On a relatively cool transition from a satellite orbit to an equilibrium glide

uvt rwe 10· DElFT ON A RELATIVELY COOL TRANSITION FROM A SATELLITE

ORBIT TO AN EQUILIBRIUM GLIDE by

B. Etkin

JUNE 1961 UTIA Report No. 75

ORBIT TO AN EQUILIBRIUM GLIDE

by

B. Etkin

JUNE 1961 UTIA Report No. 75

.

-'

.

The research reported herein was sponsored by the United States Air Force under Contract AF 49(638)-761 monitored by the Air Force Office of Scientific Research of the Air Research and Developrnent Commando

Miss Jean C. Mackworth rendered valued assistance in carrying out many hand computations. Machine cornputations were carried out on an IBM 704 by Mr. G. Galipeau.

One family of solutions is presented for a mom.tonic_ transi-tion from a circular orbit to an equilibrium glide. It is performed at high lift coefficient and low LID, and terminates at a match point where the prescribed conditions of the glide are met. The vehicle then continues on the glide at. a higher value of LID. The high drag required during transition entails the use of a large auxiliary light-weight drag body, which is jettisoned upon arrival at the match point. The presence of this drag body results in a reduction of the total heat load on the vehicle itself. The exact amount of this reduction depends on the details of the design, but it can be typically of the order of 3/4 in an entry with peak acceleration of the order of 7 g. As a re sult, the maximum average vehicle temperature can be made

relatively low, with consequent advantages in the structural design, and in the thermal protection of the payload.

..

I Il. lIl. IV.

V.

VI. VII. T ABLE OF CONTENTS SYMBOLS INTRODUCTION

APPROXIMATE SOLUTION FOR THE TRANSITION PATH

2. 1 Flight Path Angle

2.2 Minimum Height

2.3 Speed

2. 4 Match Point

2. 5 Resultant Load Factor

2. 6 Horizontal Range 2. 7 Time of Flight

EXACT SOLUTIONS FOR THE TRANSITION PATH

EFFECT OF THE PARAMETER (CLS/m)

THE MATCHING EQUILIBRIUM GLIDE

REDUCTION OF THE TOTAL HEAT LOAD ON THE VEHICLE

EXAMPLE OF THE MOTION AND HEATING

7.1 Motion 7.2 Heating Page ii 1 2 2 3 3 3 5 5 7 8 10 12 15 16 16 18

VIII. CONCL USION 20

REFERENCES 21

APPENDIX A - Correction of the Numerical Results 22

c • CL CD D Dv E F f g h J L m mI n n* q Q r R S s T t specific heat lift coefficient drag coefficient SYMBOLS

drag of whole system drag of vehicle only total energy of vehicle

resultant external force acting on system

contribution of skin friction to total vehicle drag local value of gravity

enthalpy

Joulels equivalent lift

mass of system mass of heat shield load factor, F!W

effective weight ratio of passenger

, local heat-transfer rate total heat transferred

radius from earth ,-centre to vehicle stagnation-point radius

reference area for CL and,CD distance along flight path temperature, (OR)

v

x y y vehic1e speed entry speed

circular satellite speed local weight of systern

distance parallel to earth's surface see Eq. 2.21

altitude see Eq. 4. 1 entry altitude

constant in density equation flight path angle

entryangle

value of

'6'

at reference altitude (400, 000 ft. ) gravitational constant

angular coordinate (Fig. 2. 1) emissivity

Stefan-Boltzrnann constant (4. 76 x 10- 13 BTU ft. -2 sec. -1

oR -4)

density of the air

constant in density equation

denotes horizontal flight, i. e.

'6

= 0 denotes the match point

I. INTRODUCTION

This report deals in considerable detail with the motion, and to alesser extent with the heating, of a family of atmospheric entry tra-jectories (Fig. 1. 1) which are suitable for effecting the transition from a close-in circular orbit to an equilibrium glide. ~ Only manned lifting vehicles are considered, so the entry angles are kept within the range which produce tolerabie decelerations (

"'6'

E ~ •

1'(' ).

The entry is pre-sumed to be initiated by the application of an impulse to the orbiting vehicle such that the initial conditions are as shown on Fig. 1. 1. - The unique feature of these entry manoeuvres is that at a certain match point

'MI the flight variables and altitude of the transition portion are also those of a shallow equilibrium glide for which the LID is higher thanin the transi-tion. The flight continues along this matching glide path by virtue of

jetti-soning a drag device, or otherwise reducing the drag.

By flying at a high value of CL, using low wing loading, and by dispensing with a horizontal coasting phase, the transition paths achieved are relatively short, and hence tend to minimize navigation errors

associated with errors in initial condition or variations in atmospheric conditions.

It turns out that the LID ratios required to produce such a match point are quite low. (Higher LID ratios lead to a skipping motion). When maximum or near-maximum lift is used, the LID of a winged con-figuration would be too high to meet the specified condition, so the requisite value is obtained by employing an auxiliary drag device, e. g. a light weight towed body. The presence of the second high-drag body provides the princi-pal attractive feature of this entry technique, for it shifts a large part of the total heat load (as much as 800/0) away from the vehicle to the drag body. Thus, from the point of view of average vehicle temperature, a. relatively cool entry can be achieved (in the example calculated, the maximum ave rage vehicle temperature is about 3300F). The peak loc al temperatures, depending on design, mayor may not show large reductions over other techniques.

The heat capacity and conductivity of the skin and structure in the stagna-tion regions would be primary factors in this connection. High values of these two parameters could lead to significant reductions in peak tempera-ture. In any event, the reduction achieved in the total heat load would ease the design problems of the vehicle. Whether the net result is a gain, af ter considering the requirements of the drag body, can only be determined from

The same entry technique -can of course be used with super-circular entry speeds, and with the same general advantages. The principal differences to be expected are larger decelerations and larger heat loads at a given entry angle. -Hence manned entry would have to take place at smaller angles than for circular speed.

detailed design studies. That question is beyond the scope of this paper. However, it should be noted that the technical feasibility of such drag bodies is not in doubt, as indicated by reports of industrial developments along these lines.

*

Il. APPROXIMATE SOLUTION FOR THE TRANSIT ION PATH

With the definitions indicated in Fig. 2.1, and assuming a non-rotating earth, the equations of motion which determine the flight path are CL

~ ~\/?.S

+m cast

(t?" -

'J)+

ffiVl.

H. -:

0 d} _

-.ö.i

Y\

t'

dA-(2. 1) (2. 2) (2. 3)

A well-known approxirnation which is suitable for shallow lifting entries, (

«

< < \),

at circular satellite speed is obtained by neglecting the gravity term in Eq. 2. 1, and neglecting

("1. -

c;} ') in Eq. 2.2. The approximate equations are then

r

(2.4)

(2.5)

(2. 6)

2. 1 Flight Path Angle

The usual exponential density variation is assumed,

( ~ ': fc

e-

~'*

),

and Eq. 2.6 is used to eliminate s. Then Eq.

2.5 yields

*

See for example: "Drag Devices Studied for Recovery Units", by M. Yaffee, Aviation Week, Aug. 15, 1960, p. 92; and "Missile Drag Balloon", Anon., Space Flight, May 1961, p. 112.

which has the solution

Since ~ E may be regarded as very large ( ~ '}E

>

15), the last term is negligible, and

Thus to this degree of approximation, '( is seen to be independent of the drag and the speed, being a function only of height and of the lift para-meter (eLS/rn).

2. 2 Minimum Height

The minimum height is obtained by setting "(

=

0 in Eq. 2.7, with the re sult

(2.8)

2.3 Speed

The equation for the speed is obtained by combining Eqs. 2.4 and 2. 5, viz. C[) D _\

_dV

1 _\

_t!V

~

-

-::. 'Ly'2..

dY

=

cl'!

CL L V whence t)

_('6'

-"6'

~

')

-::

L

(V/v~)

L

or I <.2.~)

v/Vr.

=

e1P. [

~

('e-'6'E'>J

The speed at the point where the flight path is horizontal, ('tÇ = 0) is denoted by Vl' Hence

(2. 10)

In order that the conditions at the point of horizontal flight shall be approximately those of an equilibrium glide. the condition for equilibrium is applied, i. e. d~ /ds

=

O.

The point at which this condition is met is called the 'match point'.

*

whence

From Eq. 2.2, with

1;

=

0 and d 1} /ds

=

0, we get

2-eLI

~

\

Vil

S

1" 'm 'f

~

I :.

~ ~

I

CL

S

=

m

Since the values of r, g, and rg do not vary by more than 10/0 during the significant. part of the transition manoeuvre, they can be taken as constants, and in particular rg

=

V E 2 . Thus we get

(2.11)

From Eq. 2. 8, we get

-r~,

(m)

",,2-~ ",,2-~

e

- -

P. oE.

o - C ~ \' L

af ter some rearrangement

~

whence Eq. 2.11 becomes,

(2. 12) which shows that the speed at the bottom of the transition path depends on

~E only.

Equation 2. 10 now serves to determine the value of D /L which

,is required in order to produce the speed given by Eq. 2. 12. Equating the two values of the speed gives

(2. 13) Thus Eqs. 2.8, 2.12, 2. 13 fix the values of Y1, V1/VE, and D/L for given values of '(' E and (eLS/m). These equations have been plotted on Figs. 2.2 and 2.3, for the following values

**

of the constants:

*

The trajectory has a point of inflection at the match point, when plotted in the xy coordinate system

**

Values corresponding to 280, 000 ft. altitude, and fitting the 1956 ARDe atmosphere there. Appendix A gives a method of correcting results given

r

₌

21. 2 x 106 ft.

~

=

4. 0 x 10- 5 per ft.

fo

=

. 00123 slugs / cu.. ft. 31. 3 fps 2

g

₌

2. 5 -Resultant Load Factor

The resultant of the aerodynamic and gravitatiónal forces acting on the system is (see Fig. 2. 1)

~

1. -=. (\... _ \IJ

~'S

"6' ') '2... -t- ( \) -

W \,'.

V\

,{,t

I

or

(2.14)

The condition for a maximum value of n is found by differentiating Eq. 2. 14.

Denoting the value of '6' at nmax by

0 ,

we find

l

t>

'2.1

?.~ (~-'te)

\ Á t 2.

~ ~

Y'~

\

-+ (

L)

e

-

'Z. _ 'L - L 't'L - L : :

0

.1 ~E - ~ -l' t) E - 1S'

This equation has been solved numerically for

't

using the values of D /L given by Eq. 2. _{13, and the corresponding values of nmax}

are plotted on Fig. 2.4.

2. 6 Horizontal Range

For small

't ,

the arc length ds is approximately the distance dx measured paral~el to the earth's surface. Thus, from Eq. 2. 6,

d~ :_~

cix

U sing the solution for '( given by Eq. 2. 7, this yields

J'<}

cl

':t..

= -

[i

_{'6'E -} C _~ S

~ -~~

11/2.

0 e

If we choose the origin for x at the point where

't

=

0, then

x:

~

-

2:.

+~-I(~)

~ IS'E '6'e

(2. 15)

We shall calculate the horizontal distance travelled af ter crossing the 400, 000 ft. altitude. The value of '6 at 400, 000 ft, with

~

=

4 x 10 - 5 is I

~o

=

L

~:

-

(C~) ~~

e-1b

J

"i or _\

~o

_ \ \ _

~0) ~L

e-H'JI

~

- L

·

YV>

~~E

For the values of eLS/m,

fa /

~

,

and

'6' E which occur in the following

calculations, the second term in the square bracket is very small, and hence

'lr..

\

(C ..

S \

~() e-\~

~E

= \ -

ï \. -;; )

f->

cr

eL. (2. 16)

. This value is substituted for '6' /'te in Eq. 2. 15, and the expansion of tanh- 1 9 in the neighborhood of 9

=

1 is used with the result

(2. 17)

whence the distance travelled parallel to the earth from 400, 000 ft. to the lowest point of the path is, for the given values of ~ .. and ~

\x

1\

=

~'ti'E ll~.

9"

-L

C~

.Jr 2.

L

'6'e

1

(2. 18) The values of xl given by this equation are plotted on Fig. 2. 5.

From Eqs. 2. 8 and 2. 17 we find th at

:(I~ ~

(1 _

1?3~)

OE '

(:>

(2. 19) With the value of ~ adopted herein, this becomes

X :

. l

(~

_

4 .3S x

\O~

)

I

'6'e:

'

(2. 20) This value may be compared with that which would result from a descent at constant angle

'f

E' i. e. ,

yielding the difference

b..X,::

x:.;'*-

'i. , :

·35

x.

\0

s

This value, which is seen to be independent of CLS/m is plotted on Fig. 2.6.

2. 7 Time of Flight

The approximate time to perform the transition from the initial height of 400, 000 ft. down to the minimum height Y1 can be calcu-lated from the approximate solutions for V and --(

From Eq. 2.9,

and fr om Eq. 2. 3

However, Y can be expressed in terms of

't

yields af ter some reduction

2.

'f

d'f

d'}

~ f..(~i-"G'I.)

whence

I.

td'r

<M). ~ - ~ (~E7..-t'L) and Eq. 2. 22 becomes

t ':

I

o

(2.22)

by means of Eq. 2. 7, which

(2.23)

(2. 24)

where 'fo is the flight-path angle at y . = 400, 000 ft., given by Eq. 2.16. By expanding the integrand in partial fractions, the integral of Eq. 2.24 can be transformed to

1

=

e::~' rn')ë~~

cl,,-

ë1.::'

T~')e;" d~

~& '(Ie

where Ei(x) and Ei(-x) are the exponential integrals, tabulated in Ref. 1. Finally, the expression for the time becomes

(2.26)

With D /L given as a function of ~ E by Eq. 2. 13, Eq. 2. 26 yields the transition time of entries at circular speed as a function of

'te.

and r~

The angle

'1'0

depends on CLS/m (eq. 2. 16) so the time t1 does also. The values corresponding to CLS/m

=

1. 0 are plotted on Fig. 2.7.

lIl. EXAC T SOL UT IONS FOR THE TRANSITION PA TH

The entry flight paths considered herein all have small values of ~ , (~<. O. 1) for which the approximations sin't

=

~

,

cos ~

=

1 are quite reasonable. However, the values of the speed loss predicted by the approximate theory are seen to be as high as 50% (see Fig. 2. 3).

Hence the approximation V2

=

rg, although good at the moment of reentry, will evidently be quite po or during the later portions of the flight paths. Furthermore, with peak load factors as low as those shown on Fig. 2.4, the neglect of the gravity term in Eq. 2. 1 is probably also questionable. A new set of equations was therefore considered, incorporating only the assumption

'«<

1,

i

.

e.

c~

î

ev1.S -

'rnd-''ls''

+

i

'M

~1.

: 0

Ct.

1.

~

V"S

+

~

('0L._'}

)~Th \j"2..dd~

=

0

~ :-"t

cL6 ..

~

::

~ e-~~

o (3. 1) (3. 2) (3. 3) (3.4)

These equations were solved by a step-by-step numerical procedure on a high-speed digital computer, for a large number of cases. The choice of values of D /L for each ~ E was guided by the approximate solution shown on Fig. 2. 3 and a trial-and-error method was used to find thevalues which led to

'6'

M ~ O. Most of the calculations were carried out for eLS/m

=

1.

However, the effect of this parameter on the results was investigated by doing a number of cases at other values, and is discussed in Sec. 4.

The value of "( corresponding to the match-point (d'f Ids

=

0)

was found in each case from the num erical solution, which. was stopped at this point. These values are plotted on Fig. 3. 1. The curves, in addition to providing the exact values of DIL which correspond to the conditions of the approximate solution, i. e. 'f_M

=

0, also provide the values of DIL

required to match equilibrium glide paths for which 't

M ') 0 (see Sec. 5). The values of

't

I't\ are seen to be only moderately sensitive to DIL. For example, an increase of DIL of 10% at '( E

=

.06 rad. increases

'( E by about 0.30. However a change of this amount at high altitude repre-sents a quite significant change in the equilibrium glide (see Sec. 5). The intercepts of the curves at

'f

_M

=

0 provide a direct comparison with the approximate solution. The re sult of this comparison, i. e. the values of

DIL for

f

M = 0 is shown in Fig. 2.3. The difference in DIL between the

exact and approximate solutions is seen to be large, the exact values being much smaller over the whole range of Y; E.

The altitude of the match point)

'1fv\)

is shown on Fig. 3.2 as a function of 'tE and DIL. The limiting line for which '(1"\ = 0 is also shown. This provides the values y

=

Y1 for comparison with the approximate solu-tion. This comparison is given on Fig. 3.3, and the difference is seen to be moderate, of the order of 8000 ft. Thus the approximate solution on Fig. 2. 2 would seem to be quite good. Figure 3.2 also shows that except for the

shallowest entries, the match point altitude is almost invariant with DIL.

The horizontal range covered between entry at 400, 000 ft. altitude and the match point is shown on Fig. 3.4. It is very sensitive to the entry angle

'6'

E, but less sensitive to DIL, although more so than

't

_{M .} The limiting values of '!t-I\ are replotted on Fig. 3.5 for comparison with the approximate solution, which is seen to underestimate the horizontal range by amounts which vary from 40 N. M. at steep entry to 300 N. M. at very shallow entries.

The time taken to reach the match point is given on Fig. 3.6.

It is seen to be quite short for steep entries, about 2 mins., increasing to only about 6 m ins. for shallow entries . The exact values for '( M

=

0 are compared with the approximate ones on Fig. 2. 7. The latter are seen to be too small by times of the order of a half minute at medium entry angles.

Figure 3. 7 shows the results obtained for the speed at the

match point. Again the results are not highly sensitive to DIL for the steeper entries, ;rE >.05. The comparison with the approximate solution is shown on Fig. 2.2. The latter is seen to be much better for speed than for DIL.

The speed loss achieved at the end of the transition manoeuvre is seen to be quite large, 50% or more for the steeper entry angles. Figure 3.9 shows how this speed loss increases with increasing peak load factor.

Values of the peak load factor occuring during the reentry transition are plotted on Fig. 3. 8. Both D /L and 'fE are important parameters. By virtue of the definition of load factor as resultant force (including gravity) divided by weight, the minimum value remains near unity. This is shown more clearly on Fig. 2.4, where the exact values corresponding to

'i' ""

= 0 are compared with the approximate solution. The latter is seen to give a good estimate of nmax, being slightly conservative. The relation between maximum load factor and speed loss is shown on Fig. 3.9, previously referred to. A loss of 50% of entry speed, or 75% of entry kinetic energy, is achieved during a transition which has a peak load factor of seven. The corresponding entry angle is about 4. 10 •

IV. EFFECT OF THE PARAMETER (CLS/m)

Equations 3. 1 and 3.2 show that the "lift parameter" CLS/m and the density occur only as a product, since the first terms of these two equations are

t

(C~ ~)~ V~­

(C~ ~)!.

V

l

+

• • •

• • •

The combined parameter may further be written as

--~'i ~ _o

e

whence and (4. 1) (4.2) d .l.~

Ir

we assume that g, ~ and CLS/m are constants, then -'t= ~L

=

n

and the solution of Eqs. 3.1 to 3.3 for given entry conditions (VE, 'tE)

at ~ ~<>O may be expressed in terms of y as

V ':: \/

(.4;

~O)

rJ

Jt",

'JE)

~:)

'f

~

't'

(-ó-

~ ~

0 )

~ ~ ~)

V

E )

D"~)

~

'::

~ (~~ ~

0 A )

~

V

E)

O'~

')

) [-' L-.J

(4.3)

The role of the lift parameter is thus seen to be simply to translate the tra-jectory vertically.

*

If CLS/m

=

1, then y

=

y. For CLS/m ~ 1, the

tra-*

This result was previously derived by J. A. F. HilI, and used by him as the basis of a set of non-dimensional solutions (Ref. 2).

jectory is displacedt

~p~~~SJ'dS}

by the amount

l

~(C

....

S

1"V'r\).

Thus if, for a given entry speed, a pair of values of 'tE and D/L are found which lead to a 'horizontal' match point, (SI"" = 0, for one value of eLS/m, then they will apply as well for all other values of CLS/m. Furthermore, the peak load factor nmax and the speed loss during the entry, as given by

V1/VE, wili also be independent of CLS/m. The approximate solution given in Section 2 is therefore correct insofar as it predicts V 1 /VE and nmax to be independent of CLS/m. The minimum height, i. e. the height at the match point, Yb is seen to be related to the lift parameter for a given ( '( E, D / L) pair by

u.

\

.

()

C \.. S

~

t

-6, - ~ l<./Vl, yy\

=

4,

=-

GdY\S • (4.4)

This result too can be obtained from the approximate solution, Eq. 2.8. As a check on the computations, this relation was tested for the solutions obtained by numerical integration. The result is shown on Fig. 4. 1 and the L. H. S. of Eq. 4. 4 is indeed seen to be a constant . .

The effects of (CLS/m) inthe exact solution on downrange time and distance are not so obvious, since these quantities have both been reckoned from the moment of crossing the 400, 000 level. The approximate solution, Eq. 2.18, shows that the law of variation of range \ XII with (CLS/m) is

(4. 5)

This relation was tested for the numerical solutions at ~ E

=

.06r, with the result shown on Fig. 4. 2. The L. H. S. of Eq. 4.5 varies from 9. 52 at CLS/m

=

O. 1 to 9.43 at CLS/m

=

8. Thus it is within 1./2% of the value at CLS/m

=

lover a range of two decades. Equation 4.5 can hence be assumed

to hold with good accuracy, and can be used to find ranges, for CLS/m different from 1. 0, from the results given on Fig. 3.4. That is, the incre-ment tQ be added to the value on the latter graph is

..

_...L

L..

Cl..

S

~'6'Ë ~ (4. 6)

The time from 400, 000 ft. to the horizontal inflection point is given in the approximate solution in terms of the exponential integrals, and a simple relation for the effect of CLS/m is not readily evident. How-ever, such a relation can be obtained as follows. An increase in CLS/m shifts the. whole trajectory upward by the amount

~~

=

~ ~ ~

This shortens the portion below the 400,000 ft. level by the amount

(4.7)

-which is the same as (4.6). The time to traverse this distance is a decrease in t1, and is approximately

~

ti'::

1.

~.6-:

- -1.."

L

CL. S

y

r.

~ '6I_E ve YY\

(4.8) This equation was tested with the numerical solutions of the exact equations for

tE

=

.06, with the result shown on Fig. 4.3. The graph of t1 vs.

~(CLS/m)

is a straight line with the slope

dil

= -16.20

d!.(CI.~/'M)

whereas that predicted by Eq. 4. 8 is -16. 15. These are within

1/30/0,

and hence Eq. 4.8 can be taken as sufficiently accurate for determining values of t1' from those given on Fig. 3.6, for values of (CLS/m) different from unity.

V. THE MATCHING EQUILIBRIUM GLIDE*

The total energy of the vehicle at any time is given by

~::. W'\(\Vl._~)

r

and that of a satellite in a circular orbit at the same height is

(5. 1)

(5. 2) where the constant? is related to the gravitational acceleration by

The drag equation can be written as

dE

~

: -

d.4.-and for "equilibrium" flight, when d

t /

ds

=

can be rewritten as

(5. 3)

(5.4) 0, the lift equation, Eq. 3.2,

(5. 5) Let

cP

be the angle subtended at the centre of the earth by a portion of the (planar) trajectory (Fig. 2. 1). Then for a shallow glide,

't

«

1, we may write

(5. 6)

* The treatment of the Equilibrium Glide given here is adapted from a lecture given at the Institute of Aerophysics by H. Multhopp of the Martin Co., on Feb. 18, 1960.

On combining Eqs. 5.4. 5.5 and 5. 6 we get the basic differential equation of the glide:

~<1>

..

I.(~\

(Es-E ')

=

0 (5. 7)

The energy ES only varies by about 1% over the altitude range of interest. and hence may be assumed to be constant with little error. Then

(5.8)

This has the solution

E :

(5. 9)

Let

4>

= 0 when E has the arbitrary value E*. Then

( E - Es') :

l

E* - Es')

e

~(~\

cp

(5. 10)

A convenient value for E* is the energy of the vehicle when it is at rest at a mean radius r. which lies in the actual range of r traversed. i. e.

t

*"

= -

J:.

"IY'\

r

(5. 11)

The energy as given by Eq. 5. 10 then varies as shown on Fig. 5. 1.

If the difference between rand r be neglected (again an error of the order of o~ly 1%) the velocity variation is found from Eq. 5.10 as

(5. 12) where V c is the circular satellite speed at the mean radius r. This speed variation is shown in Fig. 5.2. It should be remarked that the point ~

=

0

(V = 0) would not be reached in the usual case when the reentry vehicle still has non-negligible energy at the surface of the earth. Zero altitude would correspond to a small negative value of

CP.

ct>

=

<Po .

The variation in height during the glide is determined from the expressions for the speed and density. together with the equilibrium condition on the lift. Thus. from Eq. 3.2 with d~

Ids

= 0 and r

=

r. and from Eqs. 5.12 and 3.4. we get af ter a little reduction

e

l\~),

ct>

from which it can be shown that

(5. 14)

·At the ground, y = 0, and ~ .

=

CPo . Equation 5. 13 yields

(5. 15)

and Eq. 5.12 gives for the velocity at the ground, ·_{V o'}

(5. 16)

The value of'( in the glide is given by

~:- ~=_.L

t1

dÄ.-

r

~cp

(5.17)

From the solution for y(

4»,

Eq. 5.14, this expression can be evaluated, and has been simplified by the aid of Eq. 5. 13 to yield

~

=

~\

(loS

f1)e-~d+

..!:..('S;i)

L. )t YY'\

~

'f f.> \:\...

cr

(5. 18).

Equation 5. 18 provides the link which connects the entry transition manoeuvre and the equilibrium glide. lf the end conditions of the transition manoeuvre (i. e. at the match point) are substituted into Eq. 5.18, the required value of DIL in the glide is obtained, viz.

(5.19)

Of course the lift equation (Eq. 3.2) must be satisfied, with d'(

I

ds = 0, for the same values of V and ~ on both sides of the match point. It follows that CLSlm remains unchanged as the match point is passed. For small

values of '6'M > (t>/L)} will be much smaller than DIL during transition. For example it may be reduced frorn about 6 during transition to about unity during the glide. Since CLSlm must remain fixed, it appears that

the most ·suitable way to achieve the sudden reduction would be to jettison a lightweight drag body at the match point. The small loss of mass would necessitate a slight reduction of CL as weU.

The angular range during the glide is given by

(5,. 20) where <Po and

~M

are the values of

cp

at ground

l~vel

and the match point respectively. The former is given by Eq. 5. 15 (it is very nearly zero) and the latter is found from Eq. 5. 13 to be

(5.21) The horizontal range is very nearly equal to r

b..cP ,

i. e. very nearly equal to -

r

4'M . By using Eq. 5. 19 in conjunction with 5.21, the approximate expression for horizontal range is

Glide Range =

...J..- (,

~

-2' CJ2

f

~

t'\)L ( \

~:J=- ~~

')

~ ~M YYI Y' ~t"\ Cl. ~

(5.22)

which shows that the range varies simply inversely as '6'1"'\ (or proportionally to

CL

ID)~

),

and depends in a more complicated way on (CLS/m) and the

match height y M .

The time of flight during the glide is obtained from

<P

t-tl'l

~

)

cp,"*,

(5. 23)

when the solution for V is substituted (Eq. evaluated, with the result

5. 12), the integral can be

(5. 24)

where

VI. REDUCTION OF THE TOTAL HEAT LOAD ON THE VEHICLE

The total energy of the vehicle at entry is ultimately dissi-pated by the heating processes at the vehicle and the drag body. The fraction dissipated at the vehicle can be significantly smaller than unity if the ratio of vehicle drag to tot al drag is small. The total energy loss during transition is (neglecting potential energy):

~E.

;:

t

m

_(Yr;,'-V

_M

2.)

During the glide. the drag is the drag of the vehicle alone. and neglecting the mass of the jettisoned drag body. the lift remains constant as the match point is passed. Hence the ratio of the drag of the vehicle alone to the total drag during transition is

(6. 2)

and the energy fraction dissipated at the vehicle during transition is

where EE =

t

mvi is the kinetic energy a~ entry.

the energy fraction dissipated during the glide. i. e.

Tne total energy fraction is therefore

(6. 3)

To this must be added

(6.4)

(6. 5)

This fraction can be calculated quite readily from the results given in Secs. 3. 4. 5. By way of illustration, Ev/EE has beencalculated as a function of ~ E for (CLS/m)

=

1. and '(~ = 0.2 and 0.5. The results are plotted on Fig. 6.1, and show a very significant reduction of the heat load carried by the vehicle for entry angles of the order of .08r (4 1/20 _{). The}

value of ~M , and hence of (D/L)g is seen to be significant but not very im portant except at the steepest entries .

VII. EXAMPLE OF THE MOTION AND HEATING 7.1 Motion

In this section a representative entry flight path is construct-ed, consisting of a transition section. followed by an equilibrium glide. The entry conditions are '( E

= .

07 r (4.010), VE

=

25,770 fps, D/L

=

6.4. and

CLS/m = 1. O. From the numerical solutions of Sec. 3, the match point is given by

Vt-'\

= 12, 587 fps.

i

M = 207.800 ft.

'I.

M = 547.5 n. m.

t~=147.2sec.

and the maximum acceleration is 7. 15 times sea-level gravity.

From Eq. 5. 19, the value of (D IL)g is found to be 1. 180, and the glide portion of the path was computed from the equations given in Sec. 5.

The characteristics of the whole trajectory are shownin Figs. 7. 1 to 7. 6. Figure 7. 1 shows the actual flight path, which. consists of a gentle pull-up to near-horizontal flight followed by a glide which steepens af ter about 200 nautical miles to a value of

t

beyond that for which the shallow-glide theory is valid. Landing would take place.about 760 n. m. from the entry point. Figures 7.2 to 7.6 show how the variables of interest change with time. Figure 7. 2 shows that the effect of the atmos-phere on the speed is not very noticeable until about 1 minute has passed, at which time the altitude is less than 300, 000 ft. Beyond the match point, the speed variation is nearly linear with time. This is because the DIL

ratio is constant, and the lift. is not varying rnuch - hence the drag and the deceleration are nearly constant.

Figure 7.3 shows that the assumption of

'6'

«1 in the glide limits the solution to approxirnately the first 300 sec. The later portions of the curves are shown dotted. However, by this time the vehicle has des-cended to about 125. 000 ft. and slowed down to about M = 3; i. e. it has entered the supersonic phase and the values of CL and LID of the hypersonic glide would no longer apply. The essential part of the entry has thus been concluded in 5 m ins.

Figure 7. 6 shows (solid line) how the accelerationJ in units

of sea-level gravity, varies wUh the time. In considering the effect on

hurnan occupants,however, the relevant quantity is actually the 'seat-reaction', not the acceleration. Thetwo are related as follows:

Let n* be the ratio of the seat reaction to the sea-level weight of the passenger. This may aptly be called.the''physiologicalload factor", nórmally equal to unity. It is readily shown that

Y\*

~

\\

'l.

1-I1 ::.

ëf

Ll'Y\

-I+GOS

e

o

where n = load factor = F

Iw :

o..-Icr

and 9 =

1

_~ 4-

'6'

+

+-

-I-,=-.APNv D

is the angle betweenthe resultant aerodynarnic force vector and the weight vector. The values of n* have been calculated and are also plotted on

Fig. 7. 6. The physiological load factor is seen to be somewhat less than n, and hence the vehicle acceleration is a slightly misleading criterion for physiological stress. The reason for this is the angle 9 between the aero-dynamic and gravity vectors. If 9

=

180°, as in ordinary level flight, when n = 0, then n* = g/ go

==

1, which is the correct limiting result.

7.2 Heating

Approximate values of the radiation-equilibrium stagnation-point temperatures were calculated for the vehicle nose and the drag body. The rate of heat-transfer to the surface is calculated from the following semi-empirical equation given in Ref. 3

(7. 1)

where

R

=

radius at stagnation point

~<;.L.

=

standard sea-level density (. 00238 slug/ft. 3)

v

c

=

circular satellite speed (26, 000 fps) ho

=

stagnation enthalpy

hw

=

enthalpy at the surface temperature h'w

=

enthalpy at T

=

3000K

A check of the values involved in this example showed that the quantity in square brackets in Eq. 7.1 could be approximated by unity, with little error in the resulting surface temperature. The effective radii at the vehicle and drag body stagnation points. were taken to be 2 ft. and 20 ft., respectively. That for the vehicle could possibly be made much larger by taking advantage of the high angle of attack (c( ~ 450_{), however, the smaller value gives an} idea of the upper limit of the temperature. The radiation from the stagna-tion point is gi ven by

+

9r

_r

=

é

(J \"

w

(7. 2)

where

t

is the emissivity and (!" is the Stefan-Boltzmann constant. A

value of

f

= 1 was used for the emissivity. The equilibrium temperatures obtained by equating the heat transfer rates given by Eqs. 7. 1 and 7. 2 were calculated and are shown on Fig. 7. 7. The peak value obtained for the smaller radius is about the melting point of steel, but lower than that of higher-temperature alloys. It is well within the capability of non-metallic materiais. The lower temperature onihe.la:r:'gerdrag body appears to be technically feasible.

The ave rage temperature of the heat-bearing structure has also been calculated. on the assumption that 500/0 of the vehicle mass is protected payload. and the remaining 50% carries the heat load. Under this condition thermal equilibrium is not reached at the time the peak stag-nation tem perature occurs. so a step-by-step solution of the transient heat equation was used. This equation is derived as follows:

The rate of heat input is assumed to be one half the rate of friction work in the vehicle boundary layer. i. e.

(7.3)

where D is the total drag of vehicle plus drag body.

The radiation is assumed to be proportional to the fourth power of the average temperature. T. and to occur from the area 2 S. Thus

~

=

f.

ClT+~S

(7.4)

Since the local radiation is truly proportional to the fourth power of local temperature. it follows that the averaging process underestimates the radiation. Hence the factor

E

should be taken to be larger than the local surface value. However. since the vehicle is in any case hypothetical. the value

é

= 1 was used as reasonably representative. Thus

+

'-ScrT

V

(7. 5)

Let the mass of the heat shield be

m'.

and its ave rage specific heat be c. Then

or

JT

dl.

-

4-_ .J.-

('M

\(~

\

c _

l SQJT

-

~

c

J :;;,

J Cl ) ~ -,.,.,' c. V (7.6) The values used for the constants are

J

=

778 ft. lb /BTU

c = 4. 5 BTU slug -1 oR

-1

(Representative of stainless steel)

and as noted previously (m/m')

=

2. The value of Slm is given by

(CLS/m)(l/CL). With a reasonable value of CL

=

.6, and (CLS/m)

=

1. 0, then Slm

=

1. 67. The friction drag fraction f/D during transition is given by

f f Dv

=

-D

Dv

D

where Dv is the drag of the vehicle alone. On passing the match point, the dropping of the dr ag body red uc e s the D / L fr om 6. 4 to 1. 18, henc e

= = 0.184

D 6.4

The value of f/Dv depends on the effective 'bluntness' of the configuration when at high angle of attack. It would appear that a reasonable

*

value to use is f/Dv = 0.015, i. e., 1 1/2% of the total drag is skin friction. Finally then

f

- = . 015 x O. 184 = . 00277

D

The equation for the transient average temprature (Eq. 7. 6) during both the transition and glide phases was solved by the Runge-Kutta method, with the results shown on Fig. 7.8. Although the average value of the equilibrium temperature is fairly high, the transient value is much lower, reaching a peak at 300 sec. of only 7800R (3200_{F). The protection of the payload would}

evidently not be a difficult matter with so low an average external tempera-ture. The main reason for the very large difference between the two tem-peratures is the low rate of heat input to the vehicle during the trans ition phase, when the auxiliary drag body is carrying 810/0 of the total heat load.

With ave rage temperatures as .low as those shown, it is evident that most of the structural elements of the vehicle could be conven-tional materials, and not require thermal protection. Furthermore, if adequate heat capacity and conduction paths are provided in the neighborhood of the stagnation regions, it may be anticipated that the peak stagnation point temperature shown in Fig. 7. 7 could be substantially reduced.

VIII. CONCLUSION

An analysis has been presented of a particular atmospheric entry program, for satellite speed, which has certain attractive features. These are a substantial reduction of the total heat load on the vehic1e, and the provision of a short, and hence relatively accurate, flight path. Similar advantages would be expected to be found for entry at supercircular speeds.

*

Analysis of the data presented by Allan and Eggers in Ref. 4, which con-tains the parameter f/D = C'rS/CDA, shows that values of the order. 015 would be possible with blunt cones.

1. 2. HilI, J. A. F. 3. Detra. R. W. Kemp, N.H. Ridell, F . R. 4. Allen, H. J. Eggers, A. J. REFERENCES

Tables of the Sine, Cosine and Exponential Integrals". Published by the Works Project Administration,

under sponsorship of National Bureau of Standards. U. S. A., 1940.

Satellite Re -Entry with Lightly Loaded Lüting

Vehicles". MIT Naval Supersonic Laboratory Tech. Rep. 429, 1959.

Addendum to Heat Trá.nsfer to Satellite Vehicles Re-entering the Atmosphere. Jet Propulsion,. 27,

12, Dec. 1957.

A Study of the Motion and Aerodynamic Heating of Ballistic Missiles Entering the Earth' s Atmosphere at High Supersonic Speeds. NACA Rep. 1381, 1958.

APPENDIX A

.Correction of the Numerical Results for Changes in ~o and

@

Proceeding in a manner similar to that in Sec. 4, we can write the basic equations 3. 1 to 3.3 as follows:

(A. 1)

(A. 2)

~ ~-~

(A. 3)

where

_"ó

~ ~-t~ (~o

Cg)

(A. 4)

The solution to these equations can be written symbolically as

\jo:.

V (4;

~) ~)

V

E .) 'tE ')

"t:

O'(Aj

~)~)

V

E) -tE)

~

:.

~ (~. ~

A

V

'f. )

) l-) \-) E ) 'E

(A. 5)

Equations A. 5 differ from Eqs. 4. 3 in that

f

0 has been combined with

(CLS/m), and hence does not appear explicitly. The effect of agiven per-centage change in

fo

is therefore exactly the same as that of an equal per-centage change in (CLS/m). These effects can be determined by the methods given in Sec. 4.

The influence of ~ is more com plex, as indicated by its presence as an explicit parameter in Eqs. A.5, in addition to its role as an "altitude shifter" shown in Eq. A.4. Same of the particular influences of

~ can be inferred, however, from the solutions in Sec. 2 . . We find

(1) The effect of ~ on the minimum height Y1 is given by Eq. 2.8. (2) The match speed V 1 depends on

0 ,

(Eq. 2. 12).

(3) The D/L required for a given entry angle depends on ~ , (Eq. 2.13).

(4) The horizontal range \ xli varies as

i

(Eq. 2. 18). (5) The time of flight varies with ~ in a complicated manner,

--r

Circular Orbit .--- ~

---"É,,~TronSilion

Polh

---V

_E

=

V

_{c " '-}A ~

Match

Point "~----L (Jettison 'chute)

--M

- -

-

_...

_...

Glide

5 ... ,

,

FIG. L 1 ENTRY FLIGHT PATH

\

\

FIG. 2.1 NOTATION ... ...

"

FT.

3'1.10 5 , ~ ~ ~ ~

Yl

2f.1J5

t

I~~~~~~ ··1

FIG. 2.2 MINIMUM HEIGHT Y

L AS A FUNCTION OF ENTRY

ANGLE (td AND LIFT PARAMETER(CLSjro). APPROXIMA TE SOL UTION

1)(105 ,

.1:---

~----+-4~

I .2

.

;) ,. 1 2 5 ₁₀ 20 _ _ ---L-_~ 50- 100 (CLS/ ro)

"

• B .6 .4 .2

o

.02 ,_-+-_ _ _ _ -+---..,_c:::::;-;;.~ I I DIL APPROX.

I

EXACT

I

V/V E APPROX.

~

V/V E EXACT

I

I

DIL VALUES ARE THOSE REQUIRED TO PRODUCE HORIZONTAL FLIGHT AT THE

MATCH POINT.

.04 .06 .08 .10

~E RADIANS FIG. 2.3 TRANSITION PATH PARAMETERS

w c 12 10 I: 8 C" 2 o APPROX. '\.

Ij

/.~EXACT

7

)

~

/

V

' /

.02 .04 .06· .08 .10

ENTRY ANGLE, rE RAD.

FIG. 2.4 VARIATION OF MAXIMUM LOAD FACTOR WITH

ENTRY ANGLE FOR Y =0

DIL

6

4

2

2400

FIG. 2.5 VARIATION OF HORIZONTAL RANGE IX11 WITH ENTRY ANGLE 'rE AND LIFT PARAMETERCLS/m

APPROXIMATE SOLUTION

'tE

rad. 2000 1600 1200 800 400

o

- - - - -_. - - - -• Ol .02 .05 .1 .2 _{CLS/m .5} 2 5 la

....l ... ~ ....l <t: U _{... 500} l-< ;J <t: Z .... >< 400 ~ 300 200 lOG o

\

\

\

\ \

\

\

~

~

~

~

EXACTC S/m=l

~

r---

L A P P R : ; - - - -

r----allCLS/m

I

.02 .04 .06 .08 .10

rE

_RAD.

FIG. 2.6 INCREMENT IN HORIZONTAL RANGE

400 ~

'~

\

350

1\\

\

~

~

EXACT

~K

!F

ZAPPROX. 250 200 150 100 50 o .02 .04 .06 .08 .10 'tE RAD.

FIG. 2.7 VARIATlON OF TRANSITlON TIME WlTH ENTRY ANGLE

.09 1.8 .0.8 1.6 ~---~---~---~---~~---~--.~-/ 1.4 1. 2 ...---+---+---+---l~/_ .06

,

tE

rad.

't~

1.0 .8 .6 .4 .2

o

5.0 5.5 6.0 6.5 7.0 7.5 DIL

FIG. 3.1 (a) VARIATION OF 1( M WITH ~ AND DIL

C S/m=l.O L 8.0 8.5 2.0 ~----~.---~---~---.---• .---~.---~---~ '(0

"

1.6 1.2 1 + + + + + i + . , 0 5

-2

V

.8 ~---l----+---~---~---+---~rl~-~/7--+----~ .03

V

.4 ~--~----4---~---+/_---/ o 2 3 4 -5 DIL

FIG. 3.1 (b) VARlATION OF ~ WITH rE AND DIL

CL s/m=l. 0 M

-3 YM~lO 300 250 200 150 100 50

o

FT. ~ • Ol

~M_=O j~!

.02 Y-Yl-

I

~I·03

_.04

I -

_.05 I .06 '\..: I \ . . : I .07 .\. I _{: .08} .09 ''OE rad.

FIG. 3.2 VARIATION OF ~ WITH '6'EAND DIL CL S/m=l EXA SOL UT ION

2 4 6 8 10

DIL

~

300

I---\~\~--L-

.

----+----+---+--

.

- + - - " - - 1

250

~~

N

VAPPROX.

~~

200

~---~~----~---4---~~~~~~r---1

EXACT 150 ~---~---4---+---+---t---1 100 ~---~---4---+---+---t---~ 50 ~---~---~---4---~---T---1 o .02 .04 .06 .08 .10

ENTRY ANGLE '6"RAD.

FIG. 3.3 VARIATION OF Yl WITH EN.TRY ANGLE

• X_M Ni1UTICAL lVIILES 1600 r---+--~----+_---4---~---~---J 1400 1200 1000 80ü 600 400 200 " c.'

" tE,

rad. \----r-.02----+---4---~---~

~---+---~---

___

.03---_+---+_---~

.05

I

\-f--=i0~

07

~'.08

~---+---+---4---~~--.09--~~---~

FIG. 3.4 HORZONTAL RANGE TO MATCH POINT FROM 400,000 FT. INITIAL ALTITUDE C S/m=l

.

L

2 4 6 8 10

DIL.

rJ:l til ...:l 1-1 ~ ...:l <r:

u

·

1-1 t-I P <r: Z

...

...

:x:

.,..

1600

\

\

1400

~.

1200 1000~---4---\~~\,\--+---+---~---~---~

800~---r---~,---+---+---~---~

600

~-+----~"'-+---4---I

~~ACT

400

1---+---+---+---+~-/-~----+---1

APPRQX. 200~---_r---~---~---~---_+---~

o

.02 .04 .06 • 08

FIG. 3.5 HORIZONTAL RANGE TO LIMITING MATCH POINT FROM 400,000 FT. ALTITUDE C

_C

/m=l

.10

~

RAD •

t M SEC.

N

"

.02 300 ~---~--- -~---~---~---~---~ ~---~---~---+-.03 200

~---~---~---\---41---'

04

I

.05

I---+---+---~-

I

~.

06

~07

_I

_{I .}

r---==:J .

08 ;-- .09 100 t---+---l---+---~---)( E' -rad •.

o

FIG. 3. 6 VARIA TION OF TRANSITION TIME WITH D/L AND

r

C S/m=l

E L

2 4 6 8

DIL

~ .02 _{rad .} .

8~----~---~

~---~---~~

.03

.7

~

.04 .6

r======= .

05

f'---

I

_.06

:1

. 5

r==--

.07 '",::s0

I

V=V

't--

.08 1 .4

'-.---

I

. 0 9

-OE

. 3 r---~---~r_---~---~---~ .2r---~---~---~---~---~

FIG. 3.7 VARIATION OF SPEED RATIO AT MATCH POINT WITH

oE

AND DIL

.1 r---~---~~---~---~---~

o

2 4 6 8 10

)IC c E

C

~ 0 ~ ₁₂ U ~

rE

rado ~ Cl ~ 0 ..:l 10 ~ P ~ H ~ ~ ~ 8 6

/

.09

v ....

.08

V

97e-·

07 ~_.

JV"

~.06

1""Oyv

05 . . ..

.~

.04 I, ~ 4 . . / 0 0 3

4.

02 2

o

2 4 6 8 10 12 DIL

FIG. 3.8 VARIATION OF MAXIMUM LOAD FACTOR WITH AND DIL C

/v

/

v

/

v

1/

50 ;ft. Cl) Cl) 0 ....:l Cl ₄₀ t<l t<l IJ" Cl) 30 10

/

~I

FIG. 3.9 VARIATION OF SPEED LOSS WITH

MAXIMUM LOAD FACTOR EXACT SOLUTION 20 o 2 4 6 8 10 12 LOAD FACTOR, Y\ Il_{Yl- 1n(C L S/m)} 10 8 6 -4 . 2 o • 1 .2 .5 1.0 2.0 5.0 w

10

..

8 6 4 2 o .1 220 200 180 160 .1 j3Y,IX

l

+l_n(Ci'/m)

I

.2 .5 1.0 2.0 5.0

FIG. 4.2 EFFECT OF (C Slm) ON HORIZONT AL RANGE t'tr=.06" L tI SEC.

----

~

----r---r---

'-.2 .5 _1.0 C Slm L 2

FIG. 4.3 EFFECT OF (CL Slm) ON TRANSITION TIME t

E=. 06

5

10

---=

o

t

-

~~-____________

~---1---~

E 5 E*~2E 5

FIG. 5.1 VARIATION OF TOTAL ENERGY E DURING EQUILIBRIUM G LIDE 2 V _2 ----..::.-.::-:,::--- V c ---~---~~

o

2

.8 • 7 .6 .5 .4 .3 .2 .1

o

~----t- -

\ - - - + - - - + - - - t - - - - t - - - 1

~

~---+---r----,'~ ~~----4----r----~

~

r.:

1 _2° ,. I ·

FIG. 6.1 FRACTION OF TOTAL ENERGY DISSIPATED AT VEHICLE C_LS/.m=1.0

.02 . • 04 .06 _{.08 .10}

~

300 ~

~

~

TRANSITION PATH

~~I

Y (1000 ft)

~,

~

_~ MATCH POINT I - _ 1

d

GLIDE PATH

I

200 r 0 'Irt =.07 (4. Ol )

~

DIL =6.4 , , , C 1S/m =1.0

, ,

V E =25,770 fps. \ _,

,

\

FIG. 7.1 EXAMPLE OF ENTRY FLIGHT PATH - - -

,

\. 100

,

1 I I I I X NAlJTICAL MILES I I

I

I

I I I ,-o 100 200 300 4<)0 500 600 700

h

25000

~

20000

FIG. 7.2 VARlATION OF SPEED WITH TIME DURING SAMPLE ENTRY

15000 I I Y (ft/sec) MATCH POINT 10000 t (SEC)

1

5000

~--"'"

60 120 180 240 300

1

-I

,

,

-

-

,

I

, -

_I

,

I I I I I I

,

I

,

I

,

I

,

,

'0

(RAD)

,

I

,

.16

i

.

7

•

FIG. 7.3 VARIATION OF GLIDE ANGLE WITH TIME DURING SAMPLE ENTRY

.08

~

J

~

7

I

"'~

_--

/

-MA TCH POINT ~

-~

o

I

I

t(SECS) 60 120 180 240 300

!

800

FIG. 7.4 VARlATION OF DOWN -RANGE DISTANCE FLOWN WITH

TIME FOR SAMPLE ENTRY

---+---1---X NAUTICAL MILES

vn

MATCHP0I7

V

600

/7

V

400

/

[7

/

V

200

/

V

t (SEC) o 60 120 180 ₂₄₀ 300. ₃₆₀

400

~

FIG. 7.5 VARIATION OF ALTITUDE WITH TIME FOR SAMPLE ENTRY

300 r---~---~~---~---~---+_---~---_+---~~---+_---~---_+---~ Y(lOOO ft) 200

r---+----t----+----l---f-+==:::::::

100 t (SEC) o I I 60 12(,

~

~ ....

...

"

...

" "

_" , ",;11---1

_,

.,

---11

n---t--+---+-+-~\\

\

_\-~-1 \ \ \ \ \ I I I I 180 240 300 360

8.0

FIG. 7.6 VARlATION OF VEHlCLE ACCELERATION AND "PHYSIOLOGICAL

LOAD FACTOR" WlTH TIME DURING SAMPLE ENTRY. _,

.

~~

.

, , / \ alg I \ 0 I ,

-,

1

\~

, 1 , 1 .. , _: I h.#\ I , I \ - I \ -I , 6.0 I

,

I \ I

,

I \ 1

,

,

_ I 1 4.0 1

1/

I 1 I I I I 1 2.0 _ I 1 1 1 1 1 1 Va/go I I I I I

I

I I t (SEC)

"

"

I I

_'"

/ / I I I

I

~-

_J

I

I

I

-o 60 120 180 240 ₃₀₀

3000

)1 \"'

ft. (VEffiCLE)

~~I/~~~~~~~~

_[7

/

TOR

/

[7

\1

"I~

V

/

-r~M~A~TC=H~PO~IN~T-~~~~4---+---~~---+--~

/

~

~

/

1

~--r~~-~~~~~

t

'~

~~

'~

2000

FIG. 7.7 RADIATION-EQUILIBRIUM TEMPERATURES AT STAGNATION \ •

~-+

__ -+-_ _ ----+--_P_O-+INT_S_OF_V-l-EH_ICLE AND DaAG BODY.

1\

1\

I

1000

TOR 1400 1200 1000 800 600 400 200

o

~--~-/~-~~\----~--~--~--~

/ \

~DIATION-EQUlLIBRIUM

TEMPERATURE

~--r---7

~~~

I

' \

t - - - / ____ _

7

~~

\

t----7-

/

TRANSIENT TEMPERATT,TRE

ti

t (SEC) I I I _l i I I I I _J I I 60 ₁₂₀ 180 ₂₄₀ 240

FIG. 7.8 AVERAGE VALUES OF RADIATION-EQUILIBRIUM AND TRANSIENT VEHlCLE TEMPERA TURES.