Successful kinetic impact into an asteroid for planetary defence

Determining the time of impact

The time of impact was computed from spacecraft clock timestamps inserted into each downlinked telemetry frame by the spacecraft’s radio. A new telemetry frame was produced every 2.9853 ms with a precision of 20 μs. Assuming that the impact occurred in the middle of the frame, the final received frame placed the time of impact to within ±half a frame period: 401,930,051.59326 ± 0.00149 s spacecraft clock time. This spacecraft clock time of impact was converted to utc. This conversion increases the uncertainty slightly: 2022-26-9 23:14:24.183 ± 0.004 s utc. This time is the utc of the impact at Dimorphos, not the time on Earth when the last telemetry frame was received. As each DRACO image is timestamped with a spacecraft clock value, the time to impact for each image can be computed with similar accuracy.

Shape modelling of Dimorphos

We built the shape model using stereophotoclinometry (SPC), a technique that has been widely used to model the shapes of small bodies. Before the impact of DART, the shape modelling team conducted tests to understand the strengths and limitations of the SPC approach given the data expected from DART. DART images pose a challenge for any image-based shape modelling technique owing to the single viewing geometry and lighting condition. SPC provides robust results despite these challenges.

Extended Data Fig. 7 illustrates the steps used to build the Dimorphos shape model. First, we used DRACO images to determine the dimensions of an initial triaxial ellipsoid. The ellipsoid was constrained by the location of the sunlit limb, the location of the limb lit by light scattered off Didymos and the position of the terminator. Together, the sunlit and Didymos-lit limbs revealed the complete extents of the X and Z axes of Dimorphos (Extended Data Fig. 8). The position of the terminator constrained the Y-axis extent. The last DRACO image to contain all of Dimorphos in the field of view was registered to the ellipsoid. Other DRACO images were registered to that image or, in the case of the highest-resolution images of the impact site, to the immediately preceding image. To correctly determine the scale of the shape model, we used the known time of impact and spacecraft speed to set the spacecraft range for each image.

Next, we pursued parallel paths (Extended Data Fig. 7). The first path used standard SPC processes to construct small digital terrain models (called ‘maplets’) using an SPC program called lithos. Maplets were tiled all across the surface of the asteroid in the areas seen by DRACO. Maplets went through several iterations to compute the maplet topography. After several iterations, the maplet ensemble was averaged to create a global digital terrain model (DTM). Areas without maplet coverage remained unchanged. This cycle was repeated with maplets of increasingly finer ground-sample distances (GSDs) in the areas covered by higher-resolution images surrounding the impact site. The finest-scale maplets had a GSD of 3 cm, comparable to the pixel scale of the final, partial DRACO image. The second path focused on matching the sunlit limb. We built maplets along the sunlit limb that were conditioned by limb points identified in the tilt-to-height integration in an SPC iteration. These maplets were made at only two GSDs owing to the coarser resolution of images that contained limb. Once the topography in the standard maplets and limb-only maplets stabilized, we united them via several SPC iterations and proceeded to build the global DTM.

The areas covered by maplets in the Dimorphos global DTM are shown in Extended Data Fig. 1 as shaded grey regions. The blue and magenta points show the locations of the sunlit and Didymos-lit limbs. The unseen side of the asteroid is roughly centred on the 90° E view. The shape model has the volume and extents reported in Table 1. The global DTM has a typical GSD of 26 cm, but that resolution is only meaningful in the areas covered by maplets.

We also used the standard maplets to construct a DTM of the impact site. The impact site is covered by the highest-resolution images, so the impact site can support finer-scale topography than the global model. The impact site DTM has a GSD of 5 cm and is shown in Fig. 3.

Owing to the short period of time for which we have resolved images of Dimorphos, the rotation pole, prime meridian and rotation rate of Dimorphos could not be updated using SPC. Instead, we used values derived by ground-based observers. In the equatorial J2000 frame, the pole values used for Dimorphos are: BODY120065803_POLE_RA = (60.936309840897856, 0, 0) and BODY120065803_POLE_DEC = (−71.674565992873852, 0, 0).

The coordinate system of the global DTM of Dimorphos has the prime meridian pointed towards Didymos, consistent with International Astronomical Union convention: BODY120065803_PM = (64.914870949788195, 724.723943017441570, 1.0840372309 × 10⁻⁶).

A planetary constants kernel (PCK) with these values is available at https://ssd.jpl.nasa.gov/ftp/eph/small_bodies/dart/dimorphos/archive/ called ‘dimorphos_s501-preimp.tpc’. Documentation for PCK files, including units for the terms listed above, is available at https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/FORTRAN/req/pck.html#Text%20PCK%20Kernel%20Variable%20Names. The north pole of Dimorphos is in the +Z direction of the shape model but oriented towards ecliptic south.

Shape model uncertainties

We quantified uncertainties in the global DTM of Dimorphos using techniques employed to estimate the uncertainties associated with the global DTM of Bennu. These techniques were also used to assess uncertainties in shape models developed during pre-impact tests done for DART. The analyses involved comparing DRACO images against the global DTM when rendered with the same lighting and viewing geometry as the image. We used three analyses described in the above references. The first method, referred to as the limb and terminator method, used image thresholding to identify the limb and terminator in the DRACO image and rendered shape model. The thresholded images were subtracted to reveal mismatches between the image and rendered shape model and any size bias in the model (Extended Data Fig. 9a–e). The second and third methods relied on analysis of corresponding surface features, or keypoints, in images and the rendered shape model, to understand errors in the overall size of the shape model (Extended Data Fig. 9f–i). In the second method, referred to as keypoint matching, the rendered image was rotated, translated and scaled to minimize differences in keypoint locations on the DRACO image and rendered shape model. The third method, referred to as keypoint distance, was based on the measured distances between all keypoints in the DRACO image and rendered shape model.

The limb and terminator assessments indicated that the model is 75 cm too small in X, Y and Z, with a limb uncertainty of 1.3–2 m. Most of the mismatch between DRACO images and the rendered shape model occurred along the limb and terminator. We expected larger uncertainties in these areas owing to the limited coverage and single lighting condition in DRACO images. The keypoints indicated that the Dimorphos shape model is 6 ± 29 cm too large and that features in the model have point-separation errors of −11 ± 20 cm. The keypoint assessments indicated that the model performed well in areas where DRACO resolved detailed surface features, which is the area of the model that is most relevant to understanding the topography of the impact site. Comparisons between shadow lengths in DRACO images and shadow lengths in shape model renderings indicated that some boulder heights are too small. This mismatch was expected based on shape modelling tests done for DART and is a consequence of constructing the shape model from a single lighting condition and viewing geometry. Shadow lengths suggest that the height of boulder 1 in the impact site DTM is about 10% too small, but the height of boulder 2 in the impact site DTM is correct. On the basis of all of these analyses, we assumed uncertainties of 2 m in the extents of Dimorphos in the X and Z directions.

Because DRACO saw a complete outline of the asteroid (Extended Data Fig. 8) and the spacecraft approach geometry was such that this outline was primarily in the X–Z plane, the shape model dimension uncertainties in X and Z are less than 1% (Table 1). Hence, the Y-axis extent is the largest source of uncertainty in the volume. Future work will refine the errors in the Y axis, but for the time being we assume that the uncertainty in Y is two times the uncertainties in X and Z (that is, 4 m).

Measuring the volume of Dimorphos and realistically estimating uncertainties on that quantity is of particular interest because it directly affects our understanding of the mass of the asteroid. In the shape modelling tests, we took scaled shape models of an ellipsoid, Itokawa, and Bennu, rendered a set of simulated DRACO images from the ‘truth’ shape models, and then used SPC to build shape models from simulated images. We compared the volumes of the models built using SPC against the volumes of truth models. The volume errors in those tests ranged from −2% to +23%. The test with the lowest volume error was for an ellipsoidal asteroid. The roughly ellipsoidal shape of Dimorphos as seen by DRACO suggests that the volume error in the Dimorphos global DTM is probably on the order of several per cent, rather than 23%, by virtue of the simplicity of the long-wavelength shape of Dimorphos. Moreover, in the tests with the largest volume errors, the terminator of the shape model—which is the primary constraint on the extent of the shape into and out of the page—did not match the terminator of the original images. In the case of Dimorphos, the terminator matched nicely between the global DTM and the rendered shape model (Extended Data Fig. 9). Given the results from the shape modelling tests, this agreement indicated that the volume errors for Dimorphos are probably on the order of several per cent. On the basis of these assessments, a volume error of 5% seems probable. But, we adopted a volume uncertainty of 10% to be conservative. This volume error is larger than the value implied by the reported uncertainties in the X, Y and Z extents, but the intent is to be conservative.

Shape modelling of Didymos

A preliminary SPC shape model of Didymos was built from DRACO data. This model led to the preliminary volume and extent of Didymos reported in Table 1. DRACO images revealed a smaller Z axis and showed that the visible portion of the X axis probably needs to be extended by a few tens of metres compared with a radar-derived shape model. We used the techniques employed to estimate shape model uncertainties described in the ‘Shape modelling of Dimorphos’ section to assess the preliminary SPC shape model of Didymos. Because the full X and Z extents of Didymos cannot be measured in DRACO images (in contrast to Dimorphos), we conservatively attributed an uncertainty of 15 m to the lengths of all the axes of Didymos.

Impact site identification

The construction of an SPC shape model produced a set of surface landmarks that were used to determine the location (at the time of each image) of the spacecraft relative to the surface in the Dimorphos body fixed frame. This procedure used the spacecraft’s Didymos relative velocity computed by the ground navigation team using a combination of radiometric tracking data (Doppler, range) and optical images of Didymos. We used this information and the shape model of Dimorphos to compute the location of the impact site. The positions of the DART spacecraft from SPC were converted to the inertial J2000 frame and corrected for light time and aberration. The velocity of the spacecraft was estimated by fitting a second-order polynomial function to these positions as a function of time. This approach is identical to that employed previously to estimate the position of the Hayabusa spacecraft relative to Itokawa. We used the locations from the last 14 complete images collected by DART to determine the spacecraft velocity. These images contained large numbers of landmarks due to their fine pixel scales, which help anchor the spacecraft position for each image. The fit residuals (that is, the difference between the fit of the spacecraft location to that determined from SPC) are <1 m (Extended Data Fig. 10).

We combined our estimated velocity and the SPC-derived spacecraft positions of the last five images to determine the impact location. Only the last five images were used to locate the impact because heater cycling on the spacecraft introduced cyclic error rates in the inertial attitude knowledge. These rates were estimated to be small at the time of impact owing to the timing of impact relative to heater cycling but at their highest approximately 30 s before impact. Using only the last few images reduced the influence of this known (and cumulative) error source. At each of these five spacecraft positions, we propagated the velocity vector until it intersected with the surface of the Dimorphos shape model. We took the mean of these positions as the impact location. We also computed the spacecraft state at a few different heights above the intersection point to determine the order in which the solar arrays and bus contacted the surface. The propagation from each of the last five images provided the same impact point to within 1 cm. The uncertainties in the impact point location (reported in Table 1) are dominated by the residuals to the fits of the spacecraft position obtained by SPC in the Y and Z directions (Extended Data Fig. 10).

Impact-angle assessments

The tilt angle of the surface relative to the impact velocity vector defines the impact angle. Because the impact site DTM resolves topography at a 5-cm GSD, the spacecraft bus would have interacted with on the order of a thousand facets of the DTM. Therefore, we calculated a mean tilt with respect to the impact velocity vector for each facet in the impact site DTM. The mean tilt calculation is based on previous work; however, we computed mean tilt with respect to the impact velocity vector, v, rather than the radial vector to a given facet to determine the impact angle.

The magnitude of the mean tilt for a facet at the impact point is the angle between v and the average normal, nav, which is the weighted average of the normal vectors of all facets in the user-defined region, nav = ∑(niAi)/∑Ai surrounding the impact point. In this study, this region was 1.5 m in radius to exceed the size of the spacecraft bus. The normal vector of each facet in the region of interest, ni, was weighted by Ai, the area of the facet projected onto a best-fit plane to the region selected to determine the surface tilt. This yields a mean tilt of acos (nav × v/|nav||v|). The impact angle, θ, which is typically defined relative to the local horizontal for planetary impacts, is given by θ = 90 − (mean tilt). The impact angle shown in Table 1 is computed from the mean tilt and θ of the facet closest to the impact point.

As discussed in the ‘Impact site identification’ section, the location of the impact site has an uncertainty of ±68 cm. To understand the range of mean tilts that DART may have encountered given the uncertainty in the impact location, we considered the distribution of mean tilt for all facets within a circle with radius 68 cm centred on the impact site (Extended Data Fig. 2). The standard deviation of this distribution was 7° (1σ), which we attributed as the uncertainty of the impact angle DART may have experienced.

Boulder counting

The longest axis of each boulder in the final full DRACO image was identified as a line (as done for asteroid (101955) Bennu). The length of the longest axis was determined from the line length and image pixel scale, assuming the last full image as a ‘flat’ scene. The total number of boulders and pebbles identified in the final full DRACO image impact site is 953 and range in size (that is, the length of the longest axis) from 0.16 m (limit of image resolution, assuming ≥3-pixel sampling) to 6.5 m. The resulting size–frequency distribution is shown in Extended Data Fig. 6.

Estimates for the density of Dimorphos

To first order, the combined system mass, Msys, was estimated using Kepler’s third law

where a and P are the pre-impact semimajor axis and the orbit period, respectively, and G is the gravitational constant. Neglecting the aspherical shapes of Didymos and Dimorphos and their associated gravitational potentials may lead to an overestimate of the system mass by about 1–2%. However, this error is negligible as the uncertainty in Dimorphos’s semimajor axis dominates the uncertainty in the system mass. Next, the bulk density of the combined system was obtained by dividing the system mass with the combined volume of both bodies

where RA and RB are the volume-equivalent radii of Didymos and Dimorphos, respectively. On the basis of the pre-impact orbit period and semimajor axis, and the volume-equivalent diameters provided in this work, we calculated a nominal system bulk density of 2,400 ± 250 kg m⁻³ using the uncertainties quoted in the previous sentence. However, to represent additional possible systematic uncertainties, we adopted a slightly larger uncertainty, which gives a system bulk density of 2,400 ± 300 kg m⁻³.

The porosity of Dimorphos was estimated as follows:

In this work, we assumed that Dimorphos’s bulk density matches the entire system’s bulk density. There are three near-Earth asteroid systems for which the satellite’s bulk density has been independently measured: 66391 Moshup, 2000 DP107 and 2001 SN263 (two satellites). Two of those satellites (Squannit and 2001 SN263 gamma) were measured to be denser than the primary and two were less dense (2001 SN263 beta and 2000 DP107 beta). For all three systems, the 1σ uncertainties for the satellite and the primary densities overlap. Furthermore, recent work estimated that the size of Squannit is about 30% larger than estimated previously, meaning that the bulk density of Squannit may be in better agreement with its primary. Given these other examples, assuming Dimorphos’s density matches the system bulk density is a reasonable starting point, although its true density could differ substantially from this value. Hera will determine the masses and densities of Didymos and Dimorphos and test the validity of this assumption.