( This page is best viewed in Landscape page format. Last Updated 03/05/2002)

General Course Information:

• sawhitmo@nps.navy.mil
   
• More Info:

WEB PAGE CURATOR: STEPHEN A. WHITMORE

• Course Description from NPS Academic Catalog
   
• Grading Policy
   
• Office Hours
   
• Class Schedule / Textbooks
   
• Weekly Messages

Downloadable Course Notes:

Week 1:

• Introduction and Course Overview
• Orbital Kinematics: Review of the Kepler's Laws
  • The n-body problem
  • The 2-body problem
  • Kepler's Laws
    • Kepler's First law, Conic Sections
      • Circular and Elliptical Orbits
      • Parabolic and Hyperbolic Trajectories
    • Kepler's Second Law
      • The "Swept-Area Integral"
      • Swept-Area Working Charts for a Unit Ellipse
      • The "Time of Flight" Graph for Unit Ellipse
      • Conservation of Angular Momentum

Week 2:

• Orbital Kinematics: Review of the Kepler's Laws (cont'd)
  • The Orbital Velocity Vector and Kepler's Third Law
    • Orbital Period as a Function of Orbit Size
    • Escape Velocity as Consequence of Kepler's Law
• Introduction of Orbital Dynamics: Solution of the Restricted Two-body Problem
  • Derivation of Kepler's Laws from First principles
    • Comparing Restricted two-body Orbital Equations to Kepler's laws
  • The Vis-Viva Equation
    • Gravitational Potential Energy
    • Kentic Energy
    • Total Orbital Energy
    • Escape velocity: a Simple Application of the Vis-Viva Equation,
      • Escape from Perigee
      • Escape from Apogee

Week 3:

• Orbit Propagation, Predicting the Spacecraft Position
  • Review of Kepler's Second law and the "Swept-Area" Intergral
  • On the Need for a more Accurate Orbit Prediction Tool
  • Derivation of Kepler's Equation
    • The Relationships between True, Eccentric, and Mean Anomaly
    • Relating Eccentric Anomaly to the Swept Area Integral
    • Kepler's Equation
    • The Orbit Propagation Algorithm
• Numerical Solutions of Kepler's Equation
  • Steepest descent Algorithm
  • Convergence Criterion
  • Starting Conditions

Week 4:

• Kelper's Equation for Parabolic and Hyperbolic Trajectories
  • Review of Kepler's laws for an Elliptical orbit
  • Parabolic trajectories
    • The Relationship between True and Mean Anomaly
    • Derivation of Barker's Equation
  • Hyperbolic trajectories
    • The Relationships between True, Hyperbolic, and Mean Anomaly
    • Relating Hyperbolic Anomaly to the Swept Area Integral
    • Kepler's Equation for Hyperbolic Trajectories
  • The Generalized Orbit Propagation Algorithm
• Numerical Solutions of Kepler's Equation for Parabolic/Hyperbolic Trajectories
  • Steepest Descent Algorithm
  • Convergence Plots
  • Starting Conditions
• Student Briefings
  • Kepler Solver Algorithm
  • Startup Methods

Week 5:

• Maneuvering in Space I: In-Plane Orbital Maneuvering
  • The Vis-Viva Equation, Re-visited
  • In-Plane Transfer Between Circular Orbits
    • e-a plots
    • "Excess Energy" transfer
    • Hohmann Transfer orbit
      • Parameters of the Transfer Orbit
      • Entering the Transfer Orbit, "Delta-Vee" 1
      • Entering the Final Orbit, "Delta-Vee" 2
      • Delta-Vee as a Function of Orbit Radius Ratio
    • Minimum-Energy Transfers between Elliptical Orbits
      • Bi-Elliptic orbital transfer
      • Comparison to Hohmann Transfer

Week 6:

• Maneuvering in Space I: In-Plane Orbital Maneuvering (cont'd)
  • The "Star wars" Intercept Problem
    • Trajectory Type as a function of Initial Delta-V
      • Conic section parameters
      • Intercept of the Outer orbit
    • Time of Flight to Intercept Point
      • Kepler's Equation
      • Time-of-Flight Plots
    • Intercept Orbit-Phasing
• Midterm Exam, In-class, Open notes, open book

Week 7:

• Describing Orbits in Three Dimensions
  • A Primer on Rotation Matrices
    • Spherical Coordinates
    • Matrix Orthogonality
    • Three-Axis rotations
      • Direction Cosine Matrices
    • Example:
      • The Range-Tracking Problem
  • The 6 Keplerian Orbital Elements
  • Inertial Coordinate Reference System
    • "First-point-of-Aries"
  • Transformation Matrices
    • Argument of perigee
    • Inclination Angle
    • Right Ascension
  • The Velocity Vector in Inertial Coordinates
• Transformation from Inertial to Earth Fixed Coordinates
  • Projection of the Orbital Position onto the Earth's Surface
    • Time and Calendars
    • Geodesy
  • Specialized orbits
    • LEO
    • MEO
    • GEO
    • Molynia
• Dynamic Three-Dimensional Orbit Simulation
• Appendix:
  • Reading the NORAD Two-line element (TLE) sets

Week 8:

• Calculating Orbital Elements from Position and Velocity Vectors
  • Angular Momentum Vector
  • Semi-Major axis and eccentricity
    • Vis-Viva Equation
    • Eccentricity vector
  • Orbit Inclination
  • Line-Of-Nodes vector
    • Argument of perigee
    • Orbit Longitude (Right Ascension)
• Maneuvering in Space II: Out-of-Plane Orbital Maneuvering
  • Direct Launch to Inclined Orbits
    • Achievable Orbit Inclinations
    • Launch to LEO
    • Velocity of The Earth's surface as a function of latitude
    • Required DV's to LEO
  • Delta-V Required for Orbital Plane Change
    • Simple Plange Change
    • Combined Plane Change
      • Combined Orbit and Plane Change Transfer Examples
        • Transfer from LEO to Geostationary Orbit

Week 9:

• Orbital Perturbations
  • Effect of Earth Oblateness
    • The Gravitational Potential Function
    • The "J2" Effect
    • Precession of the Line-of-Nodes
    • Rotation of the Argument of Perigee
    • Sun-Synchronous Orbits
  • Higher-order Perturbations
    • Effect of Sun, Moon, Gas Giant Planets
    • Atmospheric Drag
  • Orbital decay
    • Vis-Viva (Energy) equation with Non-Conservative Forces Acting
  • Variation of Atmospheric Density with Altitude
    • "B-Star" Ballistic Coefficients
    • "H-dot" for Low Earth Orbit
    • Numerical Integration of the Energy Equation
• Relating Delta-V to Consumed Propellant
  • Newton's Laws
    • Conservation of Momentum
    • Derivation of the "Rocket Equation"
  • Total and Specific Impulse
    • Worked Examples
      • Shuttle
      • Ariane 5
      • "Venture Star"
  • Propellant Mass Fraction and the Available Delta-V
  • Calculating the Fuel Budget for an Orbital Phasing
    Maneuver of a GeoStationary Satellite
    • Design of Phasing Orbits
    • Fuel Budgeting
    • Burn Times

Week 10:

• In-Plane Orbital Maneuvering with Continuous Thrust
  • Orbital Plane Polar Coordinate System
    • Position Vector
    • Velocity Vector
    • Acceleration Vector
  • Newton's second law
    • Resolution of Forces
    • Rate of Change of Vehicle Mass
  • Equations of Motion
  • Initial Conditions
    • Position Vector
    • Velocity Vector
  • State-Vector Equations
  • Numerical Integration Algotithm
    • Prediction Step
    • Correction Step
  • Examples
    • Spiral Orbit
      • Larger T, Larger e
      • Solving for the Final Orbit
      • A Simple Control Strategy
    • Continuous Thrust Geostationary Transfer Orbit
      with Continuous Thrust Magneto-Dynamic
      Plasma Thruster (MPD)
    • DV and Mass Budget Analysis
      • Comparison to Conventional Propulsion GTO
  • Benefits of Electric Propulsion
    • High Isp
    • Low Propellant Consumption
• A Complex Example
  • Aerodynamic Orbital Transfer of the X-37 Orbital Maneuvering Vehicle

Week  11:

• Three Body Problems
  • Generalized Three-BodyProblem
    • Chaotic Nature of the General Three-Body Problem
  • The Restricted Three-Body Problem
    • Mass of Spacecraft is Negligible
    • Two-Body and Perturbation Terms
    • Planetary Sphere of Influence
• The Patched-Conic Approximation for Planetary Transfer
  • 
    
  • Patched Conic Approximation
    • Allows the Restricted Three Body Problem to be
      Closely approximated by two independent Two-Body
      Problems (Keplerís laws can be used again)
    • Outside of SOI, Spacecraft orbits sun (ignore earth)
    • Inside  of SOI, Spacecraft orbits earth (ignore sun)
  • Example, Hohmann Transfer from LEO to Jupiter
    • Hyperbolic Departure from LEO
    • Elliptical Transfer
    • Arrival at Jupiter
    • Capture by Jupiter
  • Gravitational Assists
    • Generalized Hyperbolic Excess Velocity
    • Orbit parameters
    • Asymptotic Approach Angle
    • Asymptotic Departure Angle
    • Departute Velocity at SOI
    • Helocentric Departure Vector
• Libration Points
  • Balance of Kenetic and Gravitational Potential Energy
    (Gravitational and Centrifugal Force)
  • L1 Libration Point
    • Force Balance
    • Necessary Condition for Libration
    • "Third Law" for L1 Point
    • Libration Equation
      • Approximate Solution
        Numerical Example
  • L2 Libration Point
    • Force Balance
    • Necessary Condition for Libration
    • "Third Law" for L2 Point
    • Libration Equation
      • Approximate Solution
      • Numerical Example
  • L3, L4, L5 (Heliocentric) Librations Points
  • Stability of the Libration Points
    • Example, SOHO Mission Orbit
  • L4, L5, Earth/Moon Librations Points
    • Force Balance
    • Necessary Condition for Libration
    • "Third Law" for L1 Point
    • Libration Equation
      • Exact Solution
      • Numerical Example

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