Next: The Superior Planets Up: Lunar-Solar Syzygies and Eclipses Previous: Example Eclipse Calculations

# Eclipse Statistics

Consider a very large collection of lunar-solar syzygies. For such a collection, we expect the lunar argument of latitude, , the lunar mean anomaly, , and the solar mean anomaly, , to be statistically independent of one another, and randomly distributed in the range to . Using this insight, we can easily calculate the probability that a new moon is coincident with a solar eclipse, or a full moon with a lunar eclipse, using Eq. (125) and the criteria (150)-(154). For a new moon we find:
 Probability of total solar eclipse: Probability of annular solar eclipse: Probability of partial solar eclipse: Probability of any solar eclipse:

For a full moon we get:
 Probability of total lunar eclipse: Probability of partial lunar eclipse: Probability of any lunar eclipse:

Thus, we can see that, over a long period of time, the ratio of the number of total/annular solar eclipses to the number of partial solar eclipses is about 9/5, whereas the ratio of the number of partial lunar eclipses to the number of total lunar eclipses is approximately 5/4. Furthermore, the ratio of the number of solar eclipses to the number of lunar eclipses is about 11/7. Since there are 12.37 synodic months in a year, the mean number of solar eclipses per year is approximately , whereas the mean number of lunar eclipses per year is about . Clearly, solar eclipses are more common that lunar eclipses. On the other hand, at a given observation site on the earth, lunar eclipses are much more common than solar eclipses, since the former are visible all over the earth, whereas the latter are only visible in a very localized region.

Table 40: Mean motion of the lunar-solar elongation. Here, , , , , and . At epoch ( JD), , , , and .
 (JD) (JD) 10,000 227.491 173.503 136.002 329.930 1,000 310.749 269.350 265.600 104.993 20,000 94.982 347.005 272.005 299.859 2,000 261.498 178.701 171.200 209.986 30,000 322.473 160.508 48.007 269.788 3,000 212.247 88.051 76.801 314.979 40,000 189.964 334.011 184.010 239.718 4,000 162.996 357.401 342.401 59.972 50,000 57.455 147.513 320.012 209.648 5,000 113.746 266.751 248.001 164.965 60,000 284.947 321.016 96.015 179.577 6,000 64.495 176.102 153.601 269.958 70,000 152.438 134.519 232.017 149.506 7,000 15.244 85.452 59.202 14.951 80,000 19.929 308.022 8.020 119.436 8,000 325.993 354.802 324.802 119.944 90,000 247.420 121.524 144.022 89.366 9,000 276.742 264.152 230.402 224.937 100 139.075 242.935 98.560 226.499 10 121.907 132.294 9.856 130.650 200 278.150 125.870 197.120 92.999 20 243.815 264.587 19.712 261.300 300 57.225 8.805 295.680 319.498 30 5.722 36.881 29.568 31.950 400 196.300 251.740 34.240 185.997 40 127.630 169.174 39.424 162.600 500 335.375 134.675 132.800 52.496 50 249.537 301.468 49.280 293.250 600 114.449 17.610 231.360 278.996 60 11.445 73.761 59.136 63.900 700 253.524 260.545 329.920 145.495 70 133.352 206.055 68.992 194.550 800 32.599 143.480 68.480 11.994 80 255.260 338.348 78.848 325.199 900 171.674 26.415 167.040 238.494 90 17.167 110.642 88.704 95.849 1 12.191 13.229 0.986 13.065 0.1 1.219 1.323 0.099 1.306 2 24.381 26.459 1.971 26.130 0.2 2.438 2.646 0.197 2.613 3 36.572 39.688 2.957 39.195 0.3 3.657 3.969 0.296 3.919 4 48.763 52.917 3.942 52.260 0.4 4.876 5.292 0.394 5.226 5 60.954 66.147 4.928 65.325 0.5 6.095 6.615 0.493 6.532 6 73.144 79.376 5.914 78.390 0.6 7.314 7.938 0.591 7.839 7 85.335 92.605 6.899 91.455 0.7 8.534 9.261 0.690 9.145 8 97.526 105.835 7.885 104.520 0.8 9.753 10.583 0.788 10.452 9 109.717 119.064 8.870 117.585 0.9 10.972 11.906 0.887 11.758

Table 41: Anomalies of the lunar-solar elongation. The common argument corresponds to , , , , and for the case of , , , , and , respectively. If the argument is in parenthesies then the anomalies are minus the values shown in the table.
 Arg. () Arg. () 000/(360) 0.000 0.000 0.000 -0.000 -0.000 090/(270) 6.289 1.327 -0.044 -2.075 -0.119 002/(358) 0.237 0.046 0.045 -0.074 -0.004 092/(268) 6.268 1.326 -0.090 -2.073 -0.119 004/(356) 0.473 0.093 0.089 -0.148 -0.008 094/(266) 6.239 1.324 -0.136 -2.067 -0.119 006/(354) 0.709 0.139 0.133 -0.221 -0.012 096/(264) 6.203 1.320 -0.181 -2.060 -0.119 008/(352) 0.943 0.185 0.177 -0.294 -0.017 098/(262) 6.160 1.314 -0.226 -2.050 -0.118 010/(350) 1.176 0.230 0.219 -0.367 -0.021 100/(260) 6.109 1.307 -0.270 -2.037 -0.118 012/(348) 1.408 0.276 0.261 -0.440 -0.025 102/(258) 6.051 1.298 -0.313 -2.022 -0.117 014/(346) 1.637 0.321 0.301 -0.511 -0.029 104/(256) 5.986 1.288 -0.354 -2.004 -0.116 016/(344) 1.864 0.366 0.340 -0.583 -0.033 106/(254) 5.915 1.276 -0.394 -1.984 -0.115 018/(342) 2.088 0.410 0.376 -0.653 -0.037 108/(252) 5.836 1.262 -0.432 -1.962 -0.114 020/(340) 2.310 0.454 0.411 -0.723 -0.041 110/(250) 5.751 1.247 -0.468 -1.937 -0.112 022/(338) 2.527 0.497 0.444 -0.791 -0.045 112/(248) 5.660 1.230 -0.501 -1.910 -0.111 024/(336) 2.741 0.540 0.475 -0.859 -0.049 114/(246) 5.562 1.212 -0.533 -1.881 -0.109 026/(334) 2.951 0.582 0.504 -0.926 -0.052 116/(244) 5.458 1.193 -0.562 -1.850 -0.107 028/(332) 3.157 0.623 0.529 -0.991 -0.056 118/(242) 5.348 1.172 -0.589 -1.816 -0.106 030/(330) 3.358 0.663 0.553 -1.055 -0.060 120/(240) 5.233 1.149 -0.613 -1.780 -0.103 032/(328) 3.554 0.703 0.573 -1.118 -0.063 122/(238) 5.111 1.125 -0.633 -1.742 -0.101 034/(326) 3.746 0.742 0.591 -1.179 -0.067 124/(236) 4.985 1.100 -0.651 -1.702 -0.099 036/(324) 3.931 0.780 0.605 -1.239 -0.070 126/(234) 4.853 1.074 -0.666 -1.660 -0.097 038/(322) 4.111 0.817 0.617 -1.297 -0.074 128/(232) 4.716 1.046 -0.678 -1.616 -0.094 040/(320) 4.285 0.853 0.625 -1.354 -0.077 130/(230) 4.575 1.017 -0.687 -1.570 -0.092 042/(318) 4.454 0.888 0.631 -1.409 -0.080 132/(228) 4.428 0.986 -0.692 -1.522 -0.089 044/(316) 4.615 0.922 0.633 -1.462 -0.083 134/(226) 4.277 0.955 -0.695 -1.473 -0.086 046/(314) 4.770 0.955 0.632 -1.513 -0.086 136/(224) 4.122 0.922 -0.693 -1.422 -0.083 048/(312) 4.919 0.986 0.627 -1.562 -0.089 138/(222) 3.963 0.888 -0.689 -1.369 -0.080 050/(310) 5.061 1.017 0.620 -1.609 -0.092 140/(220) 3.799 0.853 -0.682 -1.314 -0.077 052/(308) 5.195 1.046 0.609 -1.655 -0.094 142/(218) 3.632 0.817 -0.671 -1.258 -0.074 054/(306) 5.323 1.074 0.596 -1.698 -0.097 144/(216) 3.462 0.780 -0.657 -1.201 -0.070 056/(304) 5.443 1.100 0.579 -1.739 -0.099 146/(214) 3.288 0.742 -0.640 -1.142 -0.067 058/(302) 5.555 1.125 0.559 -1.778 -0.101 148/(212) 3.111 0.703 -0.620 -1.082 -0.063 060/(300) 5.660 1.149 0.537 -1.815 -0.103 150/(210) 2.931 0.663 -0.596 -1.020 -0.060 062/(298) 5.757 1.172 0.511 -1.849 -0.106 152/(208) 2.748 0.623 -0.571 -0.958 -0.056 064/(296) 5.847 1.193 0.483 -1.881 -0.107 154/(206) 2.562 0.582 -0.542 -0.894 -0.052 066/(294) 5.929 1.212 0.453 -1.911 -0.109 156/(204) 2.375 0.540 -0.511 -0.829 -0.049 068/(292) 6.002 1.230 0.420 -1.938 -0.111 158/(202) 2.184 0.497 -0.477 -0.764 -0.045 070/(290) 6.068 1.247 0.385 -1.963 -0.112 160/(200) 1.992 0.454 -0.441 -0.697 -0.041 072/(288) 6.126 1.262 0.348 -1.985 -0.114 162/(198) 1.798 0.410 -0.404 -0.630 -0.037 074/(286) 6.176 1.276 0.309 -2.006 -0.115 164/(196) 1.603 0.366 -0.364 -0.561 -0.033 076/(284) 6.218 1.288 0.269 -2.023 -0.116 166/(194) 1.406 0.321 -0.322 -0.493 -0.029 078/(282) 6.252 1.298 0.227 -2.038 -0.117 168/(192) 1.207 0.276 -0.279 -0.423 -0.025 080/(280) 6.278 1.307 0.184 -2.051 -0.118 170/(190) 1.008 0.230 -0.235 -0.354 -0.021 082/(278) 6.296 1.314 0.140 -2.061 -0.118 172/(188) 0.807 0.185 -0.189 -0.283 -0.017 084/(276) 6.306 1.320 0.094 -2.068 -0.119 174/(186) 0.606 0.139 -0.143 -0.213 -0.012 086/(274) 6.308 1.324 0.049 -2.073 -0.119 176/(184) 0.404 0.093 -0.095 -0.142 -0.008 088/(272) 6.302 1.326 0.003 -2.075 -0.119 178/(182) 0.202 0.046 -0.048 -0.071 -0.004 090/(270) 6.289 1.327 -0.044 -2.075 -0.119 180/(180) 0.000 0.000 -0.000 -0.000 -0.000

Table 42: Dates and fractional Julian day numbers of the first new moons of the years 1900-2099 CE.
 01/1/1900 2.41502e+06 18/1/1950 2.4333e+06 06/1/2000 2.45155e+06 23/1/2050 2.46983e+06 20/1/1901 2.41541e+06 07/1/1951 2.43365e+06 24/1/2001 2.45193e+06 12/1/2051 2.47018e+06 09/1/1902 2.41576e+06 26/1/1952 2.43404e+06 13/1/2002 2.45229e+06 02/1/2052 2.47054e+06 28/1/1903 2.41614e+06 15/1/1953 2.43439e+06 02/1/2003 2.45264e+06 19/1/2053 2.47092e+06 17/1/1904 2.4165e+06 05/1/1954 2.43475e+06 21/1/2004 2.45303e+06 08/1/2054 2.47128e+06 05/1/1905 2.41685e+06 24/1/1955 2.43513e+06 10/1/2005 2.45338e+06 27/1/2055 2.47166e+06 24/1/1906 2.41724e+06 13/1/1956 2.43549e+06 29/1/2006 2.45377e+06 16/1/2056 2.47201e+06 14/1/1907 2.41759e+06 01/1/1957 2.43584e+06 19/1/2007 2.45412e+06 05/1/2057 2.47237e+06 03/1/1908 2.41794e+06 19/1/1958 2.43622e+06 08/1/2008 2.45447e+06 24/1/2058 2.47275e+06 22/1/1909 2.41833e+06 09/1/1959 2.43658e+06 26/1/2009 2.45486e+06 14/1/2059 2.47311e+06 11/1/1910 2.41868e+06 28/1/1960 2.43696e+06 15/1/2010 2.45521e+06 03/1/2060 2.47346e+06 30/1/1911 2.41907e+06 16/1/1961 2.43732e+06 04/1/2011 2.45557e+06 21/1/2061 2.47385e+06 19/1/1912 2.41942e+06 06/1/1962 2.43767e+06 23/1/2012 2.45595e+06 10/1/2062 2.4742e+06 07/1/1913 2.41977e+06 25/1/1963 2.43806e+06 11/1/2013 2.4563e+06 29/1/2063 2.47458e+06 26/1/1914 2.42016e+06 14/1/1964 2.43841e+06 01/1/2014 2.45666e+06 18/1/2064 2.47494e+06 15/1/1915 2.42051e+06 02/1/1965 2.43876e+06 20/1/2015 2.45704e+06 06/1/2065 2.47529e+06 05/1/1916 2.42087e+06 21/1/1966 2.43915e+06 10/1/2016 2.4574e+06 25/1/2066 2.47568e+06 23/1/1917 2.42125e+06 10/1/1967 2.4395e+06 27/1/2017 2.45778e+06 15/1/2067 2.47603e+06 12/1/1918 2.42161e+06 29/1/1968 2.43989e+06 17/1/2018 2.45814e+06 05/1/2068 2.47639e+06 02/1/1919 2.42196e+06 18/1/1969 2.44024e+06 06/1/2019 2.45849e+06 23/1/2069 2.47677e+06 21/1/1920 2.42234e+06 07/1/1970 2.44059e+06 24/1/2020 2.45887e+06 12/1/2070 2.47712e+06 09/1/1921 2.4227e+06 26/1/1971 2.44098e+06 13/1/2021 2.45923e+06 01/1/2071 2.47748e+06 28/1/1922 2.42308e+06 16/1/1972 2.44133e+06 02/1/2022 2.45958e+06 20/1/2072 2.47786e+06 17/1/1923 2.42344e+06 04/1/1973 2.44169e+06 21/1/2023 2.45997e+06 08/1/2073 2.47822e+06 06/1/1924 2.42379e+06 23/1/1974 2.44207e+06 11/1/2024 2.46032e+06 27/1/2074 2.4786e+06 24/1/1925 2.42418e+06 12/1/1975 2.44242e+06 29/1/2025 2.46071e+06 16/1/2075 2.47895e+06 14/1/1926 2.42453e+06 01/1/1976 2.44278e+06 18/1/2026 2.46106e+06 06/1/2076 2.47931e+06 03/1/1927 2.42488e+06 19/1/1977 2.44316e+06 07/1/2027 2.46141e+06 24/1/2077 2.47969e+06 22/1/1928 2.42527e+06 09/1/1978 2.44352e+06 26/1/2028 2.4618e+06 14/1/2078 2.48005e+06 11/1/1929 2.42562e+06 28/1/1979 2.4439e+06 14/1/2029 2.46215e+06 03/1/2079 2.4804e+06 29/1/1930 2.42601e+06 17/1/1980 2.44426e+06 04/1/2030 2.46251e+06 22/1/2080 2.48079e+06 18/1/1931 2.42636e+06 06/1/1981 2.44461e+06 23/1/2031 2.46289e+06 10/1/2081 2.48114e+06 07/1/1932 2.42671e+06 25/1/1982 2.44499e+06 12/1/2032 2.46324e+06 28/1/2082 2.48152e+06 25/1/1933 2.4271e+06 14/1/1983 2.44535e+06 01/1/2033 2.4636e+06 18/1/2083 2.48188e+06 15/1/1934 2.42745e+06 03/1/1984 2.4457e+06 20/1/2034 2.46398e+06 07/1/2084 2.48223e+06 05/1/1935 2.42781e+06 21/1/1985 2.44609e+06 09/1/2035 2.46434e+06 25/1/2085 2.48262e+06 24/1/1936 2.42819e+06 10/1/1986 2.44644e+06 28/1/2036 2.46472e+06 15/1/2086 2.48297e+06 12/1/1937 2.42855e+06 29/1/1987 2.44683e+06 16/1/2037 2.46507e+06 04/1/2087 2.48333e+06 01/1/1938 2.4289e+06 19/1/1988 2.44718e+06 05/1/2038 2.46543e+06 23/1/2088 2.48371e+06 20/1/1939 2.42928e+06 07/1/1989 2.44753e+06 24/1/2039 2.46581e+06 11/1/2089 2.48406e+06 09/1/1940 2.42964e+06 26/1/1990 2.44792e+06 14/1/2040 2.46617e+06 30/1/2090 2.48445e+06 27/1/1941 2.43002e+06 15/1/1991 2.44827e+06 02/1/2041 2.46652e+06 19/1/2091 2.4848e+06 16/1/1942 2.43038e+06 04/1/1992 2.44863e+06 21/1/2042 2.46691e+06 09/1/2092 2.48516e+06 06/1/1943 2.43073e+06 22/1/1993 2.44901e+06 11/1/2043 2.46726e+06 27/1/2093 2.48554e+06 25/1/1944 2.43112e+06 11/1/1994 2.44936e+06 30/1/2044 2.46764e+06 16/1/2094 2.48589e+06 14/1/1945 2.43147e+06 01/1/1995 2.44972e+06 18/1/2045 2.468e+06 06/1/2095 2.48625e+06 03/1/1946 2.43182e+06 20/1/1996 2.4501e+06 07/1/2046 2.46835e+06 25/1/2096 2.48663e+06 22/1/1947 2.43221e+06 09/1/1997 2.45046e+06 26/1/2047 2.46874e+06 13/1/2097 2.48699e+06 11/1/1948 2.43256e+06 28/1/1998 2.45084e+06 15/1/2048 2.46909e+06 02/1/2098 2.48734e+06 29/1/1949 2.43295e+06 17/1/1999 2.4512e+06 04/1/2049 2.46945e+06 21/1/2099 2.48772e+06

Table 43: Lunar-solar eclipse functions. The arguments of , , and are , , and , respectively. , , and take minus the values shown in the table if their arguments are in parenthesies.
 Arg. () Arg. () 000/360 3.128 0.856 0.267 (180)/(180) 002/358 3.126 0.855 0.267 (178)/(182) 004/356 3.120 0.854 0.267 (176)/(184) 006/354 3.111 0.851 0.266 (174)/(186) 008/352 3.097 0.847 0.265 (172)/(188) 010/350 3.080 0.843 0.263 (170)/(190) 012/348 3.059 0.837 0.261 (168)/(192) 014/346 3.035 0.830 0.259 (166)/(194) 016/344 3.007 0.822 0.257 (164)/(196) 018/342 2.975 0.814 0.254 (162)/(198) 020/340 2.939 0.804 0.251 (160)/(200) 022/338 2.900 0.793 0.248 (158)/(202) 024/336 2.857 0.782 0.244 (156)/(204) 026/334 2.811 0.769 0.240 (154)/(206) 028/332 2.762 0.755 0.236 (152)/(208) 030/330 2.709 0.741 0.231 (150)/(210) 032/328 2.652 0.726 0.227 (148)/(212) 034/326 2.593 0.709 0.222 (146)/(214) 036/324 2.530 0.692 0.216 (144)/(216) 038/322 2.465 0.674 0.211 (142)/(218) 040/320 2.396 0.655 0.205 (140)/(220) 042/318 2.324 0.636 0.199 (138)/(222) 044/316 2.250 0.615 0.192 (136)/(224) 046/314 2.173 0.594 0.186 (134)/(226) 048/312 2.093 0.573 0.179 (132)/(228) 050/310 2.010 0.550 0.172 (130)/(230) 052/308 1.926 0.527 0.165 (128)/(232) 054/306 1.838 0.503 0.157 (126)/(234) 056/304 1.749 0.478 0.149 (124)/(236) 058/302 1.657 0.453 0.142 (122)/(238) 060/300 1.564 0.428 0.134 (120)/(240) 062/298 1.468 0.402 0.125 (118)/(242) 064/296 1.371 0.375 0.117 (116)/(244) 066/294 1.272 0.348 0.109 (114)/(246) 068/292 1.172 0.321 0.100 (112)/(248) 070/290 1.070 0.293 0.091 (110)/(250) 072/288 0.967 0.264 0.083 (108)/(252) 074/286 0.862 0.236 0.074 (106)/(254) 076/284 0.757 0.207 0.065 (104)/(256) 078/282 0.650 0.178 0.056 (102)/(258) 080/280 0.543 0.149 0.046 (100)/(260) 082/278 0.435 0.119 0.037 (098)/(262) 084/276 0.327 0.089 0.028 (096)/(264) 086/274 0.218 0.060 0.019 (094)/(266) 088/272 0.109 0.030 0.009 (092)/(268) 090/270 0.000 0.000 0.000 (090)/(270)

Next: The Superior Planets Up: Lunar-Solar Syzygies and Eclipses Previous: Example Eclipse Calculations
Richard Fitzpatrick 2010-07-21