i s the convergence of the spectrum in (4.2). O t h e r t e s t s c o n c e r n t h e a c c u r a c y o f t h e new t e c h n i q u e w i t h respect to the "exact" computation of ...
ORBIT INTEGRATION BASED UPON INTERPOLATED GRAVITATIONAL GRADIENTS E.J.O. S c h r a m a
Afde1ing der Geodesie D e l f t U n i v e r s i t y of Technology December, 1984
A f s t u d e e r s c r i p t i e b i j de A f d e l i n g d e r G e o d e s i e
Uw kenmerk
Uw brief van
Ons kenmerk
Datum 12-11-84
Delft, Thijsseweg 11 Doorklesnummer (015) 78
Onderwerp Af studeervoordracht E. Schrama.
A l l e personeelsleden, studenten en andere belangstellenden z i j n
uitgenodigd
voor mijn afstudeervoordracht g e t i t e l d : O r b i t i n t e g r a t i o n based upon interpolated gravitational gradients. De voordracht z a l plaatsvinden op Maandag 19 november a.s. 15.30 h r . i n c o l l e g e z a a l B.
E.J.O. Schrama.
10.000-iunl 1982
Algemeen telefoonnummer T.H. (015) 78 91 11 Correspondentieadres: Postbus 5030,2600 GA DELFT
822140
Overzicht s n e l l e baanintegratie technieken
E. Schrama.
D i t a f s t u d e e r v e r s l a g g e e f t een beschouwing over een s n e l l e baani n t e g r a t i e m e t h o d e voor s a t e l l i e t b a n e n op e r g l a g e hoogten. (200 km. o f m i n d e r ) . Het u i t e i n d e l i j k e d o e l van h e t onderzoek i s een methode t e v i n d e n d i e enige v e r l i c h t i n g g e e f t van de problemen d i e kunnen a l s men "Cowell's method" t o e p a s t . Het p o t e n t i a a l coëfficiënten model voor lage hoogte i s n . l . e r g ged e t a i l l e e r d , t/m graad en orde
180, (Rapp,198l). Voor e l k e i n t e g r a -
t i e s t a p dienen a f g e l e i d e n van de p o t e n t i a a l f u n c t i e ( i n bolcoördinaten) t e worden geëvalueerd. D i t komt er op neer d a t h e t ( p r a c t i s c h
gezien)
b i j n a o n m o g e l i j k i s banen op deze hoogten t e i n t e g r e r e n gebruikmakend van zwaartekrachtsmodellen
d i e c a . 32000 coëfficiënten b e v a t t e n , t e n -
z i j men g e b r u i k maakt van s p e c i a l e hardware zoals v e c t o r o f a r r a y p r o cessors, ( i n de c o n c l u s i e wordt er over h e t g e b r u i k van deze nieuw t e verwachten o n t w i k k e l i n g e n gesproken.) De i n t r o d u c t i e bespreekt h e t g e b r u i k van de nieuwe t e c h n i e k i n de Fysische
Geodesie i n r e l a t i e met s a t e l l i t e t o s a t e l l i t e t r a c k i n g en
g r a d i o m e t r i e . Verder z a l e r aandacht besteed worden aan h e t g e b r u i k van deze nieuwe t e c h n i e k t i j d e n s h e t verwerken van s a t e l l i e t w a a r n e m i n g e n . Het e e r s t e hoodstuk g e e f t een o v e r z i c h t van een a a n t a l b a s i s
eigen-
schappen van b o l f u n c t i e s (Legendre f u n c t i e s ) en numerieke problemen d i e kunnen
o n t s t a a n t i j d e n s de b e r e k e n i n g
ervan. Verder worden er een
a a n t a l aanverwante problemen besproken d i e b e t r e k k i n g hebben op de ber e k e n i n g van de a f g e l e i d e n v / d p o t e n t i a a l f u n c t i e z o a l s r e c u r s i e f o r mules voor een s n e l l e b e r e k e n i n g van g o n i o m e t r i s c h e f u n c t i e s . Het tweede h o o f d s t i i k gaat over de b e w e g i n g s v e r g e l i j k i n g e n . Het i s een systeem van tweede orde d i f f e r e n t i a a l v e r g e l i j k i n g e n w e l k , gegeven een s t a r t p o s i t i e en s n e l h e i d , g e i n t e g r e e r d kan worden t o t een s t a t e -
v e c t o r op e l k w i l l e k e u r i g moment. Het derde hoodstuk bespreekt het g e b r u i k van i n t e r p o l a t i e t e c h n i e k e n d i e g e b r u i k t kunnen worden om de b e w e g i n g s v e r g e l i j k i n g e n s n e l l e r op t e l o s s e n dan door de c o n v e n t i o n e l e p o t e n t i a a l f u n c t i e t e e v a l u e r e n . Het v i e r d e h o o f s t u k g e e f t een u i t e e n z e t t i n g over afwegingen d i e een b r u i k e r moet maken om de nieuwe t e c h n i e k t o e t e passen. Het
ge-
uiteindelijke
d o e l i s een programma op t e z e t t e n dat o p t i m a a l w e r k t . Het v i j f d e h o o f d s t u k g e e f t de r e s u l t a t e n van programma's d i e voor deze s t u d i e z i j n geschreven en g e b r u i k maken van v e r s c h i l l e n d e i n t e r p o l a t i e methoden. Een r e a l i s t i s c h e v e r g e l i j k i n g i s gemaakt met
de
conventionele
i n t e g r a t i e methode; h i e r u i t worden c o n c l u s i e s g e t r o k k e n voor de h a a l b a a r h e i d van deze t e c h n i e k . De u i t e i n d e l i j k e c o n c l u s i e s geven een afweging voor h e t g e b r u i k van
de
t e c h n i e k met t e verwachten hardware ( p a r a l l e l , v e c t o r p r o c e s s o r s ) . Een
appendix bevat de F o r t r a n programma's' en s u b r o u t i n e s . Een k o r t commen-
t a a r voor het g e b r u i k van e l k e s u b r o u t i n e word gegeven.
In het k o r t : Het u i t e i n d e l i j k e d o e l was, te
zoals a l eerder genoemd, een s n e l l e r e methode
onderzoeken voor b a a n v o o r s p e l l i n g s b e r e k e n i n g .
Een
( a a n t a l ) programma's
i s o n t w i k k e l d ( F o r t r a n 77 op een Amdahl V7B) welke s a t e l l i e t b a n e n van genoemde t y p e zo'n
l6 maal s n e l l e r i n t e g r e r e n dan de bestaande programma-
t u u r . Daarvoor moet e c h t e r een p r i j s worden b e t a a l d : er moet e e r s t twee dimensionaal
het
r o o s t e r worden aangemaakt met
a f g e l e i d e n van de
een poten-
t i a a l i n b o l f u n c t i e s . I n een gedane s i m u l a t i e was
de f i l e d i e h e t r o o s t e r
b e v a t t e 1500 t r a c k s g r o o t hetgeen overeenkomt met
30 Megabyte. De
ge-
b r u i k e r z a l een afweging moeten maken t u s s e n de v o o r d e l e n van deze t e c h n i e k (een a a n z i e n l i j k e v e r s n e l l i n g van de r e k e n t i j d ) en de nadelen ( h e t a l l o c e r e n van een b e h o o r l i j k g r o t e f i l e op d i s k . ) Opgemerkt d i e n t t e worden dat de i n de s i m u l a t i e g e b r u i k t e methode
zodanig
i s opgezet dat de g r o o t t e van de r o o s t e r f i l e m i n i m a a l
was,
I n een w e r k e l i j k e s i m u l a t i e s t u d i e o f i n een proces van s a t e l l i e t waarnemingsverwerking kunnen ( u i t e r a a r d ) andere i n t e r p o l a t i e t e c h n i e k e n r o o s t e r s g e b r u i k t worden.
en/of
18-10-84
Fast o r b i t
integration
E.
Schrasa
ABSTRACT
This report presents a fast numerical method for integrating s a t e l l i t e o r b i t s cf v e r y low a l t i t u d e s (about 200 Km. or less)« The p u r p o s e o f t h i s i n v e s t i g a t i o n i s to overcome the problems which arise if a low orbit is i n t e g r a t e d d i r e c t l y by C o w e l l ' s m e t h o d . T h i s method r e q u i r e s the evaluation of a total gravity model each time an integration step i s done. The p o t e n t i a l c o e f f i c i e n t model to use f o r low o r b i t s i s very d e t a i l e d , up t o d e g r e e and order 180, ( R a p p , 1 9 8 1 ) . For each integration step the derivatives of the potential function (in spherical c o o r d i n a t e s ) have t o be evaluated. This means t h a t i t is almost i m p o s s i b l e to i n t e g r a t e o r b i t s at t h e s e h e i g h t s using g r a v i t y models c o n t a i n i n g some 32000 c o e f f i c i e n t s , unless one uses special hardware such as array or vector p r o c e s s o r s , ( i n t h e c o n c l u s i o n more i s s a i d a b o u t t h e u s e o f them) . The i n t r o d u c t i o n d i s c u s s e s the use of the new t e c h n i q u e i n t h e f i e l d of p h y s i c a l g e o d e s y i n connection to s a t e l l i t e to s a t e l l i t e tracking and g r a d i o m e t e r e x p e r i m e n t s . Further on a t t e n t i o n w i l l be p a i d t o t h e u s e o f t h i s t e c h n i q u e i n o r b i t d e t e r m i n a t i o n and s a t e l l i t e observation processing followed by p a r a m e t e r e s t i m a t i o n . The first section introduces some basic properties of spherical harmonics and n u m e r i c a l problems in computing them. F u r t h e r on some additional problems are discussed r e l a t e d to computation of the d e r i v a t i v e s of the p o t e n t i a l f u n c t i o n such as r e c u r s i o n f o r m u l a s which compute q u i c k l y s i n e s and c o s i n e s i n one run. The s e c o n d s e c t i o n d e s c r i b e s the e q u a t i o n s of m o t i o n . They are e x p r e s s e d by a system of s e c o n d order d i f f e r e n t i a l equations which, given the epoch position and velocity c o m p o n e n t s , may be i n t e g r a t e d t c obtain the position and v e l o c i t y a t any o t h e r t i m e . T h i s d i r e c t i n t e g r a t i o n i s known a s C o w e l l ' s method and i t i s a t e c h n i q u e u s e d i n t h i s s t u d y . The t h i r d s e c t i o n d i s c u s s e s t h e u s e o f i n t e r p o l a t i o n m e t h o d s t o be u s e d t o s o l v e t h e e q u a t i o n s o f m o t i o n i n a f a s t e r way than the d i r e c t e v a l u a t i o n of the c o n v e n t i o n a l potential function formulas. The f o u r t h section discusses program c o n s i d e r a t i o n s t o be made by t h e user i n a p p l i c a t i o n s of the new t e c h n i q u e . The goal i s to get a program which runs optimally. The f i f t h s e c t i o n shows the r e s u l t of a program s p e c i a l l y written for t h i s study u s i n g v a r i o u s i n t e r p o l a t i o n methods. A realistic conparison is done with the conventional i n t e g r a t i o n t e c h n i q u e . F u r t h e r on conclusions a r e drawn f o r -
1
-
Fast
18-10-84 this
new
orbit integration
E.
Schrama
technique.
A section with of the new processors).
conclusions i s introduced, d i s c u s s i n g t h e use technique i n future hardware. (parallel
The a p p e n d i x contains t h e l i s t i n g s of subroutines p o s s i b l y useful i n other programs. A short explanation w i l l given for each r o u t i n e l i s t e d . In
short:
The f i n a l goal of t h i s study was, a s mentioned before, to f i n d a f a s t e r technique f o r o r b i t p r e d i c t i o n . A program has been developed (Fortran-77 on an Amdahl 4 7 0 V7b) which integrates satellite o r b i t s o f the mentioned type some 16 times faster than the conventional orbit prediction p r o g r a m s . T h e p r i c e t o be p a i d i s a pre m i s s i o n computation which initializes a two dimensional grid containing functions i n s p h e r i c a l harmonics. I n an actual simulation the used grid allocated 1500 t r a c K s on d i s K which e q u a l s some 30 m e g a b y t e s . T h e u s e r s h o u l d c o n s i d e r t o a s s i g n s u c h a l a r g e f i l e which s p e e d s up t h e p r o g r a m by a f a c t o r o f some 16, or t o u s e t h e c o n v e n t i o n a l r o u t i n e s . I t should be reroarKed that t h e technique used during the simulation minimizes the disk storage. In a real world s a t e l l i t e o b s e r v a t i o n processing or s i m u l a t i o n s t u d i e s other i n t e r p o l a t i o n t e c h n i q u e s o r g r i d s c a n a l s o be u s e d .
- 2 -
18-10-84
Fast
orbit integration
E.
Schrama
Introduction
The use of fast integration techniques is of great importance for any method that uses dynamic models p r e d i c t i n g movements o f s p a c e c r a f t s . N o r m a l l y one uses the Cowell method f o r i n t e g r a t i o n of satellite orbits; other (also a n a l y t i c a l ) methods have been developed which are, howeverp l e s s r e l e v a n t f o r the kind of o b s e r v a t i o n methods we are talking about. A typical feature of analytical t h e o r i e s i s t h a t t h e f o r m u l a s one g e t s a r e u s u a l l y d e v e l o p e d for z o n a l c o e f f i c i e n t s o n l y , and t h e y a r e o n l y c o m p l e t e up to low degree and o r d e r potential coefficients. Other i n t e g r a t i o n t e c h n i q u e s s u c h a s t h e Enke method a r e i n f a c t a combination o f an a n a l y t i c a l t h e o r y and a n u m e r i c a l m e t h o d . C o w e l l however i s c o m p l e t e l y n u m e r i c a l l y o r i e n t e d . A d i s a d v a n t a g e of the C o w e l l method i s t h a t c o n s i d e r a b l e computing t i m e i s needed to compute a s p a c e c r a f t t r a j e c t o r y , e s p e c i a l l y when h i g h d e g r e e and o r d e r p o t e n t i a l c o e f f i c i e n t models a r e used. On p r o c e s s i n g s a t e l l i t e observations i t i s n e c e s s a r y to r e c o n s t r u c t the p o s i t i o n s of the s a t e l l i t e o b s e r v e d . Often a number of i t e r a t i o n s are required in order to d e r i v e an estimate of the i n i t i a l position and v e l o c i t y components that matches the t r a j e c t o r y computed w i t h the observations done t o t h e s a t e l l i t e i n q u e s t i o n . In
our
case
the e n v i s a g e d
- satellite - satellite
observation
techniques
are:
to s a t e l l i t e t r a c k i n g . gradiometry.
In s a t e l l i t e to s a t e l l i t e t r a c k i n g the o b s e r v a t i o n type i s a v e l o c i t y d i f f e r e n c e o b s e r v a t i o n b e t w e e n two satellites. In p r i n c i p l e two o b s e r v a t i o n m e t h o d s are f e a s i b l e : I n the f i r s t s e t u p two s a t e l l i t e ? a r e i n t h e same, l o w , o r b i t behind each o t h e r (low-low). In t h e s e c o n d one t h e r e i s a high s a t e l l i t e which orbit i s very w e l l known o b s e r v i n g a low satellite which "feels" the detailed s t r u c t u r e s of the gravity field (high-low) . In gradiometry a new k i n d of o b s e r v a t i o n i s introduced: the second d e r i v a t i v e s of the p o t e n t i a l f u n c t i o n . How this t y p e of m e a s u r e n t w i l l be d o n e i s s t i l l a q u e s t i o n because of the instrumental problems arising in building a g r a d i o m e t e r . The o r b i t s however will also be v e r y low s o t h a t t h e s o f t w a r e must have t h e c a p a b i l i t y of r e c o n s t r u c t i n g the movement of the s p a c e c r a f t making this kind of observations. Another
use
is
that
in simulation -
3
-
studies for
gravity
18-10-8 4
Fast o r b i t
integration
E.
Schrama
research m i s s i o n s . These s i m u l a t i o n s , for the reconstruction of t h e high frequency part i n the g r a v i t y f i e l d s , are only f e a s i b l e i f o r b i t i n t e g r a t i o n c a n be conveniently.
- ^ -
18-10-84
Fast
orbit
integration
CHAPTER Bssic
E.
Schrama
1
p r o p e r t i e s of S p h e r i c a l Harmonics.
A p o t e n t i a l f u n c t i o n f o r t h e a t t r a c t i n g mass o f t h e e a r t h i s d e f i n e d i n s u c h a way t h a t i t r e p r e s e n t s a c o n v e n t i o n a l fielde In a conventional force field two m a i n integral equations hold: t h e f i r s t one i s t h e S t o k e s e q u a t i o n , t h e second one i s t h e Gauss e q u a t i o n * I f they a r e combined under the condition that one describes the f i e l d outside the greatest sphere surrounding the a t t r a c t i n g body, the resulting condition i s :
(l.l)
d i v ( g r a d ( V ) ) =0
(Laplace condition)
The s o l u t i o n i n s p h e r i c a l given by:
^ (1.2)
Vp
coordinates
^
f o r the outer
space i s -
rH-l
=
where: Vp
= potential
a t point
P
^^pj'^pj'^ = s p h e r i c a l coordinates of point P = mean e q u a t o r i a l r a d i u s Q"M = g r a v i t a t i o n a l
constant
^m^^nm" p o t e n t i a l c o e f f i c i e n t s (normalized) p 0 - normalized Legendre f u n c t i o n s of degree n hrr) a n d o r d e r m.
While i n t e g r a t i n g using C o w e l l ' s method i t i s n e c e s s a r y t o evaluate d e r i v a t i v e s of the p o t e n t i a l function with r e s p e c t to r a d i u s , c o - l a t i t u d e and l o n g i t u d e ( 1 . 3 ) , ( 1 , 4 ) , ( 1 , 5 )
- 5 -
18-10-8ti
Fast
orbit
integration
E.
Ihe computation of these fornulas requires algorithms f o r the v a r i o u s p a r t s from which they up« -Computation of
Legendre
Schrama
special are build
Polynomials.
Legendre polynomials a r e defined o r t h o g o n a l , which means:
i n such
a way
that
they
are
• \a (1.6)
co5mAp Cos Up Sc'ri
(TiApStA Up J
P
Ll•
o
' Lk L l .
where
The b a r a b o v e t h e Pnm f u n c t i o n i n d i c a t e s t h a t the Legendre polynomials are normalized. The computation of normalized functions appears to have numerical advantages above the denormalized functions, especially for high d e g r e e and order. Recursion formulas tend to be more s t a b l e in the normalized situation. The r e c u r s i o n formulas i n the non-normalized form a r e :
(1.7)
(1.8)
(1.9)
(1 .10)
Ihe
P ^ ( t )
-.^^
normalization
may
é
be
P C O
-
n-hm-l
done u s i n g t h e
- 6
-
p
following
formula:
18-10-84
Fast
orbit
E.
integration
Schrama
(1.12) ( 0 + m) j J
numerical caus e s evaluatie n of t h i s formula The direct This is for when corapu t i n g the fa c t o r i a l s . problems, of GEODÏN (a version the c u r r e n t instance a reason wh y Uni v e r s i f y of T e c h n o l o g y for p r o g r a m a v a l i a b l e a t t he D e l f t instrument tracking calibration, orbit det e r m i n a t i o n , p r e d i c t i o n s and satellite operational geodet i c parameter estimation) cannot above degree co mpute L e g e n d r e f u n c t i o n s and o r d e r 4 7, F o r o u r s t u d y t h e r e i s a g r e a t i n t e r e s t i n t h e h i g h o r d e r e f f e c t s s o t h a t s p e c i a l c a r e was t a k e n t o d e v e l o p algorithm s capable numericalll y " f r i e n d ly" of evaluating Legendre f u n c t i o n s up order 180. T h i s is to d e g r e e and a c h i e v e d by s u b s t i t u t ing the norm a l i z a t i o n f o r m u l a (1.12) i n s i d e the t o 1.11) . D o i n g s o one r e c u r s i o n f o r m u l a s (1.7 gets:
(1.13)
fcjo^t)
(1.14)
p
(1.15)
P Ct) on
(1.16)
=
'
(o-ssume
f=co54>)
(fe)
P
L
(6) =
2n
( ^ n 4 j ) ^
J
(= P
(
(1.17)
(n-m) See
f o r example Computing
fz/)~2>) (n-f-m)
J
"
(Hobson,1965) .
the d e r i v a t i v e s
of
the Legendre
The derivatives are obtained normalized r e c u r s i o n formulas:
- 7
by
-
functions.
differentation
of
the
Fast
16-10-84
(1.19)
1 ^
orbit
E.
integration
Schrama
.„36/firn
(1.20)
(1.21)
!Ani!^^ 9 b
(jr)-i)(zn+i) 1
(1.22)
/•-/.An Chs-m-Q
For LEGFDN
(2.n-é-i) (n-m-i)
"j'/^
u s e i n t h e program written for this was u s e d , s e e ( C o l o m b o , 1 9 8 1 ) .
Recursion
formulas for
9è
study
c o m p u t i n s s i n e s and
subroutine
cosines
A l o t o f computer installations do n o t h a v e arithmetic vector processors f o r evaluating p o l y n o m i a l s f o r non l i n e a r functions such as s i n e s , cosines and o t h e r f u n c t i o n s . U s i n g the D e l f t U n i v e r s i t y ' s c o m p u t e r an Amdahl 4 7 0 V 7 b . i t was n e c e s s a r y t o m i n i m i z e t h e n u m b e r of e v a l u a t i o n s o f s i n e s a n d cosines. Fortunately sines and c o s i n e s c a n be computed considerably f a s t e r using the following i d e n t i t i e s :
(1.23)
sin(a+b)
= s i n ( a ) . c o s (b)
+ c c s (a) . s i n ( b )
(1.24)
c o s ( a + b)
= c o s ( a ) . c o s (b)
- s i n (a) . s i n ( b )
Let
a=x and b=m.x
(1.25)
then:
s i n (m+l)x
= s i n (x) . c o s (m.x) + c o s (x) . s i n (ro-x)
(1.26) c o s (m+l)x
= c o s (X) . c o s (m.x) - s i n (x) . s i n (m.x)
This
i s d o n e by r o u t i n e
RECGON l i s t e d
- 8 -
i n the Appendix.
Fast
18-10-84
orbit
Chapter The
The
d y n a m i c model
equa t i o n s
E.
integration
for
Schrama
2 orbit
integratipha
of motion.
A fundamental part o f t h e program i s t h e c o m p u t a t i o n o f p o s i t i o n s and velocities of a s a t e l l i t e g i v e n an initial p o s i t i o n and v e l o c i t y . The d y n a m i c s of t h i s situation are e x p r e s s e d by the so c a l l e d equations of motion. I n GEODYN there i s an additional requirement for variational partitials, which are the p a r t i t i a l derivatives of t h e instantanious orbital elements with respect to the parameters a t epoch. T h i s i s used f o r parameter estimation and tracking instrument calibration. In this study a s i m p l i f i e d m o d e l was u s e d : (2.1)
F = g r a d (V)
w h e r e F i s t h e a c c e l e r a t i o n v e c t o r and V t h e g r a v i t a t i o n a l potential expressed by f o r m u l a (1,2). F o r more r e a l i s t i c o r b i t p r e d i c t i o n programs t h e r e a r e a number of p e r t u r b i n g f o r c e s which a r e l e s s r e l e v a n t f o r t h i s study. Summarized the main p e r t u r b a t i o n s a r e :
- Atmospheric drag c o r r e c t i o n (Fd) > ( T h i s r e q u i r e s t h e u s e of an a t m o s p h e r i c such a s J a c c h i a 1965 o r any other) - Radiation maybe t h r e e )
pressure (F*). corrections: Solar
this
density
consists
of
model
two (or
-a)
Direct
pressure.
-b) the
I n d i r e c t pressure of t h e sun l i g h t r e f l e c t e d by e a r t h . T h i s i s known a s A l b e d o r a d i a t i o n .
-c) A hypothetical i n d i r e c t pressure r e f l e c t e d by t h e moon.
of sun
light
-Gravity disturbances of masses other than t h e e a r t h ' s mass, such as. Lunar, S o l a r , Venus, Martian, Jupiter and S a t u r n g r a v i t a t i o n s . ( F m ) ( T h i s r e q u i r e s t h e u s e of ephemeris t a b l e s and/or dynamic models for disturbing masses) •rEarth
tides
(Ft) .
-Other
forces,
such
a s : (R)
-earth's infrared radiation -atmospheric p a r t i c l e drag -
9 -
pressure,
18-10-84
Fast
orbit
integration
E.
Schrama
-Poynting Robertson e f f e c t -YarKovsky and Schach e f f e c t s -Electromagnetic forces - D r a g from i n t e r p l a n e t a r y d u s t - F o r c e s due t o m e t e o r o i d i n p a c t The
complete dynamical
(2.2)
F = g r a d (V)
This last considered
+ Fd
model
becomes:
+ F« + Fro + F t + R
part (Fd + F* + in this context.
Evaluation
Fro
+ Ft
•
R)
will
not
be
o f F = g r a d (V)
The p o t e n t i a l f u n c t i o n V g i v e n by (1«2) i s w r i t t e n i n the spherical coordinates radius, longitude, and co-latitude (zero at north pole and j f a t the south pole). Ihe acceleration vector F i s given i n rectangular coordinates meaning t h a t a c o n v e r s i o n h a s t o be d o n e from s p h e r i c a l to rectangular coordinates:
(2.3.a)
Stln (jjp c o s A p
(2.3.b)
Sen
(2.3.C)
and
Zp^TpCOScj^p
the i n v e r s e
(2.4.a)
ScnAp
Tp =
transform
by:
fT^^T^^TT^ ccUr. (Sp/X^)
(2.U.b)
.
(2.4.c)
cf>^ = ^Ur,
r\/x/^Jf/'/2p)
Using (2.4.a) , (2.a.b), (2.4.C) by d i f f e r e n t i a t i n g respect to x,y,and z i t can e a s i l y be s h o w n e q u a t i o n s o f m o t i o n become:
- 10 -
them that
with the
Fast
18-10-84 (2.5«a)
1^
X =
(2.5.b)
(2.5.C)
2
" ^ •^r
orbit
E.
integration
s c n ^ CQSA + ^
cos ^.cosA
S e n cj) SCO A
Cqs4> 3 c n A
+ 1^
c.oS
'^^
r
3(|)
Scn(|? r
_
_
Schrama
3V 9V; 3A
scn\ co^A rs4>)^
In the program t h r e e a d d i t i o n a l equations are a d d e d . They a r e needed to i n t e g r a t e the v e l o c i t i e s to yield positions. I n the f i r s t s e t u p of t h e program t h e s e l a s t e q u a t i o n s were written in spherical coordinates. Unfortunately t h i s caused p r o b l e m s when integrating polar orbits because of a quick v a r i a t i o n o f the l o n g i t u d e near the p o l e s . B e c a u s e of t h i s it i s advisable to integrate in rectangular coordinates i n s t e a d of s p h e r i c a l c o o r d i n a t e s .
The t r a n s f o r m a t i o n ~ ~ t o an
of t h e i n e r t i a l c o o r d i n a t e e a r t h f i x e d one
Irame
A problem with the s i x f i r s t order d i f f e r e n t i a l equations i s t h a t t h e y a r e o n l y v a l i d i n an i n e r t i a l c o o r d i n a t e f r a m e . The p o t e n t i a l f u n c t i o n , and therefore also i t s derivatives, are expressed in earth fixed c o o r d i n a t e s . So a conversion h a s t o be p e r f o r m e d e a c h t i m e t h e p o t e n t i a l f u n c t i o n i s u s e d from t h e i n e r t i a l c o o r d i n a t e system. The c o n v e r s i o n c o n s i s t s o f t h e : - Earth's rotation - P r e c e s s i o n and nutation - P o l a r motion
-The E a r t h r o t a t i o n c o n s i s t s o f s e v e r a l p a r t s . F i r s t l y t h e r e i s an o f f s e t t e r m i n d i c a t i n g the i n i t i a l d i f f e r e n c e between an e a r t h f i x e d and an i n e r t i a l coordinate frame (STD50), s e c o n d l y t h e r e i s a v e l o c i t y term modelling the v a r i a t i o n in r a d i a n s per s e c o n d b e t w e e n an e a r t h f i x e d and an inertial frame. (OMT50). Finally there is a correction term e x p r e s s i n g t h e r a t e i n which rotation velocity varies. (The earth's rotation rate i s decreasing steadily) , (OMQ50) . P u t t i n g a l l p a r t s t o g e t h e r one gets: (2.6.1)
T
= 8480
(2.6.2)
TT
(2.6.3)
Elong
=
+
TIME
(STD50+{OMT50+OMQ50-T)-T)*PI/180 = Ilong
+
TT
E l o n g and I l o n g . a r e r e s p . t h e l o n g i t u d e and t h e i n e r t i a l c o o r d i n a t e f r a m e . - 11
-
in
the
earth
fixed
18-10-84
Fast
orbit
F.
integration
IIMF equals t o t h e number o f d a y s added c o u n t e d f r o m 21 m a r c h 1 9 7 3 OOh 00m OOs. The c o n s t a n t s h a v e t h e f o l l o w i n g v a l u e s :
by
Schrama
day f r a c t i o n s
STD50 = 100 . 0 7 5 5 4 2 OMT5 0 = 360 . 9 8 5 6 4 7 3 3 5 OMQ50 = . 2 9 E - 1 2 - P r e c e s s i o n and N u t a t i o n . T h e s e e f f e c t s a r e due t h e o b l a t e n e s s o f t h e r o t a t i n g e a r t h while i t i s a t t r a c t e d by t h e s u n a n d moon. I tcauses a change i n t h eo r i e n t a t i o n of t h e a x i s of t h e e a r t h with r e s p e c t t o an i n e r t i a l c o o r d i n a t e frame and c o n s i s t s of a long p e r i o d i c term ( p r e c e s s i o n ) and s e v e r a l short periodic t e r m s ( n u t a t i o n ) . I n GEODÏN t h e H o o l a r d ' s s e r i e s a r e u s e d t o d e s c r i b e p r e c e s s i o n and n u t a t i o n e f f e c t s o f t h e e a r t h a x i s , see (Woolard,1953) . These disturbances a r every s m a l l and therefore less relevant for this study. They c a n be eliminated using matrix r o t a t i o n formulas, s e e (HeisKanen and M o r i t z , 1 9 6 7 ) . - Polar motion. T h i s i s t h e movement o f t h e e a r t h c r u s t r e l a t i v e t o t h e a x i s of r o t a t i o n of theearth. This effect consists of several periodic terms: -1 P e r i o d i c terms ( v a r i a t i o n s i n the order of days) w h i c h a r e t h o u g h t t o be d u e t o m a s s m o v e m e n t s i n s i d e o r on t h e e a r t h , -2 Long p e r i o d i c terms years) c a u s e d by o t h e r Known a s P o l a r w a n d e r .
(variations i n theorder physical phenomena. T h i s
In t h e program written for t h i s study only r o t a t i o n was a p p l i e d f o r t h e f o l l o w i n g reasons: -The r o t a t i o n o f t h e e a r t h and can usually n o t be programs.
i s t h e most eliminated
the
of i s
earth
significant variation i n orbit prediction
-Precession, nutation and P o l a r motion a r e s m a l l e f f e c t s . T h e y c a n be s w i t c h e d o f f i n roost o r b i t p r e d i c t i o n p r o g r a m s . T h e r e f o r e i t i s p o s s i b l e t o compare t h e i r r e s u l t s with our technique. Ihe routine earth.fixed
c a r r y i n g o u t t h e t r a n s f o r m a t i o n from i n t h e program i s c a l l e d I N T F I X .
- 12 -
inertial to
18-10-64 The
integrator
Fast
orbit
used
to
E.
integration
solve
the
equations
of
Schrama
motion
Various integrators are available i n any mathematical program l i b r a r y such as I M S L o r NAG. (the l a t t e r is less well known t h a n the f i r s t o n e , NAG stands for Numerical A l g o r i t h m Group) For the p roblem at hand i t is necessary to have an i n t e g r a t o r with the following c h a r a c t e r i s t i c s : -1 High required
order integration accuracy.
-2 Capable differential
method
because
of integrating sets of equations simultaneously.
first
of
the
order
-3 A minimum of f u n c t i o n evaluations for each step ( b e c a u s e of the complexity o f the potential function which d e r i v a t i v e s are evaluated at each integration step) . -4 Stable stepsize.
solution
for
an
accurate
enough
chosen
The i n t e g r a t o r u s e d f o r t h i s s t u d y was an Adams B a s h f o r t h / Adams M o u l t o n p r e d i c t i o n c o r r e c t i o n method o f o r d e r 8. T h i s Bultistep method needs a Runge Kutta Fehlberg order 8 i n t e g r a t o r to s t a r t g e n e r a t i n g the f i r s t 8 p o i n t s . For t h i s i n i t i a l integration 13 f u n c t i o n c a l l s a r e done each step. For every following step two function evaluations are r e q u i r e d . The r e a s o n to use t h i s i n t e g r a t o r i s i t s use in NISOP, another o r b i t p r e d i c t i o n program a v a i l a b l e at the D e l f t U n i v e r s i t y of T e c h n o l o g y . (NISOP s t a n d s f o r N u m e r i c a l Integration Satellite Orbit Predictor). A lot of research has a l r e a d y been spent on t h e problem of what k i n d of i n t e g r a t o r and s t e p s i z e w i l l be n e e d e d to compute o r b i t s . These studies are however beyond the scope of this investigation.
- 13
-
18-10-84
Fast
orbit
E. Schrama
integration
Chapter
3
Interpolation
methods.
Introduction Many i n t e r p o l a t i o n m e t h o d s a r e known, a l l with s p e c i f i c properties,, see e.g. ( M o r i t z and Sunkel,1978) F o r our problem we s h a l l u s e a t h r e e d i m e n s i o n a l g r i d with nodes c o n t a i n i n g t h e d e r i v a t i v e s of t h e p o t e n t i a l f u n c t i o n V with respect to radius, co-latitude and longitude. The p o t e n t i a l f u n c t i o n V i s computed up t o , f o r i n s t a n c e d e g r e e and o r d e r 1 8 0 , l i k e ( 1 . 2 ) , s e e (Rapp,1981) . The i d e a i s t o i n t e r p o l a t e the d e r i v a t i v e s of the p o t e n t i a l f u n c t i o n V in any p o i n t w h e r e t h e s a t e l l i t e m i g h t be s o t h a t t h e e x p e n s i v e e v a l u a t i o n of (1.3) , (1.4) and (1.5) i s a v o i d e d and replaced by a much c h e a p e r i n t e r p o l a t i o n . In the first instance this chapter is ment tö be mathematically oriented describing interpolation methods w h i c h w i l l be used i n the folowing p a r t . The n e x t c h a p t e r s w i l l d e s c r i b e program c o n s i d e r a t i o n s and e x p l a i n w h a t g r i d , type of i n t e r p o l a t i o n a . s . o . i s u s e d . The
interpolation
methods used
in
this
The i n t e r p o l a t i o n method(s) used have the f o l l o w i n g characteristics: -
study. for
this
program
E v a l u a t i o n o f a minimum o f a d j a c e n t g r i d High a c c u r a c y F a s t c o n s t r u c t i o n of t h e interpolant.
must
points.
Therefore the f o l l o w i n g method i s used. It is b a s e d upon central difference formulas in combination with Taylor polynomials. G i v e n an i = - 2 ...
equally
spaced
line
with
function
0
0
0
0
f(-l)
f (0)
f(l)
f(2)
0 f(-2)
( 3 . 1 ) I t i s f ( i ) =f ( x + i * h ) . I f one o n l y t a k e s i n t o a c c o u n t t h e raid r i g h t a n d l e f t p o i n t t h e n one finds:
(3.2)
f,
values
f (i) :
2
=
^
hli
„ V
i
l
- 14
^
-
point
J l L Ü L
and
+
.
the
first
18-10--8 a
Fast o r b i t
integration
E.
Schrama
(3.3)
after addition
and s u b s t r a c t i o n
one
gets;
(3.4)
(3.5)
f. -
(3.6)
(f-,
f
,
(3.7) Substituting
(3.8)
( 3 . 6 ) and
f (x+Ax)
=
frx)
(3.7) i n t o
the Taylor
+
i.
polynomial
d H
•f
...
o gives
with
( 3 . 6 ) and ( 3 . 7 )
h where u s e :
with:
(This
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