I

I

LAMS-2632

CJC-14 REPORT COLLECTION

REPRODUCTION

COPY

I

LOS ALAMOS SCIENTIFIC

OF THEUNIVERSITYOF CALIFORNIAo LOS

LABORATORY

ALAMOS NEW MEXICO

MISSION ANALYSIS FOR NUCLEAR PROPULSION

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LEGAL NOTICE

Thisreportwas preparedas an accountofGovernment sponsoredwork. NeithertheUnitedStates,northe

Commission,noranypersonactingon behalfoftheCommission:

A. Makesanywarrantyorrepresentation, expressed

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LAMS-2632

PHYSICS

(TID-4500,16thEd.)

LOS ALAMOS SCIENTIFIC LABORATORY

OF THE UNIVERSITYOF CALIFORNIA LOS ALAMOS NEW MEXICO

REPORT WRITTEN June1961

REPORT DISTRIBUTED: November 10,1961

MISSIONANALYSIS FOR NUCLEAR PROPULSION

by

RalphS.Cooper

—u

—’ -’ Contractw-7405-ENG. 36 withtheU. S.AtomicEnergyCommission

e$j

~-h All LAMS reports are informal documents, ueually prepared for a special pur-

:Go; . .. —-

s~-,

pose and primarily prepared for uee wItbin tbe Labomtory rather than for

general dietrlbution. This report bae net been edfted, reviewed, or verified

*===& for aoouracy. All I.AMS reports express the views of the authors as of tbe

.===2ES0 time they were wrftten and do not necessarily reflect the opinfona of the Los

;g;! Alamoa Soientffio Laboratory or the final opfnion of the authors on the subject.

8~m “~

o

~C9

.,-. —

ABSTRACT

This report, originally prepared for a UCLA

in Space Propulsion, presents elementary methods

formance of high thrust rockets with emphasis on

Short Course on Advances

for determining the pernuclear pzwpulsion.

EWqple models are used to describe tie vehicles and to evaluate

mission requirements. Performance comparisons, both in general

specific missions, are made of various P~PUlsiOn system uSti

values for the system parsmeterso

iii

the

and for

typicsl

co-s

Abstract

Introduction

Vehicle Description and Perfoxmmce

Mission Requirements

Circux@anetary Operations

Lunar Operations

Interplanetary Operations

Illustrations and Examples

Engines and Vehicles

General Comparisons

Earth to Orbit

Lunar Exploration

Interplanetary Missions

Appendices

A. Stage Optimization

Be Low Thrust Operations

Symbols and Units

References and Bibliography

v

Page

iii

1

1

9

9

15

15

a

a

24

30

33

34

35

41

46

48
Introduction

Although the over-aXl optimization of a space vehicle requires

detailed knowledge and integration of all the subsystems and flight

objectives, a simple model can serve to give a good estimate of the

perfoxmmnce as weld as an understanding of the significance of the vehicle parameters. The simple models are particularly suited to prelim.

inary design and advanced systems where the parameters (e.g., engine

weight and specific @y,d.se) are not well known ad to comparisons amng

different classes of propulsion systems. We shall make simplifying assumptions concerning the vehicle components and concerning the flight

kinetics and will point out their limitations snd more detailed methods.

Vehicle Description and Performance

In field free space, we can apply the law of conservation of momentum to a rocket of mass m, which ejects a small mass dm at an exhaust

velocity v= with respect to the vehicle. Using V for the vehicle velocity, we have

Vedm = -mdV (1)

-1-
Integrating

J

m(t) ~

AV(t) =V(T)-V =-V

o e -iii

m

o

= v= In ,&J .

Note that this is independent of the rate of

program). By specifying the &rust progrsm,

to get tie distance (x)t; in particular, for

dm/dt = a

m

v(t) =Vo+velIl~ mo-ctt

which leads to

acceleration (or thrust

one can integrate further

the common case of constant

If we consider the burnout time

the familiar rocket equation

(2)

. m- 7

x(t) ‘Xo‘VJ+V41 -LY:d‘n*] “

M

AV =veln-Q

%

when the thrust ceases, Eq. (2) gives

(3)

(4)

(5)

-2-
The?

includes

plicity,

ratio Mo/~ is termed the “mass ratio)” R. The burnout mass

engine) tankage) etc., in sddition to the payload ● For simwe shall divide this “dead” weight into two categories labeled

Me and Mt. Mt will represent the tank mass which is considered separately for several reasons. We shsll be interested in the effects of

dropping useless tankage and in carrying fuel to orbit for refuel purposes. Mt will be assumed to be a fixed fraction of the propellant mass

Mp, which is true for sufficiently large tanks and accurate enough for

our p-se. Thus

‘t f=~.

P

Since M. = ~+Mp

l!he

and

‘t = f(Mo - MJ = f(l -

remainder of the dead weight,

control, holdover propellant,

$MO.

including engine, structure,

etc., will be lumped into Me

(6)

(7)

guidance

and assumed to be proportional to the gross vehicle weight (including upper

stages and payload):

Me

e

‘q”

The payload ~ is thus given by

(8)

-3-
Eq.

%=%- Mt-Me

[

l+f =Mo ~-

1

e-f.

Letting y = l$/Mo, the payload fraction for a stsge, and using

(5) to replace R, we have

(9)

(lo)

%

-AV/ve

y.r =(l+f)e -f-e. (u)

o

For an n stsge vehicle, the over-all payload fraction is the pzwduct of the yi’s for each stage:

(?)=i [(’+fi)e-Avi’v:-fi-‘J

n

(12)

where the sum of the AVi is the total mission velocity requirement AVT.

For brevity, we will occasionally drop the A. Eqs. (11) and (12) will

form the basis for most of our anslyses and co~utations. ‘lbevehicle

parameters f~, Ci, ~d Vi can be determined or guessed from !mre detailed studies. The AVi will be delxnmined from the total velocity

increment (including losses) required to perform a given tission, which

will be divided sarong

at appropriate points

One can optimize

error, but an elegant

the stages to optimize the payload or have stsging

(e.g., termination of a phase of the mission).

the multistage payload fraction only by trial and

method devised by Dr. R. H. Fox does this by

-4-
finding a single not of a polynomial

in Appendix A. If we assume the ssme

stages but ellow for different vslues

equation. This method is presented

.

kind of propulsion (v:) for all

of f. (to account for tank size or

.L

insulation) and ei (for different size engines or

ratio for upper stages), then we can optimize the

choosing

lower thrust to weight

payload fraction by

Avi(optimum) =

%+ve~n(-)-:~~n(-)(1,)

With equal parameters for all stsges,

AVT

AVi(optimum) = ~

and

%

[

-Vdnve - ~

1

n

<=

(1 + f)e -f.

(14)

(15)

We can perform an approximate optimization of the number of stsges

(which is generally ns V~Ve) to obtain

O

% -

(1+1 .4f+20ge)v#Te

~

=e ●

opt.

(16)

This equation, the derivation of which is given in Appendix A , is useful

for making gross comparison between different propulsion systems and estimating the payload fraction for “difficult” (high VT) missions. Often

-5-
one wishes to compare “dry” or manufactured weights (Md) as an estimate

of vehicle costs. In view of the uncertainties of G and f, a crude but

sufficiently precise estimate is

‘d = (c + f)Mo

or

‘d kV~ve

—=(e+f)e

%

(17)

(18)

where

k=(l

To the same

+ 1.4f + 209E). (ma)

approximation, we can estimate the propellat vol~e per

unit payload to be

Vol

[

‘V’ive - ~

~=pp +(1-e-f) e 1 (19)

-1

large VT, we can dmP the (-1).

parameters upon the payload and

where p is the propellant density. For

We can determine the effects of the

each other (exchange ratios) by differentiation of Eq. (I-1). of special

interest is the value of specific @ulse (Isp) which is the exhaust velocity divided by g, the gravitations. constant. ‘DIUS

-6-
w’#Mo)

[

= a+” (1 + f)e

-V/gI~p

31Sp Sp

-E- f1-1

-v/gI~p

= (1 + f)e

Writing this in terms of the payload

B = v/gIsp,

we have

and letting

Since pe-P is a very slowly varying function in the range of

practice (.6 <P <1.5 or for nuclear propulsion, 15,000 <V

(20]

(a)

interest in

<40,000

ft/see), we can replace it by its average over the interval.which is

0.3~2. This value multiplied by the factor (1 + f) is equal to e-l for

f --0.05, and thus a convenient approximation is

a% M.

~“~ “

For nuclear Ha propulsion neglecting dissociation, I

Sp is proportional to the square root of the absolute te~erature (T) of the exit

gas, and this leads to

(22)

-7-
&- % ‘c)

T

== ● (23)

The effect of changes in the mission velocity requirement V, such

as to account for

3%

T=-

Similarly for the

am

the earth’s mtation$ for -role) is

-v/ve

(1+ f)Moe (1 + f)”.

=,— .

Ve veR

component weights:

(24)

(25)

One can make the anslysis more detailed by separately considering

tie varioW components. For exsmple> what we have l~ed fib Me iS

frequently broken down into reactor, pressure shell, pump> nozzle, and

thrust structure; and relations for the component weights written in

terms of

We shall

but will

flow rates, pressures$ reactor size and void fraction, etc.

not go into such detail here (for examples, see references 1, 2)

make some comments concerning them. !lhereactor itself usually

constitutes the bulk of the engine weight. At temperatures of interest

and H2 as propellent,~out 50 wunds of thrust are developed per megawatt

(Mw) of reactorpowere Nuclear engines described in the open literature

-8-
average about 2 pounds/Mw, making

for a thrust equal to the initisl

the thrust to weight ratio (T/W.)

their contribution to e equsA to .@

gross weight. For ground launching,

must be at least 1.2 in practice$ while

lower values are optimum for ~er stages, and O.2 is quite satisfactory

for orbitsl vehicles. For the latter$ the low thrust requirement greatly

offsets the large reactor weight per unit thrust.

One can also compute tank weights in terms of tank pressure ~d geometry and materiel pzmperties. This is complicated by the possible need

for insulation, structure, or additiond. pressure for SUPPOfiti the PaYload2 interstate materisl$ etc. Hydrogen has a very low density which

leads to relatively high values for f> the tsm.kfraction. These r~ge

fxwm .03 to .10 in the open literature which illustrates the uncertainty

associated with advanced systems.

Mission Requirements

Circumplanetary Operations

We shall first present the results of simple dyasmics for impulsive

velocity changes and then refine these with suitable additions snd corrections which may be included by altering the mission velocity requirement VT.

Since nuclear propulsion is more appzmpriate for difficult missions}

we will skip sounding and grmnd-to-ground missions (which are discussed

in references 1 and 3) and begin with g?xmnd-to-orbit missions. The velocity required for a circular orbit about the earth is given by

-9-
.=.{!+ .yIj- (27)

where

f3e= earth’s

r = earth’s

o

gravitational

radius

r = orbit radius

‘e = ger~$ ~otier form of

constant (surface acceleration)

the gravitational constant.

To go into an orbit at tie earth~s surface would require avelocity

of 25j$100ft/sec. In practicej one must go to altitudes of 100 miles

(Vc = 25,600 ft/see, but the mission velocity requirement is higher to

account for the potential energy increase) or more to reduce atmospheric

friction so the orbit can be maintained. In addition> the rocket must

fire over an appreciable period of time (i.e., non-inuxilsively),start

verticsJJy, and

orbit. Solving

follow some trajectory through the atmosphere and into

this problem with the effects of an air drag, the earth’s

rotation, gravity, etc. is formidable and is discussed in references 1, 3.

For exsmple, one might integrate the equation of motion with an added

term for gravity, which represents a major portion of tie correction.

The result would be similar to Eq. (3), i.e.,

m

V(t)=Vo+veln~ .

-oat

gt G

o

(28)

where= is the time averagedvslue of the sine of the angle between

the velocity and the horizontal.

-1o-
The gravitational term would depend upon the trajectory and acceleration but is usually of the order of seversl thousand ft/sec. me

xutationof the earth can result in as much as +1500 ft/see, while atmospheric drsg is much less important, especially for large boosters.

The atmospheric pressure reduces the thrust (and thus the effective

exhaust velocity) for liquid propellant engines to about 9@ of their

vacuum value at sea level. ‘Ibisloss could be included in the mission

velocity or accounted for by using an average atmospheric value for Ve

for ground launched stages. The over-sJJ velocity requirement (including

losses) for a low earth

and we shall use 32,000

tions.

orbit typicslly ranges from 30,000 to 33,000 ft/see,

ft/sec as a representative value for our computaVelocity requirements for transfer to other circular orbits will depend upon the path, but the method which usually* requires the minimum AV

is the Hohmann transfer ellipse (Figure 1) which is tangential to each of

AV1

Figure 1. Hohman,nTransfer

*

E.g., see “Two impulse maneuvers” in reference 3, Chapter 8.

-11-
the circular orbits. Two impulses are required, at the ellipse’s extremums and parellel to the velocity. The first velocity change (AV1)

for transfer from a circular orbit rl to an elliptic orbit with apogee

r. = cm. is

z

where

Cular

(29)

vcl is the circular velocity at rl. To transfer to the new cirorbit requires

Thus the total AV required for transfer between circular orbits is

AV1+AV2= {(w~ 1-

)}

(1-a) -~v

F m

cl

which holds for &Ll a > 0.

In particular, the total ideal velocity necessary

ground to a circular orbit at x earth radii (x = r/ro)

(30)

(31)

to go from the

is

(32)

l?orescape from the ground, x + co and

-12-
AV1 escape = v-= ~~=36,600ft/sec.

In genersl, escape from any circular orbit requires (@ - 1) times

the orbital velocity. Surprisingly, in Eq. (32)} AV1 is not monotonic

with x but has a shsllow maximum at x = 15.6 (altitude = 58,4oo miles).

Achieving a “stationary” orbit at

than escape.

In Figure 2, we show results

22,000 miles requires 2300 ft/sec more

for ascent to a 100 mile circular orbit

and Hohmann transfer to higher circular orbits. The perigee (vp) and

apogee (va) velocities are given for the elliptic transfer orbits as well

as the circular velocities. Conservation of momentum gives v = ma.

P

Also presented in Figure 2 is the total velocity requirement VI

C+E for

circular orbit at altitude (h) and subsequent escape, showing that the

most economical assembly orbit for interplanetary flight is a low one.

Not much penalty is exacted for orbits up to about 500 mile altitude.

These results, derived for @ulsive motion in a conservative field, can

be used for inward or outward transfers. The velocity requirement for

other types of maneuvers, e.g., rotation of the plane of the orbit, can

be found from the vector difference of the initial and final vehicle

velocities. These AV~s can be large, in the above ex~ple

AV =2VC sin:

to zwtate the plane of the orbit through an angle

low thrust to weight ratios for orbital stsges in

-13-

(33)

Q. We shall consider

an appendix where we

.
LLL

xl

:

-14-
show values of T/W. of

Lunar Operations

the order of .1 b .2 are acceptable.

The moon has an aversge orbit

it elmost at infinity with respect

ing it, as can be seen from Figure

radius of 238,000 miles, which places

to the velocity requirement for reach2. ~is v~ue is 36,000 ft/sec (not

including losses) compared to 36,600 for escape. From its gravitational

constant (g# 5.30 ft/sec2) and radius (1080miles), we find its escape

and low orbit velocities to be 7783 ft/sec and 5500 ft/see, respectively.

Circum.lunarflights require very little more than the velocity necessary

for impact. Orbital maneuvers

for going from escape to a low

this mission only as difficult

Interplanetary Operations

can be computed as for the earth. The All

lunar orbit is --2300ft/see, which makes

as a stationary (22,000 mile) earth orbit.

While the same simple dynamics applies,

sumptions are necessary, and some additional

We shell assume the planets to have circular

a few more simplifying aseffects should be noted.

concentric coplanar orbits

smd consider the gravitational effects only of the single major body

affecting the vehiclets motion during each phase of the trip. For interplaneta~ voyages, we are interested in the gravitations constant

of the sun, and the ea@’s orbital radius (1 A.U.) is a convenient

unit as is tie earth~s velocity (VE = 97,800 ft/see). Eqs. (2g) through

(31) hold for transfer between orbits, where the vehicle remsins free of

planetary influence at either end of tie transfer. Howwer, just as the

-15-
total velocity increment for escape from the earth is less for departure

from a low orbit, it is nmre economical to make interplanetary departure

from a low orbit. We can evsluate the velocity requirement and demonstrate the above

Let Vml be

essentially free

statement.

the actual velocity required for the vehicle when it is

of the earth’s attraction but still at the ssme distance

from the sun, i.e., having “escaped” from tie earth (Figure 3). We can

then use Eq. (29) to compute

orbital velocity required to

with apogee

Figure 3

the excess velocity (v=) above the earthts

place the vehicle in an elliptical.orbit

‘2 = crrE;

v ‘vm~v~ (~~-+e (34)

ex

Using a low earth orbit as a sta?.%ingand reference Pofit$ = can

cslculate the totsl AV required if escaPe is accomplished sePmately~ i.e.>

-16-
AV (escape) + vu

(w- l)VCO + vex. (35)

On the other hand, if we give the vehicle a single impulse in a low orbit

to a velocity Vh, we can obtain the velocity of the vehicle sfter escape

(v=) from conservation of energy;

USing Eq. (27) and rearranging

‘h =

‘lhusthe one @ulse velocity requirement

(36)

(37)

which can

AV1=vh-vco= dV:+2V2 _

co

easily be shown to be less than

analysis can be easily extended to orbits

(38)

that given in Eq. (35). The

at intermediate s2titudes

which are of interest in capture operations. At the apogee of the

transfer orbit} the vehicle can be injected tito the target planet%

orbit with a AV2 of the foxm given in Eq. (30), and very little mre

is required to go into a very high orbit about the planet. Results

for AV1 and AV2 are shown in Figure 4. Should lower oibits be desired,

-17-
-18-
a conibinedsingle @ulse capture would be more economical just as in

tie escape case. One can use Eq. (38), written in a nmre general way>

AV (capture) = Vh - v (desired orbit)

= wv (desired orbit).

As a specis3 case of a circular caWmre orbit,

AV (capture) = w

from which one can find the circular

capture and which is

(39)

-rK

P

T

(40)

orbit radius which mitizes AV

(41)

If r is not smaller than tie planet radius, then AV (capture) can be

0.707 times the vslue computed without the planet’s gravitational effect.

‘lhisis also of interest when returning to earth, as the returning

passengers and cargo can be picked w and returned to earth by a vehicle

from earth, and thus be relieved of carrying propellant for this maneuver

on the entire round trip.

‘Iheminimum energy (Hohmann) transfer has several drawbacks; it can

be started only at predetermined widely separated times and M used in

-19-
both directions, forces a particular waiting time at the target planet.

Furthermore, the traasit times are quite long, especially to the outer

planets, and the miss distances at the target planets are a sensitive

function of the injection velocity. ‘lImsthere is much interest in

non-minimum energy missions or as they are nmre frequently termed “fast

transfer” missions. While these also can be treated analytically under

the assumptions we have made, there exists a two parsmeter fsmily of

transfer orbits from which one must select on the basis of desired

transit and stay ttie~ available AV vs. payload etc. Zhese missions

are treated in varying emounts of detail in several of the references,

and only a few examples will be presenlad here to give a feeling for

the quszrtitiesinvolved.

Hohmann transfer to Mars from a low earth orbit requires -12,000

ft/sec for the earth impulse phase and ~000 ft/sec for capture in a

1000 mile sltitude Martian orbit. ‘he transit time is 250 days, but

can be reduced to half that by increasing the earth escape AV by only

2000 ft~sec. However, the intersection of tie transfer orbit witi ws’

orbit becomes steep, and the capture velocity requirement increases by

16,000 ft/see. one can minimize the total AV (escape + capture) for a

ftied transit time by varying both the magnitude and direction of the

escape @ulse such that the vehicle’s path mskes an angle with the

earth’s orbit.

&other exsn@e is of probes to the outer planets. Minimum energy

transfers to Jupiter, Saturn, and Uranus require 2.9, 6, and 16.1 years,

-20-
respectively (md AVSS of 21,500 to 27,000 ft/see). Increasing the AV

to --29,000ft/sec would give the vehicle solar system escape velocity

and decrease the transit times to 1.2, 2.7, and 6.9 years, respectively.

To summarize, we present a table (I) listing the appzmximate velocity requirements for a variety of missions.

Illustrations and Examples

Engines and Vehicles

We shall.give some examples from the unclassified literature of

coxqponentweights for nuclear propulsion systems, includhg some equations for estimating them.

The reactor weight, which ususJly is the bulk of the engine weight,

depends upon the reactor type$ operating pressure> power leve12 void

fraction, materisls, heat transfer, etc. and cannot be evsluated from

a shqple equation. Instead we will give some examples for graphite reactors, including core and reflect, excluding pressure shell, external

structuxe, controls, pump, and nozzle. These give 1.6,000 pounds for a

16,600 Mw reactor (reference h) and4$O00 pounds for 3000 Mw (reference 2)

with both reactors operating at dOOO psi inlet pressure. Both are *1

pound/Mw or about 1 pound of engine per 40 pounds of thrust. Detailed

formulae are given in reference 2 for pump, pressure shell, and nozzle

weights, but here we shsdl present only an exsmple to give a feeling for

their relative importance.

-21-
Approximate Mission Velocity Requirements

Earth Satellite and Cislunar Operations

Partisl AV ‘V!lbtsl

ft/see ft/see

Mission (with losses)

Low Earth Orbit 32,000

Ideal velocity 26,000

Gravitation losses 5,000

Drag and pressure losses 1,000

Escape

Low orbit to escape

24 Hour (ZZL,000mile) orbit

Low to high orbit

Lunar Orbit

Earth escape to lunar orbit

Lunar Landiw

Ideal lunar escape velocity

43,000

III.,000

45,000

13,000

45,000

2,000

52,000

8,000

Lunar Round !h?ip

(Parabolic ern?thre-entry) 60,000

(Circular earth re-entry) ~o,ooo

Interplanetary Operations

Plsnetary

‘i” E” ‘obes High Orbit Escape Manned Trip

Time Capture Velocity AV From Earth

Planet AV Years AV, ft/sec ft/see Orbit, ft/sec

Mercury 18,200 .27 31,700 11,600

Venus 11,500 .38 8,300 ;~ag 60-9w& ft/sec

Maxs 11,600 .7 8,500 60-9okL@ ft/sec

Jupiter 20,000 2.9 18,500 195;000

Saturn 23,900 600 17,900 IL6,000

Uranus 26,200 16.1 15,300 68,000

Solar Escape 2g,ooo -- -“ --

-22-
TypicaJ.Engine

Power, 4500 Mw Engine

‘Ihrust,‘2L0,000pounds Reactor

Flow Area, 3.3 ft2

Flow Rate, 340 pounds/see I&essure shell.

Exit Mach No., 0.4 Nozzle

Exit Pressure, 42o psi

!Ihepump, pressure shell, and nozzle weights are roughly pmpational.

to tie operating pressure, while the reactor is insensitive to it.

The propellant tanksge weight is significant and can be computed

from the tank pressure (pt), material density (pt), and tensile strength

(a). For a s~ere, this gives

‘t =t% f=q==

Weight, pounds

9000

7000

700

900

400

which is independent of tank size. Depending qpon the choice of mate.

rials, this equation gives values of f between .02 and .08 for H2. However, in typicsl ca6es h practice, the interstage structures, fittings,

insulation, etc. can add up to 5@ of this value to give the entire

fuselsge weight fraction and thus at

is required to pin this value down.

a set of vehicle parameters for each

(Table II).

least a preliminary vehicle design

For our pwgposes, we shall select

of several propulsion systems

-23-
— ——

Table 11

Vehicle Parameters

Pxwpulsion Isp, sec v@ ftjsec f e pp, lbs/ft3

LOX-RP 300 9,660 .02 .02 63

LOX-H2 420 13,500 .04 ●02 17

Nuclear 860 27,700 .10 .04 4.3

.06*

*

Ground launched stages 0111.xwhere an average value is required, use 0.05.

General Ccxqparisons

Using the assumed parameters and Eqs. (16), (18), ~d (19)3 ~ cm

make rough estimates of the payload fraction, the ratio of hardware weight

to payload, and the propellant volume per unit payload. Significat

quantities are listed in Tsble 111 and results in Figures 5, 6, and 7.

Table III

Quantities for General Propulsion System Comparisons

ve/k (e + f) l/p (l-E-f)

LOX-RP 8,900 ft/sec .04

LOX-H2 12,100 .06

Nuclear 2L,600 .15

These equations allow direct comparisons snmng

compare the gross weight of LOX-H2 and nuclear

payload:

●015 f#/lb

●W5

.20

the systems. For exaqple,

pzwpulsion for the same

-24-
N?, 11#ft/sG

.$

.1 .$

.25-
-26-
-27=
M. (them.)

M.

=

(nut.)

Id$jve

e (them.)

kv~ve

e (nut.)

V (k/vc)-(kn/v~)

eTce

=

Thus the advantage of nuclear propulsion increases

locity, which is the result of higher specific impulse.

with mission velhere is the

sane dependence upon VT for the dry weight and volume, but the chemical

systems have initial.advantages due to their lower dry weight fraction

and higher propellant density:

Md (them.) V~27,500

= .4e

‘d (

nuc.)

and

Vol. (them.)

V~27,500

vol. (nut.)

= .27e ●

These have crossover points at 25,000 ft/sec and 36,000 ft/sec respectively, ebove which the nuclear system is

will detezmine the choice of a propulsion

reliability, safety, availability ground

superior. No single parsmeter

system; all factors$ including

support$ over-all cost, future

potential.,etc. will be involved. Our aim here is to make

isons which follow from the vehicle and mission parameters

inputs to the decision-making pzucess.

those cQarand are partisl

The above co~arisons are for vehicles using only a single type of

propulsion for all stsges. Composite vehicles using different propulsion

-28-
schemes in different stages can have smaller vslues of dry weight or

volume. Most of these two quantities for a given vehicle are in the

first stage, and the use of chemical propulsion for it mi~t materially

reduce these for the vehicle as a whole. We do not have a simple technique for minimizing structure or volume of multistage co~site vehicles

though they would be useful. Results of such optimization should be

used with caution} as we have indicated> because of the many other factors involved. However, the degree of sensitivity of tie optimized

quantity to changes from the optimum is valuable and csn usually be ob.

tained through analytical or numericsl techniques. Choosing the correct

subsidiary conditions or restrictions is also ~rtant in performing

optimization anslyses. For exsmple$ in optimizing the tJxrust/initial

weight, one gets very different results if the vehicle weight is fixed

for one calculation and the engine size for another. Either method

might be the correct one under particular circumstances● Finally, the

results, though analytically correct, might be ignored for a reason not

contained in the analysis; for exsmple, a stage or propulsion system

with poorer performance might be chosen because of its availability,

simplicity, etc. A case in point is combined nuclear-chemical propul.

sion where the propellant fuel is first heated in a nuclear reactor and

after partial.expansion is burned with an oxidizer. For the H2-02sYstem>

this gives an increased performance under certain conditions (reactor size

fixed and of a size too small for the stage considered). ‘J!heH2 issuing

from the reactor might have a specific ixqpulseof 800 seconds and a

-29-
temperature of -2500°co ‘Iheaddition and coxtibustionof the oxygen does

not raise the temperature much but does supply energy to raise the combustion products to about the same temperature. However> tiey have MU*

higher mlecular weight than the H2 resulting in higher thrust and lower

specific ixqpulse. The mixture ratio and duration of oxidizer flow can be

varied to optimize tie performance of a given stage. While this scheme

might give better performance (based on some criterion) than a simple

nuclear engine plus a separate LCIX-HOstsge, it is the author’s personal

L

opinion at the present time that the complexities of such

not worth the possible performance gain.

The missions, as well as the vehicle parameters, can

confusion lf results for one are carelessly generalized.

an engine are

be a source of

One can easily

choose tie mission such that smll differences in the vehicle parameters

make the difference between no payload and substantial payload or such

that one or anotier of various systems is superior. One should keep in

mind the old admnitio~ “Figures don?t lie, but liars do figure.” Remembering this, let us continue with (figuring) further illustrations.

Ea~ to Orbit

We ~all present results for a variety of vehicles titended for the

low earth ofiit mission. We will examine solid LOX-RP and LOX-H. boosters

with LOX-H2 or nuclear propelled upper stages,and a

vehicle. (For the solid rocket, Isp = 250 seconds,

For ease of visualization, we will-coml?a~ vehicles

c

single stage nuclear

e = .02, f = .05.)

designed to place

-30-
200,000 pounds in orbit. Since opt~zation on a gross weight/payload

basis would lead to an all nuclear stege, the booster velocity increments

are arbitrarily taken to be equsl to tie boost exhaust velocity (i.e.,

mass ratio = e) where the second stage is

Table IV

nuclear.

Chemically Boosted Nuclear Stages

200,000 Pounds in Low Earth Orbit

Weights in lF Pounds

Volumes in ld f~

Solid LOX-RP LCRK-H.

M.

‘d

Volume

First Stage

‘d

Volume

Second Stage

M.

‘d

Volume

2080

185

100

108

12

655

‘n’

88

1810

130

96

60

18

606

70

78

1520

la

113

68

57

490

53

56

The results show there is relatively little difference in

wei~t, dry weight, and volume among these cases. Notice that

the dry weight and volume lie in the nuclear stsge. mere are

gross

much of

clearer

differences if we compare the previous cases witi a single stage nuclear

vehicle and a two stage chemical vehicle.

-31-
Table V

Chemical and Nuclear Vehicles for Low Earth Orbit Missions

200,000 Pounds in Orbit

AV = 32,000 ft/sec

Nuclear LOX-H 22 LOX-RP> LOX-H2

No. of Stages 1 2

Mo> K? lbs 1070 3000 4300

Md, 1P lbs 126 181 217

Vol ● , ld ftJ 170 154 IJ.2

The single nuclear stsge has the smallest gross weight and largest

propellant volume, while the all-chemical vehicles have significantly

largergross weights and dry weights. The engine for nuclear second

stages requires power levels of 10,000 to 12,000 Mw, while the ground

launched single stsge requires at least 30,000 Mw. Chemical engines in

the million pound thrust class are being developed, while the power level

of the first nuclear engine to be flight tested is reported* to be of the

order of 1000 Mw (50,000 pounds thrust). Such a difference ~ the rel>

tive states of the art could influence the choice of vehicles. Other

factors involved in the all-nuclear vehicle are the air-scattered radiation to the payload and the launch site and the possible greater

safety and reliability of a single stage. It might be possible to return the nuclear engine or the entire

“’Robert E. L. Admson, Nucleonics 19$

—

stage for reuse, though msny

56 (APZ%L, 1961).

-32-
technical.pxmblems would have to be solved with high reliability for this

to be practicsl.

Lunar Exploration

A round trip b the lunar surface witi

earth’s atmsphere requires 60,000 ft/sec.

parabolic re-entry

The optimum nuniber

of the

of stages

are ~ and 3 for LOX-HO and nuclear propulsion respectively ~d in PracG

tice fewer can be used with small weight penalty. Here there is a great

difference on all counts between the nuclear and chemical vehicles with

intermediate results for cowsite vehicles

Table VI

(Table VI).

Lunar Exploration Vehicles

Payload = 20,000 Pounds

AV = 60,000 ftjsec

LCK-H2 Nuclear ‘m-H2$ Nuc ., LCK-H2

No. of Stsges 4 2 3

Mo, lC$ lbs 3440 406 1130

Md, K? lbs 194 60 82

vol. Prop., 1d ft3 190 74 97

The use of a single nuclear stage reduces the gross weight by a

factor of 3 and the dry weight and volume by a factor of 2, and tiese

factors are increased to 8 and 3, respectively, for an all-nuclear vehicle.

-33-
Interplanetary Missions

The same AV (60,000 ft/see) could apply for an interplanetary reconnaissance mission, starting from a low earth orbit and returning to a high

orbit. Here the gross weight is nnre significant than dry weight as the

cost of placing weight in orbit is high, and low thrust/weight ratios

(--.1)can be used with small performance penalty. Extensive exploration,

even of the inner solar syst-e~will require higher AV’s, to 100,000 f%/sec.

Co~arison of conventional systems csn be made from Figures 5 to 7, but the

competition may be snmng mm? advanced systems, e.g.~ 1000-16oo second I

Sp

nuclear heat exchangers, nuclear bomb propulsion, and nuclear electric

(ion, arc, plasma, etc.) schemes. The latter have high Isp’s (2000 to

10,000 seconds) but very low thrusts. Our method of analysis is not completely applicable to low thrust systems (T/W~ 10-3) where the criticsl

parameter is the specific weight of the power generating eqtipment. The

low &rust systems require long times ( - mmrths) for escape from a low

earth orbit which leads to higher gravitational losses and much time in

the radiation belts. Thus it apyears that even when electric propulsion

is operational, the high thrust nuclear heat exchanger will have important

areas of application.

-34-
First we can

V#eS igeg, that

~=~/MomdP=

Y= [

(1

APPENDIX A

Stage Optimization

show that the optimum number of stages

the mass ratio for each stage be about

V#Ve~ Eq. (15) of the text becomes

Assuming f, e << 1, we can drop the f in the first

to assuming all the dry weight is pxmportional to gross

rearrange to

and expand

Y=e

[

-Pl1

(e + f)epin n

approximately to obtain

Ywe (

-f31- ~(e + ~)ef3/n)

●

NOW dy/dn = O yields n . p . V@e ●

is

e.

approxima+~ly

Defining

(Al)

term (equivalent

vehicle weight),

(A.2)

(A.3) I

-35-
Differentiation of (Al) exactly plus a numerical solution for n

gives nsp. We can use this fact to derive Eq. (16) of the text. Let

u be the terms in the square brackets in Eq. (Al) and maximize y with

respect to n:

i

&=g=oa

dndn

‘Ibisleads to

f3(u+f+6)=o ulnu+—

n

or

ulnu=(l+f)alna

where

(A.4)

(A.5)

(A06)

We solve (A.~) by letting

x=

aflna

l+lna

whence

aflna

‘Sa+l+lna

u = a - x and expanding tie left side$ giving

(A.7)

(A.8)

and, using the definition of U$

a=(f+e)(l+ Ins) .

f

(A,9)

-36-
(me analysis to this point is due to

which we know to be approximately e

-1

+ 5, leading to

and

a=ri%

where

b=a+ae”

Fmm (A.6) and (A.1.1),

-@/n Dr. K. Brueckner.) Since a = e ,

for opthized n, let us write a = e

-1

Il(O@xnum)=--- f+lne :-b :f3(l+b++b2)

In a

which gives n (optimum) nmre exactly h terms of ~, f, and e. Thus

[ 1

Y(oPttiUIIl)= (l+ f)a-f. cn

[

(1 + f) 1

#3/ine(l-b)

=~- f-e ●

This is of tie form

(A.1O)

(A.12)

(A.13)

(A.14)

-k-f

Y=A

-37-
or

lny= -@k(f,e).

We can evaluate k under various assumptions concerning f and e.

{.

f=e<<l

k- (1+ 4.32f) = (1+ 4.32e)}

{

G<< f<<l

k ~ (1.33f + 3.18e)}

(A.15)

(A.1601)

(A.16.2)

{-

f<<c <<l )

(A.3.6J)

k- (1+ 1.72f+ 2.726)0

Since this is only an appmcimate limiting cas~that sXl forms give similar results

vehicles, we

the relative

h

and that in practice, e s f for boti chemical and nuclear

choose a compmnise which is correct for e = f snd reflects

importance of the two parameters, i.e.,

Y s -(I + 1.42f + 2.9~)p. (A.17)

If the stsges have different propulsion systems and thus different

vshes for e, f, snd Ve for each stage, the stage maSS ratios can be ~-

justed b obtain a maximum AVT for a fixed payload/gross weight ratio by

a ~thod developed by Dr. Robert FOX of Ucm (reference 5 in Bibliography).

-38-
We shall present the method without proof. Choose the smallest of the

exhaust velocities (vi) to be Vo. For each stege, calculate

V.-v

-i -o

Ci =--y- ●

(A018)

compute

(A.19)

where y is the chosen value of the payload fraction. Solve for q the

polynomial equation

-Q=O. (A.20)

A good trial vslue is q = Q1/n) snd Dr. Fox has shown that there is only

one reel positive mot of Eq. (20), which we shall label ~. men

(Ci + fi) V.

‘j,=~”~ (A.2L)

and

n

‘T = z

i=l ‘i=& ‘n(’i~~if~ J “

(A.22)

A31 anslysis with VT

similar to (A.20).

fixed aud y optimized leads to an irrational equation

Eqution (A.20) cen be solved analytically for two or

is the most that are of interest h practice, since

-39-

three stages, which
clifficult missions requiring many stsges will have several natunal stagx

steps (e.g., earth orbit or escape$ 1~ l~ting or Pl~e~ aPP~~~

etc.). Furthemnore, where there is a lmge ~ference ~ tie P~~~rs~

particularly Ve, the high performance stage wiU tend to “swallow” the

others. In the interesting case of a chemically boosted nuclear stage

optimized for a low earth orbit mission$ tie ti~c~ st%e ~1 entirelY

disappear.

-40-
APP~DIX B

Low Thrust Operations*

Here we evaluate tie effects of using low thrust to weight ratios

for orbital operations. me advant~e of lower engine weight is offset

to some extent by increased gravitational losses, and usually an optimum

vslue can be found. In practice, thrust/weight ratios or reactor power

levels considerably lower than the optimum can be used witi smell performance losses.

The equations for motion in a gravitational

dvr V:

w(v) -g (k)*

x-T=- m e

r

or

d(rve)

-M(v)r,

dt = m e.

field:

(Bol)

(B.2)

w

(where subscripts r and Q refer to radial and azimuthsilcomponents, and

go is the field constant at the initial orbit radius r.) can be solved

analytically only in specisl limiting cases. The high thrust limit gives

the fsniliar results obtainable from the direct vector addition of velocities. Let us introduce a tiensionless energy parameter X, which is

*

lhe analysis and results are based on tie work of Brueckner et sl. in

“Topics of ‘hmst Orbit Optimization (Los Alsmos Scientific Laboratory).

-41-
the final total energy (with respect to a

far fmm the attracting body) in tenus of

finsl mass in

zero

the initial orbit. Thus

body at rest and infinitely

the kinetic energy of the

Kinetic Energy + Potentisl Energy

7,-I ● (B.3)

$ m;

For a body in tie initisl circular orbit, h = -1 and for escape

final.velocity, X = O.

In tie high thrust (impulsive) limit, the velocity increment AV

or

siqply related to X;

Vh = V. + AV (~ulsive)

2

‘h h.—.2

V2

o

AV (iqpulsive) = (~= - l)vO.

The mass ratio R is given by

AV(iqpulsive)/va

Re=

!Ihe following

interest.

.

●

with

is

(B04)

(B.5)

(B06)

(B.7)

table illustrates the significance of X for cases of

-42-
TableB.1

High ThrustOperationsand Significanceof X

Low EarthOrbit

In Genersl V. = 25,200 ft/sec

h VJVO AIJ/vo AV(@pulsive), ft/sec TypicalMission

o < 2 C@-- 1) 10,400 Escape

1 fi (V - 1) 18,400 MercuryProbe

2 2 1 25,200 SaturnProbe

The low thrustlimit is slso solublefor X ~ O (missionsup to sad includingescapewith zem finslvelocity). Once out of the earth~s field,

the rest of the operationswouldbe the ssme as the iqpulsivelimit (no

gravitational.osses) excepttiat now the Suntsfieldmust slsobe considered. For X ~ O, the equationscan be solvedto give the mass ratio:

v

+ (1.-)

R(low thrustljmit)= e e (B.8)

In particular,k = O

vo/ve

R(low thrustescape)= e .

This is equivalentto a velocityrequirementof V. co~ared to (fi - 1)Vo

= .414 v. for the imulsivelimit.

respectively,

Similarly,for h = 1, the valuesare

for low and high thrustlimits.
‘logain an understandingof the intermediatecases,let us exsmine

the mass ratioas a functionof thrust/initialweightfor X = O and 1,

startingin a low earthorbit (v. = 25,200ft/see)and assumingap?mpellant exhaustvelocityequalto V. (I~p= 783 see). We shallgive results

(FigureB.1) based on the thrustbeing kept parsllelto the velocity,

whichhas been shownby Baker (IAM3-2403)to be closeto optimum. ‘Ihe

resultsshow that the mass ratioincreasesslowlywith decreasingT/W

until aboutT/W = 0.2,whereqponit risesfaster. These curves,together

with the weight-thrustrelationshipfor the engine,would give the optimum

power level and off-optimumlosses.

Anotiertechniquefor utilizinglow thrust/weightratioswith less

gravitationalloss is to operatethe engineseveraltimesonly nesr the

perigeeof the orbit;thus increasingonly the apogeeuntil escapeis

reached. Thishas the disadvantageof requiringseweralenginecycles,

passesthnmgh the radiationbelts,and more difficulttimingproblems.

-44-
16

-45-

Oi h-l
SYMBOLSAND UNIT’S

Equationsin the text are generallytruewith any consistentset of

units● We have used engineeringunits (pxnds, feet, seconds)in our

numericalexsmples.

SYMBOLS

e

f

g

I

Sp

k

K

M

m

n

P

R

r

r

o

T

t

v

v

c

2.~8 ..●

tankagefraction

gravitationalconstant,ft/sec2

specificimpulse(sec)

a structuralparszneter

gravitationalconstant

dependingqpon

(ft3/sec2)

fande

mass,ususllyof vehiclesand components(lbs)

mass, general(lbs)

nmiberof stages

pressure(lbs/in2)

mass ratio

orbit radius(ft)

planetradius(usu.dly earth, ft)

absolutetenpxature (‘R or ‘K)

time (see)

velocity(ft/see)

circularorbitvelocity

-46-
‘E

Ve

v

All

Vol●

x

Y

a

P

e

Q

b

c

d

E

e

i

L

m

n

velocityof earth in its orbit

propellantexhaustvelocity,ft/sec

vehiclevelocity,also shortfor AV (ft/see)

changein vehiclevelocity;SLSOmissionvelocityrequirement

propellantvolume,ft3

distance(ft);also orbit radius/planetradius

payloadfraction

propellantflow rate (lbs/see);also ratioof orbitradii

a parameter,V/ve

engine(plusstructure)fraction

angle

density,lbs/ft3

tensilestrength,lbs/ti2

SUBSCRIIICE

burnout

circularorbit,also chemicsl

& (~~actured weight)

Earth

engine(and structure)

stageindex

payload

moon

nuclear

o initial,

P propellant

ii tankage

-47-
1.

2.

3*

4.

59

6.

7*

REFERENCl!S*AND BIBLIOGRAPHY

Bussard,R. W., and DeLauer,R. D., —.Nucle~ Rocket

York,McGraw-Hill(1958).

Johnson,P. G., end Smith,R. L.$ “~

Parametersfor OrbitalLaunchNuclear

Optimization

Propulsion,New

of Powel@ant

Rockets,” NASA-TN-D-675(FebIIWY, 1961).

Seife~ H. S.

Sons (1959).

(Editor),Space Technology,New York, John Wiley and

New&rd, J. J., and kWOy$ M. M.$ “Considerationsin the Designof

a NuclearRocket,“ NuclearScienceand mineering 1$ 377 (1960)●

Fox, R. H., “StageOptimizationof tie GeneralM-StageRocket,”

AdvancedDevelopmentNote No. 24, Livermore(Universityof CaliforniaRadiationLaboratory)(August,196o).

Ehricke,K. A., ——SpaceFlight,VO1● IS ~iro~ent ad Celestial

Mechsnics,Princeton,D. Van Nostrand(1960)(Vol.11, Dynamics,

~d Vol. III, SpaceOperations$ti PreP~ation).

Ehricke,K. A., “A systemsAnalysisof Fast MannedFlightsto Venus

ad Mars,” ConvairAstronauticsReportAZM-072(March,1959).

*

See annotationat end.

-48-
8. Lawden, D. F., Advmces ——in Sx=ce —.science IS ~aPter I> Ac~emic press>

New York (1959).

9. Duga, J. F., “Analysisof TrajectoryParametersfor Probe and RoundTripMissionsto Mars,“ NASA-’IN.D28L(June,196o).

10. Vertregt,M., “InterplanetaryOrbits,”J. BritishInterplanet~ Sot.

~, 326 (1958).

u.

12.

130

tain

Severalpaperson multistagerocketoptimization:

Schurmann, E. H., Jet. Prop. 27, %3 (1957) —

Ten Dyke,R. P., ibid28, 338 (1958) —

Hall,H. H. and Zambelli,E. D., ibid—28, 463

Subotowicz,M., ibid=, 460 (1958)

Weisbord,L., ibid~ 445 (1959)

Builder,C. H., MS Journal—29, 497 (1959)

Cobb,E. A., ibid 31, 342 (1961). —

Jurovics,S., “OptimumSteeringprogramfor the

Vehicleinto a CircularOrbit,”ARS Journal—31,

(1958)

Entry of aMultistsge

518 (1X1) .

Ruppe,H. 0., “MinimumEuergyRequirementsfor Space Travel,ABMADSP-TR-4-59 (November, 1959).

References1, 3,

numerousfurther

devotedto its topic.

Annotation

and 6 are the most valuable

references.Reference1 is

and =tensive and concurrentlytie only book

Reference3 containsseveralchaptersof interest

.49.
on mission studies. Krafft Ehrickecs trilogywill probablybecomethe

standardwork in this fieldwhen it is available.Until then,his chapter in reference3 shouldsufficefor a brief reviewof interplanetary

operations.

-50-