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    Re: Plumb-line horizon vs. geocentric horizon
    From: Trevor Kenchington
    Date: 2005 Feb 10, 10:04 -0400

    Jim wrote:
    > The geoid varies from the ellipse around the world by up to + 73 m in New
    > Guinea, and - 105 m off the coast of southern India. Those would be
    > significant heights of eye.
    Not heights of eye.
    The issue is the deflection of the plumb line, which is the same angle
    as the tilt of the horizon (relative to a plane tangent to some assumed
    ellipsoid). If calculated from the geoidal height, that tilt would be
    the arctangent of the difference in geoidal heights at the observer's
    location and at his horizon, divided by his horizon distance. Unless the
    observer is high up a mountain, his horizon will not be far away and the
    geoidal height there will be very similar to the geoidal height at his
    own location. As Pierre reported, a deflection of several tenths of a
    minute is about the largest anywhere on the planet.
    > But is it necessary to consider this in
    > splitting-hair celestial navigation (putting aside the obvious point that
    > the differences are not practical for most CN)?
    If you seek precision of better than 0.5 miles, then a deflection of
    greater than a few tenths of a minute needs a correction. If you only
    need your position to within 5 miles, I expect that you can ignore
    deflection anywhere on the World Ocean. I guess that most of us are
    satisfied with the latter when actually on a boat.
    > Are the Nautical Alamanc's
    > data referenced to the earth's ellipse, or to a model based on the geoid?
    > Canada's geoid is only +/- 5 cm off the center of the earth (Natural
    > Resources Canada website today).
    Neither. My understanding is that the Almanac's data are referenced to a
    spherical world. If it was otherwise, we would have to enter the tables
    with an argument based on our latitude (if referenced to an ellipsoid)
    or our actual position (if referenced to the geoid). However, since the
    light a celestial body reaches the Earth as parallel rays (neglecting
    parallax for a moment), the angle between those rays and a line drawn at
    45 degrees to the Equator is the same angle, whether that line is drawn
    through the centre of the Earth (as in the definition of geocentric
    latitude) or some 12 miles away (as in the definition of geoidal latitude).
    That leaves the effects of non-sphericity on parallax, particularly of
    the Moon. Since the Nautical Almanac provides no corrections, I assume
    that those effects are negligible (meaning less than 0.1 minutes of arc).
    As to "Canada's geoid": Do we not use NAD83 or WGS84? Since they were
    designed around satellite systems, the centre of the ellipsoid certainly
    should coincide (very closely) with the centre of gravity of the planet.
    Nice that have got it within a couple of inches!
    > Here is my current understanding (my own words, not an authority):
    > "Astronomical latitude is the  angular distance between the plumb line and
    > and the celestial equator. Since the celestial equator is coincident with
    > the geoid's equator, then astronomical latitudes are the same as terrestrial
    > latitudes.
    Not quite -- because of deflection of the plumb line.
    > Geodetic latitudes plotted on an accurate model of the geoid will
    > very closely match astronomical latitudes. This is the critical connection
    > between a sextant observation, the astronomical data in the Almanac, and
    > your chart: If your chart is based on a sound model of the geoid in the
    > region you are sailing, then a latitude determined from a sextant sight will
    > be nearly the same as the latitude on your chart.".
    I think that is right -- though you have, of course, assumed away all of
    the other errors in the determination of latitude by sextant.
    > But I am still not clear on the coordinate reference model on which the
    > Nautical Almanac is based.
    See above.
    Trevor Kenchington
    Trevor J. Kenchington PhD                         Gadus{at}iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
                         Science Serving the Fisheries

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