COS,cj> y,(cp) = a-^jtl)-(i+{e^) COS f-je^. cos zcf -h SoCfstnc^. W.Ccj» = ( 2 - e â4e :-ie :)-(2 -4-e .-4 -|e^-^^e ,^)cpsc(> 4- sm ^(^ â je.-c^^ cos (Æ> ...... Page 213 ...
A MULTI VARIABLE APPROACH TO PERTURBED AEROSPACE VEHICLE MOTION
P a u l u s Theodorus L e o n a r d
Maria
v a n Woerkom
A DISSERTATION PRESENTED TO THE FACULTY OP PRINCETON UNIVERSITY IN CANDIDACY POR THE DEGREE OP DOCTOR OP PHILOSOPHY
RECOMMENDED POR ACCEPTANCE BY THE DEPARTMENT OP AEROSPACE AND MECHANICAL
JUNE , 1972
SCIENCES
Copyright
by
P a u l u s Theodorus L e o n a r d M a r i a van Woerkom 1972
ii
. Por t h o s e who
are seen i n g l i m p s e s ,
or d w e l l i n l a n d s u n e x p l o r e d , o r l i v e i n p o e t s ' songs.
Rabindranath
iii
Tagore
ABSTRACT The
I n v e s t i g a t i o n s presented
on the development of u n i f o r m l y
in this dissertation
v a l i d , asymptotic,
focus
approxi-
mate s o l u t i o n s to n o n l i n e a r , o r d i n a r y d i f f e r e n t i a l
equations
containing a small parameter. Systematic
use
i s made of t h e M u l t i V a r i a b l e Approach.
T h i s method i s based on the extension
"method of e x t e n s i o n " ,
of the Independent v a r i a b l e i n t o a m u l t i
a l domain of g e n e r a l l y n o n l i n e a r , independent
involving dimension-
" c l o c k s " , and
e x t e n s i o n of the dependent v a r i a b l e i n t o an a s y m p t o t i c The r e q u i r e m e n t of u n i f o r m v a l i d i t y
of t h e a s y m p t o t i c
series. solution
d e t e r m i n e s the e x p l i c i t dependence of c e r t a i n f u n c t i o n s c u r r i n g i n the a s y m p t o t i c and
the e x p l i c i t
by The
invoking
s e r i e s on a c e r t a i n number of c l o c k s ,
dependence of t h o s e
independent v a r i a b l e ; any
oc-
c l o c k s on t h e
remaining indeterminacy
original i s removed
the c o n c e p t of " r e s t r i c t i o n of number of clocks'.'
method i s shown to g e n e r a l i z e v a r i o u s well-known asymp-
t o t i c perturbation The
techniques.
M u l t i V a r i a b l e Approach i s a p p l i e d t o t h r e e
t a n t problems i n a e r o s p a c e
v e h i c l e m o t i o n , v i z . : t h e motion
of the l i n e a r , harmonic o s c i l l a t o r w i t h s l o w l y frequency;
impor-
varying
the o r b i t a l motion of a s p a c e v e h i c l e i n the
e q u a t o r i a l plane of an a e r o s p a c e
of an o b l a t e p l a n e t ; and
t h e o r b i t a l motion
v e h i c l e around a s p h e r i c a l l y
iv
symmetric p l a n e t .
and
p e r t u r b e d by l i f t -
spherically for
and
drag f o r c e s g e n e r a t e d
symmetric atmosphere s u r r o u n d i n g
both c o n s t a n t d e n s i t y atmosphere and
d e n s i t y atmosphere. The compared a n a l y t i c a l l y through
by
a
that planet -
quasi-exponential
approximate s o l u t i o n s obtained
and
are
numerically with s o l u t i o n s obtained
a p p l i c a t i o n of d i f f e r e n t
techniques. A
critical
r e v i e w of r e l a t e d a n a l y s e s and/or r e s u l t s a v a i l a b l e
in
liter-
a t u r e , i s presented f o r f u r t h e r comparison. The
t h r e e problems o u t l i n e d above a r e f o r m u l a t e d
mathe-
m a t i c a l l y i n t h e form o f t h e l i n e a r , harmonic o s c i l l a t o r t u r b e d by n o n l i n e a r t e r m s . The
M u l t i V a r i a b l e Approach i s
e q u a l l y a p p l i c a b l e to d i f f e r e n t i a l e q u a t i o n s of c l a s s e s ; s e v e r a l examples a r e d i s c u s s e d .
V
different
per-
ACKNOWLEDGEMENTS
This
i n v e s t i g a t i o n v/as s u p p o r t e d
f i n a n c i a l l y by t h e
O f f i c e o f Space S c i e n c e and A p p l i c a t i o n s ,
NASA
Headquarters,
under NASA Grant NGR 31-001-152. Mr,J.W.Haughey was t h e NASA Program Monitor. F i n a n c i a l support was a l s o p r o v i d e d through t h e Fulbright
- Hays
U,S, Government Program
, i n t h e form o f
a t r a v e l g r a n t . The I n s t i t u t e o f I n t e r n a t i o n a l E d u c a t i o n was the Program Monitor. The through
c o m p l e t i o n o f t h e i n v e s t i g a t i o n was made p o s s i b l e t h e cooperation of s e v e r a l persons. I n p a r t i c u l a r ,
the a u t h o r w i s h e s t o acknowledge : Her M a j e s t y J u l i a n a , Queen o f The N e t h e r l a n d s , f o r h a v i n g granted the opportunity f o r uninterrupted Professor
Dr. P,M,Llon, f a c u l t y a d v i s o r ,
research; f o r continued
I n t e r e s t , t r u s t , and s u p p o r t ; Professor
Dr, W.A,Sirlgnano, and Dr, P , T , 6 e y l i n g o f The B e l l
Telephone L a b o r a t o r i e s ,
f o r deepening
t h e a u t h o r ' s under-
s t a n d i n g o f t h e s u b j e c t . They, a s w e l l a s Dr, J,M.Gormally and Dr. R . P r l n g l e , J r . , s u g g e s t e d
Improvements i n t h e manu-
script; Mrs.
A l e x a n d r a B . S h u l z y c k i and Miss Mary C . A l l a n , f o r s k i l f u l
a s s i s t a n c e i n numerical
studies;
vi
Miss Rosemary E . H e i n z , f o r p a i n s t a k i n g l y d r a f t i n g t h e many figures; Miss P r a n c e s
Allison, forefficient
secretarial assistance;
S e n o r i t a I r i s Arbona T o r r e s , f o r t y p e w r i t i n g t h e b e t t e r p a r t of t h e m a n u s c r i p t ; Mrs. T h e r e s a
Buttenbaum, f o r p r o v i d i n g
a s s i s t a n c e throughout t h e a u t h o r ' s
administrative
years at Princeton
University. Some s u g g e s t i o n s made by Dr. R.V.Ramnath a t t h e i n i t i a l s t a g e of t h e i n v e s t i g a t i o n , a r e a p p r e c i a t e d . S p e c i a l t h a n k s a r e due t o t h e a u t h o r ' s u n f a i l i n g confidence
was a c o n s t a n t
source
parents,
whose
of e n c o u r a g e -
ment and s t r e n g h t . T h i s d i s s e r t a t i o n c a r r i e s Number 10'17-T i n t h e r e c o r d s of t h e Department o f A e r o s p a c e and M e c h a n i c a l Princeton University.
vii
Sciences,
TABLE OF CONTENTS Page TITLE
PAGE
ABSTRACT
. .
1 .
•
Iv
ACKNOWLEDGEMENTS
vll
TABLE OF CONTENTS
viii
L I S T OF FIGURES
CHAPTER ONE. 1.1 1.2 1.3 l.il CHAPTER TWO.
2.1 2.2 2.3 2.4 2.5 2.6 2.7 CHAPTER THREE.
3.1 3.2 3.3 3.4. 3.5 3.6 3.7 3.8 3.9 3.10
.
xl
INTRODUCTION
1.1
Background Dissertation objectives D i s s e r t a t i o n synopsis References
1.1 1.9 1.°10 1.12
DEVELOPMENT OP THE MULTI VARIABLE APPROACH
2.1
Introduction Fundamental c o n c e p t s and t h e One V a r i a b l e E x p a n s i o n Method Two V a r i a b l e - and M u l t i V a r i a b l e E x p a n s i o n Methods Hybrid Multi V a r i a b l e Expansion Methods G e n e r a l i z e d M u l t i p l e S c a l e s Approach M u l t i V a r i a b l e Approach References HARMONIC OSCILLATOR WITH SLOWLY VARYING FREQUENCY . . .
.
Introduction One V a r i a b l e E x p a n s i o n Method Two V a r i a b l e E x p a n s i o n Method G e n e r a l i z e d M u l t i p l e S c a l e s Approach M u l t i V a r i a b l e Approach Some r e s u l t s from l i t e r a t u r e Comparison References Figures Appendices
vlll
2.1 2.5 . 2.11 2.15 2.22 2.28 2.33
3.1 3.1 3.4 3.6 3.16 3.21 3.30 3.33 3.37 3.39 3.44
CHAPTER POUR.
4.1 4.2 4.3 4.4 4.5
4.6 4.7 4.8 CHAPTER P I V E . 5.1 5.2' 5.3 5.4 5.5 5.6 5.7 5.8 CHAPTER S I X .
6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 CHAPTER SEVEN.
7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9
DEVELOPMENT OP THE THREE DIMENSIONAL EQUATIONS OP MOTION POR AEROSPACE VEHICLES
4.1
Introduction ^• G r a v i t a t i o n a l - and aerodynamic a c c e l e r a t i o n vectors j D e r i v a t i o n of t h e e q u a t i o n s of motion 4.13 Two d i m e n s i o n a l motion ' • ^9 Remarks on n u m e r i c a l i n t e g r a t i o n 4*23 References ;'-24 Pigures 2'29 Appendices 4.33 THE EQUATORIAL PROBLEM
.
.
,
.
5.1
Introduction M u l t i V a r i a b l e Approach A n a l y t i c a l comparison Numerical comparison Summary References Pigures Appendices
5.1 5.5 5.13 5.25 5.27 5.28 5.31 5.36
AERODYNAMICALLY PERTURBED PLIGHT THROUGH A CONSTANT-DENSITY ATMOSPHERE . . .
6.1
Introduction M u l t i V a r i a b l e Approach D i s c u s s i o n of r e s u l t s Comparison w i t h l i t e r a t u r e Numerical comparison Summary References Pigures Appendices
6.1 6.3 6.10 6.13 6.l6 6.19 6.20 • 6.21 6.29
AERODYNAMICALLY PERTURBED PLIGHT THROUGH A VARIABLE-DENSITY ATMOSPHERE . . .
7.1
Introduction M u l t i V a r i a b l e Approach D i s c u s s i o n of r e s u l t s Comparison w i t h l i t e r a t u r e Numerical comparison Summary References Pigures Appendices
7.1 7-3 7.14 7-17 7.26 7.29 7.30 7.32 7.43
ix
CHAPTER EIGHT. 8.1 8.2 8.3 8.4
SUMMARY AND RECOMMENDATIONS Summary Further Investigations Recommendations References
L I S T OP FIGURES F i g u r e No. 3.1
Caption F a s t time for cu^
Page
(p and t r a n s f o r m e d /^--Uy
fast
3.2
Function
3.3
Exact
3.4
E n v e l o p e s o f d e v i a t i o n s A U-^^^ , and A Umv ( S = 0.025)
3.5
- i - lit)- "^-^^-^^ , where X s o l u t i o n Ug (£ = 0.025)
Envelopes of d e v i a t i o n s and
A U„v
time (Po / ' x=e^^^
3.39 3.39 3.40
> 3.41
a U^-y , '^^Ugms ?
( £ = 0. 1 )
3.42
3.6
Oscillatory, ballistic
3.7
Angle o f a t t a c k OC v e r s u s
4.1
C o o r d i n a t e systems
4.2
Aerodynamic a c c e l e r a t i o n v e c t o r s
4.3
Comparison o f d e n s i t y model ( 4 . l 4 ) w i t h R e f . 4 . 2 9 . Match p o i n t s a t 200 and 300 km. a l t i t u d e ( y = 0 . 8867 X 10 9 m.)
4.31
Comparison o f d e n s i t y model ( 4 . 1 4 ) w i t h R e f . 4 . 2 9 . Match p o i n t s a t 6OO and 700 km. a l t i t u d e ( y = 0.3797 X 10^ m.)
4.32
5.1
Phase p l a n e
5.31
5.2
Quasi-elliptic
5.3
Exact
5.4
E n v e l o p e s o f d e v i a t i o n s AV/ov , /^My^ , aW„v , A Wp^ii, , and A Wg^^ (£ = 0 . 0 1 , e ^ = 0.1)
5.34
E n v e l o p e s o f d e v i a t i o n s A Wov j a Wtv , A W„v > AWp^ip^ , and A W g ^ ^ ( £ = 0 . 0 1 , e„ = 0.9)
5.35
4.4
5.5
flight
3.43
time t
3.43
4.29 Q^o
diagram trajectory
solution
(£ = 0 , 0 1 ,
xi
4.30
5.32 e^ = 0.9)
5.33
F i g u r e No. 6.1
6.2
Caption E f f e c t of the s m a l l angle f u n c t i o n of c e n t r a l a n g l e tricity Exact solution (£ = 10-\
6.3
6.4
6.6
6.7
Kg v e r s u s
= 10-2, C l / C p
E f f e c t of l i f t 10-\
7.1
7.2
7.3
7.4
7.5
7.6
6.21 central
angle
= 0)
6-22
angle 6.23
on
e^ = 10'^
Cl/Co
= 1)
.
6.25
D e v i a t i o n A K versus c e n t r a l angle ( £ = 10-^ e^ = 10-2, c u / c p = 0 )
6.26
Deviation
a U versus
Deviation ( d = 10"^
central Cl/Cp
angle
= 0)
6.27
AU versus c e n t r a l angle = 10'^ Cu/co = 0; R e f . 6 . 1 )
6.28
B e h a v i o r o f e ' I ^ ( ^ ) as a f u n c t i o n of t h e parameter ^
7.32
B e h a v i o r of I^^C^) a s a f u n c t i o n o f t h e p a r a m e t e r s m and i
7.33
Exact solution ( £ = lO"'', e ^ =
K^. v e r s u s c e n t r a l a n g l e 10-2, y = i.33i|6, c i / c p = 0 )
7.34
E x a c t s o l u t i o n U^ v e r s u s c e n t r a l a n g l e (£.= 10-\ = 1 0 ' ^ ^ = 1.3346, Ct/cp = 0 )
7.35
E f f e c t o f l i f t on K ^ ( £ = 10-\ e^^ = 1 0 - ^ ^ = 1.3346, ci/co
=1)
7.36
= 1)
7.37
E f f e c t o f l i f t on K e (£.= 10-^ e^^ = 10"^^^= 1.3346,. cu/cd = - D
7.38
E f f e c t of l i f t
on Ue
(£,= 10-\ e^^= 10-^ (?= 1.3346, ci/cp
7.7
6.24
E f f e c t o f l i f t on Ua ( £ = lO-'', e^ = 10'', C l / C p = 1 )
(
O
unrWmly
m
t
Ta-t The tude
symbol
i s t h e c l a s s i c a l Bachmann-Landau o r d e r o f magni-
symbol ( e . g . , R e f . 2 . 7 ) . The
sequence
( 2 . 3 ) i s c a l l e d a uniformly v a l i d
to t h e e x a c t s o l u t i o n
Xt^^)
a c c u r a t e to the order N i f the
d i f f e r e n c e between t h e e x a c t s o l u t i o n and remains
of t h e o r d e r Y
u n i f o r m l y i n t h e time Then, one
where t h e sum The r i g h t
approximation
as sio
the asymptotic
sequence
( t f i x e d ) ; mathematically
domain.
can w r i t e :
21
is
called
t h e N-th
order approximation
hand s i d e i s known as t h e P o i n c a r e ' a s y m p t o t i c
t o )Cc^;ê). expan-
s i o n of X(é5Ê)with r e s p e c t t o t h e a s y m p t o t i c sequence Returning to ( 2 . 2 ) , viewed
:
t h e power s e r i e s
a s an a s y m p t o t i c e x p a n s i o n
I n view of ( 2 . 5 ) s
one
arrives
f o r m u l a t i o n of t h e N-th
In
£,•
i n t h e s e n s e of
will
. now
be
PolncareV
a t t h e f o l l o w i n g well-known
order approximation
Ha©
2.7
to the exact s o l u t i o n
:
containing
the sequence o f gauges
approximation Introduces t h e domain
t^Co,T]
. The
an e r r o r o f the o r d e r £.
. prom t h e P o i n c a r e - C a u c h y theorem
i n f e r s t h a t t h e domain of u n i f o r m v a l i d i t y decreases now
be
towards z e r o . The
reformulated
uniformly It The
throughout
i n c r e a s e s as
one Ê
c o n d i t i o n of u n i f o r m v a l i d i t y
( 2 . 4 ) can
:
i n t e-co^r] (n = 1 , 2 , . . . , N ) .
f o l l o w s t h a t t h e r a t i o )(n/x„.i (^) must r e m a i n bounded i n t . c o n d i t i o n of u n i f o r m v a l i d i t y
a "boundedness The
can t h e r e f o r e a l s o be
condition".
e x p a n s i o n ( 2 . 6 ) i s known under s u c h names as
fundamental p e r t u r b a t i o n
expansion, i n i t i a l l y
expansion, Poincare' expansion, Poisson
valid
t h e method o f a p p l i c a t i o n i s c a l l e d
:
perturbation
e x p a n s i o n , and
I n t h i s d i s s e r t a t i o n i t i s r e f e r r e d t o as t h e One s i o n , and
called
so
on.
V a r i a b l e Expan-
t h e One
Variable
E x p a n s i o n Method. A p p l i c a t i o n of t h e method i s s t r a i g h t f o r w a r d . A t t e m p t i n g f i n d an a p p r o x i m a t e s o l u t i o n t o substitute
( 2 . 1 ) accurate
( 2 . 6 ) i n t h e r i g h t hand s i d e and
£ = 0 . C o l l e c t i n g terms c o n t a i n i n g them e q u a l
to the order
l i k e powers o f £
and
t o z e r o , r e s u l t s i n a s y s t e m of N d i f f e r e n t i a l fundamental assumption t h a t
solved exactly for £
=0,
2.8
to completely
setting equa-
( 2 . 1 ) can
the s y s t e m b f N d i f f e r e n t i a l
solved r e c u r s i v e l y , leading
E*^ ,
expand t h a t s i d e about
t i o n s . I n v i e w of t h e
can be
to
be
equations
determined
e x p r e s s i o n s f o r XnCb v a l u e o f £.
( " = 0,1,2,...,N
s u c c e s s i v e l y ) * Por a given
, t h e e r r o r i n v o l v e d i n t h e a p p r o x i m a t i o n w i l l be N-f 1
of t h e o r d e r £ o
^
0
f o r s m a l l enough v a l u e s o f t ; u s u a l l y , f o r
(Ve) .
I n many c a s e s , t h e e r r o r w i l l grow t o l a r g e r v a l u e s a s t i n c r e a s e s . T h i s s e c u l a r t y p e o f n o n u n i f o r m l t y may o r may n o t be o f i m p o r t a n c e , depending
on t h e time domain o f i n t e r e s t . I n
t h e s t u d y o f t h e p e r t u r b e d motion o f p l a n e t s . i n t h e s o l a r
system,
f o r example, t h e domain o f u n i f o r m v a l i d i t y may be o f t h e o r d e r of s e v e r a l d e c a d e s , w h e r e a s ,
I n t h e s t u d y o f t h e p e r t u r b e d motion
of f a s t - o s c i l l a t i n g p a r t i c l e s , t h e domain o f u n i f o r m might
validity
be o f t h e o r d e r o f m i l l i s e c o n d s o n l y . The l a t t e r time
span
may be f a r t o o s m a l l from t h e p o i n t o f view o f a p h y s i c i s t , w h i l e t h e former time span may be q u i t e s a t i s f a c t o r y
from t h e
point of* view o f an a e r o s p a c e e n g i n e e r , b u t a g a i n u n s a t i s f a c t o r y from t h e p o i n t o f v i e w o f an a s t r o n o m e r . Thus a r o s e t h e d e s i r e t o improve t h e One V a r i a b l e E x p a n s i o n Method, i n o r d e r t o I n c r e a s e t h e domain o f u n i f o r m v a l i d i t y solution. A significant
of the approximate
s t e p i n t h a t d i r e c t i o n was made through
t h e i n t r o d u c t i o n o f t h e Two V a r i a b l e E x p a n s i o n Method, t o be presented i n the next
Section.
* I t s h o u l d be noted t h a t t h e r e a r e no a p r i o r i r e a s o n s why t h e d i f f e r e n t i a l e q u a t i o n ( 2 . 1 ) s h o u l d be l i n e a r f o r 6 = 0 . An example o f a n o n l i n e a r " o n e , o c c u r r i n g i n f l u i d i s g i v e n i n Ref.2.8,p.170.
2.9
mechanics,
Por more d e t a i l e d
d i s c u s s i o n s of the elements
of a s y m p t o t i c
t h e o r y , t h e r e a d e r i s r e f e r r e d t o t h e fundamental
studies
of
Poincare'', p u b l i s h e d i n t h e A c t a M a t h e m a t i c a ( R e f s . 2 . 4 , 5 ) and i n h i s t h r e e volumes "Les Methodes N o u v e l l e s
..."(Ref.2.6),
In /
2.9
p a r t i c u l a r V o l . 1 ; furthermore, to the c o n c i s e study of E r d e l y l as w e l l as t o t h e books by C o l e ^ * 7 and Van
' ,
Dyke^'-^^, both 2
c o n t a i n i n g worked-out s i m p l e e x a m p l e s . The
s t u d i e s by G o r m a l l y
11
'
must a l s o be mentioned. I n l i t e r a t u r e d e a l i n g with aerospace v e h i c l e
motion,
several i l l u m i n a t i n g applications are a v a i l a b l e ; i n p a r t i c u l a r , the determination of : 1. t h e t r a j e c t b r i e s o f s a t e l l i t e s o b l a t e n e s s and
drag
around
( G e y l i n g ^ * - ' - ^ * ^ ^ ) , by
the E a r t h , perturbed a magnetic
(Geyling^*"'"^, and Westerman^'"''^), and by s o l a r 2. t h e t r a j e c t o r y
o f a low t h r u s t
field
pressure(Levin^'-^^)
s p a c e v e h i c l e around
the
Sun
( W e s s e l l n g ^ * -^7) j 3 . the t r a j e c t o r y
of a r e - e n t r y v e h i c l e t h r o u g h
atmosphere (Hanin^*''•^, and 4. t h e t r a j e c t o r y
the E a r t h ' s
Shen^*-"-^);
o f an a n t i - a i r c r a f t
p r o j e c t i l e , p e r t u r b e d by
2.20
atmospheric
drag
(Milenski
V
);
i n a d d i t i o n t o S e c t i o n s 3 . 2 , 5 . 3 . 2 , a n d Appendix 6 A o f dissertation.
2.10
by
this
2 , 3 Two
V a r i a b l e - and
M u l t i V a r i a b l e E x p a n s i o n Methods
A recently introduced,
p o w e r f u l and
elegant
asymptotic
e x p a n s i o n method, d e s i g n e d t o I n c r e a s e t h e domain of u n i f o r m validity, The
i s the Two
V a r i a b l e E x p a n s i o n Method
method i n v o l v e s the i n t r o d u c t i o n of two
a " f a s t t i m e " to = t
associated with
v a r i a t i o n s i n the dependent
perturbed
"times"
;
Variable
the unperturbed,
( m o t i o n - ) v a r i a b l e ; and
» associated with
- bI
distinct
, a l s o o c c u r r i n g i n t h e One
E x p a n s i o n Method, and
(Refs.2.7,21).
the slow m o d i f i c a t i o n
fast
a "slow t i m e " of the
motion t h r o u g h t h e i n f l u e n c e of t h e p e r t u r b a t i o n
unterms.
Mathematically formulated : N
%a;E) where
=
X(é...:£)
= -
H
e-^-VnCU
d e n o t e s t h e dependence on the two
H-
Ö
t i m e s éot=^
A p p l i c a t i o n of the method i s s t r a i g h t f o r w a r d . e x p a n s i o n ( 2 . 8 ) i n ( 2 . 1 ) and terms c o n t a i n i n g
expand
l i k e powers of
£,
..
around
and
on
s o l v e d r e c u r s i v e l y . The
to f o l l o w s
i^z^et'
Substitute
s e t t i n g them e q u a l
to
equations
e x p l i c i t dependence of
from i n t e g r a t i o n of t h e
e q u a t i o n f o r ) ( ^ . The
and
£. = 0 . C o l l e c t i n g
z e r o , r e s u l t s i n a s y s t e m of N p a r t i a l d i f f e r e n t i a l t h a t can be
(2.9)
(n-th)
differential
e x p l i c i t dependence of X^^ oi^ t^ f o l l o w s
from the boundedness c o n d i t i o n :
(to,t)
bounded
( R e f . 2 . 2 1 s t a t e s t h a t e a c h X« s h o u l d this i s incorrect).
•
2.11
|or
i eZo.o^)
be bounded; i n g e n e r a l ,
(2.g)
G e n e r a l l y , t h e N-th o r d e r a p p r o x i m a t i o n t h u s remains
uniformly v a l i d
i n t h e domain
obtained
^ ^OCi^jCRefs . 2 . 2 1 , 2 2 ) .
T h i s c o n s t i t u t e s a c o n s i d e r a b l e Improvement o v e r t h e domain o f uniform v a l i d i t y Method. I f I t
a s s o c i a t e d w i t h t h e One V a r i a b l e E x p a n s i o n
t u r n s out t h a t t h e r a t i o Xn+
c a n a l s o be accomodated w i t h Por
the case w < o
leads to i d e n t i c a l
, and t h e c a s e o f i m a g i n a r y
f u n c t i o n tvcsif)-
e.
s o l u t i o n I s obtained.
mate s o l u t i o n s f o r t h i s p a r t i c u l a r f r e q u e n c y
Thus, f o r t h i s c a s e , t h e equation
9
' ,
suggested
The a p p r o x i -
f u n c t i o n a r e com-
pared with t h e exact s o l u t i o n , both a n a l y t i c a l l y
coceif) — e
tu
some minor changes i n t h e a n a l y s i s .
the s p e c i a l frequency
by Cheng and Wu^'\an e x a c t
y o
and n u m e r i c a l l y .
(3.1) i s s u b j e c t e d t o ( R e f . 3 . ^ ) :
Ufo) =1 0
»
(o) —
1
oiU> I t may be noted t h a t t h e f r e q u e n c y
f u n c t i o n s u g g e s t e d by
Cheng and Wu i s not t h e o n l y one l e a d i n g t o an e x a c t s o l u t i o n to
( 3 • 1 ) . O t h e r such f u n c t i o n s a r e , e.g.,
3.2
cs.a)
where 0(,.
»ft» 'V,»
a r e c o n s t a n t s , and m I s a p o s i t i v e
The f o r m e r two f u n c t i o n s l e a d t o e x a c t B e s s e l f u n c t i o n s , while the l a t t e r terms of t h e c o n f l u e n t
alert
l e a d s t o an e x a c t
of turning points
3.3
wave
o n e - d i m e n s i o n a l harmonic
) . P o r any c h o i c e o f f r e q u e n c y
to the occurrence
solution i n
i s r e l a t e d to the Schrodinger
f o r the unperturbed, l i n e a r ,
oscillator
s o l u t i o n s i n terms o f
h y p e r g e o m e t r l c f u n c t i o n or i n terms o f t h e
Weber f u n c t i o n ( t h e e q u a t i o n equation
integer.
f u n c t i o n one must be ( v i d e S e c t i o n 3.7
).
3.2 One V a r i a b l e E x p a n s i o n Method A c c o r d i n g t o t h e One V a r i a b l e E x p a n s i o n Method, p r e s e n t e d i n S e c t i o n 2.2, one assumes N
an a p p r o x i m a t e s o l u t i o n o f t h e form :
n=o I n view o f t h e a s s u m p t i o n t h a t U)
i s analytical i n £
, one may
= 0 :
expand tu' i n a T a y l o r s e r i e s around S
which may be r e w r i t t e n a s : lAJ^(eif)
=
Y L
'^n
(E(ff
.
(3.1,)
n = 0
where t h e c o n s t a n t
i s defined as :
cecff
S u b s t i t u t i n g the expansions (3.3)
and (3.^) i n ( 3 . 1 ) ,
combining terms c o n t a i n i n g l i k e powers o f £
, and s e t t i n g them
equal to zero, y i e l d s t h e f o l l o w i n g system of d i f f e r e n t i a l e q u a t i o n s , t o be s o l v e d r e c u r s i v e l y ; e"
i
terms
-j-
[J„
=
0
CS.S")
-|
Ut
—-diLPl]^
C3.(()'
é f terms
:
1 ^
ci (f^ ' and s o on. The zeï'oeth o r d e r term to (3.5)
t
•
•••
•
i s g i v e n by t h e c o m p l e t e
'
3.,'^.
solution
where Q,^ b, a r e i n t e g r a t i o n c o n s t a n t s . The to
first
(3.6).
on y
o r d e r term Uj i s g i v e n by t h e c o m p l e t e s o l u t i o n
The dependence o f t h e r i g h t hand s i d e óf t h i s
becomes c o m p l e t e l y
equation
known a f t e r s u b s t i t u t i o n o f t h e e x p r e s -
s i o n f o r U, -(-
Ul =
^ {ci> COS f
-
-f-
(3.9)
Lf>)
i),
d if' y i e l d i n g t h e complete s o l u t i o n :
(3-9) _ ^
(|» | ( Q . c o s
- j - b„S/n I f ) - f
Thus, t h e g e n e r a l , f i r s t
One n o t e s
v a l i d i t y \elJi/\)J , i s s e e n t o i n c r e a s e mono-
. I n view of ( 3 . I D , t h e system
t o t h e c a n o n i c a l form o f R e f . 3 . 5
iR
^
1.
dcj)/
U(o)
^
to^
=
o
clip
/
^
I)
difo
^ 3.6
(0) ,
-
i
: -
0
(3.1-2) i s
( 3 . 1 5 ) can be
The
problem
and
slow v a r i a b l e s
s o l v e d by u s i n g
(J)^andand(^^ (A as t h e
fast
C3 .„~ ^i(f^)
where c _
LO
Ca(0) •
CO'h
Co(0)
aj(o)-l
4The
expression f o r
UP
wco)| .
tocl(|> . Sin
IjD
a g r e e s f u l l y w i t h R e f s . 3 . 2 and 3.5.
Por t h e s p e c i a l c o n d i t i o n s ( 3 . 2 ) , t h e s e e x p r e s s i o n s to
reduce
: ^^f
rr
•
2/,
£V The
s e c u l a r term I n Ug d e s e r v e s
frequency
s p e c i a l a t t e n t i o n . Por t h e
f u n c t i o n ( 3 . 2 ) , one f i n d s t h a t f o r I n c r e a s i n g if
transformed
fast
time ifo r e a c h e s a l i m i t v a l u e :
I n o t h e r words, a s t h e " o l d " f a s t t i m e y? grows w i t h o u t the "new" f a s t
time
that I s I n i t i a l l y v a l l d
bound,
r e a c h e s an upper bound a s y m p t o t i c a l l y
( P i g . 3 . 1 ) . One I s t h u s c o n f r o n t e d
after restriction
, the
w i t h an a s y m p t o t i c
expansion
I n t h e e x t e n d e d domain ( and t h a t ,
( ^o,t ^
t i m e s . The r e g i o n " o f u n i f o r m
), becomes v a l l d validity 3.:V3 '
f o r much l a r g e r
f o l l o w s from c o n d i t i o n
( 3 . 2 9 ) ; one h a s , i n g e n e r a l :
where t h e c o n s t a n t Ci,z(0) and (3.2),
reflects
t h e magnitudes of t h e
i s of t h e o r d e r one.
this
Por t h e s p e c i a l
frequency f u n c t i o n
(if) i s s k e t c h e d i n P i g . 3 . 2 . A maximum o c c u r s , with ^ c f ) - f ^ —
=: ^
the c o n d i t i o n of uniform v a l i d i t y
which
coefficients
can be w r i t t e n a s :
The b e h a v i o r of (|?*:3|^/3
^
OCi)
for
. I t follows that
i s s a t i s f i e d i n the r e g i o n :
c o n s t i t u t e s an Improvement o v e r t h e r e g i o n ( 3 . 1 0 ) o b t a i n e d
by t h e One
V a r i a b l e E x p a n s i o n Method.
Summarizing t h e most I m p o r t a n t 1. The Two cannot
V a r i a b l e E x p a n s i o n Method, employing l i n e a r
be a p p l i e d t o e q u a t i o n
Kevorkian"^
(3.1);
^
i s determined
a p r i o r i through
t h e n o b t a i n a z e r o e t h o r d e r approximate and
o r d e r approximate
determined
a
require-
two
through
authors
the
(2.9;3.23).
K e v o r k i a n - C o l e a n a l y s e s have been extended
a second
(j)^ ,
s o l u t i o n , whose depen-
slow t i m e s i s d e t e r m i n e d
c o n d i t i o n of uniform v a l i d i t y 2. The
time
not I n t r i n s i c t o t h e a s y m p t o t i c method. The
dence on both f a s t
times,
n o n l i n e a r t i m e s a r e needed.
and C o l e ^ ' ^ employ a n o n l i n e a r , f a s t
whose dependence on ment
a s p e c t s of t h i s S e c t i o n :
s o l u t i o n . The
here, to
second o r d e r term i s
c o m p l e t e l y though a p p l i c a t i o n o f t h e c o n c e p t
" r e s t r i c t i o n o f number of c l o c k s "
(Section 2.6).
3.14
yield
of
3. The second o r d e r term i s found t o c o n t a i n
a quasi-secular
term; i . e . , a term which becomes unbounded i n t h e cf^ -domain, but
which r e m a i n s bounded i n t h e
-domain.
The a s s o c i a t e d
r e g i o n o f unlfórm v a l i d i t y h a s been d e t e r m i n e d .
3.15
3»4 G e n e r a l i z e d M u l t i p l e S c a l e s Approach It
was p o i n t e d
out t h a t d i r e c t
a p p l i c a t i o n o f t h e Two
V a r i a b l e E x p a n s i o n Method t o t h e problem o f t h e l i n e a r , o s c i l l a t o r with slowly varying frequency, Cole did succeed
I n o b t a i n i n g an Improved
falls.
harmonic
K e v o r k i a n and
zeroeth
order
approxi-
m a t i o n U^((^„^ 7 ^ i n s t e a d o f r e p e a t i n g t h e e n t i r e a n a l y s i s , o n l y t h e most i m p o r t a n t s t e p s a r e I n d i c a t e d ( i n t r o d u c i n g minor m o d i f i c a t i o n s ) . Ramnath c o n s i d e r s t h e d i f f e r e n t i a l e q u a t i o n *
The
connection
b e t w e e n • ( 3 . 1 ) and ( 3 . 4 5 ) i s g i v e n by : UJ
»
* Ref.
3.7,
:
equation
(3.2.12)
3.16
here
Following Ref. 3.7,
one e x t e n d s
y a \ \ )
1 ^
t h e time domain o n l y : yen,,)
where t h e c l o c k s Xo^i a r e g i v e n by :
Substitution i n (3.45) differential parameter
X
l e a d s t o the f o l l o w i n g system of t h r e e
e q u a t i o n s , o r d e r e d a c c o r d i n g t o powers o f t h e l a r g e
X '
terms :
t
| ^
terms :
jk
^
/V terms :
-j-
COCTp) V
—
2i
-
1^-
O O
OMp
Zl 0
O'^j")
=
where
One now s e e k s a s o l u t i o n t o (3.47'^) i n t h e p a r t i c u l a r form :
C l e a r l y , t h i s form s a t i s f i e s t h e e q u a t i o n o n l y i f t h e c l o c k r function
s a t i s f i e s the equation ;
l e a d i n g t o t h e i m a g i n a r y c l o c k Lt
As
and
are linearly
i n (3.48)
Independent
o f t h e form :
with respect to the f a s t
c l o c k Vi , e a c h i s used s e p a r a t e l y t o g e n e r a t e t h e e x p l i c i t
depen-
dence o f t h e unknown f u n c t i o n s o(f on t h e i r argument. S u b s t i t u t i o n i
of each y. I n t h e X
equation y i e l d s
3.17
:
C o l l e c t i n g r e s u l t s and r e s t r i c t i n g a l o n g t h e p h y s i c a l (Ref. 3.7) 2
yields
line
: Ca
Cs.s-o")
e i »i 'i
' which
CO
00
c a n be r e w r i t t e n a s C3
Using t h e connection solution
. COS
Co -de
X
—
Ciy
( 3 * 4 6 ) , one f i n a l l y o b t a i n s t h e a p p r o x i m a t e
:
w 1 ' COS i
CO- cltj' —
(3.5-0*)
C
where Cj^^ a r e i n t e g r a t i o n c o n s t a n t s . The e x p r e s s i o n i s I d e n t i c a l t h e ones o b t a i n e d i n R e f s . 3 . 2 and 3 . 5 .
to
Some remarks c o n c e r n i n g t h e e r r o r a s s o c i a t e d w i t h t h e above approximation
a r e a p p r o p r i a t e . The s o l u t i o n
(3.50) s a t i s f i e s the
A* and. A e q u a t i o n s , but n o t t h e A d e q u a t i o n . U s i n g < 3 . 4 6 ) , i t f o l l o w s t h a t t h e £" and 2 but not
the £
Initially.
connection
e^équations a r e s a t i s f i e d , 2
e q u a t i o n . T h i s l e a d s t o an e r r o r o f t h e o r d e r £
I n t e r e s t i n g l y enough, t h e i n i t i a l c o n d i t i o n s do n o t
I n t r o d u c e an e r r o r , a s t h e f u n c t i o n y
h a s n o t been .expanded i n
an a s y m p t o t i c s e r i e s . U s i n g , t h e n o r m a l i z e d f r e q u e n c y
( üJ (o) = i
),
one h a s :
y(o)
y.
—
y(o)
4- y,co)
^
èlc) :^f\hlco)^ -kV^-CCi-C)
d
-h c,
AicCo)>^(oU
~ l c c < -I-
hence :
Oa)
—
=
Ca) - C ^ C A )
(3.5-0
X 3.18
The
initial
conditions
and
y^ a r e s e e n t o l e a d t o e x p r e s s i o n s
f o r C| c o n t a i n i n g terms of t h e o r d e r one n o t a t i o n , o f t h e o r d e r one expansions special
and
£
and '/^ ; o r , i n t h e
(techniques using
l e a d t o c o n s t a n t s of t h e o r d e r one
:
which a g r e e s c o m p l e t e l y w i t h t h e z e r o e t h o r d e r term of t h e above e r r o r s t u d y
mate s o l u t i o n ) g i v e n by
: with the f i r s t
( or, i n
order
approxi-
(3.40).
Summarizing t h e most I m p o r t a n t 1. The
the
1
± view
asymptotic
o n l y ) . Por
c a s e ( 3 . 2 ) , one deduces from (3.50,51)
£
a s p e c t s of t h i s S e c t i o n :
dependent v a r i a b l e i s not expanded i n t h e form of an
aymptotlc
s e r i e s . Assuming a l i n e a r
slow c l o c k Vg and
a non-
l i n e a r f a s t c l o c k TTi , t h e e x p l i c i t dependence o f t h e f a s t on t h e o r i g i n a l explicit
time i s d e t e r m i n e d
through
clock
c o n s t r u c t i o n . The
dependence of t h e approximate s o l u t i o n on t h e slow c l o c k
i s determined
through
e x p l o i t a t i o n of the l i n e a r
Independence
of t h e components o f t h e complete s o l u t i o n t o t h e l o w e s t 2 0 V , £ ) equation.
order
2. The
approxi-
>
s o l u t i o n o b t a i n e d d i f f e r s from t h e K e v o r k i a n - C o l e
mation ( f o r g e n e r a l frequency
f u n c t i o n and g e n e r a l i n i t i a l
t i o n s ) i n that the i n t e g r a t i o n constants a l s o c o n t a i n f i r s t c o r r e c t i o n terms.( K f a c t i o n of t h e two
, £^ ) . T h i s f a c t , t o g e t h e r w i t h t h e
condiorder satis-
lowest order d i f f e r e n t i a l equations, leads to
the c o n c l u s i o n t h a t the s o l u t i o n obtained c o n s t i t u t e s a
first
o r d e r a p p r o x i m a t e s o l u t i o n , c o n t a i n i n g an e r r o r o f t h e o r d e r o f
3.19
X" , or
£
. Por t h e s p e c i a l c o n d i t i o n s ( 3 . 2 ) , t h i s
agrees completely with the f i r s t s o l u t i o n o b t a i n e d by thë Two
solution
o r d e r p a r t of t h e a p p r o x i m a t e
V a r i a b l e E x p a n s i o n Method.
3.2Ö
3.5 M u l t i V a r i a b l e Approach The
M u l t i V a r i a b l e Approach ( S e c t i o n 2.6) I s now
t o t h e problem o f t h e p e r t u r b e d yield
harmonic o s c i l l a t o r
(3.1), to
a s e c o n d . o r d e r approximate s o l u t i o n f r e e o f s e c u l a r t e r m s .
Several nonlinear
c l o c k s a r e u s e d ; they
are constructed
ously using the c o n d i t i o n of uniform v a l i d i t y . l y s i s presented ing
applied
linear
i n the previous
Unlike
S e c t i o n , no a s s u m p t i o n
rigorthe anaconcern-
Independence o f t h e s o l u t i o n - c o m p o n e n t s i s needed;
t h i s property
makes t h e method e q u a l l y a p p l i c a b l e t o l i n e a r and
nonlinear d i f f e r e n t i a l
equations,
a s i s shown i n C h a p t e r s
Five
through Seven. As t h e f r e q u e n c y citly, and
f u n c t i o n contains the product
i t i s d e s i r a b l e to Introduce
slow c l o c k s
:
expli-
t h e f o l l o w i n g system o f f a s t
y •r 0
where t h e f a s t y
and s l o w c l o c k s
, w h i l e t h e slow c l o c k
^o^,,,^.., a r e n o n l i n e a r f u n c t i o n s i s a linear
t u r n s o u t , however, t h a t t h i s p a r t i c u l a r treatment
of the d i f f e r e n t i a l equation
f u n c t i o n of ^
. I t
c h o i c e of c l o c k s f o r the
of the s e c u l a r type
(3.1),
i s equivalent to a similar
choice of c l o c k s f o r the treatment
an a s s o c i a t e d d i f f e r e n t i a l
equation
the d e f i n i t i o n
0=
transforms
of
of
of the matching type. Indeed,
(3.1) to :
cl 02 and
the i n i t i a l conditions a r e transformed
J(o) zr
O
»
—
3.21
(o) -
to ;
-i
O.siv)
(3.53,54)
C l e a r l y , problem
leads t o nonunlformlties of the ( S a n d r l - )
matching t y p e . The t r a n s f o r m a t i o n 0 = ey?was I n t r o d u c e d
t o show
t h a t t h e M u l t i V a r i a b l e Approach c a n be u s e d t o d e a l w i t h ential and
equatlorts l e a d i n g t o n o n u n l f o r m l t i e s o f t h e s e c u l a r t y p e
of t h e matching type The
alike.
c l o c k s ^o,
o f t h e o r d e r o f y> ,
One t h e n must
r e w r i t e the equation as : r.2
4where t h e new f u n c t i o n
£
.. i s o f t h e o r d e r one f o r (j>
. This re-ordered equation c l e a r l y asymptotic approximation, v a l i d t i o n o f t h e One V a r i a b l e zeroeth
f o r large
y?
to a different . Indeed,
E x p a n s i o n Method ( e . g . ) l e a d s
and ( 3 . 8 7 )
Increases exponentially between ( 3 . 8 5 )
(j>
difference
and t h e o t h e r a p p r o x i m a t e s o l u t i o n s
solutions also Increases
to a
f o r l a r g e (j) . N o t i n g t h e s i m i l a r i t y
( i g n o r i n g U^^^ ) , i t t h e n f o l l o w s
small
shows t h a t t h e i r
applica-
(3.87)*.
o r d e r term which i s p r e c i s e l y o f t h e form
Comparison o f ( 3 . 8 5 )
for
leads
of the order
obtained
that t h e e r r o r of each of these
exponentially
f o r l a r g e ij? ( a s w e l l a s
, see ( 3 . 8 5 ) ) .
* A more c o m p l i c a t e d example o f t h e need t o r e - e v a l u a t e t h e o r d e r o f magnitude o f t h e terms o f t h e d i f f e r e n t i a l
equation
as t h e Independent v a r i a b l e I n c r e a s e s , i s p r e s e n t e d by S h i and 3 11 Eckstein-"
, who a n a l y z e d low t h r u s t s p a c e v e h i c l e
e x h i b i t i n g a s i n g u l a r i t y near the escape point an i n f i n i t e v a l u e
f o r the r a d i a l distance 3.34
trajectories
(i.e.,
at a finite
predicting time).
More g e n e r a l l y , the r e a s o n
f o r the occurrence
validity
i n the decrease
for large ^
lies
towards z e r o ; f o r t h e p a r t i c u l a r c h o i c e
turning point at i n f i n i t y : For other
one
may
deals with
, such
expect
results
a much
the accuracy
of t h e v a r i o u s
/,
slower
approximations
Dt» UTV
values for
» UsM-i 3 UMV '
yield
^^tme r e p r e s e n t a t i v e
have been p l o t t e d i n Pigs.3.3-5. Pig.3.3 shows
behavior
the
s o l u t i o n o v e r s e v e r a l I n t e r v a l s of c^? , f o r
of t h e e x a c t
= 0.025. F i g s . 3 . 4 , 5 show t h e e n v e l o p e o f t h e
between the n u m e r i c a l
v a l u e of the exact
"deviation"
s o l u t i o n and
the
numeri-
cal
v a l u e of e a c h of the t h r e e a p p r o x i m a t e s o l u t i o n s ; i . e . , o f
for
each of the v a l u e s
= 0.025 and
£
£ = 0.1.
I t i s seen
the M u l t i V a r i a b l e Approach ( U^,^ ) y i e l d s t h e most a p p r o x i m a t i o n . As if mations decreases
I n c r e a s e s , the accuracy roughly
e x p o n e n t i a l l y , as p r e d i c t e d .
the o s c i l l a t i o n d i e s out s e c u l a r . The
and
and
BESY s u b r o u t i n e s
r e s p . These subroutines
on an IBM
approxi-
Initially,
purely
360/9I compu-
f o r t h e c a l c u l a t i o n of
can be used o n l y f o r £ )^
they must be r e w r i t t e n i f s m a l l e r v a l u e s of £
3.35
that
Increases,
t h e growth becomes almost
c a l c u l a t i o n s were e x e c u t e d
t e r , u s i n g BESJ
^
:
accurate
of a l l t h r e e
t h e e r r o r s d i s p l a y a s h o r t p e r i o d o s c i l l a t i o n ; as
and y
a {e/p .
as to= e
some more d e t a i l , a computer program has been w r i t t e n t o
numerical
£
(3.2)
oj
growth. I n order to study
in
of the frequency
choices
or a l t e r n a t i v e l y , l e t t i n g ^-j» - t o , one error
of nonuniform
are used.
0.02;
The
above o b s e r v a t i o n s l e a d t o a word o f c a u t i o n c o n c e r n i n g t h e
I n t e r p r e t a t i o n of t h e concept of
of uniform
validity.
asymptotic expansions, uniform v a l i d i t y
f o l l o w i n g term'of t h e e x p a n s i o n to t h e p r e v i o u s term for
a l l (j'eCo.co')
the expansion
I f t h i s t u r n s out t o be t r u e
(compare S e c t i o n 3 . 5 ) ,
shows, however, t h a t , a l t h o u g h has been o b t a i n e d
i s obtained i f each
i s a s y m p t o t i c a l l y s m a l l compared
(Section 2.2).
"uniformly v a l i d
I n the theory
t h e n some a u t h o r s
f o r a l l t i m e s " . The example "uniform
(Sections 3 . 4 , 5 ) ,
validity
call
(3.1,2)
f o ra l l times"
the error Involved i s i n
g e n e r a l a f u n c t i o n o f time and may i n c r e a s e beyond bounds, t h e r e by d e s t r o y i n g t h e u n i f o r m
validity
after
finite
times
( ^ C^"^).
T h i s shows t h a t one may c o n c l u d e whether o r n o t a n a s y m p t o t i c expansion i s "uniformly v a l i d "
only a f t e r
h a v i n g c a l c u l a t e d an
I n f i n i t e number o f terms ( N-> ^-o ) . U s u a l l y , t h i s ble.
P o r a f i n i t e number o f terms t h e n , a c o n c l u s i o n about t h e
domain o f u n i f o r m tical
c a n be drawn o n l y a f t e r
either analy-
differential
equation
( R e f s . 3 . 6 , 7 ) , or a f t e r
numeri-
c o m p a r i s o n o f t h e a^)^)roximate s o l u t i o n w i t h t h e n u m e r i c a l l y
determined (see
validity
s t u d y o f t h e growth o f t h e p r e v i o u s l y n e g l e c t e d terms i n
the o r i g i n a l cal
i s not f e a s i -
"exact" s o l u t i o n of the o r i g i n a l
a l s o Chapter
Eight).
3.36
differential
equation,
3.8 References
3.1
L a n g e v l n , P . , and De B r o g l l e , M . , e d s . L a T h e o r i e du Rayonnement e t l e s Quanta. C o n s e l l s c l e n t l f i q u e de P h y s i q u e sous l e s a u s p i c e s de M.E.Solvay, a B r u x e l l e s , 1 9 1 1 , G a u t h l e r - V l l l a r s , P a r i s , 1 9 1 2 , pp. 4 4 6 - 4 5 0 .
3.2
C o l e , J.D. P e r t u r b a t i o n Methods I n A p p l i e d M a t h e m a t i c s . P u b l i s h i n g Company,Waltham, Mass., 1968.
3.3
Blalsdell
Morse,P.M., and Peshbach,H. Methods o f T h e o r e t i c a l P h y s i c s , P a r t s I and I I . H l l l Book Company,Inc.,New Y o r k , 1 9 5 3 .
Mc Graw-
3.4
Cheng,H., and Wu,T.T. "An Aging S p r i n g " . S t u d i e s i n A p p l i e d M a t h e m a t i c s , V o l . X L I X , No.2,June 1 9 7 0 , pp. 1 8 3 - 1 8 5 .
3.5
Kevorkian,J. "The T w o - V a r i a b l e E x p a n s i o n P r o c e d u r e f o r t h e Approximate S o l u t i o n of C e r t a i n Non-Linear D i f f e r e n t i a l Equations". L e c t u r e s i n A p p l i e d M a t h e m a t i c s , V o l . 7 : Space M a t h e m a t i c s , Part 3 , J.B.Rosser, ed., American Mathematical S o c i e t y , 1966, pp. 2 0 6 - 2 7 5 .
3.6
Ramnath,R.V. A M u l t i p l e Time S c a l e s Approach t o t h e A n a l y s i s o f L i n e a r Systems. Ph.D. d i s s e r t a t i o n , Department o f A e r o s p a c e and Mechanical S c i e n c e s , Princeton U n i v e r s i t y , Princeton,N.J., 1 9 6 7 ; a l s o : R e p o r t APPDL-TR-68-60, USAP P l i g h t Dynamics L a b o r a t o r y , W r i g h t - P a t t e r s o n APB., Ohio, O c t o b e r I 9 6 8 .
3.7
Ramnath,R.V., and S a n d r l , G . "A G e n e r a l i z e d M u l t i p l e S c a l e s Approach t o a C l a s s o f L i n e a r D i f f e r e n t i a l Equations". J o u r n a l of Mathematical A n a l y s i s and A p p l i c a t i o n s , Vol.28,No.2,November 1 9 6 9 , pp. 3 3 9 - 3 6 4 .
3.8
Watson,G.N. A T r e a t i s e on t h e T h e o r y o f B e s s e l F u n c t i o n s . Cambridge U n i v e r s i t y P r e s s , 1948.
3.37
Second e d i t i o n
!
3.9
Allen,H.J. Motion of a B a l l i s t i c M i s s i l e A n g u l a r l y M i s a l i g n e d w i t h t h e P l i g h t Path upon E n t e r i n g t h e Atmosphere and i t s E f f e c t upon Aerodynamic H e a t i n g , Aerodynamic L o a d s , and M i s s D i s t a n c e . NACA TN 4048, O c t o b e r 1957.
3.10
Curtlss,H.C.,Jr. An A n a l y t i c a l Study of t h e Dynamics of A i r c r a f t i n Unsteady P l i g h t . Ph.D. d i s s e r t a t i o n . Department of A e r o s p a c e and Mechanical S c i e n c e s , Princeton U n i v e r s i t y , Prlnceton,N.J., 1965; a l s o : T e c h n i c a l Report 6 5 - 4 8 , USAAVLABS, O c t o b e r I 9 6 5 .
3.11
S h l , Y . Y . , and E c k s t e i n , M . C . "An Approximate S o l u t i o n f o r A s c e n d i n g Low T h r u s t T r a j e c t o r i e s w i t h o u t S i n g u l a r i t y " . AIAA J o u r n a l , V o l . 5 , N o . 1 , J a n u a r y I 9 6 7 , pp. 1 7 0 - 1 7 2 .
3.12 Ehrenfest,P. "On a d i a b a t l c changes of a s y s t e m i n c o n n e c t i o n w i t h t h e quantum t h e o r y " . P r o c e e d i n g s , K o n i n k l i j k e Akademie van Wetenschappen t e Amsterdam. Vol.XIX,No.3,1917,pp. 5 7 6 - 5 9 7 . •1
3.38
3.39
env.
I0-'
3.42
Local Horizontal
Ascending Flight ( / o < 0 ) Descending Flight (/o>0) Pig.3.7
Angle of a t t a c k
0< v e r s u s
3.43
time t
A p p e n d i x 3A : O s c i l l a t o r y , B a l l i s t i c Consider
t h e motion
Flight
i n t h e v e r t i c a l plane o f a b a l l i s t i c
r e - e n t r y v e h i c l e t h r o u g h a v a r i a b l e - d e n s i t y atmosphere. With t h e a s s u m p t i o n s o f an a x i a l l y
symmetric body, s m a l l angle o f a t t a c k
o(, , z e r o a e r o d y n a m i c d a m p i n g , l i n e a r i z e d a e r o d y n a m i c moment z.
• d
angle
, constant v e l o c i t y
, and c o n s t a n t f l i g h t
(Pig.3.6), the equation of motion
around
path
the center of
g r a v i t y o f t h e v e h i c l e c a n be w r i t t e n as :
where M d e n o t e s t h e r o t a t i o n a l moment a r o u n d g r a v i t y , S t h e aerodynamic r e f e r e n c e a r e a , reference lenght,
t h e moment o f i n e r t i a ,
the center of t
t h e aerodynamic «
— r
, and o f e q u a t i o n s
t h r o u g h o u t t h e domain
0^(^-4=^
£} ) t h r o u g h o u t t h e d o m a i n ( 3 C - 2 , 4 ,6^ ) ( e r r o r o f o r d e r s} ) •
P a r e n t h e t i c a l l y , i t may be n o t e d t h a t t h e h a r m o n i c l a t o r problem
oscil-
( 3 . 1 ) can' a l s o be s o l v e d by t h e M e t h o d o f A v e r a g i n g .
•One must t h e n t r a n s f o r m e q u a t i o n ( 3 . 1 ) t o t h e s t a n d a r d
form
(compare e q u a t i o n (3A-5)) :
+ where
1
U clU
One o b s e r v e s I m m e d i a t e l y from order
£
that the p e r t u r b a t i o n term £ grows _ T t o o r d e r one as I n c r e a s e s f r o m z e r o t o o r d e r 1/
o r , e q u l v a l e n t l y , as (j? i n c r e a s e s f r o m z e r o t o o r d e r |-i,n. ^ .• T h i s result
i s i n accordance w i t h those o b t a i n e d from d i f f e r e n t
siderations
(compare e q s . ( 3 C - 2 t h r o u g h 6 ) ) .
region of uniform v a l i d i t y
i s g i v e n by :
3.49
con-
Thus, here a g a i n , t h e O ^ (f
^
^
.
;
H.1
CHAPTER I V DEVELOPMENT OF THE THREE DIMENSIONAL EQUATIONS OP MOTION FOR AEROSPACE VEHICLES ^1.1 I n t r o d u c t i o n I n the previous
Chapter, s e v e r a l asymptotic
perturbation
methods have been a p p l i e d t o t h e problem o f t h e harmonic
oscil-
l a t o r w i t h s l o w l y v a r y i n g f r e q u e n c y . The approximate s o l u t i o n s obtained have been compared a n a l y t i c a l l y and n u m e r i c a l l y ,
there-
by r e v e a l i n g a r e l a t i v e s u p e r i o r i t y o f t h e M u l t l V a r i a b l e Approach. The
same t u t o r i a l
framework i s now s e t up w i t h r e s p e c t
t o the
p e r t u r b e d m o t i o n o f unpowered, l i f t i n g aerospace v e h i c l e s around a massive, o b l a t e , r o t a t i o n a l l y symmetric p r i m a r y
body surrounded
by an atmosphere. In. a c t u a l p h y s i c a l s i t u a t i o n s , i n p a r t i c u l a r m o t i o n around the E a r t h , one has i n a d d i t i o n t o o b l a t e n e s s -
and aerodynamic
p e r t u r t : ( a t i o n 0 , o, whole clariH o f perfcurbatlono due t o phenomena such as : d e v i a t i o n s from r o t a t i o n a l symmetry o f t h e p r i m a r y body; g r a v i t a t i o n due t o non-primary bodies;' t i d a l a t t i t u d e motions; electromagnetic sure; r e l a t i v i t y
friction;
effects; solar radiation
pres-
e f f e c t s , and so on. Whether o r n o t any o f these
a d d i t i o n a l p e r t u r b a t i o n s must be t a k e n i n t o account depends on the p h y s i c a l s i t u a t i o n encountered. For n e a r - E a r t h m o t i o n of, r e l a t i v e l y compact aerospace v e h i c l e s , t h e above c l a s s o f p e r t u r b a t i o n s may u s u a l l y be n e g l e c t e d done i n t h e p r e s e n t
study.
(e.g.,Refs.4.1-3); t h i s i s also
To study t h e m o t i o n o f t h e v e h i c l e , a s u i t a b l e frame o f r e f e r e n c e must be i n t r o d u c e d . The s i m p l e s t choice
i s an I n e r t i a l ,
planeto-
c e n t r i c frame, v/hose o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f mass o f the p r i m a r y body ( p l a n e t ) . The g e n e r a l , t h r e e d i m e n s i o n a l of t h e v e h i c l e w i t h r e s p e c t t o t h i s frame o f r e f e r e n c e
motion
then
f o l l o w s from t h e v e c t o r e q u a t i o n :
— s u b j e c t t o t h e i n i t i a l c o n d i t i o n s V (o) r a d i u s v e c t o r i s denoted by r
dr and
dO)
. Here, t h e
( o r i g i n a t i n g from t h e c e n t e r o f t h e
frame o f r e f e r e n c e ) , t h e t o t a l a c c e l e r a t i o n v e c t o r by a
, the
g r a v i t a t i o n a l v e c t o r by Q.. > and t h e aerodynamic a c c e l e r a t i o n v e c t o r by The
.
a n a l y t i c a l study o f (^.1) o f f e r s g r e a t c h a l l e n g e s t o
a p p l i e d m a t h e m a t i c i a n s . I n g e n e r a l , no exact
s o l u t i o n can be ob-
t a i n e d ; one must be s a t i s f i e d w i t h an approximate s o l u t i o n , e i t h e r a n a l y t i c a l or numerical.
I t o f t e n t u r n s out t h a t c e r t a i n t r a n s -
f o r m a t i o n s o f v a r i a b l e s may s i m p l i f y t h e a n a l y s i s , o r make t h e s o l u t i o n o b t a i n e d more t r a n s p a r e n t f o r p h y s i c a l i n t e r p r e t a t i o n . Three such t r a n s f o r m a t i o n s o f t h e independent v a r i a b l e come immed i a t e l y t o mind : 1. The t r a n s f o r m a t i o n t ( V ) from time t o v e l o c i t y , has found a p p l i c a t i o n i n s e v e r a l s t u d i e s (e .g. ,Ref s . ^ . ^1-7). T h i s t r a n s f o r m a t i o n i s s u i t a b l e o n l y i f t h e f u n c t i o n a l dependence between t and V I s m o n o t o n i c , such as occurs i n d r a g dominated f l i g h t , or on escape t r a j e c t o r i e s . I n t h e p r e s e n t
l\.2
study o f p e r t u r b e d
Kepler m o t i o n , a s i n g u l a r i t y
(dV/dt = 0) i n t h e t r a n s f o r m a t i o n
occurs a t l e a s t t w i c e p e r r e v o l u t i o n , making t h i s t y p e o f t r a n s formation unsuitable. 2. Another such t r a n s f o r m a t i o n : t ( r ) from t i m e t o r a d i u s . I s sometimes used (e.g.,Refs.^.8,9). T h i s t r a n s f o r m a t i o n i s s u i t a b l e • f o r monotonlcaliy ascending or descending
f l i g h t . Again, I n t h e
p r e s e n t study o f p e r t u r b e d Kepler m o t i o n , a s i n g u l a r i t y ( d r / d t = 0 ) i n t h e t r a n s f o r m a t i o n occurs a t l e a s t t w i c e p e r r e v o l u t i o n , making t h i s type o f t r a n s f o r m a t i o n t o o , u n s u i t a b l e . 3. The t r a n s f o r m a t i o n t ( ^ ) from t i m e t o " e v o l u t i o n a n g l e " i s used i n t h e p r e s e n t s t u d y . For t h r e e d i m e n s i o n a l m o t i o n , t h e angle (j> I s i n t e r p r e t e d as t h e c e n t r a l angle i n t h e i n s t a n t a n e o u s o r e v o l v i n g o r b i t a l p l a n e . Thus, t h e s o l u t i o n t o t h e e q u a t i o n s o f m o t i o n becomes g e o m e t r i c a l l y i n t e r p r e t a b l e . I t should be p o i n t e d out
that a singularity
(dj^/dt = 0 ) i n t h e t r a n s f o r m a t i o n occurs
for
l o c a l l y v e r t i c a l f l i g h t . For i n i t i a l l y n o n - v e r t i c a l
and s m a l l v a l u e s o f t h e aerodynamic l i f t
flight
p e r t u r b a t i o n as c o n s i d e r e d
i n t h e p r e s e n t s t u d y , t h e f u n c t i o n a l dependence between t and (j> I s monotonic , and t h e r e f o r e
suitable.
Of course, t r a n s f o r m a t i o n o f t h e independent v a r i a b l e i s n o t a b s o l u t e l y necessary, as i t s monotony makes i t a s u i t a b l e S t u d i e s i n v o l v i n g t i m e as t h e independent e.g.,Refs.4.10,ll,12,32.
^.3
variable.
v a r i a b l e a r e found
In,
In
a d d i t i o n t o t r a n s f o r m a t i o n of the independent v a r i a b l e , i t may
be d e s i r a b l e t o t r a n s f o r m
the dependent v a r i a b l e . Three
important
cases employing such t r a n s f o r m a t i o n (some i n v o l v i n g s i m u l t a n e o u s t r a n s f o r m a t i o n of the independent v a r i a b l e ) are : 1. Gauss' f o r m u l a t i o n of Lagrange's p l a n e t a r y
equations. This i s
b a s i c a l l y a v a r i a t i o n o f parameters f o r m u l a t i o n . I t i n v o l v e s f i r s t order, nonlinear d i f f e r e n t i a l equations, describing dependence o f t h e o s c u l a t i n g o r b i t elements on t i m e
six
the
(e.g.,Refs.
13-15). 2. The
displacement f o r m u l a t i o n , d e s c r i b i n g t h e p o s i t i o n of
vehicle r e l a t i v e t o a Cartesian
coordinate
the
system which moves w i t h
a f i c t i t i o u s v e h i c l e i n t h e u n p e r t u r b e d , two d i m e n s i o n a l o r b i t . The
p o s i t i o n Is described
by a l i n e a r i z e d system o f t h r e e
coupled,
second o r d e r d i f f e r e n t i a l e q u a t i o n s , c o n t a i n i n g t h e c e n t r a l angle ( or time ) of the f i c t i t i o u s v e h i c l e as Independent v a r i a b l e (e.g. , R e f s . i J . 1 3 , l ^ , l 6 , 1 7 ) . 3. The
harmonic o s c i l l a t o r f o r m u l a t i o n , employing a s p h e r i c a l
coordinate
system. I n t r o d u c i n g a new
i n v e r s e of the r a d i u s , and a n g l e , one
varla'ble p r o p o r t i o n a l t o t h e
transforming
from time t o e v o l u t i o n
o b t a i n s a coupled system o f two f i r s t o r d e r and
two
second o r d e r , n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s , c o n t a i n i n g
the
e v o l u t i o n angle as Independent v a r i a b l e . T h i s system i s e q u i v a l e n t t o t h a t of two harmonic o s c i l l a t o r s c o n t a i n i n g s m a l l l i n g p e r t u r b a t i o n terms. Examples of t h e a p p l i c a t i o n o f
coup-
this
method t o two d i m e n s i o n a l m o t i o n are found e.g., t h r e e d i m e n s i o n a l m o t i o n i s t r e a t e d i n t h i s way
i n Refs.4.18-20 ; e.g.,
i n Refs.
4.13,14,21,22. I n t h i s d i s s e r t a t i o n , t h e harmonic o s c i l l a t o r f o r m u l a t i o n i s adopted.
I t has t h e advantage o f immediate g e o m e t r i c a l d e s c M p t l o n
of t h e m o t i o n o f t h e v e h i c l e as f u n c t i o n o f t h e e v o l u t i o n a n g l e ; moreover, i t leads t o b o t h convenient m a t h e m a t i c a l a n a l y s i s a c c u r a t e r e s u l t s . Expressions
f o r t h e g r a v i t a t i o n a l - and
aerody-
namic a c c e l e r a t i o n v e c t o r s i n (4.1) are d e r i v e d i n S e c t i o n T r a n s f o r m a t i o n o f (4.1) t o t h e harmonic o s c i l l a t o r i s c a r r i e d out i n S e c t i o n 4.3.
Pour cases o f two
m o t i o n are deduced I n S e c t i o n 4.4.
and
4.2.
formulation
dimensional
Remarks on t h e n u m e r i c a l
inte-
g r a t i o n o f t h e e q u a t i o n s o f m o t i o n are p r e s e n t e d i n S e c t i o n 4.5'.
4.5
4.2 G r a v i t a t i o n a l - and Aerodynamic A c c e l e r a t i o n V e c t o r s
4.2.1 Coordinate
Systems
The g r a v i t a t i o n a l - and aerodynamic a c c e l e r a t i o n v e c t o r s must be s p e c i f i e d c o m p l e t e l y b e f o r e t h e e q u a t i o n s
o f m o t i o n (4.1) can
be s o l v e d . To reach t h i s g o a l , t h r e e u n i t v e c t o r systems a r e now I n t r o d u c e d , each d e f i n i n g a c o o r d i n a t e
system,(Pig.4.1).
The b a s i c c o o r d i n a t e system i s an i n e r t i a l , p l a n e t o c e n t r i c c o o r d i n a t e system, d e f i n e d by t h e i n e r t i a l , p l a n e t o c e n t r i c u n i t v e c t o r system
Tv.,y^z
f o l l o w s : t h e o r i g i n o f t h e system c o i n -
cides w i t h t h e c e n t e r o f mass o f t h e p r i m a r y body; t h e u n i t v e c t o r Tz
l i e s along t h e a x i s o f r o t a t i o n a l symmetry o f t h e p r i m a r y
body, and i s d i r e c t e d towards t h e N o r t h Pole ( f i x e d w i t h r e s p e c t to i n e r t i a l
space); t h e u n i t v e c t o r Tx i s p e r p e n d i c u l a r t o Tz ,
i t s d i r e c t i o n Is f i x e d w i t h respect t o i n e r t i a l vector
i s g i v e n by ïy =
X Tx
space;
The r a d i u s V
•
the unit i s defined
w i t h r e s p e c t t o t h i s system. The a n a l y s i s i s s i m p l i f i e d i f two moving c o o r d i n a t e
systems
are I n t r o d u c e d , d e f i n e d by t h e f o l l o w i n g u n i t v e c t o r systems : 1. t h e moving, o r b i t a l s p h e r i c a l u n i t v e c t o r system Ty.^h,e ' o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f mass o f t h e v e h i c l e , and r - ^
7 - Z A ^ ^ »
——
7 .
_
7 ^ 7
-
U%
ir
(H.Z)
2. t h e moving, e q u a t o r i a l s p h e r i c a l u n i t v e c t o r system 1^^^ ^ ; I t s o r i g i n c o i n c i d e s w i t h t h e c e n t e r o f mass o f t h e v e h i c l e , and J- ^
iz X Tr
COS X where X
denotes t h e l a t i t u d e .
^
Ü
=
Tr X Lu
^ ^
Truncation of the series a f t e r the
2.6X10
term t h u s leads t o a v e r y
s m a l l e r r o r compared t o t h e two l e a d i n g terms. I n view o f t h e o b j e c t i v e s o f t h i s d i s s e r t a t i o n , t h e p o t e n t i a l (4.5) may t h e r e f o r e be approximated
to: r
where t h e p o l y n o m i a l e x p r e s s i o n f o r
(©4) has been I n t r o d u c e d , and
where "J^ i s t h e "oblateness c o e f f i c i e n t " . I t may be noted t h a t t h e above p o t e n t i a l i s i d e n t i c a l t o t h e p o t e n t i a l g e n e r a t e d by a p o i n t mass and a quadrupole o r i g i n , r = 0. a
o f c o n s t a n t moment, b o t h l o c a t e d a t t h e
One now o b t a i n s f o r t h e g r a v i t a t i o n a l v e c t o r : 7
=
V
u
COS X
c)U
7
tH.8)
-
where t h e p a r t i a l d e r i v a t i v e s f o l l o w i m m e d i a t e l y from
(4.7).
4.2.3
Aerodynamic A c c e l e r a t i o n
Vector
The aerodynamic a c c e l e r a t i o n i s due t o t h e m o t i o n o f t h e v e h i c l e t h r o u g h t h e atmosphere s u r r o u n d i n g
t h e p r i m a r y body. I n
g e n e r a l , one has : a^-
+
cTu
t h e aerodynamic
factor,
reference
area S, and t h e v e h i c u l a r mass m (an i l l u m i n a t i n g study o f t h e e x p e r i m e n t a l e v a l u a t i o n o f these q u a n t i t i e s I s found I n Ref.^,24). The l i f t
a c c e l e r a t i o n v e c t o r i s o f a somewhat more
compli-
cated f o r m , a l l o w i n g an a r b i t r a r y bank angle "y^ ( F i g . 4.2) : dr 2 • dt _ where Ct
denotes t h e l i f t •>
M —
c o e f f i c i e n t , and where :
H
( a n g u l a r momentum)
F i n a l l y , one o b t a i n s f o r t h e aerodynamic a c c e l e r a t i o n a^—
Qo +
at
=
a, l ,
-\- a,
-h Qu ^
vector c^-ii-)
where Qp and ÖI are s p e c i f i e d by ( 4 . 1 0 , 1 1 ) , and where a^-^^^j^ f o l l o w immediately f r o m a^.t . I t remains t o s p e c i f y t h e d e n s i t y f u n c t i o n ^ and t h e aerodynamic c o e f f i c i e n t s C l p o c c u r r i n g I n 4.8
(4.12).
4 . 2 . 4 A n a l y t i c a l D e n s i t y Model I n o r d e r t o be a b l e t o t r e a t the problem
of aerodynamlcally
p e r t u r b e d m o t i o n a n a l y t i c a l l y , an a n a l y t i c a l e x p r e s s i o n f o r t h e d i s t r i b u t i o n o f t h e atmospheric
density ^
around
body must be i n t r o d u c e d i n e q u a t i o n ( 4 . 1 2 ) . The
the primary
s i m p l e s t model
t h a t comes t o mind i s t h a t o f a s p h e r i c a l l y symmetric,
steady
s t a t e d e n s i t y d i s t r i b u t i o n . Such a model i s a s t r o n g s i m p l i f i c a t i o n o f the a c t u a l d e n s i t y d i s t r i b u t i o n , as i t n e g l e c t s such phenomena as
: atmospheric
r o t a t i o n , o b l a t e n e s s o f the atmosphere,
s o l a r f l u x and r a d i a t i o n d i s t u r b a n c e s , t h e r m a l atmospheric
con-
d u c t i o n , and so on ( e . g . , R e f s . 3 . 1 5 > 2 5 , 2 6 ) . A n a l y t i c a l models t h a t take such phenomena I n t o account
do e x i s t ; o f t e n , however, t h e s e
models are e x t r e m e l y awkward t o use i n a n a l y t i c a l s t u d i e s . I n t h i s d i s s e r t a t i o n , a s o - c a l l e d "design d e n s i t y model" i s c o n s i d e r e d . I t i s o b t a i n e d from t h e a c t u a l d e n s i t y d i s t r i b u t i o n by a v e r a g i n g w i t h r e s p e c t t o l o n g i t u d e , l a t i t u d e , and t i m e , l e a d i n g t o t h e f u n c t i o n a l form :
Some o f these models are l i s t e d i n Appendix 4A. Each model i s matched t o t h e averaged, a c t u a l d e n s i t y p r o f i l e a v a i l a b l e i n l i t e r a t u r e , a t some a l t i t u d e
( r = r^ ) , u s u a l l y t h e p e r l c e n t e r o f
t h e u n p e r t u r b e d o r b i t . A number o f c o n s t a n t s a v a i l a b l e i n t h e model a l l o w s matching
at other a l t i t u d e s
( r = r, , ,
) as w e l l .
The models p r e s e n t e d i n Appendix 4A c l e a r l y I n v o l v e d i f f e r e n t a c c u r a c i e s over t h e a l t i t u d e r e g i o n under c o n s i d e r a t i o n ; t h e most
4.9
r e s t r i c t i v e model I s t h e l i n e a r one, t h e most g e n e r a l models a r e the power law model and Bruno's model.(The e f f e c t o f e r r o r s
In
the d e n s i t y model on t h e p r e d i c t e d m o t i o n o f s a t e l l i t e s has been I n v e s t i g a t e d by s e v e r a l a u t h o r s f o r t h e case o f n e a r - c i r c u l a r s a t e l l i t e o r b i t s ; e.g., see R e f . 4 . 2 7 ) . I t i s now proposed t o use a d e n s i t y model o f t h e type r e l a t e d t o Bruno's model, by t a k i n g f ( r ) = Y = c o n s t a n t ;
(4.13)
thus :
The model a l l o w s m a t c h i n g t o t h e averaged d e n s i t y p r o f i l e a t two p o i n t s . One o f these p o i n t s
(r„ ) should be l o c a t e d near o r a t t h e
i n i t i a l p e r l c e n t e r , as t h e p e r t u r b a t i o n e f f e c t s a r e most
signifi-
cant i n t h a t r e g i o n . As t h e p e r l c e n t e r descent r a t e i s r a t h e r low ( R e f . 4 . 1 5 , p . 1 5 5 ) , t h i s c h o i c e o f matching p o i n t w i l l f a c t o r y f o r long times. distance higher orbits,
remain s a t i s -
The second p o i n t s h o u l d be t a k e n a t some
a r above t h e f i r s t
p o i n t ; i t must, o f c o u r s e , n o t be
t h a n t h e i n i t i a l apocenter. Thus, f o r s m a l l - e c c e n t r i c i t y A r must be t a k e n a c c o r d i n g l y
enough, f o r l a r g e - e c c e n t r i c i t y o r b i t s ,
small. I n t e r e s t i n g l y a r must a l s o be t a k e n
r a t h e r s m a l l , as accuracy o f t h e d e n s i t y model counts most I n a r e g i o n o f o n l y about 100 km. w i d t h
(e.g.,Ref.4.28) j u s t above
p e r l c e n t e r , where t h e main p e r t u r b a t i o n e f f e c t s a r e g e n e r a t e d . Numerical s t u d i e s have been c a r r i e d o u t t o compare t h e accur a c y o f t h e proposed model ( 4 . l 4 ) , t o a d e s i g n model p r e s e n t e d by
4.29 P a e t z o l d and Zschorner * I n Pigs.4.3,4
. Some r e p r e s e n t a t i v e
r e s u l t s are shown
( A r = 100 km., R = 6378.2 km,, p e r l c e n t e r a t
4.10
200
km. and 600
km. r e s p e c t i v e l y ) . Por h i g h p e r l c e n t e r
altitudes,
t h e accuracy o f t h e model I s v e r y good over a wide a l t i t u d e
region.
Por lov/ p e r l c e n t e r a l t i t u d e s , t h e accuracy i s good over a somewhat l i m i t e d r e g i o n c o n t a i n i n g t h e two match p o i n t s ; t h r o u g h c a r e f u l choice o f t h e two match p o i n t s , t h e e f f e c t s of t h e d e n s i t y I n a c c u r a c y on t h e m o t i o n o f t h e v e h i c l e may
average o u t , however.
One must bear i n mind t h a t i n t h e course cf t i m e , " u p d a t i n g " ( i . e . , r e p e a t i n g t h e matching p r o c e s s ) o f t h e parameters o f t h e d e n s i t y model may be necessary, i n o r d e r t o m a i n t a i n accuracy w i t h r e s p e c t t o t h e averaged d e n s i t y model.
4.11
4.2.5
Aerodynamic C o e f f i c i e n t s O r b i t i n g aerospace v e h i c l e s g e n e r a l l y f l y a t a l t i t u d e s
above
about 140 km. One may t h e n assume (Refs.4.15,24,30) t h a t t h e m o t i o n takes p l a c e i n t h e f r e e m o l e c u l a r f l o w regime. an a p p r o p r i a t e , v a r i a b l e drag c o e f f i c i e n t
Studies c o n s i d e r i n g
are found i n Refs.
2 8 , 3 0 . Most s t u d i e s t r e a t t h e drag c o e f f i c i e n t
4.26,
as a c o n s t a n t ; t h i s
s i m p l i f i e s t h e a l g e b r a c o n s i d e r a b l y . S t u d i e s employing
a
lift
c o e f f i c i e n t a r e v i r t u a l l y n o n - e x i s t e n t , as i t i s u s u a l l y assumed t h a t space v e h i c l e s tumble
(Refs.4.30,31), thereby averaging out
the e f f e c t s o f l i f t . The p r e s e n t study assumes a g e n e r a l l y nonzero, c o n s t a n t coefficient
lift
( a p p l i c a b l e , e . g . , t o t h e m o t i o n o f space s h u t t l e s and
of s t a b i l i z e d s a t e l l i t e s ) , and a nonzero, c o n s t a n t drag
4.12
coefficient.
4.3 D e r i v a t i o n o f t h e Transformed E q u a t i o n s o f M o t i o n The
e q u a t i o n s o f motion
harmonic o s c i l l a t o r
( 4 . 1 ) a r e now t r a n s f o r m e d t o t h e
formulation.
S u b s t i t u t i n g t h e r e l a t i o n s Fj = H Ü
(4.2), In the
and f = T
angular momentum ( 4 . 1 1 ) , y i e l d s : 6Ï
~
Define t h e " e v o l u t i o n a n g l e "
(j) ^ ( S e c t i o n 4 . 1 )
:
r2 where i t i s assumed t h a t t h e m o t i o n i s c a l c u l a t e d from t = 0 on Equation
( 4 . 1 5 ) t h e n becomes :
S u b s t i t u t i o n o f t h e a c c e l e r a t i o n v e c t o r s ( 4 . 8 ) and ( 4 . 1 2 ) i n t h e equations o f motion
( 4 . 1 ) leads t o :
Taking t h e t i m e - d e r i v a t i v e o f t h e a n g u l a r momentum y i e l d s : dg dt T r a n s f o r m i n g from t i m e t t o t h e e v o l u t i o n angle ^
according t o
( 4 . 1 6 ) , and t a k i n g t h e i n n e r p r o d u c t w i t h Th , leads t o t h e s c a l a r equation : dhi
_
dip
jn! L \A
[
c o a ^
}iia,.T,)
-1-
del
cH.i^)
J
0^
S i m i l a r l y , a p p l y t h e t r a n s f o r m a t i o n ( 4 . 1 6 ) and s u b s t i t u t e (4.18) i n t h e d o u b l e , o u t e r p r o d u c t TkXfrKX ^ 1 ; t h i s gives t h e vector equation : du
_ _ r l ^f c o s A
c r v , . r x ) -1- au\u
(4.20)
A s i m i l a r d i f f e r e n t i a l e q u a t i o n i s found by t a k i n g t h e d e r i v a t i v e Lq zz
of
Iw X
substitute
ly
w i t h r e s p e c t t o t h e e v o l u t i o n angle
, and
(4.17,20); t h i s g i v e s :
S i m i l a r l y , t a k i n g t h e d e r i v a t i v e o f (4.17) w i t h r e s p e c t t o Cj) , and s u b s t i t u t i n g
(4.21), gives :
At t h i s stage o f t h e a n a l y s i s , i t i s c o n v e n i e n t t o i n t r o d u c e the
variables : Ic ~
^/H^
a n g u l a r momentum f u n c t i o n
U
Vy*
radius function
Zi
The above d i f f e r e n t i a l e q u a t i o n s t h e n t r a n s f o r m t o
dlr — — d Cj)
r
-7—
Le
ciü _ dcf
_ A
dtf
u.cosX- i n CÜ.Ix) - I - a k V u èo( J
I
00'.
J CW.24)
d(f^ dt
UM
c)o(
_
/ÏT
-
2A'fu.cosA. ^ ( r , . r , ) uM • èok
4.14
-
Oe] J
A d i f f e r e n t i a l e q u a t i o n f o r u ( ^ ) i s more d i f f i c u l t t o d e r i v e . Transforming the d e r i v a t i v e d r / d t r e l a t i o n s "r^rT^
with the a i d of the
, t = t ( y > ) , and subsequent use o f ( 4 . 2 4 ) ,
leads t o an e x p r e s s i o n o f t h e form n where ..j. ^ ^
_
7-
Lb
are f u n c t i o n s o f U
, It , Cn h /sir\
Sin 2£/
( f ) ^ zz
i
have
been accounted f o r . The e q u a t i o n s (4.4?) agree i n substance w i t h those presented i n R e f . 4 . 4 l .
4.22
V
4.5 Remarks on Numerical
Integration
To v e r i f y the accuracy
of a n a l y t i c a l approximations
equations o f motion presented
I n the previous Sections,
t o the numerical
I n t e g r a t i o n must be r e v e r t e d t o whenever no e x a c t , a n a l y t i c a l solution is available
or s u i t a b l e f o r comparison.
The n u m e r i c a l i n t e g r a t i o n o f system (4.1) can be
performed
a c c o r d i n g t o s e v e r a l methods (Refs.4.47,48). Cowell's method, involving d i r e c t i n t e g r a t i o n i n Cartesian coordinates, i s favored i f t h e g r a v i t a t i o n a l and aerodynamic p e r t u r b a t i o n s ( £^
resp
)
are o f t h e o r d e r of one or l a r g e r ( " s t r o n g " p e r t u r b a t i o n s ; examples : boost and r e - e n t r y t r a j e c t o r i e s ) . Encke's method, i n v o l v i n g
inte-
g r a t i o n of t h e d i f f e r e n t i a l e q u a t i o n s f o r t h e s p a t i a l d i f f e r e n c e between t h e a c t u a l t r a j e c t o r y and an u n p e r t u r b e d , r e f e r e n c e
tra-
j e c t o r y , i s f a v o r e d i f the p e r t u r b a t i o n s are s m a l l e r t h a n t h e order o f one, or i f they are moderate and a c t i n g over a r e s t r i c t e d segment o f t h e t r a j e c t o r y
(examples : l u n a r and
interplanetary
t r a j e c t o r i e s ) . The method o f v a r i a t i o n o f parameters, continuous
involving
" r e c t i f i c a t i o n " of a reference o r b i t , i s favored i f
t h e p e r t u r b a t i o n s are v e r y s m a l l compared t o one low t h r u s t - , s m a l l drag-, and o b l a t e n e s s
( examples :
influenced motion).
I n t h e p r e s e n t s t u d y , t h e p e r t u r b a t i o n s are assumed t o be very s m a l l : oé parameters may
£„ ^
I , Thus, t h e method o f v a r i a t i o n o f
be employed w i t h p r o f i t . The harmonic o s c i l l a t o r
f o r m u l a t i o n o f t h e equations o f m o t i o n , developed i n t h i s Chapter, i s s u i t a b l e par e x c e l l e n c e f o r d i r e c t a p p l i c a t i o n o f t h e method o f v a r i a t i o n o f parameters;
v i d e Chapters Six and
4.23
Seven.
4.6
References
4.1
Spitzer,L.,Jr. "Perturbations of a S a t e l l i t e O r b i t " . Jour-nal of the B r i t i s h I n t e r p l a n e t a r y Society,Vol.9,No.3,May 1950 ,pp.131-136.
4.2
Roberson,R.E. " O r b i t a l Behavior o f E a r t h S a t e l l i t e s " , P a r t I . J o u r n a l o f thé F r a n k l i n I n s t i t u t e ^Vol.264,No.3,September 1957,PP•181-202.
4.3
Kaula,W.M. C e l e s t i a l Geodesy. NASA TN D-1155, March
4.4
4.5
1962.
Chapman,D.R. An Approximate A n a l y t i c a l Method f o r S t u d y i n g E n t r y P l a n e t a r y Atmospheres. NASA TR R-11, 1959. C i t r o n , S . J . , andMelr,T.C. "An A n a l y t i c a l S o l u t i o n f o r E n t r y i n t o P l a n e t a r y AIAA Journal,Vol.3,No.3,March 1965,PP.470-475.
into
Atmospheres".
4.6
Nayfeh,A.H. "Comments on'An A n a l y t i c a l S o l u t i o n f o r E n t r y i n t o P l a n e t a r y Atmospheres'". AIAA J o u r n a l , V o l . 4 , N o . 4 , A p r i l 1966,p.758.
4.7
Citron,S.J. "Reply by Author t o A.H.Nayfeh". AIAA J o u r n a l , V o l . 4 , N o . 4 , pp.758-760.
4.8
Brogllo,L. "Lois de s i m i l i t u d e dans l e c a l c u l des t r a j e c t o i r e s de r e n t r e e e t de 1 ' a b l a t i o n f r o n t a l e des e n g i n s " . A s t r o n a u t i c a A c t a , V o l . 7 , Pasc.1,1961,pp.21-34.
4.9
Willes,R.E.,Francisco,M.C.,Reld,J.G., andLim,W.K. An A p p l i c a t i o n o f Matched A s y m p t o t i c Expansions t o H y p e r v e l o c i t y P l i g h t Mechanics. AIAA Paper No.67-598,I967.
.4.10
T l n g , L . , and Brofman,S. "On Take-Off from C i r c u l a r O r b i t by Small T h r u s t " . Z e l t s c h r l f t für angewandte Mathematik und Physik,Band 44,HeftlO/11, 1964,pp.417-428"
4.11 Brofman,W. "Approximate A n a l y t i c a l S o l u t i o n f o r S a t e l l i t e O r b i t s Subj e c t e d t o Small T h r u s t or Drag". AIAA J o u r n a l , V o l . 5 , N o . 6 , June 1967,pp.1121-1128.
4.24
4.12
Connor,M.A. " G r a v i t y Turn Through t h e Atmosphere". J o u r n a l o f S p a c e c r a f t and Rockets,Vol.3,No.8,August I 9 6 6 , p p . I 3 0 8 - I 3 I I .
4.13
Geyllng,F.T. " P e r t u r b a t i o n Methods f o r S a t e l l i t e O r b i t s " . The B e l l System T e c h n i c a l J o u r n a l , V o l . X L I I I , N o . 3 ,May 1964,pp.847-884.
4.14
G e y l i n g j F . T . , and Westerman,H.R. I n t r o d u c t i o n t o O r b i t a l Mechanics. Addlson-Wesley Reading,Mass.,1971.
4.15
King-Hele,D. Theory o f S a t e l l i t e O r b i t s i n an Atmosphere. London,1964.
Publ.
Co.,
Butterworths,
4.16
B r e a k w e l l , J . V . , and Roberson,R.E. O r b i t a l and A t t i t u d e Dynamics. AIAA Recorded L e c t u r e s e r i e s No.2, August 1969, P r i n c e t o n , M.0.
4.17
Cowley,J.R. The E f f e c t o f t h e Subsolar Atmospheric Bulge on S a t e l l i t e Re-Entry L a t i t u d e s . SUDAAR No.36O, Department o f A e r o n a u t i c s and A s t r o n a u t i c s , S t a n f o r d U n i v e r s i t y , S t a n f o r d , C a l i f o r n i a , October I 9 6 8 .
4.18
Roberson,R.E. " E f f e c t o f A i r Drag on E l l i p t i c S a t e l l i t e O r b i t s " . J e t Propulsion,Vol.28,No.2,February 1958 , p p . 9 0 - 9 6 .
4.19
Zee,C.H. " T r a j e c t o r i e s o f S a t e l l i t e s under t h e I n f l u e n c e o f A i r Drag". Progress i n A s t r o n a u t i c s and A e r o n a u t i c s , V o l . l 4 : C e l e s t i a l Mechanics and A s t r o d y n a m i c s , V.G.Szebehely,ed., Academic Press,New York,1964,pp.101-112.
4-.20 • Newton,R.R. "Motion o f a S a t e l l i t e i n an Atmosphere o f Low G r a d i e n t " . ARS Journal,Vol.32,No.5,May I962,pp.770-772. 4.21
Kevorkian,J. "The Two-Variable Expansion Procedure f o r t h e Approximate S o l u t i o n o f C e r t a i n Non-Linear D i f f e r e n t i a l E q u a t i o n s " . L e c t u r e s i n A p p l i e d Mathematics,Vol.7 : Space Mathematics, P a r t 3 , J,B.Rosser,ed.,American M a t h e m a t i c a l S o c i e t y , 1966, pp.206-275.
4.25
4.22
P e t t y , C M . , and Breakwell,J.V. " S a t e l l i t e O r b i t s about a P l a n e t w i t h R o t a t i o n a l Symmetry". J o u r n a l o f t h e F r a n k l i n I n s t i t u t e , V o l . 2 7 0 , N o . 4 , O c t o b e r I96O, pp.259-282.
4.23
Kozai,Y. "Numerical R e s u l t s on t h e G r a v i t a t i o n a l P o t e n t i a l o f t h e E a r t h from O r b i t s " . The Use o f A r t i f i c i a l S a t e l l i t e s f o r Geodesy, G.Veis,ed.,North-Holland P u b l i s h i n g Company, Amsterdam, I963,PP.305-315.
4.24
King-Hele,D.G., and Walker,D.M.C. "Upper-Atmosphere D e n s i t y d u r i n g t h e Years 1957 t o I 9 6 I , determined from S a t e l l i t e O r b i t s " . Space Research I I , H.C. van de H u l s t e t a l . , e d s . , N o r t h - H o l l a n d P u b l i s h i n g Company,Amsterdam, 196I,pp.918-957•
4.25
Broglio,L. "Lo S t u d i o D e l l ' A l t a Atmosfera Mediante I I S a t e l l i t e San Marco I I " . L ' A e r o t e c h n i c a M i s s l l l E Spazio,Vol.50,No.1, f e b b r a i o 1971,PP.9-18.
4.26
Benson,R.H., Flelschman,E.F., and H i l l , R . J . " E a r t h - O r b i t a l L i f e t i m e and S a t e l l i t e Decay". A s t r o n a u t i c s & Aeronautics,Vol.6,No.1,January I968 ,pp.38-45.
4.27
Karrenberg,H.K., L e v i n , E . , and Lewis,D.H. "Variation of S a t e l l i t e Position with Uncertainties i n the Mean Atmospheric D e n s i t y " . ARS Journal,Vol.32,No.4, A p r i l 1962,pp.576-582.
4.28
Hunzlker,R.R. " E f f e c t s o f t h e V a r i a t i o n o f Drag C o e f f i c i e n t on t h e Ephemeris o f E a r t h S a t e l l i t e s " . A s t r o n a u t i c a A c t a , V o l . 1 5 , No.3,February 1970,pp.I6I-I67.
4.29
Paetzold,H.K., and Zschorner,H. "The S t r u c t u r e o f t h e Upper Atmosphere and i t s V a r i a t i o n s a f t e r S a t e l l i t e O b s e r v a t i o n s " . Space Research I I , H.C. van de H u l s t e t a l . , e d s . , N o r t h - H o l l a n d P u b l i s h i n g Company, Amsterdam, I96I,pp.958-973.
4.30
Groves,G.V. "The I n f l u e n c e o f t h e Upper Atmosphere on S a t e l l i t e O r b i t s " . Mathematische Methoden der Himmelsmechanik und A s t r o n a u t i k , E.Stiefel,Herausgeber,Bibliographisches Institut-Mannheim, 1966,pp.147-170.
4.26
4.31
Sentman,L.H., and Nelce,S.E. "Drag C o e f f i c i e n t s f o r Tumbling S a t e l l i t e s " . J o u r n a l o f S p a c e c r a f t and Rockets,Vol.4,No.9,September 196?,PP.12701272.
4.32
Porster,K. " S a t e l l i t e Dynamics f o r Small E c c e n t r i c i t y I n c l u d i n g Drag and T h r u s t " . AIAA Journal,Vol.1,No.11,November I 9 6 3 , pp.2621-2623.
4.33
PitzpatrickjP.M. P r i n c i p l e s o f C e l e s t i a l Mechanics. Academic Press,New Y o r k , 1970.
4.34
Billik,B. "Survey o f Current L i t e r a t u r e on S a t e l l i t e L i f e t i m e s " . ARS Journal,Vol.32,No.11,November I 9 6 2 , p p . I 6 4 l - l 6 5 0 .
4.35
Kork,J. " S a t e l l i t e Lifetimes i nE l l i p t i c Orbits". Journal of the Aerospace Sciences.Vol.29,No.11,November 1962,pp.1273-1290, 1299.
4.36
Izsak,I.G. " P e r i o d i c Drag P e r t u r b a t i o n s o f A r t i f i c i a l S a t e l l i t e s " . The A s t r o n o m i c a l Journal,Vol.65,No.6,August I96O ,pp.355357.
4.37
Brogllo,L. "A General Theory on Space and Re-Entry S i m i l a r T r a j e c t o r i e s " . AIAA J o u r n a l , V o l . 2 , N o . 1 0 , O c t o b e r 1964,pp.1774-1781.
4.38
Westerman,H.R. "Secular E f f e c t s o f Atmospheric Drag on S a t e l l i t e O r b i t s " . The A s t r o n o m i c a l Journal,Vol.68,No.6.August I963,pp.382-384.
4.39
Westerman,H.R. "On S a t e l l i t e L i f e t i m e s " . The A s t r o n o m i c a l J o u r n a l . V o l . 6 8 . No.6,August 1963,pp.385-38F;
4.40
Bruno,C. Secular P e r t u r b a t i o n s o f an E a r t h S a t e l l i t e . Seminar, Department o f Aerospace and Mechanical S c i e n c e s , P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , N . J . , March 12,1970.
4.41
Anthony,M.L., and PosdlokjG.E. "Planar Motions About an Oblate P l a n e t " . ARS J o u r n a l . V o l . 3 1 , No.9,September I96I,pp.1225-1232.
4.27
4.42
Anthony,M.L., and Perko,L.M. " V e h i c l e M o t i o n I n t h e E q u a t o r i a l Plane o f a P l a n e t : a Second Order A n a l y s i s I n E l l l p t l c l t y " . ARS J o u r n a l . V o l . 3 1 , No.10,October I 9 6 I , p p . I 4 l 3 - l 4 2 1 .
4.43
Klng-Hele,D.G. "The e f f e c t o f t h e e a r t h ' s o b l a t e n e s s on t h e o r b i t o f a near s a t e l l i t e " . Proceedings o f t h e Royal S o c i e t y , S e r i e s A, Vol.247, 19 58,pp. 119^2.
4.44
Brenner,J.L. "The M o t i o n o f an E q u a t o r i a l S a t e l l i t e o f an O b l a t e P l a n e t " . B a l l i s t i c M i s s i l e and Space T e c h n o l o g y , V o l . I l l : Guidance, N a v i g a t i o n , T r a c k i n g , and Space P h y s i c s , D.P.LeGalley,ed. , Academic Press,New York,I960,pp.259-289.
4.45
Brenner,J.L. "The E q u a t o r i a l O r b i t o f a Near-Earth S a t e l l i t e " . ARS J o u r n a l , Vol.32,No.10,October I962,pp.I56O-I563.
4.46
Dallas,S.S. P r e d i c t i o n o f t h e P o s i t i o n and V e l o c i t y o f a S a t e l l i t e A f t e r Many R e v o l u t i o n s . T e c h n i c a l Report 32-1267, J e t P r o p u l s i o n L a b o r a t o r y , C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f o r n i a , A p r i l 1970.
4.47
Baker,R.M.L.,Westrom,G.B.,Hilton,C.G.,Gersten,R.H., A r s e n a u l t , J . L . , and Browne,E.J. E f f i c i e n t P r e c i s i o n O r b i t Computation Techniques . A s t r o d y n a m i c a l Report No.3, U n i v e r s i t y o f C a l i f o r n i a , Los A n g e l e s , June 1959.
4.48
Conte,S.D. "The Computation o f S a t e l l i t e O r b i t T r a j e c t o r i e s " . Advances i n Computers,Vol. 3,F.L.Alt and M.Rublnof f ,eds.. , Academic Press,New York,I962,pp.1-76.
4.28
North Pole
Pig.4.1
Coordinate systems 0 = c e n t e r o f mass o f p r i m a r y P = l o c a t i o n o f v e h i c l e , a t d i s t a n c e r from 0 OAC
= equatorial
plane
APB
= p r o j e c t i o n of t r a j e c t o r y Instantaneous radius r
L = longitude,
on sphere w i t h
\ = latitude,
»
1 = Instantaneous
inclination
4.29
10-"*
10-'^
IO-'2
I0-"
p{kg/m^)
Appendix 4A : Some S t a t i c D e n s i t y Models
Type
Mathematical
Ref.
Model
Linear where o({ = c o n s t a n t , > 0
Porster
Hyperbolic h 20 Newton^'"^^
= constant, > 0
where
Power law* where
Yi,?,?,
Pltzpatrlck^*33
~ constant
Klng-Hele^'J-5,24 Pltzpatrlck^•33 Billik^.34 Kork^-35
Standard Exponential where
Averaged Exponential
^l = c o n s t a n t , > o
where
and
Zee^-19
= c o n s t a n t , >• 0, 2Tfn4i) r cIq
r„ —
^ly) (varies seculai
J iltn.
* T h i s model g e n e r a l i z e s t h e models used by B i l l i k ^ ' 3 ^ ,
Kork^'35^
I z s a k ^ ' 3 ^ , and B r o g l l o ' * ' 3 7 , ,
continued
4.33
Appendix 4A-contlnued
Type
M a t h e m a t i c a l Model
Ref.
Modified Exponential* where
= constant, > 0
Roberson^ • -^^
Bruno Model where
= "K-o +
XiV
4-
XiV^
Bruno^-^0
Xo,j^2 = constant Geyllng'^*!^ and Westerman^'38>39 employ t h e f a c t o r R"^ I n s t e a d o f y;~
, where R denotes t h e p l a n e t a r y e q u a t o r i a l r a d i u s .
4.34
Appendix 4B : The V e c t o r System f o r ^r'.e,^ The v e c t o r system w r i t t e n below can be solved f o r
ir>^Q^y, (.(j))
p r o v i d e d t h e s o l u t i o n s K,U,o(,(^) t o system (4.34,36) a r e known.
cli
d Ü dtf Cl. str?
1 -4- T M f Si d t f /
(MB-2)
'0
d (P
XdlLf dtf; subject t o the i n i t i a l conditions :
(MB-4)
CO) =
4^
I t may be noted t h a t any one o f t h e above t h r e e v e c t o r
differential
e q u a t i o n s may be r e p l a c e d by t h e v e c t o r e q u a t i o n (4.2) :
4.35
Appendix 4c : Q u a n t i t i e s Depending on K,U. and o( .
Dimension
Relation
Quantity
Radius r
m.
Time
sec.
t 0
Angular Momentum H
mVsec.
Radial V e l o c i t y
m/sec.
Normal V e l o c i t y
m/sec.
Total Velocity
m/sec.
Longitude L
' ^ - ^ ' - ( ^ ^ ' ' a .
^
Leo.
(rad.)
c >
Latitude
X
(rad.)
Ac(f)=oC
Inclination i
(rad.)
4.36
CHAPTER V THE EQUATORIAL PROBLEM 5•1 I n t r o d u c t i o n Consider t h e n o n l i n e a r , second o r d e r d i f f e r e n t i a l
equation
( 4 . 4 5 ) , which d e s c r i b e s t h e t r a j e c t o r y o f a space v e h i c l e I n t h e e q u a t o r i a l plane o f an o b l a t e p l a n e t . W i t h t h e d e f i n i t i o n s :
the d i f f e r e n t i a l e q u a t i o n can be w r i t t e n I n t h e normal form :
é
f
Let t h e I n i t i a l p o s i t i o n o f t h e v e h i c l e c o i n c i d e w i t h an apse of t h e t r a j e c t o r y , and. l e t t h e I n i t i a l v e l o c i t y be and
( 5 . 1 ) , one a r r i v e s a t t h e f o l l o w i n g I n i t i a l
. Using
(4.36)
conditions f o r the
d i f f e r e n t i a l equation f o r W : W(0) =
where
1^ —
L
^
^
V>^/(M/ro) — 1 + ^ 0
£0)
=
>0
C^.S)
0
(velocity
By c o n s i d e r i n g t h e case o f u n p e r t u r b e d m o t i o n
parameter)
(£ = 0 ) , i t
i s r e a d i l y shown t h a t t h e K e p l e r e c c e n t r i c i t y e^ i s r e l a t e d t o t h e v e l o c i t y parameter ft^
a c c o r d i n g t o e^^ =
1 j.
(S".W)
Thus, i n t h e case o f u n p e r t u r b e d m o t i o n , t h e parameter r e g i o n 0< L
< 1 y i e l d s e l l i p t i c motion w i t h the i n i t i a l p o s i t i o n a t
apocenter;
I, = 1
y i e l d s c i r c u l a r motion; the region
i
-2 y i e l d s b o l i c motion.
hyper-
Equation
(5.2) a l l o w s s e v e r a l more p h y s i c a l I n t e r p r e t a t i o n s .
For example, I t d e s c r i b e s t h e t r a j e c t o r y o f a s l o w l y moving, i n f i n i t e l y n m a l l mano i n a weak, o p h o r i c a l l y Hymmetric, o t a t l c , E i n s t e i n g r a v i t a t i o n a l f i e l d generated ("Schwarzschlld
by a heavy, s p h e r i c a l mass
e x t e r i o r s o l u t i o n " ; s e e Appendix 5A). I t a l s o
d e s c r i b e s t h e o s c i l l a t o r y m o t i o n o f a mass hung from a s p r i n g w i t h a " s p r i n g c o n s t a n t " t h a t v a r i e s l i n e a r l y w i t h a m p l i t u d e , and of t h e v e r t i c a l component o f t h e o s c i l l a t i o n s o f t h e bob o f a s p h e r i c a l pendulum ( e . g . , R e f . 5 . 7 ) . Moreover, i t i s d i r e c t l y
related
t o t h e " s a t e l l i t e e q u a t i o n " ( R e f . 5 . 8 ) , which d e s c r i b e s t h e t r a j e c tory of a l i g h t ray i n the r e l a t l v i s t l c g r a v i t a t i o n a l
field
mentioned above (Appendix 5B). Some c h a r a c t e r i s t i c s o f t h e s o l u t i o n a r e o b t a i n e d r e a d i l y from t h e phase plane diagram ( P i g . 5 . 1 ) . The two s i n g u l a r p o i n t s i n t h e diagram a r e g i v e n by : Ws, =: ^ ~ ^^-^^
I-f
£
+
OCS')
(center) (5-. 5-)
The
I n i t i a l p o i n t , ? , l i e s on t h e a b s c i s s a , a t a d i s t a n c e io
t h e o r i g i n . I n view o f t h e assumption o f s m a l l £
from
, i^—Od),
and
P l i e s always f a r t o t h e l e f t o f t h e saddle p o i n t : o< Vprzl^
^s»'
The d i f f e r e n t i a l e q u a t i o n f o r W y i e l d s t h e f i r s t i n t e g r a l :
5.2 9
where t h e c o n s t a n t
i s determined t h r o u g h t h e i n i t i a l c o n d i t i o n s :
and i s r e l a t e d t o t h e t o t a l energy p e r u n i t mass, (Appendlx5C) :
o-r P
, through
t-
Each t r a j e c t o r y i n t h e phase plane i s c h a r a c t e r i z e d by a v a l u e o f the energy parameter IPi") =. cot ^ Wst
E^ . The energy e q u a t i o n (5.6) i s now r e w r i t t e n •^(W-Wt)(W-(>0x)(\A/-U)3) >
Wst é
UOi
where t h e values o f ^t,z,\ Ch)
^
Wsa.
»
:é
are g i v e n i n Appendix 5C. For
g e n e r a l v a l u e s o f £ ^ o , t h e r e g i o n s o f r e a l m o t i o n are : sup(o,U3t) é \J ^ toa , ancf to, ^ WI n view o f t h e assumption
of small E
, the p h y s i c a l l y
realizable
t r a j e c t o r y goes t h r o u g h t h e p o i n t s (10^,0) and ( U 3 t , 0 ) , where o n l y t h e p a r t W > o should be c o n s i d e r e d . Thus, F i g . 5 . 1 i n d i c a t e s t h a t ; u;t= LOi- Ws< leads t o c i r c u l a r m o t i o n , 0 < UJt < Ws( leads t o q u a s i - e l l i p t i c uji — o cot < 0
motion,
leads t o q u a s i - p a r a b o l i c m o t i o n , leads t o q u a s i - h y p e r b o l i c m o t i o n .
The branch g o i n g t h r o u g h (u;j ,0) corresponds
t o an ever t i g h t e n i n g
s p i r a l t r a j e c t o r y around t h e c e n t e r o f mass r = 0; i n t h e problem o f e q u a t o r i a l m o t i o n , t h i s branch does n o t correspond t o any p h y s i c a l l y r e a l i z a b l e motion.
5.3
A necessary c o n d i t i o n f o r t h e e x i s t e n c e o f f i n i t e elliptic by
) t r a j e c t o r i e s I s g i v e n by
0 é £-Ch)
o; o r . I n view o f
uj< > o
s u f f i c i e n t c o n d i t i o n i s g i v e n by
which y i e l d s an i n e q u a l i t y c o n d i t i o n f o r
(5.5), ,
. Pig.5.2 shows a
g e n e r a l , f i n i t e t r a j e c t o r y i n p o l a r c o o r d i n a t e s ( see a l s o Appendix
5D). I n p a r t i c u l a r , c i r c u l a r o r b i t s are r e a l i z e d i f (Pig.5.1) ;
Ws, =
L
> thus. I f
I -
f. 4-
e 4-
0(6') ;
a p p r o x i m a t i n g t h i s e q u a t i o n f o r s m a l l e c c e n t r i c i t y e^^ , one
obtains
I n terms o f p h y s i c a l parameters :
This r e s u l t was approximate
a l s o o b t a i n e d by Anthony and P o s d l c k * ,
from
s o l u t i o n t o the d i f f e r e n t i a l e q u a t i o n f o r W.
an
The
e x i s t e n c e of c i r c u l a r o r b i t s can be e x p l a i n e d by assuming t h a t t h e a n g u l a r r a t e o f t h e apses o f t h e o s c u l a t i n g e l l i p s e * * t h e a n g u l a r r a t e o f the space v e h i c l e . The at t h e apo-
equals
v e h i c l e t h e n remains
or p e r l c e n t e r o f t h e o s c u l a t l h g e l l i p s e . Pig.5.1
shows t h a t apo- and p e r l c e n t e r r a d i i are c o n s t a n t . The
resulting
o r b i t o f t h e space v e h i c l e i s t h e r e f o r e a c i r c l e . F i n a l l y , i t may equator
be noted t h a t t h e t r a j e c t o r y i n t e r s e c t s t h e
( c o l l i s i o n ) when r = R; t h i s y i e l d s t h e c o n d i t i o n f o r
n o n i n t e r s e c t l o n or f r e e m o t i o n
:
O ^ V
. R
S e c t i o n 5.2 o u t l i n e s t h e d e r i v a t i o n o f a second o r d e r a p p r o x i mate s o l u t i o n f o r W t h r o u g h a p p l i c a t i o n o f t h e M u l t l V a r i a b l e Approach, I n S e c t i o n 5.3,an e x t e n s i v e comparison w i t h p e r t i n e n t r e s u l t s l i t e r a t u r e i s presented
. Numerical
S e c t i o n 5.4, * Ref.5.10,equation (62)
comparison i s presented i n
' ; ** Appendix 5D, e q u a t i o n
5.4
.
(5D-3).
from
5.2 M u l t l V a r i a b l e Approach
5.2.1 System o f d i f f e r e n t i a l
equations
F o l l o w i n g t h e procedure o u t l i n e d I n S e c t i o n 2.6, assume an approximate s o l u t i o n o f t h e form : W ( r > »
I
^\^"fM....,...;o =
Where t h e c l o c k s tj)„ -
£ƒ»
^"-^""f-
>
^
cr.io)
Oer')
^o,i,3,i,:- a r e g i v e n by : ,
=
( ) . ( c f ) cl(f
( i = 1 . 2 , S , . . . , N ) iS-.li)
S u b s t i t u t i n g t h e above e x p r e s s i o n s i n system b i n i n g terms c o n t a i n i n g l i k e powers o f £
(5.2,3), com-
, and s e t t i n g them e q u a l
t o z e r o , y i e l d s t h e f o l l o w i n g system o f coupled d i f f e r e n t i a l
equa-
t i o n s , t o be s o l v e d r e c u r s i v e l y : e" terms
:
4- U, =
terms
:
4-
e^terms :
hl±
\^
and so on; and where
1
Wt =:
^^-^2)
-
^
|
^
-
( )* zz
5.5
—
-
#
2^ " ^ f
"
L^-t^)
+
S i m i l a r l y , the i n i t i a l c o n d i t i o n s are transformed t o :
£" terms : V/^ (o) =
1 +•
e* terms
O
:
E^erms :
Wi (0) =
WiCO) = O
^
7 ^ CO) = : O
Co)
1 ( o ) - ^ ( o } -o
It
» | ^
(o) +- i ( o ) . ^ c o ) 4 -
isr.ié)
öcft
J(o)-^(o)-0
and so on.
5 . 2 . 2 S o l u t i o n o f t h e z e r o e t h o r d e r system ( £° t e r m s ) I n t e g r a t i o n o f (5.12) w i t h respect t o t h e f a s t clock i m m e d i a t e l y y i e l d s t h e e x p l i c i t dependence o f Wo
where C,^o(>o((pdi
-2c,(i-P
^ )
Sin c o ( „ - o ( o 4-
i)jf^^-cosc ( l i n e a r c l o c k )
where Ci.oTf Ccfi.ï^... )
and Co^oTo Ccfj^...) a r e unknown f u n c t i o n s o f t h e i r
arguments, and w h e r e y U
i s a "compatibility constant".
I n t e g r a t i n g the d i f f e r e n t i a l equation f o r i n t o account r e q u i r e m e n t WaCtf< =
arrives
on
be
involving
( S e c t i o n 2.3).
d e p e n d e n c e o f W^^^ on (^g^ , a n d algebraic
and second
dependence o f
( 5 . 3 9 ) may
E x p a n s i o n Method,
clock
considerable
order,
of s o l u t i o n
Variable
of the e x p l i c i t
i^g , i n v o l v e s
i s resolved
solution
t h e Two
c l o c k ^. = y a n d t h e s l o w
explicit
nacy
Method
domain o f u n i f o r m v a l i d i t y
Increased
of
Expansion
order
labor.
As o n l y
equations
are used,
i s n o t known. T h i s
the concept
of " r e s t r i c t i o n
a t the second
order
the
indetermio f number
approximate
:
where
WiCc^;e)-
( 1 4 - ^ e : ) - ( t - f 4eo)
'1 +
te.
tH-il&o^(|)L.cos](t-£)y +
I •je.1
4 - { e , ( l - e , - l - {el)-
cos
5.17
zif
c-os. 2
H- l ^ e f . c o s i f
(S-.40)
The f i r s t o r d e r term c o n t a i n s a s e c u l a r t e r m , which t h e domain o f u n i f o r m v a l i d i t y t o :
Comparing ( 5 . ^ 0 ) w i t h the slow c l o c k
(5.39),
literature
o ^ C|? « ~£_i\e.„\
i t i s seen t h a t t h e i n t r o d u c t i o n
r e p l a c e s t h e f a s t s e c u l a r term i n
s e c u l a r term. The expressions
limits
of
by a slow
f o r Wj are i d e n t i c a l . A search o f
d i d n o t d i s c l o s e t h e e x i s t e n c e o f any s i m i l a r a n a l y s i s
and/or r e s u l t s .
5.18
5.3.^ G e n e r a l i z e d Two V a r i a b l e Expansion Method .Dallas^'lö, I n h i s comprehensive s t u d y o f t h e e q u a t o r i a l problem, a p p l i e d t h e G e n e r a l i z e d Two V a r i a b l e Expansion Method, with Jlo
= 1,
= 0 ( S e c t i o n 2 . 4 . 2 ) . The a v a i l a b i l i t y o f t h e
a r b i t r a r y s t r e t c h i n g c o n s t a n t s co^ a l l o w s g r e a t e r freedom i n suppressing s e c u l a r t e r m s , such as t h e one t h a t a r i s e s i n Wi t h r o u g h a p p l i c a t i o n o f t h e Two V a r i a b l e Expansion Method ( see equation
(5.40))*.
By a n a l y z i n g t h e z e r o e t h o r d e r t h r o u g h t h i r d o r d e r e q u a t i o n s , D a l l a s o b t a i n s second o r d e r approximate
solutions f o r the radius
f u n c t i o n W, t h e " r a d i a l v e l o c i t y f u n c t i o n " dW/dy> , and t h e t i m e T, as f u n c t i o n s o f t h e independent
variable
. The i n i t i a l c o n d i t i o n s
are a r b i t r a r y **. Using t h e i n i t i a l c o n d i t i o n s ( 5 - 3 ) and c a r e f u l l y the
retracing
a n a l y s i s t o e l i m i n a t e t h e a m b i g u i t y i n COo.i.a , o^^e f i n d s t h e
f o l l o w i n g expression f o r the radius f u n c t i o n : Wccfie) =
UctfiÊ)^
V/o-l- e U t
e'XJt-^- ÖC£^)
where :
t +• e. • COS
* See comment i n Ref.5•18,p.84, l a s t
paragraph.
** The values f o r t h e c o n s t a n t s coo.i.t i n Ref.5.18 a r e ambiguous; t h i s may l e a d t o erroneous
r e s u l t s , as t h e c o n s t a n t s e^^^^j a r e
r e s t r i c t e d t o be p o s i t i v e s e m i - d e f i n l t e ; see Ref.5•18,p.102. 5.19
and where*:
(f =
[i - E - ^(i
As no s e c u l a r terms o c c u r , t h e s o l u t i o n I s u n i f o r m l y v a l l d f o r
I t t u r n s o u t t o be c o n v e n i e n t ( S e c t i o n 5.4) t o w r i t e down the s o l u t i o n t h a t one would have o b t a i n e d I f t h e a n a l y s i s o f Ref. 5.18 would have been r e s t r i c t e d t o z e r o e t h o r d e r t h r o u g h second order equations only. This s o l u t i o n i s c a l l e d t h e " f a i r
comparison
s o l u t i o n " , as i t a l l o w s f a i r comparison w i t h t h e M u l t i V a r i a b l e s o l u t i o n , w h i c h was o b t a i n e d by a n a l y z i n g t h e same system o f z e r o e t h o r d e r t h r o u g h second o r d e r e q u a t i o n s ; t h u s : V^(fi£) =
Wo -H ^ W i
4-
where
Oce')
(^.42) Wo^j a r e g i v e n by ( 5 . 4 l ) , and where :
» Ref.5.18 g i v e s a p l u s - s i g n i n f r o n t o f t h e
t e r m . Comparison
w i t h e q u a t i o n ( 5 . 3 3 ) as w e l l as c h e c k i n g t h e a l g e b r a o f Ref.5.18 confirms t h a t t h i s i s a p r i n t i n g
5.20
error.
5.3.5
L l n d s t e d t Method The phase p l a n e diagram
(Pig.5.1) I n d i c a t e s t h a t the exact
s o l u t i o n i s p e r i o d i c . The L l n d s t e d t method i s t h e r e f o r e
well-
s u i t e d t o g e n e r a t e approximate s o l u t i o n s t o t h e e q u a t o r i a l Pirst
problem.
o r d e r approximate s o l u t i o n s are g i v e n by Anthony
and
Posdlck^'-'-^, and by Kyner^'-^^. They can be t r a n s f o r m e d t o y i e l d W ^ ^ ^ as f o r m u l a t e d i n ( 5 . 4 l ) , w i t h
=
Cf [ t - £ -h Ü^e')
A second o r d e r approximate s o l u t i o n i s g i v e n by Anthony Perko5.20j i t
can be t r a n s f o r m e d t o y i e l d
( 5 . 4 l ) , b u t where ^
^o.i^i
as f o r m u l a t e d i n
as g i v e n by (5.42) must be
An a t t e m p t by Anthony
and P o s d i c k ^ *
and
substituted.
t o apply the L l n d s t e d t
method t o t h e g e n e r a l problem o f t h r e e d i m e n s i o n a l m o t i o n , met w i t h . f a i l u r e , as e x p e c t e d , as such m o t i o n i s i n g e n e r a l n o n p e r i o d i c (see
a l s o R e f . 5 . 2 2 ) . The s e c u l a r t e r m i n
o n l y i n t h e s p e c i a l cases o f i n i t i a l l y initially
o f Ref.5-21 d i s a p p e a r s
e q u a t o r i a l m o t i o n and o f
p o l a r m o t i o n . I n t h e case o f i n i t i a l l y
e q u a t o r i a l motion
( e q u a t o r i a l p r o b l e m ) , t h e i r r e s u l t s reduce t o t h e e a r l i e r o b t a i n e d first
o r d e r approximate s o l u t i o n s mentioned
above.
The a t t e m p t o f G h a f f a r l 5 « 2 3 t o o b t a i n a f i r s t
order approxi-
mate s o l u t i o n a l s o deserves a t t e n t i o n . His o b j e c t i v e i s t o o b t a i n a first
o r d e r approximate s o l u t i o n f o r W((|';£) - W(y);0), which i s
a measure o f t h e r e l a t l v l s t l c c o r r e c t i o n o f t h e Newtonian
motion
of a space v e h i c l e around a s p h e r i c a l l y symmetric p r i m a r y body (Appendix 5A), I n s p i t e o f t h e s i m p l i c i t y
of the problem, the
a n a l y s i s c o n t a i n s t h r e e fundamental a l g e b r a i c e r r o r s
5.21
(Ref.5.24).
Thus, t h e a n a l y t i c a l r e s u l t s and t h e i r subsequent n u m e r i c a l e v a l u a t i o n are erroneous. P a r e n t h e t i c a l l y , i t may be p o i n t e d o u t t h a t c o r r e c t e x e c u t i o n o f t h e a n a l y s i s would g i v e t h e f i r s t
order
result*: l/cf-.e) — Wccf ;o)
where :
(Jj =
=
^ [i-t
e.'Ccosc^-
^
c o s y?) -f-£| (
Ö(e')'
Compare w i t h Ref.5.23,equation ( 2 0 ) , and a l s o p.8.
5.22
1
)
+
5.3.6
Averaging Methods Carr e t a l . ^ * ^ ^ a p p l i e d t h e K r y l o v - B o g o l l u b o v Method Of Aver-
aging, t o y i e l d a f i r s t
o r d e r approximate
s o l u t i o n , which i s
i d e n t i c a l t o t h e z e r o e t h o r d e r p a r t o f t h e Two
V a r i a b l e approximat
s o l u t i o n ( 5 . 4 0 ) , and t o t h e z e r o e t h o r d e r p a r t o f t h e L l n d s t e d t f i r s t o r d e r approximate
s o l u t i o n o b t a i n e d by Anthony and
Posdick^'
and by Kyner^*-"-^ ( S e c t i o n 5.3.5). Kyner^*'^^ and Lass and Solloway^*^^ a p p l i e d t h e
Bogoliubov-
M i t r o p o l s k y M o d i f i e d Method o f A v e r a g i n g , t o y i e l d an Improved f i r s t o r d e r approximate
s o l u t i o n * , which i s I d e n t i c a l t o t h e L l n d -
s t e d t f i r s t o r d e r approximate
s o l u t i o n o b t a i n e d by Anthony and
Posdlck5-10 and by Kyner^'-^^ ( S e c t i o n 5.3.5). Brenner**
o b t a i n e d a second o r d e r approximate
s o l u t i o n by
means o f t h e K r y l o v - B o g o l l u b o v Method o f A v e r a g i n g . Por g i v e n the
,
c o r r e s p o n d i n g v a l u e f o r W must be determined t h r o u g h n u m e r i c a l
i t e r a t i o n , which makes t h e form o f h i s s o l u t i o n l e s s d e s i r a b l e , and a n a l y t i c a l comparison d i f f i c u l t . The a n o t h e r , second o r d e r approximate
same paper c o n t a i n s
s o l u t i o n * * * , obtained through
a n a l y t i c a l i t e r a t i o n . Upon r e w r i t i n g * * * * t h i s s o l u t i o n , i t i s found t h a t t h e z e r o e t h o r d e r and f i r s t o r d e r p a r t s agree w i t h .
* T r a n s f o r m a t i o n o f n o t a t i o n and i n i t i a l c o n d i t i o n s r e q u i r e s some c a r e . **Ref.5.12,pp.279-28l.
***
Ref.5.12,pp.274-278.
*»** The use o f m^ and m^ i n t h e i n i t i a l c o n d i t i o n s o f Ref.5.12 i s somewhat c o n f u s i n g . D i r e c t a n a l y t i c a l comparison can be made by s e t t i n g m^ = mg = approximate
0 . A n a l y t i c a l comparison w i t h t h e second o r d e r
s o l u t i o n o f Ref.5.13 i s more d i f f i c u l t , as t h e r e m^
and mj are not a r b i t r a r y anymore.
5.23
e.g., t h e r e s u l t s o f Anthony and Perko^'^O ( S e c t i o n 5.3.5)» and t h a t a l l c o e f f i c i e n t s i n t h e second o r d e r p a r t o f t h e s o l u t i o n are erroneous - by consequence, so are t h e c o e f f i c i e n t s i n t h e second o r d e r p a r t o f t h e s o l u t i o n p r e s e n t e d on t o p o f p.274 o f the same paper.
5.24
5•4 Numerical Comparison Reviewing t h e r e s u l t s o b t a i n e d i n t h e p r e c e d i n g S e c t i o n s , one observes t h a t an exact s o l u t i o n has been f o u n d , i n a d d i t i o n t o v a r i o u s approximate s o l u t i o n s . I n o r d e r t o study t h e accuracy of c e r t a i n of these approximate s o l u t i o n s
i n some d e t a i l , a com-
p u t e r program has been w r i t t e n t o y i e l d n u m e r i c a l v a l u e s f o r t h e following solutions
:
1. t h e exact s o l u t i o n
:
•
, see
(5.35);
2. t h e second o r d e r , One V a r i a b l e approximate s o l u t i o n : , see
(5.39);
3. t h e second o r d e r . Two V a r i a b l e approximate s o l u t i o n : W ^ . ^ , , see
(5.40);
4. t h e second o r d e r . G e n e r a l i z e d Two V a r i a b l e approximate s o l u t i o n :
5. t h e " f a i r comparison s o l u t i o n " :
\rv'
(5.41);
W , see
(5.42);
6. t h e second o r d e r , M u l t i V a r i a b l e approximate s o l u t i o n : W^^ , see
(5.33).
Some r e p r e s e n t a t i v e r e s u l t s have been p l o t t e d i n Pigs.5.3-5. Pig.5.3 shows t h e b e h a v i o r o f t h e exact s o l u t i o n over s e v e r a l i n t e r v a l s of (jP , f o r £
= 10 , and e^ = 0.9. Pigs . 5. 4 ,5 show t h e
envelope o f t h e " d e v i a t i o n " between t h e n u m e r i c a l v a l u e o f t h e exact s o l u t i o n and t h e n u m e r i c a l v a l u e o f t h e f i v e solutions
l i s t e d above; i . e . , o f
A Wj,^ = W^^ -
approximate , and so on
(compare ( 3 . 9 1 ) ) ? f o r t h e combinations o f v a l u e s :£ = 10 ,6^ = 0.1, and £ = 10 , Oo = 0 . 9 . ( I t may c o u p l e d , as £
=
be noted t h a t a c t u a l l y £ and e^ are
/(l+Co)").
The IBM
c a l c u l a t i o n s were executed I n double p r e c i s i o n on an
360/91 computer, u s i n g t h e POLRT s u b r o u t i n e f o r t h e
of the r o o t s u)t,2,3
i n Wg , t h e GEL 1 s u b r o u t i n e f o r t h e
of the complete e l l i p t i c
calculation calculation
i n t e g r a l o f t h e f i r s t k i n d K(o(), and t h e
JELP 1 s u b r o u t i n e f o r t h e c a l c u l a t i o n o f t h e J a c o b i a n
elliptic
f u n c t i o n sn. I n b o t h cases ( P i g s . 5 . 4 , 5 ) i t i s found
t h a t t h e One V a r i a b l e
approximate s o l u t i o n ( A W ^ , , ) generates the l a r g e s t d e v i a t i o n , and t h e G e n e r a l i z e d
Two V a r i a b l e approximate s o l u t i o n ( A Wg^-y )
generates t h e s m a l l e s t one.
The o t h e r s o l u t i o n s d i s p l a y d e v i a t i o n s
of about t h e same magnitude f o r s m a l l Oo ; f o r i n c r e a s i n g e^ , these d e v i a t i o n s I n c r e a s e
: t h e l e a s t f o r the M u l t i V a r i a b l e
approximate s o l u t i o n ( A W^y )» and t h e most f o r the Two V a r i a b l e approximate s o l u t i o n ( t W^-y ) . I t must now be r e c a l l e d ( S e c t i o n 5.3.4) t h a t t h e G e n e r a l i z e d
Two V a r i a b l e approximate s o l u t i o n was
o b t a i n e d by c o n s i d e r i n g t h e t h i r d o r d e r e q u a t i o n
{
terms) i n
a d d i t i o n t o t h e lower o r d e r e q u a t i o n s , w h i l e t h e o t h e r s o l u t i o n s were o b t a i n e d by c o n s i d e r i n g t h e z e r o e t h o r d e r t h r o u g h second equations
only. This explains t h e comparatively
A WgTv . A " f a i r " Two
Generalized
V a r i a b l e a n a l y s i s would t a k e i n t o account t h e z e r o e t h
A. Wp^j^ than
small d e v i a t i o n
comparison would be o b t a i n e d i f t h e
t h r o u g h second o r d e r e q u a t i o n s
order
order
o n l y ; t h e r e s u l t i s r e p r e s e n t e d by
. I t i s seen t h a t t h e d e v i a t i o n a W^^ i s s l i g h t l y
larger
A Wp^^^ f o r s m a l l e^ (e^, = 0 . 1 ) , b u t i s t h e s m a l l e s t o f a l l
d e v i a t i o n s f o r l a r g e e^ (e^, = 0 . 9 ) . A l l d e v i a t i o n s i n c r e a s e as e^, Increases; the e f f e c t i s t h e smallest f o r
5.26
A W^.,
3'5 Summary A second o r d e r M u l t i V a r i a b l e approximate s o l u t i o n I s d e r i v e d f o r t h e e q u a t o r i a l problem. A l l c l o c k s t u r n o u t t o be l i n e a r .
This
was expected, as t h e phase plane diagram i n d i c a t e s t h a t t h e m o t i o n under . c o n s i d e r a t i o n ( 0 4 £ = 0
Quasi-elliptic
AjAy= pericenter
Flg.5.2
=|a|9
5D)
5.35
Appendix 5A : R e l a t l v l s t l c M o t i o n o f a Small Mass In h i s theory of general r e l a t i v i t y ,
E i n s t e i n proposed
e q u a t i o n s d e s c r i b i n g t h e space-time g r a v i t a t i o n a l f i e l d
(e.g.,
R e f s . 5 . 1 , 2 ) . S c h w a r z s c h i l d ^ ' ^ o b t a i n e d an exact s o l u t i o n t o t h e E i n s t e i n f i e l d e q u a t i o n s f o r t h e case o f a s p h e r i c a l l y g r a v i t a t i o n a l f i e l d , generated
symmetric
by a s i n g l e , s p h e r i c a l mass M,
whose c e n t e r c o i n c i d e s w i t h t h e c e n t e r o f t h e s p a t i a l c o o r d i n a t e s . T h i s s o l u t i o n , known as t h e " S c h w a r z s c h i l d
exterior
solution"*.
I s t h e most g e n e r a l s p h e r i c a l l y symmetric s o l u t i o n t o t h e f i e l d equations I n empty space. In the following I t
I s assumed t h a t t h e g r a v i t a t i o n a l
field
I s weak, and/or t h a t t h e f i e l d p r o p e r t i e s a t l a r g e d i s t a n c e s o n l y are c o n s i d e r e d . The s p a t i a l c o o r d i n a t e s if* , ö , and ^, can t h e n be I n t e r p r e t e d as t h e E u c l i d i a n s p h e r i c a l c o o r d i n a t e s
: radius,lati-
t u d e , and l o n g i t u d e . Consider t h e m o t i o n o f a s l o w l y moving body o f I n f i n i t e l y s m a l l mass. I n t h e f i e l d mentioned above. One can t h e n show** t h a t the s p a t i a l t r a j e c t o r y t r a v e r s e d occurs I n a f i x e d plane c o n t a i n ing
t h e c e n t e r of mass M, w h i l e t h e r e l a t i o n between r a d i u s r and
the I n - p l a n e c e n t r a l angle
(P
I s d e s c r i b e d by equation' ( 5 . 2 ) , where
(dimensionless)
where H denotes t h e a n g u l a r momentum, f t h e u n i v e r s a l g r a v i t a t i o n a l c o n s t a n t , and c t h e v e l o c i t y o f l i g h t .
* Ref.5.1,pp.177-183. *»Ref.5.1,p.
173.
5.36
Por t h e case o f t h e r e l a t i v i s t i c m o t i o n o f t h e p l a n e t Mercury around t h e Sun, one f i n d s f o r t h e s m a l l parameter
£, , a f t e r
s u b s t i t u t i o n o f t h e p r o p e r n u m e r i c a l values ( R e f . 5 . 4 ) : -8
5.37
Appendix 5B : S a t e l l i t e Equation and t h e T r a j e c t o r y o f a L i g h t Ray Struble^'Ö d e f i n e s t h e " s a t e l l i t e
e q u a t i o n " as :
T h i s e q u a t i o n can be shown t o be r e l a t e d t o e q u a t i o n ( 5 . 2 ) t h r o u g h l i n e a r t r a n s f o r m a t i o n o f t h e dependent and t h e independent variable :
C o n s i d e r i n g t h e motion o f a l i g h t r a y i n t h e r e l a t l v l s t l c g r a v i t a t i o n a l f i e l d mentioned i n Appendix 5A, one can show* t h a t the
s p a t i a l t r a j e c t o r y t r a v e r s e d occurs i n a f i x e d plane c o n t a i n -
ing
t h e c e n t e r o f mass M, w h i l e t h e r e l a t i o n between t h e r a d i u s r
and t h e i n - p l a n e c e n t r a l angle
i s d e s c r i b e d by e q u a t i o n
(5B-1),
where : \^/ —
J
£ —
3. • t ' ^—
(dimensionless)
For a l i g h t r a y whose apse touches g e n t i a l l y ) , one has ( R e f . 5 . 4 ) : r^ the
the solar surface ( t a n -
= Rg„^ = 7X 10 m. S u b s t i t u t i n g
proper n u m e r i c a l v a l u e s f o r f j M , and c, one t h e n f i n d s f o r t h e - / s m a l l parameter £ : £ = 6.4X10 ^ < 1 .
* Ref.5.1,p.188.
5.38
Appendix 5C : The .Energy E q u a t i o n M u l t i p l y i n g e q u a t i o n (5.2) by ' ^ " ^
and I n t e g r a t i n g ,
yields
the energy e q u a t i o n :
where t h e constant^''is r e l a t e d t o t h e i n i t i a l c o n d i t i o n s t h r o u g h :
The
Using
k i n e t i c energy p e r u n i t mass can be w r i t t e n as :
t h e p o t e n t i a l energy p e r u n i t mass can be w r i t t e n as :
(4.8),
r
'o(=o
Adding ( 5 0 - 2 , 3 ) , one f i n d s t h a t t h e t o t a l energy E^- = E^^^^ + E is proportional —
I
t o t h e c o n s t a n t E^ a c c o r d i n g t o : r
^
and i s t h e r e f o r e a l s o c o n s t a n t . The
energy e q u a t i o n i s now r e w r i t t e n as :
where cüt^i.^Üe)
(
C0
C o n s i d e r i n g s m a l l values f o r t h e parameter the r e g i o n The
coi ^ W
a)^ £ only,one f i n d s
must be excluded as p h y s i c a l l y
exact v a l u e s o f u^t^a.s
f o l l o w from
5.39
(5C-2,5) :
that
realizable.
v/hlch can be approximated as :
2e
^
5.40
Appendix 5D : Angular S h i f t o f t h e Apses The phase plane diagram
indicates that the solution f o r W i s
p e r i o d i c , l e a d i n g t o c o n s t a n t p e r l - and apocenter a l values o f 1^ and STT
r a d i i . Por gener-
, t h e a n g u l a r p e r i o d d i f f e r s from t h e v a l u e
( p e r i o d o f t h e u n p e r t u r b e d m o t i o n ) . One can e a s i l y d e r i v e
an exact e x p r e s s i o n f o r t h e d i f f e r e n c e i n a n g u l a r perlo.d, by c o n s i d e r i n g t h e s t r u c t u r e o f t h e exact s o l u t i o n ( 5 . 3 5 ) ; t h e angle t r a v e r s e d between any one p e r l - o r apocenter and t h e n e x t , i s g i v e n
where K(oi)' •
Por s m a l l values o f t h e parameter £
, one may expand* K(o()
i n a power s e r i e s i n £ , y i e l d i n g t h e a n g u l a r apse s h i f t p e r revolution : A
iet =
ACf-STT
-
2ÏÏ€
[i
OCe)}
•
CS-D-2)
Por t h e e q u a t o r i a l problem one t h e n h a s , i n terms o f t h e Kepler o r b i t elements a and e^ , t o t h e lowest o r d e r :
i n d i c a t i n g an advancement o f t h e apses ( P i g . 5 . 2 ) , v a r y i n g i n t h e same sense as t h e e c c e n t r i c i t y e^ , and v a r y i n g i n I n v e r s e sense of t h e semi-major a x i s ( o r mean d i s t a n c e ) a. T h i s e x p r e s s i o n agrees w i t h a r e s u l t o b t a i n e d by King-Hele**;however, h i s c o n c l u s i o n must be amended : t h e apse s h i f t c e r t a i n l y depends on t h e i n i t i a l e c c e n t r i c i t y , as i s borne o u t c l e a r l y by t h e l a s t e x p r e s s i o n i n (5D-3).
* Ref .5. 9,p. 297.
Ref. 5. H , e q u a t i o n ( 8 7 ) 5.41
Por
the
relativistic
which I n d i c a t e s
the
was
found f o r the
the
gravity
fM
fM
= ^TTa^TK
u n p e r t u r b e d motion. T h i s
the
advancement of the
The
revolution,
=
the
through the
familiar
period
of
first
a r r i v e d at
for by
^ ( 5 D - 3 ) ) l s v a l i d to the desired,
one
one
out
the
expansion
must d e t e r m i n e
p e r t u r b e d m o t i o n , which c o n s t i t u e s
of the
lowest order only;
must c a r r y
a first
the
order
u n p e r t u r b e d p e r i o d Tj^ .
apse r o t a t i o n g i v e n by
( 5 D - 4 , 5 ) i s r e l a t i v e to a
of axes which undergo p a r a l l e l t r a n s p o r t on
literature,
y i e l d s the w e l l - k n o w n e x p r e s s i o n
(5D-1) to h i g h e r o r d e r ; i n a d d i t i o n ,
correction
e^^ as
^
r e s u l t ( as w e l l as
p e r i o d of the
a and
5 where T^^ d e n o t e s t h e
apses per
i f h i g h e r o r d e r terms a r e of
eliminated
:
A This
s i m i l a r l y (App.5A) :
problem. I n p h y s i c s
is usually
the
Einstein^'^
has,
same q u a l i t a t i v e dependence on
equatorial
factor
t h i r d K e p l e r law
problem, one
c e n t r a l mass M.
See
also
Ref.5.6.
5.42
along the
triad
world-line
6.1 CHAPTER V I AERODYNAraCALLY PERTURBED FLIGHT THROUGH A CONSTANT-DENSITY ATMOSPHERE 6.1 I n t r o d u c t i o n In equations
S e c t i o n 4.^.2, a system
of coupled,nonlinear
was d e r i v e d which d e s c r i b e s t h e t w o - d i m e n s i o n a l motion o f
a l i f t i n g aerospace
v e h i c l e around a s p h e r i c a l l y
body, through an atmosphere s u r r o u n d i n g
symmetric
the present Chapter,
primary
t h a t body. The v e h i c l e i s
assumed t o f l y a t z e r o bank-angle and a t c o n s t a n t In
differential
angle-of-attack.
t h e s p e c i a l c a s e of motion t h r o u g h
a c o n s t a n t - d e n s i t y atmosphere i s c o n s i d e r e d . T h i s c a s e c o n s t i t u t e s a reasonable
a p p r o x i m a t i o n t o t h e a c t u a l c a s e o f motion t h r o u g h a
v a r i a b l e - d e n s i t y atmosphere i f one c o n s i d e r s t r a j e c t o r i e s a t h i g h a l t i t u d e s , where t h e d e n s i t y g r a d i e n t i s s m a l l , o r n e a r - c i r c u l a r t r a j e c t o r i e s , where t h e v a r i a t i o n s i n t h e n o n d i m e n s i o n a l r a d i u s are s m a l l . Using
t h e M u l t i V a r i a b l e Approach, a f i r s t
order
approx-
imate s o l u t i o n i s d e r i v e d I n S e c t i o n 6.2 and i s i n t e r p r e t e d i n S e c t i o n 6.3. I t i s compared a n a l y t i c a l l y w i t h some r e s u l t s l i t e r a t u r e i n S e c t i o n 6.k.
from
F i n a l l y , t h e approximate s o l u t i o n i s
compared n u m e r i c a l l y w i t h t h e s o l u t i o n o b t a i n e d i n t e g r a t i o n of the equations
through
numerical
o f motion i n S e c t i o n 6.5-
F o r t h e s p e c i a l c a s e o f a c o n s t a n t - d e n s i t y atmosphere ( = 1 ) , t h e e q u a t i o n s n o n l i n e a r system
dK
o f motion (k.^Z)
reduce t o the coupled,
:
_ 2£ K
,2
1-h
V2
.0 d j (.6.1)
.4 + U =
K- e ^ | n . f i - f t r
where K d e n o t e s t h e n o n d i m e n s i o n a l a n g u l a r
momentum f u n c t i o n , U t h e
n o n d i m e n s i o n a l r a d i u s f u n c t i o n , y? t h e i n - p l a n e the
c e n t r a l a n g l e , ^u/cp
l i f t - t o - d r a g r a t i o , and £ t h e drag p e r t u r b a t i o n p a r a m e t e r . F o r
s i m p l i c i t y i n n o t a t i o n , t h e s u b s c r i p t i n £ h a s been d e l e t e d . Let
the I n i t i a l
p o s i t i o n of the v e h i c l e coincide with a
perlcenter-^- o f t h e t r a j e c t o r y , and l e t t h e i n i t i a l v e l o c i t y be ¥,^^;
K^,U^
on
Lf, ;
(f^ on (j) .
In view of the presence of an e r r o r of the order one i n the second order system ( £ terms ) , only the zeroeth order and f i r s t order systems are a v a i l a b l e f o r a n a l y s i s . I n other words, no improvement can be obtained over the r e s u l t s obtained t h u s f a r ; one must set N = 1 i n equations
(6.5-6).
Invoking the concept of " r e s t r i c t i o n of number of c l o c k s " , one then has :
leading to the constancy of the f o l l o w i n g f u n c t i o n s :
0 ' o
(^.28)
in
,U
and ^ V c o ( t h e p a r a m e t e r ^
these equations,
a s no s m a l l a n g l e a p p r o x i m a t i o n has been.made;
the e q u a t i o n s numerical
does not o c c u r
a r e e x a c t ) . I n t e g r a t i o n o f (6C-3) y i e l d s t h e
values f o r the slowly varying
F i n a l l y , K(y>) and Hip
are obtained
6.33
functions
through
k,a,b(^).
(6C-1).
In
CHAPTER V I I AERODYNAMICALLY PERTURBED'PLIGHT THROUGH A VARIABLE-DENSITY ATMOSPHERE 7.1 I n t r o d u c t i o n The the one
a n a l y s i s presented presented
I n t h i s C h a p t e r I s an e x t e n s i o n
I n Chapter S i x , i n that a
quasi-exponential
d e n s i t y model i s u s e d i n s t e a d of a c o n s t a n t s p e c i f i c a l l y . I n s t e a d of using
^
= 1, one
d e n s i t y model. More uses
d e n s i t y model
( 4 . l 4 ) , which can be w r i t t e n i n terms of U as f o l l o w s
^(U) The
=
:
u- e
9
a n a l y s i s r u n s e s s e n t i a l l y a l o n g the same l i n e s as t h e
f o r the c o n s t a n t
one
d e n s i t y model.
The . a l g e b r a I n v o l v e d
i n the determination
approximate s o l u t i o n t o t h e e q u a t i o n s considerably
of
of a f i r s t
order
of motion i s s i m p l i f i e d
by n e g l e c t i n g t h e e f f e c t o f the s l o w c l o c k s on
the
r a d i u s f u n c t i o n o c c u r r i n g i n t h e exponent of t h e d e n s i t y model. I n o t h e r words, one
s u b s t i t u t e s the u n p e r t u r b e d s o l u t i o n (£ =
i n t h e exponent, l e a d i n g t o t h e e x p r e s s i o n = where for slowly
j —
lie
( c o n s t a n t ) . The
:
^
f
a p p r o x i m a t i o n i s q u i t e good
c o n t r a c t i n g n e a r - c i r c u l a r o r b i t s . The
same I s
f o r e c c e n t r i c o r b i t s as long as t h e p e r l c e n t e r d e s c e n t remains s m a l l , f o r i n t h i s c a s e t h e l a r g e density given ( 7 . 2 ) ; and generates
by (7.1)
is essentially
i t i s p r e c i s e l y t h i s region t h e main p e r t u r b a t i o n
the d e n s i t y p r o f i l e
(7.2)
0)
true
rate
pericenter-region
equal
t o t h e one
given
by
of the t r a j e c t o r y t h a t
effects. Physically
speaking,
i s t h a t e n c o u n t e r e d a l o n g an
mediate t r a j e c t o r y l y i n g c l o s e t o both t h e u n p e r t u r b e d
intertrajec-
t o r y and
the a c t u a l t r a j e c t o r y . Model ( 7 . 2 ) l e a d s to an
ment over t h e r e s u l t s of v a r i o u s a u t h o r s model the one
who
t a k e as
Improve-
density
d e s c r i b i n g t h e d e n s i t y v a r i a t i o n a l o n g the
unpertur-
bed t r a j e c t o r y . S u b s t i t u t i o n of
( 7 . 2 ) i n the e q u a t i o n s
y i e l d s the c o u p l e d , n o n l i n e a r
of motion
(4.42)
system :
CP C/.3)
VU clcfJ where the s u b s c r i p t i n £^ has S i x , the s y s t e m I s
Kfo)=
{ y
been d e l e t e d . F o l l o w i n g
s u b j e c t e d to the i n i t i a l
»^
U(o) =U(o)
1I
ft
..
conditions
Chapter :
g ( o ) -= °O 3^^°^
^l-"^^
I
where : The
M u l t i V a r i a b l e Approach i s a p p l i e d t o s y s t e m
S e c t i o n 7,2, y i e l d i n g a f i r s t
order
s o l u t i o n I s d i s c u s s e d i n S e c t i o n 7.3.
(7.3,4) i n
approximate s o l u t i o n . Related
The
r e s u l t s from
a t u r e a r e d i s c u s s e d i n S e c t i o n 7.4. F i n a l l y , the n u m e r i c a l
literaccu-
r a c y of the M u l t i V a r i a b l e approximate s o l u t i o n I s d i s c u s s e d S e c t i o n 7.5,
7.2
in
7.2 M u l t l V a r i a b l e Approach
7.2.1 System o f d i f f e r e n t i a l Following
equations
S e c t i o n 6.2.1, a p p l i c a t i o n of t h e M u l t l
Approach l e a d s t o t h e f o l l o w i n g
system o f d i f f e r e n t i a l
Variable equations,
t o be s o l v e d r e c u r s i v e l y :
terms
O
c)Cfo
s'
terms
Hi,
-
2
c)Cj)
^^'^ s u b s e q u e n t l y the e x p r e s s i o n s
f o r K,, ,\J^
( 7 . 8 ) , y i e l d s , a f t e r some a l g e b r a i c l a b o r , t h e
d i f f e r e n t i a l e q u a t i o n f o r Uj
-h
Ut
=
Fo
4-
essentially
,
,and
following
:
sin tf, 4-
F,-
cos
cp, +
where
Fc^ ((|,,,^...) =
- t •I;• ^
h • I w > ^ ^Cm-2) I ^ , ^
The
i n f i n i t e s e r i e s i n (7.25) c o n v e r g e s u n i f o r m l y
It
i s r e a d i l y s e e n t h a t t h e p a r t i c u l a r i n t e g r a l of itself, will
Equation
(Appendix 7E),. , and
hence
c o n t a i n terms t h a t a r e s e c u l a r w i t h r e s p e c t t o (^^
( 7 . 1 3 ) shows t h a t U,, I s bounded f o r a l l
7.8
.
Therefore,
the
condition
increasing
of uniform v a l i d i t y . The
e v e r , by r e q u i r i n g
(7.14) w i l l
be v i o l a t e d f o r
s e c u l a r t e r m s i n Uj. c a n be that
suppressed,how-
t h e "resonance g e n e r a t i n g p a r t "
f o r c i n g t e r m s i n (7.25) be e q u a l t o z e r o . T h i s r e s u l t s
In view of (7.26), t h i s requirement leads t o t h e system o f d i f f e r e n t i a l dependence o f
