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To Orbit…and Beyond (Intro to Orbital Mechanics)


To Orbit and Beyond (Intro to Orbital Mechanics) Scott Schoneman 6 November 03 Agenda Some brief history - a clockwork universe? The Basics What is really going on ... – PowerPoint PPT presentation

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Title: To Orbit…and Beyond (Intro to Orbital Mechanics)

To Orbitand Beyond(Intro to Orbital Mechanics)
  • Scott Schoneman
  • 6 November 03

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

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
  • 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)?

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

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

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
  • 3) Period2 is proportional to (average
  • 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.

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
  • 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
  • Also developed differential and integral
    calculus, derived Keplers three laws, founded
    discipline of fluid mechanics, etc.

  • 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

The Basics andTwo-Body Motion
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
  • Illustration from Principia

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

Gravitational Force - Simplified(Two Bodies, No
  • Newtons 2nd Law
  • Newtons law of universal gravitation (assuming
    point masses or spheres)
  • Putting these together

The Gravitational Constant
  • G is one of the less-precisely known numbers in
  • 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)

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

Conic Sections - Characteristics
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)

The Classical Orbital Elements(aka Keplerian
  • Also need a timestamp (time datum)

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

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
  • Interplanetary
  • Used to transfer between planets the Sun is the
    central body
  • Typically large eccentricities to do the transfer

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

Example Ground Tracks
  • LEO sun-synchronous ground track

Example Ground Tracks
  • Some general orbit information can be gleaned
    from ground tracks
  • Inclination is the highest (or lowest) latitude
  • 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

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

Perturbations Reality is More Complicated Than
Two Body Motion
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

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

Applications of J2 Effects
  • Sun-synchronous Orbits
  • The regression of nodes matches the Suns
    longitude motion (360 deg/365 days 0.9863
  • 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

Atmospheric Drag
  • Along with J2, dominant perturbation for LEO
  • 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
  • For circular orbits, its an evenly-distributed

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

Applications of Drag
  • Aerobraking / aerocapture
  • Instead of using a rocket, dip into the
  • 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)

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
  • Sun-Earth L1 has been the destination for several
    Sun-science missions (ISEE-3 (1980s), SOHO,
    Genesis, others planned)

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

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