Main

Double Asteroid Redirection Test (DART) was a planetary defence mission to demonstrate the feasibility of using a kinetic impactor to change the trajectory of an asteroid1. The impact was successful and highly effective, resulting in a reduction in Dimorphos’s orbital period around Didymos, which was initially 11 h and 55 min, by 33 ± 1 min (ref. 2). The LICIACube Unit Key Explorer (LUKE) instrument onboard the cubesat3 captured images of the system between 29 and 320 s after impact to reveal filamentary streams of ejecta and other complex patterns that expanded for several kilometres from the impact site4. Moreover, the dramatic brightening of the Didymos system by solar illumination of released impact ejecta was observed by ground- and space-based telescopes2,5,6 for many weeks after the impact.

The obtained 33 min reduction in the binary orbital period2 implies a momentum transfer to Dimorphos that exceeded the incident momentum of the DART spacecraft by a factor β that ranges from 2.2 to 4.9 depending on the mass of Dimorphos7, which is not currently known but will be measured by the ESA Hera spacecraft in early 20278. The β parameter is defined as the ratio of the target momentum increment after the impact to the impactor momentum, in the direction of the net ejecta momentum, and is related to the additional thrust from the production of impact ejecta7,9. β strongly depends on impact conditions (impact velocity and impact angle1) and target material properties, such as strength, porosity, bulk density and target surface structure10,11,12.

Information about the spacecraft, the impact location and the impact angle are well understood for the DART impact1 (Extended Data Fig. 1 and Extended Data Table 1). However, the mass and surface properties of Dimorphos are still ambiguous. It was not possible to directly measure the mass and bulk density of Dimorphos with DART or LICIACube. Instead, these parameters were estimated from the total mass of the binary system, as derived from Dimorphos’s orbit13, and updated volume estimates of Didymos and Dimorphos provided by DART (ref. 1). The estimated bulk density of Dimorphos ranges from ρB = 1,500 to 3,300 kg m−3 (refs. 1,7).

The surface material properties and subsurface structure of Dimorphos were also not directly measured. However, these target parameters are vital for understanding the impact process and for transforming the kinetic impactor method from a full-scale experiment by DART into a well-understood and reliable mitigation technique for planetary defence. Moreover, the material properties of Dimorphos provide information about the origin and evolution of the Didymos–Dimorphos system as well as the overall characteristics of rubble-pile asteroids and binary asteroid systems.

In this work, we simulate the DART impact numerically and compare the results with observations to infer the properties of Dimorphos. We performed numerical simulations of the DART impact using the shock physics code Bern SPH (refs. 14,15) over a range of assumed sets of material properties and interior structures for Dimorphos. We represented the DART spacecraft as a low-density spherical projectile of equivalent mass (‘Projectile’ section in Methods) while the impact velocity and angle were kept fixed. We simulated the asteroid’s response to the DART impact for up to 1 h after the time of impact using the numerical approach developed in refs. 16,17 to model a late-stage, low-speed deformation (‘Modelling approach for the late-stage evolution’ in Methods). Bern SPH’s fast-integration scheme has been validated against laboratory experiments18 and was recently successfully applied to model the impact of the Hayabusa2 Small Carry-on Impactor17. Due to the relatively short timescales modelled, the rotation of Dimorphos around Didymos’s and Didymos’s gravity were not accounted for.

We obtained realistic configurations of boulders for our rubble-pile targets from simulations of the gravitational collapse of a cloud of spherical particles with a predefined size–frequency distribution (SFD)1,19. To closely replicate the topography described by ref. 1, we then selectively removed particles positioned near the surface.

To explore a large possible range of boulder volume fractions (0 to 50 vol%), we replaced some of these boulders with matrix material when we built our target asteroid (Extended Data Fig. 2). We explicitly modelled only boulders larger than 2.5 m in diameter, and the space between boulders was filled with matrix material. Boulders smaller than 2.5 m were removed from the SFD because they were too small to be resolved individually. Thus, components smaller than 2.5 m were considered part of the matrix, which was modelled as a granular material with low but limited cohesion. This approach created an asteroid whose interior structure is like that of its surface and whose structure overall is consistent with a gravitational collapse. For the purpose of this study, other deep interior structures were not considered.

The bulk porosity of Dimorphos results from macroporosity between individual rocks and boulders as well as microporosity within rocks. An analysis of reflectance spectra of Didymos indicates that the best meteorite analogues for boulders on Dimorphos are L/LL ordinary chondrites20,21,22, which have grain densities of ~3,200–3,600 kg m−3 and low microporosities of ~8–10% (ref. 23). Using the method described by ref. 24 (‘Macroporosity calculations for Dimorphos’ in Methods), we calculated macroporosities of 38 ± 3% from the boulder SFD in the last complete image taken by DART (ref. 1) and 34 ± 4% from the global SFDs measured on Dimorphos19 (Extended Data Fig. 3). The derived macroporosity was primarily determined by the boulder SFD (Extended Data Fig. 4) and the sphericity/roundness of the boulders (Extended Data Fig. 5). Importantly, the macroporosity estimation is largely independent of the assumed minimum boulder size. In our simulations, the initial micropososity within boulders was fixed at 10% and the initial porosity of the matrix (macroporosity + microporosity) was varied between 35% and 65% (Extended Data Fig. 6). Both the porosity in the boulders and in the matrix were modelled using the P-α porosity compaction model14 with parameters as summarized in Extended Data Table 2.

Based on laboratory measurements of meteorite falls, the average tensile strength of ordinary chondrites is 24 ± 11 MPa, and there is no statistical difference between L and LL types23. In all our simulations, the initial material properties of the boulders were kept the same and we employed the tensile strength and fracture model as described in refs. 14,15, with parameters corresponding to a tensile strength YT ≈ 20 MPa for a ~2 cm specimen. For the boulders on Dimorphos, the average YT ≈ 10 MPa (Table 1). However, based on previous impact studies (for example, ref. 18), we did not expect the impact outcome to be very sensitive to the boulder tensile strength.

Table 1 Table of fixed and varied target parameters
Full size table

The response of the target matrix material to shear deformation is described by a simple pressure-dependent strength model25,26. The ability of a material to resist different types of stresses is an indicator of its strength. Granular materials, for instance, may exhibit considerable shear strength due to the presence of van der Waals forces and the particles’ inability to separate or slide over each other due to interlocking mechanisms27,28,29. Here we focus on the influence of the shear strength at zero pressure, commonly known as cohesion (or cohesive strength). Another important term in the strength model is the coefficient of internal friction. Although this parameter cannot be directly determined, it is possible to relate it to the angle of repose (Methods) and bound the range of plausible values by making reasonable assumptions. The angle of repose of low-cohesion materials has been measured to be θ = 22° (f = 0.4) for glass beads30, θ = 30° (f = 0.55) for quartz sand31 and θ = 35–45° (f = 0.7–0.9) for lunar regolith32.

On Dimorphos, for a cohesion lower than Y0 ≈ 4 Pa, the impact occurs in the gravity-dominated regime in which crater growth is halted by the asteroid’s small gravity rather than its cohesion33. Therefore, we first model impacts into cohesionless rubble piles (with Y0 = 0 Pa but with a coefficient of internal friction of f = 0.55 (ref. 31), which is equivalent to θ ≈ 30°). Given our other assumptions regarding material properties, we consider that these models of impacts into targets with no cohesion provide an upper limit on the possible momentum enhancement that can be achieved from the DART impact for a given asteroid mass.

The momentum enhancement β was calculated using two distinct methods. For the first method, β was calculated by summing the momentum over all the SPH particles with ejection velocities larger than the escape velocity vesc. For a given impact, the magnitude of ejecta momentum in the direction of the net ejecta momentum is given by

$${p}_\mathrm{ej}= \left| \sum {m}_\mathrm{e}\mathbf{v}_\mathrm{ej}\right| ,$$
(1)

where me and \(\mathbf{v}_\mathrm{ej}\) are the mass and velocity vector of individual SPH particles, respectively. The pej calculation takes the gravitational influence of Dimorphos into account. However, it does not account for the gravitational influence of Didymos. The second method, described in ref. 34, tracks the velocity magnitude of the asteroid centre of mass postimpact by summing the momentum of all material that remained below the escape velocity after the reaccumulation of the ejecta. The absolute difference in β resulting from the two calculation methods was used to calculate the error of our reported β values.

Results

First, we varied the boulder volume fraction (the volume fraction of objects >2.5 m in size) within the target between 0 and 50 vol%, while keeping the asteroid volume constant (Table 1). As a result, the mass and the bulk density of the asteroid varied with the boulder packing. Our simulations show that β is relatively insensitive to the boulder volume fractions up to ~30 vol% (Fig. 1a). For boulder volume fractions larger than about 30 vol%, the number of boulders much larger than the projectile close to the impact point is high enough that the crater efficiency and, subsequently, β are drastically reduced by boulder interlocking35 and possibly armouring36, which hinder crater growth. These results indicate that at least the surface and shallow subsurface of Dimorphos have a low volume fraction of boulders larger than 2.5 m (less than ~40 vol%), which is consistent with the last few images sent by DART before impact1.

Fig. 1: Momentum enhancement β as derived from SPH simulations.
figure 1

a–d, Continuous black lines show the β dependence on Dimorphos’s bulk density ρB derived from dynamical simulations7 (1σ uncertainty is shown by the dashed lines). Data points that cross a solid black line are consistent with the measured β from the DART impact. The uncertainty for each simulated data point (vertical bars) was calculated from the absolute difference in β calculated from two different methods (‘Momentum-enhancement calculations’ in Methods). a, β as a function of ρB for cohesionless targets with the same dimensions as Dimorphos1 and with boulder volume fractions ranging from 0 vol% (no boulders larger than 2.5 m) to 50 vol%. For fixed target volume (0.00181 km3) and fixed matrix porosity (ϕ0 = 45%), the mass and bulk density of Dimorphos vary with the boulder packing. b, β as a function of ρB for cohesionless targets with the matrix porosity varying between 35% and 65% and two boulder packings: 21 and 30 vol%. ρB was calculated for a fixed asteroid volume and varies with matrix porosity and boulder packing. c, β as a function of matrix cohesion (Y0) for the DART impact into targets with varying coefficient of internal friction (f = 0.40 or 0.55), an assumed grain density ρg = 3,200 kg m−3 and 30 vol% boulder packing. The horizontal line shows β derived from dynamical simulations for a target with ρB = 2,160 kg m−3 (minimum density consistent with the results in a and b). d, β as a function of Y0 for the DART impact into targets with f = 0.40–0.70, ρg = 3,500 kg m−3 and 30 vol% boulder packing. The horizontal line shows β derived from dynamical simulations for a target with ρB = 2,360 kg m−3 (maximum density consistent with the results in a and b).

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Our simulation results for the DART impact on a cohesionless surface of Dimorphos also provide a means to constrain the bulk density of the asteroid using the measured β (Fig. 1b), assuming a grain density in the range 3,200–3,500 kg m−3. For a body with a fixed volume and grain density (Table 1), the bulk density is influenced by its porosity. The dominant effect of additional target porosity is a reduction in the bulk density of the target rather than a reduction in ejecta from compaction of the pore space. For example, decreasing the bulk density of Dimorphos will increase the overall crater size and decrease the asteroid’s mass (for a fixed volume) and escape velocity. This allows a greater total volume of ejecta to escape, but the ejecta mass is similar. Increasing the bulk density has the opposite effect. The consequence is that the total momentum of escaping ejecta measured constrains the target bulk density and porosity: for a cohesionless surface of Dimorphos, the upper bound on β (~3.6) implies that the bulk density of Dimorphos is less than the current best estimate of the asteroid’s bulk density of 2,400 kg m−3 (ref. 1). Thus, Dimorphos is probably more porous and, therefore, may have a rubble-pile structure throughout the whole body.

Since the surface strength of Dimorphos is poorly constrained, for a fixed boulder distribution (30 vol%), matrix porosity (ϕ0 = 45%) and grain density (ρg = 3,200 or 3,500 kg m−3), we varied the matrix material cohesion (Y0 = 0–50 Pa) and coefficient of internal friction (f = 0.4–0.7) (Table 1 and Methods). Several possible combinations of cohesion, coefficient of internal friction and bulk density could result in the observed deflection and account for the observed momentum enhancement (Fig. 1c,d). Despite this non-uniqueness, it is possible to bound the range of plausible values by making reasonable assumptions since f = 0.4 is a lower limit for geological materials. For a target with f = 0.4 and ϕ0 = 45%, the cohesion on the surface of Dimorphos is probably lower than ~50 Pa (Fig. 1c,d). However, lower bulk densities (ρB < 2,000 kg m−3) or higher matrix porosities (ϕ0 > 55%) would require higher cohesion to match the observations (Fig. 1).

Ejecta curtain opening angle and morphology

In our simulations of the DART impact into Dimorphos-like rubble-pile targets, we observed the temporal changes of the ejecta cone opening angle and studied the dependences on target properties. We found no notable dependences of ejecta cone opening angle on the friction coefficient of the targets. This finding contrasts with the strong dependence of the cone opening angle on the coefficient of internal friction that is found in simulations37 of the DART impact into homogeneous planar targets. Our present simulations of impacts into Dimorphos-like rubble piles found that the ejecta cone opening angle and ejecta mass depend on target cohesion. Figures 2 and 3 compare results for cohesionless targets (Y0 = 0 Pa) with those for cohesive targets (Y0 = 500 Pa).

Fig. 2: LICIACube images of the expanding ejecta cone compared with SPH simulation results.
figure 2

a, Image acquired by the LUKE instrument onboard LICIACube at a distance of ~76 km, 160 s after the impact. b, Zoomed-in image of Dimorphos and impact ejecta. The approximate outline of the asteroid is shown in yellow. The ejecta curtain exhibits undulations, filamentary patterns and shadows. c, Bern SPH simulation of the DART impact into a cohesionless, rubble-pile Dimorphos-sized target (with f = 0.55 and ϕ0 = 45%) at T = 160 s. Due to the limited visualization domain, only a portion of the ejecta are shown. d, Simulation of the impact into a rubble-pile Dimorphos-sized target with Y0 = 500 Pa (β = 2.26 ± 0.28). e, Image acquired by LUKE at a distance of ~71 km, 178 s after the impact. f, Zoomed-in image of Dimorphos and impact ejecta. g, Same as c but at T = 178 s. h, Same as d but at T = 178 s. The optical depth of the ejecta cone is not computed for this comparison between the observations and simulation output. Panels a,b,e,f adapted from ref. 4, Springer Nature Limited.

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The ejecta plume for the cohesionless target (Fig. 2c,g) is more massive than that for the cohesive target (Fig. 2d,h). For both cases, the fastest ejecta, released shortly after the impact with velocities higher than a few tens of m s−1, form a cone opening angle ω ≈ 90° (Fig. 3a,b). Such fast ejecta are influenced by the spacecraft geometry (for example, refs. 38,39), which is highly simplified in this study. On the other hand, slower ejecta, released at late times after the impact (hundreds to thousands of seconds) with velocities vesc < v < 5 m s−1, form a wider ejecta cone angle of ~140° (at 1 m s−1, Fig. 3a) for the cohesionless target. For the cohesive target (Y0 = 500 Pa), crater growth ceases about 100 s after the impact, before the crater grows large enough for the ejection angle to be influenced by target curvature. In this case, the maximum ejecta opening angle is ~120° (at 1 m s−1, Fig. 3b). On the other hand, for the low-cohesion targets, the mass ejected at low velocities (vesc < v < 10 m s−1) greatly exceeds the low-velocity ejecta mass from the cohesive target (Y0 = 500 Pa), implying a larger cratering efficiency and crater growth continuing to later times, resulting in a wider cone opening angle influenced by target curvature.

Fig. 3: Ejecta cone opening angle.
figure 3

a, Cone opening angle derived from cohesionless (Y0 = 0 Pa) SPH simulations (shown in Fig. 2c,g) in the N–S direction (x = 0 plane) and in the E–W direction (z = 0 plane). b, Cone opening angle derived from SPH simulations with Y0 = 500 Pa in the N–S direction (x = 0 plane) and in the E–W direction (z = 0 plane). The cone opening angle derived from observations is plotted for comparison in both a and b. ω = 135 ± 4° was measured from LICIACube data based on the opening angle at the base of the cone at T ≈ 170 s, resulting in ejecta velocities in the range of a few cm s−1 to a few tens of m s−1 (ref. 4). ω = 125 ± 10° was measured from HST data for ejecta in the range 1–10 m s−1 (ref. 5). Temporal evolution measurements of the observed ejecta cone were not possible due to the limited observation window.

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The characteristics of the ejecta plume observed by LICIACube provide constraints regarding the target cohesion. At time after impact T = 160 s, LICIACube’s LUKE acquired images show ejecta concentrated into rays, which cast shadows along the ejecta cone (Fig. 2a,b). At T = 178 s, the bottom of the ejecta cone and the surface of Dimorphos are obscured by the shadow cast by the ejecta (Fig. 2e,f and Extended Data Fig. 7). The shadow observed at T = 178 s implies that crater growth and the release of low-speed ejecta continued to that time, consistent only with low-cohesion target cases, for example, ref. 40.

Images obtained from LICIACube (ref. 4) and Hubble Space Telescope (HST) observations5 revealed a wide ejecta cone angle, estimated to be ω ≈ 115–139° (Extended Data Table 3). These observations were used to determine the ejecta opening angle at specific times, up to 3 min after the impact for LICIACube (ref. 4) and up to 8 h after the impact for HST5. To compare simulations results with observations of the ejecta opening angle at a specific time after impact, we determined the implied ejecta velocity at the base of the visible cone in the LICIACube images4 (Fig. 2e,f) using its distance above the surface and time after impact (Fig. 3).

Overall, we found that to qualitatively reproduce the amount of material observed in the ejecta cone (Fig. 2c,g) as well as the observed cone opening angle of up to ω ≈ 139°, Dimorphos’s surface cohesion must not exceed ~500 Pa. From our suite of numerical simulations with the assumed boulder packing, matrix porosity and grain density summarized in Table 1, we found that the target case with f = 0.55, ρB = 2,200 kg m−3 and Y0 less than a few pascals is consistent with the target mechanical properties inferred from surface geology41 and produces a β value (Fig. 1), excavation timescale (Fig. 2) and ejecta opening angle (Fig. 3) most consistent with observations.

Deformation

Observations from the first few hours after impact imply that more than 1.3–2.2 × 107 kg of ejecta were released from the DART impact (equivalent to 0.3–0.5% of Dimorphos’s mass, assuming a bulk density of 2,400 kg m−3)6. Our simulation results for weak (Y0 < 50 Pa), Dimorphos-like targets (f = 0.55 in Fig. 1c) show that the amount of ejected material could be as high as 1% of Dimorphos’s mass (Fig. 4a). Moreover, up to 8% of Dimorphos’s mass may have been displaced or ejected at below the escape velocity of Dimorphos. In all impact scenarios simulated here, the DART impact does not produce a conventional impact crater but instead causes global deformation of the target (Fig. 4b).

Fig. 4: Global deformation of Dimorphos.
figure 4

a, Total target mass ejected with speeds above vesc and total target mass displaced or ejected below vesc, normalized by the initial target mass MT for a Dimorphos-like target with f = 0.55, ϕ0 = 45% and cohesion levels between Y0 = 0 and 50 Pa. b, Two-dimensional slices (taken at x = 0 in the y–z plane and at z = 0 in the x–y plane), at T ≈ 1 h after the impact. The boulder material is shown in yellow and the matrix material is shown in blue. The red contours show the initial target profile before the impact. The black arrows show the impact direction.

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The outcome of the impact in terms of the postimpact target morphology is highly sensitive to the target cohesion. For a cohesionless target, the ratio of the major to intermediate axes a/b could have changed from the reported pre-impact value of 1.02 ± 0.02 (ref. 1) to as much as 1.2. Such a large change in the a/b ratio is detectable with the highest-quality postimpact light curve data13,42.

A global deformation of Dimorphos would have modified the gravitational field between Didymos and Dimorphos with important implications for its orbit. The shape change would cause an additional perturbation to Dimorphos’s orbit, on top of those caused by the spacecraft momentum and ejecta recoil, and this effect can account for a few seconds to several minutes of the observed orbit period change (~33 min), depending on the magnitude of the deformation43. Any deformation would alter Dimorphos’s mass distribution and affect its postimpact rotation state (for example, refs. 44,45).

Implications for binary asteroid system formation

Our numerical simulations suggest that Dimorphos is probably a rubble-pile asteroid with a bulk density comparable to or lower than that of Didymos. Our calculations based on the observed boulder SFD indicate that the macroporosity estimate for the surface of Dimorphos (~35%) is approximately twice the value obtained for the surface of Ryugu, as determined through the same method24,46, but is comparable with the macroporosity of Itokawa24,47. However, note that this estimate is only a rough approximation due to the limited data currently available. The upcoming Hera mission will be able to provide better constrains.

Our findings serve as crucial evidence regarding the origin of Dimorphos as a secondary in a double asteroid system. To maintain its structural stability given its rapid rotation period of 2.26 h, the primary, Didymos, probably requires a higher cohesive strength, estimated to be of the order of tens of pascals48. This level of cohesion can be attributed to van der Waals forces acting between the fine regolith grains49 or to a coherent inner core50. However, our best-fitting scenarios indicate that Dimorphos, the satellite of Didymos, exhibits a cohesive strength of less than a few pascals. This observed disparity in cohesive strength between Didymos and Dimorphos suggests a potential scarcity of fine grains within Dimorphos’s structure as well as a weak and fragmented internal structure.

The material properties estimated in our study support the hypothesis that Dimorphos formed through rotationally or impact-induced mass shedding and subsequent reaccumulation from Didymos. The accretion of orbiting mass shed from Didymos occurs over a period of several days to years51, during which fine grains tend to escape from the system due to solar radiation pressure52. As a result, the accreted satellite, Dimorphos, has limited fines and cohesion.

Although the mechanical properties of Dimorphos resemble those of Ryugu and Bennu (for example, refs. 17,53,54,55), these findings may not be applicable to single, small, S-type asteroids and specifically to their moons. The implications of our study may extend beyond Dimorphos and provide valuable insights into the formation processes of similar small S-type binary asteroid systems.

Moreover, since the DART spacecraft probably caused global deformation of Dimorphos, we can infer that similarly formed asteroid moons are easily reshaped and their surfaces are relatively young16. Overall, the findings of this study provide valuable information for understanding the formation and characteristics of binary asteroids and will inform future exploration and asteroid deflection efforts.

Methods

Macroporosity calculations for Dimorphos

The bulk porosity of Dimorphos results from macroporosity between individual rocks and microporosity within rocks. The macroporosity of a granular assembly is affected by the particle SFD, interparticle friction and the packing history56. Numerical simulations of gravitational aggregation29,50,56, which is one of the probable formation mechanisms for Dimorphos-like secondaries51, can be used to estimate the initial packing of the boulders within the asteroid.

We use the method described by ref. 24 to calculate the surface macroporosity from the SFD of boulders at the impact location1 and the global boulder SFD on the illuminated side of Dimorphos19 (Extended Data Fig. 4a). We used a cumulative Weibull (Rosin–Rammler) distribution24,57,58 to represent the two SFDs:

$$N(D)={N}_{{\mathrm{T}}}\exp(-3{(D/\lambda )}^{\beta })/\beta ,$$
(2)

where D is the boulder size and NT, λ and β are constants (Extended Data Fig. 4a). We found that the fitting constants NT, λ and β vary with the assumed minimum boulder diameter Dmin, as seen in Extended Data Fig. 4b,c,d.

We used Monte Carlo code to generate a Gaussian distribution for the particle bulk density (including the approximate microporosity), the initial boulder packing, the boulder sphericity and the boulder roundness. For the bulk density calculations, we assumed a grain density of 3,500 ± 100 kg m−3, with microporosity ~10% (ref. 23), giving a mean value of 3,220 kg m−3 (Extended Data Fig. 5a). Based on the pkdgrav simulations described in ‘Numerical model’, we considered an initial packing with a mean value of 39% (Extended Data Fig. 5b). Preliminary estimates of the sphericity and the roundness of boulders were derived from the last complete image taken by the DRACO camera onboard DART59, giving mean values of 0.74 and 0.56, respectively (Extended data Fig. 5c,d). Other observational output constraints are given in Extended Data Table 3.

Numerical model

In this work, we used the Bern SPH shock physics code14,15,60. The code was originally developed by refs. 60,61 to model the collisional fragmentation of rocky bodies. It was later parallelized62 and further extended by refs. 14,15 to model porous and granular materials. The most recent version of the code includes a tensile fracture model60, a porosity model based on the P-α model14, pressure-dependent strength models15 and self-gravity. The Bern SPH code has been validated in a number of studies (for example, refs. 63,15) and benchmarks against other codes (for example, ref. 64).

We modelled the DART impact and DART-like impacts into rubble-pile ellipsoidal targets composed of different distributions of spherical boulders embedded into a matrix material (Extended Data Fig. 7). We tracked the evolution of the ejecta and of the target for up to 1 h after the impact. Due to the relatively short timescales modelled, the rotation of Dimorphos around Didymos and Didymos’s gravity were not accounted for. To obtain realistic configurations of boulders, we used the rubble-pile model with Dimorphos’s surface boulder SFD, which was generated from the gravitational accumulation simulations using the N-body code pkdgrav (ref. 65).

Projectile

The DART spacecraft bus was approximately 1.2 × 1.3 × 1.3 m in size, with structures extending to approximately 1.8 × 1.9 × 2.6 m (ref. 1). Additionally, the spacecraft featured two large solar arrays, each measuring 8.5 m in length. The spacecraft weighed 579.4 ± 0.7 kg at impact1. Studies of the influence of the spacecraft geometry on cratering39 show that a simple solid representation underestimates the surface area of the impact compared to a ~20 m complex projectile. Moreover, the penetration depth of a solid sphere is larger than for the spacecraft itself. However, ref. 38 showed that the projectile geometry affects only the very early, fast ejecta generated from within the coupling zone. Therefore, due to resolution constraints, here we simplify the projectile geometry. To match the penetration depth of the spacecraft bus, we modelled the projectile as an under-dense aluminium sphere, as opposed to a solid sphere (radius r ≈ 0.52 m and bulk density δ = 1,000 kg m−3) with the equivalent mass of the DART spacecraft. For the large cratering efficiencies studied here, for which the crater cavity grows many times larger than the DART spacecraft, we considered that a spherical projectile is a reasonable approximation for a DART-like impact.

Material model

We modelled both the boulders and the matrix material using the Tillotson equation of state for basalt61,66, with modified initial grain densities of ρg = 3,200 and 3,500 kg m−3.

For the response of the matrix material to shear deformation, we applied a simple pressure-dependent strength model, typical of geological materials25,26, with asymptotes to a certain shear strength at high pressures. The Lundborg strength model describes the yield strength as:

$$Y={Y}_{0}+\frac{fP}{1+fP/({Y}_\mathrm{dm}-{Y}_{0})},$$
(3)

where P is pressure, f is the coefficient of internal friction and Ydm is the limiting strength at high pressure. For the weak asteroid materials considered in this study, we used a constant cohesion Y0 with a strain-based weakening model that prevents artificial clumping (like the approach used in ref. 67). Our model uses a linear relation between cohesion Y0 and total strain ϵtot, and it is assumed that for ϵtot ≥ 1, cohesion is lost. When modelling the matrix material, the tensile strength was defined by extrapolating the yield strength (versus pressure) curve (equation (3)) to intersect the pressure axis. Additionally, we limited the maximum negative pressure to Pmin ≥ −Y0. To compute the coefficient of internal friction f from the measured angle of response ϕ (refs. 32,68), we followed the relation described in ref. 64:

$$f=\frac{2\sqrt{2}\sin(\phi )}{3-\sin(\phi )}.$$
(4)

We varied the cohesion Y0 between 0 and 50 Pa and the coefficient of internal friction f between 0.4 and 0.7.

Tensile strength of boulders

Based on laboratory measurements of meteorite falls, the average tensile strength of ordinary chondrites69 is 24 ± 11 MPa, with no statistical difference between the L and LL types23. However, measurements of individual specimens span from a few megapascals up to ~100 MPa. In all our simulations, the initial material properties of the boulders were kept the same and we employed the tensile strength and fracture model as described in refs. 14,15, with parameters corresponding to a tensile strength YT ≈ 20 MPa for a ~2 cm specimen. For the boulders on Dimorphos, the average YT ≈ 10 MPa.

Porosity model

The initial porosity of the boulders was fixed at 10% (refs. 70,71), whereas the porosity of the matrix (which includes boulders smaller than 2.5 m) was varied between 35% (25% macroporosity + 10% microporosity) and 65% (55% macroporosity + 10% microporosity). Both were modelled using the P versus α porosity compaction model72,73. The full description of the P versus α model implemented in the Bern SPH code is given by ref. 14. Here we used a simplified version of the P versus α model with a single power-law slope defined by the solid pressure Ps, elastic pressure Pe, exponent n, initial distension α0 and distension at the transition from the elastic regime αe:

$$\alpha (P)=\left\{\begin{array}{ll}1,\quad &\,{{\mbox{if}}}\,{P}_\mathrm{s} < P.\\ ({\alpha }_\mathrm{e}-1){\left(\frac{{P}_\mathrm{s}-P}{{P}_\mathrm{s}-{P}_\mathrm{e}}\right)}^{n}+1,\quad &\,{{\mbox{otherwise.}}}\,\end{array}\right.$$
(5)

We assumed that αe = α0. The input parameters from the matrix and for the boulders are summarized in Extended Data Table 2. Without crush-curve measurements of the surface material, our input parameters were informed by experimental quasi-static crush curves of Earth analogues and lunar regolith. We defined crush curves that have crushing strengths higher than the crushing strengths for sand74,75 but lower than the crushing strength for lunar regolith76,77. Extended Data Fig. 6 compares our P versus α crush curves with the experimental quasi-static crush curves for lunar dust76, gypsum (50% porosity)78, Lane Mountain No. 70 sand (44% porosity)75 and Ottawa sand (30–45% porosity)74.

Modelling approach for the late-stage evolution

Due to the vastly different timescales of the shock-wave propagation and crater formation, it is difficult to numerically model the entire crater formation on small, weak asteroids (~100–1,000 m). To ensure numerical stability, the maximum time step in a shock physics code is limited by the Courant criterion. In SPH, the Courant criterion requires that the time step dt is smaller than the simulation resolution divided by the speed of sound in the target, cs. To model DART-like impacts on Dimorphos, the maximum time step must therefore be Δt < resolution/cs ≈ 10−4 s. On the other hand, the crater formation and ejecta reaccumulation time in the gravity regime can last up to a few hours. Here we applied a transition to a low-speed medium in the shock physics code calculation.

At a time ttransition after the initial shock has passed, movement of the target material is governed only by low velocities, vmaterial ≪ cs, and we can switch to a low-speed medium (using a fast time integration scheme). In this step, we applied a simplified Tillotson equation of state in which all energy-related terms are set to zero. The remaining leading term of the equation of state is governed by the bulk modulus P = A(ρ/ρ0 − 1), which also determines the magnitude of the sound speed. At ttransition = 0.5, 20 and 500 s, we used A = 359 MPa, 3.59 MPa and 35.9 kPa, respectively. The shear modulus was also reduced proportionally. The Bern SPH fast-integration scheme has been validated against laboratory experiments18 and was recently successfully applied in modelling the impact of the Hayabusa2 Small Carry-on Impactor17.

Rubble-pile model of Dimorphos

To obtain realistic configurations of boulders, we used the rubble-pile model with Dimorphos’s surface boulder SFD, which was generated from the gravitational accumulation simulations using the N-body code pkdgrav65. Extra particles near the surface were removed to more faithfully represent the target topography reported by ref. 1. To explore a large possible range of boulder mass fractions, we removed some of these boulders from the pkdgrav output when we built our SPH models. We defined four different boulder distributions, with ~21, 30, 44 and 50% of the target volume occupied by boulders (Extended Data Fig. 2c). Boulders smaller than Rmin = 1.25 m (2.5 m in diameter) were removed from the SFD due to being too small to be resolved individually. We assumed that they were part of the matrix material that was used to fill the voids between the larger boulders.

Impact site and impact angle

DART impacted Dimorphos at 8.84 ± 0.45° S, 264.30 ± 0.47° E, and the impact angle was 17 ± 7° from the surface normal1. The impact was on a boulder-covered terrain, which was mapped using the global digital terrain model from ref. 1 (Extended Data Fig. 1). Using Delaunay triangulation, convex hulls were created for the three largest boulders at the site: a 6.5 m boulder (Atabaque Saxum), a 6.1 m boulder (Bodhran Saxum) and a smaller, 4 m boulder (Caccavella Saxum). These boulders were incorporated into all SPH simulations. The solar arrays made contact with Dimorphos’s surface before the spacecraft bus; however, the majority of the impact energy was transferred through the bus, which made up most of the spacecraft’s mass at the time of impact1.

Momentum-enhancement calculations

In a high-velocity impact event, the change in momentum of the asteroid ΔP is amplified by the momentum of impact ejecta that escapes the gravitational attraction of the target body10. The DART impact changed the orbital period around Dimorphos by ~33 min and was directly measured from ground-based observations2. This change in orbital period corresponds to a change in the orbital velocity of about 2.7 mm s−1.

The momentum transferred to the target body relative to the incident momentum of the spacecraft, quantified by the momentum-enhancement factor β, is defined by7:

$$M\Delta \mathbf{v}=m\mathbf{U}+m(\beta -1)(\hat{E}\bar{U})\hat{E},$$
(6)

where M is the mass of Dimorphos, \({{\Delta }}\mathbf{v}\) is the change in Dimorphos’s orbital velocity due to the impact, m is the mass of the DART spacecraft, \(\mathbf{U}\) is DART’s velocity relative to Dimorphos’s at impact and \(\hat{E}\) is the net ejecta momentum direction.

The full equation needed to determine the momentum-enhancement factor β from observations is derived in ref. 9 and further discussed in ref. 7. We define β as the total momentum change of the asteroid ΔP divided by the magnitude of the impactor momentum mU:

$$\beta ={{\Delta }}P/(mU)=1+\frac{{p}_\mathrm{ej}}{mU},$$
(7)

where pej is the magnitude of the vector sum of the momentum of the ejecta.

The value of β depends on both the target material properties and the impact conditions (for example, refs. 11,16,34,38,79,80). A value of β ≈ 1 implies that the impact ejecta made a negligible contribution to the total momentum change, whereas β > 2 means that the momentum contribution from the impact ejecta was larger than the momentum imparted by the impactor directly.

To affirm the validity of our simulation results, we calculated β using two distinct methods. For the first method, β was calculated by summing over all the SPH particles with ejection velocities larger than the escape velocity vesc. vesc was calculated by summing over all the SPH particles in the simulation. However, we did not consider the gravitational influence of Didymos.

For a given impact, the ejecta momentum is given by

$${p}_\mathrm{ej}= \left| \sum {m}_\mathrm{e}\mathbf{v}_\mathrm{ej} \right| ,$$
(8)

where me and \(\mathbf{v}_\mathrm{ej}\) are the mass and velocity vector of individual SPH particles, respectively. Note that in some of the tests and analysis presented in the Methods, β is computed with respect to the along-track direction (y) for simplicity. In the impact simulations presented here, we were able to track the impact ejecta, and we performed the β calculation after long periods (up to 1 h after the impact). Therefore, the pej calculation takes the gravitational influence of Dimorphos into account.

The second method, described in ref. 34, tracks the velocity of the asteroid centre of mass postimpact by summing the momentum of all material that remained below the escape velocity after the reaccumulation of the ejecta. The absolute difference in β resulting from the two calculation methods was used to calculate the error of our reported β value for each simulation.

Resolution tests

So that we could model numerically the very long timescales required to see the impact effects, most of the SPH simulations presented here had a limited spatial resolution of 5.6 × 106 SPH particles. Due to the low resolution employed here, the very fast (v/U > 10−1) ejecta was under-resolved, which may cause an overestimation or underestimation of the total ejected mass and ejecta momentum at high speeds.

We performed resolution tests with impacts into a cohesionless target with 30 vol% boulders, f = 0.55 and ϕ0 = 45% using two resolutions: 5.6 × 106 SPH particles and 14.5 × 106 SPH particles. The low-resolution runs overestimated the amount of mass ejected at high velocities, M(v/U > 10−2), by a few percent (Supplementary Fig. 1a). In addition, the cumulative ejecta momentum at speeds lower than v/U < 1e3 was overestimated by about 6% in the low-resolution run (Supplementary Fig. 1b).

To further study the effects of resolution, we also computed the data from impacts into cohesionless targets with 30 vol% boulders, f = 0.55 and ϕ0 = 45% but where only ~15% of the target (in the y direction) was modelled. These targets with a reduced domain were modelled with 5 × 106 SPH particles and 10 × 106 SPH particles (Supplementary Fig. 2). We see no clear difference compared to Supplementary Fig. 1.

Parameter studies

Target curvature and impact location

For a fixed homogeneous, ellipsoidal target (Extended data Fig. 2, with ρg = 3,500 kg m−3, Y0 = 0, f = 0.55 and ϕ0 = 45%), we considered DART-like impacts at different locations. Location 1 is at x = 0 m and y = 0 m in the middle of the a and b axes of the asteroid. The impact was vertical. Location 2 is at x = 0 m and z = 0 m in the middle of the a and c axes. The impact was vertical. Location 3 is the DART impact location at x = −8.31 m and z = −12.69 m (ref. 1). Location 4 is the DART impact location with the three boulders at the impact site (Extended Data Fig. 1). Both locations 3 and 4 used the DART spacecraft velocity vector as described in ref. 1 (the impact was modestly oblique).

Supplementary Fig. 3a shows the mass–velocity distribution of the ejecta from impacts at locations 1–4. Supplementary Fig. 3b shows the cumulative ejected momentum as a function of normalized vertical ejection velocity (in the +z direction for location 1 and in the −y direction for locations 2–4). For this target scenario, a vertical DART-like impact at location 1 produced βz ≈ 3.82 and the impact at location 2 produced βy ≈ 3.74. At location 3, βy ≈ 3.14, which means that the impact location and the impact angle reduced pej(y) by about 18%. At location 4, which has the same geometry as location 3 but has the three boulders at the impact site, βy ≈ 3.06. These results suggest that the three boulders at the impact location had a minimal effect on β, at least for our assumed impactor shape.

Boulder packing

For a fixed target shape (Extended Data Fig. 2) and material properties (ρg = 3,500 kg m−3, Y0 = 0 Pa, f = 0.55 and ϕ0 = 45%), we investigated the effects of the volume fraction of boulders larger than 2.5 m within the target on the ejecta mass–velocity distribution (Supplementary Fig. 4a) and the momentum–velocity distribution (Supplementary Fig. 4b). The boulders were modelled as spheres. We simulated the DART impact into targets with five different boulder volume fractions: 0, 21, 30, 44 and 50 vol%. All runs included the three boulders at the impact site, including the target with 0 vol% boulders. We found that the DART impact into targets with a boulder packing of 21 and 30 vol% have almost the same ejected mass and ejecta momentum compared to a homogeneous, matrix-only target scenario (within 10%). Moreover, impacts into targets with a boulder packing higher than ~44% show a clear reduction in the ejecta mass and ejecta momentum. For a target with boulder packing of 50 vol%, the cumulative ejecta momentum was reduced by about 40% compared to a similar impact into a homogeneous, matrix-only target. We attribute this reduction in ejecta to a lower cratering efficiency caused by boulder armouring36 and interlocking35. However, the DART impact excavated material only from the upper approximately 15 m, and our models cannot rule out the possibility that the interior of Dimorphos may be more densely packed.

For the same impact scenario, we varied the arrangement of the boulders while keeping the volume boulder packing constant at 44 and 50 vol%. The SFD distribution and the boulders at the impact site were the same. Our results show that the random nature of the boulder distribution causes a variation in the cumulative ejecta momentum of up to 10% (Supplementary Fig. 5).

Grain density

Supplementary Fig. 6 shows the momentum–velocity distributions from the DART impact into Dimorphos-like targets with two grain densities, ρg = 3.2 and 3.5 kg m−3, for targets with no boulders (0 vol%, Supplementary Fig. 6a) and targets with 30 vol% boulder packing (Supplementary Fig. 6b). For the DART impact scenario, the grain density influences only the low-velocity ejecta (v/U < 10−4).

Cohesion and coefficient of internal friction

The normalized ejected mass at speeds greater than v as a function of the normalized ejection speed v/U is shown for the DART impact into targets with varying cohesion (between 0 and 50 Pa; Supplementary Fig. 7a) and for targets with a varying coefficient of internal friction (between 0.40 and 0.70; Supplementary Fig. 8a). Supplementary Figs. 7b and 8b show the cumulative ejected momentum as a function of ejection speed.

For the range of target cohesions investigated here, only the ejecta slower than ~1 m s−1 was affected, which means that the effects are seen in terms of β and the amount of target deformation but not in terms of the morphology of ejecta at the time of the LICIACube observations (Fig. 2).

Ejecta mass and cone orientation

Supplementary Fig. 9 shows the mass of ejecta that escapes Dimorphos (v > vesc) and contributes towards the β calculation (Fig. 1).

Based on the analysis of the HST images taken within T + 8.5 h, which show ejecta structures moving away from Didymos at speeds greater than 1 m s−1, the ejecta cone centre is directed towards (right ascension (RA), declination (dec.)) = (141 ± 8°, 25 ± 6°) or (120 ± 9°, 10 ± 7°)5. The direction of the ejecta cone with projected velocities between a few and ~500 m s−1 from LICIACube images4 is (RA, dec.) = (137 ± 9°, 19 ± 12°). An analysis of both HST and LICIACube data gives the direction of the ejecta cone as (RA, dec.) = (138°, +13°) with an uncertainty of 15° (ref. 7). Our analysis reveals that at T = 160 s, the direction of the ejecta momentum vector is consistent with the observed cone direction within 1σ uncertainty (Fig. 3a and Supplementary Fig. 10), which suggests that our simulation provides a reasonable estimate of the direction of the plume. However, our analysis of all ejecta, with speeds ranging from the escape velocity (v ≈ vesc) to km s−1, as analysed at the end of the ejecta production (T ≈ 2,000 s), predicts a slightly different direction of (RA, dec.) = (150.5 ± 4°, 12 ± 5°) (Supplementary Fig. 10). This indicates that the direction of the observed segment of the ejecta plume may not necessarily reflect the true direction of the momentum-enhancement vector, as computed from the entire plume7, but may be subject to a bias based on the velocity range at which it is observed.