Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization

A novel method to compute all critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus is presented. The problem is attacked as a global optimization problem, solved by a rigorous global optimizer based on Taylor models. Thus, thigh enclosures of the stationary points are obtained. The embedded capability of the method of delivering high-order Taylor expansions is then used to analyze how uncertain orbital parameters affect the position of the stationary points and the associated distance values. Sample orbital sets and Apophis asteroid are used as test cases.     

Gravity assist space pruning based on differential algebra

In this paper a differential algebra version of the gravity assist space pruning algorithm is presented. The use of differential algebraic techniques is proposed to overcome the two main drawbacks of the existing algorithm, i.e., the steep increase of the number of function evaluations with the number of planets involved in the transfer, and the use of a bounding procedure that relies on Lipschitzian tolerances. Differential algebra allows us to process boxes in place of grid points, and to substitute pointwise evaluations of the constraint functions with their Taylor expansions. Thanks to the particular instance of multi-gravity assist problems dealt with, all the planet-to-planet legs can be treated independently, and forward and backward constraining can be applied. The proposed method is applied to preprocess the search space of sample interplanetary transfers and it also serves as a stepping stone towards a fully rigorous treatment of the pruning process based on Taylor models.     

The insurance industry and IDNDR: Common interests and tasks

Insurance, as a financial hedge against loss, is closely concerned with the problems presented by natural disasters. In many insurance markets, natural perils are covered to a greater or lesser extent by property insurance and various other branches of insurance. This entails making a close study of such perils in order that the premium requirements on the one hand and the loss potential on the other may be correctly estimated. To that end, insurers can draw upon extensive loss data and analyses from all parts of the world.Insurance has an appreciable influence on the behaviour of the public and of industry in the matter of preparedness. By making the right use of the tools of insurance technique, especially deductibles, the insured can be motivated to take preventive measures. After a disaster, the insurance industry gives swift financial help and has, for the most part, an efficient loss settlement organization at its disposal for the purpose.     

Asteroid close encounters characterization using differential algebra: the case of Apophis

A method for the nonlinear propagation of uncertainties in Celestial Mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis’ close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis.