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    UT1 vs. mean time in old almanacs
    From: Paul Hirose
    Date: 2010 May 25, 21:17 -0700

    I did a little science project to compare the modern time scale UT1 to
    the "mean time" in 18th and 19th century UK Nautical Almanacs. This
    cannot be done directly in almanacs before 1834 because their time scale 
    is Greenwich apparent time. However, a connection to mean time is 
    possible via the almanac's values for the equation of time at apparent 
    noon each day.
    
    For example, the 1772 almanac says the equation of time on February 4 is 
    14 minutes 22.9 seconds, to be added. Therefore, if we set a modern 
    program to 12:14:22.9 in UT1 (the modern equivalent of mean time), the 
    Greenwich hour angle of the apparent Sun should be almost zero. How 
    small is "almost"? That's what I wanted to test.
    
    The procedure is to obtain UT1 as explained above, then obtain TT
    (Terrestrial Time) by adding the ∆T from the table in the modern
    Astronomical Almanac. Both TT and UT1 are needed to compute the GHA of
    the Sun by a modern method:
    
    1. Extract the geocentric geometric Sun position in the ICRS
    (International Celestial Reference System) from the JPL DE406 ephemeris.
    
    2. Apply light time and aberration to obtain the apparent position in 
    the GCRS (Geocentric Celestial Reference System). The initial position 
    is incorrect because you haven't allowed for light time, so repeat Step 
    1 with almanac time minus the approximation of light time. Iterate until 
    the position converges to the desired accuracy.
    
    3. Transform the GCRS coordinates to the ITRS (International Terrestrial
    Reference System) via the SOFA subroutine iau_C2T06A. This applies the 
    latest IAU models for precession, nutation, and Earth rotation angle. 
    Converting the result to spherical coordinates yields the Sun's GHA, 
    which should be zero if the modern computation is perfectly compatible 
    with the almanac.
    
    At first I thought of testing each almanac on the same dates. However,
    to avoid systematic bias, the plan I finally adopted was to use five
    dates selected via random numbers from my calculator. For each almanac I
    picked a different set of dates. In several cases there's noticeable 
    clustering, but it would be poor procedure to exclude dates because 
    "they don't look random enough". Most people greatly underestimate the 
    tendency of random events to come in clusters.
    
    For each date I show the error (modern value minus almanac value) in
    the GHA and RA of the Sun. GHA equals GST (Greenwich sidereal time) 
    minus the Sun's right ascension. So, if the GHA and RA errors have 
    identical magnitudes and opposite signs, it means the modern and almanac 
    values for GST agree perfectly, the GHA error being purely due to a 
    discrepancy in RA.
    
    Originally I planned to test almanacs about 20 years apart, but later I
    added more data, especially in the 1700s.
    
    In the tabulations below, the errors are seconds of time.
    
    1767 Jan  7: LHA  0 s, RA -1 s
    1767 Jan 17: LHA  0 s, RA  0 s
    1767 Jan 22: LHA  0 s, RA  0 s
    1767 May 13: LHA +1 s, RA  0 s
    1767 Oct 31: LHA -2 s, RA -1 s
    
    Just five years later the almanac takes the values to a tenth of a
    second of time, so I do likewise.
    1772 Feb  4: LHA +.1 s, RA -.2 s
    1772 Sep 25: LHA  .0 s, RA -.2 s
    1772 Oct 16: LHA -.1 s, RA -.2 s
    1772 Oct 27: LHA  .0 s, RA -.4 s
    1772 Dec  2: LHA +.3 s, RA -.4 s
    
    1775 Jan 22: LHA +.1 s, RA -.3 s
    1775 Feb  5: LHA +.4 s, RA -.5 s
    1775 Mar  3: LHA +.5 s, RA -.5 s
    1775 Nov 13: LHA -.1 s, RA -.2 s
    1775 Dec 21: LHA -.1 s, RA -.1 s
    
    1777 Jan 31: LHA +.1 s, RA -.1 s
    1777 May  7: LHA +.6 s, RA -.8 s
    1777 Jun 13: LHA +1.2 s, RA -1.3 s
    1777 Sep 28: LHA +.7 s, RA -.8 s
    1777 Oct 27: LHA +.5 s, RA -.7 s
    
    1781 Jan  3: LHA  .0 s, RA -.2 s
    1781 Sep 13: LHA +.5 s, RA -.8 s
    1781 Sep 25: LHA +.7 s, RA -1.0 s
    1781 Oct 21: LHA +.7 s, RA -.8 s
    1781 Oct 29: LHA +.5 s, RA -.8 s
    
    1790 Jan  2: LHA +.5, RA -.8
    1790 Jan 15: LHA +.5, RA -.8
    1790 Feb 17: LHA +.7, RA -.9
    1790 Apr 18: LHA +1.0, RA -1.4
    1790 Apr 27: LHA +.9, RA -1.4
    
    For some reason now forgotten, I did 10 points in the 1792 almanac.
    1792 Feb  4: LHA -.1, RA -.2
    1792 Mar  1: LHA  .0, RA -.4
    1792 Jun 10: LHA +.5, RA -1.1
    1792 Jun 23: LHA +.7, RA -1.0
    1792 Aug  1: LHA +.6, RA -.8
    1792 Sep 21: LHA +.2, RA -.6
    1792 Oct 13: LHA -.1, RA -.3
    1792 Oct 31: LHA -.1, RA -.4
    1792 Dec 15: LHA -.3, RA .0
    1792 Dec 26: LHA -.3, RA .0
    
    1809 Feb 24: LHA +.4, RA  .0
    1809 Mar  3: LHA +.4, RA  .0
    1809 Apr 11: LHA +.3, RA  .0
    1809 Jun 21: LHA +.2, RA  .0
    1809 Aug 11: LHA +.3, RA  .0
    
    1830 Jan 18: LHA +.4, RA +.4
    1830 Mar  8: LHA +.4, RA +.5
    1830 May  4: LHA +.3, RA +.5
    1830 Jun 19: LHA +.2, RA +.7
    1830 Nov 13: LHA +.3, RA +.4
    
    In 1834 the almanac introduced several big changes on the recommendation
    of an Astronomical Society of London committee. One that affected my
    computations was the change to mean time as the main time scale of the
    tables. So my GHA error for the later almanacs equals the Sun GHA at
    1200 UT1 (according to the modern algorithm), minus the equation of time
    listed in the almanac for mean noon.
    
    The Preface for 1834 summarizes the almanac's history, and ends with the
    committee report that explains the rationale for the almanac changes. 
    Or, in one case, a non-change: they thought about tabulating lunar 
    distances for every hour, but decided to leave it at every three hours.
    
    http://www.archive.org/stream/nauticalalmanac30offigoog#page/n10/mode/2up
    
    The online reader at archive.org can be troublesome. My Internet
    Explorer 8 won't run it at all. Fortunately, the committee report was
    included again in the next year's almanac, and may be read online at Google:
    
    http://books.google.com/books?id=L_QNAAAAQAAJ&pg=PR15
    
    
    1851 Feb  2: LHA +.30 s, RA +.18 s
    1851 Apr 15: LHA +.27 s, RA +.19 s
    1851 Apr 16: LHA +.28 s, RA +.18 s
    1851 Sep 20: LHA +.37 s, RA +.09 s
    1851 Dec 12: LHA +.31 s, RA +.17 s
    
    1861 Mar 16: LHA +.43 s, RA +.06 s
    1861 Jul 28: LHA +.36 s, RA +.13 s
    1861 Sep  1: LHA +.36 s, RA +.14 s
    1861 Oct 15: LHA +.33 s, RA +.16 s
    1861 Nov  2: LHA +.27 s, RA +.21 s
    
    1864 Jan  7: LHA -.05 s, RA  .00 s
    1864 Jan 22: LHA -.03 s, RA  .00 s
    1864 Jul  1: LHA -.03 s, RA -.02 s
    1864 Aug 12: LHA -.02 s, RA -.03 s
    1864 Oct  7: LHA -.03 s, RA -.03 s
    
    The errors in 1864 suddenly decrease about tenfold! Based on what the
    Preface says, I think this is due to the introduction of Leverrier's
    tables. It's no statistical fluke either. This accuracy is maintained in
    later years.
    
    1866 Jan  7: LHA -.03 s, RA  .00 s
    1866 Jan 24: LHA -.04 s, RA  .00 s
    1866 Jul 14: LHA -.01 s, RA -.01 s
    1866 Jul 17: LHA -.02 s, RA -.02 s
    1866 Aug  5: LHA -.03 s, RA -.01 s
    
    1868 Jan 26: LHA -.04 s, RA +.02 s
    1868 Feb 28: LHA -.05 s, RA +.02 s
    1868 Jun 27: LHA -.01 s, RA -.02 s
    1868 Nov  7: LHA -.01 s, RA -.03 s
    1868 Nov 24: LHA  .00 s, RA -.03 s
    
    1890 Jan 25: LHA +.01 s, RA -.02 s
    1890 Aug  9: LHA -.02 s, RA -.03 s
    1890 Aug 20: LHA -.01 s, RA -.05 s
    1890 Sep  9: LHA +.02 s, RA -.05 s
    1890 Nov  4: LHA +.02 s, RA -.06 s
    
    
    To summarize, in these tests I found the UT1 time scale consistent, plus
    or minus a few tenths of a second, with mean time in late 18th century
    almanacs. In the 1860s this improved to a few hundredths.
    
    Could these discrepancies in the Sun's GHA be reduced with better ∆T
    values? I don't think so. Remember, GHA = GST - RA. Well, Greenwich
    sidereal time is a function of UT1 and (practically) independent of TT,
    so the GST at mean noon is unaffected by any change in ∆T.
    
    That leaves RA. A position from a JPL ephemeris is (practically) a
    function of TT, and TT is obtained by adding ∆T to mean time, so ∆T does
    affect the Sun's RA at mean noon. But the Sun only moves one degree per
    day, so a 1-second difference in TT equals only .003 second (time)
    difference in RA. A .3 second error in GHA would be equivalent to 100
    seconds of ∆T, far greater than the uncertainty for the late 1700s.
    
    -- 
    I filter out messages with attachments or HTML.
    
    
    
    
    

       
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