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    Predicting the time of a desired azmuth
    From: Peter Smith
    Date: 1998 Oct 28, 3:11 PM

    During Dan's well-earned vacation from the Silicon Sea cruise, we've
    not had any practical exercises.  Here is an example of a problem
    that I've worked at sea many times, but have always wondered if
    there is a better solution.
    How do you predict when the Sun's azmuth will be a particular value?
    The simplest case is predicting meridian passage -- local apparent
    noon -- and that is a fairly straightforward calculation involving
    only the almanac.  A more complex case is predicting when the Sun's
    azmuth will be perpendicular or parallel to your course (or more
    likely, your Intended Track).
     -- A sight taken directly abeam will yield a Line Of Position (LOP)
        parallel to your track, giving you a measure of course error.
        It can be useful in confirming that you are keeping clear
        of some lateral hazard (reef or shoal) in the same way a danger
        bearing is used in piloting.
     -- A sight taken directly ahead or astern gives you an LOP perpen-
        dicular to your course/track and thus distance since your last
        fix and/or distance to your destination.
     -- The two sights combined give a particularly useful running fix.
    I am off the US east coast. All times are GMT-4 (Eastern Daylight Time)
    and all courses and azmuths are in degrees TRUE. The date is 16-Jun-1998.
    My destination is Buzzards Bay entrance tower at 41d 23.8'N  71d 02.0'W.
    My morning star sights gave a 04:39:00 fix at 39d 48'N  72d 14'W. My
    course and speed made good are estimated as 030d at 5 knots.  Since I
    will be approaching this often-foggy destination at night, I would like
    a good distance-off measure during the day in case I can't get a fix by
    evening star sights.  When will the Sun's azmuth be 210d, parallel to
    my intended track?
    Solution by Ho-229 and Nautical Almanac:
    Step 1: Rough estimate of time and DR position
        An azmuth of 210d will occur sometime not too long after Local
        Apparent Noon (LAN), so as a first approximation, I project my
        position forward to 13:00, getting 40d 24'N, 71d 47'W. I figure
        the time of the Sun's meridian passage at this longitude:
            71d 47'  Estimated longitude
           -60d 00'  Time zone meridian (GMT-4)
            11d 47'  Angular distance from time meridian to local meridian
            00h47m   Meridian angle converted to time
            12:01    Local Apparent Time of meridian passage (from almanac)
            12:48    Zone Time of local noon
        So far, so good.  Had LAN been at or before the time I selected,
        (13:00) I would advance the DR/EP another hour and use that in
        the subsequent calculations.
    Step 2: Use HO-229 to find LHA that approximates the desired azmuth
         40d 24'  N  Estimated latitude
         71d 47'  W  Estimated longitude
         23d 21.3'N  Sun's declination at 13:00
        150d         Desired azmuth of Sun
        Our latitude and the Sun's declination are both North, so we will
        use the "latitude SAME name as declination" pages and look in the
        40d latitude column. Our desired normalized azmuth (Zn) is 210d,
        corresponding to a tabulated (i.e., unnormalized) azmuth of 150
        (N210E = N150W; put another way, we know the Sun must be West of
        us to give the desired azmuth, so the LHA measured West will be a
        small number, and the rule at the top of the page for North
        latitudes and LHA<180 is Zn=360-Z). Since the Sun's declination
        is between 23d and 24d, we look for entries where Z for 23d and
        24d declination straddle 150d. On the page for LHA 10d we see:
            LHA 10d, 350d
            Dec    ...     40d
             .              Z
             23    ...    150.6
             24    ...    149.3
        So, an LHA of 10d will give Z~150d and thus Zn~210d (at 40dN and
        declination between 23d and 24d).
    Step 3: Figure out when the Sun's LHA will be 10d at our position
        There are probably several ways of doing this. Here, I add the time
        it takes the Sun to move 10d West to the previously computed value
        for the Sun's meridian passage at my estimated position.
        00h40m  10d LHA converted to time
        12:48   Zone time of meridian passage (from Step 1)
        13:28   Approximate Zone Time when Sun's LHA will be 10d
    Inaccuracies in this method:
     -- The estimated position may be off, but an updated DR & EP as the
        time approaches should indicate if a re-apraisal is necessary
     -- The latitude and declination we enter the tables with are rounded
        to the nearest 1d
     -- The LHA we get out of the table is only given to 1d
    Still, experience has shown this method to be accurate to within five
    minutes if the estimated position is reasonably good.
     -- Is there a better way of doing this using HO-229?
     -- What are some other methods?  Has anyone written a calculator or
        computer program to solve this?  Can navigational calculators like
        the Celsticomp spit this right out?  Is there an efficient solution
        for non-programmable calculators?
        My gut feeling is that since your estimated latitude and longitude
        are changing linearly, and the first derivative of any of the azmuth
        formulae will give a linear approximation of the trend of Zn, one
        could work a solution for a time near the expected time, then solve
        for the exact time. (Of course, I was shakey in calculus 25 years
        ago, and now wouldn't even attempt to differentiate something like
                                   cos d * sin LHA
    	Z = arctan ---------------------------------------
                       cos L * cos d - sin L * cos d * cos LHA
        so this is only a suppostion on my part.)
    Peter Smith -- psmith@wellspring.us.dg.com
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