orbit determination from three angles observations in the presence of perturbations

165 Pages·2013·0.85 MB·English

University of Baghdad
College of Science
Department of Astronomy and Space

ORBIT DETERMINATION FROM THREE ANGLES
OBSERVATIONS IN THE PRESENCE OF
PERTURBATIONS

A Thesis
Submitted to the Department of Astronomy and Space
College of Science
University of Baghdad

In partial fulfillment requirements of the
Degree of Doctor of Philosophy
In Astronomy and Space

By
AHMED KADIR IZZET ZAINAL

Supervised

Dr. Akram Y. AL saati Dr. Abdulrahman H. ALmohammedy

April 1, 2007 Rabi' Awwal 14, 1428
ﺩﺍﺪﻐﺑ ﺔﻌﻣﺎﺟ
ﻡﻮﻠﻌﻟﺍ ﺔﻴﻠﻛ
ءﺎﻀﻔﻟﺍﻭ ﻚﻠﻔﻟﺍ ﻢﺴﻗ

ﺕﺎـﺑﺍﺮﻄـﺿﻻﺍ ﺩﻮﺟﻮﺑ ﺕﺍﺪـﻫﺎـﺸﻣ ﺙﻼﺛ ﻦﻣ ﺕﺍﺭﺍﺪﻤﻟﺍ ﺪـﻳﺪﺤﺗ

ءﺎـﻀـﻔﻟﺍﻭ ﻚـﻠـﻔﻟﺍ ﻢـﺴـﻗ ﻰﻟﺇ ﺔـﻣﺪـﻘﻣ ﺔـﺣﻭﺮـﻁﺃ
ﻡﻮـﻠـﻌـﻟ ﺍ ﺔـﻴـﻠـﻛ
ﺩﺍﺪـﻐـﺑ ﺔـﻌـﻣﺎـﺟ

ﻞﻴﻧ ﺕﺎﺒﻠﻄﺘﻣ ﻦﻣ ءﺰﺟ ﻲﻫﻭ
ءﺎﻀﻔﻟﺍﻭ ﻚﻠﻔﻟﺍ ﻡﻮﻠﻋ ﻲﻓ ﺔﻔﺴﻠﻓ ﻩﺍﺭﻮﺘﻛﺪﻟﺍ ﺔﺟﺭﺩ

ﻞﺒﻗ ﻦﻣ
ﻞـﻨـﻳﺯ ﺕﺰـــﻋ ﺭﺩﺎــﻗ ﺪـــﻤـﺣﺍ
ﻑﺍﺮﺷﺈﺑ

ﺭﻮﺘﻛﺪﻟﺍ ﺭﻮﺘﻛﺪﻟﺍ
ﻱﺪﻤﺤﻤﻟﺍ ﻦـﻴـﺴﺣ ﻦـﻤـﺣﺮـﻟﺍ ﺪﺒـﻋ ﻲـﺗﺎـﻋﺎﺴﻟﺍ ﺲﻧﻮـﻳ ﻡﺮـﻛﺃ

۲۰۰۷ ﻥﺎﺴﻴﻧ ۱ ۱٤۲۸ ﻝﻭﻷﺍ ﻊﻴﺑﺭ ۱٤
Abstracts

The roots of angles only orbit determination extend back to the time of
Laplace late 1700; most of today’s astrodynamics community has been
uncertain as to applicability of angles only estimation techniques for accurate
earth satellite orbit determination.

A new procedure has been developed for the general solution of the
minimal angles-only problem which an orbit is determined from three line of
sight observations. The procedure basis assumes two of the unknown ranges and
using Lambert problem to find the middle position; Newton-Raphson used to
obtain the solution. Multiple solutions are completely new, in such a way the
solutions already existing are not obtained again.

This procedure avoids the limitations of the classical method (Laplace,
Gauss) and a number of test problem have been explained for different celestial
object and earth satellites.

Two types of perturbations ( oblateness of earth and third body) effects on
earth satellites orbital elements are discussed and a comparison is made between
them.

The ground trace of an artificial satellite, affected by perturbations due to
oblateness of the earth, and the moon and the sun as a third body is determined
throw one period.
Acknowledgements
I wholeheartedly appreciate the unwavering support of my advisors Dr.
Akram Saati and Dr. Abdulrahman Al mohammedy. It is an honor to have
worked with men of such brilliance, dedication, and vision. I am also indebted
to head of department Dr. Ali Talib for his understanding when I needed him
the most. I appreciate fellow graduate student Salman Zedan for helping me
My family father mother brothers and my wife’s were always willing to
provide input on design decisions and refused to ever allow me to become
discouraged. I thank them all for their assistance

Ahmed
List of symbols
symbols Definitions
a Semi-major axis
a acceleration
c
a Earth radius
e
c Speed of light
E Eccentric anomaly
e Eccentricity
f flatting
G Gravitational constant
H Elevation
J Bessel function
k
L Motion constant
M Mean anomaly
n Mean motion
N Number of solution
p Half semi latus rectum distance
P Period of motion
d
p Momentum
m
P Radiation solar power
O
R Observer position vector
r Orbital position vector
r Turning point of motion
±
T Time periapase passage
U Potential energy
V Velocity
V Radial velocity
R
V Transfer velocity
T
I
α Declination
δ Right ascension
Θ Subtended angle
sb
ι Inclination
λ Longitude
ρ Ranges (slant range vector)
ρ Satellites distance from moon
m
ρ Satellites distance from sun
s
υ True anomaly
Φ Perturbation function
ω Arguments of periapse
Ω Right ascension of ascending node

II
List of tables

Table Page
Definition
number number
1.1 Classical orbital elements 28
5.1 data and results for minor planet 683 lanza 93
5.2 data and results for minor planet 40 Harmonia 94
5.3 data and results for Longs orbit 96
5.4 data and results for McCuskeys 97
5.5 data and results for swifts comet 1880v 98
5.6 data and results for minor planet Ceres 99
5.7 data and results for parabolic orbit 100
5.8 data and results for minor planet 1986 PS1 102
5.9 data and results for minor planet 1990 TG4 103
5.10 data and results for minor planet 1986 PT 104
5.11 data and results for minor planet PZ1 105
5.12 data and results for minor planet PO 106
5.13 data and results for minor planet 1973 SC2 107
5.14 data and results for minor planet 1966 (834) 108
5.15 data and results for minor planet 1966 (834) 109
5.16 data and results for periodic comet (HMP) 110
5.17 data and results for periodic comet (HMP) 1948 111
5.18 data and results for Mars 112
5.19 data and results for Venus 113
5.20 data and results for artificial satellite 1 115
5.21 data and results for artificial satellite 2 116
5.22 data and results for artificial satellite 3 117
5.23 data and results for artificial satellite 4 119
5.24 data and results for artificial satellite 5 120

III
Lists of figure

Figure Page
Definition
number number
1.1 Ellipse with axis 4
1.2 Keplers second law 5
1.3 Elliptical orbit 5
1.4 The right ascending declination inertial coordinate system 9
1.5 Earths curvature 10
1.6 Velocities versus trajectory 11
1.7 Conic section geometry 13
1.8 Elliptical geometry 13
1.9 Definition of argument of periapse (ω) 17
1.10 Elliptical geometry 17
1.11 Introduction of the auxiliary angle E 21
1.12 Hyperbolic reference area 21
1.13 Position and velocity vectors 22
1.14 Inclination tilt 24
1.15 Right ascension of the ascending node 25
1.16 Vernal equinox 26
1.17 Argument of perigee 27
1.18 Orbit in space 29
1.19 The deviation of the earth into time zones 34
1.20 The rotating earth in an inertial coordinate system 35
1.21 The geometric model of the earth 36
1.22 A cross section of the adopted earth model 37
1.23 A cross section of the adopted earth model with geodetic 38
latitude as a variable
2.1 Representation of orbit determination 43
2.2 Vector triangle relation dynamical center 45
2.3 A heliocentric orbit 45
2.4 Triangular area 49
5.1,a,b,c Perturbation by oblateness of earth on earth satellite 1 122
5.2,a,b,c Perturbation by oblateness of earth on earth satellite 2 123
5.3,a,b,c Perturbation by oblateness of earth on earth satellite 3 124
5.4,a,b,c Perturbation by oblateness of earth on earth satellite 4 125
5.1,d Perturbation by third body on earth satellite 1 126
5.2,d Perturbation by third body on earth satellite 2 126
IV
5.3,d Perturbation by third body on earth satellite 3 126
5.4,d Perturbation by third body on earth satellite 4 126
5.1,e,f Perturbation by third body and oblatness of earth on 127
earth satellite 1
5.2,e,f Perturbation by third body and oblateness of earth on 127
earth satellite 2
5.3,e, f Perturbation by third body and oblateness of earth on 128
earth satellite 3
5,4,e,f Perturbation by third body and oblateness of earth on 128
earth satellite 4
5.5 a Perturbation by oblateness of earth on earth satellite 5 129
5.5 b Perturbation by third body and oblateness of earth on 129
earth satellite 5
5.5 c Perturbation by third body and oblateness of earth on 129
earth satellite 5
5.6 Perturbation by third body and oblateness of earth on 130
earth satellite in one period

V
Contents
Contents page number

Chapter one: introduction to orbital mechanic 1
1.1 History of laws of motion 1
1.1.1 Early cosmology 1
1.1.2 Astronomy 2
1.2 Definition of the orbit determination process 7
1.3 Orbital motion 9
1.4 Orbit geometry 12
1.5 The two body problem 13
1.6 The orbit 15
1.7 Other conics 19
1.8 Relation between speed and velocity 21
1.9 Coordinate reference system and orbital elements 23
1.10 The orbit in space 29
1.11 The orbital elements and the initial condition 31
1.12 Physical working units and constants 32
1.13 Stations coordinate 36
1.14 Review 39

Chapter two: method of orbit determination 43
2.1 Angle only orbit determination 44
2.2 Laplaces method 46
2.3 Gausses method 47
2.4 Range-iteration method 51

Chapter three: the theory 54
3.1 Lamberts orbital boundary – value problem 54
3.2 New procedure 58
3.3 Elements conversion to and from position and velocity 64

Chapter four: perturbations 71
4.1 Introduction 71
4.2 Enke- type equations of motion 73
4.3 Lagrange and Gaussian perturbation equation 75
4.4 Perturbation on artificial earth satellite 81
4.4.1 Oblateness perturbations 81
4.4.2 Lunisolar perturbations 86
4.4.3 Atmospheric drag 90
4.4.4 Radiation pressure 91
4.4.4 Relativistic effect 91
VI