Abstract
Orbit uncertainty propagation usually requires linearized propagation models [1–3] or full nonlinear Monte Carlo simulations [4]. The linear assumption simplifies the problem, but fails to characterize trajectory statistics when the system is highly nonlinear or when mapped over a long time period. On the other hand, Monte Carlo simulations provide true trajectory statistics, but are computationally intensive. The tools currently used for the robust detection and prediction of planetary encounters and potential impacts with Near Earth Objects (NEO) are based on these two techinques [5–7], and thus suffer the same limitations. A different approach to orbit uncertainty propagation has been discussed by Junkins et al. [8,9], in which the effect of the coordinate system on the propagated statistics is thoroughly analyzed; however, the propagation method was based on the linear assumption and the system nonlinearity was not incorporated in the mapping. An alternate way to analyze trajectory statistics by incorporating higher-order Taylor series terms that describe localized nonlinear motion was proposed by Park and Scheeres [10]. Their appoach is based on proving the integral invariance of the probability density function via solutions of the Fokker–Planck equations for diffusionless systems, and by combining this result with the nonlinear state propagation to derive an analytic representation of the nonlinear uncertainty propagation. This method is limited to systems derived from a single potential. Differential algebraic (DA) techniques are proposed as a valuable tool to develop alternative approaches to tackle the previous tasks. Differential algebra provides the tools to compute the derivatives of functions within a computer environment [11–13]. 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 n. 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 hand side at several integration points. Therefore, carrying out all the evaluation 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 improve the Monte Carlo simulation approach by replacing thousands of integrations with evaluations of the high order expansion of the flow, reducing the computational time significantly. This algorithm is applied to the prediction of Apophis planetary encounter and potential impact taking into account its measurement uncertainties. The availability of high order maps in space and time and intrinsic tools for their inversion are then exploited in an algorithm that reduces the computation of the minimum distance from the Earth of all the asteroids belonging to the initial uncertainties cloud to the simple evaluation of polynomials. The second part of the paper deals with the rigorous study of Apophis close encounter by means of Taylor models (TM). The first methods introduced to perform validated integrations of dynamical systems [14] were based on the use of interval analysis, originally formalized by Moore in 1966 [15]. The main idea beneath this theory is the substitution of real numbers with intervals of real numbers; consequently, interval arithmetic and analysis are developed in order to operate on the set of interval numbers in place of the real numbers. This turned out to be an effective tool for error and uncertainty propagation, as both the numerical errors and the uncertainties can be bounded by intervals and rigorously propagated using interval analysis. Several codes, which implement a variety of features to improve the performances of naive interval algebra based on interval analysis, have been implemented for the rigorous integration of ODE [16– 19]. Unfortunately, all of them produce an unacceptable overestimation of the solution when applied to Solar system dynamics [20]. The reasons for such an overestimation are the so-called dependency problem and wrapping effect [21]. Taylor model integrators have shown to be a powerful tool for the validated integration of ordinary differential equations as they successflully address both these problems [22,23]. The Taylor Model approach combines high-order multivariate polynomial techniques and the interval technique for verification. In particular, it represents a multivariate functional dependence f by a high order multivariate Taylor polynomial P and the remainder bound interval I. The n-th order Taylor polynomial P captures the bulk of functional dependency. Because the manipulation of those polynomials can be performed by operations on the coefficients where the minor errors due to their floating point nature are moved into the remainder bound, the major source of interval overestimation is removed. Thus, the overestimation only occurs in the remainder bound, the size of which scales with order n of the width of the domain. When applied to the verified integration of ODE, the relationships between the state vector at a generic time t and the initial conditions are expressed in terms of a Taylor model (P, I) and a tight enclosure for the action of the differential equations on an extended region is then provided. The TM-based integrator implemented in COSY VI [22] has been already succesfully exploited for the long-term rigourous integration of asteroids motion [24]. In this paper, an improved version of the integrator that exploits dynamic domain decomposition, automatic step size control, and a flow operator based on Lie derivatives [25], is applied to the more challenging task of Apophis close approach rigorous integration. The paper is organized as follows. The models developed to describe Apophis dynamics and to evaluate the planetary ephemerides are illustrated first. The improved version of the Monte Carlo simulation together with the minimum close encounter distance algorithm are then presented. The last part of the paper is devoted to the presentation of the rigorous integration of Apophis flyby. Conclusions end the paper.