The Antimatter Photon Drive A Relativistic Propulsion System

Transcription

1 The Antimatter Photon Drive A Relativisti Propulsion System Darrel Smith & Jonathan Webb Embry-Riddle Aeronautial University Presott, AZ 8630 This paper desribes a propulsion system that derives its thrust from eletron-positron annihilation. It also desribes how a spaeraft equipped with this engine an be used to launh manned missions to Mars and Jupiter in as little as 3.8 and 0.8 days respetively. Tehnial problems assoiated with the antimatter engine and potential solutions are also disussed. Throughout this paper, this engine is referred to as the Antimatter Photon Drive or APD). Nomenlature a aeleration the speed of light in a vauum m/s) e + positron e eletron g gravitational aeleration 9.8m/s ) I sp speifi impulse m p mass of propellant ṁ mass flow rate of propellant M i initial mass of spaeraft M f final mass of spaeraft p proton p antiproton t t total time of flight relative to the earth t o total time of flight relative to a passenger in the spaeraft V hange in veloity β v/ γ gamma ray, also γ β µ + positively harged muon µ negatively harged muon π + positively harged pion π negatively harged pion π o neutrally harged pion ν µ muon neutrino ν e eletron neutrino ν µ muon antineutrino ν e eletron antineutrino m e mass-energy of an eletron 0.5 MeV) Introdution The effiient energy prodution resulting from matter-antimatter annihilations has been disussed sine antiprotons were first produed in 955. Methods for onverting pp annihilation energy into thrust to Ph.D Researh Advisor, College of Arts and Sienes Undergraduate Student, College of Engineering Copyright 00 by the Amerian Institute of Aeronautis and Astronautis, In. All rights reserved. develop advaned roket motors ontinues to spark the imaginations of physiists and engineers. 3 Motor designs using antiprotons inlude intertial onfinements fusion ICF) i.e., using the resulting muons to atalyze D T fusion), as well as venting the resultant harged partile through magneti nozzles to reate high speifi impulses with low thrust. One major problem arises when trying to onstrut a roket engine based on pp annihilation. Muh of the energy is onverted into mass whih redues the momentum of the outgoing partiles, thus reduing the thrust. The most typial reation that ours in pp annihilation near rest) is pp nπ s ) where n number of pions π +, π, and π o s) are produed in the final state. There are typially 3-7 pions produed in pp annihilations near rest. The amount of kineti energy released in the annihilation is 876 MeV i.e., m proton ) minus the rest mass energy of the pions 40 MeV for eah pion). The pions have a long enough deay length that most of their kineti energy an be used for momentum transfer i.e., thrust). Table shows the mean deay lengths βγτ) for pions and muons resulting from pp annihilations and subsequent pion deays. The mean pion momentum is 350 MeV/ for a mean number of 5 pions in the final state. The momentum is onverted into the relativisti fator βγ and this is multipled by the fator τ, the mean lifetime of the partile multiplied by the speed of light. The produt of βγ times τ gives the mean deay lengths shown in Table. In the design of any pp engine, the pion momentum must be aptured before a signifiant fration of them deay. If the pion momentum π +, π ) is not extrated within 4 meters, then 63% of them will deay into muons, whose smaller momentum an then be extrated. The momentum released in neutrinos ν s) annot be aptured beause of their extremely low ross-setion. Finally, the π o s rapidly deay into of 6

2 Deay Proess τ βγτ π ± µ ± ν µ 7.80m 9.4m π o γγ 5.nm 0.6nm µ ± e ± ν µ ν e 659m 88m Table This table shows the mean deay length of pions and muons assuming a mean pion and muon momentum of 350 MeV/ and 30 MeV/ respetively. two gamma rays 67 MeV eah) and have a high effiieny for imparting momentum. The radioative deay of pions π +, π ) shown in Table puts onstraints on the size of the absorber. A signifiant fration of the pp enter-of-mass energy is onverted into mass m π 40 MeV/), thus reduing the momentum transfer to the absorber. However, there is another form of antimatter annihilation that produes 00% gamma rays resulting in a muh improved momentum transfer, and likewise more thrust. This is the eletron-positron annihilation proess e + e γγ. ) The eletron e ) positron e + ) annihilation produes two gamma rays 0.5 MeV eah) 99% of the time. Sine none of the enter-of-mass energy is onverted into mass, the full enter-of-mass energy is available for thrust where p E/ for γ rays. The gamma rays are eletrially neutral and emitted in random diretions. Beause they are eletrially neutral, it is impossible to fous them with eletromagneti fields, whereas the harged π + and π partiles from pp annihilation an be foussed to improve the momentum transfer. Various Roket Fuels The energy and propulsive advantages of antimatter are shown in Table. 4 So far, the most effiient and yet unattained hemial roket propellants release J/kg of energy. Furthermore, the table shows that 35 U fission releases nearly times more energy than hemial reations, D 3 He fusion produes over times more energy, while matter-antimatter fusion produes over times more energy than hemial roket propellants e.g., metastable helium). Matter-antimatter annihilations yield the highest speifi impulse and jet power of any propulsion system. Furthermore, it an yield speifi impulses on the order of 0 7 seonds with thrust in the mega-newton range see Fig. ). It stands out as the most effiient of all known propulsion systems. Theory of eletron-positron propulsion A spaeraft equipped with an APD engine will annihilate eletrons and positrons to produe gamma rays that will be aptured in a paraboli dish Fig. ). Yields from Various Energy Soures Fuels Energy Release Converted Mass J/kg) Fration Chemial LO/LH Atomi Hydrogen Metastable Helium Nulear Fission 35 U Nulear Fusion DT 0.4/0.6) CAT-D T.0) D 3 He 0.4/0.6) p B 0./0.9) Matter-Antimatter Table Energy released by various fuels. Weight ompositions orrespond to mixture ratios for fusion fuels, for example, 0.4/0.6) for the D T fuels. 4 The e + e beams are direted towards a ollision point at the fous of the paraboli dish plaed at the extreme end of the spaeraft. Eah annihilation will release two γ-rays bak-to-bak where one or both photons will impat the paraboli dish depending upon the extent of the dish. Two possibilities are onsidered when the photon impats the paraboli dish. In the first ase, the photons are ompletely absorbed; while in the seond ase, they are totally refleted. Sine the γ rays are neutral and annot be foussed, the hoie of a totally refleting paraboli dish beomes the obvious hoie. While a totally refleting dish for γ rays at these energies is beyond urrent tehnology, this partiular design is onsidered beause it desribes the maximum thrust available from e + e ollisions. Theory Veloity Boosts Using Lorentz Transformations In this setion, the relativisti veloity of the spae raft is alulated as a funtion of its fuel. As the spaeraft reeives suessive boosts from one of the two γ rays, it is boosted ontinuously from inertial frame to inertial frame. For the purpose of this derivation, it is assumed that the spaeraft is moving with an initial veloity V with respet to the earth s inertial frame. A small amount of mass dm is annihilated to produe the γ rays used to propel the roket. As a result of the boost, the spaeraft s mass is redued by dm while its veloity inreases to V + dv. The Lorentz transformation relating veloities between the earth s frame and the of 6

3 The hange in the spaeraft s veloity dv ) in the spaeraft s inertial frame an be alulated using onservation of momentum. If a spaeraft of mass M annihilates a mass dm e.g., an e + e pair) it will emit eah of the photons with a momentum p dm/). Conservation of momentum in the spaeraft s inerγ II I III θ γ II Fig. This figure shows the speifi thrusts resulting from various propulsion engines. 4 spaeraft s frame is v v + V + v V 3) where v is the veloity of the spaeraft measured in the inertial frame traveling at veloity V with respet to earth s inertial frame. The hange in the spaeraft s veloity in its own inertial frame dv ), an be related to the hange in veloity observed in the earth s inertial frame, dv by using eq. 3. The relationship between dv and dv is dv V ) dv 4) where V is the instantaneous veloity of the spaeraft and v <<. Conservation of Momentum 3 of 6 Fig. Paraboli shield that ideally reflets gamma ray photons tial frame an be written as 0 Mdv p 5) where one of the photons strikes the paraboli absorber at the end of the spaeraft. For this alulation, it is assumed that the paraboli dish Fig. ) extends from π π suh that only one of the two photon impats the dish. Furthermore, we initially onsider the more realisti ase where the photon is ompletely absorbed. To obtain the momentum omponent aligned with the diretion of motion of the spaeraft, the quantity p should be multiplied by os θ whih is: os θ π π/ π/ os θdθ π. 6) If the photon ould be refleted, the os θ term would be modified to inlude the reoil momentum of the photon. In the perfetly refleting ase where the paraboli dish extends from π π, os θ beomes + π. Returning to the ase where the photon is absorbed and not refleted, Eq. 5 an be written as: ) dm 0 Mdv π. 7) Solving for dv, we find that dv dm ) ) M π. Using the relationship dm dm we an rewrite dv as dv dm ) ) 8) M π Combining Eq. 8 with Eq. 4, we an finally write the hange in the spaeraft s veloity measured in the earth s inertial frame) as a funtion of the propellent mass expended in the spaeraft. v dv ) π ) dm M 9) This differential equation an be solved by separation of variables. v dv 0 v ) Mf dm 0) π M i M

4 β v/ v x- x+ -α π β v/ v x- x+ -α α Fig. 3 Relativisti veloity of a spaeraft where α is the fration of the spaeraft s mass used for e + e annihilation. This assumes that the final state γ rays are ompletely absorbed on a paraboli dish extending from π/ +π/. Integrating both sides, the following relation is obtained: ) + v ln ) v π ln Mi. ) M f A running parameter α is defined to relate the initial and final masses of the roket M f α)m i ), where 0 α <. The parameter α represents the fration of the spaeraft s mass that is used for fuel i.e., the mass fration of e + e pairs to produe the desired thrust). Substituting this into Eq., the veloity as a funtion of fuel mass is where v Mi M f ) x x + ) ) π π α ) 3) The veloity of the spaeraft as a funtion of α is shown in Fig 3. Totally refleting paraboli dish To investigate the veloity profile for a totally refleting paraboli dish, the os θ term must be realulated. Equation 6 is modified to inlude refleted γ rays in the following way: os θ θ θ 0 + os θ) dθ 4) where the fator of takes into aount the momentum transfer along the axis of symmetry due to a perfetly elasti reoil of the γ ray. To ontinue this alulation, the parabola is extended to inlude the dashed portion Fig. ) to maximize the number of γ rays refleted from the parabola. Furthermore, the parabola is split into three regions as shown in Fig.. In regions I and III, only one γ ray strikes the paraboli dish. If one of the γ rays is refleted in region II, then both photons will α Fig. 4 The veloity as a funtion of α, the mass fration used for fuel. This is for a paraboli refletor that extends from 5π/6 +5π/6. strike the paraboli dish, thus enhaning the performane of the engine. Summing up the ontributions from all three regions for a parabola extended from 5π/6 5π6, the fator os θ beomes.985, where.000 would represent a parabola extending out to infinity. Substituting this fator into Eq. 7 in plae of the /π, and arrying out the same alulation as before, the veloity beomes ) x v 5) x + where Mi M f ).985 ) ) α The veloity as a funtion of α is shown for the extended paraboli refletor in Fig. 4. As expeted, the veloity profile for the ompletely refleting parabola Fig. 4) produes faster veloities with less antimatter when ompared to the non-refleting parabola Fig. 3). Spaeraft performane A spaeraft equipped with a matter-antimatter engine will have a speifi impulse on the order of 0 7 s. Beause of the large power output from the matterantimatter fusion, it is possible to reate many meganewtons of thrust. 5 7 Theoretially the thrust is limited by the flow rate of positrons and eletrons into the foal point. If the e + and e injetion ours at veloities less than 50 m/s, positronium atoms will form and annihilate with a high effiieny. This results in a mass injetion rate ṁ of 4.03 gm/s. From Newton s seond law, the thrust an be related to the speifi impulse. The relationship between fore and hange in momentum an be simply written as F ṗ Ma 7) where M is the instantaneous mass of the spaeraft and a its aeleration. For the purpose of this disussion, it is assumed that the aeleration, a, is onstant. 4 of 6

5 Sine the rate-of-hange in momentum is derived from photons, it is also possible to relate Eq. 7 to the fore of a photon, namely F ṗ Ė 8) where Ė is the power transferred to the spaeraft. To alulate the fration of energy that ultimately propels the roket, the ratio.985/ must be inluded i.e., the os θ term for the extended parabola). The.985 fator is due to the large fration of photons being refleted off the extended parabola 5π/6 +5π/6), while the fator of represents an idealized situation where all the photon energy is aptured by a parabola extending from π π, a physial impossibility. Using Einstein s relationship between mass and energy, the power along the axis of symmetry an be written as ).985 Ė ṁ. 9) Combining equations 7-9 the fore along the axis of symmetry beomes F ṁ). 0) If 4.03 gm/s of eletrons and positrons are annihilated at the fous of the parabola, this results in a power output of J/s. Substituting this value into Eq. 0, the total photon fore is.0 MN. This value of the thrust an be used in Eq. 7 to obtain the thrust, and then used to determine the speifi impulse by using the following equation I sp Ḟ mg ) where ṁ is the mass flow rate of eletrons and positrons and the aeleration of the spaeraft is held onstant g. Combining Eqs. 0 and, the speifi impulse beomes I sp g. ) Substituting values for and g, the speifi impulse is alulated to be s. This is the speifi impulse for a ompletely refleting paraboli dish that extends from 5π/6 +5π/6. Mission Parameters To best display the advaned performane of an APD, the minimum times of flight for rendezvous with Mars and Jupiter are alulated. To alulate the travel times for both missions, a Diret Trajetory Optimization Method DTOM) trajetory ode was used. 8 In both ases the spaeraft starts with C 3 0 i.e., a helioentri orbit with the same position and veloity vetors as the earth). In both ases, the spaeraft s orbit is transfered to helioentri orbits of the respetive planets i.e., C 4 0 and C 5 0). Further energy is required to insert the spaeraft into a planetary parking orbit. Although the V needed to insert the spaeraft into this orbit is not shown in the following results, the energy and time required is insignifiant ompared to the rest of the mission. Planet M roket M propellent Travel time Mars 400 Mt.336 Mt 3.84 days Jupiter 400 Mt Mt 0.8 days Table 3 Calulated travel times for a Martian and Jovian rendezvous assuming a propellant mass flow rate of 4.03 gm/s. In Table 3 it is shown that the spaeraft an reah Mars and Jupiter in as little as 3.84 and 0.8 days respetively. Although this is very time effiient, slower veloities and longer travel times would still be aeptable. While this is well-suited for travel within our solar system, the APD would also be apable of interstellar missions., 3 Unlike the rendezvous times within the solar system, the spaeraft must oast for most of the trip to effiiently use its fuel. The time required to travel to Alpha Centauri at relativisti veloities is shown in Table 4. If 90% of the mass is used for fuel, then one-way travel times of 5. years an be ahieved. Likewise, astronaut aging is signifiantly redued to.65 years for the same trip. M roket M propellent Veloity t t years) t o years) 400 Mt 53.9 Mt Mt 70 Mt Mt 360 Mt Table 4 Calulated travel times for a journey to Alpha Centauri using the APD engine. The time intervals for an earth observer t t) and an astornuat t o) are shown for three different fuel frations. The differene between t t and t o are due to relativisti effets. Conlusion With the use of an APD engine, it is forseeable that in the mid-to-distant future, relativisti manned interplanetary and interstellar travel is tehnially feasible. One of the major hurdles to overome will be the tehnology required to produe and store large masses of 9, 0 harged antimatter. While this has been aomplished for small quantities of antimatter, any nearterm solutions will require breakthrough tehnologies that are apable of storing many kilograms of positrons or antiprotons. 5 of 6

6 Referenes ) Gaidos, G., Lewis, R.A., Meyer, K., Shmidt, T., Smith, G.A. 998). AIMStar: Antimatter Initiated Mirofusion for Pre-ursor Interstellar Missions. Conferene on Appliations of Thermophysis on Mirogravity and Breakthrough Propulsion Physis. STAIF-99 Albuquerque, NM. January 3- February 4, 999. ) Gerrish, H.P., Martin, J.J., Meyer, K.J., Shmidt, G.R., Smith, G.A. 999). Antimatter Prodution for Near-term Propulsion Appliations. Laboratory for Elementary partile Siene. Pennsylvania State University. University Park, PA. 3) Chakabarti, S., Dundore, B., Gaidos, G., Lewis, R.A., Smith, G.A., 998). Antiproton-Catalyzed Mirofission/fusion Propulsion Systems for Exploration of the Outer Solar System and Beyond. The 34 th Joint Propulsion and Power onferene, AIAA Paper , Cleveland, OH. July 3-5, ) Kammash, Terry 995). Fusion Energy in Spae Propulsion. Washington, D.C.: Amerian Institute of Aeronautis and Astronautis. 5) Brown, Charles 99). Spaeraft Mission Design. Washington D.C.: Amerian Institute of Aeronautis and Astronautis. 6) Hill, Philip G. 99). Mehanis and Thermodynamis of Propulsion. nd edtion) New York: Addison-Wesley Publishing Coompany. 7) Wiesel, William 989). Spaeflight Dynamis. New York: MGraw-Hill, In. 8) Kluever, D.A., 997). Optimal Low-Thrust Interplanetary Trajetories by Diret Method Tehniques, Journal of Astronautial Sienes, Vol. 45, No. 3, pg ) Cassenti, Brie 000). Mass Prodution of Antimatter for High Energy Propulsion. AIAA, Journal of Propulsion and Power, Volume 6, Number. 0) Hotzsheiter, M.H., Lewis, R.A., Mithell, E., Rohet, J., Smith, G.A. 999). Prodution and Trapping of Antimatter for Spae Propulsion Appliations. Laboratory for Elementary Partile Siene. Pennsylvania State University. University Park, PA. 6 of 6