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Low-thrust trajectories are becoming increasingly important in the design of space missions. The technological developments that allowed ion propulsion to become a competitive option for missions such as Smart-1 and Deep Space 1 may be soon available for many other advanced propulsion systems that promise to deliver long duration thrust with high specific impulses and high reliability. Methods used to find the optimal trajectories for spacecraft equipped with low-thrust capabilities, are based on direct and indirect approaches highly dependent on the initial guess solution. These approaches, though successful in the detailed design of trajectories, fail in providing the designers with a tool to efficiently explore a large number of options. With this respect the use of analytical methods able to create solutions to the low-thrust problem, such as that suggested by Markopoulos, could make a difference. In his Ph.D. thesis Petropoulos introduced for the first time a new analytical solution to the problem of low-thrust trajectories. His solution, based on the use of an exponential sinusoid, is based on an inverse dynamical calculation that leads to analytical expressions both for the thrusting history and for the derivative of the polar anomaly, whenever tangential thrust is assumed. His discovery added up to the small list of analytical solutions available for this increasingly important problem. Moreover, his shape based method, inspired many other researchers. Vasile et al. tried to solve the problem of constraint violations at the trajectory boundaries (likely to happen with the exponential sinusoid) by introducing shapes defined by a larger number of parameters and by using an equinoctial element formulation. Their method is able to represent with higher fidelity a generic low-thrust trajectory at the cost of an increased number of free parameters. In the same direction is the work by Patel et al. where an attempt is made to optimise a large class of shapes given in cartesian coordinates. These last approaches constitute a compromise that allow a search through

Lambert's Problem for Exponential Sinusoids