LONG TERM PREDICTION OF
HIGH ALTITUDE ORBITS
by
Sean Kevin Collins
B.S.A.E.,
S.M.,
University of Virginia
(1974)
Massachusetts Institute of Technology
(1977)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENT FOR
DOCTOR OF PHILOSOPHY
at the
Massachusetts Institute of Technology
1981
Signature of
by
MARCH 1981
Massachusetts Institute of Technology
Author
Department of Aeronautics
& Astronautics
March 1981
Approved by
Walter M.
Chairman of 'Thesis Committee
Richard H.
Approved by
Hollister
Battin
Thesis Advisor
John P. Vinti
Approved by
Thesis Adv 'sor/
Approved by
Paul J. Cefola
Thesis Advisor/
Manuel Martinez-Sanchez
Accepted by
Cliairman,
Departmental Doctoral Committee
Accepted by
Harold Y.
Wachman, Chairman, Departmental
Graduate Committee
ARCHIVES
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAY 5
1981
LIBIRM S
0
LONG TERM PREDICTION OF
HIGH ALTITUDE ORBITS
by
Sean Kevin Collins
Submitted to the Department of
Aeronautics and Astronautics on
30 March 1981 in partial fulfillment
of the requirements for.the Degree
of Doctor of Philosophy
ABSTRACT
This
thesis
develops
a
first
order
semi-analytical
theory, based on the Generalized Method of Averaging and
making extensive use of recursive algorithms, for the rapid
and accurate calculation of the secular and long period
changes
in
the elements
of a high altitude
caused by the action of the sun and moon.
designed
to assist
the
mission
analyst
testing
the
long
term
stability of
region
above
synchronous
perturbations are a major
and. stability.
satellite
orbit
The theory is
concerned
with
selected orbits
in
the
altitude
where
"third
body"
determinant of orbital lifetime
A
representation
of
the
third
body
disturbing
potential in satellite orbital elements is essential to the
development.
Non-singular equinoctial orbital elements are
used as part of a unified
artificial
singularities
approach to the elimination of
in
the
satellite
dynamical
equations for near-circular and near-equatorial orbits.
potential is
the satellite
The
derived with respect to the reference frame of
to. minimize the analytical complexity of the
first order theory.
Special functions are employed wherever
possible to modularize the analytical structure with a view
The
computation in a numerical program.
towards efficient
potential retains the parallax factor to an arbitrary power
and no assumptions are made on the geometry of the third
body orbit.
The potential is expanded into the mean
longitudes of the satellite
and the disturbing body so that
resonance can be
studied.
2
01
An averaging theory, based on the Generalized Method
of Averaging, is developed for removing frequencies from the
satellite dynamical equations depending on rapidly varying
linear combinations of the satellite and third body mean
longitudes.
The method for obtaining the analytical forms
of the averaged equations of motion and the short periodic
recovery functions is detailed.
The third body averaging theory has been numerically
implemented and compared against Cowell integration for five
The principal conclusion of this thesis is
test orbits.
that the first
order semi-analytical third body theory can.
be used to accurately predict the long term motion of a high
makes
it
that
efficiency
with
an
satellite
altitude
analysis
superior
to
conventional
mission
decisively
techniques.
Thesis
Title:
Supervisor:
Walter M. Hollister
Associate Professor of Aeronautics and Astronautics
John P. Vinti
Thesis Supervisor:
Title:
Lecturer of Aeronautics and Astronautics
Thesis Supevisor:
Richard H. Battin
Title:
Adjunct Professor of Aeronautics and Astronautics
Associate Department Head, C. S. Draper Laboratory
Thesis Supervisor:
Paul J. Cefola
Title:
Lecturer of Aeronautics and Astronautics
Section Manager, C. S. Draper Laboratory
3
6
ACKNOWLEDGEMENTS
I wish to express my profound appreciation to the
members of my doctoral thesis committee, whose support and
This includes Procounsel have made this work possible.
fessor Walter M. Hollister, who served as chairman, Dr.
who introduced me to astrodynamics,
H.
Battin,
Richard
Celestial
of
whose vast knowledge
Vinti,
Dr. John P.
Mechanics inspired me, and Dr. Paul J. Cefola, who acted as
I would like to emphasize my gratitude
principal advisor.
guidance
and technical
for his patience
Cefola
to Dr.
I am also indebted to Wayne
throughout my doctoral program.
during
D. McClain (CSDL) for his sage advice and criticisms
Thanks are extended to Dr.
all phases of this thesis.
Mr.
Slutsky (CSDL),
Dr. Mark S.
Proulx (CSDL),
Ronald J.
Taylor
and Captain Stephen P.
Bobick (MIT/CSDL)
Aaron F.
suggestions and
for numerous helpful discussions,
(USA)
Leo W. Early
Mr.
to
extended
also
is
Appreciation
support.
exploited
constantly
was
GTDS
of
knowledge
whose
(CSDL)
during the software development.
I would like to thank Captain (Dr.) Andrew J. Green
The software
achiever.
a friend and prolific
(USA/ARMOR),
implementation of the third body theory developed in this
thesis was greatly aided by work that he performed at CSDL
Special thanks go to Captain
as an MIT doctoral student.
friendship
for his reliable
(USAF)
Shaver
S.
(Dr.) Jeffrey
and encouragement.
I wish to express appreciation to Mr. Bruce Baxter
(The Aerospace Corporation) for a useful exchange of ideas
regarding the concepts developed in this thesis and for the
suggestion of research directions.
Karen M.
reserved for Ms.
Thunderous applause is
to ensure the comSmith (CSDL) , who forsook a normal life
I am very grateful to her for an
pletion of this document.
effort that matches the highest professional standards.
My wife, Susie, deserves the lion's
Her unwavering understanding and
accolades.
the reasons for my success.
share of the
compassion are
I would like to thank the Charles Stark Draper LaboMassachusetts for providing financial
ratory of Cambridge,
of my docfor the entirety
support and research facilities
program at MIT.
toral
4
0
TABLE OF CONTENTS
1.
2.
INTRODUCTION............................
........16
1.1
Previous Work..................
........20
1.2
Overview.......................
.........23
THE THIRD BODY DISTURBING POTENTIAL.............. 26
2.1
Derivation of the Third Body
Disturbing Potential in
Inertial Coordinates.................... 27
2.2
Transformation of the Third Body
Potential from Inertial Coordinates
to Sate.llite Coordinates...............
.35
2.2.1
Reference
Frame Choice and
Implications............................ 36
2.2.2
Transformation of the Third Body
Potential to Satellite Orbital
Coordinates............................. 39
2.2.3
Induced Dependence of Third Body
Orbital Elements on Satellite
Orbital Elements....................... 81
3.
2.2.3.1
The Meaning of h' and k'...............81
2.2.3.2
The Meaning of
X'......................86
ISOLATING LONG TERM MOTION IN THE SATELLITE
DYNAMICAL EQUATIONS..............................
3.1
The Generalized
Method of
Averaging...............................
3.1.1
94
An Averaging Theory for Satellites
Moving Under the
Influence
of a
Disturbing Body........................
3.1.2
91
96
A First Order Averaging Theory......... 110
5
TABLE OF CONTENTS
(cont.)
Page
Chapter
4.
MATHEMATICAL STRUCTURE OF A DYNAMICAL
THIRD BODY MODEL FOR THE LONG TERM
PREDICTION OF SATELLITE ORBITS USING
NUMERICAL METHODS............................. .. 118
4.1
0
First Order Averaged Equations
of Motion for Third Body
Perturbation......................... .. 126
4.1.1
Criteria for Retaining Terms in
the Averaged Equations of Motion..... .. 134
4.2
Mathematical
0
Form of the
Periodic Recovery Functions.......... .. 137
4.3
Formulation of the Third Body
Theory for Numerical Computation..... .. 151
4.4
Calculation of Special Functions....
4.4.1
Calculation of Zm
4.4.2
Calculation of Jacobi Polynomials
n,r
..................
.. 176
.. 176
0
and Their Partial Derivatives
by Recurrence........................ .. 180
4.4.3
Calculation of the Coefficients
Cr, Dr and Their Partial
m
m
Derivatives by Recurrence............ .. 183
4.4.4
Calculation of the Coefficients
Ar,m, Br,m and Their Partial
s,t
s,t
Derivatives.......................... .. 193
4.4.5
Recurrence Relations
for the
Third Body Hansen Coefficient
Kernel K-n-l,r and
S
4.5
its Derivative....
..
201
Restriction of Indices in the
Third Body Theory.................... .. 211
6
0
TABLE OF CONTENTS
(cont.)
Page
Chapter
5.
NUMERICAL VERIFICATION OF THE FIRST ORDER
THIRD BODY THEORY............................... 219
5.1
Initialization of the Averaged
Equations of Motion............ ...-...221
The Computation of Third Body
5.2
Ephemerides....................
....
Analysis of the Numerical Resul ts.
5.3
228
231
...
5.3
.1
233
IUE Test Case..................
....
6.
5.3
.2
ISEE Test Case.................
5.3
.3
VELA Test Case.................
273
5.3
.4
STRATSAT Test Case.............
285
5.3 .5
Lunar Resonance Test Case......
296
CONCLU S IONS AND
FUTURE WORK........
6.1
Conclusions...............
6.2
Future Work...............
Appendix A.
...
262
. ... ... . 306
. ... ... . 310
. ... ... . 312
Computation and Storage of the
Newcomb Operators in the Third Body
Theory.................................... 316
Appendix B.
Software Implementation of the
Third Body Theory......................... 322
List of References..................................... 337
7
LIST OF FIGURES
Page
Figure
2-1
Geometry of the Third Body Problem............... 28
2-2
Orientation of the Satellite Reference
Frame with Respect to the Inertial
Reference Frame................................. 40
2-3
0
Orientation of the Third Body Position
Vector with Respect to the Satellite Frame...... 42
2-4
Geometry for the Rotation of Surface
Spherical Harmonics............................. 49
4-1
Admissible Values of the Index m vs.
4-2
Admissible Values of the Index r
5-1
Osculating
t ..........
214
vs. s..........217
Semi-Major Axis Comparison
within the 60 Day PCE Fit Span for the IUE
Orbit/Semi-Analytical versus Cowell.............239
5-2
Osculating
Eccentricity Comparison within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell...................240
5-3
Osculating Inclination Comparison within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell...................241
5-4
Osculating
Mean Longitude Comparison within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell....................242
5-5
Osculating Semi-Major Axis Differences
within the 60 Day PCE Fit Span for the IUE
Orbit/Semi-Analytical minus Cowell..............243
5-6
Osculating Eccentricity Differences within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical minus Cowell....................244
8
LIST OF FIGURES
(cont.)
Page
Figure
5-7
Osculating Inclination Differences within
the 60 Day PCE
Fit Span for the IUE Orbit/
Semi-Analytical minus Cowell....................245
5-8
Osculating Mean Longitude Differences
within the 60 Day PCE Fit Span for the IUE
Orbit/Semi-Analytical minus Cowell..............246
5-9
Comparison of the Mean and Osculating
Semi-Major Axis Histories for the 3 Year
Integration of the IUE Orbit/AOG versus
Cowell..........................................248
5-10
Comparison of the Mean and Osculating
Eccentricity Histories for the 3 Year
Integration of the IUE Orbit/AOG versus
Cowell..........................................249
5-11
Comparison of the Mean and Osculating
Inclination Histories for the 3 Year
Integration of the IUE Orbit/AOG versus
Cowell..........................................250
5-12
Differences Between the Mean and Osculating
Semi-Major Axis Histories for the 3 Year
Integration of the IUE Orbit/AOG minus
Cowell..........................................251
5-13
Differences Between the Mean and Osculating
Eccentricity Histories for -the 3 Year
Integration of the IUE Orbit/AOG minus
Cowell..........................................252
5-14
Differences Between the Mean and Osculating
Inclination Histories for the 3 Year
Integration of the IUE Orbit/AOG minus
Cowell..........................................253
9
LIST OF FIGURES
0
(cont.)
Pag
Figure
5-15
Osculating
Semi-Major
Axis Comparison
for
the 3 Year IUE Integration/Semi-Analytical
versus Cowell...................................255
5-16
Osculating Eccentricity Comparison for
0
the 3 Year IUE Integration/Semi-Analytical
versus Cowell...................................256
5-17
Osculating
Inclination Comparison for the
3 Year IUE Integration/Semi-Analytical
versus Cowell...................................257
5-18
Evolution of the Mean Eccentricity for
the
100 Year AOG Prediction of the IUE
Orbit.........................
5-19
.................. 259
0
Evolution of the Mean Inclination for
the 100 Year AOG Prediction of the IUE
Orbit......................
5-20
.....................260
Comparison of the Mean and Osculating
0
Semi-Major Axis Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
5-21
Cowell................................... .......
Comparison of the Mean and Osculating
266
Eccentricity Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
Cowell.................................... .......
5-22
267
Comparison of the Mean and Osculating
Inclination Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
Cowell..............................................268
0
10
0
LIST OF FIGURES
(cont.)
Page
Figure
5-23
Comaprison of the Mean and Osculating
Longitude of Ascending Node Histories for
the 8 Year Integration of the ISEE Orbit/
AOG versus Cowell................................269
5-24
Comparison of the Mean and Osculating
Argument of Perifocus- Histories for the
8 Year Integration of the ISEE Orbit/AOG
versus Cowell...................................270
5-25
Comparison of the Mean and Osculating
Semi-Major Axis Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell..........................................278
5-26
Comparison of the Mean and Osculating
h
Element Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell.............................................279
5-27
Comparison of the Mean and Osculating k
Element Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell..........................................280
5-28
Comparison of the Mean and Osculating p
Element Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell..........................................281
5-29
Comparison of the Mean and Osculating q
Element Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell...........................................282
11
0
LIST OF FIGURES
(cont.)
Figure
5-30
Page
Comparison
of the Mean and Osculating
Inclination Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell..........................................283
5-31
Comparison
of the Mean and Osculating
Semi-Major Axis Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG
versus Cowell...................................291
5-32
Comparison
of the Mean and Osculating
Eccentricity Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG
versus Cowell...................................292
5-33
Comparison of
the Mean and Osculating
Inclination Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG
versus Cowell...................................293
5-34
Comparison of
the Mean and Osculating
Radius of Perifocus Histories for the
8 Year Integration of the STRATSAT Orbit/
AOG versus Cowell...............................294
5-35
Comparison of
the Mean and Osculating
Semi-Major Axis Histories for the 5 Year
Integration of the Lunar Resonance Test
Orbit/AOG versus Cowell..........................300
5-36
Comparison of
the Mean and Osculating
Eccentricity Histories for the 5 Year
Integation of the Lunar Resonance Test
Orbit/AOG versus Cowell.........................301
12
0
LIST OF FIGURES
(cont.)
Page
Figure
5-37
Comparison of the Mean and Osculating
Inclination Histories fot the 5 Year
Integration of the Lunar Resonance Test
Orbit/AOG versus Cowell.........................302
B-i
Subroutine Interaction Diagram for the
Third Body Software.............................333
13
LIST OF TABLES
Table
2-1
(m,r)
Form of the Function S2n
(P ,q')......
2-2
Form of
2-3
the Function S(nl,r)(a,S,y)......
2n
Functional Form of the Third Body
Elements, h' and k'.....................
3-1
Harmonic Contributions of the
Trigonometric Argument tX + sX.......... ........
4-1
93
Nonzero Poisson Brackets of the
Equinoctial Elements....................
4-2
0
Form of the First Order Averaged
Equations of Motion for Third Body
Perturbation............................
4-3
Partial
Derivatives of the Third Body
Mean Longitude with Respect
to the Mean
Satellite Elements p,q..................
5-1
Epoch Osculating Elements for the
IUE Test Case................................ ...
5-2
233
PCE Perturbation Models for the IUE
Test Case.................................... ...
236
5-3
Epoch Mean Elements for the IUE Test Case.... ...
238
5-4
Epoch Osculating Elements for the
0
ISEE Test Case............................... ...
5-5
262
PCE Perturbation Models for the ISEE
Test Case.................................... ...
264
5-6
Epoch Mean Elements
...
265
5-7
Comparison of Cowell and AOG Execution
Times for the Eight Year ISEE Integration.... ...
272
5-8
0
for the ISEE Test Case...
Epoch Osculating Elements for the
VELA Test Case............................... ...
273
0
14
0
LIST OF TABLES
(cont.)
Page
Table
5-9
PCE Perturbation Models for the
VELA Test Case.................................. 275
5-10
Epoch Mean Elements for the VELA Test Case...... 275
5-11
AOG Perturbation Model for the
Ten Year VELA Integration....................... 277
5-12
Comparison of Cowell and AOG Execution
Times for the Ten Year VELA Integration........
5-13
284
Epoch Osculating Elements for the
STRATSAT Test Case.............................. 285
5-14
PCE Perturbation Models for the
STRATSAT Test Case.............................. 287
5-15
Epoch Mean Elements for the STRATSAT
Test Case.......................................
5-16
AOG Perturbation Model for the
288
Eight Year
STRATSAT integration............................ 289
5-17
Comparison of Cowell and AOG Execution
Times for the Eight Year STRATSAT Integration... 295
5-18
Epoch Osculating Elements for the
Lunar Resonance Test Case....................... 296
5-19
PCE Perturbation Models for the Lunar
Resonance Test Case............................. 297
5-20
Epoch Mean Elements for the
Lunar
Resonance Test Case.............................
5-21
298
AOG Perturbation Model for the Five Year
Lunar Resonance Integration..................... 299
5-22
Comparison of Cowell and AOG Execution
Times for the Five Year Lunar Resonance
Integration..................................... 303
B-1
Datasets and Region Sizes Required for the
Various Third Body Options......................
15
332
Chapter 1
Introduction
The
order
and
in
elements
the
Generalized
of
recursive
of
term
long
stability
synchronous
altitude
Method
a high
the
sun and moon.
where
of
Averaging
the
for
long
period
Earth satellite
The theory
concerned with
the
in
orbits
selected
of
new
and
altitude
of
first
a
algorithms,
secular
asset to the mission analyst
valuable
above
use
by the action of
caused
the
the
is
thesis
this
calculation
and accurate
changes
on
extensive
making
rapid
based
theory,
of
result
central
"third body"
is
a
testing
region
perturbations
are a major determinant of orbital evolution and lifetime.
The third
work of semianalytical orbit theory.
involve
of
removal of periodic
the
Parameters
formulation
size
averaged
integrated
with
an
equations
expanded
computational expense.
purely
since
the
from
a Variation
dynamical
satellite
of a conventional numerical
resulting
of
Semianalytical methods
components
of
frame-
These periodic components unnecessarily restrict
equations.
the step
(VOP)
the
within
developed
body theory is
analytical
it permits
body models
into
the
of
step
integration.
motion
size
at
then
can
greatly
The
be
reduced
This approach is superior to the use
General
Perturbations
incorporation
of
(GP)
realistic
techniques
disturbing
a numerical orbit prediction program while
16
0
minimizing
[1] .
complexity
analytical
the
Furthermore,
artificial singularities associated with GP theories arising
from
[2]
arguments
critical
and
introduced
not
are
non-canonical variable sets may be employed.
The
theory
of
generality
in
derived
this
provement over previous
thesis
represents
work.
For
the
a
body
third
semianalytical
the
remarkable
the
time
first
imfol-
lowing conceptual components are unified in a single theory:
1)
Method of Averaging
Generalized
In
long
term dynamics
an
are
motion
of
equations
contained
by
developed
area
by
Generalized
unambiguously
and
defines
furnishes
the
a
an order by order basis.
17
approximation
[4].
order
the
an
was
the
later
and
Known
of
in
[3]
straightforward
for obtaining the averaged
the
The rigorous mathema-
Averaging,
of
to
high preci-
oscillations
Mitropolsky
Method
the
Bogoliubov
and
Krylov
in
this
non-linear
of
extended
theory
of
structure
averaged
approximation
sion equations of motion.
tical
the
theory,
orbit
semianalytical
(GMA)
as
the
formalism
averaging
protocol
equations of motion on
0
To
a
specified
provides
also
Averaging
averaged
elements.
possible
through
an
Method
of
approximation
to
the
in
of
the
terms
is
approximation
This
the
Generalized
elements
orbital
precision
high
the
order,
construction
of
made
analytical
functions of the averaged elements that represent
short periodic variations
the
Short
trajectory.
sion
to
essential
the
in the high preci-
periodic
initialization
of
functions
the
are
averaged
equations of motion given high precision orbital
elements.
2)
Third
Body Resonance
For the case where the satellite and the disturbare
body
ing
commensurable
nearly
in
their mean
motions, long period terms arise in the satellite
dynamical
third
body
resonance,
with
the
tesseral
gravity
track.
in
the
potential
Third
averaged
analogous
is
harmonics
caused
body
phenomenon,
This
equations.
by
the
a repeating
resonance
equations
resonance
to
of
of
terms
motion
called
are
Earth's
groundincluded
derived
in
this thesis.
18
0
3)
Non-Singular Orbital Elements
The
terms
of
ments.
This choice
bility
for
orbits
that
near-circular
or
characteristic
is
numerical
circumvents
ele-
orbital
equinoctial
non-singular
in
formulated
are
equations
VOP
satellite
insta-
equatorial
near
VOP
Keplerian
of
equations.
4)
Special Functions
the
third
permits
theory
body
and coefficients
functions
The use of special
an
in
com-
extremely
pact and modular analytical structure.
5)
General Disturbing Body Model
No
number of
the
third
predic-
a 'numerical orbit
in
body terms available
tion program
on
made
are
assumptions
restrictions
and no theoretical
are
placed on the orbital eccentricity of either the
satellite or
of
the disturbing
input
program
parameters
tailor the disturbing
ments
of
a
is
nature of
explicit
quire
extensive
third
to
body
reprogramming
lities.
19
to the
the
This
"hard-wired"
theories
to
to
require-
orbit.
satellite
contrast
in
attribute
sufficient
is
body model
particular
Specification
body.
extend
that
re-
capabi-
0
Recursive Computation
6)
The use of recurrence relations for the numerical
of
computation
cated
special
disturbing
rapidly
body
starting
from
functions
models
allows
to
be
compli-
evaluated
determined
simply
initial
values.
1.1
Previous Work
Major
researchers
have
contributions
the
to
study
of
made
been
semi-analytical
by
several
techniques
for
determining the evolution of high altitude satellites moving
theories
applicable
intervals
are
satellite
to
over
of
averaged.
Each must
a
the
general
the motion.
and disturbing
of
averaging
period
dynamics
in
will
the
its
assumed
of
variables
to remove all
time
the satellite
averaging
These theories
the
criticism,
extended
over
angular
be judged on
satellite
double
fast
and disturbing body
components
as
mission analysis
based on successively
equations
VOP
Most of the third body
influence of a third body.
under the
high
are
the
frequency
called
own merits.
double
However,
independence
of
the
body phase angles for the purposes
exclude
averaged
the
presence
equations
of
any
of motion
long
induced
by third body resonance.
20
0
A
third
by
developed
body
Gauss
is
acceleration
perturbation
secular
the
which
in
[5]
are
average over the mean anomaly of
The
eliminated.
of
system
short periodic varia-
Gaussian VOP equations from which all
tions
disturbing
a
produce
to
averaged
doubly
body
third
was
theory
the disturbing body is seen to reduce to the purely geometrical exercise
of
ring of
that
third
matter
available
to
modernize
lunar
In
satellites.
numerical
could
orbital
of
and
effort
in
Musen
[7]
given
to
the
in
involves
elements
over
high
terms
precision
the
numerical
an
appropriate
if
on
of
a
21
long
high
set
to
[61
term
altitude
possibility
of
of
element
how Halphen's
method
non-singular
averaged equations
elements
orbital
at
epoch
One technique, used by Smith
high
of
way will
interval
precision
However,
interval.
in this
the
theory,
Keplerian
original
other
any
Musen
the
the
remove
average
the
contamination
by
study
Initialization of the
averaged elements obtained
periodic
to
explained
later
remains a challenging problem.
[81,
rederived
perturbations
an
or
version of Gauss'
programmed
solar
elements.
motion
was
and
reformulated
be
modified
[51,
instability
formulation,
orbit of
quadrature
numerical
A
Halphen
notation
of
effects
by
technique.
attributed
an elliptical
the
along
is distributed
computed
is
anomaly
attraction of
the
subsequent average over the satellite mean
The
body.
computing
is
the
contain short
not
precisely
A
chosen.
more
effective
method,
employed
[ 91
by Baxter
,
on a double angle harmonic analysis of high precision
relies
element histories to identify short periodic components.
Ash
[10]
a
developed
body
third
averaged
double
theory in Keplerida elements, based on Gauss' concept, which
was
used
to
of
high
altitude
Halphen's
interpret
physically
method
satellite
by
starting
the
numerical
orbits.
It
differs
from an infinite
sion for the third body potential rather
integration
series expan-
than directly from
the perturbing acceleration which is closed form.
average
rather
over
satellite
which
expressions
third
It
can be
the
satellite
VOP
that
are
eccentricity
this
equations
in
the
leaving
complex.
Furthermore,
is
to
assumed
assumption eliminates
representing
the
analytically
recognized,
not
unnecessarily
body orbital
shown
performed
Also,
Special functions imbedded
equations
are
is
orbit
than numerically.
averaged
literal
the
the
from
be
terms
zero.
from
dynamical
significant
contributions.
Sridharan
Lidov
theory
[12]
Seniw
[11]
to derive an explicit
based
expansion
and
on
of
the
the
first
term
followed
of
double
averaged
third
in
Legendre
polynomial
disturbing
22
the development
the
acceleration.
body
The
Gaussian VOP
tions
are
formulated
in
terms
the
lacks the
theory
high
altitude
able
to
Keplerian
moving
in
The
the
orbital
ele-
truncated
force
to handle
flexibility
orbit.
satellite
satellites
of
of the dramatically
As a consequence
ments.
model
equations and the short periodic recovery func-
general
is most
theory
region
a
below
applic-
synchronous
altitude.
general
theory
A
more
[13].
It
includes an analytical
based
on
Kaula's
model
in
extensive
use of
special
been
double
relations to simplify computation.
assumptions
are
made
to
facilitate
panying
short
periodic
body
disturbing
makes
theory
The
and attendant recurrence
Furthermore, no a priori
truncation
the
However,
averaged equations of motion.
Cook
averaging capability
[14].
functions
by
developed
third
the
elements
Keplerian
potential
for
has
deyelopment,
is
there
thereby
no
of
the
accom-
making
the
initialization of the double averaging theory from high precision orbital elements an extremely difficult and uncertain
task.
1 .2
Overview
The Generalized
Method of Averaging
is
applied to the
conservative VOP equations describing the motion of an Earth
satellite under the influence of a third body.
23
Chapter
In
third body
the
2,
disturbing
potential
developed in non-singular orbital elements using
taken
is
and
satellite
as
the
the
disturbing
both
can
resonance
that
so
body
for
variable
angular
fast
satel-
the
The mean longi-
lite orbit plane as the frame of reference.
tude
is
the
be
accomodated.
Chapter
In
Method of
3, the Generalized
Averaging
is
used to create an averaging theory applicable to a system of
rotating
rapidly
two
containing
equations
differential
The
The theory is then specialized to first order.
angles.
formal
method
tions
of
the
for obtaining
motion
functions
periodic
short
the
and
equa-
first order averaged
is
presented.
Chapter 4 details the results of applying the averaging
on
theory
the
disturbing
form
mathematical
the
short periodic
tives,
are
given.
values
for
all
atives
are
3 to
of Chapter
a system of VOP
potential
of
the
special
presented.
including
recurrence
functions
in
all
of
Truncation
their
The
motion
and
partial
relations
and
2.
Chapter
equations
averaged
functions,
The
derived
equations based
and
partial
procedures
derivastarting
derivfor
the
dis.turbing body model are discussed.
240
0
The
and
equations
interfaced
Determination
Draper
bed
a
System
Laboratory
program,
lation
with
expressed
GTDS
algorithms
version
(GTDS)
in
for
the
between
theory
high
developed
precision
in
this
of
the
at
4 are
Goddard
the
short
programmed
Trajectory
Charles
Massachusetts.
numerical
auxiliary perturbation models.
sons
Chapter
modified
Cambridge,
furnishes
in
As
integrators,
periodic
thesis
are
a
and
presented
the
test
interpo-
variations
Speed and accuracy
integration
Stark
and
compariaveraging
in Chapter 5
for selected orbits.
Chapter
areas
6
formulates
conclusions
for future investigation.
25
and
suggests
some
0
Chapter 2
The Third Body Disturbing Potential
This
potential
chapter
in
formulates
terms
of
the
third
satellite
body
orbital
disturbing
coordinates.
Mathematical operations on the potential produce a system of
high
precision
equations
form
Variation
the
of
Parameters
cornerstone of
equations.
third
the
body
These
averaging
theory developed in this thesis.
The
tial
potential
orbital
elements
numerical
instability
orbits
that
formulated
is
in
The equinoctial
is
as
expressed
part
of
an
non-singular
approach
for near-circular
characteristic
terms
in
of
the
of
elements are classically
a
=
a
h
=
e sin(w + IQ)
k
=
e cos(w + IQ)
p
=
tanI(i/2) sing
q
=
tanI(i/2)
=
M + w + IQ
cosQ
26
circumvent
and near-equatorial
satellite
classical
to
equinoc-
VOP
equations
Keplerian
elements.
defined as
[15 ]
a,
where
i,
e,
Keplerian
and
elements
I
is
The retrograde factor is present to
the retrograde factor.
eliminate a
the
M are
w, Q,
in
singularity
element
the
for orbits with
set
It is discussed in detail in
an inclination of 180 degrees.
Section 2.2.
2.1
Section
in
tial
inertial
disturbing
to
potential
the
poten2.2
Section
coordinates.
rectangular
transformation of
the
details
third body
the
derives
equinoctial
orbital elements.
Derivation of the Third Body Disturbing Potential in
2.1
Coordinates
Inertial
The
expression
an
with
Newtonian
strictly
may
be
referred.
metry.
body
r
the
disturbing
XYZ
Figure
R
with
respect
to
2-1
depicts
the
denotes
the
(S)
represent
respectively
and the disturbing body
ter of the central body.
difference
r
-
r',
begins
In
acceleration.
the
implies
this
formulation,
potential
body
third
position
three
geo-
central
The vectors
the positions of
(D) measured
motion
body
the
of
a
postulate
all
which
(C) relative to the origin of coordinates.
and r'
lite
vector
The
the
of
for
frame
inertial
an
of
development
the satel-
from the cen-
The vector d, which is simply the
establishes
27
the
relative
separation
of
Figure 2-1.
Geometry of the Third Body Problem
s0
S
z
00
0
x
28
S
the satellite and the third body.
Another useful parameter,
soon to be employed, is the angle * between r and r'.
commonly
referred
to as
elongation
the geocentric
in
It is
Earth
centered orbital analyses.
of
dynamical
equations describing the motion of the three body
system may
on
Based
the
geometry,
stated
a
set
be written as
m (R +r)
--GMM
r r-
G m
m
dd
(2-1)
and
SG
-r3
where
the
double
dot
Mmr+ G Mm'
r' 3
notation
(2-2)
indicates
the
second deriva-
tive with respect to time and
ml
mass of satellite (kg)
mass of the third body
M
G
(kg)
mass of the central body
E universal constant
29
(kg)
of gravitation
(km 3 /kg-sec2)
0l
From eq.
(2-1)
mass
the
of
(2-2) the inertial accelerations per unit
and
body
central
and
satellite
respectively
are
given by
R +
rM
G- 1~
r
d
r
(2-3)
d
3-7
0
and
=
Subtracting eq.
Gm
r
(2-4)
3r'
(2-4) from eq. (2-3) yields
_
r
GmI
+
-T r
r
G (M
3
+ m)r-Gm
m
r-
d
r
Eq.
(2-5)
(2-5)
3L)3
+
may be simplified by defining the quantities
y'
=
(2-6)
G(M + m)
(2-7)
= Gm'
Transposing
(2-5) and
the
first
invoking eqs.
term
+
right
hand
(2-6) and (2-7) leads
d
..
r
the
of
of
eq.
to
r'
+
-r
r3-d3
side
r3
3
(2-8)
30
0
Eq.
a
disturbing
body
third
the
by
forced
as
body
central
about
satellite
a
of
motion
the
represents
(2-8)
acceleration
a
=
potential
eq.
(2-9)
function.
This
can be
satellite
to the
the
conservative,
are
forces
respect
with
(2-9)
r 1
by
force represented
the gradient,
scalar
d
r'
-- 3
3 -+-
gravitational
Since
specific
d
(
-'
function
expressed
position,
of the
as
of a
satellite
coordinates has the form
U'
U11
=
(-
y'
-
r'-E
-
.
)
(2-10)
r
Subsequent
of eq.
analysis
require
Manipulation
(2-10).
d2
will
=
d
=
r2
d
-
-
=
2r
a
more
convenient
begins by recognizing
- (r -
(r - r')
- r' + r,
31
2
version
that
r')
(2-11)
a
If
is
$,
elongation,
geocentric
the
into
introduced
eq.
6
in
distance
of
the
relative
2rr'
cos
* + r' 2-1/
inverse
the
then
(2-11),
eq.
(2-10) is seen to be
1
(r 2
=
developed
in
orthogonal
the
of
terms
(2-12)
eq.
coordinates,
orbital
2
(2-12)
into
of the third body potential
the expansion
To facilitate
satellite
-
more naturally
is
Legendre
0
polynomials
defined by [16]
1)-1/2
2hX +
(h 2 -
-
I hn P ( X)
n=0
n
(2-13)
0
where
Legendre polynomial
Pn(X)
of degree
0
n and
argument X
parameter less than or equal
h
Eq.
be
(2-12) may
by factoring out r'
in
placed
2
correspondence with
{(
eq.
(2-13)
so that
2
=
to 1
)
(
-2
cos
s
)c
- 1/2
*+
1}
(2-14)
32
0
Comparing
with
(2-14)
eq.
(2-13)
eq.
yields
identifica-
the
tions
h=(
(2-15)
r)
and
X
=
(2-16)
cos *
(2-14) to take the final form
This allows eq.
00
1
d
r
n
)
n0
p
(2-17)
(Cos
Note that this representation assumes that the satellite
of
describes
the
an
perturbing
required.
body.
However,
obviously violated.
eq.
(2-12).
Only
orbit
interior
for
This
with
ensures
an exterior
respect
the
the orbit
1
as
condition
is
(r/r'
that
orbit,
to
)
<
In such a case r 2 must be factored from
interior
sidered here.
33
satellite
orbits
will
be
con-
(r'
3
- r)/r'
U1
as
(r/r'
iU
=
r
-r
2)
Further
eq.
recognizing
(2-18)
becomes,
1=
U'
6
cos * yields
n (cos
F- )n n
n0 (r
(-
-
nT
r
n=0
(2-10) and rewriting
(2-17) into eq.
Substituting eq.
PO(cos
that
*)=1
and
potential
position
with
on
solely
respect
vector.
$,
to
(2-19)
Pn (cos *) ]
the
gradient
in cartesian coorof
elements
the
Accordingly,
0
0
The satellite equations of motion
depend
*)=cos
6
n
[1 +
n=2
dinates
Pi(cos
(-18)
terms,
cancelling
after
s2 ~]
Cos
the
of
leading
the
the
disturbing
satellite
term
in
eq.
(2-19) will not contribute to the satellite motion by virtue
of
having
no
satellite
dependencies.
In view of
this
the third body disturbing potential may be rewritten in
fact
the
commonly used form
,
)
2
n=2
34
n
Pn (cos *)
(2-20)
0
Transformation of the Third Body Potential from Iner-
2.2
tial Coordinates to Satellite Coordinates
formulation
VOP
Lagrangian
The
tion problem requires
partial
of
predic-
orbit
the
derivatives of the disturbing
potential with respect to the orbital elements of the satellite.
It is therefore highly desirable to express the third
(2-20), directly in terms
body potential, represented by eq.
of
,
One
elements.
satellite
identities
standard
it
in
to
tends
special functions
such
ignore
a
the
in the satellite
by
typified
method
Kaula
application
tedious
trigonometry
spherical
Although
manipulations.
correct,
but
straightforward
a
entails
approach,
is
presence
of
theory and
[14]
of
and
algebraic
in
principle
mathematical
the reduction
in analytical complexity that derives from their use.
alternate
An
technique
involves
employing
a powerful
transformation theorem for spherical harmonics under a rotation of
tial
has
coordinates
which
been
is
to provide an expression
more
substantially
used with
success
compact.
by McClain
[17]
and will be adopted in the ensuing analysis.
35
for the potenThis
technique
and Cefola
[18]
]
Reference Frame Choice and Implications
2.2.1
0
potential
body
third
the
transforming
Before
to
satellite orbital elements, a reference frame must be chosen
for
must
numerically
be
a
is
potential
The
(2-20).
cosine of the geocentric elongation in
the
computing
invariant under a change of coordinates
Hence, the selection of a
that leaves the datum unaffected.
venience
tical
can be made principally as a matter of con-
frame
reference
or
practicality
of
form
therefore
and
quantity
scalar
eq.
the
provide
to
or
potential
a
more
computational
the
to reduce
analy-
elegant
load in a numerical orbit prediction program.
In
be made
terms,
broad
absolute
between
and
a
of
choice
the
relative.
reference
frame
may
Absolute
frames
are
the equato-
inertial reference systems commonly oriented in
rial
plane
of
the Earth
or
in
Within
plane.
the ecliptic
the context of developing the third body disturbing function
such
frames
the
have
advantage of
maintaining
dence of the third body orbital elements
elements.
turbing
body elements
from an
external
with
the notion
affected
by
the
are
ephemeris.
that
the
movement
constant
simply
This
of
reverse.
36
the
satellite
third
body
step
the dis-
parameters
decoupling
integrated
indepen-
from the satellite
Hence, at each integration
orbital
the
is
taken
consistent
elements
are
than
the
rather
On
the debit
side of
the
ledger,
the
use of
an abso-
lute frame may be undesirable from the viewpoint of a numerical
In
program.
potential
torial
the
of
expansion
the
Earth's
gravity
in equinoctial variables, as referred to the equaframe
inertial
[18 ],
inclination
an
is
function
introduced relating the satellite orbital plane to the equatorial system.
a
second
results
in the
tion,
potential
duct
However, in the case of third body perturbafunction
inclination
be
introduced.
This
appearance of an additional summation in the
and extra computational
could
is
a
reduction
in
numerical prediction program.
37
complexity.
the
overall
The end pro-
efficiency
of
a
U1
Selecting a relative reference frame in the satellite
or disturbing body orbital plane serves to eliminate one or
the
The
summation.
the
the
of
other
an
is
result
then
to
referred
were
in
derived
third
body
be
respect
to
equations
chained
the
to
the
Hence
reference.
satellite
of the
the motion
through
will
body
dynamical
frame of
be
must
elements
body
third
with
motion
in
the satellite
inertial
an
the
the disturbing
is
that
However,
absolute space.
if
For example,
elements of
frame
a
of
using the satellite reference system,
is developed
the orbital
form
simpler
analytically
to be induced dependencies.
potential
associated
the
An obvious drawback would appear
potential.
disturbing
and
functions
inclination
frame
inertial
Similarly,
frame.
if
the third body reference frame is chosen, then the elements
of
the
orbit
satellite
will
consequently
to
related
be
the
integrated
through the motion of the disturbing body
of
characteristic
This
non-inertially
referred
demonstrating
problem,
a
the
satellite
frame
it
the
leaves
frame
relative
is
selected
satellite
framework exists for
to
for
the
inertial
the purpose
third
in
the
chosen.
In
particular,
to develop
the
potential
is
elements
system of integration.
38
not
does
averaging
of
power
elements
[1.
frames
elements
Accordingly,
dynamical reference frame.
system
relative
preclude their use since an analytical
relating
defined and
be non-inertially
will
expressed
directly
of
body
the
since
in
the
Body Potential to
the Third
of
Transformation
2.2.2
Satel-
lite Orbital Coordinates
eq.
The transformation of
the
(2-20) to
equinoctial
elements of the satellite begins by expressing the cosine of
the geocentric elongation as the dot product,
where
=
symbol
1 denotes
the
cos *
1
r
- 1
r
a unit
1
tors
f,
g,
frame, the
cos(L) f + sin(L) g + 0 w
the true
L is
coordinates
r has the form,
=
where
In
vector.
satellite reference
measured with respect to the
unit vector
(2-21)
.
,
w
are
(2-22)
longitude of the satellite and the vec-
the
in Figure
unit vectors depicted graphically
frame
satellite
referred
inertially
2-2
and defined
by
f
=
272
[1
p2 , 2pq, -2pI]
+ q
(2-23)
1+p +q
T
g
=
=
2
2
2
1
2
1+p +q
[2pqI,
39
(1
+ p
-
q )I,
2q]
(2-24)
0
Figure 2-2.
Orientation of the Satellite Reference Frame
with Respect to the Inertial Reference Frame
z
A
f
W1
9V
y0
L-f+W+ 12
x
40
w T
[2p,
-2q,
(1 -
are
the
equinoctial
p2
(2-25)
q2)I]
-
1+p2 +q2
The
variables
the
specify
p
of the
orientation
the inertial frame.
a
circumvent
q
and
in
for
factor
is +1 when the orbital
inclination
is
the
inclinations
with
motion
the
frame
to
relative
The retrograde factor, I, is present to
singularity
orbits
satellite
that
elements
180*.
In
of
inclination
between
of
equations
satellite
The
degrees.
180
is 0*
either
and -1 when
value
may
be
used.
The third body unit vector is given by
1r'=
cos6 cosa f + cos6 sina g + sin6 w
(2-26)
where a and 6 are the right ascension and declination of the
third body relative to the satellite frame.
shown
in
Figure
2-3.
Substituting
eqs.
The geometry
(2-22)
and
is
(2-26)
into eq. (2-21) yields
cos* = cos6 cosa cos(L) + cos6 sina sin(L) + sin(0) sind
(2-27)
41
Figure 2-3.
Orientation of the Third Body Position
Vector with Respect to the Satellite Frame
A
w
r
AA
fg
42
the
Apply
cos(y)+
trigonometric
standard
eq.
sin(x) sin(y),
cost
sin(O)
=
identity
cos(x-y)
=
cos(x)
(2-27) becomes
sinS + cos6 cos(a -
L)
(2-28)
At this point the Addition Theorem for spherical harmonics
cosS
should
cos( a -
be
L)]
into simple
third body coordinates.
decomposes
It
invoked.
functions of
Pn[sin(O)
the
sin6
+
and
satellite
The Addition Theorem has the form
[19]
Pn [sin(y) sin(y')
=
n [sin(y)]
+ cos(y)
cos(y')
cos(x -
x')]
=
Pn [sin(y')]
n
+ 2
(nm)! P
[sin(y)]
P
[sin(y')]
cos m(x
-
x')
m=1
(2-29)
43
The
function
defined by
is
Pnm(z)
associated
the
Legendre
function
4
(16]
a
(
=
n (z
-
z2)m/2
P (z)
(2-30)
dzm
identifications
If one makes the
sin(y-)
= 0
sin(y')
= sinS
cos(y)
= 1
cos(y')
=
0
cos6
x
a
x'
L
0
then eq.
(2-29)
becomes
Pn(cos *)
n
+ 2
m=1
=
Pn (0) Pn (sin)
(n-n
P
(n+n)! nrnm(0)
44
Pnm(sin3)
cos [m( a-L)]
(2-31)
From eq.
(2-30) it
is clear that
PnO(z)
Hence eq.
(2-31)
may be rewritten as
Pn0
=
Pn (cos *)
+ 2 n
M=1
(2-32)
P n(z)
=
(n-m)!
PnO(sinS)
(2-33)
nm(0) Pnm(sino) cos[m(a-L)]
Def ining
m = 0
K
m
allows
the
m > 0
2
summation
in
eq.
(2-33)
to
begin
at
zero,
producing
Pn (cos*)
=
n
I
m= 0
K
Km
(n-)
(n+m)!
nm
(0) P
nm
(sin6) cos[m( a-L)]
(2-34)
45
the function Vnm is
If
V
n,m
=
defined
(n-m)!
(n+m)!
to be,
(2-35)
P nm (0)
0
then eq.
(2-33) may be restated as,
0
n
Pn (cosM)
n
=
Substituting
eq.
0
KM Vnm Pnm (sinS)
m n~m=n
(2-36)
into
eq.
cos[m(a-L)]
(2-20)
(2-36)
produces
the
intermediate form of the third body potential,
U'
KmVm
=
n=2
Pnm (sind)
cos[m(a-L)]
m=0
(2-37)
46
0
In order to use the rotational transformation theorem
for
e j[m( a-L)
in)
Re{n
=
(-Er)
ro
n=2
is
(2-37)
Doing so yields
rewritten in complex notation.
T
eq.
in
potential
the
harmonics,
spherical
(sin(s
K mV nmP
m=O
(2-38)
where
Re
sion
and
{ } denotes
j
V'".
=
the
real part of
Breaking
apart
the bracketed
the
expres-
complex exponential
produces
U'
1
= Re 1
nm (sin6)je
emm
[Km V
m=O
n=2
(2-39)
surface
The
will now
the third
be
rotated from the
body frame.
transformation
harmonics
spherical
under
A theorem exists
simple
origin is held fixed.
Pnm(sin6)ejma,
satellite reference
of any linearly
a
harmonics,
which describes
independent
rotation
of
frame
into
the
set of spherical
coordinates
when
the
With appropriate changes of nota-
tion, the theorem is formally stated by[20],
47
nm
=
m
(cos
r=-n
(n-r)! S(m,r)
(n-m)!
2n
- P
(cosO') ejr'
(2-40)
The unprimed quantities
of
a
point
as
measured,
frame.
The primed
dinates
of
the
8 and E are the angular coordinates
angles
same
from
O'and
as
point
from
measured
referred axes of the third body frame.
satellite
spherical
the
are
E'
the
of
axes
the
the
coor-
satellite
The parameters
p,a,T
are related to the Euler angles defining a rotation of coordinates from the third body reference frame to the satellite
frame.
These parameters will be discussed
Eq.
cal
(2-40) is now used to express the surface spheri-
harmonics,
spherical
frame.
Pnm(sind )ejma,
as
a
expressed
in
the
harmonics
This is
done by
identifying
ties shown in Figure 2-4.
the
shortly.
colatitude
in
linear
third
combination
body
of
reference
the. geometrical
quanti-
The parameter 6, generally called
spherical
coordinates,
is
measured
from
the positive w axis of the satellite coordinate frame to the
48
Figure 2-4.' Geometry for the Rotation of Surface
Spherical Harmonics
A
W
A
w '
A
g
%N
{f, g, w } = satellite frame unit vectors
A
f
f^',,
IA
f'
49
third body frame unit vectors!
a
disturbing
the
plane
of
third
body declination,
The angle
right
E in eq.
ascension,
6,
through
In
the
coordinate
the
related
0 =
relation
the place of
(2-40) holds
a.
is
It
body orbit.
the
7r/2-6.
third body
the
system
to
of
the
per-
turbing body, the angle 0' has the value w/2 since the third
A
body orbital
plane
parameter,
',
corresponds
the
to
The angle, L',
third body, L'.
the w' axis.
is normal to
true
Finally,
the
of
the
longitude
is measured from the f' axis
in the plane of the perturbing body orbit.
Making
indicated
the
substitutions,
eq.
(2-40)
becomes
P
nm
(sin6) eJma
r=-n
(n-r)! S(mr)
(n-m)!
- nr (0)
2n
(pyajT)
jrL'
(2-41)
I
Making the definition
V
n,r
=m
(0)
(n-r)! P
nr
(n-m)!
(2-42)
50
i
eq.
(2-41) may be rewritten as
n
e ma
(sine)
P
,
n,r
r=-n
ejrL'
(m,r) (paT)
2n
(2-43)
U' = Re{l
1
00
1
(2-39) produces,
(2-43) into eq.
Substituting eq.
n
n
n
(r)
1 n =2
,e jrL'
Km n,m r=-n
M=0
2n
nr
-jmL}
(2-44)
The
known
explicit
quantities
dependence
will
Courant and Hilbert
now
[20],
be
of
S
,r) (p,,T) on
determined.
As
defined
S (r,r)(prg,T) has the form
2n
51
in
a
S(m,r)
s2n
( p,a, T)
- Um
,r
(T) exp{-j[(r+m)p +
(r-m)a]
}
(2-45)
6
a
where
n+m
U(m,r)
2n
- 2F
n-m)
n+r
(-n-r, n+1-r;
sinT ) m-r
(COS T) -m-r (cost)
(sin'r
1-m-r;
,
cos 2
a
m + r < 0
(2-46)
U6
and
01
01
0
52
0
2n
(cosT)m+r
n+m)
n+r
U(m,r)
(sint)r-m
n-r
SF 1(r-n,
1+m+r;
n+r+1;
cos 2T
(2-47)
r + m > 0
The
expression
in
function
the
2
F 1 (a,b;c,4cos 2 T)
argument
(cos
2
T)
is
a
( )
and
e
hypergeometric
is
the binomial
p,
a,
coefficient.
In
related
the
(2-43),
view of eq.
to a rotation
satellite
orbital
coordinates defining
rotation matrix
from
a
the quantities
from the third
plane.
They
the
third body
frame, viz.,
53
body orbital
are
four parameter
and
actually
plane
to the
are
to
orthogonal
representation
frame
T
of the
satellite
a
q2+q
M
=
1
q2-q
2
2(qlq
2(qlq 2 -q q )
3 4
-
2(q1 q 3 +q q )
2 4
2 +q 3 q 4
)
2(qlq
3 -q 2 q 4 )
2 2 2
2
q 4 +q 2 -q 1 -q 3
2(q 2q 3 +qq4 )
2(q 2 q 3 -q1 4)
2
2
2
2
q4+q 3~91~-q2
0
/
(2-48)
g
0
Is
where v
+ q 2 + q32 + q 2and
q
=
0
=
sinT sina
(2-49)
q2
v sinT Cosa
(2-50)
q
V COST
qi
-v
3
V
The
parameters qi,
parison of eq.
sinp
COST COSP
q 2,
q 3,
0
(2-51 )
(2-52)
q 4 are determined through a com-
(2-48) with the rotation matrix
0
54
0
,2
1-p' 2+
M
2
2p'q'I'
(1+p'2 -q' 2 )I'
2p'q'
p'
-2q'
1+p' +q'
(1-p' 2-q, 2 )V
-2p' I2q'
(2-53)
The
p'
quantities
q'
are
elements
equinoctial
the
the
of
orientation
the
specify
and
third
reference
body
that
frame
Hence, they are related to
relative to the satellite frame.
the mutual inclination of the third body and satellite orbit
The
planes.
set
ordinarily
mutual
+1
inclination
singularity
moves
to
in
opposite
Instead,
in
the
but
180*.
is
plane
of
the
assumes
This
third
the
The
value
of
body
singularity
when
an
the
orbit,
is
when
-1
circumvents
of motion
which
factor,
retrograde
the
the equations
direction.
it
is
I'
symbol
not
is
the
apparent
satellite
but
in
the
dynamical.
appears because the satellite frame was used to
develop the disturbing potential.
Taking
cases
of
eq.
into
(2-53),
account
both
the
the parameters,
found to be,
55
direct
and
retrograde
of
eq.
(2-48)
qi,
are
4
[(1
q,=
4
p
+
=
(2-54)
I') + (1 + I')q']
-
(2-55)
a
q
=
(2-56)
p
2
a
q
=
+
+ I') + (1
[(1
-
(2-57)
I')q']
a
Also, it is evident that
v
=
(1
+ p,
2
+ q,2)1/2
(2-58)
a
Now
it
is
that
possible
the
parameters,
to determine
the elements p'
and q' .
eqs.
and
into
(2-49)
the
qi,
have
dependence
Substituting eqs.
(2-50) yields
been
of
identified,
p,a, and
(2-54)
an expression
and
T on
(2-55)
a
for sinT,
after squaring and adding, viz.,
a
sint
=
2
(1 + p, 2 + q, )-1/2
(p'2 + q,2)(1+I')/4
(2-59)
Automatically,
IM
cosT must be
56
41
4
cost
(1 + p' 2 + q, 2 )-1/
=
(p, 2 + q1 2
2
(1-I')/4
(2-60)
The
through
of
plus
or
minus
(+)
designation
over
the parameter set will satisfy
of the rotation matrix given in eq.
being
adopts
eqs.
(2-54)
(2-57) does not imply that an arbitrary distribution
signs
nations
in
of
(-,-,-,+)
the
signs
will
and
former
produce
(+,+,+,-).
convention
(2-53).
the conditions
Only two combi-
the
desired
The
analysis
although
result,
that
those
follows
the
other
would
and
(2-59)
into eq.
be
valid.
Substituting
(2-49)
and solving
sin a
eqs.
(2-54),
(2-58)
for sina yields
(1-I)
I
2 Vp'
+
12
I)q
+ q'
(2-61)
57
a
Substituting
eqs.
(2-55),
(2-58)
and
(2-59)
into eq.
(2-50)
4
produces
Cosa(1
+ I')p'
2 /'p'
2
Similar
manipulations
the desired
forms of
=
sinp
with
(-2-62)
eqs.
(2-51)
sinp and cosp,
-
6
+ q'2
and
(2-52)
provide
6
viz..,
( 1
I'I)p
(1__
(2-63)
a
2 /p,2 + q'2
a
and
a
cosp
=
(1 + I') Vp' 2 + q' 2 + (1
2
-
)q
p 2 + q'
6
(2-64)
6
58
01
Making use of eqs.
exponential in eq.
exp{-j[(r
through
(2-64) ,
the complex
(2-45) is seen to be
+ m)p +
(- 1)mr
=
(2-61)
(r
-
m)a.}
=
2
(p'2 + q1 2 )(m-r)/ (p'I' + jq') r(p
+ jq')-m
(2-65)
The
will
dependence
now
first.
be
of
U nr) (T)
determined.
Substituting eqs.
The
on
range
(2-59) and
59
the
elements
r<-m will
be
p'
and
q'
considered
(2-60) into eq.
(2-46),
4
U (m, r)
n+m
(n+)
[(1+p
,2+q, 2)- 1/2 (p,2+q,2 ) (1 -I'1)/4 1-m-r
1
(p, 2 + q12 )( +I
2 + q,2 -1/2
- (1 + p,
[-n-r,
n+1-r;
(P, 2 +
1-m-r;
)/4
m-r
a
,2) (1-I')/2
2
2
I
(1l+p'2 +q,2
r < -m
a
a
(2-66)
a
After some manipulation,
eq.
(2-66)
becomes
6
(m,r)
U2n
=
(_,)n+m (n~)
(1+p,2+q'
2 )r(P2+q,2)(I'm-r)/2
,
IM
(1-I' )/2
S2F 1[-n-r,
n+1-r;
1-m-r;
2
(1 + p' 2
+ q' )
I
IM
r
< -m
(2-67)
0
60
01
40
The
hypergeometric
gonal
tion
Jacobi
function
polynomials
may
be
replaced- by
P (a,b) (x)
n
by
the
the
using
orthodefini-
[16]
P (a,b)
I
_
(x)
ta+a)
2F 1
(-,
+a+b+1;
a+1;
)
(2-68)
if
X = n+r,
1
a = -m-r,
and b = m-r.
Hence,
(-m-r,m-r) (-m-rm-r)1-x
X)
2 F (-n-r,
n+l-r;
1-m-r;
)
n+r
(2-69)
The
argument
of the Jacobi
polynomial,
x,
may be
determined
by solving the algebraic equation
1-x
(p
2
+
(1 + p'
(12
+ q')
The solution is given by
61
)/2
(2-70)
4
,2
- p' 2 .
2
(1 + p' + q' 2 )
_(1
-
(2-71)
41
a
is
(2-71)
of eq.
An interpretation
made by :..amining the dot
product representation of the rotation matrix in eq.
(2-53),
viz.,
f')
(f
-g')
(f
- w')
(g
- f')
(g
- g')
(g
- w')
(w
f')
(w
- g')
(w
w')
(f
a
M
=
(2-72)
61
The
body
columns
frame
frame.
of
eq.
(2-72)
vectors
basis
Comparison
of
the
the same element of eq.
are
as
the
components
coordinatized
(3,3)
element
of
in
eq.
the
of
the
third
satellite
(2-72)
6
with
(2-53) shows
61
A
0.
p
A
'
2
_q12
'
= (1 -+p
2
1 + p' + q
1
(2-73)
0
62
0
0
The
dot
third
product
body orbital
of
the
unit
planes,
normals
denoted
in
to
the
eq.
satellite
(2-73),
and
is
equal
to the cosine of the mutual inclination and is evidently the
negative of the x in eq.
(2-71).
duct by the symbol y,
(2-69)
eq.
P (-m-r,m-r) (-)
1
n-r)
Representing the dot procan be rewritten
F
as
(-n-r,n+1-r;1-m-r;)+Y
n+r
(2-74)
Replacing
the hypergeometric
function
in
eq.
(2-67)
by eq.
(2-74) produces
U(mr)
(-1)n+m (1+p,2 +q2)r
(p12 +q1 2 )(I'm-r)/ 2 .
S +(-m-r,m-r)
n+r
r
Using
< -m
the identity for Jacobi polynomials
63
(2-75)
[171
a
P(a,b)
n
transforms eq.
-_)
(2-75)
U (,r
=
(-1)n
p(b,a)
n
(2-76)
(x)
into
41
a
2
(-1)mnr (1+p,2+q1 )r (P, 2 +q
2
) (I'm-r)/2,
, p(m-r, -m-r)
n+r
6
r
(2-77)
< -m
09
Substituting
eqs.
an expression
S
(
p',q')
(2-77)
and (2-65)
into eq.
(2-45) provides
for S (m,r) (p'q')
2n
=
(1+p, 2 +q
)r (p, 2 +q
2
0
2
)[(I'+1)m/2]-r
n+ r
r
< -m
0
CT)
01
2-78)
64
0
Examination of the direct and retrograde
cases of eq.
(2-78)
leads to the more compact expression,
S (mr)(p,,q,)
r (1 + p,
=(I')
jq,
(p
2
+ q1 2 )r
(m-r,-m-r)
n+r
,)I'm-r
r < -m
(2-79)
A similar procedure is used to determine the function
S (m,r) (p',q')
when
2n
portant
eq.
proviso.
r
>
Whereas
-m.
However,
the
there
hypergeometric
is
an
function
imin
(2-47) is valid over the entire range r > -m, care must
be taken to ensure that the Jacobi polynomial representation
is also valid.
r
> -m
into
In fact, it is necessary to break the range
the
subranges -m
the requirement.
A different
corresponding
each
strated.
(2-47)
to
Substituting
< r < m and r > m to satisfy
Jacobi
subrange.
eqs.
polynomial
This
(2-59)
will
and
leads to, after some simplification,
65
is
now
(2-60)
required
be
demon-
into
eq.
-
+ p,2 +
(
m,r)
U 2,2
(P,
2
+ q'
2
) (r-I'm)/2
61
(Y)
, p(r-m,r+m)
n-r
a
(2-80)
The
validity
determined
by
of
the
the
Jacobi
polynomial
restriction
satisfy the constraints
that
the
in
eq.
two
(2-80)
is
superscripts
6
[16]
r - m > -1
(2-81)
+ m > -1
(2-82)
r
a
6]
Since r and m can only be
integers, the constraint relations
may be rewritten as
r - m > 0
(2-83)
r + m > 0
(2-84)
a
66
40
Only
r
>
eqs.
(2-83)
m
will
and
satisfy
the
(2-84).
As
valid
over
the
range
(2-65)
into
eq.
(2-47) yields
S (mr)
(p',qI)
r
a
> m.
inequality
result,
eq.
Substituting
m-r (1+p, 2+q
(. 1 )
(' '
(p' I '+jg' )
2
relations
(2-80)
eqs.
)-r (P 2 +q
)-m
2
is
in
only
(2-80) and
) (1-I')m/2 .
(r-m,r+m)
(
r > m
)
(2-85 )
Examination of the direct and retrograde cases of eq.
(2-85)
allows the more compact expression
S mr)(pq,)
(_.fl m-r
=r
+
(
(P
r
> m
67
2
(1 + p' 2 + q 1 )-r
j'I)r-I'm
()
P(r-mr+m)
n-r
(2-86)
4
a form
Finding
range,
-m
< m, is
< r
(2-47)
of eq.
sub-
remaining
the
by using
accomplished
the hypergeometric
formation of
over
a
function, [16],
linear
trans-
viz.,
c-a-b
2
F 1 (a,b;c;z)
=
(1
-
a
2 F 1 (c-a,c-b;c;z)
z)
(2-87)
After
eq.
matching
the
parameters
in
4
eq.
(2-47)
with
those
a
of
(2-87), one obtains
4
2
F 1 (r-n,n+r+1;1+m+r;cos2
=
(sin2 T)m-r
2
6
T
F 1 (1+m+n,m-n;1+m+r;cos2
T
6
(2-88)
61
Recalling
function
that
may
be
the
first
two
interchanged
arguments
and
of
applying
a
eq.
hypergeometric
(2-68),
leads
to
6
68
09
(r~m~-m)((n+m)
n+r
p(rmrm_
n-r
(n+r)
n-m
m-r P (r+m,m-r)
n-m
(sin2
(2-89)
Finally,
substituting
Jacobi polynomial
eq.
identity of eq.
(-)r-m
P(r-mr+m)(y)
n-r
(2-59)
for
sinT
(2-76), eq.
(n+m)!(n-m)!
(n+r)!(n-r)!
and
using
the
(2-89) becomes
(1 +p12 +q1 2 )r-m
(p12 +q 12 )(1+I')(m-r)/ 2 p(m-r,r+m) (
n-m
(2-90)
Replacing
P nrm,r+m) (Y)
hand
of
side
eq.
(2-90)
in
yields
plification,
69
eq.
the
(2-86)
result,
by
after
the
some
right
sim-
a
S (m,r)
2n
2
r (n+m)!(n-m)! (1+p12 +q1
(n+r) !(n-r) !g
()
,)
m
,
,-q
m-I'r P(m-r,r+m)()
,P
(p-jq
)fl
Prn-m
0
-m < r < m
A
collection
of
S (mr) (p',q')
the
final
(2-91)
expressions
for
the
function
is given in Table 2-1.
hand sides of the
The right
Lagrangian VOP equations
contain partial derivatives of the disturbing potential with
respect to the
satellite
tial
the
inertially referred p and q
orbit.
variables
The
is
Earth's gravity potential
already
p and q elements.
elements
formulated
By contrast,
in
in
directly
of
the
equinocterms
of
an inclination function
that depends on the relative quantities p' and q' was intro-
duced when the third body disturbing potential
with
respect
(p',q')
set
of
to
the
satellite
frame.
The
must be related to the element pair
intermediate
so
quantities,
that
the
was derived
element
pair
(p,q) through a
dynamical
par-
70
40
Table 2-1. Form of the function S(m,
S '
2n
- r p(m - r, - m - r)
,
jq,)I'm
n+r
+ p,2 + 2)r (p
0 (p', q')
(p, q').
r < -m
S
2n
S
2n
, )r (n + m)( (n - m)!
(n + r)! (n - r)!
((m,
r) (p',
'( (p', q')
=
(P)r(m-
1+
p,
2
2
+ q, )-
(p,-
j qI)m
- I'r ,(m - r, r + m)
n - m
2
r( 1 +p,2 + q, )-r (p'+ jq' ,')r - l'm p(r - m, r + m)
n -r
r
71
> m
that
recognizing
under
may
a
a
change of
appear
taken.
The
product
is
be
can
derivatives
tial
dot
coordinates
The
different.
problem
is
numerically
although
direction
the
invariant
form
analytical
cosines
of
by
solved
the
third
body orbital plane unit normal with respect to the satellite
6
reference frame are formally represented by the dot products
^
a
f
A
A
=
g
- w'
(2-93)
=
w
- w'
(2-94)
y
The
right
^ 14
=
hand
aw
sides
(2-92)
of eq.
(2-92)
through
(2-94)
form
the
third column of the dot product rotation matrix that relates
the
third
verified
body
by
to the
frame
looking
at eq.
its counterpart in eq.
a
=
frame.
satellite
(2-72).
Matching
can
be
that column
to
This
(2-53) leads to
2p'
(2-95)
1+p' +q'
72
4
-2q
1+p'
(2-96)
+q'
2
(1-p,
_q, 2)11
,2
1+p' 2
Eqs.
(2-97)
(2-95) through (2-97) are the result of using satellite
frame coordinates to compute the direction cosines.
trast,
if
inertial
coordinates
are
used
to
By con-
compute
the
direction cosine dot products, then the elements p and q are
introduced naturally
lite
frame.
Hence,
(p',q')
space
body
potential
would
allow
through
and
the
a,
8
the unit
and
(p,q) space.
inclination
dynamical
y
vectors of the satel-
form
the
bridge
between
Thus, transforming.the third
function
partial
to
direction
derivatives
to
cosines
be
taken
through the chain rules,
as(m,r)
2n
ap
_
+
as(mr)
2n -
9a
as(mr)
2n
ay(-8
3a
ap
as(mr)
+ __2n
a0
5p
+
ay
(2-9 8)
73
a
and
a
as(m,r)
2n
as(m,r)
3a
2n.
Da
_
aq
+
as (m, r)
2n
aq
aa
01
as(m,r)
+
2n
37
9q
09
(2-99)
The transformation from p'
manipulating eqs.
and q'
to a,
(2-95) to (2-97).
8, and
Eq.
y is
made by
(2-97) leads
to the
61
relation
(1+ p'
2
+ q' 2
2
1+I y
(2-100)
g
61
Using
eq.
(2-100)
in
conjunction
with eqs.
(2-95)
and
(2-96)
produces the set of identities
(P' -
jq'I')
=
'il)
1 + I'-Y
___+
(2-101)
74
01
(p'
=
jq')
-
yields
Table
2-2
jOI')
1 + V' y
through
(2-100)
after
in
used
the
analysis
body
frame
the
which
the
to
cosines.
direction
into Table 2-1
The
and
follows,
the
the
aid
To
relating
should
newly
both
formulae in Table 2-2
matrix
frame
satellite
for
regard
implementation.
numerical
Using
(2-103)
(2-103)
.simplification
the direct and retrograde cases.
are
(2-102)
y
ca
(P I + jq'I')
Substitution of eqs.
'
(a +
1 +I
be
the
third
converted
developed
in
to
identities,
the result is
1+I'y-a 2
M
1
-
+-
- aa
a(y+I')
a(1+I 'y)
UI+Y-s25,
a (1+I'y)
-(+-f)I'y)
y(1+I 'y)
/
(2-104)
75
a
Table 2-2. Form of the functior. S(m, r)
2r
(1 +
l'Y)-I'm
6
(a+ j Al')I'm - r ,(m - r, - m-r)
n+r
s(m, r)
,)r
s(m, r)
2n
, r 2-m (n + m)! (n - m)! (1+ 'Y) 'r
(n +
r ! (n - rl
2n
-m
m - I'r ,(m - r, r + m)
n- m
< r< m
09
s(m,
2n
r)
,,)r
m -mr 2-r (1 + V'y) I'm (a,-
r
Il r-
I'm ,(rn -r- m, r + m)
> m
6-
76
0
09
At present the potential is expressed in terms of the
true longitude.
high
This formulation is particularly useful for
eccentricity
orbital
times
eccentricity
when
essential
changing
mensurable
it
than
is
closed
to
the
is
mean
true
in
and
true
longitude
motion is nonlinear.
collateral effect of
case
disturbing
choice
whose
retains
the
there
are
However,
the
Since
the mean longitude
of
it
longitude,
motions.
a more natural
the
since
form.
mean
satellite
their
derivative
motion,
nance
in
the
orbits,
in
This
choice.
resonance where
time
satellite
X,
can
of
third
body
the
is equal
for
be
body
are
com-
unperturbed
to the
the study of
dependence
an
mean
reso-
on the mean
Using the mean longitude also has the
simplifying
the structure of
the short
periodic recovery functions that are discussed in Chapter 3.
The
mean
transformation
longitude
is
made
from
through
the
true
longitude
the
Hansen
to
expansion
the
which
has the form [18]
n.
(
a
)emL0
t=-
77
n,m
t0
jt(
2-105)
4
where
the
index t
is
an integer
and the modified
Hansen co-
I
efficients,
Yn,m,
t
are
in
polynomials
the
eccentricity
defined by
61
Yn,m
t
=
(k +
a=0
t
Y ,m
t
=
(k -
Newcomb
(2-107)
are
recurrence
2
-nm (h + k2
a
0
X n,m
in
eqs.
(2-106)
constants which are governed
For
a
(+t-m,2
further
discussion
and
by
U1
by simple
of
these
U1
operators see Appendix A.
Multiplying
0
(2-107)
>11m
operators,
relations.
(2-106)
0
>
numerical
(h 2 + k 27
< m
jh)t-m
t
The
Xn,-m
a+m- t,a
jh)m-t I
numerator
and
denominator
of
eq.
(2-44)
(r/a)n produces
01
78
01
n
U'
=Re
r
-S (
r)
=2
n
e jrL,
,0
(
n
r
= K V
(a FrM7-0
m n,m r=-n
m
n,
()n e-jmL
(2-108)
Applying
the
Hansen
expansion
of
eq.
(2-105)
n
VK
M-0
V
m n,m
with
appro-
priate changes yields
n
U'
=
(a
Re { 1
(F
n= 2
-S
(m,r)
(a,6,y)
ejrL'
'
t=-O
t
,-m
n
Y9v
nn,r
r=-n
(k,h) ejtX}
(2-109)
Similarly,
multiplying
(2-109)
(r'/a')n+l
by
numerator
and
using
and
eq.
denominator
(2-105)
with
quired notation changes leads to the final result
79
of
eq.
the
re-
a
n
U'
=Re
a
-
(2)a
n=2
n
I
m=0
Y)
- S(m,r)
2ns=-
t=--
n
,m
Km
r=-n
-n-1,r
s
(k'
n,r
6
h'
0
(2-110)
- Yn,-m (k,h) exp[j(tX + sV')]}
t
Compare
with
this
the
frame which contains
potential
derived
in
the
0
equatorial
the additional inclination function and
corresponding summation, viz.,
0
n
U'
=
Re{ --
.
I
n=2
y
n
n
m=0
r=-n
g *(ms) (p, q
s=-n
Y
- n,r
u=-W
(k,h)
t=--
,r
mnr)
0
y-n-1
u
s(k',h')
exp[j(tX + uX'
(2-111)
80
0
The
asterisk
primed
(*)
represents
parameters
elements
the
of
reference
frame
complex
eq.
(2-111)
body
as
computed
are not
to
be
in
third
and
the
The
of
geometry
third
body
satellite
the
equinoctial
the
in
confused
with
equatorial
the
third
Body Orbital Elements on
Induced Dependence of Third
Satellite
are
The
(2-110).
body elements of eq.
2.2.3
conjugate.
Orbital Elements
primed
the
parameters
perturbing
potential
frame,
was
these
in
body
eq.
(2-110)
describe
However,
orbit.
developed
with
parameters
are
since
respect
not
to
the
the
the
inertially
referred.The next sections discuss the functional dependence
of the third body elements on the satellite elements that is
induced
These
by
the
results
choice
are
of
a
non-inertial
necessary before
eq.
reference
frame.
(2-110) can be used
to develop the satellite equations of motion.
2.2.3.1
The Meaning of h' and k'
The elements h' and k' are formally defined by
k'
=
e'
- f'
(2-112)
81
a
A
h
The
=
body
third
and
be computed
vector can
eccentricity
position
inertial
(2-113)
- f'
e'
(v') 1 ,
and
(r') 1
vectors,
velocity
from the
through the formula
(e')
(-
=
1
The
symbol
vector
- I
magnitudes
reference
frame.
universal
gravitational
The
the
P*
parameter
is
expressed
unnecessary
is
are
they
constant
-
vector
notation
The
since
(r')1
'-I' (v(r')')
~(2-114)
*
( )i indicates that
coordinates.
inertial
7)
-
v'
of
independent
is
the
sum of
and the
of
product
the
in
for
the
the
masses
of the central body and the perturbing body.
The
vectors
in
eqs.
(2-112)
and
(2-113)
must
be
expressed in a common coordinate system for the dot products
to
be
performed.
The
first
two
columns
A
matrix
in
satellite
eq.
(2-104),
give
the
f' and
of
the
rotation
A
g'
unit
vectors
in
0
(S) coordinates, viz.,
82
0
A
(f'T)
[1+I'y-a2
1+I 'y
-a,
-a( y+I' )
(2-115)
(g T)
=
[-ac I',
+y-a2I'
-a(1+I
Y)]
(2-116)
The eccentricity is transformed from inertial coordinates to
satellite coordinates by the operation
I
(e')S
(fT
(gT
=
(e_')I
(WT
(2-117)
Eq.
(2-117)
can be rewritten
83
in
terms
of dot products
as,
(e'
the
Using
=
transpose
(g)
- (e'
(w)
(e
operator
(2-118)
scalar
the
to denote
product
and recognizing that the order of the vectors is immaterial,
eqs.
(2-112)
and
may
(2-113)
be written
in
frame
satellite
coordinates as
k'
=
(f'
h'
=
(g' T)
S
(e')
-S
(2-119)
(e')
(2-120)
and
Finally,
eqs.
substituting eqs.
(2-119) and
and k'.
(2-115),
y = 1,
into
For the special case
the third body frame
with those of the satellite.
and k'
and (2-118)
(2-120) yields the final expressions for h'
They are given in Table 2-3.
of a = a = 0 and
for h'
(2-116)
In
that case,
axes coincide
the expressions
assume the well known forms
84
0
k'
-
h'
=
1
1±I'y {-
+I'y-
'
2
A
ctM[(f),
- a(y + I')
(g)
-
w
SS
e
e
[(w)-
(e')I]}
00
UL
1+I'y
I
Table 2-3.
+ [I' + y -
2
[(g)-
(e')] -
(1 + Iy)[(w)
Functional Form of the Third Body Elements,
h' and k'
4
kA
k'=
(f)1
S (e' )
(2-121)
and
a
0
A
h'
- (e')
=(g)
(2-122)
a
2.2.3.2
The Meaning of
The
mean
'
longitude
of
the
body,
third
',
is
deter-
a
mined from Kepler's equation in equinoctial elements
a
=
where
F'
is
the
F' - k' sin F' + h'
third
body
(2-123)
cos F'
eccentric
longitude
defined
a
through the relations [15]
cos F
=
2
k' + (1 + k' V)
a' /1 -
X'
01
-
h' 2 -
k,
2
(2-124)
0
0
86
sin F'
h'
=
+ (1
h' 2
-
a' /1
Y'
-
h'
a
-k'
(2-125)
The parameter a' has the meaning
(2-126)
=
'
1 + /1 -
h'
2
-
k,2
while X' and Y' are formally represented by
X
=
r'
-
(2-127)
=
r' -
(2-128)
and
87
a
The
procedure
leading
decomposition of
third
body
to
Table
any vector
that
Hence,
orbit.
2-3
by
is
lies
applicable
in
analogy,
the
X'
plane
and
Y'
to the
of
the
may
61
be
computed according to
4
X=
{[1 + Vy - a2
1
[(f)
0(r')
I]
-
0
- (r')I]
a$[(g) 1
-
-
a(y +
')[M) - (r')I]}
0
(2-129)
0
and
Y
1+I'y
+
[I'
- (r')1 ] +
+ y -
a21']
0
-
(1 + I'y)
[w
- (r')
]}.
(2-130)
88
0
40
2.2.4
Steps for the Evaluation of Third Body Quantities
The
following
steps
are
body parameters a, 8, y, h',
1)
used
to
k' and
compute
' in eq.
the
third
(2-110):
Given the position and velocity components of the
third
body
in
(v') 1 ,
compute
vector,
(e') 1 ,
normal
inertial
coordinates,
from
(2-114)
eq.
and
eccentricity
body
third
the
(r')1
and
the
unit
body orbital plane from
to the disturbing
the cross product,
(W)
2)
Given
=
the
construct
(w)1
Unit
p
and
the
unit
[(r') 1
q
(V)I]
X
elements
inertially
vectors
of
(2-131)
of
the
satellite,
referred
the
(f)1 ,
satellite
(g)1 ,
reference
frame using eqs. (2-23) through (2-25).
3)
Compute
the
the
direction
disturbing
respect
to
the
body
satellite
through (2-94).
89
cosines,
unit
a,
a,
orbital
frame
from
and
normal
eqs.
y,
of
with
(2-92)
4)
Compute
the
Table 2-3
third
body
using the results
elements
h'
of steps
and
(1)
k'
from
through
(3).
5)
Compute
the third
body
mean longitude,
X',
from
the expressions presented in Section 2.2.3.2.
90
0
Chapter 3
Isolating Long Term Motion in the Satellite
Dynamical Equations
The
previous
body disturbing
elements.
chapter
potential
the
complex
that
{
Re
=
the
quantity
depend
satellite
precision
form
in non-singular
and
the
of
the
third
equinoctial orbit
slow
directly
into
constants
elements
equinoctial
equations
body.
are
the
(31)
ej(t+
consolidates
disturbing
the
dynamical
potential
*t,s
on
* to'
t
s=-
t=-c
tors
a
The result has the structure
U'
The
developed
A
system
and
fac-
of
both
of
high
obtained
by
Lagrange
Planetary
substituting
equa-
tions.
The
harmonic
content of the high precision satellite
equations
is
determined
angles
+
sX' - as
tX
Contributions
to
the
well
by
as
the
the
satellite
linear
more
motion
combination
slowly
of
varying
stemming
from
fast
$t,s*t,s
are most noticeably a result of variations in the third body
orbital parameters.
Examples are the thirty day oscillation
91
a
in
the
lunar
eccentricity
perigee
advance.
several
classes
The
and
the
8.9
combination
year
tX
+
period
of
lunar
contributes
sX'
frequencies, each of which has
of dynamical
an analogous counterpart in the theory of a satellite moving
under the influence of the Earth's non-spherical gravity potential.
These classes are summarized in Table 3-1.
of
a
satellite
s'olely
to
tion.
In
to
of
equations
the
restrict
mission analysis,
constraints
over
satellite.
On
comparison
In
either
other
the
to
case,
a
numerical
integraprior
a
geometrical
satisfy
lifetime
operational
to
be
the
goal
may
time
span
that
is
of
the
utility
anticipated
knowledge
will
orbit
hand,
height over
a
serves
and
the goal may be to verify
the projected
the
minimum perigee
in
satellite
a
that
launch
of
size
step
of
the
long
term
the
in
information
dispensable
is
motion
illustration
excellent
harmonic
the
of
much
where
case
an
provides
analysis
Mission
the
of
ensure
quite
a
long
satellite.
evolution
of
the satellite orbit is sufficient.
92
6
a
w
0
w
w
4p
w
CONSTRAINTS ON
INDICES IN
tx + sx'
PERIOD OF TRIGONOMETRIC
ARGUMENT, tX + sX'
P
= SATELLITE PERIOD
P' = DISTURBING BODY PERIOD
ZONAL ANALOGS
t # 0, s = 0
P/ItI,
"M-DAILY"
ANALOGS
t
TESSERAL
ANALOGS
t # 0, s
DESIGNATION
L~J
0
=
0, s # 0
0
P'/
s
t = 0, s
Table 3-1.
=
0
IsI,
= 1,2,3,...
s
=
1,2,3,...
P/ It + sP/P'
Itl
SECULAR
iti
=
1,2,3,...
= 1,2,3,...
00
w
w
REMARKS
DEPENDS SOLELY ON
SATELLITE MEAN
MOTION
DEPENDS SOLELY
ON THIRD BODY
MEAN MOTION
RESONANT TERMS
(i.e., t+sP/P'~ 0)
PRODUCE
OSCILLATIONS WITH
LONG PERIODS IN
RELATION TO THE
SATELLITE PERIOD
DOUBLE AVERAGED
TERM
Harmonic Contributions of the Trigonometric Argument tX + sX'
w
4
high
tions
equations
of
X and
X'.
arising
from
Methods
for
precision
rapidly
varying
combina-
smoothing
$t,s
are outside
thesis.
the scope of this
The Generalized Method of Averaging
3.1
The
technique
the
in
use
study
of
the
theory,
context
of
satellite
remove
short
periodic
motion
that
restrict
differential
developed
was
from
by
[4 ],
In
the
is
employed
to
the
equations
of
method
oscillations
an
eliminate
oscillations.
non-linear
is
by Mitropolsky
and extended
Krylov and Bocoliubov [ 3 ],
of
system
formalism
mathematical
The
equations.
to
used
be
a
from
(GMA)
Averaging
can
which
frequencies
dynamical
of
Method
Generalized
asymptotic
for
on
high
the
from
components
frequency
dynamical
based
theory,
averaging
Method of Averaging, that can be used to re-
the Generalized
move
an
derives
chapter
This
I
integration.
mean
elements
expense.
the
short
averaged
the
The resulting
may
be
integrated
of
of
the
numerical
equations of motion
averaged
at
reduced
in
computational
The GMA also allows for the analytical recovery of
periodic
at
the
Hence,
an
variations
orbit generator.
high precision orbital elements
time
size
step
the
averaged
is
output
approximation
of
the
to
the
available at each output
generator
orbit
points
by
evaluating
6
the
94
4
periodic
corrections
depends
on
Chapter
5
the
to
order
the
of
demonstrates
mean
the
how
elements.
asymptotic
this
The
accuracy
averaging
approximation
theory.
is
used
to
initialize the averaged equations of mot-ion.
The Generalized
used
for
the
high
determination of
,
221.
of
fast
satellite
average
is
central
body
algorithms
depend
essential
have
rate
in
the
short
to
on an
the
single
of
area
where
frequencies
an
that
body,
central
the
a
the
examples
to remove
[21
remove
cases
in
arc
orbits
been based
some
found
of
already been
satellite
Researchers
to algorithmic ef f iciency
is
[23 1 .
is an analogous consideration for high altitude
There
moving
satellites
and
averaging
of
operation,
rotation
has
usually
have
However,
perturbations
the
prediction
appropriate.
averaging
on
method
the
variable.
not
additional
precision
of Averaging
low and medium altitude
Efficient
application
Method
under
the
influence
of
the
sun
and
moon
when extensive mission analysis and very long term stability
studies
are
induced
by
step
.
size
attempted.
the
of
Similarly,
Variations
in the
in
orbit
moon moving
numerical
the
its
integration
apparent
motion
to
of
at
satellite
can
restrict
most
the
motion
sun
3.5
the
days
about
[23]
the
Earth can place an upper bound on the step size of around 45
95
a
days.
Any
averaging goperation
that
removes
the
satellite
.
variable
fast
from
the
An
of
motion,
but
ignores
the motion of a third body
caused by
periodic variations
inappropriate.
equations
satellite
the
of
average
effective
is
dynamical equations must be based on eliminating the rapidly
varying
in
members
of
combination
the
where
{#t,s}
set
the
a
is
$t,s
linear
and third body mean longitudes
satellite
given by,
The
an averaging
section derives
following
the
(3-2)
tx + sX'
tos=
$t,s-
frequencies associated with the composite angles,
3.1 .1
An Averaging
Theory
removes
that
Averaging,
of
Method
Generalized
based on
theory,
Under
Moving
for Satellites
the
Influence of a Disturbing Body
Given
eq.
(3-1),
evolution
the
the equations of
the
motion describing
by a third body
perturbed
for a satellite
represented
schematically
as
potential
in
orbital
have the
form [15]
6
da.
00
(a.a
S-
k=1
j$
00
Re
t=-0o s=-co
k 9ak
i
e
= 1,...,5
t
t,s
S
(3-3)
96
61
and
dt
-
n(al)
6
1
-
a
k=1
(a6,ak)
0
00
t=-oo
s=--
Re{
k
*tsejt,sI
(3-4)
where
(aiak)
is
a
Poisson
bracket
and
n(al)
is
the
mean motion.
satellite
The asymptotic formalism of the Generalized Method of
Averaging
Hence,
presence
depends on the
the
high
precision
of
dynamical
a
small
equations
parameter,
are
v.
required
to have the form,
da
d.
=
v F
(a,
{ t
i
(a, ) };a'(a)
to's_
-
=
1,...,5
and
97
_
(3-5)
4
dA_
n(a1 ) + v F 6 (a
=
,
s(a,X)};a'(a))
(3-6)
where
4
a
a
E
vector
of
orbital
elements;
the
slowly
varying
satellite
a
A
aT =
[a,h,k,p,q]
3mean longitude of the satellite
4
a'(a)
vector of satellite
ing
orbit;
t,s
parameters
a'T =
dependent slowly varyof
the
perturbing
body
a
[a' ,h',k',ary]
(a,A)E tX + sX'(a)
a
A'(a)
satellite
dependent
mean
longitude
of
the
third body
0
98
The
dependence
orbital
satellite
Chapter
of
to
2
$t,s
and
on
reflects
elements
develop
a'
the
the
the
disturbing
slowly
decision
varying
potential
in
the
averaging
theory
developed in this chapter can be specialized to the
case of
satellite
non-inertial
The
in
made
frame.
a potential developed in
coordinates by neglecting
inertial
the induced dependencies.
In
two
the
analysis
subsets.
{ag,m}.
{Pj,k}.
One
other
The
The
so
determined
is
that
the
follows,
is
set
the
averaged
set
is
broken
into
rapidly
varying
angles,
slowly
varying
angles,
of
of
equations
members
no
that
of
{$t,s)
motion
of
are
{aIm}
to
be
appear.
Accordingly, the assumed form of the averaged dynamics is
da.
N
v
Ai
i
=
(a,
{pj,k(a,~X)};a'(a))
1,...,5
and
99
(3-7)
4
-
N
) +
n(a
-
1
v A6,p
Vj,k
4
h a'(a))
(a,)
p=
(3-8)
a
where
(-)
the symbol
denotes the mean and
4
y.
(a,X)
j,k --
jX
=
+ kX'
(3-9)
(a)
a
The
existence
of
a
parameter
that
of
Method
Generalized
transformation
near-identity
connects
element
the mean
assumes
Averaging
in
space
the
to
the
small
the
high
a
The transformation equations are
precision element space.
N
VP n
a. +
=
a
(a, {~yX
a
(a,~1) }; a' (a))
P=1
i
where
ni,p
periodic
interval
in
is
a
the
[0,27r].
(3-10)
= 1,...,6
function
that
members
of
is
the
required
set
to
be
{5',m(a,T)}
jointly
on
the
Also,
a
100
68
a
L,m
The
as
yet
objective
The
precision
is
to
Ai,p
of
equations
(3-11)
create expressions
and
is
approach
to
to
Ti,p
in
motion
relating
functions
both
develop
in
the
which
are
the high
of
sides
of
terms
transformation,
identity
near
LX + mA'(a)
=
--
undetermined
known.
The
(a,)
elements.
mean
the
with
conjunction
assumed form of the averaged equations of motion, is used to
the
transform
sides
hand
left
the
of
precision
high
The right hand sides of the osculating equations
equations.
are expanded in a Taylor series about the mean elements with
the
aid of the near identity transformation.
the
transformed
order
relating
all
remove
functional
dependence
form
small
expressions
set
the
on
The
then
equated
to
parameter
Ai,p
functions
Ai,p.
of
are
are
and
ni,p
to
yield
recovery
periodic
produce
to
averaged
then
{~X,m}
on an
to
the
function
then known.
nli,p is
To
the
begin,
differentiated
there
the
These
quantities.
equations
the
in
basis
order
by
expressions
known
dynamical
Both sides of
with
near
respect
identity
to
time.
is no explicit time variation of
orbital
elements
and
that
transformation
It
is
assumed
that
the slow third body
whatever change occurs
the motion of the satellite orbital plane.
101
is
comes from
Accordingly
a
da.
an.
5
N
da.
1t
9a
an.
a
1
_
p=1
an.
5
aa'
-
,p
+
q
q=1
n=1
9an
(Xm) am
+
n
+t~
an
=
a
an.
da
i
n
(xm)
a
at
m
1,...,6
(3-12)
0
The
with
respect
to
of
consequence
potential
of
derivatives
partial
in
the
expressing
is
not
are
the
disturbing
body
third
the
frame which
a coordinate
elements
orbital
satellite
the
parameters
body
disturbing
the
reference
system of the numerical integration.
The
function
of
Ai,p
is
averaged
the
the
assumed
form
eq.
(3-12).
The result
now
introduced
equations
0
by
of
substituting
motion
into
0
is
61
40
40
102
40
da.
N
dt
p=
1
N
VP A.
I
+
lip
p= 1
an.
+
( ,,m)
a
irp
3a I'
i
+
3an
5
an
n=1
aa
+
aa'
an.
5
X
N
9
q=1
a
q
n
r= 1
3an
n, r
+
N
an.
+
( -tm)
X
I
[(xn + mn' )
r
r=1
i
= 1,...,5
and
103
A 6 ,r
(3-13)
4
dX
N
=N
I
n + P=1
=
d
dt
A6,p
v
(,m)
-,m
aan
aX,m
{
vP
5
I
+
(X,m)
_6(,
aaXm
q=1
an
aa'
q
ar=
9a
-n + mn') +
4
+
6,p
,p
N
]
I
vrA n rI
n
[
r=1
4
r
N
[
an
an 6,
an6p+
+
[
n=1
p=1
3a
an
+
5
I
+
v
A]6
41
(3-14)
where
n'
and
(3-14)
viz,
[15]
is
the
disturbing
can be
expressed
body
mean
as a pure
motion.
power
Eqs.
series
(3-13)
in
v,
I
I
104
a
a
da.
an.
N
dt
1
VP
{A.
P=
+
(.
(Xm)
',p
+ri +mn')
+
3a1
p-1
5
+
+
an
I[ I,p-w
n=1
w=1
5
an.
aan
1,p-w
Ba'
q=1
q
I
( Irm)
an.
2
xI'm
3aa
(£,m)
+
an
aa'
+
n
an.
P-1
w=1
+
A6,w} + O(
a.-
i
= 1,...,5
and
105
v
N+ 1
(3-15)
a
d
-
N
+
dt
5
p-1
+
w1j
5
an
+
mn')
P.3"M
g
P
+
q
+
am
6,p-w
Tm6
Dan
( A~m) am
Dan
aa'
q=1
(Zn+
(tXm)
a
6,p-w
1
+
{A 6,p1+
an6
w=1
n1-1
v
a
_q ]- An,w
+
an
a
an
P-1
_6,p-w
a
+
w=1
(Xm)
A6,w} + O(vN+1)
a
(3-16)
6
The
summation on w is
not
performed
when
p
=
1
so
that
no
contribution is made at first order.
The
of
motion
right hand
are
Substituting
now
the
hand sides of eqs.
sides of the high precision equations
expressed
in
transformation
(3-5) and
terms
equations
(3-6) yields
106
of
mean
elements.
into
the
61
right
is
a
a
da
-i
dt
N
=
vF[
a+
VP
I
p=
'n
1
N
N
I
(
ts
+
X
VP
P= 1
n
+
I
p=1
VP
n6,p
'
N
a
(a +
X
VP
n
~-P
P= 1
)]
i
=
1,...,.5
(3-17)
and
dAX
N
= n(a
+
v
I
1
P=
n
)
+
N
N
N
+ vF[a+
V
p ,
t,s
VP
-. +
P=
a
,~\+
=
vp6,p
P=
N
a'
(a +
P= 1
vP n
)]
(3-18)
107
4
where
.p
is
a
vector
of
periodic
recovery
functions
asso-
4
with
ciated
the
in
Expanding
elements.
satellite
slow
a
Taylor series about the mean elements,
a
da.
=
v F
{its
(,
a'(a)
+X)
5
+ Vn=
n=1
+
{
n!
5
Da'
1
q=1
aa
k
N
pI
k= 1 p= 1
(a,,
VP n
k,p
(Dak
+
t,s
(t
(t,s) aak
a
ts
q
a
+
N
I
1
p=
t
vp n 6
s)
a
0t Fs
}n
F.
1
a = a
to's
i
= 1,...,5
ts
t"
(3-19)
a
6
and
6
6
108
a
=X
dt
+ v F6
Nn
0
=
n=1
(
t
a))
),)}
5
1
}n
-3a
n(a)1
n=1
N
k=1
5
a'
I
q= 1
aa
+
a
a
+
at-
n!
+v
+
a
1,p
n
(t,s)
k
p=1
Dak
4"s
N
aq
k
) +
v2
t s6,p t
p=1
4trs
(ts)
}n
F6
(3-20)
The right hand sides of eqs.
osculating
hand
sides
follows
Equating
element
of
that
eqs.
these
them
and
rates.
(3-15)
(3-19) and
The
same
and
(3-16).
express'ions must
matching
is
(3-20) represent the
true
the
right
Consequently,
be equal
coefficients
of
of
to each
like
it
other.
powers
of
the small parameter leads to equations that relate the A and
n- functions
to
the
known
F
function
basis.
109
on
an
order
by
order
4
In many instances, the first order approximation produces
results
of
complexity.
striking
A first
accuracy
order
with
formulation
minimal
analytical
of the averaged
equa-
tions of motion and periodic recovery functions will now be
developed
to provide
the basis
for the numerical demonstra-
tion.
3.1.2
A First Order Averaging Theory
Using the methods described in the preceding section,
the
set
of
first
order
equations
for
Ai,p
and
ni,p
are
found to be
Fi(a,
AA.
+
{~t, 5 Qa.,,X)}; a'
_,
J1jl
1
(Zn + mn')
(X ,m)
-,X
_'
Ia
(a))
I
6
a' (')
+
(a, {~lm(a,X)};
i = 1,...,5
a'
(a))
(3-21)
110
a
and, recognizing that n(a 1 ) = 1
3
n
a1
+ F6
SA
+
n-
,1
(a,
- -X)}; a' (a) +
j,k(afX) }; as
(Xn + mn')
(X,m)
I-
-3/2
al
}; a' (a))
- '
6
1/2
an 6 1
_ '
=
(a))
+
(a, {a Xm (a,
Ax)}; a'(a))
(3-22)
111
The
by
desired expressions
averaging
successively
of
member
on
dependence
all
the A
the
are
determined
and
(3-22)
over
interval
[0,27r]
to
varying
rapidly
the
functions
(3-21)
eqs.
{a t ,m} on
set
the
for
remove
first
At
angles.
each
order, the slow elements of the satellite and the third body
held
are
of
independence
the
over
fixed
is
angles
the
interval
averaging
and
The
assumed.
mutual
periodic
recovery functions are required to be jointly 2w periodic in
angles
the
of
set
the
the
are
set
Hence
{at,m}.
then the following properties hold
f
11
f
0
2w
2
da
1
[-
2
2
]
1,
f
0
...
0
da
]
,m
n-1,mn-i
{
elements
-
inrmn'
of
[151
I
(in
(Lm)
daX
the
{a1,m1
by
denoted
if
,m
-
+ mn
,
Da3 m
0
n mn
i = 1,... ,6
(3-23)
and
112
0
1
[ 2xi
f
2w
0
...
2w
2w
2w
1
f
...
0
f
1[27r
0
daxnmn
da n-1mn-1
3
n
2
-
ni1i }da
1
tam
] -
=0
(3-24)
The right hand sides of the averaged equations of motion are
then seen to be
A i ,1(a2, {ii. k( a,,X)}I;
27t
- 7r
- da
a' (a) )
0
X ,m1
2w
2w
[L
-.
1[
da Xn-1,Im
F
(a,{t,
,
};a' (a) ) -
0
0
]..
f
n- 1
i
] dan ,mn
= 1,...,6
113
(3-25)
4
Eq.
(3-25)
shows
that at first
order,
the
averaged equations
of motion are obtained by substituting mean quantities
the
right
hand
of
sides
equations, followed
by a
the
high
removal of
precision
into
satellite
terms depending
all
on
any of the rapidly rotating phases.
Now
recovery
that
the
functions
Ai,i
are
functions
known,
the
partial
=l F
-A9
by
given
are
the
periodic
differential
equations,
(in
+ mn')
(Z , m)
aM
X
i
= 1,...,5
(3-26)
and.
(
Zm)
+mn') 6,1
aa
,m
(
6
F
A
,1
3
2
n
a
1 1
(3-27)
114
a
If
the
notation
sides
hand
right
[I
] ( ,m)
of
eqs.
is used
and
(3-26)
to denote
(3-27)
a term in the
that
depends
on
72 ,m,
(in + mn')
=
aa~,m
(,m)
i
(Zm)
[Fi - A il]
(Lm)
= 1,...,5
(3-28)
and
6,1
(Yn + mn')
(Z,m)
3~IM
al
(Im)
(3-29)
For
a
(3-29)
given
pair
(Z*,m*) ,
the
solutions
are
115
to eqs.
(3-28)
and
4
T).1-
~
Ti,1
f
- m
F -A ,
Z*n+m*n'
]
d+
+ f
4
i = 1,...,5
(3-30)
4
and
4
1
161
f
Z*n+m*n'
6'
F
[(F
-
3
7
A 6 1)
2
'
n
~~
a
l 1]
'
(1*m*)
4
- ~+
(3-31)
where
of
f({at,m}')
fast
forward
angles
to show
and
g({a,,m}')
that
excludes
that
each of which depends
these
are
of
functions
-t*,m*.
It
functions must be a
is
the
set
straight-
sum of terms
on exactly one of the angles
116
4
in
the
I01
set
{[a,ml'
tical
to
(3-31).
be
Further,
that
of
the
each
term
leading
has
term
a
form that
of
eqs.
is
(3-30)
idenand
This shows that the periodic recovery functions may
separated
into
a sum of
functions,
the form
117
'ni,1,
,m,
that
have
4
Chapter 4
Mathematical Structure of a Dynamical Third
Body Model for the Long Term Prediction of Satellite
Orbits Using Numerical Methods
The
goal
numerical
orbits
tool
for
to
construct
this
thesis
for
the
long
term
third
body
perturbation
which
3
and
is
of
furnish
Chapters
2
chapter
synthesizes
the
analytical
these
prediction algorithm that
prediction
of
is
flexible
satellite
significant.
components.
components
is
a
into
an
This
orbit
numerically efficient and
free
from singularities.
I
In
Chapter
It
derived.
noctial
the
is
orbital
satellite
2,
the
formulated
elements
dynamical
potential
is
expressed
governed
by
recurrence
eliminate
the
calculations.
respect
frame
in
terms
of
non-singular
in order to avoid
equations.
in
terms
relations.
inefficiency
equi-
singularities in
Wherever
of
potential was
possible,
functions
This
was
with
associated
which
done
I
the
are
to
explicit
Furthermore, the potential was developed with
to the
choice
third body disturbing
This reference
satellite reference frame.
eliminates
a
summation
and
the
corresponding
118
a
function.
inclination
the
third body elements
on
the
satellite
side-effect of
A
h',
k',
orientation
and
this choice
X' acquire
elements
p
a
is that
dependence
and
q.
The
potential is restated here in the functional form,
U' = Re{
I
t=-
sP1F~
-e
ts
1
s=-I
[a;
a',h'(p,q),k'(p,q),a,O,y]
}
(4-1)
where
Re{ }
real part of a complex quantity
aT
{a,h,k,p,q}
tS[p,q, X]
tX + sX'(pq)
J
119
4
and *t,s has the structure
h'(p,q),
a',
an= 2
* Ysn-1,r
5
n
nI
r=0
-
[k'(pq),
a,O,y]
k'(p,q),
n
L
KV
r=-n
h'(pq)]
Yn,-m
t
=
Vm
n ,r
S (m, r)
2n
(~,Y
(k,h)
(4-2)
The definitions of the various blocks in eq. (4-2) are found
6
in Chapter 2.
The high precision satellite equations for third body
perturbations
are
potential of eq.
obtained
by
substituting
the
1
disturbing
(4-1) into the expressions,
4
6
da
=k-
1
(al,ak)
k=1
Ta~
k ak
6
i
= 1,...,5
(4-3)
a
120
a
and
6
dX
n -
t
(X,ak)
x
k=1
(4-4)
- k
where
[a
,,a2 ,a 3 ,a4 ,a s a 61
E
[a,h,k,p,q,XI
third
U's
body
disturbing
potential
A
table
orbital
of
the
E
Poisson bracket
n
=
satellite mean motion
non-zero
elements is
in Table 4-1.
(ai ,ak)
Poisson
found in
brackets
Reference
in
16 and is
The result of the substitution is
121
equinoctial
reproduced
4
I
Table 4-1. Nonzero Poisson brackets of the equinoctial elements.
(a, )
=
(X, h)
=
(, k)
=
-ks
(, p)
=
(X, q)
(h, k)
-2as 1
(h, p)
=
-kps5
(h, q)
=
-kqs 5
(k, p)
=
hps5
-Ps 5
(k, q)
=
hqs 5
=
-qs
(p, q)
=-
=
-sl S3
4
5
I
4
s2s5
where
na2
s2=
s3
=
s4
-
S4
s5
41
1
Si
1+p 2 +q
2
1 - h 2 -k
2
S
sg3
5l S
3
2 S----2s3
122
6
a.
i16
+
=
ni.
6
-
Re{
CO
I
5
k=1
t=-co0
(ai,') it t,s
5
t=-00 s=-00
i =
6
t's
+
_
e, t,s
(a ,p) Pt, s + (aiq) Qt,s
-Re{
where
(a.,a ) -2
1
k)a
k
e
ts
}
1,.,6
(4-5)
i6 is the Kronecker delta function defined by
i
0
6
i6
=
=
1,...,5
(4-6)
;
i = 6
Also,
Ptrs
* t,s
~9h'
ak'
+
a3k'
+
j * t
s
t
(4-7)
and
123
,p
4
ah,
*ts
ah'-
-q-
+
3*t,s
-3k'
Wk'
31 - 3q
j.
41
3q
t~s
(4-8)
a
The
form of eq.
(4-5) hinges on the auxiliary expressions,
61
*t 'ws
*t Is
3a
+
ast
is
3
+
3Y
ths
ay
0
(4-9)
a
and
t
s
3*t s
3q-9
Da
q
*t's
+
a~
36
-- a
1t
Y
s
-3Y
aq
0
(4-10)
The
partial
Table 2-3.
derivatives
3(h',k')/D(p,q)
are
obtained
from
The specific results are given in Section 4.3.
124
0
0
The
operators
3/3a,
the
explicit
appearance
inclination
Pt,s
orbital
function
of
the
the
frame
through
the
on
*t,s
solely
direction
a,y).
The
third
body
elements
orientation
When the disturbing
through
cosines
(a,
S(mr)2 n
connect
plane.
act
9/3y
Qt,s
and
inertial
a/Da,
of
the
potential
in
the
functions
the
to
satellite
is developed
with respect to an inertial frame, Pt,s and Qt,s vanish.
In
Chapter
developed,
of
for
3, a
the
differential
that
with
equations
much
equations.
are
fast angles.
larger
step
averaging
of frequencies
that
the averaged equations
a
order
elimination
combinations of two
is
first
theory
from a system
produced
by
can be numerically integrated
size
than
the
high
precision
Initialization of the averaged equations is made
theory
that allows
recovery of the short periodics
at the
output
by
linear
The benefit of the theory
of the
possible
was
the
feature
for the
times of the
numerical integration.
The
results
high
of
applying
precision
mathematical
explored.
of
remainder
first
order
satellite
form
of
Finally,
this
the
the
chapter
averaging
equations
partial
recursive
prediction algorithm is detailed.
125
in
discusses
the
methods
the
eq.
(4-5).
derivatives
nature
of
to
The
is
fully
the
orbit
4
4.1
Order
First
Averaged
Equations
Motion
of
for
Third
a
Body Perturbation
The first order averaging
3
transforms
a
system
of
theory developed
high precision
in
Chapter
differential
equa-
a
tions of the form,
da.
=
v F
(a,
i
{$,s (a,X); a' (a))
= 1,...,5
(4-11)
61
and
dX
=
n + v F
' It
dta
s(a, A)};
a'
(a))
(a))'
(4-12)
a
into
a
system of
averaged
equations
of motion
in mean
ele-
6
ments having the form
da.
=
V A
{ij,k (a,
(
i
=
1
126
,
' (a))
(4-13)
and
dX
=
-
dt6,1n + vA
~-
jk k
11j,
'
-
-
(a,
-
a'
(a))
(4-14)
where ni is the mean mean motion defined by
n
=
n(a)
1/2 -
=
and
-3/2
(4-1 5)
{Pj,k} is a set of slowly varying mean angles.
The
transformation
hand sides of eqs.
of eq.
(4-5).
(4-11)
comparing
the
is
begun
by
and
(4-12)
to the right hand
right
sides
This leads to a representation for the high
precision forcing
function, viz.,
127
4
v F
S 5a
=
-
Re{
s=-0
t=-0
+ (a., X)
-
Re{
jt $
I
[
X
-
4
t,s +
k
_
t,s
s I
(a ,ak
k= 1
-
[(a ,p) Pt,s
+
I
(ai,q)
Qt,s
t=-- s=--
]
te
4
i = 1,...,6
(4-16)
4
Mean elements are then substituted directly in place of high
precision satellite elements in eq.
0
v
FI
=
-
00
5
Re{[
(a
t=-ew s=-co
. (a , A) Jt$t,s]e
- Re{
a
(4-16) to yield
ikas
k=1
t
+
Baak
t s
6
[(a.,p) P
1=-s-
i
=1
128
a
t,s
,6
+ (a.,q) Q
0 t,s
i
]
e
t,s
0
(4-17)
6
01
the
of
the
interval
and
satellite
the
averaging
{Ft,s}
taken
stipulation
of
In
an
to
set
{ct,s}
on
for
the
slow
held
the
the
set
in
The
independent.
in
that
terms
qt*,s*
are
unaffected
average over a cycle of that angle.
Hence, one
of
over
ensures
angle
mean
elements
fixed
angles
the
mutually
be
to
the
are
body
addition,
independence
contain
expressions
order,
first
disturbing
interval.
are
not
produces
At
vAi,3.
functions,
do
[0,27]
the
of
respect
with
(4-17)
members
varying
rapidly
each
eq.
averaging
Successively
vFi
by
that
an
is concerned
with evaluating integrals of the form
I
-
f
-
2w
0
+
(a,)j
1
5ts
Re{[
(a1 ,ak)
k=1
*t*s*
0
2
t*s*
3ak
}
ts
jtt*
__
Re([(a ,p)Pt*,s* +
i
=
dcpt*,s*
(a1 ,q)Qt*s*
1,...,6
129
s*
-
dt*s*
(4-18)
According
integral
also
some
to
the
of
eq.
rules
of
(4-18)
vanishes.
The
second
first
is
integral
identically zero, although the reasons for this deserve
consideration.
definitions
for
f2 Tr
1
Eqs.
#~t,s
into eq.
definitions
(4-7)
and
+
_
S,
integrals
and
can
*t*,s*-
be
____*
t
*
Differentiating
of
if
t*s*
',}
-
dct*,s*
returning
this
(4-19)
partial
the
are
these
+
ap
e
the
the form,
1,...,6
=
9~t*,s*/aq
by
_s
]
vanish
shown
provide
Substitution
t*,s*
i
These
(4-8)
(4-18) produces integrals oi
Re{[(a,p)j
(aq)j -
and
Ut's-
0
This
averaging, the
first order
independent
to
relation
the
derivatives
of
$t*,s*.
expression
with
respect
for
to
-p
and q leads to,
130
01
d
-
*
[t*X + s*X'(p,q)]
=
s*
4-)ap
(4-20)
and
_'__
=_-
q
a
[t*X +
=
s*X' (p,q)]
-
s*
aq
3q
(4-21)
The mean
as the
longitude of the third
sum of a dynamical
A' (t,p,q)
=
'
body,
A',
may be thought
term and a geometrical
ar
(t -
DYNAMICAL
t 0) +
X'
0
term,
of
viz.,
(p'q)
GEOMETRICAL
(4-22)
where
131
4
t
E
current time
to
m
reference time
reference -alue of the mean longitude
X'O
The
dynamical
as
X' changes
third
the
is reference frame independent
It
in its orbit.
body moves
how
term describes
since it depends only on the elapsed time from a reference
and
third
the
on
semi-major
body
X' as
resulting
orbital
vectors,
plane map
f'
and
in
in
in
the
invariant
the
the variation of
satellite
the orientation of
a rotation
into
g',
term reflects
changes
from
Variations
elements.
plane
and
p
the
q
satellite
body unit
of
the
third
of
the
disturbing
body
The rotation changes the datum with respect to which
orbit.
X'0
the geometrical
is
which
At a point in the third
under coordinate transformations.
body orbit,
axis
is
measured,
causing
X'
to
vary
in
turn.
Differentation of eq. (4-22) with respect to p and q yields
the intuitively satisfying results
0
(4-23)
132
0
and
a'
ax10
aq
aq
Substituting
eqs.
(4-23)
(4-24)
and
(4-24)
into
eqs.
(4-20)
and
(4-21) leads to the desired relations
t*,*
ap
-s*
ax' 0
(4-25)
t*,*
=s*
ax' 0
(4-26)
and
__
__
_
133
I
t*,s*/3p
Hence,
angle
by
change
At
measured.
'0 /a4
and
that
in
the
first
order,
the int,.egral of eq.
preceding
for
procedure
averaged
{Ft,s}
that
motion
to
map
be
into
the
the
for
on
from
which
'
is
Da' 0 /p
derivatives
averaging
the
caused
simply
leads
to
vAi,1
functions
of
an
motion.
retained
in
interval,
extremely
of
Those
the
members
so
of
vAi,1.
At
first
in
equations
index
set
s
eq.
and
those index pairs gives
first
angles
the
t
simple
the
averaged
on
summations
the
produce only
expressions
the
specific
Restricting
{(ts)}.
(4-17) to
result
equations
are
line
partial
over
depend
is
$t*,s*
of
the
not
(4-19) vanishes.
obtaining
order
do
reference
constant
are
The
of
3 t*,s*/Dq
variation
The
*t*,s*-
the
and
in
the defining
terms
order,
depending on other indices cannot contribute to the averaged
equations of motion.
0
A
philosophy
for
retaining
index
pairs
in
the
averaged equations of motion will now be developed.
4.1.1
Criteria
for
Retaining
Terms
in
the
Averaged
Equa-
tions of Motion
Averaging
ing
dynamical
is
merely
frequencies
a
formal
from
a
technique
system
of
for
eliminat-
differential
134
6
to
is
It
equations.
judiciously
retained
in
third
period
double
averaged
problem,
dynamics in
of
that
motion,
can be
entails
are
so
be
to
that
the
realized.
identifying
the
For
long
(4-17).
terms.
linear
this
analyst
the mission
frequencies
averaging
kernel
dynamical
averaged
trivial
eq.
task of
equations
advantages of
body
The
the
those
specify
the
computational
the
therefore
of
eq.
(4-17)
consists
of
the
These are terms which depend on
combination
of
the
satellite
the
and disturbing
body mean longitudes, viz.,
If
no further
represent
as
transformations
the
irreducible
in
embodied
j0,0.
step
admissible
Accordingly,
(4-27)
0
t=0,s=0
size
additional
are made,
minimum
Hence,
for
terms
of harmonic
they
a
then these dynamics
determine
the
maximum
integration.
numerical
considered
information,
for
inclusion
in
the averaged equations of motion should have periods that do
not
restrict
dictated
satisfying
by
the
the
such
integration
double
step
size
averaged
a requirement
135
could
to
less
dynamics.
be
than
that
Candidates
introduced
in
the
4
the
when
case
the
and
satellite
third
commensurable in their mean motions.
N
where
and
geometrical
repeat
long
This phenomenon, known
N
N'
are
(4-28)
N'
of
linear combinations of
the
X and
third
to
disturbing
satellite
in the
that
states
(4-28)
satellite
of
revolutions
variations
periodic
Eq.
integers.
relationship
every
nearly
may be expressed mathematically as
as third body resonance,
n
are
body
motion
will
body
body.
the
Since
occur
6
for
X' that are slowly varying, one
obtains the auxiliary condition
S
If
two
body
=
relations
are
(4-29)
0
tx + sx'
used,
then
eq.
(4-29)
can
be
0
rewritten
in
terms of the mean motions of the
satellite and
third body, viz.,
$t's=
tn + sn'
~
136
0
(4-30)
The
union
of
eq.
(4-30)
eq.
(4-28) leads to the constraint relation
tN + sN'
with
the
statement
of
resonance
0
=
in
(4-31)
which provides a filter for the values of t and s that give
rise
to
included
eq.
long
in
the
(4-31)
averaging
terms
period
averaged
in
eq.
equations
(4-17)
of motion.
is a generalized constraint that
a
as
special
Hence
case.
are found by restricting the indices
satisfy eq.
motion are
4.2
Given
(4-31).
this,
the
seen to have the form found
be
Notice that
includes double
functions
vAi,1
(4-17) so as to
in eq.
the
could
that
averaged
equations
of
in Table 4-2.
Mathematical Form of the Periodic Recovery Functions
Those terms
in
eq.
tribute periods to the
unecessarily
tion.
(4-17)
for which tN + sN'
satellite motion
restrict the
step
that
are
* 0 condeemed to
size of a numerical
integra-
In accordance with the methods detailed in Chapter 3,
the amplitudes of these periodic variations can be recovered
as
a
orbit
function
of
generator
the
mean
through
element
the
integrals
137
output
evaluation
of
of
the
the
averaged
indefinite
I
6
Table 4-2. Form of the first order averaged equations of motion for third body perturbation.
5
(aii, ak) A
Re
S5i6-
k=1
5) PO,0 + (ai
00 + (
k
0,0
I
DOUBLE AVERAGED TERMS
cc
- Re
cc
5
(ail ak) A
E2E
t=-o=
s=-0o
k=1
ak
5) rP
~ +(ail X)it t s+(ail
5
tN + sN' = 0
t # 0, s, 0
jot3,s
+ (i;,ii
(4-32)
5,s
RESONANT TERMS
a40 ,ah'
0
0,
h'
aji
aPo ak'.
ak' I a5 '
0
'
0,0 ah'
ah'
ai
q
ak'
g
4
61
138
61
00
V,1
t=-o
0
v0 i ,1,t,s
s=-
tN+SN'#0
1
f Re{[
-
t=-
S=--
tn+sn'
5
I
k=1
-
k
t
(a ,a
s
+
aa k
tN+ sN '#0
+
(.,~\)
jt
~p
+
(a.,
+ (ai .,)
)P
i = 1,...,5
and
139
] e
Its}d~ts
(4-33)
4
=
v n6,1l
0
o
v n6,1,t,s
t=-co s=-m
4
tN+sN' *0
co
s1n
s=co
t=-"
a
5
f [Re{[
tn+sn'
0
I (X
k=1
,ak)
aak
t, s
+
tN+sN' #0
a
+
(
,P)Ptf
+
(,~q)
5 ,,
S + 3.
] e t
a
V
nl 1 ,1,t,,s]
ts
a
(4-34)
a
where
V,n
Ah
Ak
0
1
v21
vn 3,1
Ap
vn
Aq
vn5, 1
Ax
vn6 ,
1
(4-35)
01
1
61
and
a
140
a
Aa.
1
Notice
that
=
a.
1
eq.
-
i = 1,...,6
a.
1
(4-34)
has
taken
into
Poisson bracket (T,T) is equal to zero.
notation
can
be
achieved
by
complex quantities
141
defining
(4-36)
account
A
that
the
simplification of
the
slowly
varying
a
and
(4-33)
Eqs.
(4-34)
may then be rewritten
as
4
1
V nl,
t=-w
f
Re(Miets.t,s
dt
s
s=-w tn+sn
4
tN+sN' *0
i = 1,...,5
(4-39)
and
4
V n
6,1
f
=
t=-cc s=-c0
[ReM
tn+sn,
s
e
6
,t,s
a
tN+sN' #0
3
n
2
-
a
v
48
dit,s
1,1,t,s
(4-40)
Expressions
varying
for
variations
the periodic
satellite
elements
forward integration of eq.
are
obtained
(4-39).
142
in
the
by
five
the
slowly
straight-
01
The result is:
61
a
vn.
=
-
i,1
t=-oo
Re{j M.
s=-oo
t1
tn+sni
1,
t,s
e
tjs
tN+ SN' #0
i = 1,...,5
The periodic
correction
for the satellite mean longitude is
determined in two steps.
V
n
=
(4-41)
From eq.
-
trn+sn,
(4-41), one finds,
Re{j Mts
,'
e
ftFs
(4-42)
Substituting eq.
(4-42) into eq. (4-40) and
143
integrating,
4
v
fl 1
6,1
3
2
=
n
a
t01
CO
Re{M
e
ts
_
g
(tn+sn')
tN+ sN' *0
4
Retj
t=-0
s=-O
M 6 ,ts e
t, s
tn+sn'
tN+sN' *0
a
(4-43)
a
Eqs..
and
(4-41)
into its
(4-43)
can be rewritten by
breaking Mi,t,s
real and imaginary parts, viz.,
a
M.
1, t,s
5
=
)
Re(M.
+
j
Im(Mi.
(4-44)
)
41
and by recalling
e jt, s
that
-
cos *t's +
j
sin
(4-45)
t s
4
After
performing
the
complex
multiplications
the real part, one obtains,
and
extracting
4
144
4
00
v
fnl
Im(M.,,
~
-
0
{
={
1
t=- s=-w
Im(M.
)
it~s
tn+sn'
Cos
+
$
tN+ sN'1#0
Re (M. ItI)
_
+
sin
tn+sn'
i
(4-46)
=5
and
3
v n6,1
t=-
{0
s =S
Re(M
n
a
1,t,s
)
tn+sn'
+ Im(M)6t s
','
tn + sn'
tN+sN'#0
3
Co
-
+
-
n
a
Im(M
_ -1,t,s
tn+sn'
)
tn + sn
+ Re(M
sin
t,s
(4-47)
145
4
in
formulated
of
problem
problems
of
a
into
functions,
to
those
to simplify
the
found
eqs.
Accordingly,
forms of which
in
the
central
are
shared
the
the
as
(4-46)
and
the
by
the
and
both
(4-47)
may
be
short
periodic
mathematically
identical
body
of
problem.
I
computational
includes
that
classes
distinct
longitude
structure
package
software
three
mean
common
The
to
perturbations,
body
the satellite
angle.
analogous
is
central
can be exploited
perturbations.
cast
variables,
fast
terms of
hour
Greenwich
flow
two
as
perturbations,
body
third
non-spherical
in
formulated
,wo
of
problem
The
The
results
are:
Zonal analogs;
t #0,
[v-lil]zonals
I
t=1
s = 0
it,0
i
cos it,0
=
1,...,6
+
Si't,
sin
t,0
(4-48)
where
I
t,o
(4-49)
=t~
and
1
146
a
[Im(M
1,t,0
tn
) -
L,t,0
Im(M
i,-t,0)
i
1,...
=
5
(4-50)
[Re (M
Si,t, 0
tn
,-t,
,t,0) + Re(Mi
0)I
1
3
C6 ,t,0
tn
+
Im(M 6 ,t 0
)
-
Im(M
1
1
Re (M6 ,t, ) + Re (M6
0
] _o 1,
t
,0
)
+
(4-51)
I
+ Im(M1 ,-t
,0
+
tn
t
s-monthly or s-yearly terms;
[v n
) - Re(M1,r-t,0
-
Im(m 6 ,-t, 0
n
a
tn
+
1 ,t,0
a
_1
t, 0
Re (M
a
-o
i,os
i
=
(4-52)
t0 )
= 0,
s
0
cos I0,s + gi,0,s sin *O,s}
1,...,6
147
(4-53)
a
where
a
(4-54)
SsX'
p0, s
4
and
c1,0,s
=
( 1m(M
1 ,0 ,s
) ]
+ -m(M
10
0
=
(4-55)
s1,0,s
= 1
Re (M1
,
6b
01
) + Re (M1,,-
(4-56)
a
ci,o,s
=
(~+Im(M,
0,s
-
i,,-s
I
i
=2,.,6
i
si,0,S
S[ Re(M,0 ,)
+ Re(M,0 ,)
]
(4-57)
6
148
4
Tesseral
t
analogs;
[v n
0
_s
* 0,
]tesseral
-
i,t,s Cos
s-i
t=-t#0
tN-sN'
-s
+
0
+ Sivt s sin it,-s
i =
...
(4-58)
,6
where
t-S
=
(4-59)
tx - sx'
and
ci,t,s
-
tn2 sn'
[
I(Mi,t,-s)
- Im(Mi,
ts
i=1,...,5
(4-60)
5.
i~t~s
=
tn-sn,
[ Re(Mi..- ) + Re(Mi.
,s1-~
~
149
~)
a
1
tn-sn'
3
2
S
6,t,s
+
Im(M
6
Re(M1r t,-s)-
-
a-
-
,t,-s)
Re(M1,-trs)
tn
-
+
sn'
4
Im(M 6 ,-t,5s)
a
(4-61)
1
_
_
tn-sn
3
-
2
a
+
+
Im(M1,t,- s)+Im(M1,t,s
tn
sn'
-
a
+ Re(M 6,t,-s
5
) +,s
(4-62)
functions
the
Eqs.
expansions
of
classes
and
variations
periodic
periodic
short
of
6
reproduces
(4-53)
(4-48),
the
three
exactly
just described
(4-47).
problem.
of
union
The
a
the
and
Fourier
are
(4-58)
for
(4-46)
eqs.
third
body
6
The
~ and
coefficients
expansion
S are
functions
of only the slowly varying elements of the satellite and the
disturbing
be
body.
calculated
averaged
explicitly
orbit
intermediate
This property allows the
the
on
and
generator
times.
The
form
coefficients to
integration
grid
interpolated
of
the
Fourier
of
at
the
any
series
150
a
developed
for
compatible
with
Early
[24]
the
third
body
interpolator
for the
perturbation
software
already
central body theory
so that
problem
is
created
by
implementa-
tion is straightforward.
4.3
Formulation
of
the Third Body
Theory
for
Numerical
Computation
Sections 4.1 and 4.2 described the formal representations of the third body averaged equations of motion and the
short
periodic
implementation
requires
a
computed
functions.
of
an
detailed
and
an
averaged
knowledge
For third
understanding
of
and
its
the
partial
mean longitude, A',
of
the
a
numerical
prediction
the
properties
methodology
of
with
the
to
It also involves establishing
the
be
their
entails an
complex
of
to
for
respect
derivatives
theory
quantities
body perturbations, this
form of the partial
elements
of
derivatives
related quantities.
tional
orbit
efficient
computation.
t,,s
However,
function
satellite
the func-
third
body
with respect to the mean equinoctial p,q
satellite.
The
nature
of
considered first.
For convenience, Tt,s is restated here
151
Tt,s
will
be
4
(-
I,
t, s
Y
n
n
Y-n-1 ,r
[k'(piq),
h'(pq)]
,r) (a,,Y)
nr
m=0 Km n,m r=-n
n=2
Ytn-
41
(k,ih)
a
(4-63)
a
The
complex
factors
inclination
coefficients
may be
t,s
of
S (m,r)
function
Y -n-1,r
formally broken
into
imbedded
and
2n
Yn,-m,
t
'
and
s
are
its real
the
the
Hansen
S
function
The
and
in
a
imaginary parts to
yield the expression,
I
S (m, r)
(a,a,
y)
=
C
(a,
a)
j D r(a,
+
a) ]
A(y)
(4-6m
(4-64)
where
depending
on
the
following definitions
relationship
of
r
to
m,
one
of
the
is used
a
152
S(m,
S2 n
, r2 r (1 +I')
r)I')Im-r
aj2(+,
Pn+r()
Cr+Dr
m
r)
A(y)
m
(4-65)
r < -m
S(m,r)
2n
-(
m-Ir
W)r
2 -m
(n+m)!(n-m)! (1 +I'Y)Ir
(n+r)!(n-r)!
P(m-r,r+m)()
n-m
Cr +D rA(y)
m
m
Ay
-m
< r
()r()m-r
(r-jOI')r-I'm
S(mr)=
2n
(4-66)
< m
2
r(
1
+IY)I'mP(r-m,r+m)
n-r
jDr
m
Cm +
A(y)
r > m
The
third
product
real
of
body
a
power
Hansen
complex
series
in
(4-67)
coefficient
polynomial
the
may
be
broken
into
a
in
h'
and
k'
and
square
of
the
disturbing
the
purely
body
orbital eccentricity, viz.,
s
=
(k' -
jn'h') |r-sI K sn-l,r
s
153
(4-68)
where
a
K- n-1 ,r
s
X-n-1
_ U
,r
e 2i
,g1i+ \Gtq
i+
a
(4-69)
q = s
-
(4-70)
r
a
(4-71)
= sign(q)
n'
61
The
expansion
Their
coefficients,
computation
Appendix
A.
The
and
X,
are
storage
are
satellite
Hansen
the
Newcomb
discussed
in
coefficients
operators.
detail
have
in
the
a
similar form
n,-m
t
_
m-tI
--
Kn, -m
t
(4-72)
0
where
a
154
69
S0
gn,-m 2
Kn,-m
t
i=0
i+2
-
2
i'
2i
(4-73)
2
q = t + m
(4-74)
n = sign(q)
(4-75)
Substituting
eqs.
(4-64),
(4-68)
and
(4-72)
into
the
expression for *t,s yields
n
I
n
a-)
n= 2
m=0
n
KmV
0
(k -
t
m
j-nh)|-m-t|}
If the designation is made
155
m
A(y)
r=-n
- K-n-1,r KnF-m {(Cr + j D r) (k'
s
m
- jnlh')
r-s I
(4-76)
a
j B r,m
Ar,m +
s,t
s,t
=
(k' -
jn'h')|r-sI
(k
-
jn
-)m-t|
I
(4-77)
a
and
the
complex
then Tt,s
is
multiplication
a
ts
n= 2
-
eq.
is
(4-76)
performed,
form
seen to have the
-
in
n
a )
a
n
n
m=0
,m
K
K-n-1,r Kn,m {(ArIn
st
t
s
r=-n
,r A(y)
-
a
C
m
-C
Br,m
s,t
a
+
j(B rm
s( t
Cr
s t D m )}
m + Ar,m
4
(4-78)
a
If
Vnm
and
Vmn,r
are
grouped
into
the
aggregate
function
61
Zm
n,r
mV
n,m
n,r
(4-79)
a
156
4
then eq.
(4-78) becomes
-
y,
sa
t s
n=2
(ta
n
n
n
I
m0 K m r=-n
s(Ar,tm m s{rt
Kn,-m
t
B st
,m Dr)
m +
m- ,
r
m
n,r A(y) Kn5
j(Br,m
st C m + Ar,m
s,t Dr)}
m
(4-80)
The
final
form
considerations.
of
is
Tt,s
The Jacobi
by
shaped
polynomial
practical
embedded
in
A(Y)
is
governed by a recurrence relation which will soon be introduced.
It is
employed
to ensure that the recurrence
advantageous
to compute Jacobi
increasing
subscript,
polynomials
it
since
is for
in
the direction of
that direction
convenient starting values can be determined.
the
summations
summation
purposes
restricted
over
of
a
of
n
eq.
is
(4-80)
are
innermost.
numerical
program,
to run over finite
that
Accordingly,
so
that
the
Furthermore,
for
the
the
must
reordered
ranges.
indices
Hence, the
*t,s used in the numerical implementation is
157
is
be
form of
01
M
Xj
r=R%1 m=0
N
x
R2
t,
Km
t
-
a
n=max(2,m,r )
,r Kn,-m {(Ar,m C
K-ns
(a/a')n Zn,r A(y)
st
B r,m Dr) +
m st m
Dr)}
(B rm
m
st
s,t C mr+Ar,m
40
(4-81 )
a
The
requires
software
implementation
of
the
derivatives:
the following partial
third
body theory
01
40
01
09
158
h partial
t,S
R2
=
aiE
at
r=R
- -n-1,r [ 2K
S
+
M
N
-
j(Brm
s,t
m= 0
dKn, -m
t
de
(a/a')
n=max(2,m,
n
A(y)
-
Ir)
{(Ar~m
Cr - Br,m Dr) +
s,t M
s,t
M
r
m + A rm
s,t DMr)}
+
+ Kn,-m
t
K
mi
{
s,t
9h,
CrAr,m 3B r,m DrB
s,t
m
M _
+
aih
j(
rm Cr
3A ,m Dr
St
m +
M
sot
ah
aih
(4-83)
Eq.
(4-83)
has been derived by employing
3 Kn,-m
t
h
d Kn,-m
t
d e
2
- 2
9e
3h
where
159
the relation
(4-84)
-2
ae
a
(4-85)
2h
=
+ - 2)
-2
6
k partial
R2
a3K
01
a
-K-n-1 ,r[
s
N
M
(a/a')n Z
r=R1 m=0
2K
n=max(2,m, Ir
dKn, -m
t
de 2
r A(Y)
-
)
61
Cr
{(Ar,m
s,t m
-
Dr) +
Br,m
s,t m
01
+
j(B r~m
s,t
+ K ,-m
t
Cr + Ar,m Dr) +
s,t m
m
9Ar,mCr
s,t m
ak
B r,m Dr
m
s,t
ak
3B r,m
+ j(
_st
+
9Ar,m Dr
m
s,t
qk
(4-86)
01
01
4
61
160
09
apartial
t,s
a
-Kn-1,r
s
-
R2
ir
M
a
M0
Kn,-m 1
t
Br,m a
+(sot
-+
+
N
n,r
m n=max(2,m, r)
A r Fm aC r
Br,m aDr
s,t m _
st
3a
a7
)
+
Ar,m 3Dr
s ,t
a
(4-87)
161
Spartial
a
t, s
8
_
R2
y.
a-'
r=R
SK- n-1,r Kn,
t
s
Br ,m
+
( s "t
m
aCr
M
m=
N
m nmax(2,m,
Ar ,m
(s
t
acr
m _
Brm
s,t
n,r
|r|)
6
3Dr
+
0
A rm 9Dr
+
s Ft a
6
(4-88)
162
4
DT t,s
=
y-,
R2
M
N
,
K
r=x
mn=max (2,m, Ir
m= 0
K- n-1,r Kn -m((Ar, m Cr
s
t
sit
+
j(Brm
s,t
m
-
n
I)
Brm
s,t Dr)
m
m
dA
+
r
r)}
m + A rm
s,t Dm
(4-89)
163
0
h'
partial
=
BhI
Kn,-m
6
2
M
r=R
m=0
N
Km
dK-n 1 ,r
2h, dms
| |)
n=max(2,m,
(Arm r
-
(aa
)
n
,r
A( y)-
Br,m Dr) +
+ j(Br~m c r + Ar,m Dr)} +
ss
m
sdt
s~t m
+
-n-,r
s
3B r,m Dr
9Ar,m Cr
m) +
m _
( st
h3h'
Mh'
3B r,m Cr
m
j(asht
3Ar,m Dr
m
st
3h'.
(4-90)
6
6
6
164
4
k' partial
M
N
Zn ,r A(y)
(a/a') n z
K
m= 0
r=R
"n=max(2,m, r
-
)
mI
Kn, -m
t
+
[2k,
j(Brm
s,t
Cr
mf
+ K-n-1r{
s
+
dK-n-1 ,r
s
de' 2
- Br,m Dr) +
C
(Ar,m
s,t m
s ,t
m
+ A rm D r)} +
s,t
DAr,m
s,t
3k'
m
Cr
m
aBr,m
ak
aBr,m Dr
M) +
s,t
3Ar,m Dr
m)
s,t
+ ak'
I
(4-91)
In
view of
eqs.
(4-7)
through
(4-10),
the
ary partial derivatives are also required:
165
following auxili-
4
ap
--
(T+I'y)
c
+ 2
'
"
'"'
~
~
a
'
4
(4-92)
I I' q k'
where
9h'
-
3q
2 I I'{(1
c
I I
1+Iv y--
+ 2
c
(4-93)
>2 + q 2
+
=
-
I'Y)
(h'
(w
- e')
y[a(f
-
- p k'} +
- e')
+
0(g
e'
(4-94)
9k'
2
1
r
-.
,.
.
,.
-
,,1
166
is
---
=
2
9q
{8k' + a(g - e')
(1I'
-
8(f - e')} +
c
I I'
(4-96)
h'
c
and
[17]
ga
2
apc
c
3a
3q
2
= I
c
[q I 8 + y]
(4-97)
p8
(4-98)
(4-99)
c
2
I (pc
-
y)
(4-100)
c
2
(4-101)
c
ay
2
(4-102)
C
167
4
partial
The
derivatives
of
longitude with respect to the mean
q
are
determined
as
follows.
the
third
body
mean
satellite elements p and
dependence
The functional
of
X' may be given by,
4
'I
=
'[(r')
(4-103)
, (v') S]
41
where
(r')s
and
are
(v')s
third
the
position
body
By the
velocity vectors in the satellite frame coordi'nates.
the partial
chain rule,
derivatives
X' with respect
of
and
to p
a
and q have the form
a
3
--
[T
3r'
(r
r'_)_
s
T
s
[3X']T
[
-
s
T
3(v')
39
(4-104)
a
and
3;1'
3
I(r'
[ 3(r
Ax']T
-
) s+[
4
3(v'
s
3'T
s05
3(r')s
(4-105)
168
4
4
4
where
the
superscript
denotes
third
body
ephemerides
are
tial
coordinate
the
generally
eqs.
system,
Since
transpose.
available
(4-104)
and
in
an
(4-105)
the
inerare
rewritten using the transformation relations:
(r' )s
=
[f
g
w]T
(4-106)
(v'
)s
-
=
[f
g
w]T (v)
(4-107)
and
where
vectors
result
(r') 1
of
of
and
the
(v') 1
third
substituting
are
the
in
body
eqs.
inertial
(4-106)
(4-104) and (4-105) is,
169
position
and
and
velocity
coordinates.
(4-107)
into
The
eqs.
4
A
AA
AL
ap- -
[ax
7 (r
)J
]T
[.2-
s
]
p
p
(r')
1 +
I
_p
A
A
A
+
[a]
[a)
a
ap
(v - )s
ap
ap
-
]w
IT (v)
6
-
(4-108)
4
and
-(r')
s
aq-
[ aq-
T
e
aq
aq
a
A
'a
3(v )s
'
aq
T (v'I
aq aq
aq
4
(4-109)
11
Analytical
the
satellite
lite p and
[17]
expressions exist
frame
for
derivatives
the partial
of
unit vectors with respect to the satel-
q elements.
As
taken
from
a
work
by
McClain
a
they are,
4
170
0'
af
=
---
2
-
(q
-
ap
I
(4-110)
g + w)
c
AC
af
2
-
-
39
c
ap
=
c
I p g
(4-111)
I
q f
(4-112)
I(p f-w)
(4-113)
=_
aq
C
=
f
W
(4-114)
aw
2
_
Making
use
4-3.
The
longitude
of
C
eqs.
partial
with
(4-115)
I g
-
aq
through
(4-110)
derivatives
respect
to
the
velocity vectors have closed
found
in
a work by Shaver
of
(4-115)
the
disturbing
leads
third
body
to Table
body
position
mean
and
form representations which are
[25].
171
They are:
0
0
Table 4-3. Partial derivatives of the third body mean longitude with respect to the mean satellite elements
p, q.
01
2
=
ap
c
ax' [q lg+w
Iqif -f
|
61
+
51T ]
T[ q |I g+
w
-lq
-T
(~)l}
(4-116)
Is
aq
[ai )S] T
2fF
S
c[ar')sJ
(v')s]
-
I -
^p- ) I-w T
I7 pg I-1(pf -W) -1g
L
lpg
0
-f
(p-w)I
lg]
(4-117)
0
0
172
01
0
axv'=
-
-n =r
+
-s
, a ,2
1
~
+ k'
(h' 1
(q'
i'f
-
1-h' -k'
n , a 12
I'
)
a
- p'
g'
')
(h'
[h'g
]
+ ks
4)
f
+
-
w'
(4-118)
and
3a'
6'/1-h'j -k'1
2
n a2
n a
-
+
ax
(h'
+
(h' 1=
ay
+
k'
aX'
k)
fI
g']
+ k'
+
+
1
g (q'Y'I'
-
p'X' )w'
(4-119)
173
4
where
it
is
all
that
understood
calculations
performed
I
[25]
in satellite frame coordinates and
8' =
are
1
(4-120)
1+/1-h'
4
(4-121)
41
2 -k'2
A
X'
=
r' cos L'
=
r
Y'
=
r'
=
r'
sin L'
-f'
(4-122)
-
4
-n' a' (h' + sin L')
,
h' 2 -
1-
(4-123)
k
4
n'a'(k' + cos L')
,s
2
/1 -
h'
-
(4-124)
k
a
G'
n'a' 2
=
/1
h
(4-125)
k
i
k' ' k
ni
ax'
= -
+ a'
+ g
Y'
I'
(4-126)
a
ay'
=
-
a'
k''fi'g
k
(X'Y' + G' )
(4-127)
174
01
ax,
=l h' S'X'
,f
(4-128)
- G')
aT'
n'S '
=
ay'
X' x'
-
2 +
a ',
a
r
3
(4-129)
_
(G'k' a'X'
-
n'a'Y, 2
2
-h2
(4-130)
=i-
al j'i'
a'
+
G'
3h'
r'
3
/1 -h'
(G'k' 8'Y'
2
+ n'a'X'Y' )
2
-k'
(4-131)
3A'
3kT
,
,
a'
_
r'
/11-h'
(G'h' a'X'
+ n'a'X'Y')
2 -k' 2
(4-132)
=-
ar
'2
3
r' 1 v1 -h'
(G'h' S'Y'
2
-
n'a'X'2
)
2
_k'1
(4-133)
175
41
4.4
Calculation of Special Functions
4
Calculation of Zm
n,r
4.4.1
Zn,r
=
m
41
is defined by
The coefficient Zm'
n,r
n
n,r
n,m
(4-134)
a
where
a
V n,rn
S(n-r)
(n+n)!
(4-135)
m (0)
n,rn
a
Vm
n,r
-
(4-136)
(n-r)
(n-m)!! P n,r (0)
01
and
the
Legendre
(4-135)
functions
functions
(4-136)
and
Z
r
n,r
Pn,m( 0 )
of
and
argument
into eq.
Pn,r(0 )
zero.
are
associated
Substituting
eqs.
a
(4-134) yields
(n-r)!
(n+rn)! P n,rn (0) P n,r (0)
(4-137)
0
0
176
61
The
indices
positive
n
and
are
m
constrained
However,
integers.
the
positive and negative values.
eq.
(4-137)
to
computational
positive r.
consider,
form
for
index
being
when r is
r
over
may
only
assume
the
both
(4-137)
r
> 0
and
0.
A
is
now
determined
for
A simple symmetry relation
compute Z m ,
n,r
run
Hence, there are two cases of
those
eq.
to
r
<
will then be used to
negative.
r _> 0
Eq.
[17]
(4-137)
is
written
in
terms
of double
factorials
by recognizing that
P
n,m
(0)
=
(-1) (n-m)/2
(n+m-1)!!
(n-m)!!
(4-138)
(0)
=
-(-1) (n-r)/2
(n+r-1)!!
(n-r)!!
(4-139)
and
P
n,r
where the definition is made
(-1)!!
=
(4-140)
1
177
4
Substituting
eqs.
(4-138)
and
(4-139)
into
eq.
(4-137)
4
yields
Zm
n,r
(n-r) !
(n+m) !
1) (n-m) /2
(n+m-1 )!!
(-1) (n-r)/2
(n-m)!!
(n+r-1 )!!
(n-r)!!
4
(4-141)
4
or after simplification
I
n[(r+m)/2]
Zm
n,r
r > 0
(n-r-1)!!
(n+m)!!
(n+r-1)!!
(n-m)!!
(4-142)
r < 0
I
a
Replacing
r
by -r
in
eq.
(4-142) produces the expres-
sion
178
6
n+[(r-m)/2]
Zm
n ,-r
(n+r-1)!! (n-r-1)!!
(n+m)!! (n-m)!!
(4-143)
Comparing eq.
The
ty.
The
(4-143)
Zm
coefficient
exploitation
calculations
in
a
(4-142)
to eq.
has
n~r
of
this
numerical
leads
an
interesting
property
program.
immediately to
It
helps
is
to
easy
properminimize
to
show
that the following relations are true
P n,r(0)
=
0 ;
|n-ri
= odd
(4-145)
Pn,m (0)
=
0
;
In-ml
= odd
(4-146)
and
179
4
Hence,
all
2P
n,r
even or
must have
all
odd.
the
if
vanishes
indices
r,
m
and
Thus the members of the triad
if a term in
the same parity
n
are
not
(r,m,n)
-corresponding
ii
to that set of indices is to be non-zero.
4.4.2
Their Partial
and
Polynomials
Calculation of Jacobi
Derivatives by Recurrence
The function A(y),
(4-65)
eqs.
m
P (E)
a
to
(4-67)
through
r,
the
contains
(y),
the forms of which can be found in
depending
on
orthogonal
the
relationship
of
polynomial
Jacobi
where
a=
n + r
m -r
=-m
r
< -m
(4-147)
r
a= n-m
m -r
S=
T=r
a
T
The Jacobi
=
=
=
+
-m
< r < m
(4-148)
m
n- r
r -m
r + m
polynomial
r
and
> m
its derivative
(4-149)
with
respect to
are governed by the standard recurrence relations
180
[161
Y
P
E
(y)
=
(a
2
a + a 3 a Y)
4
E(S T)ay
T)
E
a
45a
0
(4-150)
and
dP( ,T)
a+l
dy
(a 22 aa + a 3
a
)
)
+
al
P
'
T)
dy+
a3
3
dP~ E
( Y)
a a
a- a
dP
,
)
a-i
dy
(4-151)
where
181
4
a + 1) ( a + E + T +
al .
S2(
a2a
=
(2a +
C
a3 a
=
(2a +
e +
+
T +
1) (2 a +
1) (C 2
T) (2a+
s+
E +
(4-152)'
T)
(4-153)
2)
T
+ 2) (2a +
E +
T +
1)
(4-154)
a4 a
=
2(a + e)(a +
t)(2a + e +
T +
Starting values for the recurrences
explicit expressions
(4-155)
2)
are
given
by the
g
[16]
a
182
01
P
, T)
0
P
T)
P
,
1
=
(4-156)
=1
[C
-
(C + T + 2)y]
T +
(4-157)
T)
0
(4-158)
0
=
dP~ C, T)
dy
4.4.3
Calculation
=
1+
the
of
(4-159)
2
Coefficients
r
CCm
coefficients
m
,
Dr
m
and
by Recurrence
Their Partial Derivatives
The
C
and
Dr
are
defined
ac-
cording to
+ jD
(a
+
jaI')I'm-r
r < -m
(4-160)
(a
+
j6)m-I'r
-m < r < m
(4-161)
(ca -
jaI)r-I'm
183
;
r
> m
(4-162)
a
This
section
develops
recurrence
relations
partial
coefficients
C ,
M Dr
m
and
their
with
to
a
6.
Starting
respect
recurrences
and
are also determined.
that
recognizing
eqs.
(4-160)
for
the
derivatives
values
for
The task
is
through
(4-162)
these
simplified
can
by
be
collapsed into two cases, viz,
C + j
m
D
(a + jgI')I'm-r
=
m
r < I'm
(4-163)
> I'm
(4-164)
and
Cr +
m
j Dr
m
(a
r
r-I'm
-
The exponents of the complex polynomials of eqs.
are
(4-164)
recurrence
positive
that
exponents.
r
indices
recurrence
corresponding
because
are
relations
the
the
same parity
than
greater
sum
must be
required
only
for
it
and
have
the
m
must
that
increasingly
has
differ
shown
parity,
coefficients
two.
integers
This
difference
of
even.
recurrence relations
The
two
by
been
same
connect
should
Hence,
zero.
since
exponents
or
to
equal
Furthermore,
relations
to
or
(4-163) and
having
is
the
follow
directly:
184
6
Case I
For clarity, it
is appropriate to rewrite eq.
(4-163)
as
CI'm-r +
Incrementing
C (I'm-r) +2 +
j
D I'm-r
=
(a + jai')I'm-r
r
< I'm
(4-165)
I'm-r by 2 leads to
j
D I'm-r)+2
+
=
(a
=
(a +
jaI')(I'm-r)+2
jaI')I'm-r
(a +
jai')2
(4-166)
Substituting eq.
(4-165) into
185
(4-166)
4
C (I'm-r)+2-+ j D I'm-r)+2 " (CI'm-r + j DI'm-r)
4
[(a2
2)
+
j 2a
W
(4-167)
a
Performing
the
complex multiplication
and
equating
the real
and imaginary parts yields the recurrence relations
I
0
0
6
The
recurrences
at the point r
of
eq.
(4-168)
and
(4-169)
are
initialized
= I'm, where
a
186
6
01
CO
(4-170)
=1
r = I'm
D
0
0
Recurrence
CI'm-r
and
(4-171)
relations
with
DI'm-r
differentiating eq.
9C Im-+
3a(
for
the
respect
partial
to
a
may
derivatives
of
found
by
be
(4-165)
jD.Im-r
(I'm -
+
r)(a + jSI') I'm-r-1
4
(4-172)
Incrementing
I'm -
aC (I'm-r)+2 +
r
by 2
jD(I'm-r)+2
=
=
(I'm-r)+21(a+jOI')(I'm-r)+1
[(I'm-r)+2] (a+j
IW)I'm-r (a+j6I')
(4-173)
Substituting eq. (4-165) into eq.
187
(4-173)
4
3C(I'm-r)+2
.3D(
+ 3
3a
I'm-r)+2
3a
,
=[(I'm-r)+2] (C I'Im-r +jD I'Im-r)
I
- (a
6
+ jaI')
(4-174)
a
After performing the complex multiplication and equating the
real
and
imaginary
parts,
one
obtains
the
recurrence
rela-
tions
a(I'm-r)+2
=
[(I'm-r) + 2]
(aCI'm-r -
I'DIm-r)
6
(4-175)
40
(I'm-r)+2
a
=
[(I'm-r)
+ 2]
(IC
(I
I,
0
r
< I'm
(4-176)
0
Again, the initialization is found at the point r = I'm
188
=
0
(4-177)
r
3Da
=
I'm
0
0
0
(4-178)
9a
In similar fashion, the recurrence relations for the partial
derivatives with respect to a are given by
3C( I'm-r)+2
-I'[(I'm-r) +
=
2
](aI'CIm-r +
aD I m-r
(4-179)
D( I'm-r)+2
=
I'
I'm-r) +
I'm
with initial conditions
189
2] ( aC I Im-r
-
WI'DIm-r)
(4-180)
a
=0 0
(4-181)
g
r = I'm
=D 0
(4-182)
6
Case II
Having rewritten eq.
Cr-I'm +
j Dr-I'm
S(a
(4-164) as
-
jaI')r-I'm
r
(4-183)
> I'm
a
the
recurrence
relations
for
Cr-I'm
Dr-I'm
and
are
found
to be
01
C(r-I'm)+2-
(a2 _
2)Cr-I'm + 2a$I' Dr-I'm
(4-184)
D(r-I'm)+2
=
-2aaI'
r
Cr-I'm + (a
> I'm
-
2)
Dr-I'm
61
01
(4-185)
190
0
with the
initial values
C
=
1
(4-186)
0
(4-187)
0r
DO0
The
recurrence
relations
for
the
a
partial
deriva-
tives are given by
3C(r-I m)+2
=
[(r-I'm) + 2](aCI
+
SI'DrI'
(4-188)
D r-I'm)+2=
[(r-I'm) +
r > I'm
The initialization is furnished by
191
2] (-aI'CrI
+
aDr-)
(4-189)
4
0
=
(4-190)
3D
r
aD0
=
I'm
0
(4-191)
0
=
I
I
Finally,
the
recurrence
relations
for
the
S partial
deriva-
a
tives have the form
aC(r-I'm)+2
=
-I'
[ (r-I'm)+2] ( IC
r-I
a
'm
(4-192)
6
r-I'm)+2
ID
=
-I' [(r-I 'm) + 2] ( aCr-I 'm + SI 'D r-I
'm)
6
r
> I'm
(4-193)
01
The initial values are
ac
=
0
(4-194)
r
D0
61
= I'm
0
(4-195)
192
0
4.4.4
definition
B r,r
s,t
the
of
Arm
As,t
Coefficients
Derivatives
Partial
and their
B r,Im
s,t__
The
the
of
Calculation
Ar,rm
s,t
coefficients
and
come from the expression
Arm +
s,t
jBr,m
s,t
k'
=
-
jn'h') Ir-s I(j
-
jh) -ur-t|
(4-196)
where
=
sign(s
-
(4-197)
r)
and
n
=
(4-198)
sign(t + m)
If the following definitions are made
193
a
e r-s I+ jf Ir-
=
(k'
r-s|
-j'h')
(4-199)
and
a
g
g
|-m-t|
+
(4-200)
-j
jhl-m-tI
4
then,
after
equating
A r,m
s,t
the
performing
real
and Br,rm
s,t
and
the
complex
imaginary
multiplications
parts,
the
and
coefficients
a
are found to be
6
6
0
194
A rm
=
e
|r-s1 9
m-t|-
~ f
|r-sj
-- t|
h I-m-
(4-201)
B
,m
s vt
-
e |r-s
h
|m-t|
+ f jr-s
9 -m-tj
(4-20 2)
From
eq.
(4-201)
and
(4-202),
relations
195
one
obtains
the
following
a
aA rm
I
e ir-s
sot
ag I-n-t|
ah|-m-tl
Ir-si
(4-203)
4
3Ar,m
st
ah
ag~1
e9
e
-m-t
e r-s
h-m
r-s|
I
4
I4
(4-204)
ae
9A r,m
s
3h'
tjr-s
h'
afr-s
~ ah
|-m-t|
9
1
h
61
(4-205)
a
3A r,,m
-A
Wk
e
e r-s
ak'
-
a
|_
9
|-m-t|
I
r-s|
k'
01
hti
(4-206)
and
41
0
0
196
9B r,rn
s t
=
aiR
e |r-s
9s -m-tI
3
r-s|
ah
(4-207)
3h -- t
s t
3R~
=
er-s
r-s
9
-i I-m-tI
3i
(4-208)
3Brm
s t
3e r-s
3 h'
h I-m-tj
+
af r
-m-t
9
I
(4-209)
3B
ak'
' m
ae
|r-s
ak
afr-s
hI-m-t
+
I
3k'
9
-m-ti
(4-210)
197
a
The
coefficients
eqs.
of
(4-199)
and
and
(4-200)
0
their
partial
Since
there
between
m
derivatives
are
and
coefficients
The
t,
the
be
computed
constraints
parity
recurrences
corresponding
techniques
the desired
no
can
to
)t the prior
are
exponents
section
may
by
recurrence.
between
designed
that
be
r
and
to
differ
used
to
s or
relate
by
1.
achieve
results:
0
e
|r-s|
+1
-
k'
e
Ir-sl
+ n' h'
f |r-sl
(4-211 )
0
f r-sj +1
e
=
1
f
=
0
-
-n'
h'
e r-s
+ k'
f ir-s|
(4-212)
(4-213)
Ir-s
=0
(4-214)
0
9
|-m-t|
+1
h |-m-t| +1
~
91-m-t|
-
h gm-t
+ n h h-m-t|
+ k h |-m-t|
(4-215)
(4-216)
0
(4-217)
-m-ti
h0
0
=
0
(4-218)
0
198
0
s +
=
r - sj
-s+
-
(|r-s|
f
+ 1) e(r-
+ 1)
s|9
(4-219)
fjr-sj
(4-220)
(420
3e 0
(4-221)
0
S=
jr-s
=
0
af0
07
D
=
(4-222)
0
-se+1
=
(
-|L+
=
-(
jr-s|
+
1)
n'
f
r-s|
(4-223)
e2r-s24
(4-224)
af
r-s|
+
1)
n'
9e0
07=
(4-225)
(425
0
r-sj
Sfo
=
=
0
0
(4-226)
199
a
ag I-m-t +1
=
( |m-t|
=
J-m-t
1)
+
g
(4-227)
3h
-m-t +1
+ 1) h I-m- ti
ak
=
(4-228)
(4-229)
0
1-m- t
=
0
ah0
=
+1=
9h=
ag
(4-230)
0
( -m-t|
-(
=
0
=
+
1)
h-t|
T
n
(4-231)
(4-232)
(4-233)
I-m-t |
h0
-m-ti
1)
+
=
0
0
(4-234)
9E
09
0
200
4
4.4.5
Recurrence
Relations
for
the
and
Kernel K'-n-lr
efficient
S
The modified
Third
its
Hansen
coefficients
(k' -
in' h')|rsI
Body
Hansen
Co-
Derivative
for
the
third
body
have the form
Y-n-1r
S
=
Mathematical expressions
polynomial
already
in
been
(4-235)
introduced.
of
computation
derivative
eq.
the
for
the
The
recurrence
function
the complex
derivatives
partial
its
are now examined.
S
calculation of
and
kernel
(4-235)
Kn-,r
relations
K
'
for
and
have
the
its
The two cases corresponding to
s # 0 and s = 0 are considered separately.
Case I;
s * 0
The
recurrence
relation
governing
the
third
body
kernel functions when the subscript is non-zero may be taken
from an 1855 Hansen manuscript
201
[26]:
4
I
I
I
where
a
=
C
-n
-
(4-237)
1
a
and
a
e' 2 )-1/2
(1 -
x'
(4-238)
a
The
evaluating
Kc.r
s
-nX-
of
initialization
3
,
the
the
power
explicit
given
in
eq.
-nk-4,
where
recurrence
(4-69)
ny
is
the
series
for
lower
c
is
accomplished
by
representation
of
=
-nt-1,
bound
on
61
the
index
n given by
a
202
61
=
Ir)
(4-239)
2
initialization
this
that
Note
max(2,m,
singularities in eq.
ensures
(4-236) for c
=
-
Ir |
that
-
the
apparent
2 and c = -4 are
never encountered in practice.
The derivative of the kernel function with respect to
e' 2 is seen to obey the recurrence relation,
dKc,r
s
_
2
de'
,2
x,
Kc, r +
s
2
{(c+2)(c+4)(2c+5)
--
dKc+1,r
de'
+ s
r
2s
-
(c+3)
(c+3)
x'
dKc+ 2 ,r
2
s
de'
[(c+2) (c+4)+2sr V1-e,
Kc+
s
-
(c+4)[(c+2) -r2
2
,r
+ s2
(c+2)
4
dKc+ ,r
s
de' 2
+
I
(4-240)
The
recurrence
is
derivative of eq.
explicitly
initialized
(4-69), viz.,
203
by evaluating
the
5
dKc,r
=
de' 2
j=0 ((j+1)
Xc,r
+1 , j+
j+-
e,2j
4
2
(4-241)
4
for c = -n
- 1,
-n
-2,
-n
-n
-3,
-4.
4
Case II; s = 0
4
If
s
is
set
equal
to
zero
in
eq.
(4-236)
then
the
following specialized recurrence relation results:
a
Kcr
0
_
(c+2)x'2
2 2
[(c+2) -r ]
(2c+5)
{c+)
Kc+1,r 0
c
-
c+3)
(
Kc+2,r}
0
I
(4-242)
6
The
zero
series
subscript kernel
which
terminates
function can be represented by a
a finite
after
hence may be computed in closed form.
tial
values
for eq.
(4-242)
number
of
terms
and
Accordingly, the ini-
are given by the simple
expres-
6]
sions
204
61
K- rlr
=
0
(4-243)
K- Irl-2,r
=
(1 / 2 )1 1 x' 2rIl+l.
(4-244)
0
0
The
derivative
to zero
in
eq.
recurrence
is
readily
setting
s
dKc+1,r
2
02
de' 2
by
(4-240)
dKc,r
,2
2
(c+2) 2
[(c+2) -r ]
K0c,r +
-
The
obtained
{(2c+5)
2
dKc+ ,r
(c+3) d '2
de'
}
0
de'2
(4-245)
initial conditions are given by
dK~0|rl-1,r
02
=
0
=
(1/2 )IrI+1 (2 Ir + 1) x,2|rI+3
(4-246)
de'
dK-I ri-2,r
dO
de'
(4-247)
205
4
4.4.6
Recurrence Relations for the Satellite Hansen Coefficient Kernel K n, -m
a
and its Derivative
t
The
modified
Hansen
coefficients
for
the
satellite
41
have the form
Y=
The
and
recurrence
(k -
jnh) -m-
relations
for
the
t Kn,m
(4-248)
kernel
function
K n,-m
t
its derivative are now discussed for t * 0 and t = 0.
6
6
Case I; t * 0
The recurrence relation governing
the satellite
nel functions when the subscript is non-zero
ker-
is given by:
61
01
01
206
Kn,-m
t
-
{(n-1) [n(n-2)
1
t2 (n-2)
_
n(n-2)(2n-3)
Kn 3 , m + n[
-
]
2tm Vi-e
2
2 -
n- 2, -m
t
Kn-4
(4-249)
The
of
initialization
evaluating
Kn,-m
t
nX+3,
tialization
in
where
the
explicit
nX
power
for
(4-73)
eq.
is
ensures
in
defined
that
accomplished
by
series
representation
of
n
nt,
is
recurrence
the
the
eq.
apparent
=
(4-239).
ny+1,
This
np+
2
,
ini-
singularity at n=2
is
not encountered.
The derivative of the kernel function with respect to
e2 is seen to obey the recurrence relation
207
4
dn, -m
t
2
*
de
t
2
1
(n-2)
{tm(n-1) X Kn-2,-m
t
+
dKn-2, -m
+
(n-1)[n(n-2) -
2tmV1 -e
t- 2
de
n(n-2) (2n-3) -
n-3, -m
-d-
-
(n-2)2 - m
2
2]Kn-4,-m
+ n[(n-2)
2 - m2
,
-n
4
dKn- ,-m
t
de
e 2
1
(4-250)
where
x
The
=
recurrence
(1 _-
is
(4-251)
2) -1/2
6
explicitly initialized
derivative of eq. (4-73), viz.,
208
by
evaluating
the
a
a
a
dKn, -m
t
d- 2
j=0
-
(j+1) Xn,-m
j+
e
1
2j
,j
(4-252)
for n
=
Case II;
n+l, n + 2 ,
n.,
t
If
n k+3.
= 0
both sides of
(4-249)
eq.
are
multiplied
by the
factor t 2 (n-2) and the index t is then set to zero,
=
0
+
Setting
n
=
-
(n-1)(n-2) Kn-2,0
m 2 ]1
[(n-2)2
+ [(- -2
n+2
in
eq.
-
- e
(4-253)
+
Kn-3,-m
(n-2)(2n-3) K
0
2) Kn- 4 ,-m
) K0
leads
to
(4-253)
the
specialized
recurrence relation:
Kn,-mK
1
n(n+1)
{n(2n+1)Kn-1 , -m
o
-
(n2
m2
-
2) Kn-2, -m
0
( 4-254)
209
4
The
zero
Hence,
subscript
the
initial
kernel
values
may be
computed
for eq.
(4-254)
in
are
closed
given
form.
by
the
4
simple expressions
I
Km,-m 0
(-1)m
m+1,-m
The
derivative
(2m+1)!!
(m+1)!
m
recurrence
+!
is
(4-255)
[2(m+1)+e 2
obtained
by
(4-256)
differentiating
I
I
eq. (4-254)
dKn, -m
0
1T-
de 2
I
dKn-l,-m
{n(2n+l)
-2
n(n+1)
de
I1
dKn(n
2
,-m
d0- 2
de(-27
- m )
-
n-2,-m
(4-257)
U1
The initial
conditions are given by
210
41
6
dKm, -m
0 .
0
(4-258)
de
dK m+1,'-m
0
1
-
(4-259)
(m+2)!
Restriction of Indices in the Third Body Theory
4.5
In a numerical
must be established
third
)m
m( (2m+1)!!
2
body
orbit prediction program, a procedure
for
disturbing
limiting
potential
the number of
the minimum
to
the
terms in
required
to
accurately predict the motion of a satellite in a particular
orbital
regime.
Once
free
truncation
parameters have
been
set, this task entails the determination of constraining relations governing the
indices of the multiple sum
T2
S2
R2
M
t=T
1
s=S
r=R
m=0
N
(4-260)
211
n=max(2 ,m,I r)
4
In
the
numerical
of
implementation
the
third
body
theory
4
derived in this thesis, the truncation parameters are:
1)
Maximum
N
power
of
the
parallax
factor,
4
(a~/a')
2)
M*
Upper
bound
on
the
satellite
Hansen
coefficient d'Alembert characteristic,
4
I-m-t|.
3)
R*
E
Upper
bound
on
the
third
body
Hansen
coefficient d'Alembert characteristic,
I.
r-s
a
The
bound
on
the
satellite
Hansen
coefficient
d'Alembert
characteristic may be mathematically expressed by
a
<
I-m-ti
(4-261)
M*
which translates directly to an inequality on m:
-M*
-
t
<
m
212
<
M* -
t
(4-262)
6
6
The index m is not allowed to take on arbitrary values since
the application of the Addition Theorem introduced the additional constraint:
0 < m
The
intersection
parallelogram
is
special
case
of
substituting eq.
M*
admissible
region of
displayed
graphically
region
<
(4-263)
(4-262) with eq.
of eq.
shaped
< N
Solving
N.
in
(4-263) leads to a
m versus
Figure
eq.
4-1
(4-262)
same
that
eq.
for
the
t
and
(4-263) yields the necessary bound:
(4-264)
solution can be derived using Figure 4-1.
(4-264)
The
for
-M* - N < t < M*
This
t.
shows
that
an
increase
in
the
Notice
number
of
multiples of the satellite mean longitude can be accomodated
only by retaining more powers of the satellite eccentricity
in the disturbing potential.
accordance
with eq.
(4-264),
With the t index restricted in
it
is
well behaved constraint on m, viz.,
213
now possible
to write
a
4
Figure 4-1.
Admissible Values of the Index m vs. t
N
M*i
-M*
\,
2144
a
The
max(
0,
on
the
bound
-M*-t ) < m < min(
third
body
Hansen
N )
M*-t,
coefficient
(4-265)
d'Alembert
characteristic takes the mathematical form:
(4-266)
Ir-sI < R*
which leads to
-R*
As
in
the
satellite
+ s < r
< R*
the
case,
(4-267)
+ s
index
r
cannot
vary
arbi-
trarily because the rotation theorem for spherical harmonics
provides the auxiliary inequality:
-N
< r
< N
215
(4-268)
4
The
shaped
case
special
of
substituting eq.
R*
N.
<
with
region
graphically
seen
is
region
The
(4-267)
eq.
parallelogram
another
S.
of
intersection
Solving
eq.
of
in
leads
(4-268)
Figure
eq.
versus
r
admissible
4-2
for
the
s
and
for
(4-267)
to
4
I
(4-268) provides the required bound:
4
With
the
s
index
restricted
in this way,
a uniformly
valid
4
constraint on r can be written, viz.,
I
I
Making
(4-265) ,
(4-260)
use
(4-269)
the
of
(4-270) ,
and
can be recast
constraints
the
of
eqs.
multiple
sum
(4-264),
of
eq.
as
I
I
41
216
41
Figure 4-2.
Admissible Values of the Index r vs. s
R*
217
4
M*
min(R*+s,N)
R*+N
4
r=max(-N,-R*+s)
s=-R*-N
t=-M*-N
N
min(M*-t,N)
m=max (0,-M*-t )
4
n=max (2, m, I r
(4-271)
I
Of
course,
necessary
for
to
a given
use
the
(4-271 ) simply shows
selection
full
range
the maximum
of N,
of
t,
M* and
s,
allowable
r,
R*
and
is
it
m.
not
Eq.
4
ranges.
I
I
I
218
4
41
Chapter 5
Numerical Verification of the First Order
Third Body Theory
The first order third body averaging theory developed
in
was
thesis
this
studies
of
results
of
orbital
a
theory
that
tests
theory
can
be
motion of a
makes
stability.
numerical
the
used
high altitude
decisively
it
This
of
The
predict
satellite
with
to
superior
the
long
term
presents
the
third
show
results
accurately
reduce
and
chapter
implementation
design.
to
substantially
mission - analysis
of
expense
computational
to
designed
an
that
the
the
body
the
long
term
efficiency
that
conventional
mission
analysis techniques.
The theory is applied to the long
orbits
five
test
and
semi-major
conditions
for
which
span
axis.
the
a
For
averaged
term prediction of
broad
range
of
each
test
orbit,
equations
eccentricity
of
initial
motion
are
determined from the high precision orbital elements at epoch
using
The
a
least
averaged
squares
equations
differential
of
motion
are
correction
then
algorithm.
integrated
and
compared for speed and accuracy against Cowell integration.
219
4
The
equations
averaging
version
(GTDS)
theory
of
the
for
are
at
the
Amdahl
470
V/8
test
furnishes
order
and
Trajectory
Charles
Cambridge, Massachusetts.
CSDL's
first
programmed
Goddard
modified
the
third
interfaced
with
Determination
Stark
Draper
body
a
System
Laboratory
in
This version of GTDS operates on
digital
facilities
computer.
GTDS
include
that
numerical
integrators, interpolation algorithms for the short periodic
coefficients, and auxiliary perturbation models.
The
consists
software
of
an
Averaged
Periodic Generator
for
routines
software
right
third
hand
GTDS.
AOG
the
perturbation
averaged
that
coefficients
third
(AOG)
body
and
interface with the
theory
a
theory
of constructing
equations
correspond
dynamics.
of
to
motion
the
The SPG
for the zonal,
times
Since
the
periodics
SPG
the
coefficients
averaged
have
a
share a great
equations
similar
many
are
of
analytical
subroutines.
220
motion
and
structure,
for
computes
tesseral
interpolated
the
double
m-daily analogs on the integration grid of the AOG.
other
Short
executive
satellite
is capable
and resonant satellite
short periodic
the
semianalytical
The
of
of
Orbit Generator
(SPG) which
the
sides
body
averaged
the
in
implementation
and
At all
by
GTDS.
the
short
the AOG
and
the
two
Nevertheless,
components
of
different
the
averaging
fields of
discussion
of
the
indices
program
software
can
flow
are
independent
be
specified
for
and
subroutine
and
each.
A
interactions
can be found in Appendix B.
Section
5.1
determination
describes
the
of
precision data.
the
AOG
method
initial
employed
conditions
for
from
the
high
Section 5.2 presents a method for computing
disturbing body ephemerides over the time
long term satellite orbit prediction.
spans required for
The numerical results
of applying the averaging theory to selected test orbits are
discussed in Section 5.3.
Initialization of the Averaged Equations of Motion
5.1
Given
elements
a
at
set
epoch,
of
a
high
precision
satellite
set
corresponding
of
orbital
initial
mean
elements for a long term integration must be determined in a
way
which
is
consistent
the
with
averaged equations of motion.
harmonic
content
of
the
This requires the elimination
of contributions to the high precision elements arising from
frequencies
equations
element
that
of
motion.
space
divergence
have
is
to
mean
been
If
the
element
apparent
removed
conversion
space
between
221
from
is
the
the
satellite
from
osculating
inaccurate,
averaged
then
a
element
4
histories
and
the
trajectory
[27].
term
long
trends
of
the
high
precision
For the test cases discussed in this chapter, the AOG
was
initialized
(PCE)
capaoility
squares
epoch
using
of
elements
to
trajectory
integration.
the
The
GTDS.
differential
mean
Precise
the
on
output
of
Elements
procedure
is
a
PCE
algorithm
correction
based
Conversion
the
of
fit
of
high
a
that
a
least
solves
for
semi-analytical
precision
I
Cowell
The PCE can be performed over any time span.
The exact initialization procedure is as follows:
1)
Given a set of initial osculating elements and an
appropriate
perturbation
model,
a high precision
trajectory is produced by Cowell integration.
At
the integration
span,
the
the
satellite
fixed
intervals
inertial
position
within
rectangular
and
components
velocity
are
of
output
to
a
file
to
represent actual observations.
222
4
2)
With
a
and
field
perturbation
compatible
epoch,
priori estimate of the mean elements at
5*0,
a
an
where
a 0 ~
(5-1)
X0 ]
k0 , p0 ,0 q
h
0
to the osculating
a semi-analytical approximation
element histories is generated over the same span
as
the
recover
first
the
must be stressed
recovery
model
of
convergence
an
is
essential
the
used
to
periodic
short
order
that
is
SPG
It
elements.
integrated mean
the
to
corrections
The
integration.
Cowell
short periodic
accurate
ensure
to
the
correction
differential
algorithm.
3)
At
each observation
employed
to
velocity
components
determined
of
step
compute
two
body relations are
from
the
the
residuals,
5b,
observations
from the actual
223
and
position
satellite
orbital
by semi-analytical methods.
observation
subtracting
time,
is
A vector
obtained
computed
observations.
elements
in
by
this
For a single
4
observation
satellite
time,
where
T,
position
and
the
six
components
velocity
are
of
computed
4
through the equations
4
S[a( T)+vn
rE ( T )
rc (T
-
,(
1
)+vn
( ) v 2,1
V 1,11
(T)
1
,...,~(T)+vni
X(T
+V
6,11
4
(5-2)
4
the
correction
to
the
mean
elements
at
epoch
is
given by the expression
4
-
a*
(A
=
T
WA)
-1
A
T
W6b
(5-3)
41
In
the
eq.
(5-3),
W is
Jacobian
of
a
X
weighting
and
X
with
matrix
and
respect
A
to
is
7*0
evaluated at the observation time, viz.,
AA
=
3X(t)
I
9X(t)
1
(5-4)
_
i*
-
i*
- 0
0
a
t= T
a
I
a
224
Eq.
can
(5-4)
expanded
be
by
chain
rule
partial
differentiation to yield:
I
9X(t)
3Zt)
a* (t)
3a*(t)
3a*(t)
aa*(t)
A
I
I
0
6x6
0
I-
aa*(t)
0
Ba_*(t)
0
6x6
a*(t)
6x6
- 0
0
aa*(t)
0
6x6
-
a*
t= -r
(5-5)
where
a*(t)
is
a*(t)
the vector of osculating
=
225
[a, h,k, p,q, ]
elements
(5-6)
4
At
first
order,
aa*(t)/a3*(t),
is
the
partial
derivative
4
given by
an
aa*(t)
I6x6
a_
=a(t I6x6 +
ga*(t)
where
matrix,
is
a
6x6
(5-7)
t~
9a*(t)
identity
matrix.
The
least
I
I
squares differential correction algorithm in GTDS
is
to
found
estimates
thesis,
of
converge
eq.
eq.
even
(5-7).
(5-7)
was
For
for
all
very
runs
approximated
identity matrix, so that eq.
crude
in
this
by
the
I
(5-5) reduces to
I
K
aX(t)
I
0
aa*(t)
aX(t)
06x6
aa*(t) '
-
A=
I
I
3-*(t)
6x6
-; 0
t=T
41
(5-8)
41
The
partial
differential
derivatives,
equation
aa*(t)/a *0 ,
are
governed
by
the
[22]
226
I
d
[a* (t)1[
-aa*(t)
a* 0
a* (t)1
a* (t)
0
(5-9)
with the initial conditions
a*(t0
The
partial
observation
integrated
derivatives
times
output
3[f*(t)]/a1*(t),
double-sided
(5-10)
6x6
a*
by
of
is
eq.
are
obtained
interpolating
(5-9).
computed
in
The
GTDS
at
on
the
the
matrix,
using
a
finite differencing algorithm where
the mean elements are varied by an amount
227
[221,
4
=
Aa*(t)
10-
(5-11)
a*(t)
4
4)
After
the
all
the
revised
observations
estimate
to
have
the
been
mean
processed,
elements
at
4
epoch is given by
4
a*
=
a* 0+
i*
(5-12)
4
This new estimate
in
replaces
the
a priori
estimate
Step 2.
4
5)
Steps
2
through
4
are
until
repeated
a
convergence criterion is met.
4
5.2
The Computation of Third Body Ephemerides
In
GTDS,
the
geocentric
position
and
velocity
I
components
of
the
sun
and
moon
are
usually
obtained
by
evaluating Chebyshev polynomials.
The coefficients of these
polynomials
ephemerides
are
computed
from
supplied
on
magnetic tape by the Jet Propulsion Laboratory (JPL) and are
stored on permanent files.
At the present time,
I
these files
can accomodate requests for third body ephemerides that fall
228
4
within
If
a period between
1 January
long numerical
integrations
to
arbitrary
very
option
choose
an
1971
and
14 January
are necessary or
epoch
of
1984.
if
the
integration
is
desired, than an extended capability is required.
For
chapter,
the
the
analytical
moon.
numerical
extended
theories
capability
for
the
theories
These
integrations
secular
are
Supplement to the ,Astronomical
on the solar tables of Newcomb
taken
provided
motion
Ephemeris
by
of
the
from
[28]
in
the
the
this
use
sun
of
and
Explanatory
and are based
and the lunar tables of
The mean equinoctial elements
Brown.
moon,
is
discussed
for the
sun and
in mean ecliptic of date coordinates, are given by:
Mean Solar Elements
a
=
km
h'=
0.01675104 sinr
k'=
0.01675104 cosr
p
0.0
1=
=
q
'=
where
149598412.7
r
is
0.0
2790.69668 + 0*.9856473354 d + 0*.000303 C 2
the
longitude
of pericenter
according to
229
which
is
computed
4
2810.22083 + 00.0000470684 d + 0*.000453 C 2
=
r
4
and
4
d
=
Julian days from 1900.0
C
=
Julian centuries from 1900.0
(C = d/36525)
a
Mean Lunar Elements
a'
=
4
km
384388.1743
h'=
0.054900489 sinr'
k'
0.054900489 cosr'
p'
=
0.0449322554 sinil
q
=
0.0449322554 cosQ
'=
270*.434164 +
g
13*.1763965268
0.001133 C 2
d -
4
where r'
is the longitude of pericenter given by,
r
=
+ 00.1114040803 d -
3340.329556
C2
00.010325
4
and
9
is
the
longitude
of
the
lunar
ascending
node
on
the
ecliptic,
a
S=
259* .183275
-
00.0529539222
d + 00.002078
C2
a
230
At
each
elements,
two
evaluation
body
mean
ecliptic
velocity
of
vectors
equatorial
the
relations
rectangular components
in
of
third
are
T
date
are
coordinates
coordinates.
then
by
transformed
rotating
Further
23*.452294 -
-
00.00000164 T 2 +
+
100.000000503 T 3
the
to
compute
the
them
depend
Julian
on
the
position
to
mean
through
and
of
date
the
mean
[28],
T -
0*.0130125
of
The
centuries
since
1950.0.
reference
system
of
the
integration.
Analysis of
This
order
number
rotations
numerical
5.3
=
is
used
equinoctial
of the position and velocity vectors
obliquity of the ecliptic given by
where
body
the Numerical Results
section describes
third body averaging
integration,
for
five
test
the
theory,
orbits.
performance
in
of
the
first
comparison with Cowell
These
test
satellite
orbits demonstrate the ability of the theory to predict long
term motion accurately and efficiently over a wide range of
orbital geometries.
231
4
For
each
computed
by
and
Cowell
the
ephemerides
the
does
test
same
of the
the third body
method
for
integration.
not
Except for the ISEE
all
orbit,
enter
both
the
any
averaging
the
Hence,
into
ephemerides
of
source
the
were
theory
of
the
comparisons.
test case, which used the JPL ephemeris,
resulss of
this
section
were
obtained
using
the
analytical ephemeris described in Section 5.2.
232
4
5.3.1
IUE Test Case
The
International
(IUE)
Explorer
Ultraviolet
orbit
provides an accuracy baseline for the third body AOG and SPG
before
initial
moving
on
osculating
to
more
elements
demanding
of
the
cases.
test
orbit
are
defined
Table 5-1.
Table 5-1. Epoch Osculating Elements
for the IUE Test Case
a=
42143.48243 km
e=
0.2353378723
Epoch = 7 March
1972
Ph.0,0m.0,0s.0
28.301165210
i = 199.61370900
Period
=0.996 day
264.69887660
270.4208456*
Radius of Perigee = 32225.5 km
Radius of Apogee = 52061.4 km
Lunar Parallax Factor (a/a')
Solar Parallax Factor
233
~
0.1
(a/a')s ~ 0.0003
The
in
4
The IUE orbit was
chosen to demonstrate the following
points:
1)
The
validity of
the PCE
procedure
for
obtaining
I
epoch mean elements from osculating elements.
2)
The
ability
of
high
precision
long
in
the
AOG
to
trajectory
comparison
to
track
over
the
the
an
time
mean
arc
span
of
a
which
is
the
PCE
of
initialization.
I
3)
The
accuracy
of
the
third
body
short
periodic
model.
I
4)
The remarkably long
integration step size that
is
permitted by the third body averaging theory.
The
initial
mean elements
for
the averaged
of motion were obtained by performing a PCE
of
high
spaced
precision
over
initialization
sixty
days.
were
osculating-to-mean
quadrature
position
to
fit
equations
to 184
and
velocity
components
The
a priori
elements
provided
by
the
GTDS
sets
evenly
for
the
numerical
conversion which used a 96 point Gaussian
average
period of the satellite
the
osculating
t29].
trajectory
over
one
The perturbation models used
234
61
in
the
PCE
are
given
in
Table
5-2.
The
truncation
parameters N, M* and R*, in the lunar and solar models, were
chosen on
the basis of experience and the relative
parameters
such
as
eccentricities.
the
third
Given
the
maximum ranges on the
by
the
4.
reflects
The
frequency
symmetry of
the present
to handle only
indices,
noted
that
models
in
the parameters
Table
truncation
improving
5-2
are
studies
numerical
the
capability
symmetrical
and
satellite
truncation
inequality relations in eqs.
Chapter
Model
body
actually
are
efficiency
(see Chapter 6).
235
parameters,
and
the
are given
(4-264) and
(4-269) of
of the
specified
t
orbital
s,
ranges
frequency
sizes of
chosen
third body
fields.
for
for
the
without
with
PCE
software
It should be
lunar and
conservative
required
the
a
sacrificing
view
solar
choices.
towards
accuracy
4
Table 5-2. PCE Perturbation Models for the IUE Test Case
1 ) AOG + SPG Perturbation
N
M*
R*
-5 <
-5 7
N
Model:
EARTH
MOON
SUN
=6
=6
=4
t < 5
s 7 5
N
=
M*=
R*=
-5 < t
-5 7 s
E
10
8
4
< 5
7 5
J2
NOTE: AOG retains only
double averaged terms
for the third body
S maximum power of the parallax factor, (a/a')
Hansen coefficient
M*
upper bound on the satellite
d'Alembert characteristic
R*
upper bound on the third body Hansen coefficient
d'Alembert characteristic
2) Cowell Perturbing Acceleration:
point mass sun-
point mass moon
J2
236
a
Since the IUE orbit is non-resonant with respect to a
body,
third
averaged
the double
only
are
terms
retained
The mean
the third body AOG for the initialization process.
elements determined in
order
solution, these
judge the quality of the converged
to
In
Table 5.3.
are shown in
the PCE
in
mean elements were used to initialize an integration of the
Short periodic recovery at the output times of the
the PCE.
approximation
to
to
the
osculating
a
Cowell
Representative
element
through
The
symbol
by
the
within
5-4.
(+),
(*) .
(*).
the
averaging
oscillation
in
are
trajectory
Whenever
of
the
the
the
over
semi-analytical
resolution
overwrites the
of
the
while
symbol
the
Cowell
which
trajectory
prediction
histories
semi-analytical
a
produced
integration
numerical
compared
span of
the sixty day fit
of motion over
averaged equations
is
Figures
is
trajectory
by
then
5-1
the
denoted
agree
histories
scale,
then
span.
same
represented
element
plot
in
shown
was
the
to
(+)
The plots demonstrate the high accuracy
orbital
the
particular,
In
theory.
eccentricity
the
semi-monthly
and
inclination,
induced by the motion of the moon, is seen to be reproduced
by
the
third
demonstration
element
body
short
of the
fit
difference plots
periodic
model.
span accuracy
in Figures
5-5
is
A more
graphic
provided
by the
through
5-8.
The
element differences are very small fractions of the elements
themselves. For example, the semi-analytical semi-major axis
237
41
from
differs
its
generated
at
point
meters
than fifteen
span.
Similarly,
degree
of
any
counterpart
within
the
by
more
no
computed
the PCE
fit was
longitude.
mean
is the
accurate
fit
sixty day
the averaging theory holds to within
Cowell
the
that
evidence
Cowell
4
10-4
Further
absence of an
4
appreciable bias in the element difference plots.
Table 5-3.
Epoch Mean Elements
for the IUE Test Case
Keplerian
Equinoctial
a = 42143.09386 km
a = 42143.09386 km
e = 0.2357026946
h = 0.2284825453
i = 28.27923713*
k
=
199.67553290
=
-0.0578920263
p = -0.0848190305
= 264.5426255*
q = -0.2372094942
7 = 14.70681220
M = 270.4886538*
4
elements
The PCE procedure adjusts the mean satellite
at epoch
to produce
generated data.
averaging
theory
the "best" semi-analytical
fit
to Cowell
As a result, the observed accuracy of
an
sufficient
to
over
the
fit
span
is
not
a
I
238
a
Osculating Semi-Major Axis Comparison within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell
Figure 5-1.
GTOS
COMPARE
PROGRAM
1000000
SATELLITE IUE
KEPLERIAN ELEMENT HISTORIES
I.------------------------------------------------------------------------------------------------42145.484
-
I
I
I
42144.884
42144.284
4*
++4
.+
.I
4.
++,
'I
-
+
I+
+
I+
I+
42143.084
-
+
4.+
4.
42143.684
+
+
+.
+
+
.
+
4+
+
+.
+.
+.
+.
*
+
-
+
I+
+.
+.
42142.484
I
+4
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++
42141.884
I
++
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42141.284
+
I
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42140.684
4
+
I
.
-
I
-.
42140.084
42139.484
0.
----
7.5-00
15.s*_00
22.-,5-0--- 3 0._0 0 ---
00
37.5S_0--- 45*.
TIME FROM YYMMOD
720307
5--s2.50o---
HHittSS IN DYS
0
()=Cowell
()=AOG & SPG
239
60.00 ---
67.5-0 --- 75.00
4
Figure 5-2.
C
Osculating Eccentricity Comparison within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell
C
GTDS COMPAREPROGRAM
SATEL.ITE
KEPLERIAN ELEMENTHISTORIES
IUE
1000000
I.--.------.---------.---------.---------.---------.---------.---------.---------.---------.---------.- I
2.354999E-1
2.352999E-1
.
I +++
I+ * ++
I
.I
I
I
I
I
2a50999-1
2.348999E-1
E
C
C
E
N
I
I
I
I
.
+
+
I
I
I
I
+
+
+.++4
+
+4
+
+4
+++
+
2.344999E-1
44
44
4 +4
++
++
C
I
T
Y
+
+++
2.342999E-1
2.340999E-1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
2.338999E-1
2.336999E-1
I
4+4+
2.346999E-1
T
R
a
is
a
2.334999E-1
I.---------------------------------------------------------------------------------------------------I
67.50
60.00
52.50
45.00
37.50
30.00
22.50
15.00
7.300
0.
4
75.00
TIME FROMYYMMDD HHMMSS IN DYS
720307
0
(+) = Cowell
a
(*) = AOG & SPG
240
C
C
Figure 5-3.
Osculating Inclination Comparison within the
60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell
GTDS COMPARE PROGRAM
SATELLITE
KEPLERIAN ELEMENT HISTORIES
I ---------.-----------------28.31499
I
+
28.30749
+
+4
+++
.+
I+
I
++
++
+
4
2+
28.29999
+
4+
+++++
+
+
++
+
I
- - - - - - - - - - - - - -
.I++
+
I
1000000
IUE
----------------------------------------------------------
I-
+
+4+4
++4
+
+++
+
28.29249
I
I
I
28.28499
I
I
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I
I
+++4
28.27749
+
I-
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I.
I
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28.26999'
I
I
I
I
I
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I
I
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I
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I
28.26249
28.25499
28.24749
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28.23999
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0.
7.500
15.00
22.50
30.00
37.50
45.00
TIME FROMYYMMD0 HHMMSS IN- DYS
720307
0-
(+) = Cowell
(*) = AOG & SPG
241
52.50
60.00
67.50
--. I
75.00
4
Figure 5-4.
4
Osculating Mean Longitude Comparison within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical versus Cowell
4
GTDS COMPARE PROGRAM
SATELLITE
EQUINOCTIAL ELEMENTHISTORIES
IUE
1000000
I.---------.---------.---------.---------.------- ------- ---.---------.---------.--- ------.---------.- I
200.0
I
160.0
I
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I
I
+
r
41
I
I
I
I
120.0
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I+
+
80.00
41
I
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I
40.00
I
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0.
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+
-40.00
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I
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-80.00
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4
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I
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I
I
-160.0
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a
-200.0
.---------.---------.---------.---------.---------.---------.---------.---------.---------.----------75.00
60.00
67.50
52.50
37.50
45.00
30.00
15.00
22.50
7.500
0.
TIME FROMYYMMD0 HHMISS
720307
0
IN DYS
I
(+)
C*)
242
=
=
Cowell
AOG & SPG
a
a
Osculating Semi-Major Axis Differences within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical minus Cowell
Figure 5-5.
STDS
15.00
COMPARE
PROGRAM
KEPLERIAN ELEMENTDIFFERENCES
I--------------------------------------------------.'
SATELLITE
---------------
-
IUE
1000000
£
I
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12.50
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10.00
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7.500
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TIME FROM YYMMOO
720307
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HHMMSS
0
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IN
DYS
---------------60.00
52.50
67.50
75.00
4
4
Osculating Eccentricity Differences within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical minus Cowell
Figure 5-6.
4
COMPARE
GTDS
PROGRAM
SATELLITE
KEPLERIAN ELEMENT DIFFERENCES
---------------.
I. ---------.-
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a
244
41
Osculating Inclination Differences within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical minus Cowell
Figure 5-7.
GTOS COMPARE PROGRAM
1000000
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245
I
I
Osculating Mean Longitude Differences within
the 60 Day PCE Fit Span for the IUE Orbit/
Semi-Analytical minus Cowell
Figure 5-8.
4
GTOS COMPAREPROGRAM
SATELLITE
EQUINOCTIAL ELEMENT DIFFERENCES
I.---------.---------.------------------.
- ------.----.
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7.500
TIME FROMYYMMDDHHIISS
720307
0
IN DYS
4
4
246
4
ensure
acceptable
mean elements
year
AOG
behavior
in Table
prediction
for
5-3
to
a
were
test
longer
used
the
integration.
to initialize
ability
of
The
a
the
three
averaged
equations of motion to predict the mean of the corresponding
high
precision
than
the
trajectory
time
span
of
truncation parameters
models
are
compare
the
plots
inclination
symbol
(*)
same
for
as
semi-major
satellite
axis
the
that
much
longer
initialization.
The
for
given
in
the
Table
axis,
through
Cowell
trajectory
output
of
since
element
inclination are
motion
of
high
the
sun.
5-12
differences
to
all
for
the
trajectory
very
interval.
is
The
5-9
5-14.
further
5-11
and
The
the
on
the
averaged
in the
high
and
the
over
the
mean of
the
difference
5-11
verification
mean
introduced by the
follows
symmetry
the
symbol
The
from the
element
and
where
dependence
closely
through
solar
Element
eccentricity
the AOG
through
5-2.
day oscillations
Evidently,
Figures
lunar and
AOG.
the
the semi-annual variations
integration
Figures
180
histories
precision
corresponding
The
is
eccentricity,
mean longitude has been removed
precision
year
and R*
constant
perturbation models.
the
PCE
Figures 5-9
mean
is
arc
semi-major
in
(+) represents
the
M*
those
the
an
the
N,
are given
represents
over
are
of
of
three
plots
found
the
in
element
the
PCE
initialization and the accuracy of the averaged perturbation
247
a
a
Comparison of the Mean and Osculating SemiMajor Axis Histories for the 3 Year Integration
of the IUE Orbit/AOG versus Cowell
Figure 5-9.
a
GTOS COMPAREPROGRAM
SATELLITE
KEPLERIAN ELEMENTHISTORIES
------------------------I
I
I
I
I+
I
a
++
4
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I
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I
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61
I------------------ ------------------------------------------------------------------------1500.
1350.
1200.
1050.
900.0
750.0
600.0
450.0
300.0
150.0
0.
TIME FROMYYMMDD HHMMSS IN OYS
0
720307
6
(+)
(*)
=
=
Cowell
AOG
6
248
6
Comparison of the Mean and Osculating
Eccentricity Histories for the 3 Year
Integration of the IUE Orbit/AOG versus
Cowell
Figure 5-10.
GTDS COMPAREPROGRAM
SATELLITE
KEPLERIAN ELEMENT HISTORIES
I.---------------------------------------------------------------------------------.----------I
1000000
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2.3700E-1
I+**
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3 00. 0 ---
450. 0 ---
600.0 ---
TIME FROM YYMMDD
720307
()=
()=
249
900.0--- 105-0.
750.0 ---
HMMS
IN DYS
0
Cowell
AOG
1200.
1350.
1500.
Comparison of the Mean and Osculating
Inclination Histories for the 3 Year
Integration of the IUE Orbit/AOG versus
Cowell
Figure 5-11.
0
GTOS COMPAREPROGRAM
IUE
SATELLITE
KEPLERIAH ELEMENTHISTORIES
I --- ------------------------------------------------------------------------------------------28.3500
28.2900
I
I
I
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I
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600.0
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TIME FROMYYMMODHMMSS
0
720307
900.0
1050.
1200.
1350.
1500.
IN OYS
(+) = Cowell
(*) = AOG
40
250
Figure 5-12.
Differences Between the Mean and Osculating
Semi-Major Axis Histories for the 3 Year
Integration of the IUE Orbit/AOG minus
Cowell
GTDS COMPARE
PROGRAM
KEPLERIAN ELEMENT DIFFERENCES
SATELLITE
IUE
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I---------.---------.---------.---------.---------.---------.---------.---------.---..------.------..-
1900.
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150.0
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450.0
600.0
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---------.---------.----
750.0
900.0
TIME FROM YYMMODHNMMSSIN DYS
720307
0
251
-----.--------------
1050.
1200.
1350.
1500.
0
0
Differences Between the Mean and Osculating
Eccentricity Histories for the 3 Year
Integration of the IUE Orbit/AOG minus Cowell
Figure 5-13.
a
GTDS COMPAREPROGRAM
SATELLITE
KEPLERIAN ELEMENTDIFFERENCES
1000000
IUE
I.---------.---------.---------.-------------------.---------.---------.-------------------------------
1.000E-3
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150.0
300.0
450.0
600.0
750.0
TIME FROMYYMMDD HNMSS
720307
0
900.0
1050.
1200.
1350.
1500.
IN DYS
0
0
252
Figure 5-14.
Differences Between the Mean and Osculating
Inclination Histories for the 3 Year
Integration of the IUE Orbit/AOG minus Cowell
GTOS
COMPARE
PROGRAM
KEPLERIAN ELEMENTDIFFERENCES
I.-----------------------------------------------------------------------900.0
.
I
**
I
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I
SATELLITE IUE
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TIME FROM YYMMOD
720307
253
HHMMSS
0
900.0
IN OYS
1050.
1200.
1350.
1500.
0
the
element history plots
the
altitude
semi-major axis
caused by third body perturbations amount to
about
of
comparison
periodic
have reproduced
resolution of
three
short
Similarly,
in
eccentricity
variation
the
induced
to
by
within
the
notably
5-10,
sun,
produce
eccentricity
the
resolution
evidence that
in
of
an
working
well.
the
mean
to
corrections
periodic
Figure
is
model
semi-annual
the
approximation
to
that
is
5-16
Figure
the
the
in
but two points
is powerful
This
periodic
short
generated
Cowell
exact
scale, at all
the plot
body
to within the
Cowell generated element,
prediction span.
year
third
the
the
axis
semi-major
mean
constant
the
to
corrections
short
that
shows
5-15
Figure
with
5-9
Figure
A
kilometer.
1
as
vary
to
axis
semi-major
much
as
by
value
mean
its
the
cause
themselves
by
lunar short
The
a large percentage of the total variation.
periodics
the
in
variations
periodic
short
to
At
through 5-17.
in Figures 5-15
the
IUE,
of
leads
AOG
the
of
precision
high
the
of
times
compare
the
at
elements
orbital
recovery
semi-analytical
A
models.
plot
scale.
A
comparison of the inclination plots in Figures 5-11 and 5-17
another example of the accuracy with which the
provides yet
semi-annual
SPG.
variations have been modelled
Hence,
indications
predictions
the
that
that
three
year
the
the third body
integration provides
accurate
extremely
include
in
long
third body models
254
term
the
first
orbital
developed in
0
0
Osculating Semi-Major Axis Comparison for
the 3 Year IUE Integration/Semi-Analytical
versus Cowell
Figure 5-15.
STDSCOMPARE PROGRAM
KEPLERIAN ELEMENTHISTORIES
42144.984
SATELLITE
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I
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150.0
300.0
450.0
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600.0
750.0
TIME FROM YYMttDO HHSS
720307
0
(+)
(*)
255
=
=
900.0
1050.
IN DYS
Cowell
AOG & SPG
1200.
1350.
1500.
Osculating Eccentricity Comparison for the
3 Year IUE Integration/Semi-Analytical versus
Cowell
Figure 5-16.
OTDS COMPAREPROGRAM
KEPLERIAN ELEMENTHISTORIES
--------------------------------------I.
---------.
- --------
SATELLITE IUE
- -------------
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I++
2.3400E-1
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+++
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2.1900E-1
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2.1300E-1
2.1000E-1
-
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0.
150.0
300.0
450.0
600.0
750.0
TIM FRMt YYMD
720307
900. 0
HHM0SS IN DYS
0
()= Cowell
()= AOG & SPG
256
1050.
1200.
1350.
1500.
Figure 5-17.
Osculating Inclination Comparison for the
3 Year IUE Integration/Semi-Analytical versus
Cowell
GTDS
KEPLERIAN ELEMENTHISTORIES
---------. ---------.
------------28.3500
I.
COMPARE PROGRAM
----
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150s.
a
can
thesis
this
spans
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be
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for
file
Cowell
The
150 time regularized
generated from an integration that used
steps
a
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possible
from
a
used
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circulation
fast
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frequencies
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of
equations
Since
the
as
the
such
angles,
satellite
made
was
This
year.
one
contrast,
In
eccentricity.
of
for
periods
a
it is not unreasonable for
size
body
is
This
satellite.
although
moderate
step
removing
third
the
of
orbit
satellite
the
size,
step
conservative
tne
of
revolution
per
was
comparison
year
three
the
argument of perigee and the longitude of the ascending node,
generally
are
quite
the
AOG
was
retained
in
perigee.
Hence, an AOG
size
has
the
the
altitude
of
considerable
8.9
year
the
IUE,
of
amplitude
of
advance
lunar
step size of one year was permitted
without loss of accuracy.
step
at
oscillation
period
shortest
long
enormous
This remarkably large integration
consequences
for
the
efficient
analysis of high altitude orbital stability over decades or
centuries.
A
one
hundred
year
AOG
prediction
of
the
IUE
orbit was made with a one year integration step size, using
the epoch
33
seconds
mean elements of Table
of
CPU
time.
The
5-3.
element
eccentricity and inclination are plotted
5-19.
The
AOG executed
histories
for
in
the
in Figures 5-18 and
At present, there are no precision benchmarks against
258
Figure 5-18.
Evolution of the Mean Eccentricity for the
100 Year AOG Prediction of the IUE Orbit
GTDS COMPAREPROGRAM
KEPLERIAN ELEMENTHISTORIES
I.-------------.---------.----------------------------------2.5000E-1.
I
I*
2.3000E-1
I
*
.
I *
I
SATELLITE IUE
1000000
----------------------------.
I
*
**
N
N
-I
2.1000E-1.
I
1.00E1 .
I
1.7000E-1
E
C
C
E
N
T
R
I
C
I
T
y
.
*I
I
*
*
*
-N
.
I
I
N*
**
*
I
1.5000E-1
I
I
*
*
**
**
N****
I
I
**
***
*
*
I
*
*
I
*
**
I
I
*
*
*
-I
I
*
*
I
I
I
I
*
**
1.3000E-1
.
1.1000E-1
.
9.OOOOE-2
I.
I. -- - - - ..
.
50I
7.0000E-2
5.0000E-2
I259
- - - - ..0E4
150+
- - - -I-.0E4
250+
TIEFOIY
72007I
- - .300+
I
Y
.0E4
400+
.0E4
500
IDHMMS
- --
- . - - -
0
6
Evolution of the Mean Inclination for the
100 Year AOG Prediction of the IUE Orbit
Figure 5-19.
GTDS
COMPARE PROGRAM
SATELLITE
KEPLERIAN ELEMENTHISTORIES
IE
1000000
----.-------. ------------------.---------.---------. I
.---.------.------.....--------...-----
45.000
*
.
42.500**
*-
40.000-
N
37.500-
CI
I
*
L
I
**
I
A
T
0
N
G-
*
'
DI
*
35.000
I
*
*
*
*
*
RI
EI*
I
-
32.500
N
E
30.000
E
S
.
-
*
I
*
27.500
.
-
25.000
.
-
32.500
20.000
-
.
0.
-
5000.
1.000E+4
2.500E+4 3.000E+4
2.800E+4
1.500E+4
TIME FROM YYMMOD HHMMSI IN DYS
0
720307
260
3.500E+4
4.000E+4
4.500E+4
5.000E
which
the
correctness
this
of
be
can
prediction
measured.
is
However, the basic regularity of the element histories
a
good indication that their deviation from the physical world
Furthermore, the 54 year period
is at least a slow process.
of
oscillation
for
inclination, predicted
integrated.
addition,
associated
Cowell
corresponding
The
5-19.
It
the
with
computer,
would
be
build-up
numerical
of
computations
make the
suspect.
261
output
time
seen
in
Figure
cannot
consuming.
error
on
[30]
by Kozai
trajectory
prohibitively
intensive
would
clearly
is
orbit,
geosynchronous
the
length
in the
a
points
over
fixed
be
In
time,
word
increasingly
0
ISEE Test Case
5.3.2
61
Sun-Earth
International
The
demonstrates
the
accuracy
when
the
satellite
first
the
of
orbit
(ISEE)
Explorer
body
third
order
S
theory
orbital
eccentricity
is
large.
The initial orbital elements are shown in Table 5-4.
Table 5-4. Epoch Osculating Elements
for the ISEE Test Case
Epoch = 7 March 1972
a = 70850 .0 km
oh.0,0m.0,0s.0
e = 0.89
i
= 29.0*
= 49.50
i
Period = 2.2 days
= 0.21*
S
M = 0.0
Radius of Perigee = 7793.5 km
Radius of Apogee = 133906.5 km
Lunar Parallax Factor,(a/a')~ 0.18
Solar Parallax Factor,(a/a')s~ 0.00047
The epoch mean elements for the AOG were obtained by
a PCE procedure over sixty days.
The a priori elements
0
for
the initialization were provided by averaging the osculating
over
trajectory
numerical
models
one
of the
period
osculating-to-mean
used
in
the
PCE
are
satellite
conversion.
shown
in Table
using
The
the GTDS
perturbation
5-5.
The
ISEE
262
0
orbit
is non-resonant
only
the
double
retained
PCE
with respect
averaged
in the AOG.
are
given
initialize
in
an
A
Table
time
was
of
the
for
These
year
the
third
Cowell
using
elements
integration
A one year
precision
performed
revolution
5-6.
eight
high
a third body
so that
body
The mean elements determined
equations of motion.
used.
terms
to
200
of
are
from the
were
used
the
averaged
to
integration step
size
integration
the
same
steps
per
time
satellite.
over
regularized
The
five
slow
was
Keplerian
element histories are shown in Figures 5-20 through 5-24 for
both
the
AOG
and
Cowell
are represented by the
are designated
(+)
within
are
the
seen
element
This
to
order
axis
point
point
of
orbital
effects.
epoch
occurs
in
400
variation
at
the
at
perigee
period,
is
the
Since
extremely
PCE
axis
of
was
set
short
the
procedure
by
by
J2
over
to
the
kilometers.
periodic
is
on
which
comparison
tended
of
the single
short
time
in
to
osculating
scale
70850
caused
the
the
the
to
where
agree
the
The mean elements
that
5-20
elements
test case,
histories
mean
corresponding
semi-major
Cowell
the mean elements
scale.
note
Figure
kilometers.
acts
two
the
will
plot in
while
the
follow
One
The
in the previous
when
(*)
closely
histories.
variation
As
resolution of the plot
semi-major
Cowell
the
(+),
symbol
a (*)
by
overwrites
predictions.
to
smooth
the
this
the
its
This accounts for the apparent discrepancy between
the single Cowell point and the predicted mean.
263
The absence
of
any
further points
selection
phenomenon.
Figures 5-21
the
lunar
the
in
evidenced
The
argument
of
is caused
and
behavior
inclination
by the
perigee which was
second
not
a point
is
5-20
oscillatory
pronounced
eccentricity
and 5-22
in Figure
at perigee
plots
of
multiple of
removed
by
the
averaging operation.
Table 5-5. PCE Perturbation Models for the ISEE Test Case
1) AOG + SPG Perturbation Model:
N
M*
R*
-5 <
-5 7
EARTH
MOON
SUN
= 5
=
=
t
s
5
4
< 5
7 5'
N
M*
R*
-6 <
-6 7
= 12
=
=
t
s
J 2 ,J3 J 4
12
6
< 6
7 6
NOTE: AOG retains only
the double averaged
terms for the third body
N
maximum power of the parallax factor, (a/a')
M*
upper bound on the satellite Hansen coefficient
d' Alembert characteristic
R*
upper bound on the third body Hansen coefficient
d'Alembert characteristic
2) Cowell Perturbing Acceleration:
point mass sun
point mass moon
3,J 4
J2 I.J
264
01
Table 5-6.
Epoch Mean Elements
for the ISEE Test Case
Keplerian
Equinoctial
a = 70405.55206 km
a = 70405.55206
e = 0.8908091435
Ih=
i = 29.24922760*
k = 0.5734846282
=
km
0.6816570336
p = 0.1984228809
49.501046560
W = 0.42472839970
q = 0.169462882
M = 359.2566209*
T = 49.18239586*
A
argument
look
of
at
the
perigee
element
and
plots
longitude
of
for
the
the
ascending
satellite
node
reveals that their circulation periods are approximately 16
years
year
and 21 years respectively.
integration step size
for
an orbit
the
low perigee
benefit
of
with
strong
height.
retaining
only
for
This indicates that a one
the AOG
is
appropriate
zonal perturbations
Hence,
the
it
arising from
would appear
double
averaged
even
that
third
the
body
terms in the AOG is not diminished by the inclusion of other
265
0
6
Comparison of the Mean and Osculating SemiMajor Axis Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
Cowell
Figure 5-20.
GTOS
COMPARE
PROGRAM
SATELLITE
KEPL.ERIANELEMENT HISTORIES
70850.00
ISEEAU
1000000
.+
I
.
I
.
I
.
I
0
70795.00
70740.00
70685.00
I
.
I
-
-
I
70630.00
I
0
70575.00
70520.00
70465.00
I
I
4
4
4
70410.00
70355.00
.I
I
+
444
+++
4
++4+++++
4
44
+4+
+4
4444
+
+4
++
+4
++
+
+
I
I
I
I
4
4+4+
+
+
++
I
I
I
I
++
I
I
70300.00
--------------.---- -----.
I.---------1500.
1000.
500.0
0.
--------.
2000.
--------
2500.
.---------
3000.
TIME FROMXYMMDD HHMMSS IN DYS
720307
0
(+) = Cowell
(*) = AOG
266
.--------.--
3500.
4000.
4500.
5000.
Figure 5-21.
Comparison of the Mean and Osculating
Eccentricity Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
Cowell
STDS
COMPARE
PROGRAM
SATELLITE
KEPLERIAN ELEMENT HISTORIES
ISEEAS
1000000
----.------ ---.------------------------------ I
-.----------------.-----
I.--------------------------
8.95000E-i
I
I
I
I
I++
I+
8.89000E-1
.+
I4*
I+
I
I
I
I
4*++
8.83000E-1
.
I
+
I+
E
C
C
N
T
R
I
C
I
T
y
8.83000K-i1
8.77000E-1
I
I
.+
I
I
4
+
+*
I
8.77000E-1
+ *a*
+
+
+
+'
**
+
+
+*
+
I
I
I
I
+*
*
+
4
+4
*
+
*
4
*
*+
*
I
I
I.
I
+
++4
+
*
+*
+ 44+
*
+w*
a *+
*4
+4
8.7000E-1
+'
++4
+
+
++
*
*4
+
I
I
I
I
+4
*
I
I
I
I
0
*
*
+
*
I
.
I
I
I
I
+4
*+
444
+
+4
8.47000K-i
.
4
+
444
+4
*
,I
4'
+4
4+4
4
4
I
I
I
I
+44
*
8.59000K-i
I
I
I
I
4
*444*4
+
*
+
+
4
I
I
I
I
I
I
I
I
8.35000E-1
-- -.-------------------------------------------------------------------------------------------.
4500.
5000.
4000.
3000.
3500.
2500.
1500.
2000.
1000.
0.
500.0
TIME FROM YYMMOO
720307
HHMMSS
0
IN DYS
(+) = Cowell
(*) = AOG
267
Comparison of the Mean and Osculating
Inclination Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
Cowell
Figure 5-22.
U]
GTOS COMPARE PROGRAM
KEPLERIAN ELEMENTHISTORIES
SATELLITE
ISEEAB
1000000
I.---------.---------.---------.-------- -.---------.--- ------.---------.---------.---------.---------.- I
45.000
.
I
I
I
I
I
++
42.500
I
*+
+*+++
+
+4++
+4,
I
I
**+
*
I
+
+
+
I
++*+
+*
+
+
40.000
I
I
+
4+
*
++4
4
+++
*
*
+
*
*
35.500
. I++
I+
I
I
I
I *
0
I
I
I*
I+
I+
I
I
I
27.500.
25.000
27.500
20.000
0.
500.0
1000.
1500.-
2000.
2500.
TIME FROM YYMMODHHMISS
720307
0
3000.
3500.
4000.
4500.
5000.
IN DYS
(+)
= Cowell
(*) =AOG
268
U]
Figure 5-23.
Comparison of the Mean and Osculating
Longitude of Ascending Node Histories for the
8 Year Integration of the ISEE Orbit/
AOG versus Cowell
GTOS COMPARE PROGRAM
KEPLERIAN ELEMENT HISTORIES
I.------------------.---------.---------.--------50.00
.+
35.00
I ++
I
I
I
.
SATELLITE
.
---------------------------------------------.
ISEEAS
1000000
I
II
I
I
+++
++,
I
I
I
I
20.00
T
I
I
I
4,
5.000
I
I
I
I
-10.00
I
I
.1
I
-25.00
I
I
I
I
.4
-40.00
I
I
I
-I
-55.00
I
I
I
I
4,,
-70.00
I
I
I
I
4,,
4,,,
-85.00
.
+++
I
I
I
I
-100.0.
I.---------------------------.---------.---------.---------.---------.---------.---------.---------.---I
0.
500.0
1000.
1500.
2000.
2500.
3000.
3500.
4000.
4500.
5000.
TIME FROMYYMMO0 HHIISS
720307
0
IN DYS
(+) = Cowell
(*) = AOG
269
Figure 5-24.
Comparison of the Mean and Osculating Argument
of Perifocus Histories for the 8 Year
Integration of the ISEE Orbit/AOG versus
Cowell
0
0
GTDS COMPARE PROGRAM
KEPLERIAN ELEMENTHISTORIES
I.-
SATELLITE
---------------------------- ---------------------------
ISEEAS
1000000
------------------ -----------------.
200.00
I
I
I
I
I
I
I
I
I
180.00
I
I
I
I
4.4
.4.
I
I
I
I
44,
0
.4.
I
I
I
I
.4
*4
.4
I
I
I
I
4.
60.000
4000
*4
I
I
I
I
I
I
I
10.000
I
I
I
I
I +
I+
I
I
I
I
*4
4.4
I
I
I
I
++
++
++4
I
I
I
I
0.
0.
500.0
#b
1000.
1500.
2000.
2500.
3000.
TIME FROM YYMMD HHIISS IN DYS
7203.7
0-
+)=Cowell
(*) = AOG
270
3,V0.
4000.
4500.
5000.
0
averaged
long
perturbation
term
models.
predictions
to
This
be
made
allows
with
a
very
efficient
realistic
force
field.
A
the
AOG
execute
comparison of
is
given
in
integration
that
the
execution
in
integration times between Cowell
Table
5-7.
approximately
over
the
eight
1/30
the
year
span.
comparison
is
time
includes
also
The
even
more
the
comparison plots described above.
271
AOG
is
time
of
It
the
should
favorable,
creation
observed
of
the
to
Cowell
be
since
and
noted
the
AOG
ephemeris
01
Table 5-7. Comparison of Cowell and AOG Execution
Times for the Eight Year ISEE Integration
Orbit Generator
Step Size
Execution Time
Cowell
200 steps/rev.,
300 seconds
time
regularized
31536000 seconds
AOG
a
9.95
seconds
(1 year)
Notes
1)
2)
3)
4)
The Cowell integrator is based on the 12th
order Adams-Bashforth Predictor /
Adams-Moulton Corrector Algorithm
The AOG uses a 4th order Runge-Kutta
integration algorithm
0
The execution time for the Cowell
integration contains overhead associated
with file creation.
The execution time for the AOG contains
overhead assciated with file creation and
0
includes the ephemeris comparison step.
S
0
0
272
5.3.3
VELA Test Case
The evolution and stability of the high altitude VELA
orbit are almost exclusively determined by the action of the
sun
and
the
perturbations
semi-major
moon.
can
axis
At
induce
that
are
this
short
on
altitude
periodic
the
order
lunar
and
variations
of
100
118230.0 km
Epoch = 8 April 1970
0.003
Oh.0,0m.0,0s.0
32.520
0.00
Period = 4.68 days
47.00
0.00
Radius of Perigee = 117875.31 km
Radius of Apogee = 118584.69 km
.Lunar Parallax Factor,
Solar Parallax Factor,
273
(a/a')y ~ 0.3075
(a/a')s ~ 0.0008
in
the
kilometers.
The initial orbital elements are shown in Table 5-8.
Table 5-8. Epoch Osculating Elements
for the VELA Test Case
solar
0
The initial mean elements for the AOG prediction were
The
5-9.
Table
in
given
argument
days.
Both periods
still
the
third
body.
in
the
long
in
SPG
are
in
the
SPG
has
resonance
the
with
circulates
critical
which
PCE
a
period
167 days, while the second multiple of the
of approximately
AOG
the
included
frequencies
argument of
The critical
priori
resonance of the VELA orbit with the
reflect the shallow 6:1
moon.
a
correction
in
used
models
perturbation
The
algorithm.
The
differential
squares
least
the
to
days.
the
as
used
were
elements
sixty
over
procedure
PCE
satellite
osculating
input
a
from
obtained
are far too short to be
only
retains
However,
it
since
comparison
period
a
to
the
the
produces
the
fit
double
resonance
terms
with
span of
included
for
be modelled
periods
the PCE.
in the
terms
averaged
should
83
around
of
are
that
Failure
to
model the effect can cause a noticeable bias in the computed
mean
elements
at
epoch.
The
mean elements
the PCE are found in Table 5-10.
274
determined
from
Table 5-9. PCE Perturbation Models
Table 5-9. PCE Perturbation Models
for the VELA Test Case
1) AOG + SPG Perturbation Model:
MOON
SUN
N
= 4
N
M* = 4
= 4
R*.= 8
R*
-4 < t < 4
-4 7 s 7 4
NOTE: AOG r etains
= 8
M* = 4
-4 < t < 4
-12 ~ t 7 12
only the do uble
averaged th ird body
terms
maximum power of the parallax factor,
N
(3/a')
M*
2
upper bound on the satellite Hansen coefficient
d'Alembert characteristic
R*
2
upper bound on the third body Hansen coefficient
d'Alembert characteristic
2) Cowell Perturbing Acceleration:
point mass sun
point mass moon
Table 5-10. Epoch Mean Elements for the VELA Test Case
Equinoctial
Keplerian
a = 118095.5461
km
a = 118095.5461
e = 0.0020558154
h = 0.0017593771
i = 32.48414636*
k = 0.0010634702
= 359.9590416*
p = -0.000208255
W = 58.889744020
q = 0.2913232484
M = 348.2477284*
7 = 47.096514020
275
km
0
to
used
were
5-10
Table
in
elements
mean
The
The
initialize a ten year AOG prediction of the VELA orbit.
numerical
integration
truncation
parameters
characteristic
the AOG prediction
are
and
are
PCE
the
in
used
those
d'Alembert
The upper bound on the
Table 5-11.
in
compiled
for
R*
M*,
than
different
slightly
N,
The
year.
one
of
size
step
a
used
for the satellite Hansen coefficient has been
increased to accomodate some growth in the satellite orbital
is not inconsistent with the model used in
third body model
the
since
PCE
the
orbit
VELA
Cowell
compare
fit
would
file
was
created
PCE.
slow
Cowell
was 2700
5-25
is
trajectory
ten
the
through
year
5-30
for
and
the
by
the
set
equinoctial
the
of
over
as
represented
while the mean output of the AOG is denoted by
symbol
(+),
(*).,
Figure
determined
Figures
in
elements
The
inclination.
plots
The
models
same
the
integration step size
comparison
are provided
integration
five
The
using
both.
by
achieved
be
same
the
epoch eccentricity of
small
very
the
Ephemeris
seconds.
the
at
the
in
used
those
new
The
span.
prediction
year
ten
the
over
eccentricty
the
5-25
mean
shows
the
that
semi-major
axis.
PCE
has
The
effectively
other
averaged
element histories show a similar ability to track the mean
of the Cowell generated elements.
the
k element
at
the end of
conditions error.
276
ten
The slight divergence in
years may
be
an
initial
Table 5-11. AOG Perturbation Model for the
Ten Year VELA Integration
SUN
MOON
N
= 4
M* = 4
N
= 8
M* = 6
R* = 4
R* = 8
N
NOTE: AOG retains only
the double averaged third
body terms.
maximum power of the parallax factor, (a/a')
M* 3upper bound on the satellite Hansen coefficient
d'Alembert characteristic
R*
upper bound on the third body Hansen coefficient
d'Alembert characteristic
Table 5-12 compares the execution times of the Cowell
The use of the first order third body
and AOG predictions.
theory
for
a
with a
ten
one year
year
run
by
step has
reduced
at
a
least
factor
the
of
execution
50.
time
Again,
it
should be noted that the AOG run time includes the ephemeris
comparison
step
so
that
the
efficiency
theory is even better than indicated.
277
of
the
third
body
0
0
Comparison of the Mean and Osculating SemiMajor Axis Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell
Figure 5-25.
0
GTDS COMPAREPROGRAM
SATELLITE
EQUINOCTIAL ELEMENTHISTORIES
.------
-------
---------
----
I. ---------------------------
1000000
VELA
'I
-----------------------------
118250.0
118215.0
118180.0
I
+
.
.
I
I
I
I
I
I
I
I
+.
118145A 0
I+
I+
+
118110.0
+. ++
++
+
I
I
I
I
+
.+
I
I
I
I
I +
IM*M
118075.0
.
I
.
+
+.
+
+
118040.0
I
I
I
I
++
I
I
4
+
+
I
'I
117970.0
I+
I
I
.
I
I
I
I
117935.0
I
I
I
I
I
117900.0
11790
0
I
I
I
I
+4+
+
I
118005.0
0
I
I
I
I
I+
I
.
--0.
--500.0
1000.
1500.
2500.
2000.
.--------------------4000.
3500.
3000.
4500.
0
is
1I---5000.
TIM FROMYYMMODHH1MMSSIN DYS
700408
0
0
(+)
(*)
=
=
Cowell
AOG
is
278
Figure 5-26.
Comparison of the Mean and Osculating h
Element Histories for the 10 Year Integration
of the VELA Orbit/AOG versus Cowell
GTDS COMPARE PROGRAM
1.300E-2
E 4INOCTIAL ELEMENT HISTORIES
w--."
SATELLITE
VELA
I
I
I
I
I
I
I
1.ZSOE-2
I
I
I
I
I
+
4.4.4.
4.
1.00OE-2
4.
+
4.
+4.
+
4.
*4.4.4.4.4.4.4.4. +
+
4.**4.
4.4.4.4.
4.
4.
+
+
4.
4.
4*4. 4.
4.,
4.4.
4.
4.
*+
+
I+*
I +
-
4.4.
4.4.4.
4*
7.500E-3
5.000E-3
1000000
-------
*4
+
*
+*++
I+
+
I
I
+
I
-2.SOOE-3
+
+*
++
.
0.
I.
-5.000E-3
+
+
I
*+
I
-5.00E3.I~
I
~
4.
*4.4
++
**
******
~~ ~~
44.
44
4
4.44.+
+*.*
+...
I
I
-I
I
a
-7.500E-3
I
I
I
I
I
I
I
I
-1.000E-z
0.
500.0
1000.
1500.
2000.
2500.
3000.
TIME FROMYYMMODHHMMSS IN DYS
700408
0
(+)
(*)
279
=
=
Cowell
AOG
3500.
4000.
4500.
5000.
Comparison of the Mean and Osculating k
Element Histories for the 10 Year Integration
of the VELA Orbit/AOG versus Cowell
Figure 5-27.
0
0
GTDS
COMPARE
PROGRAM
SATELLITE
EQUINOCTIAL ELEMENTHISTORIES
1000000
VELA
01
1.500E-2
I
I
I
I
1.300E-2
*1
I
I
I
I
I
1.100E-2
4.4.4.4.4.4.4.4. 4.
4.4.
4.4. 4.
4.
4.4.4.
4.
9.OOOE-3
4.
+
4.
4.4.4.4
+*.
4
4.4.
4.
4.
I
I
I
I
4.4.4.
4.4.4.
4.
I
I
I
I
4.
4.4.4.
4.4.
4.
*.4.
4.4.
7.000E-3
I
I
I
I
4.4.4.
k
4.4.
4.
5.000E-3
4.
.+*
4
+.
*.44
+*
4+ +.
3.000E-3
.+
I
4
4+.+4.
+.44.
I+
II+
1.000E-3
*++
+
I*
. +
+ +
++
+
*.
4
I
I
*.4
+
4.4*.+
+
+
+.
4++.
+.4.
4
.4+.
-3.000E-3
-1.000E-3
.4
.
.
4
4.
..
+
I+
I
+.4.
0
4.,
-5.000E-3
I.- 0.
----.-------.-1000.
500.0
..
.----------.------1500.
2000.
2500.
TIM FRO" YY1OD
700408
3000.
.---3500.
.----.-----.
4000.
4500.
5000.
HHMMSS IN OYS
0
0
()=Cowell
(*) = AOG
280.
Figure 5-28.
Comparison of the Mean and Osculating p
Element Histories for the 10 Year Integration
of the VELA Orbit/AOG versus Cowell
GTDS COMPARE
EGUINOCTIAL EtEMENTHISTORIES
I----------.0.
PROGRAM
SATELLITE
.
--------------------
VELA
-------------------------------.
.++
I
I
I
++~
.
I
I
I
I
I
I
I
I
*++4
++.
I
-1.5000E-2
1000000
++
+*
++
+
-3. 0000E-2.
4,
4.
-4.5000E-2
-6.0000E-2
I .
.
-7.5000E-2
.
++
4.,
4.*
4.,
r)
-9.0000OE-2Z
4.,
-1.0500E-1
.
-1. 2000E-1
.
++,
++,
-1.3500E-1
-1.5000E-1
I-----------------.---0.
500.0
1000.
-------------.
-------------------------.---------------------------.
1500.
2000.
2500.
3000.
3500.
4000.
4500.
TIME FROMYYMMDD
700408
HHMM3S IN OYS
0
()=Cowell
(*)
281
=
AOG
I
5000.
0
Figure 5-29.
Comparison of the Mean and Osculating q
Element Histories for the 10 Year Integration
of the VELA Orbit/AOG versus Cowell
0
0
GT6S
COMPARE PROGRAM
EQUINOCTIAL ELEMENT
HISTORIES
I. -- -
SATELLITE
------------------------------
--.---------.------
---
---------------------------.
VELA
1000000
--------
3.0000E-1.
I
++++*+++++++4+++++++*
I+++
I
I
I
I
I
+4
+++
0]
2.4000E-1
2.2000E-1
I.
2.0000E-1.
I-
CT
1.8000E-1
I
I
I
I
1.6000E-1
1.40OOE-1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1.2000E-1
1.OOOOE-1
0.
500.0
1000.
1500.
2500.
2000.
3000.
TIME FROMYYMtOD0 HHMMSS
700408
0
3500.
4000.
4500.
5000.
IN DYS
0
(*)
=
=
Cowell
AOG
282
09
Figure 5-30.
Comparison of the Mean and Osculating
Inclination Histories for the 10 Year
Integration of the VELA Orbit/AOG versus
Cowell
GTDS COMPAREPROGRAM
SATELLITE
KEPLERIAN ELEMENTHISTORIES
I.---------------------.
I
I
32.500
+++++++.4.* +
1000000
VELA
------------------------------------------------------
I
I
I
I
+ * ++++4.*
++++.4+*+
+4
.*+
I
I
I
I
*4.
31.500.
I
I
I
I
4.4.
30.500
.
29.500
.
++
I
I
I
I
++
I
I
I
I
28.500
I
I
I
I
4.
27.500
26.500
25.500
24.500
I
I
I
I
I
I
I
23.500
0.
500.0
--------------------.--------.---.-.-1000.
2500.
2000.
1500.
--------------- ------3000.
TIME FROMYYMMODHHMMSS IN OYS
700408
0
+)=Cowell
(*) = AOG
283
3500.
4000.
4500.
.
5000.
0
Table 5-12.
Comparison of Cowell and AOG Execution Times
for the Ten Year VELA Integration
Orbit Generator
Step Size
Execution Time
Cowell
2700 seconds,
fixed step
482 seconds
AOG
31536000 sec.
9.63 seconds
0
(1 year)
0
Notes
1)
2)
3)
4)
The Cowell integrator is based on the 12th
order Adams-Bashforth Predictor /
Adams-Moulton Corrector algorithm.
The AOG uses a 4th order Runge-Kutta
integration algorithm.
The execution time for the Cowell
integration contains overhead associated
with file creation
The execution time for the AOG contains
0
overhead associated with file creation and
includes
the ephemeris
comparison
step.
0
284
U1
5.3.4
STRATSAT Test Case
The
STRATSAT
orbit
is
discussed in this thesis.
presented
in
Table
During
this
5-13.
period,
to very large values.
to predict
its
epoch
case.
The
ability of
demanding
test
case
Extremely
strong
to decay within
the
lunar
ten years
eccentricity
grows
rapidly
is
evolves radically with respect to
demonstrated
the third body
by
the
theory
STRATSAT
5-13. Epoch osculating Elements for the
STRATSAT Test Case
a= 2 11868.8 km
Epoch = 21 March 1985
oh.0,0m.0,0s.0
e= 0 .001
i
=
90.0*
=
180.00
) =
M
=
Period = 11.23 days
0.00
0.00
Radius of Perigee = 211656.93 km
Radius of Apogee = 212080.67 km
Lunar Parallax Factor,
285
test
to converge for
high values of the parallax factor is also demonstrated.
Table
of
The capability of the third body AOG
an orbit which
conditions
most
The initial orbital elements are
perturbations cause the orbit
epoch.
the
(a/a')k ~ 0.55
0
The initial mean elements for the AOG prediction were
osculating
satellite
The
quantities.
given
PCE
a
from
obtained
in Table
procedure
elements
The
sixty
frequencies
The
a
priori
included
are
PCE
the
in
used
days.
the
as
used
were
models
perturbation
5-14.
over
in the SPG
reflect the shalluw 5:2 resonance of the STRATSAT orbit with
the
The
moon.
critical
of
argument
period of approximately 233 days, while the second
of
the
critical
which
retains
the
However,
prevent
circulates
with
multiple
a period of about
These periods are too short to be included in the
116 days.
AOG
argument
a
has
resonance
the
a
only
resonance
bias
in
elements determined
the
the
double
should
be
mean
modelled
elements
at
in
epoch.
from the PCE are given in
286
lunar
averaged
the
terms.
SPG
The
to
mean
Table 5-15.
9
Table 5-14. PCE Perturbation Models for the
STRATSAT Test Case
1) AOG + SPG Perturbation Model:
MOON
N
= 12
M* =
6
R* =
8
-4 < t <
-10 7 s 7
4
NOTE: AOG retains only the
double averaged third body
terms
10
N
maximum power of the parallax
M*
upper bound on the satellite Hansen coefficient
d'Alembert characteristic
R*
E upper bound on the third
2) Cowell Perturbing Acceleration:
287
(/a')
body Hansen coefficient
d'Alembert characteristic
point mass moon
factor,
0
Table 5-15. Epoch Mean Elements for the
STRATSAT Test Case
0
Equinoctial
Keplerian
a = 211134.9014
a = 211134.9014 km
km
e = 0.0044788188
h = -0.0022599939
i = 90.11755585*
k = 0.0038668133
= 180.0375365*
p = -0.000656481
= 149.6579277*
q = -1.002053629.
X = 178.67126280
M = 208.97579860
The
initialize
mean
an
elements
year
eight
Table
in
The
numerical
year.
The
truncation
parameters
found in Table
N,
5-16.
R*
M*,
The
size
for
to
STRATSAT
the
of
a step
used
used
were
5-15
prediction
AOG
integration
orbit.
prediction are
0
of one
the
maximum power
AOG
of
the parallax factor has been conservatively chosen to ensure
the
on
the
satellite
characteristic
to
The upper bound
accuracy of the eight year prediction.
accomodate
has been allowed
the
orbital eccentricity.
anticipated
288
d'Alembert
coefficient
Hansen
to
assume
growth
its
of
maximum value
the
satellite
Table 5-16. AOG Perturbation Model for the Eight Year
STRATSAT Integration
MOON
N
= 15
M* = 15
R* =
8
NOTE: AOG retains only the double
averaged third body terms.
N
maximum power of the parallax factor, (a/a')
M*
upper bound on the satellite Hansen coefficient
d'Alembert characteristic
R*
on the third body Hansen coefficient
d'Alembert characteristic
2 upper bound
289
0
used
150
time
5-31
eccentricity,
through
Figure
accurately
eccentricity
of
radius
0.8
that
of
the
per
plots
of
the
5-32.
Cowell
over
the
revolution
for
are
the
shown
perigee
attention should
in Figure
follow
from an integration
inclination
The
Special
eccentricity plot
to
comparison
and
5-33.
5-34.
steps
regularized
Ephemeris
satellite.
axis,
was created
file
The Cowell compare
be
is
in
Figures
plotted
focused
The AOG
(*)
integration
eight year
semi-major
the
on
is observed
(+) up
prediction
to
thesis
can
be
reliably
used
to monitor
the
an
span.
The plots indicate that the third body theory developed
this
in
in
evolution
and stability of rapidly evolving orbits.
Table
and Cowell
5-17
compares
predictions.
the
execution
times
The execution time of
of
the
the AOG
AOG
for
an eight year run is seen to be at least a factor of 14 less
than that of the corresponding Cowell integration.
290
09
Figure 5-31.
Comparison of the Mean and Osculating SemiMajor Axis Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG versus
Cowell
COMPARE PROGRAM
OTDS
KEPLERIAN ELEMENTHISTORIES
213450.
SATELLITE
STRATSAT 1000000
.
I
I
I
I
+
213020.
212590.
I
I
212160.
44
I+
211730.
4
I
211300.
++,
+I
.
210440.
+4
4
+
+
+
4
+++
4
4
+4
+
I
I
I
+
4
+
210870.
+
+
+
I
I +
. +
+++
++
4
+
+
4
+4
4
4
4
+ +
+
+
+
+
.
4
+
4,+
+4
++
I
+
+
210010.
4
+
I
I
I
I
209580.
I
I
I
I
I
I
I
I
209150.
0.
500.0
1000.
1500.
2000.
2300.
TIM FROM YYMMD0
850321
(+)
(*}
291
=
=
3000.
HN'NSS
0
IN DYS
Cowell
AOG
3500.
4000.
4500.
5000.
0
Figure 5-32.
0
Comparison of the Mean and Osculating
Eccentricity Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG versus
Cowell
0
GTDS COMPARE PROGRAM
KEPLERIAN ELEMENTHISTORIES
I
I.----.-----.----------------
~~~~~------------------------------------------------*---------
8.0000E-1
SATELLITE STRATSAT 1000000
---------- ----------- :---------.I
--
--------- ----
--
.
I
+
4
I
I
4+*
I*
7.2000E-1
I
I
I
I
6.4000E-1
I
I
I
I
4*
5.6000E-1
E
C
C
E
N
T
R
I
C
I
T
y
0
I
I
I
I
4.8000E-1
I
I
I
+*
+4*
4.0000E-1
I
I
I
I
++
-I
3.2000E-1
.
I
I
I
I
44
44*
2.4000E-1
.
I
I
I
I
1.6000E-1
.
I
I
I
I
4,,
4,
8.0000E-2
I
+
I+++++++*
I
I
I
I
4,,,,,
4*44
k,,,,
4
0.
I.---.
-
0.
500.0
--------------
1000.
---- ---
--
1500.
-
2500.
2000.
---
--
3000.
3500.
-
-
-
4000.
4500.
-
5000.
TIME FROM YYMMD HH?01SS IN DYS
850321
0
(+)
(*)
=
=
Cowell
AOG
0
292
41
Figure 5-33.
Comparison of the Mean and Osculating
Inclination Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG versus
Cowell
GTDS COMPARE PROGRAM
KEPLERIAN ELEMENT HISTORIES
SATELLITE
I.--------------------.---------.-- -------.------------------.-
STRATSAT
1000000
-----------------------------------.I
120.00
I
I
I
I
I
I
I
116.00
I
I
I
I
4*
I
112.00
I
I
I
I
108.00
10
00.
I
I
I
I
4*
.
4,
100.00
4*
*4*
r
44*
44*
96.000
44*
444**4
92.000
.
++ ++++4
88.000
84.000
80.000
I.
.0.
~ ~~~~~ .----.----.------.-
500.0
1000.
1500.
----.
2000.
2500.
TIME FROM YYMMDD
850321
HHM55S IN DYS
0
(+)
= Cowell
( *)
= AOG
293
-----.---------.---------.---------.-
3000.
3500.
4000.
4500.
5000.
Figure 5-34.
0
Comparison of the Mean and Osculating Radius
of Perifocus Histories for the 8 Year
Integration of the STRATSAT Orbit/AOG versus
Cowell
GTDS COMPAREPROGRAM
SATELLITE
HISTORIES OF PERIFOCAL RADIUS
2.1170E+5
I
+I
STRATSAT
1000000
I
---------------------------------------------.
-
I.---------.------------------.-----------------.
++++
1.9482E+5
I
I
I
I
4*
1.779'.E45
I
I
I
I
4*
4,
**
1.6106E+S
44*
I
I
I
I
4*
44*
1.4418E+5
4*
I
I
I
I
4*
4,
1.2730E45
4*
I
I
I
I
4*
1.1042E.5
I
I
I
I
4*
4*
93540.
I
I
I
I
*
4*
76660.
I
I
I
I
I
*
44*
59780.
*
I
I
I
I
4*
4
4
4E900.
I.--------------0.
500.0
---.
1000.
-
1500.
----- - ----.
2000.
2500.
-
-.
------
3000.
---------.---------.
--------.
3500.
4000.
4500.
5000.
TIME FROMYYMMODHH1MtSS IN DYS
850321
0
(+) = Cowell
(*) = AOG
01
294
01
Table 5-17. Comparison of Cowell and AOG Execution
Times for the Eight Year STRATSAT Integration
Orbit Generator
Cowell
Step Size
Execution Time
150 steps/rev.,
170 seconds
time
AOG
regularized
31536000 seconds
(1 year)
12.44 seconds
Notes
1)
2)
3)
4)
The Cowell integrator is based on the 12th
order Adams-Bashforth Predictor /
Adams-Moulton Corrector algorithm.
The AOG uses a 4th order Runge-Kutta
integration algorithm.
The execution time for the Cowell
integration contains overhead associated
with file creation.
The execution time for the AOG contains
overhead associated with file creation and
includes the ephemeris comparison step.
295
Lunar Resonance Test Case
5.3.5
0
This
test
case
the
examines
5:2
are given
resonance
with
in Table 5-18.
the moon.
The
the
third
The initial orbital
body AOG to accomodate resonance terms.
elements
of
capability
The orbit is
in a near
argument
crisical
of
the
resonance has a period of approximately 1340 days, while the
second
multiple
period of 670
of the critical
argument
circulates
a= 2 09831.6 km
for the
0
Epoch = 21 March 1985
Oh.0,0m.0,0s.0
e= 0 .001
90.0"
= 90.0*
3
a
days.
Table 5-18. Epoch Osculating Elements
Lunar Resonance Test Case
i=
with
Period = 11.07
days
= 0.0*
M = 0.00
Radius of Perigee = 209621.76 km
Radius of Apogee = 210041.43 km
Lunar Parallax Factor, (a/a')k = 0.546
296
0
The initial mean elements for the AOG prediction were
obtained
from
satellite
PCE.
a
PCE
elements
over
were
The perturbation
included
these
terms
used
as
model
days.
the
used
The
a priori
in
the
PCE
osculating
input
is
the
to
given
in
were
the
not
AOG.
also
Care
was
modelled
taken
in
the
to
ensure
SPG.
that
The mean
elements determined from the PCE are given in Table 5-20.
Table 5-19. PCE Perturbation Models for the
Lunar Resonance Test Case
1) AOG + SPG Perturbation
Model:
MOON
NOTE: AOG retains the double
averaged terms and the resonant
terms corresponding to the index
N
= 12
M* =
6
R* = 8
-4 < t < 4
-10
7
s 7
N
M*
10
pairs (t=2,s=-5),
(t=-2,s=5),
(t=4,s=-10), and (t=-4,s=10).
These index pairs are deleted
from the SPG.
maximum power of the parallax factor,
2
(a/a')
upper bound on the satellite Hansen coefficient
d'Alembert characteristic
R*
2
in
Resonant terms with periods of 670 days or over
Table 5-19.
were
sixty
upper bound on the third body Hansen coefficient
d'Alembert characteristic
2) Cowell Perturbing Acceleration:
point mass moon
297
6
6
the
Epoch Mean Elements for
Table 5-20.
Lunar Resonance Test Case
Equinoctial
Keplerian
=
km
= 209702.4301
209702.4301
km
e = 0.0049611077
h = 0.0023951282
i = 90.09639511*
k = -0.0043446461
3
= 90.045887870
p =
= 61.08692413*
q = -0.0008022429
I
M = 298.16803290
1.001683508
= 89.3008449*
0
The
Table
in
elements
mean
were
5-20
used
to
initialize a five year AOG prediction of the lunar resonance
the
averaged
equations of motion used a step size of 115 days.
The AOG
test
model
is
in
shown
Hansen
satellite
numerical
The
orbit.
been
increased
five
year
to
Table
5-21.
coefficient
allow
prediction
for
The
on
the
characteristic
has
growth
over
the
compare
file
was
upper
d'Alembert
eccentricity
The
span.
of
integration
Cowell
bound
generated from an integration that used 150 time regularized
steps per revolution of the satellite.
Ephemeris comparison
298
0
plots
are
for
the
provided
histories
(*)
semi-major
axis,
in
Figures
are
in general
eccentricity
5-35
through
agreement
and
inclination
5-37.
with
the
motion observed in the Cowell generated histories
The
AOG
long
term
(+).
Table 5-22 compares the execution times of the Cowell
and AOG predictions.
five
year
run
is
seen
The execution
to be
at
least
time of the AOG for the
a
factor
of
3.5
less
than that of the corresponding Cowell integration.
Table 5-21.
AOG Perturbation Model for the Five Year
Lunar Resonance Integration
MOON
NOTE: AOG retains the double averaged
terms and the resonant terms
N = 12
M* = 12
corresponding to the index pairs
(t=2,s=-5), (t=-2,s=5), (t=4,s=-10),
R* =
and
8
N
2
M*
2 upper bound
(t=-4,s=10).
maximum power of the parallax factor, (a/a')
on the satellite Hansen coefficient
d'Alembert characteristic
R* 2 upper bound on the third body Hansen coefficient
d'Alembert characteristic
299
0
Comparison of the Mean and Osculating SemiMajor Axis Histories for the 5 Year Integration
of the Lunar Resonance Test Orbit/AOG versus
Cowell
Figure 5-35.
61
0
GTDS COMPAREPROGRAM
KEPLERIAN ELEMENTHISTORIES
210650.
.
210170.
+
+
I
I
I+
I*
209690.
.* +
I+*
4
+
**+
I*
.
4.
*
*+++4.
*+4
209210.
+
*
+
. +
.++
+
*.
4.4+
+
4+
*
208730.
I
*
*-
.+
4.
I
I
I
I
209210.
4.
4.
*.
*
I
I
I
I
+
4.
+
+
+
4.*
+.
+
4.
+
4.
~.4.
+
4.
*.
.
I
I
207770.
4.
4.
I,
I
208290.
4.
4.
4
+
+.
+
+.
+.
I
I
I
I
4.+
*.4*+
4.4
+
+.
4.4.
I
+ +
I
+.
+4+
I
I
I
I
206330.
I
I
I
I
I
I
I
I
205850.
I.-- ------ ---------.---------.---------.-------- ----------
0.
250.0
500.0
730.0
1000.
1250.
TIME FROMYYMMODHHMtSS
850321
0
-------- -------------------------- -
-
1500.
1750.
2000.
2250.
I
2500.
IN DYS
(+) = Cowell
(*) = AOG
0
300
Comparison of the Mean and Osculating
Eccentricity Histories for the 5 Year Integration
of the Lunar Resonance Test Orbit/AOG versus
Cowell
Figure 5-36.
GTDSCOMPARE PROGRAM
KEPLERIAN ELEMENTHISTORIES
S.OOOOE-1
I
4.5000E-1
+
+*
*
I
.*
+I
+I
I
I
4.0000E-1
*+
* *
E
*++
TI**+
C TI**4+
NI
3 .5000E-1
03.OOOE-1
5
0
2 .OOOOE-1
E
E
*++
.*
I.+
+ +I+
**
++++
1.500E-1
TI
I
I
* +
I.
I
I+
2.SOOOE-1
I
*4+
+I
I
R
**+
"
-
*++
+
I
4
I
I
4
*
44
I
I ++4
+
-I
EI
I802
40I
44
1.0O00E-1
I.- - - - - -- - - - --0+
I
I
I+
-
-
- -
-
-
-- -
--
4-
- -
-
.
--
-
-
-*-
-
-"
-
-
-I"
" -
-
-
I
*++4
+*
I
4*4**
I
I
-
I
I-- - - - -
44
------- --
()=
()=
301
Cowell
AOG
-- - - - --
a
Comparison of the Mean and Osculating
Inclination Histories for the 5 Year Integration
of the Lunar Resonance Test Orbit/AOG versus
Cowell
Figure 5-37.
a
a
COMPAREPROGRAM
6TDS
KEPLERIAN ELEMENTHISTORIES
.-----------------------------------------------------------
I. -------- ---------.--------------
91.5000
91.1500
+
+
I
I
I
I
+***
+*
4
I
+
+
+
I
+
*
N+
4++
4
+
+
+
+
I
I
I
I
++
4
+
++*N
*+
+
90.4500
4+
+4+*
+
I
+
+*
+*
*N
+++N+
++++*NNN
+
+
+
+
+
4
4++
4
IN*N
.+ +
I+
I.
*
4 +
I
I
I
I
4
*44
I
*4
+4
+
I
89.7500
61
**
*
+
a
I
I
I
I
*
I+*
I *+4
90.1000
6
I
I
I
*
**
**
NNNN4
+
*
*
89.4000
4
*
4+
4+
0s
89.0500
I
I
I
I
88.7000
4
I
I
I
I
88.3500
I
I
I
I
I
I
I
88.0000
.
0.
250.0
500.0
750.0
1250.
1000.
1500.
1750.
2000.
2250.
2500.
TIME FROM YYMMODHHMMSS IN DYS
850321
0
0s
(+)
(*)
=
=
Cowell
AOG
01
302
40
Table
5-22. Comparison of Cowell and AOG Execution Times
for the Five Year Lunar Resonance Integration
Orbit Generator
Step Size
Execution Time
Cowell
150 steps/rev.
time regularized
111
AOG
9936000 seconds
seconds
32 seconds
(115 days)
Notes
1)
2)
3)
4)
The Cowell integrator is based on the 12th
order Adams-Bashforth Predictor /
Adams-Moulton Corrector algorithm.
The AOG uses a 4th order Runge-Kutta
integration algorithm.
The execution time for the Cowell
integration contains overhead associated
with file creation.
The execution time for the AOG contains
overhead associated with file creation and
includes the ephemeris comparison step.
303
0
This test case has demonstrated the capability of the
third
body
Table 5-22
to
resonant
to
shows
Cowell
comparison
incorporate
integration
to
resonance
that the efficiency
is
the previous
terms that were
of motion
to
AOG
terms.
of the AOG with respect
substantially
test
cases.
included
in
This
the
degraded
in
because
the
is
averaged
have periods that are still too short
the double averaged terms.
However,
The result is
equations
in relation
an unnecessary
restriction of the integration step size.
An
attempt
with only
was
the double
made
to perform
averaged
with the resonance modelled
semi-major
PCE
should
axis
have
plot
in
PCE
initialization
terms retained
in the SPG.
Figure 5-35
converged
a
would
in
the AOG and
Inspection of the
indicate
that
to a constant semi-major
the
axis of
around 208395 kilometers, which seems well removed from the
exact
the
resonance
PCE
caused
value
procedure
by
of
did
"overshoot"
207830
not
in
For
correction algorithm.
kilometers.
This
converge.
the
least
Nevertheless,
was
squares
an iteration
of
the
probably
differential
PCE,
it
is
possible that a correction to the mean semi-major axis was
computed
resonance
that placed
with
the
the orbit in
moon.
Since
a region of nearly exact
the
tesseral
analog
short
periodics are numerically ill-conditioned
in
the vicinity of
exact
of
small
resonance
because
of
the presence
304
divisors,
the
semi-analytical
for
the
subsequent
residuals
between
osculating
orbit
generated
trajectory
was
iteration
the
and
Cowell
the
data
fit
the
so
the
to
approximation
were
span
Hence,
erroneous.
semi-analytical
the
over
as
large
to
prevent convergence of the PCE.
It
is
desirable
model
to
all
sharpest
the
but
commensurabilities as short periodics, while retaining only
the
double
averaged
terms
of
averaging
can
benefit
study
of
constrained
initialization
with respect
the
in
be
AOG,
exploited.
procedures
to the
that
solution
the
full
Accordingly,
the
so
that
are
point
more
is required
to avoid the problems encountered in this test case.
305
highly
Chapter 6
Conclusions and Future Work
The
goal
of
this
thesis
semi-analytical
third
body
dict
term
evolution
the
orbits
long
both
has
theory
accurately and
that
of
with
been
to
construct
can be
high
used
altitude
an efficiency
a
to presatellite
that
exceeds
the capability of conventional mission analysis techniques.
The
averaged
functions
mathematical
equations
was
of
based
development
motion
on
the
and
of
short
application
the
periodic
of
the
Method of Averaging to the Lagrangian form of
cision
satellite
body disturbing
essential
equations.
potential
to the
A
in
tial was performed in Chapter
given
in
factor
eq.
(2-110).
to an arbitrary
The
The
orbital
approach
the
to
the
satellite
potential
near-equatorial
were
elimination
dynamical
orbits.
third
elements
for
retains
was
the poten-
the
power and no assumptions
elements
the
2, with the final result being
the geometry of the third body orbit.
tial
Generalized
the high pre-
orbital
analysis
body
recovery
representation of
satellite
development.
third
used
of
The
306
made
on
Non-singular equinocas
artificial
equations
are
parallax
for
potential
part
of
a
unified
'singularities
near-circular
was
developed
in
and
with
to
respect
the
the reference
order
first
The potential
efficient computation in a numerical program.
was
also expanded
the satellite
of
longitudes
into the mean
to
towards
view
a
with
theory.
possible
wherever
structure
analytical
the
the
employed
were
functions
modularize
of
complexity
analytical
Special
frame of the satellite to minimize
and the disturbing body so that resonance could be studied.
of
Substitution
Planetary equations
leads to a system of satellite equations
of motion that depend
lite
and
theory,
the
on the
on
based
of
Method
the
right
hand
sides
in
eq.
(3-25),
motion,
shown
average
of
the
high
respect
to
each
of the
of
the
are
precision
rapidly
of
linear combinations
of
At first
quantities.
formulated
in
The
obtained
equations
varying
terms
of
periodic
short
the
mean
307
equations
averaged
osculating elements have been replaced
mean
for
equations
the satellite and disturbing body mean longitudes.
order,
satel-
Averaging,
the
of
components
motion depending on rapidly varying
the
an averaging
3 developed
Generalized
undesired
of
removal
the
fast angles of
two
Chapter
third body.
the
Lagrange
the
into
potential
the
of
multiple
a
by
motion
angles,
of
where
with
all
by the corresponding
recovery
elements
functions,
output
by
the
a
averaged
equations
evaluation
of
the
of
motion,
indefinite
are
obtained
integrals
in
eqs.
from
the
(3-32)
and
(3-33).
Chapter 4 synthesized the results of Chapters 2 and 3
to produce mathematical expressions for th±e numerical imple-
averaged
the
of
mentation
of
equations
series
Fourier
of
m-daily
in
derivatives
Section 4.3.
of
elements
is
shown
for
the
4.2
third
the
of
h',
j
Table
in
for the
The
variazonal
tesseral,
The
with
investigated
the
4-2.
periodic
short
potential
of
forms
respect
to
in detail
in
Of particular interest are the partial derivak'
and
and l.
to
form
body potential.
related quantities were
satellite
elements
the
The
theory.
Section
of
analogs
partial
tives
motion
coefficients
tions were developed
and
body
third
with
'
These
dynamical
the
respect
expressions
system of
to
the
mean
the
relate
satellite
third
through
reference
evolution of the satellite orbital plane and are
body
the
the result
of having chosen a non-inertial frame to develop the potential.
Section 4.4 contained a complete discussion of recur-
rence relations
for the computation of special functions
auxiliary quantities.
body
theory
relations on
were
Truncation
introduced
the indices of
in
parameters
Section
4.5.
for
and
the third
Constraining
the third body potential
were
derived on the basis of these parameters.
308
0
The
of
choice
over
the
third
body
disturbing
of
structure
the
of
elements
analytical
orbital
the
the
on
dependence
a
assumed
body
third
compact
potential
body
disturbing
the
of
associated
the
and
more
the
for
exchange
In
summation.
the
elimination
the
to
function
inclination
satellite
the
led
potential
of
development
the
for
frame
inertial
frame
reference
satellite
non-inertial
the
orbital elements of the satellite which had to be accounted
of
equations
the
their
partial
only
theory
t
are
eq.
s
recovery
advantage was in
non-inertial
third
body
frame
elements
and
to be computed
once for each evaluation of the averaged element rates
of
use
periodic
are known and have
derivatives
the
proof
respect
to
was
with
not periodic.
(4-19)
equations
of
are
the
that
the
mean
the
for
of
derivatives
partial
satellite
order
first
elements
p
and
i
Hence, averaging integrals of the form in
This
vanish.
contributions
frame
coordinate
relative
a
to the
relating
central result
A
the short periodics.
and
the
of
forms
analytical
the
in
formulated
potential
the
since
averaged
However, the net computational
functions.
favor of
short
the
in
and
motion
the
in
taken
were
derivatives
partial
when
for
means
retained
motion
for
in
terms
constraint of eq. (4-31).
309
that
the
that
no
first
do
residual
order
not
secular
averaged
satisfy
the
Chapter
5
described
of
implementation
The
theory.
results
third
first
order
body
software
third
the
components:
the
the
Orbit
Averaged
of
a
numerical
body
averaging
consisted
Generator
(AOG)
of
two
and
the
Short Periodic Generator (SPG).
The capability to construct
a semi-analytical
to high precision
ephemerides
is
algorithms.
sion
of
approximation
provided
combination of
epoch
(PCE)
mean
initialization
elements
procedure
from
osculating
long
term
integrations of
the averaged
Long
term
predictions
five
using
the
the AOG
and SPG
This capability was used in the Precise Conver-
Elements
accurate
by a
satellite
AOG
and
of
compared
test
against
to
produce
elements
for
equations of motion.
orbits
Cowell
were
performed
integration
for
speed and accuracy.-
6.1
Conclusions
The
principal
conclusion of
this
thesis
is
that the
first order semi-analytical third body theory can be used to
accurately predict
satellite
superior
further
(on
the
with
to
an
the long
efficiency
conventional
conclusion
order
of
term motion of a high altitude
is that
one
that
mission
analysis
the large
year)
makes
it
decisively
techniques.
integration step
permitted
by
the
third
A
sizes
body
310
0
are
theory
models
compatible
other- - averaged
with
that might be included
in
a
general
results
verify
perturbation
analysis
mission
program.
The
of
capable
numerical
test
predicting
the
mean
is
tory over a time span that
of
a
high
very long
provide
a
highly
accurate
AOG
the
is
trajec-
precision
comparison to the
in
The SPG was also
span of the initialization process.
to
that
of
representation
found
the
third
body short periodic variations.
The excellent agreement of long term AOG predictions,
based
Cowell
on
the
PCE-generated
precision
ephemeris
epoch
mean
validates
elements,
both
the
with
the
and
SPG
AOG
algorithms.
The semi-analytical third body theory was shown to be
successful in predicting the long term motion of high altitude satellites over a broad range in orbital geometry.
theory
produced very
fine results
for high
satellite orbital eccentricity and for
parallax
factor.
The
capability
to
of
the
large values of the
model
resonance in the AOG was also demonstrated.
311
values
The
third
body
a
Several
conclusions
behavior
recursive
processes
be
used
speed
to
drawn
regarding
in
computation
well
appear
body
In par-
"stand-alone"
found
in
investigations
issue
stability
have
recurrences
in
derivation
along
rapidly
parallax
MACSYMA
package
manipulation
powers
is
4.4.6,
using
The
[31].
since
the
quite
of
since
factor
the
their
converges
even
when
the
interpolator
Finally,
large.
of
Hansen's
theory
the
parallax
symbolic
resolution
some
by
relations,
the
attention
Furthermore,
of
supported
recurrence
little
received
is
the
required
was
1855.
factor
and
4.4.5
Sections
of
sub-
the third
and
conclusion
This
behaved.
third
non-zero
of the satellite
coefficients
scrir'ted Hansen
the
for
relations
recurrence
the
the
the
body theory are stable for the test cases examined.
ticular,
the
First,
body software.
third
of the
numerical
can
concepts are applicable to the computation of the third body
short periodic coefficients, since the coefficients meet the
slowly varying.
requirement of being
6.2
Future Work
Several
suggested
the
by
analysis,
developed
in
areas
of
the present
research.
refinement,
this
investigation
future
thesis.
and
are
of
algorithms
concerned
exploration of new algorithmic approaches.
312
been
are concerned with
Some
application
Others
have
with
the
the
accuracy
At
present,
coupling
an
terms in
of
interactions
and
averaged
the
with
of
solar
and
lunar
non-spherical
of
neglecting
integration
is
often
difficult
propagation
of
small
errors
Orbit
Averaged
the
substantially longer
Generator
be
of
taken
the
with
over
a
long
a
on
term
the
from
conditions.
initial
initializing
span
data
day fit
each
perturbations.
distinguish
the
the 60
than
test cases in Chapter 5.
to
arise from the
this problem could be provided by
into
Insight
body
terms
to
in
can
perturbations
these
effect
The
of motion remains
terms
central
order
second
including
equations
Coupling
question.
unresolved
other
importance
the
theory.
body
third
order
first
the
limits of
determine
to
required
is
testing
numerical
Further
span used
that
is
for
the
This would allow greater advantageerror
PCE
smoothing
Any
properties.
residual discrepancies between the resulting semi-analytical
trajectory
and
Cowell
integration
could
then
such
second
terms
order
would
become
more
more
The effect
realistically attributed to second order terms.
of
be
important
as
the prediction span is lengthened.
Continuing with the question of model accuracy, orbit
determination
long
arcs
of
tests
using
actual
the
tracking
semi-analytical
data
for
theory
high
and
altitude
satellites would provide another stringent end-to-end test.
313
A careful
through algorithmic optimization.
are. possible
software
body
third
to the
Improvements
examination
of
the flow of indices in the third body theory could lead to a
more completely recursive software architecture which places
reliance
less
Z ,r
coefficient
in
discussed
for
are not
the
computing
These.
4.4.1.
Section
implemented
in
work to date
the
Furthermore, while
software.
current
functions.
special
helpful
be
are easy to derive, but
recurrences
the
would
relations
Recurrence
of
storage
the
on
has demonstrated both accuracy and efficiency, these results
have
usually
been
various indices.
indices
these
achieved
with
conservative
values
of
the
Automatic algorithms to rationally select
and
axis
semi-major
arbitrary
an
for
would be quite valuable.
eccentricity
to
capability
The
stability
map
parametrically
regions for high altitude satellite orbits can be of immense
the study of orbits
for
value
strong
lunar
and
in the
solar perturbations
STRATSAT class, where
can
cause
rapid
decay
The first
if initial conditions are not judiciously chosen.
order third body theory can provide this capability with an
efficiency
test
orbits
that was not previously
can
the time that it
trajectory.
mission
be
generated
in
possible.
averaged
A great
element
many
space
in
would take to integrate one high precision
Having
constraints,
found
a
a mean
orbit
corresponding
that
set
satisfies
of
the
osculating
314
0
could
elements
not,
If
tions.
they could be
used
input to
priori
as a
be suf-
might
injection condi-
world
real
to determine
accurate
ficiently
elements
These osculating
periodic recovery.
short
analytical
using
computed
be
then
a
PCE which would refine the estimate of the real world orbit.
the
was interior
satellite orbit
that
assuming
By
body.
can be
expression
the
to the orbit of
body,
orbit
extend
the
applica-
development
The
satellite
the
derivatives
satellite
in the
region.
translunar
in
to
polynomial
bility of the third body theory to satellites moving
tial
third
exterior
is
Legendre
a new
which will
developed
body,
under the assumption that
the satellite
of the disturbing
the orbit
inverse of
the
the disturbing
and
was performed
(2-17),
eq.
for
expansion
between the satellite
the distance
given in
polynomial
Legendre
The
of
of
the
elements with
orbital
into
elements
orbital
introduces
frame
reference
third body
disturbing poten-
third body
the
equations
partial
respect
of
to
motion.
The cross coupling of these partial derivatives in a second
order
averaging
tial
in
is
This
complexity.
(3-20).
theory
lead
can
to significant
illustrated
clearly
in
eqs.
analytical
(3-19)
and
Hence, the representation of the
third body poten-
eq.
(2-111), may offer
inertial
substantial
coordinates,
advantages
if
an
seen
in
attempt
second order terms.
315
is
made
to
develop
Appendix A
Computation and Storage of the Newcomb
Operators in
theory,
the
functions
computed
(4-236).
non-zero
using
the recurrence
Hansen
and
the
relations
body
third
coefficient
body
are
(4-249)
and
disturbing
in
eqs.
kernel
Similarly, the derivatives of the kernel functions
using
are computed
and
satellite
the
for
subscripted
the
of
implementation
software
the
In
the Third Body Theory
The
(4-240).
calculation
the
of
recurrence relations
recurrences
initial
values
in eqs.
the
require
from
power
the
(4-250)
explicit
series
expressions:
Satellite
000
n, -m
K%
ti=0
-
I0 X
n, -m
i+
-
e
2i
(A-i)
9,i+_ q -
and
316
0
dKn, -m
t
(j+1)
de 2
j=0
- 2j
Xn,-m
j+ -
+1,
e
j++1
(A- 2)
where
q
=
(t
+ m)
(A-3)
Disturbing Body
K-n-1,r
s
i=o
X-n-1 ,r
i+ . 2
,12i
(A-4)
and
dK-n-1
s
de' 2
,r
1 1,r
x-nj+
j=0
+
+q+1,
j+
e ,2j
1
(A-5)
317
a
where
a
q
The
series
=
s -
(A-6)
r
coefficients,
and are governed
Xu,v,
are
by the recurrence
the
Newcori
relations
operators
[17]:
a =0
4p X
p,
0
u~~v
u,v+1uv+
2(2v - u) X
=
1 0 + (v
p-
,
u) X
p-2,0
(A-7)
p = 0
4a X
0, a
=
-2(2v
+ u)
Xu, -1
0, a-1
-
(v +
u) Xu, 0, o-2
(A-8)
318
0l
p > 1,
4
a = 1
(p
+ 1)
X
p,1
=
2(2v -
+
+ (v
-
u)
u, v+ 1 p- 1 'l
2(2v + u)
(v U) xu,v+2 + 2(2p u)1
2 -
xu, vp ,0
1
+
u,v
u)Xp 1.
(A- 9)
p = 1,
4(1
a > 1
+
a)
X
1,a
=
2(2v -
-
(v + u)
u)
X
u,v+1 0,a
2(2v + u)
u,v-2 + 2(2a -
1, a-2
-u)
2 2-u)v
X
1 ,a-1
X
r
0, a-1
(A-10 )
319
6
p
> 1,
a > 1
0
4( p +
Xurv
p, a
a)
=
+
(v
-
Xuv+1 p-1 , a
u)
2(2v -
2(2v + u)
Xu v-1 +
p, a-1
v+2
u,v-2
(v + u) Xp v-2 +
a-2+
p--2,p,
u)
1p-2,a
+ 2(2p + 2a -
4
-
u)
_
X
p-i , a-i
(A-11)
The
for
conditions
initial
these
recurrence
relations
are
given by,
0
u,v
0
-
u
X1,0
=v
Xu ,v
0 ,1
The
Newcomb
identically
(A-12)
1
u
(A-1 3)
72
v -
operators
u
2
with
(A- 14)
negative
subscripts
are
zero.
320
0
0
Newcomb
The
relations
recurrence
are
and
in
stored
the
block data
FORTRAN
compiled
The
by
"offline"
computed
B).
Appendix
(see
subprograms
are
operators
of
versions
these subprograms are linked to GTDS at run time, along with
the
three
initialize
to
software,
body
third
the
of
rest
dimensional Newcomb operator arrays.
there
accessed
to
operator
Newcomb
are
explicitly
the
develop
each
for
functions
kernel
body
disturbing
and
satellite
be
can
that
arrays
time,
present
the
At
value of the index n, up to and including n = 20.
For
operator
a. given
are
arrays
value
of
valid
for
construct kernel
the
satellite
20
Newcomb
with
coefficients
Hansen
to
up
characteristics
d'Alembert
n,
and
functions with subscripts
used
be
can
that fall
to
within
the range,
-41
< t < 41
(A-15)
Likewise, for a particular value of n, the disturbing
coefficients
can
that
be
fall
used
with
to
within
are
arrays
operator
Newcomb
body
d'Alembert
characteristics
functions
kernel
construct
for
valid
up
with
to
Hansen
10
and
subscripts
the range,
-31
< s < 31
3.21
-
(A-16)
0
Appendix B
Software Implementation of the Third Body Theory0
The
for
been
and
orbital
to the
is
subroutine
SPANAL
for
which
time
has
averaged
perturbations.
the
independent
(SPG)
governs
for
coefficients
periodic
elements
double
constructs
which
Generator
Periodic
and
mean
both
body
of motion
The
for the third body Averaged Orbit
ANAVR
rates
element
for
third
resonant
(AOG)
Short
GTDS
subroutine
executive
Generator
The
into
implemented
equations
averaged
the
the sho-t periodic corrections
non-resonant
GTDS
for
theory
is
the
averaged
perturbations.
by
driven
computation
the
of
GTDS
short
averaged
analytically
perturbations.
B.1
Subroutine Descriptions
This
implement
EVAL,
all
the
of
the analysis
temporary
modified
section
body
third
theory.
the subroutines
of
this
updates.
to
describes
thesis
The GTDS
incorporate
the
the
subroutines
With
the
were expressly
and
were
which
exception
written
included
in
of
to test
GTDS
as
subroutine EVAL was extensively
analytical
ephemeris
discussed
in Chapter 5 and is thus included in the list of third body
322
Wherever appropriate, references to the thesis
subroutines.
link the software
text that
included
to the analysis are
in
the subroutine descriptions.
The twenty-four subroutines that follow form the core
of
the
body
third
generate
the right hand
They
software.
averaging
used
are
to
sides of the averaged equations of
motion for double averaged third body perturbations.
1)
AAPRIM:
Computes
n
including
third
2)
AVEXEC:
Computes
Tt,s
its
averaged
orbit
to
(4-86)
stores
= N, where N
body Legendre
and
and
(5/a' )n
up
to
is the maximum degree
and
of the
polynomial expansion.
the
partial
derivatives
generator
(4-91)].
and
real
[eqs.
Computes
imaginary
for
the
(4-81)
to
sines
and
parts
third
of
body
(4-83) ,
eqs.
cosines
of
tT+sX' for the third body resonance implementation.
3)
BROLUN:
Computes
the mean equinoctial
elements
of
the
Moon in mean ecliptic of date coordinates using Brown's
theory [Section 5.2].
4)
BROSOL:
Sun
in
Computes the mean equinoctial elements of the
mean
ecliptic
of
date
Newcomb's theory [Section 5.2].
323
coordinates
using
a
5)
CMRDMR:
Computes
parts
of
[eqs.
(4-168)
the
6)
to
DUBFAC:
complex
to
stores
the
polynomials
(4-171),
real
and
Cr
M (a, 0)
+
(4-184)
eqs.
derivatives with respect
its partial
(4-175)
and
(4-182) ,
Computes
eqs.
(4-188)
to
to
imaginary
j
D
m (a,,)
(4-187) ] and
a and
a
[eqs.
to (4-195) ].
and stores double factorials
required
for the averaged orbit generator and the short periodic
generator.
7)
EPRIM:
Computes
disturbing
[Table
2-3] .
and k'
with
[eq.
8)
body
(4-92)
the
Sun
numerical
Computes
orbit
and
eqs.
rotation
to
partial
elements
frame
of
the
coordinates
derivatives
elements
the satellite
p
of
h'
and
q
to (4-96) ].
and
the
in
using
matrices
body
k'
satellite
position
Moon
and
the
(4-94)
integration
coordinates
in
to
respect
,
h'
Computes
Computes
EVAL:
the
fixed
velocity
reference
the
for
components
for
of
the
system
analytical
ephemeris.
converting
coordinates
and
inertial
mean
of
1950.0 coordinates to true of date coordinates.
9)
FACT:
Computes and stores factorials
averaged orbit generator
required
for the
and the short periodic generator.
324,
0
10)
HANSTO:
Computes
satellite
Hansen
subscript
zero,
11)
[eqs.
functions
(4-254)
to
to h and k
of
(4-256)
].
satellite
the
of
the
stores
kernel
derivatives
functions with respect
kernel
to
coefficient
Kn,-m
and
form
closed
partial
the
Computes
in
[eqs.
(4-257)
(4-259)].
subscript
kernel functions of
Hansen coefficient
disturbing body
zero,
[eqs.
KO-n-l,r'
the
stores
and
form
closed
in
Computes
HANTDO:
(4-242)
to
(4-244)].
Computes the partial derivatives of the disturbing body
12)
kernel
functions
with
(4-245)
to (4-247)
].
Computes
HKSAT:
the
parts
of
[eqs.
(4-215)
and
(4-218)
]
HPKP:
of
the
(4-211)
JACPOL:
real
and
imaginary
(k
-
jh)
-m-t
derivatives
(4-277) to (4-234) ].
Computes and stores the real and imaginary parts
complex
to
and k'
Computes
derivative
with
(k'
polynomials
(4-214) ]
respect to h'
14)
k'
partial
its
and
with respect to I and li [eqs.
13)
and
polynomials
complex
to
the
stores
[eqs.
h'
to
respect
and
-
jn'h')
partial
its
[eqs.
derivatives
with
(4-219) to (4-226) ].
[eqs.
and stores the function
respect
Ir-si
to
Section 4.4.2].
325
Y
[eqs.
(4-65)
A(y)
to
and its
(4-67),
a
15)
MEXEC:
Executive
over m
summation
for the
routine
in
0
16)
Executive
NEXCF:
over
summation
for the
n
in
Hansen
satellite
the
when
used
is
that
t,s
routine
coefficient kernel functions are of subscript zero.
17)
Computes
NOD:
the nutation matrix for a rotation from
S
mean of date coordinates to true of date coordinates.
18)
Computes
OBLIQ:
matrix
rotation
the
transforms
that
0
ecliptic coordinates to equatorial coordinates.
19)
coordinates.
precession
Computes
the
matrix
of
inverse
to
transform
date
of
mean
to
coordinates
1950.0
of
mean
the
Computes
PRECES:
0
precession
the
matrix.
20)
REXCF:
Executive
-that
*t,s
is
summation
routine
for
the
when
the
disturbing
used
over
r
in
Hansen
body
coefficient kernel functions are of subscript zero.
21)
THIRD:
the
Computes
disturbing
the
body
disturbing
body
eccentricity
semi-major
vector
in
9
axis,
inertial
coordinates and the direction cosines of the disturbing
body unit orbit normal with respect to the unit vectors
of the satellite frame
[Section 2.2.4].
326
0
0
22)
Computes the components of
XYZPOS:
the disturbing body
position vector in mean ecliptic of date coordinates.
23)
Computes the components of the disturbing body
XYZVEL:
velocity vector in mean ecliptic of date coordinates.
24)
Computes
ZCOEFF:
and
coefficient
the
(4-142)
[eqs.
Zm
n,r
(4-144) ].
order
In
averaged
to
generator
orbit
body
third
model
or
model
to
corrections to the mean elements, the
in
resonance
the
short
following
the
periodic
subroutines
are required:
25)
the
Transfers
ASSGN:
the
containing
contents
body
third
of
Newcomb
the
common
operators
blocks
into
a
single common block.
26)
NEXNEW:
4
for the summation over n in
when
used
is
that
't,s
routine
Executive
the
satellite
Hansen
coefficient kernel functions have non-zero isubscripts.
27)
REXNEW:
4 t,s
Executive
that
is
used
coefficient kernel
routine for the summation over r in
when
the
disturbing
body
Hansen
functions have non-zero subscripts.
327
a
28)
SATNEW:
Computes
satellite
Hansen
non-zero
subscript
have
by
coefficient
expansions
[eq.
derivatives
of
(4-249) ].
the
29)
SATONE:
Computes
satellite
Hansen
one
subscript
(4-250)
the
coefficient
with
beginning
n =
to
respect
with
are
relations
recurrence
stores
and
values
the
of
that
have
expansions.
of
the
satellite
h
and
k.
explicitly
The
re-initialized
instability
to avoid
11
with
functions
kernel
derivatives
partial
that
].
(4-252)
initial
from
of
partial
the
from the Newcomb operator
functions
kernel
to
the
operator
functions
kernel
recurrence
by
starting
been constructed
Computes
Newcomb
Computes
satellite
respect to h and k [eqs.
values
initial
the
from
stores
functions
kernel
from
starting
constructed
been
and
recurrence
for
high
values of n when the kernel function subscript is one.
30)
Computes
THDNEW:
by
recurrence
and
stores
the
disturbing body Hansen coefficient kernel functions of
non-zero
have
been
from
constructed
expansions
[eq.
derivatives
of
from
starting
subscript
(4-236) ].
the
disturbing
with respect to h' and k'
[eqs.
values
initial
the
Newcomb
Computes
body
the
kernel
(4-240) and
that
operator
partial
functions
(4-241) ].
328
0
The
relations
recurrence
names
having
third
body
and SATONE,
body
that
is
n
are 'loaded under
expansion.
polynomial
software
incorporates
is,
recurrence
THDNEW,
in
functions
There
NEWn.
are
the
for
relations
19
of
the
to initialize
disturbing
kernel
body
these
form
the
having
loaded under names
presently
are
these
Likewise,
Newcomb operator block data subprograms, used
the
the
The
of
19
through NEWS20.
NEWS2
of
degree
the
Legendre
potential
subprograms,
SATNEW
where
function
kernel
satellite
form NEWSn,
the
third
present
in
which
subprograms,
the
initialize
to
used
are
data
block
operator
Newcomb
subprograms
corresponding to NEW2 through NEW20.
The subroutine which is specifically required for the
third
implementation
resonance
body
the
in
averaged
orbit
generator is:
31)
Computes the mean longitude
THDLAM:
for
use
in
the
resonant
[Section 2.2.3.2].
the
third
body
eqs.
The
(4-118)
to
averaged
the
body
third
orbit
generator
Computes the partial derivatives of
mean
p and q satellite
of
longitude
elements
(4-133)
[eqs.
with
respect
(4-116)
and
to
the
(4-117),
].
subroutines which
are
specifically
required
the third body short periodic implementation are:
329
for
32)
SPTHDB:
Computes
coefficients
and
33)
(4-38),
DBLANG:
for
eqs.
Computes
the
third
analytical
short
body perturbations
(4-48)
to (4-62)
periodic
[eqs.
(4-37)
].
the third body mean longitude
and its
partial derivatives with respecL to p and q.
A
schematic representation of the overall subroutine
structure for the third body AOG and SPG
is given in Figure
B-1.
At
present
partitioned
data
the
sets
body
third
on
the
software
CSDL Amdahl
470
resides
V/8
which
in
are
cataloged under the following names:
Non-Resonant Software
FORT
(1) SKC1756.GTDS.UPDATE.
LOAD
(FORT)
(2) SKC1756.GTDS.SP.
(
0
LOAD
0
330
0
Resonance Software
FORT
(3) SKC1756.GTDS.RESON.
~LOAD)
FORT
(4)
SKC1756.GTDS.SP.RESON.
I
~LOAD~
Satellite
Hansen Coefficient Newcomb Operators
FORT
(5)
SKC1756.GTDS.NEWSAT1.
LOAD
FORT
(6)
SKC1756.GTDS.NEWSAT2.
LOAD
Third Body Hansen Coefficient Newcomb Operators
FORT
(7) SKC1756.GTDS.NEWTHD1.
LOAD
FORT
(8) SKC1756.GTDS.NEWTHD2.
LOAD
331
W
Table
B-1
third
body
indicates which data sets
option
and
are needed
minimum
the
region
for
size
a given
that
is
0
required for the resulting updated GTDS load module.
Third Body Optic-
Data Sets Required
by Identifying no.
(see text)
Required
Region Size
0
(1)
AOG without
resonance
1052K
AOG+SPG without
resonance
(1),(2),(5),(6),,
(7),(8)
4976K
AOG with
resonance
(3),(4),(5),(6),
(7),(8)
4980K
AOG+SPG with
resonance
(3),(4),(5),(6),
(7),(8)
4980K
0
Table B-1.
Datasets and Region Sizes Required for the
Various Third Body Options
0
332
0l
9
0
9
0
W'
WAUA-
SHORT PERIODIC GENERATOR
AVERAGED ORBIT GENERATOR
*
Figure B-i.
EXISTING GTDS SUBROUTINE
Subroutine Interaction Diagram for the Third Body Software
(page 1 of 4)
L~J
*
Figure B-i.
rl
EXISTING GTDS SUBROUTINE
Subroutine Interaction Diagram for the Third Body Software
0
(page 2 of 4)
0
p
S
5
9
5
5
5
5
5
B
D
DUBFAC
Figure B-i.
HKSAT
rCMRDMR
AAPRI
HPKP
Subroutine Interaction Diagram for the Third Body Software
FACT
(page 3 of 4)
LA)
SATONE
SATNEW
HANSTO
F
Figure B-l.
e0
ZCO EF
JACPOL
.Subroutine Interaction Diagram for the Third Body Software
0
(page 4 of 4)
00
0
00
0
References
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J.,
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340
0
BIOGRAPHY
Sean Kevin Collins was born 21 February 1952 on the
Fort Belvoir military reservation in Virginia to John M. and
He attended Fort Hunt High School in
Gloria 0. Collins.
In June of 1974 he
Alexandria, Virginia from 1966 to 1970.
High Distincwith
Degree
of
Science
Bachelor
received the
tion. in Aerospace Engineering at the University of Virginia
Having entered the graduate
in
Charlottesville, Virginia.
of the Massachusetts Institute of Technology in
school
September of 1974, he was granted a Master of Science degree
He
in Aeronautics and Astronautics in February 1977.
At present Mr.
July 1980.
married Susan Jane Howling in
is
a Draper Fellow at the Charles Stark Draper
Collins
Laboratory in Cambridge, Massachusetts.
341