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    Re: One-body fix
    From: Geoffrey Kolbe
    Date: 2009 Apr 27, 07:00 +0100

    
    At [NavList 8049], Peter Hakel wrote
    :
    ==========================
    
    Thank you, Frank, for the detailed discussion of the rapid-fire fix from
    multiple observations of a single body during a very short time
    interval.  Coincidentally I have just been looking into the rather academic
    problem of obtaining a fix from the altitude and azimuth of a body. I
    understand that this is not done in practice because azimuth is
    difficult/impossible to measure to sufficient accuracy.  Nevertheless, I
    would like to know just how accurately one can measure azimuth in the
    field/ocean and what portable devices (besides compasses), are available
    for this (if any).
    
    I hereby display the solutions to this "one-body fix" problem, if only as a
    curiosity.  I expect that these have been worked out before but I haven't
    found them anywhere so far (references would be much appreciated).  The
    known quantities are the GP (Dec, GHA), observed altitude (Ho), and azimuth
    to GP (Zn).  From the navigation triangle we can calculate the LHA and then
    the Latitude:
    
    ==========================
    
    If your position is unknown, the problem with measuring azimuth is knowing
    where True North is. If you can see Polaris, measuring azimuths on land is
    not a problem, just use a theodolite. Though for latitudes higher than
    about 50 degrees it would be difficult to get your eyeball behind the
    telescope, unless you have a 'broken' theodolite. Too, the errors in
    measuring azimuths go as (the error in how level your instrument is) times
    (the tan of the altitude), so the chosen bodies need to be low in the sky.
    
    Given a reasonable 30x telescope on an average theodolite, the surveying
    books say you should be able to see Polaris on a clear day. On a trip to
    Egypt last November, I spent a frustrating afternoon trying to see Polaris
    in a reasonably clear sky. No luck. I think it has to be a very clear sky
    to see this second magnitude star in the daytime.
    
    The other thing is that azimuths are not like position lines, which are
    equally useful regardless of the azimuth of the celestial body. Consider a
    one-body fix on Polaris, for example. The altitude of Polaris will give you
    your latitude (pretty much) and the azimuth of Polaris tells you.... what?
    Regardless of where you are in the Northern Hemisphere, the azimuth of
    Polaris is going to be pretty much the same and tells you not much more
    than you are in the Northern Hemisphere! So a one-body fix on Polaris will
    not be much use to fix your position. Now consider the sun rising in the
    East or setting in the West at the equinoxes. The local time of this event
    will give you your longitude - but anybody anywhere on that longitude will
    see the same event (sunrise in the East or sunset in the West) at the same
    time, so an azimuth of the sun will not tell you much.
    
    So, for useful azimuth measurements, use bodies low in the sky, avoid
    bodies at the Cardinal points.
    
    Geoffrey Kolbe
    
    
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