Orbital Mechanics 101

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Orbital Mechanics 101

• Jessie McCartney

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Agenda

• What is an orbit?
• Types of Earth orbits
• Launching
• Space Environment
• Orbit Perturbations
• Attitude Determination
• Maneuvering
• Orbit Determination
• Satellite End of Life

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What is an orbit?

• Orbits are the result of a perfect balance
  between the forward motion of a body in space,
  such as a planet or moon, and the pull of gravity
  on it from another body in space, such as a large
  planet or star. Northwestern University.

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What is an orbit?

• An orbit is a regular, repeating path that one
  object in space takes around another one. An
  object in an orbit is called a satellite. A
  satellite can be natural, like the Earth or the
  Moon. It can also be man-made, like the Space
  Shuttle or the ISS.NASA

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What is an orbit?

• In physics, an orbit is the gravitationally
  curved path of one object around a point or
  another body, for example the gravitational orbit
  of a planet around a star--Wikipedia

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Basic Orbit Equations

• Circular Orbit Velocity
• Circular Orbit Period
• Escape Velocity (free-fall)
• µGMgravitational parameter

Escape velocity interactive
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Keplers Three Laws

• Planets move around the Sun in ellipses, with the
  Sun at one focus

1. The square of the orbital period of a planet is
   proportional to the cube of the mean distance
   from the Sun.

Keplers 3rd Law
Keplers 1st and 2nd Law

• The line connecting the Sun to a planet sweeps
  equal areas in equal times.

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Types of Earth Orbits

• LEO (Low Earth Orbit)
• 520-1,500 km altitude
• Orbital period ninety minutes

• MEO (Medium Earth Orbit)
• 20,000 km altitude
• Between LEO and GEO
• Orbital period 5-6 hours

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Types of Earth Orbits

• HEO (High Earth Orbit or Highly Elliptical Orbit)
• 40,000 km altitude (at perigee)
• Large dwell time over one hemisphere
• Orbital period 12-24 hours

• GEO (Geosynchronous or Geostationary Earth Orbit)
• 36,000 km altitude
• Orbital period 24 hours (matches Earths
  rotation)

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LEO
Low Earth Orbit (LEO) Orbiting at an altitude of
600-1,000 km.
Path of Satellite
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Example Ground Trace
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Launch vehicles

• The most common launch vehicles are rockets. They
  are referred to as Expendable Launch Vehicles
  (ELVs).
• Other launch methods include air-launching like
  Pegasus.

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Launch vehicle limitations

• LatitudeLaunch site
• Mass / Power
• Shape/Configuration
• Space available

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Post-Launch Space Environment Impacts to vehicle

• Vacuum?outgassing, exposure
• Debris
• Magnetic fields
• Solar radiation
• Cosmic Rays
• High Energy Particles
• ?South Atlantic Anomaly
• ?Single Event Upsets

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Orbit Perturbations Reality is More Complicated
Than Two Body Motion
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Sources of Orbital Perturbations

• Several external forces cause perturbation to
  spacecraft orbit
• 3rd body effects, e.g., sun, moon, other planets
• Unsymmetrical central bodies (oblateness caused
  by rotation rate of body)
• Earth Requator 6378 km, Rpolar 6357 km
• Space Environment Solar Pressure, drag from
  rarefied atmosphere
• Reference C. Brown, Elements of SC Design

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Relative Importance of Orbit Perturbations
Reference SpacecraftSystems Engineering, Fortesc
ue Stark

• J2 term accounts for effect from oblate earth
• Principal effect above 100 km altitude
• Other terms may also be important depending on
  application, mission, etc...

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Orbital Perturbation Effects Regression of Nodes
Regression of Nodes Equatorial bulge causes
component of gravity vector acting on SC to be
slightly out of orbit plane
This out of orbit plane component causes a slight
precession of the orbit plane.
The resulting orbital rotation is called
regression of nodes and is approximated using the
dominant gravity harmonics term, J2
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Orbital Perturbation Rotation of Apsides
??
Rotation of apsides caused by earth oblateness is
similar to regression of nodes. The phenomenon
is caused by a higher acceleration near the
equator and a resulting overshoot at
periapsis. This only occurs in elliptical
orbits. The rate of rotation is given by
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Atmospheric Drag

• Along with J2, dominant perturbation for LEO
  satellites
• Can usually be completely neglected for anything
  higher than LEO
• Primary effects
• Lowering semi-major axis
• Decreasing eccentricity, if orbit is elliptical
• In other words, apogee is decreased much more
  than perigee, though both are affected to some
  extent
• For circular orbits, its an evenly-distributed
  spiral

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Attitude Determination and Drag Profile

• Attitude determines which end is pointing forward
  and hence the drag profile
• Deltas on yaw, pitch, and roll axes
• Spin Stabilized (stable rotation about one axis)
• Gyro Stabilized/Three axis stabilized using
  reaction wheels
• Unstabilized (tumbling space rock)

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Maneuvering Impacts to Orbital Position

• Fuel calculation (mass of slosh-y tank)
• Thruster operation
• Orbital position impact and propagation of errors

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Orbit Determination

• Coordinate Systems
• Keplerian
• Cartesian
• Ephemeris
• Orbit Tracking

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Kepler Elements

• a is the semi-major axis of the orbit
• e is the orbits eccentricity
• i is the orbits inclination with respect to the
  central bodys plane
• ? is the argument of perigee
• O is the right ascension of the ascending node
• ? is the spacecrafts true anomaly.

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Earth-Centered Cartesian Coordinates
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Cartesian Elements

• Earth-Centered Inertial (ECI)-- Inertial, in this
  context, simply means that the coordinate system
  is not accelerating (rotating).
• Earth-Centered Fixed (ECF) or Earth-Centered
  Earth-Fixed (ECEF) ECEF is a non-inertial system
  that rotates with the Earth. Its origin is fixed
  at the center of the Earth.

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Ephemerides

• Reference epoch (day and time)
• Plus six constants of integration (initial
  conditions)
• Keplerian (a, e, i, ?, O, v) elements for
  classical coordinate system
• Cartesian coordinates (x, y, z, xdot, ydot, zdot)
  for ECI or ECF systems
• Kepler elements are easier for humans to work
  with (spot orbital variations more quickly)
• Cartesian elements are easier for computers to
  work with
• Most modern systems will easily convert between
  the two

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Ephemeris example
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Orbit Tracking

• Accuracy dependant on frequency of position
  updates
• Depends on download speed, how long satellite is
  in view of ground station(s), readability of
  transmitted information
• If equipped with GPSR (GPS receiver), the GPS
  position can be equated to truth and the change
  in satellite position determined from there

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Orbit tracking for the ISS
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Satellite End-of-Life

• Controlled re-entry
• Like launch, but much more stressful and in
  reverse
• Dont hit the humans!
• Uncontrolled re-entry
• Primarily used for vehicles that will burn up in
  the atmosphere, or that are no longer operable

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Backup Slides

• In no particular order

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Atmospheric Drag

• Effects are calculated using the same equation
  used for aircraft
• To find acceleration, divide by m
• m / CDA Ballistic Coefficient
• For circular orbits, rate of decay can be
  expressed simply as
• As with aircraft, determining CD to high accuracy
  can be tricky
• Unlike aircraft, determining r is even trickier

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Principal Orbital Perturbations

• Earth oblateness results in an unsymmetric
  gravity potential given by where ae
  equatorial radius, Pn Legendre Polynomial
  Jn zonal harmonics, w sin (SC
  declination)
• J2 term causes measurable perturbation which must
  be accounted for. Main effects
• Regression of nodes
• Rotation of apsides

Note J21E-3, J31E-6
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Part II The CubeSat Standard

• The CubeSat is a 10x10x10cm, 1kg public
  picosatellite design specification proposed by
  Stanford and Cal Poly San Luis Obispo
  universities in the USA.
• To date, low-earth orbit (LEO) CubeSat missions
  have had typical lifespans of 3-9 months.
• Cost to complete a CubeSat mission (inception to
  launch to operation to end-of-life) ranges from
  lt100,000 to 1,500,000, depending on a variety
  of factors.
• Working from a standard promotes rapid
  development and idea sharing
• Picosatellites are already a hot topic in
  aerospace. Worldwide interest is focused on
  CubeSats in particular, partly because they are
  becoming a de facto standard.

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Prograde vs. Retrograde

• Prograde
• Any orbit in which the spacecraft moves from west
  to east
• Usual direction of rotation in our Solar System.
• Only a handful of objects orbit or rotate in the
  opposite direction
• Retrograde
• Any orbit in which the spacecraft moves from east
  to west
• This is the less usual direction in the Solar
  System however, it is not impossible.
• For example, Venus has retrograde spin and some
  comets notably comet Halley, which was
  encountered by ESAs Giotto spacecraft in 1986
  also has a retrograde orbit.

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Spacecraft Horizon

• Spacecraft horizon forms a circle on the
  spherical surface of the central body, within
  circle
• Spacecraft can be seen from central body
• Line of sight communication can be established
• Spacecraft can observe the central body

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Hohmann Transfer

• Hohmann transfer is the most efficient transfer
  (requires the least DV) between 2 orbit assuming
• Only 2 burns allowed
• Circular initial and final orbits
• Perform first burn to transfer
• to an elliptical orbit which just touches
• both circular orbits
• Perform second burn to transfer
• to final circular GEO orbit

Initial Circular Parking Orbit
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Systems Engineering

• Looking at the Big Picture
• Requirements What Does the Satellite Need to Do?
  When? Where? How?
• Juggling All The Pieces
• Mission Design Orbits, etc.
• Instruments and Payloads
• Electronics and Power
• Communications
• Mass
• Attitude Control
• Propulsion
• Cost and Schedule

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Spacecraft Design Considerations

• Instruments and Payloads
• Optical Instruments
• RF Transponders (Comm. Sats)
• Experiments
• Electronics and Power
• Solar Panels and Batteries
• Nuclear Power
• Communications
• Uplink/Downlink
• Ground Station Locations
• Frequencies and Transmitter Power

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Spacecraft Design Considerations(Contd)

• Mass Properties
• Total Mass
• Distribution of Mass (Moments of Inertia)
• Attitude Control
• Thrusters Cold Gas and/or Chemical Propulsion
• Gravity Gradient (Non-Spherical Earth Effect)
• Spin Stablized
• Magnetic Torquers
• Propulsion
• Orbit Maneuvering and/or Station Keeping
• Chemical or Exotic
• Propellant Supply

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Spacecraft Design Considerations(Contd)

• Cost and Schedule
• Development
• Launch
• Mission Lifetime
• 1 Month, 1 Year, 1 Decade?

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NASA Earth-Observing Satellites
Low Earth Orbit Orbiting at an altitude of
600-1,000 km.
Ascending Orbit The satellite is moving South to
North when that portion of the orbit track
crosses the equator.
Sun-Synchronous The satellite is always in the
same relative position between the Earth and Sun.
Descending Orbit The satellite is moving North
to South when that portion of the orbit track
crosses the equator.
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Satellite Inclination
Low Inclination Orbit (often near 57º-- Space
Shuttle) no polar coverage
High Inclination or Polar Orbit (near
90º) virtually complete global coverage
Equator
Inclination The
position of the orbital plane relative to the
equator. For near-polar orbits, typically about
97º.
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