AA 4362 Astrodynamics

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AA 4362Astrodynamics
Maneuvering in Space I In-Plane Orbital
Maneuvering
Week 5 Vallado Chapter 6, Sections 6.1, 6.2, 6.3
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The Vis-ViVa Equation (revisited)
Total (Mechanical) Energy of the Satellite For
a satellite orbiting in a gravity field outside
of the atmosphere (no external drag) there are
no dissipative forces acting, thus the total
energy of the satellite is constant throughout
the orbit
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Total (Specific) Mechanical Energy
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Total (Specific) Mechanical Energy
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Vis-Viva Equation
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Vis-Viva Equationfor all of the conic sections
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Applications of the Vis-Viva equation
Escape Velocity Where is the best place
in an elliptical orbit to escape from a planet?
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Escape Velocity
Escape Velocity Where is the best place
in an elliptical orbit to escape from a planet?
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Escape Velocity
Escape Velocity Where is the best place
in an elliptical orbit to escape from a planet?
Minimum DV Is always! at perigee
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Which leads us to In-Plane Orbital Transfer
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Feasible Transfer Orbits
Transfer Orbit MUST Intersect BOTH Inner and
Outer Orbits
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Feasible Transfer Orbits (contd)
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Excess-energy transfer orbit
Wasted DV
Shaded region of aT, eT phase plot
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Optimal-energy transfer Orbit
Two curves Are tangential In space
Intersection of the Curves on the aT, eT phase
plot
Why, well it is a Constrained
optimization Problem and the answer Is a bit
subtle but here goes
Target Orbit
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Why is Hohmann Transfer Optimal For DV
DV2
DV1
Burn 1
Burn 2
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Why is Hohmann Transfer Optimal For DV (contd)
Velocity Of target Orbit
Velocity Of initial Orbit
Velocity Vector after burn 1
Velocity Vector after burn 2
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Why is Hohmann Transfer Optimal For DV (contd)
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Why is Hohmann Transfer Optimal For DV (contd)
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Why is Hohmann Transfer Optimal For DV (contd)
Recall that from the Keplers Third law
derivation
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Why is Hohmann Transfer Optimal For DV (contd)
Or plugging in the Vis Visa equation
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Why is Hohmann Transfer Optimal For DV (contd)
Since e lt 1 . Is always positive
Thus for a given aT, the minimum DVT occurs at
the minimum Allowable eccentricity
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Why is Hohmann Transfer Optimal For DV (contd)
Optimal Solution Occurs Along the feasible region
boundary
Thus For Optimal Transfer The equalities hold
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Why is Hohmann Transfer Optimal For DV (contd)
Now Look at Orbital Energies
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Why is Hohmann Transfer Optimal For DV (contd)
Substituting in from the Vis-Viva equation
Boundary Value _at_ Burn 1 Curve
Boundary Value _at_ Burn 2 Curve
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Why is Hohmann Transfer Optimal For DV (contd)
Substituting in from the Vis-Viva equation
Minimum transfer energy will be at
minimum Eccentricity value along the
feasibility boundary
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Why is Hohmann Transfer Optimal For DV (contd)
Optimal Solution Occurs Along the feasible region
boundary
Intersection of the boundary curves
Thus For Optimal Transfer The equalities hold
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Why is Hohmann Transfer Optimal For DV (contd)
Solving for aT, eT
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Why is Hohmann Transfer Optimal For DV (contd)
Solving for aT, eT
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The Hohmann Transfer
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The Hohmann transfer

• Most fuel efficient method
• All velocity changes are tangential
• (change velocity magnitude but not direction)
• Between circular or (aligned elliptical orbits)
• Takes longer than other less efficient transfers
• Tangential elliptical transfer orbit
• (example Geosynchronous Transfer Orbit GTO)

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Hohmann Transfer Calculating the DVs
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Hohmann Transfer Calculating the DVs
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Hohmann Transfer Calculating the DVs
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Hohmann Transfer Calculating the DVs
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Hohmann TransferTotal DV
-1
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Hohmann Transfer Problem Solved!
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ExampleDV required for Hohmann Transfer from
GEO to LEO
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ExampleDV required for Hohmann Transfer from
GEO to LEO (contd)
Compute Orbit ratio
Compute Normalized DV
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ExampleDV required for Hohmann Transfer from
GEO to LEO (contd)
Compute Initial Orbit Velocity
Compute Required DV
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Example Problem 2
Orbital Rendezvous
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Transfer Orbit parameters
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Transfer Orbit period
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Angular Velocity
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Angular Velocities (within initial orbits)
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Lead angle
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Interceptor Wait time
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