A high order impact probability computation tool for earth resonant returns of near earth objects

This thesis introduces the combination of Differential Algebra and Automatic Domain Splitting for uncertainty propagation and impact probability computation for resonant returns of Near Earth objects. Uncertainty propagation and estimation in nonlinear systems poses relevant challenges to astrodynamicists. Present-day approaches mainly refer to linearized propagation models or full nonlinear Monte Carlo simulations. Classical tools used for robust detection and prediction of planetary encounters and potential impacts of Near Earth Objects (NEO) are based on these techniques. Alternative methods rely on the use of Differential Algebra (DA). Differential Algebra supplies the tools to compute the derivatives of functions within a computer environment. Consequently, the Taylor expansion of the flow of ordinary differential equations can be obtained by carrying out all the operations of any explicit integration scheme in the DA framework. DA has already proven its efficiency in the non-linear propagation of uncertainties within different dynamical models. Nonetheless, the accuracy of the method drastically decreases in highly nonlinear dynamics. Examples of this kind can be found in the propagation of asteroids motion after a close encounter with a major body. This thesis introduces the concept of Automatic Domain Splitting to overcome the described issues and applies it to the problem of uncertainty propagation of NEO motion and the computation of the impact probability of resonant returns. During the integration of the initial conditions, the polynomial vector representing the current state is monitored. When the nonlinearities cause the high order terms of the polynomial to grow, integration is paused and the current domain is split along one variable. A maximum number of splits is imposed, and the final result is a set of polynomial state vectors, each describing a portion of the initial uncertainty domain. Potentially hazardous subdomains are then automatically identified among all the domains whose integration was stopped before the resonant return reaching the maximum number of splits. The identification is carried out by performing a pointwise integration and detecting the points that impact the Earth. The impact probability is evaluated by integrating the probability density function over all the small volumes corresponding to the impacting points. The asteroid 99942 Apophis is used as test case: this asteroid will have a close encounter with the Earth in 2029. Propagating the estimates of Apophis' state available in 2009, this close encounter paves the way to a possible resonant return in 2036. More specifically, Apophis' equinoctial parameters on 18th of June 2009, as retrieved from the Near Earth Object Dynamics Site in September 2009, are set as initial conditions. The developed algorithm is used to assess the impact probability of Apophis at its resonant return in 2036.