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    One-body fix
    From: Peter Hakel
    Date: 2009 Apr 26, 11:11 -0700
    Re: [NavList 8049] The rapid-fire fix

    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:

    sin LHA = sin Zn * cos Ho / cos Dec

    sin Latitude = ( sin Ho * sin Dec  -  cos Ho * cos Dec * cos LHA * cos Zn )  /  ( 1  -  cos Ho * cos Dec * sin LHA * sin Zn )

    The denominator in the Latitude expression can vanish under very special circumstances, in which the result is Latitude = 0.  I don't think we need to worry about cos Dec = 0 in the first formula.  The LHA is hereby considered to be between -180 and +180 degrees rather than the conventional 0 - 360 degree range.  An additional step in getting the LHA from the arcsine is needed when Ho < 0, i.e. for bodies below the horizon. In the end, Longitude = - ( GHA + LHA ).


    Peter Hakel



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