Journal o/TM British lnlerplanetary Society, Vol. 43,pp. 537-550, 1990.

MULTIPLE PROPULSION CONCEPT FOR INTERSTELLAR FLIGHT: GENERAL THEORY AND BASIC RESULTS

GIOVANNI VULPETTI Telespazio SpA per le Comunicazioni Spaziali, Via Tiburtina 965, 00156 Rome, Italy.

The two-dimensional theory on the multiple propulsion mode developed by the author in 1979 is extended to the four-dimen­sional case. Basic numerical results are discussed in the light of the conceptual and technology advances occurred over the last decade. A further generalisation and significant improvements and refinements have been added to the former theory.

1. INTRODUCTION

This paper has the ultimate aim of studying the limits of the space propulsion capabilities to be expected as far as current knowl­edge of the physical laws allows. The first objective is to carry out and then specialise the Multiple Propulsion Mode (MPM) equations to examples involving recent technology develop­ments and propulsion concepts.

Basic Physics made significant advances in the eighties. The Grand Unification Theories (GUT) and Supergravity have been contributing to the understanding of the structure of the Universe as a whole. Very impressive experimental results have supported such theories; though, several predictions such as the magnetic monopole,the proton decay and the Rigg boson have no confir­mation, there might be an amazing application to space travel, possibly rendering the rocket obsolete. The GUT explain all fundamental forces in terms of messenger particles: gluons for strong interaction, photons for electromagnetic interaction, the W and Z (a type of photon) particles for weak interaction, (envisaged) gravitons for gravitation. We have no control over such particles other than usual photons, to a limited extent. Consequentlythecharacteristicsofthehigh-energymacroscopic phenomena must be accepted as such. We can only try to arrange several of such "elementary" interactions in a propulsion system to control the final energy released, but we cannot control the features of the interacting particles, on which that energy de­pends, to make our engine simpler, more effective and reliable. Moreover, although the GUT foresee seven additional dimen­sions, they seem to be ''rolled-up" with sizes extraordinarily smaller than an atomic nucleus. We cannot sense such dimen­sions at all, in practice. Furthermore, no evidence of tachyons exists today. As far as the amount of energy density available in Nature, we have no true scientifically-based idea of how to utilise the incredible energy of the quantum vacuum.

These facts indicate that spaceflight, over most of the next century, will be based essentially on the same principles applic­able today, with two major extensions:

(1) space-time in a true relativistic sense

(2) nucleon-antinucleon annihilation energy utilisation.

Multiple Propulsion may then consist of arranging those propulsion modes which would exhibit the highest degree of performance among those ones analyzed hitherto:

(1) pure-rocketmode (PRM) (scientifically analysed since the last century [1]),

(2) photon sail/collector mode (PSM),

(3) ramjet-like mode (RLM) (first envisaged by Bussard [2]).

These three modes, if simultaneously present in a starship, will generally interact with each other.

In 1979 the author presented the MPM concept in two-dimen­sional (one space+ time) form [3]. Key points were the interac­tion between modes and a control law bringing the MPM starship to an enhanced rocket achieving high terminal speed with a propulsion ratio really low. Such last feature has been pointed again in [4]. This current work presents generalisations (three space+ time) and changes in an attempt to give a unified theory of the powered flight, from the classical Tsiolkovsky rocket equation to the relativistic starship motion. This theory could be also utilised to analyze the performance of some new propulsion system for the desired missions.

2. MPMBASICASSUMPTIONS

To find out the MPM equations, a number of assumptions are necessary. Most hypotheses are general enough in their nature to be hard interpreted as limitations to the theory. The MPM as­sumptions are as follows:

(1) no particular kind of exothermal processes occurring in­side the ship's body is referenced

(2) no particular class of ship's engine is referenced

(3) the starship may exhibit PRM, PSM and RLM simulta­neously or switch easily from one mode to another

( 4) the starship can capture matter from the space environment and by interaction with onboard fuel, exhaust it, besides

537

Giovanni Vulpelli

absorbing incidental electromagnetic radiation - partially or completely - and converting it into energy useful for propulsion

(5) the starship is able to reflect completely or partially the incident radiant energy in order to be accelerated or decel­erated; the electromagnetic momentum is presumed to be parallel to the ship vector velocity

(6) a negligible interaction between incident radiation and exhausting matter beams is assumed

(7) an onboard accumulation of mass from outside (when point 4 applies) is allowed

(8) non-propulsive utilisation ofabsorbed lightis allowed

(9) the overall thrust is due to three beams: the pure-rocket beam (PRB or k-jet), the photon beam (PHB or c-jet), the ram beam (RB or m-jet). From an energy balance view­point these will be usually coupled.

(10) the ship's motion is generally four-dimensional; thus, the jets at point-9 are allowed to be directed differently; curved trajectory arcs in free-field but using thrust are then possible.

The consequences of such assumptions become clearer as theirtranslation into mathematical tenns proceeds. However, the above points create a complex history of energy and momentum in the starship. Each propulsion mode will be characterised by a set of efficiency parameters entering the MPM motion.

3. PROPULSION MODE PARAMETERS

The next step in the MPM equations is to define the characteristic quantities for each propulsion mode. This is made in this section and the related nomenclature specified.

Pure-Rocket Mode: Ma = activemass Mi £

inert mass fraction of Ma converted into a utilisable energy through an exothermal reaction. It refers to the total energy of all reaction products (particles), except negligible-interaction ones, if any. It is an effective value which depends also on the way k-jet is ob­tained

a = fraction of e going to particle's rest mass s = fraction of (1 - e) lost into space at zero total three­

momentum (ZTM) in the ship frame (SF)

£K = (1 - a)e fractional kinetic energy of the reaction particles

Photon Sail/Collector Mode

Wo =

Ws

power - as measured in Galactic Frame (GF) - at the photon emission system (PHES) located in the de­parture star-system. PHES is assumed at rest in GF

power - as measured in GF - impinging the sail system. This sail, a collecting/reflecting surface, is

One can think of specifying Ws, which affects the ship dynamics, and hence

calculating the power of the GF source through the source dependent equation

Wo -Wo (Ws) to be appropriately developed

538

assumed to be controllable in direction. This power level depends on the ship-PHES distance, the interstellar medium absorption and the beam focusing characteristics (which in turn affects the sail area).*

TJ - fraction of PHES power - as measured in SF -absorbed by the ship sail. The (I - T]) fraction is assumed to be reflected. Diffuse-reflection and de­layed diffuse emission are not considered here. All (1 - T]) fonns the c-jet.

µ = fraction of TJ effectively added as kinetic power to thek-jetand/ortothem-jet µ= Omeanseither: both k-jet and m-jet are not active, TJ = 0, no support to the matter beam occurs.

v = the fraction of TJ radiated at ZTM in SF. One can think of a partial non-propulsive utilisation of the quantityT] (1 - µ);then TJ (1- µ-v)representsthe effective non-propulsive energy fraction on-board.

Ramjet Mode

dMe = scooped elementary mass in SF; 'Yv dMe = energy input to ship, 'Yv being the Lorentz factor of

the starship velocity V

n = . fraction of 'Yv dMe ''trapped" on-board. fraction of 'Yv dMe lost at ZTM outside the ship in SF.

For simplicity we may use: dMu = (l - l - n) dMe representing the rest-mass unclergoing reaction.

l, n should be zero for a quite effective ram system; n > 0 would mean a positive ship's mass variation due to the scooped mass (e.g. particles trapped or absorbed after interaction with the fuel onboard)

e

r

fraction of dMe converted into a utilisable energy; it refers to the total energy of all products except the interaction-negligible particles. It is an effective value depending on the way them-jet is formed.

fraction of e going to particle rest-mass.

The ram-mode so pictured has no counterpart of the rocket­mode parameters. Such a number is necessary if and only if the scooped mass interacts with some onboard fuel entailing s > 0 (e.g. antimatter). In these environments, however, one parameter sis sufficient to describe the interaction-energy losses from both rocket and ram modes.

Block diagrams for each mode (figs. 1,2 &3) are useful to follow the energy history in the starship.

The above set of parameters may be regarded as a sort of propulsion state vector, the minimum set of values necessary and sufficient to describe the dynamical effects of every modes.

4. DEFINITIONS AND CONVENTIONS

The formalism of Special Relativity is used to describe the energy-momentum history for the starship powered by MPM. The following metrics of the flat space-time manifold are adopted:

Multiple Propulsion Concept for Interstellar Flight: General Theory and Basic Results

' - - - -- - - - . -- -- -- - - -- - - - - -- -- -- - - -- - - - -- -- - -- -- -- - - - - - - -- - -- - - -- -- -{ M-JET

I A I

l MASS LOST PROPULSIVE INfOSPACE KINETIC ENERGY

(I - E) S dM, (I - 0) E dM,

' · - -- -

DEPLETION OF MASS CONVERTED ACTIVE INfO PROPULSION

MASS dM, ENERGY E dM,

"

RESIDUAL MASS REST ENERGY OE dM

(I - E) (I • S) dM,

4:'f. v-.... .<""~·A<

DEPLETION OF INERT MASS

dM,

Fig. 2 Energy sharing for a photons sail/collector system.

PSM ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' I I

Fig. 1 Energy sharing for a pure-rockets ystem. ' I I I I I

I

' ' I I I

I

K-JET

Z.T.M. RADIATED COMPONENT

\JTI Ld;

PHOTON BEAM ABSORBED

NON-PROPULSIVE COMPONENT

(I - v - µ )1J Ldr

~

ENERGY IN SF ~A Ti L dr

~

(1- l]) Ldr

~¥a11A6

M-JET

KINETIC El'<"ERGY TO THE

OTHER JETS µ 11 Ldr

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Giovanni Vulpetti

TOTAL ENERGY TRAPPED ON·

BOARD

ZThl LOST ENERGY

ly, dM, n 7. d.\!,

(~ RESTMASS t- COMPONENT !; (l·l·n) dM ' . ~-:

~,ih,%Q",r;r,,~z7..7Y" ,-,---:n7,_

f] ENERGY !NPlIT f TOTHESTARSHIP ·~; 1y dM. ,. L&M:.?,,;;!,k;J.J;f2Jj.X%~

):l KINETIC v COMPONENT 5, <1.· l) dM,'

tJ t·l.b-1;Jj}J.,!1/./.;;,r/,j1;.t,~?J:!J I

~

MASS C01'VERTED INTO uTILISABLE

ENERGY c dM;

REST-MASS re dM.'

"'~,(J. -- ~ ~

~ ICL""ETIC ENERGY ~ (l·r) e dM;

~

~\z"~ ~ M·JET

dM,' • (l · l·n) dM,

PSM

Aro -1

r-1 ]

A =AMi>= _\

1

, A,(J)=l, ... ,4 (01)

Any four-vector is written in such a way the space-like com­ponents first appear. The speed oflight and the light year are set to unity. In passing from ship frame to galactic frame the follow­ing rank-two tensor (boost) is necessary:

<l>~cx. = [ ET 'YvYJ 'Yv Y 'Yv

(02)

In (02) Y. denotes the instantaneous starship velocity in GF. The terms of the matrix 3 are reported in any good textbook of Relativity (5 & 6], together with the set of properties it satisfies. Properties of E in the current context will be recalled when appropriate and d r will denote the elementary proper time of the ship the instantaneous rest-mass of which is M.

A four-vector is generally here identified by capital Latin letter super(sub)-scripted by Greek indices, whereas the corre­sponding three-vectoris denoted by the same letterunderscored, unless otherwise specified. Four-vectors are generally used in this paper with a reference to their contravariant components. Differentiation of four-vectors is particularly meaningful with respect to the ship's proper time. Quantities primed are generally referred to the SF in contrast to the corresponding unprimed quantities, defined in GF, unless the reverse is explicitly men­tioned, e.g. for exhaust velocities.

540

RESIDUAL REST-MASS (I-<:) dM;

· C.u .. A'JJ11~'J Fig. 3 Energy sharing for a generalised ramjet system.

5. ENERGY-MOMENTUM HISTORY FOR PROPULSION MODES

The fundamental point for carrying out the MPM motion equa­tion is to determine the elementary variations of the four-momen­tum of the propulsion modes in the ship frame. For the flow diagrams of figs. 1, 2, 3 are also taken into account. Furthermore, a control volume u is imagined the surface of which exactly envelops the starship' s physical contours, including electromag­netic fields for scooping, ejecting and/or reflecting charged particles.

5.1 Pure-RocketSystem

Let ufk) = 'YK [Q(k) lf be the rocket jet velocity four-vector in SF, y being the Lorentz factor. The k-jet four-momentum vari­ation is given by:

dP'C}) - ['YkQ(k)dMp]- UC})dM (03) V~ - 'YkdMp - lK p

where dMp = dMi +OE dMa represents the total rest-mass ejected in the proper interval dr by PRM. Such mass is given a four-velocity m'tc). Outside u one has a ZTM

dP'f/c,l) = [(1-E)~dMa] (04)

where Q = [0 0 O]r. The rocket-mode needs nothing else for the moment.

Multiple Propulsion Concept/or Interstellar Flight: General Theory and Basic Results

5.2 Photon Sail/Collector System

This propulsion mode is a little more complex with respect to the previous one. In SF the radiant energy rate, Ldepends on the ship velocity. According to the assumption 5 of sect. 2 it is shown from Relativity that

L=Ws (l-V)/(l+V) (05)

where V is the magnitude of Y.. The GF rate of energy striking the sail/collector is W=(l-V)Ws for a receding ship, as it is our case. This assumption causes no loss of generality. In fact, it is sufficient to exchange signs in Eq. (5) to deal with an approach­ing ship.

During the elementary proper time the sail/collector is struck by photons the four-momentum of which is dP'a,._>; dP'a+) denotes the four momentum of the reflected photons. The total change in SF is therefore

d p ·a > = d p·a+>- d p·a-> = [-<2-~ ~ ~~ ~NJ (06)

The effect of the photon momentum variation is sensed on­board as a force parallel to Y.. From fig. 2 it is apparent that, whereas the c-jet is given an energy (1-11) L dr, the amount 11 L dr can be in principle utilised for the other two jets and for the non-propulsive ship systems. As a consequence of this partial utilisation, in SF one observes a ''leakage of photons" as a radiating sphere of four-momentum

d p·a.l) = [ v 11 ~ dr] (07)

Equation (10) takes on this simple expression because in GF the relative velocity between spaceship and interstellar medium is the opposite of the spaceship velocity. In another class of inertial reference frames, where the interstellar medium is not at rest, the spaceship velocity can no longer enter Eq. (10) this way.

It should be noted that vfe> is a contravariant four-vector, namely, the four vector of the elementary scooped mass, al­though its content is identical to the covariant velocity of the spaceship. Denoting the four-vector (in SF) of the m-jet by ufm), the change of the four-momenttun of the external matter processed and exhausted amounts to:

dP'~m) =[Ym!l..(m)dm] = ufm) dm (11) ymdm

where dm = (1-e+re) (1-l-n) dMe, while at ZTM outside v there is a loss of four-momentum equal to:

d P ··re.I) = [1 rv ~Me] (12)

According to fig. 3, there takes place in general a non-zero contribution to the ship mass given by

(d M>e = n 'Yv d Me (13)

Equations (11) through (13) represent what essentially char­acterise the ramjet propulsion mode in SF. Note that the set e=O and l=n=O means a pure braking of the ship. If n= 1 the SF force is halved; however, mass is accumulated on-board for a future use. l ~ 0 means a braking with a reflection coefficient 1-l/2.

Finally, the four-momentum variation ofthec-jetis expressed 5.4 Ship System by:

dP'fc> = dP'a+> = [(1-11)Ldr(-~IV)] (1-11) L dr

(08)

dP'fc> dP'(c)a. = 0 as it must be. In contrast, dP'a,> dP' (L)a. ~ O because Eq. (6) is merely a difference be­tween two beams always (unless 11 = 1) different in direction. In other words, the left-hand side of Eq. 6 does not represent a photon beam for 11 < 1. However, it can correspond to the four-momentum variation of a fictitious rest-mass beam as one can recognise from the four-vector definition.

The ship' s mass undergoes a variation due to the interaction with and the processing of the external radiant energy:

(d M)L = (1 - v-µ)TJ L dr (09)

as it follows easily from fig. 2.

5.3 Ramjet System

In this mode the spaceship receives a matter four-momentum input and, after processing it, releases a four-momentum output. The four-momentum of the captured (scooped) elementary mass into the control volume during dr is expressed by

The spaceship is a non-inertial frame of reference where basic sensors such as accelerometers measure a three-dimensional acceleration during d r. One can picture an external force respon­sible of the acceleration measured on board in such a way the GF force and the SF force components along the ship's instantaneous velocity are equal to each other in both direction and magnitude. Thus, the ship's body is given the elementary four-momentum.

dP 'fs> = [Md!!~T] (14)

where as is the opposite of the accelerometer output and dM the total ship mass variation in d r .

6. FOUR-MOMENTUM CONSERVATION

All the basic elements are now to hand to proceed toward the MPMmotionequations, using the four-momentum conservation law in both SF and GF.

6.1 Total Four-Momentum in SF

d P'fe> = -[-'Yv Y.. d Me] _ YvdMe --~)dMe (10) In the ship-frame the four-momentum balance due to the differ­ent systems analyzed in sect. 5 can be written as follows:

541

Giovanni Vulpelti

d p • fl> + d p ·fk.l) + d p ·a> + d p ·a.l) + d P 'fe> + d P 'fe.o + d P 'fm) +

d p ·rs> = [ ~ J = aa (15)

where the different rows correspond to the contributions related to the rocket-mode, photon-mode, ramjet-mode and ship system, respectively. These terms have been expressed in sect. 5. The fourth components of Eq. (15) give the explicit value of dM, namely, the rest energy variation of the spaceship. By simple substitution we get:

Y·I=Y·Is (22)

Therefore, the four-thrust acting on the ship is expressed by

T1" = '/v[r~Y] + MV°" (23)

The thrust I and its associated power are expressible by inserting Eq. (19) into Eqs. (21-22). Recalling the following identity [6] ,

:a: (.) = <rv -1)[ ( · ) · y 1 f/V2 + (.) (24)

where in our case (.) coincides with Mas, the thrust magnitude d M = - '/k (d M1 + ae d Ma) - (1-e)s d Ma + 11 L dr - Yrl L dr + '/v d Me -'Im dm - lyv d Me (16) can be computed directly by utilising the invariance of a four­

vectormagnitude: Figures 1,2,3 also contain the interaction of matter beams in terms of kinetic energy. Neither beams give energy to the c-jet but both could receive photon energy. We can thus write:

('/k - l)d Mp + ('/m - l)dm = (1 - a)e d Ma + (1 - r)e d Mu + ('/v - l)d Mu + µ11 L dr (17)

By combining Eqs. (16-17) and making dMu explicit we obtain

dM = -dM;-[(1-e)s+e]dMa + nyvdMe + (1 - µ - v)T) L dr (18)

The most important feature of Eq. 18 is that, in general, dM can be of any sign depending on the number and the strengths of the modes being operated.

The space-like part ofEq. (15) may now be processed. Divi­ding by the elementary time and collecting stream terms at the right-hand side results in:

Mas = -yk Q(k)M!' + (2-11~ L YIV (l9) - -'Ym Q(m)ffl - '/v f Me

where the dot denotes differentiation with respect to the ship proper time. Equation ( 19) is formally obvious. However, it must be coupled with Eq. (17) to remove the seeming arbitrariness about the jet velocity magnitudes of the matter beams.

Equations (17 - 19) provide information necessary to consider the ship's four-momentum variation in the galactic frame.

6.2 Total Four-Momentum in GF

The ship's four-momentum variation in GF is given by:

d p rs) = d <MV'"> = <I> f3 a. d p ·rs) =

[ M 'E g_s dr + '/v f dM J (20)

M '/v( g_s· f ) dr + '/v dM

Let Mas =Is the force measured as inertial in SF; it gives rise, when transformed to GF, to

'[ = 8 [slyv (21)

In addition, it is known that

542

2 2 ~ ·2 2 ·2 T1°Ta. = '/v[('.[ · f) -r] + M =-Ts+ M

or, more explicitly, by using Eqs. (18, 19, 22) again

r = T;1y~ + <Is. f)2

Ts. V = -yk( U(k)' f) Mp + (2 -11)L V - - -'Ym ( U(m) · f) m - 'Yv V2 Me

When LI IY., Eqs. (26 & 26a) immediately give

(25)

(26)

(26a)

T= Ts='Yk U(k)Mp+ (2-11)L+'/m U(m)m-rv V Me (26b)

a well-known result in a one-dimensional analysis. Equations 26 and the left one in 26b are formally independent of the propulsion modes. Specialised equations arise when expressions such as Eqs. (19 & 26a) are available.

So far, only the rightmost side of Eq. (20) has been handled. Such equation, formally, is the motion equation we are looking for. Equation (20) can be transformed in such a way to achieve two goals:

( 1) an explicit differential equation for the ship four-velocity,

(2) a general expression to which some property of invariance may be added.

The right-hand side of Eq. (18) can be arranged into three physically different terms as follows:

d J = d M; + e d Ma (27a)

d N = (1 - µ) 11 L dr + (l + n) '/v d Me (27b)

d Q = (1 - €) s d Ma + VT) L dr + l 'Yv d Me (27c)

Equation (27a) represents all coming-from-ship total energy exhausted outside at non-ZTM in SF. Equation (27b) corre­sponds to the energy processable on-board for non-propulsive purposes; part of this energy is lost at Z1M in SF and is represented by Eq. (27c). Therefore, Eq.(18) can be cast into the form

dM = - dJ - dQ + dN (28)

Note that clJ and dN contain kinetic energy terms in addition to rest-mass terms. Only to the dQ term can a four-momentum

Multiple Propulsion Concept/or Interstellar Flight: General Theory and Basic Results

be associated in agreement with Eq. (15). Thus, in GF we have

d p (Q) a. = <I> fl a. [ £] = ya dQ (29)

The swn of the four-momenta of the jets in Eq. (15) is given by

d P ''(;eis)= Vfk)dMp + L/fm)dm - \tfe)dMe + (30) U~) (1-11) L dr - vf+>L dr

where U a. [-VIV] a. [VIV] <-> = T · u <+> = -1

are the unit four-vectors of µie c-jet and the coming photon beam, respectively. Note that dP /is does not generally equal one pro duct of a four-velocity and a mass-energy variation. This is due to the presence of two different photon beams in Eq. (30) (see also sect. 7).

The conservation of the four-momenta variations in GF together with Eqs. (29-30) results in the following arrangement

MdvA + t/-(dM + dQ)=-<I>µA.dP't1/eis) (31)

Equation (31) can be considered the general equation of motion for space jet-propulsion involving zero-rest mass and/or non­zero-rest-mass particles. In particular, any of the propulsion modes considered can be singled out. Also, the dynamics result­ing from any two-mode or three-mode interactions can be cor­rectly computed.

Because <I>~ is a function of the components of ya, only one four-velocity in GF appears in Eq. (31 ). The mass-energy source tenns can be identified - as a whole - in the ship mass variation, the ejected energy and the energy lost into space at zero total three-momentwn in the ship frame of reference, respectively. Conversely, these considerations could be also taken as starting points to be coupled with explicit tenns in the thrusting modes of interest.

To revert to Eq. (30), the energy associated to the jets acting simultaneously is given l;>y:

dE (jets)= "fkdMp + Ym dm + (1-11)Ldr (32)

The multiple propulsion mode is characterised by energy released by matter transported on-board and energy entering the ship body from outside. The total energy involved during dr can be simply expressed as:

dEt = dM;+EdMa+(l-E)sdMa+ "(vdMe + L dr = d E (jets)+ dQ +dB

(33)

where dB = (1 - µ - v) 11 L dr + n Yv dMe is the energy re­tained on-board. Equation (33) is obtained by using Eqs. (17, 27c, 32).

The efficiency of MPM can be expressed by the fraction of the jets energy with respect to the overall energy. Analogous to the PRM , is a definition of mass-energy utilisation efficiency as follows:

'Ila = d M (jetsY d Et (34a)

where dM (jets) = dMp + dm. Either11m or 110. or both, can vary with the ship proper time.

Equation (31) can be then written as follows:

Md VA.+ 0 [dM + (1 -11m) dE (JetsY'llm - dB]= - <I> µA. d P ' t1/eis) (35)

In the case of a pure-rocket system there is a set of special interrelationships:

d M()ets) = d Mp dB=O

d P '~els)= U (k) a. d Mp dE(Jets)= "(k dMp= d Mt+ Ed Ma

dM=-(ykdMp+ d Q)=-dE1

(36)

'llm = - d E(jetsY dM = "(k 'Ila= (Q + M)IM = - "(k Mp!U

With the aid of these equations, Eq. (35) is simplified to

d 0 = -11m [0-<I> µA. U ~yykl dM/M (37)

which represents the relativistic extension of the classical rocket equation with the mass utilisation efficiency as an explicit term. Under the non-relativistic and one-dimensional motion condi­tions, it is easy to recognise that both the space-like and time-like parts of Eq. (37) give the same information that is:

d V = - 11m U (k) dMI M (38)

namely, the Tsiolkovsky rocket equation in differential form.

7. THE EQUIV ALENTPURE-ROCKET

A multiple-propulsion spaceship can consist of several fun­damental thrusting modes activated during its flight. When the motion is simultaneously governed by more than one mode, one may expect that the combined propulsions exhibit a general performance somewhat higher than a pure rocket's. Otherwise one would use the PR mode only.

By comparing Eqs. (31 & 37), this means that

- [ ya dQ +<I> fl a. d P '~ets)] = [ (1 -11m) ya+ 'llm <I> fl a. ufl*l 'Y*] dM

(39)

The right-hand side of Eq. (39) indicates that the fictitious rocket-ship is equivalent to the real spaceship by exhibiting the same velocity, mass variation and utilisation efficiency. To do this the rocket-ship should have an exhaust four-velocity lfi in SF. The fact that this is conceptually possible can be proved by casting the left-hand side ofEq. (30) into the general form:

d P ''(;eis) = d (M1 U (J) a.) (40)

'llm = d E (jetsY d Et (34) where dMJ may be thought as a certain ejected mass endowed with a certain four-velocity ~ in SF such that Eq. (40) is satisfied. If and only if the three-dimensional velocity li(J) is Similarly, a rest-mass efficiency can be defined as:

543

Giovanni Vulpetti

constant in both direction and magnitude, then the right-hand side of Eq. ( 40) is equal to ~.This, in tum, is meaningful only if some of the jet-matter modes is present in MPM. The value of d.MJ is detennined by the condition of ~ V(J'µ = 1 applied to Eq. 40. Note that ~ V<±_p.= 0.

Thus, we have

V (J) A. = dP 'tersy[dP 'Uets) dP '(jets}<I] 112 (41)

The denominator of Eq. (41) provides dMJ. Equation (41) implies that one or more beams in Eq. (30) have to be modulated in order to yield ~constant throughout the flight.By definition of rocket equivalence one can write:

d P 'bets) = [-11m J!..* dM] -11m dM = -110 U • A. dM (42)

which is cobsistent with Eq. (39) if and only if11m is expressed by the last of Eqs. (36). The space-like part is therefore made explicit into:

Yk Q(k) dMp + Ym Q(m) dm + Yv f dMe (43) - (2 - 11) ~/V) L dr = - 110 Y• Y..• dM = - Is dr

In general, two vector controls (li(K) and li(m>), and a number of scalar controls (L d r, dMa, dMi, dMe) are presentin Eq. ( 43 ). To make the physics more significant,itis necessary to maximise the function

7°' Ta. + /... K (Q(k). Qcm>) (44)

with respect of li(K) and li(m), K being the implicit form of Eq. ( 17) and 'A.a Lagrange multiplier. Although long calculations are necessary to find the maximum of the above function, the final result is as simple as significant

Q(k) = Q(m) = Q (45)

Therefore, the jet velocities of the two matter beams are to be made equal to each other to maximise the four-thrust magnitude. Equation ( 45) does not mean that U is independent of time; in addition, !JIU could be different from± YJV.

Recalling Eq. (17) once again, the magnitude of U can be obtained from

_1

= ( 0-a)edMa+[(l - r)e+yv - l]dMu+ (46

) Yu µ 11 L drfl (dMp + dm)

which therefore expresses the magnitude of the vector con­trols as function of the scalar ones. Note that the right-hand side is the ratio between all available on-board propulsive energy and all rest-mass of the matter beams. Inserting Eqs. ( 45 & 46) into Eq. (43), the relationship between .U and.U• is made explicit

U = [ -11m V• dM -yv f dMe + (2-11)(f/ V) L dr] (47) - l (dM;+ e dMa + µ 11 L dr + YvdMu)

To calculate the scalar product in Eq. (41), namely, dMJ, produces:

(dM J ) = 11 m 2

(1 - U • 2 )(d.M)2 ( 48)

which can be also expressed in terms of U by Eq. 47. In addition, equalling the space-like and time-like parts ofEqs. ( 40 & 42), produces

544

Y1 !}_(J)dM1 = -11m!l• dM (49a)

y1dM1 = -11mdM (49b)

Combining Eqs. ( 49a) and ( 49b) results in

!}_(J) = !l• (50)

li(J) can, in turn, be read in terms of .U. the common four-vel­ocity of the k-jet andm-jet, throughEq. (47). Equations (47, 48 & 50) exhibit a common significant feature: they contain the equivalent-rocket jet velocity .U• in their right-hand sides. There­fore, the actual exhaust velocity of both matter jets and the overall momentum of all beams can be read in terms of .U• . .U could be time-varying even if Eq. (50) compels .U• to be a constant vector. From what said after Eq. (42), it is possible to write explicitly

Y• 110 = 11m = 1 + dQ/dM (51)

which entails

(dM 1 2) = (dM + dQ)2/y • 2 = 11 m2 (dM)2/ y • 2 (52)

dM1 = -11odM (53)

full consistent with the rocket equivalence. By means of .U• and 11m (or 110) the motion of the equivalent rocket is completely detennined through Eq. (37).

The set of control equations consists of Eqs. (46,47,50,51). They are eight in number and contain fourteen controls: !I•. !I. !IO» Ma, Mi, L, Me, 11m· Thus, there are six degrees of fre~dom. The information content in the momentum rate dP'{fetsydr has been already used in Eqs. (47 & 34). Equation (53) may be utilised as a test of internal consistency for the final solution.

Because for the pure-rocket mode the degradation of the effective exhaust speed is due to the mass lost at ZTM, two of the degrees of freedom are used as follows:

L = (Q + L0 ) iL(r)

Me = (-M - Mi)f(V) ie (r)

iL = 0,1

ie = 0,1

(54)

(55)

where f(V) is an appropriate function independent of the way the incoming flux may be controlled. Lo, assumed to be in general dependent on the SF time, can be considered a sort of bias which augments the pure rocket performance. The dimen­sionlessquantitiesiL (r) and ie (r )represent switching piece-wise functions allowing mode activation/disactivation in order to optimise the flight with respect to some index. In fact, as already noticed in [3], one can have a thrusting/coasting with respect to a certain propulsion mode.

The effect of such things on the overall solution will be seen as the related steps proceed. Equations (51, 54 & 55) are first used to find L, Q and Me as function of Ma. Recalling the ex­pressions 18 and 27c and arranging, one carries out

-ief{ 11 iL [e(v - s (1-µ))Ma + (1-v- µ)Lo] -Me = [E(l - s) + (1 + µ 11 iL - 11 iL)S] Maf (56)

l{l + Ae AL+ 11 iL [(1 - µ) l A.e - v (1 + l A.e)lf

L = iL [(1-e)sMa +Lo+ lyvMe]l 'A.L (57)

Multiple Propulsion Concept for Interstellar Flight: General Theory and Basic Results

Q = [(1 - E) S Ma + VTt iL Lo+ l 'Yv Me]h.L = - (1 -Tim) M (58)

where AL = 1 - v TJ iL Ae = 'Yvf(v) ie

NeitherMe nor L depend on Mi, but on the ship's speed. The photon sail propulsion mode takes on a special role in the MPM. In fact, AL= 0 is not allowed inasmuch as it would be meaning­less to absorb all incoming photon energy first and then radiate it completely at ZTM. Consequently, the photon mode must be either present actively ( 1 ~'AL> 0) or truly absent (AL= 1).

Three other degrees of freedom are used to set II• to some vector, constant according to Eq. (50). Thus, nine equations have beenemployedandL,Me, Q, !J:., _!I(J)areknown. Wbatremains to do is to determine the behaviours of Mi, Ma, and .II. The last degree of freedom is spent in assuming Ma to be known. More generally ,Ma ( r) can be determined by optimising it with respect to some performance index, e.g. the total energy spent, the flight time, the simultaneity ofreceived information in multiple-probe missions, and so on. This is a desirable feature of the equivalent pure-rocket mode whereupon this concept allows an overall optimisation of the fictitious rocket flight to be made similarly to a real pure-rocket flight.

The programmes Mi and U can be obtained by Eqs. ( 46 & 47).

U = [~+ O? Q• -y~ f_Me+ (2-TI) ~IV)LJ - /(Mi+EMa+YvMu + µTIL)

(59)

'Yu =<Mi + E Ma+ '}'v Mu+ µTl L)l(:Mj + oEMa + m) (60)

where Me, L and Q are provided by Eqs. (56, 57 & 58) respectively.

By considering the Lorentz factor of the right-hand side of Eq. (59) and equalling it to the right -hand side of Eq. (60), one obtains an equation of Mi as function of Ma, II• . Y, .II, Y and the parameters of the propulsion modes . Substituting Mi back to Eq. (59), one gets llexplicitly.

8. COMPLETION OF THE ROCKET-EQUIVALENT SOLUTION

Equations (56, 57 & 58) together with Eq. (18) can be cast into the form

Me = f(V) (X' e Ma + cpeLo) ie = fXeMa; Xe~ 0, 'Pe~ 0 (61)

L = (X'L Ma+ 'PL Lo) iL =XL Ma; XL~ 0, 'PL~ 0 (62)

M =(X's - Xi) Ma+ <psLo = (Xs - :Xi) Ma; XI~ O,cps ~ 0 (63)

Q = X'Q Ma + <l>QLo = XQ Ma; XQ ~ 0, <l>Q ~ 0 (64)

where (X' e, <pe ) and (X 'L. <pL ) are independent of both Ma and Lo; Mi/Ma =x1; X's and cps are also functions of x· e and :X'L Note that Xe and XL contain the switching functions. Acting on Eqs. (59 & 60) produces

2 2 -V• Xi +2Bx1+C=O (65)

The coefficients ofEq.(65) are given below:

01 = (1-o) e, 02= (1 +a) e, b1 = (1 - l- n)(yv-1 + e -re)f(V)

h2 = (1 - l - n)("{v + 1 - e +re) j(V), c1 = µT), d1 = yvf{.V), di= 2 -11

2 B=(x.r+xa> u * +(c1 -di!!.•· y1V)v.+(b1 +d1 !!.•· Y)~+a1

2 ~--2 c =(cl - d 211. L + [(01 + 02) Cl + 2 <x.r + XQ) di!!.• . y1v ]XL+

[(b1 + b2) Cl+ 2Vd1 d2] XL Xe+ [ a1 b2 +bi a2 - 2(Xs + XQ) di Q• · ~ J Xe -

2 2 2 ( )2 2 (d I V - b1 b2)X e + a1a2 - Xs + X Q U •

Inserting the parameters :x (.) and Eqs. (61-64) into Eq. (59) results in

[(:XJ-XQ-Xs) Q• -YvXefY.+ (2-Tl)XLfNJ V= (66) - t[x1+E+(l-l-n)yvfXe+lillXL]

Eq. (58) can be read explicitly as

Tim = 1 - XQl(X; - Xs) (67)

Gathering Eqs. (51, 63 & 67) results in

-T1oM = (:Xi - XQ - Xs) Mal'Y• (68)

Because the left-hand side ofEq. (68) is the invariant magnitude of the four-vector dP ·~rsy dr and Maly• is apparently invari­ant, it follows that this MPM policy is accomplished in such a way to render the quantity

X ( ¥_, iL, ie) = X1 - XQ - Xs (69)

an invariant. The fact that x depends on Y should not surprise because, recalling the remark after Eq. (10), such velocity must be viewed here as the relative one between the interstellar medium and the spaceship, which is plainly an invariant. Anal­ogous considerations bold for the PSM mode inasmuch as the light source is at rest in GF. x is of interest because it is a propulsion invariant.

With all necessary time behaviours it is now possible to conceive a multiple propulsion spaceship as dynamically-evol­ving as a pure rocket vehicle with a performance higher than the real rocket system operating in the multiple combination. Some significant pure rocket dynamics properties are therefore applic­able to an MPM working in the rocket-equivalent mode. Some of these may be particularly useful in mission feasibility evalu­ation. Equation (37), in particular, bas a time-like component which expands into

dyv = 'Yv'Tlm (Q• · Y.)dMIM (70)

Therefore, the GF rate of the ship's total energy results in

dEldt = M+T1m(Q•·Y.>M = M+I" Y. (71)

545

Giovanni Vulpetti

The rightmost hand ofEq. (71) follows easily from Eqs. (22) and the last equality in Eq. (43). Combining Eq. (71) with the definition of (ship) kinetic energy, say, K gives

dK =Tim (Jl..•. ~Mdt+Mdt-Mdr (72)

Equation (72) can be usefully integrated under the hypotheses of rectilinear motion (11• 11 -Y) with M = const. and L const. One then obtains explicitly

MJYJ- Mo"(o = Ts (SJ- So)+ M (lj- to) (73)

Equation (73) is a significant motion integral showing the basic interrelationship between mass and mass rate (SF), path (GF), flight time (GF), velocity (GF) and thrust (SF) of a spaceship powered by a rocket-equivalent propulsion mode.

9. DISCUSSION OF RESULTS

This section discusses a number of flights to highlight multiple­mode behaviours and the advantages with respect to a pure­rocketmode. Such examples could be considered as basic results.

9.1 Interstellar Flight Optimisation Code

The optimisation process underlying the flights to be discussed in this paper will be extensively presented in another paper on Multiple Propulsion Concept. the driven algorithm is a robust version of the Levenberg-Marquardtmethod.

A computer code named SMAC (Starship Mission Analysis Code), written in FORTRAN-77 and running completely in double-precision (IEEE Floating Point Standard) on a full 32-bit personal computer with MS-DOS 3.3 or UNIX System-V oper­ating system, has been set up by the author to deal with a three/four-dimensional starship trajectory profile caused by any physical combination of single modes and multiple-mode (equi­valent rocket formulation) propulsion. The current version (A.10) of this code is composed of about 4500 lines and requires 400 KBytes of real memory. The execution time for the cases

FUSION·BASEO MULTIPLE-PROPULSION SINGLE srAGE

0.9

0.8

0.71 ~ RU

0.6+ MN A I L T

0.5t I S s

0.4t ~ 0.3

0.2

0.1

00 28,8 )7 .6 86.4 115.2 SHIP-TIME (day)

546

2

3.5

3

2.5

2

1.5

0.5

0 144 0 28.8

discussed below range from 0.2 to 4 minutes. Version A.IO is able to deal with up to four thrusting and two coasting phases in order to cover an extremely broad class of out-of-solar-system and interstellar flights. The user can select both the performance index out of four objective functions meaningful for starflight and the control parameters out of admissible thirty-one; up to eight equality constraints and seven inequality constraints on state and control parameters, respectively, can be included in the optimisation process. The major purpose of SMAC is to have a unified research tool, capable to be augmented in performance as experience accumulates, for correctly studying missions be­yond Pluto. Version A.10 is able to deal with the gravitational field of moving stars,especi"11.y useful when the spaceship speed is sufficiently low such as in the early phases of the acceleration from the departure star-system and the final ones of a decelera­tion close to the target star-system.

9.2 Multiple Propulsion Examples

Two kinds of flight are presented, the related results being grouped by using normalised units. The speed of light, the sidereal year and the spaceship initial mass have been set to one. The distance unitis the lightyear. Unless explicitly specified, the same units are used in presenting results.

9.2.1 High Terminal Speed Flights

The first example consists of a 20-tonne single-stage ship powered by a D-He3 fusion engine augmented by a microwave beam of 48 TW in GF (1 TW =1000 GW) to generate a dynamics equivalent to a rocket of exhaust speed equal to 0.93. Since no loss of energy at ZTM has been assumed in this example, the mass/energy utilisation efficiency equals one. The profiles shown in Figs. 4.1 regard the achievement of a cruise speed of 0.5 with a propulsion mass ratio equal to 1.80, the 44% of the exhausted mass being inert. To do this,· having specified that the photon power is constant in GF (then in SF the power decreases according to Eq. (5)), the inert mass is modulated according to the behaviour of Fig. 4.2. The fusion power and the

cm factors

57.6 86.4

SHIP-TIME (day)

115.2 144

Fig. 4 Multiple Propul­sion Mode (equivalent rocket): one-thrusting­phase single-stage vehicle endowed with fusion­based rocket augmented by photons mode. The photon beam power is spe­cified in the galactic frame. Figure 4.1: behaviours of state variables vs. the ship time; Figure 4.2 : beha­vious of the chi factors (see definitions in the text), matterbeamexhaustspeed and thrust vs. the ship time.

Multiple Propulsion Concept for Interstellar Flight: General Theory and Basic Results

1.00

0.90

O.IO

0:10

t; 0.60

5 0 o.so w

~ 0.40 ::;:

" 0 :z. O.JO

0.20

0.10

O.OD

0.000 0.050

ANTIMAT!'ER-BASI!D MULTIPLE-PROPULSION SING!Jl,.STAGI! SPACESHIP

O. IOD 0.150 0.200 o.~

SHlP TIME (year)

T'Tl 0.)()() 0.350

l.lO

l.00

2.lO

~ 2J)O

~ z 0

~ 0. 0

l.lO

" 0. 1.00 0 ~ ::; < ~ O.lO

" 0 >:

D.00

l.

AN11MATIE.R.-BASI!D MULTIPUl-PROPULSION SlNGl.£.STAGI!: CIU - fac101t

l.

x. I..

--h~ •lamjocl j I I I I ii ii I j I I Ii I ii ii I I ii ii I I ii j I I I ii I ii I I I i I I ii ii I I I ii I I ii I I I II I I Ii I I i I

0.000 0.050 0.100 o.uo 0.200 0.2.SO 0.300 0.) }() SHIPllMI! (year)

Fig. 5 Multiple Propulsion Mode (equivalent rocket): two-thrusting-phase single-stage vehicle endowed with ll!ltimatter-based rocket augmented by photons and ramjet modes. The photon beam power is specified in the galactic frame. Figure (5.1): behaviours of state variables vs. the ship time; Figure (5.2): behaviours of the chi-factors (see definitions in the text) vs. the ship time.

I.~

1.00

0.95

ANllMATreR·DASIJO MIJUlPU!,.PROP\Jl.SION SINOL.11-STAGB 1: Mus Utilisatioo P.fficicocy 2: 1.lauerllcamSpecd

0.90~

D.85 O>

(2)

0.80

0.75

0.70

0.65

0.60

0.55

"""' ' ... 0.50 I I I I I I I I I I I I I I I I I I I I I I I I Iii i Ii I Iii o ii;';'.",',".':':' I I I I I I I I I I I I I I I I I I I I I I I I I I

D.000 Q.l)jO 0.100 o.uo 0.200 0.150 D.lOD 0.350 SHIPTlMI! (,real)

Fig. 6 Matter beam exhaust speed and mass/energy utilisation efficiency vs. the ship time for the flight of fig. 5. Note that ( 1) the ramjet phase is successive to the photon phase. (2) that both phases of thrusting are made to be equivalent to a photon rocket.

photon power are of the same order of magnitude. For compari­son, a pure-rocket fusion-based vehicle at the max specific impulse allowed by the nuclear fusion with the same mass ratio would exhibit a final speed of0.051, starting from zero speed as it is in the example of fig. 4. Alternatively, a mass ratio of 548 is required to achieve one halfthe speed oflight. No realistic flight would be possible with a starship of 20 or even 200 tonnes.

Figures 5.1, 5.2 and 6 are the analogous ones of Figs. 4 for an antimatter-based multiple-propulsion ship with additional capa­bility. Since the antimatter rocket exhaust speed is potentially much higher than that of a fusion vehicle, here e = 0.675~ 0.26 if the pion energy is utilised [ 4], it is possible to add a second propulsive phase characterised by a ram-augmented rocket [7]. Such a combined mode seems to have some chance more with

respect to the fusion-based ram; in fact, it is not possible to prime a rammode by the nuclear fusion [8]. In terms of terminal speed, the first phase (rocket+ photons) exhibits 0.43; then, the photon mode, active for about 30 days at a power level of 35 TW (GF), is switched off (the distance from the solar system is 0.018) and the ramjet.mode (of field intake area= 8000 Km2 in an environ­ment of 5 proton/m3

) activated while the rocket-mode active mass rate is halved (from 0.001 to 0.0005 kg/s). Ramjet and rocket are active for about 90 days. The presence of the antimat­ter-matter annihilation mode plus appropriate additional modes, for low and high ship speeds, allow the ship dynamics to be made equivalent to a photon rocket, although with a mass/energy efficiency less than one especially in the ram mode as shown in fig. 6. The outcomes of the second acceleration phase, where an i£Jefficiency of n=l=O.l for the ram mode was assumed, are a terminal ship speed of0.70 and a final mass ratio of 3.11; 52% of the total onboard propellant consists ofinertmass.

For comparison, a single-stage matter-antimatter rocket, with the same history of the inert/active mass sharing (that is an increasing variable jet speed), would accelerate for 120 days up to 0.58, of which 0.26 would be gained after the first 30 days. The rocket mass ratio would increase up to 4.26 in order to achieve a speed of0.70.

Such examples confirm the recognised importance of a photon component for efficaciously propelling a starship [3,9,10,11]. Moreover, the ram mode may perhaps be viewed better as a slow decelerating device during long quasi-co'ttings.

9.2.2 Pro:xima Centauri Orbiter Missions

One may envisage a trajectory to Proxima Centauri, given by way of example only. The flight entails no ram-mode, according to the considerations of sect. 9.2.1. The photon mode, when present, is envisaged only to transfermomentum to the ship body; namely, TJ = 0. (Absorption of photon energy by the onboard propellant will be considered in another paper.)

Figures 7 .1 and 7 .2 show the mass and speed time-evolution of a 20-tonne two-stage ship powered by antimatter engines to reach Proxima Centauri in 15 years. Flight figures are reported

547

Giovanni Vulpetti

0.8

0.6+

0.4+

I 0.2

0

0

0.9

0.8

0.7 •

0.6 t-

0.5 t-

0.4 t 0.3

0.2 t 0.1

0

0

--I I

0.9

0.8

N 0 RU MN A 1 LT I S s E D

PROXIMA CENTARUI ORBITER MSSION

WITH PURE-ROCKET ANTIMATTER ENGINE

mass

ncce lcmtion phnse flight time (SF) - 15 yr initiol moss - 20 10nnes

12 2" 36

SHIP TIME (day)

PROXIMA-CENTAURI ORB!TE.R-MISSION Willi FUSION-BASED MULTIPLE PROPULSION

N 0 R M A L I s E D

u N I T s

9.25

initial mass • 20 IODDC$

scage jettisoning

lllllS•

lwo MCCClenu.ioo ~· (rodet + photon)

18.5

SHIP TIME (day)

27.75

speed

initial mass - 20 IOnnes flight time [SF] - 15 yr

scage jettisoning

37

48 I

l 0.3 I (scage jettisoning)

0.25 +""" ' deceleration phase

0.2 + ~

N

~ 0.15 + ~ u

MN A I LT 1 S

0.1 + s

0.05

0

2

0.8

0.7

0.61

o~r

0.4 t-

0.3 ~

0.2

0.1

0 0

I

0

I! D

II 22 SHIP TIME (day)

flight time (SF) - 35 yr

jettisoning during coasting

N 0 RU MN A 1 L T 1 S s E D

dccelcnuioo phase (rocket)

aw•

•peed

33

20 4() 60 SHIP TIME (<Illy)

2 PROXIMA~CENTAURI ORBITER-MISSION WJTII ANITM~·BASED MULTIPLE PROPULSION

0.6 suge jettisoning

44

80

0.7 ~lo~ mass

N 0 RU MN A 1 LT 1 S s

0.6i N 0 RU

0.5T MN A I LT

0.4T ~ s E D

0.3

0.2

0.1

548

15.5

two aettleration phases (rockets + photons)

31

SHIP TIME (day)

46..5 62

0.4

I! D

0.3

0.2

0.1 speed

o I I 11 f' I I 111 I I I I 11 I I I 11 I I I I 11 11 I I I 11 I I I 11 l' t 0 28.S 57

SHIP TIME (day) 85.S 114

I

Fig. 7 Prox.ima Centauri orbiter mission by means of a two-stage pure-rocket propulsion system basedonmatter-antimatterannihi-lation. The flight is optimised with respect to the delivered mass with a prefixed transfer time. Mass and speed behaviours vs. the ship time are shown for the acceleration and deleration phases in fig. (7 .1) and fig. (7.2),respectively.

Fig. 8 Prox.ima Centauri orbiter mission by a three-stage multiple propulsion system: two stages are powered by nuclear fusion and photon beam; the braking stage is based on a fusion pure-rocket. The flight maximises the delivered mass with a prefixed transfer time. Mass and speed behaviours vs. the ship time are shown for the ac­celerationanddeceleration phases in fig. (8.1) and fig. (8.2), respec­tively.

Fig. 9 Proxima Centauri orbiter mission by a three-stage multiple propulsion system: two stages are powered by matter-antimatterand photon beam; the braking stage is based on an antimatter pure­rocket. The flight maximises the delivered mass with a prefixed transfer time. Mass and speed be­haviours vs. the ship time are shown for the acceleration and de­celeration phases in fig. (9.1) and fig. (9 .2),respectively.

Multiple Propulsion Concept/or Interstellar FUght: General Theory tllld Basic Results

TABLE 1: Normalised propulsion flows (CHI factors, see eqs. 61-64 ), mass utilisation efficiency, beam exhaust speed, initia acceleration and thrusting times (ship frame )for the orbiter missions of figs. 7 through 9. (Two significant digits are reported).

Stage Thrusting Initial Stage Mass Utilisa- Beam Exhaust Nonnalised Propulsion Flows: Times.Flight- Acceleration tion Efficiency Speed Inertial Flow Photon Flow

Time (g) (1lm) (C) (Xt) (XL) (yr)

a. Fig. 7: Antimatter Acceleration Stage, Antimatter Deceleration Stage

1st 0.13 2.37 0.97 0.33 8.51 0 2nd 0.12 1.56 0.96 0.37 6.5 0

15

b. Fig. 8: Fusion plus Photon Acceleration Stages (2), Fusion Deceleration Stage

1st+ 2nd 0.10 1.04 1.0 0.046 2.56 1.61 3rd 0.24 0.22 1.0 0.069 0.59 0

35

c. Fig. 9: Antimatter plus Photon Acceleration Stages (2), Antimatter Deceleration Stage

1st 0.079 1.48 0.91 2nd 0.090 1.55 0.92 3rd 0.31 0.63 0.95

15

in Table (la). The flight is devised to maximise the mass at the target. The ship mass ratio results in 5.57. The dry mass of the first stage is 3 tonnes, jettisoned at the end of the acceleration phase; that of the second stage is 1 tonne. A payload of 1.26 tonnes will then orbit Proxima. Such masses are reasonable extrapolations of what has been investigating in antimatter pro­pulsion since the end of seventies. A detailed bibliography can be found in (12].

The flight of figs. 8.1 and 8.2 regards a 20-tonne three-stage ship powered by a photon-augmented fusion-rocket multiple system. Significant figures are reported in Table lb. The final mass is slightly heavier than the previous case although the payload may result in a lower value because a fusion engine should weigh more than an antimatter one of the same class. The flight time has been increased to 35 years, otherwise no reason­able flight is possible. A photon beam of 36.5 TW sent to the ship for 37 days, together with a non-modulated fusion beam, allows the ship dynamics to be made equivalentto a pure-rocket of0.95 exhaust speed. The braking phase is accomplished by the fusion rocket only. The D-He3 overall consumption amounts to about 8.2 tonnes. The ship mass ratio, excluding a 3-tonne total jetti­soning, is as low as 6.7. All above figures should be within a

0.52 2.73 0.73 0.48 3.33 0.93 0.40 5.29 0

foreseeable technology on a solar-system scale, including a fully robotics and light payload to be orbited at an appropriately short distance from the target star. .

Finally, Table le, figs. 9.1. and9.2displaythemainresultsof a max-final-mass trajectory for a photon-augmented antimatter­rocket multiple system. The equivalent rocket has an exhaust speed of 0.95. As for the fusion-based flight, this ship has three stages weighing 20 tonnes altogether, and, similarly to the anti­matterpure-rocket, the flight time is also 15 years. The GFpower levels during the two accelerating phases are 19.7 and 8.4 TW, respectively.

The advantage with respect to the pure-rocket of fig. 7 is the lowerpropulsionmassratio: 2.95 against5.57. Consequently this decelerating stage has an initial mass of 11. 7 tonnes against 5 .1 tonnes; assuming the same technology and engine class, a de­livered payload of 4.5 tonnes, against about 1.3 tonnes.

This perfonnance and the saving oflarge amounts of antimat­ter trade off against the increased technological complexity due to the photon component (i.e. the ultra-high power light sources in the solar system) will undoubtedly affect the real feasibility of the mission.

REFERENCES

l. K.E. Tsiolkovsky, "Collected Works", Vol. 1 (1951), Vol. II (1954), Izd. AkademiiNauk, Moscow (Papers and Books from 1895to1930).

2. R.W. Bussard, "Galactic Matter and Interstellar Flight",ActaAstronautlca, Vol. 6, No. 4, pp. 179-194, 1960.

3. G. Vulpetti, "Multiple Propulsion Concept: Theory and Performance", JBIS 32, pp. 209-214, June 1979.

4. G. Vulpetti, "Antimatter Propulsion for Space Exploration",JBIS39, No. 9, SepL 1986.

549

~

Giovanni Vulpetti

5. C. Moller, '"The Theory of Relativity", Oxford University Press, London, 1976.

6. G. Vulpetti, "Relativistic Astrodynamics: Non-Rectilinear Trajectories for Star Exploration Flights",JB/S34, pp. 477-485 (1981).

7. A.A. Jackson, "Some Considerations on the Antimatter and Fusion Ram­Augmented Interstellar Rocket", JBIS 33, pp. 117-120, 1980.

8. T.A. Heppenheimer, "On the Infeasibility oflnterstellar Ramjets", JBIS 31, pp.222-224, 1978.

9. R.L Forward, ''Round-Trip Interstellar Travel Using Laser-Pushed Light­Sails", JSR 21, pp. 187-195, 1984.

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10. R.L Forward, "Starwisp: An Ultra-Light Interstellar Probe", JSR 22, pp. 345-350, 1985.

11. -D.P. Whitmire and A.A. Jackson, ''Laser Powered Interstellar Ramjet", JBIS 30, pp. 223-226, 1977.

12. R.L Forward, "Antimatter Science and Technology Bibliography" (Aug. 1987 update), pp. 686-752, in "Antiproton Science and Technology W orlt­shop Proceedings", RAND Corporation, USA, (6-9 Oct. 1987), World Scientific Publishing, Singapore, 1988.

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