ebook img

The optimisation of low-acceleration interstellar relativistic rocket trajectories using genetic algorithms PDF

0.56 MB·
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview The optimisation of low-acceleration interstellar relativistic rocket trajectories using genetic algorithms

The optimisation of low-acceleration interstellar relativistic rocket trajectories using genetic algorithms 7 1 0 Kenneth K H Fung✩ ,a,b, Geraint F Lewisa, Xiaofeng Wub 2 aSydney Institute for Astronomy, School of Physics, A28, The Universityof Sydney, NSW n 2006, Australia a bSchool of Aerospace, Mechanical and Mechatronic Engineering, J07, The Universityof J Sydney, NSW2006, Australia 8 2 ] h p - Abstract e c a A vastwealthofliterature exists onthe topic of rockettrajectoryoptimisation, p s particularly in the area of interplanetary trajectories due to its relevance to- . s day. Studies onoptimising interstellarand intergalactictrajectoriesare usually c i s performed in flat spacetime using an analytical approach,with very little focus y h on optimising interstellar trajectories in a general relativistic framework. This p [ paperexaminestheuseoflow-accelerationrocketstoreachgalacticdestinations 1 inthe leastpossibletime, withageneticalgorithmbeing employedfortheopti- v 0 misation process. The fuel required for each journey was calculated for various 3 types of propulsion systems to determine the viability of low-accelerationrock- 0 0 ets to colonise the Milky Way. The results showed that to limit the amount 0 . of fuel carried on board, an antimatter propulsion system would likely be the 2 0 minimum technological requirement to reach star systems tens of thousands of 7 1 light years away. However, using a low-acceleration rocket would require sev- : v eral hundreds of thousands of years to reach these star systems, with minimal i X time dilation effects since maximum velocities only reached about 0.2c. Such r a transit times are clearly impractical, and thus, any kind of colonisation using low acceleration rockets would be difficult. High accelerations, on the order ✩ Correspondingauthor. Email address: [email protected], [email protected], [email protected] (Kenneth KHFung✩ ,a,b,GeraintFLewisa,XiaofengWub) Preprint submitted toActa Astronautica February 2, 2017 of 1g, are likely required to complete interstellar journeys within a reasonable time frame, though they may require prohibitively large amounts of fuel. So fornow,itappearsthathumanity’sultimate goalofagalacticempire mayonly be possible at significantly higher accelerations,though the propulsion technol- ogy requirement for a journey that uses realistic amounts of fuel remains to be determined. Keywords: Interstellar trajectory optimization, general relativity, genetic algorithm, Milky Way 1. Introduction Awealthofliteratureexistsonoptimisingspacetrajectories,inparticularin- terplanetarytrajectoriesduetoitsapplicationinthenear-future. Amajorityof the research focuses on optimising trajectories for a specific propulsion system, 5 rather than for a general propulsion system that utilises the rocket equation. Solar sails appear to be the favourite propulsion candidate for trajectory opti- misation due to the fact that there is no fuel consumption, hence considerably simplifyingtheanalysis: [1]usedbasiccalculustooptimisethesolarsystemexit speed for a spacecraft using a solar sail; [2] optimised interplanetary solar sail 10 trajectories with respect to the flight time using particle swarm optimisation; [3, 4] used evolutionary neurocontrol to optimise low-thrust interplanetary tra- jectories; [5] used sequentialquadratic programmingto optimise the flight time forasmallspacecrafttoreachtheedgeoftheheliosphereusingsolarandnuclear electricpropulsionsystems;and[6]usedageneticalgorithmtooptimisethefuel 15 consumption during orbital transfers. However, solar sails are not practical for interstellartravelsince they require a constantexternalsource ofenergy,which isnotalwayspresentintheexpanseofinterstellarspace. Researchconductedin optimisinginterstellartrajectorieshavemostlybeenperformedwithinaNewto- nian model, thereby simplifying the analysis by ignoring the relativistic effects 20 of time dilation. The discovery that time is relative has raised many interesting discussions, 2 and has produced a plethora of literature on its effect on interstellar travel. Within the scientific community, many authors have examined the effects of time dilation whilst travelling interstellar and intergalactic distances, though 25 all but a few of the calculations were performed in flat spacetime. [7], [8], and [9] considered the effect of an expanding universe when traversing intergalactic distances, and showed that a constant acceleration is necessary if one wishes to reachnearby galaxies within human lifetimes (though this is sensitive to the cosmological parameters used). In the currently favoured cosmological concor- 30 dance model, a rocketeer accelerating at a constant rate of g = 9.81ms−2 is able to reach 99% of the way to the edge of the universe well within a human lifetime [9], though upon return, many billions of years would have passed for those living on Earth. Optimising an interstellar trajectory is an extremely complex and difficult 35 task, and producing the correct solution may not always be possible. Almost all attempts consider either a Newtonian or special relativistic approach, as a general relativistic approach compounds the difficulty of the task. [10] derived the optimality conditions for rocket trajectories in general relativity, though it wasdonefromananalyticalapproachanddidnotconsideranyspecifictrajecto- 40 ries. To date, very little researchhas been performed on optimising interstellar trajectories in a general relativistic framework. 2. Theory 2.1. General Theory of Relativity The assumption of the constancy of the speed of light c means that space 45 and time could be unified into a single coordinate system where the position of a particle is xα = (t,x,y,z). In the curved spacetimes of general relativity, a rocketeerwillexperience kinematic andgravitationaltime dilation. The proper time τ they experience is dependent on their velocity magnitude as well as the local spacetime geometry, described by a metric tensor g . αβ 3 The equations of motion of a travellerin curved spacetime in Einstein sum- mation convention is given by d2xα dxβ dxγ =−Γα +aα (1) dτ2 βγ dτ dτ where aα = (at,ax,ay,az) is the four-acceleration (the relativistic analogue of three-acceleration), and the Christoffel symbols Γα describes the local space- βγ time geometry [11], 1 ∂g ∂g ∂g g Γδ = αβ + αγ − βγ (2) αδ βγ 2 ∂xγ ∂xβ ∂xα (cid:18) (cid:19) The parameters of a particle are related through two normalisationconditions: g uαuβ =−c2 (3) αβ g uαaβ =0 (4) αβ 50 whereuα =(ut,ux,uy,uz)isthefour-velocity(therelativisticanalogueofNew- tonian three-velocity). 2.2. Milky Way Mass Model To model the effect of spacetime curvature due to the mass of the Milky Way, the static weak field metric will be used, which describes the spacetime geometry in a weak, time-independent, gravitational field, such as that of the Milky Way [12]. The static weak field depends on the Newtonian gravitational potential Φ, and is described by the metric 2Φ 2Φ 2Φ 2Φ g =diag −c2 1+ , 1− , 1− , 1− (5) αβ c2 c2 c2 c2 (cid:18) (cid:18) (cid:19) (cid:19) ThegravitationalpotentialduetotheMilkyWaygalaxyismadeupfromthe gravitational effects of the bulge, disk, and dark matter halo. The Miyamoto- 55 Nagai disk, Hernquist bulge, and Navarro-Frenk-White potential models are used to model the gravitational influence of the galactic disk, bulge, and halo, 4 respectively. The potential of the Miyamoto-Nagaidisk [13] is given by GM d Φ =− (6) d 2 x2+y2+ r + z2+b2 d d r (cid:16) p (cid:17) where Md = 10×1010M⊙ is the mass of the disk, rd = 6.5kpc is the scale length of the disk, and b =0.26kpc is the scale height of the disk [14]. d The potential of the Hernquist Bulge [15] is given by GM b Φ =− (7) b x2+y2+z2+r b p 60 whereMb =3.4×1010M⊙ is the massofthe bulgeandrb =0.7kpc is the scale length of the bulge. The potential of the Navarro-Frenk-White Halo [16] is given by GM x2+y2+z2 h Φ =− ln +1 (8) h x2+y2+z2 p rh ! p where M is the mass of the halo and r is the scale length of the halo. The h h massandscalelengthsarecalculatedfromthevirialmassM ofthehalo,which v is the enclosed halo mass at the virial radius R . The exact size of a galaxy v is difficult to quantify as the halo mass density extends out continuously into intergalacticspace,andthe virialradiuscanbethoughtofastheradiusbeyond whichthehaloblendsintothebackgroundmatterintheuniverse. FortheMilky Way, the virial mass is roughly Mv = 150×1010M⊙ [17]. The virial radius is calculated from the virial mass using [18] 2M G 1/3 v R = ≈294.5kpc (9) v H2Ω ∆ (cid:18) 0 m th(cid:19) where H = 70.4×10−3kms−1Mpc−1 is the Hubble constant, Ω = 0.3, and 0 m ∆ = 340 is the Hubble constant, matter density of the universe, and over- th 5 density of dark matter compared to the average matter density, respectively 65 [19]. The mass and scale lengths of the dark matter halo are related to the virial massandvirialradiusviathedarkmatterhaloconcentration,whichisdescribed by the halo concentration parameter c , approximated by [20] h −0.13 M c ≃9.6 v (1+z)−1 ≈12 (10) h 1013M⊙ (cid:18) (cid:19) where z is the redshift, which is zero for host dark matter halos. The mass [21] and scale length [22] of the halo are then given by M Mh = ln(c +1)v− ch ≈91.4×1010M⊙ (11) h ch+1 R v r = ≈24.5kpc (12) h c h The gravitationalpotential of the Milky Way is then the sum of eachof the individual components. 2.3. Relativistic Rocket For a relativistic rocket, the proper acceleration a (i.e. the acceleration as experienced by the traveller) is related to the rate of change of mass of the rocket [23] by 1 τ v τ 1 dm e adτ =− dτ (13) c c m dτ Z0 Z0 wherev istheeffectiveexhaustvelocityofthepropellants. Iftherocketexpends e all its fuel after a proper time of τ , then it is straightforwardto show that f 1 τf m =m exp a(τ)dτ (14) 0 r v (cid:18) e Z0 (cid:19) 6 where m is the final mass of the rocket. If the mass of the fuel is m , then r f m =m +m , and hence 0 f r 1 τf m =m exp a(τ)dτ −1 =m β (15) f r r v (cid:20) (cid:18) e Z0 (cid:19) (cid:21) where β is the fuel-to-empty rocket mass ratio. Smaller values of v will result e 70 in a larger value of β, and hence more fuel will be required as expected. To compute the rocket trajectories around the Milky Way, we need to pre- scribe anaccelerationfour-vectorfor Equation(1). Giventhe magnitude of the properacceleration,a,wedeterminethecomponentsofthefour-accelerationby settingthethrustvector. Weletthespatialcomponentsofthefour-acceleration be ai =(ax,ay,az)=a (a ,a ,a ) (16) s x y z where a , a , and a are the components of a unit vector, so that x y z ai =a (a xˆ+a yˆ+a ˆz)=|ai|(a xˆ+a yˆ+a ˆz) (17) s x y z x y z and a2 +a2+a2 =1 (18) x y z The values of at and a are determined via the normalisation conditions from s Equations (3) and (4), leading to g g at =−ak ii tt (19) g g [g k2+g (ut)2] tt ii ii tt r g a =aut tt (20) s g [g k2+g (ut)2] ii ii tt r 7 where k ≡uxa +uya +uza (21) x y z and the metric terms are 2Φ g =−c2 1+ (22) tt c2 (cid:18) (cid:19) 2Φ g =1− (23) ii c2 2.4. Genetic Algorithms When approaching an optimisation problem, many types of optimisation methods canbe employed,eachoffering unique advantagesas wellas disadvan- tages. [24] explored various numerical optimisation methods commonly used 75 in trajectory problems, and compared the benefits of each; [25] explored differ- ent optimisation methods to solve several spacecraft trajectory problems, and concluded that the most accurate solutions are produced by a combination of different solvers. Since the exact mass and size parameters of the Milky Way are still open 80 to debate, a simple optimisation algorithm will suffice since we are only con- cernedwithcalculatinganapproximateoptimaltrajectory. Ageneticalgorithm was chosen due to its simplistic and heuristic nature, and are computationally inexpensive to run since they also do not depend on any derivatives and their respectivematriceslike mostother optimisationmethods. [26]exploredthe use 85 of genetic algorithms in astronomy and astrophysics to solve a variety of prob- lems, and demonstrates their simplicity and robustness when compared with conventionaloptimisation techniques. To simulate the genetic process, we start with an initial sample size and gradually evolve it toward the optimal solution [27]: 90 1. Arandompopulationisfirstconstructed,representingthefirstgeneration of the species. 8 2. The fitness of each member of the species is evaluated, with the fittest member of the species is passed onto the next generation (elitism). 3. The rest of the population is created by selecting the fittest members of 95 thefirstgeneration(selection)usingalinearprobabilitydistribution,with breeding also occurring between the fittest members (crossover). 4. Amutationisrandomlyintroducedintocertainmemberstocreategenetic diversity. 5. Steps 2 to 5 are repeated for each subsequent generation. 3. Method 100 3.1. Boundary Conditions TheSunorbitsthegalacticcoreoftheMilkyWayatadistancer ofroughly 0 8.5kpc, velocity v of about 220kms−1 [28], and an orbital period of roughly 0 220Myr [29]. The Sun’s orbit is roughly elliptical, and oscillates up and down 105 relative to the galactic plane [30]. We will assume that the Sun lies and stays within the galactic plane. The initial coordinates of the Sun are set at xα =(0,0,−r ,0) (24) 0 and the initial four-velocity is uα =(ut,−v ,0,0) (25) 0 where ut is determined from Equation (3) to be c2+g [(ux)2+(uy)2+(uz)2] ut =± − ii (26) s gtt A rocket arriving at a galactic destination (x ,y ,z ) must also have the f f f correct orbital velocity of the star system. Galactic rotation curves show that the orbital velocity as a function of radial distance is roughly constant outside 9 the galacticbulge[31]. Forthe Milky Way,the orbitalvelocityremainsroughly constant around 220kms−1 outside a radius of about 3kpc. Thus, the rocket must arrive at its final destination with a final speed v of 220kms−1, with f four-velocity components ux =v sinφ sinθ (27) f f f f uy =−v cosφ sinθ (28) f f f f uz =−v cosθ (29) f f f where the final inclination angle θ and azimuthal angle φ are given by f f y φ =tan−1 f (30) f x f z θ =cos−1 f (31) f x2 +y2 +z2 f f f q The equations of motion are integrated using Matlab’s ode45 solver, which is a non-stiff solver that utilises an explicit fifth-order Runge-Kutta method. The Christoffel symbols were calculated from an m-file developed by [32], 110 which is available on the MathWorks File Exchange website. The code was analysed before being extensively tested on some common spacetime metrics, and successfully reproduced their corresponding Christoffel symbols. 3.2. Implementing the Genetic Algorithm The equations of motion require four input variables from the rocket: the magnitude a of the four-acceleration, and the three components of the unit thrust vector. To reduce the number of variables that need to be solved, the unit thrust vector of the rocket is parameterised in spherical coordinates using 10

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.