Coordinate Systems

How positions are described: where the origin is, how the axes point, and what kind of coordinates are used. See also ENAE 441: Reference Frames.

Defining a Coordinate System

1. Center: location of the origin
   1. Center of the Earth / gravitation (orbit center)
   2. Observation site on the surface
   3. A satellite
2. Orientation: principal (first) axis, polar (third) axis
   1. For Earth-centered: the polar axis is the north pole axis
   2. For orbit-centered (perifocal): angular momentum
   3. Satellite-centered: angular momentum
3. Type: Cartesian, polar
   1. Cylindrical
   2. Spherical
   3. Distance

Spherical and Cartesian Coordinates

• Spherical to Cartesian (Matlab sph2cart, AstroPy spherical_to_cartesian):
• Cartesian to spherical (Matlab cart2sph, AstroPy cartesian_to_spherical):
  • with the two-argument arctangent
  • : azimuthal angle in the equatorial plane; : elevation above it

Earth-Centered Frames

• Curtis Fig 1.18: Earth-centered inertial frame and an Earth-centered noninertial frame embedded in and rotating with the Earth
  • along the vernal equinox direction, along the spin axis
  • through the Greenwich meridian; a surface point is located by east longitude and north latitude , with a local East / North / Zenith frame drawn at
• Two Earth-centered reference frames:
  1. Inertial: principal axis is fixed to the vernal equinox
  2. Rotating (body fixed): principal axis points to the prime meridian
• Celestial sphere (directions in space) vs Earth sphere (points on the ground)

Geocentric Equatorial Frame

• Curtis sec. 4.2
• Geocentric equatorial , also called ECI:
  • = principal axis, points to the vernal equinox
  • = polar axis
  • Equinox, or ascending node (where the Sun crosses the equator going north)
• Polar angles:
  • Right ascension
  • Declination
• The axes slowly move because of the precession of the equinoxes, nutation, and polar motion (see Precision Reference Frames)

Earth-Centered Earth-Fixed Frame

• ECEF: rotates with the Earth
• Polar angles:
  • Longitude
  • Latitude

Transformation Between ECI and ECEF

• Sketch: in the equatorial plane, the prime meridian is at angle from , and a local meridian is a further (east longitude) beyond it
• : right ascension of the prime meridian = Greenwich sidereal time (GST)
• : right ascension of the local meridian = local sidereal time
• Transform by rotating between the two frames (about the common polar axis by )
• as a function of time: Sidereal Time

Lunar Coordinate Frames

• 
  • Earth obliquity ; lunar orbital inclination to the ecliptic
  • Lunar obliquity (to its orbit plane), to the ecliptic
• Non-rotating frame: principal axis defined by the cross product of the Earth’s mean rotational pole of J2000 with the Moon’s rotational north pole
• Rotating frames:
  • PA: principal axis frame, based on the Moon’s moments of inertia
  • ME: mean Earth (older)