Optimal Kinetic-Impact Geometry for Asteroid De&#64258;ection
Authors: Jiao Yifei , Cheng Bin , Baoyin Hexi 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Introduction&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Kinetic impact is recognized as an e&amp;#64256;ective and feasible method to defend most of medium-size asteroid threats with a warning time of years to decades [1]. For a kinetic deflection, the momentum transfer is influenced not only by the target material properties, but also by the impact geometries including the asteroid shape, the specific location or direction to impact, as well as the orbital geometries of both the impactor and the asteroid. A good understanding of these geometric e&amp;#64256;ects, which have not been well investigated in existing literature, is of high importance to the design of kinetic deflection missions, especially in the approaching stage of the spacecraft [2][3]. Here we propose a new approach to improve the kinetic deflection efficiency, considering both the impact geometry and orbital geometry.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Asteroid Delta-V Hodograph by Kinetic Impact&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;For a given orbital geometry, the spacecraft is capable to strike any location or region on the target asteroid in the same traveling velocity at the end of its flight time, with almost no extra fuel consumed. To evaluate the velocity change distribution with different impact locations, we introduce the asteroid &amp;#916;v hodograph, which is described by the terminal point of &amp;#916;v vector with its origin fixed to the COM of the target asteroid. For each impact location, we assume a local oblique impact in which the impact angle is determined by the impact velocity direction and the surface slope, as shown in Fig.1.&lt;/p&gt;
&lt;p&gt;&amp;#160;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;The delivered impulse is then expressed as a function of the surface normal vector&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;where &amp;#946; and &amp;#947; represents the momentum transfer efficiencies in the directions of surface normal and downrange tangent [4]. When substituting a spherical-coordinate expression of the unit normal, the &amp;#916;v hodograph is obtained as&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;where the specific values of &amp;#946; and &amp;#947; depend on the impact angle (&amp;#960;/2-&amp;#952;). Note that the profile is unique regardless of the asteroid shape. Furthermore, the hodograph can be simplified to a spherical surface when supposing that the two efficiencies keep constant.&lt;/p&gt;
&lt;p&gt;Fig.2 shows a simplified application of &amp;#916;v hodograph to deflect a spherical asteroid in 2D. Considering that the required impulse direction is along the orbital velocity of the asteroid [5], the optimal impact geometry corresponds to an off-centered location at &amp;#952;=&amp;#945;/2, where the orbital geometry angle is &amp;#945;, rather than a centered impact case. We simulated oblique impacts into rocky material in different incident angles with a commercial code, LS-DYNA SPH, and evaluated the two efficiencies semi-analytically using the method in Ref.[4]. The more realistic Delta-V hodograph should be a distortion of the simplified spherical shape.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Optimal Location/Direction to Impact&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;Recall that the hodograph has a unique profile regardless of the asteroid shape. However, different asteroid shapes correspond to various &amp;#916;v distribution on the hodograph. Fig.3 presents the deflection results with a polyhedral model of the asteroid Bennu [6], which has a near-spherical shape. With increasing &amp;#945;, the preferable impact area deviates from the center to the leading side, and the di&amp;#64256;erence of the effective impulse (along the orbital velocity of the asteroid) becomes more significant when striking into di&amp;#64256;erent locations. For &amp;#945;=45&amp;#176; and 60&amp;#176;, the effective &amp;#916;v of certain locations (yellow area) could be 33% to 50% more than a centered impact, while it may reduce by half in some other locations (dark green area). Therefore, it is crucial for the spacecraft to select the impact site appropriately according to the &amp;#916;v hodograph when approaching its target. In smaller impact geometry angle cases, however, the effective impulse appears to be less sensitive to surface locations, so that changing the impact location would not a&amp;#64256;ect the deflection distance too much.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;For elongated asteroid shapes, however, the available &amp;#916;v distribution on the unique hodograph cannot cover the complete profile, but changes periodically with the asteroid rotation. It is feasible to select the rotational phase of the target asteroid at the moment of impact to improve the effective impulse, called &amp;#8220;direction optimization&amp;#8221; in this work. Fig.4 presents the results to deflect an Itokawa-like polyhedral asteroid [7] from different impact directions with the same orbital geometry &amp;#945;=45&amp;#176;. An arbitrary impact direction, if not appropriate, could decrease the deflection magnitude significantly. However, in a preferable impact direction, the effective impulse is well improved and shows less sensitivity to the locations.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Acknowledgements&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;This work is supported by the National Key R&amp;D Program of China (2019YFA0706500).&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;[1] Council, N. R., Defending Planet Earth: Near-Earth-Object Surveys and Hazard Mitigation Strategies, The National Academies Press, Washington, DC, 2010.&lt;/p&gt;
&lt;p&gt;[2] Feldhacker, J. D., Syal, M. B., Jones, B. A., Doostan, A., McMahon, J., and Scheeres, D. J., &amp;#8220;Shape dependence of the kinetic deflection of asteroids,&amp;#8221; Journal of Guidance, Control, and Dynamics, Vol. 40, No. 10, 2017, pp. 2417&amp;#8211;2431.&lt;/p&gt;
&lt;p&gt;[3] Brack, D. N., and McMahon, J. W., &amp;#8220;E&amp;#64256;ects of Momentum Transfer Deflection E&amp;#64256;orts on Small-Body Rotational State,&amp;#8221; Journal of Guidance, Control, and Dynamics, Vol. 43, No. 11, 2020, pp. 2013&amp;#8211;2030.&lt;/p&gt;
&lt;p&gt;[4] Raducan, S., Davison, T., and Collins, G., &amp;#8220;Ejecta distribution and momentum transfer from oblique impacts on asteroid surfaces,&amp;#8221; Icarus, Vol. 374, 2022, p. 114793.&lt;/p&gt;
&lt;p&gt;[5] Vasile, M., and Colombo, C., &amp;#8220;Optimal impact strategies for asteroid deflection,&amp;#8221; Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 858&amp;#8211;872.&lt;/p&gt;
&lt;p&gt;[6] Nolan, M., Magri, C., Howell, E., Benner, L., Giorgini, J., Hergenrother, C., Hudson, R., Lauretta, D., Margot, J., Ostro, S., et al., &amp;#8220;Asteroid (101955) Bennu shape model V1. 0,&amp;#8221; NASA Planetary Data System, 2013, pp. EAR&amp;#8211;A.&lt;/p&gt;
&lt;p&gt;[7] Gaskell, R., Saito, J., Ishiguro, M., Kubota, T., Hashimoto, T., Hirata, N., Abe, S., Barnouin-Jha, O., and Scheeres, D., &amp;#8220;Gaskell itokawa shape model v1. 0,&amp;#8221; NASA Planetary Data System, 2008, pp. HAY&amp;#8211;A.&lt;/p&gt;