intro to astrodynamics - utah state university
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MAE 5540 - Propulsion Systems
Intro to Astrodynamics
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Kinematics versus Dynamics
• Up to now we have mostly dealt with orbital motions from a kinematics point of view … I.e. Kepler’s lawsWere used simply as descriptors of orbital motion
• Kepler's laws are a reasonable approximation of the motions of a small body orbiting around a much larger body in a 2-body universe
… but there are no Physics (I.e. Isaac Newton)Involved
• Kepler derived his laws of planetary motion by Empirical observation only.
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Summary: Kepler’s Laws
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Kepler's Laws (cont'd)
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Kepler's Laws (cont'd)
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Time of FlightFinal position
Time of flight
t2
t1
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Kepler's Laws (cont'd)
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Kepler's Laws (cont'd)
Haven’t reallyproven this! Yet.
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Time-of-Flight Graphs
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Time-of-Flight Graphs (cont’d)
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Velocity Vector, re-examined(concluded)
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Angular Velocity of Spacecraft (cont’d)
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Isaac Newton
• Sir Isaac Newton used his new calculus and laws of motion and gravitation to show that Kepler was right.
• One day in 1682 he came up to his friend, Edmund Halley, and casually mentioned to him that he'd proved that, with a 1/r2 force law like gravity, planets orbit the sun in the shapes of conic sections.
• This undoubtedly took Halley aback, as Newton had just revealed to him the nature of the Universe (at least the Universe as it was known then).
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Newton …
•Halley then pressed Newton to publish his findings, but he realized that he'd forgotten the proof.
• After struggling to remember how he had proved the theorem, he published his work and it later appeared in full form in his classic work: Philosophiae Naturalis Principia Mathematica -- commonly known as the Principia --published in1687.
• OK … let walk down Newton’s path to enlightenment!
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You’ve seen it before …..
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Total Specific Energy• First Calculate Radius of Curvature of Ellipse
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Total Specific Energy (2)
• Next Calculate Radius of Curvature of Ellipse at Perigee
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Total Specific Energy (3)
eT constant everywhere in orbit
• Now look at perigee condition force balance
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Total Specific Energy (4)
eT constant everywhere in orbit
• Now look at perigee condition force balance
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Total Specific Energy (5)
eT constant everywhere in orbit
… at perigee conditions
• Sub Into Energy Equation
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Total Specific Energy (6)
eT constant everywhere in orbit
• Similarly @ Apogee condition force balance
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Total Specific Energy (5)
eT constant everywhere in orbit
… at perigee conditions
• Sub Into Energy Equation
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(6)
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Orbital Energy Review
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Orbital Energy
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•
Total …
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Orbital Energy
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Postscript:Escape Velocity
Postscript II: Escape Velocity
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Postscript:Escape Velocity (cont'd)
Postscript II: Escape Velocity
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Postscript:Escape Velocity (cont'd)
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Postscript II: Escape Velocity
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How about for a parabolic trajectory?
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How about for a parabolic trajectory?
• Orbital Energy is with regard to an escape trajectory!
•Circular, Elliptical Orbit à eT < 0•Parabolic (Escape) Trajectory à eT = 0•Hyperbolic Trajectory à eT >= 0
See Appendix 2.3.2 for Hyperbolic Trajectory Proof44
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Vis-Viva Equation for All the Conic-Sections
See Appendix 2.3.2 for Hyperbolic Trajectory Proof45
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Appendix 2.3: Total Specific Orbital EnergyAlternate Derivation
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Appendix 2.3.2: Total Specific Orbital Energyfor Hyperbolic Trajectory
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How about for a hyperbolic trajectory?
• Conservation of Energy and Angular Momentum still hold …. So ….
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Hyperbolic Energy• Recall, from the “First Law” derivation
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Hyperbolic Energy (continued)• At Perigee, n=0
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Hyperbolic Energy (continued)
And for a hyperbola
• Substituting into energy equation
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Hyperbolic Energy (continued)• Substituting into energy equation
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Hyperbolic Energy (continued)
• But from the General Form for the Conic section
hyp
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Hyperbolic Energy (continued)
Evaluating at perigee
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Hyperbolic Energy (continued)• Substituting into the Energy equation
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Hyperbolic Energy (concluded)
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