To Orbit…and Beyond (Intro to Orbital Mechanics)

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To Orbitand Beyond(Intro to Orbital Mechanics)

• Scott Schoneman
• 6 November 03

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Agenda

• Some brief history - a clockwork universe?
• The Basics
• What is really going on in orbit Is it really
  zero-G?
• Motion around a single body
• Orbital elements
• Ground tracks
• Perturbations
• J2 and gravity models
• Drag
• Third bodies

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Why is this important?

• The physics of orbit mechanics makes launching
  spacecraft difficult and complex Its difficult
  to get there! (with current technology)
• Orbit mechanics touches the design of essentially
  all spacecraft systems
• Power (shadows? Distance from Sun?)
• Thermal ( )
• Attitude Control (disturbance environment)
• Propulsion systems (launch, orbit maneuvers -
  indirectly affects structures)
• Radiation environment (electronic design)
• All of the above can affect software
• Practical problems
• Where will the satellite be when can I talk to
  it?
• When will it see/not see its mission target?
• How do I get it to see its mission target or
  ground stations (attitude, propulsion maneuvers)?

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Earth-Centered Sun-Centered

• The Universe must be perfect! All motion must be
• based on spheres and circles (Aristotle)
• Ptolemy (c. 150 AD) worked out a system of
• epicycles, eccentrics and equants based
  on
• circles
• Fit observations for many centuries

• Copernicus (1543) published his sun-centered
  universe
• Mathematical description only
• Described retrograde motion well, but still used
  circles and epicycles to fit observational details

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Observations Ellipses

• Tycho Brahe (1546 - 1601) Foremost observer of
  his day
• Most accurate and detailed observations performed
  up to that time

• Johannes Kepler (1571 - 1630)
• Used Tychos observations in attempt to fit his
  sun-centered system of spheres separated by
  regular polyhedra
• Could not fit the observations to systems of
  circles and spheres
• Resorted to other shapes, eventually settling on
  the ellipse

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Keplers Laws

• Kepler made the leap to generalize 3 laws for
  planetary motion
• 1) Planets move in an ellipse, with the sun at
  a focus
• 2) The motion of a planet sweeps out area at
  a constant rate
• (thus the speed is not
  constant)
• 3) Period2 is proportional to (average
  distance)3
• The harmony of the worlds
• My aim in this is to show that the celestial
  machine is ...... a clockwork
• Note that these were purely EMPIRICAL laws -
  theres no physics behind them.

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Halley and Newton

• Edmond Halley (1656 - 1742) sought to predict the
  motion of comets, but couldnt fit modern
  observations with older comet theories
• Suspected inverse-square law for force, but
  sought Newtons help
• Helped Newton (technically financially) publish
  Principia

• Isaac Newton (1643 - 1727) proved inverse-square
  law yields elliptical motion
• Published Principia in 1687, bringing together
  gravity on Earth and in space (between the Sun,
  planets, and comets) into a single mathematical
  understanding
• Also developed differential and integral
  calculus, derived Keplers three laws, founded
  discipline of fluid mechanics, etc.

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• Albert Einstein
• Showed that Newton was all wrong (or at least not
  quite right), but we wont talk about that.
• (Newton is close enough for most engineering
  purposes)

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The Basics andTwo-Body Motion
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Newtons Mountain

• The knack to flying lies in knowing how to throw
  yourself at the ground and miss. (paraphrased)
  - Douglas Adams
• Orbit is not Zero-G - There IS gravity in space
  - Lots of it
• Whats really going on
• You are in FREE-FALL
• You are always being pulled towards the Earth (or
  other central body)
• If you have enough sideways speed, you will
  miss the Earth as it curves away from beneath
  you.

• Illustration from Principia

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Gravitational Force

• Newtons 2nd Law
• Newtons Law Of Universal Gravitation (assuming
  point masses or spheres)
• Putting these together

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Gravitational Force - Simplified(Two Bodies, No
Vectors)

• Newtons 2nd Law
• Newtons law of universal gravitation (assuming
  point masses or spheres)
• Putting these together

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The Gravitational Constant

• G is one of the less-precisely known numbers in
  physics
• Its very small
• You need to first know the mass and measure the
  force in order to solve for it
• You will almost always see the combination of
  GM together
• Usually called m
• Can be easily measured for astronomical bodies
  (watching orbital periods)

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Conic Sections

• Newton actually proved that the inverse-square
  law meant motion on a conic section

http//ccins.camosun.bc.ca/jbritton/jbconics.htm
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Conic Sections - Characteristics
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Ellipse Geometry

• Most Common Orbits are Defined by the Ellipse

• a semi-major axis
• e eccentricity e / c ( ra - rp )/ ( ra rp
  )
• Periapsis rp , closest point to central body
  (perigee, perihelion)
• Apoapsis ra , farthest point from central
  body (apogee, aphelion)

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The Classical Orbital Elements(aka Keplerian
Elements)

• Also need a timestamp (time datum)

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State Vectors

• A state vector is a complete description of the
  spacecrafts position and velocity, with a
  timestamp
• Examples
• Position (x, y, z) and Velocity (x, y, z)
• Classical Elements are also a kind of state
  vector
• Other kinds of elements
• NORAD Two-Line-Elements (TLEs) (Classical
  Elements with a particular way of interpreting
  perturbations)
• Latitude, Longitude, Altitude and Velocity
• Mathematically conversion possible between any of
  these

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Orbit Types

• LEO (Low Earth Orbit) Any orbit with an
  altitude less than about 1000 km
• Could be any inclination polar, equatorial, etc
• Very close to circular (eccentricity 0),
  otherwise theyd hit the Earth
• Examples ORBCOMM, Earth-observing satellites,
  Space Shuttle, Space Station
• MEO (Medium Earth Orbit) Between LEO and GEO
• Examples GPS satellites, Molniya (Russian)
  communications satellites
• GEO (Geosynchronous) Orbit with period equal to
  Earths rotation period
• Altitude 35786 km, Usually targeted for
  eccentricity, inclination 0
• Examples Most communications satellite missions
  - TDRSS, Weather Satellites
• HEO (High Earth Orbit) Higher than GEO
• Example Chandra X-ray Observatory, Apollo to the
  Moon
• Interplanetary
• Used to transfer between planets the Sun is the
  central body
• Typically large eccentricities to do the transfer

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Ground Tracks

• Ground Tracks project the spacecraft position
  onto the Earths (or other bodys) surface
• (altitude information is lost)
• Most useful for LEO satellites, though it applies
  to other types of missions
• Gives a quick picture view of where the
  spacecraft is located, and what geographical
  coverage it provides

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Example Ground Tracks

• LEO sun-synchronous ground track

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Example Ground Tracks

• Some general orbit information can be gleaned
  from ground tracks
• Inclination is the highest (or lowest) latitude
  reached
• Orbit period can be estimated from the spacing
  (in longitude) between orbits
• By showing the visible swath, you can estimate
  altitude, and directly see what the spacecraft
  can see on the ground
• Example swath

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Geosynchronous and Molniya Orbit Ground Tracks

• GEO ground track is a point (or may trace out a
  very small, closed path)
• Molniya ground track hovers over Northern
  latitudes for most of the time, at one of two
  longitudes

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Perturbations Reality is More Complicated Than
Two Body Motion
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Orbit Perturbations

• J2 and other non-spherical gravity effects
• Earth is an Oblate Spheriod Not a Sphere
• Atmospheric Drag
• Third bodies
• Other effects
• Solar Radiation pressure
• Relativity

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J2 Effects - Plots

• J2-orbit rotation rates are a function of
• semi-major axis
• inclination
• eccentricity

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Applications of J2 Effects

• Sun-synchronous Orbits
• The regression of nodes matches the Suns
  longitude motion (360 deg/365 days 0.9863
  deg/day)
• Keep passing over locations at same time of day,
  same lighting conditions
• Useful for Earth observation
• Frozen Orbits
• At the right inclination, the Rotation of Apsides
  is zero
• Used for Molniya high-eccentricity communications
  satellites

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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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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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Applications of Drag

• Aerobraking / aerocapture
• Instead of using a rocket, dip into the
  atmosphere
• Lower existing orbit aerobraking
• Brake into orbit aerocapture
• Aerobraking to control orbit first demonstrated
  with Magellan mission to Venus
• Used extensively by Mars Global Surveyor
• Of course, all landing missions to bodies with an
  atmosphere use drag to slow down from orbital
  speed (Shuttle, Apollo return to Earth,
  Mars/Venus landers)

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Third-Body Effects

• Gravity from additional objects complicates
  matters greatly
• No explicit solution exists like the ellipse does
  for the 2-body problem
• Third body effects for Earth-orbiters are
  primarily due to the Sun and Moon
• Affects GEOs more than LEOs
• Points where the gravity and orbital motion
  cancel each other are called the Lagrange
  points
• Sun-Earth L1 has been the destination for several
  Sun-science missions (ISEE-3 (1980s), SOHO,
  Genesis, others planned)

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Lagrange Points Application

• Genesis Mission
• NASA/JPL Mission to collect solar wind samples
  from outside Earths magnetosphere
• Launched 8 August 2001
• Returning Sept 2004

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Third-Body Effects Slingshot

• A way of taking orbital energy from one body ( a
  planet ) and giving it to another ( a spacecraft
  )
• Used extensively for outer planet missions
  (Pioneer 10/11, Voyager, Galileo, Cassini)
• Analogous to Hitting a Baseball Same Speed,
  Different Direction