Effects of interstellar drag on minimum-time trajectories with thrust-limited relativistic rockets

An analysis is presented of the effects of drag due to the presence of interstellar matter which acts on a relativistic thrust-limited rocket traveling a minimum-time trajectory between fixed points with zero initial and final velocities. Two rockets are considered: an ideal photon rocket with an effective exhaust velocity equal to the speed of light (c) and an ideal nuclear-fusion rocket with an effective exhaust velocity of 0.0893c. The mathematical problem is outlined, and a two-point boundary-value problem is derived using the necessary conditions for a minimum-time trajectory. Solution of the latter problem indicates that the main effect of interstellar drag is that a singular arc (where thrust equals drag) may have to be incorporated into the minimum-time trajectory. It is concluded, however, that no singular arcs occur with realistic drag coefficients and that the effects of drag are very small for a flight time of 40 years and a maximum acceleration of 1 light year per year per year (as recorded on the rocket).