# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Spherical earth model vs. ellipsoid**

**From:**Mike Wescott

**Date:**1999 Mar 15, 6:37 PM

My apologies if this has been seen already, but I think my previous attempt was lost. Lu Abel wrote: > As my mother used to say, "you're in the right church, but sitting in > the wrong pew." Actually we're both in the wrong pew (I oversimplified), But I think I'm closer than you :-) > Latitude and longitude are defined as angles with respect to the > earth's center. But Mike is definitely right that our horizon will not > be perfectly perpendicular to the line of L/Lo eminating from the > earth's center. I would correct his diagram by labelling the line > perpendicular to the local horizon "altitude" and not "latitude." No. Not exactly anyway. What follows is derived from App X, "Geodesy for the Navigator", Vol 1. of the 1984 Ed of Bowditch, and from the Glossary in Vol 2. See also the attached figure. The geoid (the black lumpy curve of fig-1) is the Earth's surface at mean sea level. I.e., it is the surface to which oceans would conform over the entire earth if the oceans were free to adjust to the combined effects of gravity and the earth's rotation. Because the distribution of the mass of the earth is not uniform. The surface of the geoid is an irregular curve. One aspect of this surface is that gravitation is everywhere equal on this surface and everywhere perpendicular to the surface. Because the geoid varies over the surface of the earth and is not easily predictable, geographers (or would that be geodicists?) have defined ellisoids to closely model the geoid. Different ellipsoids have been defined for different puposes or different areas. An ellipsoid that models North America well will not work as well in Austrailia, for example. One such ellipsoid is reresented by the red curve, and red axes in fig-1. Note that the axes do not necessarily coincide with the celestial axes (shown in black). Terrestrial Latitude (green) is the angle a ray from the center of the earth makes with the celestial equatorial plane. Geocentric Latitude (red) is the angle a ray from the center of the standard ellipsoid makes with the equatorial plane of the standard ellipsoid. Astronomic Latitude (black) is the angle a normal to the geoid makes with the celestial equator. This "normal" is the direction a plumb-bob will point. Astronomic Latitude is what is measured by a sextant or theodolite. Geodetic Latitude (magenta) is the angle a normal to the standard ellipsoid makes with the equatorial plane of the ellipsoid. Longitudes are similarly defined. What is used on charts? Geodetic coordinates. What do we measure with a sextant (et al.)? Astronomic coordinates. What's the difference? Not much, because the ellipsoids are defined to minimize measurement errors over some specific region (or globally, as with WGS-84) maximum "deflections of the vertical", as such differences as called, are less than 0.5' if Bowditch can be believed. So, where do Geocentric and Terrestrial coordinates come into play? In short, they don't. > Let's clearly understand that with celestial observations, we're trying > to use observations made at the earth's surface and almanac data to > deduce our L/Lo. Agreed. > But L/Lo have to be defined with respect to the earth's center, or else > how do we end up with the situation where one may move slightly more or > slightly less than a nautical mile when one makes a one minute change > in latitude near the equator or north pole? Because the surface of the Earth and the Geoid are closer to an ellispoid than to a sphere this situation will occur for any of these definitions of Latitude. And if you look at the mathematics behind the Mercator Projection and behind "Table 6: Length of a Degree of Latitude and Longitude", you'll find the assumption that Latitude is Geodetic Latitude, defined by the perpendicular to the tangent plane at the surface of the ellipsoid. > When I made my earlier comment on this thread, I did recognize that the > oblateness of the earth causes the local horizon to be not precisely > perpendicular to a line drawn from the center of the earth, but I chose > not to cloud a discussion of whether elipsoidal models affect celestial > fixes with that relatively minor point. > Unfortunately, my trig and calculus are sufficiently rusty that I can't > dash off an estimate of the error induced by the earth's oblateness. > Anybody?? The difference between Geocentric and Geodetic Latitude (using WGS-84) is 0 at the poles and at the equator. The difference maximizes at about 45d latitude with a difference of 11.5 minutes: Geodetic Geocentric Diff in Latitude Latitude minutes ======== ========= ========= 0.000 0.000 0.0 15.000 14.904 5.8 30.000 29.834 10.0 45.000 44.808 11.5 60.000 59.833 10.0 75.000 74.904 5.8 89.000 88.993 0.4 90.000 90.000 0.0