Andrew F Cheng

Harrison F Agrusa

Brent W Barbee

Alex J Meyer

Tony L Farnham

Sabina D Raducan

Derek C Richardson

Elisabetta Dotto

Angelo Zinzi

Vincenzo Della Corte

Thomas S Statler

Steven Chesley

Shantanu P Naidu

Masatoshi Hirabayashi

Jian-Yang Li

Siegfried Eggl

Olivier S Barnouin

Nancy L Chabot

Sidney Chocron

Gareth S Collins

R Terik Daly

Thomas M Davison

Mallory E Decoster

Carolyn M Ernst

Fabio Ferrari

Dawn M Graninger

Seth A Jacobson

Martin Jutzi

Kathryn M Kumamoto

Robert Luther

Joshua R Lyzhoft

Patrick Michel

Naomi Murdoch

Ryota Nakano

Eric Palmer

Andrew S Rivkin

Daniel J Scheeres

Angela M Stickle

Jessica M Sunshine

Josep M Trigo-Rodriguez

Jean-Baptiste Vincent

James D Walker

Kai Wünnemann

Yun Zhang

Marilena Amoroso

Ivano Bertini

John R Brucato

Andrea Capannolo

Gabriele Cremonese

Massimo Dall'ora

Prasanna J D Deshapriya

Igor Gai

Pedro H Hasselmann

Simone Ieva

Gabriele Impresario

Stavro L Ivanovski

Michèle Lavagna

Alice Lucchetti

Elena M Epifani

Dario Modenini

Maurizio Pajola

Pasquale Palumbo

Davide Perna

Simone Pirrotta

Giovanni Poggiali

Alessandro Rossi

Paolo Tortora

Marco Zannoni

Giovanni Zanotti

Momentum transfer from the DART mission kinetic impact on asteroid Dimorphos

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Momentum transfer from the DART mission kinetic impact on asteroid Dimorphos

The NASA Double Asteroid Redirection Test (DART) mission performed a kinetic impact on asteroid Dimorphos, the satellite of the binary asteroid (65803) Didymos, at 23:14 UTC on 26 September 2022 as a planetary defence test [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] . DART was the first hypervelocity impact experiment on an asteroid at size and velocity scales relevant to planetary defence, intended to validate kinetic impact as a means of asteroid deflection.

Here we report a determination of the momentum transferred to an asteroid by kinetic impact. On the basis of the change in the binary orbit period [START_REF] Thomas | Orbital period change of Dimorphos due to the DART kinetic impact[END_REF] , we find an instantaneous reduction in Dimorphos's along-track orbital velocity component of 2.70 ± 0.10 mm s -1 , indicating enhanced momentum transfer due to recoil from ejecta streams produced by the impact [START_REF] Li | Ejecta from the DART-produced active asteroid Dimorphos[END_REF][START_REF] Ahrens | Hazards Due to Comets and Asteroids[END_REF] . For a Dimorphos bulk density range of 1,500 to 3,300 kg m -3 , we find that the expected value of the momentum enhancement factor, β, ranges between 2.2 and 4.9, depending on the mass of Dimorphos. If Dimorphos and Didymos are assumed to have equal densities of 2,400 kg m -3 , β σ = 3.61 (1 ) -0.25 +0. 19 . These β values indicate that substantially more momentum was transferred to Dimorphos from the escaping impact ejecta than was incident with DART. Therefore, the DART kinetic impact was highly effective in deflecting the asteroid Dimorphos.

Observations from the NASA Double Asteroid Redirection Test (DART) spacecraft on approach found Dimorphos to be an oblate spheroid with a boulder-strewn surface, and the spacecraft struck within 25 m of the centre of the figure [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] . Ejecta from the DART impact were observed in situ by the Italian Space Agency's Light Italian Cubesat for Imaging of Asteroids (LICIACube) spacecraft, which performed a flyby of Dimorphos with a closest approach about 168 s after the DART impact [START_REF] Dotto | LICIACube-The Light Italian Cubesat for imaging of asteroids in support of the NASA DART mission towards asteroid (65803) Didymos[END_REF] . The impact ejecta were further observed by Earth-and space-based telescopes, revealing ejecta streams and dust tails similar to those seen in active asteroids thought to be triggered by natural impacts [START_REF] Li | Ejecta from the DART-produced active asteroid Dimorphos[END_REF][START_REF] Tancredi | Lofting of low speed ejecta produced in the DART experiment and production of a dust cloud[END_REF] . Ground-based telescopes and radar determined that the DART impact reduced the binary orbit period by 33.0 ± 1.0 (3σ) min (ref.

2).

As a planetary defence test mission, a key objective of DART is to determine the amount of momentum transferred to the target body relative to the incident momentum of the spacecraft, quantified by the momentum enhancement factor β (for example, refs. 4,7,8), which is defined by the momentum balance of the kinetic impact,

M m m β ∆ = + ( -1)( ^⋅ ) ^. (1) v U E U E
Here, M is the mass of Dimorphos, ∆v is the impact-induced change in Dimorphos's orbital velocity, m is DART's mass at impact, U is DART's velocity relative to Dimorphos at impact and E ˆ is the net ejecta momen- tum direction. v M∆ is the momentum transferred to Dimorphos, U m is DART's incident momentum and the final term in the equation is the ejecta's net momentum written in terms of the spacecraft incident momentum. In this formulation, β is the ratio of actual imparted momentum to the impactor's momentum in the direction of the net ejecta momentum. Although previous works have defined β using the impactor's momentum in the surface normal direction [START_REF] Rivkin | The Double Asteroid Redirection Test (DART): planetary defense investigations and requirements[END_REF][START_REF] Feldhacker | Shape dependence of the kinetic deflection of asteroids[END_REF] , we elect to use the ejecta direction instead as our reference for the result to be independent of the surface topography. These definitions are equivalent in the case in which the ejecta direction is in the surface normal direction. A β value near 1 would indicate that ejecta recoil had made only a negligible contribution to the momentum transfer. A β > 2 would mean that the ejecta momentum contribution exceeded the incident momentum from DART.

The full v ∆ cannot be determined with the available information [START_REF] Richardson | Predictions for the dynamical states of the Didymos system before and after the planned DART impact[END_REF] , but its component along Dimorphos's orbital velocity direction, referred to as the along-track direction, can be estimated from available data including Dimorphos's orbit period change. To express β in terms of the along-track component of ∆v, we take the scalar product of (1) with the unit vector e ˆT in the along-track direction. Solving for β yields,

β = 1 + (∆ ⋅ ˆ) -( ⋅ ˆ) ( ˆ⋅ )( ˆ⋅ ˆ) . ( 2
) M m T T T v e U e

E U E e

For the remainder of this work, we refer to the along-track component of Dimorphos's velocity change, v e ∆ ⋅ ˆT, as Δv T . Figure 1 shows the geometry of the DART impact, including the nominal ellipsoidal shapes used for Didymos and Dimorphos in our analysis, and the nominal orientations of U, e ˆT and Ê at the time of impact.

The major unknowns in calculating β are Δv T , M and E ˆ. We first use a Monte Carlo approach to produce a distribution for Δv T consistent with the measured period change that incorporates the various uncertainties involved. We sample many possible combinations of Didymos system parameters, including the ellipsoid shape extents of the asteroids, pre-impact orbit separation distance between the two asteroids' centres of mass (that is, Dimorphos's pre-impact orbit radius), pre-and post-impact orbit periods, and net ejecta momentum direction E ˆ. We use the full two-body problem code (General Use Binary Asteroid Simulator (GUBAS) [START_REF] Davis | Doubly synchronous binary asteroid mass parameter observability[END_REF] , Methods) that implements coupled rotational and orbital dynamics to numerically determine Δv T for each sampled combination of input parameters. Coupled dynamics are necessary because of the non-spherical shapes of Didymos and Dimorphos and their close proximity relative to their sizes. A range of values for M is generated by combining the volumes of the sampled ellipsoid shape parameters with values for Dimorphos's density. The Monte Carlo approach is summarized by Extended Data Fig. 1. As Dimorphos's density has not been directly measured and has a large uncertainty, we treat it as an independent variable and uniformly sample a wide range of possible values between 1,500 and 3,300 kg m -3 , a range that encompasses the 3σ uncertainty [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] . Using a technique modified from that of ref. 12 (Methods), we apply observations of the ejecta by means of Hubble and LICIACube data to obtain a preliminary measurement of the axis of the ejecta cone geometry (Extended Data Figs. 2 and3). The cone axis direction is identical to E ˆ assuming the ejecta plume holds the momentum uniformly, and we find E ˆ points towards a right ascension and declination (Dec) of 138° and +13°, respectively (Extended Data Fig. 2). We assign a conservative uncertainty of 15° around this direction. Finally, β also depends on DART's mass and impact velocity, as well as Didymos's pole orientation [START_REF] Thomas | Orbital period change of Dimorphos due to the DART kinetic impact[END_REF] . Those quantities have negligibly small uncertainties relative to those of the other parameters discussed previously and are therefore treated as fixed values (not sampled). See Methods for additional details on the Monte Carlo analysis, Extended Data Table 1 for a list of parameters and uncertainties, and Extended Data Table 2 for the covariances that were used.

We find that v σ ∆ =-2.70± 0.10 (1 ) T mm s -1 , on the basis of the observed impact-induced period change of -33.0 ± 1.0 (3σ) minutes and the shapes and separation of Didymos and Dimorphos 1,2 . Figure 2 shows the distribution of Δv T values from the Monte Carlo analysis, along with the fitted mean and standard deviation. The resulting spread of β values as a function of Dimorphos's density, calculated by means of equation ( 2), is presented in Fig. 3, along with linear fits for the mean β versus density trend and its 1σ confidence intervals. The linear-fit slope is expressed as a scale factor on the ratio of density to the nominal value of 2,400 kg m -3 (ref. 1). For that nominal Dimorphos density, at which Dimorphos and Didymos would have roughly equal densities, β = 3.61 -0.25 +0.19 with 1σ confidence. The mean β ranges between 2.2 and 4.9 as a function of density across the range of 1,500 to 3,300 kg m -3 and, overall, β ranges between 1.9 and 5.5 with 3σ confidence.

Our result for β is consistent with numerical simulations [START_REF] Jutzi | Hypervelocity impacts on asteroids and momentum transfer. I. Numerical simulations using porous targets[END_REF][START_REF] Syal | Deflection by kinetic impact: sensitivity to asteroid properties[END_REF][START_REF] Cheng | Asteroid impact & deflection assessment mission: kinetic impactor[END_REF][START_REF] Raducan | The role of asteroid strength, porosity and internal friction in impact momentum transfer[END_REF][START_REF] Raducan | The effects of asteroid layering on ejecta mass-velocity distribution and implications for impact momentum transfer[END_REF][START_REF] Rainey | Impact modeling for the Double Asteroid Redirection Test (DART) mission[END_REF][START_REF] Kumamoto | Predicting asteroid material properties from a DART-like kinetic impact[END_REF][START_REF] Stickle | Effects of impact and target parameters on the results of a kinetic impactor: predictions for the Double Asteroid Redirection Test (DART) mission[END_REF][START_REF] Owen | Spacecraft geometry effects on kinetic impactor missions[END_REF][START_REF] Luther | Momentum enhancement during kinetic impacts in the low-intermediatestrength regime: benchmarking and validation of impact shock physics codes[END_REF][START_REF] Decoster | Statistical significance of mission parameters on the deflection efficiency of kinetic impacts: applications for the next-generation kinetic impactor[END_REF] and laboratory experiments [START_REF] Walker | Momentum enhancement from a 3 cm diameter aluminum sphere striking a small boulder assembly at 5.4 km s -1[END_REF][START_REF] Flynn | Hypervelocity cratering and disruption of porous pumice targets: implications for crater production, catastrophic disruption, and momentum transfer on porous asteroids[END_REF][START_REF] Durda | Laboratory impact experiments with decimeter-to meter-scale targets to measure momentum enhancement[END_REF][START_REF] Flynn | Momentum transfer in hypervelocity cratering of meteorites and meteorite analogs: implications for orbital evolution and kinetic impact deflection of asteroids[END_REF][START_REF] Hoerth | Momentum transfer in hypervelocity impact experiments on rock targets[END_REF][START_REF] Walker | Momentum enhancement from aluminum striking granite and the scale size effect[END_REF] of kinetic impacts, which have consistently indicated that β is expected to fall between about 1 and 6. However, non-unique combinations of asteroid mechanical properties (for example, cohesive strength, porosity and friction angle) can 1 | Schematic of the DART impact geometry on Dimorphos. The preimpact orbit is shown with a solid line around Didymos. The dashed line sketches the orbit change due to the impact. Orbits are drawn roughly to scale. The positive pole direction of Didymos is h ˆ (pointing down in the bottom panel). DART's incident direction is U ˆ, the net ejecta momentum direction is Ê (which points to a right ascension (RA) and declination (Dec) of 138° and +13°, respectively), and the direction of Dimorphos's orbital motion, referred to as the along-track direction, is ˆT e . The relative positions of the Sun and the Earth are also indicated. The upper panel shows the view from Didymos's negative pole direction, whereas the lower panel provides a perspective view. Scale bar, 1 km. produce similar values of β in impact simulations [START_REF] Stickle | Effects of impact and target parameters on the results of a kinetic impactor: predictions for the Double Asteroid Redirection Test (DART) mission[END_REF] . Future studies that combine estimates of β with additional constraints from the DART impact site geology 1 and ejecta observations [START_REF] Li | Ejecta from the DART-produced active asteroid Dimorphos[END_REF][START_REF] Dotto | LICIACube-The Light Italian Cubesat for imaging of asteroids in support of the NASA DART mission towards asteroid (65803) Didymos[END_REF] will provide greater insight into Dimorphos's material properties. In addition, ESA's Hera mission [START_REF] Michel | The ESA Hera mission: detailed characterization of the DART impact outcome and of the binary asteroid (65803) Didymos[END_REF] is planned to arrive at the Didymos system in late 2026. By measuring Dimorphos's mass and other orbital properties, Hera will allow us to significantly improve the accuracy and precision of the β determination.

DART's impact demonstrates that the momentum transfer to a target asteroid can substantially exceed the incident momentum of the kinetic impactor, validating the effectiveness of kinetic impact for preventing future asteroid strikes on the Earth. The value of β from a kinetic impact is key to informing the strategy of a kinetic impactor mission (or missions) to mitigate a future asteroid impact threat to Earth [START_REF] Statler | After DART: using the first full-scale test of a kinetic impactor to inform a future planetary defense mission[END_REF] . Should β turn out to be greater than two across a wide range of asteroid types, it would mean important performance improvements for kinetic impactor asteroid deflection missions. If β > 2, as opposed to β ≅ 1, then the same sized kinetic impactor could deflect a given asteroid with less warning time, or deflect a larger asteroid with a given warning time than it otherwise could.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-023-05878-z.

Methods

Numerical determination of Δv T and β

Several parameters affect the value of β as presented in equation ( 2): Δv T , M and E ˆ. The along-track velocity change, Δv T depends on orbit period change, pre-impact semimajor axis and the shapes of Didymos and Dimorphos, whereas M depends on Dimorphos's shape and bulk density (which was not measured). E ˆ is the only parameter that is directly observed, but it still has considerable uncertainty. Thus, there are 12 total unknown input parameters: three axis lengths for Didymos's ellipsoidal shape (A x , A y , A z ), three axial lengths for Dimorphos's ellipsoidal shape (B x , B y , B z ), Dimorphos's bulk density ρ B , the pre-impact orbit semimajor axis a pre , pre-impact orbit period P pre and post-impact orbit period P post , and two angles to define the ejecta momentum direction vector (E ˆ). Extended Data Table 1 lists these input parameter values and their uncertainties, along with additional known quantities needed to calculate β. To account for this large set of input uncertainties, we use a Monte Carlo approach in which 100,000 possible cases are generated by randomly sampling the input parameters within their uncertainties. We assume the DART spacecraft mass, DART impact velocity vector and Dimorphos's orbital velocity direction (referred to as the along-track direction) are all known precisely because their uncertainties are negligibly small compared to the uncertainties of the other input parameters.

The pre-impact orbit semimajor axis, pre-impact orbit period and post-impact orbit period are sampled as a multivariate Gaussian distribution using the mean values and covariance matrix from the 'N22+' solution (refs. 35 and 36 of ref. 2; Extended Data Tables 1 and2). This accounts for the small correlations between those three parameters. The physical extents of Didymos and Dimorphos from ref. 1 are sampled uniformly, as those uncertainties are not Gaussian (Extended Data Table 1). β depends strongly on Dimorphos's mass, but the mass is poorly constrained because Dimorphos's bulk density has not been directly measured [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] . Therefore, we treat density as the independent variable, sample it uniformly and report β as a function of Dimorphos's density.

For each Monte Carlo sample for the Didymos system, a secant search algorithm (a finite-difference Newton's method) described in ref. 37 is first used to compute the density of Didymos required to reproduce the sampled pre-impact orbit period, given the sampled pre-impact orbit semimajor axis, body shapes and Dimorphos's density. Then, a second secant search algorithm is used to determine the Δv T required to achieve the sampled post-impact orbit period. We match the pre-and post-impact orbit periods because these are directly measured by ground-based observations and are thus the best-constrained parameters of the system 2 . Given the non-Keplerian nature of the Didymos system, we use the GUBAS to numerically propagate the binary asteroid dynamics. GUBAS is a well-tested full two-body problem (F2BP) code that can model the mutual gravitational interactions between two arbitrarily shaped rigid bodies with uniform mass distributions [START_REF] Davis | Doubly synchronous binary asteroid mass parameter observability[END_REF][START_REF] Hou | Mutual potential between two rigid bodies with arbitrary shapes and mass distributions[END_REF] . GUBAS has been benchmarked against other F2BP codes 39 and used extensively in previous dynamical studies of the Didymos system (for example, refs. 10,37,40). Finally, the mass of Dimorphos is calculated from its ellipsoidal shape and Dimorphos's density. This mass, along with the computed Δv T and sampled net ejecta momentum direction, are provided as inputs to equation (2) to calculate the value of β corresponding to each of the 100,000 realizations of the system. For a discussion on estimating E ˆ, see the Ejecta plume direction section below. The process described herein is summarized graphically in Extended Data Fig. 1.

The convergence criteria on both secant algorithms are set such that the simulated orbit period matches the desired orbit period to an accuracy ten times better than the uncertainty on the measurements themselves. The numerical simulations measure the average orbit period of Dimorphos in an inertial frame over 30 days to account for small fluctuations in the mutual orbit period resulting from spinorbit coupling [START_REF] Meyer | Libration-induced orbit period variations following the DART impact[END_REF] . Our selection of 100,000 as the number of samples to use in the Monte Carlo analysis was informed by calculating an estimated minimum necessary number of samples from the Central Limit Theorem and then testing sample sizes near that estimated value. The β estimate results are well converged with 100,000 samples.

In the numerical simulations, both Didymos and Dimorphos are modelled as triaxial ellipsoids with physical extents from ref. 1. Images from DRACO and LICIACube showed that both Didymos and Dimorphos have an oblate spheroid shape [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] . There is no advantage to using more sophisticated shape models while the internal mass distributions of the bodies are unknown. Instead, the ellipsoidal approximation allows for easy sampling of a range of plausible moments of inertia as a proxy for different internal density distributions. For example, given the current uncertainties in Didymos's physical extents [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] , sampling over the given range of ellipsoidal shapes results in a range of plausible second-order gravity terms (analogous to the spherical harmonic terms J 2 , C 22 and so on), which play an important role in the system's dynamics due to the tight separation of the binary components. Neglecting their shapes and assuming Keplerian dynamics results in v σ ∆ =-2.86± 0.095 (1 ) T , whereas GUBAS's second-order gravity model finds v σ ∆ =-2.70 ±0.10 (1 ) T . Although fourth-order dynamics influence higher-order dynamical effects [START_REF] Agrusa | The excited spin state of Dimorphos resulting from the DART impact[END_REF][START_REF] Meyer | Libration-induced orbit period variations following the DART impact[END_REF] , we find that it comes with a significantly higher computational cost yet plays a negligible role in determining Δv T . A smaller batch (due to increased computational cost) of roughly 4,000 runs was conducted with fourth-order dynamics, which resulted in v σ ∆ =-2.68± 0.10 (1 ) T , indicating the second-order dynamics model is appropriate for determining Δv T given the current uncertainties in the orbit solution and body shapes. This result was also independently verified using analytical models, accounting for Didymos's gravitational quadrupole, which agreed within a few percent of the second-order numerical results, as expected given their dynamical approximations.

We do not sample the rotation period of Dimorphos, as it is assumed to be equal to the pre-impact orbit period before the impact, with reasoning as follows. A measured orbit semimajor axis drift directed inwards [START_REF] Scheirich | Preimpact mutual orbit of the DART target binary asteroid (65803) Didymos derived from observations of mutual events in 2003-2021[END_REF] indicates the system is evolving under the influence of the binary Yarkevsky-O'Keefe-Radzievskii-Paddack effect [START_REF] Ćuk | Effects of thermal radiation on the dynamics of binary NEAs[END_REF] , which requires a secondary in near-synchronous rotation. Furthermore, radar images constrain Dimorphos's spin period to be within 3 h of the synchronous rate [START_REF] Naidu | Radar observations and a physical model of binary near-Earth asteroid 65803 Didymos, target of the DART mission[END_REF] . Recent models for tidal dissipation in binary asteroids suggest that any free libration would dissipate on 100-year timescales [START_REF] Meyer | Energy dissipation in synchronous binary asteroids[END_REF] , making any substantial free libration unlikely given the timescales for excitation mechanisms such as close planetary encounters and natural impacts [START_REF] Richardson | Predictions for the dynamical states of the Didymos system before and after the planned DART impact[END_REF] . Furthermore, Dimorphos's pre-impact eccentricity is constrained to be less than 0.03 (refs. 35,41), putting the maximum possible forced libration amplitude 44 at around 0.5°. Although Dimorphos's rotation state is not precisely determined by DART, this body of evidence suggests that Dimorphos was probably in near-synchronous rotation and on a nearly circular orbit before the DART impact.

Our model further assumes all momentum is transferred instantaneously, because earlier work showed the time duration of the momentum transfer has a negligible effect on the resulting dynamics [START_REF] Richardson | Predictions for the dynamical states of the Didymos system before and after the planned DART impact[END_REF] . The instantaneous torque on Dimorphos due to DART's slightly off-centre impact 1 is also neglected as the corresponding change in Dimorphos's rotation state is small compared to that arising from exciting Dimorphos's eccentricity and libration by the impact [START_REF] Richardson | Predictions for the dynamical states of the Didymos system before and after the planned DART impact[END_REF][START_REF] Agrusa | The excited spin state of Dimorphos resulting from the DART impact[END_REF][START_REF] Michel | European component of the AIDA mission to a binary asteroid: characterization and interpretation of the impact of the DART mission[END_REF] . Finally, the effects of reshaping and mass loss due to cratering and ejecta are also neglected, as these effects are expected to be smaller in magnitude than the current roughly 1 min uncertainty on the post-impact orbit period [START_REF] Nakano | NASA's Double Asteroid Redirection Test (DART): mutual orbital period change due to reshaping in the near-earth binary asteroid system (65803) Didymos[END_REF] and will remain poorly constrained until the Hera mission characterizes the Didymos system in 2027 (ref. 30). We leave these higher-order effects for future work once the post-impact orbit solution is refined further.

Ejecta plume direction

We use observations of the ejecta plume to determine the ejecta momentum direction Ê. The conical ejecta plume was imaged by the LICIACube LUKE camera [START_REF] Dotto | LICIACube-The Light Italian Cubesat for imaging of asteroids in support of the NASA DART mission towards asteroid (65803) Didymos[END_REF] and the Hubble Space Telescope (HST) [START_REF] Li | Ejecta from the DART-produced active asteroid Dimorphos[END_REF] . We apply a technique used to derive cometary spin poles [START_REF] Farnham | A McDonald Observatory study of comet 19P/Borrelly: placing the deep space 1 observations into a broader context[END_REF] to estimate the orientation of the ejecta cone axis. Although it is possible to have an asymmetric distribution of ejecta momentum (mass and velocity) within the cone, we assume the cone to be axially symmetric. The approach applies the ejecta cone's bright edges (if captured in an image) to compute the apparent direction of the cone axis projected onto the sky, which is assumed to be the middle of the edges.

For a LICIACube observation, the projected cone axis defines the LICIACube-axis plane in inertial space that contains the line-of-sight and the projected axis. The cone axis can lie anywhere in this plane. The analogous plane HST axis is defined from early HST images of the plume (those taken within 2 h after the impact) that show similar radial velocity to the ejecta in the LICIACube images, indicating it is probably the same ejecta material observed on a larger spatial scale. The intersection of these planes defines the cone axis orientation in three dimensions, but unfortunately the LICIACube-and HST-axis planes are nearly parallel. Thus, these observations do not provide a unique solution but they constrain the axis orientation to a narrow swath of the sky (Extended Data Fig. 2). However, LICIACube LUKE images resolved the ejecta cone and the cone morphology over a large range of viewing angles during the flyby, further constraining the cone axis orientation [START_REF] Li | Ejecta from the DART-produced active asteroid Dimorphos[END_REF] . During the approach to Dimorphos, the cone was pointed towards LICIACube, with the ejecta obscuring parts of Dimorphos. During recession from Dimorphos, the cone pointed away from Dimorphos and revealed it in silhouette (Extended Data Fig. 3). The tightest constraint on cone orientation would have come from closest-approach images, with the cone axis in the plane-of-sky, as the cone transitioned from pointing towards the observer to pointing away. Unfortunately, Dimorphos and the ejecta cone were outside the LUKE field of view for 13 s around closest approach, and we lack images from the transition.

The resolved LICIACube images are used to eliminate portions of the swath in Extended Data Fig. 2 where the observed cone morphology is inconsistent with those axis orientations. For example, half of the swath is rejected because the axis would be pointed in the opposite direction of what was observed. We also exclude orientations in which the cone would point too close to the line-of-sight during the approach or recession of LICIACube. We find the axis orientation to be (right ascension, Dec) = (138°, +13°). We assign conservative uncertainties of roughly 15° in all directions based on the angular extents of region 5 in Extended Data Fig. 2.

  orbit U, E and e T ˆÛ , E and e T ˆFig.

1 Fig. 2 |Fig. 3 |

 123 Fig. 2 | Probability distribution of Δv T , the along-track component of the change in Dimorphos's velocity induced by DART's impact, generated by our Monte Carlo analysis that samples over input parameter uncertainties.The histogram consists of 100,000 Monte Carlo samples and is normalized to an area of unity. A Gaussian fit to the distribution indicates a mean Δv T of -2.70 mm s -1 with a standard deviation of 0.10 mm s -1 .

Extended Data Fig. 2 |

 2 Ejecta cone orientation lies in the swaths of sky (black lines) defined by HST and LICIACube observations. The light-blue envelope outlines the axis position uncertainty in the direction measured in the sky plane. Red lines divide the along-plane swaths into regions that are excluded based on cone morphology in LICIACube images: 1) and 2) are excluded because the ejecta cone would point in the opposite direction from what is observed; 3) is excluded because the axis would lie too close to the sky plane; 4) is excluded because the axis would lie too close to the line-of-sight; and 5) is the expected region for the axis orientation. The yellow dot denotes the best solution (RA,Dec) = [138°,+13°] with the dark-blue envelope showing the extent of possible solutions. The red square is the direction of the incoming DART trajectory [128°,+18°] and the green triangle shows Dimorphos's velocity vector [134°,+5°]. The LICIACube swath is defined for the +175 s image shown in Extended Data Fig. 3. Extended Data Fig. 3 | Two LICIACube LUKE images showing the ejecta morphology that were used to reduce the possible axis orientation solutions. The left panel shows an approach observation, 156 s after impact, with the ejecta in front of and partially obscuring Dimorphos. The right panel shows the ejecta morphology after close approach, 175 s after impact, Dimorphos silhouetted against the ejecta cone. The images show the red channel from frames LICIACUBE_LUKE_L2_1664234220_00005_01 and LICIACUBE_LUKE_L2_1664234239_01003_01. The bright object in the upper corner of each image is Didymos.

Acknowledgements We thank S. Marchi, K. T. Ramesh, P. Sanchez, S. Schwartz, J. Steckloff, M. B. Syal and G. Tancredi for their comments on the manuscript. M.Z., I.G., D.M. and P.T. acknowledge D. Lubey, M. Smith and D. Mages for the useful discussions and suggestions regarding the operational navigation of LICIACube. This work was supported by the DART mission, NASA contract no. 80MSFC20D0004. S.D.R. and M.J. acknowledge support from the Swiss National Science Foundation (project number 200021_207359). S.D.R., M.J., R.L., P.M., N.M. and K.W., acknowledge the funding from the European Union's Horizon 2020 research and innovation programme, grant agreement no. 870377 (project no. NEO-MAPP). P.M. acknowledges support from CNES, ESA and the CNRS through the MITI interdisciplinary programmes. N.M. acknowledges support from CNES. E.D., V.D.C., E.M.E., A.R., I.G., P.J.D.D., P.H.H., I.B., A.Z., S.L.I., J.R.B., G.P., A.L., M.P., G.Z., M.A., A.C., G.C., M.D.O., S.I., G.I., M.L., D.M., P.P., D.P., S.P., P.T. and M.Z. acknowledge financial support from Agenzia Spaziale Italiana (ASI, contract no. 2019-31-HH.0). S.E. acknowledges support through NASA grant number 80NSSC22K1173. S.C. and J.D.W. acknowledge support from the Southwest Research Institute's internal research programme. G.S.C. and T.M.D. acknowledges support from the UK Science and Technology Facilities Council grant no. ST/S000615/1. F.F. acknowledges funding from the Swiss National Science Foundation (SNSF) Ambizione grant no. 193346. J.-Y.L. acknowledges the support provided by NASA through grant no. HST-GO-16674 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract no. NAS 5-26555, and the support from NASA DART Participating Scientist Program, grant no. 80NSSC21K1131. R.N. acknowledges support from NASA/FINESST (grant no. NNH20ZDA001N). J.M.T.-R. acknowledges financial support from project no. PID2021-128062NB-I00 funded by MCIN/AEI (Spain). Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the US Department of Energy, National Nuclear Security Administration under contract nos. DE-AC5207NA27344 and LLNL-JRNL-84276. Simulations were performed on the YORP and ASTRA clusters administered by the Centre for Theory and Computation, part of the Department of Astronomy at the University of Maryland. M.Z., I.G., D.M. and P.T. wish to acknowledge Caltech and the Jet Propulsion Laboratory for granting the University of Bologna a license for an executable version of MONTE Project Edition S/W.

Data availability

The dynamical simulations were carried out using GUBAS, which is publicly available on Github (https://github.com/meyeralexj/gubas). The Dimorphos orbital velocity direction vector components presented in Extended Data Table 1 were computed using the dimorphos_s501. bsp and sb-65803-198.bsp (Didymos) data files available at https://dart. jhuapl.edu/SPICE_kernels/spk. The DART incident velocity vector components presented in Extended Data Table 1 were computed using those two files in combination with the DART_2022_269_1241_ops_v01_impact. bsp data file, available at the same URL. Data availability at that URL is planned until summer 2023, after which those data may be found at https://naif.jpl.nasa.gov/naif/data.html.

Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Author contributions

Competing interests

The authors declare no competing interests.

Additional information Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41586-023-05878-z. Correspondence and requests for materials should be addressed to Andrew F. Cheng. Peer review information Nature thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Reprints and permissions information is available at http://www.nature.com/reprints. Extended Data Fig. 1 | Outline of algorithm used to calculate β. The Monte Carlo variables are outlined by dashed lines and are defined as follows: preimpact orbit period P pre ; pre-impact orbit semimajor axis a pre ; Didymos ellipsoid extents A x , A y , A z ; Dimorphos ellipsoid extents B x , B y , B z ; Dimorphos density ρ B ; post-impact orbit period P post ; and net ejecta momentum direction E ˆ. First, a secant algorithm iterates Didymos density ρ A to match P pre . Next, another secant algorithm iterates the along-track change in Dimorphos's velocity Δv T to match P post . Finally, M is calculated using the ellipsoid extents and density of Dimorphos, and then combined with Δv T and E ˆ to calculate β.

Extended Data Table 2 | Covariance matrix

The covariances used to sample semimajor axis, a pre , and pre-and post-impact orbit periods. The orbital solution from refs. 2,36 fits the pre-impact mean motion at the impact epoch, . The covariance matrix is constructed using units of km for a pre and rad/s for n pre and Δn.


Jian-Yang Li

Masatoshi Hirabayashi

Tony L Farnham

Jessica M Sunshine

Matthew M Knight

Gonzalo Tancredi

Fernando Moreno

Brian Murphy

Cyrielle Opitom

Steve Chesley

Daniel J Scheeres

Cristina A Thomas

Eugene G Fahnestock

Andrew F Cheng

Linda Dressel

Carolyn M Ernst

Fabio Ferrari

Alan Fitzsimmons

Simone Ieva

Stavro L Ivanovski

Theodore Kareta

Ludmilla Kolokolova

Tim Lister

Sabina D Raducan

Andrew S Rivkin

Alessandro Rossi

Stefania Soldini

Angela M Stickle

Alison Vick

Jean-Baptiste Vincent

Harold A Weaver

Stefano Bagnulo

Michele T Bannister

Saverio Cambioni

Adriano Campo Bagatin

Nancy L Chabot

Gabriele Cremonese

R Terik Daly

Elisabetta Dotto

David A Glenar

Mikael Granvik

Pedro H Hasselmann

Isabel Herreros

Seth Jacobson

Martin Jutzi

Tomas Kohout

La Fiorangela

Forgia

Monica Lazzarin

Zhong-Yi Lin

Ramin Lolachi

Alice Lucchetti

Rahil Makadia

Elena Mazzotta Epifani

Patrick Michel

Alessandra Migliorini

Nicholas A Moskovitz

Jens Ormö

Maurizio Pajola

Paul Sánchez

Stephen R Schwartz

Colin Snodgrass

Jordan Steckloff

Timothy J Stubbs

Josep M Trigo-Rodríguez

Ejecta from the DART-produced active asteroid Dimorphos

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Some active asteroids have been proposed to be formed as a result of impact events [START_REF] Jewitt | The asteroid-comet continuum[END_REF] . Because active asteroids are generally discovered by chance only after their tails have fully formed, the process of how impact ejecta evolve into a tail has, to our knowledge, not been directly observed. The Double Asteroid Redirection Test (DART) mission of NASA [START_REF] Daly | Successful kinetic impact into an asteroid for planetary defence[END_REF] , in addition to having successfully changed the orbital period of Dimorphos [START_REF] Thomas | Orbital period change of Dimorphos due to the DART kinetic impact[END_REF] , demonstrated the activation process of an asteroid resulting from an impact under precisely known conditions. Here we report the observations of the DART impact ejecta with the Hubble Space Telescope from impact time T + 15 min to T + 18.5 days at spatial resolutions of around 2.1 km per pixel. Our observations reveal the complex evolution of the ejecta, which are first dominated by the gravitational interaction between the Didymos binary system and the ejected dust and subsequently by solar radiation pressure. The lowest-speed ejecta dispersed through a sustained tail that had a consistent morphology with previously observed asteroid tails thought to be produced by an impact 4,[START_REF] Snodgrass | A collision in 2009 as the origin of the debris trail of asteroid P/2010 A2[END_REF] . The evolution of the ejecta after the controlled impact experiment of DART thus provides a framework for understanding the fundamental mechanisms that act on asteroids disrupted by a natural impact [START_REF] Jewitt | The asteroid-comet continuum[END_REF][START_REF] Tancredi | Lofting of low speed ejecta produced in the DART experiment and production of a dust cloud[END_REF] .

The Hubble Space Telescope (HST) observed the ejecta once every 1.6 h during the first 8 h after the DART impact (Extended Data Table 1) at the viewing geometry shown in Fig. 1. The image collected at about T + 0.4 h (Fig. 2a) shows diffuse ejecta with several linear structures and clumps (concentration of materials ejected at similar velocities) spanning nearly the entire eastern hemisphere of Didymos. After about T + 2 h, the initial, diffuse dust cloud had mostly dissipated and an overall cone-shaped ejecta morphology emerged with the edges of the hollow cone shown as two linear features (l7 and l8) because of the optical depth effect. The ejecta cone showed many distinct morphological features (Fig. 2b-f), some of which are visible in several images between T + 3 and T + 10 h and extending to nearly 500 km from the asteroid. These features moved radially away from the asteroid at constant speeds of a few to about 30 m s -1 as projected in the sky (Extended Data Table 2). The radial expansion motion of these features suggests that this material is directly ejected from the Didymos system without being appreciably influenced by the gravity of the system or by solar radiation pressure. On the basis of the position angles (angle measured from the north towards the east) of the cone and a simple model (Methods), we find that the observed ejecta cone is consistent with a three-dimensional opening angle of 125º ± 10º and centreline at a position angle of 67º ± 8º which is almost parallel to the incoming direction of the DART spacecraft. The observed ejecta cone is wider than the ejecta produced by the vertical impact cratering experiments on granular media [START_REF] Cintala | Ejection-velocity distributions from impacts into coarsegrained sand[END_REF][START_REF] Housen | Ejecta from impact craters[END_REF] , although wider opening angles could be explained by the curvature of the target surface [START_REF] Schultz | Impacts of hemispherical granular targets: Implications for global impacts[END_REF] and the angle of internal friction of the target [START_REF] Raducan | Ejecta distribution and momentum transfer from oblique impacts on asteroid surfaces[END_REF] as well as the geometry of the projectile [START_REF] Raducan | Influence of the projectile geometry on the momentum transfer from a kinetic impactor and implications for the DART mission[END_REF] .

The ejecta of Dimorphos were different from the ejecta of comet 9P/Tempel 1 produced by Deep Impact [START_REF] A'hearn | Deep Impact: excavating comet Tempel 1[END_REF] , a previous planetary impact experiment of comparable scale (Extended Data Fig. 1a-c). Both experiments delivered similar momentum to their targets. The Deep Impact spacecraft carried 80% more kinetic energy than the DART spacecraft but the 6-km diameter nucleus of Tempel 1 (ref. 12) was considerably more massive than the 151-m diameter Dimorphos 2 . At the scale of the HST, the Deep Impact ejecta were diffuse and mostly featureless, The symbol 'x' marks artifacts due to, for example, residual cosmic rays, frame boundaries, background objects and defective pixels. The times correspond to the mid-observation time of each image. Black lines mark diffraction spikes. All images are displayed with the same logarithmic brightness scale. Sky north is in the upward direction and the east is to the left. The yellow arrows point to the direction of the Sun, the cyan arrows the heliocentric velocity direction of Didymos and the red arrows the direction of the DART spacecraft at impact, all projected in the sky plane at the time of observation. The HST had a pointing drift during the exposures of some images, causing a smear of about 4-7 pixels in the first four images (before T + 5.0 h) and about 14 pixels in the T + 6.6 h image, all along the northeast-southwest direction (Methods). The drift widens the tail and the two diffraction spikes orthogonal to the direction of the drift. Most features are much larger than the length of the drift; we added uncertainties to account for the effect of this drift in our measurements. Many features are visible, including linear features (l1-l12), an arc (arc 1), a circular feature (c1), blobs (b1-b3) and a tail. The ejecta cone is marked by linear features l7 and l8. Scale bars are 200 km at the distance of Didymos.

expanding at an average speed of about 100 m s -1 and a maximum speed of about 300 m s -1 (refs. 13,14). This difference in ejecta morphology is probably because of the different target compositions and subsurface structures. Whereas Tempel 1 has a highly porous subsurface [START_REF] Richardson | A ballistics analysis of the Deep Impact ejecta plume: determining Comet Tempel 1's gravity, mass, and density[END_REF] composed of fine-grained dust and is rich in volatiles [START_REF] Sunshine | The distribution of water ice in the interior of Comet Tempel 1[END_REF]17 , the bouldery surface and potential rubble-pile interior of Dimorphos 2 could perturb the ejecta curtain and produce inhomogeneous structures in the ejecta [START_REF] Ormö | Boulder exhumation and segregation by impacts on rubble-pile asteroids[END_REF][START_REF] Kadono | Pattern of impact-induced ejecta from granular targets with large inclusions[END_REF] .

From approximately T + 0.7 days to T + 2.1 days, the ejecta features composed of slower dust escaping at less than around 1 m s -1 emerged from the base of the ejecta cone (Fig. 3a-d). The ejecta during this stage were characterized by curved ejecta streams in the north (s1) and south (s2), some small curvilinear features (l16-l19) between them and a slight wrapping of these features around Didymos. The gravity of Didymos, which accounted for 88% of the gravitational potential of the binary system at the impact site, slowly distorted the shape of the original ejecta cone and created different morphologies for s1 and s2. The dust ejected from the original northern cone edge (l7) was in close proximity to Didymos (Fig. 1). As suggested by numerical simulation predictions [START_REF] Ferrari | Ejecta formation, early collisional processes, and dynamical evolution after the DART impact on Dimorphos[END_REF][START_REF] Yu | Ejecta cloud from the AIDA space project kinetic impact on the secondary of a binary asteroid: I. Mechanical environment and dynamical model[END_REF] , this dust was accelerated by Didymos and the trajectories were bent before escaping the binary system, forming the northern curved stream s1 (Extended Data Fig. 2). The end of s1 near the asteroid contains relatively slow particles, the trajectories of which were bent more than those of the relatively fast particles farther away, causing the near end to shift clockwise about Didymos, resulting in an 18º twist of s1. By contrast, most of the dust in the original southern cone edge (l8; Fig. 2) was launched away from Didymos. Thus, these trajectories are less affected by the gravity of Didymos, leading to a less curved southern stream (s2) with its near end slowly wrapping around the asteroid over time (Fig. 3a-f). The small curvilinear features between the two streams (l16-l19) were probably composed of dust ejected from the front or back of the hollow ejecta cone, behaving more or less similar to either of the two curved streams and slightly rotating from the original radial direction.

Beyond the gravitational influence of the Didymos system, solar radiation pressure naturally separates particles of different sizes along the sunward-antisunward direction because small particles are accelerated faster than large particles [START_REF] Burns | Radiation forces on small particles in the solar system[END_REF] . The northern stream (s1), situated roughly orthogonal to the sunward direction, increasingly widened to form the observed wing-like shape, with a diffuse antisunward edge and a relatively sharp sunward edge (Fig. 3f-j). This sharp edge indicates a cut-off in the largest particle size of the ejecta. Because the southern stream was nearly aligned towards the Sun, those particles were first slowed by solar radiation pressure before eventually being turned towards the antisunward direction. Starting from T + 4.7 days, the particles moving at different speeds and directions in s2 because of the inhomogeneous distributions of dust in the ejecta were separated into individual features (l20-l24; Fig. 3f). These particles reached maximum projected sunward distances of up to 150-200 km. All of these individual features (l20-l24) and the small curvilinear features (l16-l18) between the two main streams were stretched along the sunward-antisunward direction over time by solar radiation pressure (Fig. 3f-i). The finer particles in features l16-l18, which were located to the north of Didymos, were pushed farther away from Dimorphos and caught up to the larger particles ejected into s1 earlier, appearing to overlap with the wing-like structure and creating a more complex pattern (Fig. 3g,h).

As a result of solar radiation pressure, a dust tail started to emerge antisunward nearly opposite to the ejecta cone at about T + 3 h. This tail quickly stretched out to a projected length of more than 1,500 km and exceeded the spatial coverage of our images (Fig. 4). Around T + 5.7 days, the narrow tail showed a relatively bright and sharp southern edge and a parallel but more diffuse northern edge (Fig. 4h). The overall morphology of the tail of Dimorphos is similar to that of P/2010 A2, an active asteroid probably triggered by an impact 4,[START_REF] Kim | Anisotropic ejection from active asteroid P/2010 A2: an implication of impact shattering on an asteroid[END_REF][START_REF] Hainaut | A2 LINEAR. I. An impact in the asteroid main belt[END_REF] (Extended Data Fig. 1d,e). The width of the tail, which is approximately 1 arcsecond, is consistent with an initial speed of the dust comparable with the orbital speed of Dimorphos, suggesting that the tail contains the slowest ejecta particles. Moreover, the early tail within T + 2 days slightly curved towards the south (Fig. 4d,e), whereas after T + 8 days the tail became slightly more fan-shaped (Fig. 4i-k). With radiation pressure sorting the particle size along the tail, the earliest tail at around T + 3 h was dominated by micrometre-sized particles, whereas centimetre-sized particles dominated the portion of the tail inside the HST field of view in the final image. The brightness profile of the tail is related to the particle-size distribution of the ejecta. Assuming a power law for the differential size distribution, we derived an exponent of -2.7 ± 0.2 for particles with radii between 1 µm and a few millimetres, and an exponent of -3.7 ± 0.2 for larger particles up to a few centimetres in radius (Extended Data Fig. 3). Ejecta particles were observed to continuously leave the Didymos system through the final images acquired after T + 15 days (Extended Data Figs. 4 and5).

Furthermore, a secondary tail appeared between T + 5.7 days and T + 8.8 days (Fig. 4i-k) but was no longer discernible on day T + 18.5 (Fig. 4l). It originated from the Didymos system and pointed about 4º north of the original tail, creating an overall fan-shaped tail morphology during this timeframe. The cause of the secondary tail is unclear, and several mechanisms will be explored (Methods and Extended Data Figs. 4 and6), although the morphologies are consistent with the previous observations of active asteroids with multiple tails [START_REF] Jewitt | The extraordinary multi-tailed main-belt comet P/2013 P5[END_REF][START_REF] Jewitt | Episodic ejection from active asteroid 311P/PANSTARRS[END_REF][START_REF] Kleyna | The sporadic activity of (6478) Gault: A YOPR-driven event?[END_REF][START_REF] Jewitt | Episodically active asteroid 6478 Gault[END_REF] . The whole evolutionary sequence of the ejecta of Dimorphos discussed above is shown in Supplementary Video 1. The DART mission clearly demonstrated that impacts can activate asteroids, consistent with previous asteroid observations [START_REF] Jewitt | The asteroid-comet continuum[END_REF] . Our observations provide a basis for reassessing the previous observations of active asteroids thought to be triggered by an impact. The evolution of the ejecta of Dimorphos suggests that the observed particle size in active asteroid tails could depend on the age of the tail, consistent with the range of particle sizes measured in the tails of active asteroid 311P/PanSTARRS [START_REF] Jewitt | Episodic ejection from active asteroid 311P/PANSTARRS[END_REF] . The lack of sub-millimetre-sized dust in the tail of P/2010 A2, therefore, could be a result of the observations occurring 10 months after impact 4,[START_REF] Snodgrass | A collision in 2009 as the origin of the debris trail of asteroid P/2010 A2[END_REF][START_REF] Hainaut | A2 LINEAR. I. An impact in the asteroid main belt[END_REF] . DART, which is a controlled, planetary-scale impact experiment, provides a detailed characterization of the target, the ejecta morphology and the entire ejecta evolution process. DART will continue to be a model for studies of newly discovered asteroids that show activity caused by natural impacts.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-023-05811-4.

Article

Methods

Observations and data reduction and processing

We used a total of 19 HST orbits (period 95 min) over about 19 days to observe the Dimorphos ejecta (Extended Data Table 1). The first orbit (orbit 0o) was before the DART impact. The second orbit to the seventh orbit (orbits 01-06) started about T + 15 min and continuously observed the ejecta except when Earth occluded the view of the target. In the next five orbits (orbits 11-15), we observed the ejecta roughly once every 12 h, and then once every day in the subsequent three orbits (orbits 16-18). In the final phase (orbits 21-24), observations were made once every 3 days. The observations concluded 18.5 days after impact. In each orbit, images were collected at several exposure levels, in which the central core of Didymos was unsaturated in short exposures and long exposures saturated Didymos to image the relatively faint ejecta and tail. All images were collected through filter F350LP (pivot wavelength 587 nm, bandwidth 149 nm) [START_REF] Calamida | New photometric calibration of the Wide Field Camera 3 detectors[END_REF] .

The observations were planned to track at the Dimorphos ephemeris rate. The tracking nominally included corrections for parallax because of the orbit of the HST around Earth and was expected to keep Didymos inside the field of view with minimal drift in the field of view for all exposures. However, owing to an as-yet unexplained tracking problem, some orbits lost the target in various numbers of exposures, and some long exposures included a pointing drift of more than ten pixels. We limited our analysis to those exposures with fewer than seven pixels of drift and occasionally used long exposures with more drift when no good images were available for the particular orbits.

Images were calibrated by the HST standard calibration pipeline at the Space Telescope Science Institute 31 . We then removed the sky background measured from a square 100-400 pixels wide and 100-300 pixels from the top right corner, depending on the image size. This area is in general 20 arcseconds away from Didymos and shows no sign of any ejecta.

Aperture photometry was measured in all short, unsaturated exposures that have been corrected for charge transfer efficiency 31 but not geometric distortion (.flc files, available from the HST data archive; see 'Data availability'). The centroid was defined by a two-dimensional Gaussian fit with a 5 × 5 pixel box centred at the photocentre. The pixel area map was used to correct pixel area variations in the image 31 . The total counts were measured with circular apertures of a radius of 1-130 pixels (0.04-5.2 arcseconds). We converted the total counts to flux density and Vega magnitude based on the photometric calibration constants (PHOTFLAM = 5.3469 × 10 -20 erg Å -1 cm -2 per electron, PHOTZPT = 26.78) provided in the image headers and the HST photometric calibration website. The total brightness of Didymos including the ejecta and the total brightness of ejecta are shown in Extended Data Fig. 4.

We used the images corrected for charge transfer efficiency and geometric distortion (.drc files) to study the morphology of the ejecta. To increase the signal-to-noise ratio of the faint ejecta features, we stacked all long exposures in each orbit because no change is visible in the ejecta morphology with each orbit. The centroid of long exposures that are saturated in the centre was determined by the cross-section of the diffraction spikes. Some long exposures with pointing-induced drift were included in the stack, but those with more than ten pixels of drift were discarded. The effects of this drift are accounted for as extra positional uncertainties to the measurements of features, which are mostly larger than the length of the drift. Cosmic rays and background stars were removed in the stacking process. Because different numbers of good long exposures were available for each orbit, the total exposure times varied from 25 to 50 s in most stacked long exposures and reached 155 s for the orbit 21 stack and 110 s for the orbit 23 stack.

Various image-enhancement techniques commonly used for studies of comets [START_REF] Samarasinha | Image enhancement techniques for quantitative investigations of morphological features in cometary comae: a comparative study[END_REF] were used to assist the identification of ejecta features, including azimuthal median subtraction and division, azimuthal and radial reprojection, different brightness stretching and the use of various colour tables. All identified features were confirmed by several techniques.

The speeds of features as projected in the image plane were estimated by assuming that all features originated from the asteroid at the time of impact and moved directly away from the asteroid. The projected distance of a feature from the asteroid and the corresponding observation time yielded the projected speed of the feature. Note that the speeds estimated this way do not represent the true terminal speeds of the features after escaping the binary system for slow ejecta (< about 1 m s -1 ) or for features affected by solar radiation pressure. The trajectories of the features in those cases are notably affected by the gravity of Didymos (Extended Data Fig. 2) or solar radiation pressure.

Ejecta cone opening angle and direction

We based our ejecta cone characteristics on the ejecta structures moving at more than 1 m s -1 in the images within T + 8.2 h (Fig. 2). These structures showed a linear motion moving away from the asteroid along the radial direction from the binary asteroid (Extended Data Table 2). Assuming that most of the ejecta dust is within a thin cone-shaped curtain, the two edges of the cone would appear as two bright rays along the radial direction from Dimorphos because of the optical depth effect when viewed from the side. Because the DART impact velocity is close to the sky plane (Extended Data Table 1), if we assume that the cone direction is close to the inverse of the DART impact velocity direction, the cone is close to being viewed from the side in the HST images and the opening angle spanned by the two edges of the cone (linear features l7 and l8) is close to its three-dimensional opening angle. This is confirmed by our derived cone geometry described in the following.

We measured the position angles of the two edges of the ejecta cone from both the original and the enhanced images (see 'Observations and data reduction and processing'). The uncertainty range of the position angles is defined by the apparent width of the linear feature. Our measurement resulted in an ejecta cone centred within 5º of the incoming direction of DART with an opening angle of about 130º. Because of the fuzziness of the ejecta rays and their slight curvature, the uncertainty of the measured position angles could be as high as ±8º, resulting in an uncertainty of the opening angles up to ±12º. Taking the mean of these two edges and the maximum value of the uncertainty yields the ejecta cone axis at a position angle of 67° ± 8° under the assumption that the ejecta cone is axisymmetric about the cone axis.

To further constrain the ejecta cone geometry, we constructed a three-dimensional numerical cone model parameterized by the direction of the cone axis in right ascension (RA) and declination (dec), as well as an opening angle, to compare with the images. We first projected the six early post-impact images (Fig. 2) in an azimuthal-radial projection and, for each image, generated a histogram of pixels brighter than 18 mag arcsec -2 along the azimuthal direction. The azimuthal bins with the highest pixel counts (except for those of the tail and diffraction spikes) define the two cone edges with approximately a Gaussian distribution. The mean and the 1σ uncertainty of the position angles of the two cone edges are derived from the histograms. On average, the northern and southern cone edges are at position angles of 4º ± 8º and 131º ± 8º, respectively, consistent with the measurements described above. We then generated simulated images from the model ejecta cone and computed the corresponding histogram following the same approach for the actual images. This histogram was compared with the measured cone edge position angles to calculate a score, defined as

∑ ∑ f s σ π x μ σ = 2 exp - 1 2 ( -) i j n j i j i i =1 2 =1 2 2        
where σ i and μ i are the standard deviation and mean of the northern or southern edges (i = 1, 2), respectively, x j is the position angle of the histogram bin j for the simulated image and s j is the pixel count in bin j. We searched the cone axis in the full range of RA and dec and the opening angle in 100º-160º for the highest score. Because the HST images alone could not determine whether the cone faced towards or away from Earth, this approach resulted in a pair of best-fit cone-axis solutions that were symmetric with respect to the image plane. We thus considered both as feasible cone-axis directions. The uncertainties of the solutions were estimated with 500 random samples of the measured cone edge position angles distributed in two Gaussians with the measured means and standard deviations. The best-fit cone-axis directions were (RA, dec) = (141º ± 8º, 25º ± 6º) and (120º ± 9º, 10º ± 7º), both with an opening angle of 125º ± 10º (1σ uncertainties). Both solutions are about 12º from the image plane, with the former pointing towards Earth and the latter pointing away.

Dynamic model of the tail

The position angle of the tail and its uncertainty were determined by the radial directions from the asteroid that define the visible boundary of the tail at the furthest point along the tail in all (short and long exposures) stacked images that contain the tail. The dust dynamics model under the influence of solar radiation pressure follows a previous model [START_REF] Burns | Radiation forces on small particles in the solar system[END_REF] , in which the motion of dust is determined by β srp . β srp , which is defined as the ratio of the solar radiation pressure force to the solar gravitational force, depends on particle radius, r, and density, ρ, as

β KQ ρr = / srp pr
where K = 5.7 × 10 -4 kg m -2 is a constant, Q pr is the radiation pressure coefficient averaged over the solar spectrum, which is usually assumed to be 1. We assumed a grain density of 3.5 × 10 3 kg m -3 for the dust in the ejecta, following the density of ordinary chondrite meteorites [START_REF] Britt | Stony meteorite porosities and densities: a review of the data through 2001[END_REF] , considering that the Didymos-Dimorphos system shows an S-type spectrum that is associated with LL ordinary chondrite material [START_REF] Dunn | Mineralogies and source regions of near-Earth asteroids[END_REF] . Pre-impact modelling suggested that the acceleration of solar radiation pressure always exceeds that of the gravitational acceleration of the Didymos system for ejecta particles smaller than 100 µm in size [START_REF] Ferrari | Ejecta formation, early collisional processes, and dynamical evolution after the DART impact on Dimorphos[END_REF][START_REF] Finson | A theory of dust comets. I. Model and equations[END_REF] . These small particles are pushed out of the binary system in less than 10 h. The gravity of Didymos is predominant within about 3 km for millimetre-sized particles and 10 km for centimetre-sized particles.

The modelling of the orientation of the tail in the sky plane follows the synchrone-syndyne approach [START_REF] Rossi | Dynamical evolution of ejecta from the DART impact on Dimorphos[END_REF] , in which synchrones are the loci of dust particles ejected with zero initial velocity at the same time but with various β srp . The measured position angles of the tail of Dimorphos coincide to within 4º of the direction suggested by the synchrones associated with the time of impact in all images, suggesting that solar radiation pressure dominates the tail formation (Extended Data Fig. 7). The small discrepancy between T + 1 days and T + 5 days is probably because of the slight apparent curvature of the tail (Fig. 4e-h), which may be related to the non-zero mean initial velocity of dust particles with respect to the binary system, inherited from the orbital speed of Dimorphos.

The non-zero initial velocity of ejecta dust causes the tail to widen. The average initial velocity of the ejecta of Dimorphos, as projected in the image plane, has a northward component, which causes the tail to widen towards the north with respect to the loci of the hypothetical zero-velocity particle (synchrone). The relatively sharp southern edge and the more diffuse northern edge are consistent with the expectation from the ejecta mass-speed relationship [START_REF] Fahnestock | Pre-encounter predictions of DART impact ejecta behavior and observability[END_REF] because the number of dust particles decreases with increasing ejection speeds. The 1-arcsecond width of the tail is consistent with an initial velocity dispersion Δv = 0.15 m s -1 , comparable with the orbital speed of Dimorphos, suggesting that the tail is primarily composed of the slowest ejecta.

The inverse proportionality of β srp with particle size means that small particles experience stronger solar radiation pressure and are pushed away from the asteroid faster after ejection than large particles. Because the duration of our HST observations is much shorter than the orbital period of Didymos around the Sun (2.1 years), the motion of particles along the tail relative to the asteroid under solar radiation pressure can be approximated by a constant acceleration motion. As the length of the tail grows, particles of various sizes spread out along the tail, with the smallest particles remaining near the far end of the tail from the asteroid and the larger particles dominating the end near the asteroid. Assuming a power-law differential particle-size distribution with an exponent of α for the tail, we derived that the brightness of the tail is expected to have a power-law relationship with the distance to the asteroid with an exponent b = -4 - α.

We extracted the brightness profiles of the tail from stacked long exposures from T + 5 h until the last stack at T + 18.5 days (Extended Data Fig. 3). The exponent α of the differential particle-size distribution was derived from the linear part of the tail-brightness profiles (in log-log space) in various images, corresponding to a range of β srp from 0.2 × 10 -4 to 8 × 10 -4 , to be between -2.2 and -3.1, with an average of -2.7 and a standard deviation of 0.2. The range of β srp indicates that particle sizes range from 1 µm to a few millimetres. In images after about T + 6 days, the tail brightness displays two regions with different power-law slopes. The inner region appears to be influenced by the particles in the curved ejecta streams that started to overlap with the tail. The outer region has best-fit slopes close to -2.7 as in the early images, whereas the slope of the inner region ranges from -3.6 to -3.9. The range of β srp for the inner region is 7 × 10 -4 to 1 × 10 -5 , corresponding to millimetre-to centimetre-sized particles. The lack of small particles in the curved streams is expected because particles with a size of 100 µm or smaller should have been removed a few hours after the impact. The apparent increasing steepness of the particle-size distribution in this size range also seems to indicate that the bulk of ejecta particles have a size cut-off of a few centimetres. If the particle-size distribution of the tail represents that of all ejecta, then a power-law index of -2.7 means that the total ejecta mass is dominated by the largest particles.

The above treatment assumes that the albedo is independent of particle size, which needs to be examined. On the basis of laboratory measurements of the phase function of micromtre-sized aerosols [START_REF] Volten | Scattering matrices of mineral aerosol particles at 441[END_REF] and millimetre-sized particles [START_REF] Muñoz | Experimental phase functions of millimeter-sized cosmic dust grains[END_REF] , along with supporting models of scattering efficiency [START_REF] Nousiainen | Can particle shape information be retrieved from light-scattering observations using spheroidal model particles?[END_REF] , the albedo of micrometre-sized particles is about 70% that of millimetre-size-grains at the phase angle of our early observations (54º). This brightness ratio is reversed at the phase angle corresponding to the final images (74º), in which micrometre-sized particles become about 16% brighter. Our calculation indicates that the small difference between the albedos of micrometre-and millimetre-sized particles changes the best-fit power-law index of the particle-size distribution by less than 2%. Our assumption of the same albedo for micrometre-to centimetre-sized particles holds.

Secondary tail

The small decrease in the overall fading rate of the total brightness of the Didymos-Dimorphos system between about T + 5 days and T + 7 days indicates an increase in the total scattering cross-section in the ejecta within 10 km of the system (Extended Data Fig. 4), partly compensating for the ejecta moving out of the photometric aperture. It is unlikely to be caused by a change in albedo for the ejecta particles. Injection of new dust particles into the ejecta was considered.

This scenario and its timing are also supported by the synchrone model (Extended Data Fig. 6), in which the projected direction of the secondary tail is consistent with the synchrones associated with about T + 5.0 days to T + 7.1 days. The narrow width of the secondary tail similar to that of the original tail suggests a low initial velocity of about 0.15 m s -1 for the dust particles. Although the Didymos binary environment could complicate the dust motion and cause deviation from the zero initial velocity assumption of the idealized synchrone model, the observed low initial velocity of the dust in the secondary tail implies limited effects.

The possible mechanisms of the secondary dust emission could include the re-impact of ejecta blocks onto Dimorphos or Didymos [START_REF] Finson | A theory of dust comets. I. Model and equations[END_REF] or large ejecta blocks disintegrating into small pieces because of spin-up or mutual collisions (S.L.I. et al., manuscript in preparation). Mass shedding from the surface of Dimorphos because of rotation is not likely given its slow rotation if its spin is tidally locked. However, mass movement and shedding from Didymos could potentially be triggered by ejecta re-impact because its fast rotation causes a net outward acceleration at its equator, although no clear indication of this has been confirmed yet [START_REF] Thomas | Orbital period change of Dimorphos due to the DART kinetic impact[END_REF] . Once the dust is lifted from the surface of Dimorphos or Didymos using these mechanisms, solar radiation pressure will quickly sweep the dust into the antisunward direction, forming a secondary tail.

Other mechanisms, such as the dynamic interaction between the slow ejecta dust and the binary system 41 , gravitational scattering for the ejecta dust when they are turned back by solar radiation pressure and pass the binary system, or photon-charged dust particles under the influence of interplanetary magnetic field [START_REF] Hartzell | Dynamics of 2D electrostatic dust levitation at asteroids[END_REF] could also result in the unusual tail morphology that leads to the appearance of a secondary tail. Our dynamics simulations suggested that a secondary dust emission is not necessary to form a secondary tail that has a morphology consistent with the one observed. However, these scenarios may not be accompanied by the increase in ejecta dust as suggested by the fading lightcurve of the Didymos system.

(Fig. 1 |Fig. 2 |

 12 Fig. 1 | Geometry of the Didymos system at the time of impact as viewed from the HST. Sky north is in the upward direction and the east is on the left in this view. The equivalent diameters of Didymos (large spheroid) and Dimorphos (small spheroid) are 761 m and 151 m, respectively 2 . The orbit of Dimorphos around Didymos before the impact, depicted by the black circle, has a semimajor axis of 1.206 ± 0.035 km 3 and an eccentricity of <0.03 (ref. 29). The sizes of Didymos and Dimorphos and their separation in the figure are to scale.The entire system lies within one pixel in the HST images. Dimorphos orbits Didymos clockwise with a speed of approximately 0.17 m s -1 . The positive pole of Didymos (also the orbital pole of the system) is represented by the blue line, pointing close to the south celestial pole and 51º out of the sky plane away from Earth. The Sun is at a position angle of 118º, represented by the orange line and the dot-circle symbol. The DART spacecraft vector is represented by the red line, with arrows, going from east to west at a position angle of 68º and within 1º of the sky plane.

Fig. 3 |

 3 Fig. 3 | Evolution of ejecta from T + 0.7 days (T + 17.8 h) to T + 18.5 days. The inset, image orientation, brightness stretch, scale bars and vector arrows are all the same as in Fig.2. The symbol 'x' marks imaging artifacts. The main characteristics of the ejecta during this time period include the curved ejecta streams (s1 and s2), linear features (l7, l11-l24), blobs (b3-b5), a circular feature (c1) and an arc (arc 2). a-g, The original north edge of the ejecta cone (l7) is still visible in images before T + 5.7 days. a-e, The early southern curved stream (s2) could be overlapped with the southern edge of the original ejecta cone (l8), which is not separately marked. g-k, The northern curved stream (s1) widened along the tail direction at about T + 5 days, forming a wing-like feature. b-f, A group of linear features (l16-l24), some of which are part of the southern curved stream (l21-l24), showed a clockwise rotation around Didymos from T + 1.1 days to T + 4.7 days. g-i, These linear features later (T + 5.7 days) stretched along the tail direction under solar radiation pressure, with those in the north of Didymos overlapping with the wing-shaped feature. h-j, A secondary tail is visible between T + 8.8 days and T + 14.9 days (also see Fig.4). The curved edge of the wing-like feature is visible in k. The question marks after the annotations of l23 & l24 in h and l22 in i mark the relatively uncertain identification of these features due to their faintness and the large changes in their positions and orientations from the previous images in the sequence.

Fig. 4 |

 4 Fig. 4 | Tail formation from the Dimorphos ejecta cloud. a-l, All frames are rotated such that the expected direction of the tail based on our dust dynamic model (Methods) is in the horizontal direction extending towards the right. All frames are displayed with the same logarithmic brightness scale. The regions outside the field of view are marked by a dark blue colour. The symbol 'x' marks imaging artifacts. The scale bars are aligned with the asteroid at one end and extend 200 km towards the tail direction. a-c, Note that the first three frames have pointing-induced drift in the plane of the sky of 5-7 pixels approximately along the direction of the vertical diffraction spikes.The drift in all other frames is smaller than two pixels. The first frame (a) in this sequence acquired at T + 0.08 days (T + 1.9 h) shows no signs of a tail. A tail was visible starting from the second frame (b) acquired at T + 0.15 days (T + 3.5 h). The tail continued to grow in a direction that is, in general, consistent with an impulsive emission of dust from Dimorphos at the time of impact. i-k, The secondary tail is visible between T + 8.82 days and T + 14.91 days, pointing at about 4º north of the original tail.

Extended Data Fig. 4 |

 4 Brightness evolutions of Didymos and the ejecta. (a) Total magnitude of Didymos in 10 km, 30 km, and 50 km radius apertures at the distance of Didymos measured from HST images as a function of time after impact. (b) Magnitude of ejecta with respect to time after impact. The black curve in both panels is the magnitude of Didymos based on the IAU HG phase function model with a G = 0.20 43 , scaled to match the observed pre-impact magnitude. The ejecta magnitude corresponds to the difference between the observed total flux and the flux from Didymos. The ejecta is brighter than Didymos for about 2.5 days after impact in the 10 km radius aperture. Extended Data Fig. 5 | Azimuthally averaged radial profiles of Didymos and ejecta. The curves are extracted from the pre-impact image (-0.1 d) and the last three images (+11.9 d, +14.9 d, and +18.5 d). The widened PSF profiles of late images suggest a slightly extended source due to ejecta dust close to the asteroid. 1 pixel corresponds to 0.04" or 2.1-2.3 km at the distance of Didymos in the last three images. Extended Data Fig. 6 | Synchrone analysis of the main tail and the secondary tail. (a) Image taken at T+11.86 days is displayed in logarithmic brightness stretch. North is up and east to the left. The features marked by "x" are artifacts from a background object and a cosmic ray hit. (b) Same image as in (a) but with synchrones corresponding to various dates overlaid. The direction of the main tail is consistent with the synchrone at impact time (T+0.0 days), and the secondary tail is consistent with the synchrones between T+5.0 and T+7.1 days. Extended Data Fig. 7 | The position angles of the tail measured from HST images. The blue circles are measured from the stacked images of the short exposures, and the orange circles are measured from the stacked images from the long exposures. The green triangles are the position angles of the secondary tail. The red dashed line is the antisolar direction, and the blue solid line is the position angle of synchrones for dust emitted at the time of impact. The tail orientation measured from the short exposures could be affected by the secondary tail due to the low signal-to-noise compared to the long exposures.

Acknowledgements

This work was supported by the DART mission, NASA contract no. 80MSFC20D0004 and by the Italian Space Agency (ASI) through the LICIACube project (ASI-INAF agreement AC no. 2019-31-HH.0). Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the NASA. J.-Y.L. acknowledges the support provided by NASA through grant HST-GO-16674 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, under NASA contract NAS 5-26555. L.K. acknowledges support from the NASA DART Participating Scientist Program, Grant no. 80NSSC21K1131. R.L., D.A.G. and T.J.S. acknowledge funding from the NASA/GSFC Internal Scientist Funding Model (ISFM) Exospheres, Ionospheres, Magnetospheres Modeling (EIMM) team, the NASA Solar System Exploration Research Virtual Institute (SSERVI) and NASA award no. 80GSFC21M0002. R.M. acknowledges support from a NASA Space Technology Graduate Research Opportunities (NSTGRO) Award (contract no. 80NSSC22K1173). P.M. acknowledges funding support from the Horizon 2020 research and innovation programme of the European Union under grant agreement no. 870377 (project NEO-MAPP), the CNRS through the MITI interdisciplinary programmes, CNES and ESA. F.F. acknowledges funding from the Swiss National Science Foundation (SNSF) Ambizione grant no. 193346. J.O. has been funded by grant no. PID2021-125883NB-C22 by the Spanish Ministry of Science and Innovation/State Agency of Research MCIN/AEI/10.13039/501100011033 and by the European Regional Development Fund "A way of making Europe". G.T. acknowledge financial support from project FCE-1-2019-1-156451 of the Agencia Nacional de Investigación e Innovación ANII (Uruguay). T. Kohout is supported by the Academy of Finland project 335595 and by institutional support RVO 67985831 from the Institute of Geology of the Czech Academy of Sciences. F.M. acknowledges financial support from grants CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033 and PID2021-123370OB-I00. Research by M.G. is supported, in part, by the Academy of Finland grant 345115. J.M.T.-R. acknowledges financial support from the project PID2021-128062NB-I00 funded by Spanish MCIN/AEI/10.13039/501100011033. We thank J. DePasquale (STScI) for generating the animation included in Supplementary Video 1.

Data availability

All raw HST data associated with this Article are archived and are publicly available at the Mikulski Archive for Space Telescopes (https://mast.stsci. edu/search/ui/#/hst/results?proposal_id=16674) hosted by the Space Telescope Science Institute. The stacked long exposures in Figs. 234are available from a website hosted at JHU/APL (https://lib.jhuapl.edu/ papers/ejecta-from-the-dart-produced-active-asteroid-dimo). Other related data are available from the corresponding author upon request.

Extended Data Table 1 | HST observations of DART impact

This table lists the times and observing geometries for each of the 19 orbits of HST observations. The parameters listed in this table only refer to usable images in each orbit. Some images were lost due to tracking problems.

Extended Data Table 2 | Selected features and their approximate plane of sky speed

The speeds reported here are averaged over all measurements in multiple images for every feature. The scatters in measured speeds are typically < 5% for each feature. The measurement is based on the approximate distance of the feature to the asteroid and the corresponding mid-observation times.

Author contributions J.-Y.L. is the principal investigator of the HST programme (GO-16674), together with co-investigators M.M.K., C.A.T., A.S.R., S. Chesley, L.K., A.F.C., E.G.F., to observe the DART ejecta. J.-Y.L. leads the effort to develop this paper. M.H., T.L.F. and M.M.K. contributed to the ejecta cone measurements and modelling and to the other ejecta evolution study. G.T. contributed to obtaining and analysing photometry data, the analysis of the formation of the tail and the comparison with active asteroids. -Y.L., P.M., R.M., J.O., M.P., C.S., J.S., P.S., S.R.S., J.M.T.-R., A.F., T. Kareta, T. Kohout, A.M., L.K., F.L.F., M.L. and H.A.W. provided comments and improvements to the paper.

Competing interests

The authors declare no competing interests.

Additional information Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41586-023-05811-4.

Correspondence and requests for materials should be addressed to Jian-Yang Li. Peer review information Nature thanks Masateru Ishiguro and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Reprints and permissions information is available at http://www.nature.com/reprints. The slope values from the outer section have β srp higher than 1x10 -4 , and those from the inner section correspond to β srp between 1x10 -4 and 1x10 -5 . The dashed horizontal line is the average -2.7 for the outer sections, and the green shaded area represents the standard deviation.

Article