# NavList:

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

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Re: Plumb-line horizon vs. geocentric horizon
From: Frank Reed CT
Date: 2005 Feb 10, 22:53 EST
Chas you wrote:
"My concept of gravity is that it is a force that orginates
from the center of mass of a body. On earth, that place
would be earth center regardless of the earth shape
model."

The gravitational force vector points to the exact center of an object only in one very specific case: an object with exact spherical symmetry. If an object can be divided up into a series of spherical shells, like the layers of an onion, then even if the shells have varying density, the gravitational force outside the object will behave *exactly* as if all of the mass of the object were concentrated at a single point at its center. In every other case, the field is more complicated and there is no net "center of gravity" from which the gravitational force seems to emanate as seen from all points around the object.

Consider a simple case that necessarily would contradict the idea that gravity emanates from the center of mass of an object. Imagine a small planetoid in the shape of a torus --the Planet Donut. If you stand anywhere on the surface of this "donut", you will feel a gravitational tug holding you on the ground. On the outside of the donut, gravity points towards the center, but on the inside surface, it points back towards the outside again. It only points towards the center of mass --an empty spot at the center of the torus-- in a few places with special symmetry. To work out the net gravitational field at any point, you would have to do an integration (easily performed by simple summing on a home computer) adding up the gravitational pull from each small bit of matter within the torus (or any other arbitrary shape).

Back to the Earth. Our planet is well-approximated to first order as a sphere. But since it is rotating and not perfectly rigid, it has been "spun out" into a an oblate spheroid with an oblateness of about 1 part in 297. This shape can be approximated to "second order" by taking a sphere a wrapping it with an equatorial torus --put a belt on the Earth. That belt of matter pulls a plumb bob towards the equator. It pulls most strongly when you're near the equator, but then it's mostly in the vertical direction so the maximum tilt of the plumb-line from this cause is in mid-latitudes.

Now for the confusing part. Satellites in orbit experience the Earth's oblateness entirely via this "belt" of matter around the Earth's center. It causes their orbits to precess (both in perigee and node) at a fairly substantial rate. Gravity is stronger near the equator because there's more mass there. Observers on the Earth's surface, however, experience an additional force which is observationally indistinguishable from an addition to gravity. This is the centrifugal effect that I mentioned in that earlier post. Because objects along the Earth's equator are being "flung out" by the Earth's rotation, the "net" observed gravitational field is weaker at the equator instead of stronger --despite the belt of mass around the equator. Fortunately, we never have to worry about this difference. Because the Earth is a semi-fluid object and because we sail on the very fluid oceans, the surface of the ocean is always perpendicular to the local plumb-line (from the net of gravity and centrifugal force). And there is a simple reason for this: if the surface of the fluid were *not* perpendicular to the local plumb line, then part of the local net gravity would be a force along the surface and would cause the fluid to move. In other words, it would create a current until the surface reached a perpendicular state.

And you wrote:
"If one considers a plane that is tangent at the
observer's location to the geoidal earth surface
and constructs a line perpendicular with the plane
there, this line would indeed miss earth center.  I
would consider that line to define the local vertical.
With the exception of the poles and points on the
equator, it would not coincide with the direction
defined by a plumb line."

That's the thing-- the line you're calling the local vertical is actually identical to the local plumb line that you could set up with a bit of plumbum on the end of a string.

-FER
42.0N 87.7W, or 41.4N 72.1W.
www.HistoricalAtlas.com/lunars
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