Spiral Lambert’s Problem

Abstract

Lambert’s problem is solved for the case of a spacecraft accelerated by a continuous thrust. The solution is based on the family of generalized logarithmic spirals, which provides a fully analytic description of the dynamics including the time of flight and involves two conservation laws. The structure of the solution yields a collection of properties that are closely related to those of the Keplerian case. A minimum-energy spiral is found, with pairs of conjugate spirals bifurcating from it. Thanks to the integral of motion related to the energy, the solutions are classified as elliptic, parabolic, and hyperbolic. The maximum acceleration reached along the transfer can be solved in closed form. The problem of designing a low-thrust trajectory between two bodies reduces to solving two equations with two unknowns. Double-time opportunity transfers appear naturally thanks to the symmetry properties of the generalized logarithmic spirals. Comparing the Keplerian and spiral pork-chop plots in an Earth–Mars example shows that the spiral solution might increase the mass fraction delivered to the final orbit, thanks to reducing of the magnitude of the impulsive maneuvers at departure and arrival.