Momentum transfer from oblique impacts

Introduction:

Earth is continuously impacted by space debris and small asteroids, and, while large asteroid impacts are very rare, they have the potential to cause severe damage. NASA's Double Asteroid Redirection Test (DART) aims to be the first mission to test a controlled deflection of a Near-Earth binary asteroid [1, 2], by impacting the smaller component of the 65803 Didymos asteroid system, Dimorphos. The impact will thereby alter the binary orbit period by an amount detectable from Earth [3].

ESA's Hera mission [3, 4], that will arrive at Dimorphos several years after the DART impact. It will rendezvous with the asteroid system and perform detailed characterisation of Dimorphos's volume and surface properties, as well as measure the DART impact outcome, such as change in the binary system orbit and the volume and morphology of the DART impact crater.

In high velocity impacts on an asteroid the change in momentum of the asteroid ΔP can be amplified by the momentum of crater ejecta that exceeds the escape velocity, which is often expressed in terms of the parameter β=ΔP/mU, where mU is the impactor momentum [5]. The amount by which crater ejecta enhances asteroid deflection-that is, the normalised momentum of the crater ejecta that escapes the gravitational attraction of the target body (β-1)-has been found to vary significantly depending on the target asteroid's properties and composition [6].

Previous numerical simulations [7, 8] have quantified the sensitivity of the asteroid deflection to target material properties. To allow for a large variety of material properties to be studied, these simulations employed a two-dimensional shock physics code with an axially-symmetric mesh geometry, which restricted the studies to vertical impacts only. However the DART spacecraft will impact the surface of Didymoon at an oblique angle [3]. Here we investigate the influence of impact angle on the ejecta momentum transfer with the aim of developing an empirical scaling relationship for β as a function of impact angle.

Numerical methods:

We used the iSALE3D shock physics code [9] to numerically simulate the DART impact in two and three dimensions. The DART spacecraft structure was modelled as a porous aluminium sphere, impacting a 20% porous, homogeneous basaltic regolith target at 7 km/s. The cohesive strength of the damaged material was 10 kPa.

Influence of the impact angle on the net momentum:

Consistent with previous laboratory-scale oblique impact experiments [10, 11] and DART impact models [12], our simulations show that the ejecta from oblique impacts displays higher speeds and lower ejection angles in the downrange direction, and lower speeds and higher ejection angles in the uprange direction of the impact.

Figure 1 compares the surface topography of a vertical DART impact, at a 90^o angle of incidence to the target plane, and an oblique DART impact at a 45^o angle. The time-frames of the oblique impact Figure 1b show a highly asymmetric ejecta distribution at early times of the cratering process <0.10 s, compared to the same times in the vertical impact (Figure 1a). The asymmetric ejecta flow becomes more symmetric as the crater grows towards its final size.

The asymmetry of the ejecta can have important implications for momentum transfer vector. The net momentum of the target after the impact is the vector sum of the impactor momentum and the crater ejecta momentum enhancement vectors. The projectile imparts an its momentum along the impact direction; that is, downrange and into the target. As most of the ejecta momentum is launched in the downrange direction, the net momentum imparted to the target, which acts in the opposite direction, acts downward and slightly uprange. The vector sum of the impactor momentum and the momentum enhancement vectors is in between the individual vectors, implying that the ejecta momentum acts to increase the vertical component of momentum transfer and reduce the horizontal downrange component.

Figure 2 shows the direction of the total momentum imparted to the asteroid, as a function of time, measured relative to the downrange horizontal direction, for four different impact angles. As the crater grows towards its final diameter (at about 1 s), the uprange direction of the ejecta momentum becomes more perpendicular to the surface. The direction of the net momentum imparted to the target also changes, from the downrange direction, towards the vertical direction. In the scenarios simulated here, for impacts into a 10 kPa target, the direction of the net momentum at the end of the crater growth is about 83^o for the 60^o impact, ~77^o for the 45^o impact and ~66^o for the 30^o impact.

In the simulations presented here, crater growth is halted by the target's strength before the total momentum direction becomes vertical. However, it is expected that with increasing cratering efficiency (e.g. decreasing strength), the ejecta momentum will make a larger contribution towards the total momentum vector. 

Towards an ejecta scaling relationship for oblique impacts:

Ejecta scaling relationships are useful to predict the ejecta distribution and momentum transfer for vertical impacts. However, most planetary impacts are oblique and stationary point-source scaling [6] becomes inadequate. An aim of this work is to develop an ejecta scaling relationship for oblique impacts, based on numerical simulation data.

Three-dimensional simulations of the DART impact at vertical, 60^o, 45^o and 30^o impact angles provide information about the ejecta mass-velocity distribution as a function of impact angle and azimuth. The momentum carried away by the ejecta from a vertical impact, β-1, can be found from integrating the mass, dM, within the radial distance range from n₁ to n₂R/a [1]. Here we apply this semi-analytical equation on a per azimuth basis and then sum each azimuthal ejecta distribution to derive β-1 values from all ejecta in oblique impact scenarios (Fig. 3).

Acknowledgements: We gratefully acknowledge the developers of iSALE (www.isale-code.de) and STFC for funding (Grant ST/N000803/1).