AA 4362 Astrodynamics

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AA 4362Astrodynamics
Kepler's laws for parabolic and hyperbolic
trajectories
Week 4 Vallado, Chapter 2, sections 2.2, 2.3, 2.4
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Keplers First Law(revisited)
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Which Trajectory?
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Keplers Second Law(revisited)
Valid for ALL conic sections
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Keplers Second Law(contd)
Valid for ALL conic sections
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Time of Flight Elliptical orbit(revisited)
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Time-of-Flight Elliptical Orbit(revisited)
Weve already done this
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Time-of-Flight Elliptical Orbit(revisited)
Simplifying Left Hand Side gives
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Time-of-Flight Elliptical Orbit(revisited)
Keplers Third law!
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Time-of-Flight Elliptical Orbit(revisited)
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TOF for Parabolic Trajectory
OK. This gives us an Idea how to define period Of
parabolic and hyperbolic trajectories
Keplers Second Law
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TOF for Parabolic Trajectory (contd)
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TOF for Parabolic Trajectory (contd)
cos(?)
Keplers Third Law for Parabolic Trajectory
rmin
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Time-of-Flight Plotfor Parabolic Trajectory
Parabolic Mean Anomaly
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Keplers Relationshipsfor a Parabolic Trajectory
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Simplifying .
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Keplers Relationshipsfor a Parabolic Trajectory
TOF ---gt Elapsed time since (until) passage of
the Orbit Perigee
TOF -- Time from perigee passage
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TOF for Hyperbolic Trajectory
Keplers Second Law
eT -- Eccentricity Of Hyperbolic Trajectory aT
-- axis parameter of Hyperbolic Trajectory
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TOF for Hyperbolic Trajectory(contd)
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TOF for Hyperbolic Trajectory(contd)
Substituting Back into the Time-of-Flight Equation
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TOF for Hyperbolic Trajectory(contd)
Evaluating The Integral Numerically Gives .
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Time-of-Flight Plotsfor Hyperbolic Trajectory
Gives a good Conceptual feel for what is
happening but is not precise enough for orbital
calculations
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Hyperbolic Anomaly
F -gt Hyperbolic Anomaly
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Keplers Equation for Hyperbolic Trajectories
OK this is going to be harder than it was for a
parabola!
Following the Circle/Ellipse Analogy, we
circumscribe an Arbitrary Hyperbola with an
Equilateral Hyperbola .I.e. a Hyperbola with
eccentricity After that the process pretty
much duplicates the Elliptical derivation
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Keplers Equation for Hyperbolic Trajectories
(contd)
F Hyperbolic Anomaly
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Scaling the HyperbolicGeometry - y coordinates
Hyperbolic similarity
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Scaling the HyperbolicGeometry - x coordinates
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Relationship of Hyperbolic Anomaly to True
Anomaly
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Relationship of Hyperbolic Anomaly to True
Anomaly (contd)
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Relationship of Hyperbolic Anomaly to True
Anomaly (contd)
Transformation from hyperbolic anomaly to
true Anomaly
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Relationship of HyperbolicAnomaly to the
Hyperbolic Area Integral
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(
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Relationship of HyperbolicAnomaly to the
Hyperbolic Area Integral (contd)
Substituting and collecting terms
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Relationship of HyperbolicAnomaly to the
Hyperbolic Area Integral (contd)
Substituting and collecting terms given
Keplers Equation For a Hyperbolic Trajectory
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Generalized OrbitPropagation Algorithm
First Determine which Conic section is
applicable
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Generalized OrbitPropagation Algorithm (contd)
Next
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Generalized OrbitPropagation Algorithm (contd)
Ellipse Parabola n ??????????
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Generalized OrbitPropagation Algorithm (contd)
Compute Initial Mean Anomaly
Ellipse Parabola ??????????
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Generalized OrbitPropagation Algorithm (contd)
Compute the Orbit Period
Ellipse Parabola ??????????
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Generalized OrbitPropagation Algorithm (contd)
Compute the Future Mean Anomaly
Ellipse Parabola ??????????
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Generalized OrbitPropagation Algorithm (contd)
Now Solve Keplers (Barkers) Equation for
new Eccentric (parabolic, hyperbolic) anomaly
Ellipse Parabola ??????????
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Generalized OrbitPropagation Algorithm (contd)
Solve for Future True Anomaly
Ellipse Parabola ??????????
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Generalized OrbitPropagation Algorithm
(concluded)
Finally, compute the length of the radius vector
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Homework 8
Extend Your Kepler Propagation and Solver
Algorithm To Be smart enough to solve all
trajectories. Elliptical, Parabolic, Hyperbolic
Allow For arbitrary M, e, and Starting value
for E Evaluate Algorithm performance for
e0. 5, e1.0, e1.5, e5 Use Methods, 1,
2, 3 as startup algorithms Count number of
iterations for convergence Draw Convergence
Plots for e1, e5 cases