The feasibility of interstellar flight is discussed. Mathematical equations for single-stage and multistage rocket propulsion are developed; velocity data and transit times are presented. The conclusions indicate that interstellar travel is theoretically feasible by utilizing known staged nuclear-energy systems

Jaffe, Leonard D.

Spencer, Dwain F.

English

NASA Technical Reports Server

Technical Report No. 32-233 
Feasibility of Interstellar Trauef 
Dwain F. Spencer 
Leonard D. Jaffe 
8 I 
JP' J E T  P R O P U L S I O N  L A B O R A T O R Y  
/ C A L I F O R N I A  1 N S T I T U T E  OF TECHNOLOGY 
P A S A D E N A .  C A L I F O R N I A  
March 15, 1962 
https://ntrs.nasa.gov/search.jsp?R=19620001503 2020-03-11T16:42:55+00:00Z
NATIONAL AERONAUTICS A N D  SPACE ADMINISTRATION 
CONTRACT NO. NAS 7-100 
Technical Report No. 32-233 
FEASIBILITY OF INTERSTELLAR TRAVEL 
Dwain F. Spencer 
Leonard D. Jaffe 
Charles W. Cole, Chief 
Engineering Mechanics Division 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
C A L I F O R N I A  1 N S T I T U T E  OF T E C H N O L O G Y  
P A S A D E N A ,  CALIFORNIA 
March 15, 1962 
Copyright@ 1962 
Jet Propulsion Laboratory 
California Institute of Technology 
JPL Technical Report No. 32-233 
CONTENTS 
................................................................................. ............................................ 1 I. Introduction 
................................................................................................ II. Basic Equations for Single-stage Rocket 2 
111. Basic Equations for Multistage Rocket 5 
IV. Examples of Velocities and Transit Times 9 
V. Conclusion 10 
Nomenclature 11 
References 17 
.................................................................................................. 
............................................................................................ 
................... .................................................................................................................. 
........................... .................................................................................................................. 
................................. ............................................................................................................ 
TABLES 
12 1. 
2. Mass ratios required for fusion rockets, E = 4 x ............................................................................ 13 
Mass ratios required for fission rockets, E =  7 x ............................................................................ 
FIGURES 
1. Over-all mass ratio required versus energy fraction for various fractions of  l ight velocity .............. 14 
2 Fractions of l ight velocity attainable for a deuterium fusion rocket versus number of stages 
.......................................................... for various dead-weight fractions (payload fraction = lo-’) 15 
3. Fractions of  l ight velocity attainable for a deuterium fusion rocket versus number of stages 
for various dead-weight fractions (payload fraction = .......................................................... 15 
4 Fractions of l ight velocity attainable for a deuterium fusion rocket versus number of stages 
.......................................................... for various dead-weight fractions (payload fraction = 16 
5. Fractions of l ight  velocity attainable for a five-stage deuterium fusion rocket versus 
over-all payload fractions for various dead-weight fractions .......................................................... 16 
iii 
JPL Technical Report No. 32-233 
ABSTRACT 
The feasibility of interstellar flight is discussed. 
Mathematical equations for single-stage and multistage 
rocket propulsion are developed; velocity data and transit 
times are presented. The conclusions indicate that inter- 
stellar travel is theoretically feasible by utilizing known 
staged nuclear-energy systems. 
1. INTRODUCTION 
The earliest studies of relativistic rocket mechanics by Ackeret (Ref. 1 and 2), Tsien (Ref. 3), 
Bussard (Ref. 4, and others made two implicit assumptions that severely limit performance of the rockets 
considered. They assumed that nuclear-energy rockets are limited to a single stage and that the available 
energy corresponds to a fixed fraction of the final vehicle mass. The latter assumption apparentiy arose from 
the thought that spent nuclear fuel would either be retained on board or dumped, rather than exhausted at  high 
velocity. These assumptions are neither necessary or desirable. 
More recently, interstellar travel has been considered by Sanger (Ref. 5) and Stuhlinger (Ref. 6). 
They realized that the limitation regarding the amount of energy available being a function of the propellant 
mass rather than the final m a s s  was unnecessary; however, they did not consider staging the vehicles as i s  
done with chemical rockets. They concluded, therefore, that interstellar travel using nuclear reactions as an 
energy source is impossible because of fundamental limitations on the amount of energy available for rocket 
propulsion. In contradiction, the analysis presented in this report shows that nuclear fission or fusion rockets 
can be considered for interstellar travel. 
1 
JPL Technical Report No. 32-233 
II. BASIC EQUATIONS FOR SINGLE-STAGE ROCKET 
The basic equations for single-stage rocket propulsion at relativistic velocities were derived by 
Ackeret and have been utilized by subsequent workers. Ackeret's work i s  inexact, however, in that he 
considers the rest m a s s  exhausted to equal the rest mass of fuel consumed. More exactly, the rest mass of 
fuel consumed i s  
M - M e ,  + E M f  f -  
where M e ,  = rest m a s s  exhausted and E M  - rest m a s s  of fuel converted to kinetic energy. The initial rest 
mass of the vehicle is 
f -  
M ,  = M f  + M b  
where M b  is the rest m a s s  of the vehicle at burnout. 
Let  
M b  
X = -  
M f  
Then 
The stage mass ratio is 
(3) 
M b  X 
This i s  simply the result obtained with a chemical propulsion system. 
To discuss the exterior energetics of the vehicle, a coordinate system fixed in space and a system 
relative to the vehicle may be used (Ref. 4, 5 and 6). Let u represent the velocity of the vehicle relative to 
JPL Technical Report No. 32-233 
the stationary system,v the velocity of the exhaust relative to the stationary system, and w the exhaust 
velocity relative to the vehicle. The exhaust velocity w is determined by the particular fuel employed and i s  
taken as a constant. By employing conservation of momentum, mass, and energy, and the Lorentz addition of 
velocities, Ackeret showed that the final vehicle velocity i s  given by 
A relationship between the exhaust velocity and the fraction of fuel converted to energy gives the 
desired form for the final velocity. In the coordinate system moving with the vehicle, the kinetic energy of 
the exhaust i s  
*ex = 
d M e ,  c2  
- dM,,c 2 (7) 
The kinetic energy results from the conversion of rest m a s s  to energy within the engine. For every increment 
dMex exhausted, EdM i s  converted to energy and from Eq- (1) this is f 
EdM,= (fi) dM,, 
Then b. (7) has the form 
Solution for w/c gives 
W 
- =  c p = T i  
3 
JPL Technical Report No. 32-233 
or, in terms of the specific impulse 
Equations (10) and (11) were also given by Sanger and Huth (Ref. 7). 
vehicle is 
The final form of Eq. 
(11) 
(6) for a one-stage 
~~ 
JPL Technical Report No. 32-233 
111. BASIC EQUATIONS FOR MULTISTAGE ROCKET 
The kinematics of multistage relativistic rockets have been treated only by Subotowicz (Ref. 8);  
however, he did not examine energy requirements. As shown in Ref. 8,  the burnout velocity un for the nth 
stage is given by 
As in the classical case (Ref. 8), optimum staging occurs for equal step mass ratios or equal step burnout 
fiactions if each step has  the same exhaust velocity. Then Eq. (13) reduces to 
un s ~ ~ ~ / ~  - 1  
where w / c  is given by Eq. (10). Then 
"n 
lim - = 1 
(14) 
(15) 
for a fixed step m a s s  ratio. Thus,  if enough stages are utilized, regardless of the exhaust velocity or mass 
ratio per stage, it  is theoretically possible to attain a final velocity near that of light. 
Another important aspect in the feasibility of interstellar travel i s  the final payload mass which can 
be delivered by a particular vehicle. Consider an n-stage vehicle with stage burnout rest mass (x M ) and 
stage structural or dead rest m a s s  (PM ) . Then the payload mass of the ith stage 
f i  
f i  
the initial m a s s  of the (i + 11th stage. 
5 
JPL Technical Report No. 32-233 
From Eq. (4) 
Now 
But 
and 
Now 
(x1 - P J  
M f  1 
Then 
( X I  - P p X 2  - P,) 
(1  + x l ) ( l  + x2)  N O 1  
- 
M03 - 
Continuation of this procedure yields the desired result 
(21) 
(22) 
M =  
P M o l  
6 
IPL Technical Report No. 32-233 
Since the step fractions pi and the stage fractions xi for optimum staging should be the same for all 
stages, Eq. (23) reduces to 
"0 (1 + X I n  
It may be of interest to determine the maximum vehicle burnout velocity for a given dead-weight . 
fraction p and desired over-all payload fraction a. Algebraic solution for x from Eq. (24) yields 
Substituting in Eq. (5) gives 
and from Eq. (14) 
(24) 
Using Eq. (IO), the final burnout velocity of the n-stage vehicle in terms of over-all payload fraction, dead- 
weight fraction, and fraction of m a s s  converted to energy, is 
7 
JPL Technical Report No. 32-233 
Figure 1 i s  a plot showing the over-all mass ratio required versus energy fraction E for various final vehicle 
velocity ratios un/c. The over-a11 mass ratio i s  given by 
8 
I 
JPL Technical Report No. 32-233 
IV. EXAMPLES OF VELOCITIES AND TRANSIT TIMES 
Following are some examples of velocities and transit times which may be attainable. The fraction 
of mass converted to energy by uranium fission is about 7 x 
obtained from Fig. 1, shows the over-all mass ratio A necessary to reach various velocities u,/c for a fission 
rocket with E = 7 x lo*. Table 2 shows the necessary m a s s  ratio for a fusion rocket with an energy con- 
version fraction E = 4 x 
for a two-way trip with deceleration at each end, the mass ratios must be raised to the fourth power. These 
values are also shown in the tables. 
by deuterium fusion, 4 x Table 1, 
If deceleration at the destination is required, the m a s s  ratios must be squared; 
Mass ratios of lo3 to lo6 seem quite feasible in principle. For unmanned probes, one-way trips with- 
out deceleration may well be adequate. Feasible velocity ratios corresponding to the mass ratios mentioned 
above are then 0.3 to 0.5 for uranium fission and 0.6 to 0.8 for deuterium fusion. The corresponding travel 
times depend on the acceleration used. If 1-g acceleration could be achieved, relativistic velocities would 
be reached within a few months and the spacecraft could then coast to its destination a t  the velocity indicated 
above. To reach Alpha Centauri at 4.3 light years, the transit times would be 9 to 14 years with a fission 
rocket and 6 to 7 years with a fusion rocket. 
For two-way trips with deceleration at each end, a s  might be required for manned missions or sample 
returns, a multistage fission rocket could reach about un/c = 0.13. However, on this basis, the 8.6-light-year 
round trip to Alpha Centauri would require 66 years. With a deuterium fusion rocket, un/c of 0.3 seems 
attainable: the round trip to  Alpha Centauri would then require 29 years. 
Figures 2 to 4 show the attainable vehicle burnout velocity as a function of the number of stages for 
An interesting feature of these payload ratios of IO-', 
curves i s  the fact that a five-stage vehicle attains nearly the maximum possible velocity increment for a 
particular payload fraction. 
and IO5 for a fusion rocket with E = 4 x 
Figure 5 displays the effect of the dead-weight fraction fl  for a five-stage fusion rocket at various 
payload ratios. The relatively small effect of the dead-weight fraction upon performance i s  a very significant 
feature in the design of this type of system. It indicates that a strong effort should be made to obtain 100% 
burnup even a t  the cost of additional structural weight. 
9 
JPL Technical Report No. 32-233 
V. CONCLUSION 
It i s  concluded that if staged nuclear-energy rockets are used, relativistic velocities can be attained 
with reasonable mass ratios. Improvements in technology would be required, but no energy sources beyond 
the known fission and fusion reactions need be employed. 
10 
IPL Tadmica/ Report No. 32-233 
NOMENCLATURE 
I 
M b  
' e x  
M f  
MP 
(Mp) 
MO 
i 
i +1 
' e x  
( M o )  
u 
V 
W 
P 
A 
E 
specific impulse 
rest mass a t  burnout 
rest mass exhausted 
rest m a s s  of fuel consumed 
payload mass 
payload mass of the ith stage 
initial rest m a s s  
initial m a s s  of the ( i  + 1)th stage 
kinetic energy of exhaust 
vehicle velocity relative to stationary system 
burnout velocity for nth stage 
exhaust velocity relative to stationary system 
exhaust velocity relative to vehicle 
dead-weight fraction 
stage structural or dead rest m a s s  for nth stage 
step fractions 
stage m a s s  ratio 
over-all m a s s  ratio 
energy conversion fraction 
rest mass of fuel converted to kinetic energy 
burnout fraction 
stage burnout rest m a s s  for nth stage 
stage fractions 
over-all payload fraction 
11 
JPL Technical Report No. 32-233 
One- way trip, 
with deceleration 
Table 1. Mass ratios required for fission rockets, E = 7 x 
Two-way trip, 
with deceleration 
Fraction of 
I i ght velocity 
u,/c 
2.0 x 102 
4.8 io4 
1.7 10’ 
8 x lo9 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
3.8 io4 
2.3 io9 
- - -  
- - -  
Required over-all mass ratio A 
One-way trip, 
without deceleration 
1.4 x 10’ 
2.2 x 102 
4.1 103 
9.0 104 
2.1 x 106 
1.0 x 108 
12 
JPL Technical Report No. 32-233 
Table 2. Mass ratios required for fusion rockets, E = 4 x 
Fraction of 
light velocity 
U n / C  
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
Required over-all mass ratio A 
One-way trip, 
without deceleration 
3.0 x 10’ 
8.9 x 10’ 
3.3 x 10’ 
1.1 x 102 
4.4 x 102 
2.3 103 
1.6 io4 
2.1 x 106 
1.4 10’ 
One-way trip, 
with deceleration 
9.0 x 10’ 
7.8 x 10’ 
1.1 io3 
1.2 io4 
1.9 io5 
5.2 x lo6 
2.6 x lo8 
- - -  
_ _ _  
Two-way trip, 
with deceleration 
13 
JPL Technical Report No. 32-233 
a 
0 
a 
a 
l- 
LL 
v) 
v) 
E 
-1 
-I 
I 
LL 
W > 
0 
a 
MASS FRACTION CONVERTED TO ENERGY 4 
Fig. 1. Over-all mass ratio required versus energy fraction 
for various fractions of light velocity 
14 
IPL lechnicd Report No. 32-233 
Fig. 2 Fractions of light velocity 
attainable for a deuterium fusion 
rocket versus number of stages 
for various dead-weight 
fractions (payload 
fraction = 10-1) 
Fig. 3. Fractions of light velocity 
attainable for a deuterium fusion 
rocket versus number of stages 
for various dead-weight 
fractions (payload 
fraction = 10-3) 
NUMBER OF STAGES n 
15 
JPL Technical Report No. 32-233 
Fig. 4. Fractions of light velocity 
attainable for a deuterium fusion 
rocket versus number of stages 
for various dead-weight 
fractions (payload 
fraction = 
NUMBER O F  STAGES n 
Fig. 5. Fractions of light velocity 
attainable for a five-stage deu- 
terium fusion rocket versus 
over-all payload fractions 
for various dead- 
weight fractions 
10-2 lo-' 
OVER-ALL PAYLOAD F R A C T  I ON 
I P L  Technical Report No. 32-233 , 
REFERENCES 
1. Ackeret, J., “Theory of Rockets,” Heluetica Physica Acta, Vol. 19, 1946, pp. 103- 112. 
2. .................... “Theory of Rockets,” Iournal of the British Interplanetary Society, Vol. 6, 1947, pp. 116 - 
123. 
3. Tsien, H. S., “Rockets and Other Thermal Jets Using Nuclear Energy,” The Science and Engineering of 
Nuclear Power, Addison Wesley, Cambridge, Vol. 11, 1949, pp. 177 - 195. 
4. Bussard, R. W., “Galactic Matter and Interstellar Flight,” Astronautica Acta, Vol. 6, 1960, pp. 179- 194. 
5. Sanger, E., “Atomic Rockets for Space Travel,” Astronautica Acta, Vol. 6, No. 1, 1960, pp. 4-  15. 
6. Stuhlinger, E., “Photon Rocket Propulsion,” Astronautics, Vol. 4, No. 10, October 1959, pp. 36, 69, 72, 
74, 76, 78. 
7. Huh,  J., “Relativistic Theory of Rocket Flight with Advanced Propulsion Systems,” A R S  Journal No. 
30, 1960, pp. 250 - 253. 
8. Subotowicz, M., “Theorie der relativistischen n-Stufenrakete,” Proceedings of the 10th International 
Astronautical Congress, London, 1959, Vol. 2, pp. 852-864. 
17 


Feasibility of Interstellar Travel

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