Vallado, Chapter 1, Pages 4048

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Vallado, Chapter 1, Pages 40-48
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The Generalized Three-Body Problem
"three body problem" is the solution of the
motion of three bodies under their mutual
attraction. General three-body motion has
chaotic properties. Even the general "restricted
three body problem" where one of the bodies is
very small--e.g. Earth, Moon and spacecraft--is
analytically Insoluble Solutions must be
generated using numerical simulation Specific
solutions exist, like the ones in which the
spacecraft is positioned at one of the Lagrange
points.
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Restricted Three Body Problem
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Restricted Three Body Problem
Ignore Mass of Space Craft
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Acceleration of Target Relative to Earth
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Acceleration of Target Relative to Earth (contd)
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Acceleration of Target Relative to Sun
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Acceleration of Target Relative to Sun (contd)
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Sphere of Influence (contd)
Define Sphere of Influence as Locus in space
where Relative Magnitude of Perturbations of
Earth on Spacecraft Relative Magnitude
of Perturbations of Sun on Spacecraft
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Sphere of Influence (contd)
Substituting in the definitions
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Sphere of Influence (contd)
Substituting in
Solution to Higher Order Vector Equation No
General Analytic Solution
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Sphere of Influence (contd)
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Special Case I (contd)
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Special Case I (contd)
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Special Case I (contd)
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Special Case I (contd)
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Special Case I (contd)
RSOI
km
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Special Case II
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Special Case II (contd)
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Special Case II (contd)
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Special Case II (contd)
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Special Case II (contd)
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Special Case II (contd)
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Special Case II (contd)

x
RSOI
km
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Sphere of Influence
Clearly --- The sphere of influence
RSOI
km
Case II
Aint a Sphere!
RSOI
km
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Sphere of Influence (contd)
Actually it looks something like This .
Commonly Used Approximation
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Sphere of Influence (contd)
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Sphere of Influence (SOI)
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Sphere of Influence (SOI)
Expanded view
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Sphere of Influence and the Patched-Conic
Approximation
Allows the Restricted Three Body Problem to be
Closely approximated by two independent Two-Body
Problems (Keplers laws can be used again)
Outside of SOI, Spacecraft orbits sun (ignore
earth) Inside of SOI, Spacecraft orbits
earth (ignore sun)
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Example Orbital Transfer Between Planets
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Planetary Data
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Departure from Earth
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Compute Parameters of Transfer Ellipse
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Compute Velocity (Heliocentric) Required at
Transfer OrbitPerihelion
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Compute Velocity of Earth Relative to Sun
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Compute Gravitational Sphere of Influence (SOI)
of Earth
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Compute Required Excess Hyperbolic Velocity at
Edge of SOI
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What DV at LEO is required to Give Vhypexcess at
SOI Edge
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What DV at LEO is required to Give Vhypexcess at
SOI Edge
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Compute Asymptotic Departure Angle for Hypersonic
Trajectory
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Compute Asymptotic Departure Angle for Hypersonic
Trajectory(contd)
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Transfer Orbit Phasing
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Departure from Earth (summary)
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Arrival at Jupiter
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Hyperbolic Approach to Jupiter
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ApHelion Velocity of Transfer Orbit
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Compute Velocity of Jovian Orbit Relative to Sun
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Compute Hyperbolic Excess Velocity at SOI
Arrival
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Compute Gravitational Sphere of Influence (SOI)
for Jupiter
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Compute Hyperbolic Semi-Major Axis for Jovian
Approach
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Compute Velocity relative to Jupiter at Closest
Approach
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Arrival at Jupiter (contd)
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Finally Compute DV for Insertion into 25,000
Altitude Orbit
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Generalized Excess Hyperbolic Velocity
Planetary Arrival (_at_SOI), Planetary Coordinate
System
I -- Angle of Planetary equatorial plane
coordinate system w.r.t. to Ecliptic plane
(Solar equatorial plane)
gt 0 at SOI trajectory is hyperbolic
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Generalized Excess Hyperbolic Velocity
I -- Angle of Planetary equatorial plane
coordinate system w.r.t. to Ecliptic plane
(Solar equatorial plane)
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Generalized Excess Hyperbolic Velocity
Determine Planetary Orbit Semi-major Axis
(Hyperbolic Vis-Viva Equation)
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Generalized Excess Hyperbolic Velocity
Hyperbolic Eccentricity Direction of
Perigee (Closest approach)
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Generalized Excess Hyperbolic Velocity
Orbital Inclination With respect to Planetary
Coordinate System
l R x V
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Generalized Excess Hyperbolic Velocity
Argument of perigee
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Generalized Excess Hyperbolic Velocity
Right Ascension
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Generalized Excess Hyperbolic Velocity
Asymptotic Approach Angle (to direction of
perigee) Asymptotic Departure Angle (from
direction of perigee)
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Generalized Excess Hyperbolic Velocity
Planetary Departure (_at_ SOI ) (Planetary
Coordinate System)
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Generalized Excess Hyperbolic Velocity
Departure Velocity at SOI
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Generalized Excess Hyperbolic Velocity
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Generalized Excess Hyperbolic Velocity
Departure Velocity at SOI
I -- Planetary equatorial plane coordinate
system to Ecliptic plane (Solar equatorial plane)
So You can see that the Patched conic gravity
assist problem although Complex is simply
just described by a series of simple calculations
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Gravity Assist Problem
Orbit selection Orbital phasing
Solution to the Boundary Value Equations
What you need to get to
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Gravity Assist Problem (contd)
Subject to the constraints
departure
and all of the other orbital vowels a,e,i,
Iterative Solution process for general
solution Method of Nonlinear-shooting
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Non -Linear Shooting Methods
Prescribed Result
Input required to Give prescribed result
estimation problem given Y, solve for X
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Non -Linear Shooting Methods(contd)
Prescribed Result
Input required to Give prescribed result
estimation problem given Y, solve for X
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Non -Linear Shooting Methods(contd)
Iterative process Once Convergence Reached
check
for feasibility