perturbation

Celestial Mechanics and Astrodynamics: Theory and Practice

  • Introduction
    • Definitions
    • History
    • Properties of Conics
      • The Ellipse, 0 < e < 1
      • The Parabola, e = 1
      • The Hyperbola, e > 1
    • Astronomical Background
    • Stability and Chaos
      • Three-Body Problem
      • Solar System
      • Resonances, Singularities and Regularization
    • Stability Determination
      • Poincaré Surface of Section
      • Hill Stability
      • Lyapunov
      • Kolmogorov-Arnold-Moser Theorem
      • Spacecraft Orbit Stability
    • Chaos Determination
    • Observational Data
      • Transit Circle
      • Photographic
  • Vectors
    • Introduction
    • Scalar Product
    • Vector Product
    • Triple Scalar and Vector Products
    • Velocity of Vector
    • Rotation of Axes
    • Angular Velocity
    • Gradient of a Scalar
    • Momentum and Energy
      • Simple Harmonic Motion
      • Linear Motion in an Inverse Square Field
      • Foucoult’s Pendulum
  • Reference Systems and Relativity
    • Reference Systems
    • Relativistic Coordinate Systems
      • Newtonian Coordinates
      • Relativistic Coordinates
      • ICRS, BCRS, GCRS
      • Geodesic Precession and Nutation
    • Reference Frames
      • Celestial Reference Frames
      • CIP and CIO
      • Equation of Equinoxes
      • Equation of Origins
      • Terrestrial Reference Frames
      • Terrestrial Intermediate Origin
      • ECEF, ECI, ECR
      • Satellite Geodesy
      • GNSS Reference Systems
    • Time Scales
    • Coordinate Systems
      • Origins and Planes
      • Horizon Reference Frame
      • Geocentric Coordinates
      • Geodetic Coordinates
      • Geographic Coordinates
      • Astronomical Coordinates
    • Kinematics of the Earth
      • Earth Orientation
      • Precession
      • Nutation
      • Polar Motion
    • Observation Effects
      • Aberration
      • Proper Motion
      • Radial Velocities
      • Parallax
      • Refraction
      • Relativistic Light Deflection
      • Space Motion
      • Tidal Effects
      • Earth Satellite Equations of Motion in GCRS
  • Central Force Motion
    • Introduction
    • Law of Areas
    • Linear and Angular Velocities
    • Integrals of Angular Momentum and Energy
    • Equation of the Orbit
    • Inverse Square Law
      • Eccentricity Vector
      • From Orbit to Force Law
    • Einstein’s Modification of the Orbit Equation
    • University of Newton’s Law
  • The Two-Body Problem
    • Introduction
    • Classical Orbital Elements
      • Osculating Orbital Elements
      • Nonsingular Orbital Elements
    • Motion of the Center of Mass
    • Relative Motion
    • The Integral of Areas
    • Elements of the Orbit from Position and Velocity
    • Properties of Motion
    • The Constant of Gravitation
    • Kepler’s Equation
      • Series Expansion
      • Differential Method
    • Position in the Elliptic Orbit
    • Position in the Parabolic Orbit
    • Position in a Hyperbolic Orbit
    • Position on the Celestial Sphere
      • Heliocentric Coordinates
      • Geocentric Coordinates
  • Orbit Determination
    • Introduction
    • Known Radius Vectors
    • Laplace’s Method
    • Gauss’s Method
    • Lambert’s Theorem
    • Parabolic Orbits, Olber’s Method
    • Circular Orbits
  • The n-Body Problem
    • Introduction
    • Equations of Motion
    • Angular Momentum, or Areal Velocity, Integral
    • Integral of Energy
    • Stationary Solutions of the Three-Body Problem
    • Generalization to n Bodies
    • Equations of Relative Motion
  • The Restricted Three-Body Problem
    • Introduction
    • Equations of Motion
    • The Jacobi Constant
    • Zero Velocity Curves
    • The Lagrangian Points
    • Stability of Motion Near the Lagrangian Points
    • Hill’s Restricted Three-Body Problem
      • Equations of Motion
      • Hill’s Equations of Motion
    • Families of Periodic Orbits
  • Numerical Procedures
    • Differences and Sums
    • Interpolation
    • Lagrangian Methods
    • Differentiation
    • Integration
    • Differential Equations
    • Errors
    • Numerical Integration
    • Numerical Integration by Runge-Kutta Methods
    • Accumulation of Errors in Numerical Integration
    • Numerical Integration of Orbits
      • Equations for Cowell’s Method
      • Equations for Encke’s Method
      • Comparison of Cowell’s and Encke’s Methods
    • Equations with Origin at the Center of Mass
  • Canonical Equations
    • Introduction
    • Canonical Form of the Equations
    • Eliminating the Time Dependency
    • Integral of a System of Canonical Equations
    • Canonical Transformation of Variables
      • Necessary Condition
      • Sufficient Condition
    • Examples of Canonical Transformations
      • Change of Variables by Means of a Generating Function
      • Conjugate Variables to Qj
    • Jacobi’s Theorem
    • Canonical Equations for the Two-Body Problem
    • Application of Jacobi’s Theorem to the Two-body Problem
      • Meaning of the Constants a
      • Variables Conjugate to Qi
      • Application to the General Problem
      • The Delaunay Variables
    • The Lagrange Equations
    • Small Eccentricity and Small Inclination
      • Small Eccentricity
      • Small Inclination
      • Universal Variables
  • General Perturbations Theory
    • Introduction
    • Variation of Parameters**
    • Properties of the Lagrange Brackets
    • Evaluation of the Lagrange Brackets
    • Solution of the Perturbation Equations
    • Case I: Radial, Transverse, and Orthogonal Components
    • Case II: Tangential, Normal, and Orthogonal Components
    • Expansion of the Third-Body Potential
      • The Factor (r|r’)2
      • The Factor P2.cos phi
    • The Earth-Moon System**
    • Expansion of the Gravitational Potential
    • Atmospheric Drag
    • Regularization of Perturbed Motion
  • Motion Around Oblate Planets
    • Introduction
    • Axially-Symmetric Gravitational Field
    • Equatorial Motion
      • The Orbital Angle and Radial Period**
      • New Orbital Elements
      • Open Orbits and the Escape Velocity
      • Circular Orbits
    • The Cid-Lahulla Approach
      • Polar-Nodal Coordinates
      • The Cid-Lahulla Radial Intermediary**
      • Comparison with Brouwer’s Approximation
    • Solution for Motion in a Cid-Lahulla Potential**
      • Main Steps Towards a Solution**
      • New Independent Variable
  • Semi analytical Orbit Theory
    • Introduction
    • Preliminaries
    • Semianalytical Models
      • The Zonal Part of the Geopotential**
      • Second-Order Effects
      • The Tesseral-Sectorial Part of the Geopotential
      • Atmospheric Drag
    • Frozen Orbits
    • Sun-synchronous and Repeat Ground-track Orbits
    • Geostationary Orbits
      • In-Plane Motion
      • Out-of-Plane Motion
      • Averaged Solution
      • The Perturbed Problem**
  • Satellite Orbit Control
    • Introduction
    • Stability and Control of Dynamical Systems
    • Impulsive and Continuous Maneuvers
    • Gravity Assist Maneuvers
      • Multiple Gravity Assists
      • Concatenation Rules**
      • Optimization of Orbits
      • Dynamic Optimization
      • Linear Orbit Control**
      • Low Earth Orbit Control
        • Altitude Correction
        • Frozen Orbit Control
        • Sun-synchronous Orbit Control
        • Repeat Ground-track Orbit Control
      • Geostationary Orbit Control
        • North-South Stationkeeping
        • East-West Stationkeeping
        • Eccentricity Correction
      • Nonlinear Feedback Control of Orbits**
      • Fixed-Magnitude Continuous-Thrust Orbit Control**
      • Comparison of Continuous-Thrust Controllers
  • Optimal Impulsive Orbit Transfers
    • Introduction
    • Modified Hohmann Transfer***
    • Modified Bi-Elliptic and Bi-Parabolic Transfers
      • Definitions
      • Modified Bi-Elliptic Transfer
        • Calculating Ycrit**
        • Evaluating the effect of X on maneuvers where Y =Ycrit
        • Extending the evaluation to include all Ycrit < Y
      • Modified Bi-Parabolic Transfer
      • Comparison Between the Modified Bi-Parabolic and the Modified Hohmann Transfers
        • Bi-Elliptic Transfer
        • Bi-Parabolic Transfer
  • Orbit Data Processing
    • Introduction
    • Principle of Least Squares
    • Least Squares Approximation
    • Orthogonal Polynomials
    • Chebyshev Series
      • Chebyshev Approximation**
      • Other Polynomial Approximations
    • Fourier Approximation: Continuous Range
    • Fourier Approximation: Discrete Range**
    • Optimum Polynomial Interpolation
    • Chebyshev Interpolation
    • Economization of Power Series
    • Recursive Filtering
    • Mean Elements Estimator
      • Initial Conditions and Parameter Values
      • Uncontrolled Orbits, Single Run
      • Orbits with No Control Inputs, Monte-Carlo Runs
      • Impulsive Maneuvers
      • Continuous Thrust
  • Space Debris
    • Introduction
    • SGP4 Propagator and TLE
    • Sizing the Debris
    • Time of Closest Approach
    • Probability of Collision
    • Calculating the Required v