Elliptical ejecta of asteroid Dimorphos is due to its surface curvature

Domain: earth_science
Published: 2025-02-14
Editorial outcome: minor_revision

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 Earth1,2. 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 future1–4. 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 missions1–4. 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 impact5–7. Recent impact physics studies suggest the dependence of β on the impactor’s properties and orientation and the target’s composition and strength8–15. 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 topography11,16–19. From a tactical perspective, imparting higher kinetic energy is preferred to maximize deflecting the target1–4. 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 scale23–25. 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 site26,27. 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) Didymos28,29. 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 (IAU_DIMORPHOS) with an impact speed of 6.145 km/s28,29. 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 pressure32–44. 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 3.61 ± 0.219546.

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
The cone geometry measurement applies images taken by HST and LICIACube, extending earlier efforts: those using LUKE images only42 and HST images only40,46 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.

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. Illuminational 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).

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 s41,42. 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°.

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, Pfl (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, Δ*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 the 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 Pfl (see “Methods”, Subsection Geometric Factor for DART). Comparison tests confirm the consistency between our approach and the iSALE-2D shock-physics code49–52 (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 Pfl (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.

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(Δ*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 m54,55. 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°) tend 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 toward positive azimuthal angles by about 45°. The net geometric factor relative to a flat surface target, Pfl, 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 range11,58.

Methods
Ejecta cone geometry
Similar to earlier work40,46, 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) give 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.

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 determine the position angles of the cone edges.

The position angle search obtain 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-liked 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:

Xk = \sum_{i=1}^{2} \sum_{j=1}^{n} \frac{1}{\sqrt{2\pi} \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=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, LCC2 and LCC3, and the Dimorphos shape model. 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. 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.

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

r = r0 (1 - cos Δ) ^ (1 / (Z - 1)) (2)

vr = α / r ^ (Z - 1) (3)

vΔ = vr / (Z - 2) * (sin Δ / (1 - cos Δ)) (4)

where r is a mass element’s radial location along a streamline, while vr and vΔ 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:

α = (g R ^ (Z + 1) / (4 Z (Z - 2))) ^ (1 / 2) (5)

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

V = K1 * (mi / ρt) * (ρt * vi^2 / Y) ^ (μ) * (ρt / ρi) ^ (2/3 + μ) (6)

where K1, μ, and Y are empirically determined parameters. mi, ri, and vi 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. ρi and ρt 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 = (3V / (2π)) ^ (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 framework81,82 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 = 1 / (κ) * (λ I1 + J2) (8)

λ = sin ψ / sqrt(3 * (3 - sin ψ)) (9)

κ = 6 cos ψ / sqrt(3 * (3 - sin ψ)) (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 = LT / LTref = (β - 1) / (βref - 1) (11)

where LT and LTref 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, LT = L · nT, where L is the ejecta momentum, and LTref = Lref · nT, where Lref is the ejecta momentum on the reference surface. P can be convertible with β for these targets, where βref 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 Pfl, while that for a spherical target is Psp. 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 Pfl’s unity value can give a direct insight into how close (far) the momentum transfer is to (from) the ideal case. However, Psp 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 MAMOS
Samples are available through the MAMOS 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:

Δm = -2.5 log (b / a) (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, Lp, as the sum of each streamtube’s mass times its surface velocity, dm v:

Lp = sum_{i=1}^{m} sum_{j=1}^{n} dm_{i,j} v_{Δ*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:

L = sum_{i=1}^{m} γ_{0,i} Lp,i (14)

where Lp,i and r0,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 r0,i was to ensure that γ is consistent over the entire azimuthal pieces regardless of the variations in r0,i. However, this equation must keep the units on both sides consistent, needing r0,i to be dimensionless. Considering an alternative form dividing r0,i by its unity length, i.e., (r0,i/1)^γ, can mitigate this issue. This way, L and Lp 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 form6,46:

β = 1 + (L · nT) / ((Esc · Lsc) · (Esc · nT)) (15)

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

β = (3.61 ± 0.2) * (ρ / ρref)^(-0.03 ± 0.02) (1σ) (16)

where ρref 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 Pfl. Running 1000 Monte Carlo simulations with Gaussian-distributed inputs offers the statistical behaviors of Pfl (Table 1).

Validation of geometric factor computation by Maxwell Z-model
Comparisons between the Maxwell Z-model and iSALE-2D simulations49–52 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 Psp.

The present iSALE-2D simulations follow the parameter settings from earlier work10,12. 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 Psp 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 sand 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 Psp. 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 MAMOS 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 Pfl. 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*E and r0 become extremely small and large, reducing numerical accuracy. To avoid this issue, the algorithm only considers Δ*E > 20°, allowing all streamtubes to cover up to about 83% of the entire volume. The Δ*E > 20° 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 = (U^2 * msc) / (2 * M) (17)

where msc is the spacecraft mass, M is the target mass, and U is the spacecraft relative impact speed. In the present study, msc and U are identical to DART’s, and msc ≪ M. On the other hand, Q*D is also a function of the target radius and strength90,91. 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 define the following equation:

Q*D = 100 * as * Q*D_0 * R^bs + 100 * bs * B * ρ * R^bs (18)

where R is the disrupting target radius, ρ is the bulk density, and Q*D_0, a, 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 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, ϵmin:

ϵmin = (k * V * Q*D)^(-1/m) (19)

where k and m are the Weibull parameters, given in Supplementary Table 3, and V is the disrupting target volume. Y at a given R is written as:

Y = ϵmin * Es (20)

where Es = 5.3 × 10^10 Pa is Young’s modulus. Combining these equations with the assumption that the Weinbull parameters and Es are size-independent yields the relationship between Y and Y(1.5 cm):

Y = (V / (1.41 × 10^-5))^(-1/m) * Y(1.5 cm) (21)

The second sample is a cohesionless target58,93. Q*D for this case gives the equation90:

Q*D = ag * R^μg * U^2 * (ρ / ρref)^μg (22)

where ag and μg 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. The interpolation function gives the following power law:

Y = ξ * R^η (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 target58,92.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper determines the geometry of the elliptical cone of the DART ejecta based on observational data and the parameters (Z, α) of the Maxwell Z-model to reproduce the ejecta cone observations. The momentum transfer efficiency of DART impact was then reproduced by assuming those parameters and a newly introduced scalar coefficient (γ); using Z, α, and γ, the ratio of the momentum transfer efficiency ('geometric factor') for impact against a flat surface to that against a curved surface in the DART case was shown, and the importance of the curvature of the body is emphasized. Then, using Z and γ, and α derived from the PI scaling law of cratering efficiency, the geometric factor for near-Earth objects of radius 50 m or less, taking into account the curvature of the shape, is presented, and strategies for kinetic deflection of NEOs are discussed.\nBased on the state-of-the-art knowledge of impact processes and strength of small bodies to date, the methodology of this study quantitatively demonstrates the effect of target curvature on geometric factors in kinetic deflection, and the discussion is extended to future strategies. The descriptions are clear, and the content is rich and stimulating. This manuscript is worthy of publication in Nature Communications. There are no essential flaws.\nSome comments are provided below to help with the revision.\n・Eqs.14 and 15\nShouldn't L on the left-hand side of Eq.14 be used instead of L_p in Eq.15? Furthermore, if both L and L_p are momenta, shouldn't r_0 be normalized by some variable on the length scale?\n・Reason for the introduction of γ\nThe observed ejecta cone is considered to be determined by the finest particles. Fine particles are considered to follow the Z-model well. On the other hand, large blocks may not be able to move along streamlines, but they carry a certain amount of momentum. Therefore, I think the deviation from the ideal Z-model reflects the presence of larger particles.\n・Minor comments\n1) 3rd page, 4th line from the bottom\nThe phrase 'without modification by the Maxell Z model' sounds less direct. How about 'ejecta cone determined from observational data is…'\n2) 4th page, caption of Fig. 1.\nThere is no description of 'SL E'.\n3) 14th page, Maxwell Z-model\nA separate description of r_0 from alpha and Z is required.\nIn Eq.3 and Eq.4, v_r and u_r must be unified.\n4) 20th page, second line after Eq.17\nm_ss should be m_sc.\n5) 21st page, just below Eq.21\ngivens -> gives\n6) 21st page, Eq.23\nIt took some time to understand the explanation of Eq.23.\nAt the very least, the authors could consider specifying somewhere that Eq.23 represents the curves in Fig.4.\n7) Table S1\nThe headline of the 'Maxwell-Z model' does not seem suitable. How about 'Transient crater radius parameters' or so?\n8) Table S3, first line\nσ_T is not used in the text.\n9) Table S3\nResults of R_Q*_D for pumice and basalt bodies should be different (but are currently the same in the table).\n10) Table S3\nThe strength of the strengthless body is not 0.01 MPa.", "reviewer_2": "The manuscript \"Kinetic deflection change due to target global curvature as revealed by NASA/DART\" by Hirabayashi et al. investigated the ejecta plume induced by DART spacecraft striking Dimorphos. They derived the elliptical ejecta cone geometry, using the Monte Carlo simulations with the observed data by both HST and LICIACube. With the momentum transfer enhancement factor and Maxwell Z-model, they concluded the importance of target curvature for the kinetic deflection. This is a highly intriguing paper that that provides insights for future missions using the images of DART's impact and modellings; the concept of employing multiple impactors for planetary defense appears sensible from a scientific standpoint. This paper will be of interest to both researchers working in this field and more general disciplines. However, I believe that some discussions would benefit from further attention before the paper is published. I strongly recommend conducting a thorough review of the Tables and making any necessary improvements.\n1. Effect of impact angle\nThere is no discussion on the effect of impact angle. Since Daly et al. (2023) indicated the impact angle of DART is 73 degree, does this manuscript also use this angle? If so, please clarify this. Alternatively, do they simply assume the vertical impact to apply the Maxwell Z-model? If this is the case, please clarify whether the assumption is valid. Although Raducan et al. (2022 Icarus) considered the flat target, they indicated the ejection angle depends on the impact angle. Although this manuscript described the consistency between the Maxwell Z-model and iSALE-2D simulations in the methods, which is good results, these are only valid for the vertical impacts, if I understand correctly. I agree that the spacecraft geometry is less effective than the ejecta cone's geometry, but I am not sure about the impact angle. I am also wondering if the impact angle might explain the twist of the ejecta cone toward the east-west direction. Please add some description of the usage of impact angle in this manuscript.\nDoes the following line suggest the importance of impact angle or target curvature?\n\"Both narrow- and wide-cone opening angles are wider than an ideal 90 opening angle of an ejecta plume on flat surfaces\"\n(Line 2 from the top on page 3)\n2. Ejecta cone's tip (apex), axis, and impact point and its angle\nI am curious about the relationship between the elliptical cone (tip (apex) and axis) and impact (impact point and its angle). My intuition is that they were in alignment, but there seems an offset between the direction of the red and blue arrows in Figs. 2 J, K, and L (cone axis and DART's direction). Is the estimated location of the ejecta cone's tip aligned with the impact point and impact angle of DART (i.e., (x,y,z) in IAU_DIMORPHOS can be estimated from (S, E) in ISU_DIMORPHOS)? What do you think about the reason for \"The angle between the ejecta cone axis and the DART anti-incident direction is 15 pm 5\"\n(last line of 2nd paragraph on page 3)?\n3. Ejecta cone edge\nFigure 3C (and its caption) confuses me. Are all of these based on the Maxwell Z-model: can the Maxwell Z-model also estimate surface-cone intersection (i.e., cone edge) in addition to excavation (r_E^*)? If the former comes from the observation, do the overlapped regions have any meaning for the ejecta cone?\n4. Tables\nTables 1 & 3 look identical. While I cannot completely judge the related part, I reviewed some numbers written in the main text. Please do check these.\nThere are some typos in Table S.1 (see below). Also, Table S.3 may lack the parameter \"k\" which would be used to derive Equation (21), if I understand correctly. Although these are supplementary tables, it would be appreciated if the authors made these complete ones.\nMy apologies but I do not have expertise in observation. Thus, I cannot fully assess the observation part (e.g., Table 2 and S.2) and some of my comments might be misguided.\nOther comments\nEjecta velocity\nIf it is possible to capture the velocity of the ejecta cone from the observed images, I think it would be great to compare it with the Maxwell Z-model's results (i.e., Eq. (3) and (4)).\nPage2, Line 7 from the bottom: LICI \"A\" Cube?\nPage 16, Eqs. (8), (9), and (10):\nSince this manuscript used alpha and beta before (streamtube speed and momentum transfer enhancement factor), these should be changed to avoid misunderstanding, unless those have the same meaning.\nPage 14 Eq (2) and Page 17 Eq. (14)\nWhat is \"r_0\"? It seems one of the Maxwell Z-model's parameter (streamline's kinematics), but less description compared to alpha and Z. Is this the same as r_E^* or a different one? Isn't it better to add this in Table 1 (or 3)?\nPage 18, \"The equivalent radius is 75 m for all cases\":\nIs there any reason to not simulate the Dimorphos size target?\n\"Epsilon-alpha compaction model\":\nIf possible, please rephrase this to avoid misunderstanding of alpha (see the above comments too).\nPage 20, 2nd line below Eq. (17): m_s\"c\" << M\n\"Q_D^* is a function of the impactor speed, target radius, and strength\": This confuses me because Eq. (17) does not contain strength. Please consider rewording this line.\nTypos in Table S.1.\nImpact speed Units km/s\"\" (not km/s\"^3\")\nBulk density Units kg/\"m\"^3\nBulk density of Impactors seems lighter than the water (107 kg/m^3), but is this true?\nDoes the \"Internal friction coefficient (damaged)\" has a unit?", "reviewer_3": "Overall: My take-away from this paper is that it is more efficient to send multiple low-energy impactors than a single high energy impactor to deflect an asteroid that has high curvature. The authors show that wide ejecta cone seen from the DART impact is actually associated with a smaller excavation radius on Dimorphos's curved surface than would be expected on a flat surface. The authors are tackling a unique problem. Cratering on a highly curved surface is far less studies that impacts on a flat surface. The authors have shown that the curvature has a significant effect on the momentum transfer along the orbit of the asteroid. This work is novel and well-written, and the results are important for the next stage in planetary defense innovation and missions, and should therefore be published. I have listed a few points that may benefit from clarification below:\nResults\n\"By definition, this distance, which also accounts for the tangential direction, should be larger than a typical crater radius measured as half the crater’s rim-to-rim distance. With this consideration, the excavation distance is overall comparable with the reported crater radius\"\nI think this could use a little clarification. Is it possible to quantify how much smaller the crater size should be than excavation radius? Stating that the excavation radius is comparable to other estimated crater sizes implies that this is quantifiable. Otherwise, what makes these two estimates \"comparable\"?\n\"The net geometric factor relative to a flat surface target, Pf l , is 44 ± 10% for Dimorphos. If the target surface is perfectly flat, the derived geometric factor leads to β = 6.9 ± 3.5(1σ ), which is still within its predicted range\" -\nthe ranges referred to here are from a different source (45) than the enhancement factor ranges listed in the beginning of the main text (citations 32 and 33).\nIf the 1 sigma error in the geometric factor is 10%, I am having trouble seeing how you obtain 3.5 as the 1-sigma error on the derived momentum enhancement factor. Based on equation 11 and the reported errors on 3.61=B, I get errors closer to -1.5 and +2.3. Perhaps I am missing something here. I checked this because a non-significant change in momentum enhancement factor based on such a large difference in along-track momentum seemed unintuitive to me.\nGeometric factors on Near-Earth Objects less than 50 m radius\n\"Applying our Maxwell Z-model approach enables quantifying how the geometric factors change due to the impact locations for 103 NEOs less than 50 m in radius observed by the MANOS (Mission Accessible Near-Earth Objects Survey) project\"\nIt might be helpful for the authors to clarify why they don't look at asteroids larger than 50m in radius. Assuming the same impact energy as from DART, how quickly does the effect of curvature fall off with size? - this might be on the forefront of the readers' minds since Ryugu and Tempel 1 impacts were brought up earlier.\nKinetic deflection strategy learned from DART\n\"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.\" -\nit also seems that the multiple deflectors launched at difference times could provide additional reconnaissance for subsequent impactors.\nThe recent paper \"Orbital and Physical Characterization of Asteroid Dimorphos Following the DART Impact\" by Naidu et al 2024, indicates a change in shape after the DART impact. How significant could this shape change affect the curvature? It seems like the shape change from squashed sphere to a more elongated squashed sphere results in both increases and decreases of curvature in different areas on the surface. A discussion around this would be beneficial, even if it is just acknowledgement that the multiple impactor method may need to take these shape changes into account between each impactor to maximize the efficacy of this proposed method.\nFigures\nFig 2 - the top row may benefit from adding borders to visually contain each perspective."}, "author_response": "Dear Reviewers,\nThank you for your thorough comments and suggestions on the current manuscript. We\nmodified it based on your input. In the manuscript, we highlighted modified sentences in\nblue. Below, we respond to the comments and suggestions listed.\nResponse to Reviewers:\nWe compiled our response to the reviewers’ comments and suggestions below.\nReviewer #1 (Remarks to the Author):\nEqs.14 and 15: Shouldn't L on the left-hand side of Eq.14 be used instead of L_p in Eq.15?\nFurthermore, if both L and L_p are momenta, shouldn't r_0 be normalized by some variable\non the length scale?\nThat is correct. L should appear on the left side of Equation (15). We updated Equation (15)\nbased on this comment. Also, thank you for the comment on r_0. We agreed. We modified\nEquation (14) to avoid misinterpretations. r_0 was specified for each azimuthal piece,\nthough gamma was kept constant over all pieces (but gamma is determined from the\nmeasured beta value). The correct formulation is provided in Equation (14) in the new\nmanuscript. The reason for using the current form of r_0^gamma was to ensure that gamma\nis consistent over the entire azimuthal pieces regardless of the variations in r_0, where r_0\nfor a given azimuthal piece was defined as r_{0,i}. The new manuscript uses r_{0,i} to define\nL properly. To fix the apparent issue about the inconsistency of the units, we added a new\nstatement that r_{0,i}^gamma is defined as a non-dimensional factor divided by a unity\nlength, i.e., (r_{0,i}/1)^gamma. In this way, we can keep the units consistent on both sides.\nReason for the introduction of γ: The observed ejecta cone is considered to be determined\nby the finest particles. Fine particles are considered to follow the Z-model well. On the other\nhand, large blocks may not be able to move along streamlines, but they carry a certain\namount of momentum. Therefore, I think the deviation from the ideal Z-model reflects the\npresence of larger particles.\nThank you for suggesting this point. Introducing gamma was meant to compensate for\nenergy loss over each streamline. The statement can be found before Equation (14). We\nagree that the particle size distributions contribute to it; however, many diUerent eUects\ncould also deviate the actual behavior from the original Maxwell Z-model (Austine et al.,\n1981; Anderson et al., 2004). We decided to generalize this point by saying energy loss and\nkeeping the statement as is. The manuscript does not discuss the influences of the boulder\nsize distribution on the ejecta cone formation, as the current technique cannot identify the\nparticle size distribution. However, such analysis has been published (Jewitt et al., 2023) or\nis currently underway, some of which were led by the authors of this manuscript (Farnham\net al., 2023).\nAustin et al. (1981), “Z-Model analysis impact cratering: An overview,” In Proceeding of the\n12th Lunar Planetary Science Conference, 197-205.\nAnderson et al. (2004), “Experimental ejection angles for oblique impacts: Implications for\nthe subsurface flow-field,” Meteoritics & Planetary Science, 39, 303-320.\nJewitt et al. (2023), “The Dimorphos Boulder Swarm,” The Astrophysical Journal Letters, 952,\nL12.\nFarnham et al. (2023), “Spatial distribution of the boulders in the DART impact ejecta: A 3-D\nanalysis,” 54th Lunar and Planetary Science Conference 2023, 2426.\n3rd page, 4th line from the bottom: The phrase 'without modification by the Maxell Z model'\nsounds less direct. How about 'ejecta cone determined from observational data is…'\nModified.\n4th page, caption of Fig. 1: There is no description of 'SL E'.\nSL E is the streamline that is farthest from the impact site along a given azimuthal piece. This\ndescription was added to the caption.\n14th page: A separate description of r_0 from alpha and Z is required. Maxwell Z-model, In\nEq.3 and Eq.4, v_r and u_r must be unified.\nThe definitions of alpha and Z are provided on p.14. “r0 controls the size of the streamline, Z\ndefines how sharply the streamline changes its direction, and alpha gives a material ejection\nspeed.” Also, Eq.3 and Eq.4 were modified.\n20th page: Second line after Eq.17, m_ss should be m_sc.\nModified\n21st page: Just below Eq.21, givens -> gives\nModified\n21st page, Eq.23: It took some time to understand the explanation of Eq.23. At the very least,\nthe authors could consider specifying somewhere that Eq.23 represents the curves in Fig.4.\nA statement was added to Figure 4. “The curves in diUerent colors, calculated using Equation\n(23), show interpolations between the catastrophic disruption thresholds for a pumice-like\ntarget and a cohesionless target (red) and a basalt-like target and a cohesionless target\n(blue).” The italic part was the newly added statement.\nTable S1: The headline of the 'Maxwell-Z model' does not seem suitable. How about\n'Transient crater radius parameters' or so?\nModified.\nTable S3: first line, σ_T is not used in the text.\nModified. This was meant to be Y.\nTable S3: Results of R_Q*_D for pumice and basalt bodies should be diberent (but are\ncurrently the same in the table).\nWe reviewed the original setting and corrected the values slightly (Table 3S). This\nmodification needed additional calculations to update Figure 4. The predicted range,\nhowever, became broader, though the trends are consistent. We added a new sentence at\nthe end of Page 10, “However, the predicted range is generally broad, given the higher\nuncertainties of model and parameters.”\nTable S3: The strength of the strengthless body is not 0.01 MPa.\nThis should be 0.01 Pa = 1e-8 MPa. The value was corrected.\nReviewer #2 (Remarks to the Author):\nEbect of impact angle, There is no discussion on the ebect of impact angle. Since Daly et al.\n(2023) indicated the impact angle of DART is 73 degree, does this manuscript also use this\nangle? If so, please clarify this. Alternatively, do they simply assume the vertical impact to\napply the Maxwell Z-model? If this is the case, please clarify whether the assumption is valid.\nAlthough Raducan et al. (2022 Icarus) considered the flat target, they indicated the ejection\nangle depends on the impact angle. Although this manuscript described the consistency\nbetween the Maxwell Z-model and iSALE-2D simulations in the methods, which is good\nresults, these are only valid for the vertical impacts, if I understand correctly. I agree that the\nspacecraft geometry is less ebective than the ejecta cone's geometry, but I am not sure about\nthe impact angle. I am also wondering if the impact angle might explain the twist of the ejecta\ncone toward the east-west direction. Please add some description of the usage of impact\nangle in this manuscript.\nAll the states of asteroids and spacecraft came from SPICE kernels; the version we used is\nd430, consistent with Daly et al. (2023). We also examined whether this impact angle\ninfluences the ejecta cone geometry but concluded that this would have no influence.\nAnderson et al. (2004) (Fig. 5 in their paper) performed oblique impact experiments and\nconfirmed that an impact angle higher than 60 deg does not give asymmetric cone geometry\nover the azimuthal direction. We added a statement supporting this idea on Page 8 (before\nthe section “Geometric factor on Dimorphos”) and a clarification about the impact angle at\nthe end of the first paragraph in the “Ejecta cone geometry” section on p.12.\n“The DART impact with slight obliquity (i.e., the 73◦ impact angle) likely does not influence\nthe ejecta cone geometry much, given recent experimental work showing that oblique\nimpacts with an impact angle of higher than 60◦ exhibit symmetric cone geometry over the\nazimuthal direction.”\n“The resulting impact angle (from the local horizon) is 73◦.”\nAnderson et al. (2004), “Experimental ejection angles for oblique impacts: Implication for\nthe subsurface flow field,” Meteoritics & Planetary Science 39, 2, 303-320.\nDoes the following line suggest the importance of impact angle or target curvature? \"Both\nnarrow- and wide-cone opening angles are wider than an ideal 90 opening angle of an ejecta\nplume on flat surfaces\" (Line 2 from the top on page 3)\nThis statement supports the importance of the target curvature. Given the earlier\ndiscussions by Anderson et al. (2004) and the 73 deg impact angle, we conclude that the\nimpact angle does not aUect the ejecta cone geometry. Again, a supporting statement was\nadded to the last paragraph in the “DART-driven impact ejecta cone geometry” section on\np.8.\n“The DART impact with slight obliquity (i.e., the 73◦ impact angle) likely does not influence\nthe ejecta cone geometry much, given recent experimental work showing that oblique\nimpacts with an impact angle of higher than 60◦ exhibit symmetric cone geometry over the\nazimuthal direction.”\nI am curious about the relationship between the elliptical cone (tip (apex) and axis) and\nimpact (impact point and its angle). My intuition is that they were in alignment, but there\nseems an obset between the direction of the red and blue arrows in Figs. 2 J, K, and L (cone\naxis and DART's direction). Is the estimated location of the ejecta cone's tip aligned with the\nimpact point and impact angle of DART (i.e., (x,y,z) in IAU_DIMORPHOS can be estimated\nfrom (S, E) in ISU_DIMORPHOS)? What do you think about the reason for \"The angle between\nthe ejecta cone axis and the DART anti-incident direction is 15 pm 5\" (last line of 2nd\nparagraph on page 3)?\nThe angle between the ejecta cone axis and the DART anti-incident direction is defined with\nrespect to the impact point (Figure 1). We encountered an issue in determining the ejecta\ncone geometry uniquely. To resolve this issue, we added an assumption that the cone axis\ncrosses the DART impact site as closely as possible (see the last statement on p.13).\nTherefore, the 15-deg angle is with respect to the impact point. Based on our earlier response,\nwe conclude that this angle is too small to consider the impact obliquity as a major source\nof the ejecta’s skewed feature.\nFigure 1. Geometry of the ejecta cone axis (yellow) and the DART trajectory (cyan). The\nimpact point is at the intersection of these two lines. The magenta dots are the intersections\nbetween Dimorphos and the ejecta cone. The brown plane slices along the equatorial plane.\nFigure 3C (and its caption) confuses me. Are all of these based on the Maxwell Z-model:\ncan the Maxwell Z-model also estimate surface-cone intersection (i.e., cone edge) in addition\nto excavation (r_E^*)? If the former comes from the observation, do the overlapped regions\nhave any meaning for the ejecta cone?\nRemove the surface-cone intersection from the plot. However, we also recognized that the\ndistance from the intersection between SL E and the surface to the vertical line along O, i.e.,\nrE* sin(ΔE*), would give useful information about the size of the crater cavity. Thus, the new\nversion of Figure 3C plots the rE* distribution and the rE* sin(ΔE*) distribution.\nTables 1 & 3 look identical. While I cannot completely judge the related part, I reviewed some\nnumbers written in the main text. Please do check these.\nRemoved Table 3.\nThere are some typos in Table S.1.\nModified. See below.\nAlso, Table S.3 may lack the parameter \"k\" which would be used to derive Equation (21), if I\nunderstand correctly. Although these are supplementary tables, it would be appreciated if\nthe authors made these complete ones.\nk is one of the Weibull parameters and is reported to be 3.e28 (cm-3) for nominal non-porous\nmaterials and 8.0e37 (cm-3) for nominal porous materials. The powers look very high, but\nthey are correct. While Equation (19) has the k value, it eventually disappears in Equation\n(21). This comes from comparing two conditions (here is the size) in Equation (20). When Es,\nk, and m are size-independent, comparing two diUerently sized targets can eliminate k. Thus,\nwe decided not to add the values of k.\nIf it is possible to capture the velocity of the ejecta cone from the observed images, I think it\nwould be great to compare it with the Maxwell Z-model's results (i.e., Eq. (3) and (4)).\nMeasuring and computing the velocity (of particles in the ejecta) is challenging in the current\nmodel. This issue comes from our scope, i.e., constraining the shape of the ejecta cone (and\nthe momentum transfer) but not the velocity. We used Equations (3) and (4) in our model\nmainly because our approach needed to determine alpha and Z. Determining these\nparameters is possible because we could track the size of the ejecta cone at the time of\nimaging. However, we do not trust the velocity derived from Equations (3) and (4) with the\nmeasured alpha and Z – the Maxwell Z-model overly simplifies velocity calculations, as\nprovided in earlier literature. Our major interest was in determining the momentum\ncalculation. While doing this calculation, we mitigated an issue by introducing the gamma\nparameter. Ongoing eUorts are to assess the particle speeds in the ejecta (Farnham et al.,\n2023; Jewitt et al., 2023).\nFarnham et al. (2023), “Spatial distribution of the boulders in the DART impact ejecta: A 3-D\nanalysis,” 54th Lunar and Planetary Science Conference 2023, 2426.\nJewitt et al. (2023), “The Dimorphos Boulder Swarm,” The Astrophysical Journal Letters, 952,\nL12.\nPage2, Line 7 from the bottom: LICI \"A\" Cube?\nCorrected.\nPage 16, Eqs. (8), (9), and (10), Since this manuscript used alpha and beta before\n(streamtube speed and momentum transfer enhancement factor), these should be changed\nto avoid misunderstanding, unless those have the same meaning.\nCorrected. alpha is now lambda, and beta is now kappa.\nPage 14 Eq (2) and Page 17 Eq. (14), What is \"r_0\"? It seems one of the Maxwell Z-model's\nparameter (streamline's kinematics), but less description compared to alpha and Z. Is this\nthe same as r_E^* or a diberent one? Isn't it better to add this in Table 1 (or 3)?\nr0 generally defines the distance of material ejection from the impact point on a flat target.\nWhen Δ = 90 deg, r = r0. rE* is the distance between the intersection of SL E and the surface\nfrom the impact point. ΔE* does not have to be 90 deg. r0 is determined with alpha and Z\nalong each azimuthal piece. We decided not to provide the values of this parameter, given\nthat (1) we anticipate that they vary significantly, (2) r0 does not provide meaningful\ninformation for our case, where the surface has curvature, and (3) rE* is a better\nrepresentation in Table 3C to give the size of the excavation along each azimuthal piece.\nPage 18, \"The equivalent radius is 75 m for all cases\": Is there any reason to not simulate the\nDimorphos size target?\nUsing the same shape with Dimorphos in iSALE causes higher simulation burdens. iSALE\nhas multiple settings. iSALE2D’s axisymmetric setting significantly reduces simulation\nburdens. Using a realistic condition requires iSALE3D, where higher simulation burdens are\ninevitable. Given that earlier studies investigated similar cases, we considered using these\nearlier results more valuable than complicated cases, particularly for validation.\n\"Epsilon-alpha compaction model\": If possible, please rephrase this to avoid\nmisunderstanding of alpha (see the above comments too).\nWe decided to leave it as is. This model (Epsilon-alpha compaction model) has been used\nin the community for a long time (Wünnemann et al., 2006). We understand the confusion;\nhowever, we also encountered challenges in rephrasing or redefining naming parameters\nand models, given the nature of this manuscript, in which we have applied various research\napproaches.\nWünnemann et al. (2006), “A strain-based porosity model for use in hydrocode simulations\nof impacts and implications for transient crater growth in porous targets,” Icarus 180, 514–\n527.\nPage 20, 2nd line below Eq. (17): m_s\"c\" << M\nCorrected.\n\"Q_D^* is a function of the impactor speed, target radius, and strength\": This confuses me\nbecause Eq. (17) does not contain strength. Please consider rewording this line.\nOur intention in this statement was that this parameter was also a function of these\nquantities. QD* is first defined as U^2/2 * msc/M but also depends on strength and target\nsize. The community has recognized this problem as critical. EUorts have been made to\nexplore this correlation (e.g., Jutzi et al., 2010; Raducan et al., 2024a). We corrected this\nsentence in the following way: “On the other hand, QD* is also a function of the target radius\nand strength.”\nJutzi et al. (2010), “Fragment properties at the catastrophic disruption threshold: The eUect\nof the parent body’s internal structure,” Icarus, 207, 54-65.\nRaducan et al. (2024a), “Lessons Learned from NASA’s DART Impact about Disrupting\nRubble-pile Asteroids,” PSJ, 5, 79.\nTypos in Table S.1., Impact speed Units km/s\"\" (not km/s\"^3\")\nCorrected.\nBulk density Units kg/\"m\"^3\nCorrected.\nBulk density of Impactors seems lighter than the water (107 kg/m^3), but is this true?\nOne of the challenges in this analysis was to model the spacecraft as an impactor correctly.\nSimply considering a cube filled with aluminum does not accurately describe a preferred\ntarget. We considered the spacecraft’s mass to be given and used its equivalent volume. The\nspacecraft’s interior is almost empty as the frame encloses it. We did not consider how such\nan empty structure influences the impact processes – however, a series of impact\nsimulation studies analyzed this influence and showed simplification is possible.\nDoes the \"Internal friction coebicient (damaged)\" has a unit?\nThe internal friction coeUicient is tan(u), where u is the friction angle. 0.55 = tan(u) gives u =\n29 deg, a typical value of loose geological materials.\nReviewer #3 (Remarks to the Author):\n\"By definition, this distance, which also accounts for the tangential direction, should be\nlarger than a typical crater radius measured as half the crater’s rim-to-rim distance. With this\nconsideration, the excavation distance is overall comparable with the reported crater radius\"\nI think this could use a little clarification. Is it possible to quantify how much smaller the\ncrater size should be than excavation radius?\nWe agreed. The statement was confusing, and also the definition was unclear. In the new\nstatement, we defined a new quantity, rE* sin(ΔE*), the distance from the intersection\nbetween SL E and the surface to the vertical line along O, which is equivalent to the crater\nradius (by definition). The statement meant to discuss our relatively large ejecta cone\ncompared to the reported crater size. This issue likely came from model simplification and\nuncertainties (to describe the complex cone geometry). We mentioned this issue, though we\nalso suggested that this discrepancy may be mitigated because numerical models predicted\na curved cone edge near the impact site, which allows smaller crater excavation to create a\nlarger ejecta cone (Figure 2). The new statement was added to the second paragraph on p.6.\n“Comparable to the crater radius, half the crater’s rim-to-rim distance, the average distance\nof material excavation from the cone axis, i.e., r∗E sin(∆∗E), is 54 ± 9 m. This result is larger\nthan the numerically predicted size, which depends on the target’s strength. Modeling the\nDART impact on Dimorphos as a cohesionless target (< 50 Pa) no longer causes a crater-like\nmorphology but generates flatter depression at a radius 25 − 55 m. The predicted ejecta\nmass suggests Dimorphos as a weak target (Ferrari et al. in review). If low-moderate material\nstrength (∼ 1 − 10 kPa) influenced the impact, it would create a crater 15-30 m in radius\n(Stickle et al., in review). The derived larger value (54 ± 9 m) may come from our assumption\nthat the cone edge is linear, while its geometry is likely complex and curved around near the\nimpact site, allowing smaller crater excavation to generate a larger ejecta cone. This\ncomplexity, culminated by model simplification and uncertainties, causes challenges in\ninferring the crater size and the target’s strength.”\nFigure 2. Curved structure of ejecta cone in Fig. 2 by Raducan et al. (2024b).\nRaducan et al. (2024b), “Physical properties of asteroid Dimorphos as derived from the\nDART impact,” Nature Astronomy volume 8, p.445–455.\n\"The net geometric factor relative to a flat surface target, Pf l, is 44 ± 10% for Dimorphos. If\nthe target surface is perfectly flat, the derived geometric factor leads to β = 6.9 ± 3.5(1σ),\nwhich is still within its predicted range\" - the ranges referred to here are from a diberent\nsource (45) than the enhancement factor ranges listed in the beginning of the main text\n(citations 32 and 33).\nWe updated the citation. Raducan et al. (2022a, the original citation) showed the beta range\non a spherical target (Figure 3). This result suggests that when cohesion is almost zero, the\nbeta value may reach about 6 (when f = 0.4, where f is the friction coeUicient). Because a\nspherical target tends to have lower beta values due to its curvature, we interpreted that our\nbeta range is comparable.\nFigure 3. Beta range for a spherical target (Raducan et al., 2022a), provided in Figure 15 in\ntheir study.\nRaducan et al. (2022b) also predicted the beta range for a flat target as a function of cohesion\nfor two diUerent porosity cases (Figure 4). Since the bulk porosity is unknown, we considered\nthe full range, i.e., beta < 7.\nFigure 4. Beta range for a flat target as a function of cohesion for 20% and 50% porosity.\nCheng et al. (2023) did not apply to this beta (6.9 ± 3.5), given that it is for a flat target, while\ntheir value is for Dimorphos, which has high curvature. The key takeaway of this manuscript\nis that the beta value changes due to curvature.\nAfter re-examining our calculation, however, we found that the value needed to be corrected.\nThe updated value is 7.2 pm 3.8. This is likely the upper end of the predicted range. We\nmodified the last sentence of the last paragraph just before the Discussion section (p.6).\n“If the target surface is perfectly flat, the derived geometric factor leads to β = 7.2 ± 3.8(1σ),\nlikely the upper end of the predicted range.”\nRaducan et al. (2022a), “Global-scale Reshaping and Resurfacing of Asteroids by Small-\nscale Impacts, with Applications to the DART and Hera Missions,” PSJ, 3, 128.\nRaducan et al. (2022b), “Ejecta distribution and momentum transfer from oblique impacts\non asteroid surfaces,” Icarus, 374, 114793.\nCheng et al. (2023), “Momentum transfer from the DART mission kinetic impact on asteroid\nDimorphos,” Nature 616, 457–460.\nIf the 1 sigma error in the geometric factor is 10%, I am having trouble seeing how you obtain\n3.5 as the 1-sigma error on the derived momentum enhancement factor. Based on equation\n11 and the reported errors on 3.61=B, I get errors closer to -1.5 and +2.3. Perhaps I am\nmissing something here. I checked this because a non-significant change in momentum\nenhancement factor based on such a large diberence in along-track momentum seemed\nunintuitive to me.\nUsing Equation (11) yields\nbeta_ref = 1. / P * (beta – 1) + 1.\nInstead of using an analytical approach, we used Monte Carlo simulations to determine the\nerrors. The code written in Matlab is given in Figure 5. The results are:\nMean beta for flat target: 7.1416\nStd beta for flat target: 3.6957\nFigure 5. Trial code for computing the error of the beta value for a flat surface case.\n\"Applying our Maxwell Z-model approach enables quantifying how the geometric factors\nchange due to the impact locations for 103 NEOs less than 50 m in radius observed by the\nMANOS (Mission Accessible Near-Earth Objects Survey) project.\"It might be helpful for the\nauthors to clarify why they don't look at asteroids larger than 50m in radius. Assuming the\nsame impact energy as from DART, how quickly does the ebect of curvature fall ob with size?\nWe followed the Decadal Survey’s recommendation that NEOs less than 50 m in radius are\nthe most probable threat and high-priority targets for rapid reconnaissance flyby missions.\nThe statement is already given in the first paragraph, just after the abstract. A new statement\nwas also added to the first paragraph of the MAMOS target section on p.9.\n“The reason for focusing on objects less than 50 m in radius comes from recommendations\nfrom publicly available reports.”\n\"Compared to mass measurement, which requires additional operational constraints,\nimaging the target even during a fast flyby can provide subicient information to infer the\ncurvature and surface conditions, significantly improving the accuracy of momentum\ntransfer.\" - it also seems that the multiple deflectors launched at diberence times could\nprovide additional reconnaissance for subsequent impactors.\nIncorporated a statement emphasizing this point in the last paragraph of the strategy section\non p.10.\n“The proposed multi-impactor scenario can also oUer additional reconnaissance before the\nplanned collisions at diUerent times. Such multiple observations enable tracking possible\nlarge-scale modifications that a target may have altered early impacts. Global shape\ndeformation is one of such modifications.”\nThe recent paper \"Orbital and Physical Characterization of Asteroid Dimorphos Following the\nDART Impact\" by Naidu et al 2024, indicates a change in shape after the DART impact. How\nsignificant could this shape change abect the curvature? It seems like the shape change\nfrom squashed sphere to a more elongated squashed sphere results in both increases and\ndecreases of curvature in diberent areas on the surface. A discussion around this would be\nbeneficial, even if it is just acknowledgement that the multiple impactor method may need\nto take these shape changes into account between each impactor to maximize the ebicacy\nof this proposed method.\nOur discussion about the curvature is for the pre-impact condition, while Naidu et al. (2024)\ndiscuss the post-impact condition. These issues can be connected, given that our\nrecommendation is not to intentionally make each impact so large to be aUected by\ncurvature. When impacts cause global deformation, they also create new curvature. We\nincorporated a statement emphasizing this point in the last paragraph of the strategy section\non p.10.\n“Dimorphos likely changed its shape significantly due to the DART impact, experiencing\nadditional orbital perturbation.”\nFigures, Fig 2 - the top row may benefit from adding borders to visually contain each\nperspective.\nVertical partitions in the top row were added to aid visualization. The updated version of\nFigure 2 is attached below (Figure 6).\nFigure 6. Updated version of Figure 2."}, {"round": 2, "reviewer_comments": {"reviewer_1": "All of the author's responses and revisions to my comments were satisfactory.", "reviewer_2": "The authors have fully addressed my comments. Given the high quality of the manuscript, I recommend it for publication.\n(Remarks on code availability)\nAlthough I am not familiar with MATLAB, I tested the code and confirmed that it ran successfully and produced the results shown in the README file.", "reviewer_3": "Thank you for addressing the comments in the response and within the paper."}, "author_response": ""}]

Decision letter