Celestial Mechanics and Dynamical Astronomy manuscript No. (will be inserted by the editor) P. Di Lizia · R. Armellin · M. Lavagna Application of high order expansions of two-point boundary value problems to astrodynamics Received: date / Accepted: date Abstract Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body prob- lem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighbor- hood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers. Keywords two-point boundary value problem, differential algebra, halo orbit, aerocapture. 1 Introduction Two-point boundary value problems (TPBVP) appear frequently in space trajectory design when solving a system of ODE with boundary conditions on both sides of the integration interval (Armellin and Topputo 2006). A typical example is the classical Lambert’s problem, in which a conic arc linking two fixed positions in a given time is to be identified in the frame of the two-body problem. Such a problem can be solved by means of efficient semi-analytical algorithms, since an analytic solution is available in the case of Kepler’s problem (Battin 1993). Even optimal space- craft control problems are reduced to TPBVP within the framework of indirect methods (Bryson and Ho 1975): the full system of equations, made up by the states and the Lagrange multipliers dynamics, must be solved for initial and final conditions derived by problem requirements. This highlights the relevance of the TPBVP in spacecraft dynamics and the wide applicability of the related methods and algorithms. P. Di Lizia, R. Armellin, and M. Lavagna Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano Via La Masa, 34 – 20156, Milano, Italy E-mail: {dilizia, armellin, lavagna}@aero.polimi.it