Low-Thrust Transfers from GEO to Earth-Moon Lagrange Point Orbits

Andrew AbrahamMoravian College, 2013

Newton’s Laws

1. If then & the object moves in a straight line

2. Action = Reaction:

Orbits: Inertial Reference Frame

Orbits: Inertial Reference Frame

Orbital Trajectories

Circular Orbit

Applications of Various Orbits

Low Earth Orbit (LEO)

Medium Earth Orbit (MEO)

Geosynchronous Earth Orbit (GEO)

Molniya Orbit (HEO)

Russia

Visible 83% of the time

Non-Chemical

vs.

Chemical

Fuel + Oxidizer

Ions + Electric/Magnetic Fields

Atmospheric Operation

Ion EngineChemical

Engine

Rocket

ACTIONRE-ACTION

Rocket Equation

𝑚𝑓

𝑚0

=𝑒− Δ𝑉𝐼 𝑠𝑝𝑔0Rocket Equation

= Change in Spacecraft’s Velocity = Efficiency of Rocket = 9.81m/s2

= Final Spacecraft Mass = Initial Spacecraft Mass

Mass vs. Specific Impulse (Isp)

Chemical Propellant: Isp= 200 - 300s %Mass = 25.0%

Low Thrust: Isp = 3000s %Mass = 89.5%

𝑚𝑓

𝑚0

=𝑒− Δ𝑉𝐼 𝑠𝑝𝑔0

Δ𝑉=3𝑘𝑚/ 𝑠

Ion Engine

Ion Engine

• Constant Thrust of 500-700mN

• About the weight of 8 quarters or 0.1lbs

• Consumes 2-8KW of electrical power from solar arrays

High Thrust (Chemical)

Low Thrust (Ion)

Orbit Maneuvers:High vs. Low Thrust

Example: LEO to GEO

• LEO ……………….Green• Low Thrust ……Red• GEO ………………Blue

• 1000Kg • 855.5Kg

• Time of Flight: 102 days

Add the Moon!?

The 3 Body Problem

Velocity…………GrayForce of …...RedForce of ……GreenForce of ……Blue

𝑚2

𝑚1

𝑚3

Make Simplifying Assumptions

Assume: Circular Restricted 3 Body Problem

1.

2. & are only influenced by each other and orbit their common center of mass in perfectly circular orbits

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Circular Restricted 3-Body Problem (CR3BP)

Define:µ≡

𝑚2𝑚1+𝑚2

Synodic Reference Frame

WARNING!!!Non-Inertial Reference Frame

(Rotating)

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CR3BP Equations of Motion

𝑈=𝑉 −12

(𝑥2+𝑦 2 )

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CR3BP: 5 Equilibrium (Lagrange) Points

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Characterization of Lagrange Points

m

m

Pendulum (Stable)

Inverted Pendulum (Unstable)

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Unstable Lagrange Point

L2

Applications:CommunicationsNavigation (GPS)Observation

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Lyapunov, Halo, and Lissajous Orbits

Image Credit: NASA

Moon

Moon

L1

L1

Earth

Earth

Halo

Lyapunov Lissajous

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Merging Low-Thrust & Halo Orbits in the Earth-Moon System

L4

L5

L3

L1 L2

moon

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Merging Low-Thrust & Halo Orbits in the Earth-Moon System

L3

Different View

L5

L4L2

L1

moon

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One More View

1000kg Spacecraft

69kg of fuel used for 60 day flight

GEO-like orbit to Halo orbit

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NASA Space Station Resupply Mission(s)

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Thank You!

Questions?

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Applications: Sun-Earth System

Wilkinson Microwave Anisotropy Probe (WMAP)

1. Solar & Heliospheric Observatory (SOHO) @ Sun-Earth L1

2. WMAP, James Webb Telescope, Plank @ Sun-Earth L2

3. Planet-X @ Sun-Earth L3

4. Trojan Asteroids @ Sun-Earth L4 & L5

Centrifugal Force

“Weightlessness”(Non-Inertial Reference Frame)

y

xFg

v

Fc

F = maF = Fg + Fc = 0

Orbits: Inertial Reference Framey

x

Fg v