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When an asteroid experiences an impact, its path is changed. How much it changes is important to know for both asteroid evolution studies and for attempts to prevent an asteroid from impacting the Earth. In an impact process the total momentum of the material is conserved. However, not all of the material is of interest, but only that remaining with the asteroid. The ratio of the change of momentum of the remaining asteroid to that of the impactor is called the momentum multiplication factor; and is commonly given the symbol β. It has been known for some time that β can be greater than unity, and in some cases far greater. That could be a significant factor in attempts to deflect an asteroid with an impact, and can also be important in the stirring of objects in the asteroid belt due to mutual impacts. The escaping crater ejecta are the source of the momentum multiplication. Housen and Holsapple (Housen, K.R., Holsapple, K.A. [2011a]. Icarus 211, 856–875) have given a recent summary of ejecta characteristics and scaling. Here we use those ejecta results to determine how β depends on the impactor properties, on the asteroid size and composition, and establish the paths and time of flight of all of the ejecta particles. The approach is to add the contribution of each element of ejected mass accounting for its initial velocity, its trajectory and whether it escapes the asteroid. The goal in this paper is to provide a theoretical framework of the fundamental results which can be used as a test of the veracity of experiments and detailed numerical calculations of impacts. A subsequent paper will present direct laboratory results and numerical simulations of momentum multiplication in various geological materials.

@article{holsapple_momentum_2012, title = {Momentum transfer in asteroid impacts. {I}. {Theory} and scaling}, volume = {221}, issn = {0019-1035}, url = {http://www.sciencedirect.com/science/article/pii/S0019103512003958}, doi = {10.1016/j.icarus.2012.09.022}, abstract = {When an asteroid experiences an impact, its path is changed. How much it changes is important to know for both asteroid evolution studies and for attempts to prevent an asteroid from impacting the Earth. In an impact process the total momentum of the material is conserved. However, not all of the material is of interest, but only that remaining with the asteroid. The ratio of the change of momentum of the remaining asteroid to that of the impactor is called the momentum multiplication factor; and is commonly given the symbol β. It has been known for some time that β can be greater than unity, and in some cases far greater. That could be a significant factor in attempts to deflect an asteroid with an impact, and can also be important in the stirring of objects in the asteroid belt due to mutual impacts. The escaping crater ejecta are the source of the momentum multiplication. Housen and Holsapple (Housen, K.R., Holsapple, K.A. [2011a]. Icarus 211, 856–875) have given a recent summary of ejecta characteristics and scaling. Here we use those ejecta results to determine how β depends on the impactor properties, on the asteroid size and composition, and establish the paths and time of flight of all of the ejecta particles. The approach is to add the contribution of each element of ejected mass accounting for its initial velocity, its trajectory and whether it escapes the asteroid. The goal in this paper is to provide a theoretical framework of the fundamental results which can be used as a test of the veracity of experiments and detailed numerical calculations of impacts. A subsequent paper will present direct laboratory results and numerical simulations of momentum multiplication in various geological materials.}, number = {2}, urldate = {2018-05-31TZ}, journal = {Icarus}, author = {Holsapple, Keith A. and Housen, Kevin R.}, month = nov, year = {2012}, keywords = {Asteroids, Collisional physics, Cratering, Impact processes, Near-Earth objects}, pages = {875--887} }

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