Sailing Towards the Stars Close to the Speed of Light
SSailing Towards the Stars Close to the Speed of Light
Andr´e F˝uzfa ∗ and Williams Dhelonga-Biarufu, Olivier Welcomme Namur Institute for Complex Systems (naXys), University of Namur,Rue de Bruxelles 61, B-5000 Namur, Belgium (Dated: July 15, 2020)We present a relativistic model for light sails of arbitrary reflectivity undertaking non rectilinearmotion. Analytical solutions for a constant driving power and a reduced model for straight motionwith arbitrary sail’s illumination are given, including for the case of a perfectly reflecting light sailexamined in earlier works. When the sail is partially absorbing incoming radiation, its rest massincreases, an effect discarded in previous works. Is is shown how sailing at relativistic velocities isintricate due to the existence of an unstable fixed point, when the sail is parallel to the incomingradiation beam, surrounded by two attractors corresponding to two different regimes of radial es-cape. We apply this model to the Starshot project by showing several important points for missiondesign. First, any misalignement between the driving light beam and the direction of sail’s motionis naturally swept away during acceleration toward relativistic speed, yet leading to a deviation ofabout 80 AU in the case of an initial misalignement of one arcsec for a sail accelerated up to 0 . × 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%), thetrip duration (between 22 and 33 years), the rest mass variation (up to 14%) and the time dilationaboard (about 140 days difference) are all presented and their variation with sail’s reflectivity isdiscussed. We also present an application to single trips within the Solar System using Forward’sidea of double-stage lightsails. A spaceship of mass 24 tons can start from Earth and stop at Marsin about 7 months with a peak velocity of 30 km/s but at the price of a huge energy cost of about5 . × GWh due to extremely low efficiency of the directed energy system, around 10 − in thislow-velocity case. PACS numbers:
I. INTRODUCTION
The discovery of the cruel vastness of space beyondthe Solar System is rather recent in human history,dating back from the measurement of the parallax ofthe binary star 61 Cygni by Bessel in 1838 [1]. Thiswas even pushed orders of magnitude further by thework of Hubble on intergalactic distances in the 1920s[2, 3]. It is also at the beginning of the XXth centurythat space exploration with rockets based on reactionpropulsion began scientifically considered by pioneerslike Tsiolkowksy, Oberth, Tsander, Goddard to namebut a few. While interplanetary travel was the primarymotivations of their work, dreams of interstellar travelappeared to these precursors coincidentally. Althoughnot theoretically impossible, interstellar travel has everbeen largely considered unfeasible, for a variety ofgood reasons. The major drawback lie in the hugegap between interplanetary and interstellar distances,initially settled by Bessel’s discovery: the distance tothe nearest star system, Alpha Centauri, is roughly tenthousand times the distance to planet Neptune, on theoutskirts of our solar system. It took about 40 years tothe fastest object ever launched by humans, the Voyager1 space probe, to reach the edges of the Solar System18 billion kilometers away, at a record cruise velocity ∗ Electronic address: [email protected] of 17 km/s. Millenia long trips would then be neededto cross interstellar distances. Since Bessel’s epoch, weknow that the gap to the stars lies in these four ordersof magnitudes, needless to mention the much larger gapto the galaxies.There have been many suggestions to go across thestars, and we refer the reader to the reference [4] foran overview, some of which could even be considered asplausible, yet unaffordable, while some others are simplynon 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-lighttravels. It seems to us that only the first category isrelevant for plausible discussions, and the current paperinvestigates maybe the most plausible (or should wesay the least doubtful) proposal among this category:directed energy propulsion. Generation ships are hugeautonomous space stations embarking a whole popula-tion at sub-light velocities and only the far descendantscan hopefully arrive at destination. The main difficultyof these proposals is to maintain (intelligent) life in theark during millenia-long trips. Even if launched on aballistic trajectory towards the stars, generation shipsstill require to embark a consequent amount of energyto maintain their internal biotopes, since there is toofew available energy in the deserts of interstellar space.A 10 GW power source operating during a 10 000 yearslong trip amounts to about five times the world annual a r X i v : . [ phy s i c s . pop - ph ] J u l energy production to be stored aboard. Spacetimedistortions like wormholes and warp drives [5–7] areindeed based on general relativity but also require exoticmatter with (strongly) negative pressures – like darkenergy – produced or gathered in large amounts andstored into extreme densities. Indeed, curving spacetimeis a situation of strong gravitational field which requireshigh compactness, a dimensional quantity given by theratio GM/ ( c L ) (with G Newton’s constant, M the restmass of the system, c the speed of light in vacuum and L the characteristic size of the system) that measuresthe strength of a gravitational system. Even withdark energy at profusion, the corresponding amountof energy that should be involved is of order c L/G ,so that producing a one-km wide spacetime distortionrequires spending about 10 the world annual energyproduction, all into such exotic avatar of matter. Finally,faster-than-light travel is not well scientifically soundsince this is forbidden by conventional physics (specialrelativity). And, most of all, this is completely uselesssince time dilation at relativistic velocities will shortentrip duration for the traveler, unless one covets atmanaging some galactic empire or some other appealingspace opera ideas. This last category should be viewedas the least serious of all, unless laws of physics bringone day considerable surprises.Let us therefore focus on relativistic reaction propulsion,which has also received many scientifically sound con-siderations. To achieve velocities close to that of light,the propulsion system must bring energies in amountsclose to that of the rest mass of the ship: E = m c .This requires to go well beyond chemical rockets to turnnaturally to concepts involving nuclear energy (fissionor fusion), antimatter rockets (direct or indirect) orbeam-powered directed energy propulsion. This last isbased on radiation pressure and consists of using theimpulse provided by some external radiation or particlebeam to propel the space ships. According to manyauthors [4, 8–17], this is maybe the most promising onefor three reasons: (1) it does not require embarking anypropellant, (2) they allow reaching higher velocities thanrockets expelling mass and submitted to the constraintsof Tsiolkowsky equation and (3) they benefit from astrong theoretical and technical background includingsuccessful prototypes [17]. However, maybe the majordrawback of directed energy propulsion lies in their weakefficiency: the thrust imparted by a radiation beamilluminating some object with power P is of order P/c .Roughly speaking, one Newton of thrust requires anillumination on the sail of at least 300 MW.The idea of using the radiation pressure of sun-light to propel reflecting sails dates back from the earlyages of astronautics and was suggested by Tsanderin 1924 (see [8] for a review of the idea). Severalproposals have been emitted for using solar sails forthe exploration of the inner Solar System (for the first proposals, see [18, 19]), reach hyperbolic orbits with theadditional thrust provided by sail’s desorption [20] oreven tests of fundamental physics [21]. Space probes likeIkaros and NanoSail-D2 have successfully used radiationpressure from sunlight for their propulsion. However,since solar illumination decreases as the square of thedistance, this method is interesting for exploring theinner solar system but not for deep space missions. Thefirst realisation of a laser in 1960 really opened the wayto consider using them in directed energy propulsion,an idea first proposed by Forward in 1962 [9], sincelaser sources are coherent sources of light with largefluxes, allowing one to consider sending energy over largedistances toward a space ship. The Hungarian physicistMarx proposed in a 1966 paper in Nature [22] the sameidea independently of Forward, together with the firstrelativistic model describing the straight motion of sucha laser-pushed light sail. This paper was quickly followedby another one by Redding [23] in 1967 that correctedone important mistake made by Marx in the forcesacting on the lightsail. Twenty five years later, Simmonsand McInnes revisited Marx’s one dimensional model in[24], extending its model to variable illuminating power,examining the efficiency of the system and how recyclingthe laser beam with mirrors could increase it.Forward’s idea of laser-pushed lightsails for inter-stellar journey has known a strong renewal of interestsince 2009 through successive funded research programsthat are still active. As soon as 2009, joint NASA-UCSBStarlight [25] investigates on the large scale use ofdirected energy to propel spacecrafts to relativisticspeeds, including small wafer scale ones. In 2016, theBreakthrough Starshot Initiative [26] has been initiatedto focus on wafer scale spacecrafts and interstellarfly-bys to Alpha Centauri with the objective of achievingit before the end of the century. A nice review onlarge scale directed energy application to deep spaceexploration in the Solar System and beyond, includingmany detailed engineering aspects, can be found in [17].In the context of these research programs, severalauthors have started modeling Solar System and in-terstellar missions based on this concept. Some basicone dimensional modeling of directed energy propul-sion, accounting for some relativistic effects as well asprospective applications for solar system explorationcan already been found in [27]. In [10], one can finda model for an single trip from Earth to Mars withmicrowave beam-powered lightsail but without givingthe details of their model. The authors of [12–14, 16]based their results on the model by Marx, Redding,Simmons and McInnes [22–24] which is only validfor one-dimensional (rectilinear) motion of perfectlyreflecting light sails. The case of arbitrary reflectancehas not been correctly considered so far: this involvesinelastic collisions between the propulsive radiationbeam and the lightsail, leading to an increase of the restmass of the last, as was already shown in [15, 28] fordifferent types of photon rockets. The authors of [12, 14]provided interesting modeling of the early accelerationphases, non-relativistic regime and power recycling inthe rectilinear motion case but did not provide the ana-lytical solutions in terms of the rapidity as we do here.[16] started from the Marx-Redding-Simmons-McInnesmodel and investigated the possibility to use high-energyastrophysical sources to drive the lightsails duringtheir interstellar journey. Finally, [13] provided a morecomplete and realistic physical model of a lightsail’srectilinear motion, including notably thermal re-emissionof absorbed radiation through Poynting-Robertson effectand a model for development costs. Unfortunately, thisauthor also only considered one single equation of motionamong the two that are given by special relativity inthat case and, doing so, it was wrongly tacitly assumedthe rest mass of the sail remains constant when theincoming radiation is absorbed, which cannot be trueunless specific constraints. We will give here the impactof sail’s rest mass variation, notably in the case withPoynting-Robertson drag. In addition, non rectilinearmotion has not been considered so far, so that the effectof a misalignement of the incoming beam and sail’svelocity on the trajectory has not been investigated yet.This paper will propose a relativistic model for nonrectilinear motion of lightsails with arbitrary reflectivity,investigates their general dynamics and provide usefulapplications to the acceleration phase of fly-by missionsto outer space or single trips within the Solar Systemusing double-stage light sail.In Section II, we recall some fundamental proper-ties of photon rockets in special relativity, rederiving therelativistic rocket equation and apply them to lightsails.We then focus on the general motion of perfectly absorb-ing ”white” sails and give semi-analytical solutions forthe particular case of straight motion. Then, the caseof a perfectly absorbing ”black” lightsail is discussedand a model accounting for both radiation pressureof the incoming beam and Poynting-Robertson dragis established. Finally, we show how one can combinethe previous cases of white and black sails to builda model for the general motion of ”grey” light sailswith arbitrary reflectivity and useful semi-analyticalsolutions for straight motion is provided. In SectionIII, we apply previous results to (i) a dynamical systemanalysis of sailing at relativistic velocities, showing howintricate this discipline will be, (ii) the accelerationphase of the Starshot mission, providing many relatedphysical quantities such as sail’s inclination, distance,proper velocity and acceleration, rest mass, time dilationaboard, efficiency and trip duration and (iii) single tripsin the Solar-System with Forward’s idea of multi-stagelightsail [8]. We finally conclude in Section IV with someemphasis on the importance of the presented results andby giving some perspectives of this work.
II. GENERAL MOTION OF RELATIVISTICLIGHT SAILS
A light sail is a spacecraft propelled by the radiationpressure exerted on its reflecting surface by some inci-dent light beam. In the case of directed energy propul-sion, this radiation is provided by some external powersource, like an intense terrestrial laser. We are inter-ested in determining the trajectory of the sail as seenfrom the reference frame of the external power source.Actually, this a geometrical problem of special relativ-ity: find the sail’s worldline
L ≡ ( X µ ( τ )) µ =0 , ··· , =( cT ( τ ) , X ( τ ) , Y ( τ ) , Z ( τ )) (with ( cT, X, Y, Z ) cartesiancoordinates in laser’s frame and τ the sail’s proper time)that satisfies the following equations of motion c dp µ dT = F µ (1)where p µ and F µ are respectively the sail’s four-momentum and the four-force (in units of power) actingon it in laser’s frame and T is the source’s proper time.It will later be useful to consider the equations of motionexpressed in terms of the sail’s proper time c ˙ p µ = f µ (2)where a dot denotes a derivative with respect to propertime τ and the four-force in these units is given by f µ = γF µ where γ = dTdτ is the well-known Lorentz factor accounting notably fortime dilation between the sail and the source. To deter-mine the worldline L , one needs to remember that thesail’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 vectoris a unit time-like four-vector in spacetime, λ µ λ µ = − − , + , + , +) for Minkowski’s metric)whose derivative, the four-acceleration ˙ λ µ is orthogo-nal to it: λ µ ˙ λ µ = 0, or in other words that the four-acceleration ˙ λ µ is a spacelike vector. The sail’s restmass m ( τ ) is defined by the norm of the four-momentum: p µ p µ ≡ − m ( τ ) c , and is in general not constant underspecific conditions due to the accelerating motion. Toemphasize this important feature, we can make use ofthe relations above to reformulate the equations of mo-tion (2) as the following system: ˙ mc = − λ µ f µ mc ˙ λ µ = f µ + ( λ α f α ) λ µ ˙ X µ = cλ µ (3)which are non-linear in λ µ · Therefore, it is obvious thata constant rest mass requires the four-force f µ to beeverywhere orthogonal to the tangent vector λ µ whichincludes the free-motion ( f µ = 0) as a trivial particularcase. In general, the rest mass is therefore not constantas is the case when absorption of radiation by the sailoccurs. We also refer the reader to [15, 28] for severalmodels of photon rockets with varying rest mass.Another important additional property arises whenone considers photon rockets [15], i.e. when the drivingpower f µ is provided by some incoming or outgoingradiation beam, which corresponds to absorption andemission photon rockets respectively. A light sail isactually a combination of both cases. During absorp-tion and emission processes, the total four-momentum p µ tot = p µ + P µ of the system rocket ( p µ ) + beam ( P µ ) isconserved, ˙ p µ + ˙ P µ = 0 or, equivalently, that f µ = − ˙ P 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 f µ P µ = 0 (4)which will allow us to build consistent models for theradiation-reaction four-force ( f µ ) = ( f T , c (cid:126)f ) (with f T the power associated to the thrust (cid:126)f ). Without lossof generality, one can write down the ansatz ( P µ ) = E γ /c ( ± , (cid:126)n γ ) where E γ stands for the beam energy, thesign ± (cid:126)n γ is a unit spatial vector pointing in thedirection of the beam propagation. Then, one finds fromEq.(4) that f T = ± c (cid:126)f • (cid:126)n γ (5)An immediate application of this property arise whenone considers the straight motion of a photon rocket, forwhich ( λ µ ) ≡ γ (1 , β, ,
0) ( β = tanh( ψ ) with ψ the rapid-ity) for a motion in the X − direction. Then, from Eq.(5),we have that f T ( τ ) = ± cf X ( τ ) and the equations of mo-tion Eq.(3) now reduce to (cid:40) ˙ mc = f T ( τ ) exp( ∓ ψ ) mc ˙ ψ = ± f T ( τ ) exp( ∓ ψ )which can be directly integrated to give m ( τ ) = m exp( ± ψ ). The rest mass is therefore higher (lower)than m = m ( τ = 0) for an absorption (emission) pho-ton rocket. This result can be put under the followingfamiliar form ∆ V = c (cid:12)(cid:12)(cid:12)(cid:12) m − m m + m (cid:12)(cid:12)(cid:12)(cid:12) (6)where ∆ V = β · c is the velocity increase from rest.Eq. (6) is nothing but a relativistic generalization of theTsiolkovsky equation for photon rockets, as obtained forthe first time by Ackeret in [29].With these general elements on photon rockets, wecan now build several models of light sails, eitherperfectly reflecting (”white”) sails, perfectly absorbing(”black”) sails or partially reflecting (”grey”) sails. A. The non-rectilinear motion of the perfectlyreflecting light sail
In this section, we will generalize the light sail modelof Marx-Redding-Simmons-McInnes [22–24] for rectilin-ear motion to any arbitrary motion of the light sail. Tocompute the sail’s trajectory from the equations of mo-tion Eqs.(1-3), one needs a model for the driving four-force 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 sailand second, photon emission achieves recoil of the sail.The case of a perfectly reflecting ”white” light sail corre-sponds to no variation of the rest mass (internal energy).According to Eq.(3), this happens under the conditionthat λ µ f µ tot = 0 where the total four-force f µ tot is the sumof 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(cid:126)n s ) (with (cid:126)n s aunit spatial vector pointing in the direction of the sail’smotion), the condition of keeping constant the sail’s restmass yields f T tot = β · c ( (cid:126)f in + (cid:126)f ref ) • (cid:126)n s , (7)which is different than Eq.(5) since (cid:126)n s (cid:54) = (cid:126)n b .Let us now focus on the time-component of thefour-force f T which represents the time variation ofsail’s energy due to the twofold interaction with thebeam. The infinitesimal variation of sail’s energy due tothe incident beam is given by dE in = ( I.A/c ) dw where w = cT − || (cid:126)R || is the retarded time, I is the intensityof the beam (in W/m ) and A the sail’s reflectingsurface. Indeed, only the radiation belonging to the pastlight cone of the sail can contribute to the four-force.Setting β = || (cid:126)V || /c = d ( || (cid:126)R || ) / ( cdT ), we have that dw = c (1 − β ) dT and f T in = γ dE in dT = P (1 − β ) γ (8)where we set P = I.A the emitted power in source’sframe. Similarly, the infinitesimal variation of sail’s en-ergy due to the reflected beam is due to the radiation ofthe future light cone: dE ref = ( P (cid:48) /c ) du = P (cid:48) (1 + β ) dT where u = cT + || (cid:126)R || is the advanced time, yielding f T ref = γ dE ref dT = P (cid:48) (1 + β ) γ, (9)where P (cid:48) = I (cid:48) .A with I (cid:48) the intensity of the reflectedlight beam.We can now turn on the thrusts (cid:126)f (in , ref) . Since thetotal four-momentum of the sail and the beam isconserved during each process of radiation emission andabsorption that constitutes the reflexion, the thrusts (cid:126)f (in , ref) must verify Eqs.(4,5), such that we have c (cid:126)f (in , ref) = f T (in , ref) (cid:126)n (in , ref) · (10)From Snell-Descartes law of reflexion, we have that (cid:126)n in • (cid:126)n s = − (cid:126)n ref • (cid:126)n s ≡ cos( θ ) · Putting Eqs.(8-10) and the above equation into the con-straint of constant rest mass Eq.(7), we find P (1 − β )(1 − β cos( θ )) = − P (cid:48) (1 + β )(1 + β cos( θ )) (11)together with the following expressions for the compo-nents of the four-force: f T tot = 2 P γβ cos( θ ) (cid:18) − β β cos( θ ) (cid:19) c (cid:126)f tot = P γ (1 − β ) (cid:18) (cid:126)n in − − β cos( θ )1 + β cos( θ ) (cid:126)n ref (cid:19) (12)In the particular case of straight motion, one has that θ = 0 ( (cid:126)n in = − (cid:126)n ref = (cid:126)n s ) and the four-force reduces to mc dγdτ ≡ f T tot = 2 P γβ (cid:18) − β β (cid:19) mc d ( γβ ) dτ ≡ cf X tot = 2 P γ (cid:18) − β β (cid:19) (13)which are the same equations of motion than in [24]. Thelast of these equations was integrated numerically in [12–14] and also used as a starting point of [16]. However,when the power of the incident radiation beam P is con-stant, there is a simple analytical solution. Indeed, re-expressing the system Eqs.(13) in terms of the rapidity ψ ( γ = cosh( ψ ) and β = tanh( ψ )) gives the single equation˙ ψ = 2 Pmc exp( − ψ )whose solution for a sail starting at rest at τ = 0 is ψ = 12 log (1 + 4 s ) (14)or, maybe more conveniently, in terms of the velocity inlaser’s frame: β = Vc = 2 s s (15)where the dimensionless time s is given by s = τ /τ c with τ c = mc /P the characteristic time of the lightsail travel. For constant power of the incident radiationbeam, the light sail takes an infinite amount of time toreach the speed of light, its terminal velocity.Let us now focus on non-rectilinear motion of thelight sail which is described by the equations of motionwith the four-force Eqs.(3,12). We recall the ansatzfor the tangent vector: ( λ µ ) = γ (1 , β(cid:126)n s ) with (cid:126)n s aunit spatial vector in the direction of the sail’s motionand 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 co-ordinates and the destination. Due to Snell-Descartesreflexion law, the three vectors (cid:126)n s , the direction of sail’smotion (also normal to the sail’s surface) and (cid:126)n in , ref thedirections of the incident and reflected radiation beamsrespectively are coplanars. We can therefore choosethe Y-axis so that this plane corresponds to the ( X, Y )plane without loss of generality. The incident radiationbeam is emitted from the source and later hits the sail,so that (cid:126)n in = (cid:126)R/ || (cid:126)R || with (cid:126)R the sail’s position vector.The direction of motion (cid:126)n s does not necessarily pointstowards the destination and one might have to use thismodel to compute course correction. Let us denote by θ the angle of incidence of the radiation beam and thesail’s surface, (cid:126)n s • (cid:126)n in = cos( θ ) = − (cid:126)n s • (cid:126)n ref (where thelast equality is due to Snell-Descartes reflexion law),and denote by φ the angle between the sail’s directionof motion and the destination (cid:126)n s • (cid:126)e X = cos( φ ) · We cantherefore work with the following ansatz: (cid:126)n Ts = (cos( φ ) , sin( φ ) , (cid:126)n T in = (cos( φ − θ ) , sin( φ − θ ) , (cid:126)n T ref = ( − cos( φ + θ ) , − sin( φ + θ ) ,
0) (16) θ = φ − arctan (cid:18) YX (cid:19) Figure 1 provides an illustration of the triplet of unitvectors (cid:126)n s, in , ref and the angles used. FIG. 1: Sketch of the perfectly reflecting light sail: the lightsource is located at the origin of coordinates, the incidentbeam hits the sail with an angle θ w.r.t. the velocity of thesail before it is reflected following the same angle, due toSnell-Descartes reflexion law Thanks to these parameterisations, the equations ofmotion Eqs.(3,12) now reduce to˙ ψ = 2 Pmc (cid:18) − β β cos( θ ) (cid:19) cos( θ )˙ θ = − (cid:16) ˙ ψ + cR sinh( ψ ) (cid:17) sin( θ ) (17)˙ R = c sinh( ψ ) cos( θ )where R = X + Y is the distance from the source tothe sail and where P , the power of the incident radiationbeam, 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 nextsection on applications. B. Relativistic motion of a Perfectly AbsorbingLight Sail
The case of a perfectly absorbing ”black” sail corre-sponds to a perfectly inelastic collision between the pho-tons of the incident radiation beam and the sail, in whichthe total energy of the beam is converted into both inter-nal and kinetic energy of the sail. As a consequence, therest mass of the sail is no longer constant. The four-force( f µ ) = ( f T , c (cid:126)f ) (with (cid:126)f the thrust) acting on the sail isnow given by the driving power: f T in = P (1 − β ) γ (18)(see previous section) and the thrust: c (cid:126)f = f T in (cid:126)n in , (19)according to Eq.(4). Therefore, the motion of the blacksail is rectilinear and directed along the direction (cid:126)n in ofthe incident radiation beam.Without loss of generality, we can identify the X-axis tothe direction of destination as viewed from the source.The vector (cid:126)n t in = (cid:126)n ts ≡ (cos( φ ) , sin( φ ) ,
0) is collinear tothe sail’s position vector and the tangent vector to thesail’s worldline is given by ( λ µ ) = (cosh( ψ ) , sinh( ψ ) (cid:126)n in )so that the corresponding equations of motion Eqs.(3)can be written down˙ R = c sinh( ψ ) (20)˙ mc = P e − ψ (21) mc sinh( ψ ) ˙ ψ = P e − ψ (1 − cosh( ψ ) e − ψ ) (22) mc cosh( ψ ) ˙ ψ = P e − ψ (1 − sinh( ψ ) e − ψ ) (23)with R is the euclidean distance to the source and φ =cst since the motion is rectilinear. Substracting the lasttwo previous equations yields simply mc ˙ ψ = P e − ψ andtherefore m ( τ ) = m e ψ (24)while the rapidity is given by integrating the simple equa-tion de ψ dτ = 3 Pm c (25)for any given function P modeling the power of theincident radiation beam.In the case of constant power P , the above rela-tions can be directly integrated to give the following solutions for the evolutions of the rest mass m and therapidity ψ m = m (1 + 3 s ) / (26) ψ = 13 log (1 + 3 s ) (27)with s = τ /τ c the proper time in units of the character-istic time τ c = m c /P and where we assumed the sailstarts at rest at τ = 0 · The velocity in laser’s frame isthen simply given by β = Vc = ˙ Rcγ = (1 + 3 s ) / − s ) / + 1which constitutes a useful relation to keep within easyreach for performing estimations.However, one should also consider that the pho-tons that have been absorbed by the black sail arethermally re-emitted through blackbody radiation, asis done in [13]. Due to sail’s motion, the re-emission isanisotropic and produces a Poynting-Robertson drag [30]on the black sail as a feedback. This drag is simply givenby c (cid:126)f P R = − P abs β(cid:126)n s where P abs = P (1 − β ) / (1 + β )is the power absorbed by the black sail. From Eq.(5)and (cid:126)n s = (cid:126)n in for the black sail, ones finds that f TP R = c (cid:126)f P R • (cid:126)n in · Accounting both for the radiationpressure from the incoming radiation beam Eqs.(18,19) and the Poynting-Robertson drag, this leads to thefollowing equations of motion for the black sail:˙ mc = P e − ψ (cid:0) − tanh( ψ ) e − ψ (cid:1) mc ˙ ψ = P e ψ − e ψ + e ψ + 1 e ψ + e ψ (28)for arbitrary power function P . It must be noticed that,even in the presence of thermal re-emission modeled bythe Poynting-Robertson drag, the rest mass of the sailis not constant, as is tacitly (and unfortunately wrongly)assumed in [12–14, 16]. These last equations for the blacksail will be used to model the acceleration phase of theStarshot mission. C. A General Model for arbitrary relativisticmotion of Non-perfect Light Sail
So far we have been considering two extreme cases of alight sail : the perfectly reflecting case (the ”white” sail),which has reflectivity (cid:15) = 1, and the perfectly absorbingone, which has (cid:15) = 0 · In the first, the rest mass of the sailis constant while in the second, inelastic collisions leadto a variation of the internal energy (rest mass). We nowneed a model for any intermediate values of the reflectiv-ity (cid:15) ∈ [0 , · Determining the sail’s trajectory requiresto integrate the vector field ( λ µ ) tangent to the sail’sworldline. However, the equations of motion are not lin-ear in terms of the tangent vector λ µ , preventing to usedirectly our previous solutions in a simple linear combi-nation. We must therefore exploit the linearity of theequations of motion with respect to the four-momentumto build such superposition. Indeed, let us write downthe four-force acting on a ”grey” sail with reflectivity (cid:15) as follows f µg = (cid:15)f µw + (1 − (cid:15) ) f µb (29)where f µw,b stand for the four-forces of the particular casesof a perfectly reflecting (white) sail (cid:15) = 1 and a perfectlyabsorbing (black) sail (cid:15) = 0, respectively. According tothe equations of motion Eqs.(2), the momentum can besplitted in the same way: p µg = (cid:15)p µw + (1 − (cid:15) ) p µb (30)so that ˙ p µw,b = f µw,b (31)˙ X µ = p µg M c (32)since the four-momentum of the grey sail is related to thetangent vector of the sail’s worldline by p µg = M c Λ µ (33)where the sail’s rest mass is given by − M c = η µν p µg p νg (34)( 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’sunknowns T ( τ ) , X ( τ ) , Y ( τ ) requires integrating the tan-gent vector field Λ µ = dX µ / ( cdτ ) which is derived fromthe four-momentum p µg ( τ ) by Eqs.(33-34). The generalsolution for the grey sail, p µg ( τ ), with reflectivity (cid:15) canbe obtained from the linear combination Eq.(30) of theparticular solutions for the white and black sails p µw,b ( τ ): p µw = m c (cosh( ψ w ) , sinh( ψ w ) (cid:126)n ts ) p µb = m b c (cosh( ψ b ) , sinh( ψ b ) (cid:126)n t in ) (cid:126)n ts = (cos( φ ) , sin( φ ) , (cid:126)n t in = (cos(Θ) , sin(Θ) ,
0) = (cid:126)R t || (cid:126)R || (35)with φ = θ + Θ (Θ = arctan ( Y /X )) and the auxiliaryvariables ψ w , m b , ψ b , θ are solutions of˙ ψ w = 2 Pm c (cid:18) − tanh( ψ w )1 + tanh( ψ w ) cos( θ ) (cid:19) cos( θ )˙ θ = − (cid:18) ˙ ψ w + c √ X + Y sinh( ψ w ) (cid:19) sin( θ ) m b = m e ψ b ˙ ψ b = Pm c e − ψ b , where m is the initial mass of the sail. The above equa-tions accounts for the effect of radiation pressure alonein the particular case of the black sail. If one adds thePoynting-Robertson drag to the model of the black sail,the last two equations must be replaced by˙ m b c = P e − ψ b (cid:0) − tanh( ψ b ) e − ψ b (cid:1) m b c ˙ ψ b = P e ψ b − e ψ b + e ψ b + 1 e ψ b + e ψ b · In total, one must solve a system of six (sevenwhen accounting for Poynting-Robertson drag) ordi-nary 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 alsospecify a model for the power of the incident radiationbeam P . We will consider here the following simplemodel for this function P , which was introduced in [14] P = P ; D < D max P (cid:18) D max D (cid:19) ; D ≥ D max (36)where D ( τ ) = (cid:0) X ( τ ) + Y ( τ ) + Z ( τ ) (cid:1) / is the time-dependent euclidean distance between the source of thepropelling radiation and the sail D max is the maximumdistance at which the beam spot encompasses the sail’ssurface. This maximum distance is related to the sail’scharacteristic size R , the one of the beam source, r ,the radiation wavelength λ by the following relation (cf.[14]): D max ≈ rR λ up to some order of unity geometrical factor dependingon the shape of the beam source. In this model, theenergy that propels the sail beyond the distance D max decays as the inverse of its distance to the source. Amore sophisticated but also more realistic model for thebeam power is the Goubau beam of [13], whose shape isqualitatively similar to the simple model of [14] used here.We can illustrate this procedure by deriving theanalytical solution for the straight motion of a grey sail,without the Poynting-Robertson drag. We found thislast of little impact in practice, though it is includedin the numerical simulations of the Starshot missionin the next section. The analytical model below cantherefore be used for quickly computing estimationsof the rectilinear trajectory. The tangent vector formotion along X is given by (Λ µ ) = γ (1 , β, , (cid:126)n Ts = (1 , ,
0) = (cid:126)n T in = − (cid:126)n T ref . The four-force actingon the grey sail is given by the decomposition Eq.(29).The white sail component of the four-force is given by f Tw = 2 P γβ − β βf Xw = 2 P γ − β β from Eq. (13). The black sail component of the four-forceincludes only the contribution of the incident radiationbeam f Tb = f T in = γ (1 − β ) P and f Xb = f Tb · Regrouping all these elements intoEq.(29), one can express the four-force acting on the greysail in terms of the rapidity ψ as f T = P (cid:0) e − ψ − (cid:15) e − ψ (cid:1) (37) f X = P (cid:0) e − ψ + (cid:15) e − ψ (cid:1) · (38)The equations of motion in this case simply reduce to˙ mc = P (1 − (cid:15) ) e − ψ m ˙ ψc = P (1 + (cid:15) ) e − ψ · or, equivalently, m = m exp (cid:18) − (cid:15) (cid:15) ψ (cid:19) (39)˙ ψ = Pm c (1 + (cid:15) ) exp (cid:18) − (cid:15) (cid:15) ψ (cid:19) (40)for arbitrary power P . In the simplest case of constantpower P , the following analytical solution can be found m = m [1 + (3 + (cid:15) ) s ] (1 − (cid:15) ) / (3+ (cid:15) ) (41) ψ = (cid:18) (cid:15) (cid:15) (cid:19) log (1 + (3 + (cid:15) ) s ) (42)for a grey sail with reflectivity (cid:15) , starting from rest withrest mass m ( s = τ /τ c with τ c = m c /P ). These resultsare consistent with the previously obtained particular so-lutions for (cid:15) = 1 and (cid:15) = 0 · III. APPLICATIONS
We propose here three applications of the original lightsail model derived in previous section. First, we presentthe general dynamics of perfectly reflecting light sails,then we apply our model to flyby missions at relativisticvelocities and finally to single trip with double-stage lightsails.
A. Sailing at relativistic velocities
Let us consider the general motion of perfectly reflect-ing ”white” light sails, as given by Eqs.(17). In thismodel, the sail is reflective on both sides of its surface.Fig. 2 presents several trajectories of light sails comingfrom infinity at τ → −∞ and passing closest to the lasersource at τ = 0 with distance R = R min for differentvelocities β and inclination θ . After the closest ap-proach, the sail is deflected by the light source depend-ing on sail’s velocity at closest approach ψ and sail’s -2 -1 0 1 2X-2-1.5-1-0.500.511.52 Y FIG. 2: Various trajectories of perfectly reflecting ”white”sails with driving power P decaying as 1 /R . A star indicatesthe location of the laser source, a square one particular po-sition of the light sail with associated reflecting surface andvector triplets (cid:126)n s, in , ref inclination θ there. A complete study of the light sail’sphase diagram is given at Figs. 3 and 4. First, thissystem admits only one fixed point ( θ = π/ , ψ (cid:48) w = 0)which is an unstable saddle point. Indeed, a linearstability analysis of system (17) indicates that the ja-cobian of the right hand side has two eigenvalues ofopposite signs ( λ , = (1 ± (cid:112) /R ) / R ≥ θ → , ψ (cid:48) w →
0) and ( θ → π, ψ (cid:48) w → − θ → π, ψ (cid:48) w →
0) for power decaying as 1 /R ). FIG. 3: Phase diagram in the plane ( ψ w , ψ (cid:48) w ) of the perfectlyreflecting ”white” sail. Left plot is for constant power P whileright plot shows the case of driving power P decaying as 1 /R .Straight lines corresponds to β = 0 and dashed lines to β =0 . Due to this configuration of the phase space, there aretrajectories starting and ending with (cid:126)n s collinear ( θ = 0)or anti-collinear ( θ = π ) to the incident beam for vanish-ing velocities β at closest approach (loop-shaped trajec-tories drawn with straight lines in Figs. 3 and 4). Butif velocity at closest approach β is large enough, trajec-tories can overcome the unstable fixed point and transitfrom the vicinity of one attractor to the other. This isthe case of trajectories shown in dashed lines in Figs. 3and 4. FIG. 4: Phase diagram in the plane ( θ, ψ (cid:48) w ) of the perfectlyreflecting ”white” sail. Left plot is for constant power P while right plot shows the case of driving power P decay-ing as 1 /R . The dot and the triangles respectively indicatethe location of the unstable saddle point ( θ = π/ , ψ (cid:48) w = 0)and the attractors ( θ → , ψ (cid:48) w →
0) and ( θ → π, ψ (cid:48) w → − θ → π, ψ (cid:48) w →
0) for decaying power). Straight lines corre-sponds to β = 0 and dashed lines to β = 0 . This singular configuration of the phase space, withtwo attractors surrounding an unstable equilibriummakes the laser-sailing at relativistic speed an intricateand delicate discipline.
B. Acceleration phase of a nano-probe interstellarmission
Let us apply our results to the Starshot mission[12–14, 16, 26]. This project aspires to send tiny lightsails of one gram mass-scale towards Proxima Centaurifor a fly-by at a cruise velocity of about 20% that oflight. The extreme kinetic energy would be providedby a gigantic ground-based laser during an accelerationphase lasting a few hours. The probes will then freelyfly toward Proxima Centauri which they should reachwithin about 20 years. Let us therefore consider a restmass at start m of one gram and a powerful lasersource emitting light with power P = 4GW. We alsoassume, following [14], that the size of the laser sourceis 10km and that of the sail is 10m so that the maximaldistance. If the laser’s wavelength is 1064nm, then themaximum distance D max until which the laser beamcompletely encompasses the sail is about 0 . τ c = m c /P of this system isabout 6 . (cid:15) , with the model derivedin previous section (including Poynting-Robertson drag).The first point to be addressed is the aiming accu-racy to reach such a far-away destination as ProximaCentauri. Indeed, at start the light sails might not beperfectly perpendicular to the incident radiation beamnor to the direction of destination. As before, let usdenote by θ and Φ the angles between the vector (cid:126)n s normal to the sail’s surface and the incident radiationbeam (cid:126)n in = (cid:126)R/ || (cid:126)R || and the direction e X from the laser’ssource to destination. One last angle is Θ which givesthe position of the light sail with respect to destination.One good news is that any small misalignment willbe quickly corrected naturally during the accelerationphase. Indeed, Fig. 5 gives the evolution of the angles A ng l e s ( a r cs e c ) (n s ,e X ) ( )(n i ,e X ) ( )(n i ,n s ) ( ) FIG. 5: Evolution of the angles θ (straight line), Φ (dottedline) and Θ (dashed line) in the very early phases of acceler-ation for (cid:15) = 0 . (cid:15) = 1. The asymptotic values are givenby θ →
0, Θ → Φ → (cid:15)θ θ, Φ , Θ characterizing the grey sail’s dynamics. Onecan see that the initial small misalignement θ (cid:28) θ = 0, ψ (cid:48) = 0) presented in the previous section. Forperfectly reflecting light sails ( (cid:15) = 1), Snell-Descartesreflexion law imposes Φ = Θ + θ at all times. Since θ goes to zero, this yields Φ → Θ → θ = θ ( τ = 0) sothat the sail’s velocity quickly aligns with the incidentradiation beam and the sail’s trajectory becomes radial,in the direction Θ ≈ θ · As illustrated in Fig. 5,we find numerically that the asymptotic direction ofmotion Θ ∞ can be fairly approximated by Θ ∞ = (cid:15)θ when one deals with a grey sail of reflectivity (cid:15) . As aconsequence, the transverse deviation at destination isgiven by Y ≈ R des (cid:15)θ with R des the distance betweenthe source and the destination. To give an idea, aninitial misalignement θ of only one arcsec results in thecase of Starshot in a deviation of 81 × (cid:15) astronomicalunits after R des = 4 . (cid:54) = 0, which leads to the same asymptotic value:Θ → (cid:15)θ .Since any small initial misalignement θ is sweptaway by the driving force, the sail’s trajectory shortlybecomes close to rectilinear. Fig.7 represents the evolu-tion of velocity in the source’s frame, proper acceleration a = c · dβ/dτ felt by the light sail and distance R to thesource for various values of its reflectivity (cid:15) . The velocityquickly saturates once the sail overcomes D max and the0 V e l o c i t y ( c ) = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 10 1 2 3 4Source's proper time T(h)050010001500200025003000 P r ope r a cc e l e r a t i on ( g ) = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 00 1 2 3 4Source's proper time T(h)00.511.522.533.544.55 D i s t an c e ( A U ) D max D max D max D max D max = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 FIG. 6: Evolution of light sail’s velocity (front panel), properacceleration (middle panel) and distance to the source (bot-tom panel) with time for five values of (cid:15) equally spaced be-tween 0 and 1. Straight lines give the results for a decayingdriving power given by Eq.(36) while dashed lines are for con-stant power driving power decays with inverse square of distance.It must be pointed out that the Starshot space probeswill experience an effective gravitational field (givenby proper acceleration) as large as 2500 that of Earthduring the first hour of acceleration toward their cruisevelocity. During this extreme first hour they will crossthe distance to D max , of about a third AU, and they willreach a distance equivalent to that of Jupiter in only4 hours. The energy cost spent by the driving source E = P .T (with T ≈
4h is the total duration of theacceleration phase) to achieve that is about 5 . × Jor 12 . E K = ( M g γ − m ) c communicated toeach probe to the source’s energy cost E . The efficiencyis about 3% for white sails ( (cid:15) = 1) and close to 27% forblack sails ( (cid:15) = 0). This might look counter-intuitive atfirst sight since perfectly absorbing sails reach a highervelocity (of about 0 . c ) than perfectly absorbing ones (ofabout 0 . c ), but this is due to the increase of the restmass when (cid:15) (cid:54) = 1 as we shall see immediately. Althoughusing perfectly reflecting white sails constitutes an(even) worse loss of energy, their main advantage liein the shorter trip duration toward Proxima Centaurias shown in Fig. 7: white sails arrive at destination inabout 22 years against 31 years for black sails, shouldthey survive to the acceleration phase and the trip (cf.also [16]).We can now conclude this analysis of the Starshot E ff i c i en cy T i m e a t a rr i v a l ( y r) FIG. 7: Efficiency (top panel) and trip duration (bottompanel) of the Starshot probes as a function of the sail’s re-flectivity (cid:15) project by giving the evolution of the Starshot probesrest mass and the time dilation aboard during theacceleration phase, as is done in Fig.8. As explainedabove, perfectly reflecting white sails have constantrest mass while perfectly absorbing ones will exhibit an1increase of their (inertial) rest mass by about 14% whichis significant enough to consider for any hypotheticmanoeuvre of the Starshot probes. Time aboard theserelativistic probes will also elapse slower, by about 2%for white sails, which represent almost a difference of137 days between on-board time and mission’s controltime at the end of the mission (22 years for (cid:15) = 1).From the point of view of the Starshot probes, thisrelativistic effect will shorten the effective trip durationby 137 days. This effect must be taken into accountto wake up on-board instrumentation and starts thescientific programme at the right local time, otherwisethe destination system will be missed by approximately4 . × AU(= 0 . × c × S a il ' s r e s t m a ss M ( m ) = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 00 1 2 3 4Source's proper time T (h)00.0020.0040.0060.0080.010.0120.0140.0160.018 T i m e d il a t i on T / - = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 1 FIG. 8: Evolution of Starshot probes rest mass (top panel)and time dilation
T /τ − (cid:15) . Straight lines in the top panelindicate the prediction for constant power C. Single trips in the Solar System with two-stagelightsails
In the context of large scale directed energy propul-sion, several interesting proposals for the explorationof the Solar System at sub-light velocities have beenmade (see for instance [17]). Among these, the idea ofreconverting onboard the laser’s illumination to drive anion thruster leads to an efficient solution for high-massmissions. We do not pretend to propose a detailed SolarSystem mission here but rather we revisit Forward’s ideaof multi-stage lightsails with the tools established in this paper.The major issue with laser-pushed lightsails, accordingto the pioneer G. Marx in [22], was the slow-down onceapproaching destination since this external propulsioncannot be reversed. That is why R. Forward suggestedin [8] to use multi-stage lightsails: the large sail that isused during the acceleration phase separates at somepoint between an outer ring and a central breaking sailto which payload is attached. After separation, the lasersource is still propelling the outer sail that stays on aaccelerated trajectory while the payload reverses its ownsail to catch the reflected light coming from the outerring and uses its radiation pressure to reduce its speed.We propose to revisit this idea with our model bygiving a detailed trajectory of single trip inside the innerSolar sytem, beyond the simple description made in [8].For the sake of simplicity, we will restrict ourselves torectilinear motion. The abscissae of the outer ring andpayload 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). Duringthe acceleration phase, the inner payload sail and theouter 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)): P o = P ; X o,p < X P (cid:18) X X o,p (cid:19) ; X ≥ X (43)where P is the power emitted by the light source lo-cated at the origin of coordinates. X stands for themaximal distance at which the light source completelyencompasses the outer sail and after which the illumina-tion decreases as the inverse of the distance squared. Atsome point the outer ring sail separates from the innerpayload ring which now uses its front reflective surfaceto collect light reflected from the outer sail to decelerate.The payload sails then enters a deceleration phase whichis driven by the following power function: P p = − P o ; ( X o − X p ) < X − P o (cid:18) X X o − X p (cid:19) ; ( X o − X p ) ≥ X (44)with P o is the power acting on the outer sail Eq.(43), X is the maximal distance at which the light reflected fromthe sail completely covers the payload’s sail located at X p and X o ≥ X p is the position of the accelerating outersail.Fig. 9 presents a typical example of a rectilinear singletrip performed with two-stage light sails. We consider apayload of rest mass m p = 20 tons, an outer sail of mass m o = 4 tons (hence a total mass of 24 tons) powered bya laser source of power P = 10GW. Following [14], if2 V e l o c i t y ( k m / s ) Outer RingPayload D i s t an c e ( A U ) Outer RingPayload
FIG. 9: Evolution of the outer and inner payload sails veloc-ities (top panel) and distance (bottom panel). A dotted lineindicates the cruise velocity of Voyager 1 probe. we assume a sail of thickness of 1 µ m and a density of1 . g/cm , then the radius of the outer sail with massof 4 tons is ∼ X of 5 AU after which theillumination starts decreasing with the inverse square ofthe distance to the source. Similarly, the outer sail isable to completely illuminate the inner payload sail upto a relative distance X = X o − X p which we chooseto be equal to 5 AU in the example of Fig. 9. Thelight source is used to both accelerate and deceleratethe double-stage spaceship, which means that it mustcompletely illuminate the outer sail over a large distance(here X = 5 AU) covering both deceleration andacceleration. If the outer sail has a radius of 954 m aswe have seen above, this could be achieved with directedenergy system of radius close to 2 km, assuming aninfrared laser of wavelength equal to 1064 nm (Nd:YAGlaser, see also [14]).After the separation, the lighter outer sail is freedfrom its heavier inner payload sail and therefore un-dergoes a stronger acceleration and a fastly increasingrelative distance to the payload. This implies thatthe duration of the breaking phase is shorter than theacceleration one and that the decay of the illuminationof the inner payload sail arrives earlier as the outersail quickly goes out of the distance X at which thelight from the outer sail completely illuminates the inner payload sail. The lighter the outer sail the moreimportant its post-separation acceleration and the moreimportant this effect.In the example illustrated in Fig.9, the accelera-tion phase of the double-stage 24 tons spaceship lastsfor 4 . X o of 7 AU whichis X = 5 AU further than the distance reached by thepayload. It takes about 7 months to the 20 tons payloadto reach a distance of 2 AU, roughly the average distanceseparating planets Earth from Mars. The total energyspent by the laser source to make this trip possible ishuge, around 2 × J or 45 thousands kilotons. Theefficiency of the directed energy system at the end of theacceleration phase is very poor, close to 10 − · IV. CONCLUSION
Among a mess of various ideas for making interstellartravel possible, some were not that fanciful that couldhave appeared on first sight. In fact, interstellar travelmight well have been invented almost 60 years ago byForward [9] and Marx [22]. Their vision was to use thethen freshly realized laser to propel reflecting sails atrelativistic velocities, towards the stars. This rather oldidea has taken a long time to be fully explored withinrelativity. Curiously, Forward, although a renownedspecialist of this discipline, did not push his ideavery far into formalism and detailed computations inthe relativistic limit in his papers on the subject. Arelativistic model for the straight motion of a perfectlyreflecting light sail was introduced by Marx in 1966 [22],then seriously corrected by Redding in 1967 [23] beforebeing revisited into its final form in 1992 by Simmons &McInnes in a pedagogical paper. Under the recent burstof interest accompanying NASA’s Starlight and Break-through Initiative and other programs in the previoustwo decades, this restricted model has served as a basisfor a variety of interesting extensions [10, 12–14, 17, 27].Unfortunately, these attempts did not revise the fun-damentals of the Marx-Redding-Simons-McInnes modelinto special relativity which lead to some incompletenessof the recent models and sometimes confusing presenta-tions. For instance, one major drawback of these recentpapers is that they tacitly assume that the rest mass ofabsorbing sails is constant, which is only true for specificconditions on the four-force, manifestly not full-filled orexplicited in these models. In addition, the non recti-linear motion of lightsail has not been investigated so far.3This paper introduced the appropriate formalismto go beyond this situation, by deriving the generalmodel for the non rectilinear motion of partially re-flecting ”grey” lightsails starting from general principlesin special relativity. As part of the family of photonrockets, the lightsails have to deal with four-forces thatobey several constraints and exhibit several interestingproperties, including the variation of their rest masswhen inelastic collisions with the incoming radiationoccur. The general model of ”grey” sails is build on acombination of two particular cases: first, the one ofperfectly reflecting ”white” sails which has a generalplanar motion and second, the one of perfectly absorbingblack sails whose motion is along the direction of theincoming radiation beam and ruled by the push ofexternal radiation pressure and the drag force fromPoynting-Robertson effect. Our model requires numer-ical integration although in simplistic cases analyticalsolutions allow crude approximations.We also presented three applications of our model.First, sailing at relativistic velocities has been shownto be intricate due to the instability of the equilibriumwhen the sail stands parallel to the incoming beam.Whatever the initial conditions outside of this unstablepoint, any lightsail will naturally relax toward twopossible positions with the velocity anti or co-linearwith the direction of the incoming beam both with c asasymptotic speed. This behaviour will allow lightsailsto spontaneously align with the incoming beam whileaccelerated, yielding a prediction for a deviation fromtheir initial aim. Second, we provide the predictions ofour model on the Starshot mission, within some genericnon-restrictive assumptions that could be easily adaptedif desired. The model accounts for initial deviations ofthe sail’s position and velocity vectors from the directionof destination, which produces non-negligible deviationsat arrival. To show this, we have provided a simpleexample with a small initial misalignement of one arcsecamplitude and our predictions on lightsail’s velocity,proper acceleration and distance are comparable tothose of previous works [12–14], although accounting fortransverse deviation and rest mass variation when thesail is partially absorbing. There are several importantpoints for mission design that have been derived here:aiming accuracy, increase of inertial mass for manoeu-vring, time dilation effect and wake-up time of theprobe’s internal systems, propulsion efficiency and theneed to seriously consider power-recycling to improve it.Third, we provide a very simple model of a single trip inthe Solar System with ton-scale two-stage lightsail andGW-scale laser, to illustrate the potential interest of thistechnique for interplanetary exploration.Of course, the results presented here are not ex-haustive and should be carefully extended to a more realistic numerical modeling prior to any directed energymission. While our model is valid for non-rectilinearmotion and general sail’s reflectivity, there are severalphysical effects that must be added to compute a precisetrajectory of such an intricate (and costly) missiontowards far-away planets or even stars. This includesmodeling of the environment: incoming beam propertiesand interaction with the sail, gravitational influences,solar system constraints (e.g. a free path from a ground-based laser and a sail), interplanetary and interstellarmedia, magnetic fields, anisotropic thermal radiation, toname but a few. Several papers have already startedinvestigating some of these effects (see for instance[13, 16] for a review), which could now benefit fromthe general model presented here. Another concern ofthe authors are telecommunication and astronavigationissues which must take into account several effects fromgeneral relativity (see [15] for a first approach in the caseof straight motion of photon rockets). Following [13],we agree that a serious effort of physical and numericalmodelling, beyond simplistic models sometimes solvedusing spreadsheets, must be made prior to any launch.We also think that small scale ground-based experimentsand flying prototypes will be necessary to calibrate themodels and adjust numerical simulations (see [13] for aninteresting suggestion).The key idea of Forward and Marx that could wellmake interstellar travel possible one day is undoubtlythe externalization of the propulsion’s energy source.Since Forward already envisions in [8], the most ob-vious primary energy source for that purpose is theinexhaustible 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 hugeand radiation beams that will be efficient for propulsionbut potentially also to other dangerous aims. Thesending of one single tiny probe, of mass one gram,toward Alpha Centauri individually requires as muchenergy as the one delivered by several kilotons of TNT.Besides this frightening energy waste, cost estimationfor such program is of the order of 10 billion dollars[13]. Of course, it is natural to wonder if the stars areworth these sacrifices, no matter their appealing beautyto the stargazer. To some respects, like exploration ofthe unknown and maybe one day our own survival, theycertainly are. Yet, it is a charming idea that the key toreach the stars could lie in the glare of their close andlovely yellow dwarf cousin. Acknowledgements
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