Earth and Sky: Coordinate Systems and the Turning Earth

Locating Places on Earth

  • Earth’s axis of rotation defines the locations of its North and South Poles and of the equator, halfway between.
  • East is the direction toward which Earth rotates, and west is the opposite; at almost any point on Earth the four directions—north, south, east, and west—are well defined, with the exception that at the exact North and South Poles, east and west are ambiguous because points there do not turn.
  • To map positions on a sphere, we use circles that play the role of a rectangular grid on a plane. A few key ideas:
    • A great circle is any circle on the surface of a sphere whose center is the center of the sphere (the greatest possible circle on a sphere).
    • Earth’s equator is a great circle.
    • Meridians are a family of great circles that pass through both the North and South Poles and are perpendicular to the equator, crossing it at right angles.
    • Any point on Earth has a meridian passing through it, which specifies its east–west location (longitude).
  • Longitude is defined as the number of degrees of arc along the equator between your meridian and the Prime Meridian (the meridian that passes through Greenwich, England).
    • The Prime Meridian has longitude 0°.
    • Longitudes are measured east or west of Greenwich from 0° to 180°.
    • Example: the time-honored clock-house benchmark of the U.S. Naval Observatory in Washington, DC has longitude 77.066^\circ\mathrm{W}.
  • Latitude is the north–south position, measured as the number of degrees north or south of the equator.
    • Latitudes range from 0° at the equator to 90° at the poles (0° to 90° north or south).
    • Example: the latitude of the Naval Observatory benchmark is 38.921^\circ\mathrm{N}; the poles have latitudes of 90^\circ\mathrm{N} (North Pole) and 90^\circ\mathrm{S} (South Pole).

Why Greenwich? The Prime Meridian and the global grid

  • Greenwich was chosen as the location for the 0° longitude due to its between-Europe-and-United-States position and its historical role in developing methods to measure longitude at sea.
  • Longitudes are measured to the east or to the west of the Greenwich meridian from 0^\circ to 180^\circ.
  • The 0° longitude line is the Prime Meridian, marking the reference from which all east/west measurements are taken.
  • Washington, DC example: longitude 77^\circ\mathrm{W} and latitude 38^\circ\mathrm{N}.

Locating Places in the Sky

  • Positions in the sky use a coordinate system analogous to Earth’s, but defined on the celestial sphere.
  • The sky is often represented as a celestial sphere with markers:
    • The north celestial pole and the south celestial pole (where the Earth’s axis would pierce the celestial sphere).
    • The celestial equator, a great circle on the celestial sphere that lies in the same plane as Earth’s equator.
  • Declination (δ) is the celestial analogue of latitude:
    • δ is measured from the celestial equator toward the north (positive) or toward the south (negative).
    • Example: Polaris has a declination close to +90°.
  • Right ascension (RA) is the celestial analogue of longitude:
    • RA is measured from the vernal equinox, the point where the Sun’s ecliptic crosses the celestial equator.
    • RA can be expressed in degrees or in units of time, because the sky appears to turn once per day as Earth rotates.
    • The full circle corresponds to 360°, which is equivalent to 24 hours of RA.
    • Therefore, 360^\circ = 24\ \text{h} and 1\ \text{h} = 15^\circ.
    • Equivalently, RA in degrees relates to RA in hours by\text{RA}{\text{deg}} = \text{RA}{\text{h}} \times 15^\circ and\text{RA}{\text{h}} = \dfrac{\text{RA}{\text{deg}}}{15^\circ}.
  • Example (Capella): The approximate celestial coordinates are given as RA and δ, but the transcript provides these values incompletely (the exact numbers are not included).
  • Visualization: One way to imagine these circles is to picture Earth as a transparent sphere with terrestrial coordinates painted on the inside; the celestial sphere is projected around you, with the terrestrial poles, equator, and meridians appearing as dark shadows on the sky-sphere.

The Turning Earth (Evidence for rotation)

  • Why do stars appear to rise and set? This apparent turning can be accounted for either by a daily rotation of the sky around a stationary Earth or by the rotation of the Earth itself.
  • Since the 17th century, it has been generally accepted that the Earth turns, but a clear demonstration was needed.
  • Jean Foucault provided an unambiguous demonstration in 1851: he suspended a 60-meter pendulum with a mass of about 25 kilograms from the Pantheon in Paris and started the pendulum swinging with a steady plane of oscillation.
    • After a few minutes, the plane of oscillation had shifted, showing that the Earth was rotating beneath the pendulum, not that the pendulum was rotating independently of the Earth.
    • This device—Foucault’s pendulum—has become a common exhibit in science centers and planetariums worldwide.

Quick references and notes

  • Poles and equator define axes used to map both Earth and sky.
  • The coordinate systems are designed to uniquely identify celestial and terrestrial objects in a manner similar to a city’s grid.
  • The conversion between angular measurements and time is a practical consequence of Earth’s rotation and the daily cycle of the sky.
  • The concepts connect foundational ideas from the earlier lecture on Observing the Sky: The Birth of Astronomy (Earth-centric view evolving into a rotating-Earth view) and set up practical tools for locating objects in both the terrestrial and celestial contexts.

Summary of key formulas and definitions (for quick study)

  • Great circle: a circle on a sphere whose center coincides with the sphere’s center.
  • Equator: a great circle equidistant from the poles; latitude 0°.
  • Meridian: a great circle passing through the poles; perpendicular to the equator; defines longitude.
  • Longitude: the angle between the Greenwich meridian and your meridian, measured along the equator; range 0^\circ\le\text{Longitude}\le 180^\circ (east or west).
  • Latitude: the angle north or south of the equator; range 0^\circ\le|\text{Latitude}|\le 90^\circ.
  • Prime Meridian: longitude 0° (through Greenwich).
  • Declination: \delta, measured from the celestial equator toward the north (positive) or south (negative), with\delta\in[-90^\circ,+90^\circ] and Polaris having \delta\approx +90^\circ.
  • Right ascension (RA): measured along the celestial equator from the vernal equinox; can be expressed in degrees or hours.
    • Relationship:360^\circ = 24\ \text{h} and 1\ \text{h} = 15^\circ.
    • Conversion:\text{RA}{\text{deg}} = \text{RA}{\text{h}} \times 15^\circ and\text{RA}{\text{h}} = \dfrac{\text{RA}{\text{deg}}}{15^\circ}.
  • Visual model: Earth is a transparent sphere; celestial sphere is the inner view showing projected coordinates.
  • Foucault pendulum: a qualitative demonstration of Earth’s rotation; a long pendulum’s plane of oscillation changes over time due to Earth rotating underneath it.