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    Re: Horizons, was Summary of Bowditch Table 15
    From: Jim Thompson
    Date: 2005 Jan 31, 08:49 -0400

    Trevor and George, Thank you for pointing me on the right path.  How does
    this summary look?
    
    Jim Thompson
    jim2{at}jimthompson.net
    www.jimthompson.net
    --------------------
    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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