Efficient Techniques for Small Bodies Uncertainties Propagation and Impact Leading Condition Identification
Armellin, Roberto; Di Lizia, P.; Bernelli-Zazzera, F.
Politecnico di Milano

Recently, several tools and techniques have been developed to allow for the robust detection and prediction of planetary encounters and potential impacts by near-Earth asteroids. These methods suffer of being either not sufficiently accurate when relying on simplifications (e.g. linear approximations) or computationally intensive when based on several accurate numerical integrations, as in Monte Carlo approach.

Differential algebraic techniques are proposed as a valuable tool to develop alternative approaches to answer the previous tasks. Differential algebra (DA) serves the purpose of computing the derivatives of functions in a computer environment. More specifically, by substituting the classical implementation of real algebra with the implementation of a new algebra of Taylor polynomials, any function f of v variables is expanded into its Taylor series up to an arbitrary order. This has an important consequence when the numerical integration of an ordinary differential equation (ODE) is performed by means of an arbitrary integration scheme. Any explicit integration scheme is based on algebraic operations, involving the evaluations of the ODE right handside at several integration points. Therefore, carrying out all the evaluations in the DA framework allows differential algebra to compute the arbitrary order expansion of the flow of a general ODE initial value problem.

The availability of such high order expansions is exploited to develop tools for the prediction of planetary encounters and potential impacts taking into account measurement uncertainties.
More specifically, an improvement of Monte Carlo simulation approach is first obtained by replacing the computationally intensive multiple integrations with evaluations of the high order expansions of the flow. This results in a significant computational time saving at the expense of negligible loss in accuracy. The availability of high order maps in space and time and intrinsic tools for Taylor map inversion are then exploited in an algorithm that reduces the computation of the minimum distance form Earth and the corresponding epoch for all the asteroids belonging to the initial uncertainties cloud to the simple evaluation of polynomials. Finally, an impact leading algorithm is developed to determine, at an arbitrary observation epoch, the initial conditions that would lead to an Earth impact.