Discussion

circular orbits

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

F  = Fg	⇒	m2v2	=	Gm1m2	⇒	v = √	Gm
		r		r2			r

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

v = √	Gm	=	2πr
	r		T
	Gm	=	4π2r2
	r		T2
	r3	=	Gm	= constant
	T2		4π2
	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
  • 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

[magnify]

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

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

[magnify]

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.

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. 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), August 1978
      ICE was originally named ISEE 3 (International Sun-Earth Explorer 3).
    • Wind, November 1994
    • Solar and Heliospheric Observatory (SOHO), December 1995
    • Advanced Composition Explorer (ACE), August 1997
    • Genesis, August 2001
    • LISA Pathfinder, 2010
    • Kuafu (夸父計劃), 2012
    • Deep Space Climate Observatory (DSCOVR), in storage for dumb-assed political reasons
      Also known as Triana (in honor of Rodrigo de Triana, the first member of Columbus' crew to spot America) and derisively called "GoreSat" and an "overpriced screen saver" by dumb-asses.
    • Global Sunshade
      A proposal to place large opaque satellites at L1 to reduce solar radiation received by earth and counteract global warming.
• L2
  • Artificial satellites in earth's shadow
    • Geotail? 1992
    • Wilkinson Microwave Anisotropy Probe, June 2001
    • Eddington, 2008 (cancelled)
    • Herschel and Planck, May 2009
    • Gaia, December 2011
    • James Webb Space Telescope, originally known as the Next Generation Space Telescope, 2014
    • Darwin, 2015
    • Terrestrial Planet Finder (TPF)?
    • New Worlds Mission (Starshade)?
    • Advanced Technology Large-Aperture Space Telescope (AT-LAST or ATLAS Telescope)?
• L3
  • Fictional "anti-earths" behind the sun
    • Journey to the Far Side of the Sun. (a.k.a. Doppelgangers) 1969
    • "Bizzaro World" — home of Bizzaro Superman
    • The original planet Vulcan?
• L4 & L5
  • Proposed location for large-scale, "cities in space" orbiting the earth
  • Jupiter has hundreds of Trojan asteroids.
    • Greek camp ahead of Jupiter
    • Trojan camp behind Jupiter
  • Mars
    • 1990 MB (5261 Eureka)
    • 1998 VF31
    • 1998 QH56 ?
    • 1998 SD4 ?
    • 1999 UJ7
    • 2007 NS2
  • Saturnian satellite groups
    • Telesto-Tethys-Calypso
    • Helene-Dione-Polydeuces
  • Neptune trojans (ahead of the planet)
    • 2001 QR322
    • 2004 UP10
    • 2005 TN53
    • 2005 TO74
    • 2006 RJ103
    • 2007 VL305
  • Earth
    • Cruithne
      I think this is from New Scientist: "Even more peculiar is the 'horseshoe orbit' in which the third body turns around the three points of equilibrium, L3, L4 and L5. Cruithne is such an object. Discovered in 1997, it is a 5 km diameter asteroid that takes 770 years to complete its horseshoe orbit. Thus every 385 years it comes to its closest point to Earth, some 15 million kilometers. Last time was in 1900, next — if you can wait — will be in 2285."

noncircular orbits

qualitative description of noncircular orbits

centripetal-gravitational forces don't balance

uses

• 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.