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# Equation of Time

At any particular observation site on the earth's surface, local noon is defined as the instant in time when the sun culminates at the meridian. However, as a consequence of the inclination of the ecliptic to the celestial equatior, as well as the uneven motion of the sun around the ecliptic, the time interval between successive local noons, which is known as a solar day, is not constant, but varies throughout the year. Hence, if we were to define a second as of a solar day then the length of a second would also vary throughout the year, which is clearly undesirable. In order to avoid this problem, astronomers have invented a fictitious body called the mean sun. The mean sun travels around the celestial equator (from west to east) at a constant rate which is such that it completes one orbit every tropical year. Moreover, the mean sun and the true sun coincide at the spring equinox. Local mean noon at a particular observation site is defined as the instance in time when the mean sun culminates at the meridian. Since the orbit of the mean sun is not inclined to the celestial equator, and the mean sun travels around the celestial equator at a uniform rate, the time interval between successive mean noons, which is known as a mean solar day, takes the constant value of 24 hours, or 86,400 seconds, throughout the year. Universal time (UT) is defined such that 12:00 UT coincides with mean noon every day at an observation site of terrestrial longitude . If we define local time (LT) as hrs., where is the terrestrial longitude of the observation site, then 12:00 LT coincides with mean noon every day at a general observation site on the earth's surface.

According to the above definition, the right ascension, , of the mean sun satisfies

 (108)

where is the sun's mean ecliptic longitude. Moreover, it follows from Eqs. (16) and (98) that the right ascension of the true sun is given by
 (109)

where is the inclination of the ecliptic to the celestial equator, the sun's equation of center, and its mean anomaly. Now, neglecting the small time variation of the longitude of the sun's perigee [i.e., setting in Eq. (95)], we can write [see Eqs. (100), (101), and (103), as well as Table 30]
 (110)

It follows that, to first order in the solar eccentricity, , we have
 (111)

where
 (112)

Now,
 (113)

represents the time difference (in hours) between local noon and mean local noon (since right ascension crosses the meridian at the uniform rate of an hour), and is known as the equation of time. If is positive then local noon occurs before mean local noon, and vice versa.

The equation of time specifies the difference between time calculated using a sundial or sextant--which is known as solar time--and time obtained from an accurate clock--which is known as mean solar time. Table 34 shows the equation of time as a function of the sun's ecliptic longitude. It can be seen that the difference between solar time and mean solar time can be as much as 16 minutes, and attains its maximum value between the autumnal equinox and the winter solstice, and its minimum value between the winter solstice and vernal equinox.

Table 30: Keplerian orbital elements for the sun and the five visible planets at the J2000 epoch (i.e., 12:00 UT, January 1, 2000 CE, which corresponds to JD). The elements are optimized for use in the time period 1800 CE to 2050 CE. Source: Jet Propulsion Laboratory (NASA), http://ssd.jpl.nasa.gov/. The motion rates have been converted into tropical motion rates assuming a uniform precession of the equinoxes of .
 Object Mercury Venus Sun Mars Jupiter Saturn

Table 31: Arc minute to decimal fraction conversion table.
 0 0.167 0.333 0.5 0.667 0.833 0.003 0.17 0.337 0.503 0.67 0.837 0.007 0.173 0.34 0.507 0.673 0.84 0.01 0.177 0.343 0.51 0.677 0.843 0.013 0.18 0.347 0.513 0.68 0.847 0.017 0.183 0.35 0.517 0.683 0.85 0.02 0.187 0.353 0.52 0.687 0.853 0.023 0.19 0.357 0.523 0.69 0.857 0.027 0.193 0.36 0.527 0.693 0.86 0.03 0.197 0.363 0.53 0.697 0.863 0.033 0.2 0.367 0.533 0.7 0.867 0.037 0.203 0.37 0.537 0.703 0.87 0.04 0.207 0.373 0.54 0.707 0.873 0.043 0.21 0.377 0.543 0.71 0.877 0.047 0.213 0.38 0.547 0.713 0.88 0.05 0.217 0.383 0.55 0.717 0.883 0.053 0.22 0.387 0.553 0.72 0.887 0.057 0.223 0.39 0.557 0.723 0.89 0.06 0.227 0.393 0.56 0.727 0.893 0.063 0.23 0.397 0.563 0.73 0.897 0.067 0.233 0.4 0.567 0.733 0.9 0.07 0.237 0.403 0.57 0.737 0.903 0.073 0.24 0.407 0.573 0.74 0.907 0.077 0.243 0.41 0.577 0.743 0.91 0.08 0.247 0.413 0.58 0.747 0.913 0.083 0.25 0.417 0.583 0.75 0.917 0.087 0.253 0.42 0.587 0.753 0.92 0.09 0.257 0.423 0.59 0.757 0.923 0.093 0.26 0.427 0.593 0.76 0.927 0.097 0.263 0.43 0.597 0.763 0.93 0.1 0.267 0.433 0.6 0.767 0.933 0.103 0.27 0.437 0.603 0.77 0.937 0.107 0.273 0.44 0.607 0.773 0.94 0.11 0.277 0.443 0.61 0.777 0.943 0.113 0.28 0.447 0.613 0.78 0.947 0.117 0.283 0.45 0.617 0.783 0.95 0.12 0.287 0.453 0.62 0.787 0.953 0.123 0.29 0.457 0.623 0.79 0.957 0.127 0.293 0.46 0.627 0.793 0.96 0.13 0.297 0.463 0.63 0.797 0.963 0.133 0.3 0.467 0.633 0.8 0.967 0.137 0.303 0.47 0.637 0.803 0.97 0.14 0.307 0.473 0.64 0.807 0.973 0.143 0.31 0.477 0.643 0.81 0.977 0.147 0.313 0.48 0.647 0.813 0.98 0.15 0.317 0.483 0.65 0.817 0.983 0.153 0.32 0.487 0.653 0.82 0.987 0.157 0.323 0.49 0.657 0.823 0.99 0.16 0.327 0.493 0.66 0.827 0.993 0.163 0.33 0.497 0.663 0.83 0.997

Table 32: Mean motion of the sun. Here, , , and . At epoch ( JD), , and .
 (JD) (JD) (JD) 10,000 136.474 136.002 1,000 265.647 265.600 100 98.565 98.560 20,000 272.947 272.005 2,000 171.295 171.200 200 197.129 197.120 30,000 49.421 48.007 3,000 76.942 76.801 300 295.694 295.680 40,000 185.894 184.010 4,000 342.589 342.401 400 34.259 34.240 50,000 322.367 320.012 5,000 248.237 248.001 500 132.824 132.800 60,000 98.841 96.015 6,000 153.884 153.601 600 231.388 231.360 70,000 235.315 232.017 7,000 59.531 59.202 700 329.953 329.920 80,000 11.788 8.020 8,000 325.179 324.802 800 68.518 68.480 90,000 148.262 144.022 9,000 230.826 230.402 900 167.083 167.040 10 9.856 9.856 1 0.986 0.986 0.1 0.099 0.099 20 19.713 19.712 2 1.971 1.971 0.2 0.197 0.197 30 29.569 29.568 3 2.957 2.957 0.3 0.296 0.296 40 39.426 39.424 4 3.943 3.942 0.4 0.394 0.394 50 49.282 49.280 5 4.928 4.928 0.5 0.493 0.493 60 59.139 59.136 6 5.914 5.914 0.6 0.591 0.591 70 68.995 68.992 7 6.900 6.899 0.7 0.690 0.690 80 78.852 78.848 8 7.885 7.885 0.8 0.789 0.788 90 88.708 88.704 9 8.871 8.870 0.9 0.887 0.887

Table 33: Anomalies of the sun.
 0 0.000 1.671 90 1.915 -0.028 180 0.000 -1.671 270 -1.915 -0.028 2 0.068 1.670 92 1.912 -0.086 182 -0.065 -1.670 272 -1.915 0.030 4 0.136 1.667 94 1.907 -0.144 184 -0.131 -1.667 274 -1.913 0.089 6 0.204 1.662 96 1.900 -0.202 186 -0.196 -1.662 276 -1.909 0.147 8 0.272 1.654 98 1.891 -0.260 188 -0.261 -1.655 278 -1.902 0.205 10 0.339 1.645 100 1.879 -0.317 190 -0.326 -1.647 280 -1.893 0.263 12 0.406 1.633 102 1.865 -0.374 192 -0.390 -1.636 282 -1.881 0.321 14 0.473 1.620 104 1.849 -0.431 194 -0.454 -1.623 284 -1.867 0.378 16 0.538 1.604 106 1.830 -0.486 196 -0.517 -1.608 286 -1.851 0.435 18 0.604 1.587 108 1.809 -0.542 198 -0.580 -1.592 288 -1.833 0.491 20 0.668 1.567 110 1.787 -0.596 200 -0.642 -1.574 290 -1.812 0.547 22 0.731 1.545 112 1.762 -0.650 202 -0.703 -1.553 292 -1.789 0.602 24 0.794 1.522 114 1.735 -0.703 204 -0.764 -1.531 294 -1.764 0.656 26 0.855 1.497 116 1.705 -0.755 206 -0.824 -1.507 296 -1.737 0.710 28 0.916 1.469 118 1.674 -0.806 208 -0.882 -1.482 298 -1.707 0.763 30 0.975 1.440 120 1.641 -0.856 210 -0.940 -1.454 300 -1.676 0.815 32 1.033 1.409 122 1.606 -0.906 212 -0.997 -1.425 302 -1.642 0.865 34 1.089 1.377 124 1.569 -0.954 214 -1.052 -1.394 304 -1.606 0.915 36 1.145 1.342 126 1.530 -1.001 216 -1.107 -1.362 306 -1.568 0.964 38 1.198 1.306 128 1.490 -1.046 218 -1.160 -1.327 308 -1.528 1.011 40 1.251 1.269 130 1.447 -1.091 220 -1.211 -1.292 310 -1.487 1.058 42 1.301 1.229 132 1.403 -1.134 222 -1.261 -1.254 312 -1.443 1.103 44 1.350 1.189 134 1.358 -1.175 224 -1.310 -1.216 314 -1.397 1.146 46 1.397 1.146 136 1.310 -1.216 226 -1.358 -1.175 316 -1.350 1.189 48 1.443 1.103 138 1.261 -1.254 228 -1.403 -1.134 318 -1.301 1.229 50 1.487 1.058 140 1.211 -1.292 230 -1.447 -1.091 320 -1.251 1.269 52 1.528 1.011 142 1.160 -1.327 232 -1.490 -1.046 322 -1.198 1.306 54 1.568 0.964 144 1.107 -1.362 234 -1.530 -1.001 324 -1.145 1.342 56 1.606 0.915 146 1.052 -1.394 236 -1.569 -0.954 326 -1.089 1.377 58 1.642 0.865 148 0.997 -1.425 238 -1.606 -0.906 328 -1.033 1.409 60 1.676 0.815 150 0.940 -1.454 240 -1.641 -0.856 330 -0.975 1.440 62 1.707 0.763 152 0.882 -1.482 242 -1.674 -0.806 332 -0.916 1.469 64 1.737 0.710 154 0.824 -1.507 244 -1.705 -0.755 334 -0.855 1.497 66 1.764 0.656 156 0.764 -1.531 246 -1.735 -0.703 336 -0.794 1.522 68 1.789 0.602 158 0.703 -1.553 248 -1.762 -0.650 338 -0.731 1.545 70 1.812 0.547 160 0.642 -1.574 250 -1.787 -0.596 340 -0.668 1.567 72 1.833 0.491 162 0.580 -1.592 252 -1.809 -0.542 342 -0.604 1.587 74 1.851 0.435 164 0.517 -1.608 254 -1.830 -0.486 344 -0.538 1.604 76 1.867 0.378 166 0.454 -1.623 256 -1.849 -0.431 346 -0.473 1.620 78 1.881 0.321 168 0.390 -1.636 258 -1.865 -0.374 348 -0.406 1.633 80 1.893 0.263 170 0.326 -1.647 260 -1.879 -0.317 350 -0.339 1.645 82 1.902 0.205 172 0.261 -1.655 262 -1.891 -0.260 352 -0.272 1.654 84 1.909 0.147 174 0.196 -1.662 264 -1.900 -0.202 354 -0.204 1.662 86 1.913 0.089 176 0.131 -1.667 266 -1.907 -0.144 356 -0.136 1.667 88 1.915 0.030 178 0.065 -1.670 268 -1.912 -0.086 358 -0.068 1.670 90 1.915 -0.028 180 0.000 -1.671 270 -1.915 -0.028 360 -0.000 1.671

Table 34: The equation of time. The superscripts and denote minutes and seconds.
 Aries Taurus Gemini Cancer Leo Virgo 00 -0728 00 +0102 00 +0330 00 -0142 00 -0627 00 -0244 02 -0651 02 +0127 02 +0321 02 -0209 02 -0630 02 -0211 04 -0615 04 +0150 04 +0310 04 -0236 04 -0631 04 -0136 06 -0538 06 +0212 06 +0256 06 -0303 06 -0629 06 +0059 08 -0501 08 +0231 08 +0241 08 -0329 08 -0624 08 +0021 10 -0424 10 +0248 10 +0223 10 -0353 10 -0616 10 +0018 12 -0348 12 +0303 12 +0204 12 -0417 12 -0606 12 +0058 14 -0312 14 +0316 14 +0143 14 -0439 14 -0554 14 +0140 16 -0236 16 +0326 16 +0120 16 -0500 16 -0538 16 +0222 18 -0201 18 +0334 18 +0057 18 -0519 18 -0520 18 +0305 20 -0128 20 +0340 20 +0032 20 -0535 20 -0500 20 +0349 22 +0055 22 +0343 22 +0006 22 -0550 22 -0437 22 +0432 24 +0023 24 +0343 24 +0020 24 -0603 24 -0412 24 +0516 26 +0006 26 +0341 26 +0047 26 -0614 26 -0345 26 +0600 28 +0034 28 +0337 28 -0114 28 -0622 28 -0315 28 +0644 30 +0102 30 +0330 30 -0142 30 -0627 30 -0244 30 +0728 Libra Scorpio Sagittarius Capricorn Aquarius Pisces 00 +0728 00 +1540 00 +1355 00 +0142 00 -1058 00 -1358 02 +0810 02 +1555 02 +1322 02 +0043 02 -1132 02 -1346 04 +0853 04 +1608 04 +1246 04 +0015 04 -1202 04 -1331 06 +0934 06 +1617 06 +1208 06 -0113 06 -1230 06 -1314 08 +1014 08 +1623 08 +1126 08 -0211 08 -1254 08 -1255 10 +1053 10 +1626 10 +1042 10 -0307 10 -1316 10 -1233 12 +1130 12 +1626 12 +0955 12 -0403 12 -1334 12 -1210 14 +1206 14 +1623 14 +0907 14 -0457 14 -1349 14 -1144 16 +1241 16 +1616 16 +0816 16 -0550 16 -1401 16 -1117 18 +1313 18 +1606 18 +0723 18 -0641 18 -1409 18 -1048 20 +1343 20 +1552 20 +0629 20 -0730 20 -1415 20 -1017 22 +1412 22 +1536 22 +0533 22 -0816 22 -1417 22 -0945 24 +1438 24 +1515 24 +0437 24 -0901 24 -1417 24 -0912 26 +1501 26 +1452 26 +0339 26 -0942 26 -1413 26 -0838 28 +1522 28 +1425 28 +0241 28 -1022 28 -1407 28 -0803 30 +1540 30 +1355 30 +0142 30 -1058 30 -1358 30 -0728

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Richard Fitzpatrick 2010-07-21