orbital mechanics i

discussion summary practice problems
  • orbital mechanics 1


    circular orbits

    Newton’s laws only. Nothing about energy or momentum. Centripetal force and gravitational force.

    Fc  =  Fg
    mv2  =  GMm
    rp rp2
    v = √ Gm

    Kepler’s third law. Derive Kepler’s third law of planetary motion (the harmonic law) from first principles.

    v = √ Gm  =  r  
    r T
      Gm  =  2r2
    r T2
    r3  =  Gm  = constant
    T2 2
    r3  ∝  T2

    The “constant” depends on the object at the focus. Although formulated from the data for objects orbiting the sun, Newton showed that Kepler’s third law can be applied to any family of objects orbiting a common body.

    orbit families

    • LEO: low earth orbit, typical altitude < 2000 km
      • space shuttle
      • space station
      • Hubble Space Telescope
      • iridium
      • remote sensing: EROS, Landsat
      • communications: email, text messaging, paging
    • MEO: medium earth orbit, typical altitude 10,000 to 20,000 km
      • Semisynchronous 12 hour period
      • GPS: Global Positioning System
    • GEO: geosynchronous earth orbit, seven earth radii, one-ninth of the distance to the moon, altitude = 36,000 km
      • Arthur C. Clarke: In 1945, while still in his late 20s, he was the first to propose the concept of using a network of satellites in the geosynchronous orbit for television and telecommunications
      • meteorology: GOES – Geosynchronous (Geostationary) Operational Environmental Satellites
      • communication:
        • signal relays for terrestrial broadcast and cable systems
        • direct broadcast satellite TV and radio
      • TDRS: Tracking and Data Relay Satellite
      • GEOs tend to drift toward a large gravity anomaly over the Indian Ocean — an area of above average gravity thought to be above a mantle subduction zone

    A “snapshot” of the Earth and about 500 of its artificial satellites generated one summer evening in 2002. Nearly all of them are GEOs or LEOs. Satellites on the ring are in geosynchronous earth orbit (GEO). Those clustered near the Earth are in low earth orbits (LEO). Scattered in between are satellites in medium earth orbits (MEO). The moon, earth’s only natural satellite, is approximately nine times farther from the Earth than the ring of geosynchronous satellites. Source: NASA.

    binary systems

    circular motion about the center of mass

    still just a balance between centripetal and gravitational force, but slightly more complicated

    lagrange points

    The three body problem. Lagrange libration points are the simplest solutions (sometimes called Lagragian points or just Lagrange points).

    Still just a balance between centripetal and gravitational forces, but now more complicated

    The five Lagrange points of the Earth-sun system. Satellites in orbit at these locations remain fixed with respect to the Earth and sun. This figure is not drawn to scale.

    L1 and L2 are approximately four times farther from the Earth than the moon. L3 is a very near the “anti-earth” point.

    L4 and L5 are at the vertex of an equilateral triangle formed with the Earth and sun. L4 leads the Earth and L5 follows.

    Paraphrase needed: Objects can settle in an orbit around a Lagrange point. Orbits around the three collinear points, L1, L2, and L3, are unstable. They last but days before the object will break away. L1 and L2 last about 23 days. L1 and L2 last about 23 days. Objects orbiting around L4 and L5 are stable because the Coriolis force keeps them spinning around the Lagrange point.

    • L1
      • Artificial satellites between the sun and earth
        • International Cometary Explorer (ICE), 1978. Originally named International Sun-Earth Explorer 3 (ISEE 3). Moved to solar orbit in 1983.
        • Wind, 1994.
        • Solar and Heliospheric Observatory (SOHO), 1995.
        • Advanced Composition Explorer (ACE), 1997.
        • Genesis, 2001. Moved to solar orbit in 2005.
        • Deep Space Climate Observatory (DSCOVR) Earth Polychromatic Imaging Camera(EPIC), 2015. Provides advanced warning of potentially adverse space weather events and monitors the environment of the Earth on a hemispheric scale. DSCOVR spent ten years locked in storage for dumb-assed political reasons. Originally known as Triana (in honor of Rodrigo de Triana, the first member of Columbus’s crew to spot America). Derisively known as the “GoreSat” and a “multimillion-dollar screen saver” by political opponents of US Vice President Al Gore. (Al Gore was a proponent of DSCOVR during the time he was a presidential candidate.)
        • LISA Pathfinder, 2015? LISA stands for Laser Interferometer Space Antenna. Originally named Small Missions for Advanced Research in Technology 2 (SMART 2). The LISA Pathfinder mission is one component of the larger LISA project to look for low-frequency gravitational waves.
        • Kuafu Project (夸父計劃), 2017? A Chinese space weather monitoring system composed of three satellites. Kuafu A will be placed at L1. Kuafu B1 and B2 will be placed in polar orbits. The project is named after Kua Fu (夸父), a giant in Chinese mythology who died trying to catch the sun. Autotranslators translate Kua Fu (夸父) as “bragadoccio” or “boast father”.
        • Global Sunshade. A proposed geoengineering project to place large opaque satellites at L1 to reduce solar radiation received by earth and counteract global warming.
    • L2
    • L3
      • Fictional “anti-earths”
        • Pythagorean counter earth behind the “central fire”, which is different from the sun. Part of their numerological belief in a 10 body universe. 10 since 1 + 2 + 3 + 4 (the sum of the harmonic series) equals 10
        • Journey to the Far Side of the Sun. (a.k.a. Doppelgangers) 1969
        • Bizzaro World, a.k.a. htraE (Earth spelled backwards). Home of Bizzaro Superman
    • L4 & L5
      • Jupiter trojans
        • 4269 in the L4 Greek camp ahead of Jupiter
        • 2432 in the L5 Trojan camp behind Jupiter
      • Venus trojans
        • 1 temporary trojan at the L4 point, 2013 ND15
      • Mars trojans
      • Saturnian satellite groups
        • Telesto-Tethys-Calypso
        • Helene-Dione-Polydeuces
      • Uranus trojans
      • Neptune trojans
        • 13 L4 trojans. The first confirmed was 385571 Otrera
        • 4 L5 trojans
      • Earth trojans
        • 2010 TK7 Earth’s first confirmed trojan asteroid
        • The twin STEREO spacecraft will spend some time around L4 and L5 but won’t enter into true Lagrangian orbits. STEREO A (for ahead) is approaching L4 and STEREO B (for behind) is approaching L5. STEREO is an acronym for Solar Terrestrial Relations Observatory. We want to encourage good relations between the sun and the Earth.
        • Spitzer Space Telescope, 2013–2015. Was in an approximate L5 orbit (an Earth trailing orbit).
      • Earth-moon (as opposed to sun-earth)
        • The L4 and L5 points in the Earth-moon system have been proposed as prime locations for large “cities in space”. The most famous proponent of such proposals was the American physicist Gerard K. O’Neill who wrote a popular book on the subject in 1977 filled with fantastic illustrations called The High Frontier: Human Colonies in Space.
    • Co-orbital states
      • Prograde, orbiting in the same general direction as the planet
        • The body goes through a cycle of catching up and falling behind the planet. This makes it appear to change direction in the planet’s frame of reference
          • Horseshoe orbit
            • 3753 Cruithne oscillates back and forth between the Earth’s L4 and L5 points (passing through L3 in the middle) once every 770 years. The orbit appears circular in the sun’s frame of reference but horseshoe shaped (or c-shaped) in the Earth’s. Every 385 years it comes to within 15 million kilometers of Earth. The last time was 1900. The next will be 2285.
            • 2002 AA292006 JY262010 SO162013 BS45,
          • Tadpole orbit
            • An apparent orbit around L4 or L5. Elliptical when viewed from the sun, but tadpole shaped (or comma shaped) when viewed from the Earth.
        • When the orbits of two bodies have different eccentricities but similar periods (1:1 orbital resonance), the one on the more elliptical orbit appears to loop around the one with the more circular orbit
      • Retrograde, orbiting in a direction opposite that of the planet
        • Trisectrix (limaçon). A type of roulette curve that crosses itself once resulting in a smaller, more elliptical loop inside a larger, more circular loop.

    noncircular orbits

    qualitative description of noncircular orbits

    centripetal-gravitational forces don’t balance


    • Molyna orbit
      Developed for coverage of areas above 60° (?) latitude. Typically uses three satellites in an unusually elliptical orbit. Each satellite rotates into the farthest point from earth, where it spends about 8 (?) hours. The satellite obeys Kepler’s second law of planetary motion, so the speed at this point is very low. If the period of the satellite is set just right, the satellite will appear relatively motionless during this period.
    • (Hohmann) transfer orbit, opportunities
      The point in an orbit where the engines are fired becomes a point in a new orbit. The burn point is an intersection between the old and new orbits, a point of common tangency in most cases. The burn must occur where the current and desired orbits intersect.

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