to the largest remnant, and most have tens or hundreds of bound clumps ranging from single particles up to clumps as large as $10^{-3}$ the mass of the primary. Given the similar size of single simulation particles with known satellites, we refer to all orbiting debris as temporary satellites.

The large number of temporary satellites found in nearly all simulations (see Table 2) suggests that additional processes related to temporary satellite survival and orbit evolution must

play an important role in removing or combining many or most of them. Therefore, in addition to the impact and reaccumulation models, the medium-term stability (on the order of 1000 days; hundreds of orbits) of temporary satellites is also mapped as a function of initial orbits and the shape and spin of the largest remnant. This uses the REBOUND and REBOUNDx simulation packages, where the primary body is modeled as a sphere having the same spin rate and $J_2$ and $C_{22}$ gravity terms resulting from the pakdgrav simulations, where these gravity terms represent the degree 2 terms of the primary’s gravity field accounting for its respective oblateness and ellipticity. This modeling provides an estimate of the dynamical lifetimes of the temporary satellites resulting from each simulation.

Thus, the process described above models a nearly catastrophic impact between asteroids, the propagation of the shock through the target, and its immediate reshaping and ejection of debris. Through the first handoff the shape of the largest remnant is maintained and allowed to continue to evolve, while its spin may be altered at the handoff and then allowed to continue to evolve over time. The second handoff, from the reaccumulation to long-term stability, freezes in the shape and spin of the largest remnant, which is now the primary body for all bound debris. The sizes of bound debris stop evolving in size, but their orbits continue to evolve.

## 2.2. Subcatastrophic Impact to Gravitational Reaccumulation of Primary and Satellites

The impact scenarios examined are typical of those experienced by large asteroids in the main asteroid belt, where this study used a subset of subcatastrophic impacts previously modeled with different tools (38), with impact speeds of 5 km s⁻¹ over a range of impact angles from 0° to 75° by 10–36 km diameter impactors onto 100 km targets. None of the impacts reached the specific impact energy required to disrupt a target, $Q_D^*$ (the value where half of the target mass is dispersed). Thus, the modeled scenarios are all subcatastrophic impacts, with approximately 50%–99% of the total mass returning to the largest remnant. During the handoff to the *N*-body phase, pakdgrav particles were given additional linear and angular momentum to represent targets with preimpact spin periods of 3, 4, 6, and 10 hr. The parameters examined were selected not to examine all possible impact scenarios, but rather to explore a wide range of post-impact shape and spin configurations of the largest remaining remnant and to permit analysis of satellite formation around them.

The *N*-body simulations show that the evolution of the system depends heavily on the angular momentum of the system, where cases with preimpact rotation or high angle of impact often produce outcomes similar to that shown in Figure 2. There is an initial asymmetric ejecta cloud, and the primary body is immediately distorted into an elongated shape. In this case a 13 km object impacts a 100 km spherical target with a 10 hr preimpact rotation at a 60° angle at a speed of 5 km s⁻¹. As a result, the primary is distorted (see Figure 2), and a tail of debris is generated that is distinct from the immediate ejecta cloud that trails the rotation of the primary and begins to clump into discrete remnants.

Strong correlations are found between the total mass and initial orbital properties of temporary satellites at the end of reaccumulation, specifically related to the shape and spin properties of the primary body. A search for bound particles
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Evolution of the target asteroid immediately following impact. The outcome of a 13 km object impacting a 100 km spherical target with a 10 hr preimpact rotation at a 60° angle at a speed of 5 km s⁻¹ shown at 0.1, 1, 3, 5, 7, and 9 hr post-impact. The particles that are part of the main primary mass are colored gray. The rest of the particles are colored by the pericenter of their instantaneous Keplerian orbit. Red colors correspond to trajectories with a pericenter less than the primary's long axis, while blue colors indicate pericenters above the primary's long axis. Dark red indicates a pericenter less than zero, which corresponds to a hyperbolic (unbound) orbit. See Figures 11 and 12 in the Appendix for visualizations of the same impact with a nonspinning target and a target initially with a 4 hr rotation period, respectively.</figcaption></figure>

and clumps was performed at the end of the reaccumulation phase for each set of parameters, where a hierarchical search for bound companions was performed on the system (Z. M. Leinhardt & D. C. Richardson 2005). Head-on impacts and lower-energy impacts (those that result in minimal reshaping) result in most ejecta leaving the surface on simple ballistic trajectories that should, without further perturbation, return to their original launch location and thus reimpact. This results in cases where temporary satellites have initial orbit pericenters smaller than the primary's long axis ($a_{pri}$; Figure 3).

This contrasts sharply with the high angular momentum cases where a significant fraction of the bound ejecta have pericenters larger than the primary's longest semiaxis, or even twice the longest semiaxis (Figure 3). The debris that reaches the highest pericenter is typically sourced from the tail of debris seen in Figure 2.

## 2.3. Temporary Satellite Properties

Observations (Figure 1) also show that there are some large asteroids with elongated shapes but moderate spin rates (>10 hr) that do not have known satellites. Furthermore, the nonspherical gravitational potential around highly elongated primaries will perturb close-in orbits (e.g., D. J. Scheeres 1999). Therefore, it is essential to compare the initial orbits of temporary satellites with stability zones around irregularly shaped spinning primaries to understand their long-term stability. Likewise, it is essential to compare the total mass of long-term stable satellites with the masses of observed

satellites to account for potential accretion processes and minimize dependence on simulation resolution.

At the end of each pkdgrav simulation (38 hr post-impact), the physical and orbital properties of the temporary satellites are tabulated (see Table 2). Specifically, we find that impacts resulting in more elongated primaries, as quantified by the ratio of their major to minor axis ($a/b$), tend to put more mass in orbit (Figure 4(a)), and these primaries also tend to have faster post-impact spins (Figure 4(b)). Remarkably, we see a sharp transition in the incidence of primaries with satellites at post-impact spin periods of 10 hr or less, matching observations.

Utilizing the shape and spin of the primary from each simulation, we use REBOUND to compute the lifetime of a test particle on an uninclined orbit as a function of its eccentricity and pericenter distance to constrain the stability of temporary satellites. Test particles are initialized at apocenter and integrated for 1000 days. The simulation is stopped if the particle impacts the circumscribing sphere of the primary or leaves the primary's Hill sphere (assuming that the primary is located at 2.5 au). The $J_2$ and $C_{22}$ moments, as well as the primary's circumscribing sphere, are defined based on the body's moments of inertia at the end of the reaccumulation stage (see Appendix). These stability maps find that for pericenters beyond ~2–3 primary radii (measured by the primaries' longest semiaxis length) satellites are for numerous orbits (Figure 5).

Not all observed elongated asteroids have satellites. While there are loss mechanisms such as impacts and tides, the
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Distribution of initial pericenters for temporary satellites. The distribution of satellite pericenters is shown in blue for a nonspinning target hit at an impact angle of 15°, where all temporary satellites have initial pericenters below 1 primary radius. This contrasts with a target with moderate spin of 10 hr and a highly oblique impact angle of 60°, shown in red (the same case as shown in Figure 2), that has significant ejecta on orbits with pericenter above 1 and even above 2 primary long semiaxis lengths.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Properties of temporary satellites from simulation outcomes. The normalized mass of satellites having initial orbits with pericenters above 1 primary long semiaxis length as a function of (a) primary post-impact axis ratio *a/b*, (b) primary post-impact rotation period, (c) mass of the largest remnant mass as a fraction of the total system mass (including escaping material), and (d) the ratio of the impact energy to the rotation-dependent catastrophic disruption threshold. The size of each point represents the total number of temporary satellites for each given simulation where the numbers spanned 0–316. The normalized mass is the total mass in satellites as a fraction of the primary’s mass. The gray zone in panel (b) shows the spin rates for all known primaries in the observed sample.</figcaption></figure>

stability analyses repeated with elongated shapes but moderate or slow rotation rate find that the initial stability of ejecta has a direct dependence on the spin rate for asteroids with identical elongated shapes (Figure 6). This effect should act to frustrate satellite formation around large and elongated asteroids that have moderate spin rates. Therefore, we conclude that a combination of both fast rotation and elongation resulting from a large impact is strongly preferred for this direct path to produce stable satellites.

## 2.4. *Mass and Density Effects, S-type Asteroids, and the Very Largest Asteroids*

There are no known satellites around large S-type asteroids. There are fewer large S-type asteroids in the main asteroid belt

(~30 compared to ~170 non-S-type asteroids), providing ~5× fewer targets for large impacts. This still leads to an expectation that a few S-type asteroids should have satellites, although it is possible that the lack of them is simply an artifact of small number statistics. If S-type asteroids required more energy to reach similar levels of disruption owing to their higher density, then an argument could be made about there also being fewer potential impactors. However, two things work against this idea. First, the power-law size–frequency distribution of asteroids has a slope of approximately –2.5 in this size range, so if twice as much energy is needed to generate an equal disruption, it would require an impactor with only 1.25 times larger size. This only decreases the available impactors by a factor of roughly 1.75 relative to C-type
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Two different impact outcomes with similar post-impact shape and total mass lost, but with different post-impact spin rates, where (a) a case resulting in a mildly elongated final shape (a, b, c = 57, 45, 40 km) but rapid 6.7 hr rotation period places numerous initial satellites into the medium-term stable regions (satellites plotted as gray circles). The impact scenario had a 13 km impactor hit at an oblique 60° impact angle into an initially 10 hr rotating target that resulted in 80% of the mass remaining in the largest remnant (case 17 in Table 1). (b) An outcome with a similarly elongated final shape (a, b, c = 59, 52, 34 km) but a final 12.2 hr rotation period that resulted in nearly all initial orbits having pericenters below 1 primary radius and almost certain to reimplement. The impactor was 7 km and impacted at a 45° angle, resulting in a final remnant with 88% of the total system mass (Case 12 in Table 1). The diagonal structures on these plots are locations of constant semimajor axis and correspond to spin-orbit resonances with the primary.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Stability maps for different spin periods. Stability maps are shown as a function of rotation period around a primary with a bulk density of 1 g cm⁻³ having the same semiaxes as the primary of Figure 5(a) (a, b, c ~ 57, 45, 40 km) corresponding to a $J_2$ and $C_{22}$ of ~0.05 and ~0.026 when normalized to the primary's long axis. At short rotation periods, the primary spins much faster than the test particles' orbit periods, and the perturbing effect of $C_{22}$ is relatively weak. As the rotation period increases, close-in orbits have shorter lifetimes and the region containing long-term stable orbits is smaller. The colors show the particle lifetime.</figcaption></figure>

asteroids. Second, there is no strong evidence for differences in the disruption laws for S- and C-type asteroids, where the energy required to disperse the denser S-type target seems to be roughly offset by the extra energy required to crush pore space in lower-density, higher-porosity materials (see M. Jutzi et al. 2019; W. F. Bottke et al. 2020).

Another possibility, or contributing factor, is that the initial stability of temporary satellites tends to be lower around a higher-density primary body. In general, a primary's elongated shape (i.e., $C_{22}$) tends to destabilize close-in orbits. But in the limit that the primary's spin is significantly faster than a satellite's mean motion, this effect becomes negligible as the perturbing effect averages out over a single orbit period. Therefore, close-in orbits around a fast-rotating, high-density primary will tend to be less stable owing to their shorter orbit periods, which are closer to the primary's spin period. We demonstrate this effect in Figure 7, where the outcome of a test particle's orbit is shown as a function of its pericenter distance and eccentricity for a primary with the same elongated shape

but having different bulk densities and spin periods. The largest stable regions for close-in orbits exist around fast-rotating, low-density primaries, and the sizes of these regions decrease with increasing density or spin period. The black line in these figures indicates the synchronous orbit, so any orbit below this line would evolve inward on secular timescales owing to tides. In particular, for a high-density primary, the destruction of satellites by tides becomes a more important issue for spin periods longer than those shown on these plots.

Finally, it is shown here that the correlation between the post-impact shape and spin is what determines likelihood for satellite production and survival. If S-type asteroids' internal structure or general rheology prohibits or frustrates the fast-spinning outcomes or production of highly elongated shapes, then this might explain the lack of observed satellites. The observed distributions of shape and spin for S-type asteroids do not indicate an obviously different set of outcomes. A K-S test of the spin frequency distributions and light-curve amplitude distributions was used to test the null hypothesis
<figure><img src="image_10.png" /><figcaption>**Figure 7.** The fates of test particles in orbit around an ellipsoidal primary as a function of its bulk density and spin period. The primary has the same semiaxis lengths as those from Figures 5(a) and 6 (a, b, c ~ 57, 45, 40 km), and the top row shows the same simulations as plotted in Figure 6. The black line indicates the semimajor axis corresponding to the synchronous orbit. Orbits below this line would tidally evolve inward owing to tides. In general, as the primary's density or spin period is increased, the periods of close-in orbits become closer to the primary's spin period, leading to smaller stable regions owing to the stronger perturbing effect of $C_{22}$.</figcaption></figure>

that C-type and S-type distributions were drawn from the same distribution. Resultant *p*-values of 0.06 and 0.28 were found for the spin frequency and amplitude distributions, respectively, so we cannot reject the null hypothesis for either and accept that their shape and spin distributions could be similar.

Notably, some of the known large asteroids with satellites are M-type asteroids, with expected, or measured, densities even greater than those typically found for S-type asteroids (B. Carry 2012). These might require even larger impactors to achieve an elongated and fast-spinning outcome. Their properties might speak to a larger class of collisions at a different epoch of solar system history outside the main belt that is discussed later. They may also provide examples of where smaller impacts on targets that gained elongated shapes and rapid rotations through other processes may still provide a pathway to form satellites.

## 2.5. Provenance of High-pericenter Debris

Here the provenance of ejected material that reaches stable orbits is traced back through the modeling to the original SPH target. Outcomes that produce numerous satellites with high pericenter preferentially source material from a few to tens of kilometers in the subsurface.

Some cases show discrete depths that preferentially place debris onto high pericenter orbits, suggesting that the specific dynamics of the impact will determine the locations from the target most likely to successfully source the satellites (Figure 8). Specifically, Figure 8 shows four cases with a preferential sourcing of material from depths of 10–20 km. A very disruptive case of a 13 km impactor hitting at a 60° angle produces this preference for sourcing from depth for all preimpact spin rates of 3, 4, and 10 hr (Figures 8(a), (b), and
<figure><img src="image_11.png" /><figcaption>**Figure 8.** Provenance of satellites. The provenance of all temporary satellites (black circles) and temporary satellites with pericenters greater than $1.5R_{\text{pri}}$ (purple) for six different simulation outcomes are used to show the normalized mass originating from different depths of the target. The solid gray line is the distribution of all mass in the parent body, and all curves are normalized to show the relative preference for satellite provenance.</figcaption></figure>

(c)). The effect is strongest at the lowest of those preimpact spin rates. The same effect is re-created at lower impact angle cases with 4 hr preimpact spin period (Figures 8(d) and (f)).

The case with the most total temporary satellites, but not with the most high-pericenter satellites, sourced the deepest material, with some satellites originating from below 35 km depth (Figure 8(c)). Meanwhile, for a scenario that produces zero satellites with $q > 1.5a_{\text{pri}}$, all of the temporary satellites were sourced very near the surface (Figure 8(e)).

### 3. Discussion

We presented simulations of asteroid impacts that resulted in a wide range of post-impact shape and spin states. As found

in the observed population, there is a correlation between satellite formation and the shape and spin of the largest remaining remnant. The formation of the nonspherical shape, combined with rapid rotation, allows some ejecta to reach orbits that will not immediately reimpact the primary and thus directly reach stable orbits as determined from long-term stability modeling.

Spherical shapes have the largest regions of stability for orbiting debris, but this work finds that the impacts that produce spherical primaries put minimal or zero ejecta onto initially high pericenter orbits that can remain stable. Meanwhile, the stable zones around elongated asteroids shrink relative to spheres, but the impacts that create such
extreme shapes preferentially launch debris onto initially high pericenter orbits. In the simulations presented here, the latter appears to provide an important pathway to satellite formation around large asteroids. This pathway is direct—no additional mechanisms are needed to change or stabilize orbits—and places enough mass in orbit to explain the observed systems.

While the total mass of satellites is similar to that estimated for observed systems, the total number of initially stable satellites is much higher in any given simulation than what is observed, often numbering in the hundreds. Simulation resolution limitations (each particle in these simulations is approximately 1 km in size) probably mean even higher total numbers in reality, or, more likely, the formation of a debris cloud that participates in the formation of a handful of stable satellites. Thus, while these simulations are able to reproduce some of the observed characteristics of multiasteroid systems, additional modeling may be required to fully understand the mechanisms that play a role in the formation and long-term stability of asteroid satellites.

Collisional evolution of the large number of satellites will depend on the orbits and size distribution of debris, where the resolutions of the models here are not sufficient to estimate the latter. Collisions are in competition with at least two other effects, where ejecta with close pericenters can be scattered and escape the system owing to interactions with the primary (D. J. Scheeres 1999). The stability simulations presented here aimed to identify ejecta that was stable against scattering, but the 1000-day timescale tested may be shorter than that required for collisional evolution to take over for some cases. If collisions act faster than a satellite can be ejected, then numerous satellites might stabilize and contribute to the population of long-term stable mass. Meanwhile, tides will act on very long timescales of tens of Myr for orbit circularization and an order of magnitude or longer for orbit expansion (P. A. Taylor & J.-L. Margot 2011).

If collisional evolution is the dominant evolutionary factor immediately after impact, then ejecta may evolve to a disk without significant loss from dynamical ejection. To explore the end-member case of no dynamical loss, we can make the simple assumption that angular momentum of all of the bound debris is conserved, and then a circular equivalent orbit ($a_{eq}$; R. M. Canup 2004) can be calculated for a resultant relaxed disk. For the systems with significant initial debris with pericenter above two primary long semiaxes the typical values for $a_{eq}$ are between 4 and 6 primary radii. If the debris damps to a disk and forms a single satellite, or just a very small number of satellites, then the timescale to tidally evolve their orbits outward could take tens to hundreds of millions of years to evolve outward toward or beyond 10 primary radii (P. A. Taylor & J.-L. Margot 2011). Notably, nearly all observed satellites are found between 4 and 14 primary radii. This scenario implies that any individual satellite may include material from numerous initial temporary satellites that went through collisional relaxation to a disk and subsequent accretion.

Catastrophic disruptions that create large families do not directly correlate with satellite formation. Some, but not all, of the asteroids with satellites are associated with asteroid families, and not all of those belonging to families have satellites. Larger asteroid families will be created in cases where less total mass is retained by the target body, which is reported here as the mass of the largest remnant (Figure 4(c)).

The simulation outcomes do not find a strict correlation between smaller largest remnants (i.e., likely larger asteroid families) and increased mass of stable temporary satellites. Rather, there are cases with similar final largest remnants with very different total masses of temporary satellites.

Instead, we find that the mass of satellites normalized by the primary mass is correlated with the specific impact energy, *Q*, of the collision when it is normalized by a rotation-dependent catastrophic disruption threshold, $Q_{RD}^*$ (R.-L. Ballouz et al. 2014; M. Jutzi 2015). Here $Q_{RD}^*$ is a simple mapping of the mass-loss outcome to the specific impact energy space, through the universal law of catastrophic disruption defined by Z. M. Leinhardt & S. T. Stewart (2012). The spread in outcomes for cases with post-impact spin periods <10 hr is due to the fact that those cases represent a wide range in specific impact energies (see Figures 4(b) and (d)). In the cases of low $Q/Q_{RD}^*$, the impact does not produce temporary satellites; however, the primary's relatively high preimpact spin is preserved following the impact.

With further asteroid discoveries in the main belt likely to come from future large surveys, it is possible that any large asteroid with a satellite may still be found to have small or dispersed associated families (see M. Brož et al. 2022b regarding recent discovery of a small family at Kalliope). However, the discovery of very large families is likely close to complete (M. Brož et al. 2013; R. Deienno et al. 2021).

There are some primaries that are so large and with such an extreme shape that they challenge the understanding of asteroid family formation and identification, as they have no large detectable family of asteroids associated (this argument goes for asteroids with no known satellites as well). The primary culprit here is Kleopatra, which is the extreme of this group personifying the “dog bone” shape (F. Marchis et al. 2016, 2021) with a ~5 hr spin rate. Its axes are measured to be 267 ± 6 km, 61 ± 6 km, and 48 ± 6 km for an astonishing *c/a* ratio of 0.18 and a clearly double-lobed morphology (M. K. Shepard et al. 2018). Furthermore, the two satellites in orbit have been measured to be tidally evolving in a way that is only possible if they are relatively young (<1 Gyr; M. Brož et al. 2021). Such extreme shapes are not created in the simulations described here and presumably only occur in even more energetic impacts. Such impacts would liberate a huge amount of mass and may require differentiated targets, which are also not considered in our study. Given the very large size of the target asteroid and the large amount of mass that would be lost, such an impact at any point in the solar system history would, presumably, result in a large family of asteroids easily detectable with the numerous family-finding algorithms (see D. Nesvorný et al. 2015). Strengthening this case is the young reported age of Kleopatra's satellites, making it even harder to hide a related enormous asteroid family.

It has been proposed that some, or all, of the main asteroid belt was implanted from beyond Jupiter or from within the regions where terrestrial planets formed, inside ~1.5 au (W. F. Bottke et al. 2006; K. J. Walsh et al. 2011; S. N. Raymond & A. Izidoro 2017; C. Avdellidou et al. 2024). Therefore, it is possible that an asteroid like Kleopatra suffered its large, reshaping impact before it was in the main belt. This general idea is a solution to the “missing mantle” problem, where very large and high-density asteroids presumed to have been differentiated asteroids could have lost
their mantles in large impacts without littering the entire asteroid belt with their differentiated remnants, which are spectroscopically distinct and easy to detect (see F. E. DeMeo et al. 2015).

If Kleopatra's satellites are young, this would mean that they did not form in the impact that distorted Kleopatra's shape and left it spinning so rapidly; rather, they formed through a different pathway. The large impact, which presumably liberated an enormous amount of material, would have happened elsewhere, outside the main belt. Then, a later, relatively small impact would have liberated ejecta that was driven toward stable orbits owing to its existing highly elongated shape and rapid spin. This would rely on a pericenter-raising dynamic, due to its extraordinary shape, and explain the lack of a large associated family and the putative young age of the satellites (M. Brož et al. 2022a). This dynamic is similar to a mass-shedding scenario, postulated for near-Earth asteroids spun up by the YORP effect whose equators are nearly in Keplerian orbits (see J. L. Margot et al. 2015; K. J. Walsh & S. A. Jacobson 2015; F. Marchis et al. 2021). Here we postulate that the extreme shape and spin of Kleopatra provide a pathway for ejecta on simple ballistic trajectories to have their pericenter quickly raised to reach stability.

The suite of simulations presented here does not test all possible impact conditions or preimpact shape and spin combinations for targets. In the interest of a manageable parameter space, the preimpact rotation of the target is always prograde with respect to the impactor (see the Appendix), the targets are always spherical, and some of the preimpact rotation rates are among the most extreme observed rotation rates. Rather, the parameter space explored here was specifically designed to produce a wide range of post-impact shape and spin outcomes for the largest remnant of subcatastrophic impacts.

Likewise, this work explores only the direct pathway to satellite formation and does not attempt to model the interactions between ejecta, which could, in some cases, lead to collisions and orbit changes even among debris initially on $q < 1R_{\text{pri}}$ orbits. Notably, Alauda and Euphrosyne belong to the largest asteroid families among the asteroids with known satellites, but they are the least elongated of the observed primaries. A possible solution for these cases is that a large mass of debris was liberated at once in their family-forming impact, giving rise to the chance of ejecta colliding with itself on short timescales and evolving their orbits (D. D. Durda et al. 2004). The pathways studied here focused on ejecta immediately reaching stable orbits, but a collisional mechanism operating rapidly could provide the same dynamical tool of raising the pericenter of initially unstable debris, making it long-term stable. Future work focusing on these mechanisms will need to confront the observations that find that not all asteroids that are members of large families have satellites. This suggests that if the satellites of Alauda and Euphrosyne formed in their family-forming event, this process is a relatively low probability outcome.

## Acknowledgments

The authors thank David Minton for a challenging question and important discussion.

This work was supported through numerous funding programs: NASA under grant 80NSSC22K0045 issued through the Solar System Workings Program (K.J.W., R.-L. B.), and UCA J.E.D.I. Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01 (HFA) CNES, the University of Tokyo (P.M.). The Czech Science Foundation has supported this research through grant 22-17783S (J.H.). M. J. acknowledges support from SNSF project No. 200021_207359.

Software: pkdgrav (D. C. Richardson et al. 2000), SPH (M. Jutzi et al. 2019), REBOUND (H. Rein & S. F. Liu 2012; H. Rein & D. Tamayo 2018), REBOUNDX (D. Tamayo et al. 2020).

# Appendix
## Appendix Information

### A.1. Applying Preimpact Rotation

The procedure is as follows: for each particle $i$, we find the difference in velocity at $t=0$ (before impact) between the spin rate used in the SPH impact simulation and the desired spin rate for our *N*-body simulation. As an example, to approximate an impact into a target rotating with period, $P$, of 6 hr using this procedure, we take the start and end states of the SPH simulations of an identical nonrotating target. Then, we map—onto the outcome of the impact into the nonrotating target—the differences in velocities between the rotating and nonrotating targets at $t=0$. The difference is then added to the particles at the handoff time, $t=t_h$, for the nonrotating case, providing the conditions for an *N*-body simulation of a collision onto a rotating target. This procedure can be done for arbitrary values of $P$, as long as individual particles have not experienced many collisions between $t=0$ and $t=t_h$. For the simulations presented in this work, this is not the case, as the particles are in a ballistic phase before the handoff. The spin of the target is always assumed to be prograde with respect to the angular momentum of the impact.

Here we provide a comparison of the evolution of the target following the handoff from SPH to pkdgrav for an impact with no preimpact rotation, one with 6 hr preimpact rotation and one with the prescription applied as described above (Figures 9 and 10). The impact was at $5 \text{ km s}^{-1}$ of a $10 \text{ m}$ impactor into a $1 \text{ km}$ target, and the $c/a$ axis ratio, rotation rate, and total accumulated number of particles converge to very similar values for the case of 6 hr preimpact rotation and 6 hr applied rotation (note in Figure 10 that the rotation rate of the nonspinning target converges to 114 hr, and it is not shown in the figure). Furthermore, Figures 11 and 12 show outcomes for the same impact as shown in Figure 2, but for cases with no preimpact rotation and 4 hr preimpact rotation period, respectively.
<figure><img src="image_12.png" /><figcaption>**Figure 9.** The outcome of a 5 km s⁻¹ impact of a 10 m impactor into a 1 km target with different preimpact rotation states. All images are following the handoff from SPH to pkdgrav. (a) A nonspinning target results in a moderately prolate object with minimal spin (see Figure 9 for the evolution of the spin rate of the largest remnant). (b) An SPH simulation with the target initially spinning at 6 hr. (c) The same impact with the spin adjusted to apply 6 hr preimpact rotation on the target 100 s after impact.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 10.** The outcome of a 5 km s⁻¹ impact of a 10 m impactor into a 1 km target with different preimpact rotation states. (a) The axis ratio *c/a* of the largest remnant as a function of time for all three cases. (b) The rotation period of the largest remnant for the 6 hr impact case and the applied 6 hr rotation case (the nonspinning target case converged to 114 hr and is not shown). (c) The number of *N*-body particles in the largest remnant over time since impact for all three cases.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Figure 11.** Evolution of the target asteroid immediately following impact. The outcome of a 13 km object impacting a 100 km spherical target with no rotation at a 60° angle at a speed of 5 km s⁻¹ shown at 0.1, 1, 3, 5, 7, and 9 hr post-impact. The particles that are part of the main primary mass are colored gray. The rest of the particles are colored by the pericenter of their instantaneous Keplerian orbit. Red colors correspond to trajectories with a pericenter less than the primary's long axis, while blue colors indicate pericenters above the primary's long axis. Dark red indicates a pericenter less than zero, which corresponds to a hyperbolic (unbound) orbit.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Evolution of the target asteroid immediately following impact. The outcome of a 13 km object impacting a 100 km spherical target with a 4 hr preimpact rotation at a 60° angle at a speed of 5 km s⁻¹ shown at 0.1, 1, 3, 5, 7, and 9 hr post-impact. The particles that are part of the main primary mass are colored gray. The rest of the particles are colored by the pericenter of their instantaneous Keplerian orbit. Red colors correspond to trajectories with a pericenter less than the primary's long axis, while blue colors indicate pericenters above the primary's long axis. Dark red indicates a pericenter less than zero, which corresponds to a hyperbolic (unbound) orbit.</figcaption></figure>
## A.2. Rebound Integration Methods

To determine the dynamical lifetimes of temporary satellites, we use the REBOUND N-body code (65, 66) along with the REBOUNDx package to include the perturbations due to the primary's $J_2$ and $C_{22}$ moments (67). The $J_2$ and $C_{22}$ moments are defined as (e.g., 68, 49, 69)

$$J_2 = \frac{C - (A + B)/2}{MR_{\text{eq}}^2}$$

$$C_{22} = \frac{1}{4} \frac{B - A}{MR_{\text{eq}}^2},$$

where $M$ is the primary's mass, $A \le B \le C$ are the primary's moments of inertia, and $R_{\text{eq}}$ is a normalizing radius that is typically set to the equatorial radius of the primary. We set $R_{\text{eq}}$ to be the circumscribing sphere of the primary defined by its DEEVE long axis $a_{\text{pri}}$. The primary's DEEVE semiaxis lengths ($a_{\text{pri}} \ge b_{\text{pri}} \ge c_{\text{pri}}$) and moments of inertia are related through the following relations:

$$A = \frac{M}{5}(b_{\text{pri}}^2 + c_{\text{pri}}^2)$$

$$B = \frac{M}{5}(a_{\text{pri}}^2 + c_{\text{pri}}^2)$$

$$C = \frac{M}{5}(a_{\text{pri}}^2 + b_{\text{pri}}^2).$$

The implementation for the primary's $J_2$ is functionally identical to that already included in REBOUNDx (67). We

added the $C_{22}$ term using REBOUNDx, where the additional accelerations felt by a test particle are

$$\ddot{x} = -\frac{GM}{r^2} \frac{R_{\text{eq}}^2}{r^2} \frac{x}{r} C_{22} \left( 15 \frac{x^2 - y^2}{r^2} - 6 \right)$$

$$\ddot{y} = -\frac{GM}{r^2} \frac{R_{\text{eq}}^2}{r^2} \frac{y}{r} C_{22} \left( 15 \frac{x^2 - y^2}{r^2} + 6 \right)$$

$$\ddot{z} = -\frac{GM}{r^2} \frac{R_{\text{eq}}^2}{r^2} \frac{z}{r} C_{22} \left( 15 \frac{x^2 - y^2}{r^2} \right),$$

where $x, y, z$ are the particle's coordinates coordinated in the primary's body-fixed frame, $r$ is the distance to the primary center, and $G$ is the gravitational constant.

The primary is assumed to be in principal axis rotation with a fixed spin rate equal to the primary's spin after the pkdgrav simulations (so small rotations about its nonprincipal axes after the pkdgrav simulations are ignored). Particle orbits are integrated using the IAS15 integrator (70). Test particles are initialized at the apocenter of their orbit with varying pericenter distances and integrated forward in time for 1000 days until they either collide with the primary or are ejected from the system. Particles are deemed to have escaped if they pass beyond the system's Hill sphere, assuming the primary is located at 2.5 au. Stability maps for all simulations are found in Figures 13 and 14.
<figure><img src="image_16.png" /><figcaption>Case 1</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Case 2</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Case 3</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Case 4</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Case 5</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Case 6</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Case 7</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Case 8</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Case 9</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Case 10</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>Case 11</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Case 12</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Case 13</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>Case 14</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Case 15</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Case 16</figcaption></figure>

**Figure 13.** Temporary satellites plotted in gray on a color scale of dynamical lifetimes as a function of orbital eccentricity and pericenter distance. The impact properties for each case can be found in Table 1.
<figure><img src="image_32.png" /><figcaption>Case 17</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Case 18</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Case 19</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>Case 20</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>Case 21</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>Case 22</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>Case 23</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>Case 24</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>Case 25</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>Case 26</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>Case 27</figcaption></figure>

<figure><img src="image_43.png" /><figcaption>Case 28</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>Case 29</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>Case 30</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>Case 31</figcaption></figure>

<figure><img src="image_47.png" /><figcaption>Case 32</figcaption></figure>

**Figure 14.** Temporary satellites plotted in gray on a color scale of dynamical lifetimes as a function of orbital eccentricity and pericenter distance. The impact properties for each case can be found in Table 1.
### A.3. Table of Simulation Outcomes

Simulation outcomes for the model parameters are presented in Table 2.

<table><caption>**Table 2**
Simulation Outcomes for the Model Parameters of Impact Speed ($V_{imp}$), Impactor Radius ($R_{imp}$), Impact Angle, the Angle of Friction of the *pkggrav* Simulation Settings, and the Preimpact Rotation of the Target ($P_{Target}$)</caption><thead><tr><th>Simulation ID</th><th><i>V</i><sub>imp</sub><br>(km s<sup>-1</sup>)</th><th><i>R</i><sub>imp</sub><br>(km)</th><th>Angle<br>(deg)</th><th>Friction<br>(deg)</th><th><i>P</i><sub>target</sub><br>(hr)</th><th><i>M</i><sub>rem</sub>/<i>M</i><sub>target</sub></th><th><i>Q</i>/<i>Q</i><sup>*</sup><sub>RD</sub></th><th><i>P</i><sub>rem</sub><br>(hr)</th><th><i>b</i> / <i>a</i></th><th><i>c</i>/<i>a</i></th><th><i>N</i><sub>sat</sub></th><th><i>N</i><sub>sat</sub><br>(<i>q</i> > 1<i>a</i><sub>pri</sub>)</th></tr></thead><tbody><tr><td>1</td><td>5</td><td>3</td><td>15</td><td>18</td><td>4</td><td>0.99</td><td>0.02</td><td>4.37</td><td>0.95</td><td>0.92</td><td>27</td><td>4</td></tr><tr><td>2</td><td>5</td><td>3</td><td>15</td><td>18</td><td>&infin;</td><td>0.99</td><td>&lt;0.01</td><td>84.7</td><td>0.98</td><td>0.89</td><td>0</td><td>0</td></tr><tr><td>3</td><td>5</td><td>5</td><td>15</td><td>18</td><td>4</td><td>0.89</td><td>0.23</td><td>7.01</td><td>0.64</td><td>0.51</td><td>166</td><td>85</td></tr><tr><td>4</td><td>5</td><td>5</td><td>15</td><td>18</td><td>&infin;</td><td>0.95</td><td>0.1</td><td>26.7</td><td>0.97</td><td>0.72</td><td>92</td><td>0</td></tr><tr><td>5</td><td>5</td><td>3</td><td>30</td><td>18</td><td>4</td><td>0.99</td><td>0.02</td><td>4.28</td><td>0.97</td><td>0.94</td><td>23</td><td>2</td></tr><tr><td>6</td><td>5</td><td>3</td><td>30</td><td>18</td><td>&infin;</td><td>0.99</td><td>&lt;0.01</td><td>47.3</td><td>0.99</td><td>0.92</td><td>12</td><td>0</td></tr><tr><td>7</td><td>5</td><td>5</td><td>30</td><td>18</td><td>4</td><td>0.88</td><td>0.23</td><td>5.98</td><td>0.75</td><td>0.52</td><td>137</td><td>55</td></tr><tr><td>8</td><td>5</td><td>5</td><td>30</td><td>18</td><td>&infin;</td><td>0.93</td><td>0.13</td><td>20.36</td><td>0.95</td><td>0.59</td><td>67</td><td>0</td></tr><tr><td>9</td><td>5</td><td>3</td><td>45</td><td>18</td><td>4</td><td>0.99</td><td>0.02</td><td>4.21</td><td>0.99</td><td>0.97</td><td>18</td><td>3</td></tr><tr><td>10</td><td>5</td><td>3</td><td>45</td><td>18</td><td>&infin;</td><td>0.99</td><td>0.01</td><td>44.5</td><td>0.99</td><td>0.94</td><td>13</td><td>0</td></tr><tr><td>11</td><td>5</td><td>7</td><td>45</td><td>18</td><td>4</td><td>0.78</td><td>0.43</td><td>5.83</td><td>0.91</td><td>0.54</td><td>257</td><td>160</td></tr><tr><td>12</td><td>5</td><td>7</td><td>45</td><td>18</td><td>&infin;</td><td>0.88</td><td>0.24</td><td>12.2</td><td>0.88</td><td>0.57</td><td>91</td><td>1</td></tr><tr><td>13</td><td>5</td><td>3</td><td>60</td><td>18</td><td>4</td><td>0.72</td><td>&lt;0.01</td><td>4.22</td><td>0.99</td><td>0.98</td><td>11</td><td>0</td></tr><tr><td>14</td><td>5</td><td>3</td><td>60</td><td>18</td><td>&infin;</td><td>0.99</td><td>&lt;0.01</td><td>51.57</td><td>0.99</td><td>0.98</td><td>10</td><td>0</td></tr><tr><td>15</td><td>5</td><td>13</td><td>60</td><td>18</td><td>4</td><td>0.72</td><td>0.56</td><td>5.86</td><td>0.64</td><td>0.49</td><td>316</td><td>120</td></tr><tr><td>16</td><td>5</td><td>13</td><td>60</td><td>18</td><td>6</td><td>0.76</td><td>0.47</td><td>5.91</td><td>0.77</td><td>0.65</td><td>177</td><td>74</td></tr><tr><td>17</td><td>5</td><td>13</td><td>60</td><td>18</td><td>10</td><td>0.8</td><td>0.4</td><td>6.75</td><td>0.78</td><td>0.69</td><td>170</td><td>81</td></tr><tr><td>18</td><td>5</td><td>13</td><td>60</td><td>18</td><td>&infin;</td><td>0.85</td><td>0.31</td><td>9.48</td><td>0.88</td><td>0.62</td><td>154</td><td>18</td></tr><tr><td>19</td><td>5</td><td>3</td><td>75</td><td>18</td><td>4</td><td>0.99</td><td>&lt;0.01</td><td>4.45</td><td>0.98</td><td>0.97</td><td>4</td><td>1</td></tr><tr><td>20</td><td>5</td><td>3</td><td>75</td><td>18</td><td>&infin;</td><td>0.99</td><td>&lt;0.01</td><td>191.4</td><td>0.99</td><td>0.98</td><td>4</td><td>0</td></tr><tr><td>21</td><td>5</td><td>18</td><td>75</td><td>18</td><td>4</td><td>0.99</td><td>0.01</td><td>5.27</td><td>0.72</td><td>0.58</td><td>57</td><td>36</td></tr><tr><td>22</td><td>5</td><td>18</td><td>75</td><td>18</td><td>&infin;</td><td>0.99</td><td>&lt;0.01</td><td>17.22</td><td>0.98</td><td>0.88</td><td>12</td><td>0</td></tr><tr><td>23</td><td>5</td><td>13</td><td>60</td><td>33</td><td>6</td><td>0.76</td><td>0.47</td><td>5.68</td><td>0.76</td><td>0.67</td><td>202</td><td>85</td></tr><tr><td>24</td><td>5</td><td>13</td><td>60</td><td>33</td><td>10</td><td>0.8</td><td>0.4</td><td>6.45</td><td>0.86</td><td>0.65</td><td>149</td><td>83</td></tr><tr><td>25</td><td>5</td><td>13</td><td>60</td><td>33</td><td>&infin;</td><td>0.85</td><td>0.31</td><td>9.59</td><td>0.9</td><td>0.56</td><td>165</td><td>19</td></tr><tr><td>26</td><td>5</td><td>5</td><td>15</td><td>0</td><td>&infin;</td><td>0.95</td><td>0.1</td><td>28.34</td><td>0.91</td><td>0.85</td><td>105</td><td>1</td></tr><tr><td>27</td><td>5</td><td>5</td><td>30</td><td>0</td><td>&infin;</td><td>0.93</td><td>0.14</td><td>20.21</td><td>0.95</td><td>0.8</td><td>75</td><td>1</td></tr><tr><td>28</td><td>5</td><td>7</td><td>45</td><td>0</td><td>&infin;</td><td>0.88</td><td>0.24</td><td>12</td><td>0.85</td><td>0.82</td><td>98</td><td>2</td></tr><tr><td>29</td><td>5</td><td>13</td><td>60</td><td>0</td><td>6</td><td>0.77</td><td>0.47</td><td>5.85</td><td>0.87</td><td>0.64</td><td>164</td><td>76</td></tr><tr><td>30</td><td>5</td><td>13</td><td>60</td><td>0</td><td>10</td><td>0.8</td><td>0.4</td><td>6.84</td><td>0.91</td><td>0.73</td><td>172</td><td>94</td></tr><tr><td>31</td><td>5</td><td>13</td><td>60</td><td>0</td><td>&infin;</td><td>0.84</td><td>0.31</td><td>9.35</td><td>0.85</td><td>0.78</td><td>123</td><td>22</td></tr><tr><td>32</td><td>5</td><td>18</td><td>75</td><td>0</td><td>&infin;</td><td>0.99</td><td>&lt;0.01</td><td>17</td><td>0.98</td><td>0.89</td><td>11</td><td>0</td></tr></tbody></table>

**Note.** The simulation outcomes for each impact are the mass of the largest remnant as a fraction of the target mass ($M_{rem}/M_{target}$), the impact energy scaled by the rotation-dependent catastrophic disruption threshold ($Q/Q_{RD}^*$), the rotation period of the largest remnant, its intermediate-to-long axis ratio ($b/a$) and its short-to-long axis ratio ($b/a$), the number of total temporary satellites formed ($N_{sat}$), and the number with orbit pericenter larger than the long semiaxis length.
ORCID iDs

Kevin J. Walsh https://orcid.org/0000-0002-0906-1761

Ronald-Louis Ballouz https://orcid.org/0000-0002-1772-1934

Harrison F. Agrusa https://orcid.org/0000-0002-3544-298X

Josef Hanuš https://orcid.org/0000-0002-2934-3723

Patrick Michel https://orcid.org/0000-0002-0884-1993

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