This paper analyzes the locally-optimal heliocentric transfer of a spacecraft propelled by an electric solar wind sail, an innovative propellantless propulsion system that generates a propulsive acceleration exploiting the momentum of solar wind particles. The potentialities of such an advanced thruster are investigated in terms of flight times required to achieve a given heliocentric orbit. The problem is addressed using a locally-optimal formulation, by minimizing a scalar performance index that depends on the time derivatives of the osculating orbital elements. The proposed algorithm gives an estimate of the globally-optimal flight time with reduced computational efforts compared to a traditional optimization approach. Also, when the performance index involves a single orbital parameter and the transfer trajectory is two-dimensional, the proposed approach provides an analytical solution to the locally-optimal control problem. The procedure discussed in the paper is used to quantify the near-optimal performance of an electric solar wind sail in some advanced mission scenarios, such as the design of a heliocentric non-Keplerian orbit for solar activity monitoring, the exploration of the Solar System boundaries, and the rendezvous with comets 1P Halley and 67P/Churyumov-Gerasimenko.

In this study, new analytical solutions to the equations of motion of a propelled spacecraft are investigated using a shape-based approach. There is an assumption that the spacecraft travels a two-dimensional spiral trajectory in which the orbital radius is proportional to an assigned power of the spacecraft angular coordinate. The exact solution to the equations of motion is obtained as a function of time in the case of a purely radial thrust, and the propulsive acceleration magnitude necessary for the spacecraft to track the prescribed spiral trajectory is found in a closed form. The analytical results are then specialized to the case of a generalized sail, that is, a propulsion system capable of providing an outward radial propulsive acceleration, the magnitude of which depends on a given power of the Sun-spacecraft distance. In particular, the conditions for an outward radial thrust and the required sail performance are quantified and thoroughly discussed. It is worth noting that these propulsion systems provide a purely radial thrust when their orientation is Sun-facing. This is an important advantage from an engineering point of view because, depending on the particular propulsion system, a Sun-facing attitude can be stable or obtainable in a passive way. A case study is finally presented, where the generalized sail is assumed to start the spiral trajectory from the Earth’s heliocentric orbit. The main outcome is that the required sail performance is in principle achievable on the basis of many results available in the literature.