orbital mechanics - faculty.kfupm.edu.sa · orbital mechanics aa236a 3 november 2008. announcements...
TRANSCRIPT
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Orbital Mechanics
AA236A
3 November 2008
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Announcements
• Nice job on the first checkout
• Rover 2 checkout
– Mon. 17 November, 3:45 pm
– GPS, SD Card, and Color/Rad sensor are only – GPS, SD Card, and Color/Rad sensor are only
major hardware components left!
• VOTE TOMORROW!!!
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Source Material
• SMAD: Chapters 6-7
• Bate, Mueller, White – “Fundamentals of
Astrodynamics”
• AA279 – Space Mechanics• AA279 – Space Mechanics
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Kepler’s Laws
1. The orbit of each planet is an ellipse, with
the sun at a focus.
2. The line joining the planet to the sun
sweeps out equal areas in equal times.sweeps out equal areas in equal times.
3. The square of the period of a planet is
proportional to the cube of its mean
distance from the sun.
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Conic Sections
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Orbit Geometry
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Elliptical Orbit Geometry
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Orbit Equations
2
2
11
310*674.6,,
kg
NmGGMaar
rr onperturbatithrust
−≈=+=+ µµ vvv
&&v
Two-Body Equation of Motion
Specific Mechanical Energy and Specific Angular Momentum
ar
vE
22
2 µµ−=−= vrh
vvv×=
arE
22−=−= vrh ×=
Trajectory Equation
( )νcos1
12
e
ear
+
−=
Orbital PeriodOrbit Type Eccentricity, e Energy, E
Circle e = 0 E < 0
Ellipse 0 < e < 1 E < 0
Parabola e = 1 E = 0
Hyperbola e > 1 E > 0
µπ
3
2a
T =
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Coordinate Systems
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Orbital Parameters
• Orbital or Keplerian Elements
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Orbital Parameters
• a, semi-major axis: size of the orbit
• e, eccentricity: shape of the orbit
• i, inclination: angle between the angular momentum vector and
Earth’s rotation axis
• Ω, longitude of the ascending node: angle between vernal equinox
direction and the point where the orbit crosses the equatorial plane direction and the point where the orbit crosses the equatorial plane
in a northerly direction
• ω, argument of periapsis: angle between the ascending node and
the orbit’s periapsis
• ν, true anomaly: angle between periapsis and the satellite’s current
location
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Orbital Parameters
• Π, longitude of periapsis
• u0, argument of latitude
at epoch
• l0, true longitude at epoch• l0, true longitude at epoch
νω
νω
ω
++Ω=
+=
+Ω=Π
0
0
l
u
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Keplerian Elements Formats
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Ground Tracks
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General Orbit Types
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Interesting Orbit Types• Polar Orbits
– LEO orbits with high inclination
and travel near the poles
• Equatorial Orbits
– Low inclination and travel near
the equator
• Geosynchronous
• Sun Synchronous
– Passes over any given point at
the same local solar time
– Approx. constant orientation
with respect to the Sun
– Orbit must precess ~1
degree/day eastward (600-800
km, ~98° inclination)• Geosynchronous
– Orbital period matches Earth’s
sidereal rotation period
– Returns to the same place in the
sky at same time each day
• Geostationary
– Circular, 0° inclination
geosynchronous orbit
km, ~98° inclination)
• Molniya
– High eccentricity
– ~12 hr period, 63.4° inclination
• GPS
– ~12 hr period, 26,600 km
• Lagrange Points
– Gravitational equilibria
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Orbit Types
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Sun Synchronous Orbit
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Orbit with Respect to the Sun
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Molniya
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GPS Satellites
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Lagrange Points• Positions where the gravitational pull of the two large
masses precisely equals the centripetal force required to
rotate with them
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L1 and Examples
• Objects at L1 orbit with same period as Earth
• Earth “weakens” the pull of the Sun and slows the orbit down
• Sun-Earth L1 good for sun observations
– SOHO: Solar and Heliospheric Observatory
– ACE: Advanced Composition Explorer
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L2 and Examples
• Objects at L2 orbit with same period as Earth
• Earth “strengthens” the pull of the Sun and speeds up the orbit
• Sun-Earth L2 good for telescope missions
– WMAP: Wilkinson Microwave Anisotropy Probe
– ACE: Advanced Composition Explorer
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L4 and L5 Examples
• Also called triangular Lagrange or Trojan points
• Only stable Lagrange Points
• Jupiter, Saturn, and Neptune have objects caught in their L4 and L5
points
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Orbit Transfer
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Orbit Transfer - How
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Orbit Transfer Methods
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Orbit Perturbations
• Will your satellite stay where you put it?
• Is there anything that will change the
satellite’s orbit?
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On-Orbit Accelerations
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Solar Pressure
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Solar Pressure
• Solar Sails
– Sailing on the solar
wind
• Can we use it in a • Can we use it in a
cube?
– Sail to the…?
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Atmospheric Drag
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Ballistic Coefficient
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The Earth is not a Sphere!
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Non-Spherical Effect
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Inclination and J2
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J2 Effects
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Orbital Debris
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Orbital Debris
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Importance of Orbits to a Mission• How does the orbit affect mission design?
– Thermal issues
– Ground tracking, satellite visibility
• Link budget
– Power generation
– Navigation and tracking
• Sun sensor visibility, eclipse times
– Propulsion, orbit maintenance– Propulsion, orbit maintenance
– Payload concerns
– Radiation amounts
– Launch site and providers
• Cost of the launch
• Are you the primary?
• Range safety
• When should you start analyzing orbits to satisfy mission
requirements?
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Software and Links• Keplerian elements online
– http://www.amsat.org/amsat/keps/menu.html
– http://celestrak.com/
• Tracking Software
– STK
– Linux– Linux
• Predict
• Sattrack
– Windows
• Nova
• Websites
– Heavens-above.com
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AA236 Guidelines
• Design for worse case orbits
– We generally don’t know what orbit we’ll get
– Consider worse case for power, thermal, and
communication
– Rule out orbits?– Rule out orbits?
• Possible delta-V capability
– Cold gas thrusters
– Electric propulsion
– Solar Sail
– What can we do with it?