We hope that this simple attempt of extending to therelativistic powered flight well-known concepts of the"classical propulsion" may stimulate other interestedauthors towards the manifold aspects of so vast a problem.

Journal of the British Interplanetary Society, Vol. 33, pp. 27-34, 1980.

NOISE-EFFECTS IN RELATIVISTIC PURE-ROCKET DYNAMICS

G. VULPETTIIstituto di Macchine e Tecnologie Meccaniche, Rome University, Italy

A one-dimensional analysis of the delivered state errors of an uncontrolled relativistic noise-corrupted pure-rocket ship is performed. The equations giving the evolution of the mean and covariance matrix of the statevector are developed under certain assumptions and numerically integrated forward with respect to the ship time.A set of numerical examples regarding nominal constant-thrust flight missions are presented with special emphasison a photon-propulsion rendez-vous flight. The general assessment of performance is focused on the final un-certainties of position and velocity. A very simple scheme of correction of errors is discussed.

1. INTRODUCTION

SINCE THE INCEPTION OF systematic theoretical studiesabout the Interstellar Propulsion & Flight the attention ofresearchers has been focussed on the deterministic aspects(at least to the author's knowledge) of the equations ofthe motion of a starship and the consequences implied.

However, the performance characteristics of the(proposed) interstellar propulsion systems can undergoirregular anomalies (analogous to interplanetary systems).As a result, the actual thrust programme will be alteredwith respect to the nominal (deterministic) profile.Because both pulsed-type and continuous-type interstellarengines are to be powered for long intervals of time,even small fluctuations in the related performance cancause significant terminal state errors.

Since studies about the possibility of interstellar flightare concentrated on the feasibility (largely 'conceptual') ofsuch drives, it may seem premature to talk about any"noise" to be considered in more realistic motionequations. Nevertheless it may be interesting from atheoretical point of view to begin by examining to whatextent thrust anomalies could affect the profile of thedynamical variables of a pure-rocket starship (or even of amultiple-propulsion ship which behaves as a pure-rocketvehicle [I, 2).

Unmodelled irregularities in the thrust-acceleration canstem from unpredictable variations of the power of anon-board nuclear reactor, the rate of the propellant flow,the frequency of the thrust duty-cycle (if any) and soforth. As an example, when a magnetic nozzle is employedto exhaust a plasma at high velocity, casual variationssuperimposed on the mass rate nominal programme cause achange of the beam's kinetic energy density; as a conse-quence, the zone of detachment between the exhaustingplasma and the magnetic lines varies in position, causing analtered value of the specific impulse. Also, when a pulsedengine is operating, the duration of the thrust pulse mightnot be sufficiently repetitive. As another example, a laser-powered rocket can undergo thrust fluctuations becausethe laser power received is altered by an unpredictableabsorption of the interstellar medium between source andvehicle. All these causes may have strong effects on thenavigation accuracy. Then, to the modelled accelerationthere is added 'an un modelled contribution (which must becompensated for a targeting success).

We are not concerned with compensation techniqueshere; we rather analyse the probabilistic behaviours of thedynamical variables related to an internally-powered ship.

2. PROBLEM STATEMENT

Let us consider a vehicle powered by a pure-rocket typepropulsion system in deep-space environments. We cansplit the actual thrust into two parts: a deterministiccomponent and a random one which we assume to bewhite noise. * More specifically, we write

dV = 1; as dr + B(V, as' r ) dv ** (I)

where ship timeship's velocity in a galactic frameon-board thrust accelerationLorentz factorBrownian motion process with unitvariance parameter [6);

r =V =as =1v =

v is assumed to be independent of V'(r >o) and as(r =0).

The resulting thrust-acceleration is a stochastic processas is the ship's velocity; another differential equation isthen necessary to account for the time variations of theacceleration:

das = As (7T,r) d r + C(as' r ) dJ1 ** (2)

where As is a deterministic function of the ship time andcertain non-random parameters 7T;J1 is a further Brownianmotion process assumed independent of v, V(r =0) andas (r=o).

Let us point out that Eqs. I, 2' are merely models of 'the behaviour of the actual (stochastic) processes. Toevaluate to what extent these models approach the realphenomena is beyond the scope of this paper. Eqs. I, 2are to be intended in the Ita sense [6). At this point, itis clear that the distance travelled by the ship is a stochas-tic process as well. If we suppose that no further noise is

* This is an extension to the relativistic case of simple modelsdesigned for the classical rocket equations [3,4,5].

** Band C are functions giving the effective amplitudes of thenoises.

27

"'

G. Vulpetti

involved, we can write

dS = V "tv dr (3)

where S(r) is the flight-path at time r. We can write tileabove equations in a vector form as follows

(4)dR = F dr + G dN

where RT = I S V as I, FT V -2 A I"tv "tv as s '

I ~v I ('T' denotingd,u transposition)II ~ ~ ~II

G dN

with the following statistics

E[dN] = 03, E[dN dNT] =11 ~ ~ ~ II dr = Q dr

Formally, dN/dr is a vector Gaussian white noise. Underthe assumptions made the vector process R is a Markovprocess (6). Eq.4 is dia.grammed in Fig.l. The compactform of Eq. 4 translates the fact that here we are going toconsider the process (S, V, as) as a whole; namely, we arenot regarding S and V as state variables separated from asonly referred to as outcome of a continuous measurementfrom which one estimates V and S. Thus, we are interestedin the unaltered evolution of the joint process denoted byR. Rather, the model we are dealing with idealises certainenvironments. We have thought of unpredictable anomaliesin the working of the pure-rocket propulsion system.These have been taken into account through a noise addedto the nominal profile of the acceleration rate in Eq.2.Because the ship velocity variation depends upon as' evenwithout noise in Eq. I, V results in a random variable.Physically, this fact is plain. Moreover, in Eq.l we haveadded another perturbation uncorrelated with the previousone. This means that the motion of the ship is "polluted"by a non-pure-rocket component. This circumstance may,for instance, happen at low speeds while the ship iscrossing a complex gravitational field (which cannot be

•

Fig.1. Block diagram of the stochastic equations of motion of arelativistic pure-rocket. Independent white noises have beeninserted into the equations of velocity and on-board accelerationrates. Ship time is taken as the independent variable.

28

detected on-board through as)' or may consist of randomerrors in computing the velocity V (e.g. by changes ofsome reference value), and so forth.

In order to simplify the subsequent procedure we makethe following additional assumptions regarding the noiseamplitudes:

B = const. c = K IAsl + H

K and H being suitable constants.

If we denote the first order probability density of thestochastic process R by p = p(Rp r), its time evolution isgoverned by the forward Kolmogorov equation (6).

ap/or = L(p) (5)

where the operator L is defined as followsn n

L(.) = -I:i a (. Fi )/a Ri + % I;;i,j a2[ . (GQGT)i]·]/1 1

a Ri a Rj (Sa)

where n denotes the dimension of R, Ri its generic com-ponent and so on. Performing derivatives we obtain

-V "tv Ps + %B2 Pvv - "t-~ as Pv + % C2 P -asas

(6)

(The p' subscripts stand for partial derivatives, as usual).Eq.6 is a complicated equation which, unless one limits toa classical solution with As = 0, does not offer a generalclosed solution. In that limit case, Eq.4 would be linearand the process R would be a Gauss-Markov process.Specifically, if So = 0 and V0 = 0, we would have (startingfrom delta-Dirac distributions) the following mean valuesand covariances

var(S) B2r3/3 + H2rS /20

B2r + H2r3/3'1= a* rs var(V)

*as = ag var(as) H2 r (7)

cov(S,V) = B2r2/2 + H2r4 /8 , cov(S,as) = H2r3/6

cov(V,as) = H2 r2 /2

where a; is the nominal constant value of the acceleration.In the general case it is unthinkable to carry out an explicitexpression for p(R,r). Although p contains any desiredinformation, nevertheless we are forced to limit the analysisonly to some significant moments. Thus, we make furtherassumptions about the density p. First, let us suppose thatthe probability density satisfying Eq.S is sufficientlysymmetric around the mean value of R so that odd R'scentral moments can be ignored. Second, we assume thatthe elements of the R's covariance matrix are so smallthat all even central moments of higher order can beneglected.

Under these supplementary hypotheses, making use ofthe differential Ito stochastic calculus [6], we will obtainthe equations of the evolution of both the expectation ofR and its covariance matrix. The nex t section is devotedto a concise explanation of the algorithm.

3. EVOLUTION EQUATIONS

The hypotheses made on p(R,r) allow us to convert

integro-differential equations giving the exact time evolu-tion of the expectation and covariance matrix of a randomvector (Ref. 6, page 137) into ordinary differentialequations of easier implementation on a computer. Sub-stantially, the procedure consists of starting from thedefinitions of mean and covariance, expanding in Taylor-series up to the second order under the above assumptionsabout the shape of the probability density and taking intoaccount the supplementary terms when a stochasticfunction is differentiated.

Denoting the mean and covariance matrix of R by Rand A respectively, we get

dR/dr = F(R)+ )121 tr(FkRA) tr(F~RA) tr(F~RA) 1 ~ (8)

where FkR (i=1,2,3) is the hessian matrix of Fi, the i-thcomponent of F. With regard to the A's evolution, fromthe definition of A we have

dAJdr = dE(RR T)/dr - (dR/dr)RT - R dRT/dr (9)

where E( . ) denotes the expectation operator. Using thesuperscripts a, ~, 'Y for partitioning, setting

RR T = I Ra: R~: R'Y I, GQGT W

it is possible to demonstrate that

dE (RRT)/dr = EI R~ : R~F: Rl F 1 + E(W)

E(RFT + FR T) + W (10)

where the subscript 'R' denotes the first order differen-tiation. Eqs. (9, 10) are integro-differential equations.Applying to them the same above-mentioned proceduregenerating Eq.8, we can obtain an ordinary differentialequation for A.

To sum up, the solving system of the evolutionaryequations is carried out as follows (in interstellar units):

x = Y 'Yy + 3Y 'Yt ~~ 12

Y = (I - y2)Z - Z ~2y - 2Y ayz

Z = As·2 3~x = 2 'Yy axy·2 2 2 2~y=-4YZ ~y+2(1-Y )ayz+B (II)

~~ = (K'IAs~ + H)2

3 2 2qxy= 'Yy ~y - 2YZ axy + (I - Y ) axz• 3axz= 'Yy ayz• _ 2 2ayz- (1 - Y ) ~z - 2YZ ayz

where we have redefined RT = I X Y Z I, the subscripts x,Y, z being referred to as the components of R respectively;~2 and a stand for variance and covariance respectively.

The system (11) is to be integrated forward startingfrom initial conditions which mayor may not be relatedto delta-functions.

4. DISCUSSION OF THE RESULTS

Eqs. 11 could be significantly integrated in the following

Noise-Effects in Relativistic Pure-Rocket Dynamics

cases of nominal programmes:

(a) constant thrust, constant effective exhaust speed;

(b) variable thrust, constant effective exhaust speed;

(c) constant ship-frame acceleration.

The corresponding nominal profiles of the proper accelera-tion rate, that is the function As (1T, r), are given by

2(a) As = [asol (1 - asorlu )2uJ sgn (aso)*

(aso denotes the initial value of the nominal as'whereas u stands for the effective nominaljet speed);

(b) As = (tiT - M/M) TIM M = M - u-1 Jr T dr, 0 0

(where M represents the nominal value of the ship'srest mass, and T the nominal thrust; differentiationis intended with respect to the ship time);

(c) As = 0

However, because most of the interstellar missions literaturenowadays concerns constant-thrust constant-jet-speedflights, we limit ourselves at present to case (a) only.**

We focus our attention largely on the final standarddeviations of position and velocity which represent theultimate consequences of a noise presence. By final wemean the values at the nominal ship-frame flight time. Astarship is envisaged to fly in an uncontrolled way until theprefixed time of flight; thereon the gained uncertainties inthe state vector are analysed.

Although a transfer trajectory to nearby stars can beconsidered as rectilinear, however one easily realises thatdelivery errors only in the line-of-flight exclude manyrealistic cases. In fact, final indeterminations on transverseaxes are decisive for targeting. This more complete analysiscould be performed in a successive phase of study byutilising the general three-dimensional relativistic rocketequation of [IJ. In that case a 9 x 9 covariance matrixwould be involved (at least), whereas our current calcula-tions have dealt with a 3 x 3 matrix. Nevertheless, a one-axis analysis can playa considerable role in "actual" casesas well. With regard to a rendez-vous flight to a nearbyplanetary system, for example, the true targets (i.e. thespace zones around the circling planets) are displaced withrespect to the approaching trajectory of the starship (seeFig.2). If at some prefixed time the cargo-ship is "split"into smaller probes to be boosted to the planets, a greatuncertainty in the final position and velocity of the "bus"may result in an off-fuel-programme of the mini-ships. Toshift the detachment time may be a solution, but couldentail a difficult reprogramming of the "swarm" of thetrajectories of the probes inasmuch as the states of theplanets would be changed (their relative positions could becritical for a successful multiple launch). The requirementsfor a fly-by flight are less stringent but affected to someextent by final uncertainties. The following discussion of

* For sake of simplicity (but losing nothing of physics, we mustemphasise) we have assumed that neither leakage (or "evaporation")of inert mass occurs from our spaceship nor that the vehicle holdsheat associated with the fraction of propellant not exhausted away.In these environments the above relation is fully valid (see [1]).Furthermore, this form still holds for a photon rocket.** Our FORTRAN listing of the code which has been designed forprocessing all three cases is available on request.

29

C. Vulpetti

po'Wered are

fr.e-fall arc

/~tran,rerl@traject ',' ....-: \ ,I tarcet

I , . .yet ••

'\ . i. \. /\ . /1 .., ).~ /1

SU>i // i~ .~*~-------j( (- .pproaching E·Di,.~·

/ /11ne-ol'-fl1ght '/"determlnation

off-nominal pointeof launching probes

Fig. 2. Change of final conditions from position indeterminationfor a multi-payload cargo flight in off-nominal rendez-vousenvironments. Note the extremely different conditions of approach-ing the planets of the target star-system.

our computer results may be viewed also in the light ofthese remarks.

The first set of results concern fast fly-by flights to theAlpha Centauri system. We have envisaged a three-stagevehicle, the main characteristic values of which are:(aso) stage = 0.03 g*t(nom~al value), Ustage = 0.?3 c_(nom. value), (Tburn) stage - 0.75 yr and coast. time -33.25 yr. In order to isolate the effects of the noise sources,at first the initial statistical conditions on path, velocity andacceleration are assumed to be delta-Dirac distributions. Wepoint out that the occurrence of the noise processes in thesecond and third stages takes place independently of thefirst and second stage's respectively. As a result, the delta-distribution of the initial acceleration is reset for thesestages. In general, the computer programme re-initialisesthe standard deviations at any subsequent stage on valuesproportional to the corresponding initial acceleration's.This is a principle followed throughout in the sequel, asallowed us by the assumed independence between thenoise and the initial value of the ship state in any phaseof flight.

Fig. 3 shows various behaviours of the final ~x versusthe initial amplitude Co for a fixed K/H ratio anddifferent values of the amplitude B. Fig. 4 is the analogof Fig. 3 with respect to the final ~y' A first glancereveals that the two noises must be restricted to very smallvalues in order to not generate exorbitant 3~x and 3~y'In particular, if a final uncertainty of 15 A. U. is required,Co must not exceed 2. x 10-6 g*/yrYz, whereas the maxi-mum B (which alters the pure-rocket behaviour) should beabout 150 m/s/yrYz. This last disturbance may take placealso when the ship coasts. Note that the long coastingamplifies the errors at burnout and, if the B-noisevanishes, the final uncertainties are strongly proportionalto Co' Taking into account changes from K/H values, wehave found for the modelled 'ship and flight

~x = 2.6725x I06 C(T = 0, B = 0, K/H = 300)~y = 3.619xlOs C(T= 0, B= 0, K/H= 300)

t 1 g*~ 0.97 g.

(A.U.) (12)(Krn/s)

30

The coefficients in (12) can be decreased by augmentingH with respect to K, on account of the different waysthey enter during the engine-noise-corrupted phase. Inaddition, Figs. 3-4 show that linearity persists with respectto the rocket-noise for increasing intervals of Co. Fig. 5gives the acceptable range of B once the final (I-sigma)state errors are specified, here chosen equal to 5 A.U. and5 Km/s respectively. Notice how it is meaningless tosimultaneously consider large tolerances in position andsmall errors in velocity. Figs. 3-4-5 also show that thiskind of mission undergoes non-linear effects only in some-what narrow ranges of the thruster-noise.

The cases discussed above represent hypotheticalexamples of perfect initial knowledge of the ship accelera-tion. Fig. 6 displays the effect of an initial indetermination~zo =1= o. Once again, the curves show two linear regionsseparated by a narrow non-linear zone. The final error inposition is strongly sensitive to ~zo which, in addition,dominates the error-at-target behaviour for very lowengine-noise. The values of ~x turn out to be prohibitiveeven for ~zo values as low as 3x I0-9 g* and increaseproportionally to them. Comparing Figs. 3 and 6 we notethat an initial indetermination in ~z can, however, becounterbalanced by a decrease of B in that the same final

10~

10

~ 0.)

";5 - - - - - - - - - - - - -- - -- - - --

8 :: 0

Fig. 3. Behaviour of the flight path standard deviation at thefinal (nominal) ship-time versus the initial amplitude of the engine-noise for a three-stage vehicle accelerating and coastin~ to the AlphaCentauri system. This amplitude is expressed in g*!yr 2 units (lg*= 0.97 g). Trends are shown for different values of the amplitudeof the pure-rocket noise.

10

K/II=]OO (I. U.)

__ ~l1~"~JO~Km~/~./~Y~r_l ~

~O~.J~ ~

10

10-7

Fig.4. Behaviour of the ship's velocity standard deviation at thefinal (nominal) ship-time versus the initial amplitude of the engine-noise for a three-stage vehicle accelerating and coasting to the AlphaCentauri system (also see Fig. 3).

102

;~•..•

10

)0 m/s/yr~

Kill = JOO (I.U.)

Fig. 5. Interrelationship between the delivered position and velocityuncertainties for a three-stage ship flying to the Alpha Centaurisystem (also see Figs. 3-4).

Noise-Effects in Relativistic Pure-Rocket Dynamics

u=) "./a/yr;1\/11=)00 (t.",)

10

10-6 10-5

c. (g.fyr~)

Fig. 6. Behaviour of the flightpath standard deviation at the final(nominal) ship-time versus the initial amplitude of the engine-noisefor a three-stage ship accelerating and coasting to the Alpha Centaurisystem. Trends are shown for different values of the standarddeviation of the initial on-board acceleration. The amplitude of thepure-rocket noise has been fixed at 3 m/s/yrV2.

error in position or velocity can be maintained.The previous analysis has also shown that the coasting

phase, although without an inherent thruster-noise, amplifiesthe uncertainties accumulated over the acceleration phase.This implicitly gives rise to a question. Are there minima ofthe errors Lx and Ly delivered at the end of the coasting-asfunction of a suitable combination of the number of stagesand coasting time for a fixed flight distance? Limiting ouranswers to flights of the cited type, Fig. 7 shows someresults for various inputs of noise. A maximum of five stageswas allowed, and the flight distance was fixed at 6 ly. Noindetermination in the stage initial acceleration was assumedin order to see "clean" effects from both pure-rocket andthruster noises. (Also indicated in Fig. 7 are the correspond-ing uncertainties in position at burnout). The values of Lxexhibit no minima, although varying over three orders inCo and two ones in B. In contrast, there is a systematicexponential-like decrease of the position error as the numberof stages increases. Such a trend has been found even forlarger intervals of B and Co (not plotted in Fig. 7) and forLzo *" 0 up to 3x10-6 g*. We. suspect that the absence ofminima depends upon the prefixed distance in a non-simple

31

G. Vulpetti

90

i. .Ix at hurnout x 10

1. • , It

.120

'" J .,0-6 e ,

II/H-'"'' (I.u.}80

70

60

2 3 5 64

NUMBER Of' STAGES

Fig. 7. Final position and velocity standard devia tions against thenumber of stages of a ship covering a fixed distance of 6 ly byapproximately sharing the burning and coasting time intervals. Alsoshown are the corresponding values at the increasing burnout times.The ranges of noise are indicated.

manner. Ly displays a different feature as the two starredcurves show in Fig. 7. The lower one increases whereas theother exhibits a "broad minimum" around the two-stagecase. From a practical point of view such trend is of a littleutility (Ly ranges here from 4 to 5 Km/s). In general, wehave noticed slight variations of Ly with the stages. Thus LXand Ly are characterised by two quite differentsensitivities with respect to pure-rocket and engine noises inaugmenting the number of stages in fixed-distance missions.Mathematically, such differences ar.~ to be ascribed largelyto the ways of varying of the rates l'l~y. In fact, fromEqs.ll, whereas during a protracted acceleration the formerincreases strongly, the growth of the latter is appreciably·mitigated by a negative term.

We would point out that in all the cases analysed everyvariance, covariance and noise square involved in the propa-gation of the moments are so small that the nominaldistance and velocity are coincident with the actual X and Ywithin seven or eight significant figures at least. In order to"detect" computed values correspondingly different (as thefirst two equations in (II) strictly predict) the noiseamplitudes would have to be taken to levels so high that thepropagated errors in position and velocity would be meaning-less for any realistic physical environments.

So far we have dealt with missions without a decelerationphase; furthermore, the exhaust speed of the multistaged

32

ship was supposed to be rather lower than the speed oflight. In order to consider flights encompassing a rendez-vous and, largely, to enhance non-linear characteristicspeculiar of a true relativistic pure-rocket, we have investigat-ed the behaviour of a hypothetical photon-ship acceleratingfor ten years (ship time), coasting ten years and deceleratingten years again. The starting values of distance, velocity andacceleration are

X=oY=o Z = 0.095

With regard to the deceleration phase we have set

Z = -0.095 s, ~ -6""'z = 1. 10 g*

We emphasise the fact that Eqs.11 are complex enough torequire a considerable machine time for analysing severaltens of photon-drives; on the other hand, any procedure ofanalytic approximation on Eqs.11 would produce relevantmistakes at high velocities. Such reasons have induced us topick out one significant case among a dozen examined. Itdisplays the typical features of a photon-rocket. Highrelativistic behaviour is assured by a cruise speed equal to0.995 c. To cite a consequence, some final "Lorentz effect"thrusting indeterminations magnify into the final coastingvalues by a factor up to I~ :::106

. Fig. 8 shows the threecorrelation coefficients evolving along with the ship time.We note a quick increase of them; after an interval of 4 yrthey approach each other very closely, thereafter theirbehaviour diverges. Cxy keeps values near unity with asmall "dip" around the first switching time. Cxz and C zare identically zero during the coasting because, regardlessof previous noises, the propulsion system is switched to anon-random value of thrust. As deceleration progresses anon-linear behaviour between distance and velocity emerges.In contrast, the respective correlations between X, Y, Zincrease as time passes. They cause a strong asymmetrybetween acceleration and deceleration. In the latter bothCxz and Cyz reach maxima at the same time, but Cxz isgenerally rather low, i.e. distance is uncorrelated with (notindependent of) thrust acceleration. The positive signs of

.ACCELERATION COASTING DECELERAT ION

1.0<1\ ,.,..._=---....~••HU

~...'"oc

cxy

C xl0xz

zo~..•...~oo

cyz

cxz

10 20SHIP TIME (yr)

)0

Fig. 8. Time-evolution of the correlation coefficients of a highlyrelativistic photon rocket (cruise speed = 0.995 c). Quasi-lineardependence and strong non-linear asymmetries emerge from thedifferent phases of flight.

,·.trIaI,,'"c.-t-"-·rJ.I,"·

J _

z.

.0

Fig. 9. Evolution of the standard deviations of position, velocityand on-board acceleration versus the ship-time for a highly relativisticphoton-rocket (cruise speed = 0.995 c). Even though very smallvalues are chosen for the amplitudes of the two noises, strongasymmetry and self-damping effects are brought about.

Cxz and Cyz are easily explained in physical terms. Theabove asymmetries are reproduced in Fig. 9 where thestandard deviations of the state components are displayedas a function of the ship time. By assumption the time-behaviour of ~z is identical in both the propulsion phases.During the acceleration ~y is seen to increase, reach amaximum and decrease towards a value close to its startingone. This sort of ~.ingular "self-damping" is caused by thefirst term of the ~t.equation which is negative during anacceleration interval. However, such a term cannotgenerally prevail over the remaining two unless the shipvelocity is sufficiently high. This can occur for a photon-rocket with an adequate propulsion mass ratio. The reverseof such behaviour characterises the deceleration intervalwhere ~y increases monotonically. The selected coastingdoes not appreciably amplify the previous uncertainty invelocity because the only present disturbance - the pure-rocket noise - has been assumed very small.

Another asymmetry in the current analysis concerns~x. Starting from a delta-distribution ~x takes on about8 A.V. (here) at the end of the acceleration, while theengine noise amplitude is passed from 9 x 10-3 to3.6 X 10-5 g*/yr'h. The ensuing growth during the coastingphase-results in 87 A.V., that is about 66% of the flight-enderror. (In considering these values one should take intoaccount the distance covered in a flight such as this,namely, almost 165 ly). These figures mean that a shorterdeceleration time - bringing to a fly-by of the star system- "yields" final errors very close to the rendezvous.

The general tren?s of ~x and ~y suggest to us that inthe environments pictured here an effective guidance may"calmly" ignore delivered uncertainties in velocity.Although a continuous control throughout a thrustingphase can keep the effects from the engine-noise at verylow levels such as those envisaged above, however manytens of A. V. in position error can be delivered. Further-more, unless the ship is relatively close to some star ofknown space-time coordinates, it will probably be verydifficult to perform on-board direct measurements of theposition of the ship in a star background with a resolutionof a few A. U. or less in order to ascertain its amount andsign. However, an on-board recording of acceleration ismandatory and the history of the following quantity

.- ts-Lla (Ll) = J (as - asn) dt/(ts - to - Ll) (13)

to

Noise-Effects in Relativistic Pure-Rocket Dynamics

can be made in the computer memory. In (13) ts is thefirst switching (ship) time whereas Ll represents an intervalbefore the nominal switching-off; asn(t) is the nominal on-board acceleration. If Ll is sufficiently short, it (Ll) will havethe sam~ sign as a (0) and the signs of ~x y( ts) are thenknown ill advance. Taking into account the foreseen enginedisturbance from (ts - Ll) to ts, it is possible to compute theshift 5 ts in order to later compensate for the position error.A possible rough scheme is outlined below. Setting

*ts = ts + 5ts , I 5ts I .;;;Ll

one determines 5 ts such that the estimate of the actualvelocity at t~ equals the nominal one at ts, that is

(14)

The dimensionless number X could be inferred from somevelocity-dependent measurement or through a guessedprobability density function for the ship velocity. Theprevious equation does not suffice for driving the positionerror to zero before the deceleration phase begins. Thus, ashift 5 tc to the nominal coasting duration tc is to be in-cluded as follows:

-r(Y) (tc + 5tc)Y + X(t~ ) = Xn(t~ + tc + 5tc)

X(t~ ) = Xn(t~ ) + X ~x(t~ )

-Y = Y(B2,5ts,5tc) (15)

In Eqs. (15) Y represents the mean velocity over thecoasting; strictly speaking, it depends upon the pure-rocketnoise acting in this phase of flight (e.g. a drag caused bythe impact of interstellar dust against the ship). If it werenegligible (as in Fig. 9), then Y equals Y(t~). Naturally, thevery simple scheme above does not include an otpimisationof the correction sequence, for instance one based on theas(t) profile. Moreover, it neglects the uncertainties in bothposition and velocity which the final thrusting unavoidablyadds. The above policy is applicable with some modificationto non-photon-propulsion ships. It is meant to approachthe interstellar guidance also through "short" changes inthe nominal switching-(ship)- times. However, we point outthat any deeper development of this topic should be per-formed by modelling deterministic inputs (controls) -pulsed or piecewise-continuous in kind - into a stochasticsystem such as that treated here or, more correctly, adimension-extended (x, y, z) evolution system.

5. CONCLUSIONS

In this paper a first simple approach has been made foranalysing the propagation of uncertainties in position andvelocity versus the proper time of an uncontrolledrelativistic starship experiencing engine and/or pure-rocketnoises. Models of delta-correlated' noises have been insertedas additive random disturbances into the pure-rocket one-dimensional equations. The equations of evolution of themean and covariance matrix of the ship's state vector werecarried out under certain hypotheses of symmetry of thefirst-order probability density. Multistaged starships underconstant (nominal) thrust have been considered as well asphoton rockets accelerating, coasting and decelerating.Analysing the errors in position and velocity at the finalnominal flight time, both linear and non-linear functionaldependences have been found in terms of the amplitudesof the pure-rocket and thruster noise at the initial time ofpropulsion. These noise values are to be confined to verysmall levels in order to keep the delivered position errors

33

Eta Carinae. A large hydrogen emission nebula in the far southern sky, with wide superimposed dust lanes. The narrow twisted lanes show wherecold interstellar matter is forcing its way into the hot gas. They are known as elephant trunks. The central star is unique, strongly variable witharray of remarkable properties. It may be a newly-formed star settling down to a stable state. (Anglo-Australian Telescope).

G. Vulpetti

at reasonable values. The effects of initial independent un-certainties in velocity and (on-board) accelerationhave been evaluated. Finally, a simple scheme of guidancehas been suggested for balancing the position indetermina-tion which accumulates over the acceleration phase ofarendezvous flight. It has been semi-quantitatively recognisedthat it is necessary to shift both the switching-off andswitching-on times.

ACKNOWLEDGEMENTS

The author is very grateful to Prof. M. Shaerf, Director,and Dr. G. Bertocchi of the Computer Center of RomeUniversity who have afforded the opportunity of implement-ing his computer code on the Center's UNIVAC 1100/20time-sharing multi-processing system.

34

REFERENCES

1. G. Vulpetti, "A Problem in Relativistic Navigation: The Three-Dimensional Rocket Equation," JBIS, 31, 344-350 (1978).

2. G. Vulpetti, "Multiple Propulsion Concept: Theory andPerformance," JBIS, 32,209-214 (1979).

3. R. A. Jacobson, J. P. McDaneli and G. C. Rinker, "Use ofBallistic Arcs in Low-Thrust Navigation," JSR, 12,138-145,(1975).

4. T. Nishimura, "Worst-Error Analysis of Batch Filter andSequential Filter in Navigation Problems," JSR, 12,133-137(1975).

5. B. D. Tapley and H. Hager, "Estimation of Unmodeled Forceson a Low-Thrust Space Vehicle," JSR, 12,592-598 (1975).

6. Andrew H. Jazwinski, "Stochastic Processes and FilteringTheory," Academic Press, New York 1970.