AA 4362: Astrodynamics

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AA 4362 Astrodynamics
Introduction to Orbital Dynamics Solution of
the Restricted Two-Body Problem (Vallado,
Sections 1.3.2, 1.3.3, 1.3.6)
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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.
Keplers laws Were 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 Keplers Laws
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Kepler's Laws (cont'd)
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Kepler's Laws (cont'd)
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Kepler's Laws (cont'd)
Havent really proven this! Yet.
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Time-of-Flight Graphs
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Time-of-Flight Graphs (contd)
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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 t heorem, 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
Newtons path to enlightenment!
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Gravity Inverse Square Law
Gravity Inverse Square Law
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Some Light Vector MathDerives the Acceleration
Law
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Satellite Angular Momentum
Torque
Velocity Vector
Angular Momentum
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Evaluating the Cross Products
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Gravitational Torque Acting on Spacecraft
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Angular Momentum is Constant!
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Integrate Acceleration Equation
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Boundary Conditions
And .
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Boundary Conditions (contd)
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Boundary Conditions (contd)
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Velocity Vector
B
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Inner Products
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Inner Products (contd)
Collecting terms . But Since
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Almost there
Polar Form of a Conic Section, Keplers First
law is Proved!
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Ellipse
From Homework 2
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Ellipse (contd)
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Elliptical Orbit Eccentricity
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Ellipse (contd)
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Elliptical Orbit Summarized(Keplers First Law)
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Keplers Second Law
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Keplers Second Law (contd)
Averaged angular momentum evaluated by
integrating around a full orbit trajectory
Constant!
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Keplers Second Law (contd)
Averaged angular momentum can also be evaluated
by integrating around a partial orbit trajectory
QED!
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Keplers Third Law (revisited)
Keplers Third Law In a Restricted Two Body
Orbit, the squares of the periods of any two
objects revolving about the sun (earth) are in
the same ratio as the cubes of their mean
distances.
QED!
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Conclusion
So NOW! We proved that
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Conclusion (contd)
Weve also already shown that
And .
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Where are we headed?