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    Re: Sextant Positions versus Map Datums?
    From: Paul Hirose
    Date: 2002 Jan 21, 1:25 AM

    What an interesting discussion. I've followed it closely and saved all
    the messages in a dedicated folder. For the first time I've had to
    really think out how our spherical Earth assumptions affect celestial
    navigation.
    
    It's my belief that the almanac and sight reduction tables are equally
    valid on both a sphere and an ellipsoid. Let's explore this with a
    thought experiment.
    
    Imagine the inside of the celestial sphere is marked with parallels
    and meridians, so an observer on Earth can look up and see them. For
    simplicity's sake, assume there's no Earth rotation.
    
    Initially, let's pretend Earth is a perfect sphere and you are at 30N
    90W. At this location set up a surveyor's transit, level it, and
    adjust the horizontal circle to read zero when the telescope is
    sighted on the celestial pole. By doing this you have aligned the
    instrument's vertical axis to the zenith, which is the intersection of
    the 30N and 90W lines on the celestial sphere. You've also brought the
    horizontal circle's zero point into the plane of the 90W celestial
    meridian.
    
    If I name a specific parallel and meridian on the celestial sphere,
    can you compute the altitude and azimuth required to direct the
    transit's scope to that spot in the sky? Yes, of course. This is
    basically the same problem we solve during a sight reduction. Like our
    imaginary lines in the sky, the coordinates in the almanac are pure
    spherical coordinates and aren't based on any datum. If we assume
    Earth is spherical too, computing altitude and azimuth is
    straightforward.
    
    But what if the world is ellipsoidal? Doesn't that make the
    computation much more difficult? No, because of the way latitude is
    defined on an ellipsoid. Mapping and navigation use geodetic latitude,
    which is the angle between a normal to the ellipsoid (e.g., a plumb
    line) at your location and the equatorial plane. (Meeus calls this
    "geographical latitude".)
    
    So if you're at 30N 90W on an ellipsoid, the 30N and 90W lines on the
    celestial sphere intersect at your zenith, just as they did on the
    spherical Earth. Leveling your transit and aligning it to north gives
    it the same orientation relative to the celestial sphere that it had
    on the spherical world.
    
    To be sure, its location has changed. But since this movement is
    negligible compared to the enormous radius of the celestial sphere,
    the apparent positions of points on the sphere don't change. Thus on
    an ellipsoidal Earth you can accurately compute Hc and Zn with
    spherical trig.
    
    But we're not quite done yet. The actual shape of the Earth, as sensed
    by a transit or sextant, is the geoid. That's an imaginary sea level
    surface perpendicular to gravity at all points. Its shape is extremely
    complex, too hard to handle in routine computations. That's why datums
    assume Earth's cross section through the poles is a perfect ellipse.
    
    Unfortunately our instruments know nothing about this, and align
    themselves to gravity. Their celestial observations produce astronomic
    latitude and longitude, which are coordinates on the geoid and aren't
    in any datum at all. (I guess you could call it "God's datum".) But
    the geodesists who create a datum always try to make it a good overall
    fit to the geoid. Astronomic coordinates should match the geodetic
    coordinates quite well.
    
    Exactly what "quite well" means, depends on the angle between a normal
    to the ellipsoid and the local vertical at that spot. The angle is
    called "deflection of the vertical", and is separated into north-south
    and east-west components. By the standards of the celestial navigator,
    deflection of the vertical is very small, but it can be greater than
    some observational errors.
    
    http://www.ngs.noaa.gov/GEOID/DEFLEC99/
    
    Click the interactive computations link and try inputting some
    coordinates. I tried that star atop the Texas capitol dome -- sum of
    N-S and E-W deflections came to 4.08 seconds, about 400 feet on the
    ground. That's how much a perfect celestial fix would differ from NAD
    83.
    
    
    Bottom line:
    
    1. The tabulated coordinates in the almanac are independent of datum.
    
    2. The correction for Moon's parallax is slightly (.2') affected by
    the non-spherical shape of Earth. The almanac's text explains how to
    compute an accurate value. The Moon correction tables don't bother
    about this, though. All other bodies are so far away, Earth may be
    considered a sphere as far as parallax is concerned.
    
    3. Sight reduction tables and formulas lose no accuracy on an
    ellipsoidal world. The plotting is slighly affected. But given the
    geometric liberties taken by the intercept method, the accuracy of
    sextant observations, and the small size of the intercepts, "one
    minute equals one mile" is close enough. Sight reduction programs
    which converge interatively on the solution, like the method described
    in the almanac, can finish with very short intercepts. Here
    non-spherocity should have virtually no effect.
    
    4. The discrepancy between the astronomic coordinates obtained from
    celestial observations and the geodetic coordinates on charts is
    slight, at least in NAD 83. However, for a perfectionist applying all
    possible corrections it can be significant.
    
    --
    
    
    paulhirose---.net (Paul Hirose)
    

       
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