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 $1500\mathrm{K}$), 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 $5.3\times {10}^4$ GW h due to extremely low efficiency of the directed energy system, around ${10}^{-4}$ in this low-velocity case.

Figure 1

Sketch of the perfectly reflecting light sail: the light source is located at the origin of coordinates, and the incident beam hits the sail with an angle $\theta$ with respect to the velocity of the sail before it is reflected following the same angle, due to the Snell-Descartes reflection law.

Figure 2

Various trajectories of perfectly reflecting white sails with driving power $P$ decaying as $1/R^2$. A star indicates the location of the laser source, and a square indicates one particular position of the light sail with an associated reflecting surface and vector triplets ${\vec{n}}_{s,\mathrm{in},\mathrm{ref}}$.

Figure 3

Phase diagram in the plane $(\psi ,{\psi }^{\prime })$ of the perfectly reflecting white sail. The upper plot is for constant power $P$ while the lower plot shows the case of driving power $P$ decaying as $1/R^2$. Straight lines correspond to ${\beta }_0=0$ and dashed lines correspond to ${\beta }_0=0.9$.

Figure 4

Phase diagram in the plane $(\theta ,{\psi }^{\prime })$ of the perfectly reflecting white sail. The upper plot is for constant power $P$ while the lower plot shows the case of driving power $P$ decaying as $1/R^2$. The dots and the triangles, respectively, indicate the location of the unstable saddle point $(\theta =\pi /2,{\psi }^{\prime }=0)$ and the attractors $(\theta \to 0,{\psi }^{\prime }\to 0)$ and $(\theta \to \pi ,{\psi }^{\prime }\to -2)$ [$(\theta \to \pi ,{\psi }^{\prime }\to 0)$ for decaying power]. Straight lines correspond to ${\beta }_0=0$ and dashed lines correspond to ${\beta }_0=0.9$.

Figure 5

Evolution of the absorptance $1-\alpha$ (a) and of the sail's temperature (b) for the following parameters: $P0=4\mathrm{GW}$, $S=16{\mathrm{m}}^2$, $C_V=1900\mathrm{J}/\left(\mathrm{kg}\mathrm{K}\right)$, and a reflectivity $\epsilon$ of $99.9\%$.

Figure 6

Evolution of the angles $\theta$ (straight line), $\varphi$ (dotted line), and $\mathrm{\Theta }$ (dashed line) in the very early phases of acceleration for five values of $\epsilon$ equally spaced between 0 and 1.

Figure 7

Evolution of the light sail's velocity (a), proper acceleration (b), and distance to the source (c) with time for five values of $\epsilon$ equally spaced between 0 and 1. Straight lines give the results for a decaying driving power given by Eq. (38) while dashed lines are for constant power.

Figure 8

(a) Evolution of time dilation $T/\tau -1$ aboard during the acceleration phase for various sails' reflectivity $\epsilon$. Straight lines indicate the prediction for constant power. (b) Trip duration of the Starshot probes towards Proxima Centauri as a function of the sail's reflectivity $\epsilon$.

Figure 9

Evolution of the Starshot probe's blackbody temperature during acceleration phase (a) and final temperature at the end of the acceleration phase as a function of the sail's reflexivity $\epsilon$ (b).

Figure 10

Evolution of the outer and inner payload sails' velocities (a) and distance (b). A dotted line indicates the cruise velocity of the Voyager 1 probe.