A solution for atmospheric entry trajectories deviating from equilibrium glide

1

JULI

A SOLUTION FOR ATMOSPHERIC ENTRY TRAJECTORIES DEVIA TING FROM EQUILIBRIUM GLIDE

by M. Hanin

'J

A SOLUTION FOR ATMOSPHERIC ENTRY TRAJECTORIES DE VIA TING FROM EQUILIBRIUM GLIDE

by M. Hanin

Manuscript received September 1965

ACKNOWLEDGEMENTS

The author wishes to acknowledge stimulating discussions with Professpr B. Etkin.

The work leading to this report was done while the author was on sabbatical leave from the Technion- Israel Institute of Technology and held a Senior Research Fellowship of the Faculty of Applied Science and Engineering, University of Toronto.

SUMMARY

An analytical solution is obtained for atmospheric ,entry trajec-tories of lifting vehicles at subcircular speeds, constant angles of attack and small initial flight-path angles. The solution represents motions whose initial conditions deviate from tho$e of an equilibrium glide. It is known from previous investigations that these motions are in general oscillatory. In the present

study, the equations of motion are linearized about equilibrium glide and then are solved asymptotically in series of descending powers of a large parameter k, where k 2 denotes the ratio of planetary radius to the atmospheric decay

length. Explicit expressions are derived for terms of the series up to the order of k- 2 .

TABLE OF CONTENTS

PAGE

SYMBOLS iv

1. INTRODUCTION 1

2. EQUA TIONS OF MOTION 2

3. OSCILLATORY SOLUTION 8

4. PARTICULAR SOLUTION 19

5. SOL UT ION TO THE SECOND ORDER 22

6. ADDITIONAL TERMS 27

REFERENCES 30

an' bn, c n (n

=

0, 1,2, .. . ) A(+) _n, A(-) _n B (j

=

1,

2,

3, 4)

(Jó

_{"C L} D E F g h h n H Hn i k L m N r S t U v v n SYMBOLS constants of solution constants of integration functions of E, Eq. (75) drag and lift coefficients dimensionless drag, Eq. (50) lift-to-drag ratio

complex phase function gravity per unit mass altitude deviation, Eq. (13)

terms of solution for h

series for oscillatory altitude terms of the series H

atmospheric density decay parameter,

j

r

f3

dimensionless lift, Eq. (50)

mass of vehic1e

dimensionless net normal force, Eq. (50) planetary radius

reference area of vehic1e time

dimensionless velocity,

V/Jgr

velocity deviation, Eq. (13)

v

w

Y

~n

f

J>o

T

Superscripts I Subscript i SYMBOLS velocity

series for oscillatory velocity terms of the series W

altitude

reciprocal of atmospheric density - decay length, Eq. (2)

flight-path angle, positive downwards initial flight-path angle parameter, k ~ i flight-path angle deviation, Eq. (13) terms of solution for

S

series for oscillatory flight-path angle terms of the series ~

dimensionless altitude,

f3

Y phase function

vehicle parameter, C LSrfo/ 2m atm~spheric density

reference density, Eq. (2) dimensionless time, t J i ï ;

constant in equilibrium glide solution, Eq. (10) circular frequency

equilibrium glide solution oscillatory solution

particular solution

1. JNTROpUCTION

Numerical solutions by Chapman(1), Campbell (2), Loh(3) and others, for atmospheric entry of lifting vehic1es at subcircular speeds and constant lift and drag coefficients, have shown that trajectories deviating from an equilibrium glide are oscillatory. In these trajectories the altitude of the vehic1e oscillates about the mean descending path with a moderate damping and increasing frequency. The oscillations occur when the initial flight-Qath angle is not as small as required for an equilibrium glide, or when the initial altitude and velocity do not satisfy the condition of vertical equilibrium, so that the aerodynamic lift and the centrifugal force due to motion about a

spherical earth do not balance closely the weight of the vehicle.

An approximate theory of the oscillatory trajectories was given by Campbell in a brief note(2). Campbell linearized the equations of motion for small deviations from the equilibrium glide. He then neglected several terms so as to obtain a simple differential equation, and deduced from this equation an expres sion for the frequency. The damping was then evaluated by taking into account some of the previously neglected terms.

The present work aims to give a more complete and systematic solution for the subcircular entry trajectories deviating from the equilibrium glide. The method of analysis employed here consistsof solving the linearized equations of motion by means of asymptotic expansions, based on the fact that the atmospheric density decay parameter k is large (k has the value 30 for Earth and has comparable magnitudes for several ot~îer planets). The result-ing solution represents the variations of altitude, velocity and flight-path angle with time in t!le form of asymptotic series in descen<fing powers of the large parameter k. This method avoids making arbitrary neglections in the equations of motion, and allows the evaluation of successively higher terms of the solution. Explicit expressiol1s are derived for terms up to the order of k- 2.

The analysis SfiOWS that the oscillatory ,;,·.haracter of the t ra-jectories is due primarily to tl'~e atmospheric density gradient and the de-pendence of lift on the density. Thus, iî the vehic1e has a higher altitude than the equilibrium-glide altitude corresponding to its velocity, the lift force is less than that required for equilibrium with the gravity and centrifugal forces because of the density gradient, so that there is a net vertical force directed downwards. Similarly, if tt1e vehicle is below the equilibrium- glide altitude corresponding to its velocity, the density gradient produces a net veTtical force directed upwards. 'The oscillatory motions during subcircular entry are therefore essentially different fr om the low- speed phugoid oscillations that are generated by the dependence of lift on the velocity. When the flight

speed is comparable with the circular satellite speed, the atmospheric density decay parameter k becomes significant in the equations of motion and thus the density gradient becomes predominant in generating trajectory oscilla.tions.

2. EQUATI0NS OF MOTION

For a lifting vehic1e traversing the atmosphere of a spherical earth in a great-circ1e plane, the differential equations of motion are

mV

~

=

mgcos~

dt dY

dt

= ..

V sin 1( (la) (lb) (lc)

Here m denotes the mass of the vehic1e, V its velocity, Y its altitude above the earth surface, '0 the flight-path angle measured fr om the local horizontal plane and taken positive downwards, r the distance from the center of earth, g the.,gravity force

on

unit mass, ~ the local atmospheric density, CD and CL are the drag and lift coefficients based on the reference area S, and t is the time. The variation of the atmospheric density with altitude is described to a good approximation (1) by the exponential law

-{3Y

=

(2)

where {3 and ,90 are constant, and thus .{3 -1 is an atmospheric decay length (mean values for Earth are (3-1

=

23,500 ft.,

..po

=

0.0027 slug/ft 3). It is permissible to disregard the variations of rand g since the distance from the center of earth changes only by a small fraction when the vehic1e moves

through the atmosphere. It is also assumed that the aerodynamic coefficients CL and CD remain constant during the motion, as they would do in hyper sonic flight at a constant angle of attack.

To state the equations of motion in a dimensionless form, we introduce the variables

u

=;=

J

~g

,

71

=

{3Y,

Jr

.

l = t -

_{. r} (3)

so that U is the velocity referred to the circular satellite speed,

'ft

is the altitude referred to the atmospheric decay length, and

1:'

is a dimensionless time based on the circular speed and the planetary radius. In virtue of (2) and (3), the equations of motion (1) become

dU

dL

=

sin?( -;t E d

n

= _ k 2 U sin '('

d'ê"

·· 2

-m-U

e

where E denotes the lift-to-drag ratio

(l is a parameter defined by

il

=

CL Sr _2m

fo

and k is the dimensionless atmospheric decay parameter, namely

k

=.r;;

(4a) (4b) (4c) (5) (6) (7)

The values of k are 30 for Earth, 32 for Venus, about 12 for Mars, and ab0ut 60 for Jupiter(3).

An approximate particular solution representing the equilibrium glide at subcircular velocities was found by Sanger(4) under the assumption

that the flight-path angle and its rate of change are negligibly small during the glide. The gravity force on the vehic1e must th en be balanced by the lift force and the centrifugal force arising from motion about a spherical earth. The equations for iJle equilibrium glide solution are obtained from (l) by neglecting the term

mV~

.·

in (lb), neglecting the longitudinal gravity component

mg sin'ó in (la), and putting sin ~ =(f, cos ({

=

1 in other terms. In dimen-sionless form, the equations are

dU d?'"

d'r!

dT=

-(8a) (8b) (8c)

where

TI,?'J. , '('

denote the equilibrium glide s0lu.tion. The equation (8b) expresses the condition of vertical equilibrium and yields a relation between the velocity and the altitude in the glide. On eliminating

'7'L,

the equations.-(8.) give dU 1-

U

2 (9a.)

- -

= d?, E

't=

2k- 2 dU (9b) U '(l-

U

2) d

C-The equilibrium glide solution for subcircular speeds is accordingly

u

= tanh

(Tc

'. E

-t)

(lOa)

'7J.

=

log(~

ij'l)

=

log

f-SinhY'E-t)J

1-

U2

(lOb)

"6

= 2k- 2 _{-, 2}

₌

2k-- 2 _{ctnh 2 (}

'~

_-

_{ê' )}

E EU (lOc)

where ~ is a constant of integration and determines the initial values of U ,

12

and_ (t, the starting point of the glide being taken at

1:

= O. The initial value Ui of the dimensionless velocity is related to

Cc

by

U· ₁ = tanh

Tc

E ( 11)

In virtue of (lOc), the equilibrium glide solution (la) is consistent with the assumptions made in its derivation provided that k is large and that the lift-to-drag ratio E is not small in comparison with unity, and excepting the final part of the trajeetory where the velocity U becomes small. The solution describes a smooth gliding descent through the atmosphere during which the vehicle decelerates gradually from its initial subcircular velocity. The flight-path angle is very small, of the order of 1/k2E, but increases as the velocity decreases. The deceleration is given by g( 1-U2 )/E and thus is relatively low.

However, the equilibrium solution (la) does not constitute a general solution for atmospheric entry since it contains only one free constant

?c

while the general case involves three freely prescribed initial values. This is due, of course, to the neglect of d~ /d~ in the equilibrium glide approximation, so that the differential equations of the glide have a lower order than the fuU equations of motion. From (la) it follows that the initial values U i '

1i

i, Yi of the equilibrium glide must satisfy the conditions

(12a)

=

(12b)

A more general solution for atmospheric entry of lifting vehic1es is obtained by considering motions that do not satisfy the initial conditions of the equilibrium glide but are sufficiently close to the glide to permit a lineari-zation of the equations of motion. Denoting by v, h, ~ the deviations of velocity, a.1titude and flight-path angle of the motion from those of the equi-librium glide, we represen.t the solution by

u

=

u

+

v,

7L

=

Î/.. +

h, (13)

and assume that

Ihl

«

1 , v

1«

ij (14a)

(14b)

No restriction is made regarding the magnitude of the ratio

I

~I

/

'lf, so that

the deviation

b

of the flight-path angle is allowed to be larger (or smaller)

than the equilibrium glide angle

"'j.

To linearize the equations of motion under the assumptions (14), we insert (13) into the fuU equations (4), put there

sin (

0'

+

~)

= (f +

f: ,

cos (

0

+ ~)

=

1 (15)

in accordance with (14b), and then substitute for

duidT,

e-~

and

d-YL./dT

from

(9a), (8b) and (8c). This gives

=

_ 2 1- (U + v ) U + v 2 _ 1-

U -

-h · d ~ U2 (U+v)e - d~ dh

=

k 2

[TI "'1 -

(U

+ v ) ( '( +

&

?

d'Z'"

.!1

..

Next we expand the expressions in bracketQ in powers of h and v/U ,observe

that in each bracket the terms independent of h and v cancel out, and finally neglect squares and products of h and v/U in comparison with their first powers according to (14a). A set of linear differential equations is then ob-tained for h, v and

b.

To express all the coefficients in terms of U , we substitute 0' from (lOc) and note that (lOc) gives with the aid of (9a)

=

-2 -2

4k (l-U ) E2 U"3

(l6)

The resulting set of differential equations for motions deviating from equi-librium glide is -,2 + -2 (l- U ) v -2 d S; U

- - +

2v

dL

- -2 - U (1 - U )h = - 4k -2(1 _ Ü 2 ₎ ---.~2-V ---. - 2 2k U (17 a) (l Tb) (l7c) I

Since U varies with

?:'

as stated in (lOa), the equations (l7) have variable bl.lt known coefficients. The terms on,the right-hand sides of (l7a) and (17b)

arise from terms that we re neg}ected in the equilibrium glide approximation,

and therefore they represent corrections to the equilibrium solution itself.

However, these correction terms are of the order of k - 2 and

thu~

are relatively

sm all.

Denoting by Ui,

12i>

tri the initial values of the velocity, altitude and flight-path angle of the motion under consideration,

àncl

allowing these values to differ from those of equilibrium glide, we have the initial conditions

at

T

= 0: v= ₍₁₈₎

are restricted in magnitude by the linearization assumptions (14). Apart from these limitations, the initial values Ui , ?ti, Yi of the motion are arbitrary.

In order to fulfil the conditions (18) a general solution is required.

In what follows, a solution of the differential equations (17) with the initial conditions (18) is obtained for large k by means of asymptotic

expansions. As the equations (17) are linear and non-homogeneous, the re-quired general solution is formed by deriving first a solution of the correspond-ing homogeneous equations (Section 3), th en a particular solution of the full equations (Section 4), a.n.d adding.

3. OSCILLATORY SOLUTION

We denote by v>:<, h>:<:, ~,:< a solution of the homogeneous differential equations associated with (17), so that

- dv >:< -2 EU - - + 2(1- U ) v>:<

d'2"

E 'U2 dh>!<_ + 2 v>:< + k 2 EU 3 >:<

=

0 d '7;' (19a) (19b) ( 19c)

When ~>:< is eliminated by substituting from (19c) into (19a) and (19b) and using (9a), the equations become

- 2 dh':< 2. Ek d ?: - U (1 - U )h>!< + EU _(20a) (20b) -2d >:C

-~

+ 2k - - - 2 EU

1-d?:

For large values of the parameter k, an asymptotic solution is obtained by putting

=

(21a)

v* = WekF (21b)

where Hand Ware series in descending powers of k

(22a)

(22b)

and where F,

Ru,

W n a~e functions of

"'2:"

but do not depend on k. The form (21) - (22) is suggested by the usual method~ Jor solving asymptotically

differential equations with large parameter 5. To find the functions F,

~

and W_n J we substitute (21) into the equations (20). As the common factor

e kF cancels out, (20a) and (20b) become respectively

E

r

_k-1_{H :dF} + k- 2 dH

l _

U(1-

U 2 )H + EUrkW dF +

dwl

L

d

-r

d

r

-J

L

1 d T d

Z-J

+ (23a) (23b)

Now weintroduce the series (22) for H and W into the equation (23) and compare the coefficients of the resulting powers of k. Sinée the term with highest power of K in (23a) is proportional to WodF/ d'r, and we suppose that F is not a constant, it follows that

W _o =0 (24)

The terms with highest power of k in (23b) yield then a differential equation for the function F, namely

.( 25)

Proceeding further with the comparison of coefficients, and taking (24) and (25) into account, we get from the equation (23a)

_2 (1 - U ) Ho - E - - W1= 0 dF d?; - -2 - dF U(1- U )H2 - E U - W3 d?:

=

EH 1 dF +

dr

EU dW2 +2(1-

d'r'

U2 )W2 + dHo

+

E

d,

Similarly, the equation (23b) gives

=

(26a) (26b) (26c)

o

(27a)

=

EU -2 _{U(1 - U )} - -2 dHo

_{d'2: -}

_{2Wl d} dF _1:

f

2

j

-2 dF dH _2 d F _ . - 2 dF 2EU - _ _ 2 + EU - - 2 + U(1-U ~ H2 d'r d , d'r d'r

=

2W1-dF ···

-d1:

(27b) (27c)

The~~ relations determine the asymptotic solution,h':<, v*, 6>:<. In fact, the

functlon F follows from (25), the functions Ho and W1 are determined by solving the pair of equations (26a) and (27a), then Hl and W2 are obtained by solving (26b) and (27b), and so on. The differential equations for the re-quired terms are of the first order only and can be integrated explicitly. The solution for ~ * is deduced fr om the equation (19c) by substituting (21) with

(22) and making use of (24) and (9a). It has the form

(28a)

~1

=

Ho dF (29a)

TI

d?:

D.

₂

=

-

-

Hl dF 1 dHo (29b) TI d.oz-

TI

d"2:'

ó.

₃

=

-

-

H2 dF 1 dH1 2W1 _3 (29c)

TI

dL

TI

d?; EU

For the function F we have from (25)

dF + .

J

_2

d'r = _1 1-U

(30)

so that there are two conjugate solutions. The initia1 va1ue of F can be chosen free1y.without 10ss of generality, and we take F

=

0 at

r

=

O. In virtue of (9a), the differentia1 equation (30) is equivalent to

-dF = +

(31)

Integrating, we find that F is given by

F= + i f.P (32)

,

where

lP

denotes the function

~ = _{E (arcsinUi - arcsin} ij )

= E arc sin (tanh

~c

) - E arcsin (anh

?"

~-

'2'")

(33)

It follows that the motions described by the solution h>:C, v>:<,

6

>:< are oscillatory

i. e. since

lP

is real and varies monotonically with

"Z:",

the factors

e kF

=

e

±

jkc1l are 0 scillating function s of time. The angular frequency

uJ

of the motion can be defined by

(34)

and we get from the equation (30)

= (35)

(2)

This expression for the frequency agrees fully with that given by Campbell

The Frequency increases during the motion and approaches the limit

-.{g{3-

.

For Earth, the period of the oscillations decreases towards the value of about

170 sec.

The amplitudes of the oscillations are represented by the

series H, W,

A

for altitude, velocity and flight-path angle respectively. The

leading terms Ho, k-1W1,

k-1~i,

of the series give accordingly the first

approximation to the amplitudes at large k. To find these terms we solve the

pair of equations (26a) and (27a). On eliminating W 1 and using (30) a single differential equation for Ho is obtained, namely

_ dHo

2EU

-ay-

+ 3H_o

=

0 (36)

3dU

=

(37)

Therefore the function Ho is

_ (±) fj3/2 _

(

!">~o

l"'c

_~\3/2

Ho - Ao -2 3/4 - Ao smh

I

(l-U) E

(38)

where

A~±)

denotes constants of integration. There are two constants,

A~+)

belonging to the solution with F = + i'f and

A~-)

to that with F = - ilP. The equations (26a) and (29a) yield now directly the functions W 1 and Al. Taking the upper and lower signs in correspondence to F

=

± ifP, we obtain

(39)

+ 'U- 1 / 2

°A(-) --~

1 0 (1- ( 2

)

-

1/4

(40)

The terms Ho, W1 and À 1 decrease during the motion since

U decrease s.

It follows that the oscillations of altitude, velocity and flight-path angle are damped, at least to the first approximation for large k. The damping is moderately strong, and the oscillations persist for several periods until they abate. As shownby the equations (38) - (40), the oscillations of altitude

diminish more rapidly than those of velocity and flight-path angle, in this

order. The equation (38) for the damping of altitude oscillations coincides

with the result of Campbell' s theory(2).

The next term Hl of the solution for the altitude is obtained from the equations (26b) and (27b). The coefficients in these equations consist

of functions that have already been found. Eliminating W2 and employing (30),

(38) and (39) we get for Hl

_ dH1 2 E U -d't"

=

+ ° h( -) + l.n.. O

fj.~

/ 2 (1 + 8E 2 + 12U2) 4E( 1-- U2) 5 /4 (41) •

.

.

This differential equation has left-hand side of the same form as (36), and ean

therefore be solved by the method of variabIe eoefficients by putting

=

whieh gives with the aid of (9a)

+

1 'A(±) 0 2 _2 1 +

8E

.

+ 12U

8EtJ

2 (1-

tJ1

3/2 =

dU

di

Henee we find on integration of (43) that the function Hl is

=

+-(±) iAo (42) (43) (44)

where A(±) denotes a pair of additional eonstants of integration. The

magni-tude of

t-f

₁ diminishes during the motion, but at small

TI

its deerease is slower

than that of Ho.

The seeond term

L::.

_{2 of the series for the oscillatory}

fiight-path angle follows now at onee on introducing the funetions Ho and Hl Lnto the

equation (29b). The resulting expression is

=

(45)

This term first deereases in magnitude and then inereases as U beeomes

small. However, its decrease does not imply that the oscillations would

diverge, sinee the solution eeases to be valid in the final part of the trajeetory

where

U

is low. As the equilibrium-glide approximation does not hold at low

veloeities, the same limitation applies to the oseillatory solution. It may also

be noted that the term

A

_{2 inereases at smal.!:}

tJ

less rapidl;y than the

equili-brium flight-path angle

~

whieh grows like U- 2 .

Some further terms of the oscillatory solution are given in Seetion 6.

The constants

A~

±) the initial conditions of the general

and A

\±)

are determined in Section 5 from solution for terms of orders kO and k- 1

respectively. The general asymptotic solution must contain th~ee arbitrary constants for each order k- n , since terms of each order have to satisfy separately three initial conditions resulting from <t8). For each n the oscillatory solution supplies two constants

A~

±) , and the third is introduced by the particular solution (Section 4).

To $tate the oscillatory solution in a real form, we add the series (21), (22), (29) taken with F = + i lp to those taken with F = - i lp , and represent the constants by

=

=

.!.

(a + 2 n 1 - (a -2 n (n

=

0, 1, 2, ... ) ib ) n (46)

where a n and bn are real constants. Denoting by k-n

h~,

k-n

v~,

:

k-n

Ó~

the terms of the resultant series, and using superscripts (+) and (-) to in-dicate the functions Hn. Wn' An that correspond to F

=

+ i lp and:f

= -

i

ti

respectively, we have

v n':<

=

W(+)e+ik~ + W( -") e -ik ~

n n'

(' ;'<

On (47)

Accordingly, the asymptotic solution describing'oscillatory deviations from equilibrium glide is

(48a)

where

h~

=

=

= -3/2 U -3/2 [

+ U

7

al cos(k lf) - _{b l sin(k /I)}

(1- U 2)3 4

=

ij

1/2 = (48c) (49a) -(49b) (49c) (49d) (4ge)

and where the functions U and

lP

are given in (lOa) and (33). Expressions

f?~ the terms h~, v-~ and b~ are inc1uded in Section 6.

can be determined by identifying the forces to which the terms of the solution are due. For this purpose we denote

L

=

ÀU 2 en., -~

N

=

L + U2 - 1

D

=

e - ""rt..

(50)

so that L represents the lift, D the drag, the weight of the vehicle is taken as unity, and N represents the net normal force arising when the centrifugal

force U2 and the lift do not balance the vehic1e' s weight. From the equations

of motion it follows that the differential equations (19) which govern the oscillatory solution have the form

,I,

(

d ~/"

( aD

)*

~D

) ,I, 0>:< - "- - + - - v + h'"

₌

₀ d?'

èlu

à?t.

-U

d F.':< +

(~~)

,I,

+ (

~~)h*

-, -v = 0 d'l:" (51) d'C

where the bars indieate quantities taken at the equilibriu.m glide N

=

O. The terms of the solution ean now be related to the forces by repeating the steps made in their derivation but u.sing the form (51) instead of (19). We find then that the equation (35) for the frequency is equivalent to

2

(g~)

₎

(52)

=

This expression shows that the oscillations are due to the variation of the net normal force with aUitude at constant velocity, that is, to the atmospheric density gradient through the dependence of lift on density. Accordingly, if the vehicle has a higher (or lower) altitude than the equilibri1.~m-glide altitude corresponding to its velocity, the lift is smaller (or larger)-than that required for equilibrium beeause of the atmospheric density gradient, henee there

results a net normal force directed downwards (or upwards, respectively), and it is this normal force that generates the oscillatory motion. To identify the main factors which produce the damping of the oscillations, we derive from (51) the differential equation (36) for the amplitude term Ho getting

1 d

2 d

'2:'

(53)

Therefore the damping of altitude oscillation is due to (1) the dependence of lift and of centrifugal force on velocity, coupled with the dependence of drag on altitude, (2) the deceleration, and (3) the time variation of the derivative of lift with respect to altitude. All the three terms in the coefficient of Ho

contribute to positive damping.

4. PARTICULAR SOLUTION

The fuU (non-homogeneous) differential equations (17) for the deviations from equilibrium glide admit a large k a particular solution

, I r ' h , v , 0 of the form h'

₌

h' ₀ + k -1 hl 1 + k- 2 , _{h 2} + (54a) vi

₌

_V I -1 1 -2 I o + k v1 + k v2 + (54b)

~'

=

k -2

bi

2 + k -3~ 3 I + (54c)

where the functions

h',

v I,

~

I are independent of k. The series for

~

I does not contain

term~ of~he

ofaers kO and k-1 since from the equation (17c) it follows that these two terms must vanish. The functions

h~

, v

~

,

b'ri

are obtained by introducing (54) into the differential equations (17) and comparing the coefficients of the resulting powers of k. From (17a) we get then

I

EU

dvo d?:

_2 I _ _~ ,

I _2 I _2 I dV1 EU + 2(1- U ) v 1 _{U(1-U )h 1}

dr

I _{_2} EU _d~ dv 2 + "2(1- U _2 )v2 I U(1-U

Similarly, the equation (17b) separates into

I 2v 0 I 2v 1 - ..:-.,'2 , U (1- U )h 1

=

0

and the equation (17c) gives

-3~;' I -2 EU 2

=

-

2vo EU -3 ~' I _2 EU 3

=

-

2v₁ EU I dho d~

,

dh 1 d~ I ) h 2

=

0 2

=

-U 4(1-'Ü

2

,

)

E2

t

+ (55b)

EÜ~/

2 (55c) (56a) (56b) (56c) (57a) (57b)

( \

Accordingly, the particular solution is determined by solving (55a) and (56a) for h

~

and v

~

, calculating

~~

from (57a), solving (55b) and (56b) for

h

i

and vf ' and so on. The procedure involves again differential equations

of the first order only. By eliminating h ~ from (55a) and (56a), the following equation is obtained

E _{2Uv o} - I

=

o

Which becomes with the aid of (9a)

+

Vo

2UdU

1- U 2

Therefore the function Vi _o

= c o( 1 - U ) -2

=

0

is

(58)

(59)

where Co denotes a constant of integration. The equation (56a) yields now the

function h~

I

ho

and the term (9a), namely

bi

2

=

=

u

(60)

~'

₂ follows from (57a) on substituting (59) and (60) and using

(61 )

,

(1 - U 2') v l

=

cl (62a) 1 2cl hl

=

U (62b)

where cl is an additional constant of integration.

The terms of the particular solution given in (59) - (62), have a simple meaning: they represent variations of the equilibrium glide due to changes in its initial condition. In fact, the derivatives of the equilibrium solution (10) with respect to its constant 1:'c are

=

=

-2 1 - U E ,4k:' 2( 1 _

Ü

2 ) E 2 U 3 2

=

EU

(63)

and thus have the same forms as the terms (59) - (62). Therefore these terms do not describe motions taking place outside the equilibrium glide trajectory. Nevertheless, it is necessary to inc1ude them in the general solutiop since the arbitrary constants that they contain are needed in order to satisfy the general initial conditions of the motion. Their presence allows for the possibility that the equilibrium glide about which the motion oscillates may have a different initial point than the chosen equilibrium solution to which the motion is re-ferred.

I I

r-'

The higher- order terms h ~, v2 and 03 of the particular solution are given in .Section 6. These tertnS il'lclude carrections to·the equilibriur,n glide approximation.

5. SOLUTION TO THE SECOND ORDER

The sum of the oscillatory solution (48) and the particular solution (54) satisfies the linearized equations of motion (17) asymptotically at large k, and contains three arbitrary constants ~, _{b n, cn for each power}

k- n . Hence the sum constitutes a general asymptotic solution for small

qeviations from equilibrium glide.

It follows that subcircular atmospheric entries flown with

constant lift and drag coefficients and deviating from equilibrium glide are described by

'lL

=

7L+

ho

*

+

ho

,

+

k-1 (hl >:<

+

hl) , (64a)

u

₌

u

+ + (64b)

=

~

+

(64c)

to the secónd order with repsect to the large atmospheric-decay parameter k. The terms inc1uded in "this solution are given in (10), (49) and (59) - (62). The solution is valid when the deviations from equilibrium glide are sufficiently small so as to fulfil the linearization requirements (14).

While the magnitudes of the leading terms for the altitude and velocity deviations in (64) do not depend on k, the leading term for the·flight-·

path- angledeviation is of the order of k- 1, and the angle '( of the equilibri

-um glide is even smaller, being proportional to k- 2 Thus the motions re

-presented by the solution (61) have flight-path angles of the order of k-1 radians. As this restriction mu~\h0ld also for the initial flight-path angle

1S'i ' we assume that (fi

=

0 (k ) and denote

'6.

1

=

-1~

k L

1 (65)

so that

Ii

is of the order of kO;, Altitude deviations arising from an initial flight-path angle are proportional to the parameter

I?

rather than to the angle 't'i itself.

The initial conditions of motion (18) applied to the solution' (64) state that

at

r

= 0

=

-1 ,..., k f ~ ₁

-(66)

where (65) and (12b) are used for ((i and 'ti. These conditions yield

separate requirements for terms of each order, since the coefficients of the

powers k-n do not depend on k at "

=

O. Comparing the coefficients, we find

that the initial conditions for the terms of the solution are

at

7:-

₌

0 h >(' ₀ + vi

₌

0

~

'

;

=

-"' vi' + h' 0

u· -

1

r:

I V 1

=

("-,

-ij 2 -0

=

'11..

.

-

7/,. 1 1

u

₁

·

2 (67)

An explicit form of the initial conditions is obtained by sub stituting the ex-pressions (49) and (59) - (62) into (67). A set of equations then results for the

Ui

3/2 a o (1- 1.\2)3/4 - 2,,-(l-U· JC 1 0 ij1/2 b o (1 ... &2)1/4 q3/2 al 2c₁ (1 -

U

.

-

2/)3/4 1 . - 2 (1 - Ui )c₁ U i 1/2b . 1 ; . (1 _ U~)1/4

-U. 1

=

+

=

=

U· -₁

U·

₁

=

=

2 + EU·₁ 2 b_o q1/? [(14 + l6E2) q2 - 1 - SE9 8E(1_fJ. 2

)5/4

1 a o [(14 + 16E )-Ui 2 - 13 - SE2] SEi\1/2 (1- Ui 2)3/4 -2 4c o (1- Ui )

EU.

3 1 where

11.

.

1 is a function of U· 1 as stated in (12a).

(6Sa) (6Sb) (68c) (6Sd) (6Se) + (6Sf)

The motion is determined, of course, by its initial values

Ui, n i ,

f1.

If these values are prescribed, the initial velocity Ui of the reference equilibrium glide is not given to start with, and a question arises regarding the choice of Ui or of the equ!yalent constant ~ for the solutioIl.

In view of the linearization restrictions, Ui shquld be chosen so as to make

the initial deviations

Ui-lh

and

?\-71i(

Ui ) small, if possible. Wh en the deviations are sufficiently small, the actual choice of Ui is immaterial.

In fact, as noted in Section 4, the particular solution represents variations

of the equilibrium glide due to small changes in

'2C

or in Ui , and therefore ,its terms compensate the general solution for such cbanges. Bence the solution

(64) is essentially insensitive to the va1~e of Ui. However, from (6Sa) ~1 (60) it follows that if the difference Ue Ui is small but comparable to 1-Ui , then the magnitude of the constant 90 will be comparable to unity, and the

altitude term h

~

of the particular solution win not be small. Since in many problems of atmospheric entry Ui is close to 1 corresponding to nearly

circular initial speeds, we must choose Ui-lh to be practically zero, for

otherwise the term h ~ would become unnecessarily large. Moreover, with

Ui taken equal to Ui the solution gives a correct initial value for the frequency of the oscillation, since

thenJg{3(1-Ui~

)coincides with the correct

value Ig{3(1-Ui 2) resulting from (52). Accordingly, we choose

u·

₁

=

U· ₁

so that

=

o

The solution (64) becomes th en

-u

=

U + k - 1 ( v ':< 1 + v I) 1

and the equations (11) and (68) for the constants of the solution teduce to

=

E arctanh(ud

-/ ) f . _

1t.

.

't.. 1 1

u.

1 / 2 b 1 0

=

(69) (70) (71a) (71b) (71c) (72a) (72b) (72c)

where U_i5 / 2 b₁ (1- U i2)1/4 cl

=

'n

i

=

log (72d) 13, -

8E~l

·

.. ,

=

- - -2 -I-·E (72e) Ui ri 2 E(1- Ui ) (72f)

(

jW i 2 ) 1- Ui2 (73)

Figures 1-3 eompare the solution (71) with some results eom-puted by numerieal inte~ration of the equations of motion and given by

Chapman(l), Campbell( ) and Loh(3). The agreement is good when the de-viationf3 from equilibrium glide are smali. When the deviations are rather large, as in Figure 2, the s01ution still describes adequately the phase

variation of the osçillations, but gives less aecuraey for the amplitude. The deerease of aeeuracy at large amplitudesean probably be attributed to the linearization on which the solution is based. Figure 4 illustrates the effects of initial altitude, flight-path angle and velocity deviations on the sub sequent motion, as ealeulated from the solution (71).

6. ADDITIONAL TERMS

Further terms of the asymptotie solution ean be derived by following the procedures putlined in Sections 3 and 4. From the equations

{~6),

(27) and (47) we find that the oscillatory terms

v~

, h; and

b

3

"

a;re

+

U

1/2 (22 + 16E2)

U

2 - 5 8E(1- U 2 ) 3/4

-3/2

U / [ a l sin(k

ti)

+ b 1 cos(kll

)1

(l_U2 )14

j

+

+ C" ,:e D 3 b o sin(k

IJ

>J

2 _2 3/4 E (l-U ) (74b)

=

-2 2

(14+16E)U -13-BE [a cos(kllJ) -_{~ b} (lil)]

1 sin k~

BEU1/2(1_U2)3/4 1 ·

+

(74c)

where _{a2 and b 2 are constants introduced by integration, and Bj denote for}

brevity

64E4 + BOE2 + 119 4 2

B ₁

=

_{B 2} = 64E + 240E +~O5

12BE 2 64E2 13 64E 4 _{+ 144E}2 + 145 B4 64E 4 + 272E2 + 273 =

=

64E2 3 128E2 (75)

I , cl

For the terrns ~, h 2 and IJ 3 of the particular solution we obtain from the

equations (55) - (57), taking Co = 0 as in (70), =

+

=

4 + 2E 2 _ (4 +-3E2 )

U

2 E2

5

arctanh

5

+

+

4(3

+

E 2 ) - 6(2

+

E2)

5

2 E 2 Ü2 (1- ( 2) -2 4c 1 (1 - U ) EU3

+

(76a) (76b) (76c)

,

where 02. is an additional constant of integration. The expressions for· v2

and h

2

take into account non-homogeneous terms of the differential equations

(17) and therefore they contain corrections to the equilibrium-glide approxi-mation. The constants a2, b 2 and c2· are determined by the initial conditions

at

'2:"

=

0 v2 >:< + v2 I

=

0 ,

=

0

(77)

Since the terms in (74) and (76) enter the solution as coefficients of k" 2 ··and

k- 3 ,th~ do not contribute significantly at large k. They become appreeiable

when U and E are low and when the deviations are relatively large, but

then the solution itself ceases to be valid. Within the validity limits of the analysis, it is sufficient to include in the solution the terms given in (71).

1. Chapman, D. R. 2. Campbell, G. S. 3. Loh, W. T. H. 4. Sanger, E. 5. Erdelyi, A. :...·REFERENCES

An approximate Method for Studying Entry into Planetary Atmospheres.

NACA Technical Note 4276, May, 1958.

Long Period Oscillations During Atmospheric Entry. ARS Journal, American Rocket Society,

Vol. 29, No. 7, pp. 525- 527, July, 1959. Dynamics and Thermodynamics of Planetary Entry. Prentice-Hall, 1963.

Raketen-Flugtechnik. R. Oldenburg, 1933. Asymptotic Expansions. Dover, 1956.

"

Figure 1 .

numerieal solution

present s olut ion , eq. (71)

Comparison with Chapman' s entry from deeaying orbits angle of de seent. 2 k

=

30 , mg/SC D

=

50 lb/ft . numerieal with E = 1 solution

[1]

for and 1 0 initial

FIG. 2

Altitude

Velocity

numerical s olut ion

present solution, eq. (71)

Comparison with a numerical solution given by

Campbell [2] for entry from GO miles

2

r

.= 0 , V.= 25750 ft/sec., mg/S= 18 lb/ft.,

1 1

C

Figure 3 .

E 2

numerical solution

present solution , eq. ( 71 )

E 1

Comparison with 2 numerical mg/ SC

_n

= 3.2 1b/ft, k= 30 ,

solutions given by Loh

[3]

fT 1= 23, 500 ft.

r

i :: 0.065 (I

o i 0.165

:: - 0.035

Figure 4. Atmospheric entries close to equilibrium glide .

(~) Effects _ of initial fligh~-path aniIe. _ _

~

•

U. = 0.990 1 Ui = 0.995 U. = 0.985 1

Figure 4. Atmospheric entries close to equilibrium glide.

(b) Effects of initial velocity .

..

...

'7

i = 14.40

1

i = 14.80

'i

i = 14.00

Figure 4. Atmospheric entries close to equilibrium glide.

(c) Effects of initial altitude.

-4

°

k=30, E=2, CLSrfo/m=7.27xlO, ri=.065, Ui=.990