A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2020 Feb 2, 18:14 -0800
The Earth has a flattening or oblateness. The pole-to-pole diameter is less than the equatorial diameter. But the Earth has all sorts of lumps and bumps, both in its physical form in terms of mountains and ocean trenches, and also in its purer gravitational form (the geoid) which still has various undulations though much smaller in scale. Deciding exactly how much flattening there is in the real Earth is a bit problematic. So let's not.
The flattening or oblateness of the Earth is one of the two key parameters in any idealized ellipsoid model of the Earth used as a basis for mapping, known as a "datum" (the other is the mean radius in meters or some other physical unit or any equivalent). The only ellipsoid or datum that matters today, of course, is the WGS84 datum. All others are obsolete, obsolescent, or confined to highly specialized fields. In the WGS84 ellipsoid, which is a pure mathematical form with no mountains, valleys, or other ups and downs, the oblateness of the Earth is 1/298.257223563 or roughly one part in 298. For a quick number, just call it 1/300.
You can do some order of magnitude numbers knowing nothing more than the oblateness ratio. For example, when you look straight down (to the nadir opposite your zenith), you're not looking at the center of the Earth unless you're on the equator or at either pole. Anywhere else your view is slightly tilted. By symmetry you might guess correctly that the maximum deflection is at 45° latitude. How big is the angular deflection? Well, it's going to be of the same order of magnitude as the oblateness of 1/300. That's an angle as a pure ratio (a.k.a. "radians") so as usual we can convert to minutes of arc by multiplying by 3438: 3438/30 = 11 minutes of arc. So as a first estimate, when you look straight down, you're looking 11' away from the line that points straight to the Earth's center in mid-latitudes. That might even be a good estimate. ;)
While one part in 300 isn't much, over the scale of the entire Earth, and when we're dealing with small details like the angles in celestial navigation, it can add up.