Algorithms for Satellite Orbital Dynamics

Preface
Introduction
Orbital Dynamics in the Solar System
Two Dynamical Systems in the Orbital Dynamics
Mathematical Models for Satellite Motion: The Perturbed Two-Body Problem [1–8]
The Two-Body Problem and Kepler Orbit
The Method of Solving the Perturbed Two-Body Problem
The Perturbed Restricted Three-Body Problem in the Motion of Deep-Space Prober
The Restricted Three-Body Problem for Circular and Elliptical Motions [9–12]
Models for the Restricted N-body Problem and the Perturbed Restricted Three-Body Problem [13, 14]
The Restricted Problem of (n + k)-Bodies [15, 16]
General Restricted Three-Body Problem
References
Contents
About the Author
1 Selections and Transformations of Coordinate Systems
1.1 Time Systems and Julian Day [1, 2]
1.1.1 Selection of Standard Time
1.1.2 Time Reference Systems
1.1.3 Julian Day
1.2 Space Coordinate Systems [2–6]
1.3 Earth’s Coordinate Systems [2, 6–10]
1.3.1 The Realization of the Dynamical Reference System and J2000.0 Mean Equatorial Reference System
1.3.2 The Intermediate Equator and Three Related Datum Points
1.3.3 Three Geocentric Coordinate Systems
1.3.4 Transformation of the Earth-Fixed Coordinate System O-XYZ and the Geocentric Celestial Coordinate System O-xyz
1.3.5 Relationship Between the IAU 1980 Model and the IAU 2000 Model
1.3.6 The Complicity in the Selection of Coordinate System Due to the Wobble of Earth’s Equator
1.3.7 Coordinate Systems Related to Satellite Measurements, Attitudes, and Orbital Errors
1.4 The Moon’s Coordinate Systems
1.4.1 Definitions of the Three Selenocentric Coordinate Systems [6]
1.4.2 The Moon’s Physical Libration
1.4.3 Transformations Between the Three Selenocentric Coordinate Systems
1.5 Planets’ Coordinate Systems
1.5.1 Definitions of Three Mars-Centric Coordinate Systems
1.5.2 Mars’s Precession Matrix
1.5.3 Transformation of the Mars-Centric Equatorial Coordinate System and the Mars-Fixed Coordinate System
1.5.4 Transformation of the Geocentric Coordinate System and the Mars-Centric Coordinate System
1.5.5 An Explanation of the Application of the IAU 2000 Orientation Models of Celestial Bodies
References
2 The Complete Solution for the Two-Body Problem
2.1 Six Integrals of the Two-Body Problem
2.1.1 The Angular Momentum Integral (the Areal Integral)
2.1.2 The Orbital Integral in the Motion Plane and the Vis Viva Formula
2.1.3 The Sixth Motion Integral: Kepler’s Equation
2.2 Basic Formulas of the Elliptical Orbital Motion
2.2.1 Geometric Relationships of the Orbital Elements in the Elliptical Motion
2.2.2 Expressions of the Position Vector "0245r and Velocity
2.2.3 Partial Derivatives of Some Variables with Respect to Orbital Elements
2.2.4 Derivatives of M, E, and F with Respect to Time t
2.3 Expansions of Variables in the Elliptical Orbital Motion
2.3.1 Expansions of Sin kE and Cos kE
2.3.2 Expansions of E, r/a, and a/r
2.3.3 Expansions of Sin F and Cos F
2.3.4 The Expansion of F
2.3.5 Expansions of (ra)ncosmf and (ra)nsinmf
2.3.6 Expansions of (ar)p, E, and (F - M) in the Trigonometric Function of F
2.4 Transformations from the Orbital Elements to the Position Vector and Velocity and Vice Versa
2.4.1 Calculations of the Position Vector "0245r(t) and Velocity (t) from Orbital Elements σ(t)
2.4.2 Calculations of the Orbital Elements σ(t) from "0245r(t) and (t)
2.4.3 Calculations of Orbital Elements σ (t0) from Two Position Vectors "0245r(t1) and "0245r(t2)
2.4.4 Method to Solve Kepler’s Equation
2.5 Expressions and Calculations of Satellite Orbital Variables
2.5.1 Two Expressions of the Longitude of Satellite’s Orbital Ascending Node
2.5.2 Expressions of Satellite’s Position Measurements from a Ground-Based Tracking Station
2.5.3 Equatorial Coordinates of the Sub-Satellite Point
2.5.4 Satellite’s Orbital Coordinate System
2.5.5 Expressions of Errors in Satellite Position
2.6 Parabolic Orbit and Hyperbolic Orbit
2.6.1 The Parabolic Orbit
2.6.2 The Hyperbolic Orbit
2.6.3 Formulas for Calculating the Position Vector and Velocity
References
3 Analytical Methods of Constructing Solution of Perturbed Satellite Orbit
3.1 The Method of the Variation of Arbitrary Constants Applied to the Perturbed Two-Body Problem
3.2 Common Forms of Perturbed Motion Equation
3.2.1 Perturbed Motion Equations Formed by Accelerations of the (S, T, W)-Version and the (U, N, W)-Version
3.2.2 The Perturbation Motion Equations Formed by ∂R⁄∂σ-Version
3.2.3 Canonical Equations of Perturbation Motion
3.2.4 Singularities in the Perturbation Equations
3.3 Perturbation Method of Constructing Power Series Solution with a Small Parameter
3.3.1 Perturbation Equations with a Small Parameter
3.3.2 Existence of Power Series Solution with a Small Parameter
3.3.3 Construction of the Power Series Solution with a Small Parameter: The Perturbation Method
3.3.4 Secular Variations and Periodic Variations
3.4 An Improved Perturbation Method: The Method of Mean Orbital Elements
3.4.1 Introduction of the Method of Mean Orbital Elements
3.4.2 The Mean Values of Related Variables in an Elliptic Motion
3.4.3 Construction of Formal Solution: The Method of Mean Orbital Elements [3–8]
3.4.4 Example
3.4.5 Two Annotations About the Method of Mean Elements
3.5 The Method of Quasi-Mean Elements: The Structure of the Formal Solution
3.5.1 Small Divisors in Expressions of Perturbation Solutions
3.5.2 Configuration of Formal Solution: The Method of Quasi-Mean Elements
3.6 Methods of Constructing Non-singularity Solutions for a Perturbed Orbit
3.6.1 Configuration of the Non-singularity Perturbation Solutions of the First Type
3.6.2 Configuration of the Non-singularity Perturbation Solutions of the Second Type
References
4 Analytical Non-singularity Perturbation Solutions for Extrapolation of Earth’s Satellite Orbital Motion
4.1 The Complete Dynamic Model of Earth’s Satellite Motion
4.1.1 Selection of Calculation Units in Satellite Orbit Dynamics
4.1.2 Analyses of Forces on Satellite’s Orbital Motion
4.1.3 Further Analyses of the Forces Acting on a Satellite
4.2 The Perturbed Orbit Solution of the First-Order Due to Earth’s Dynamical Form-Factor J2 Term
4.2.1 The Perturbed Orbit Solution of the First-Order in Kepler Orbital Elements [1–5]
4.2.2 The Non-singularity Perturbation Solution of the First Type
4.2.3 The Non-singularity Perturbation Solution of the Second Type
4.3 The Perturbed Orbit Solution of the First-Order Due to Earth’s Ellipticity J2,2 Term
4.3.1 The Perturbed Orbit Solution of the First-Order in Kepler Orbital Elements
4.3.2 The Non-singularity Perturbation Solution of the First Type
4.3.3 The Non-singularity Perturbation Solution of the Second Type
4.4 Additional Perturbation of the Coordinate System for the First-Order Solution
4.4.1 The Cause of the Additional Perturbation of the Coordinate System [3, 8]
4.4.2 The Additional Perturbation Solution in Kepler Orbital Elements
4.4.3 The Non-singularity Additional Perturbation Solution of the First Type
4.4.4 The Non-singularity Additional Perturbation Solution of the Second Type
4.4.5 Selection of Coordinate System and Related Problems
4.5 The Perturbation Orbit Solution Due to the Higher-Order Zonal Harmonic Terms Jl (l ≥ 3) of Earth’s Non-spherical Gravitation
4.5.1 General Expression of the Perturbation Function of the Zonal Harmonic Terms Jl (l ge3)
4.5.2 The Perturbation Solution of the Zonal Harmonic Jl (l ge3) Terms
4.5.3 The Non-singularity Perturbation Solution of the First Type by the Zonal Harmonic Terms Jl ( l ge3 )
4.5.4 The Non-singularity Perturbation Solution of the Second Type by Zonal Harmonic Terms Jl ( l ge3 )
4.5.5 The Perturbation Solution of the Main Zonal Harmonic Terms J3 and J4 in Kepler Elements
4.6 The Perturbation Solution Due to the High-Order Tesseral Harmonic Terms Jl,m (l ≥ 3, M = 1, 2, ⋯, l) of Earth’s Non-spherical Gravitation
4.6.1 The General Expression of the Perturbation Function of the Tesseral Harmonic Terms Jl,m (l ≥ 3, M = 1, 2, ⋯, l)
4.6.2 The Perturbation Solution Due to the Tesseral Harmonic Terms Jl,m ( l ge3,m = 1 - l )
4.6.3 The Non-singularity Perturbation Solution of the First Type Due to the Tesseral Harmonic Jl,m ( l ge3,m = 1 - l ) Terms
4.6.4 The Non-Singularity Perturbation Solution of the Second Type Due to the Tesseral Harmonic Jl,m ( l ge3,m = 1 - l ) Terms
4.6.5 The Perturbation Solution Due to the Tesseral Terms, J3,m (M = 1, 2, 3) and J4,m (M = 1, 2, 3, 4) in Kepler Elements
4.6.6 The Non-singularity Perturbation Solution of the First Type Due to the Tesseral Harmonic Terms J3,m (M = 1, 2, 3) and J4,m (M = 1, 2, 3, 4)
4.6.7 The Non-singularity Perturbation Solution of the Second Type Due to the Tesseral Harmonic Terms J3,m (m = 1, 2, 3) and J4,m (m = 1, 2, 3, 4)
4.7 The Perturbed Orbit Solution Due to the Gravitational Force of the Sun or the Moon
4.7.1 The Perturbation Function and Its Decomposition
4.7.2 The Perturbation Solution Due to the Gravity of the Sun or the Moon
4.8 The Perturbed Orbit Solution Due to Earth’s Deformation
4.8.1 Expression of the Additional Potential of Tidal Deformation
4.8.2 Effect of the Main Term in the Additional Tidal Deformation Potential (the Second-Order Term of l = 2) on a Satellite Orbit
4.9 Post-Newtonian Effect on the Orbital Motion
4.9.1 The Post-Newtonian Effect
4.9.2 Perturbation Solution Due to the Post-Newtonian Effect
4.9.3 Other Post-Newtonian Effects on the Earth’s Artificial Satellite Motion
4.10 Perturbed Orbit Solution Due to the Solar Radiation Pressure
4.10.1 Calculation of Radiation Pressure
4.10.2 Two States of Radiation Pressure Perturbation
4.10.3 The Perturbation Solution Due to Radiation Pressure
4.10.4 The Non-singularity Perturbation Solution of the First Type Due to the Radiation Pressure
4.10.5 The Non-singularity Perturbation Solution of the Second Type Due to the Radiation Pressure
4.11 Perturbed Orbit Solution Due to Atmospheric Drag
4.11.1 Damping Effect: Atmospheric Drag
4.11.2 Atmosphere Density Model
4.11.3 Atmospheric Rotation and the Expression of Atmospheric Drag
4.11.4 Structure of the Perturbed Solution Due to the Atmospheric Drag
4.11.5 The Non-singularity Perturbation Solution by the Atmospheric Drag
4.12 Orbital Variations Due to a Small Thruster
4.12.1 The Perturbation Solution Due to an (S,T,W)-Type Thrust
4.12.2 The Non-singularity Perturbation Solution Due to an (S,T,W)-Type Thrust
4.12.3 The Perturbation Solution by a U-type Thrust
4.12.4 The Non-singularity Perturbation Solution Due to a U-type Thrust
References
5 Satellite Orbit Design and Orbit Lifespan Estimation
5.1 Sidereal Period and Nodal Period [1–3]
5.1.1 The Transformation Between the Sidereal Period Ts and the Nodal Period Tφ
5.1.2 The Anomalistic Period
5.2 Orbital Characteristics of Polar Orbit Satellite [2, 3]
5.2.1 Basic Theories
5.2.2 Preservation of Polar Orbit
5.3 Existence and Design of Sun-Synchronous Orbit [2–5]
5.3.1 Conditions of Forming a Sun-Synchronous Orbit
5.3.2 Sun-Synchronous Orbits for Different Celestial Bodies
5.4 Existence and Design of Frozen Orbit [2–5]
5.4.1 Basic State of Frozen Orbit
5.4.2 Basic Equations of a Possible Frozen Orbit
5.4.3 A Particular Solution of Eq. (5.40): The Frozen Orbit
5.4.4 Stability of Frozen Orbit
5.4.5 Frozen Orbit for Other Celestial Bodies
5.4.6 Characteristics and Applications of Satellite Orbit with a Critical Inclination
5.5 Existence and Design of Central Body Synchronous Orbit
5.5.1 Basic State of Central Body Synchronous Satellite Orbit [2, 3, 5]
5.5.2 Existence and Evolution of a Central Body Synchronous Satellite (Earth, Mars)
5.6 Estimation and Calculation of Satellite’s Lifespan Due to the Mechanism of Gravitational Perturbation
5.6.1 Definition and Mechanism of a Low Orbit Satellite Lifespan Due to Gravitational Perturbations [6–10]
5.6.2 Overview of Low Orbit Satellite Lifespan for Earth, the Moon, Mars, and Venus
5.6.3 Evolution Characteristics and Lifespans of Orbit with a Large Eccentricity [2, 9]
5.6.4 Evolution Characteristics and Lifespans of High Earth Satellite Orbit [6, 10]
5.6.5 Key Points About Estimating Satellite Orbit Lifespan Due to Gravitational Perturbations
5.7 Estimation and Calculation of Satellite Orbit Lifespan in the Perturbed Mechanism of Atmospheric Drag
References
6 Orbital Solutions of Satellites of the Moon, Mars, and Venus
6.1 Characteristics of Gravitational Fields of Earth, the Moon, Mars, and Venus
6.1.1 Basic Characteristics of Earth’s Gravity Potential
6.1.2 Basic Characteristics of the Moon’s Gravity Potential
6.1.3 Basic Characteristics of Mars’s Gravity Potential
6.1.4 Basic Characteristics of Venus’s Gravity Potential
6.2 Perturbed Orbital Solution of the Moon’s Satellite
6.2.1 Selection of Coordinate System
6.2.2 Mathematical Model for the Perturbed Motion of the Moon’s Satellite
6.2.3 The Numerical Solution for the High Precise Orbital Extrapolation
6.2.4 The Analytical Perturbation Solution of the Moon’s Satellite Orbit
6.2.5 Additional Perturbation of Coordinate System [5, 6]
6.2.6 Applications of Analytical Orbital Solution in Orbital Design
6.3 Perturbed Orbital Solution of Mars’s Satellite
6.3.1 Selection of Coordinate System
6.3.2 The Mathematical Model of Perturbed Motion for a Mars’s Satellite
6.3.3 The Analytical Perturbation Solution of Mars’s Satellite Orbit [7, 8]
6.4 Perturbed Orbital Solution of Venus’s Satellite
6.4.1 The Perturbation Function of Venus’s Non-Spherical Gravity Potential
6.4.2 The Structure and Results of the Analytical Perturbation Solution
References
7 Orbital Motion and Calculation Method in the Restricted Three-Body Problem
7.1 Selection of Coordinate System and Motion Equation of a Small Body
7.1.1 The Motion Equation of a Small Body in the Barycenter Inertial Coordinate System
7.1.2 The Motion Equation of a Small Body in the Synodic Coordinate System
7.2 Jacobi Integral and Solution Existence of the Circular Restricted Three-Body Problem
7.2.1 Jacobi Integral in the Circular Restricted Three-Body Problem
7.2.2 Existence of Solution of the Circular Restricted Three-Body Problem
7.3 Calculation and Application of the Libration Point Positions of the Circular Restricted Three-Body Problem
7.3.1 Conditions of Existence for Libration Solutions
7.3.2 The Positions of the Three Collinear Libration Points
7.3.3 Two Triangle Libration Points
7.3.4 Dynamical Characteristics of the Five Libration Points
7.3.5 Characteristics and Applications of the Stability of the Five Libration Points
7.3.6 Calculations and Applications of Libration Points in the Restricted Problem of (2+2) Bodies
7.4 Orbit Design for Formation Flying of Satellites and Companion-Flying in the Exploration of Asteroids
7.4.1 The Principle of Satellite Formation Flying
7.4.2 The Problem with the Eccentricity in Orbit Design of Formation Flying of Satellites
7.4.3 Extension of the Principle and Related Orbit Design Method of Satellite Formation Flying
7.4.4 Orbital Problem of Companion Flying in Asteroid Exploration
7.5 Geometric Characteristics of Libration Point Orbit and Orbit Determination
7.5.1 Geometric Characteristics of Libration Point Orbits
7.5.2 Analysis of Forces on a Prober’s Motion in a Libration Orbit
7.5.3 Orbit Determination and Forecast Method of Libration Point Orbit
7.5.4 Orbit Determination of Libration Point Orbit and Precision Examination of Short-Arc Forecast [22]
7.5.5 Orbital Transformation Between the Two Coordinate Systems for a Libration Point Orbit Prober
References
8 Numerical Method for Satellite Orbit Extrapolations
8.1 Basic Knowledge of Numerical Method in Solving the Motion Equation
8.1.1 Basic Principles of Numerical Method in Solving Motion Equation
8.1.2 Basic Concepts
8.2 Conventional Singer-Step Method: The Runge–Kutta Method
8.2.1 The Fourth-Order RK Method (RK4)
8.2.2 The Runge–Kutta-Fehlberg (RKF) Method
8.3 Linear Multistep Methods: Adams Method and Cowell Method
8.3.1 Adams Methods: Explicit Methods and Implicit Methods
8.3.2 Cowell’s Method and Størmer’s Method
8.3.3 Adams-Cowell Method
8.4 Key Issues in Applications of the Numerical Method in Orbital Dynamics
8.4.1 Selections of Variables and Corresponding Basic Equations
8.4.2 Singularity Problem
8.4.3 Homogenization of Step-Size
8.4.4 Control of the Along-Track Errors
8.5 Numerical Calculation of the Right-Side Function
8.5.1 The Perturbation Acceleration of the Zonal Harmonic Term "0245F1(Jl)
8.5.2 The Perturbation Acceleration of the Tesseral Harmonic Term "0245F(Jl,m)
8.5.3 The Recursive Formulas of Legendre Polynomials, Pl(µ) and the Associated Legendre Polynomials Pl,m(µ), and Their Derivatives [15, 16]
8.5.4 The Perturbation Acceleration of the Tidal Deformation "0245F( k2 ,J2,m )
8.6 The Role of the Hamiltonian Method in the Orbital Evolution
References
9 Formulation and Calculation of Initial Orbit Determination
9.1 Formulation of Orbit Determination
9.2 A Review of Initial Orbit Calculation in the Sense of the Two-Body Problem
9.2.1 Basic Conditions for Initial Orbit Determination
9.2.2 Construction of the Basic Equation for an Initial Orbit
9.3 Initial Orbit Determination for Perturbed Motion
9.3.1 Construction of the Basic Equation for Initial Orbit Determination
9.3.2 Initial Orbit Determination Using Angle Data Over a Short-Arc
9.3.3 Initial Orbit Determination Using (ρ, A, h) Data or Navigation Information
9.3.4 Examination of Orbit Determination Method Using Actual Measurements
9.3.5 Initial Orbit Determination When a Deep-Space Prober is on a Transfer Orbit
9.3.6 Initial Orbit Determination Using Space-Based Angle Measurements (α, δ)
9.3.7 A Brief Summary of Initial Orbit Determination
References
10 Precise Orbit Determination
10.1 Precise Orbit Determination: Orbit Determination and Parameter Estimation
10.2 Theoretical Calculation of Measurement Variables
10.3 Calculation of Transformation Matrixes
10.3.1 Matrix Y("0245r,)
10.3.2 Matrix ( ("0245r,)σ )
10.3.3 State Transition Matrix Φ
10.4 Estimation of the State Variable: Calculation of Precise Orbit Determination
10.4.1 Certainty of Solution in the Orbit Determination
10.4.2 Process of Calculating Solution in the Orbit Determination
10.5 The Least Squares Estimator and Its Application in Precise Orbit Determination
10.5.1 Estimation Theory and a Few Commonly Used Optimal Estimation Methods
10.5.2 The Least Squares Estimator
10.5.3 Two Processes of the Least Squares Estimator
10.5.4 Least Squares Estimator with a Priori State Value
10.6 Orbit Determination by Ground-Based and Space-Based Joint Network and Autonomous Orbit Determination by Star-To-Star Measurements
10.6.1 Outline of Space-Based Network of Orbit Tracking and Determination
10.6.2 Basic Principles of the Orbit Determination of Ground-Based and Space-Based Joint Network
10.6.3 The Rank Deficiency in the Autonomous Orbit Determination by Start-To-Star Measurements
References
Appendix A Astronomical Constants
Appendix B Formulas of Mean Ephemeris of Major Celestial Bodies in the Solar System
Mean Orbit Elements of Major Planets
Orbit Elements of the Moon
Another Calculation Method of Orbit Elements of Major Planets
Appendix C Orientation Models of Major Celestial bodies in the Solar System
References