Sailing Towards the Stars Close to the Speed of Light

We present a relativistic model for light sails of arbitrary reflectivity undertaking non rectilinear motion. A reduced model for straight motion with arbitrary sail's illumination is also 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 increases, an effect discarded in previous works. Is 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 misalignement between the driving light beam and the direction of sail's motion is naturally swept away during acceleration toward relativistic speed, yet leading to a deviation of about $80$ AU in the case of an initial misalignement of one arcsec for a sail accelerated up to $0.2\times c$ toward Alpha Centauri. Then, the huge proper acceleration felt by the probes (of order $2500$-g), the tremendous energy cost (of about $13$ kilotons per probe) for poor efficiency (of about $3\%$), the trip duration (between $22$ and $33$ years), the rest mass variation (up to $14\%$) and the time dilation aboard (about $140$ days difference) are all presented and their variation with sail's reflectivity is discussed. We also present an application to single trips within the Solar System using Forward's idea of double-stage lightsails. A spaceship of mass $24$ tons can start from Earth and stop at Mars in about $7$ months with a peak velocity of $30$ km/s but at the price of a huge energy cost of about $5.3\times 10^4$ GWh due to extremely low efficiency of the directed energy system, around $10^{-4}$ in this low-velocity case.


I. INTRODUCTION
The discovery of the cruel vastness of space beyond the Solar System is rather recent in human history, dating back from the measurement of the parallax of the binary star 61 Cygni by Bessel in 1838 [1]. This was even pushed orders of magnitude further by the work of Hubble on intergalactic distances in the 1920s [2,3]. It is also at the beginning of the XXth century that space exploration with rockets based on reaction propulsion began scientifically considered by pioneers like Tsiolkowksy, Oberth, Tsander, Goddard to name but a few. While interplanetary travel was the primary motivations of their work, dreams of interstellar travel appeared to these precursors coincidentally. Although not theoretically impossible, interstellar travel has ever been largely considered unfeasible, for a variety of good reasons. The major drawback lie in the huge gap between interplanetary and interstellar distances, initially settled by Bessel's discovery: the distance to the nearest star system, Alpha Centauri, is roughly ten thousand times the distance to planet Neptune, on the outskirts of our solar system. It took about 40 years to the fastest object ever launched by humans, the Voyager 1 space probe, to reach the edges of the Solar System 18 billion kilometers away, at a record cruise velocity * Electronic address: andre.fuzfa@unamur.be of 17 km/s. Millenia long trips would then be needed to cross interstellar distances. Since Bessel's epoch, we know that the gap to the stars lies in these four orders of magnitudes, needless to mention the much larger gap to the galaxies.
There have been many suggestions to go across the stars, and we refer the reader to the reference [4] for an overview, some of which could even be considered as plausible, yet unaffordable, while some others are simply non physical. We propose here to classify the (many) proposals of interstellar travel into four categories: (1) relativistic reaction propulsion, (2) generation ships, (3) spacetime distortions and (4) faster-than-light travels. It seems to us that only the first category is relevant for plausible discussions, and the current paper investigates maybe the most plausible (or should we say the least doubtful) proposal among this category: directed energy propulsion. Generation ships are huge autonomous space stations embarking a whole population at sub-light velocities and only the far descendants can hopefully arrive at destination. The main difficulty of these proposals is to maintain (intelligent) life in the ark during millenia-long trips. Even if launched on a ballistic trajectory towards the stars, generation ships still require to embark a consequent amount of energy to maintain their internal biotopes, since there is too few available energy in the deserts of interstellar space. A 10 GW power source operating during a 10 000 years long trip amounts to about five times the world annual arXiv:2007.03530v2 [physics.pop-ph] 14 Jul 2020 energy production to be stored aboard. Spacetime distortions like wormholes and warp drives [5][6][7] are indeed based on general relativity but also require exotic matter with (strongly) negative pressures -like dark energy -produced or gathered in large amounts and stored into extreme densities. Indeed, curving spacetime is a situation of strong gravitational field which requires high compactness, a dimensional quantity given by the ratio GM/(c 2 L) (with G Newton's constant, M the rest mass of the system, c the speed of light in vacuum and L the characteristic size of the system) that measures the strength of a gravitational system. Even with dark energy at profusion, the corresponding amount of energy that should be involved is of order c 4 L/G, so that producing a one-km wide spacetime distortion requires spending about 10 26 the world annual energy production, all into such exotic avatar of matter. Finally, faster-than-light travel is not well scientifically sound since this is forbidden by conventional physics (special relativity). And, most of all, this is completely useless since time dilation at relativistic velocities will shorten trip duration for the traveler, unless one covets at managing some galactic empire or some other appealing space opera ideas. This last category should be viewed as the least serious of all, unless laws of physics bring one day considerable surprises.
Let us therefore focus on relativistic reaction propulsion, which has also received many scientifically sound considerations. To achieve velocities close to that of light, the propulsion system must bring energies in amounts close to that of the rest mass of the ship: E = m 0 c 2 . This requires to go well beyond chemical rockets to turn naturally to concepts involving nuclear energy (fission or fusion), antimatter rockets (direct or indirect) or beam-powered directed energy propulsion. This last is based on radiation pressure and consists of using the impulse provided by some external radiation or particle beam to propel the space ships. According to many authors [4,[8][9][10][11][12][13][14][15][16][17], this is maybe the most promising one for three reasons: (1) it does not require embarking any propellant, (2) they allow reaching higher velocities than rockets expelling mass and submitted to the constraints of Tsiolkowsky equation and (3) they benefit from a strong theoretical and technical background including successful prototypes [17]. However, maybe the major drawback of directed energy propulsion lies in their weak efficiency: the thrust imparted by a radiation beam illuminating some object with power P is of order P/c. Roughly speaking, one Newton of thrust requires an illumination on the sail of at least 300 MW.
The idea of using the radiation pressure of sunlight to propel reflecting sails dates back from the early ages of astronautics and was suggested by Tsander in 1924 (see [8] for a review of the idea). Several proposals have been emitted for using solar sails for the exploration of the inner Solar System (for the first proposals, see [18,19]), reach hyperbolic orbits with the additional thrust provided by sail's desorption [20] or even tests of fundamental physics [21]. Space probes like Ikaros and NanoSail-D2 have successfully used radiation pressure from sunlight for their propulsion. However, since solar illumination decreases as the square of the distance, this method is interesting for exploring the inner solar system but not for deep space missions. The first realisation of a laser in 1960 really opened the way to consider using them in directed energy propulsion, an idea first proposed by Forward in 1962 [9], since laser sources are coherent sources of light with large fluxes, allowing one to consider sending energy over large distances toward a space ship. The Hungarian physicist Marx proposed in a 1966 paper in Nature [22] the same idea independently of Forward, together with the first relativistic model describing the straight motion of such a laser-pushed light sail. This paper was quickly followed by another one by Redding [23] in 1967 that corrected one important mistake made by Marx in the forces acting on the lightsail. Twenty five years later, Simmons and McInnes revisited Marx's one dimensional model in [24], extending its model to variable illuminating power, examining the efficiency of the system and how recycling the laser beam with mirrors could increase it.
Forward's idea of laser-pushed lightsails for interstellar journey has known a strong renewal of interest since 2009 through successive funded research programs that are still active. As soon as 2009, joint NASA-UCSB Starlight [25] investigates on the large scale use of directed energy to propel spacecrafts to relativistic speeds, including small wafer scale ones. In 2016, the Breakthrough Starshot Initiative [26] has been initiated to focus on wafer scale spacecrafts and interstellar fly-bys to Alpha Centauri with the objective of achieving it before the end of the century. A nice review on large scale directed energy application to deep space exploration in the Solar System and beyond, including many detailed engineering aspects, can be found in [17]. In the context of these research programs, several authors have started modeling Solar System and interstellar missions based on this concept. Some basic one dimensional modeling of directed energy propulsion, accounting for some relativistic effects as well as prospective applications for solar system exploration can already been found in [27]. In [10], one can find a model for an single trip from Earth to Mars with microwave beam-powered lightsail but without giving the details of their model. The authors of [12][13][14]16] based their results on the model by Marx, Redding, Simmons and McInnes [22][23][24] which is only valid for one-dimensional (rectilinear) motion of perfectly reflecting light sails. The case of arbitrary reflectance has not been correctly considered so far: this involves inelastic collisions between the propulsive radiation beam and the lightsail, leading to an increase of the rest mass of the last, as was already shown in [15,28] for different types of photon rockets. The authors of [12,14] provided interesting modeling of the early acceleration phases, non-relativistic regime and power recycling in the rectilinear motion case but did not provide the analytical solutions in terms of the rapidity as we do here. [16] started from the Marx-Redding-Simmons-McInnes model and investigated the possibility to use high-energy astrophysical sources to drive the lightsails during their interstellar journey. Finally, [13] provided a more complete and realistic physical model of a lightsail's rectilinear motion, including notably thermal re-emission of absorbed radiation through Poynting-Robertson effect and a model for development costs. Unfortunately, this author also only considered one single equation of motion among the two that are given by special relativity in that case and, doing so, it was wrongly tacitly assumed the rest mass of the sail remains constant when the incoming radiation is absorbed, which cannot be true unless specific constraints. We will give here the impact of sail's rest mass variation, notably in the case with Poynting-Robertson drag. In addition, non rectilinear motion has not been considered so far, so that the effect of a misalignement of the incoming beam and sail's velocity on the trajectory has not been investigated yet. This paper will propose a relativistic model for non rectilinear motion of lightsails with arbitrary reflectivity, investigates their general dynamics and provide useful applications to the acceleration phase of fly-by missions to outer space or single trips within the Solar System using double-stage light sail.
In Section II, we recall some fundamental properties of photon rockets in special relativity, rederiving the relativistic rocket equation and apply them to lightsails. We then focus on the general motion of perfectly absorbing "white" sails and give semi-analytical solutions for the particular case of straight motion. Then, the case of a perfectly absorbing "black" lightsail is discussed and a model accounting for both radiation pressure of the incoming beam and Poynting-Robertson drag is established. Finally, we show how one can combine the previous cases of white and black sails to build a model for the general motion of "grey" light sails with arbitrary reflectivity and useful semi-analytical solutions for straight motion is provided. In Section III, we apply previous results to (i) a dynamical system analysis of sailing at relativistic velocities, showing how intricate this discipline will be, (ii) the acceleration phase of the Starshot mission, providing many related physical quantities such as sail's inclination, distance, proper velocity and acceleration, rest mass, time dilation aboard, efficiency and trip duration and (iii) single trips in the Solar-System with Forward's idea of multi-stage lightsail [8]. We finally conclude in Section IV with some emphasis on the importance of the presented results and by giving some perspectives of this work.

II. GENERAL MOTION OF RELATIVISTIC LIGHT SAILS
A light sail is a spacecraft propelled by the radiation pressure exerted on its reflecting surface by some incident light beam. In the case of directed energy propulsion, this radiation is provided by some external power source, like an intense terrestrial laser. We are interested in determining the trajectory of the sail as seen from the reference frame of the external power source. Actually, this a geometrical problem of special relativity: find the sail's worldline L ≡ (X µ (τ )) µ=0,··· ,3 = (cT (τ ), X(τ ), Y (τ ), Z(τ )) (with (cT, X, Y, Z) cartesian coordinates in laser's frame and τ the sail's proper time) that satisfies the following equations of motion where p µ and F µ are respectively the sail's fourmomentum and the four-force (in units of power) acting on it in laser's frame and T is the source's proper time.
It will later be useful to consider the equations of motion expressed in terms of the sail's proper time where a dot denotes a derivative with respect to proper time τ and the four-force in these units is given by is the well-known Lorentz factor accounting notably for time dilation between the sail and the source. To determine the worldline L, one needs to remember that the sail's four-momentum p µ is related to the tangent vector λ µ ≡ dX µ cdτ and the sail's rest mass by p µ = m(τ )·c·λ µ (τ ), in general with variable rest mass m· The tangent vector is a unit time-like four-vector in spacetime, λ µ λ µ = −1 (with a signature (−, +, +, +) for Minkowski's metric) whose derivative, the four-accelerationλ µ is orthogonal to it: λ µλ µ = 0, or in other words that the fouraccelerationλ µ is a spacelike vector. The sail's rest mass m(τ ) is defined by the norm of the four-momentum: p µ p µ ≡ −m 2 (τ )c 4 , and is in general not constant under specific conditions due to the accelerating motion. To emphasize this important feature, we can make use of the relations above to reformulate the equations of motion (2) as the following system: which are non-linear in λ µ · Therefore, it is obvious that a constant rest mass requires the four-force f µ to be everywhere orthogonal to the tangent vector λ µ which includes the free-motion (f µ = 0) as a trivial particular case. In general, the rest mass is therefore not constant as is the case when absorption of radiation by the sail occurs. We also refer the reader to [15,28] for several models of photon rockets with varying rest mass.
Another important additional property arises when one considers photon rockets [15], i.e. when the driving power f µ is provided by some incoming or outgoing radiation beam, which corresponds to absorption and emission photon rockets respectively. A light sail is actually a combination of both cases. During absorption and emission processes, the total four-momentum p µ tot = p µ + P µ of the system rocket (p µ ) + beam (P µ ) is conserved,ṗ µ +Ṗ µ = 0 or, equivalently, that f µ = −Ṗ µ . Then, since the rest mass of the photon always vanishes P µ P µ = 0, taking its derivative d(P µ P µ )/dτ = 0 yields which will allow us to build consistent models for the radiation-reaction four-force (f µ ) = (f T , c f ) (with f T the power associated to the thrust f ). Without loss of generality, one can write down the ansatz (P µ ) = E γ /c(±1, n γ ) where E γ stands for the beam energy, the sign ±1 refers to absorption and emission processes respectively and n γ is a unit spatial vector pointing in the direction of the beam propagation. Then, one finds from Eq.(4) that An immediate application of this property arise when one considers the straight motion of a photon rocket, for which (λ µ ) ≡ γ(1, β, 0, 0) (β = tanh(ψ) with ψ the rapidity) for a motion in the X−direction. Then, from Eq. (5), we have that f T (τ ) = ±cf X (τ ) and the equations of motion Eq.(3) now reduce to which can be directly integrated to give m(τ ) = m 0 exp(±ψ). The rest mass is therefore higher (lower) than m 0 = m(τ = 0) for an absorption (emission) photon rocket. This result can be put under the following familiar form where ∆V = β · c is the velocity increase from rest. Eq. (6) is nothing but a relativistic generalization of the Tsiolkovsky equation for photon rockets, as obtained for the first time by Ackeret in [29].
With these general elements on photon rockets, we can now build several models of light sails, either perfectly reflecting ("white") sails, perfectly absorbing ("black") sails or partially reflecting ("grey") sails.
A. The non-rectilinear motion of the perfectly reflecting light sail In this section, we will generalize the light sail model of Marx-Redding-Simmons-McInnes [22][23][24] for rectilinear motion to any arbitrary motion of the light sail. To compute the sail's trajectory from the equations of motion Eqs. (1)(2)(3), one needs a model for the driving fourforce f µ that the radiation beam applies on the sail. The propulsion of a reflecting light sail is twofold: first, photon absorption communicates momentum to the sail and second, photon emission achieves recoil of the sail. The case of a perfectly reflecting "white" light sail corresponds to no variation of the rest mass (internal energy). According to Eq.(3), this happens under the condition that λ µ f µ tot = 0 where the total four-force f µ tot is the sum of the four-forces due to incident and reflected beams f µ in and f µ ref .
Since we have that (λ µ ) = γ(1, β n s ) (with n s a unit spatial vector pointing in the direction of the sail's motion), the condition of keeping constant the sail's rest mass yields which is different than Eq.(5) since n s = n b .
Let us now focus on the time-component of the four-force f T which represents the time variation of sail's energy due to the twofold interaction with the beam. The infinitesimal variation of sail's energy due to the incident beam is given by dE in = (I.A/c)dw where w = cT − || R|| is the retarded time, I is the intensity of the beam (in W/m 2 ) and A the sail's reflecting surface. Indeed, only the radiation belonging to the past light cone of the sail can contribute to the four-force.
where we set P = I.A the emitted power in source's frame. Similarly, the infinitesimal variation of sail's energy due to the reflected beam is due to the radiation of the future light cone: where P = I .A with I the intensity of the reflected light beam.
We can now turn on the thrusts f (in,ref) . Since the total four-momentum of the sail and the beam is conserved during each process of radiation emission and absorption that constitutes the reflexion, the thrusts f (in,ref) must verify Eqs. (4,5), such that we have From Snell-Descartes law of reflexion, we have that Putting Eqs.(8-10) and the above equation into the constraint of constant rest mass Eq. (7), we find (11) together with the following expressions for the components of the four-force: In the particular case of straight motion, one has that θ = 0 ( n in = − n ref = n s ) and the four-force reduces to which are the same equations of motion than in [24]. The last of these equations was integrated numerically in [12][13][14] and also used as a starting point of [16]. However, when the power of the incident radiation beam P is constant, there is a simple analytical solution. Indeed, reexpressing the system Eqs. (13) in terms of the rapidity ψ (γ = cosh(ψ) and β = tanh(ψ)) gives the single equatioṅ ψ = 2P mc 2 exp(−2ψ) whose solution for a sail starting at rest at τ = 0 is ψ = 1 2 log (1 + 4s) (14) or, maybe more conveniently, in terms of the velocity in laser's frame: where the dimensionless time s is given by s = τ /τ c with τ c = mc 2 /P the characteristic time of the light sail travel. For constant power of the incident radiation beam, the light sail takes an infinite amount of time to reach the speed of light, its terminal velocity.
Let us now focus on non-rectilinear motion of the light sail which is described by the equations of motion with the four-force Eqs. (3,12). We recall the ansatz for the tangent vector: (λ µ ) = γ(1, β n s ) with n s a unit spatial vector in the direction of the sail's motion and introducing the rapidity ψ as γ = cosh(ψ) and β = tanh(ψ). Let us consider the X-axis as the line between the light source located at the origin of coordinates and the destination. Due to Snell-Descartes reflexion law, the three vectors n s , the direction of sail's motion (also normal to the sail's surface) and n in,ref the directions of the incident and reflected radiation beams respectively are coplanars. We can therefore choose the Y-axis so that this plane corresponds to the (X, Y ) plane without loss of generality. The incident radiation beam is emitted from the source and later hits the sail, so that n in = R/|| R|| with R the sail's position vector.
The direction of motion n s does not necessarily points towards the destination and one might have to use this model to compute course correction. Let us denote by θ the angle of incidence of the radiation beam and the sail's surface, n s • n in = cos(θ) = − n s • n ref (where the last equality is due to Snell-Descartes reflexion law), and denote by φ the angle between the sail's direction of motion and the destination n s • e X = cos(φ)· We can therefore work with the following ansatz:  Thanks to these parameterisations, the equations of motion Eqs. (3,12) now reduce tȯ where R 2 = X 2 + Y 2 is the distance from the source to the sail and where P , the power of the incident radiation beam, is an arbitrary function. The sail's rest mass m is constant in the case of a perfectly reflecting "white" sail. These equations will be investigated further in next section on applications.

B. Relativistic motion of a Perfectly Absorbing Light Sail
The case of a perfectly absorbing "black" sail corresponds to a perfectly inelastic collision between the photons of the incident radiation beam and the sail, in which the total energy of the beam is converted into both internal and kinetic energy of the sail. As a consequence, the rest mass of the sail is no longer constant. The four-force (f µ ) = (f T , c f ) (with f the thrust) acting on the sail is now given by the driving power: (see previous section) and the thrust: according to Eq.(4). Therefore, the motion of the black sail is rectilinear and directed along the direction n in of the incident radiation beam.
Without loss of generality, we can identify the X-axis to the direction of destination as viewed from the source. The vector n t in = n t s ≡ (cos(φ), sin(φ), 0) is collinear to the sail's position vector and the tangent vector to the sail's worldline is given by (λ µ ) = (cosh(ψ), sinh(ψ) n in ) so that the corresponding equations of motion Eqs. (3) can be written downṘ = c sinh(ψ) (20) with R is the euclidean distance to the source and φ = cst since the motion is rectilinear. Substracting the last two previous equations yields simply mc 2ψ = P e −2ψ and therefore while the rapidity is given by integrating the simple equation for any given function P modeling the power of the incident radiation beam.
In the case of constant power P , the above relations can be directly integrated to give the following solutions for the evolutions of the rest mass m and the rapidity ψ m = m 0 (1 + 3s) with s = τ /τ c the proper time in units of the characteristic time τ c = m 0 c 2 /P and where we assumed the sail starts at rest at τ = 0· The velocity in laser's frame is then simply given by which constitutes a useful relation to keep within easy reach for performing estimations.
However, one should also consider that the photons that have been absorbed by the black sail are thermally re-emitted through blackbody radiation, as is done in [13]. Due to sail's motion, the re-emission is anisotropic and produces a Poynting-Robertson drag [30] on the black sail as a feedback. This drag is simply given by c f P R = −P abs β n s where P abs = P (1 − β)/(1 + β) is the power absorbed by the black sail. From Eq. (5) and n s = n in for the black sail, ones finds that f T P R = c f P R • n in · Accounting both for the radiation pressure from the incoming radiation beam Eqs. (18,19) and the Poynting-Robertson drag, this leads to the following equations of motion for the black sail: for arbitrary power function P . It must be noticed that, even in the presence of thermal re-emission modeled by the Poynting-Robertson drag, the rest mass of the sail is not constant, as is tacitly (and unfortunately wrongly) assumed in [12][13][14]16]. These last equations for the black sail will be used to model the acceleration phase of the Starshot mission.

C. A General Model for arbitrary relativistic motion of Non-perfect Light Sail
So far we have been considering two extreme cases of a light sail : the perfectly reflecting case (the "white" sail), which has reflectivity = 1, and the perfectly absorbing one, which has = 0· In the first, the rest mass of the sail is constant while in the second, inelastic collisions lead to a variation of the internal energy (rest mass). We now need a model for any intermediate values of the reflectivity ∈ [0, 1]· Determining the sail's trajectory requires to integrate the vector field (λ µ ) tangent to the sail's worldline. However, the equations of motion are not linear in terms of the tangent vector λ µ , preventing to use directly our previous solutions in a simple linear combination. We must therefore exploit the linearity of the equations of motion with respect to the four-momentum to build such superposition. Indeed, let us write down the four-force acting on a "grey" sail with reflectivity as follows where f µ w,b stand for the four-forces of the particular cases of a perfectly reflecting (white) sail = 1 and a perfectly absorbing (black) sail = 0, respectively. According to the equations of motion Eqs. (2), the momentum can be splitted in the same way: since the four-momentum of the grey sail is related to the tangent vector of the sail's worldline by where the sail's rest mass is given by (η µν being the Minkowski metric with signature (−, +, +, +)). Since the trajectory of the sail is given by L = (cT (τ ), X(τ ), Y (τ ), Z = 0), finding the trajectory's unknowns T (τ ), X(τ ), Y (τ ) requires integrating the tangent vector field Λ µ = dX µ /(cdτ ) which is derived from the four-momentum p µ g (τ ) by Eqs. (33-34). The general solution for the grey sail, p µ g (τ ), with reflectivity can be obtained from the linear combination Eq.(30) of the particular solutions for the white and black sails p µ w,b (τ ): with φ = θ + Θ (Θ = arctan (Y /X)) and the auxiliary variables ψ w , m b , ψ b , θ are solutions oḟ where m 0 is the initial mass of the sail. The above equations accounts for the effect of radiation pressure alone in the particular case of the black sail. If one adds the Poynting-Robertson drag to the model of the black sail, the last two equations must be replaced bẏ In total, one must solve a system of six (seven when accounting for Poynting-Robertson drag) ordinary differential equations for the unknown functions (ψ w , θ, (m b ), ψ b , X, Y, T ) of the sail's proper time τ .
In order to determine the sail's trajectory, one must also specify a model for the power of the incident radiation beam P . We will consider here the following simple model for this function P , which was introduced in [14] where D(τ ) = X 2 (τ ) + Y 2 (τ ) + Z 2 (τ ) 1/2 is the timedependent euclidean distance between the source of the propelling radiation and the sail D max is the maximum distance at which the beam spot encompasses the sail's surface. This maximum distance is related to the sail's characteristic size R, the one of the beam source, r, the radiation wavelength λ by the following relation (cf. [14]): D max ≈ rR 2λ up to some order of unity geometrical factor depending on the shape of the beam source. In this model, the energy that propels the sail beyond the distance D max decays as the inverse of its distance to the source. A more sophisticated but also more realistic model for the beam power is the Goubau beam of [13], whose shape is qualitatively similar to the simple model of [14] used here.
We can illustrate this procedure by deriving the analytical solution for the straight motion of a grey sail, without the Poynting-Robertson drag. We found this last of little impact in practice, though it is included in the numerical simulations of the Starshot mission in the next section. The analytical model below can therefore be used for quickly computing estimations of the rectilinear trajectory. The tangent vector for motion along X is given by (Λ µ ) = γ(1, β, 0, 0), such that n T s = (1, 0, 0) = n T in = − n T ref .
The four-force acting on the grey sail is given by the decomposition Eq. (29). The white sail component of the four-force is given by (13). The black sail component of the four-force includes only the contribution of the incident radiation beam Regrouping all these elements into Eq.(29), one can express the four-force acting on the grey sail in terms of the rapidity ψ as (37) The equations of motion in this case simply reduce tȯ or, equivalently, for arbitrary power P . In the simplest case of constant power P , the following analytical solution can be found for a grey sail with reflectivity , starting from rest with rest mass m 0 (s = τ /τ c with τ c = m 0 c 2 /P ). These results are consistent with the previously obtained particular solutions for = 1 and = 0·

III. APPLICATIONS
We propose here three applications of the original light sail model derived in previous section. First, we present the general dynamics of perfectly reflecting light sails, then we apply our model to flyby missions at relativistic velocities and finally to single trip with double-stage light sails.

A. Sailing at relativistic velocities
Let us consider the general motion of perfectly reflecting "white" light sails, as given by Eqs. (17). In this model, the sail is reflective on both sides of its surface. Fig. 2 presents several trajectories of light sails coming from infinity at τ → −∞ and passing closest to the laser source at τ = 0 with distance R 0 = R min for different velocities β 0 and inclination θ 0 . After the closest approach, the sail is deflected by the light source depending on sail's velocity at closest approach ψ 0 and sail's  Figs. 3 and 4. First, this system admits only one fixed point (θ = π/2, ψ w = 0) which is an unstable saddle point. Indeed, a linear stability analysis of system (17) indicates that the jacobian of the right hand side has two eigenvalues of opposite signs (λ 1,2 = (1 ± 1 + 4/R)/2 ; R ≥ 0). This unstable equilibrium point is surrounded by two attractors (θ → 0, ψ w → 0) and (θ → π, ψ w → −2) ((θ → π, ψ w → 0) for power decaying as 1/R 2 ).   4: Phase diagram in the plane (θ, ψ w ) of the perfectly reflecting "white" sail. Left plot is for constant power P while right plot shows the case of driving power P decaying as 1/R 2 . The dot and the triangles respectively indicate the location of the unstable saddle point (θ = π/2, ψ w = 0) and the attractors (θ → 0, ψ w → 0) and (θ → π, ψ w → −2) ((θ → π, ψ w → 0) for decaying power). Straight lines corresponds to β0 = 0 and dashed lines to β0 = 0.9 This singular configuration of the phase space, with two attractors surrounding an unstable equilibrium makes the laser-sailing at relativistic speed an intricate and delicate discipline.

B. Acceleration phase of a nano-probe interstellar mission
Let us apply our results to the Starshot mission [12-14, 16, 26]. This project aspires to send tiny light sails of one gram mass-scale towards Proxima Centauri for a fly-by at a cruise velocity of about 20% that of light. The extreme kinetic energy would be provided by a gigantic ground-based laser during an acceleration phase lasting a few hours. The probes will then freely fly toward Proxima Centauri which they should reach within about 20 years. Let us therefore consider a rest mass at start m 0 of one gram and a powerful laser source emitting light with power P 0 = 4GW. We also assume, following [14], that the size of the laser source is 10km and that of the sail is 10m so that the maximal distance. If the laser's wavelength is 1064nm, then the maximum distance D max until which the laser beam completely encompasses the sail is about 0.3AU. The characteristic time τ c = m 0 c 2 /P 0 of this system is about 6.2 hours. We can now study the acceleration phase of such spacecrafts starting from rest as a function of the sail's reflectivity , with the model derived in previous section (including Poynting-Robertson drag).
The first point to be addressed is the aiming accuracy to reach such a far-away destination as Proxima Centauri. Indeed, at start the light sails might not be perfectly perpendicular to the incident radiation beam nor to the direction of destination. As before, let us denote by θ and Φ the angles between the vector n s normal to the sail's surface and the incident radiation beam n in = R/|| R|| and the direction e X from the laser's source to destination. One last angle is Θ which gives the position of the light sail with respect to destination. One good news is that any small misalignment will be quickly corrected naturally during the acceleration phase. Indeed, Fig. 5 gives the evolution of the angles θ, Φ, Θ characterizing the grey sail's dynamics. One can see that the initial small misalignement θ 1 quickly vanishes after the start of the acceleration phase. This is due to the vicinity of the attractor (θ = 0, ψ = 0) presented in the previous section. For perfectly reflecting light sails ( = 1), Snell-Descartes reflexion law imposes Φ = Θ + θ at all times. Since θ goes to zero, this yields Φ → Θ → θ 0 = θ(τ = 0) so that the sail's velocity quickly aligns with the incident radiation beam and the sail's trajectory becomes radial, in the direction Θ 0 ≈ θ 0 · As illustrated in Fig. 5, we find numerically that the asymptotic direction of motion Θ ∞ can be fairly approximated by Θ ∞ = θ 0 when one deals with a grey sail of reflectivity . As a consequence, the transverse deviation at destination is given by Y ≈ R des θ 0 with R des the distance between the source and the destination. To give an idea, an initial misalignement θ 0 of only one arcsec results in the case of Starshot in a deviation of 81 × astronomical units after R des = 4.4ly long trip. Our model allows also accounting for an initial error in aiming to destination, i.e. Θ 0 = 0, which leads to the same asymptotic value: Θ → θ 0 .
Since any small initial misalignement θ 0 is swept away by the driving force, the sail's trajectory shortly becomes close to rectilinear. Fig.7 represents the evolution of velocity in the source's frame, proper acceleration a = c · dβ/dτ felt by the light sail and distance R to the source for various values of its reflectivity . The velocity quickly saturates once the sail overcomes D max and the It must be pointed out that the Starshot space probes will experience an effective gravitational field (given by proper acceleration) as large as 2500 that of Earth during the first hour of acceleration toward their cruise velocity. During this extreme first hour they will cross the distance to D max , of about a third AU, and they will reach a distance equivalent to that of Jupiter in only 4 hours. The energy cost spent by the driving source E 0 = P 0 .T (with T ≈ 4h is the total duration of the acceleration phase) to achieve that is about 5.4 × 10 13 J or 12.9 kiloton (the equivalent of one Little Boy-class A-bomb) per Starshot probe! One can therefore ask what is the efficiency of such an energetic waste. This is given in Fig. 7 by examining the ratio of the total kinetic energy E K = (M g γ − m 0 )c 2 communicated to each probe to the source's energy cost E 0 . The efficiency is about 3% for white sails ( = 1) and close to 27% for black sails ( = 0). This might look counter-intuitive at first sight since perfectly absorbing sails reach a higher velocity (of about 0.2c) than perfectly absorbing ones (of about 0.15c), but this is due to the increase of the rest mass when = 1 as we shall see immediately. Although using perfectly reflecting white sails constitutes an (even) worse loss of energy, their main advantage lie in the shorter trip duration toward Proxima Centauri as shown in Fig. 7: white sails arrive at destination in about 22 years against 31 years for black sails, should they survive to the acceleration phase and the trip (cf. also [16]).
We can now conclude this analysis of the Starshot project by giving the evolution of the Starshot probes rest mass and the time dilation aboard during the acceleration phase, as is done in Fig.8. As explained above, perfectly reflecting white sails have constant rest mass while perfectly absorbing ones will exhibit an increase of their (inertial) rest mass by about 14% which is significant enough to consider for any hypothetic manoeuvre of the Starshot probes. Time aboard these relativistic probes will also elapse slower, by about 2% for white sails, which represent almost a difference of 137 days between on-board time and mission's control time at the end of the mission (22 years for = 1). From the point of view of the Starshot probes, this relativistic effect will shorten the effective trip duration by 137 days. This effect must be taken into account to wake up on-board instrumentation and starts the scientific programme at the right local time, otherwise the destination system will be missed by approximately 4.7 × 10 3 AU(= 0.2 × c × 137days). In the context of large scale directed energy propulsion, several interesting proposals for the exploration of the Solar System at sub-light velocities have been made (see for instance [17]). Among these, the idea of reconverting onboard the laser's illumination to drive an ion thruster leads to an efficient solution for high-mass missions. We do not pretend to propose a detailed Solar System mission here but rather we revisit Forward's idea of multi-stage lightsails with the tools established in this paper.
The major issue with laser-pushed lightsails, according to the pioneer G. Marx in [22], was the slow-down once approaching destination since this external propulsion cannot be reversed. That is why R. Forward suggested in [8] to use multi-stage lightsails: the large sail that is used during the acceleration phase separates at some point between an outer ring and a central breaking sail to which payload is attached. After separation, the laser source is still propelling the outer sail that stays on a accelerated trajectory while the payload reverses its own sail to catch the reflected light coming from the outer ring and uses its radiation pressure to reduce its speed.
We propose to revisit this idea with our model by giving a detailed trajectory of single trip inside the inner Solar sytem, beyond the simple description made in [8].
For the sake of simplicity, we will restrict ourselves to rectilinear motion. The abscissae of the outer ring and payload X o,p indicate their distance to the laser source.
To determine the trajectory of the double-stage light sail, we make use of equations of motion Eqs.(39, 40). During the acceleration phase, the inner payload sail and the outer ring sail are bound and evolve together at distance X o = X p and with total rest mass m tot = m o + m p . The light source illuminates the outer sail with power P o given by the following function (see also Eq.(36)): where P 0 is the power emitted by the light source located at the origin of coordinates. X 1 stands for the maximal distance at which the light source completely encompasses the outer sail and after which the illumination decreases as the inverse of the distance squared. At some point the outer ring sail separates from the inner payload ring which now uses its front reflective surface to collect light reflected from the outer sail to decelerate. The payload sails then enters a deceleration phase which is driven by the following power function: with P o is the power acting on the outer sail Eq.(43), X 2 is the maximal distance at which the light reflected from the sail completely covers the payload's sail located at X p and X o ≥ X p is the position of the accelerating outer sail. Fig. 9 presents a typical example of a rectilinear single trip performed with two-stage light sails. We consider a payload of rest mass m p = 20 tons, an outer sail of mass m o = 4 tons (hence a total mass of 24 tons) powered by a laser source of power P 0 = 10GW. Following [14], if we assume a sail of thickness of 1µm and a density of 1.4 g/cm 3 , then the radius of the outer sail with mass of 4 tons is ∼ 954m. We also assume the laser source is capable to illuminate the outer sail with constant power up to a distance X 1 of 5 AU after which the illumination starts decreasing with the inverse square of the distance to the source. Similarly, the outer sail is able to completely illuminate the inner payload sail up to a relative distance X 2 = X o − X p which we choose to be equal to 5 AU in the example of Fig. 9. The light source is used to both accelerate and decelerate the double-stage spaceship, which means that it must completely illuminate the outer sail over a large distance (here X 1 = 5 AU) covering both deceleration and acceleration. If the outer sail has a radius of 954 m as we have seen above, this could be achieved with directed energy system of radius close to 2 km, assuming an infrared laser of wavelength equal to 1064 nm (Nd:YAG laser, see also [14]).
After the separation, the lighter outer sail is freed from its heavier inner payload sail and therefore undergoes a stronger acceleration and a fastly increasing relative distance to the payload. This implies that the duration of the breaking phase is shorter than the acceleration one and that the decay of the illumination of the inner payload sail arrives earlier as the outer sail quickly goes out of the distance X 2 at which the light from the outer sail completely illuminates the inner payload sail. The lighter the outer sail the more important its post-separation acceleration and the more important this effect.
In the example illustrated in Fig.9, the acceleration phase of the double-stage 24 tons spaceship lasts for 4.2 months, after which it reaches a cruise velocity of about 30 km/s, almost doubling the current record of 17 km/s hold by the Voyager 1 probe. The separation of the payload from the outer ring sail then occurs and a breaking phase decelerates the payload for about 3 months. After that, the illumination from the outer sail quickly fades away, the outer sail has reached a velocity of about 160 km/s and a distance X o of 7 AU which is X 2 = 5 AU further than the distance reached by the payload. It takes about 7 months to the 20 tons payload to reach a distance of 2 AU, roughly the average distance separating planets Earth from Mars. The total energy spent by the laser source to make this trip possible is huge, around 2 × 10 17 J or 45 thousands kilotons. The efficiency of the directed energy system at the end of the acceleration phase is very poor, close to 10 −4 ·

IV. CONCLUSION
Among a mess of various ideas for making interstellar travel possible, some were not that fanciful that could have appeared on first sight. In fact, interstellar travel might well have been invented almost 60 years ago by Forward [9] and Marx [22]. Their vision was to use the then freshly realized laser to propel reflecting sails at relativistic velocities, towards the stars. This rather old idea has taken a long time to be fully explored within relativity. Curiously, Forward, although a renowned specialist of this discipline, did not push his idea very far into formalism and detailed computations in the relativistic limit in his papers on the subject. A relativistic model for the straight motion of a perfectly reflecting light sail was introduced by Marx in 1966 [22], then seriously corrected by Redding in 1967 [23] before being revisited into its final form in 1992 by Simmons & McInnes in a pedagogical paper. Under the recent burst of interest accompanying NASA's Starlight and Breakthrough Initiative and other programs in the previous two decades, this restricted model has served as a basis for a variety of interesting extensions [10, 12-14, 17, 27]. Unfortunately, these attempts did not revise the fundamentals of the Marx-Redding-Simons-McInnes model into special relativity which lead to some incompleteness of the recent models and sometimes confusing presentations. For instance, one major drawback of these recent papers is that they tacitly assume that the rest mass of absorbing sails is constant, which is only true for specific conditions on the four-force, manifestly not full-filled or explicited in these models. In addition, the non rectilinear motion of lightsail has not been investigated so far. This paper introduced the appropriate formalism to go beyond this situation, by deriving the general model for the non rectilinear motion of partially reflecting "grey" lightsails starting from general principles in special relativity. As part of the family of photon rockets, the lightsails have to deal with four-forces that obey several constraints and exhibit several interesting properties, including the variation of their rest mass when inelastic collisions with the incoming radiation occur. The general model of "grey" sails is build on a combination of two particular cases: first, the one of perfectly reflecting "white" sails which has a general planar motion and second, the one of perfectly absorbing black sails whose motion is along the direction of the incoming radiation beam and ruled by the push of external radiation pressure and the drag force from Poynting-Robertson effect. Our model requires numerical integration although in simplistic cases analytical solutions allow crude approximations.
We also presented three applications of our model. First, sailing at relativistic velocities has been shown to be intricate due to the instability of the equilibrium when the sail stands parallel to the incoming beam. Whatever the initial conditions outside of this unstable point, any lightsail will naturally relax toward two possible positions with the velocity anti or co-linear with the direction of the incoming beam both with c as asymptotic speed. This behaviour will allow lightsails to spontaneously align with the incoming beam while accelerated, yielding a prediction for a deviation from their initial aim. Second, we provide the predictions of our model on the Starshot mission, within some generic non-restrictive assumptions that could be easily adapted if desired. The model accounts for initial deviations of the sail's position and velocity vectors from the direction of destination, which produces non-negligible deviations at arrival. To show this, we have provided a simple example with a small initial misalignement of one arcsec amplitude and our predictions on lightsail's velocity, proper acceleration and distance are comparable to those of previous works [12][13][14], although accounting for transverse deviation and rest mass variation when the sail is partially absorbing. There are several important points for mission design that have been derived here: aiming accuracy, increase of inertial mass for manoeuvring, time dilation effect and wake-up time of the probe's internal systems, propulsion efficiency and the need to seriously consider power-recycling to improve it. Third, we provide a very simple model of a single trip in the Solar System with ton-scale two-stage lightsail and GW-scale laser, to illustrate the potential interest of this technique for interplanetary exploration.
Of course, the results presented here are not exhaustive and should be carefully extended to a more realistic numerical modeling prior to any directed energy mission. While our model is valid for non-rectilinear motion and general sail's reflectivity, there are several physical effects that must be added to compute a precise trajectory of such an intricate (and costly) mission towards far-away planets or even stars. This includes modeling of the environment: incoming beam properties and interaction with the sail, gravitational influences, solar system constraints (e.g. a free path from a groundbased laser and a sail), interplanetary and interstellar media, magnetic fields, anisotropic thermal radiation, to name but a few. Several papers have already started investigating some of these effects (see for instance [13,16] for a review), which could now benefit from the general model presented here. Another concern of the authors are telecommunication and astronavigation issues which must take into account several effects from general relativity (see [15] for a first approach in the case of straight motion of photon rockets). Following [13], we agree that a serious effort of physical and numerical modelling, beyond simplistic models sometimes solved using spreadsheets, must be made prior to any launch. We also think that small scale ground-based experiments and flying prototypes will be necessary to calibrate the models and adjust numerical simulations (see [13] for an interesting suggestion).
The key idea of Forward and Marx that could well make interstellar travel possible one day is undoubtly the externalization of the propulsion's energy source. Since Forward already envisions in [8], the most obvious primary energy source for that purpose is the inexhaustible amount of energy radiated by the Sun. But this means it has to be collected by huge amounts, probably on a planetary scale, and converted into huge and radiation beams that will be efficient for propulsion but potentially also to other dangerous aims. The sending of one single tiny probe, of mass one gram, toward Alpha Centauri individually requires as much energy as the one delivered by several kilotons of TNT. Besides this frightening energy waste, cost estimation for such program is of the order of 10 billion dollars [13]. Of course, it is natural to wonder if the stars are worth these sacrifices, no matter their appealing beauty to the stargazer. To some respects, like exploration of the unknown and maybe one day our own survival, they certainly are. Yet, it is a charming idea that the key to reach the stars could lie in the glare of their close and lovely yellow dwarf cousin.