AA 4362 Astrodynamics

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AA 4362Astrodynamics
Solving Kepler's (Barkers) Equation for
parabolic and hyperbolic trajectories
Week 4 Vallado, Chapter 2, sections 2.2, 2.3, 2.4
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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
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Generalized Formsfor Keplers Equation
Ellipse Parabola ??????????
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Ellipse
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Ellipse (concluded)
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Parabola
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Parabola (continued)
True Mean Anomaly
Iteration Index

Current estimate
Refined estimate
Current estimate of mean anomaly
Steepest Descent
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Parabola Convergence plot(near perigee)
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Parabola Convergence plot(far from perigee)
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Parabolic Solver, Startup Condition
Approximate Solution as n --gt p

n
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Parabolic Solver, Startup Condition
Example M 10.0

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Parabolic Solver, Startup Condition
Example M 0.1

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Hyperbola
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Hyperbola (concluded)
True Mean Anomaly
Iteration Index
Current estimate
Refined estimate
Current estimate of mean anomaly
Steepest Descent
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Hyperbola, Numerical example
Example
n, Approx 115?
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Hyperbola, convergence plot
true anomaly near perigee
n, Approx 115?
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Hyperbola, convergence plot
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Hyperbola, convergence plot
true anomaly far from perigee
n, Approx 131.8?
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Hyperbola, convergence plot
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Soooo we need a reliablestartup algorithm
So where do we start? with a really Simple
ad-hoc solution .. When M 0, we simply
kick it off zero by adding or subtracting e
(Vallado Algorithm)
F0
F0
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Ad-Hoc Start Up Algorithm
Me 16.148
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Homework 9
Develop a Higher order Taylor Series
method Similar to that Developed for the
Elliptical Kepler Startup Hint
0
0
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Homework 9 (contd)
Find single real root of resolvent cubic
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Homework 9 (contd)
Real Root
Compare convergence properties of this startup
method Versus Ad-Hoc startup algorithm for a
variety of e, M
F0
F0