File Table-S15.2020.txt of Package python-skyfield

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File Table-S15.2020.txt of Package python-skyfield (Revision 337fd352b3c700d2c80d2d733acb5e84)

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Table S15: The Polynomial Coefficients for DT -720.0 to 2019.0   v. 2020
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Row         Years                  Polynomial Coefficients
  i      K_i    K_{i+1}        a_0         a_1         a_2         a_3
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  1    -720.0    -100.0   20371.848   -9999.586     776.247     409.160
  2    -100.0     400.0   11557.668   -5822.270    1303.151    -503.433
  3     400.0    1000.0    6535.116   -5671.519    -298.291    1085.087
  4    1000.0    1150.0    1650.393    -753.210     184.811     -25.346
  5    1150.0    1300.0    1056.647    -459.628     108.771     -24.641
  6    1300.0    1500.0     681.149    -421.345      61.953     -29.414
  7    1500.0    1600.0     292.343    -192.841      -6.572      16.197
  8    1600.0    1650.0     109.127     -78.697      10.505       3.018
  9    1650.0    1720.0      43.952     -68.089      38.333      -2.127
 10    1720.0    1800.0      12.068       2.507      41.731     -37.939
 11    1800.0    1810.0      18.367      -3.481      -1.126       1.918
 12    1810.0    1820.0      15.678       0.021       4.629      -3.812
 13    1820.0    1830.0      16.516      -2.157      -6.806       3.250
 14    1830.0    1840.0      10.804      -6.018       2.944      -0.096
 15    1840.0    1850.0       7.634      -0.416       2.658      -0.539
 16    1850.0    1855.0       9.338       1.642       0.261      -0.883
 17    1855.0    1860.0      10.357      -0.486      -2.389       1.558
 18    1860.0    1865.0       9.040      -0.591       2.284      -2.477
 19    1865.0    1870.0       8.255      -3.456      -5.148       2.720
 20    1870.0    1875.0       2.371      -5.593       3.011      -0.914
 21    1875.0    1880.0      -1.126      -2.314       0.269      -0.039
 22    1880.0    1885.0      -3.210      -1.893       0.152       0.563
 23    1885.0    1890.0      -4.388       0.101       1.842      -1.438
 24    1890.0    1895.0      -3.884      -0.531      -2.474       1.871
 25    1895.0    1900.0      -5.017       0.134       3.138      -0.232
 26    1900.0    1905.0      -1.977       5.715       2.443      -1.257
 27    1905.0    1910.0       4.923       6.828      -1.329       0.720
 28    1910.0    1915.0      11.142       6.330       0.831      -0.825
 29    1915.0    1920.0      17.479       5.518      -1.643       0.262
 30    1920.0    1925.0      21.617       3.020      -0.856       0.008
 31    1925.0    1930.0      23.789       1.333      -0.831       0.127
 32    1930.0    1935.0      24.418       0.052      -0.449       0.142
 33    1935.0    1940.0      24.164      -0.419      -0.022       0.702
 34    1940.0    1945.0      24.426       1.645       2.086      -1.106
 35    1945.0    1950.0      27.050       2.499      -1.232       0.614
 36    1950.0    1953.0      28.932       1.127       0.220      -0.277
 37    1953.0    1956.0      30.002       0.737      -0.610       0.631
 38    1956.0    1959.0      30.760       1.409       1.282      -0.799
 39    1959.0    1962.0      32.652       1.577      -1.115       0.507
 40    1962.0    1965.0      33.621       0.868       0.406       0.199
 41    1965.0    1968.0      35.093       2.275       1.002      -0.414
 42    1968.0    1971.0      37.956       3.035      -0.242       0.202
 43    1971.0    1974.0      40.951       3.157       0.364      -0.229
 44    1974.0    1977.0      44.244       3.199      -0.323       0.172
 45    1977.0    1980.0      47.291       3.069       0.193      -0.192
 46    1980.0    1983.0      50.361       2.878      -0.384       0.081
 47    1983.0    1986.0      52.936       2.354      -0.140      -0.165
 48    1986.0    1989.0      54.984       1.577      -0.637       0.448
 49    1989.0    1992.0      56.373       1.648       0.708      -0.276
 50    1992.0    1995.0      58.453       2.235      -0.121       0.110
 51    1995.0    1998.0      60.678       2.324       0.210      -0.313
 52    1998.0    2001.0      62.898       1.804      -0.729       0.109
 53    2001.0    2004.0      64.083       0.674      -0.402       0.199
 54    2004.0    2007.0      64.553       0.466       0.194      -0.017
 55    2007.0    2010.0      65.197       0.804       0.144      -0.084
 56    2010.0    2013.0      66.061       0.839      -0.109       0.128
 57    2013.0    2016.0      66.920       1.007       0.277      -0.095
 58    2016.0    2019.0      68.109       1.277      -0.007      -0.139
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The above table of polynomial coefficients enables evaluation of DT in 
seconds (s) and its derivative (the length of day lod) in milliseconds 
(ms) for any epoch between $-720$ and 2019.  It is not valid outside the 
specified range of years.

For the year and fraction Y, extract the coefficients a_0, a_1, a_2,
a_3 from row i, where K_i <= Y <= K_{i+1} and form
     t   = (Y - K_i)/(K_{i+1} - K_i), where 0 <= t < 1, and 
thus calculate
     DT  = a_0 + a_1 t + a_2 t^2 + a_3 t^3  seconds
     lod = (a_1 + 2 a_2 t + 3 a_3 t^2) / (K_{i+1}-K_i) / 0.36525 ms 
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These coefficients reproduce the spline approximation discussed by
L.V. Morrison, F.R. Stephenson, C.Y. Hohenkerk and M. Zawilski, in
their latest paper entitled ``Addendum 2020 to `Measurement of the 
Earth's Rotation: 720 BC to AD 2015'' published in the Royal Society 
Proceedings A, 478, 2021, see https://doi.org/10.1098/rspa.2020.0776.

Details of the original analysis is published in Royal Society 
Proceedings A, 472, 2016, at https://doi.org/10.1098/rspa.2016.0404

All the data is also available from the website of HM Nautical Almanac
Office at http://astro.ukho.gov.uk/nao/lvm/.
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File Table-S15.2020.txt of Package python-skyfield (Revision 337fd352b3c700d2c80d2d733acb5e84)

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