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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 <J, of the energy M. into nonzero rest-mass particles cepts relevant to high energy density source rockets. in order to make a (generally additional) controllable Basic concepts and equations of special relativity can ejection beam. The <J value can be generally composed be found in a number of excellent textbooks, e.g. [2]. of two tenns of different physical origin. We denote them by u,P and <J;nd• where "sp" and "ind" stand for spon­ taneous and induced, respectively. The fonner tenn rep­ resents the fractional rest mass of particles directly pro­ *Paper presented at the 33rd Congress of the International duced. In fact, a complex reaction generally yields massive Astronautical Federation, Paris, France, 26 September-2 Oc­ tober 1982. products. On the other hand, some reaction products tSenior Scientist, Space Mission Analysis Division, F.B.l.S. would be completely Jost if not reconverted into charged 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.
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