Discuss with respect to celestial navigation Earth as a spheroid-compression

Of course. This is an excellent and fundamental topic in the theory of navigation. Let’s discuss the Earth as a spheroid with respect to celestial navigation, breaking it down from the simple model to the more complex reality and its practical implications.

1. The Simple Model: The Perfect Sphere

For introductory teaching and most day-to-day calculations in celestial navigation, the Earth is assumed to be a perfect sphere. This simplification is incredibly useful because:

• Simple Geometry: All great circles (like the equator and lines of longitude) are perfect circles of the same radius.
• Constant Nautical Mile: One minute of arc along any great circle is equal to one nautical mile. This is the very definition that underpins our distance measurements and chart scales.
• Simple Latitude: Latitude is simply the angle at the center of the Earth between a point on the surface and the equatorial plane.

For a navigator using a sextant at sea, aiming for an accuracy of 1-2 nautical miles, the spherical model is almost always good enough. The small errors it introduces are typically less than the inherent errors of the observation itself (timing, sextant reading, atmospheric refraction).

2. The Accurate Model: The Oblate Spheroid (and its “Compression”)

In reality, the Earth is not a perfect sphere. Due to its rotation, centrifugal force causes it to bulge at the equator and flatten at the poles. This shape is called an oblate spheroid.

• Compression: This “flattening” or “compression” is the key characteristic. The Earth’s polar radius is shorter than its equatorial radius.
  • Equatorial Radius: ~6,378.1 kilometers (3,963.2 miles)
  • Polar Radius: ~6,356.8 kilometers (3,949.9 miles)
  • The Difference: The diameter at the equator is about 43 km (27 miles) greater than the pole-to-pole diameter.
• The Model: The standard model used for all modern mapping, GPS, and navigation is the World Geodetic System 1984 (WGS-84), which defines the Earth as a specific oblate spheroid.

3. How Does the Spheroid Shape Affect Celestial Navigation?

While the spherical model is a useful approximation, the underlying mathematics and reference systems of modern navigation are built upon the more accurate spheroid model. Here are the specific effects:

a) The Definition of Latitude (The Most Important Effect)

On a sphere, the line of gravity (a plumb line) points directly to the center of the Earth. On a spheroid, this is not true except at the poles and the equator. This leads to two different definitions of latitude:

• Geocentric Latitude ($\phi’$): The angle at the Earth’s center between the equatorial plane and the point on the surface. This is the “simple” spherical definition.
• Geodetic Latitude ($\phi$): The angle between the equatorial plane and a line drawn perpendicular (normal) to the surface of the spheroid at that point.