Elliptical ejecta of asteroid Dimorphos is due to its surface curvature

Abstract

Kinetic deflection is a planetary defense technique delivering spacecraft momentum to a small body to deviate its course from Earth. The deflection efficiency depends on the impactor and target. Among them, the contribution of global curvature was poorly understood. The ejecta plume created by NASA’s Double Asteroid Redirection Test impact on its target asteroid, Dimorphos, exhibited an elliptical shape almost aligned along its north-south direction. Here, we identify that this elliptical ejecta plume resulted from the target’s curvature, reducing the momentum transfer to 44 ± 10% along the orbit track compared to an equivalent impact on a flat target. We also find lower kinetic deflection of impacts on smaller near-Earth objects due to higher curvature. A solution to mitigate low deflection efficiency is to apply multiple low-energy impactors rather than a single high-energy impactor. Rapid reconnaissance to acquire a target’s properties before deflection enables determining the proper locations and timing of impacts.

Introduction

Planetary defense is an international effort to mitigate threats of small-body collisions with Earth^1,2. Among its key planetary defense technologies is kinetic deflection, in which a spacecraft collides with a hazardous body to change its trajectory and eliminate or reduce the risk of its impact on Earth³. Kinetic deflection is a practical approach when targets are less than about 250 m in radius if the encounter with Earth is several decades in the future^1,2,3,4. In contrast, near-Earth objects (NEOs) less than 50 m in radius are the most probable threat and high-priority targets for rapid reconnaissance flyby missions^1,2,3,4. The momentum transfer enhancement factor, known as β, is a well-used parameter, in its simplest form, the total momentum imparted to the target by the spacecraft and ejecta normalized by its momentum before impact^5,6,7. Recent impact physics studies suggest the dependence of β on the impactor’s properties and orientation and the target’s composition and strength^{8,9,10,11,12,13,14,15}. Local topography is another contributor to kinetic deflection. When the crater formation is at a local scale on the target body, momentum transfer is only affected by local topography^{11,16,17,18,19}. From a tactical perspective, imparting higher kinetic energy is preferred to maximize deflecting the target^1,2,3,4. However, when a higher kinetic energy impact generates a larger impact crater relative to the target body, the target’s global curvature becomes important to the excavation process²⁰. How a target’s global curvature controls kinetic deflection remains unexplored.

Without the context of planetary defense, two previous spacecraft missions conducted relatively small impact experiments compared to the sizes of their target bodies. NASA’s Deep Impact mission targeted Comet 9P/Tempel 1, a Jupiter family comet about 6 km across²¹. The impacting spacecraft, with a mass of 364 kg at an impact speed of 10.3 km/s collided with this cometary nucleus²¹ and produced a crater with a radius of 100 ± 10 m²², which is only 4% of the comet’s equivalent radius. Deep Impact suggests the strong dependence of ejecta plume geometry and morphology on impact obliquity and target properties at a local scale^23,24,25. Similarly, JAXA’s Hayabusa2 mission conducted a Small Carry-on Impactor (SCI) experiment on the carbonaceous asteroid (162173) Ryugu²⁶. The 2 kg copper projectile impacted at a speed of 2 km/s, creating a 7.3 m radius crater (only 2% of the asteroid’s equivalent radius); it appears that ejecta formation was affected by the presence of a large boulder beneath the impact site^26,27. These impacts were too small to impart significant momentum transfer at a realistic scale for planetary defense and be affected by their targets’ global curvatures.

NASA’s Double Asteroid Redirection Test (DART) mission was the first full-scale planetary defense demonstration mission that deliberately crashed a spacecraft into Dimorphos, the smaller satellite of the binary system (65803) Didymos^28,29. On September 26, 2022, at 23: 14: 24 UTC, the DART spacecraft (579 kg at the time of impact) successfully impacted at 8.84° S and 264.30° E on Dimorphos in the body’s fixed frame (ISU_DIMORPHOS) with an impact speed of 6.145 km/s^28,29. The spatial extent of the pre-impact Dimorphos was 179 m × 169 m × 115 m, indicating a relatively oblate shape³⁰. This impact caused a mutual orbit period change of Dimorphos relative to Didymos of  − 33.0 ± 1.0(3σ) min³¹. After the collision, the ejecta plume evolved with time, exhibiting a cone-like shape that grew initially but later decayed gradually, a major part of which joined as part of the tail mainly driven by solar radiation pressure^{32,33,34,35,36,37,38,39,40,41,42,43,44}. β estimates ranged between 2.2 and 4.9 (depending on the assumed bulk density of the asteroid, which remains unknown but will be constrained by ESA’s Hera mission⁴⁵); for a bulk density of 2.4 g/cc, β was determined to be $3.6{1}_{-0.25}^{+0.19}$⁴⁶.

Here, we show in detail the contributions of an asteroid’s global curvature to kinetic deflection by using the recently obtained data from DART. Measuring the geometry of the DART impact-driven ejecta first applies images taken by Hubble Space Telescope (HST)⁴⁰ and the Light Italian Cubesat for Imaging of Asteroids (LICIACube)’s camera, LUKE (LICIACube Unit Key Explorer)^41,42. The contribution of Dimorphos’s global curvature to the DART momentum transfer is determined using the measured cone geometry. Furthermore, an analysis applies an established approach for the DART impact case to interpret the global curvature’s contribution to kinetic deflection for NEOs less than 50 m in radius.

Results

Ejecta cone geometry

The cone geometry measurement applies images taken by HST and LICIACube, extending earlier efforts: those using LUKE images only⁴² and HST images only^40,46 with the assumption of circular cone geometry, and those using LUKE images only to constrain the elliptical cone geometry⁴⁷. The present approach differs from theirs, using both HST and LICIACube images. The results are consistent with theirs but give the uncertainties of the elliptical cone geometry, which is critical for the following momentum transfer computation.

The parameters describing the cone geometry include the wide-cone opening angle, θ₁, the narrow-cone opening angle, θ₂, the azimuthal angle from Dimorphos’s north at the impact site defining the wide-cone orientation, ϕ, and the cone axis direction described using Right Ascension, RA, and Declination, DEC (Fig. 1). The J2000 International Celestial Reference System (later known as J2000) is the baseline coordinate frame defining the cone axis direction, RA and DEC. The analysis uses a time range at 160–195 s after the impact (T + 160–195 s) within a distance of a few kilometers from Dimorphos’s surface (i.e., LICIACube images) and, equivalently, at T + 0.41–8.2 h within a few hundred kilometers (i.e., HST images). The ejecta cone geometry observed by both HST and LICIACube consists of particles with consistent ejection speeds of 0.5–20 m/s. Applying these durations comes from the timing of LICIACube’s closest approach (CA) with Dimorphos at T + 168 s when the ejecta had its cone geometry well-established. The time range of the HST images no longer aligns Dimorphos’s location relative to Didymos’s with the ejecta cone (Fig. 1a), causing complex ejecta motion near Dimorphos’s surface⁴⁰. However, this near-surface ejecta behavior does not affect the cone geometry at the spatial scale of the HST images.

Fig. 1: Ejecta cone and impact cratering flow fields.

The arrows in light blue show $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$ in J2000, while those in green (x, y, z) are the Dimorphos-fixed frame (IAU_DIMORPHOS). The red arrows give the local frame at the impact site. z^″ is the DART impact direction, x^″ is orthogonal to Dimorphos’s north and z^″, and y^″ is orthogonal to these axes. a and b Cone geometry and Dimorphos. The cone’s perimeter defines its edge. c Slice in light red representing a plane used for the Maxwell Z-model. The red curve over Dimorphos represents the intersection between the body and the slicing plane. d Illustrations of streamlines defined as SL. An example streamtube is a region between SL1 and SL2. SL E, given in dark blue, is the streamline farthest from the impact site along a given azimuthal piece.

Monte Carlo simulations determine the geometry of the elliptical cone (see “Methods”, Subsection Ejecta Cone Geometry). Constraints are the position angles (eastward angles from the sky north in J2000) of the cone edges, i.e., the perimeters of the cone seen by both HST and LICIACube. The solution for the cone edge geometry is θ₁ = 133 ± 9°, θ₂ = 95 ± 6°, ϕ = 28 ± 17°, RA = 141 ± 4°, and DEC = + 20 ± 8° (Table 1). Both narrow- and wide-cone opening angles are wider than an ideal 90° opening angle of an ejecta plume on flat surfaces⁴⁸. The determined azimuthal angle for rotation, ϕ = 28 ± 17°, suggests the cone’s wide-cone opening direction is preferentially parallel to Dimorphos’s north-south direction. Illumination alignments between the cone edges and the terminator on Dimorphos at T + 175 s and T + 178 s constrain the location of the ejecta cone’s tip (apex) as (x, y, z) = (− 4 ± 6, − 3 ± 9, 9 ± 10) m in the Dimorphos body-fixed frame (later known as IAU_DIMORPHOS), in which x is toward Didymos, z is Dimorphos’s north, and y is orthogonal to these axes (Fig. 1) (see “Methods”, Subsection Ejecta Cone Geometry).

Table 1 Properties determined by cone measurements and the Maxwell Z-model approach

Figure 2 shows the derived cone geometry consistent with that observed by HST and LICIACube. HST images capture the ejecta features extending celestial north and southeast at T + 18,147 s (= T + 5.0 h). LICIACube, on the other hand, observed Dimorphos from different views throughout its CA, which was T + 168 s^41,42. At T + 160 s, the CubeSat faced the approach side on which the ejecta plume was opening. The spacecraft rapidly approaches the cone and passes at T + 162 s. At T + 170 s, it looks at one side of the ejecta cone. At T + 195 s, the spacecraft observes the entire departure side, mainly looking at the ejecta cone. The angle between the ejecta cone axis and the DART anti-incident direction is 15 ± 5°.

Fig. 2: Observed and simulated ejecta cone edge geometries.

a–d Didymos-Dimorphos orientations seen from observers in J2000. The versions of the Didymos and Dimorphos shape models are Didyv003⁹⁶ and Dimov004³⁰. The green arrows give IAU_DIMORPHOS as a reference. e–h Images taken by observers (HST in e and LICIACube in f–h). The HST image captures the global structure of the ejecta cone, evolving toward the north (N) and southeast (SE), and the tail extended toward the northwest. All images are centered at Dimorphos. i–l Simulated cone geometry in light yellow seen from observers (HST in i and LICIACube in j–l). The ejecta cone, illustrated as a yellow cone with frames in black, exhibits its geometry differently in HST and LICIACube images. The cone observed by HST (i) does not change over the considered time, while that by LICIACube changes dynamically. LICIACube first sees the front side of the cone (j), where Dimorphos is hidden behind it. Once passing through it, the spacecraft observes its backside behind Dimorphos (k and l). The red arrows show the direction of the cone axis, while the blue ones represent the DART incidence angle. Column (a, e, i) shows HST measurement at T + 18,147 s = T + 5.0 h. Columns (b, f, j), (c, g, k), and (d, h, l) give LICIACube measurements at T + 160 s, T + 170 s, and T + 195 s, respectively.

Momentum transfer

How Dimorphos’s global curvature contributes to DART’s momentum transfer is quantified using the geometric factor, the ratio of the ejecta momentum on a curved surface target to that on a flat surface, P_fl (see “Methods”, Subsection Geometric Factor). Monte Carlo simulations apply the Maxwell Z-model, an empirical, first-order approximation model, to determine the kinematics of subsurface flow fields and ejecta’s ballistic trajectories (see “Methods”, Subsection Maxwell Z-model). A material flow follows a single curved trajectory called a streamline, which is a function of the counterclockwise angle from the impact incident direction, Δ, and two kinematic parameters, α and Z, representing ejecta’s speed and geometry, respectively. A streamtube is a spatial envelope sandwiched by two streamlines. Materials ejected at speeds less than the Dimorphos escape speed (about 9 cm/s) do not influence momentum transfer and are eliminated from consideration. This condition defines the farthest contribution of material excavation, giving the streamline-surface intersection from the impact site, ${r}^*_{E}$, and the resulting angle from the incident direction, ${\Delta }_{E}^*$ (Fig. 1d). Figure 1d illustrates how a streamtube enclosed by two streamlines, SL1 and SL2, changes its trajectory and the ejection direction due to surface curvature.

The algorithm uses the measured β⁴⁶ as a constraint to determine the geometric factor. The terminal velocities at the surface points of streamtubes give a set of ballistic locations of ejected surface elements at T + 175 s. Comparing these ballistic trajectories with the measured ejecta cone geometry in all azimuthal directions uniquely determines α and Z. The method then finds a positive scalar, γ, to make the simulated β consistent with the measured one and determine P_fl (see “Methods”, Subsection Geometric Factor for DART). Comparison tests confirm the consistency between our approach and the iSALE-2D shock-physics code^49,50,51,52 (see “Methods”, Subsection Geometric Factor Validation). Monte Carlo simulations, accounting for the above simulation steps, input the cone geometry measurements (θ₁, θ₂, RA, DEC, ϕ) to give the statistical behaviors of Z, α, γ, and P_fl (Table 1 and Fig. 3). The results show Z = 2.9 ± 0.4 (Fig. 3a), α = (3.1 ± 2.2) × 10⁻⁴ hm^(Z+1)/s (Table 1), where hm are hectometers (= 100 m). Because Z inversely correlates with α, Z becomes low (high) when α is high (low) (Fig. 3a, b). Lower (higher) Z distributions characterized by the extrema cause shallow (steep) ejection.

Fig. 3: Variations provided with error bars in Maxwell Z-model parameters and projected cone geometry onto Dimorphos’s surface with azimuthal angle from Dimorphos’s north.

a Variations in Z. b Variations in α. c Variations in excavation range from the impact point, ${r}_{E}^{*}$, in blue, while those from the cone axis, ${r}_{E}^{*}\sin {\Delta }_{E}^{*}$, in red. d Variations in geometric factor along each piece in an azimuthal direction. Uncertainties are provided in 1σ by computing the standard deviations of the whole samples per azimuthal angle bin. Source data are provided as a Source Data file.

The ejecta cone, determined from observational data, opened widely at azimuthal angles (ϕ) of about 0° (Dimorphos north) and about ± 180° (Dimorphos south), indicating that the cone extends along almost the Dimorphos north-south direction (Fig. 3c). The slight shift toward a positive azimuthal angle indicates the wide-cone opening direction’s slight twist toward the east-west direction, ϕ = 28 ± 17°. The range of material excavation derived by the Monte Carlo approach using the Maxwell Z-model, on the other hand, becomes longer in the east-west direction but shorter in the north-south direction. This is because the flow field extent in the north-south direction is limited by Dimorphos’s higher curvature in this direction than in the east-west direction. This result contrasts with earlier numerical work suggesting surface curvature might not contribute significantly to the ellipticity of an impact crater⁵³, mainly because of Dimorphos’s high curvature in the north-south direction.

Comparable to the crater radius, half the crater’s rim-to-rim distance⁴⁸, the average distance of material excavation from the cone axis, i.e., ${r}_{E}^*\sin ({\Delta }_{E}^*)$, is 54 ± 9 m. This result is larger than the numerically predicted size, which depends on the target’s strength. Modeling the DART impact on Dimorphos as a cohesionless target (< 50 Pa) no longer causes a crater-like morphology but generates flatter depression at a radius of 25−55 m^54,55. The predicted ejecta mass suggests Dimorphos as a weak target (Ferrari et al. in review⁵⁶). If low-moderate material strength (about 1−10 kPa) influenced the impact, it would create a crater 15−30 m in radius (Stickle et al., in review⁵⁷). The derived larger value (54 ± 9 m) may come from our assumption that the cone edge is linear, while its geometry is likely complex and curved around near the impact site, allowing smaller crater excavation to generate a larger ejecta cone⁵⁴. This complexity, culminated by model simplification and uncertainties, causes challenges in inferring the crater size and the target’s strength.

The geometric factor relative to a flat surface target in a given azimuthal direction becomes lower but more variable along the wide cone direction (Fig. 3d). The east-west direction (ϕ of about  ± 90°) tends to have higher geometric factors with lower uncertainties. This direction offers a flatter surface, providing positive momentum transfer and, thus, positive geometric factors. On the other hand, because of high curvature approximately along the north-south direction (ϕ of about 0° and  ± 180°), ejected materials tend to depart toward Dimorphos’s anti-along track direction. The Dimorphos-south direction (ϕ of about  ± 180°) tends to have negative geometric factors, indicating ejecta’s contribution to negative momentum transfer. Again, the ejecta cone’s twist causes the shift of lower geometric factors towards positive azimuthal angles by about 45°. The net geometric factor relative to a flat surface target, P_fl, is 44 ± 10% for Dimorphos. If the target surface is perfectly flat, the derived geometric factor leads to β = 7.2 ± 3.8(1σ), likely the upper end of the predicted range^11,58.

Discussion

DART-driven impact ejecta cone geometry

While the elliptical cone geometry likely results from many factors, Dimorphos’s high curvature in its north-south direction is the major contributor. At the time of impact, DART’s solar panels faced the Dimorphos north-south direction at azimuthal angles of ϕ = − 2°, 178°²⁸, and the spacecraft geometry likely created an elliptical crater excavation, causing complex ejecta cone geometry⁸. Also, these solar panels hit two approximately-six-meter-sized boulders at the impact site, Bodhran (ϕ of about 0°) and Atabaque (ϕ of about 145°)²⁸. Recent work suggests that the ejection angle would become shallow in the direction where a boulder exists^59,60. However, given that the ejecta cone encloses almost half of Dimorphos’s body at T + 175 s, the crater likely grows large enough that the contribution of the spacecraft’s geometry to the ejecta evolution is no longer significant⁵⁴. Earlier experimental work suggested the crater growth recovers the ejection steepness when the excavation extent exceeds the boulder size⁵⁹. Because the excavation range is tens of meters (Fig. 3c) at this time, the influence on the ejecta cone geometry of both the spacecraft’s solar panels (each  < 10 m) and the pre-existing meter-sized boulders is minimal. Given the elliptical cone, the ejecta plume does not directly cross Didymos, in contrast to the earlier suggestion that the ejecta might directly hit the asteroid based on the circular ejecta cone assumption⁴².

The fact that the measured wide-cone dimension twists toward the east-west direction by 28 ± 17° is still under investigation. With increasing curvature, an ejecta plume becomes shallower relative to the initial surface²⁰, so the wide-cone opening dimension should be ideally parallel to the Dimorphos north-south direction. While no decisive explanation exists, this twist may relate to the complex ejecta morphology. The observed ejecta morphology is highly heterogeneous^41,42,61, including complex ray structures, clump and boulder distributions, and hole-like features lacking materials. Small particles and boulders interact in the ejecta cone, flowing around them like obstacles, leading to directional inhomogeneities in the ejecta plume. The interactions between small particles and larger boulders complicate the ejecta plume structure⁶². Boulders at various speeds^33,61 would change the ejecta plume structure with time. Such interactions within the ejecta plume, which may be attributed to structural heterogeneity in Dimorphos¹⁹, might be responsible for deviations in the ejecta cone geometry from the ideal cone geometry. The DART impact with slight obliquity (i.e., the 73° impact angle) likely does not influence the ejecta cone geometry much, given recent experimental work showing that oblique impacts with an impact angle of higher than 60° exhibits symmetric cone geometry over the azimuthal direction⁶³. Regardless of the wide cone’s slightly twisted orientation, its preferred orientation toward the Dimorphos north-south direction does not change.

Geometric factor on Dimorphos

The derived geometric factor, P_fl= 44 ± 10%, suggests that the contribution of ejecta momentum to momentum transfer decreases by a factor of larger than two compared to the same impact on a flat surface. This low geometric factor results from Dimorphos’s higher curvature in its north-south direction. Based on the asteroid’s pre-impact extents, the ratio of the semi-minor axis (c) to the semi-major axis (a) is 0.64³⁰. Because Dimorphos’s higher curvature in its north-south direction causes a shorter excavation range, the flow fields beneath the surface are short-lived. Therefore, excavated materials cannot change their flow directions well enough to achieve a higher ejection angle from the surface before launch. Such trends are so extreme in some directions that ejecta momenta possess anti-along track components, giving negative momentum transfer (Fig. 3d). These mechanisms noticeably reduce the net momentum transfer. The observed clumps and boulders ejected from Dimorphos may also contribute to the off-track momentum transfer but not significantly to the along-track component⁶¹. If the DART impact-driven crater were small relative to Dimorphos’s size, local morphological features would be more influential than global curvature. Deep Impact and SCI represent small impact conditions relative to the target size.

Given its limited settings, our Maxwell Z-model approach only gives first-order approximations of the flow fields. However, it offers insights into the DART impact on Dimorphos’s curved surface. The variations in Z over different azimuthal angles, i.e., Z = 2.9 ± 0.4, are similar to those observed for a flat surface^63,64,65, and its mean is consistent with the model’s ideal case, i.e., Z = 3⁴⁸. This finding indicates that regardless of Dimorphos’s high curvature, the typical value of Z for a flat surface target can reproduce the momentum transfer with a new scalar parameter, γ. One caveat is that the model parameters are set to be constant throughout the ejecta evolution. Earlier reports suggest Z increases with Δ⁶⁶. Z is approximately 2 when Δ is about 0°, meaning non-rotational flows, while Z ≥ 4 when Δ ≥ 75°. Thus, when Δ is high, flow rotation also becomes high, giving a higher ejection angle from the surface horizon.

Geometric factors on near-earth Objects less than 50 m radius

The low geometry factor of the DART impact due to Dimorphos’s north-south high curvature suggests necessary considerations of a target’s curvature to predict kinetic deflection accurately. When the crater size is large relative to a target’s size, a more practical case due to a typical desire for higher kinetic deflection, global curvature influences the flow fields and, thus, the cone geometry. The geometric factor decreases if the curvature is high, leading to a lower momentum transfer. Applying our Maxwell Z-model approach enables quantifying how the geometric factors change due to the impact locations for 99 NEOs less than 50 m in radius observed by the MANOS (Mission Accessible Near-Earth Objects Survey) project^67,68. The reason for focusing on objects less than 50 m in radius comes from recommendations from publicly available reports^1,2,3,4. Limited observation constraints add an assumption to the present study that each NEO sample is a prolate body with identical semi-intermediate and semi-minor axes (see “Methods”, Subsections Light curve samples from MANOS and Geometric factors for MANOS samples). Each body is set to have two impact locations: one along the semi-major axis and the other along the semi-minor axis. Because the impacts at both locations are vertical with respect to the local surface, the impact point along the semi-major axis has higher curvature.

Monte Carlo simulations, using the established Maxwell Z-model combined with the π-scaling relationship^24,69,70 (see “Methods”, Subsection Transient Crater Radius), determine the statistical trends of the geometric factors of all samples (see “Methods”, Subsection Geometric Factor for MAMOS Data). Impactors are assumed to have the same impact energy as DART. The analysis concept is to apply the derived Maxwell Z-model parameters from the DART study while α is independently determined for each sample. One key parameter for the α computation is the strength of each sample. While the strength is unknown, their spin states may constrain their lower bound. Some NEOs spin exceptionally rapidly, while others do not. Depending on their spin states, some objects should have mechanical strengths to stay structurally intact⁷¹. A semi-analytical stress analysis computes the minimum cohesive strength for each body to keep the structure intact, which is used for the α computation (see “Methods”, Subsection Minimum Cohesive Strength).

Figure 4 shows the distributions of the geometric factors of the samples. The geometric factor positively correlates with the target radius and strength (Fig. 4a, b). Given a constant impact energy identical to the DART impact, higher strength, and a larger target radius make the transient crater radius small relative to the body⁶⁹, reducing the curvature effect and thus making the geometric factors higher. The transition zone from zero geometric factors (= 0%) to high geometric factors (>70%) is narrow. This zone also overlaps the predicted catastrophic disruption threshold, which defines an impact condition when the final target mass is less than half the original (see “Methods”, Subsection Catastrophic Disruption Threshold). However, this predicted range is generally broad, given the higher uncertainties of models and parameters. The standard deviation increases when the geometric factor decreases because the uncertainty is more sensitive to the variations in the empirical parameters when an impact affects a larger volume in a target.

Fig. 4: Geometric factor distributions of 99 small NEOs measured by the MANOS project^67,68.

a and b Mean and standard deviation of the geometric factor for an impact along each target’s semi-minor axis. c and d Mean and standard deviation of the relative geometric factor of an impact along the semi-major axis to one along the semi-minor axis. All panels give the geometric factor distributions in percentile as a function of diameter and minimum strength. All negative outcomes in Panels (a and b) are truncated and noted as zero geometric factors. The curves in different colors, calculated using Equation (23), show interpolations between the catastrophic disruption thresholds for a pumice-like target and a cohesionless target (red) and a basalt-like target and a cohesionless target (blue). For the cohesionless target, we assume a minimal strength of 0.01 Pa. The dashed, solid, and dotted lines give bulk densities of 1000 kg/m³, 2000 kg/m³, and 4000 kg/m³. If target asteroids stay on the right side of those thresholds, they do not experience catastrophic disruption. Source data are provided as a Source Data file.

Figure 4c and d show the distribution of the relative geometric factor, defined as the difference between an impact along the semi-major axis and an impact along the semi-minor axis. The results show lower relative geometric factors along the semi-major axis than the semi-minor axis for all samples. This is because impacts along the semi-major axis encounter higher curvatures. The variation of a relative geometric factor with strength and radius appears more random, indicating the geometric factor’s dependence on the curvature. This finding suggests the importance of a target’s curvature for kinetic deflection. Therefore, selecting a flatter impact point on the target is crucial to enhancing momentum transfer.

Kinetic deflection strategy learned from DART

The increase in the encounter speed is reported to make such single impacts with high kinetic energy more feasible⁷². Another concept may be to disrupt a target body as part of kinetic deflection⁷³. However, our finding suggests that scenarios employing a single impactor having higher kinetic energy are not ideal because the efficiency of momentum transfer changes due to global curvature (Fig. 5). A flatter surface target offers higher momentum transfer for a single impact with a given kinetic energy due to higher changes in the subsurface flow fields. Alternatively, a lower kinetic energy impactor results in a smaller crater that is less affected by global curvature, increasing momentum transfer efficiency. With these trends to enhance momentum transfer, employing multiple, smaller impactors is a better solution than having a single, large impactor. In the multiple-impactor scenario, each impactor has a smaller kinetic energy than one in the single-impactor scenario. Still, the net kinetic energy can be comparable when all impactors collide with the target. Practically, the multiple-impactor scenario can send impactors at different times and aim at flatter surface points on the target, maximizing the net momentum transfer (Fig. 5). However, the crater size should not be too small because local boulders and topography can become more influential on the ejecta plume formation, as seen by earlier experimental tests⁶², changing the trends of momentum transfer. This condition applies to Deep Impact and Hayabusa2, which observed complex ejecta formation^22,26.

Fig. 5: Schematics for kinetic impact-driven momentum transfer depending on different scenarios.

Impactors with two kinetic energies add the same net kinetic energy to a target by performing multiple impacts. Net momentum transfer changes due to orientation and kinetic energy per impactor. A larger impactor may add higher kinetic energy at once, possibly causing the resulting ejecta cone to be affected by an asteroid’s curvature and have a lower kinetic deflection efficiency. Multiple smaller impactors hitting lower-curvature sides of targets may increase a net kinetic deflection efficiency. The ID of the attached LICIACube image is liciacube_luke_l0_1664234219_00112_01.fits.

Constraining the physical properties of a target object before kinetic impact is essential. While no details have been reported for global curvature⁴ in the past, this study proposes it as a key contributor that can easily change the efficiency by a factor of a few. The practical approach is to visit the object in situ and conduct key measurements as much as possible before deciding the timing of kinetic deflection. A rapid response to this demand after identifying a potential threat is not yet a mature technology⁴. Compared to mass measurement, which requires additional operational constraints, imaging the target even during a fast flyby can provide sufficient information to infer the curvature and surface conditions, significantly improving the accuracy of momentum transfer. The proposed multi-impactor scenario can also offer additional reconnaissance before the planned collisions at different times. Such multiple observations enable tracking possible large-scale modifications that a target may have altered early impacts. Global shape deformation is one of such modifications. Dimorphos likely changed its shape significantly due to the DART impact^74,75, experiencing additional orbital perturbation^70,76,77. Demonstrating capabilities to acquire such properties by a rapid reconnaissance mission is strategic to achieving sophisticated advances in kinetic deflection technologies⁷⁸.

Methods

Ejecta cone geometry

Similar to earlier work^40,46, the present measurements apply J2000 to describe the cone axis direction. In this subsection, “north" and “south" mean the celestial north and south, respectively. The ejecta projected onto the view plane of HST and LICIACube images (Table 2) gives the position angles of the cone edges. For each image, the maximum counts of bright pixels identify the directions of the linear features. Images are categorized as the side view if obvious bright features exist; otherwise, they are sorted out as the front view for earlier time stamps or the back view for later time stamps. HST images only show the cone’s side view⁴⁰, while LICIACube images change the view with time⁴⁷. The sorting process of LICIACube images intentionally accounts for the front and back sides because they narrow the range of the cone orientation. The dynamical states of the asteroids and spacecraft come from the DART mission-driven SPICE kernels with a version of d430. The resulting impact angle (from the local horizon) is 73°²⁸.

Table 2 Position angle measurements using LICIACube LUKE and Hubble Space Telescope (HST) images used in the present analysis

Five LICIACube images taken at 160–195 s after the DART impact (T + 160–195 s) capture the ejecta plume at speeds of about 20 m/s. These images also enable measuring Dimorphos’s center of the figure, compared with the simulated rim locations generated by the Dimorphos shape model. Six HST images taken at T + 0.41–8.2 h⁴⁰ offer the ejecta geometry consisting of ejected particles with comparable speed to those observed by LICIACube. Removing multiple sources unrelated to the cone edges, including diffraction spikes and a tail that evolved due to solar radiation pressure⁴⁰, effectively determines the position angles of the cone edges.

The position angle search obtains the position angle at the bright-pixel count’s peak as a mean and the peak’s width as a 1σ uncertainty (Table 2). Finding the width is not challenging because the peak is a single set with a Gaussian-like distribution without overlapping with other peaks. Averaging out the measurements of all HST images reduces the number of data samples to a single sample, as HST’s view plane does not change significantly. For LICIACube images, the samples at the earliest and latest epochs capture the front and back sides of the ejecta cone.

The cone geometry determination process uses the measured position angles to obtain the cone’s geometric parameters (DEC, RA, ϕ, θ₁, θ₂), assuming the ejecta cone geometry is time-invariant. The following formula gives the score for the k-th image:

$${X}_{k}={\sum }_{i=1}^{2}{\sum }_{j=1}^{n}\frac{1}{\sqrt{2}{\sigma }_{k,i}}\exp \left\{-\frac{{({x}_{k,j}-{\mu }_{k,i})}^{2}}{2{\sigma }_{k,i}^{2}}\right\}$$

(1)

where n is the number of pixels in the image, and i represents the northern and southern edges. x_k,j is the position angle of a simulated pixel j, μ_k,i is the mean of the measured northern or southern edges, and σ_k,i is its standard deviation. The final score is a sum of X_k(k = 1, . . . , 6). The score of the k-th image becomes zero if the view conditions are inconsistent in the simulated and measured cone geometries. For example, such a condition includes when the simulated cone geometry predicts either front or back views while the measured cone is in the side view. Monte Carlo simulations consider 500 cases by varying the measured position angles based on Gaussian uncertainties.

Determining the location of the cone apex uses two LICIACube images, LCC 2 and LCC 3, as they offer the side view of the ejecta cone and Dimorphos’s shape, partially illuminated by the sun. Extraction of the alignments between the ejecta cone edge regenerated by the measured geometric parameters and the terminator on Dimorphos (Fig. 6) identifies the cone apex location with 1σ uncertainties. One issue is that using LCC2 and LCC3 does not sufficiently constrain the apex location along the out-of-plane direction uniquely. To avoid this issue, the cone apex determination assumes the cone axis crosses the DART impact site as closely as possible. Monte Carlo simulations account for 30,000 cases by parameterizing the measured cone’s geometric parameters (Table 1) to determine the case where the cone axis is the closest to the DART impact site. Simulations are rejected if the simulated cone edge is outside the 1σ uncertainties of the measured alignments. 1000 sets of this routine give the apex location in IAU_DIMORPHOS as (− 4 ± 6, − 3 ± 9, 9 ± 10) m.

Fig. 6: Cone edge alignments determined using LICIACube images.

a 09-26T23:17:19 (LCC2). b 09-26T23:17:22 (LCC3). Table 2 defines Frame IDs. The cyan lines illustrate the alignments of the southern ejecta cone edge, while the yellow lines define the northern cone edge. The solid line is the mean alignment, while the dashed lines give cone edge variations, which defined 1σ uncertainties in this work. Both panels show the cone orientation in J2000. Source data are provided as a Source Data file.

Maxwell Z-model

The basic version of the Maxwell Z-model consists of three kinematic equations⁷⁹:

$$r={r}_{0}{(1-\cos \Delta )}^{\frac{1}{Z-2}}$$

(2)

$${v}_{r}=\frac{\alpha }{{r}^{Z}}$$

(3)

$${v}_{\Delta }={v}_{r}(Z-2)\frac{\sin \Delta }{1+\cos \Delta }$$

(4)

where r is a mass element’s radial location along a streamline, while v_r and v_Δ are its radial and tangential speeds, respectively. These quantities vary with Δ, a counterclockwise angle from the impact incident direction (Fig. 1). For a vertical impact on a flat surface, the downward direction is Δ = 0°. r₀, α, and Z represent a streamline’s kinematics. r₀ controls the size of the streamline, Z defines how sharply the streamline changes its direction, and α gives a material ejection speed. Among the kinematic parameters, α also correlates with gravitational acceleration, g, and the transient crater radius, R, which can be determined using the π-scaling relationship⁸⁰:

$$\alpha={\left\{\frac{g{R}^{2Z+1}}{4Z(Z-2)}\right\}}^{\frac{1}{2}}$$

(5)

Computation of transient crater radius

The π-scaling relationship provides the transient crater volume, V⁶⁹:

$$V={K}_{1}\left(\frac{{m}_{i}}{{\rho }_{t}}\right){\left\{\left(\frac{g{r}_{i}}{{v}_{i}^{2}}\right){\left(\frac{{\rho }_{t}}{{\rho }_{i}}\right)}^{-\frac{1}{3}}+{\left(\frac{Y}{{\rho }_{t}{v}_{i}^{2}}\right)}^{\frac{2+\mu }{2}}\right\}}^{-\frac{3\mu }{2+\mu }}.$$

(6)

where K₁, μ, and Y are empirically determined parameters. m_i, r_i, and v_i are the impactor’s mass, radius, and speed, respectively. The used values for the parameters are provided in Supplementary Table 1. Y is the strength parameter. ρ_t and ρ_i are the bulk densities of a target and an impactor, respectively. The following equation then provides the transient crater radius using the transient crater volume:

$$R={\left(\frac{3V}{\pi }\right)}^{\frac{1}{3}}$$

(7)

This parameter is hypothetical in this study, given that it is usually defined for an impact on a flat surface⁶⁹ and should be larger than the excavation range, ${r}_{E}^*$.

Computation of minimum cohesive strength

Equation (6) needs the strength parameter, Y, as an input to determine the transient crater volume in Equation (7). This study assumes that this strength parameter is comparable to the minimum cohesive strength, i.e., the lowest cohesive strength that keeps the original object structurally intact. This assumption leads to an upper bound of the crater size derived by the π-scaling relationship (see “Methods”, Subsection Transient Crater Radius). Semi-analytical framework^81,82 computes the minimum cohesive strength using a target’s ellipsoidal shape, bulk density, and spin state. Linear elasticity gives a semi-analytical stress field when the body’s rotation is in the principal axis mode. This process is independent of Young’s modulus. Using the same notation with the strength parameter, Y, for the minimum cohesive strength formulates the following Drucker-Prager yield criterion:

$$Y=\frac{1}{\kappa }(\lambda {I}_{1}+\sqrt{{J}_{2}})$$

(8)

$$\lambda=\frac{2\sin \psi }{\sqrt{3}(3-\sin \psi )}$$

(9)

$$\kappa=\frac{6\cos \psi }{\sqrt{3}(3-\sin \psi )}$$

(10)

where I₁ and J₂ are pressure and shear stress invariants, respectively, and ψ is the angle of friction, which is fixed at ψ = 35°⁸³. This provides the spatial distribution of the minimum cohesive strength over the entire body. The process selects the maximum value of Y over the object, which usually appears at the center at a short spin period⁸². The π-scaling relationship uses this computed Y to determine the transient crater radius, R.

Definition of geometric factor

The geometric factor, P, defines the ratio of the along-track momentum carried by ejecta on a curved surface to that by a vertical impact on a geometrical reference:

$$P=\frac{{L}_{T}}{{L}_{Tref}}=\frac{\beta -1}{{\beta }_{ref}-1}$$

(11)

where L_T and L_Tref are the along-track ejecta momenta that form on the curved and reference surfaces, respectively. These scalars are also equivalent to the following vector expressions, L_T = L ⋅ n_T, where L is the ejecta momenta on the curved surface and n_T is the along-track unit vector, and similarly, L_Tref = L_ref ⋅ n_T, where L_ref is the ejecta momentum on the reference surface. P can be convertible with β for these targets, where β_ref is β for a reference surface. The reference surface can be arbitrary, but the simplest one may be a flat surface, which generally offers the highest ejecta momentum among any convex surface.

Two reference surfaces considered in this work are a flat-surface target for the DART impact on Dimorphos and kinetic impacts on the MAMOS samples and a spherical target for the comparison test using iSALE-2D. The geometric factor for a flat surface target is P_fl, while that for a spherical target is P_sp. Using a spherical target for the comparison test results from applying well-established and calibrated simulation results from iSALE-2D. On the other hand, using a flat surface, which generally gives the highest efficiency among convex surfaces, can offer simple diagnostics of the momentum transfer efficiency. For example, reading P_fl’s unity value can give a direct insight into how close (far) the momentum transfer is to (from) the ideal case. However, P_sp does not give such insightful views easily because it can be larger than unity even when the ejecta momentum does not reach the flat surface case.

Light curve samples from MANOS

Samples are available through the MANOS light curve campaigns, which offer samples’ spin periods, sizes, and shapes. The project sampled 308 NEOs over 4.5 years⁶⁸. Many samples are smaller targets less than 50 m in radius, unlike those cataloged in the Asteroid Lightcurve Database, which archives larger objects in general⁸⁴. 99 sample objects have full and partial light curve data over their spin periods⁶⁸. Each sample’s shape is assumed to be a biaxial ellipsoid, where its semi-intermediate and semi-minor axis are equal. When the semi-major axis is a, and the semi-minor (intermediate) axis is b, this ratio becomes b/a. Using an available relative amplitude, Δm, results in b/a, given the following equation:

$$\Delta m=-2.5\log \left(\frac{b}{a}\right)$$

(12)

Geometric factor of DART impact on Dimorphos

Slicing the asteroid’s body parallel to the DART incident direction makes thin pieces over all azimuthal directions, each slice later called an azimuthal piece. Each azimuthal piece defines a volume at an azimuthal angle, extending to the normal direction to the DART incident direction, and thus looks like a wedge. While the number of azimuthal pieces is arbitrary, 100 pieces give acceptable computational accuracy without adding significant computational burdens. The Maxwell Z-model approach determines α, Z, and a new scaling parameter controlling the net momentum, denoted as γ. A developed scheme uniquely determines α and Z for each azimuthal piece by comparing the measured cone geometry with the spatial distribution of ejected surface materials in 10,000 streamtubes in each azimuthal piece at 177 s, the mean time of imaging for LCC2 through LCC4.

The determined α and Z for all azimuthal pieces give the ideal momentum, L_p, as the sum of each streamtube’s mass times its surface velocity, $dm{{{{\bf{v}}}}}_{{\Delta }^*}$, where dm is the streamtube’s mass and ${{{{\bf{v}}}}}_{{\Delta }^*}$ is the ejection velocity:

$${{{{\bf{L}}}}}_{p}={\sum }_{i=1}^{m}{\sum }_{j=1}^{n}d{m}_{i,j}{{{{\bf{v}}}}}_{{\Delta }^*i,j}$$

(13)

where i and j are the indices representing an azimuthal piece (1 ≤ i ≤ m) and a streamtube in one azimuthal piece (1 ≤ j ≤ n). However, Equation (13) is unrealistic because this does not account for energy loss within each streamtube. For model simplicity, introducing γ yields the actual ejecta-carried momentum:

$${{{\bf{L}}}}={\sum }_{i=1}^{m}{r}_{0,i}^{\gamma }{{{{\bf{L}}}}}_{p,i}$$

(14)

where L_p,i and r_0,i are the momentum and r₀ of an azimuthal piece of i, and γ takes the same value over the entire azimuthal pieces. The reason for introducing γ as a power of r_0,i was to ensure that γ is consistent over the entire azimuthal pieces regardless of the variations in r_0,i. However, this equation must keep the units on both sides consistent, needing ${r}_{0,i}^{\gamma }$ to be dimensionless. Considering an alternative form dividing r_0,i by its unity length, i.e., ${({r}_{0,i}/1)}^{\gamma }$, can mitigate this issue. This way, L and L_p have consistent units.

Given the geometric parameters, cone apex location, β, α, and Z, the algorithm iteratively determines γ such that Equation (15) satisfies under an error threshold for γ of 0.01%. This process first computes the simulated β using its form^6,46:

$$\beta=1+\frac{{{{\bf{L}}}}\cdot {{{{\bf{n}}}}}_{T}}{({{{\bf{E}}}}\cdot {{{{\bf{L}}}}}_{sc})\cdot ({{{\bf{E}}}}\cdot {{{{\bf{n}}}}}_{T})}$$

(15)

where E is the unit vector of the net ejecta momentum, and L_sc is the momentum carried by the spacecraft. Supplementary Table 2 gives the values of the Dimorphos along-track direction (n_T) and the DART incident direction. The process then uses a Newton method to obtain γ by comparing the simulated β with the measured one⁴⁶:

$$\beta=(3.61\pm 0.2)\frac{\rho }{{\rho }_{ref}}-0.03\pm 0.02(1\sigma )$$

(16)

where ρ_ref is the reference bulk density fixed at 2400 kg m⁻³, and ρ is the considered bulk density, which is 2, 400 ± 900(1σ) kg m⁻³.

Each Monte Carlo simulation also calculates the ejecta momenta on a reference surface. Computing it uses the same inputs, including the azimuthal variations in α and Z. However, it does not perform the iterative scheme to compute γ but uses the value obtained above. Applying the derived ejecta momenta on both curved and reference surfaces to Equation (11) results in P_fl. Running 1000 Monte Carlo simulations with Gaussian-distributed inputs offers the statistical behaviors of P_fl (Table 1).

Validation of geometric factor computation by Maxwell Z-model

Comparisons between the Maxwell Z-model and iSALE-2D simulations^49,50,51,52 reveal that both models give consistent geometric factors relative to a spherical target (Fig. 7). In this test, nine iSALE-2D simulations with different biaxial ellipsoids offer variations in β, assuming that each target’s along-track direction corresponds to an impactor’s anti-incident direction. The target dimension is 2a × 2b × 2c, where b = c. The simulations parameterized b/a, which ranged between 0.4 and 2.0 with an increment of 0.2. The equivalent radius is 75 m for all cases. To mimic the DART impact condition, each case assumes a low-density impactor modeled as a sphere with a radius of 1.2 m and a mass of 580 kg. The impact speed is 6 km/s, and the impact site is along the axis. We determine β for each case to obtain P_sp.

Fig. 7: Result comparison between Maxwell Z-model and iSALE-2D simulations.

The x-axis is the aspect ratio of the biaxial ellipsoid target with dimensions of 2a × 2b × 2c, where b = c, and the y-axis is the geometric factor relative to a spherical target, P_sp, in percentiles. The aspect ratio is defined as b/a. The black line with error bars resulted from Maxwell Z-model simulations, while the red line was from iSALE-2D simulations. The Maxwell Z-model’s error bar gives 1σ uncertainties. Source data are provided as a Source Data file.

The present iSALE-2D simulations follow the parameter settings from earlier work^10,12. The impactor’s material behavior follows the Tillotson equation of state (EOS)⁸⁵ and the Johnson-Cook strength model for aluminum⁸⁶. The target’s behavior follows the Tillotson EOS for basalt⁸⁷ with a modified grain density of 3500 kg/m³, which corresponds to the average grain density of L/LL chondrites⁸⁸. The current version of iSALE-2D sets a simple pressure-dependent strength model to define the target’s shear strength⁵⁰, with a cohesive strength of 1 Pa and a coefficient of internal friction of 0.55. The target’s porosity is 45% at the initial condition, and its behavior follows the ϵ − α compaction model⁵¹. All parameters are available in Supplementary Table 1. Given impact scaling relationships⁸⁹, the iSALE-2D results may recreate impact behaviors given in the range of Z  = 2 − 3, where Z is the Maxwell Z-model kinematic parameter.

The Maxwell Z-model approach explores the statistical trends of P_sp by considering Gaussian-based inputs to the model based on our earlier analysis for Dimorphos (Z = 2.932 ± 0.406 and γ = 0.731 ± 0.120) and those from the literature of the π-scaling relationships (K1= 0.22 ± 0.02 and μ = 0.47 ± 0.07). These π-scaling parameters assume impacts on dry sands and rocks²⁴. The impactor’s bulk density, another input parameter in the model, is fixed at constant at 1925 kg/m³ for all cases to make this test consistent with the iSALE-2D runs. The strength parameter, Y, is set to be 1 Pa to mimic vertical impacts on cohesionless targets. The analysis performs 1200 runs with Gaussian-distributed inputs for each b/a case. Some unrealistic solutions exist, giving extremely high or low P_sp. Such solutions come from parameter conditions at the tails of their distributions or simply ill-defined numerical values for the parameter conditions. The post-processing steps remove any solutions being higher than 200% or lower than − 200%, removing 15−25% of all solutions, having no significant impact on the statistical trends of our results.

Geometric factors for MANOS samples

Simulations for each sample perform two geometric factor computations. The first computation considers an impact along the semi-minor axis, while the second one simulates that along the semi-major axis. All simulations assume vertical impacts with the same impact scale as the DART impact, leading to axisymmetric ejecta momenta. The assumption is that the along-track direction corresponds to an impactor’s anti-incident direction. With these simulation settings, impacts along the semi-major axis experience a higher curvature than those along the semi-minor axis. The simulation scheme is the same as the geometric factor computation for the DART impact on Dimorphos, which determines P_fl. However, there are two differences. First, this scheme uses the parameters determined from the DART impact case, except for α. Second, α comes from Equation (5) because it depends on R and g, inconsistent with the DART impact case. R is determined using the π-scaling relationship, where Y comes from the minimum cohesive strength. Key inputs in the π-scaling relationship are compiled in Supplementary Table 2.

Monte Carlo simulations give the statistical behavior of each sample’s geometric factor. The bulk density in this test ranges between 1000 and 4000 kg/m³ to cover its uncertainties. This distribution varies Y, R, g, and thus α for each case. Each MAMOS sample has 1200 simulations for one impact scenario, i.e., 2400 simulations for both scenarios. One issue is that when the transient crater is too large, the computation of the ejecta momentum accumulates numerical errors. This is because Δ and r₀ become extremely small and large, reducing numerical accuracy. To avoid this issue, the algorithm only considers ${\Delta }_{E}^* > 2{0}{\circ }$, allowing all streamtubes to cover up to about 83% of the entire volume. The ${\Delta }_{E}^* > 2{0}{\circ }$ constraint likely underestimates the geometric factor. However, such a case makes the geometric factor unrealistic and is rejected by the allowed geometric factor range anyway. For one impact scenario of each sample, this sorting process yields about 85% of simulations that satisfy a geometric factor ranging between − 200% and 200 %.

Catastrophic disruption threshold (${Q}_{D}^*$)

The catastrophic disruption threshold, ${Q}_{D}^*$, defines the specific impact energy per mass required to disperse half of the target material mass, which is given as:

$${Q}_{D}^ \ast=\frac{{U}^{2}}{2}\frac{{m}_{sc}}{M}$$

(17)

where m_sc is the spacecraft mass, M is the target mass, and U is the spacecraft relative impact speed. In the present study, m_sc and U are identical to DART’s, and m_sc ≪ M. On the other hand, ${Q}_{D}^*$ is also a function of the target radius and strength^90,91. However, no adequate disruption threshold formula covers the parameter range considered in this study. Accepting this issue, our approach uses two samples with different strengths at different target radii to draw the correlations. In this test, each sample has three bulk density cases, 1000 kg/m³, 2000 kg/m³, and 4000 kg/m³.

The first sample is a target with higher strength⁹². ${Q}_{D}^*$ for a high-strength material defines the following equation:

$${Q}_{D}^ \ast=10{0}^{{a}_{s}}\times {Q}_{0}{R}_{{Q}_{D}^*}^{{a}_{s}}+10{0}^{{b}_{s}}\times B\rho {R}_{{Q}_{D}^*}^{{b}_{s}}$$

(18)

where ${R}_{{Q}_{D}^*}$ is the disrupting target radius, ρ is the bulk density, and Q₀, a_s, b, and B are the empirical parameters. Applying an earlier study using Smooth-Particle Hydrodynamics (Bern SPH)⁹² yields two high-strength materials. One is a basalt-like target, and the other is a pumice-like target. The parameters used here are based on impact simulations with an impact speed of 5 km/s and a target radius of 1.5 cm⁹², given in Supplementary Table 3. Using Equation (18) determines ${R}_{{Q}_{D}^*}$ at kinetic energy imparted by the DART-like impactor.

The earlier targets⁹² were 1.5 cm radius, provided with their strengths at this size, Y(1.5 cm). Re-scaling Y(1.5 cm) to Y requires the static failure threshold for a specimen failing at the smallest strain, ϵ_min:

$${\epsilon }_{min}={(k{V}_{{Q}_{D}^*})}^{-\frac{1}{m}}$$

(19)

where k and m are the Weibull parameters, given in Supplementary Table 3, and ${V}_{{Q}_{D}^*}$ is the disrupting target volume. Y at a given ${R}_{{Q}_{D}^*}$ is written as:

$$Y={\epsilon }_{min}{E}_{s}$$

(20)

where E_s = 5.3 × 10¹⁰ Pa is Young’s modulus. Combining these equations with the assumption that the Weinbull parameters and E_s are size-independent yields the relationship between Y and Y(1.5 cm):

$$Y={\left(\frac{{V}_{{Q}_{D}^*}}{1.41\times 1{0}^{-5}}\right)}^{-\frac{1}{m}}Y(1.5cm)$$

(21)

The second sample is a cohesionless target^58,93. ${Q}_{D}^*$ for this case gives the equation⁹⁰:

$${Q}_{D}^ \ast={a}_{g}{R}_{{Q}_{D}^*}^{3{\mu }_{g}}{U}^{2-3{\mu }_{g}}$$

(22)

where a_g and μ_g are empirical parameters, and U is the impact speed (Supplementary Table 3). These quantities are based on Bern SPH simulations using an impact speed range of 3–9 km/s and an impact mass of 500 kg⁵⁸. In this scaling relationship, Y is fixed at 10⁻² Pa.

Interpolating these two samples yields a correlation between Y and ${R}_{{Q}_{D}^*}$. The interpolation function gives the following power law:

$$Y=\xi {R}_{{Q}_{D}^*}^{\eta }$$

(23)

where ξ and η come from the constraint that this scaling function must cross the data samples above. The approach considers two scaling functions: the combination of a pumice-like target and a cohesionless target and that of a basalt target and a cohesionless target^58,92.