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    Re: Error in Nautical Almanac Polaris SHA Feb-June?
    From: Trevor Kenchington
    Date: 2004 Apr 17, 22:17 +0000

    Jim,
    
    Look at it this way:
    
    The lowest SHA, during 2004, for any given star can be expressed as X
    degrees plus Y minutes, where Y is anything between 00.0 and 59.9. The
    highest SHA for that same star can be expressed as X degrees plus Y+Z
    minutes, where Z is a rather small number. (They are not called "Fixed
    Stars" without good reason.) The tabulated values in the Almanac are, of
    course, Y, Y+Z or something in between.
    
    Since Z is always small, Y+Z can only exceed 59.9 if Y is close to that
    value. Hence, tabulated values above 59.9 are rare.
    
    To calculate an expected number of them, in 2,076 table entries, you
    would need a probability distribution for the values of ZZ -- the
    difference between the minimum SHA for each star and its SHA in each
    month. Then you could assume that the 173 stars should have values of Y
    uniformly distributed between 00.0 and 59.9 (i.e. one star per 0.35
    minutes of SHA) and you could figure out the probability of each of
    those 173 having one or more values of ZZ large enough to drive its Y+ZZ
    above 59.9.
    
    Then again, it might be easier to add up the number of table entries
    that exceed 59.9!
    
    
    The other question is why should Polaris be the one star to stand out in
    this respect? The answer to that one is that, for a star located very
    close to the celestial pole, a very small apparent motion can carry a
    star across many minutes of SHA.
    
    For an analogy: Someone at the U.S. Amundsen/Scott base at the South
    Pole can take a short after-dinner stroll around the pole and pass
    through all 360 degrees of longitude in a few minutes. A similar walk in
    Equatorial regions would not see you pass through as much as 0.1 minutes
    of longitude. The large changes in longitude of both Polaris and someone
    at the Earthly pole are, of course, simply artefacts of the arbitrary
    (though very convenient) human decision to divide spheres using
    meridians which converge at the poles.
    
    
    Trevor Kenchington
    
    
    Jim Thompson wrote:
    
    > However, now that I look more closely at those star SHA/Declination tables
    > on pages 268-273, almost none of the other stars have a similar situation.
    > In the 2004 NA only 18 out of 2,076 SHA entries exceed 59.9'!  I would have
    > thought that based on random chance alone a far greater proportion would
    > range over two adjacent whole degree values.  What causes this seeming
    > coincidence?
    >
    > One way to think of the problem is this:
    > - Assume that the whole minutes portion does not exceed 84.9 (in fact
    > Polaris is the largest such value in 2004: 84.6)
    > - There are 84 x 10 = 840 possible values for the minutes portion, ranging
    > from 00.0' to 84.9'.
    > - The proportion of values larger than 59.9 in the range of numbers from
    > 00.0' to 84.9' is (84.9-59.9)/84.9 = 25/84.9 = 0.29, or 29%.
    > - There are 12 months in a year and 173 stars, for a total of 2,076 entries.
    > - Based on random chance alone, 0.29 * 2,076 = 602 entries would be 60' or
    > more.
    >
    > But given that SHA changes in very small increments for each star, then I
    > have not correctly stated the odds of a star having an SHA that ranges over
    > two whole degree portions.
    >
    > At this point in my thinking I run out of gas, in part because I'm beat
    > after teaching a marine radio course all day and the Carlsberg Light is
    > kicking in, and in part because my understanding of probability math is
    > weak.
    >
    > What am I missing?
    >
    > Jim
    
    
    --
    Trevor J. Kenchington PhD                         Gadus{at}iStar.ca
    Gadus Associates,                                 Office(902) 889-9250
    R.R.#1, Musquodoboit Harbour,                     Fax   (902) 889-9251
    Nova Scotia  B0J 2L0, CANADA                      Home  (902) 889-3555
    
                         Science Serving the Fisheries
                          http://home.istar.ca/~gadus
    
    
    

       
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