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    Almanac Calculations
    From: Gordon Talge
    Date: 1997 Mar 11, 20:07 EST

    I have recieved several messages asking about my polynomial calculations.
    Where did I get the info, and so on. I have typed up a little history
    of the almanac and  what I did. Please note I have left out alot
    of history. The Explan. Supplement can fill you in on alot more.
    >From about 1898 to 1983, the position of the Sun were taken from
    Simon Newcomb's  "Tables of the Motion of the Earth on its Axis and
    around the Sun",  Astronomical Papers Vol. VI, Tables of the Four
    Inner Planets. These tables give the position of the Earth in
    Elliptic Latitude and Longitude, ie the Earth's position on the
    elliptic. This volume also contains the other 3 inner planets.
    The positions of the outer planets and moon where taken
    from Hill and Brown.
    Jean Meeus' book, "Astronomical Formulae for Calculators" uses a
    cut down version of Newcomb's Theory as do most computer programs.
    A paper by T.C. Van Flandern and K.F Pulkkinen called "Low-
    Precision Formulae for Planetary Positions", The Astrophysical
    Journal Supplement Series, 41:391-411, 1979 November is also a cut
    down version of Newcomb's theory, but not as cut down as Meeus'.
    William-Bell will send you a copy of the paper for about $5.00. The
    USNO reprint service will send it to you for FREE. See their web
    Newcomb's theory is what you would call an analytic theory. In the
    1970s and 80s it seems that NASA, the USNO and JPL determined
    through need and observations, that a replacement had to be found
    for Newcomb's almost 100 year old theory. JPL came up with its
    developmental ephemeris DE200/LE200. DE part for the planets and LE
    for the moon.
    DE200/LE200 is based on numerical integration of the fundamental
    differential equations governing the motions of the moon and
    planets. Since the output from the integrator was very large they
    use chebyshev polynomials constructed in a special way to cut down
    on the size. DE200/LE200 gives the barycentric ( center of mass )
    rectangular coordinates of the planets and the geocentric
    rectangular coordinates of the moon.
    You can get the complete DE200/LE200 from the internet at
    ftp://ftp.navigator.jpl.nasa.gov, along with software to convert
    the ascii files to binary and to read the resulting ephemeris.
    It covers from 1599-2169. It is the basis and background ephemeris
    for the Astronomical Almanac and Nautical Almanac since 1984.
    The barycentric coordinates were chosen because they work well with
    the integrator. To be useful they need to be converted to more
    familiar coordinates. Page B36-B41 in the Astronomical Almanac
    outlines and gives an example of such a procedure.
    This procedure has also been coded in Fortran and C, called NOVAS
    by the USNO and is on their web site.
    Other theories are also around. Meeus' book, "Astronomical
    Algorithms" uses a cut down version of VSOP87 and EPL2000. Which
    are the French answers to DE200/LE200. VSOP87 is for the planets
    and EPL2000 is for the Moon. VSOP87 and EPL2000 are analytic
    theories which consists of hundreds of terms that need to be
    evaluated to give complete accurate positions. The complete theory
    ( not just Meeus' cut down version ) is on the internet at
    ftp://ftp.bdl.fr. The site of the French, Bur. des Long.
    In 1997 the USNO started publishing "The Almanac for Computers"
    which gave power series polynomials for the positions of the Sun,
    Moon, and Navigational planets for the year for the navigator. They
    also gave chebyshev series polynomials for the Sun, Moon, planets,
    etc for the astronomer. They tried to match the accuracy of the
    Nautical Almanac and Astronomical Almanac.
    These polynomials were for use with programmable and non-
    programmable pocket calculators.
    When pcs became common, they came out with the "Floppy Almanac" and
    later "MICA". The Floppy Almanac is like the A / C and MICA is just
    chunks of JPL's DE200/LE200. In 1991 they ceased publication of
    A/C. This information is no longer available from them as far as I
    know.  BTW, they also dropped the FA and MICA.
    I had always wondered how the polynomials were calculated in the
    now out of print A/C. Since I now had access to DE200/LE200 and
    some of the other tools, I thought I would try and duplicated some
    of their results.
    Here are some of the procedures I followed.
    1) Using DE200/LE200  and the USNO NOVAS routines, I read in the
    positions of the Sun for 00h 00m 00s UT every day for about 40
    2) I calculated the GHA, Dec, SD for the Sun for each day.
    3) Using Lagranage interpolation, I calculated the positions for
    special normalized times as required by chebyshev approximation.
    4) I calculated the chebyshev approximations for the GHA, Dec and
    SD for  a 32 day period.
    5) I then cut down the chebyshev approximations to 6 terms and
    converted them to power series.
    For the planets, it was almost the same as the Sun, except that
    there is not SD.
    For the Moon, it was a little harder. The polynomials are valid for
    the Moon for only 8 days at a time and there are 8 terms in the
    power series.
    The Moon's position was read in from DE200/LE200 every 3 hours for
    8 days and the GHA, Dec, HP and SD were calculated.
    My reference for all of this is the 1991 edition of A/C and
    "Chebyshev Polynomials in Numerical Analysis" by L. Fox and I.B.
    Parker. Oxford University Press, 1968.
    I used the djgpp port of the GNU G77 Fortran compiler off the
    Some observations: Try as I might, I could not exactly duplicate
    the numbers in the 1991 A/C. I got someone over at the USNO to
    answer my e-mail. I found out that  for the Navigation positions
    they did not use DE200 and that I should  compare my approximations
    with the original DE200. IE compare the approximations with what I
    was approximating, not with something else. When I did that, they
    matched up.
    I kept the polynomials small to be able to enter them into a pocket
    calculator by hand. The validity could be increased by making the
    polynomials longer.
    The polynomials are not a replacement for the Nautical Almanac,
    they are a duplication of some of the results.
    -- Gordon Talge
                                        (. .)
             |  Gordon Talge WB6YKK            e-mail: gtalge{at}XXX.XXX  |
             |  Department of Mathematics      QTH: Loma Linda, CA    |
             |  Mt. San Jacinto College        Lat.  N  34? 03.1'     |
             |  San Jacinto, CA                Long. W 117? 15.2'     |

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