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    Polynomial Almanac
    From: Gordon Talge
    Date: 1996 Sep 02, 15:45 EDT

    This is probably my last post on polynomials approximations. Thanks to
    everyone who responded with feed back and tried them out. I learned alot
    about approximations and the Nautical Almanacs.
    For example, sometimes the Air Almanac and the Nautical Almanac have
    different values listed for the same date and time.
    Sept 19, 1996 00h 00m 00s the Air Almanac has for the Sun:
    GHA 181d 32.5min Dec. N 1d 27.4min
    and the Nautical Almanac has:
    GHA 181d 32.6min Dec. N 1d 27.5min.
    ------------------------------------------
    The Sept poly gave GHA 181d 32.5min and Dec N 1d 27.4min
    The Floppy Almanac, ICE and MICA sometimes give slightly different answers,
    so I decided not to sweat it, if my polynomials didn't match up to exactly
    0.1 min on everything.
    Gordon
    
    Since getting some feed back from the list, I have revised my polynomial
    for the Sun, and have added one for the Moon.
    The Moon has a longer polynomial and is valid for only about a week.
    Power Series for the Moon
    Dates : Sep. 1 to Sep. 8 1996
              A = 4.0   W = 1
                  Moon                Moon           H.P.      S.D.
                  GHA                 Dec.
    Term
      0          1710.0627          17.8118         0.9213      0.2510
      1          1392.4493           3.5367        -0.0416     -0.0113
      2             1.2898          -6.8970         0.0206      0.0062
      3             0.3901           0.1067         0.0026     -0.0003
      4             0.0828           0.3447        -0.0026     -0.0028
      5            -0.1331          -0.0606         0.0006      0.0033
      6            -0.0059           0.0075         0.0000      0.0015
      7             0.0122          -0.0002        -0.0001     -0.0022
    sums         3104.1479          14.8496         0.9008      0.2454
    The power series for the Moon is used exactly the same way explained below for
    the Sun, except that their are 8 terms instead of 6 in the Moon power series.
    Example:   Moon  Sept 4 17h 32m 16s
              t = 4.730740741
              x = ( ( t - 1 ) / 4 ) - 1  = -0.067314815
    this is the argument for the power series.
        GHA = 1616.335961    or removing multiples of 360 we get
           GHA = 176.3359610 degs  or ** 176 degs 20.2 mins  **
     Using the same argument for the Dec. H.P. and S.D. we get
        Dec. =  17.5425 degs or ** N 17 degs 32.5 mins  **
        H.P. = 0.9242 degs or   ** 55.5 mins  **
        S.D. = 0.2518 degs or   ** 15.1 mins  **
    ------------------------------------------------------------------
    ------------------------------------------------------------------
    Power Series for the Sun
    Dates : Sep. 1 to Oct. 1 1996
             A = 16.0  W = 1
                   Sun                  Sun           Sun
                   GHA                  Dec.          SD
    Term
      0          5941.3640             2.2319         0.2652
      1          5761.4229            -6.1847         0.0012
      2             0.0046            -0.0946         0.0001
      3            -0.0506             0.0653         0.0000
      4            -0.0051            -0.0004         0.0000
      5            -0.0024            -0.0008         0.0000
    sums        11702.7334            -3.9833         0.2665
    Dates: Oct 1 to Nov 1 1996
             A = 16.0   W = 1
                   Sun                  Sun            Sun
                   GHA                  Dec.           SD
    Term
      0          5943.6509             -9.2726        0.2675
      1          5760.8108             -5.8514        0.0012
      2            -0.3099              0.2859        0.0000
      3            -0.0427              0.0749       -0.0001
      4            -0.0004              0.0002        0.0000
      5            -0.0018             -0.0009        0.0000
    
    sums        11704.0969            -14.7639        0.2686
    Dates:  Nov 1 to Dec 1 1996
    
             A = 16.0  W = 1
    
                   Sun                  Sun            Sun
                   GHA                  Dec.           SD
    Term
      0          5943.7628            -18.9951        0.2696
      1          5759.2133             -3.9009        0.0010
      2            -0.4361              0.7241       -0.0001
      3             0.0191              0.0727        0.0000
      4             0.0066             -0.0042        0.0000
      5            -0.0008             -0.0012        0.0000
    
    sums        11702.9574            -22.1046        0.2705
    The way to use these series is first convert UT into decimal then
       t = d + 24 /UT,     where d is the day of the month.
      Next, get x where x in between -1 and +1. ie [-1,+1] using the formula
        x =  ( (t - W ) / A ) -1, use x as the argument to evaluate the polynomial.
    f(x) = a0 + a1*x + a2 * x^2 + a3 * x^3 + a4 * x^4 + a5 *x^5 .
     This can be better evaluated as
     f(x) = a0 + x*(a1+x*(a2+x*(a3+x*(a4+x*a5)))).
    Once the GHA, Dec. or S.D. is obtained, remove any multiples of 360 degrees
    and convert to degrees and mins. or in the case of the S.D. mins, round off
    to the nearest .1 min
    Example:  GHA for Sept 18th  7h 28m 19s  UT
     7h 28m 19s = 7.471944444 hours or  7.471944444 / 24 = 0.311331019 parts of
     a day.
    Since d= 18 we have for t, t= 18.311331019 days
     x = (( 18.311331019  - 1 )/ 16 ) -1   or x = 0.081958189
    This is what we use for the argument of the polynomial.
    Evaluating the GHA polynomial at 0.081958189 we get
    GHA = 6413.559790 .  Removing multiples of 360 degrees we get
               293.5597896d or  ** 293d 33.6 mins  **
    So the GHA for Sept 18th 7h 28m 19s is 293d 33.6min
      By the 1996 Air Almanac I get for Sept 18th 7h 20mins
               291d  28.8 min
               The correction for 8min 19s is 2d 4.8min
               so the   **  GHA is 293d 33.6 min   **
    Using the same argument for the Dec. Series I get
                1.724183513 or  **  N 1d 43.5   **
      (Note: N is + and S is - )
        The Air Almanac give  N 1d 43.6 for 7h 20min and
                              N 1d 43.4 for 7h 30min
      Since 28min 19sec is between 20 and 30mins and is closer to 30 but
        still less and the dec is going down, N 1d 43.5 seems resonable.
        I don't know what the Nautical Almanac says with it's d correction.
    I have tried to fit the polynomial to give the proper value with an
    error of not more then 0.1 min
    The series is ONLY valid for the month stated, NOT before and NOT after.
    The sums at the bottom are NOT used in calculations, they are used to
    check that you entered the coefficients correctly.
    What I did was use a programable calculator to evaluate the polynomial
    by loading in the coefficients into the storage registers and then
    running it through.
    The reference ephemeris used is JPL's DE200 which is the background basis for
    the Astronomial Almanac and Nautical Almanacs.
    -- Gordon
    
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