Maximum Terminal Velocity of Relativistic Rocket*
···~-..,,~-·~·
ActaAstronautica Vol. 12, No. 2, pp. 81-90, 1985 0094-5765185 $3.00 + .00 Printed in Great Britain Pergamon Press Ltd.
MAXIMUM TERMINAL VELOCITY OF RELATIVISTIC ROCKET*
G1ov ANNI VULPETTit Telespazio, SpA per le Comunicazioni Spaziali, Via A. Bergamini 50, 00159 Rome, Italy
(Received 23 April 1983; revised version received 4 May 1984)
Abstract-The maximum terminal velocity problem of the classical propulsion is extended to a relativistic rocket assumed broken down into active mass, inert mass and gross payload. A fraction of the active mass is converted into energy shared between inert mass and active mass residual. Significant effects are considered. State and co-state equations are carried out to find the exhaust speed optimal profile. A first major result consists of a critical value of inert mass. Beyond it both true and effective jet speeds increase with time. Below it the true jet speed profile is reversed. At criticality, the best control consists of both velocities constant in time. A second meaningful result is represented by an interval of inert mass outside which no optimal control exists. Numerical results are discussed with particular emphasis to current concepts of antimaner propulsion.
NOTATION 2. STATEMENT OF THE PROBLEM
a alpha Any spacecraft (S/C) endowed with a pure rocket P beta propulsion system can be broken down into three main y gamma systems in tenns of mass: the active mass from which e epsilon the propulsive energy is extracted, the inert mass to a sigma A. small lambda which this energy is generally transferred and then ex A capital lambda hausted and the gross payload. A key point is to establish r tau the mass-energy utilization history in the SIC. In [3] a LI capital delta 14-parameter model is considered. Those parameters ac x chi count for effects such as mass jettisoning, leakage, non iJ partial differentiation In natural logarithm propulsive energy, nozzle spreading and so forth. That -+ arrow model contains specifications relevant to an antimatter ~ double arrow propulsion system concept. In the present context we simplify that model by re taining only three key parameters. They are redefined 1. INTRODUCTION here. The problem of maximizing the tenninal velocity of a Figure 1 shows the mass-energy distribution for thrust space vehicle has been dealt with extensively in litera ing. It is valid for both continuous-mode and pulsed ture. A particular attention has been devoted to both mode propulsions. The continuous-mode is referenced electric and nuclear propulsions for interplanetary and here for greater clarity. out-of-solar-system missions. Here we cope .with this The fraction e of the active mass M. is converted into problem in a general way. Our approach is independent energy by some exothennic reaction. This fraction e , of the particular spacecraft one could consider, provided encompasses the rest and kinetic energy of all products only that the power source for propulsion is on-board. but the interaction-negligible particles. The inert mass Therefore we will not consider vehicles receiving mo M, receives a fraction of this energy and then it is ejected. mentum and/or energy from outside. These last types Sometimes the active and inert mass physically coincide. of propulsion have.been included in [1]. Also, the inert mass could be absent. The relativistic fonnulation of this problem is consid A special situation would consist of the (optional) ered here. This is the driving purpose of such a paper. capability of a propulsion system to reconvert a fraction, In addition, a second goal consists of introducing con say AA 12:2-B 81 82 G. VULPETI! ~ active moss pr<>pUsive ii CROSS I residual kinetic energy (1-c)(1-s) PAYLOAD (1-a)cM I:, Mo 0 moss lost ACTIVE moss converted or discarded MASS into o into space STORED utilizable energy IJ E Tl (1-c)sM 0 Mo r;Mo INERT MASS rest-energy STORED acM Mi 0 I - Fig. I. Spacecraft mass breakdown and energy utilization history. massive particles (e.g. high-energy gamma rays trans Introducing the variable x = dMJdM., the kinetic formed into electron-positron pairs). The fraction of the energy eqn (3) allows us to immediately obtain: total energy of these particles going to rest mass upon transformation is just the term aind· Therefore, the overall U 2 = I - [(X + ae)l(X + e)]2. (4) ejection beam can contain "sub-beams" originated from the inert arid active mass. From the basic equation: In high-energy exothermic reactions massive particles can be the final products of some chain of secondary dE, + dEP + dE1 = 0, (5) reactions. In intermediate phases a fraction of the total energy may be quite Jost because some particles have we can obtain the rocket motion equation in differential very low interaction cross sections (e.g. neutrinos) or form: they are hard directable. The fraction e is an effective value depending upon the energy distribution in the chain. y:dv = -(UdMIM)(X + e)l[X + e These considerations introduce the "jettisoning" factor, +(l - e)s] ~ -U, dMIM. (6) say s, into the current model; s represents the fraction of the mass (l - e)M. lost into space at zero total mo The second fraction in the right-hand side of eqn (6) mentum with respect to the SIC frame. can be referred to as the utilization efficiency relative to , I In order to write the mass-energy conservation Jaws the total mass released outside the ship. It equals one we refer to an inertial heliocentric frame (HF) and the when e = l ors = 0, being in general time-dependent set of the instantaneously inertial ship frames (SF) as as well as U. Note that whereas U is independent of the usually defined in special relativity. We have in differ value s, the above efficiency does not depend upon a. ential form (the speed of light has been set equal to Only the effective speed, U., depends on both. one) [I]: We maximize dE, = M Vy~ dV + y, dM, ~ ,~ dEP = (l - UV) Yu y, dMP, (l) J = y; Vdr. = tanh- 1 (V) , (7) dEi = (I - e) s y, dM., i0 ~ '! where dMP = dM, + a e dM •. The symbol M denotes subject to the integral constraint: the ship (rest) mass, V its velocity, Uthe true jet speed. ('1 . The subscripts s, p, j stand for ship, propellant and Jo X M. dr = M,, (8) jettisoning, respectively. The Lorentz factor is denoted by y0 . Equations (l) are valid in HF. In SF the mass energy conservation is to be read as follows: and the differential constraint, from eqn (2): dM = -(I - e) s dM0 - e dM. - dM,, (2) M = - [(I - e)s + (X + e)] M~. (9) (I - a) e dM. = (Yu - I) dMP. (3) The proper propulsion time r1 is fixed. In eqns (7) Maximum terminal velocity of relativistic rocket 83 through (9) the dot represents differentiation with respect alytically as function of known quantities such as M 0 to the proper time r. and M1. As a matter of fact, H does not contain time ~ . Ifwe assume Ma = cons!. = Malr1 and assigned, the explicitly; then, inserting A from eqn (13) into the H only degree of freedom of the system is x. However, expression (12) and considering H = H0 , one obtains: for analytical purposes it is easier to take as control variable y. which is related to x through (a + P - PX)l[M (X2 - 1)"2] 1 2 = (a + P - PX0)/[M0(Xfi - 1) ' ], (17) (y. - l)(x + ae) = (I - a)e, (10) where we have set X = y., a = (1 - a)e, P = as easily obtainable from eqn (4). Thus the state equation (1 - e)s. (9) can be recast into the form: In principle, if X were known, eqn (17) would give ! 0 I the optimal X as function of M. Equation (11) would be l i M = -Ma [(l - e)s the only one to be integrated. However, X0 cannot be + (1 - a)ey.f(y. - I)]. (II) generally obtained in closed form as function of e, a and s, values characteristic of a given propulsion system. Then, by inserting eqn (4) into the definition of y. the Therefore, starting from a guess X0 , the basic system to Hamiltonian H of the current problem can be expressed be integrated is composed of eqns (11) and (15) plus as follows: their respective variant trajectories. Successive differ X A !' ential corrections bring 0 to converge (note that 0 is :I HI Ma = (I - a)e[(y. + l)/(y. - I )]"21M arbitrary). + A.[(1 - a)e y.f (y. - 1) - e] The integral constraint (8) could be chosen equal to -A [(I - e)s + (1 - a)e y.l(y. - l)], (12) zero, that is no inert mass is considered. From eqn (10) one immediately concludes that the true jet speed is ii: where the expressions in y. are easily obtainable com constant (and equal to the speed of light if a is zero). ~ ~ bining eqn (4) and the definition of y•. The multipliers By a physical analogy, one could then ask whether an IL A. and A are time-independent and time-varying respec optimal time-independent profile may occur under more !!:1': general conditions. For eqn (15) admits the solution I~ I tively. His different from zero because r1 is assigned. tt; y. = cons! only if the term in brackets vanishes. One value of y. is acceptable: 3. OPTIMAL EQUATIONS No (sub-C) limit on the exhaust velocity is set. Be y~ - I = afP. (18) cause H contains neither a linear control nor any limit on it, the problem is nonsingular and the optimal program From a pure mathematical viewpoint, eqn (18) can be is given by aH I ay. = 0. This control equation can be simply obtained by equating the right-hand side of eqn arranged into the form: (17) to zero. Physically, equation (18) implies that there is a critical value of M;. say M,,, which ultimately de A - ). = [(y. - l)/(y. + 1)] 112 /M. (13) termines the trend of the optimal profile. We get from eqn (10): After differentiating eqn (13) with respect to time, noting i = 0 identically and making explicit the mass M,c = M.[s - e(s + a)]. (19) adjoint equation A == '-- aH I aM, we arrive at the fol lowing differential equations: One realizes that for an actual M, < M,c the profile of U is reversed, namely, the initial values of U are greater A = ((1 - a)e Ma1M 2J than its final ones. In contrast, U, never decreases, as 112 x [(y. + l)l(y. - 1)] , (14) expected. If M,c < 0, we have a classical profile (i.e. time-increasing) for any actual M,. y. = (Mal M) ((1 - a)e (15) What is the physical cause of a critical x, say x*, x (y. + 1) - (1 - e)s(y~ - l)]. which equals [s - e(s + a)]? Recalling eqn (10) Jet us note that the fractional (rest) mass accelerated up to the Finally, the proper acceleration is obtained from eqn speed U is x + ae. This value, in critical conditions, (6) as function of the control as follows: results simply in: 2 V = y; .[(l - a)e M0 /M} [(y. x* + ae = s(l - e). (20) 112 + 1)/(y. - 1)] • (16) The right-hand side of eqn (20) is the total fractional Equations (11 ), (14 )-(16) represent the optimal sys energy Jost into space. Therefore, if one adds to the tem which provides the best control to maximize the active mass flow (ae) an amount of extra (inert) mass final velocity of a relativistic ship. such that the overall Joss of energy into space is exactly Some remarks are in order. First, only eqns (11) and compensated, the optimal exhaust speed is a constant (15) are coupled. This means that, if the initial value of profile. One can recognize that only for a high-e reaction y. were known, one could easily carry out the optimal the value of s may be sufficiently high to entail x* profile. However, it is not possible to express y.(O) an- significantly positive. In Jow-e environments any actual 84 G. VULPETTI s approaches zero and x* is nonpositive. This is essen Table 1. Major energy sources tially why "classical engines" such as electric devices exhibit a unique optimal trend of specific impulse for MAX USEFUL max-AV. ENERGY EFFECTIVE Let us examine the optimal equations at criticality. TYPE (MJ/Kg) YIELD Here the solution is in closed form. From equations ( 10), (11), (15), (16) one can carry out: CHEMICAL (LH2/ 15 1.7 E - 10 LOX) FREE RADICAL 220 2.4 E - 09 U = 2 [a(a + 2P)]" /(a + p), (21) (H + H-+ H1) METASTABLE ATOM 480 5.3 E - 09 3 M = M0 - M.(a + 2P)t, (22) (He ) NUCLEAR FISSION 8.2 E + 07 9.1 E - 04 NUCLEAR FUSION 3.9 E + 08 4.3 E - 03 LI tanh- 1 V = (l 2p1a)- 112 In (R), (23) + PROTON- 9.0 E + 10* 0.55 - 1 ANTIPROTON where R denotes the propulsion mass ratio. In general, ANNIHILATION RisexpressedbyM.f[M - M, - eM. - s(l - e)M.]. 0 *Per kilogram of matter-antimatter combined. Equations (21) and (23) show that both U and U, are time-constant. A special case is M, = M,, = 0 (then s = ea/(l - e)). Equation (23) results in: tive essentially for the very high energy density that nucleon-antinucleon annihilation can offer. However, LI tanh- 1 V = [(l - a)/(l + a)] 112 In (R). (24) the involved problems are the most challenging in theory and technology. The (rest) mass-flow rate can be expressed as follows: Figure 2 shows significant (mean) values of a proton antiproton annihilation-at-rest reaction. A low-energy ae (25) MP= M., proton-antiproton annihilation largely produces a number of neutral and charged pions. The neutral pion decays whereas the on-board acceleration is given by: into two high-energy gamma rays; the charged pions 2 112 decay into muons and neutrinos; the muons, in tum, give a,= (M.IM)e(l - a ) • (26) electrons, positrons and neutrinos as final products. Be In these environments the true jet speed results in: cause of such temporal sequence of events, the energy efficiency of an antimatter engine would be strongly u = (l - 0'2)112. (27) dependent upon the type of particles which ultimately furnish the propulsive energy, especially if the annihi Where a equals zero (then s = 0), one would again lation products were to be exhausted with no additional find the ideal photon rocket. component of ordinary matter[7-10]. The energy recon version concept could improve this efficiency[lO]. 4. POTENTIAL APPLICATIONS S. NUMERICAL RESULTS Extending the max-terminal velocity problem to rock ets for which the effective energy conversion yield is A computer code has been implemented to solve for high, the classical solution generalizes into three possible the general optimal equations. Besides the usual printout, regions of operation. In the overcritical region the rel a three-dimensional graphic representation output option ativistic control repeats the classical one. In the under provides the analyst for single-graph and multiple-graph critical region the true jet speed program is reversed; in displays. If a single case is selected, the scaling factors contrast, the effective jet speed increases as burning pro are explicitly written. When several cases are to be com gresses. Finally, in the critical regime both optimal con pared, the critical mass is reported. Title and label ex trols are time-constant. These are strict results. planation can be found below each graph frame. The critical environment is driven by eqn (19). It has The multiple-case figures presented here are charac no practical importance in classical propulsion (electro terized by the projections of the optimal behaviours on magnetic and nuclear systems included) because the the coordinate planes. The indicated scales are normal amounts of propellant involved in the corresponding mis ized by the following quantities (the optimal solution can sions envisaged are considerably greater than the related be set in a dimensionless form): the prefixed thrusting critical masses. Thus, the question arises what might be time, the initial ship mass, the maximum value of the a possible area of future application. Table l displays inert mass on active mass ratio. Velocities are expressed the values of the energy yield for the main exothermic in speed-of-light unit. Path is expressed in lightyear. processes in nature. Figure 3 shows the optimal profile concerning a space In the last two decades, several concepts for fast in craft which, ideally, exploits the whole energy released terplanetary and interstellar missions to nearby stars have in the n:.iclear fusion. The fractional kinetic energy avail been developed. Recently, investigators in the USA and able amounts to 0.004 [Because the whole fuel mass is Europe have performed studies about antimatter pro ejected, we set e = l and a = 0.996. so that pulsion[ 4-10). This propulsion concept is highly attrac- a = (l - a)e = 0.004. In addition, in the ideal case Maximum terminal velocity of relativistic rocket 85 MEAN PATH1 I 111 ! IOO&. I Cll ,... 111ri1 soM .... I 11lf ~ 0 t 0 t t .;) n~..... Ka ,K i' Fig. 2. Energy sharing following the average reaction of proton-antiproton annihilation at-rest. The symbol n denotes a pion, K' does charged kaons (K-mesons), K~ and K2 indicate the two states, of short and long life respectively, by which a neutral K-meson can be described. Pions and kaons represent the nonresonant primary product of a nucleon-antinucleon annihilation. " s = 0. More generally, the case of an incomplete ejec Figure 4 regards a spaceship endowed with a propul tion of the fusion mass can be dealt with by decreasing sion system capable of utilizing the energy of the charged e and increasings appropriately.] The critical mass equals pions. The neutral pions have already decayed in gam - a M 0 • The optimal control is therefore of classical mas supposed quite lost in this example. The quantities type. In order to have an increasing jet speed we must i; = 0.618 and em = 0.23 represent the total energy and add inert mass. A modulated xis then allowed, as shown the rest-energy, respectively, of the charged pions in in Fig. 3. An interesting effect has been emphasized here units of annihilation energy (the m!ss of two protons, in the behaviour of x and, in tum, in the U's. The namely, about 1877 MeV). The propulsion kinetic en terminal xis very low. If we augment M; beyond a certain ergy available is (I - a)e = 0.388 (about JOO times value, say M1., the mass-flow rate ratio crosses zero and the best fusion engine). The value of s equals one. The takes on negative values. This has no physical meaning. value of the active mass has been chosen rather high The upper limit on M; is to be ascribed to the fact that (0.5) for a greater clarity of graphic presentation. The Mp ,.,, aeM 0 in this type of engine. In contrast, in a nu ship is a relativistic spacecraft. The inert mass has been clear electric propulsion (NEP) system (where the active varied to show the regions of optimality and other effects. mass does not participate in the jet) the propellant flow In Fig. 4 the undercritical, critical and overcritical rate can in principle be made arbitrarily low, as it is behaviours are shown. The effective jet speed is prac known. A fission NEP S/C is characterized by the fol tically insensitive to the inert mass value unless it is lowing model values: i; = 0.001, a = 0, s = 0. The comparable with the active mass. Particularly interesting M;, value is identically zero. is the convexity of x(r). The sign of such convexity is The energy density of a power source is chiefly de constant with respect to both time and inert mass. termined by the factor a relevant to the exothermic re Figure 5 displays the optimal trend in the energy region actions the power plant utilizes. A very high a, the high where the charged pions are decayed, but the muons are est one known at present, pertains to the annihilation not. The available propulsive kinetic energy is a = 0.312; process. Figures 4 through 7 present the optimal behav however, the total energy drops significantly on account iours of envisaged matter-antimatter powered space of neutrinos produced by pion decay. This means a strong ships. The first three diagrammes refer to a;nd = 0. A increase of the critical mass. As a consequence, case (a) partial reconversion of the gamma energy into relativistic exhibits a negative flow rate ratio. These considerations electron-positron pairs is considered in the fourth one. can be repeated for the behaviour (a) in Fig. 6. There The basic energy source is the proton-antiproton anni the optimal trends are relevant to an antimatter engine hilation at rest. The values of e and a are reported from[9]. which exploits the energy of the electrons and positrons 86 G. VULPETII I I I: 8 I tit11 I .• 5tifl9 0 tl'I'" 0 0. . tO f :,.. o.8 .g o. 6 ::0- o. 4 " .p o.2 ~ o:i. '"°o .g .o.11o~ i O!O ..s·a - .o~ = cJ' "~ <:::> 0 ~ c,...""~--i <:::::."'>""::. 3> ~~.,. 0 C? ···-(lime, true jet, flows) - (time.elf. jet.mass) --- (time.ship vel.) -·-(time.path) Ua=0.250 lli=0.250 s=0.000 (s CJ r) a 1.0000 0.996 1.000 Fig. 3. Ideal fusion engine spacecraft. A modulation of the true jet speed is obtained by adding inert mass to the basic flow of active mass. This option may result of practical importance if some constraint were to limit the active mass on-board for a certain mission. I I I o.8 fl! o.6 Z. .g o.4 ii o.2 l~ ... !1- o.4 .g ,, CV Q. .1- "~ ~ "CS' ~-..:., \.S"'~"i, ... ~e. (time, true jet,llowsj - (time,elf. jet,mass) --- (time, ship vel.) -·-(time.path) Critica/lli = 0.076 ll~=0.500 s=l.000 (t CJ r) a 0.6180 0.372 1.000 Mi varied (a] Mi=0.019 [b) Mi=0.076 (c) Mi=O.JOO Fig. 4. Antimatter propulsion spacecraft utilizing the energy of the charged pions. The gamma and neutrino energy is lost. Maximum terminal velocity of relativistic rocket 87 from the muon decay. Figures 5-6 (and other similar 20% to 30%[10]). Because of the gamma conversion, results not reported in this paper) show the existence of assumed to be accomplished by means of light particles a lower Jimit of M,, say M 11 , at which the initial x is (i.e. electrons and positrons), the critical mass decreases exactly zero. M,,, in high-e environments, is the coun down to a negative value. The max-velocity is then terpart of M,. in low-e environments. Such values are achieved by means of a classical-type control. solutions of the following equation: Finally, the current formulation of the max-LIV prob lem can be further generalized by allowing the sum, say m, of the active and inert mass to be fixed and leaving I'oI'f..M, + ai; Ma) = a (Yo If Mo - Yt I'o M1), (28) Ma free. The control consists of Ma and x(r). Although unknown, Ma is assumed to be constant during the burn where we have set r = (a + p - py). ing. The problem is partially singular because the Ham Equation (28) has been obtained by- combining eqns iltonian is linear in Ma and nonlinear in x (or y.). Instead (I 0), (15), (17) and integrating by variable separation. of dealing again with the analytical problem, we have Ii M;i corresponds to Yo = 11 a whereas M,,. does to y1 = examined a number of cases by running the previous i: 1 la. The other endpoints, respectively, can be obtained computer code several times. Focusing our attention on from eqn (17). Equation (28) is to be solved numerically. the antimatter propulsion once again, we have obtained For the set of values of Fig. 5 we get M,, = 0.033 M0 • the graphic output shown in Fig. 8. The value The interval [M,,, M,J represents the admissible band m = Ma + M, = 0.5 has been selected. The starred I for the inert mass to have a physically acceptable optimal / marks in Fig. 8 represent parabolic least-square fits giv r control. The trend of such control is driven by the sign ' ing an approximate location of the maxima, namely, of the difference M, - M,,. Let us notice that for a NEP max[max-LI V] values. A number offeatures can be drawn. spacecraft M,,-.. 0 and M,. -.. 1, that is the admissible The absolute maxima fall in the overcritical region. They range is ideally the whole vehicle mass. This would are very wide and broaden as the propulsion energy explain why this "band effect" has not been noted in increases. As a consequence, it is not useful to augment classical propulsion. the active mass beyond a certain level. This would limit ij: 'I Figure 7 represents an envisaged matter-antimatter S/ to a certain extent the antimatter amount and its complex I C where 50% of the gamma energy is utilised for pro management on-board. This sort of saturation effect takes pulsion (this most probably is a limit in effecting the place because at low M, the jet speed variation throughout i!1.l'.i reconversion process. Realistic values could range from the thrusting is lower than at high M,. 11" ti"t1'" 1)5 ti,,90 ~ 1.0 ttir 0 i?1 oJ3 ell . ~ o. 6 -0"'- -0 o.4 ~ 0- o.2 tO o. C1! {;?~ 06• ... ~ o.'* .g "' o.:z '\~ ~ <;:::.,<:>... ...... (lirne,true jel,llow1) - (tirnt,ell. jel,rnau) ---- (lirne,1hip vel.) --· (li1111,polh) Crillcal.. Mi" 0.170 lla•0.500 s•l.000 (co T) • 0.4860 0.3~8 1.000 Mi •ari~d (o) lli•0.011 (b) lli•0.076 (c) llim0.300 Fig. 5. Antimatter propulsion spacecraft utilizing the energy of the muons. The gamma and neutrino energy is lost. 88 G. VULPETII tiff'8 ,,5tiri9 0 I ,I t1'1'" 0 0. . ', :p II' " stiiP I:. ,I 1.0 o. i3'~ o.6 ... j'j ,,,. o.4 .g i 0.2 '\~ ~ <:::>'l-' ....<-.s 'o ~ c:::> ·,..::. ~ ·,..:. .._bo ~·'?.,. .... ~e. ~.... ·-·- (time,lrue jel,llau) - (lime.ell. jel,maas) ----(lime.ship vel.) --· (time,poth) Crltlcallli = 0.41S llo=0.500 s•l .000 (1 ' r) • 0.1700 0.000 1.000 ~i varied [a) lli=0.019 [b) lli=0.076 [c) lli=O.JOO '•• ' Fig. 6. Antimatter propulsion spacecraft utilizing the energy of the electrons and positrons. The gamma and neutrino energy is lost. ,1', . tiff's 0 ttir1Jst1ri90 . tO 0 . ~ o.a ~ o.6 .a .g ::0- 1.0 o. ~ ifc:; !1- o.4 .g 2 z.(0 o. '\~ '6' ~-~ 'I'~~ ;;p -::.~.,. ~o• ······· (time,true jet,llau) -- (time,ell. jel,mass) ----(time.ship vel.) -·-·(lime.path) Crltlcallli = -0.02 Ya=0.500 s=l.000 (1 a r) • 0.8070 0.286 1.000 Iii 1aried (a) lli=0.019 [b) lli=0.076 [cl lli=O.JOO Fig. 7. Antimatter propulsion spacecraft utilizing the energy of the charged pions and one-half of the gamma energy amount (compare with Fig. 4). ·1 1i1 Maximum terminal velocity of relativistic rocket 89 r!: ·,I ------~------~------1 I , . 0 •n•r1v•O.tfl I J•ttisaf.000 I I 0.9 reeen11•0.312 I 0 0.1 •n•riv•0.807 J•ttis=t.000 I :::.... 0.1 T'econv•0.286 I I I 0 I 0.6 •n•riv=0.486 I ~ J•ttis=f. 000 e- 0.6 rsconv=0.358 -:a h. 0.4 IC:! i. c::::i 0.3 )f c( lj :!E ,, ~ I ··1'0. , 16 0 ·S.o I I I I I I I I I I •ACTIVE MASS Ol5 INERT MASS Fig. 8. Antimatter propulsion spacecraft: the active mass is allowed to vary, but the sum of inert and active mass is fixed. I 6. CONCLUSIONS again from the general equations under a few simple conditions (see Appendix). A general model of pure rocket has been presented. It has been applied to the field-free maximum terminal velocity problem extended to relativistic environments. 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VULPETII APPENDIX: OPTIMAL ANALYTICAL SOLUTION which is the well-known profile. In addition, eqn (10) gives: 112 2 Here we specialize the optimal equations of Section 3 to U = (2e/x) ::? X = Xo(M/Mo) • (A-4) the case s = 0, u = 0 and e 112 112 U(t) = [R- (R• - R- 112) tit.], (A-7) UU=2eM.IM, (A-1) U + 112 where V = (2e M. t.t(M0 - M1)) and t• is the thrusting time. M = -2 e M.tU2. (A-2) Note that eM. is just the effective power released by the nuclear reactor. The time profile of x can be obtained from eqns (A-7) By a direct substitution of e M. from (A-2) into (A-1) we and (A-4). In the current context, as well as in Stuhlinger's, have: the propellant is assumed to be fully accelerated. This entails U = U,. The present profile can be generalized in a wider NEP :·i U!U -M!M ::? U UoM 1M, (A-3) context such as the subject dealt with in [12]. I = = 0 '.I ,I . ! f •,1i ·'! ~ :1 .:'! Add-on to Appendix-A Although some relationships are independent of the particular units system one may choose, however in general one should remind: c1 1 year 1 and replace: 2 Maa M c and Uxx U c x 0, f .OR. t t year[SI] in order to restore the equations in the paper to SI M M 0 22Maa tbb2 M t f RU 1, 2 0 MMUMMMff0 0 0 0 2 M a U(),/ t U0 t Uff U 0 M 0 M U 0 R UM00 M Mt() 0 2 M a 1 2 t UM00 M t M t URVR 1 aabb , 1 2MRMR00 1 2 1 V R 1 UR1 where denotes the average over the burning time 0,tb .