mgutheses.inmgutheses.in/png/T 1488/T 1488.pdf · GOVERNMENT OF INDlA DEPA RrT,VF.N'I' OF SP.4C:E ', VIKRAM SARABHAI SPACE CENTRE THIRUVANANTHAPURAM - 695 022 TELEGRAM : SPACE TELEPHONE

Analytical and Numerical Predictions for Near-Earth's Satellite Orbits with

KS Uniform Regular Canonical Equations

THESIS SUBMITTED TO

MAHATMA GANDHI UNIVERSITY KOTTAYAM, KERALA

FOR THE AWARD OF THE DEGREE OF

D O ~ ~ O F @ M ! . O S ~

MATHEMATICS UNDER THE FACULTTY OF SCIENCE

M. Xavier James Raj

Applied Mathematics Division Vikram Sarabhai Space Centre Thiruvananthapuram - 695 022.

January 2007

Dedicated to

My Ever Loving Parents

(Late) Shri V. Michael,

(Late) Smt. Amburoseammal

Brother

(Late) Shri M. Raja Retnam

DECLARATTON

1 hereby declare that the entire work recorded in the thesis has been

carried out by me under the supervision of Dr. R. K. Sharma, Head, Applied

Mathematics Division, Vikrarn Sarabhai Space Centre, Thiruvanmthapuram

and no part of it has been submitted for the award of any degree or diploma

of any Institution previously.

M. XAVIER JAMES RAJ

GOVERNMENT OF INDlA DEPA RrT,VF.N'I' OF SP.4C:E

', VIKRAM SARABHAI SPACE CENTRE THIRUVANANTHAPURAM - 695 022

TELEGRAM : SPACE

TELEPHONE . 047 1 -2565629

FAX :0471- 2704134

E-mail : rk-sharma@vssc. gov.in

APPLIED MATHEMATICS DIVISON

Dr R. K. SHARMA

ScientistEngineer-G

HEAD

CERTIFICATE

This is to certify that the thesis entitled "ANALYTICAL AND NUMEFUCAL

PREDICTIONS FOR NEAR-EARTH'S SATELLITE ORBITS WITH KS

UNIFORM REGULAR CANONICAL EQUATIONS'is the research work carried

out by Shri M. Xavier James Raj under my guidance. The content of the thesis, either

partially or fully, has not been considered for the award of any other degree or diploma

by any other University or Institution in India or abroad.

R. K. SHARMA

Supervising Guide

m&flm* INDIAN SPACE RESEARCH ORGANISATION

Abstract

The thesis deals with analytical and numerical orbit predictions for near-Earth satellite

orbits using KS uniformly regular canonical equations of motion with Earth's flattening

and atmospheric drag forces. Chapter 1 deals with general introduction and review of

available literature on satellite motion along with the derivation o f KS uniformly regular

canonicat elements. Chapters 2 and 3 deal with long-term orbit predictions numerically

with Earth's zonal and tcsseral harmonic terms. Comparisons are made with real satellite

data. A new analytical theory for short term orbit predictions with J2. J3 and J4 is

presented in chapter 4. Only two of the nine equations are solved analytically to compute

the state vector and time, due to symmetry in the equations of motion. The theory is

found better than three other theories. In Chapters 5, 6 and 7, new analytical theories for

long-term motion of near Earth satellite orbits with air drag are developed with three

different atmospheres with constant density scale height. Series expansions are carried

nut up to third order terms in eccentricity and atmospheric flattening parameter c.

Numerical comparisons of semi-major axis and eccentricity up to 1000 revolutions

obtained with the present solutions and two other analytical theories show the superiority

of the present theories.

Keywards:

Obalteness, zonal and tesseral harmonic terms, KS unifonnl y regular canonical

elements, near-Earth satellite, air drag, spherical atmosphere, oblate

atmosphere, diurnally varying atmosphere, bilinear relation.

Acknowledgements

I am deeply indebted to Dr. R. K. Sharma, Head, Applied Mathematics Division, Vikram

Sarabhai Space Centre, Thiruvananthapuram for his constructive ideas, invaluable

guidance, excellent comments, continuous support and inspiring discussions. which had a

great influence on the studies carried out for this research work meant for the thesis.

My special gratitude is to Dr. G. Madhavan Nair, Chairman, Indian Space research

Organization for permitting me to pursue this research work.

I sincerely thank Dr. 8. N. Suresh, Director, Vikrarn Sarabhai Centre and Dr. V.

Adimurthy, Associate Director, Vikram Sarabhai Centre for reviewing this work from

time to time. I would like to thank Dr. P. V. Subba Rao, former Head. Applied

Mathematics Division, Vikram Sarabhai Centre; Dr. A. R. Acharya, former Deputy

Director, Vikram Sarabhai Centre; Shri Madan Lal, Deputy Director, Vikram Sarabhai

Centre and Shri S. R. Tandon, Group Director, Aero Flight Dynamics Group, Vikram

Sarabhai Space Centre for encouraging me to pursue the studies. I also thank all my

colleagues in Applied Mathematics Division, specially Dr. A. K. Anil Kumar for their

full support and co-operation.

I would like to express my deepest appreciation to my wife Srnt. 13. Josephine Mary

Xavier and to our three sons Roshan Xavier, Rofan Xavier and Rajan Xavier for their

sacrifices and their love and affection towards me. I am happy to acknowledge my

brother Shri M. Benjamin Dhas and sisters for their silent recognition of my et'forts and

continued support. I will place my thanks to my in-laws for the encouragement given to

me in doing this research work.

This acknowledgement will not be complete, if I do not remember my loving friends,

colleagues and relatives.

M. Xavier James Raj

LIST OF TABLES

LIST OF FIGURES

LIST OF APPENDICES

NOMENCLATURE

CHAPTER 1 INTRODUCTION

Historical background

General Introduction

Perturbed Equations of motion

Fictitious Time / New Independent Variable

Levi-Civita Matrix

KS Transformation

Wamiltonian's Equations of motion

Canonical Elements

Canonical equations of motion

Canonical Equations of Motion in Fictitious time

Separation of Jacobi's equation

KS Uniformly Regular Canonical Elements

Drag force in terms of KS uniformly regular canonical elements

Bessel functions of imaginary argument I,(z)

Conclusion

CHAPTER 2 LONG TERM QRBIT PREDICTIONS WITH EARTH'S OBLATENESS

2.1 Introduction

2.2 Perturbations and tegendre Polynomials

2.3 Initial conditions

2.4 Numerical Integration

Orbits

2.5 Sun-synchronous orbit

2.6 Near Critical inclination orbits

2.7 Comparison with JRS- 1 A orbital data

2.8 Conclusions

CHAPTER 3 LONG TERM ORBIT PREDICTIONS WITH EARTH'S FLATTENING 67-89

3. t Introduction 67

3.2 Earth's flattening perturbations 68

3.3 Derived Legendre Functions and Normalized Geopotential Coefficients 69

3.4 Computational Procedure 7 0

3.5 Numerical results 7 I

3.6 Comparison with IRS-IA orbital data 73

3.7 Conclusions 74

CHAPTER 4 SHORT TERM ORI3ITAL THEORY with Jz, J3 and Jq

4.1 Introduction

4.2 Equations of Motion

4.3 Analytical Integration

4.4 Comparison with other solutions

4.5 Numerical results

4.6 Conclusions

CHAPTER 5 ORBITAL THEORY WITH AIR DRAG: SPHERICALLY SYMMETRICAL EXPONENTIAL ATMOSPHERE

5.1 Introduction

5.2 Model for air density

5 -3 Equations of motion

5 -4 Analytical integration


5.6 Conclusion

CHAPTER 6 ORBITAL THEORY WITH AIR DRAG: OBLATE EXPONENTIAL ATMOSPHERE

6.1 Introduction


6.3 Equations of motiotl


6.5 Numerical Results

6.6 Conclusion

CHAPTER 7 ORBITAL THEORY WITH AIR DRAG: OBLATE DIURNALLY VARYING ATMOSPHERE 157-185

7.1 Introduction


7.3 Analytical integration


7.5 Conclusion

BIBLIOGRAPHY 186- 1 99

cal -ical Predict-'s a t e 0 - m - LIST OF TABLES

Table 2.1 Earth's Zonal harmonic terms 52

Table 2.2 Initial Conditions (Position, Velocity and Orbital Parameters) 53

Table 2.3 Variation o f Time & semi-major axis with Earth's zonal harmonics after 100 revolutions

Table 2.4 Variation of eccentricity with Earth's zonal harmonics after revolutions 55

Table 2.5 Bilinear relation and Energy equation after 100 revolutions with J2 to J3h 56

Table 2.6 Initial Osculating & Mean orbital elements (Case E) 57

Table 2.7 Comparison of Observed & Predicted Values of a real satellite 58

Table 3.1A GEM-T2 Normalized Coeff~cients for Zonal harmonics 75

Table 3.1 B GEM-T2 Normalized Coefficients Sectorials and Tesserals 76

Table 3.2 Initial Conditions (Position, Velocity and Orbital Parameters) 80

Table 3.3 Bilinear relation x 1 O8 after 22 hours 8 1

Table 3.4 Variation in orbital parameters after 22 hours 82

Table 3.5 Differences in orbital parameters due to Tesseral harmonics 83

Table 3.6 Comparison of Observed and Predicted Values (IRS- 1 A) 84

Table 4.1 Initial conditions (Position, Velocity and Orbital Parameters) I02

Table 4.2 Comparison with other solutions 103

Table 4.3 Comparison of time and semi-major axis with KS theory 104

Table 4.4 Comparison of eccentricity and inclination with KS theory 105

. . -~~umerhl&r- for N-'s # -

Table 5.13 Decrease in semi-major axis for Hp = 220 km and e = 0.05 afier 1000 revolutions 129

Table 5 . I4 Decrease in eccentricity for Hp = 220 km and e = 0.05 after 1000 revolutions 129

Table 6.1 Decrease in semi-major axis and eccentricity far Hp = 200 km, e = 0.05 and i = 85" up to 100 revotutions 149

Table 6.2 Decrease in semi-major axis Hp = 250 km, e = 0.05 and i = 25' up to 1000 revolutions 149

'Table 6.3 Decrease in eccentricity for Hp = 250 km. e = 0.05 and i = 25" up to 1000 revolutions 149

Table 6.4 Decrease in semi-major axis for Hp = 250 km and i = 1 5" after I000 revolutions 150

Table 6.5 Decrease in eccentricity for H, = 250 km and i = 15" after I000 revolutions 150

Table 6.6 Decrease in semi-major axis H, = 250 km and e = 0.1 after 1000 revolutions 150

'I'able 6.7 Decrease in eccentricity for Hp = 250 km and e = 0. I after 1000 revolutions 151

Table 6.8 Decrease in semi-major axis for e = 0. I and i = 1 So after I000 revolutions

'Table 6.9 Decrease in eccentricity for e = 0.1 and i = 1 5" after 1000 revolutions

Table 7.1 Decrease in semi-major axis upto 1000 revolutions for Hp = 200 km, e = 0.1 and i = 55" 152

lable 7.2 Decrease in eccentricity upto 1000 revolutions for Hp=200km,e=O.i and i=55'

'I'able 7.3 Decrease in semi-major axis after 500 revolutions for e =O. 1 and i =. 85' 179

Table 7.4 Decrease in eccentricity after 500 revolutions for e=0.1 and i=8S0

Table 7.5 Decrease in semi-major axis after 500 revolutions for Hp = 200 krn and i - 35" 180

Table 7.6 Decrease in eccentric,ity after 500 revolutions for Hp = 200 km and i = 35'

Table 7.7 Decrease in semi-major axis after 500 revolutions for klp = 175 km and e == 0.1 181

Table 7.8 Decrease in eccentricity after 500 revolutions fur Hp = 175 krn and e = 4 . 1

P w o n s fur -1lite a- - LIST OF FIGURES

Figure 2.1 Variation of of mean eccentricity, argument of perigee and inclination

Figure 2.2 Variation of of mean eccentricity and argument of perigee for i = 60'

Figure 2.3 Variation of of mean eccentricity and argument of perigce for i = 63O.2

Figure 2.4 Variation of mean eccentricity and argument of perigee for i = 65"

Figure 2.5 Variation of long period (T) of mean eccentricity with inclination upto J24

Figure 2.6 Difference between observed and predicted values of semi-major axis

Figure 2.7 Difference between observed and predicted values of Eccentricity

Figure 2.8 Difference between observed and predicted values of Inclination

Figure 2.9 Difference between observed and predicted values of right ascension of ascending node 63

Figure 2.10 Difyerence between observed and predicted values of argument of perigee 64

Figure 2.1 1 Difference between observed and predicted values o f Mean anomaly 64

Figure 2.12 Difference between observed and predicted values of Mean argument 65

Figure 2.1 3 Difference between observed and predicted values of perigee height 65

Figure 2.14 Difference between observed and predicted values o f apogee height 66

Figure 2.14

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Values of Bilinear relation

Difference between observed and predicted values of semi-major axis

Difference between the observed and predicted values of eccentricity

UitYerence between the observed and predicted values of inclination

Difference between the observed and predicted values of right ascension of' ascending node

Difference between the observed and predicted values of argument of perigee

Difference between the observed and predicted values of Mean anomaly

Difference between the observed and predicted values of perigee height

Difference between the observed and predicted values of apogee height

Values of Bilinear relation

Comparison of numerically and analytically computed values of variation in semi-major axis during one revotution (case A)

Comparison of numerically and analytically computed values of variation in eccentricity during a revolution (case A)

Comparison of numerically and analytically computed values of variation in inclination during a revolution (case A)

Comparison of numerically and analytically computed values of variation in semi-major axis during a revolution (case 0)

Figure 4.5 Comparison of numerically and analytically computed values of variation in eccentricity during a revolution (case D) 110

Figure 4.6 Comparison of numerically and analytically computed values of variation in inclination during a revolution (case D) 110

Figure 4.7 Difference between numerically and analytically computed values of position vector during a revolution (case D) 1 1 1

Figure 4.8 Comparison of differences between numerically and analytically computed values of semi-major axis (case A ) 1 1 1

Figure 4.9 Comparison of differences between numerically and analytically computed values of eccentricity (case A) 112

Figure 4. t 0 Comparison of differences between numericatly and analytically computed values of inclination (case A)

Figure 4.1 I Comparison of differences between numerically and analytically computed values of semi-major axis with respect to inclination(case C)

Figure 4.12 Comparison of differences between numerically and analytically computed values of eccentricity with respect to inclination (case C)

Figure 4.1 3 Comparison of differences between numerically and analytically computed values of inclination with respect to inclination (case C)

Figure 5.1 Di f'ference between numerically and analytically computed values of semi- major axis up to 1000 revolutions. 130

Figure 5.2 Difference between numerically and analytically computed values of eccentricity up to 1000 revolutions. 130

Figure 5.3 Difference between numerically and analytically computed values of semi- major axis after 1000 revolutions with respect to perigee height.

Figure 5.4 Difference between numerically and analytical1 y computed values of eccentricity afker I000 revolutions with respect to perigee height.

Figure 5.5 Difference between numerically and analytically computed values of semi-major axis aRer 1000 revolutions with respect to eccentricity.

Figure 5.6 Difference between numerically and analytically computed values of eccentricity after I000 revolutions with respect to eccentricity.

Figure 5.7 Difference between numerically and analytically computed values of semi-major axis after 1000 revolutions with respect to inclination.

Figure 5.8 Difference between numerically and analytically computed values of eccentricity after 1000 revolutions with respect to inclination.

Figure 6.1 Difference between numerically and analytically computed values of semi-major axis up to I000 revolutions 1.52

Figure 6.2 Percentage errors between numerically and analytically computed values of semi-major axis up to I000 revolutions 152

Figure 6.3 Difference between numerically and analytically computed values of eccentricity up to I000 revolutions. 153

Figure 6.4 Percentage errors between numerically and analytically computed values of eccentricity up to 1000 revolutions. 153

Figure 6.5 Difference between numerically and analytically computed values o f semi-major axis after I 000 revolutions with respect to eccentricity.

Figure 6.6 Difference between numerically and analytically computed values of eccentricity with respect to eccentricity after 1000 revolutions.

Figure 6.7 Difference between numerically and analytically computed values of the semi-major axis with respect to inclination after t 000 revolutions.

for M ~ ' s - 'igure 6.8 Difference between numerically and analytically computed

values of eccentricity with respect to inclination after 1000 revolutions. 155

Figure 6.9 Difference between numerically and analytically computed values of the semi-major axis with respect to perigee height after I000 revolutions. 156

Figure 6.10 Difference between numerically and analytically computed values of eccentricity with respect to perigee height after I000 revolutions.

Figure 7.1 Difference between numerical and analytical values of semi-major axis up to 1000 revolutinns. 182

Figure 7.2 Difference between numerical and analytical values of eccentricity up to 1000 revolutions 182

Figure 7.3 Difference between numerical and analytical values of semi-major axis with respect to perigee height after 500 revolutionsl83 ..

Figure 7.4 Difference between numerical and analytical values of eccentricity with respect to perigee height after 500 revolutions. 183

Figure 7.5 Difference between numerical and analytical values of semi-major axis with respect to eccentricity afier 500 revolutions. 184

Figure 7.6 Difference between numerical and analytical values of eccentricity with respect to eccentricity after 500 revolutions. 184

Figure 7.7 Difference between numerical and analytical values of semi-major axis with respect to inclination after 500 revolutions. 1 85

Figure 7.8 Difference between numerical and analytical values of eccentricity with respect to inclination after 500 revolutions. 185

LIST OF APPENDICES

Appendix 4.1

Appendix 6.1

Appendix 7.1

Appendix 7.2

Appendix 7.3

Appendix 7.4

Nomenclature

a : Semi-major axis

A : Effective area of the satellite

Al . B1 : Canonical drag forces

BN : Ballistic coefficient

c : Ellipticity of the atmosphere

C u : Drag coef'ficient - -

( ;,.)I, - sf,,,?, : Normalized geopotential coeficient

: Aerodynamic drag force per unit mass

: Eccentricity

: Eccentric anomaly

: True anomaly

: Negative of the total energy

: Negative of the Kepler energy

: Hamiltonian

: Average density scale height

: Apogee height above the Earth surface

: Homogenous Hamiltonian

: Perigee height above the Earth surface

: Inclination

: Bessel functions of imaginary argument

J ,I : Earth's nth zonal harmonic term

111 : Initial inclination

I . : 1,evi-Civita matrix

I_ 7 : 'I'ransformation matrix of L

m : Mass of the satellite

M : Mean anomaly

N,,,,, : Normalizing factor

: Generalized momenta

: Perturbations other than V

: Legendre polynomial of degree n,

: Normalized Legendre functions

: Radial distance

: Earth's equatorial radius

: Apogee radius

: Perigee radius

: Initial perigee radius

: New independent variable

: Generating function

: Time

: Position vector in KS uniform regular canonical variables

: Velocity vectors in KS uniform regular canonical variables

: Velocity

: Perturbation due to oblateness

Velocity at initial perigee

: Position components

: Position vector

: Velocity components

: Velocity vector

: Right ascension of the Sun

: Right ascension of the center of the day time bulge

: KS canonical elemet~ts

: Declination of the Sun

: Declination of the center of the day time bulge

Kronecker delta functions

r a n d s !atuiCe orb- - : Rotational rate of the atmosphere about the Earth's axis

: Right ascension of ascending node

: Argument of perigee

: Atmospheric density

: Average density

: Density at initial perigee

: Radial distance from the Earth's centre to the surface of oblate spheroid

: Equatorial radius of oblate spheroid

: Ellipticity of Earth

: Rotational rate of the Earth

: Geocentric latitude

: Geocentric longitude

: Gravitational constant

: Time element

: Decrease in semi-major axis

: Decrease in eccentricity

A t r

Canonical Equations

1. i Historical background

An accurate orbit prediction of the Earth's artificial satellites is an important requirement

for mission planning, satellite geodesy, spacecraft navigation, re-entry and orbital

lifetime estimates. The accurate prediction of satellite decay time some months or years

ahead remains one of the most difficult and intractable problems of orbital mechanics,

chiefly because the lifetime depends strongly on the future variations in air density,

which are at present not accurately predictable. A sate1 lite moving under the gravitational

field of Earth will be influenced by various forces, known as the perturbing forces such as

the shape of the Earth, the Sun's radiation, the resistance of the atmosphere. gravitational

attraction due to the Sun and the Moon, the Earth's magnetic field etc. The combined

effects of all these perturbations cause the motion of the satellite to deviate from the two

body motion. Far the near Earth satellite orbits, the forces due to the Earth's flattening

and the air drag are responsible to bring the satellite back to Earth. 'Thus inclusion of

these effects as the perturbing forces becomes very important for precise orbit

computation of the near Earth satellite orbits.

Historically, the motion o f the Moon had played a very prominent role in the study of the

perturbations and very useful mathematical methods for orbit computations were

evolved. Its djffict~lt orbit challenged scientists seeking solutions for perturbed motion. In

1690, Isaac Newton laid the basis for determining the Moon's orbit with his law of

gravitation. He derived much of his Principia from studying the Moon's motion. Clairaut

(1 71 3- 1765) continued the study, along with d' Alernbert ( I 7 17-1 783) and Euler ( 1 707-

Chapter-1 Introduction 1

t 783). Newton explained most of the variations in the Moon's orbit, except the motion of

the perigee. In 1749 CIairaut found that the second order perturbation terms removed

discrepancies between the observed and theoretical values which Newton hadn't treated.

Then, about a century later, the full explanation was found in one of Newton's

unpublished manuscripts [ I ] .

Laplace ( 1 749-1 827) brought his special form of mathematical elegance to the solution of

the Moon's motion. First, he used the true longitude (h) as the independent variable in the

equations of motioh. Then, he explained the secular acceleration of h - it depends on the

eccentricity of the Earth's orbit, which changes over time. Adams showed that this

approximation was within 6' of the observed values. Roy [2] suggested that this

discrepancy i s related to tidal friction. As is often the case, the mathematical masters

provided the answer hundreds of years ago.

- + Peier .A. Hanstn ( 1 795- 1874) restructured much of Laplace's work and developed tables

that were used from 1862 to about 1922. Newcomb (1 835-1909) made some empirical

corrections to Hansen's tables. and his modified results were used after 1883. DeIaunay

(1 8 16- 1872) i s credited with developing the most complete algebraic solution For the

Moon's motion until the age of modem computers. He published his results in La theorie

du mouvement de la tune (The Theory of Lunar Motion) during 1860-1867. It was

accurate enough to predict the Moon's motion to one radius over a period o f 20 years.

The ability to precisely model the Moon's motion remained somewhat elusive as

measuring devices became more accurate. Hill ( 1 838-1 91 41, significantly improved

Lunar theory by introducing the theory o f infinite matrices. He also refined theories for

the motions of Jupiter and Saturn. Brown (1866-1938) followed the improvements of

Hill's theory. Brown spent most o f his life studying the motion of the Moan. He

published very accurate tables in1908. They replaced Hansen's tables in about t923 and

were used for decades afterward. Brown produced a lunar theory consisting of more than

1500 terms. Deprit reproduced Delaunay's Lunar theory with computers and symbolic

manipulation. He found some errors in Delaunay's theory and therefore improved its

Chapter-I Introduction

accuracy. Battin [3] highlighted that Euler was the first to present the variation of

parameters (VOP) method. The first practical orbit-determination technique was an

application of VOP proposed by Lagrange in 1782. He formed what we know today as

the Lagrange Planetary equations.

1.2 General Introduction

For calculating the ephemeris of an artificial Earth's satellite, it has became necessary to

use extremely complex force models i n order to match with the accuracy, which is

consistent with the present day operational requirements and observational techniques.

The effects of the atmosphere are difficult to determine, since the atmospheric density,

and hence the drag undergoes large fluctuations. To predict the orbit precisely, a

mathematical model representing these forces must be sefected properly for integrating

the resulting differential equations of motion .The options for mathematical solutions can

be classified as analytical, semi-analytical and numerical.

The analytical solution can he obtained by general perturbation techniques. General

perturbation techniques replace the original equations of motion with an analytical

approximation that captures the essential character of the motion over some limited time

interval and which also permits analytical integration. Such methods rely on series

expansions of the perturbing accelerations. In practice, we truncate the resulting

expressions to allow simpler expressions in the theory. The trade-off speeds up

computation but decreases accuracy. Unlike numerical technique, analytical methods

produce approximate, or "general" results that hold for some limited time interval and

accept any initial input conditions. The quality of the solution degrades over-time, but the

numerical solution also degrades- at different rates and for different reasons. Analytical

techniques are generally more difficult to develop than numerical techniques, but they

often lead to a better understanding of the perturbation source.

Semi-analytical techniques combine the best features of numerical and analytical

methods to get the best mix of accuracy and efficiency. The result can be a very accurate.

s for -Earth's S a s r

relatively fast algorithm which applies to most situations. But semi-analytical techniques

vary widely. We choose a semi-analytical technique mainly for its ability to handie

varying orbital applications, its documentation, and the fidelity and the number of force

models it includes. Most semi-analytical techniques have improved accuracy and

computational efficiency, but the availability of documentation (including very structured

computer code) and flexibility are often important discriminators.

The numerical methods use special perturbation techniques. The perturbations are

deviations from a normal, idealized, or undisturbed motion. Special perturbation

techniques numerically integrate the equations of motion including all necessary

perturbing accelerations. Because numerical integration is involved, we can think of

formulations which produce a specific or special answer that is valid only for the given

data (initial conditions and force-model parameters). Although numerical methods can

give very accurate results and often establish the "truth" in analysis, they suffer from

their spec,ificity, which keeps us from using them in a different problem. Thus, new data

means new integration, which can add lengthy computing times. Persona! computers now

compute sufficiently fast enough to perform complex perturbation analysis using

numerical techniques. However, numerical methods suffer from errors that build up with

truncation and round-ot'f due to fixed computer word length. These errors can cause

numerical solutions to degrade as the propagation interval lengthens.

In literature there are large numbers of analytical solutions, which describe the effects of

atmospheric drag on the motion of an artificial satellite in the gravitational field of an

oblate Earth. Newton 141 was the tlrst scientist who studied the effect of drag on the orbit

of a satellite. He showed that a body acted on it, by an inverse square gravitational

attraction and moving in an atmosphere with density proportional to I/r follows a

contracting equiangular spiral path. This was the general estimation for the next 250

years. Singer [S ] developed a semi-analytical method to evaluate first the lifetimes of

circular orbits and then by assuming an impulsive deceleration at perigee, he estimated

the lifetimes for elliptic orbits. His choice of the profile of the air density was quite close

to that which was later established for times of low solar activity. Henry 161 discussed

lifetimes of the elliptic orbits. A semi-numerical method based on the classical

Newtonian equations was presented by Davis, Whipple and Zirker [7]. All these works

were carried out before the launching of the artificial satellite Sputnik in October 1957.

Modern analysis o f perturbations centered on events of the late 2oth centaury. In

particular, the launch of the Sputnik-l satellite by Soviet Union on October 4, 1957

sparked tremendous interest in space. Although the pioneers had laid the foundation for

many of the required analysis, tremendous technical gaps existed when applying these

resutts to the modern small satellite. Indeed, much of the analysis had been formulated

tor celestial objects and distant planets. Satellites orbiting near Earth presented some new

challenges.

The launching of the satellite Sputnik-! provoked several further attempts at methods o f

predicting sate1 lite I ifetime. Large number of papers appeared on determining density

from the changes in the orbital period of a satellite.

In 1959, Kozai published his work at the Srnithsonian Astrophysical Observatory and the

Harvard College Observatory in Cambridge, Massachusetts. The perturbations of the six

orbital elements o f a close Earth orbiting satellite moving in the gravitational field of the

Earth without air assistance were derived as functions of mean elements and time. No

assumptions were made about the order of magnitude of eccentricity and inclination. The

coef'ficient of the second harmonic of the potential was assumed to be a small quantity of

the first order and that those of the third and fourth harmonic were of second order. The

results included periodic perturbation of the first order and secular perturbation of up to

second order 181. Kozai's approach had remarkable insight and provided the basis for the

first operational, analytical approaches for determining satellite orbits. But because it

didn't include drag, the results were very limited, especially because most of the early

satellites operated almost entirely within the drag envelop (the atmospheric region that

strongly afyects a sate1 lite's orbit).

Chapter-1 Introduction

Prod/- for N e a r - P f m Rmulac Canonical

The same year, Brouwer [9] developed a satellite theory for military planners and

operators. He published the results in the very same issue of the Aslronomical journal in

which Kozai's results appeared. His approach provided the solution of the main problem

for a spheroidal Earth potential limited to the principal term and second harmonic which

contained the small factor Jz. The solution was developed in powers of J2 in canonical

variables by Von Zeipel method. The periodic terns were divided in two classes: the

short-periodic terms contain the mean anomaly in their arguments: the arguments of the

long-periodic terms are multiples of the mean argument of perigcc. 'The results were

obtained in closed form. The solution did not apply to orbits near the critical inclination

of 63.4 degrees. Brouwer also gives contributions due to J3. Jq and Js in the same paper.

Although Brouwer's ideas were very similar to those of Kozai, he used different method.

In 1 96 1 , Brouwer and Hori [lo] extended the original work to include the effects of drag.

But new theories and applications continued to develop. As the computational throughput

of machines grew, the complexity of the theories increased. During the mid 1960's and

I 970's several different contributors developed satellite theories based on the VOP

formulation-loosely based on the perturbation technique known as the method of

uveri~ging.

'The next decade saw a unique semi-analytical theory from a team of scientist led by Paul

Cefola. Kemarkably, one of the technical inspirations of their Draper Semi analytical

Satellites Theory (DSST) came from the work of Hansen in 1855 [ I I]. Cefola found

Hansen's epic work on expansions for modeling elliptical motion. Recent studies by

Barker et a]. [12] have shown this particular approach to be considerably more accurate

then existing analytical theories with comparable (or better) computing speed.

Methods of increasing power were given by Groves [13], Sterne [ I 41 and King-Hele [ 151.

Theories detining the variation of orbital period, perigee distance and eccentricity with

time were first developed by Nonweiler 1161 and then by King-Hele and Lestie [ I 71. The

latter work was developed in to a very important book [ I 81. The book was considerably

improved latter [ I 91 by adding new researches in the area of satellite motion under the


effect of air drag force and some other perturbing forces. Perkins [20] used perturbation

solutions to obtain lifetime estimates and results were given for the cases of exponential

and power law-density variation. Parkyn [21] integrated analytically Lagrange's

planetary equations 1221 and obtained the variations in 'a' and 'e' in terms of modified

Bessel functions [23]. The effect of non-stationary atmosphere on the orbital inclination

was considered by Bosanquet 1241, Vinti [25] and also by Merson and Plimmer 1261.

Using a power-law variation of density with height, Michelsen [27) formulated the orbital

variations during a satellite's whole lifetime while per-orbit changes were investigated by

Parkyn 1281 and many others.

'The variational equations that describe the motion of an artificial satellite about the center

o f the Earth are usually expressed in terms of the classical orbital elements ( semi-major

axis 'a', eccentricity 'e', inclination 'i', right ascension of ascending node 'IZ', argument

of pedigree 'w' and mean anomaly 'M'). The dynamical system of a satellite motion

perturbed by both atmospheric drag and gravitational attraction is non-linear, non-

conservative in form and the integration of the system, in general i s analytically

intractable. Some of the early studies and analytical difficulties for the coupled problem

were addressed by de Nike [29]. Suitable perturbation m h o d s in celestial mechanics to

drive an approximate solution with desirable accuracy are (i) the Vun Ziepel method [30,

3 1 , 321 (ii) the two variable asymptotic expansions 133, 341 (iii) the Lie series 135, 361

and (iv) the general theory of method of averaging [37, 381. Morison [38,39] has showed

that both the Von Zipel method and the two variable asymptotic expansions are special

cases of general theory of method of averaging. Hori [40], Kame1 [4 11 and Choi and

Tapley [421 have discussed the extension of Lie series to non-canonical systems. Shinad

1431 discussed the equivalence of the Von Zeipel method and Lie series. Watanabe [44]

and Ahmed and Tapley 1451 discussed the equivalence of the method o f averaging and

Lie transform.

After Brouwcr and 1Iori 1101 published their work, Lane [46], Lane and Cranford [47],

Zee [481, Barry and Rowe 1491, Willey and Piscane [50], Chen [51]. Watson et al 1521,


Santora [53], Muller et a1 [54], Hoots [55 , 561 and Vihena de Moraes [57] have obtained

analytic solutions. Lane [46] used a non-rotating spherical power function density model

1581. The theory was carried out to the same order as that of Brouwer-Hori theory and

had same limitations. The use of power function density model removed the convergence

problem. Lyddane 1591 and Davenport [60] examined the problems of small eccentricity

and small inclination for Brouwer's [9] drag free solution and gave their modifications to

remedy the singularities. Lane and Cranford 1471 improved Lane's [46] theory by

reformulating the theory to eliminate the sin divisors. It is also an extension of L,yddane-s

[59] modification for a drag-free solution to remove the additional small divisors prrrsenr

in the drag terms. The complete explicit solutions for all six orbital elements were given.

Though the numerical results were not included in the theory, truncated version of their

theory [61] was used in the NOKAD operational system for many years. Zee 1481 used a

set of dimensionless variables derived from the spherical coordinates and an averaging

method [62] to obtain a first-order singly averaged dynamical system. In this theory, only

the second zonal harmonics and a non rotating spherical exponential density model was

taken in to consideration and was restricted to small values of eccentricity.

Barry and Rowe 1491 used a Fourier series expression for the density model in which the

coefficients were determined by the Jacchia 1970 model [63] . The solutions were

obtained through first-order periodic and second order secular effects for Earth's

oblateness (.I2, J3 and Ja) and first-order drag effects. Lorell and Liu 1641, Liu and Alfred

[65], Liu [66] and Slutsky and McCtain [67] proposed the drag-free theories in an

anatytical form by the averaging of conservative perturbations. W i l ley and Pisacane [50]

extended the Lane theory by introducing a power function density model with a quadratic

instead of a linear density scale height. Their complete solutions and detailed anaIysis are

given in [68]. Chen [5 I ] introduced a modified exponential density function to take the

atmospheric oblateness and the diurnal variation of density in to consideration. The

values of parameters: density, density scale height and measure of the amplitude of'day-

to-night variation in density are computed using Jacchia 1971 density model [69]. Chen

assumed that the drag perturbing force is a second order quantity and the associated


perturbing terns are expanded in power series of 'e' and retaining terms up to e2. He then

extended his drag free analysis [70j with 32, J3 and Jd and obtained a second order

solution for the dynamical system with the combined effect of Earth's oblateness and a

rotating atmosphere using two-variable asymptotic expansions. Watson et al [52]

introduced an analytical iterative method to avoid the complexity of the oblateness - drag

effects illustrated by Brouwer and Hori [IO] and Sherrill [71]. In their method, the efTect

of Earth's oblateness was accounted for by the Vinti spheroidal theory [72j. Two test

cases were given for the long-term decay predictions. In both cases the predicted

lifetimes were with in 4% of the true vaIues with out the use o f Vinti differential

correction algorithm [73] .

Santora [53] included the oblateness and diurnal density effects of the atmosphere on the

orbits of small eccentricity. The average changes in 'a' and 'x = ae' due to drag in one

revolution were obtained i n terms of the modified Bessel functions. He used Kozal's [8]

drag free solution to improve the determination of decay rates. Muller et al [54j presented

an analytic theory to improve the short and long-periodic, secular effects of Jz and higher

order zonal harmonics, secular and drag effects. Moots [55] used the gravitational and

atmospheric models as used by Lane [46] and arrived at an improved analytical solution.

A numerical comparison done with a slightly modified Lane and Canford theory [47]

using the same reference orbits showed a noticeable improvement in accuracy. Vilhena

de Moraes [57] extended Ferraz Mello's 1741 solution to include the atmospheric drag

and solar radiation ef'fect. He adopted the transformation suggested by Ferraz Mello 1751

and the Delauny angular variables were modified to avoid the appearance of Poisson's

terms. He then applied the method of variations of arbitrary constants and successive

approximations to obtain the coupled solution. Hoots 1561 used his solution and Liu's

singly averaged variation equations [76] to arrive at an analytical solution for the

dynamical system with J2, J3 and J4 and drag by choosing a rotating empirical density

model. The constant parameters and the assumed solutions were determined as in 1551.


tical a n d i - B l 3 - The analytical expressions of the density models not only lack the dynamic representation

of the atmosphere but also may not provide accurate values for the density. An alternative

approach of the coupled problem is to adopt a combination of general and special

perturbation technique, referred to as a semi analytical method. This method enhances

efficiency through the use of analytical techniques whenever possible and makes

sufficient numerical methods to permit the inclusion of an empirical density model

without using series expansions. Well-known and commonly used models are Jacchia

1964 [77], Jacchia 1970 1631, Jacchia I97 1 1761, Jacchia 1977 [78] and MSlS 78 [79] ,

Barlier et al. LBO], MSIS-86 1811 and MSIS-90 [82]. The empirical density models [63.

76, 791 treat the density as a function of altitude above the surface of an oblate Earth,

longitude, latitude, solar flux, geo-magnetic index and time. In an analytic version of

Jacchia 1977 model [78], de Lafontaine and Hughes (831 known as the global analytical

model (GAM) avoids the large memory space requirements of the tabular models and the

extensive computer time needed by the numerical models. h addition to static variations

(including flattening of the atmosphere), the GAM accounts for the solar activity, geo-

magnetic activity, diurnal, semi-annual and seasonal-latitudinal cycles of density

variations. These variations are graphically illustrated in de Lafontaine [84] and de

Lafontaine and Marnmen [85] . A comparison o f the Jacchia 1977 model [78] with

accelerometer density data [86] concludes that the root-mean-sq uared e m r in Jacchia

density model is around 10%. MSIS-90 uses analytic models to model the lower altitudes

to account for disturbances such as solar activity, magnetic storms and daily variations as

well as latitude, longitude and monthly variations. The application of these and some

other empirical atmospheric density models in orbital mechanics in the real world has

been discussed by Liu et a1 [86].

Representative works in analytic approach are by Pimm [87], Kaufman and Dasenbrock

[88], Barry, Pim and Kowe [891, Dallas and Khan [90], Wu et at [91], Lidov and

Solov'ev 1921, Green and Cefola 1931 and Liu and Alford [94]. Pimm 1871 developed a

semi-analytic. long-term orbit theory that used Simpson's method to evaluate the

averaged drag effects for a rotating atmosphere, Dit'ferent analytic solutions [7. 951 ofthe


Numeri- for Near -Em's

Earth's gravitational perturbation due to J2 to .I4 zonal harmonics were adopted to obtain a

combined solution by means of a fourth-order Runge-kuta (R-K) method. The Jacchia

1964 atmospheric density model [77] was used for the analysis. The historical

Smithsonian Astrophysical observatory (SAO) mean orbital elements, determined by the

SAO Differential orbit improvement programme published by the SAO special report

1841 were used as the database. Kaufman and Dasenbrock [87] formulated a lengthy

semi-anal yt ic solution t'or analyzing both lunar and terrestrial orbiters. Dallas and Khan

[90] developed a semi-analytical theory using the singly averaged differential equations

i n terms of parameters valid for all eccentricities less than 1. Wu et aI [91] used an

averaging technique similar to that of Kozai 171 and developed a second order semi-

analytic theory to include the perturbations due to nun-spherical Earth atmospheric drag,

third-body efyects and the solar radiation pressure.Two rotating empirical density models,

CIRA 1972 [96] and Jacchia 1977 [78] are considered in their theory. In their analysis, all

perturbing forces are treated as second order quantities except J2. Solov'ev [97) described

semi-analytic drag free theory (including non spherical Earth and third-body

perturbations) using singly averaged Delauny variables and the Von Zeipel method.

Lidov and Solov'ev 1921 extended the theory to include the atmospheric drag for a high

eccentricity resonant orbit with an exponential density function. Cefola ct al [98]

discussed a semi-analytic approach to include the Earth's oblateness. third-body

perturbations, atmospheric drag and solar radiation pressure using the generalized method

of averaging. Green and Cefola 1931 assumed Fourier series expansion for the short-

periodic variations and developed a semi-analytic solution. The coefficients for the drag

variations are determined using a method similar to that o f Lutsky and Uphoff 1991 by

numerical quadrature technique. Using Liu's singly averaged drag free equations [IOO].

which include JZ, J1 and j,, Alford and 1,iu [94] developed a semi-analytic long-term orbit

theory using a generalized method of averaging with the assumption that the drag force is

a second order quantity. The procedure is to extend a system of first order differential

equations for a set of well-defined mean orbital elements to include the drag eftect due to

a rotating atmosphere. Numerical results [ I O I ] for the long term decay predictions vcrsus

satellite orbital data obtained from SAO special reports demonstrated that a semi-analytic


theory could provide a means to estimate the orbital decay history and lifetime with good

accuracy and efficiency. Liu and Alford [94] extended their long-term solutions to

include the computation of the fast variable, mean anomaly, and use of an initial orbit

determination algorithm [ 1 021 so that accurate short-term ephemeris can be generated.

The stroboscopic method developed by Roth and used in different applications 1103,

104, 105, 1 061 expresses the variation-of-parameter (VOP) equations with the true

anomaly as the independent variable and time as a dependant variable. In most serni-

analytic theories 187, 1071, Wagner, Douglas and Williamson [ I 081 and Alford and L i u

I1091 carry out their numerical averaging with respect to the true anomaly although the

propagation equations of the mean orbitat state always rely on the mean anomaly. The

computation of the mean anomaly from the true anomaly is an explicit operation. De

Lafontaine [84] discusses this method to be an extension of stroboscclpic method.

Theories, which discuss the complete transformation of the first order in Jz, will exhibit a

second order secular mean anomaly error due to initialization procedure. Lyddane and

Cohen i l l O] have demonstrated this fact by recovering the second order quantities ctue to

52' in semi-major axis. Later Hreakewell and Vagners [l I I ] also investigated the problem

and concluded that accuracy may be kept to the second order by either including thc

mean motion with the aid of the energy integral or fitting an orbit theory to data over

many revolutions. A theoretical determination o f the projected area A, and of the drag

coefficient Cn, is also a very involved field. The most relevant papers on the aerodynamic

properties of the satellites are those by Cook [ l 12, 1 131, Nocilla [ I 141, and Jastrow and

Pearse [ I 151. Drag coefficients were investigated in King-Hele [18], Cook [ I 12. 1 131,

Williams [I 16, 11 71 and Nocilla [ I 141. The effects of uncertainty and variation in CD are

treated in Hunziker [ f 181 and a discussion of the thermal accommodation coefficient and

other related parameters are found in King-Hele 11 81, Ladner and Ragsdale [ I 191 and

Wachmen [120J. The works in the area of orbital mechanics and the numerous related

fields are provided in Szebehely 112 11 and methods of orbital determination are explained

in Escobal [ I 22, 1231.

Chapter-l Introduction

Sterne [I241 and Ewart [I251 evaluated the changes in orbital elements at the end of one

revolution in a general form. Davies [I261 evaluated the effects of atmospheric

oblateness. Santora [ 1 27, 1281 studied the combined effect of atmospheric oblate ness

and the day-to-night variation with low eccentricity orbits. The effects of atmospheric

rotation and of geomagnetic and solar activity on the accuracy of prediction of satellite

position are studied in 1129). Swinerd and Boulton [I301 presented a more

comprehensive atmospheric model that combines atmospheric rotation, oblateness and

the daytime bulge. 'J'hey determined the perturbations over one revolution with third

order accuracy and are an improvement over Santora's. King-Hele [ I 3 1 1 used graphical

approximation to consider the dynamic density variations of the atmosphere. The elr'ect

of atmosphere on both satellite's orbital inclination and right ascension of ascending node

were evaluated fully by Sterne [ I 241 and by Cook and Plirnmer [ 1 321. Subsequent works

for an oblate atmosphere was by Cook [I33], an atmosphere with H varying with height

was by King-Hele and Scott [ I 341 and an atmosphere with day-to-night variation was by

King-Hele and Walker 1 1 351. Later, using the same atmospheric model, Cook and King-

Hele [136] evaluated the effects on near circular objects. Results with oblate atmosphere

with diurnal variation for near-circular orbits were derived by Swinerd and Houltan

[137]. They further considered the effects of the variation of density scale height nith

altitude [13 11. In another paper [138] the change in argument of perigee during one

revolution for near-circular orbits with an oblate diurnally varying atmosphere was

studied. A theory for high eccentricity orbits in a spherically symmetrical atmosphere

was developed by King-Hele [I 391. King-Hele [140] evaluated the effects of meridional

winds on orbits of small eccentricity for a spherical atmosphere. Results were derived for

high eccentricity orbits by King-Hele and Walker [I411 in an oblate atmosphere. An

improved theory for determining changes in Satellite orbits caused by meridional winds

was developed by King-Hele and Walker [ I 421. Works during the 1970's were reviewed

in the thesis of de Lafontaine, which also includes discussion of the sources o f error.

There was a more general review by de Lafontaine and Garg [143].


The KS total-energy elements equations 11441 is a very powerful method for numerical

solution with respect to any type of perturbing forces as the equations are less sensitive to

round-off and truncation errors in the numerical integration algorithm - Merson [ I 451,

Graf et a1 [1461, Sharma and Mani [147]. An orbital frequency based on the total energy

gives more accuracy to in-orbit position calculations; the equations are everywhere

regular in contrast with the classical Newtonian equations, which are singular at the

collision of the two bodies. The equations are smoothed for eccentric orbits, because

eccentric orbit anomaly i s the independent variable. Due to symmetry in these equations,

only two of the nine equations were solved analytically to get the complete solution.

Sharma [I481 generated analytical expressions for short-term orbit predictions with zonal

harmonics .I2 by the method of series expansions. The study was continued with J1 and J4

by Sharma [I491 and Sharma and James Raj [I501 up to J6 terms using KS element

equations. In an attempt to compute more accurate short-periodic terms due to .IZ. even

for very high eccentricity orbits, Sharma [ 15 11 integrated the KS element equations

analytically in close form in eccentricity and inclination.

Sharma [I521 made an attempt to get an analytical solution using an analytical oblate

exponential atmospheric density model by series expansion, which include up to

quadratic terms in eccentricity and c, a small parameter depending on the ellipticity of the

atmosphere. Sharma [ I 53 J extended the work in a wide range of eccentricity by including

the third order terms. Sharma [I541 developed a third order non singular solution with an

oblate atmospheric model by including the effect of diurnal bulge. Other studies carried

out by Sharma were generation of non singular analytical theories with a spherical

symmetrical atmosphere [I551 and oblate exponential atmosphere [I%], for high

eccentricity satellite orbits.

A particular canonica! form of the KS differential equations, known as KS uniform

regular canonical equations, where all the ten elements are constant for unperturbed

motion and the equations permit the uniform formulation of the basic laws of elliptic,

parabolic and hyperbolic rnoiion (Stiefel and Scheifele, [144j, p250) are found to provide

P r e d i w for NsarcEarth's SatQUite Orbits wfth K S r m - accurate short as well as long term orbit predictions numerically, with Earth's zonal and

tesseral harmonic terms (Sharma and James Raj, [ I 57, 158, 1 591). These equations were

utilized by James Raj and Sharma, [ 1601 to generate an analytical solution for short term

orbit predictions in close form in eccentricity and inclination, with respect to Earth's

zonal harmonic terms Jz. J3 , Jq. The solution was a significant improvement over the

analytical solution with US elements by Sharma, [I51 1. The J2 solution was found to be

better than the solutions of Engles & Junkins 11 6 11 and Jezewski I t 621. The analytical

solution with .Iz provided a metre level accuracy after 1 800.0009 seconds in the case of a

ballistic trajectory of high eccentricity (0.9 1) with the initial altitude ot' I00 km 11 631.

Motivated by the improvement with the KS uniform regular canonical equations in the

analytical orbit predictions with .I*, J 3 , J4, the canonical forces were also included in these

canonical equations of motion to obtain the analytical orbit decay equations with the

atmospheric drag force, which i s the most important perturbing force for re-entry and

lifetime studies of near-Earth orbits.

Chapter 1 deals with some of the historical developments available in the literature Sor

predicting satellite motion using numerical, analytical and semi-analytical methods. We

have also highlighted some of the works in the relevant field, which helped us to bring

out the present theory explained in this thesis. We have also described the dit'ferent steps

used in deveIoping the theory for K S uniformly regular canonical element equations, for

predicting the motion of an artificial Earth satellite. For continuity sake, some of the basic

definitions involved in the theory are also touched upon. The perturbing aerodynamical

forces, which are likely to act on an artificial satellite, are also discussed. Since, the

present studies also consist of integrals in the form of tnodified Bessel's functions, the

integral representation of the Bessel's function o f the first kind and of imaginary

arguments is also included with its properties. The chapter ends with the advantages of

the KS uniformly regular canonical element equations for predicting the motion of an

art i f ic ial satellite over the Newtonian equations of motion.

Orb-

Canonical EElrrations

Chapter 2 deals with long-term orbit predictions numerically using KS uniform regular

canonical element equations with Earth's zonal harmonic terms. Using the recurrence

relations of the Legendre polynomials [ I 641, any number of zonal harmonic terms can be

included. However, we have computed the orbits by including the terms up to J3& to

study the effect of higher zonal harmonic terms. The validation is made by comparing the

predicted values of the orbital elements with the observed values of satellite IRS-]A for

up to more than 1200 revolutions (87 days time). The detailed numerical study was also

carried out for Sun-synchronous orbit and the orbits near critical inclination. The results

were published in thejournal 'Earth, Moon and planets' [158j.

Chapter 3 deals with the long-term orbit predictions numerically Earth's zonal and

tesseral harmonic terms, Using the properties of the modified Legendre polynomials, any

number of Earth's zonal and tesseral harmonic terms are included. However, we have

computed the orbits by including the terms up to J19.1Y, using GEM T2 Earth's gravity

model 1 t 65 (. The validation i s made by comparing the predicted values of the orbital

elements with the observed values of satellite IRS- I A for up to more than 650 revolutions

(47 days time). The study was published in the 'Proceedings of 44'h ISTAM Congress'

11591.

In Chapter 4, a new non singular analytical orbital theory for short term orbit predictions

using KS uniformly regular canonical element equations with Earth's zonal harmonic

terms J2, J j and .I4 is developed, The superiority of the developed analytical theory i s

found out by comparing the result obtained by the present theory using Jz effect with the

theories of Engles and Junkins [ I 6 13, Jeweski 11 621 and Sharma 11 5 1 1. The study was

published in 'Advances in Space Research' 11601.

In Chapter 5. a new non singular analytical theory for the motion of near Earth satellite

orbits with the air drag effect is developed for long-term motion in terms of the KS

uniform regular canonical elements, by assuming the atmosphere to be symmetrically

spherical with constant density scale height. Series expansion method is employed and

terms up lo third order terms in eccentricity are considered. Only two of the nine


Orbits with KS Uniform Reoular w n i c a l E a u a w

equations are solved analytically to compute the state vector and change in energy at the

end of each revolution, due to symmetry in the equations of motion. Numerical

comparisons of the important orbital parameters semi-major axis and eccentricity up to

1000 revolutions, obtained with the present solution, with KS elements analytical

solution and Cook, King-HeIe and Walker's theory 11661 with respect to the numerically

integrated values, show the superiority of the present solution over the other two theories

for a wide range of eccentricity, inclination and perigee height. 'l'he study was published

in 'Planetary and Space Science' Journal [l671.

Chapter 6 describes the development of a non singular analyticat theory for orbit

predictions using KS uniformly regular canonical element equations by assuming the

atmosphere to be oblate exponential with constant scale height. A third order solution is

obtained by integrating the KS uniformly regular canonical element equations

analytically. The terms up to third order terms in 'e' and 'c' are included in the series

expansions. Numerical comparisons of the important orbital parameters semi-major axis

and eccentricity up to 1000 revolutions, obtained with the present solution, with KS

elements [I541 analytical solution and Swinard-Boulton 11301 with respect to the

numerically integrated values, show the superiority of the present solution over the other

two theories for a wide range of eccentricity, inclination and perigee height. The study

was presented in 56th International Astronautical Congress held at Fu kuo ka, Japan

during October 2005 [ I 681.

In Chapter 7, a new non singular analytical theory is developed for orbit predictions with

a complex force model of oblate exponential atmosphere with day to night variations. A

third order analytical solution is obtained by analytical integration in terms of modified

Bessel's functions. Comparison of the present analytical solution with the analytical

theories of KS and Swinard-Boulton up to 1000 revolutions fbr a wide range of

eccentricity, inclination and perigee height reveals the superiority of the present theory

over the other two analytical theories. The study was accepted for publication in

'Planetary and Space Science' Journal 11 691.


* 1.3 Perturbed Equations of motion

The Newtonian equations of motion of a particle with mass ' m ' attracted by a central

body af mass ' M ' at a distance ' r ' with respect to a coordinate system centred at M is

given by

+ ? + ~ T = o ; p = k " ~ + m ) ( I .3.\) I'

where k' is the universal gravitational constant.

Forces nthcr than the attraction of the central body of mass M. like atmospheric

resistance, light pressure, third body attraction or shape and size of the central body may

act on the particle of mass m. All such additional forces may be represented by a single

forcePacting per unit mass of the particle, which is called the perturbing force. The

perturbing puteutial V depending upon the tirne't' and position 2 of the particle,

generate a perturbing force given by the gradient of the perturbing potential V . P

contains the forces which cannot be derived from the potential V. 'l'he perturbed

equations of motion are

1.4 Fictitious Time / New Independent Variable

l'he equations of motion (1.3. I ) and (1.3.2) are singular, near collision of the two bodies

(at r = 0). Hence the equations are not suitable for numerical integration for small values

of r. To avoid these singularities, we can regularize the equations of motion. 'The basic

idea for performing this regularization is to compensate the intjnite increase of the

velocity i at collision by multiplying by an appropriate scaling factor, which vanishes at

collision. Obviously, such a factor is distance r itself. So, a new independent variable's' is

introduced in such a way that the velocity with respect to s is rk .


. dr dr 1.e. - = r . - .

ds dt

The new variable 's' is called the tictitiaus time defined by the following three equivalent

expressions.

The transformation from the ordinary time 't' to the fictitious time 's' is perfbrmed by

So, by using the fictitious time 's' the equations o f motion (1.3.2) can be written as

where the prime indicates the differentiation with respect to 's'.

The kinetic energy of the particle per unit mass is given by

where v is the velocity of the particle and ( , ) denotes the scalar product of the two

inserted vectors.

We have

r? = ( Z, -7 ),

Differentiating (1.4.3) with respect to t, we get . -.

? = ( 2 . i ) *

By using ( 1 -3.21, we get

Using ( I .4.4). we get

Using ( 1.4.3). the above equation become

The terms inside the parenthesis of the left side is called the Kepler energy and it is

denoted by -hk.

The negative energy h is given by the relation h = hk - V.

Using ( 1 -4.3, the above equation become

and this leads to the law of energy given by


Now by Eq.(1.4.3), we have

v2 = (i,i),

Thus kq.(1.4.6) becomes

Thus the energy relation ( I .4.7) becomes

Now, Eq.(1.4.8) becomes

1.5 Levi-Civita Matrix

The transformation of 2 in hvo dimensionat physical plane into a new two dimensional

parametric plane i? defined by r = u2 is given in terms of complex variables by the

mapping

xl + i x2= (ul + iu~}' . ( 1.5. I )

This transformation, known as Levi-Civita's transfurmation 11 701 i s equivalent to

2 7 X I = uj -uz-. ~ 2 ' Zulu2. ( 1.5.2)


By differentiation, the matrix relation

is obtained. This appears to be in more compact form by introducing Levi-Civita matrix

Therefore, by using the above Lcvi-Civita matrix (L-matrix), Eqs. ( 1.5.2) and ( 1.5.3) can

be written as

2 = L(u)G, ( 1 S . 5 )

2' = 2 L(ii)Zi'. ( I .5.k)

The L-matrix satisfies the following properties

i) L( Zi ) is orthogonal

:. L'(ii)L(ii) = (ut, 2 ) = P' ,

i i ) The elements of L( G ) are linear and homogenous functions of the parameters u, . I

i.e. ~ ( 2 ) = L(G').

iii) The first column o f L ( G ) is the position vector of ii.

Finally, the following two rules are valid for any two vectors ii and V in the parametric

plane.

I.(:) c' = L(?)($), ( 1 S.8)

(ii,Zi) L($)vt - 2(Zi.?) L(Ei)? + (v ,vt)L(G).Ei=O. ( I S .9)


1.6 KS Transformation

'The generalization of Levi-Civita matrix to a 4-vector composed of ul, ul, us, u4 is

defined by

In this matrix. the upper left hand corner is the previously described Levi-Civita matrix.

lJsing the matrix (1.6. I) , a vector (xi. xz; x3) in the physical space is transformed to a 4-

vector 2 by adding its fourth component of value zero. Hence, a similar transformation

to (1 -5 .5 )

2 = L(u')ii ( 1 h.2)

is defined.

This may be explicitly written as 2 2 2 2

XI = U1 - U2 - u3 + Uq , X: = 2 (u1 U2- U3 uq),

X3 ' 2 (u, U3 + u2 uq).

'This generalization of Levi-Civita transformation is called KS (Kustaanheimo-Stiefel)

transformation 11 71 1 and the above matrix L ( G ) detjned in (1.6.1) is called the KS

matrix. The US matrix satisfies all the three properties, satisfied by the L matrix,

provided in section I .5 . Using ( I .5.9), we obtain the following relation for two vectors u,

and v,.

U ~ V I - U ~ V ~ + u ~ v ~ - u ~ v ~ = O , (, 1 h.4)

which is known as the bilinear relation and it plays a fundamental role in the present

studies.


This KS transformation regularizes [172] the non-linear Kepler equations of motion

(1.3.1) into the following four second order linear differential equations with constant

coefficients

The Eqs.( 1 h . 5 ) in the perturbed case can be written as

which satisfies the following bilinear relation

UJ Q I - u ~ Q z + u ~ Q ~ - u I Q 4 = 0 .

From Eqs. ( 1 -6.3) and (1.6.7), we know that the perturbing potential V becomes a

function of the independent variables t, u,, u2, u3, u4 and

av av ax -- - -> au, z a S a., , j=I ,2 ,3 ,4 .

Using ( I .6.3), we get

Chapter-l Introduction

av The vector - appearing in (1 A.7) has a vanishing fourth component, therefore we may 32

write

Comparing (1 h.9) and (1.6. lo), we get

Now ( 1 -6.7) i s reduced to

L - A

Eq. ( 1 -6.6) can be written in terms of Kepler energy hk as

It has been established by Stiefel and Scheifele [I441 that a much better numerical

precision i s achieved, if the total energy h given in ( 1 -4.10) is used instead of the Kepler

energy hk . In terms of h, we have

Chapter- 1 Introduction

1.7 Hamiltonian's Equations of motion

The equations

are called HamiItonian's equations of motion, where (x,, pi), i = 1,2,3, . . . ..,n are canonical

elements.

1.8 Canonical Elements

Let (xi, p,), i = 1, 2, 3 ,..., n be pairs of conjugated variables and H, PI, Xi for i = 1 , 2,

3 ...., n be given functions of arguments t, xi, p, for i = 1,2,3 ,....., n. Then the system of

differential equations

for the unknown functions xi(t), pi(t), i = 1,2,3, ....., n is said to be canonical, if H is

flamiltonian and Pi, X I are canonical forces.

1.9 CanonicaI equations of motion

The Newtonian equations of motion (1.3.2) are differential equations of second order,

which can be written as a system of twice as many equations of first order. 'The canonical

form is a special form of such a system of first order-Let


for Nerarcm's Orbits with ICS Uniform R W

'Then Eq.(1.4.3) becomes

Hence, the Hamiltonian of the system, which is a function of time t, Xand i is

'I'he motion is now described by

i3H i. = - X aH I I I ' j~ I =---Pi, for i = I . 2 , 3

a!), ax, where XI and PI are calted canonical forces. So, the canonical equations of motion ( 1 -9.4)

are same as the Newtonian equation of motion (1.3,2). This implies that the Hamiltonian

M absorbs al l the forces derived from the potential V; where as the remaining forces are

cousidered as the canonical forces Pi and Xi .

1.10 Canonical Equations of Motion in Fictitious time

The equations of motion in the rectangular coordinate system (x , , x2, x3) under the

perturbed potential V, given in (1 -3.2) are

In the canonical form, the above equations of motion can be written as

with the Hamiltonian

whurc xk and pk are generalized coordinates and momenta. respectively. r is the distance

o f the particle from the central body, t is the time and p is the gravitational constant.


Earth's SamWe Orb- Canonical Eauation~

Adding the negative of total energy p, to the Hamiltonian H, we obtain homogeneous

flamiltonian

with the equations o f motion (1.8.1) as

dt Applying the ti~ne transformation = r , in the Harniltonian (1.10.3). we get the new

ds

with the equations of motion

where the function xo is equat to t.

Employing the canonical KS transformation given by

- X, = X,,

to equations ( I . 10.4), we obtain the new Hamiltonian

The bilinear quantity

is a first integral of the new canonical equations o f motion [ 144. p235 1

- - - - i.e. l ( p , X ) = 0

Hence the Hamiltonian in (1.10.5) reduces to

'The basic canonical system with respect to the fictitious time s is obtained by utilizing the

canonical transformation

l and applying the scaling factor -to the Hamiltonian (1.10.6), the resulting Hamiltonian

4

where

and the new canonical variables arc

The final transformation can be written as

xo = 2 uo (= t),

On any solution, the value of p, i s the negative physical energy and the value of H is

zero.

The canonical equations

dzi, aH dw, d H -=- - - --- (k = 0.1.2,3.4) ds dw,'ds 3% '

corresponding to the Hamiltonian (1.10.7) are the equations of a perturbed harmonic

oscillator.

I .i 1 Separation of Jacobi's equation

By canceling the unimportant constant r a n d the perturbing potential V in (1.10.7), we 4

are faced with the unperturbed Hamiltonian

1 " , - C ( w : + w ~ . ; ) .

2 k - l


cal P w i o n s for 0- ~ . q

W i c a l E-

Let 2 n-1

be the Jacobian of the equation and assume

s = S,,(u,) + S , ( u , ) + Sz(u2) + 5 3 ( u 3 ) + "SI1u4)

This transforms the Eq. ( 1 .1 1.2) into

The tirst separation-step provides

and thus reduces (1.1 1.3) to

The second separation-step gives

Firstly it follows

and second t y

The remaining two separation-steps yield the three ordinary differential equations

Chapter-i Introduction

Solving for the variables u k , wk , we get

Substituting ( I . 1 I . 16) in to (1.1 1.1) we get the unpertiirbed 1-Iamittonian as

t,I,= CY,+LY,+Q, +a,. ( 1 . 1 1.17)

and the corresponding canonical equations of motion are

d a , - aH I I - - 0 fork=0,1 ,2 ,3 ,4 . d~ ap,

Showing that P,,P,,P,,P, vary linearly with s, where as P,, and a, ,a, ,a,,a,

remain constant during pure Kepler motion. The ten variables a,,P, are elements.

Obvious1 y the a, ,a,, a,, a, play the roIe of amplitudes and P, ,P,, P,, flmI play the role

o f the oscillator. As before a,= w, is the squared frequency or the negative of the

t physical energy. As seen firstly u,, = - and from Eq.(I . l 1 -16). p,, i s a time element, 2

The corresponding perturbed Hamiltonian corresponding to (I .11.7) in terms of the

clements a, ,p! is

with the canonical equation of motion

d k t i nu, g7- --=-..- -- - . (k = 0, 1. 2 , 3. 4.) ds o'P, ' ds I?@,


1.1 2 KS Uniformly Regular Canonical Elements

The generating function

through the canonical transformation

'I'ransforms the Harniltonian (1 .1 1.19) to the form

with the canonical equations of motion are

Solving for uk and wk, we get

corresponding elements are as follows


From equations ( I . 1 2.2) and (1.12.3), it is seen that all the ten canonical elements a,, a, are constant in unperturbed motion. Moreover these equations hold good for circular,

elliptic or hyperbolic orbits. This is the uniformly regular set of canonical elements.

The bilinear relation in ( 1 -6.4) can be written in terms of the above elements as

%PI -a,Pz + f f , PJ -a, P4 = 0. (1.12.6)

The transtbr~nation from the KS uniformly regular variables u k and wk to the position and

velocity vectors are performed by the following relation obtained from Eqs. (1.5.5) and

The time etement r with respect to the physical time t is given by

T = / + ( u , i i l ) /h .

Differentiating the a h v e equation with respect to s, we get

r ' = r + ( J , i in)/h+(u",u")h-h'/h2.


Substituting (G,i?) = r in (1.6.14), we get

From the energy Eq. (1.4.1 O), we get

Substituting ( I .l2.10) and (1 -12.1 1 ) in (1.12.9), we get

Substituting h = 2 AO, 6' = G / r in the above equation, we obtain

1,13 Drag force in terms of KS uniformly regular canonical elements

If Ak and Bk are the canonical forces attached to the system of equations ( I . 11.2) and

( I . I 1-41. then the canonical equations of motion are (Stiefel and Scheifele, [144, ~ 2 5 0 1

dp, d f r -- - da, = -- + A,, (k = 0, 1,2.3 .4) (1.13.1) aH

4 ds l?ah ds d a ,

where

If 6 is the aerodynamic drag force per unit mass on a satellite o f mass rn (King-Hele)

[ 1 53. then

r - ,- kVo = (v.D) , r T 2 w . 1 = L ($ID, (j=1.2.3.4) (1.i3.3)


where, 6 = F A C,'d m, F = f ' - %

A as I ) Jv 12, p is the atmospheric density, A is r;b

the rotational rate of the atmosphere about the Earth's axis, r is the initial perigee

radius, io is the initiat inclination, v is the velocity at the initial perigee, Cu and A are, Po

respectively. the drag coefficient and the effective area of the satellite.

If we consider only the canonical forces, then the Equations of motion (1.13. I ) become

2 Substituting L' = - i and the expressions of R from Eq. (1.13.4) in Eq. ( 1.13.3), we

1.1

I We = - p ~ r l ~ 1 3 and 2

Substituting r = a ( I - e cos E), r I / '= p ( I + e ME E) and the expression of w, tiom Eq.

(1.12.4) in the above Eqs. (1.13.6), we get

and

1 W,= - --p8r/?/ ( I + r cos E)(P, cos(&s) + a , &sin(& i)).

2

for j = I ,2,3,4. (1-13.7)

Finding the partial derivatives of uk with respect to ai. P, from Eq. ( I . 12.4), wc get


for N-te Orbits with KC

3% - - - cos (&s) if j = k

Jff ,

= 0 . otherwise

du, = O and 84,

du, - - i j ' j = k 3 4 &

= 0 , otherwise . ,fur both j and k = 1,2,3,4.

Substituting ti = 2daos, in the expressions of Wi in Eqs.(1.13.7) and using the partial

derivatives from (1.13.8). in Eqs. (1.13.2) and simplifying, we get

I AO- - p 6 r l ~ l p(1 + e m s E),

4

where


S?, - P f'2j + a Qzj + e (p Poj - a QOj)/2,

S1,= FL Poj+ a QYI+ e (p Pq - a Qzj)/2.

S21= e (P Poj - a Q,j)/2,

S 3 j = ~ P I I + ~ Q I , .

ST,- e (p PI, - a Q1,)/2,

ss, = p P;, ,

Shj = C tl P!,.

For j -1 .2 ,3 ,4 ,

Utilizing the following series expression for I J 1 up to third order terms in eccentricity

where the coefficients v, 's are

vo = - ( c - e2/4)/2, 2

v l = 1 - e /4,

v2 = ~ 1 2 , 2 v;=e /3. i v j = e- /8.

Simplifying Eq. (1.13.9) after substituting the value of I C 1 from the @ ( I .l3.10) in the

equations of motion ( I . l3.5), we get

I;;,, +4, cosE+F,, m Z E + F , , m 3 E + ~ , c o s 4 E + ~ , s i n E + I ; ~ , s i n 2 E +

& 4 u (1.13.12) !<,sin3E+&,sin4E+F;,,E

for i =1,2.3 ,....... 8 with a,+4=pj for j =1,2,3,4.

where

D o = B o + B l e / 2 ,

DI = B I + e ( Ba+B2/2),

D z = B z + e ( B I + B 3 ) / 2 ,

D3 = B3+ e 8212 ,

BO= 1 + e 2 / 4 .

B l = e ( 1 + 3 e 2 / 8 ) , 2 Bz = e 1 4 , 3 Bj = e- 1 8.

Foj - BO SO, + ( BI S I , + B2 S Z ~ ) 1 2,

Flj" RoSl.,+ B I SoJ+ ( S2;+ B2SiJ)/2,

F ~ j = B n s 2 ~ + B:!Sr!i+ SI , ( B I + B3)~ '2 ,

F3, = B B ~ S ( ~ + ( 51 Slj+ 8 2 % ) ) 12,

F4j=B,,S4+S3j(B1+B3)/2,

F5.i = Bo S3j + (B I SdJ - BZ S3j) 1 2,

F6j = Bo S4, ?i s3j ( HI - B3) / 2,

F7j = ( B ~ SdJ+ BBZ 53,) 12?

F R ~ = (Hz S4;- B3 S3,) 1 2 ,

FBj = B2Ss, -t B IS,, 12.

By utilizing the fact that E = 2daus, the above equations ( I . 13.1 I ) and (1.13.12) become

for i =1,2,3 ,......, 8 with a J r 4 = PJ,(j = 1.2,3,4) (1.13.14)

1.14 Bessel functions of imaginary argument l.(z)

The integrals are evaluated using the integral representation of the Bessel function of the

first kind and of imaginary argument


a/ P- for ~ m r - - ~ p b l b - 1

1, ( r ) = - 2j exp(z cos E ) cos nE dE , 2iT 0

= 2 R ~ = O m! (n + m)!

'The function follows the following recurrence relations

In-, (4 + I.,,, (4 = 21, , (~) ,

.I '. (z) + 1. ( 2 ) = zI,,-, (21

21'. ( z ) - I,, ( 2 ) = 21 ,?*, (2 ),

dlll where 1 ', denotes - . a5

1.15 Conclusions

Advantages of the KS uniform regufar canonical clement equations are as follows:

i) Regularizes the non linear equations of motion into linear equations of motion.

ii) Instabilities associated with solving the two body conic equations are eliminated.

i i i ) An orbital frequency based on the total energy gives more accuracy to

calculations of in orbit positions.

iv) Equations are less sensitive to round off and truncation errors in the numerical

integration algorithm.

v) Accuracies of the numerical computations can be examined through the bilinear

relation.

vi) Only two of the equations are needed to solve analytically to obtain to state vector

due to the symmetry in the equations of motion.

vii) All the ten elements (a, , fl, ) are constant in the unperturbed motion.

viii) The equations of motion are valid for elliptic, parabolic and hyperbolic orbits.

ix) Any type of forces can be modeled easily in the equations of motion.

Thus, the KS uniformly regular canonical elements are very much suitable for numerical

as well as analytical orbit predictions for satellite mution.

Analvtical and Numerical Predictions for Near-Earth's Satellite Orbits with KS Uniform Reaular

Canonical Equations

WITH EARTH'S OBLATENESS

2.1 Introduction

In this chapter a detailed numerical study is carried out using the KS uniformly regular

canonical equations of motion with respect to Earth's oblateness [157, 1581. 'The KS

uniformly regular canonical elements, which are ten in numbers, are constant in the

unperturbed motion and even in the perturbed motion the substitution is straightforward

and elementary due to the transformation laws being explicit and closed expressions. For

a detailed numerical study, we have developed orbit computation software 'UOBLAT' by

including Earth's oblateness in the KS uniformly regular canonical elements. Utilizing

the recursion formulas of Legengre's polynomials, we have included any number of

Earth's zonal harmonics J, in the software. A fixed step size fourth order Runge-Kutta-

Gill method [I731 is employed for numerical integration of the KS uniformly regular

canonical equations of motion.

To study the effect of the higher zonal harmonics J, and integration step-size variation,

we considered 4 test cases A, B, C' and D covering a large range of semi-major axis and

eccentricity and carried out numerical computations up to Jjg terms. Bilinear relation and

energy equations are used as checks to find out the accuracies of the numericat

integration. To carry out detailed study of a Sun-synchronous orbit, computations are

made tor a 900 km height near-circular Sun-synchronous satellite orbit (case E) for a long

duration of 220 days time (over 3000 revoIutions) and the necessity of including more

number o f Earth's zonal harmonics is noticed. The same orbit with change in inclination

is utilized to study the et'fect o f Earth's zonal harmonics on orbits which are near to the

Chapter-2 Long Term Orbit Predictions with Earth's Oblateness 43

critical inclination (63.43 degrees). The mean eccentricity (em) is found to have long

periods of 459.6, 6925.1 and 1077.6 days, respectively. Sharp changes in the variation of

mean argument of perigee (w,) near the minima of em are noticed. The values of w, are

found to be very near to -f- 90 degrees in the extrema of em. The same orbit is utilized to

study the effect of variation of inclination from 0 to 180 degrees on long period (T) of en,.

T is found to increase rapidly as the inclination approaches the critical inclination. To

find the effectiveness o f the KS uniformly regular canonical equations for long term orbit

predictions, comparison of the predicted values of the orbital elements with the actual

observed orbital elements of IRS-I A satellite ( 1 988-21 A) for over 1200 revolutions is

made.

In section 2.2 we have developed the equations of motion using KS uniform regular

canonical elements by including any number of Earth's zonal harmonic terms using

Legendre polynomials. From the initial conditions (position and velocity vectors), the

calculation of the initial conditions for KS uniform regular canonical elements are

provided in section 2.3. The numerical results are discussed in section 2.4. In section 2.5,

the study related to Sun-synchronous orbit is described. In section 2.6, the study about the

long term behavior of orbits near critical inclination is presented. The results from the

numerical integration are compared with the observed values of IRS-IA satellite in

section 2.7. Conclusions are drawn in section 2.8.

2.2 Perturbations and Legendre Polynomiats

In t h i s chapter, we are assuming that the only perturbing force acting on an artificiaI

satellite is, due to Earth's gravitational field with axial symmetry. In which case the

perturbing potential V in the equations (1.12.2) is

with

" 3 cosy =-, r

Chapter-2 Long Term Orbit Predictions with Earth's Oblateness

A n a l - , ! for I U N Unl-

J, = nth zonal harmonic term of Earth,

P, = Legendre polynomial of degree n,

R = Earth's equatorial radius.

With respect to V in (2.2.1), we have developed an orbit computation package

'UOBLAT' through the KS uniformly regular canonical equations of motion provided in

av (1.8.10). For computation of V and - with respect to Legendre polynomial of any

3%

degree n. we have utilized the following recurrence formulas of Legendre's polynomials:

n P,(x) = (2n- 1) x P,-1 (x) - (n- 1 ) Pn-*(x), (2.2.2)

with the starting values

Po(x) = 1 , PI (x)= x

and

P,' (x) = xpn-,' (4 t- raP,-, (x),

with the starting value

P,~ (X) = 0 .

2.3 Initial conditions

Knowing the initial position and velocity vectors Jc and 2 at the instant t = 0, we

compute

Then the perturbing potential V can be computed using (2.2.1) and (2.2.2). The energy is

computed as

The initial KS uniformly regular canonical position vector ui can be found either i'rom


or from

Further. the KS uniformly regular canonical velocity vector w, can be computed as

follows

I I W ~ = - ( Z ~ , X - - U ~ ~ + U ~ ~ ) , W ~ = - ( - ~ , X + Z ~ ~ ~ J + U ~ ~ ) .

2 - 2

Adopting these initial values for (ui, wi) and s = 0, we obtain the initial values of the KS

uniformly regular canonical elements (a, ,P , ) using (1.13.2).

2.4 Numerical Integration

For the numerical integration of the differential equations (1 . I 2.3) using the perturbing

potential V in (2.2. I ) , we have employed a fixed step size fourth-order Runge-Kutta-Gill

method. The computations were carried out using the recurrence relation (2.2.2) and

(2.2.3) up to the Earth's zonal harmonic terms J36. The values of the Earth's zonal

harmonics up tn .ish, which are provided in Table 2.1, are taken from fiough [1741.

Chapter-2 Long Term Orbit Predktions with Earth's Oblateness 46

for N- 0- - Detailed numerical computations were carried out for 6 test cases A, B, C, D, E and F,

whose initial conditions (position and velocity vectors) along with the orbital parameters

are provided in Table 2.2. Case A is a low earth orbit with small semi-major axis and

eccentricity, case B is with medium semi-major axis and eccentricity, case C has a

sufficiently large semi-major axis and eccentricity, case D is a highly eccentric with very

large semi-major axis. case E is a typical near-circular Sun-synchronous orbit and case F

is IRS- I A sateltite data.

7'0 know the ef'fectiveness of the Earth's zonal harmonics with respect to integration step

sizes, the numerical integration of the equations of motion provided in (1.12.3) has been

carried out up to Jz, Jh, Jls and JJ6 for the first four test cases A, B, C and D with the

integration step-sizes of 36, 48, 96, 120, 144, 180 and 360 steps I revolutions. Tables 2.3

and 4 provide the values of the important osculating orbital parameters semi-major axis

and eccentricity for the 4 cases A to D after nearly 100 revolutions. It may be noticed

from the Tables 2.3 and 2.4 that a larger integration step-size of 36 steps/revolution is

suff~cient to provide accurate osculating semi-major axis and eccentricity for the cases A,

0 and C, even after 100 revnlutions. However, for the high eccentricity case D, as can be

noticed from the Tables 2.3 and 2.4. a reasonably smaller integration step-size of 180

steps/ revolution is necessary for accurate computations after 100 revolutions. It has been

noticed that the other orbital parameters are also accurate with the above numerical

integrations.

The accuracies of the numerical integration are checked through the bilinear relations

provided in (1.6.4) and (1.12.6), which are satisfied by the KS unifonnly regular

canonical variables (u,, w;) and elements (ai, Pi). In our computations, we have noticed

that the value obtained from the L.H.S. of the bilinear relation (1.6.4) turns out to be the

negative of the value obtained from L.H.S. of (1.12.6). Table 2.5 provides the values

obtained from the bilinear relation (1.12.6) after 100 revolutions with respect to the zonal

harmonics Jz to JJ6. The table also provides the difference between the initial energy and

the energy at the instant of computations from the energy equation provided in ( 1 -4.7)


aRer 100 revolutions. These values serve as a good test for the accuracies of the

numerical integration. This is a clear indication that the uniformly regular canonical

equations provided in ( I . 12.3) could be used effectively with respect to the force model

considered. It may be pointed out that the bilinear relation and energy differences do not

remain constant during a revolution. Further it is noticed that the KS uniformly regular

canonical elements ai and pi have more uniform variations and less amplitudes than the

corresponding orbital parameters a, e, i . C2, w and M during a revolution and provides

better accuracies during numerical integration.

2.5 Sun-synchronous orbit

To study the effect of higher zonal harmonics in Sun-synchronous orbit, we have

generated mean orbital elements for a 900 krn near-circular Sun-synchronous orbit (case

E) for 220 days (nearly 3078 revolutions) using the software 'UOBLAT'. Its initial

osculating orbital elements, chosen for the study along with the mean orbital elements are

given in Table 2.6. The conversion of the osculating orbital elements to mean orbital

elements are carried out through the theory of Chebotarev's [I751 first-order short-

periodic variations due to .I2. The mean orbital elements are generated for this case E with

J 2 to J24 terms up to 220 days time (nearly 3078 revolutions). It is noted that the mean

semi-major axis (a,) remains nearly constant, white the mean right ascension of

ascending node (n,) varies almost linearly during the 220 days time. Figure 2.1 depicts

the variation of mean eccentricity (e,), argument of perigee (w,) and inclination (i,) up

to 220 days time. It can be easily noted that the eccentricity and inclination have long-

periodic terms of period 1 1 9.9 days and occur almost at the same time. A slight increase

in the peak value of inclination is alsu noticed. It i s also noticed that the extrema of these

variations occur, when the argument of perigee is near to f 90 degrees. The argument of

perigee varies rapidly near the minimum of em and i,. Variation of em and w, terms up to

Jz and Jg is also shown in the tigure to show the effect of higher zonal harmonic terms.

The figure depict that J2 has no long periodic effect on e,,.

Chapter-2 Long Term Orbit Predictions with Earth's Oblatwness

for NaaPE8rth's - 2.6 Near Critical inclination orbits

'To show that the higher zonal harmonic terms have significant effect near critical

inclination (63.43 degrees) orbit, we have generated mean orbital elements for three cases

with i = 60,63.2 and 65 degrees. The other initial osculating orbital elements are same as

that of case E. Figure 2.2 depicts the variation of em and w, for i = 60 degrees with terms

up to J24 for 600 days time. Variation of these parameters up to J3 tenns is also shown in

the figure to know the effect of higher zonal harmonics. It can be easily noticed that the

variation of em with up to J24 terms is much higher than that of J3 tenns and the extrema

occur at different times when their respective w, are very near to k 90 degrees. Figure

2.3 depicts the variation of e,, and m, for i = 63.2 degrees with terms up to 524 for 3750

days time. Variation of these parameters up to J4 terms for 500 days time is also shown in

the figure to show the effect of higher zonal harmonics. Here also, it is noticed that the

extrema of em occurred, when om are very near to + 90 degrees. Figure 2.4 depicts the

variations of em and om for i = 65 degrees with terms up to J2 and J24 for 650 days time.

As can be seen from the figure, the variation of these parameters are quite different for

the terms up to J4 and Jz4, showing the significant effect of higher zonal harmonic terms.

Also, w, is found to be very near to +190 degrees at the time of extrema o f em. Though

we are using the words 'very near to t_ 90 degrees' at the time of extrerna of e,,, due to

the nature of our studies, however from tl-te large number of computations with different

inclinations, we strongly feel that w, is 'r 90 degrees at the extrema of em. As can be seen

from the figures 2.1 to 2.4, the long periodic terrns in em have quite large periods (459.6,

6925.1 and 1077.6 days for 60, 63.2 and 65 degrees inclination, respectively). As we

approach the critical inclination (63.43 degrees), this period increases rapidly showing

the difficulties involved in solving the critical inclination problem, when Earth's zonal

harmonic terms are considered as perturbing force.

Computations are carried out to study the effect of orbital inclination for long-period (T)

of em of case E for the force model consisting of Earth's zonal harmonic tenns up to JZ4.

Figure 2.5 provides thc variation of log T, when the inclination varies from 0 to I80


degrees. As seen from the figure that T increases rapidly as we proceed towards the

critical inclinations (63.43, I 1 6.57 degrees) from the right o f 0 degree and left of I80

degrees. Though, we evaluated Tat 63.2 degrees of inclination, it increases sharply as we

proceed towards the critical inclination. Again from 90 degrees, as we proceed towards

the critical inclination, T increases rapidly. It may be noticed that T at 90 degrees of

inclination is much higher than at 0 or 180 degrees of indination.

2.7 Comparison with LRS-IA orbital data

1'0 find the effectiveness of KS uniformly regular canonical elements. we have compared

the predicted values of the orbital parameters of the satellite LRS- I A ( 1 988-2 1 A) with the

observed values for 87 days time with respect to the initial epoch of 13Ih July 1988, 0.0

UT. Table 2.9 provides the observed and predicted values of the orbital parameters' with

.I2 to Jjh terms for 87 days time. From the Table 2.9, it is noticed that the observed values

matches reasonably well with the predicted values up to 87 days time.

Figures 2.6 to 2.1 2 depict the differences between the observed and predicted values of

the orbital elements a, e, i , 51, to, M with J2 to J6 and .IT to J3h and u (w+M) for 47 days of

time, for which the continuous observed orbital data at one day interval was available.

From figure 2.6, it is noticed that the predicted values of the most important orbital

parameter 'semi-major axis', which is a measure of energy is having the maximum

difference of 79 meters with J2 to .I3& and 119 meters with Jz to J16 fmm the observed

values during the 47 days time. As seen from the figure 2.7, the difference between the

observed and predicted values of eccentricity with Jz to Jj6 are better than J2 to Jh terms.

From figure 2.8, it is seen that the difference between the observed and predicted values

of inclination is having a small secular growth. In figure 2.9, the difference between the

observed and predicted values of R aIso having a secular growth and it is lesser with J2 to

J J h than JZ to J h terms. As can be noticed from Figures 2.10 and 2.1 I. it is clear that the

observed and predicted values of w and M with J2 to J36 gives better prediction than J2 to

Jh terms. Though larger differences are noticed in these values at some instant of times, it


for I V N - is clearly seen from figure 2.12 that the observed and predicted values of Mean argument

u match reasonably well. Figures 2.13 and 2.14 depict the differences between the

observed and predicted values of the perigee height and apogee height. From all the

figures 2.6 to 2.14, it i s clearly seen that the predicted values match well with J2 to JJ6

than Jz to Jg terms with the observed values. Figure 2.1 5 depicts the values obtained from

the bilinear relation up to 47 days time, which is used as a check for numerical

computations. The maximum value i s seen to be about 1.0 x lo-' around 25 days time,

which is quite small, showing reasonably good accuracies of the numerical computations

during the entire duration of 47 days.

2.8 Conclusions

The KS uniform regular canonical equations with Earth's oblateness perturbations

provide an efficient and accurate integration method for orbit computations, even for long

durations. Usage of Legendre polynomials and i ts recurrence relation to compute Earth's

potential and its partial derivatives economizes the computationai procedures and time,

which is useful for inclusion o f higher order zonal harmonic terms. Inclusion of large

number of zonal harmonic terms in the Earth's potential becomes necessary for accurate

orbit predictions for near circular satellite orbits. Near the critical inclination, the effect of

oblateness is very prominent on some of the orbital parameters of near circular orbits and

long-periodic terms have very large period.

C h a p t e r 2 Long Term Orbit Predictions with Earth's Oblateness

Table 2.1

Earth's Zonal harmonic terms

Chapter-2 Long Term Orbit Predictions with Earth's Oblatemess

~nalvtlcal and Numerical Predktions for NearEarth's SatelIIte Orbits with KS U n i h m R-ular Canoni-1 EauatrQar

Table 2.2 Initial Conditions (Position, Velocity & orbitaI parameters)

i Variables Case ! I

i A B c i D 1 E ( s u n - S ~ C ~ T O ~ O U S ) / F (IRS-I A) j

Chapter2 Long Term Orbit Predictions with Earth's Oblatenea 53

A na / ' l n N u vtrca s d mer k nl Pred i COO - n s for Nwr-Earth's Sate I IiteOr bi ts w ith KS Uniform Rwular Canon i c~,t&vatio ns

Table 2.3 Variation of Time and semi-major axis with Earth's zonal harmonics after 100 revolutions

Chap ter-2 Long Term Orbit Predictions with Earth's Obla teness 54

[ Case Zonal harmonics

upto / A l J 2

Time

(msd)

6.3 164823

Number o f steps / rev.

1 6705.95 l 3 i 6705.95 1 3 i 6705.95 13

I 6705.95 13

180

6705.95 1 3

36 48 96 1 44 I

Analvtical and Numerical Predlctions for N ~ r E a r t h ' S Satellite Qrbits with KS Uniform Reuular Canonical Eauatioa

Table 2.4 Variation of eccentricity with Earth's zonal harmonics after 100 revolutions


Case Zonal harmonics

Number of steps / rev. 36

0.0141452

0.0137773

0.01 37480

0.0137346

0.03833 13

i upto J2

i i I J I ~

- B 1 J2

1 Jn

1 J I R

48

0.0141452

0.0137773

0.01 37480

0.0137346

0.03833 13

C

536

J2

96

0.0141452

0.0137773

0.0137480

0.0386024 1 0.0386024

I -b 1 J I R

136

i J2

I I

J36

0.0386252

0.0386274

0.1741792

0.1 742966

0.1743018

0.1743017

0.949763 1

144

0.0386024

0.0386252

0.0386274

0.1741792

0.1742966

0.1743018

0.1743017

0.9500468

0.0386024

0.0386252

0.0386274

0.1741792

0.1 742966

0.1743018

0.1743017

0.9500467

0.0386252

0.0386274

0.1741792

0. I742966

0.1743018

0.1743017

0.9500356

0.01 37346

0.0383313

I80

0.01 37346 1 0.01 37346 ' 0.0 137346

0.0386024 1 0.0386024

0.0386252 1 0.0386252

360

0.0141452 0.0141452

0.0137773 0.0137773

0.01 37480 I , 0.0 137480 I

0.03833 13

0.0386274

0.1741792

0.1742966

0.1743018

0.1743017

0.9500467

0.0141452

0.0137773

0.0137480

0.03833 13 1 0.038331 3

0.0386274

0.1741792

0.1742966

0.1743018

0.1743017

0.09500468

Analvtical and Numerical Predictions for Near-Farthrs Rat~ilite Orbits with KS Unifprm Regular Canonical Eauatlons

Table 2.5 Bilinear relation and Energy equation after 100 revolutions with Jz to JS

1 I Steps / I Bilinear relation x 10" 1 Energy Equation x lo-' I

Chapter-2 Long Term Orbfit Pmdjctions with Earth's Oblateness 56

rev. I I

Cases

for -r-Fa&hrs wte Orb- - Table 2.6

Initial Osculating & Mean orbital elements (Case E)


e

i (deg.)

0.00063

99.033

0.00072

99.09 1

Table 2.7

Comparison of Observed & Predicted Values of a real satellite (Case F) - - -

OscuIatinp: orbital Parameter 1 i (a%-) 1 C2 (deg) ( ~(deg.)

0 - Observed values P - Predicted values

Chapter-2 Long Term Orblt Predictions with Earth's Oblateness

Figure 2.1 Variation of mean eccentricity, argument of perigee and inclination

Figure 2.2 Variation of mean eccentricity and argument of perigee for i = 60"


Figure 2.3 Variation of mean eccentricity and argumeot of perigee for i = 63O.2

/-- --- " - -- -- - XI ., ,,-'

I ,,<' n ;?at8

-, -- \ / i jC"

I W I N A T 10 FI = 6 7:- -. ,A ,- - . - 1

,,,' .. -- ..- ' ' ., , --,, , f

O o S b 1 - d

1, r ) o j 4

(> C 3 0 1 2 -

$

-.. -, .

I , jm:

. -- . A

/' UP T O T2, TLHM's

,/ - I

/' - UP YO J~~ T e r n s ;pnc> !

,3 0040 r I I I H O I ? ' It

- U P TO .& TERMS 1; ; :3

\ \

11 2 0 !

1 - -- 1

---, . I

. I COO4 \.\ -,,, - - , %. ., .

I \, /' . : r "+, t ./-'

>.>

- . J*" I ,. . ,< 1:)' T > J2" ye l?MS <> <>4:)02!

I .i, 165 .HW ud 5 SCK) - 7 5 ~ -- ";;q 0 .. . . - I cx:

'im-'' -

Figure 2.4 Variation of mean eccentricity and argument of perigee for i = 65'


Figure 2.5 Variation of long period (T) of mean eccentricity with indination upto J 2 q

Chap-2 Long Term Orbit Predictions with Earth's Oblateness

I Predialons for -'s p RM&

Time in days

Figure 2.6 Difference between observed and predicted values of semi-major axis

Figure 2.7 Difference between observed and predicted values of eccentricity

Chapter-2 Long Term Orbit Predictions with E r t h ' s Oblateness 62

Tiwe i t h days

Figure 2.8 Difference between observed and predicted values of inclination

10 15 20 25 35 40 45 Time in tlnys

Figure 2.9 Difference between observed and predicted values of right ascension of ascending node

Chapter-2 Long Term Orbit Predictions with Earth's Oblatenes

* 0 5 10 15 20 25 30 35 40 15 Tima in days

Figure 2.10 Difference between observed and predicted values of argument of perigee

Figure 2.1 1 Difference between observed and predicted values of Mean anomaly


- for N e ! !

Figure 2.12 Difference between observed and predicted values of Mean argument

Figure 2.13 Difference between observed and predicted values of perigee height


Figure 2.14 Difference between observed and predicted values of apogee height

Figure 2.1 5 Values of Bilinear relation

Chapter-2 Long Term Orbit Predictions with Earth's Obtateness

Analvtical and Numerical Predictions for Near-Earth's Satellite Orbits with KS Uniform Reqular

Canonical Equations

for N e a r - E a a

LONG TERM ORBIT PREDICTIONS WITH EARTH'S FLATTENING

3.1 Introduction

A satellite orbiting the Earth experiences accelerations due to a wide range of physical

causes. One of the acceleration is due ta the Earth's flattening (zonal and tesseral

harmonic terms). This effect plays an important role in studying the motion of satellite

orbits with eccentricity < 0.2. whose perigee is more than 600 km, as the effect

atmospheric drag reduces considerably for such orbits. In chapter 2, we had a detailed

study of the KS uniformly regular canonical elements, to predict the long-term orbit

computations with Earth's oblateness by including the Earth's zonal harmonics up to J36

terms [ I 57, 1581. Since the shape of the Earth is tri-axial, its inclusion in the perturbing

potential of the Earth is essential far better orbit computations.

In this chapter, we have studied the effect of Earth's flattening by utilizing the KS

uniformly regular canonical elements, to predict the long-term orbit computations. In the

software 'UOBLAT', we have also included the Earth's tesseral harmonic terms, which

define the shape of the Earth more accurately. Utilizing the recurrence formulas for

derived Legendre functions and normalized geopotential coeff?cients, we are able to

include any number of Earth's zonal and tesseral harmonics terms. However, we have

crnployed a 19 x 1 9 Earth's model of GEM-T2 [ 1651 for the present studies.

For a detailed numerical study, three test cases with different semi-major axis,

eccentricity and inclination are chosen to assess the effkct of the tesseral harmonic terms.

It is noticed that even with the inclusion of tesseral harmonic terms the computational

Chapter-3 Long Term Orbit Predictions with Earth's Flattening 67

for Near-- KS - accuracies are maintained with the inclusion of higher order terms in the force model with

respect to integration step sizes. The bilinear relations provided in (1.6.4) and (1.12.6) are

used as a check for the accuracies of the numerical integration carried out in the study. To

study the effectiveness of the method, we have compared the predicted values of the

orbital parameters of the Indian satellite IRS-1 A ( 1 988-2 1 A) with the observed values for

47 days time. Earlier observed secular growth with only Earth's zonal harmonic terms

[157, 1581 in the orbital parameters i and fl is found to decrease with the inclusion of

tesseral harmonic terms.

In section 3.2, we have developed the equations of motion using the KS uniform regular

canonical elements with the Earth's zonal and tesseral harmonic terms. In section 3.3, we

have discussed the Earth's perturbations due to zonal and tesseral harmonics. In section

3.4, we discussed about derived Legendre functions and normalized geopotential

coefficients. In section 3.5, we have discussed about the computational procedures

adopted. The numerical integration and results obtained from the solutions are provided

in section 3.6. In section 3.7, the numerical results are compared with the observed data

of the Indian satellite IRS-1 A. We have concluded the chapter with section 3.8.

3.2 Earth's flattening perturbations

In the present analysis, we assume the forces acting on an artificial satellite are those due

to the Earth's flattening (zonal and tesseral harmonics), in which case

[{c. . , c o s m / l + S... s i n m a } P,?, (s in #)],(3.2.I) r n = 2

here R is the mean equatorial distance, 4 i s the geocentric latitude, h is the longitude, r is

the distance of the satellite form the central body (Earth), and C,, and S,., are

dimensionless constants known as gravity coefficients for zonais, sectorial and tesserals

harmonic terms and P,.,, represent the set of Lengendre and associated Legendre functions

for including the Earth's zonal and tesseral harmonic terms.

Chapter-3 Long Term Orbit Predictions wjth Earth's Flattening

a forb KS-

with

= a - CB, - B(d - t , ) ] ,

A 2 tan a = -, A 3

where 0 is the rotational rate of the Earth.

Wc have utilized the following recurrence formulas [I441 for the inclusion uf higher

order zonal and tesseral harmonic terms using P,,,,.

Pn>, (sin 4) = (2n - 1)co s #P,,, (sin +) ; for n=m

- -- (2n -1) (n- 1) sin #Pn-,,m (sin 4) - - <,-,,,,,,(sin #) ; for m=O n PI

=(2n - I)~~s@P,-, . , - , (sin #) ; otherwise (3.2.2)

with

Po+,) ( s i n @ ) = 1 .Q,

3 P, , , , (s in 4 ) = s in # = -, r

P,., ( s i n 4 ) = co s #.

3.3 Derived Legend re Functions and Normalized Geopotential

Coefficients

The implementation of V given in (3.2.1) results in some computational difficulties. One

of the difficulty i s the range of the magnitudes of the parameters P,.,; C,,, and S,?,, as n

increases. So, to avoid the large variations in the magnitudes of these parameters,

uormalized Legendre functions have been proposed ( 1 761, which have a more graceful

change in the exponent.

Chapter3 Long Term Orbit Predictions with Earth's Flattening

The normalizing factor is defined such that the normalized spherical harmonics will have

the mean square value of one on the unit sphere. The normalized Legendre functions are

defined such that the product of the gravity coefficients and the corresponding Legendre

functions remain constant.

- - where c.,,. C',,,,,, and .q,,,,, arc the normalized functions and geowential coefficients,

respectively.

The typical normalizing factor is taken to be

N ,,,,, = [(n - m)!(Zn + 1)(2 -a,,,) /(n + rn)!]' ' '

and thus. we achieve (3.3. I )

where a,, is the kronecker delta functions which is 1 when m = 0, otherwise 0.

3.4 Computational Procedure

To integrate the differential equations ( I . 12.3) with the perturbation (3.2. I ) , we have to

compute the right hand side of the equations (1.12.3). For that, we proceed as follows:

where

C f l , , cos m/l + S ,., sin d}

34 r ,,=, sin#) -mtm4e1., (sin@)}

Chapter-3 Long Term O h i t Predictions with Earth's Flattening

_ - - F(! J Z[rn (s.,, cas mi. + c.,,, sin m ~ } P,, (sin +)I, 3/Z rn=2 m = ~

d r -= - 2 u i c o s & S , a a i

with

- - - ( - 0 s J- a 0 9 for i = 1,2,3,4. d a i

and

3.5 Numericai resuits

The numerical integration of the KS uniformly regular canonical equations of motion

(1.12.3) has bcen carried out with a fixed step size fourth order Runge-Kutta-Gill method

with the perturbing function V provided in (3.2.1) by including Earth's zonal and tesseral

harmonic terms up to J19,1r) . The values of the normalized Earth's zonal and tesseral

harmonic terms are taken from GEM 'P2 11651, which are provided in Tables 3 . la and

3. lb. Three test cases A, B and C whose initial position and velocity vectors along with

the osculating orbital parameters and epoch considered are provided in Table 3.2, which

are selected for detailed nurneric,al study. The epoch is chosen as 13th July, 1998 for the

Chapter-3 Long Term Orbit Predictions with Earth's Flattening 7 1

cases A and C whereas I st March, 1998 is taken for the case £3. Case A corresponds to a

low perigee, small eccentricity and high inclination orbit, while case B corresponds to a

near-circular Sun-synchronous orbit with altitude of 722 km and inclination of 98.377

degrees. Case C is of slightly higher eccentricity (0.35) and large semi-major axis with

perigee and apogee heights of 422 and 7787 km, respectively, and inclination of 30

degrees.

To test the sensitivity of the ~iirmerical integrator with respect to the force model. we have

considered step sizes of approximately 48, 72, 96, I20 and 240 stepslrevolution for

numerical integration up to 22 hours time in the present study for cases A, B and C. The

terms up to Jh.", JII).o, j19.0, Jb.,b, .IIo,~o and Jl9,rs are considered in the force model. Table

3.3 provides the bilinear relation in (1.6.4), which is a check for the accuracies. with

respect to the earlier chosen force models. As in the case of oblateness perturbation, here

also it is noticed that the values of the bilinear refation in (1.12.6) provides the negative

values of the bilinear relation in (1.6.4). By noticing the values of the bilinear relation

from the Table 3.3 it can be seen that the computational accuracies are maintained with

higher-order terms in the force model with respect to integration step size. From Table

3.3, it is also observed that a high integration step size of 240 steps/revolutions is

sufficient for the numerical computations. Table 3.4 provides the variations in osculating

orbital parameters: semi-major axis, eccentricity, inclination, right ascension of ascending

node, argument of perigee and mean anomaly, after 22 hours with respect to change in

force model for the three cases A, B and C. It may be seen from Table 3.4 that the semi-

major axis increases for case A and decreases for cases B and C with EMh's flattening.

From Table 3.4, it is also noticed that the eccentricity increases for the cases A and B and

decreases for the case C, while in inclination the variation i s considerable for small

eccentricity cases A and B, but it is negligible for high eccentricity case C with the

inclusion of tesseral harmonic terms. As can be seen from Table 3.4 that the tesseral

harmonic terms decrease the variations in the semi-major axis, except for the cast C with

the inclusion of higher order terms up to 519,~g. From Table 3.4, i t i s noted that the


for Near-EarLh 's KS W&m R e a u k - variation in is reduced and w is increased by including the tesseral harrnonic terms for

cases A and B, while in the high eccentricity case C, the effect is negligible. It is also

noticed that the mean anomaly increases with increase in tesseral harmonic terms for

cases A and C, while for case B it decreases.

Table 3.5 provides the change in the orbital parameters due to the tesseral harmonic terms

Jb,br J10.10 and Jlrl.lq alone after 22 hours of time. From Table 3.5 it i s noticed that the

effect of Earth's tesseral harmonic terms is more in the semi-major axis for cases A (129

meters), B (88.6 meters) and C ( 1 8 meters). It is noticed that with the inclusion of higher

order tesseral harmonic terms. the effect in the important orbital parameter .semi-major

axis' increases for the case A, whereas in the case o f B it decreases then increases slightly

and for the case C there i s an increase and then it decreases. For cases A. H and C the

tesseral harmonic terms reduce the eccentricity.

3.6 Comparison with IRS-I A orbital data

To find the effectiveness of the KS uniformly regular canonical equations with Earth's

zonal and tesseral harmonic terms, we have compared the predicted values of the orbital

elements with the observed values of IRS-IA orbit from 13th July 1998 onwards For 40

days time. The initial conditions are provided in Table 3.2 (case Dl, which are

corresponding to the observed values. The KS uniformly regular canonical equations of

motion are integrated numerically with a step sizes of 240 stepslrevolutions with Earth's

force model of 5 19.19 and J IY,O for 47 days (660 revolutions approximately). Table 3.6

provides the observed and predicted values of the osculating orbital parameters for 1. 6,

12, !8. 25. 30 and 40 days of time with J19,14 and J19,". Figures 3.1 to 3.8 depict the

differences between the predicted and observed values of a, e, i, Q , w, M, H, and Ha,

respectively. It may be noticed from Table 3.6 and Figures 3.1 to 3.8 that overall

accuracies in the computations o f the orbital elements improved with the inclusion of the

tesseral harmonic terms up to 25 days (350 revolutions approximately) duration. 'l'he

Chapter-3 Long Term Orbit Predictions with Earth's Fiattenlng 73

maximum deviations during this period in the semi-major axis are 45 and 80 meters with

J19,is and J19.o; while for e, i, 52 , 0, M, Hp and Ha , the deviations with Jlq,lg and Jls?~ are

0.00003 and 0.000044; 0.00295 and 0.003 15 degrees; 0.0039 and 0.0099 degrees; 1 -7 and

4.1 degrees; 1.7 and 4.1 degrees; 245 and 345 meters; 205 and 330 meters, respectively.

Figure 3.9 depicts the L.H.S of the bilinear relation (1.6.4) with JIgTo and The values

are of the order of 1 o - ~ . It is noticed that the changes in the orbital parameter; with 240

and 360 stepslrevolution are negligible with J19,19 force model for the entire duration of 47

days (660 revulutic~ns approximately).

3.7 Conclusions

Inclusion of tesseral harmonic terms in the perturbation V in KS uniform regular

canonical equations improved the accuracy of orbit predictions for long term orbit

computations. Usage of Legendre and associated Legendre knctions and its recurrence

relations is useful in including Earth's higher order zonal and tesseral harmonic terms in

the potential function effectively. Inclusion of higher order flattening terms in the Earth's

pote,ntial becomes necessary for accurate orbit predictions for near circular satellite orbits.

Comparison of the predicted values of the orbital parameters with the observed values of

IRS-IA satellite over a longer duration shows that the method provides accurate orbit

predictions even for longer duration.

Chapter-3 Long Term Orbit Predictions with Earth's FIaCtenlng

Analvtical and Numerical Predictions for Near-Earth 's Satellite Orbits with KS Uniform Reaular Can~nlcal Eauatlons

Table 3.1A GEM-T2 Normalized Coefficients for Zonal harmonics (Units of lo6'

Chapter-3 Long Term Orbit Predictions with Earth's Flattening

Index Value

-484.1652998

0.0900847

0.0340918

0.021 1398

-0.0101581

N

2

7

12

17

22

M

0

0

0

0

0

-0.0064182

0.0051093

0.0021499

0.0002937

0

0

0

0

Index

28

Value

0.9570331

0.0483835

0.0429873

0.0086686

-0.0241859

N

3

8

13

18

23

32

37

42

47

M

0

0

0

0

0

29

34

44

49

0

0

0

0

-0.0008836

-0.0057588

0.0002269

0.0000776

0

0

3 8 - % - ~ ~ ? ~ ~ ~ 0

0

Index Value

0.5399078

0.0284403

-0.0208746

-0.0048120

0.0010847

N

4

9

14

19

24 -------

-0.0027771

0.002244 1

0.0013010

0.0008891

M

0

0

0

0

0

33

43

48

0.0076193

-0.0047918 -

0.00 1 105 1

-0.0008216

Index

0

0 -

417- 0

0

30 31

Value

0.0686883

0.0549673

0.0008078

0.0199685

0.0069648

N

5

10

I5

20

25

35

45

50

M

0

0

0

0

0

0

0

0

Value

-0.1496092

-0.0519374

-0.0069674

0.0095754

0.0009484 ---

Index

N

6

11

16

21

26

0.0078141

0.00206 10

0.0020158

0.0002472

M

0

0

0

0

0

36

46

Predictions for N e a p E a ' s S- Orbik with KS uniform -0niral Eauatiom

Table 3.1B GEM-T2 Normalized Coefficients Sectorials and Tesserals (Units of 10')

Chapter3 Long Term Orbit Predictions with Earth's Flattening 76

Analvticai and H u r n ~ r i a l Predl&jons for Neap&a* Sate - . I l i t ~ Orbits with fI$ Uniform Reoular Canonical Eauation~


Analvti-I and Nurnerira f Pwdicti qns for Neat-Earth's &Ssleiliti= Orbits with KS Uniform R e a d e r Canonical Eauations


Analvtical and EJum~rlcal P r e d i M f o r H h ' s Satellite Orbits with KS Uniform Rwuiar Canonical Eauatlon~


rcal P r e d i f o r - Orb~ ts with KS- W n i c a l -

Table 3.2 Initial conditions

(Position, Velocity and Orbital Parameters)

Chapter-3 Long Term Orbit P ~ d i c t i o n s with Earth's FIa ttening

Parameter

x (kmj

Y (km)

z (km)

?i (km / sec)

y(km/sec)

i(krn/sec)

a (km)

e

i (deg)

fl (deg)

rn (deg)

M (deg)

[Jp (km)

Ha (km)

Period

(Min)

Epoch

considered

A

1 .O

-573 .O

-6553.0

7.9

0.0

0.0

678 1 .00 1 82

.02993662

85.00271 8

0.0

269.70904

0.282 12

199.8365

605.837 1

92.62

13.07-1998

Case

C

1.0

-5888.972

-3400.0

8.9

0.0

0.0

10482.46127

.35 12974 1

30.0

0.0

269.97602

0.0 1457

42 1.8348

7786.7578

178.01

13.07-1998

B (Sun-

synchronous)

6389.4339

62.9080

-3097.1 175

-3.2 126489

-3.1.244661 3

-6.6530608

7 1 00.60473

.00002353

98.377

184.6571

26.34844

1 79.8 1 096

722.2 727

722.6068

99.24

01-03.1998

D

(IRS-I A)

1 1 10.9458

-677.9023

-71 78.3814

-0.4792287

-7.3382249

0.6274580

7273.52823

.0032 1994

99.058338

267.04298

74.01013

200.99428

871.9429

91 8.7836

102.89

1 3.07-1 998

Table 3.3 Bilinear relation x 10' after 22 hours

Chapter-3 Long Term Orbit Prcdldions with Earth's Flattening

Case

A

R

C

Stepslrev.

{approx.)

48

72

96

120

240

48

72

96

120

240

48

72

96

120

240

Earth's harmonics J , , upto

J 19-0

6.69

0.88

0.2 1

0.069

0.00216

-1.72

-0.23

-0.054

-0.018

-0.00055

23.88

3.18

0.76

0.25

0.0078

Jl0,lo

6.44

0.85

0.20

0.066

0.00207

-1.73

-0.23

-0.054

-0.018

-0.00055

23.77

3.16

0.75

0.25

0077

J6,0

6.64

0.88

0.2 1

0.068

0.00214

-1.65

-0.22

-0.052

-0.017

-0.00052

23.82

3.17

0.75

0.25 ppp

0.0078

J19,19 - 6.50

0.86 -

0.20

0.067

0.00209

-1 -77

-0.23

-0.056

-0.018

-0.00056

23.77

3.16

0.75 -

0.25

0.0077

J6,6

6.44

0.85

0.20

0.066

0.00207

- 1.70

-0.22

-0.053

-0.017

-0.00054

23.72

3.16

0.75

0.25

0.0077

~ O , O

6.69

0.88

0.21

0.069

0.00215

-1.73

-0.23

-0.054

-0.018

-0.00055

23.86

3.17

0.76

0.25

Table 3.4 Variation in orbital parameters after 22 hours

Chapter-3 Long Term Orbit Predictions with Earth's Flaftenlng

Case

A

B

C

Orbital

elements

a (km)

c x l o "

i x 1 0 " (deg)

R x ] 0 3 ( k g )

w (deg)

M(deg)

a (km)

e x l o 4 i x l o " A

R x 10 (dcgj

rn (deg)

M(deg)

a (km)

e x 1 0

i x 1 0 4 (dep)

0 x 1 0 ~ (deg)

o(deg)

M (deg)

19.381 7

23.279

67.82

-640.59

-3.1 875

74.6176

-5.8199

9.9390

34.64

924.82

-124.1 59

163.665

-3.35 1 1

-6.408 1

95.88

-1819.71

2.9046

150.93 1

harmonics

J I o,o

19,3832

23.283

67.82

-640.09

-3.202 1

74.6321

-5.8 196

9.9689

34.64

924.31

-124.254

163.570

-3.3490

-6.4073

95.95

-1820.11

2.9047

150.93 1

Earth's

~6,6

19.2596

23.1 13

87.71

-438.92

-3.241 1

74.7167

-5.73 14

9.7596

10.44

924.24

-124.552

163.295

-3,3365

-6.4375

95.07

-1819.74

2.9050

150.938

519-19

19.2538

23.093

99.32

-638.45

-3.2632

74.7410

-5.7320

9.7590

8.68

923.74

- 124.933

162.906

-3.3524

-6.4479

94.82

-1820.4

2.9056

150.944

(J,,,) upto

JIO.IO I ~ 1 9 . u

19.2570

23.108

85.25

-638.39

-3.2635

74.7434

-5.7323

9.7800

9.47 ~~~-~~

923.86

-124.653

163.188

-3.3472

-6.4432

95.1 1

-1820.1

2.9050

1 150.942

1 9.3826

23.28 1

67.82

-640.05

-3.1 992

74.6300

-5.8200

9.9567

34.64

924.25

-124.2 I4

163.610

-3.3496

-6.4078

95.94

-1820.3

2.9054

150.93 1

Table 3.5 Differences in orbitai parameters due to tesseral harmonics


Case

A

I3

C

Parameter

a(m)

e x 10'

i x 10' (deg)

12 1 o3 (deg)

o x [ o2 (deg) - -

M

a(m)

e x 10"

i x lo4 (deg)

fl x 1 o3 (deg)

ia x 10' (deg)

M

J I ~ , I ~ - J I ~ , ~ I

-128.81

-1.885

20.50

1.60

-6.40

0.1 I I1

8 8.06

1 -977

-25.96

0.49

-28.04

-0.704 - --

Js,~ - Js,o

-122.18

-1.661

19.89

1.67

-5.37

0.099 1

88.55

1 -794

-24.20

-0.58

39.27

-0.370

a (m)

e x 10'

i x lo4 (deg)

R x lo3 (deg)

(0 x lo2 (deg)

h.I we&

J I O , ~ O - JIO,O

-126.18

-1.752

17.43

1.70

-6.14

0.1 113

87.29

1.889

-25.17

0.55

-60.14

-0.382

14.62

-0.294

-0.81

-0.03

0.04

0.0063

18.00

-0.359

-0.84

-0.02

0.03

0.0 1 05

-2.77

-0.40 1

-1.12

-0.04

0.02

0.0 I26

Table 3.6 Comparison of Observed and Predicted Values (IRS-1 A)

I Days ( Case I Osculating Orbital Parameter

Chapter3 Long Term Orbit Predidions with Earth's Flattening

1

6

12

18

25

30

35

40

0 -

/

0

PT

PZ

0

PT

PZ

0

PT

PZ

0

PT

PZ

0

PT

PZ

0

PT

PZ

M (deg.)

196.61

1 96.64

196.74

1 17.05

1 16.47

1 17.50

117.81

1 17.63

1 18.71

329.96

329.58

329.5 1

21 1.10

2 10.95

2 10.49

128.04

127.61

1 28.40

182.04

181.75

181.17

44.58

43.27

39.13

tesseral terms

a (km)

7274.048

7274.047

7274.093

729 1.047

729 1.06 1

7290.983

7273.91 7

7273.9 14

7273.925

7289.282

7289.263

7289.327

728 I .254

728 1.277

728 1.3 18

7285.643

7285.732

7285.638

i (deg.)

99.0565

99.0579

99.0580

99.044 1

99.0465

99.0474

99.0562

99.0568

99.0580

99.0484

99.0485

99.0484

99.05 15

99.053 1

99.0534

99.0459

99.0497

99.0507

e

.0033 19

.003320

.003318

.OO 1462

.OO 1459

-00 1 465

.00092

.(I0093

.00094

.OO 1 19

.00121

.OO 12 1

.0015 8

.OO 1 60

.00162

.00091

.00094

.00097

99.0504

99.0556

99.0564

99.0446

99.0487

99.0482

terms, PT -

.00112

-001 10

.OO 1 1 1

.00041

.00042

,00042

with zonal

0

PT

PZ

0

PT

P7,

R (deg)

268.023 1

268.0227

268.0231

272.94 1 0

272.9402

272.9422

278.8356

278.8363

278.8396

284.7392

284.7408

284.7446

29 1.6090

29 1.6 1 28

291.61 88

296.5361

296.5386

296.545 8

7276.522

7276.557

7276.504

7289.6 10

7289.527

7289.642

w (deg.)

62.14

62.1 1

62.01

60.23

60.86

59.80

32 1.75

322.09

320.9 1

1 1.87

12. I2

12.18

12.36

1 7.04

17.47

18.42

19.01

1 8.04

301.4395

30 1.445 1

30 1.4529

306.3618

306.3686

306.3767

Predicted with Observed, PZ - Predicted

243.08

243.42

243.94

299.12

299.83

304.48

zonal &

Fig u re - 3.1 Difference between observed and predicted values of semi-major axis

0.05 1 I 1 I I I I ! I

19 x 19 Model : 0.04 - - - -

Figure 3.2 Difference between the observed and predicted values of eccentricity

Chapter3 Long Term Orbit Predictions with Earth 3 Flattening

Figure 3.3 Difference between the observed and predicted values of inclination

Time i lr days

Figure 3.4 Difference between the observed and predicted values of right ascension of ascending node


Figure 3.5 Difference between the obsenied and predicted values of argument of perigee

Figure 3.6 Difference between the observed and predicted values of Mean anomaly


525 I I I 7 I 1 I I I I - - 19 x 19 Model 1 : i

Figure 3.7 Difference between the observed and predicted values of perigee height

450 1 1 I I I T I I I

19 K 19 Modal : : 19 x 0 motlel j 1 1 .

r m QI

0 5 15 20 25 30 35 40 45

Figure 3.8 Difference between the observed and predicted values of apogee height


Figure 3.9 Values of Bilinear relation


Analvtical and Numerical Predidions for Near-Earth's Satellite Orbits with US Uniform Resular

Canonical Eaua tions

with Jz, J3 and J4,

4.1 Introduction

In the previous chapters 2 and 3, it is found that the KS uniformly regular canonical

equations, when integrated numerically with respect to a simple numerical integrator (4th

order Runge-Kutta-Gill method) are found to provide very accurate orbit predictions with

complex force models of Earth's zonal and tesseral harmonic terms even for long

durations. Engels & junkins [16 1 1, Jezewski [ 1 621, and Sharmn ( 1 5 I ] evolved analytical

solutions with J2 for short-term orbit predictions with different formulations. In this

chapter, we have evolved an analytical solution with Jz using KS uniform regular

canonical equations 11441, which is found to be more accurate than the above three

analytical solutions. Though the dominating short periodic variation for near-Earth orbit

is due to the Earth zonal harmonic term J2, inclusion of higher zonal harmonic terms are

essential for accurate short term orbit predictions as well as for generation of accurate

mean elements. As the variation in semi-major axis due to J 3 and J d for low perigee

height and low eccentricity orbits vary up to 100 meters during a revolution and for high

eccentricity orbits up to 10 km, as well as other orbital parameters also vary significantly,

inclusion of these terms is necessary for ktter orbit prediction. In the present study, we

have also generated analytical solutions with .I3 and J4 terms for short-term orbit

predictions.

A numerical experimentation with the analytical solution over a wide range of orbital

parameters semi-major axis, eccentricity and inclination has been carried out. The results

obtained from the analyticat expressions in a single step during one revolution match

Chapter4 Short rerm Orbital Theory with It J3 and J4 90

quite well with numerically integrated values. Comparison of the present solution with

the numerical integration and with other analytical solutions shows the superiority of the

present analytical solution over a wide range of orbital parameters: semi-major axis,

eccentricity and inclination.

In section 4.2, we have developed the analytical equations of motion using KS uniform

regular canonical elements with the Earth's zonal harmonic terms J2 to J4. In section 4.3.

we have analytically integrated the equations of motion with J2 to J4 terns. The

comparisons of the results obtained from the present analytical solution with other

anatytical solutions are provided in section 4.4. Numerical results obtained from the

theory are provided in section 4.5. Conclusions are drawn i n section 4.6. Coefficients

occurred in the analytical solution of equations of motion are provided in Appendix 4.1.

4.2 Equations of Motion

The KS uniform canonical equations of motions (1.12.3) with a perturbing potential V

are

dpi dH dai aH -- - I- -=-- ; for i = 1,2,3,4 ds h i ' ds dfli

and the modified time z is given by the equation (1.12.1 2) is

d z - 1 dh (ii, $1 (4.2.2) ds 4&

where

When perturbation due to Earth's oblateness only are considered in V, then

Chapter4 Short Term Orbital Theory with 3* fa and J4

with

Therefore, considering the terms up to n = 4, the equations of motion (4.2. I ) and (4.2.2)

become

For computing the Eq.(4.2.5), we are having the following partial derivatives

Applying (4.2.7) in (4.2.5) and considering only Jz, J3 and J4 terms, the equations of

motions become

for i = 1,2, 3,4, 5, 6, 7, 8 such that q,=,4j, for i = 1,2,3.4. (4.2.8)

and the time equation (4.2.6) becomes

where A = 2&; and the coeficients anj, b,,,, L+ MTIJk. PYk and QnJk are provided in

Appendix 4.1 .

Chapter-4 Short Term Orbital Theory with J3, J g and k


In (4.2.5) and (4.2.6), the radial distance r can be expanded as r = do ( l - q cos F ). where

J - ~ I Q ,

[:=A - y,wherrtany = d , / d , .

Let

, ~ c o s " ' F s i n ~ F A,) = - dF

(1 - ecos F ) "

Therefvre, for integrating (4.2.5) and (4.2.6) analytically, we obtain the following

integrals

cos A 10 ,,,lo=/-ds=[cosahn - s i n a ~ ~ * l ] / d , " ,

I*

sin A 01 I , ~ ~ = ~ ~ d , y = [ ( c o s a h , ~ -s ina~, '" l /d ," , r

sin 2 A ln2"= j-3 = bin 2a(2hn2~ - i l n o O ) - 2cos2a ~ , " ] / d , , " ,

r

sin 3 A I,,"= I-dr = b a s 4 a ( 4 ~ n 2 1 - A?,") + sin 3 a ( 4 A , , ' ~ - I A , , ' o ) ] I ~ , " ,

r

Chapter-4 Short Term Orbital Theory with J t , A and Jd 93

I , , j~= j sin 5 A

r n 4 s

= [sin 5a (1 61\,," - ~oA," + 5 ~ ~ ' ' ) - cos5a (I 6 A - 1 2 ~ ~ : -+A,~"' )] 1 d,".

where

,lnW = [ - q sin F + (2n - 3 ) ~ $ , - (n - 2)~ :~ : . ,jilr PI > 1 (.-1)q +"-I I

UI 1 ,j , Jor n > 1 " ( I - a"-'

# = ( I - y c o s ~ ) , q = I - e ' ,

Hence equations (4.2.1 I) and (4.2.12) are integrated analytically to give

11 " " a,, n+!-1

hak = -- x Jtl (2)" z - 1 (L..i ,: + Mny I , . i4.3.l) JJ;;; n - 7 d=o d,, I=o

,fork = 1,2,3,4,5,6,7,8

such that cr,,, = PI ,/hr I -. 1,2.3,4 and

with 1,'; = A':, fbr at1 n and i.

Chapter-4 Short Term Orbital Theory wIth Ja J3 and J4

At s = 0, the initial values of a,,,& are computed using the equations (4.2.7) and

(4.2.8). Then at s = sl, we obtain A a , and AP, using the equation (4.3.1 ). Hence we find

the final a, and ,B, as

a, = u,, + Aa,

P, = P,,, + AP, (4.3.3)

Then the position and velocity vectors u, and wi in KS uniformly regular canonical

variables can bc obtained using (1.12.4), which can be transformed in to the position and

velocity vectors using ( I . 12.7) and (1.12.8). Then the osculating orbital parameters at this

point can be easily comptlted.

Corresponding Ar can be computed using the equation (4.3.2). Then the time can be

obtain from

t = AT - (u, .w)l(2a,) .

4.4 Comparison with other results

As described in section 4.3. ti, compute A a , , Ap, and A T , we have programmed the

equations (4.4.1) and (3.4.2) in IBM / IRS 6000. Then the corresponding KS uniformly

regular canonical elementsa, and p, and are computed through (4.3.3). 'The resulting K S

uniform regular canonical position and velocity vectors are computed and transformed

into time using Eq.(4.3.4) and the state vectors using Eqs.(1.12.7) and (1.12.8).

'to find the effectiveness of the present analytical solution, comparisons are made with

the analytical theories of Engels and Junkins 11651, Jezewski [I661 and Sharma [ 1471.

We have considered the example 1 of Jezewski, which is a case of a ballistic trajectory of

high eccentricity 0.91 with flight duration of 1800.0009 seconds with Earth's zonal

harmonic term J2. 'rabte 4.1 pmvides the initial conditions and the final position vector

obtained from six different solutions. The first solution by Bond [I671 is a KS

numerically integrated 12-element formulation that includes the entire Jz perturbation.

Chapter-4 Short Term Orbital Theory with J* J3 and J d 95

for -'s - The third solution by Engels and Junkins [I651 is an analytical non-iterative solution to

first order in J2 of the Keplerian Lambert problem. The fourth solution by Sharma [ 1471

is a solution obtained from the analytical integration of the KS element equations (KS)

with Jz. The fifth solution by Jezewski [I661 is a uniform analytical solution from a non-

canonical approach correct to first order in J2. The second solution labeled as Xavier and

Sharrna (KSC) is the solution obtained in a single step from the analytical theory 11481

presented in this chapter with J2. The sixth solution labeled as Jz = 0 is a classical two

body solution and is included to indicate the effect of Jz perturbation. It may be noted

that our solution is very much comparable with the exact (Bond) solution and i s superior

to that o f Engels and Junkins, Jezewski and Sharma. The seventh one labeled as N UM (J2

to J4) is the solution obtained by the numerical integration of KS uniformly regular

canonical element equations with J2 to J d and the last is the solution with the present

analytical theory with J2 to J4. It should be noted from the Table 4.1, that the numerical

and analytical solutions match quite very well with Jz to J4 terms as well.

For a detailed numerical study, tive test cases A, B, C D and E are chosen with

eccentricities of 0.004, 0.0294, 0.337, 0.73 and 0.92, respectively. The perigee height of

these cases A to E is 200 Km with the apogee heights of 253, 605, 6879, 36470 and

1 62 194 km, respectively. 'The semi-major axes for the test cases A to E are 6605. 678 1 ,

9918, 24713 and 87575 km, respectively. Case A is a near circular orbit. case I3 is

slightly elliptical orbit, case C is an eccentric orbit, case D is a GTO type of orbit and

case E is a highly eccentric orbit. The other parameters chosen are: high inclination of

85', right ascension of ascending node as 0" argument o f perigee as 270" and mean

anomaly as 0'. The numerical values V U M ) presented in this report are generated using

the software 'UOFILAT' whose methodology and validation are reported in chapter 2.


For a detailed numerical study. we have considered 5 test cases, cases A to E, whose

initial state vector i and ,were provided in Table 4.2 along with the resulting orbital

Chapter-4 Short Term Orbital Theory with 3, and 34 96

parameters and periods. The values of the constants Earth radius (R), gravitational

constant (p) and Earth's zonal harmonic terms (Jz, J3 and J4), which are used in the

computation are also provided in Table 4.2.

Tables 4.3 and 4.4 provide the time and variations in the important osculating orbital

parameters a, e and i with J3 as well as .I4 for cases B and E with respect to the initial

conditions, obtained with the present solution (KSC) in a single step and the numerically

integrated values (NUM) with tixed step size 4* order Range-Kutta Gills method with

step size of about 1 degree in eccentric anomaly and with the analytical solution of K S

elements (KS) after 100, 200 and 350 degrees in eccentric anomaly. From the Tables 4.3

and 4.4, we notice that the present analytical solution (KSC) matches quite well with the

numerical values (NUM) and are better than the analytical solutions with KS elements

(KS) during nearly one revolution (350~ eccentric anomaly) in all three parameters a, e

and i with J3 as well as J4. We have also found that the other orbital parameters n, ro and

E obtained by this analytical solution (KSC) also match quite well with numerical values

(NUM) and are better than the analytical solution with KS elements (KS). Table 4.5

provide the KS uniformly regular canonical elements for the four test cases A, B. D and E

obtained by the analytical solution (KSC) with J2 to J4 in single step and numericat

integrated values (NUM) through a Axed step size of about 1 degree in eccentric anomaly

for nearly half and one revolution. 'The corresponding time and orbital parameters are

provided in Table 4.6. From these Tables 4.5 and 4.6, it is noticed that the analytical

solution (KSC) match very well with the numerical solution (NUM).

Comparisons of analytical (ANALYTICAL,) and numerical (NUMERICAL) solutions for

the important orbital elements: semi-m~jor axis, eccentricity and inclination during a

revolution for the two test cases A and D are shown in Figures 4.1 to 4.6. From these

Figures 4.1 to 4.6, it is observed that the maximum difference between the numerical and

analytical solutions for a . e and i art. less than 67 meters, 7 . 2 ~ 1 0 ~ and 2.7~10.' degrees,

respectively for case A; 99 meters, 9 . 5 ~ 1 o ' ~ and 2 . 9 ~ 1 om6 degrees, respective1 y for case D

with J2 to Jq, Figure 4.7 provides the differences between analytical and numerical values

Chapter-4 Short Term Orbital Theory with Jt J3 and J4 97

of the position vector x, y, z during a revolution for the case D. The maximum

differences of -9,4 and 2 1 meters are noticed in the position vector during a revolution.

Comparison of results from analytical solution with KS uniformly regular canonical

elements (KSCANO) and KS elements (KS) with respect to numerically integrated

values for the important orbital elements: semi-major axis, eccentricity and inclination

during a revolution was carried out for the test case A. Figures 4.8 to 4.10 depict the

differences between the numerical I y integrated values and the analytically computed

values from K S uniformly regular canonical elements (ANAL) as well as KS elements

(ANAL I ) for the semi-major axis, eccentricity and inclination during a revolution for the

case A. Figures 4.1 1 to 4.13 depict the differences between the numerical solution with

respect to the analytical solutiotls of KS uniform regular canonical elements (KSCANO)

and KS elements (KS) over a wide range of inclination from 0 to 90 degrees for the

chosen timings of W.f and 77.3 minutes, where the differences between the numerical

and both the analytical solutions are noticed to be high. From the Figures 4.8 to 4.13, it is

observed that the differences between the numerical solutions with respect to KS

elements (KS) are higher than that of the KS uniform regular canonical elements

(KSCANO) during a revolution for all the three parameters a, e and i. From the Tables

4.2 to 4.6 and Figures 4.1 to 4.13, it is clear that the present analytical solutions matches

quite well with the numerical solution and i s better than some o f the existing solutions for

a revolution over a wide range of orbital parameters a, e and i .

4.6 Conclusions

An analytical solution for the short term orbit prediction of satellites, in a closed form

with J2 to 54 has been obtained in terms of KS uniform regular canonical elements. Due to

symmetry in the equations o f motion, only one of the eight equations needs to be

integrated analytically to generate the state vector. Numerical results indicate that the

solution is quite accurate over a wide range of orbital parameters: semi-major axis,

eccentricity and inclination. Comparisons with the other analytical solutions show the

superiority of the present theory ovcr the other analytical theories.

Chapter-* Short Term Orbital Theory with J t J* and J4 98

Appendix 4.1

Chapter-* Short Term Orbital Theory with I* I3 and Jq

for -Earth's KS -

For i = 1,2,3,4 For i = 5,6,7,8

L2im = -L201 = = LdOl = PI /(2&), L,,, = L , , , = L,,, = L,,, = a, 12,

Mzo, -M,, , - -a, 12, M,,, = M,,, = 9 1 /(2&),

L 3, = -L,, = P, / (2JK), L,,, =L, , , =a,/2),

M301 =a, 1 2 , )v3tjl - 4 /(2&)

Then forall i = 1,2,3,4, 5,6,7, 8

ho = LdlO = P3,,Lkm + 0.5(P3, ,LkOl + Q3, 1 1 4 u , ),

L / t l = L411 = P 7 1 0 L k 0 1 + P 3 1 1 L k 0 0 3

M , l l = mill = p31UMk01 + ~ 3 1 l L h f 1 o ,

L j ~ ~ = L112 = 0 . 5 ( P 3 i I L k o ~ - !231iMk01 ),

= M 4 1 2 = 0 .5 (P>~ l M k n l + g3I 1 LkOl 9

f o r j = 2 . k = 3 , j = 3 . k = 2

L,?u = LQO = P22uL100 + O . ~ ( P ~ ~ I L , O I + Q2:1Mlnl 1-

Chapter-4 Short Term Orbital Theory with Ja J3 and Je

All other Lijk, Mijk, and Q i j k 'S are zero.

Chapter-4 Short Term Orbital nteory with J* J3 and Jq

Table 4.1

Comparison with other solutions Initial conditions:

Position components (6478,0,O) km ; Velocity components (7, I , 3) Ms.

Orbital elements:

Semi-major axis = 6222.02 km, eccentricity = 0.91, inclination = 7 I .565".

Coasting time = 1800.0009 seconds.

Chapter-4 Short Term DrMtal Theory with fa b and J4

$1. No.

- --

I

2

3 - - --

4 -- - -

5 - -- - --

6

7

8

Method of solution

Bond

Xavier and S h m a

(KSC)

Engles and Junkins

Sharrna (Ks) .---- ---

Jezewski - --

J2=0

ANAL (Jz to J q )

NUM(J2 to J q )

Position components (km)

x 1

10970.929

10970.929

10970.928

10970.930 -

10970.687 --

10980.077

10970.924

10970.924

-- x2

1435.480

1435.480

1435.480 - -

1435.480 - -

1 435.642 -

1435.926

143 5.980

1335.980

- -

x j

4304.95 1

4304.95 1

4304.95 1 .- --

4304.953

4304.898 - 4307.777

4304.937

3304.937 ---

P- for '- 1 ' I - Table 4.2

Initial conditions (Position, Velocity and Orbital Parameters)

Values of constants used:

K = 6378.135 km,

J2 = 1 .OR263 x 1 I)-',

Variable

X I (km)

X? (km)

x~ (km) --u

i, {km/ A )

p = 398600.8 km3 s-~,

J 3 = -2.53648 x 10.' and Jq = - I .52 x 10.'

Chapter4 Short Term Orbital Theory with J* 33 and J4

1

CASE

A

0.0

-573.3 132

-6553.0 -

7.8

.k2 (km / ,5 )

x , (km / s)

a (km) -"

e

i @%I a (deg)

w (deg)

M (deg)

lip (ktn)

B

0.0

-573.3132

-6553

7.9

0.0

0 .O

6781.0568

0.0299404

85.0

0.0

27.0

0.0

0.0

0 -0

6604.6526

0.0040307

85.0

0.0

270.0

0.0

H, (km)

1' (tn n ts)

C

0.0

-573.3 132

-6553

9.0

0.0

0.0

991 7.5357

0.336727

85.0

0.0

27.0

0.0

89.065 92.657 --

D

0.0

-573.31 32

-6553.0 ,-.--

10.25

0.0

0.0

247 1 3.274

0.733826

85.0

E

0.0

-573.31 32

-6553.0

10.8

0 .o

0.0

87575.384

0.9248872

85 .O -

0.0

27.0

0.0-

0.0

27.0

0.0

Table 4.3 Comparison of time and semi-major axis with KS theory

Chapter-4 Short Term OrMtal Theory with Ja J3 and 1,

Para-

meter

Time

(mts)

a ( m )

After

E

100

200

350

100

200

350

Met-

hod

NUM

KSC

'- KS -

N[JM

KSC

KS

N U M

KSC

KS

N t l M

KSC

KS

Num

K SC ---

KS

N U M

KSC

K S

B

25.29331936

25.2933 1936

25.2933 1936

5 1.60702 10 I

51.60702101

51.60702102

90.12460896

90.12460896

90. I2460895

23.682

23.682

23.682

48.60 1

48.60 1

48.602

3.077

3.077 -- 3.077

- -

E

270.96903605

570.96903605

570.9690360 1

2604.796 1741

2604.7961741

2604.7961742

4289.5 139925

4289.5139925

4289.5 139924

5269.5 10

5269.529

5269.540

5269.479

5269.478

5269.485

5959.543

5959.522

5959.676

CASE

B

25.293108864

25.293 108864

25.293 I08864

5 1 A06359704

51.606359704

51.606359703

90.1 23626673

90.123626673

90.1 23626676 - "

- 14.465 - -

- 14.465

- 14.465

-1 1.422

- 1 1.422 -.-.A- .

- 1 1.423

-2.894

-2.894

-2.894

- - - - - - - . . - - E

570.892417790

570.8924 17790

370.89247 1 166

2604.424357 190

2604.424357190

2604.424357191

4288.9045623 16

4288964X3I6 4288.904562306

-301 4.497 - - . - - -

-30 14.500

-30 14.503 ... "

-30 14.503 . - . - -30 14.50 1

-30 14.500 -

-35 8 1 -863

-358 1.846 -- --" -

-3581.782

Table 4.4 Comparison of eccentricity and inclination with KS theory

Chapter-4 Short Term Orbital Theory with Jtr J3 and J4

Parameter

e x lo7

ix107

(deg)

Epoch of

comparison

E (deg)

100

200

350

' 100

200

350

Method

NUM

KSC

K S

NUM

KSC

KS

NUM

KSC

KS

NUM

KSC

K S

Num

KSC

KS

NUM

KSC

KS

- J3

- .- J4

I3

43.176

43.176

43.1 76

22.8 17

22.8 14

22.8 16

5.676

5.676

5.677

81.045

81.045

8 I .045

176.206

176.206

176.206

10.520

10.522

10.524 I

CASE

E

45.1858

45.1858

45.1 859 -- -

45. t 839

45.1839

45.1839

51.159

51.159

51 .I60

59.097

59.097

59.099

59.142

59.144

59.144

65.020

65.020

65.022

B

-29.02 15

-29.0214

-29.0214

-33.7252

-33.7253

-33.7248

-5.01 2

-5.012

-5.01 2

-49.1 04

-49.1 04

-49.104

-37.153

-37.1 53 -

-37.155

-9.944

-9.945

-9.946

E

-25.8501

-25.8501

-25.8501 .-

-25.8502

-25.8502

-25.8502

-30.662

-30.662

-30.66 1

-33.857

-33.857

-33.856

-33.855

-33.854 -"L-

-33.854

-4 1.944

-4 1.944

-4 1.943

Table 4.5 Comparison of analytical and numerical values of KS uniformly regular canonical

elements with Jz to Jq

Chapter-4 Short Term Orbltal Theory with I& JJ and 3 4

C

a

s

e

- A

B

D

E

~~p Rev

0.5

1.0

0.5

j.0

0.5

1.0

0.5

Met-

hod

NUM

KSC

N l l M

KSC

N U M

KSC

N U M

KSC

NUM

KSC

N-UM

K SC

NUM

NlJM 1.0

KS Uniformly Regular canonical element

P4 XI OZ

-4.62 '

-4.62

-9.09

0.15

-4.51

-4.52

-9.08

0.05

-2.90

-2.90

-7.02

0.04

-2.67

-2.67

-6.64

0.06

P2

19.452

19.452

19.374

19.496

19.702

19.702

19.624

19.744

25.281

25.581

25.529

25.617

26.957

26.957

26.909

26.992

PI

224.062

224.062

223.999

223.654

226.922

226.922

226.866

226.529

294.194

294.194

294.1 24

293.916

309.947

309.947

309.876

309.685

P3

222.867

222.867

222.488

222.825

225.725

225.725

225.345

225.674

292.869

292.869

292.601

292.803

308.581

308.581

308.332

308.516

-a,

57.253

57.251

57.262

57.350

57.259

57.257

57.267

57.350

57.336

57.336

57.33 1

57.350

57.342

57.342

57.337

57.350

a3

57.1 15

57.1 17

57.217

57.132

57.1 18

57.120

57.212

57.13 1

57.136

57.136

57.150

57.132

57.137

57.173

57.145

57.132

a2

5.008

5.009

5.029

4.998

5.008

5.008

5.027

4.998

4.999

4.999

5.002

4.998

4.998

4.998

5.000

4.998

- a 4

XI 02

1 . I

1.2

2.34

0.0002

1 . I

1 . I

2.16

.00002

0.23

0.23

0.209

.00002

0.15

0.15

0.0472

.00004

Table 4. 6 Comparison of analytical and numerical values of

orbital elements with Jz to Jq

Chapter4 Short Term Orbital Theory with Ja J$ and J4

C

a

s

e

A

B

D

E

Met-

hod

N U M

KSC

NIJM KSC

NUM

KSC

NUM

KSC

NUM

KSC

NUM

KSC

NUM

KSC

NII M

KSC'

Time

(Mnts)

44.483

44.483

88.970

88.949

46.407

46.407

92.8 19

92.799

325.62

325.62

65 1.70

65 1.69

2236.9

2236.9

4477.4

4477.4 I L L -

a (km)

6605.189

6605.158

6604.668

6604.665

6783.564

6783.538

678 1.059

6781.058

24901.04

2490 1.07

2471 3.31

247 13.3 I

89988.35

89988.60

87576.54

87576.54

e

0.00799

0.00800

0.00403

0.00403

0.03371

0.03372

0.02994

0.02994

0.73589

0.73589

0.73383

0.73383

0.92690

0.92690

0.92489

0.92489

Orbital

i (deg)

85.0001

85,000 1

85.0000

85.0000

85.0003

85.0003

85.0000

85.0000

85.0025

85.0025

85.0000

85.0000

85.0025

85.0025

85.0000

85.0000

E ldeg)

179.610

179.580

357.665

357.503

179.836

1 79.835

359.604

359.525

179.912

179.9 12

359.823

359.802

"179.849 '

179.849

360.000

359.987 u

Element

Q(deg)

359.977

359.977

359.953

360.000

359.978

359.978

359.955

360.000

359.992

359.992

359.984

360.000

359.994

359.994

360.000

360.000

w rdeg)

269.592-

269.6 18

270.738

271.073

269.855

269.86 1

269.789

270.046

269.956

269.956

269.91 4

270.002

269.964

269.964

270.000

270.003

for

Figure 4.1 Comparison of numerically and analytically computed values of variation in semi-major axis during one revolution (case A)

T i r n c ( M i n t r tes)

Figure 4.2 Comparison of numerically and analytically computed values of variation in eccentricity during a revolution (case A)

Chapter-4 Short Term Orbital Theory with Ja J3 and J4

h's -

Figure 4.4 Comparison of numerically and analytically computed values of variation in semi-major axis during a revolution (case D)

- - . . . -

I d

1

,*,/- --+. -. % .

FI 0 -5

i X .- i

- I 0

Chapter-4 Short Term Orbital Theory with J* Ja and J4

- 1 5 r

- 2 0 '

I

i I A Y h I , ' t " l I C A i ,

. t

, . N I I M E R I C A L 7 J

( N C I M - A N A L ) ~ * I 0n0.0 -.. .A" --.. . - . ---- -- , . -y " " -" " 7 . , - . , -I 0 1 0 2 0 90 40 5 0 BCp 7 0 8 0 90

7'imc in in t e s ) Figure 4.3 Comparison of numerically and analytically computed values

of variation in inclination during a revolution (case A)

- P- for N m h ' s #

Figure 4.5 Comparison of numerically and analytically computed values of variation in eccentricity during a revolution (case D)

Figure 4.6 Comparison of numerically and analytically computed values of variation in inclination during a revolution (case D)

Chapter-4 Short Term Orbital Theory with J f , J3 and J ,

' i ' lME ( M I N U T E S )

Figure 4.7 Difference between numerically and analytically computed values of position vector during a revolution (case D)

Chapter-4 Short Term Orbital Theory with Ja J3 and J4

z n o A..--_... -.

1 7 5 f7 - . . A N A L 1

* * A N A L . 15U

/ I

1 2 5 '\ - 1 0 0 i \

m i

w 1.

7 5 LU I I- 50

1, I

LLt r w 25 - / ' . - -- m 0

- 2 5

- 5 0

- 0 5

- 100

----.- a .

- * \ i x , -1 -\,- -

I

-125 1 ,,-.L--77 - - - - - . . . 0 1 0 20 , 3 0 4 0 5 0 li O 7 U t i 0 9 u

'I'irrlc. (h1il-l ~r t c s )

Figure 4.8 Comparison of differences between numerically and analytically computed values of semi-major axis (case A)

for N m h ' s Satellite KS -

Figure 4.9 Comparison of differences between numerically and analytically computed values of eccentricity (case A)

Figure 4.10 Comparison of differences between numerically and analytically computed values of inclination (case A)

Chapter-4 Short Term Orbital Theory with Jl 1 and Jq 112

Inc%ina+ion (degrees]

Figure 4.1 1 Comparison of differences between numerically and analytically computed values of semi-major axis with respect to inclination(case C)

- 2 5 0 0 10 2 0 30 4 0 5 0 6 0 7 0 8 0 9 0

Inclination [degrees)

.. > .

& S { l , f i - 7 7 . 1 ? ' 1 , \ ) I . . . ... i......... L

Figure 4.12 Comparison of differences between numerically and analytically computed values of eccentricity with respect to inclination (case C)

Chapter-4 Short Term Orbital Theory with I* J3 and J4 113

- 7 5 0 0 X Q 2 0 30 4 0 5 0 6 0 7 0 8 0 90

Inclination [degrees)

Figure 4.13 Comparison of differences between numerically and analytically computed values of inclination with respect to inclination (case C)

Chapter-4 Short Term Orbital Theory with J* 1 3 J 4 and

~ s r

Canonical Eauations

ORBITAL THEORY WITH AIR DRAG: SPHERICALLY SYMMETRICAL EXPONENTIAL ATMOSPHERE

5.1 Introduction

The effect of air drag on a satellite orbit is quite different from that of the gravitational

field. Since the air density decreases rapidly as the height above the Earth increases, a

satellite in an orbit of appreciable eccentricity is affected mostly by drag within a small

section of the orbit, where it i s closest to Earth. Therefore the effect of air drag is to retard

the satellite as it passes the perigee, with the result that it does not swing out so far from

the Earih at the subsequent apogee passage. Hence, the apogee height will be reduced

while the perigee height remains almost constant t i l l the orbit contracts and becomes

more nearly circular. As a result, both the orbital elements 'a' and 'e' decrease steadily,

while the perigee radius 'a(l -e)' decreases very slowly. lf the Earth is considered as

spherical, an initial circular orbit will remain circular and the air drag will reduce semi-

major axis on ty, at an increasing rate, so that the satellite spirals in.

If the atmosphere is stationary and spherical in form, semi-major axis and eccentricity

will be the only elements to change under the influence of air drag. However, in reality at

the relevant heights of 120-2000 km, the atmosphere rotates approximately, but not

exactly at the same speed as the Earth. 'Phis rotation subjects the satellite to small

sideways forces, which slightly alter the orientation of the orbital plane, leading to a

small but slowly increasing changes in orbital inclination and small periodic changes in

'R'. In addition, the atmosphere is oblate like Earth, and the consequent symmetry in

C ha pter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphere 1 1 5

for N e a r - N r m - drag can alter 'w' by an amount, which is usually small but can be large for near circular

orbits.

In this chapter, we take a simple model for air density by assuming that the density

depends on the distance from the Earth's centre and varies exponentially with the

distance. We have developed a new non singular anaIytical theory for the motion of an

artificial satellite in near Earth orbit with air drag in terms of the KS uniformly regular

canonical elements by a series expansion method, by assuming the atmosphere to be

symn~etricatly spherical with constant density scale height [ I 67, 1771. The terms up to

third order in eccentricity are retained. Only two of the nine equations are solved

analytically to compute the state vector and change in energy at the end of each

revolution, due to symmetry in the equations of motion. For comparison purpose these

equations are integrated numerically with a fixed step size fourth order Runge Kutta Gill

method with a small step size of half degree in eccentric anomaly. Numerical

experimentation with the analytical solution for a wide range of perigee altitude,

eccentricity and orbital inclination has been carried out up to 1000 revolutions. The

results obtained from the analytical expressions match quite we1 l with the numerically

integrated values and show improvement over the results obtained from the third order

theories of Cook, King-Hele and Walker [ I 661 and Sharma [ I 531.

Section 5.2 deals with the air density model considered. Section 5.3 deals with equations

of motion with the assumed atmospheric model. Section 5.4 deals with the analytical

integration of the equations of motion. In section 5.5 numerical results are discussed. The

conclusions arrived from the study are presented in section 5.6.


If we take the simplest model for air density, assuming that the density p depends solely

on the distance r from the Earth's centre and varies exponentially with r with constant

density scale height H [ I 91, then

Chap ter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponentla1 Atmosphere 1 1 6

P = & > < I exP ( ( r p , - w H 1 , (5.2.1)

where p," is the density at the initial perigee point, r is the initial distance from the P ,I

Earth's centre.

Substituting r, = a0 ( I -eo) , P = I/H and x = a.e in (5.2.1) we obtain the expression of p U

in a spherically symmetrical atmosphere of constant scale height H as

p = p ,,, e x p ( p ( a , - a - x , , ) + p x c o s E ) . (5.2.2)

5.3 Equations of motion

The KS uniformly regular canonical equations of motion with air drag perturbation

given in Eqs. ( 1 . 1 3.13) and (1.13.14) are

F,, + F , , c o s E + F , , c o s 2 E + F , , c o s 3 E

+ F,, c o s 4 E -+ F,, s i n E + F,, s in 2 E + , d E 8

F, , sin 3 E + F, , sit1 4 E + F,, E I for i =1,2,3 ,......, 8 with ~ j + 4 = pJ, Ij = I ,2?3,4). (5.3.2)

When we substitute the spherically symmetrical atmosphere p given in Eq. (5.2.2) in the

above Eqs. (5.3. I ) and (5.3.21, we get

d c! 2 = K __S i ' ( 0 " - u - r , , 1

d E 8 J " a a ,

[ D , + D , c o s E -1- D , c o s 2 E + D , c o s 3 E ] e B X C " $ "

F,, , + F , , c o s E + F , , c o s 2 E -t F , , c o s 3 E

+ F, , c o s 4 E + F, , sin E + F , , s i n 2 b; I eP"""' ,

+ F, , sin 3 E + F,isin 4 E + F,>,E

fur i = 1,2,3.. ....., X with q t 4 = f3, ,Cj = 1,2,3,4) (5.3.4)

Chapter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphere 1 17

I Predieions for KS Unl- - 5.4 Analytical integration

The change in Q) and ai during one revolution, Aa,, say, are obtained by integrating the

Eqs. (5.3.3)and(5.3.4)from E = Oto2n:.

We note that the nonzero integrals in the resulting expressiuns are of the form o f linear

combinations of the modified Bessel functions

1, ( 2 ) = ] exp(z cos E ) cos nE dE. 21c 0

Integrating the equations of motion (5.3.3) and (5.3.4) from 0 to 2n and using Eq. (5.4. I ) ,

we obtain the changes after one revolution as:

with a,,? - p, for i = 1,2,3,4. (5.4.3)

5.5 Numericai results

To compute ( ~ k , P k (k = 0, 1 , 2, 3, 4) at the end of each revolution, we have programmed

Eqs. (5.4.2) and (5.4.3) in double precision arithmetic on an IBM RS 1 6000. Once a k , @k

are known, uk, wk can be computed from Eqs. (1.2.5), and the state vectors x and x are

computed using the relations provided in Eqs. (1.12.7) and (1.12.8), which are then

converted in to the orbital elements. The value of the ballistic coefficient b, = ~ / C D A i s

chosen as 50.0 kg/rn2, and the Earth's equatorial radius and p are taken as 6378.135 km

and 398600.8 km3 s'l, respectively. ClRA (1972) mean atmospheric density model is

employed to compute the values of the density and density scale heights. Eqs. ( 1.13.5) are

C ha pter-5 Orbital Theory with Air mag: Spherically Symmetrical Exponential Atmosphere 1 1 8

for N-te Orb- KS Uaiform.Rwular - -

numerically integrated (NUM) with the spherical atmospheric model p provided in

(5.2.2) with a small step size of half degree in the eccentric anomaly, with a fixed step

size fourth order Runge Kutta Gill method to obtain the numerical solutions. Detailed

numerical simulations are carried out with respect to orbits with variations in eccentricity,

inclination and perigee height to study the effect of the drag force considered. The values

of cu, IT and E are taken as 60, 30 and 0 degrees, respectively. 'The accuracies of the

numerical computations examined with the help of the bilinear relation provided in

( t .4.4.) and ( 1.8.13) in the KS uniformly regular canonical elements are found to be very

satisfjctory. The effectiveness of the present analytical solution is established by

comparing the results obtained by the present solution with other analytical snlutinns over

a wide range of perigee height, eccentricity and inclination.

Tables 5.1 and 5.2 provide the decrease in the important orbital parameters semi-major

axis (a) and eccentricity (e) after 50 revolutions along with the percentage errors [= 100

x (NUM-ANAL) / NUM], with respect to the analytical solutions o f the present theory

[I671 which we represent by KSC, third order KS theory [I531 represented by KS and the

extended theory of Cook - King Hele & Walkar 11661 represented by CKW, for the orbits

having e = 0.05 and i = 30' with variation in perigee height from 135 to 220 km. It may

be seen that the present solution provides better estimates of semi-major axis and

eccentricity than provided by the other two theories. As expected. the percentage errors

decreases with the increase in perigee height. It is interesting to note that the present

solution gives only 70 metres difference after 50 revolutions in the semi-major axis for a

very low perigee height (Hp) of 135 km, whereas the other KS and C K W theories

provides the differences of 204 metres and 760 metres, respectively. Tables 5.3 and 5.4

provide the decrease in semi-major axis and eccentricity after 50 revolutions for Hp = 200

km and e = 0.05 with change in orbital inclination from 1 to 90 degrees. It should be

noted that the percentage errors are almost constant in all the cases. Here also, it is seen

that the present solution provides better estimates for semi-major axis and eccentricity

than the other two theories. Tables 5.5 and 5.6 provide the decrease in semi-major axis

and eccentricity up to 100 revulutions for H, = 200 km and e = 0.05 and i = 85 degree. In

Chapter-5 Orbital Theory with Alr Drag: Spherically Symmetrical Exponential Atmosphere 1 19

Earth's # - this case also is seen that the present solution provides better estimates for semi-major

axis and eccentricity than the other two theories. From the Tables 5.1 to 5.6, it i s evident

that the present analytical solution provides better estimates for the important orbital

parameters: semi-major axis (a) and eccentricity (e) for a wide range of perigee height,


To find out the effectiveness of the present analytical solution for long term orbit

predictions, we have carried out computations up to 1000 revolutions f i r the orbits

having a wide range of perigee height, eccentricity and inclination. Tables 5.7and 5.8

provide the decrease in semi-major axis and eccentricity along with the percentage errors

up to 1000 revolutions for the orbit having Hp = 220 km and e = 0.05 and i = 85 degree.

obtained from KSC, KS and C K W theories. It is noticed that the maximum decrease in

the semi major axis and eccentricity after 1000 revolutions are 184.882 km and 0.0239,

respectively. It may be noted that the accuracy of the present analytical solution does not

change significantly with the increase in revolution number, where as in the other two

solutions the inaccuracies increase with the increase in the revolution number. Tables 5.9

and 5.10 provide the decrease in semi-major axis and eccentricity after 1000 revolutions

for the orbits having e - 0.1 and i = 30' with variation in perigee height korn 1 75 to 300

km. As expected, it is noticed that the maximum decrease in the semi ma.jor axis and

eccentricity after 1000 revolutions are noted for the low perigee height of 175 km, and

are 5 1 1.82 km and 0.0658, respectivefy. Tables 5.1 1 and 5.12 provide the decrease in

semi-major axis and eccentricity after 1000 revolutions for H, = 250 km and i = 25

degree with change in eccentricity from 0.025 to 0.15. As expected, the maximum

decrease in the semi major axis and eccentricity after 1000 revolutions are noted for the

low eccentricity of 0.025 and are 127.672 km and 0.0152, respectively. Here also, it i s

seen that the present analytical theory provides better estimates of semi-major axis and

eccentricity than the other two analytical theories for the wide range of eccentricities

considered. Tables 5.13 and 5.14 provide the decrease in semi-major axis and eccentricity

after 1000 revolutions for H, = 220 km and e = 0.05 with change in orbital inclination

from I to 90 degrees. It i s noticed that the change in semi-major axis and eccentricity

Chapter -5 Orbital Theory with Air Drag: SpherJcally Symmetrical Exponential Atmosphere 1 20

increases with respect to increase in orbital inclination. The maximum decrease in the

semi major axis and eccentricity after 1000 revolutions is noted for i = 90 degrees and are

188.34 km and 0,0243, respectively. Here also, it is seen that the present analytical theory

provides better estimates for semi-major axis and eccentricity than the other two

analytical theories. From the Tables 5.7 to 5.14, we have seen that the present analytical

solution provides better estimates than the other two solutions for the important orbital

parameters: semi-major axis and eccentricity, after I000 revolutions over a wide range of

perigee height, eccentricity s ~ ~ d inclination.

Figures 5.1 and 5.2 depict the differences between the numerically integrated and

analytically computed values of semi major axis and eccentricity up to 1 OOO revolutions

for the orbit having H p = 220 km and e = 0.05 and i = 85 degree, obtained fro111 all the

three analytical solutions. Figures 5.3 and 5.4 depict the above differences atler 1000

revolutions for the above orbit having e = 0.1 and i = 30' with variation in perigee height

from 175 to 300 km, obtained from all three analytical solutions. It is observed that for

lower perigee heights ( I 75.2 10 km) where air drag effect is more, the KSC solution

shows significant improvement in accuracy over the other two solutions. Figures 5.5 and

5.6 provide the above differences after 1000 revolutions for the orbit having Hp = 250 krn

and i = 30 degrees with variation of eccentricity from 0.025 to 0.2. The accuracy of semi

major axis computation is of the same order, when the initial eccentricity i s up to 0.13

and the accuracy o f eccentricity computation i s o f the same order when the initial

eccentricity is up to 0.065. Figures 5.7 and 5.8 depict the above differences after 1000

revolutions for the orbit having H, = 220 km and e = 0.05 with variation of inclination

from I to 90 degrees. The KSC solution provides better estimated of semi major axis and

eccentricity for all the inclinations from 1 to 90 degrees. From the figures 5.1 to 5.8, i t is

seen that the present analytical solution provides better estimates than the other two

analytical solutions even after 1000 revolutions for the drag perturbed orbital parameters:

semi major axis and eccentricity over a wide range o f initial perigee height, eccentricity

and inclination, which shows the superiority of the present theory over the other theories..

C hapter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphere 1 2 1

5.6 Conclusion

The KS uniformly regular canonical element equations are integrated analytically by

series expansion method with air drag force, by assuming a spherically symrnetrical

exponential atmosphere with constant density scale height. A non singular solution up to

third order terms in eccentricity is obtained. Only two of the nine equations are solved

analytically to compute the state vector and change in energy at the end of each

revolution, due to symmetry in the equations of motion. Comparison of the present

solution with the KS elements analytical solution and modified Cook, King-Hele and

Walker theory with respect to the numerically integrated values over a wide range of

eccentricity, perigee height and orbitat inclination, show the superiority of the present

solution over the other two theories.

Chap ter-5 Orbital Theory with Air rag: Spherically Symmetrical Exponential Atmosphere 1 22

Table 5.1 Decrease in semi-major axis after SO revolutions for e = 0.05 and i = 30'

1 HP 1 Decrease in semi-major axis (km) Percentage error - I (km) ( NUM I KSC I 1

KS I CKW I KSC / KS I CKW

Table 5.2 Decrease in eccentricity after 50 revolutions for e = 0.05 and i = 30'

Chap terd Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphere 1 23

Table 5.3 Decrease in semi-major axis for Hp = 200 km and e = 0.05 after 50 revolutions

Table 5.4 Decrease in eccentricity for Hp = 200 km and e = 0.05 after 50 revolutions - -- --

i

(deg)

I

15

30

45

60

75

90

C ha pter-5 Orbital Theory with A i r Drag: Spherically Symmetrical Exponential Atmosphere 1 24

i

(deg)

1

t 5

30

45

60

75

90

Decrease in semi-major axis (km) Percentage error

KSC -----

-0.002 1

-0.002 1

-0.002 1

-0.0022

-0.0023

-0.0023

-0.0024

CKW

10.6732

10.7303

10.8994

11.1712

1 1.5307

1 1.9568

12.423 1

NUM KS

Decrease in eccentricity x lo3

pp

10.6726

10.7297

10.8987

11.1705

1 1.5300

1 1.9561

12.4223

NUM

1.388 10

1.39553

1.4 1754

I .45292

Percentage error

KS

-0.0052

-0.0052

-0.0053

-0.0055

-0.0057

-0.0059

-0.006 1

1.49971

1.55517

1.61586

CKW

-0.0057

-0.0057

-0.0058

-0.0060

-0.0062

-0.0064

-0.0067

10.6728

10.7299

10.8990

11.1708

1 1.5302

1 1.9563

12.4226

KSC

1.3881 5

1.39558

1.41 759

f .45297 --

CKW

-0.0039

-0.0039

-0.0040

-0.0041

KSC

-0.0033

-0.0034

-0.0034

-0.0034 p- -

10.673 1

10.7302

1 0.8993

11.17lI

1 1.5306

1 1.9567

1 2.4230

KS

-0.005 1

-0.005 1

-0.0052

-0.0054

KS

1 -388 1 7

1.39560

1.4 1 76 1

1.45299

-0.0042

-0.0043

-0.0045

CKW

1.388 15

1.39559

1.4 1760

1.45298

-0.0056

-0.0058

-0.0060

1.49976 -0.0035

1.55522

1.61592

1.55526

1.61595

1.55523

1.61593

-0.0036

-0.0037

Table 5.5 Decrease in semi-major axis for Bp = 200 km, e = 0.05 and i = 85'

up to 100 revolutions

Table 5.6 Decrease in eccentricity for Hp = 200 km, e = 0.05 and i = 85" up to 100 revolutions

Rev.

No.

1

5

10

25 --

50

75

100 -

C ha pter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphee 1 25

-Rev.

No.

1

5

10

25

50

75

100 L

Decrease in semi-major axis (km) - -

Percentage error

CKW

0.23667

1.18458

2.37228

5,95432

11.98909

18.10769

24.3 1400

NUM

0.23671

1.13481

2.37273

5.95537

11.99095

18.11010

24.3 1666


CKW

0.01 98

0.0194

0.0 190

0.0 177

0.0155

0.0133

0.01 10

KSC

0.00072

0.0069

0.0064

0.005 1

0.0025

-0.003

-0.0037

- -- Percentage error

KSC

0.23670

1.18473

2.3 7258

5.955507

11.99065

18.11017

24.3 1757

K S

0.01 16

0.0113

0.01 09

0.0098

0,0077

0.003

0.0025

CKW

3.083 5

t 5.4349

30.9131

77.6 1 04

156.3359

236.2211

3 17.3169

NUM

3.0848

1 5.4.4 13

30.9258

77.64 14

156.395 1

236.3056

31 7.4236

KSC

0.0249

0.0249

0.0249

0.0248

0.0246

0.0242

0.0235

KS

0.23669

1.18468

2.37247

5.85479

11.99003

18.10915

24.3 1 606

KSC

3.0841

15.4375

30.9181

77.622 1

156.3566

236.2484

317.3491

KS

0.030 1

0.030 1

0.0301

0.030 1

0.0296

0.0292

0.0285

KS

3.0839

1 5.4367

30.9165

77.6 18 I

156.3487

236.2366

317.333 1

CKW

0.04 1 9

0.04 14-

0.0410

0.0398

0.0378

0.0358

0.0336

for b o r n

Table 5.7 Decrease in semi-major axis for H, = 220 km, e = 0.05 and i = 85"

up to 1000 revolutions -


Rev.

No.

1

10

100

250

500

750

1 000

C ha pterd Orbitat Theory with Air Drag: Spherically Symmetrical Exponential Atmosphem 126

Percentage error Decrease in semi-major axis (km)

Rev.

No.

1

10

100 -

250

506

750

1000

CKW

-0.003- '

-0.003

-0.004

-0.004

-0.002

-0.007

-0.0 10

KSC

-0.001

-0.001

-0.001

-0.001

-0.001

-0.0003

0.0003

Percentage error

KS

-0.003

-0.003

-0.004

-0.004

-0.005

-0.005

-0.006

CKW

0.14335

1.4356 1

14.5770

37.4452

78.950 1

126.559

1 84.900

NUM

0L14334

1.43556

14.5764

37.4436

78.9459

126.550

1 84.882

Eccentricity x lo3

CKW

-0.003

-0.003

-0.003

-0.003

-0.004

-0.006

-0.008

NUM

0.0 1 846

0.18486

I .87836

4.83039

lo. 1995

16.3600

23.8622

KSC

-0.003

-0.003

-0.003 -

-0.002

-0.002

-0.00 1

-0.001

KSC

0.14334

1.43558

14.5766

37.4440

78.9465

126.550

184.88 1

KS

-0.003

-0.003

-0.003

-0.004

-0.004

-0.005

-0.006

KS

0.14335

1.43561

14.5766

37.4450

78.9493

126.556

184.893

KSC

0.0 1 846

0.18486

1.87841 - --

4.8305 1

10.1997

16.3603

23.8624

KS

0.0 1 846

0.1 8486

1 .a7842

4.83058

10.2000

16.3609

23.8637

CKW

0.0 1 846

0.18486

1 -8784 1

4.83056

10.2000

16.36 10

23.864 1

Table 5.9 Decrease in semi-major axis for e = 0.1 and i = 30' after 1000 revolutions

Table 5.10 Decrease in semi-major axis for e = 0.1 and i = 30U after 1000 revoiutions

H~

(km)

1 75

1 8 5 ."

200 - - ..

220

240

260

300

C ha pter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphee 1 27

H P

(km)

175

185

200

220

240

260

3% - - -

Decrease in semi-major axis (km)

NUM

528.182

330.169

197.494

1 12.224

68.4006

43.55 19

19.1966

Percentage error

KSC

0.006

0.002

0.001

0.0004

0.0002

0.0%01

0.00008 --

-. ,.

Decrease in eccentricity x 10"

NUM

68.003

41.470

24.305

13.593

8.1988

5.1798

2.2568

Percentage error

CKW

528.66 1

330.247

1 97.504

I 12.222

68.3978

43.5497

19.1954

KSC

528.1 50

330. 141

1 97.492

1 12.224

68.4004

43.55 19

19.1966

KS

-0.084

-0.041

-0.022

-0.01 0"" -.

-0.004

-0.00 1

0.002

KSC

0.005

0.001

-0.000 1

-0.0007

-0.0009

-0.00 1

-0.001 - "

KS

528.625

330.305

197.538

1 12.235

68.4035

43.5524

19.1962

CKW

-0.09 1 - -0.024

-0.005

0.002

0.004

0.005

0.006

-, .

CKW

68.064

4 1.478

24.306

1 3.593

8.1985

5.1 796 -,

2.2566

KSC

68.000

4 1.469

24.305

13.593

8.1989

5.1 700

2.2568

KS

-0.085

-0.038

-0.0 16

-0.003

0.004

0.008 .

0.01 1 .

KS

68.06 1

41.485

24.309

13.594

8.1985

5.1794 ? -,

2.2565

CKW

-0.089

-0.02 1

-0.003

0.002

0.004

0.005

0.005

Table 5.11 Decrease in semi-major axis for a = 250 km and i = 85" after 1000 revolutions

Table 5.12 Decrease in eccentricity for a =250 km and i = 85" after 1000 revolutions

e

0.025

0.05

0.075

Chap ter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphem 1 28

e

0.025

0.05

0.075

0.1

10.15


NUM

127.672

68.816

58.428

0. I 54.339

6 . i2.482 .-

Percentage error

KSC

-0.001

0.000

0.000 - .."

0.002


KSC

127.673

68.8 16

58.428

54.339

52.48 1

NUM

14.35 1

8.770

7.272

6.486

5.656

Percentage error

KS

-0.004

-0.002

-0.002

-0.002

0.004

KS

127.677

68.818

58.429

54.340

52.480

CKW

-0.006

-0.002

0.0003

0.005

0.031

KSC

14.35 1

8.770

7.272

6.486 -

5.656

CKW

-0.005

-0.002

0.001 - . - -

0.005

-0.032

KSC

-0.002

-0.001

-0.001 - - - - - - - -0.001

0.001

CKW

127.680

68.817

58.427

54.336-0.0002

52.466

KS

-0.005

-0.002

0.001 - - - - - - - -

0.006

0.035

KS

14.352

8.770

7.272

6.486 --

5.656

CKW

14.352

8.770

7.272

6.486

5.656

Table 5.13 Decrease in semi-major axis for Hp = 220 krn and e = 0.05 after 1000 revolutions

Table 5.14 Decrease in eccentricity for Hp = 220 km and e = 0.05 after 1000 revolutions

i

(deg)

I

15

30

45

60

75

90

Chapter-5 Orbital Theory with Air Drag: Spherically SymmetrlEal Exponential Atmosphere 129

Percentage error

i

(deg)

1

15

30

45

60-

75

90


KSC

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

KS

-0.004

-0.004

-0.005

-0.005

-0.005

-0.005

-0.006

NIJM

152.921

153.991

157.1 91

162.429

169.549

178.309

188.340

Eccentricity x lom3

CKW -

-0.007

-0.007

-0.007

-0.007

-0.008

-0.009

-0.010

KS

152.927

153.998

157.199

1 62.437

169.558

178.3 18

. 188.351

KSC

152.920

153.99 1

157.19 1

162.429

169.549

178.308

188.339

NUM

19.7625

19.9004

20.3122

20.9857

21.8999

23.0220

24.3033

Percentage error

CKW

152.931

1 54.002

157.202

162.44 1

169.563

178.324

188.359

KSC

-0.0009

-0.0009

-0.0009

-0.0009 - -0.0009

-0.0009

-0.0009

KSC

19.7627

19.9006

20.3124

20.9859

21.9001

23.0222

24.3035

KS

-0.005

-0.005

-0.005

-0.005

-0.006

-0.006

-0.006

KS

19.7635

1 9.90 14

20.3 1 33

20.9868

2 1.90 1 1

23.0234

24.3049

CKW

-0.006

-0.006

-0.006

-0.006

-0.067

-0.008

-0.009

CKW

1 9.7636

1 9.90 1 6

20.3 134

20.9870

2 1.90 1 4

23.023 8

24.3054

Orb- -

Figure 5.1 Difference between numerically and analytically computed values of semi- major axis up to 1000 revolutions.

Figure 5.2 Difference between numerically and analytically computed values of eccentricity up to 1000 revolutions.

C ha pter-5 Orbital Theory with A i r Drag: Spherically Symmetrical Exponential Atmosphere 1 30

Figure 5.3 Difference between numerically and analytically computed values of semi- major axis after 1000 revolutions with respect to perigee height.

Figure 5.4 Difference between numerically and analytically computed values of eccentricity after 1000 rev01 utions with respect to perigee height

C hapter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exponential Atmosphere 131

Cha

Initial eccentricity

Figure 5.5 Difference between numerically and analytically computed values semi-major axis after 1000 revolutions with respect to eccentricity.

0 -5 I I I 1 I I I 1

E $ 0 2 - - - - - - > --..---- P " 0.1 -----.:-..----:-------- - - - - - - - : - - - - - - - - : - - - - - - - a a *- , -'-.dm- U .- - a - - - - - - ,*--.+7=-- - - - - - - -, - _ _ - _ - - . - _ - - - - - u LT-:i:: - :

# E w ' J KS : ; 4.1 -.---..: ---...--: -......- 1 - - - - - - - ;.- C ; - 2 - A-'--- . - - a - .---- - - L - - - a -- 5.L '

s . '% .- 0.2 --.---; -.------: --------: -- - - '--: - - - - - - - A m - I*<--'--..- ' - - - L - - - - ---.

T.' U s

, . I . '* r

5 a 4.4 ------; - - - - - - - - ;----..--- 5 - - - - ---; - - - - - - - :- -. --. --;-. - - - - - >.-. . - - -

: '%.,

4.5 -- - --. :- - - - - - - :- - - - - - - -' : '\ , - - - - - - - - r - - - - . - " . - " . - - . - T - - - . 2 - - - , - - - . - - - - r - " - - - \T

0.6 I I 1 I 1 I I I 0.025 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2


Figure 5.6 Difference between numerically and analytically computed values eccentricity after 1000 revolutions with respect ta eccentricity.

pter-5 Orbital Theory with Air Drag: Spherically Symmetrical Exportentiat Atmosphere

with

Figure 5.8 Difference between ournerieally and analytically computed values uf eccentricity after 1000 revolutions with respect to inclination.

C ha pter-5 Orbital Theory with Air Drag: Spherically Symmtrlcal Exponential Atmosphere 133

15

10

E - YI

-- 8 5 L 0 '16 E .- o al VI E

I I I I I I I 1

/ Hp-i225 k m i e -)mi 1

--------; -----.- 1 . - - - - - - ' r - - - - - - - : - - - . - - . ' .-------; ------. ' ; - - -mx.-&.!

:KS : i -.-.-.-i i-.-.-.-!-.-.-.-r - - - - - - , : - - - - - - - : . - - - - - : - - - - - - : - - . - - . : - - ~ . - L : - ~ . - L ~ " : ~ ~ ~ ~ ~ ~ . : L ~ ~ L - ~ -

--------:--- ---. j - - - - - - - j - - - - - - - : ~ ~ c - - i ~ ~ ~ ~ ~ ~ ~ ; lllL1ll iiiiiiii; *-

Q1

k n

-10

-75

-.-. -.-I -.-_ . - . *c !-*-----:-.- ' CKW : , .---;-.q.- ._ ' : --.-._, : --------:-..-..-' ' - - - - - - - : - - - - - - - ; - - - - - - - ' T - - - - - - - : - - - - - ' .Ly--___;-__-__- , ---5 -..

1 I 0

I I I 1 I I fO 20 30 # 50 60 70 80 90

Initial indination (deg)

Figure 5.7 Difference between numerically and analytically computed values semi-major axis after 1000 revolutions with respect to inclination.

Analvtical and Numerical Predictions for Near-Earth's Satellite Orbits with KS Uniform Reqular

Canonical Equations

1 Pm&&m for M . a i S ~ r M ' s i

ORBITAL THEORY WITH AIR DRAG: OBLATE EXPONENTIAL

ATMOSPHERE

6.1 Introduction

In this chapter we have generated a new non-singular analytical solution for long-term

orbit predictions with the KS uniform regular canonical equations with air drag force by

assuming the atmosphere to be oblate with constant density scale height, up to third order

terms in eccentricity. The density of the atmosphere depends on its height above the

Earth's surface, which is an oblate spheroid. Consequently the atmosphere is also oblate,

and in the region where the theory is most often used (between the altitudes of 150 and

400 km), the surfaces of constant density tend to be spheroid and of approximately follow

the same ellipticity as of the Earth. We developed a new theory on the assumption that

the density is constant on the surfaces of spheroids of fixed ellipticity E whose axes

coincide with the Earth's axis. The model, with the ellipticity E which is assumed to be

equal to the Earth's ellipticity, 0.00335, is almost equivalent to an atmosphere with

density at a given height above the Earth is constant. Here we assumed that the density

varies exponentially with height above the oblate spheroid. A small parameter c

depending upon the ellipticity of the atmosphere is introduced; whose value normally

does not exceed 0.2 during the orbital life of a satellite except in the final stage of a

satellite's life, in the density function. Numerical experimentation with the analytical

solution for a wide range of perigee altitude, eccentricity and orbital inclination has been

carried out up to 1000 revolutions. The results obtained from the analytical solution

match quite well with the numerically integrated values. We have also compared our

Chupf~r-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

I P r e d ~ & ~ u s . - far Nmw&rth's SaMiUe Or-

numerical results with the third-order KS theory [I531 and the third order Swinerd &

Bouiton theory [130]. All the three analytical theories compare well. However, the

present analytical theory is found to have some superiority over the other two analytical

theories.

In section 6.2, we introduced the expression for the air density at any point on a satellite's

orbit in an oblate atmosphere. The density function i s obtained up to third-order terms in

e and c. Also we transformed the geocentric latitude to argument of latitude in the

expression for air density. Section 6.3 contains the equations of motion with oblate

atmosphere. 'The change in the KS uniformly regular canonical elements. during one

revolution is obtained in section 6.4 in terms of the moditied Bessel functions. Terms up

to third-order terms in eccentricity and c, a parameter dependent on flattening of the

atmosphere, are retained. Also, the change in energy using the KS uniformly regular

element during a revolution is formulated in terms of the modified Bessel functions. The

changes in the other seven KS uniformly regular canonical elements are found by just

changing the initial conditions, as the KS uniformly regular canonical element equations

are symmetric. The analysis of the numerical results obtained from the newly developed

analytical theory and comparison of the results with other analytical results are provided

in section 6.5. We have concluded the chapter in section 6.6. The expressions for the

coefficients in section 6.3 are provided in section 6.7 as Appendix 6.1.

Section 6.2 deals with the air density model considered for the study. Section 6.3 deals

with equations o f motion with the atmospheric model considered. Section 6.4 deals with

the analytical solutions obtained from the analytical integration of the developed

equations of motion. In section 6.5 we discussed with numerical results obtained from the

theory and the chapter ends with the conclusions arrived from the study, which was

provided in section 6.6. Appendix 6.1 contains the coefficients appearing in the

developed analytical solutions of the equations of motion.

C'kuprer-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere


For any given geocentric latitude 4, the air density is proportional to exp I- r IH). Here 'r'

d r is the distance from the Earth's centre and H = - p-, the density scale height, is kept

d p

as constant. 'The radial distance cr from the Earth's centre to the surface of an oblate

spheroid of equatorial radius a~ and small ellipticity E i s given by

c = o, ( I -&-sin ~ + O ( F ' ) ) , (6.2.1 )

where cp is thc geocentric latitude and 0(e2}, less than 0.03, is negligible, at: is so chosen

that the spheroid defined by (6.2.1) passes through the initial perigee point of the satellite

distant r{,,, from the Earth's centre and at latitude #pO . Then (6.2. I ) becomes

1 - &sin # 0 = r,,(,

1 - s sin ' In the spheroid (6.2.2) the air density is pp,, and the density varies exponentially with

height above the spheroid, so that

P = P , e x p ( - P ( r - d ) , (6.2.3)

here p,,,, is density at the perigee, P = 1 1 H.

Now we transform 4. the latitude of the satellite with the angular position of the satellite

in its orbit using the following relation

sin y = s i n i sin (w + f ) . (6.2.4)

where i, w arld f are the orbital inclination, argument of perigee and true anomaly,

respectively. Substituting Eqs. (6.2.2) and (6.2.4) in Eq. (6.2.3), the air density p at any

point (r,O) of the orbit is

p = p,,, e ~ p { - p ( v - $ ~ , ) - t c c o s 2 ( ~ + f ' ) - c c o s 2 r 0 , + O ( C F ) ) .

(6.2.5)

where

Chapter-6 Orbltal Theory wIth Air Drag: Oblate Exponential Atmosphere

In Eq. (6.2.5) expanding up to third order terms in c, we may write

exp (c c.os 2(w +f)) = I + c cos 2(w +j') +

Substituting (6.2.6) into (6.2.5) and then converting the true anomaly 0 into the eccentric

anomaly E and expanding with r = a (I-ecosE), the density p in (6.2.5) can he written as

p = kexp(afl(1 -ecosE))

[ [ 1 + c cos 2f w + E)) + ce(cos(2cll-t 3@ - cos(2ro + A'))

1 + - c"3 cos 2(m + E) + cos 6(0 + El ) ) + 0(ce3,c2e2. cb, cJ)j, 24

where

k = p , , exp ( ,B r,, - c cos 2~0" )

Simplifying (6.2.7), we get

p = k exp{ap(l- e cos E ) )

[So + S, cos21u + EE)+ S, cos(2w + 2 E ) + S,cos(2o1+ 3 E ) (6.2.8)

+S , cos(4m + 3 E ) + S , cos(2w i- 4 E ) + S, cos(4w + 4 E )

+S, cos(4w + 5 E ) + S8 cos 6(w + E ) ] ,

where

Chapter-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

for w . .

6.3 Equations of motion

The K S uniformly regular canonical equations of motion with air drag perturbation

given in Eqs. (1.13.13) and ( I . 13.14) are

F,, + F , , c o s E + F 2 , c o s 2 E + F 3 , c o s 3 E

+ F,, cos 4 E + F,, sin E + F,, sin 2 E + F 7 , s i n 3 E + F8,s in4E + FqiE

for i = I ,2,3 ,......, 8 with ajd= Pj, (j = I ,2,3,4) . (6.3.2)

When, we substitute the value of p given in Eq. (6.2.8) in the above Eqs. (6.3.1) and

(6.3.2), we get

G, cos jE + H,,, cos(2o + kE) + k=-2

9

K , cos(4o + mE) I V,, cos(6o + H E ) n=3 1

Chapter-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphem

, mr j E + a,, sin j ~ ) + 1'0

JN,,,,,,, cos(4w + mE) + M,,,, sin(4o) + m E ) ) 1 1 1 - - - I l y

for i=l ,2 ,3 ,4 .5 ,6 ,7 ,8 . (6.3.4)

where the coefficients Cij, Hk, K,, Vn, Pji, Qji, Rk,, Lk, Nmi, Mml, T,, and Cni are provided in

the Appendix 6. I .


The explicit form of the perturbations, .hi is obtained by integrating Eqs.(6.3.3) and

(6.3.4) with respect to E from O to 271. We note that the non zero integrals in the resulting

expressions are o f the form of linear combinations o f the modified Bessel functions

The following integrals are also used for the analytical integration of Eqs. (6 .3 .3) and

(6.3.4).

2i cos (nto k m E ) dE = cos nml,.

C'hupter-6 Orbital Theory wIth Alr Drag: Oblate Exponential Atmosphere

'1 sin (nw + mE) dE = sin nwl , . 0

Noting that the non-zero integrals in the resulting expression of the equations of motion

(6.3.3.) and (6.3.4) are o f the form of linear combinations of the modified Bessel

functions (6.4.1) and integrating them from 0 to 271 using the Eqs.(6.4.2), (6.4.3) and

(6.4.4), we obtain the changes in Auo and Aai after one revolution as:

A u O = p7', [Go t G,CO s 2 0 + G ? C U S 4~ + G ~ C U s 6011, (6.4.5)

H,, + H,, cos 20) + Hz, cos 4 0 + H3, cos 6011, ALE, = 7;

+HA, sin 2m+ Hs, sin4m+H6, sin601

with a l+4= pi for i = 1, 2, 3, 4, (6.4.6)

where

ChupIer-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

for N- Or-h Kq -

The expressions for the coefficients F,i's are provided in section 1 . 1 3.

6.5 Numerical Results

To compute a k . P k (k = 0 - 4) at the end of each revolution, we have programmed

equations (6.3.1) and (6.3.2) in double precision arithmetic on an IBM RS/6000 computer

available. Once ak, Pk are known, uk, wk can be computed from Eqs. (1.8.3), and the state

vectors x and i are computed from Eq.(t .4.4), which are then converted in to the orbital

elements. Detailed numerical simulations are carried out with respect to orbits with

perigee height of 200 and 250 km, where the effect of drag force is significant. The

values of a, fi and E are taken as 60'. 30" and 0°, respectively.

Tables 6.1 to 6.8 provide the decrease (A) in semi-major axis (a) and eccentricity (e)

obtained with the numerical integration (NUM), from the present third order analytical

solution represented by KSC, from the third order KS theory represented by KS and from

the third order Swinerd & Boulton theory represented by SB. The numerical integration is

Chapter-6 Orbital Theory with Air Drag: Oblate Expanential Atmosphere 141

carried out by integrating the KS element equations of motion with a small step size of

half degree in the eccentric anomaly, with a fixed step size fourth order Runge-Kutta-Gilt

method by using the complete expression of the density p given in (6.2.5). CIRA 1972

[96] atmospheric density model is employed to compute the values of the density and

density scale heights at perigee altitude (H,). The % H theory of King-Hele and Scott

[I341 i s used for computing the density scale height H. The value of E, A, the ballistic

coefficient b, = rn/c~A. the Earth's equatorial radius and p are taken as 0.00335, 1.2 and

50.0 kg/m2, 6378.135 krn and 398600.8 kmJ sL2, respectively. It is observed lhat the

results obtained by numerically integrating the KS element equations of motion match

very well with the numerically integrated values obtained from the KS uniformly regular

canonical element equations of motion over a wide range of orbital parameters.

Table 6.1 provides Aa and Ae obtained with the numerical integration (NUM) and from

the analytical theories KSC, KS and SB up to 100 revolutions for the orbit having I i p =

200 km, e = 0.05 and inclination = 85". The bilinear relations (1.6.4) and (1.12.6)

satisfied by the KS uniformty regular canonical equations are used as a check for all the

computations carried out with the present theory. It is seen that semi-major axis decreases

by 24.3 km, and eccentricity decreases by 0.003 t 7 after I00 revolutions. The results from

all the 3 theories compare quite well with the numerically integrated values. 'Tables 6.2

and 6.3 provide the decrease in semi-major axis and eccentricity along with the

percentage errors obtained from all the three analytical theories up to 1000 revolutions

for the orbit having Hp = 250 km, e = 0.05 with the inclination of 25', obtained from

KSC, KS and SB. It is noticed that the maximum decrease in the semi-major axis and

eccentricity after 1000 revolutions are 67.9 km and 0.000866, respectively. It may be

noted that the accuracy of the present analytical solution does not change significantly

with the increase in revolution number, where as in the other two solutions the

inaccuracies increase with the increase in the revolution number. Tables 6.4 and 6.5

provide the decrease in semi-major axis and eccentricity along with the percentage errors

after 1000 revolutions for H, = 250 km and i = 15 degree with change in eccentricity

from 0.025 to 0.15. As expected, the maximum decrease in the semi ma*jor axis and

C'h~pier-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere 142

eccentricity afier 1000 revolutions are noted for the low eccentricity of 0.025 and are

122.5 krn and 0.0139, respectively. Here also, it is seen that the present analytical theory

provides better estimates of semi-major axis and eccentricity than the other two analytical

theories for the wide range of eccentricities considered. Tables 6.6 and 6.7 provide the

decrease in semi-major axis and eccentricity along with the percentage errors after 1000

revolutions for H, = 250 km and e = 0.1 with change in orbital inclination from 1 to 90

degrees. It is noticed that the change in semi-major axis and eccentricity increases with

respect to increase in orbital inclination. The maximum decrease in the semi major axis

and eccentricity after 1000 revolutions is noted for i = 90 degrees and are 61.1 km and

0.0073, respectively. Here also, it is seen that the present analytical theory provides better

estimates for semi-major axis and eccentricity than the other two analytical theories.

Tables 6.8 and 6.9 provide Aa and Ae with the percentage errors after 1000 revolutions

for the orbit with e = 0. I and i = 15" with perigee height varying from 175 to 250 km.

The maximum decrease in tho parameters semi-major axis and eccentricity i s 5 1 1.8 km

and 0.0658, respectively, when Hp is 175 km. It i s noted that the present KSC theory

match very well with the numerical values (NUM) than the other two theories at a11 the

perigee heights, From all the tables 6.1 to 6.9, it i s noticed that all the 3 theories provide

good accuracies; however, it is observed that the present KSC theory provides better

matching with the numerical solutions than the other two theories.


analytically computed values and the percentage errors for semi-major axis up to 1000

revolutions for the orbit having H, = 250 km, e = 0.05 and inclination of 8 5 O degree. It i s

easily noted that the KSC theory is found to have less difference than the other 2 theories.

It is also noticed that the % error is lesser with KSC theory than the other two theories.


analytically computed values of eccentricity and the percentage errors obtained from all

the three analytical theories with respect to the numerically integrated values of the of

eccentricity up to 1000 revolutions for the same orbit with all the three theories. tlere.

also it is noticed that the present KSC theory provides better result than the other two

C'hapter-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere 143

for -'s p- - theories. Figures 6.5 and 6.6 depict the differences between the numerically integrated

and analytically computed values of the important orbital parameters semi-major axis and

eccentricity for the orbit having Hp = 250 krn and i = 15 degrees with eccentricity varying

from 0.025 to 0.1 8. Here also it is found that the present KSC theory is more consistent

than the other two theories. Figures 6.7 and 6.8 depict the differences between the

numerically integrated and analytically computed values of the important orbital

parameters semi-major axis and eccentricity for the orbit having Hp = 250 km and e = 0.1

degrees with inclination varying from 1 to 90° degrees. Here also it is noted that the KSC:

theory gives better estimates ofthe semi-major axis and eccentricity at all the inclinations

considered. Figures 6.9 and 6.10 depict the differences between the numerically

integrated and analytically computed values of the important orbital parameters semi-

major axis and eccentricity for the orbit having e = 0.1 and i = 15" with variation in

perigee height from 175 to 250 km. The change in semi-major axis is found to be less

with KSC theory than the other two theories for all the H, under consideration.

6.6 Conclusion

The KS uniformly regular canonical element equations are integrated analytically by a

series expansion method with air drag force, by assuming an analytical oblate

exponential atmosphere with constant density scale height. A non-singular solution up to

third-order terms in eccentricity and c, a parameter dependent on flattening of the

atmosphere, is obtained. Only two of the nine equations are solved analytically to

compute the state vector and change in energy at the end of each revolution, due to

symmetry in the equations of motion. Comparisons of the present solution, third order K S

theory and third order Swinerd & Boulton theory with the numerically integrated values,

show that the present solution provides better estimates of semi-major axis and

eccentricity than the other two theories over a wide range of eccentricity, perigee height

and inclination.

C'hupier-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

for -h's -- - Appendix 6.1

Chapter-fi Orbital Theory with Air Drag: Oblate Exponential Atmosphere

to or- -

chapter-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

Chiipfcr-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

P- for Near-- KS U- -

Uhuptrr-6 Orbital Theory with Air Drag: Oblate Exponentbl Atmosphere

Table 6,l Decrease in semi-major axis and eccentricity for

Hp = 200 km, e = 0.05 and i = 85" up to 100 revolutions

Table 6.2 Decrease in semi-major axis Hp = 250 kin, e = 0.05 and i = 25' up to 1000 revolutions

Rev.

No.

1

5

10

25

50

' 75

100


Rev.

NO.

100

200

500

1000


Chapter-6 Orbital Theory with Air Drag: Oblate Exponentfa1 Atmosphere

T-- - -- Eccentricity x I 0

NUM

3.0848

15.44 13

30.9258

77.6414

156.39:

236306

3 I 7.424

Rev.

No.

100

200

500

1000

NUM

0.23671

1.1 848 1

2.37273

5.95537

1 1.9909

18.1 10 1

24.3 167

- Decrease in semi-major axis (km)

KSC

3.0841

1 5.4375

30.91 g1-- . - 2

77.6221

'--c5-

3 17.349

- . - - Percentage error

KSC

0.23670

1 . 1 8473

2.37258

5.95551

1 1.9907

18.1 102

24.3 1 76

SB

6.306

12.704

32.498

67.874

KSC

-0.0007

-0.0009

-0.00 I7

-0.0037

/--. -- - . 1 .86.569

- - - - . . - -

Decrease in Eccentricity x 1 o5

KS

3.0839

1 5.4367

30.9 1 65

77.61 81

156.349

238.248--236.237

3 1 7.333

KS

6.306

12.704

32.498

67.874 .

NUM- ' -KC

- - - -. - - Percentage error

SB

3.0835

15.4349

30.913 1

77.6104

156.336

236.22 1

3 17.3 17

KS

0.23669

1 . I 8468

. -

KS

-0,OO 14

-0.00 1 7

-0.0028

-0.005 1

6.306

1 2.704

32.497

67.87 1

NUM

8.028

16.178

41.414

SB

0.23667

I. 1 8458

- - - - - - SR

-0.00 1 2

-0.00 15

-0.0024

-0.0044 ----. -

0.306

12.704

32.497 -

67.873 --- . -

86.568 86.568 86.569

KSC -

8.028

16.178

41.414

KS

8.028

16.178

41.414 --

SB

0.00 1 8

0.0016

0.0008

-0.00 1 0 J

KSC

0.00 1 5

0.0014

0.0008

-0.0003

2.37247 1 2.37228

5.85479 ' 5.95432 -

1 1.9000 1 1.989 1 --- 1 IB.IOL12

24.3 1 61

SB

8.028

16.178

41.414

KS

0.0023

0.0020

0.0011

-0.0008

Table 6.4 Decrease in semi-major axis for H, = 250 km and i = 15' after 1OOO revolutions

Table 6.5 Decrease in eccentricity for H, = 250 km and i = 15" after 1000 revohtions

e

0.025

0.050

0.1

0.1 5

Table 6.6 Decrease in semi-major axis H, = 250 km and e = 0.1 after 1000 revolutions

Percentage error Decrease in semi-major axis (km)

E

0.025

0.050

0.1

1 (deg) I-KSC 1 KS SB 1 KSC I KS 1 SB

SB

-0.0 13 --

-0.003

0.005

0-032 .. . ,- J

NUM

122.48

67.465

53.432

51.656

KSC

-0.01 1

-0.001

-0.00002

0.002

Percentage error

mi Decrease in sem i-major axis (krn)


KS

-0.01 4

-0.003

-0.003

0.005

Decrease in Eccentricity x lo3

KSC

-0.006

-0.00 1

-0.001

0.00 1

Percentage error 1

KSC

122.49

67.466

53.432

KS

-0.009

-0.002

0.007

0.036

NUM

13.866

8.601

6.378

0 15 5 567 I-.." --.- -

KS

122.50

67.467

53.433

KS

13.867

8.601

6.378

5.567

SB

-0.000

-0.002 ,--

0.005

0 . ~ 3 3 . - . 1

KSC

13.867

8.601

6.378

5.567

SB

122.49

67.467

53.429

51.655 1 5 1.654 .-

SB

13.867

8.601

6.378

5.565

5 1.640

Table 6.7 Decrease in eccentricity for H, = 250 km and e = 0.1 after 1000 revolutions

Table 6.8 Decrease in semi-major axis for e = 0.1 and i = 15" after 1000 revolutions

I

(deg)

5

30

60

90

1 HP 1 Decrease in semi-major axis (km) Percentage error -7 / (kmm) 1 NUM / KSC 1 KS SB / KSC KS SB

I

Table 6.9 Decrease in eccentricity for e = 0.1 and i = 15" after 1000 revolutions

Decrease in Eccentricity x 1 o3 NUM

6.354

6.460

6.793

7.306

Percentage error

Chapier-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphere

KSC

-0.001

0.0003

0.005

0.010

4 (km)

175

200

225

250

KSC

6.354

6.460

6.793

KS

0.007

0.008

0.020

0.029 7.306 1 7.304 ( 7.303

KS

6.354

6.459

6.792

SB

0.005

0.008

0.027

0.042 -.- .

Decrease in eccentricity x I o3

SB

6.354

6.459

6.792

NUM

65.806

23.855

1 1.71 1

6.378 -.

-. Percentage error

KSC

-0.01

-0.004

-0.003

-0.001

KSC

65.815

23.855

1 I .711

6.378

KS

-0.09

-0.02

-0.004

-0.002

KS

65.868

23.859

1 1.71 1

6.378 --

- SB I

.- -0.09 1

-0.04 I - -0.004 I

-0.003 -A

SB

65.867

23.856

1 1.71 1

6.378 - --

Figure 6.1 Difference between numerically and analytically computed values of semi-major axis up to 1000 revolutions

Figure 6.2 Percentage errors between numerically and analytically computed values of semi-major axis up to f 000 revolutions

C'hupter-6 Orbital Theory with Air Drag: Oblate Exponentlal Atmosphere

I Q - ~ 9 I I I I I I 1 I I

0 100 200 3Q0 600 500 MI0 TOO 800 900 1000 R e v o h ~ t l o ~ ~ 1a181n ber

Figure 6.3 Difference behveen numerically and analytically computed values of eccentricity up to 1000 revolutions

I .5 I I 1 I I I I I I

0 100 200 300 400 500 600 700 800 900 1000 Revolurlo~l ~na~ln ber

Figure 6.4 Percentage errors behveen numerically and analytically computed values of eccentricity up to 1000 revolutions.

Chapter-6 Orbital Theory with Air Drag: Oblate Exponential Atmosphwe

- I P- for N e a N R ~ &

Figure 6.5 Difference between numerically and anaIyticalIy computed values of semi-major axis after 1000 revolutions with respect to eccentricity.


Figure 6.6 Difference between numerically and analytically computed values of eccentricity with respect to eccentricity after 1000 revolutions


Figure 6.7 Difference between numerically and analytically computed values of the semi-major axis with respect to inclination after 1000 revolutions.

Figure

Chapter-6 Orbital Thcwy with Air Drag: Oblate Exponential Atrnosphem

Figure 6.9 Difference between numerically and analytically computed values of the semi-major axis with respect to perigee height after 1000 revolutions.

Figure 6.10 Difference between oumerically and analytically computed values o f eccentricity with respect to perigee height after 1000 revolutions.

Chapter-6 Orbital Theory with Air Drag: Oblan Exponentla1 Atmosphere

A c r

Canonical Eauations

ORBITAL THEORY WITH AIR DRAG: OBLATE DIURNALLY VARYING

ATMOSPHERE

7.1 Introduction

In this chapter, we have developed a non singular analytical theory in terms of uniformly

regular KS canonical elements with air drag using oblate diurnally varying atmosphere

with constant density scale height. The series expansion method i s utilized to generate the

analytical solution and terms up to third-order in eccentricity and c are retained. Only two

of the nine equations are solved analytically to compute the state vector and change in

energy at the end of each revolution, due to symmetry in the equations of motion and

computation for the other equations ([f44], p91) is made by changing the initial

conditions. For comparison purpose KS elements equations are integrated numerically

with a fixed step size fourth-order Runge-Kutta-Gill method with a small step size of half

degree in eccentric anomaly. Numerical experimentation over a wide range of perigee

altitude, eccentricity and orbitaI inclination has been carried out up to 1000 revolutions.

The numerical results obtained with the analytical solutions match quite well with the

numerically integrated values and show improvement over the third-order theories of

Swinerd & Boulton [ I 661 and Sharma 11 541.

Section 7.2 deals with the air density model. Section 7.3 deals with the analytical

integration of the developed equations of motion. Numerical results are discussed in

section 7.4. The conclusions of the study are presented in section 7.5. The coefficients

appearing in the equations of motion are provided in Appendix 7.1, Appendix 7.2,

Appendix 7.3 and Appendix 7.4.

Chapter-7 Orbltal Theory with Air Drag: Oblate Diurnally Varying Atmosphere


If the density p is assumed to be vary sinusoidally with a, where @ is the geocentric

angular distance from the direction o f the density maximum, then after Santora [ I 271, we

write

p = p,, (1 + F cos a) exp ( -P( r - 0) ‘) . (7.2.1)

with

I - E sin2@ 0 = r f-1 F = - , "f - P1nm 1

1 p=- . ('1. I - s sin' B, f + I Pmin Ho

where E i s the ellipticity of the Earth, po is the average density on the reference spheroid.

when @ = 90°, Ho and 0 are average density scale height and geocentric latitude,

respectively. The ellipticity of each of the oblate atmospheric surfaces is assumed to be

the same as the ellipticity of the Earth.

The density p in (7.2. I ) can be written as [ I 301

[1+ ccos 2(w + E) + ce{cos(Zw + 3E) - cos(2o~+ E ) ) 1 2

+-c e (cos2w -4cos2(w+E)+3cos2(cll+2E)) 4

1 2 2 3 + - c3 (3 cos 2(w + E ) + cos 6(cl, + E ) ) + (Z(CF, ce' , c e , c e, c4)], (7.2.2) 24

with

A = sin &sin i sin w + cos SH ( cos (a - ag) cos w - cos i sinla - a ~ ) sin ru ),

R = sin 2iBsin i cos co - cos { cos (SZ - a ~ ) sin (11 + cos i sin(52 - un) cos w 1.

Chapter-7 Orbital Theory with Air Drag: Oblate Diurnally VaryIng Atmosphere

where (a, , 6,) and (ae , SB) are the right ascension and declination of the Sun and the

center of the day time bulge.

Expanding the terms of F A and F B in (7.2.2) up to third power in e, we get

Then the equation (7.2.2) becomes

1+ FA(C, +C, cosE+C,cos2E+C3cos3E+~ w s 4 ~ ) P= P," W r r a ( m E - l ) )

+FB(C, sin E +C, sin2E+C3 sin3E+C4 sin4E) 1 (7.2.3)

where the coefficients S, for i = 0,. . .,8 are provided in Section 6.2 and

Simplifying (7.2.3), we get

P,)" p = -exp (-a P (1 - cos E ) ) 4 - -

IN- c o s i E + q s in iE] + 1-0 I

1-5

C [ N . ~ 0 ~ { 2 w + ( j - 8 ) E ) + P . s i n ( ( 2 w + ( j - 8 ) E ) ] ,=s J J

21 +C [ N ~ cos (4#+(k-I8)EJ+ Pk s in ( (4w+(k- l8 )E) ]+ k-17

[ N ~ ~ ~ s ( 6 w + (m -26)E) + Pm sin((4m + (rn - 2 6 ) E ) l m=ZU -

where the coefficients N, ' s and PI ' s for i = 0,. .. . . .,36 are provided in Appendix 7.1.

Chapter-7 Orbital Theory with Air Drag: Oblate Diurnally Varyfng Atmosphem

After some algebra, the resulting Eqs. (7.2.4) and (7.2.5) are integrated from 0 to 27c

using the Eq. (7.3.1). We obtain the changes in Aao and Aai after one revolution as:

Lo+L,cos2w+L,cos4w+L,cos6w ha, = pJ , + L, sin 2w + L, sin 4w + L, sin 6 0

M,, + M , , c o s 2 u , + M , , c o s 4 w + M 3 , c o s 6 ~ A a i = J ,

+M,, sin 2u + M,, sin 4u-t M,, sin ~ C L )

fori =1,2,3 ,.... 8 with aji-4 = f3j for j = 1,2,3,4.

where, Jr and the coefficients Lj and Mii, for j = 0. 1. 2, 3, 4, 5 , 6 are provided in

Appendix 7.4.


To compute ak, P k (k = 0, 1,2, 3 , 4 ) at the end of each revolution, we have programmed

equations (7.3.2) and (7.3.3) in double precision arithmetic on an IBM RS/6000

computer. Once ak, Pk are known, uk, wk can be computed using (1.12.4), and the state

vectors x and x are computed from the relations in (1.12.7) and (1.12.8), which are then

converted in to the osculating orbital elements. 'Fhe value of the ballistic coefficient b, =

m/CnA is chosen as 50.0 kg/m2, where CD, m and A are. respectively, the drag

coefficient, mass and effective area of the satellite. The values of RE, E, A, p are taken as

6378.135 km, 0.00335, 1.2, 398600.8 km3 s - ~ , respectively. Jacchia ( 1 977) atmospheric

density model i s employed to compute the values of density and density scale height.

Arbitrarily, 22 August 1995 is chosen as the initial epoch. The value of 10.7 cm solar flux

(F10.7) is used as 150.0 and averaged geomagnetic index (A,) is taken as 10. The KS

element equations of motion with the Jacchia (1977) atmospheric density model are

numerically integrated WUM) with a small step size of half degree in the eccentric

anomaly (E), with a tixed step size fourth order Runge-Kutta-Gilt method to obtain the

numerical solutions. '1'0 study the effect of the drag force, detailed numerical simulations

are carried out with respect to orbits with variations in eccentricity, inclination and

Chapter-7 Orbital Theory with Air Drag: Oblate Diurnally Varylng Atmosphere

perigee height. The values of argument of perigee (o), right ascension of ascending node

(SZ) and E are taken as 60, 30 and 0 degrees, respectively.

Tables 3.1 and y.2 provide the decrease in semi-major axis (a) and eccentricity (e) along

with the percentage errors obtained through the KS numerical integration denoted by

NUM, from the present anaIytical solution represented by from the KS theory

represented by KS and from Swinerd & Boulton theory denoted by SB, up to 1000

revolutions for the orbit with Hp = 200 km, e = 0.1 and i = 55". It may be noted that the

decrease in the semi-major axis and eccentricity after 1000 revolutions art: 2 13.0 km and

0.0266, respectively. It may be noted that the KSC theory has minimum percentage error

with respect to other two theories up to 1000 revolutions. Tables3.3 and7.4 provide the

same information after 500 revolutions for the orbit with e = 0.1 and i = 85" with

variation of perigee height from 165 to 225 km. It may be noted that the decrease in the

semi-major axis and eccentricity after 500 revolutions for 165 km perigee height are,

3 15.9 krn and 0.0401, respectively. In this case also. it is noticed that the present theory

has the minimum percentage error. Tables7.5 and3.6 provide the same information after

500 revolutions for the orbit with Hp = 200 km and i = 35" with variation of eccentricity

from 0.05 to 0.2. It may be noted that the decrease in the semi-major axis and eccentricity

after 500 revolutions for e = 0.05 are 12 1.6 km and 0.0 160, respectively. In this case also,

it is noticed that the present theory provides minimum percentage error. Tables 7.7 and

7.8 provide the same information after 500 revolutions for the orbit with Hp = 175 km

and e = 0.1 with variation of inclination from 5 to 85 degrees. I t may be noted that the

decrease in the semi-major axis and eccentricity after 500 revolutions for i = 40 are 1 99.9

km and 0.0249, respectively. Here also, it is noticed that the KSC theory provides

minimum percentage error.


analytically computed values with respec,t to the present theory, KS theory and St3

theory of semi-major axis and eccentricity upto I000 revolutions, for the orbit with Hp =

200 km, e = 0.1 and i = 55". The differences in SB theory become negative to positive

162 Chapter-7 Orbital Theory with Air Drag: Oblate Diurnally Varying Atmosphere

after 450 revolutions and then gradually increasing up to 4 km after 1000 revalutions. It

is noticed that KSC theory provides lesser differences in both the orbital parameters:

semi-major axis and eccentricity than KS theory. Figures3.3 andT.4 depict the same

information after 500 revolutions, for the orbit with e = 0. I and i = 8S0with variation of

perigee height from 165 to 225 km. It is noticed that the differences in SB theory

becomes positive to negative with respect to higher Hp after 500 revolutions. Here also, it

is noticed that KSC theory provides less difference in both a and e than K S theory.

Figures 7.5 and 7.6 depict the same information after 500 revolutions, for the orbit with

Hp = 200 km and i = 35" with variation of eccentricity from 0.05 to 0.2. Here also, it is

noticed that KSC theory provides lesser differences in both a and e . Figures3.7 and#.&

depict the same information after 500 revolutions, for the orbit with Hp = 175 km and e =

0.1 with variation of incfination from I to 89 degrees. In this case aIso, it is noticed that

KSC theory provides less difference in both a and e . From the above 8 Tables and 8

Figures, we find that the present analytical solution matches quite well with the numerical

solution and is superior to the other two theories over a wide range of perigee height,


7.5 Conclusion

The KS uniformly regular canonical element equations, integrated analytically by a series

expansion method up to third-order terms in eccentricity with air drag force, by assuming

an analytical oblate exponential atmosphere with diurnal variation, provide an accurate

non singular theory for eccentricity up to 0.2. Comparison of the theory with the third

order K S theory and third order Swinerd & Boulton theory, shows the supremacy of the

present solution.

Chapter7 Orbital Theory with Air Drag: Oblate Diurnally Varying Atmosphere

Appendix 7.1

Na = 2FA Ck SO, for k = 0,1,2,3,4;

N s = F A ( C z S 1 +C3 Szf C4S3),

Nb=FA(C?Sl +C3 S2f C4S3),

N7 = FA C4 S1,

Ns = FA(Ci S1+ Cz Sz + C3 Sj + C4 SS),

Ns = FA (2CoS + C 1 S2+C2S3+C3SS)+S I ,

N lo = FA (2CoS2+Cl S I+C I S3+C2S5)+S~,

N 1 I = FA (2CoS3+Cl Sz+C I Ss+ClS I)+&,

N = FA (2CoS5+C1 S3+C~S2+C3S I)+ Sj,

N I 3 = FA (C 1 S ~ + C ~ S ~ + C ~ S Z + C ~ S ~ ) ,

N ,4 = FA (C2S5+C3S3+C4S2),

N I = FA (C3S5+C4S3),

Nit, = FA C4Ss,

N17 = FA C4S4,

N 1 8 ' FA (C3 S4 f C4 S6).

N I L ] = FA (C2S4+C3S6+C4S7),

Nzo FA (C2S6+C3S7+C~S4),

N2 1 = FA (2CoS4+C I S6+C2S7)+S4,

N22 = FA (2CoS6+C 1 Sa+Cl S7)+S6,

N23 = FA (2CnS7+C 1 S ~ + C ~ S ~ ) + S T ,

NZ4 = FA (C S7+C2S6+C3S4),

hZ5 = FA (C2Sy+C3S6+C4S4),

N z ~ = FA ( C ~ S Y + C ~ S ~ ) ,

N27 - FA CdSx,

N,, = FA C32-mSg, for m = 28,29,30,3 I ;

Nj? = 2 FA Co S8 ,

N, = FA CkmJ2 SR, for n = 33,34,35,36.

l',, = 0.0

Chapter-7 Orbial Theory with Air Drag: Oblate Diurnally Varylng Atmosphere

for - P,=2FBCjSSo, forj=1,2,3,4;

P j = - F B C 4 S I ,

P 6 = - FB (C3S1+ C4 S2),

P7=- FB (C2S1 +C3 S2f C4S3),

PI(= - FB (C1 Si +Cz Sz + C3 Sg + CqS.5).

P y = - FB ( CI S2+CZS3+C3SS),

P,o= FB ( C1Si - C1S3- CzSs ),

[Il = FB ( CIS* - Cis5 + ClSi ),

P12" FB ( C I S > f C2S2 + C ~ S I ),

P13 = FB (C1S5+C2S3+C3S2+C4Si),

Pl 4 = FB (CZSS+C~S~+C~SZ),

Pis = FB (C3S5+C4S3),

PI(,= FB C4Ss,

P I 7 = - FB C4S4,

P I X = - FB(C3S4+C4S6) ,

- FB (C2S4+C3S6+C4S7).

Pz0 = - FB (C2S6+C3S7+C S4),

Pzl = - FB ( C1Sb+C2S7),

P22 = FB ( C Sq - C I S7),

= FB ( C Sh+ClS4),

P24 ' FB (C1 S7+C2S6+C3S4),

P ~ s = FB ( C Z S ~ + C ~ S ~ + C ~ S ~ ) ,

P2b' FB (C3S7+C4S6),

P2-, = FB C4S7,

P,,, = - FB C32.,Ss, for m = 28,29,30,31;

Pj2 = 0.0

P,, = FB Cn.32Sg, for n = 33,34,35,36.

Chapter-7 Orbital Theory with Air Drag: Oblate Diurnally Varying Atmosphere

Appendix 7.2

Chapter-7 Orbital Theory with Alr Drag: Oblate Diurnally Varying Atmosphere


Chapter-7 Orbital Theory with Air Drag: Oblate ~iurnally Varying ALmosphere


Appendix 7.3


Chapter-7 Ohitat Theory with Air Drag: Oblate Diurnally Varying Atmosphere

Earth's S -



Orb-

Appendix 7.4

Chapter-7 Orbital Theory with Air Drag: Oblate Diurnally Varying Atmosphen

for ff-Sa#dl,W Or&!tu&b KS Uniform Rsqylar -


Table 7.1 Decrease in semi-major arb upto 1000 revolutions for

Hp = 200 KP, e =0.1 and i = 55'

Table 7.2 Decrease in eccentricity upto 1000 revolutions for

Hp = 200 km, e =0,1 and i = 55'

HP

(km)

I

I0

2 5

50

100

250

500

750

1000

Chapter-7 Orbital Theory with Air Drag: Oblate murnally Varying Atmosphere

HP

(km)

1

10

25

Decrease in semi-major axis (km) - - ---

Percentage error

NUM KSC

~ierease in eccentricity x lo3

KS

0.1725

1.726

4.3 19

8.666

1 7.476

442985

95.248

15 1.708

213.738

SB

-0.299 .- -

-0.299 -- -

KSC

0.143

0.144

-- 0.1719

- - . 1.720

NUM

0.02074

0.2076

0.5195

1.043

1 1.585

18.578

-

Percentage error

SB

0.1 724

I .725

4.3 16

8.660

17.461

44.91 1

94.776

149.970

209.041

KS --

-0.370 - - -

-0.370 -

- 0.1716

-- -- . - 1.717

- --

KSC

0.120

0.120

0.1 17

0.1 12

0.102

0.066

-0.002

-0.083

-0.177

0.140

0.1 35

0.123

0.086

0.01 7

-0.064

-0.1 58

KSC

0.02072

0.2073

0.5189

1.042

2.104

5.435

1 1.585

18.593

26.41 4 -- --

-0.363 -0.340

-0.364 -0.38 1

-0.365 -0.464

-0.367 -0.429

-0.368

-0.367 0.643 -

4.303 -

8.634

17.412 -- - 44.8 I9

94.897

151.152

212.962

KS

-0.361

-0.361

-0.364

4.297

8.622 " --

17.390 .- -- -

44.780

94.88 1

15 1.249

21 3.299

-0.3 70

KS

0.02082

0.2083

0.5214

1.047

2.1 13

5.458

1 I .628

18.646

26.464

SB

-0.299

-0.299

-0.3 18

-0.299 -

SB

0.02080

0.2082

0.52 12

1.047

2.1 14

5.464

1 I .635

18.595

26.1 98 "-

-0.370

-0.370 -0.286

-0.370

-0.368

Table 7.3 Decrease in semi-major axis after 500 revolutions for e = 0.1 and i = 85"

Table 7.4 Decrease in eccentricity after 500 revolutions for e = 0.1 and i = 85'

HP

(km)

165

175

190

Chapter-7 Orbital Theory with Air Drag: Oblate DTurnally Varying Atmosphere

98.480

75.14 1 75.157 75.269 75 220 -*

5 1.55 I 5 I .609 5 1.685 0.020 -0.093 -0.239 -.

HP

(krn)

165

175

190

200 -

210

225


NUM

315.935

214.702

13 1.662

Percentage error

KSC

0.026

-0.103

-0.084

Decrease in eccentricity a lo3

KSC

31 5.853

214.923

13 1.772

NUM

40.135

26.832

16.203

.-

Percentage error

KS

-0.223

-0.309

-0.27 1

12.045 -

9.383

6.221

KSC

-0.03 1

-0.162

-0.1 37

KS

3 16.630

215.365

132.01 9

SB

5.855

2.244

0.523

KSC

40.147

26.875

16.225

SB

297.438

209.883

130.973

12.057

9.087

6.222

KS

-0.1 86

-0.283

-0.248

KS

40.2 10

26.907

16.243 ---

SB

4.349 - .

1.047 - - -0.232

SB

38.390

26.545

16.240

12.068 -

9.142

6.225

12.101

9.082

6.254

-0.069

-0.023

-0.149

-0.073

- -0.535

-0.530

Table 7.5 Decrease in semi-major axis after 500 revolutions for Hp = 200 km and i = 35"

Table 7.6 Decrease in eccentricity after 500 revolutions for Hp = 200 km and i = 35'

e

0.05

0.075

0.1

0.1 25

0.15

0.175

0.2

- - - - - -. . r - i i )Dei ieGii i i i i i t=i 1 o3 Percentage error --l

Chapter7 Orbital Theory with Air Drag: Oblate Diurnally Varying Atmosphere


(km)

0.05 - 0.075

0.1

0.125

0.15

0.175

0.2

NUM

121.643

1 00.702

93.182

90.404

90.127

91.493

94.114

Percentage error

KSC

0.132

0.1 42

0.143

0.14 1

0.1 36

0.13 1

0.126

KSC

121.482

1 00.559

93.049

90.276- '

90.005

91.373

93.995

NUM

16.045

KS

-0.219

-0.259

-0.309

-0.362

0.415 -

-0.466

-0.5 14

KSC -

0.156

0.140

0.1 57

0.151

0.143

0. I35

0.128

K S

121.910

1 00.963

93.470

90.73 1

90.50 1

91.919

95.834

SB

0.564

0.33 1

0.244

0.207

0.1 96

0.2 1 1

0.247

-- KSC

16.020

SB

120.957

1 00.368

92.955

90.2 17

89.950

91 -300

93.88 1

KS

-0.23 5

-0.273

-0.3 17

-0.360

-0.399

-0.432 '

-0.458

KS

16.083

1 SB

-0. I67 - ---

-0.283

-0.3 13

-0.3 16

-0.305

-0.276

-0.23 I

12.862

11.367

10.476

9.887

9.464

9.184

SB

16.072

12.897

1 1.403

10.513

9.927

9.528

9.142

12.842

I 1.349

1 0.460

9.873

9.465

9.172

12.899

1 1.402

10.05 1

9.91 8

9.504

9.205

Table 7.7 Decrease in semi-major axis after 500 revolutions for Hp = 175 km and e = 0.1

Table 7.8 Decrease in eccentricity after 500 revolutions for Hp = I75 krn and e =0.1

i

(deg)

5

15

25

40

55

70

85


HiJ

(km)

5

15

25

40

55

70

85


NUM

195.466

196.211

197.3 12

199.85 1

203.692

208.779

2 14.702

Percentage error

KSC

0.068

0,053

-0.033

-0.128

-0.1 70

-0.146

-0.1 02

7- -- Eccentricity x 10

KSC

195.332

196.153

197.377

200.107

204.038

209.083

2 14.923

NUM

24.334

24.432

24.578

24.912

25.412

26.070

26.832

- -- .-

Percentage error

KS

-0.123

-0.145

-0.247

-0.365

0.432

-0.406

-0.308

KSC

0.115

0.097

-0.009

-0.132

-0.200

-0.1 95

-0.1 62

KS

195.705

196.549

197.799

200.580

204.57 1

209.627

2 15.365

SB

2.140

2.126

2.050

1.969

1.969

2.072

2.244

KSC

24.307

24.430

24.580

24.945

25.463

26.12 1

26.875

SB

191.283

192.077

193.268

195.91 6

199.682

204.453

209.883

KS

-0.129

-0.152

-0.254

-0.368

-0.426

-0.389

-0.283

KS

24.366 -

24.476

24.640

25.004

25.520

26.171

26.907

- SB

1.005 - -. -

0.99 1

0.9 14

0.832

0.825

0.91 6

1.067

SB

24.090 - .

24.195

24.353

24.705

25.203

25.83 1

26.545

0 100 200 300 400 500 600 700 800 900 1000 Revolt~tio~m Number

Figure 7.1 Difference between numerical and analytical values of semi-major axis up to 1000 revolutions.

Revalut io~~ Number

Figure 7.2 Difference between numerical and analytical values of ecentricity up to 1000 revolutions

Chapter 7 Orbital Theory with Air Drag: Oblate Diurnally Varying Atmosphere

Figure 7.3 Difference between numerical and analytical values of semi-major axis with respect to perigee height after 500 revolutions.

l .Z 1 I I I I

Illtical Perigee height (km)

Figure 7.4 Difference between numerical and analytical values of eccentricity with respect to perigee height after 500 revolutions.

Chapter-7 Orbital Theory with Air Drag: Oblate Diurnal1 y Varying Atmasphere

Figure 7.5 Difference between numerical and analytical values of semi-major axis with respect to eccentricity after 500 revolutions.

Figure 7.6 Difference between numerical and analytical values of eccentricity with respect to eccentricity after 500 revolutions.

Chapter7 Orbital mewy with Air Drag: Oblate Diurnally Varylng Atmosphere

Figure 7.7 Difference behveen numerical and aoa?ytical values of semi-major axis with respect to inclination after 500 revolutions.

Figure 7.8 Difference behveen numerical and analytical values of eccentricity with respect to inclination after 500 revolutions.

Chapter-7 Orbltal Theory with Air Drag: Oblate Diurnally Varying Atmosphere

BIBLIOGRAPHY

1. David A.Vallado, 2001 , "Fundamentals of Astrodynamics and Applications, Second edition, Space technology library Microcosm Press, California.

2. Roy, A.E., 1988, "Orbital Motion", John Wiley & Sons, New York.

3 . Battin, R.H., 1987, "An Introduction to Mathematics and Methods of Astrodynamics". AlAA Education Series, New York.

4. Newton, I . , 1687, "Philosophiae Naturalis Principia Mathematics", Koy. Soc.. London.

5 . Singer. S.F., 1956, "Studies of a Minimum Orbital Unmanned Satellite of the Earth (MOUSE), Part I]", Astronautics Acta, 2, 125- 144.

6. Henry, I.G., 1957, "Lifetimes of Artificial SateIlites of the Earth", Jet Propulsion. 27.2 1 -24.

7. Davis, R.J., Whipple, F.H., and Zirker, J.B., 1956, "Scientific Uses of Earth Satellites", Chapman and Hall, London, 1 -22.

8. Kozai, G., 1959, "The Motion of a Close-Earth Satellite", Astron. .I., 54. 367-377.

9 Brouwer,D., 1959, "Solution of the Problem of Artificial Sateffite Theory Without Drag", Astron. .I., 64. 378-397.

10. Brouwer, D., and Hori,G., 196 I , "Theoretical Evaluation of Atmospheric Drag Effects on the Motion of an Artificial Satellite", The Astron. J. , 66, No.5, 193- 225.

I I . Hansen, P.A., 1 855, "Expansion of the Power of the Radius Vector with the Sine or Cosinus of a Multiple of the True Anomaly in Terms of Series", AbhandIungen der Koniglichen Sachsischen Gesellshaft fur Wissenschaft, 2(3), 1 83-281.

1 2. Barker, W.N ., Casali,S.J., and Wallner,R.N., 1 995, "The Accuracy of General Peturbations and Semianalytic Satellite Ephemeris Theories", AAS-95-432.

13. Groves, G.V., 1958, "Effects of the Earths Equatorial Bulge on the Lifetimes 01. Artificial Satellites and its Use in determining Atmospheric Scale-Heights", Nature Lond., 181, 1055.

14. Steme, T.E., 1958, "Formula for Inferring Atmospheric Density from the Motion of Artificial Earth Satellites", Science, 127, 1245.

15. King-Hele, D.G., 1959, "Density of the Atmosphere at Heights between 200 km and 400 km, from Analysis of Artificial Satellite Orbits", Nature Lond., 183, 1 224- 1227.

1 6 . Nonwelier, T.R. F . , 1 958, "Perturbations of Elliptic Orbits bt Atmospheric Contact", J. Brit, Interplan, ~ O C . , 16, 368-379.

17. King-Hele, r1.G.. and Leslie, D.C.M., 1958, "Effect of Air Drag on the Orbit of the Russian Earth Satellite 1957 a: Comparison of Theory and Observation", Nature Lond., 181. 1761 -1 763.

18. King-Hele, D.G.. 1964, "Theory of Satellite Orbits in on Atmosphere", Butterworths, I .ondon.

19. King-Hele, D.G., 1987, "Satellite Orbits in an Atmosphere; Theory and Applications", Blackie, G tasgow and London.

20. Perkins, F.M., "An Analytical Solution for Flight Time of Satellites in Eccentric and Circular Orbits", Astronautica Acta, 4, Fasc. 2, 1958, p.1 13.

21. Parkyn, D.G., 1958, "Elliptic Orbits in a Frictional Atmosphere", Amer. J . Phys. 26,436-440.

22. Brouwer, D., and Clemence, G.M., 196 1 , "Methods of Celestial Mechanics". Academic Press, New York, NY.

23. Watson, C.N., 1944, '.A treatise on the Theory on Bessel Functions". 2"" edn. University Press, Cambridge.

24. Bosanquet, C.H., Nov. 1958, "Change of Inclination of a Satellite Orbit", Nature. 182, No.4648, p. 1533.

25. Vinti, J.P., 1959, "Theory of the Effect of Drag on the Orbital Inclination of an Earth Satellite", J , Res. Nat. Bur. Stand., 62 pp. 79-88.

26. Merson, R.H.. King-Hele, D.G., and Plimmer, R.N.A., Jan.1959, "Changes in the Inclination of Satellite Orbits to the Equator", Nature, 183, No.4656,239-240.

27. Michelson, H.F. 1959, "Orbit Decay and Prediction of the Motion of Artificial Satellites", Advances in Astronautical Sciences, 4.

28. Parkyn, U.G., Jan. t960, Satellite "Orbits in an Oblate Atmosphere", J. Geophys. Res., 65,9- 17.

29. De Nike, J . , 1956, "The Effect of the Earth's Oblateness and Atmosphere in a Satellite Orbit", J. of the Franklin Institute, Monograph No,2, pp.79-88.

30. Von Zeiphel, H., 191 6 a, "Recherches sur le Mouvernent des Petites Planets". Arkiv.Mat.Astron., Fys.11, No. 1.

3 I . Von Zeiphel, EI., 19 16 b, "Recherches sur le Mouvement des Petites Planetes", Arkiv.Mat.Astron., Fys. 11, No.7.

32. Von Zeiphel, H.. 1917, "Recherches sur le Mouvement des Petites Planetes". Arkiv.Mat.Astron., Fys.12, No.9.

33. Cole, J.D., and Kevorkian. I., 1963, "Uniformly Valid Approximations for Certain Nonlinear Differentiat Equations", Proceedings of the 1961 Internationat Symposium on Nonlinear Diff. Eqn. and Nonlinear Mechanics, J.P. La Salle and S. Lefschetz, eds.. Academic Press, New York, NY., 1 13-120.

34. 'Kevorkian, J . , 1 962, "The Variable Expansion Procedure for the Approximate Solution o f Certain Nonlinear Differential Equations", Rept. SM-42620, Douglas Aircraft Co.

35. Hori, G., 1966, ""Theory of General Perturbation with Unspecified Canonical Variables", Publ. Astron. Soc. Japan, 18,287.

36. Deprit, A., 1 969. "Canonical Transformations Depending on a Small Parameter". Celestial Mechanics. 12

37. Kyner, W.T., 1966, "Averaging Methods in Celestial Mechanics", The theory of Orbits in the Solar System and in Stellar Systems, Edited by G. Contopoulous. Academic Press, London and New York.

38. Morrison, J.A., 1966, "Generalized Method of Averaging and the von Zeipel Method", Methods in Astrodynamics and Celestial Mechanics, Edited by R.L.Durncombe and V.G. Szebehely, Academic Press, New York and London.

39. Morrison, J.A., 1966, "Comparison of the Method of Averaging and the Two- Variable Expansion Procedure", SIAM Review, 8, No.1

40. Hori, G.1, I 97 1 , "Theory of General Perturbation for Non-Canonical System". Publ.Atron.Soc., Japan, 23, 567.

4 I . Karnel, A.A., 4 ( 1 97 1 ), "Lie Transforms and the Hamiltonization of Non Hamiltonian System". Celestial Mechanics, p.397.

42. Choi, J.S., and Tapiey, B.D., 7 (1973), "An Extended Canonical Perturbation Method", Celestial Mechanics, 77-90.

43. Shinad, H., 2(1970), The Equivalence of von Ziepel Mappings and Lie Transforms", Celestial Mechanics, 1 1 4- 120.

44. Watanabe, N., 1974, "Equivalence of the Method of Averaging and the l,ie Transform", Sci.Rep. 1, Vol.LVI I, 1 1-22.

45. Ahmed, A.H., and Tapley, B .D., 1982, "Equivalence of the Generalized Lie-Hori Method and the Method of Averaging", AIAA paper 82-7 1 , AlAA 2oth Aerospace Science Meeting, Orlando, Florida.

46. Lane, M.H., 1965, "The Development of an Artificial Satellite Theory Using a Power-Law Atmospheric Density Representation", A1 A A paper 65-35, AIAA 2"d Aerospace Sciences Meeting, New York, Jan 25-27.

47. Lane, M.H., and Cranford, K. H ., 1 969, "An Improved Analytical Drag Theory for the Artificial Satellite Problem", AIAA paper 69-225, AIAAIAAS Astrodynamics Conference, Princetown NJ.

48. Zee, C.H., 3 (19711, "Trajectories of Satellites under the Combined Influences of Earth Oblateness and Air Drag", Celestial Mechanics, 148- 168.

49. Barry, B.F., and Rowe, C.K., 1972, "Long-Term Orbit Prediction Using Two- Variable as Asymptotic Expansions and the Automated Manipulation Capabilities of the FORMAC Language", AIAA paper 72-938, AIAAIAAS Astrodynamics Conference, Palo Alto, CA.

50. Willey, R.,E., and Pisacane, V.L., March-June 1974, "The Motion of an Artificial Satellite in a Nonspherical Gravitational Field and an Atmosphere with a Quadratic Scale Height", J. Astron. Sciences. Vol.XX l I , No.546,230-243.

5 1 . Chen, S.C.H., May 1974, "Ephemeris Generation for Earth Satellites Considering Earh Oblateness and Atmospheric Drag", Northrop Services, Int., Huntzville AL, M-240-1239.

52. Watson, J.S., Mistretta, G.D., and Bonavito, N.L., 1 1 (1974), "An Analytic Method to Account for Drag in the Vinti Satellite Theory", Celestial Mechanics. 1 45- 147.

53. Santora, F.A., 1974, AlAA paper 74-839, "Satellite Drag Perturbations in an Oblate Atmosphere with Day-to-Night Density Variation", AIAA Mechanics and Control o f Flight Conference, Anaheim CA.

54. Muller, A., Scheifele, G., and Starke, S., 1979, "An Analytical Orbit Prediction for Near-Earth Satellite", A1 AA papper 79-1 22, 1 7" Aerospace Science Meeting, New Orieans LA.

55 . I-ioots, F.R., 23 (1981), "Theory of the Motion of an Artificial Earth Satellite", Celestial Mechanics, 307-363.

56. Hoots, F.R., 1982, AlAA paper 82-1409, "An Analytical Satellite Theory Using Gravity and a Dynamic Atmosphere", AIAAlAAS Astrodynamics Conference, San Diego CA.

57. Velhena de Morals, R., 25 ( 1 981 ). "Combined Solar Radiation Pressure and Drag Effects on the Orbits of Artificial Satelljtes", Celestial Mechanics, p28 1-293.

58. Lane, M.H., Filtzpatric, P.M., and Murphy, J.J, March 1962, ''On the Representation of Air Density in Satellite Deceleration Equations by Power Functions with lntegral Exponents", Project SPACETRACK, Air Proving Ground Center, Eglin AFB Fh, Tech. Doc. Rept.No. APGC-TDR-62-15.

59. Lyddane, R.H., "Small Eccentricities or Inclinations in the Brouwer Theory of the Artificial Satellite", Satellite", Astron, J, Vo1.68, p.555.

GO. Davenport, P.,1965, "Modification of Brouwer's Solution for the Artificial Satellites to Include Small Eccentricities and Inclinations", Document X-543-65- 23, Goddard Space Flight Center, Greenbelt MD.

61. Lane, M.H., and Hoots, F.R., 1979, "General perturbations Theories Derived From the 1965 Lane Drag Theory", Project SPACETRACK Report No.2, Aerospace Defense Command, Peterson AFB CO.

62. Bogoliubov, N.N., and Mitropolsky, Y.A., 1961, "Asymptotic Method in the Theory of Nonlinear Oscilations", New York, Gorden and Beach.

63. Jacchia, L.G., 1970, "New Static Models of the Thermosphere and Exosphere with Empirical Temperature Profile", SAO SP-3 13.

64. Lorell, J., and Liu, A., 1971, "Methods of Averages Expansions for Artif cia1 Satellites Applications", NASA CR 1 18267.

65. Liu, J.J.F., and Alford, R.L., 1973, "Transformations and Variations Equations for an Artificial Satellite about an Oblate Earth", TN-240-1 199, Northrop Services, inc.. Huntsville, Alabama.

66. Liu, J.1 .F., Jan 1974, "A Second-Order Theory of an Artificial Satellite Under the influence of the Oblateness of the Earth", Proceedings of the twelfth AIAA Aerospace Sciences Meeting, Washington, D.C., Paper No.74- I I 6.

67. Slutsky, M., and Clain, W.D., Aug.1982, "The First-Order Periodic Motion of an Artificial Satellite due to Third-Body Perturbations", Proceedings of the AAS/A IAA Astrodynamics Specialists Conference, Lake Tahoe, Nevada, Paper N0.81-106.

68. Willey, R.E., and Pisacane, V.L., Jan 1974, "The Motion of an Artificial Satellite in a Non-spherical Gravitational Field and an Atmosphere with a Quadratic Scale Heighty, Appl. Phy. Lab., John Hopkins Univ. Rept. TG- 1236.

69. Jaccia, L.G., 197 1 , "Revised Static Models of the Thermosphere and Exosphere with Empirical Temperature Profiles", SAO SP-332.

70. Chen, S.C.H., July 1972, ''The Motion of an Artificial Satellite around an Oblate Ptanet", Northrop Services, inc., Huntsville, Alabama, TR-242- 1 1 14.

71. Sherril, J.J., 1966, "Development of a Satellite Drag Theory Based on the Vinti Formulation", Ph.D dissertation, University of California, Berkley.

72. Vinti, J.P., 1969, "Improvement of the Spheroidal Method for Artificial Satellite", , Astron. J. 74, No. I, 25-34.

73. Walden, H., and Watson, J.S., 1967, "Differential Corrections Applied to Vinti's Accurate Reference Orbit with inclination of the Third Zonal Harmonics", NASA TN D-4088.

74. Ferraz Mello, S., 1963, "Action de la Pression de Radiation sur le Mouvement d'un Satellite Artificial de la re", Proc. 1 4 ~ Intern. Astonaut, Congress, Paris, Vol. IV, PWN, Warsaw.

75. Ferraz Mello, S., 25 (19811, "Elimination of Secular Terms Generated by the Coupling of Perturbations", Celestial Mechanics, 293-296.

76. Liu, J.J.F. and Alford, R.L., 1980, "Semi -analytic Theory for a close-Earth Artificial Satellite", .I. Guidance and Control,Vot.3, No.S, 304-3 1 1.

77. Jachhia, L.G., 1964, "Static Diffusion Models of the Upper Atmosphere with Empirical Temperature Profiles", SAO SP- t 70.

78. Jachhia, L.G., 1977, "Thetmospheric Temperature, Density and Composition: New Models", SAO SP-375.

for Orbik with KS U- - 79. Hedin, A.E., et al .June 1977, "A Global Thermospheric Model Based on Mass

Spectometer and incoherent Scatter Data, MSIS 1 , N2 Density Representation", J . Geophysics Research, 82, No. 16.

80. Barlier, F.C., Berger, J.L., Falin, et al., 1978, "A Thermospheric Model Based on Satellite Drag Data", Ann. Geophys., 34, 9-24.

81. Hedin, A.E.. 1987, "MSIS-86 Thermospheric Model", J . Geophys. Res., 921A5) 4649-4662.

82. Hedin, A.E., 1991, "Extension of the MSIS Thermosphere Model into the Middle and Lower Atmosphere", J, Geophys.Res., 96(A2), 1 159-1 172.

83. De Lafontaine. J., and Hughes, P.C., 1983,"An Analysis Version of Jacchia's 1977 Model Atmosphere", Celestial Mechanics, 29, 3-26

84. De Lafontaine, J., 1986, "The Derivation of a Second-Order Satellite Trajectory Predictor in Terms of Geometrical Varibles", Acta Astronautics, 13, No.8, 481- 489.

85. De Lafontaine, J . , and Mammen, R., 1986, "Orbit Lifetimes Prediction and Safety Considerations", Space Safety and Rescue 1984-85, AAS Science and Technology Series, 64.

86. Liu, I.J.F., France, R.G., and Wackernagel, H.B., 1983, "An Analysis o f the Use of Empirical Atmospheric Density Models in Orbital Mechanics", SPACETRACK Report No.4, Space Command, Peterson Air Force Base, Colarado.

87. Pimm, R.S., Aug. 17-1 9, 197 1 , "Long-Term Orbital Trajectory Determination by Super-Position of Gravity and Drag Perturbations", AAS Paper 7 1 -376, AASlAlAA Astrodynamics Specialist Conference, Ft Lauderdale FL.

88. Kaufman, B., and Dasenbrock, R., 198 1 , "Long 'Term Stability of Earth and Lunar Orbiters: Theory and Analysis", Acta Asronautica, 8. No.5-6, 6 1 1-623.

89. Barry, B.F., Pimrn, R.S., and Rowe, C.K., 197 1. "Techniques of Orbital Decay and Long-Term Ephemeris Prediction for Satellites in Earth Orbit", NASA CR 121053.

90. Dallas, S.S., and Khan, I . , 1976, "The Singly Averaged DitTerential Equations o f Satellite Motion for O<e<l", A i AA paper 76-830. AIAAlAAS Astrodynamics Conference, San Iliego CA.

for -- 91. Wu, D.D., Wang, C.B., and Tong, F., 1978, "A New Semi-analytical and Semi

numerical Method for Computation of the Second-Order Perturbation of Artificial Satellites", Acta Astronautics, Sinica, 19, No.2, I3 1-1 5 1 .

92. Lidov, M.L., and Solov'e V, A.A., 1979, "Atmospheric Perturbations on the Motion of a Satellite in a High-ementricity Orbit", Cosmic Research, 16, No.16, 640-65 1.

93. Green, A.J., and Cefola., June 25-27, 1979, "Fourier,-Series Formulation of the Short Periodic Variations in Terms of Equinoctical Variables", AAS paper 79- 1 33, AASlAIAA Astodynamics Conference, Prov incetown MA.

94. t iu, J.J.F., and Alford, R.L., 1973, "Semi-analytic Theory for a Close-Earth Artificial Satellite", TN-240- 1 199, Northrape Services. Inc., Huntsville, Alabama.

95. Arsenault, J .K. , Enright, J.D., and Purcell, C.. 1 964. "General Perturbations 'Techniques for Satellite Orbit Prediction Study", 1'ech.Doc.Report.No.AL-TDR- 64-70. Vols. 1 and 2.

96. CIRA 1 972, "COSPAR IntemationaI Reference Atmosphere", Akademic - Verlag..

'37. Solov'ev, A.A., 1974, Institute of Applied Mathematics, Academy of Sciences of the USSR, preprint 86-87.

98. Cefola, P.J., Long, A.C., and Holloway, G.Jr., 1974, "The Long-term Prediction of Artificial Satellite Orbits", AIAA paper No.74-170, AIAA 12''' Aerospace Science Meeting, Wash. DC.

99. l,utsky, D., and Uphoff, C., Jan. 1975, "Short-Period ic Variations and Second- Order Numerical Averaging", AIAA paper No.75-11, AIAA 13' Aerospace Science Meeting, Pasadena CA.

100. Liu, J.J.F., Nov.1974, "Satellite Motion about an Oblate Earth", AIAA 5.12, NO.] I , 151 1-1516.

101. Liu, J.J.F., April 1975, "Orbital Decay and Lifetime Estimation7','I'N-240-1441. Northrop Services,Inc., Huntsville AL.

102. Arsenault, J.L., "Ford, K.C., and Koskela, P.E., 1970, "Orbit Determination Using Analytic Partial derivatives of a Perturbed Motion ". AIAA J., 8, No. I , 4- 12.

103. Roth, E.A., 1973, "Fast Computation of High Eccentricity Orbits by the Stroboscopic Method", Celestial Mechanics, 1.8, 245-249.

104. Roth, E.A., Oct. 1984, "The Gaussian Form of the Variation-of-the Parameter Equations Formulated in Equinoctial Elements", Proceedings of the 35th Congress of the International Astronautical Federation, Lusanne, Switzerland, Paper No. IAF 84-336.

105. Roth, E.A., 1985, "Application of the Stroboscopic Method to Equinoctial Elements", Acta Astronautica, 12, No.2,71-80.

106. Roth. E.A., 1985, "Perturbations by Air Drag Based on a Power Law", Proceedings o f the 3dh Congress of the International Federation, Stockholm, Sweden, Paper No. 85-250.

107. Stusky, M., and MC Clain, W.D., Aug.1981, "The F'irst-Order Short -Periodic Motion of an Artificial Satellite due to Third-body Perturbations". Proceedings of the AASIAIAA Astrodynamics Specialists Conference, Lake Tahoe, Nevada, paper No. 8 1.106.

108. Wagner, C.A.,Douglas, B.C., and Williamson, R.G., 1974, "The ROAD Programme", NASA TM-X-70676.

lU9. Alfbrd, R.L., and Liu, J.J.F., 1974, "The Orbital Decay and lifetime (LIFTIM) Prediction Program", M-240- 1278, Northrop Services, Inc., Huntsville, Alabama.

110. Lyddane, R.H., and Cohen, C.J., 1962, "Numerical Comparison Between Brouwer's theory and Solution by Cowell's Method for the Orbit of an Artificial Satellite", Astron. J. 67, 1 76-1 77.

1 1 1. Breakwell, J.V., and Vagners, J. 1979, "On Error Bounds and Initialization in Satellite Orbit Theories", Celestial Mechanics, 235-264.

I f 2. Cook, G.E., 1960, "The Dynamic Drag of Near-Earth Satellites", Ministry of Aviation, Aeron, Res.Counci1 Current Papers, C.P. #523.

1 13. Cook. G.E., 1965,"Satellite Drag Certificates", Planet. Space Sci., 13. 929-946.

1 14. Nocilla, S., 1972, "Theoretical Determination of the Aerodynamic Forces on Satellites", Astronautica Acta. 17, p.245.

1 15. Jastro, R., Pearse, C.A., 1957, Atmospheric Drag on the Satellite", J. Geophys. Res.,62,No.3,p.413.

for Orbits - 1 16. Williams, R.R., Aug. 1972, "Atmospheric Drag Cross Section of Spacecraft

Solar arrays", J. SpacecraR, 9, No.8 p. 565.

I 1 7. Williams, R.R. "Drag Coefficients for Astronautical Observatory Satellites", J .Spacecraft, 12, No.2, 74-78.

118. Hunziker R.R., 1970, Effects of the Variation o f Drag Coefficient on the Ephemeris of Earth Satellites". Astronautics Acta, IS, p. 16 1 .

119. Ladner, J.E., Kagsdale, G.C., Jan. 1964, "Earth Orbital Satellite I,ifetimeM, NASA TN D-1995.

120. Wachman, H.Y ., Jan. 1962, "The Thermal Accomrnodatinn Coefficient : A Critical Survey", ARS J., 32, p.2

I 2 1 . Szebehcl y, V., 1978, "Celestial Mechanics during the Last Two Decades", AIAA paper 78-8, ALA A 16" Aerospace Science Meetings, Hunstville, Alabama.

t 22. Escobal, P.R., 1965, "Methods of Orbit determination". New York, Wiley.

123. Escobal, P.R., 1968, "Methods of Astrodynamics", New York, Wiley.

1 24. Sterne, 'V.E., 1960, "An introduction to Celestial Mechanics", Ch.4, Interscience, New York.

125. kiwart, D.G., 1962, "The Effect of Atmospheric Drag on the Orbit of a Spherical Earth Satellite", J.Bri t. Interplan, Soc., 18, 269-272.

126. Davies, M.J., 1963, "The Dynamics of Satellites", Springer, Berlin, 1 1 1 - 1 22.

127. Santora, F.A., 1975, "Satellite Drag Perturbations in an Oblate Diurnal Atmosphere", AIAA J . , 13, 1212-1216.

128. Santora, P.A., 1976, "Drag Perturbations of Near-Circular Orbits in an Oblate Diurnal Atmosphere", AIAA J. , 14, 1 196-1 200.

129. Eliasberg, P.E., Kugaenko, B.V., 1976, "Effects of Upper Atmospheric Density Variations on Artificial Earth Satellites Orbits", Space Res. XVL, Academic Veralg, Berlin.

130. Swinerd, G.G., and Boulton , W.J., 1982, "Contraction of Satellite Orbits in an Oblate Atmosphere with a Diurnal Density Variation", Proc. Roy. Soc.. A 383. 127-145.

Predi- for

13 1 . King-Hele, D.G., 1978, "Methods for Predicting Satellite Orbital Lifetimes", J . of the Brit. interpl. Sac., 31, 181-196.

132. Cook, G. E., and Plimmer, R.N.A., 1960, "The Effects of Atmospheric Rotation on the Orbital Plane of a Near-Earth", Proc. Roy. Soc. A, 258,5 1 6-528.

133. Cook, G.E.. 196 1 a, "Effect of an Oblate Rotating Atmosphere on the Orientation of a Satellite Orbit". Proc. Roy. Soc., A 261,246-258.

134. King-Hele, D.G., and Scott, D.W., 1969, "The Effect of Atmospheric Rotation on a Satellite Orbit, when Scale Height Varies with Height", Planet. Space Sci., 17, 2 17- 232.

135. Cook. G.E., and King-Hele, D.G., 1965, "The Contraction of Satellite Orbits under the Influence of Air Drag. V. with Day-to-Night Variation in Air Density", Phil. Trans. R. Soc. Land. A 259.33-67.

136. Cook, G.E., and King-Hele, D.G., 1968, "The Contraction of Satellite Orbits under the Influence of Air Drag. VI. Near Circular Orbits with Day-to-Night Variation in Air, Density", Proc.Roy. Soc., Lond. A 389, 153- 170.

137. Swinerd, G.G., and Boulton, W.J., 1983, "Near -Circular Orbits in an Oblate Diurnatly Varying Atmosphere", Proc.Roy. Soc., A 386,55-75.

138. Boufton, W.J., 1983, "The Advance of the Perigee of a Satellite in a Ratating Oblate Atmosphere with a Diurnal Variation on Density". Proc.Roy. Soc.. A 389, 433-444.

139. King-Hele, D.G., 1962, "The Contraction of Satellite Orbits under the lniluence of Air Drag. Ill, High Eccentricity Orbits (0.2<e < I ) " . Proc.Roy. Soc.. A 267, 531-557.

140. King-Hele, D.G, 1966, Proc.Roy. Soc., A 294,261-272.

4 King-Hele.D.G.,and Walker, D.M.C., 1976, Proc.Roy. Soc.,A350,281-298.

142. King-Hele, D.G., and Walker, D.M.C., 1987c, An Improved 'Theory for Determining Changes in Satellite Orbits Caused by Meridional Winds, Proc. Roy. Soc., A 41 1. 19-33.

143. de Latbntaine, J., and Garg, S.C., Sept. 1979, "A Review of Satellite Lifetime and Orbit Decay Prediction", U?'IAS Review No.43.

1 44. Stiefel, E.L., and Scheifele, G., 1 97 1, "Linear and Regular Celestial Mechanics", Springer-Veralg, Berlin, Heidelbelg, New York.

145. Merson, R.H.., 1 975, "Numerical Integration of the Differential Equations of Celestial Mechanics", RAE-TR-74184.

146. Graf, Jr.0.F.. Mueller, A., and Starke, S., 1977, "The Method of Averages Applied to Differential Equations", NASA-CR- 15 1607.

147. Shama, R.K., and Mani L., 1985, Indian J. Pure Appl. Math., 16, 833-842.

148. Sharma. R . K . , 1989, "Analytical approach using KS Elements to Short-'l'erm Orbit Predictions including J?", Celestial Mechanics and Dyn. Astron., 46. 321- 333.

149. Sharma, R.K., 1 993, "Analytical Short-Term Orbit Predictions with Jj and .I4 in terms of KS Elements". Celestial Mechanics and Dyn. Astron., 56, 503.

150. Sharma, R.K. , and Xavier James Raj, M., 1997, "Analytical Short-'r'erm Orbit Predictions with J2 to .I6 in terms of KS Elements", Indian J . Pure Appl. Math., 28 (9), 1155-1 175.

15 1. Sharma, R.K., 1997, "Analytical Integration of K-S Elements Equations with 32

for Short-Term Orbit Predictions", Planetary and Space Science, 45, No. I 1 , 148 1 - 1486.

152. Sharma, R.K. . I99 1 , "Analytical Approach using KS Elements to Near-Earth Orbit Predictions Including Drag", Proc. Roy. Soc., Lond.,A 433, 12 1 - 130.

153. Sharma, K.K., 1992, "A Third - Order Theory for the EfTect of Drag on Earth Satellite Orbits". Proc. Roy. Soc., Lond., A 438,467-475.

154. Sharma. R.K. , 1997, "Contraction of Satellite Orbits using KS Elements in an Oblate Diurnally Varying Atmosphere", Proc. Roy. Soc., Lond., A 453, 2353- 2368.

155. Sharma, R.K. , 1998, "Contraction of High Eccentricity Satellite Orbits using KS Elements with Air Drag", Proc. Roy. Soc., Lond., A 454, 168 1 - 1689.

156. Sharma, K.K. , 1999, "Contraction of High Eccentricity Satellite Orbits using KS Elements in an Oblate Atmosphere", Advances in Space Resarch., 23,693-698.

157. Sharma, R.K. and Xavier James Raj, M., 1986, "On orbit prediction with KS uniform regular canonical equations with oblactness", Proceedings of3lst ISTAM Congress, 25-53.

158. Sharma, R.K. and Xavier James Raj, M,, 1988, "Long-term orbit computations with KS uniformly regular canonical elements with oblactness", Earth, Mi)on and PIanels, 32.63- 178.

159. Xavier James Raj. M. and Sharma, R.K., 1999, "Orbit prediction with Earth's flattening: KS carton ical elements", Proceedings of 441h Congress of' the lvrdicrn Sucieiy of dheorcticul and applied mechunics, 1 15-1 23.

160. Xavier James Raj, M. and Sharma, R.K., 2003, "Analytical short-term orbit prediction with J2, J 3 , J4 in terms of K-S uniformly regular canonical elements", Advances in ,Ypuce Reseurch, 31,20 1 9-2025.

161. Engels, R.C. and Junkins, J.L, 1981, "The gravity perturbed Lambert probjem: A K-S variation of parameters approach, C'elestiaE Mechanics, 24, 3-2 1 .

162. Jezewski, D.J.A., 1983, "A non-canonical analytic solution to the J2 perturbed two-body problem", C~'~.lrstiul Mechanics, 30, 343-361.

163. Bond,V.R., 1974, "The uniform regular diflerential equations of the K S transformed perturbed two body problem", Celestial Mechanics, 10,303-3 19.

164. Daniel J.Fonte, Jr., Ronald J-Proulse and Paul J.Cefola, Implementing a 50 x 50 gravity field model in an orbit determination system, AAS 93 693, 2369-2389.

165. Marsh, J.G., Lerch, F.J., Putney, B.H., Felsentreger, T.L. and Sanchez. B.V., 1990, "The GEM-T2 gravitational model", Journal of Geophysical Resarch, 95, 22043-22053.

166. Cook, G.E., King-Hele, D.G. and Walker, D.M.C., 1960, "The contraction of satellite orbits under the influence of air drag. Part I. With spherically symmetrical atmosphere", Proc. R. Soc. Lond. A 257,224-249.

167. Xavier James Raj, M. and Sharma, R.K., 2006, "Analytical orbit predictions with air drag using KS uniformly regular canonical elements" Plane6ar?, and Spuce Science, 5 4 3 10-3 1 6.

168. Sharma, R.K. and Xavier James Raj, M., 2005, "Analytical orbit predictions with oblate atmosphere using KS uniformly regular canonical elements", 56"' IAC, Fukuoka, Japan, IAC-C l .h.04.

169. Xavier James Raj, M. and Sharma, R.K., 2006, Planetary and Space Science, accepted for publication.

170. Levi-Civita, T., 1906, "Sur la Resolution Qualitative du Problem Restraint des Trios Corps", Acta Math., 30, 305-327 (Italian).

1 7 1 . Kustanheimo, P., Stiefel, E.L.. 1 965, "Perturbation Theory of Kepler Motion based on Spinor Regulrization", J. Reine Angew. Math., 218, 204-2 19.

t 72. Stiefel, E.L., 1973, "A Linear Theory of the Perturbed Two-body Problem (Regularization)". Kecent Advances in Dynamical Astronomy, Ed.B .D.Tapley and V. Szebehely, 3-20, D. Reidel Company.

173. Sastry, S.S., 1985, "Introductory Methods of Numerical Analysis", Prentice-Hal l of India Pvt.Ltd.. New Delhi.

174. Hough, M.E., 198 1 , Celestial Mechanics 25, 1 1 1 - 136.

175. Chebotarev,G.A., 1964, "Motion of an artificial Earth satellite in an orbit of small eccentricity", AIAA Journa t 2, 203-208.

176. John, B. Lundberg and Bob E.Schulz,988, "Recursion formulas of l..egendre functions for use with non singular geopotential models", Journal of Guidence, l l , 3 1 - 3 8 .

177. Xavier James Raj, M . and Sharma, R.K., 2006, 'CELESTIAL MECHANICS Recent Trends', NAROSA publishing house, 160-1 68.

178. Fominov, A.M., NASA TTF-17108, 1976.

179. Burs ,L. and Pal,A ., Astr. Nachr. 303,269-275, 1982.

180. Brookes, C.J., Proc. R. SOC. London, A 437.46 1-474, 1992.

18 1 . Nair, L.S. and Sharma, R.K., 2003, "Decay of satellite orbits uses K-S elements in an oblate diurnally varying atmosphere with scale height depending on altitude", Advances in *ace Research, 31,20 1 1 -20 1 7.

Documents

mgutheses.inmgutheses.in/png/T 1488/T 1488.pdf · GOVERNMENT OF INDlA DEPA RrT,VF.N'I' OF SP.4C:E ', VIKRAM SARABHAI SPACE CENTRE THIRUVANANTHAPURAM - 695 022 TELEGRAM : SPACE TELEPHONE