Sailing towards the stars close to the speed of light

We present a relativistic model for light sails of arbitrary reflectivity and absorptance undertaking nonrectilinear motion. Analytical solutions for a constant driving power and a reduced model for straight motion with an arbitrary sail's illumination are given, including for the case of a perfectly reflecting light sail examined in earlier works. When the sail is partially absorbing incoming radiation, its rest mass and temperature increase, an effect completely discarded in previous works. It is shown how sailing at relativistic velocities is intricate due to the existence of an unstable fixed point, when the sail is parallel to the incoming radiation beam, surrounded by two attractors corresponding to two different regimes of radial escape. We apply this model to the Starshot project by showing several important points for mission design. First, any misalignment between the driving light beam and the direction of the sail's motion is naturally swept away during acceleration toward relativistic speed, yet leads to a deviation of about 80 A.U. in the case of an initial misalignment of 1 arc sec for a sail accelerated up to 0.2c toward Alpha Centauri. Then, the huge proper acceleration felt by the probes (of order 2500 g), the tremendous energy cost (of about 13 kt per probe) for poor efficiency (of about 3%), the trip duration (between 22 and 33 years), the temperature at thermodynamic equilibrium (about 1500K), and the time dilation aboard (about 160-days difference) are all presented and their variation with the sail's reflectivity is discussed. We also present an application to single trips within the Solar System using double-stage light sails. A spaceship of mass 24 tons can start from Earth and stop at Mars in about seven months with a peak velocity of 30 km/s but at the price of a huge energy cost of about 5.3×10^{4} GW h due to extremely low efficiency of the directed energy system, around 10^{−4} in this low-velocity case.