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    Tahanea problem
    From: Mal Misuraca
    Date: 1997 Dec 17, 8:03 AM

      The solution:
       Looking in the back of the Nautical Almanac, you see a list of 273 stars
    with their sidereal hour angles and declinations month by month for 1997.  It
    occurs to you that one of those stars may pass directly overhead or very close
    to Tahanea.  If it is a prominent star, and one that passes overhead the
    island during your nighttime, there will be a specific moment in GMT when the
    star will be directly overhead the island and therefore will serve as a beacon
    directing you to the island.
       Figure out which star, and when it is over the island, and point the boat
    at the star at that instant, and you are on course for the island.
      Congratulations to Richard Stofer and Paul Hirose, who figured this out and
    named the brightest star in the winter sky, Sirius, as the likely candidate.
    Sirius passes just about seven miles north of Tahanea every night in winter
    time.  Sirius cannot very easily be missed, both because of its intrinsic
    brightness (-1.6 mag) and because it lies very close to the winter
    constellation Orion, known to us all.
      Both Richard and Paul got the steps right.  First, pick the star that seems
    to do the job---by looking for a star whose N or S declination matches or
    nearly matches the landfall you're looking for.  Next, determine the time of
    sunrise and sunset at your DR (daily pages adjusted for arc to time).  Next,
    determine when the star will be overhead the landfall (GHAA + SHA = GHA star =
    geographical position).
      In this case, Sirius is overhead Tahanea, actually a few miles north, at
    12:10:06 GMT on November 24 and about 3 min 56.4 sec earlier each night
    thereafter.  Point the boat at Tahanea at 12:10:06 on November 24, and you
    have a bearing to the island.  Make a rough estimate how many degrees Sirius
    lies from your own zenith (zenith distance) at that moment, and at 60 nm per
    degree, estimate your distance from the island.  You now have a heading from
    your compass and a rough distance to the island.  Repeat each night at the
    appropriate time.
      Paul makes the good point that as the island draws near to your zenith,
    taking a bearing on it proves more and more challenging, as anyone who has
    tried to take very high altitude sights of the sun will attest.  Thus, a
    bearing from about 600 nm out, when the island is 80 deg above the horizon, is
    the final bearing for heading.
      As it happens, Sirius lines up with Procyon to set a course to Tahanea.
    Procyon is another very bright star in the same area of the sky as Orion and
    Sirius.  Thus, in this case you could put Procyon directly at your back,
    Sirius dead ahead, and your heading to Tahanea would be confirmed.
      Neither Richard nor Paul mentioned the extra hazards of the approach to the
    Pallisers and Tahanea, and you would certainly want to take those into
    account.  This is a series of very low-lying reefed islets, hardly to be seen
    from a long way off.  Contrast this problem, for example, with emergency
    navigating to, say, Oahu.  The star Kornephoros, or Beta Hercules, passes
    directly overhead Kaneohe Bay on the windward side of that island in the
    summertime.  Oahu, like the rest of the Hawaiian Islands, is a tall island and
    should give fair warning as we close on it, very much unlike Tahanea.  There
    are islands close to Tahanea, adding to the hazard.  We would therefore keep
    the closest watch for the island and attempt to make landfall during daylight
    hours, perhaps even heaving to in order to assure ourselves of not running up
    on a reef.
      We might even decide to go on to Tahiti, if we don't need our friend's water
    casks that much.  What star might we use for that purpose?
      Is there any other method we might use during daylight hours to help us find
    Tahanea in November?  More about this in about a week.
      Mal Misuraca
      Passage East
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