A possible compromise between rocket and atmospheric braking

< X H

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von KAR:M:AN INSTITUTE

FOR FLUID DYNAMICS

TECHNICAL NOTE 17

A POSSIBLE COMPROMISE BETWEEN ROCKET

AND ATMOSPHERIC BRAKING

by

L. MOULIN

RHODE-SAINT-GENESE, BELGIUM

JUNE

1964

VTH

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-von KAR MAN INSTITUTE FOR FLUID DYNAMICS

TECHNICAL NOTE 17

A POSSIBLE COMPROMISE BETWEEN ROCKET AND ATMOSPHERIC BRAKING

by

La MOULIN

JUNE

1964

The research reported in this document has been sponsored by the Air Force Office of Scientific Research, through the

ABSTRACT

An attempt has been made to optimize the problem o~ recovering an interplanetary vehicle through the earth's atmosphere. Optimum conditions are defined as the ones which would minimize the dead weight, which includes the fuel requ~­

red for eventual rocket braking outside the atmosphere, and the mass which is ablated for heat protection during the flight into the atmosphereo It is shown that when chemicalor nuclear propulsion is considered, pure atmospheric braking is always the best solution,but when electrical propulsion can be used,

an optimum compromise between partial rocket and atmospheric braking may exist, depending upon the respective qualities of the propulsion system and ablating materialo

TABLE OF CONTENTS Nomenclature I Int rodu ct i on 11 Impulse-type propulsion 1. Basic equations 2. Orbit correction 3. Optimum conditions 111 Low thrust propulsion

1. Basic equations

2. Approximate solution IV Flight into the atmosphere

1. Entry corridor

2. Ablative heat shield V Results

1. Final mass ratio

2. Impulse type propulsion 3. Low thrust propulsion VI Conclusions Ref'erences Fi gure s page 1

4

4

5 8 9 9 10 15 15 15 17 17 17 18 22 25

NOMENCLATURE

c ratio of effective exhaust velocity to circular velocity at sea level

e orbit excentricity

E(k.~) incomplete elliptic integral of the second kind F(kt~) incomplete elliptic integral of the first kind go acceleration of gravity at sea level

h non dimensional angular momentum I specific impulse

m ratio of mass at burn out to initial mass m. initial mass

~

M mass of the vehicle

Md total amount of mass dissipated during recovery U non dimensional specific energy

V velocity

VE ratio of entry velocity to local value of circular velocity

w ratio of velocity to circular velocity at sea level Y flight path angle (positive below local horizon)

r

non dimensional velocity increment

ê angle between thrust and velocity vector (positive below velocity vector)

ç heat of ablation

~ ratio of ablated mass to mass at entry ~ hyperbolic exentric anomaly

P ratio of radius .vector to planet's radius

cr ratio of distance along the trajectory to planet's radius

T acceleration of the system of propulsion

Subscripts

E

i

o

values at entry into the atmosphere

values at the point where propulsion ~s initiated values measured on the initial trajeetory

1.

Ia Introduction

In most general terms, the problem of recovering safely a space vehicle at the surface of aplanet can be viewed as the one of dissipating the total amount of kinetic energy which the vehicle has in the direct vicinity of the planet. One may indeed, consider that the small value of the velocity which is allowed for landing is weIl negligible compared to the velocity of the vehicle on the approaching trajeetory.

If the planet ~s surrounded by no atmosphere. the only solution to the problem is to use rocket braking. But if the planet is surrounded by an atmosphere, advantage is taken of i t to convert the kinetic energy into heat. The mass of fuel

which would be otherwise required is then saved, what contributes to increase the payload of the vehicle.Unfortunately,part of the dissipated heat, which is equivalent to the kinetic energy, ~s transferred to the vehicle itself by convection and radiation. Therefore, the net gain in weight which is achieved by using the atmosphere for energy conversion ~s not equal to the saving ~n fuel, since an additional penalty must be paid in the form of heat shield, to proteet the vehicle against aerodynamic heat ing.

The purpose of the present work is to investigate whether or not an intermediate technique would not offer better advan-tages as suggested by H.J. Allen

in

ref.l. Considering a given vehicle on a given trajeetory, approaching some planet sur-rounded by an atmosphere~ one can foresee that a partial rocket braking outside the atm~sphere would alleviate the heating

problem as a consequence of a lower entry velocity into the atmosphere, but at the cost of fuel expenditureo If one

20

considers all the existing possibilities. starting from one given initial trajectory~ one can generate two curves, as shown in figol t for the fuel consumption and the mass of the heat shield; i t is indeed obvious that braking down to lower velocities would require more fuel, whilst the mass of the heat shield would be accordingly reduced.

It is then Ie git imate t 0 consi de r that the ove raIl expeodà:lle deadweight iof a recoverable vehicle would be made of two ' com-ponents. fuel and heat shield respectively~ which vary in opposite direction with respect to the entry velocity into the atmosphereo Optimum conditions for recovery can consequent-ly be defined by the criterion that the total deadweight should be minimized in terms of the velocity at entryo

Numerous parameters are obviously involved in such a problem, but their number can actually be reduced to a few by a proper analysis. First of all~ the problem can be split into two partso The first one will concern the flight outside the atmosphere, where the problem then consists in transfer-ring the vehicle from its initial trajectory to another one which would hit the atmosphere at lower velocity~ but still matching entry corridor requirementso The second part would

consist of the analysis of the flight into the atmospherei to define entry corridor and best body shapes. Both solutions can be lumped together by using the convenient but arbitrary concept of the "top" of the atmosphere. which can be set at a proper altitudeo The purpose of the both analysis should

obviously be to minimize the deadweight component which th~y

cover, i.eo fuel consumption for the first part and weight of heat shield for the secondo

30

avoid an overwhelming series of machine computati ong and

therefore approximate analytical solutions have been preferred for the investi gation of the problem outside the atmosphere o

For the second part, use has been made of data whi ch are already available in the literature for entry corri dor and heat shield calculations o

The anal ysis of orbital transfer had to be divi ded ioto two parts~ depending upon the system of pro~ulsi~n which is consideredo If the case of chemical and nuclear propulsion can be approached on the classical grounds of the impulse assumption, low-thrust electrical propulsion9 howevert must

be treated by other methods o Both cases are considered in the subsequent secti onso

To somewhat simplify the analyt ical approach, the scope of the problem has been reduced to two body confi gurations with inverse square law gravitational fi elds; in addition~ i t has been considered that all trajectories woul d remain co-planar with the initial one o

IIo Impulse Type Propulsion

10 Basic Equations

In the case of two body problem. Kepleri an orbits can be described by the following equations

2U

h

=

PW cosY

which are the specific energy and angular momentum integrals respectivelyo In the above equations w is the ratio of the actual velocity to the value of the circular velocity at t he surf~ce of the planet, p Lhe ratio of the radius vèctör tb the , rad~us' Of the .planèti)and.. ythe a.ngle betwean the velocity vector and·itbe olljball"h'or:i:toû'IF-Throilghöa;t tbe -an.f,si.ä~ th:~Fa.ngle ·y i ' 'balten pbsitive'bèlcw the local horizono

The excentricity e of the orbit can be related to t he two fundamental integrals by the relationship

e2

=

1 + 2Uh2

The top of the atmosphere will be set arbitrary at a distance P

E from the center of the planet o The definition of the entry corridor wi~l consist of specifying the limiting values of the entry angle Y

E, which correspond respectively to undershoot and overshoot boundaries. for a particular value of the entry velocity wEo By virtue of Eqs (201) and (202)i

this is equivalent to deffne the values of energy U_E and angu-lar momentum hE at the top of the atmosphere o The latter

quantities are subsequently used as boundary conditions for the analysis of the flight outside the atmosphereo

20 Orbit Correction

In general® the vehicle will be on a trajectory of energy U and angular momentum h~ and must be transferred on a particular trajectory (UE~ hE)o To avoidi for practical casess a process of long duration, which would result from using a multi-impulse technique~ it has been considered here that the correction should be achieved by one impulse only~ although this might result in a larger fuel consumption (refo2)0

Transfer is achieved by producing a velocity increment ~V at an angle w with respect to the local horizon j as

indica-ted in figo2o By geometrical considerations~ one obtains the

fOllowing relationships r2+ 2rw cos{w-y ) + 2U ! ! ! prcosw

=

h - h 2 ! 2U

=

0 2

where subscripts 1 and 2 refer to initial and final trajecto-ries respectivelyo The quantity r is the non dimensional

velocity increment~ referred to the value of circular velocity at the surface of the planeto

It has been established in refo3 that the problem ~s subjected to a condition of possibility, which restricts the domain of possible variations of Po Physically~ the condition

states that the point of the first orbit at which transfer takes place must be in a domain bounded by the apogee and

can be written

<p <

For initial and final orbits entirely fixed_t the system of Eqs (204), (205) contains 3 unknowns p ~r ~w and optimiza-tion is consequently possibleo The easiest way to approach the solution is to state the problem in a slightly different waY0 and consider that the initialorbit is not entirely determinedo One must in practice regard the energy as inexorably fixed by the particular mission in space. but the value of the angular momentum can be more or less chosen arbitraryo By imposing then the particular value of the radius vector at which trans-fer is expected to take placet h l can be regarded as the

parameter of optimization o

Eliminatipg the w between Eqs (2 04) and (205). one

obt ains

r

4 _{+ 2}

_r

_{2 f(h ) + g(h ) = 0 (2 0}

₇

₎

I 1 with 2hl h2

₄

f( hl ) = _{P P} 2 (UI + U₂ ) g( hl )

4(u

_{l- U2 )2} 8(h2 -hl )2 8 _{(h 2 -h}l) (UI h₂ -U₂ hl ) = + _p3 + p2 (2.8)

For a given value of p, the condition of optimum can be derived from Eq(207) as

or, deriving Eqs (2 07) and (208) with respect to h

1 and

substituting ~n Eq (209)

together with the condition

- 1 1+PU 2

1 + pU 1 )

to secure a rea1 va1ue of

r

0

(2 010)

(2011)

E1iminating r bet ween Eqs (2 .7) and (2.10) yie1ds the

optimum va1ue of hl" One obtains the simp1e re1ationship

w 2

-(2.)

w 2 or using Eg (202) (2013)

which indicates that optimum conditions are obtained by

tangentia1 impu1seso One can easi1y verify that Eq (2 012)

automatica11y satisfies condition (2011)0 The va1ue of the

v~locity increment

r

is then easi1y eva1uated and can be

wri tten as

r

= (2.14)

where upper s~gns correspond to decreasing energ~es, lower signs

80

30 Optimum conditions

Having obtained optimum conditions for a particular

value of the radius vector at the point of tr~nsfer~ the latter

parameter can now be variedo From Eq (2015) one obtains easily

which is always positive o Consequently lowest possible fuel

consumption will be obtained by tangenti al transfer at the

perigee of the initial orbit o In the present problem~ since

the initialorbit is expected to cut the top of the atmosphere ~

the smallest possible value of p is actually PEo Since braking

only will be considered. the optimum val ue of

r

will be given

by

(2016)

The corresponding angular momentum requi red for the initial

orbit will be given by

The ratio m of the mass at burn out to the initial mass i s then

given by the simple relati onship

r

vs

m

=

exp {-

g I

}

9

0

IIIo Low Thrust Propulsion

1 0 Basic equations

For low thrust propulsion, a simplified solution has been sought i) assuming that the value of the thrust would be a con-stant~ set at a constant angle to the velocity vectoro With the

notations of fig 3i the non dimensional equations of motion can

be written as m dw2 2 do ~ ₌ do dm

"dä

=

=

-m sl.ny + T cos Ó p2 sin y T c

where m l.S the ratio of the actual value of the mass to the

initial one, T the ratio of thrust to initial wei ght measured

at sea level~ c the ratio of effective exhaust velocity to circular velocity at sea leveli 0 the ratio of distance along the flight path to planet9 _{s radiuso}

The above system of equations can be transformed by introducing the definitions (201), (202)~ (203) of energYil angular momentum and excentricityo One obtains easily

dU

100 dh dcr de 2

-

dcr

=

20 Approximate solution .!.e. cos ( Y+ Ó ) mw 4 Th = -mw [ h 2 .,... (pw -1) cos Ó -pw

u

PS in Ysin

ó1

(307)

Since U~ hand e are invariants ~or _{a kleperian orbit e}

and since T ~ ~or electrical propulsion₉ can be regarded as a

small quantity, the above system o~ equations immediately

suggests the use o~ a perturbation method to obtain an

appro-ximate soluti on ~or the problem 0 Inte gration requires t 0

distinguish between the di~~erent types of orbitso For the present purposese the analysis can be restricted to the case

of hyperbolic orbitso The detailed analysis has been developed

in ref04 and leads to the following resultso Introducing the

hyperbolic excentric anomaly E; z by the relationship

e cosh E;, = 1 + 2U p

the following differential equation can be derived~ for quantities measured on the unpertubed orbit, denoted by

the subscript 0 ~

_ 1/2

+ 2U o [e 2cosh20 E;, 0 - iJ ':J

=

Ilo

in the system of equations (3.5) to (307) and eliminating (J

with the aid of Eq (3 09) results into a system of equations which can be rather easily integratedo The result of the

inte-gration obtained in ref.4 can be written as follows :

) + 't

c{l-m =

---u

= U - 'tcoso [HU;) - H{E;.)]

0+ 2U l.

o

In the above equations, the subscript 0 has been

d~opped for E;, which s t i l l represents the excentric anomaly measured on the initialorbit; E;. is the p~rticular value at

l.

the point where propulsion is startedo

The auxiliary functions are defined by the following relation-ships ~

f{ E;) = _. E; ,- e ₀ sinh~

E; 1/2

H ( E;) =

f

[e 2 coshn

-IJ

dn =

0 0 . 1/2 - e [ ( 1_-1-) F (k,

~)

o 2 -E (k, ~) - tanhE;{e 2cosh2E;-1 e 0

]

e 0 0 0 (3015) E; _1/2 1 E;

1 ( E;) =

_J

[eo 2cosh 2 n

-IJ

dn=

-

-

[F~k, ~)] (3.16) e

0 0 0

120

3 1 e cosh~ 1/2

=

2COSh- (eOcosh~)-2{1- 0

4

) (eo2cosh2~-1)

(3018)

In the double signst the upper signs corresponds to the case

of a vehic1e moving initially towards perigee~ lower signs to

an initia1 motion in the opposite directiono

Argument and modulus of the elliptic integrals are

respectively given by -1 1 ~

=

sinh

_(cOSh~)

(3019) k

₌

1 (3020) e 0

For given intial and final orbits® the system of

equations (3011) to (3013) contains four unknowns miö~~ and ~.o

l.

One of these parameters is consequently free and can be .used

for optimization of the fuel consumption~ given by the quantity

( l-m) 0

It has been shown in ref05 that optimization of fuel

consumption was obtained for tangential thrust~ ioeo~

=

00 _for

increasing energies and Ö

=

1800 _{for decreasing energieso}

Knowing that. optimization can be pushed one step

further~ as done above for impu1se-type propulsion0 by considering that the value of the

angular momentum of' the ini tial orbi t may to some extent. be left arbitrary, whilst the value of the energy is fixede for one particular missiono Putting Icosól

=

1 in Eqs (3.11) to

(3.13). there still remain four unknowns. namely mi h.~ and

o

~. 0 The initial angular momentum h can now be regarded as

1 0

a free parameter for the next step in optimizationo

The discussion of this particular problem (ref.5)

shows that in the case of a vehicle moving initially towards perigee, the value of' h should be selected in such a way that

o

10w thrust propulsion would terminate as close as possible to perigee; in the case of initial motion in opposite direction. h should be such that propulsion be initiated at perigeeo

o

For the present purposes, the only case of' interest is the motion of a vehicle towards perigee, and propulsion is

strictly used with the purpose of decreasing the energyo Consequently, upper signs should be used into the equations. and the first part of' the optimization shows that one should have

cosó

=

-1

or retrothrust o The second part of the optimization indicates

that propulsion should terminate at perigeeo Since the excentric anomaly ~ is a parameter which is measured on the initial

orbit, the latter condition can only be f'ulfilled approximately by defining the terminal point as the one for which ~

=

00

In view of this, the system of equations (3011) to (3013) can be written as

T (e sinh~ .-~.) (2U )3/2 0 ~ ~ o 2 1/ 2 e -1 hE = [-

~U]

+ o T - - - - r /- g( C 0e ) (2U ) 5 2 ~ 0 o

which must be solved numerica11y for (l-m), e end~. as unknowns o

150

IVo Flight into the Atmosphere

10 Entry corridor

To define the boundaries of the entry corridor, it has

been considered for the present applications~ that a lifting

vehicle would be used, with a lift to drag ratio of 05.

posi-tive at undershoot boundary, and negative at overshoot

bounda-ryo The undershoot boundary corresponds to a maximum

decelera-tion of 10 g's. Moreover. the top of the atmosphere has been

arbitrarily set at a non dimensional radius of ~E

=

1 0020

The data for these boundaries have been calculated

trom re~.6, and are represented on fig o4 0 The values ·of energy

and angular momentum U

E and hE required at the top of the

atmosphere have been evaluated for the mean value of Y

E between

the two corridor boundarieso Consequently, all parameters for entry conditions become a function of the entry velocity V

E onlyo

2 0 Ablative heat shield

For heat shield calculations_t use has been made of data

which are already available in the literatureo Since atmospheric

entry has been considered to take place at very high speeds _t

radiative heating must play an important role o Since small nose

radii appear to be then a better solution to the problem,

conical shapes can be regarded as realistico

Values of the ratio of the mass dissipated byablation.

to the initial mass, are already available in the literature

for such conical shapes, and are given in terms of the entry velocity, for different types of ablating materialso Data for

teflon and quartz can be found in refo7~ those relative to graphite have been given in refo8

VoResults

10

Final mass ratio

The symbol m used in the previous sections represents

the ratio of the mass af ter propulsion~ Mb ~ to the initial

00

mass m.o Denoting by ~ the ratio of ablated mass to the

~

initia~ mass at entry~ ioeo Mboo~ and by Md the total amount

of mass which has been dissipated~ ioeo fuel consumed plus mass

ablated, one easily est ablishes the rel ationship

m_~

-the ratio m, originated by propulsion~ iSt af ter optimization~

a function of the energy of the initialorbit U _o ~ of the

specific impulse I of the system of propulsion~ of the level

of acceleration T delivered by t he system, and of the selected

entry velocity in the atmosphere VEo The ratio ~ is a function

of the entry velocitYt and of the characteristics of the

abla-tive material which can be represented by the heat of ablation

ç. Therefore~ Eqo

(

501)

can be more explicity written as

which shows that finallYt f ive parameters are left in the

problem, the influence of which must be explored numericallyo

2. Impulse type propulsion

T cannot be considered in the problemo The value~ of mare

evaluated from Eqs (2018) and (20l~); values of l:I are taken

from refs 7 and 80 Ca1culations have been carried out for three particular va1ues of the energy of the initialorbit,

U _o = 106, 2, 204, for ablating materials as teflon, quartz

and graphite. and for two particular values of the specific

impulse I

= 3501>700 secs. which are representative of chemica1

and nuc1ear propulsion respectivelyo

The values of the ratio of dissipated mass to initial

mass are given in figc5 to 7 c

Results show clearly that the amount of mass dissipated

is kept to a minimum if no rocket braking is applied and

consists on1y of the amount of mass ablatedo Moreover, it appears that there is very litt1e difference between the curves for different ablating materialso In fact, specific impulses of that order are too low, and consequently fuel expenditure is too 1a.Ige to compete with atmospheric brakingo

It indi cate s ' that the est imate d curve for fue1 consumpt ion in figa1 always has a 1arger order of magnitude that the one for ab1ationc

30 Low thrust propulsion

The calculations ~ave been carried out for identica1

conditions, in the case of low thrust propulsiono The ratio m must now be computed with the aid of the system of equations

(305) to (307) 0 Same values as before have been kept for the

energy of the initialorbit; the specific impulse has now been set successively at 3,000, 5,000, 7,000 and 10,000 secs to be representative of various types of electrical propulsion systemsi

-

-190

the level of acceleration has been allowed to vary between

-2

-4

10 and 10 0

The first salient feature is that for the conditions

which have been explored, the mass ratio m turns out to be

practically independent of the acceleration T 0 Only the last

decimal places are affected by a change in T e and the

respec-tive curves for m cannot be differenciated on a graph at

reasonable scaleo However, low thrust propulsion must be

initiated at a larger distance from the earth if the

accelera-tion is smaller, which is physically obvi ouso The values of

the ratio of dissi pated mass to initial mass are represented

on figs08 to 10, which are then independent of the acceleration

of the propulsion systemo

Large di fferences now exist between the different

ablating materiaIs, which indicates that for the higher

speci-fic impulses considered, fuel consumption becomes of the

same order of magnitude than ablatiooo

Considering first the case of teflon~ one sees that

partial rocket braking outside the atmosphere always leads

to a better solution than pure atmospheric braking·o' Even for

a specific impulse of 3,000 secs onlYt a minimum appears

clearly on the curves,for a lower entry velocity than the one

one would have without rocket brakingo The trend is more and more definLte for larger values of the specific impulseo

An ablating material like quartz exhibits different

propertieso Results indicate that rocket braking is not

advantageous except for large specific impulses and large

200

Under such circumstances only~ moderate rocket braking could

ba efficently used o

For graphite on the other hand~ the results clearly indicate that rocket braking should not be considered~ pure atmospheric braking always, offering the best solutiono

It should be pointed out that the values of pare actually underestimated0 since it has been assumed in refs 7 and 8 that

the body remained conical all the timeo A change in body shape

resulting from ablation of the nose would result in larger

values of Pt and make the final curves look more like those

for teflon 0

If one considers those cases where rocket braking is

applicable 0 one should remember that the values of the angular

momentum of the initialorbit are imposed by the optimization process o Knowing then both energy and angular momentum of the

best initial orbitt one may calculate what its perigee radius

should bet which is now obviously a function of the

accelera-tion T_o The non dimensional values of the perigee of the

ini-tial orbit have been evaluated for values of the initial energy equal to 106, 108 and 2, and are represented in terms of the entry velocity on figso 11 to 130

The conclusions one may draw are that perigee radius of the initialorbit must be larger if one intends to hit the

atmosphere at a lower velocitYII and for a given velocity at entrYt should ~e larger if the acceleration delivered by the system of propulsion is lower o Moreovert for a fixed value of

the entry vel.ocity. perigee radius must be sma,llerif the initial energy is largero In any case, perigee radii are vell outside velo city is requiredo Consequently. one salient feature of

210

low thrust rocket braking outside the atmosphere is that the return trajectory towards the earth should be selected so as to deliberately miss the atmosphere. to allow for brakingo This in fact changes the boundary conditions which would normally be imposed on the analysis of transfer from one p!'anet to earth 0

220

VloConclusi ons

To opti mize the recovery of an interplanetary vehicle~

the dead weight should in general be regarded as made up of

two components~ the fuel which would be required for eventual

rocket braking~ and the amount of mass which is ablated during

the flight in the atmosphereo Since both components vary in

opposite di rection with the entry velocity~ the overal l dead weight can be expected to have a minimum in terms of the ent ry velocity int o the atmosphereo One may split t he problem in two

parts~ one related to calcul ation the trajectories outside t he

atmosphere. the other one devoted to the study of the fli ght

within t he atmosphere o Both parti cular analyses should be

conducted so as t o define conditions whi ch minimi ze the amount of mass whi ch has to be di ssipated in each phase o

For the present purposesi approximate analytical

techniques had to be devel oped for low thrust propu~sion~ in

order to avoid numerous numerical calculati onsoThose solutions

are suited for optimization of the fuel consumpti ono Data on

ablation shields have been taken from t he l iterature for

conical body shapes whi ch represent arealistic solution for

entry at very large velocities, weIl in excess of earth

parabolic velocityo

It can be concluded from the present analysis that

pure atmospheric braking should always be favoured, for entry velocities up to 20 km/ sec" if chemicalor nuclear propulsion

systems are consideredo Earth bound and l unar missions can

reasonably be included in that categoryo

230

optimum compromise may exist between rocket and atmospheric

brakingo Optimum conditions are strongly dependent of t he

properties of the ablating material~ and independent~ as far

as the mass ratio is concerned~ of the level of acceleration whi ch can be delivered by the propulsion systemo

For ablative materials having the properties of te,flon. i t appe ar's that for ent ry veloei ty up to 20 kmi se c iI part i al

rocket braking should always be. usedil even for speci fic impulses

as low as 3i OOO secs o For materi als like graphite~ pur~

atmospheric braki ng is always the best solut ion o Materi als

like quartz appear to be intermediate casest whereby the

optimum solution tends to t he one for teflon~ when the specific

impulse ~s hi_{gh il to} the one for graphite when the specific impulse is lowo

In such cases where rocket braking would be retained

as an optimum solutionil attention must be paid to the fact

that the initial trajectory of the vehicle must be selected

so as to deliberately miss the earth9_{s atmosphere o ConsequentlYil}

the choice of a return trajectory does not depend only on the

entry corridor requirements i but also on the possible

compro-mise between rock et and atmospheric brakingo Since low thrust

propulsion is very likely to be used for fast interplanetary travelil a return trip can be viewed as a low thrust powered

flight. ranging from a gi ven planet down to the vicinity of

the earth₉ where thrust reversal would be achieved before

atmospheric entryo

From all this~ i t can be concluded that the decision of use or no use of partial rocket braking outside the

materials or high speciric heat! such as graphiteo Ir such materials become of current use. pure atmospheric braking turnsout to always be the best solutiono If not, i t appears that partial rocket braking might be round desirablei since i t would orrer the further advantage or bringing the problem or atmoshperic entry·at very high speeds closer to current technologyo

250

REFERENCES

10 Allen HoJo Hypersonic Aerodynamic Problems of the Future -Paper presented at the AGARD meeting Brusselsi

Belgium. April 3-6, 19620

2. Barrar RoB. Two Impulse Transfer vs One - Impulse transferg Analytic Theory - AIAA Journal Vol 1 n° 1

January 1963 pp 65-680

30 Moulin L. Recovery of Satellites and Space Vehicles Training Center for Experimental Aerodynamics Technical Note n° 9. 1962

40 Moulin Lo

5. Moulin L.

Approximate analytical solution for low thrust propulsion in space - AF EOAR Grant 63-39

Scientific Report n° 1 - December 1963t also

VKI TN 16i May 1964 0

Approximate analysis of low thrust transfer between coplanar hyperbolic orbits

AF EOAR Grant 63-39 Scientific Report n° 2. May 1964. also VKI TN 17 May 19640

60 Chapman DoRo~ Kapphahn AoKo Table of Z functions for atmo-sphere entry analyses - NASA TR R-I06~ 19610

70 Allen HoJo. Seiff A •• Winowich Wo Aerodynamic heating of conical entry vehicles at speeds in excess of earth parabolic speed - NASA TR R-185, Deco1963

8. Allen HoJo Gas Dynamics problems of space vehicles o Proceedings of the NASA-Univo conference on the Scienee & Techn of Space Exploration - NASA SP-ll Vol 2 pp 251-2670

Mass ...."

-Fig. 1 / heat shield ..", / Entry Vetocity

Schematic Variation of deadweight Components

-local horizon

w

Fig. 2

Fig. 3

Parameters of low thrust propulsion

-~

j....--t

~

~

V

V

-"L=

100 tb

/ft

2

CDA

IL/DI

=

0.5

1.4

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Fig. 5 - I mpulse- type propulsion - Ratio of total dissipated

mass to initial mass

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V.K.I. TN 17 AF EOAR GRANT

63-39

SR-3

von Karman Inst1tute tor F1uid Dynamios,

Jun.

1964.

A POSSIBLE COMPROMISE BETWEEN ROCKET AND

ATMOSPHERIC BRAKING, by L. MOULIN.

An attempt has been made to optimize the prob-lem ot recovering an interplanetary vehic1e

through the earth's atmosphere. Optimum condi-tions are detined as the ones whioh wou1d min-imize the dead weight, whioh inoàudes the tuel required for eventual rooket braking outside

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e~~ JO sel~llvnb eAI~oedsa~ e~~ uodn ~uIPued

-ep '~slxe ~vm ~uI~v~q ol~e~dsom~v puv ~e~oo~

1vI~~vd uee.~eq eSlmo~dmoo mnml~do uv ' pesn

eq uva uOistndo~d tvoi~~Oet8 ue~. ~nq 'UOi~

-ntos ~seq e~~ s~v~tv si ~ui~v~q ol~e~dsom~v

e~nd 'pe~e.lsuoo SI uOl91ndo~d ~ve1onu ~o 1VO

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pe~v1qv BI qOI~. 9BV. 8~~ puv 'e~e~d90m~v eq~

the atmosphere, and the mass whioh is ablat.d for heat proteotion dur1ng the flight, into the atmosphere. It 1s shown that when ohe*i-cal or nuolear p~opuls1on is oonsidered, pure

atmospherio braking is always the best solu-tion, but when eleotrioal propulsion can be used, an opt1mum oompromise between partial rooket and atmospherio braking may exist, de-pending upon the respeotive qua11ties of the propulsion system and ablating material.

V.K.I. TN

17

AF EOAR GRANT

63-39

SR-3

von Karman Institut. tor Fluid Dynamios,

Jun.

1964.

A POSSIBLE COMPROMISE BETWEEN ROCKET AND

ATMOSPHERIC BRAKING, by L. MOULIN.

An attempt has been made to optimize the

prob-lem ot reoov.ring an interplanetary vehiole

through the earth's atmosphere. Optimum

condi-tions are detined as the ones whioh would

min-imize the dead weight, which inoludes the tuel

required tor eventual rooket braking outside

the atmosphere, and the maas .hioh ia ablated

tor heat protection during the tlight, into

the atmosphere. It is shown that when ohemi-oal or nuclear propulsion is oonsidered, pure

atmospherio braking is always the best

solu-tion, but when eleotrical propulsion oan be

used, an optimum compromise between partial rooket and atmospheric braking may exist,

de-pending upon tàe respective qualities ot the propulsion system and ablating material.

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8tOlq8A ~~v~8uvtd~a~ul uv ~ul~eAooe~ JO met

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