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Trevor and George, Thank you for pointing me on the right path.  How does
this summary look?

Jim Thompson
jim2@jimthompson.net
www.jimthompson.net
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Outgoing email scanned by Norton Antivirus

Where is the center of the earth?

Most images in celestial navigation texts and my web pages show the earth as
a perfect sphere, placing the terrestrial equator, celestial equator, and
celestial horizon through the dead center of that sphere. While this
simplicity makes learning easier, and in fact is not far off the truth, the
spherical model does not represent the real shape of the earth. The
difference is very small, but needs to be considered. See article 204 in
Bowditch 2002.

The simple spherical model says that a horizon is a curve on the celestial
sphere carved by a plane that is perpendicular to a line drawn from the
center of the earth through your position on the surface of the earth. It is
convenient, and not far off the truth, to think that the local vertical line
that you measure with a sextant runs through the center of the earth, but in
fact the horizon planes are perpendicular to the direction of gravity (the
plumb line), which does not necessarily follow that path. Bowditch 2002:
"Horizontal, adj. Parallel to the plane of the horizon; perpendicular to the
direction of gravity.". Since you look at the sea horizon to measure the
vertical angle to a celestial body with a sextant, and since the sea horizon
is very nearly perpendicular to the direction of gravity, except for tiny
variations in some parts of the planet, then the vertical angle you measure
with a sextant is also perpendicular to gravity. Sextants automatically
measure on the local vertical (direction of gravity) from the local visible
horizontal (horizon), which is 90? to the local vertical. The local vertical
is a plumb line, not the line to the center of the earth.

The angular difference between the plumb line (direction of gravity) and the
line to the geometric center of the earth is near zero at the equator and
poles, but varies to a maximum of  about 00? 11' arc at 45? Latitude,
because the earth is not a perfect sphere. The difference between the two
lines would result in up to about 11 miles of error (11' of arc) in the
mid-latitudes if charting was done based on the geometric center of the
earth. Modern charting methods use datums and mathematical models that more
closely approximate the real shape of the earth, so the difference between
the local vertical based on the direction of gravity and the local vertical
based on the direction to the center of the charting model is very little.

All this is not important for celestial navigators because the angular
difference between the plumb line and a line to the center of whatever
mathematical model is used to describe the shape of the earth for charting
in the navigator's region is not signfiicant enough to cause problems in
most practical celestial navigation. Manually obtained CN positions
generally are no more precise than 1-2 miles, but the difference between the
two vertical lines is much less than that. However mechanical celestial
navigation systems that obtain positions to within a few meters do need to
consider the difference in angle between the plumb line and the line drawn
to the center of the local charting model. For example, that small
difference could lead to serious imprecision in a computerized system that
uses an automatic star tracking device to precisely measure star altitudes
above a visible horizon, if that system was trying to achieve a precision of
a few meters.

Jim

   

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