Time-Optimal Heliocentric Transfers With a Constant-Power, Variable-Isp Engine

We study minimum-time heliocentric transfers for a spacecraft propelled by an electric thruster that draws constant electrical power P while continuously varying its exhaust speed (variable Isp). The vehicle is assumed to depart from and arrive on circular heliocentric orbits (i.e., initial and final velocities match the local circular velocity at the respective radii). First, we derive an analytic solution of the one-dimensional, gravity-free brachistochrone and discuss how a finite exhaust-speed ceiling modifies the solution, producing a boost–coast–brake structure. Next, we formulate the full planar Sun-field optimal-control problem, derive two closed-form first integrals, and show that the indirect formulation reduces to a seven-dimensional boundary-value problem. Finally, we present a practical numerical continuation strategy that obtains a coarse feasible endpoint via global optimization and then refines it by homotopy and Powell’s local solver. Numerical examples for a 1GW engine with an initial/dry mass of 3000 t→1000 t demonstrate Earth–Jupiter-class transfers in roughly 200–220 days that commonly exploit a solar Oberth pass. Reproducible code and data are available at the project repository.