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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 $1/86,400$ 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 $0^\circ $. If we define local time (LT) as $LT = UT- \phi(^\circ)/15^\circ$ hrs., where $\phi$ 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, $\bar{\alpha}$, of the mean sun satisfies

\begin{displaymath}
\bar{\alpha} = \bar{\lambda},
\end{displaymath} (108)

where $\bar{\lambda}$ 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
\begin{displaymath}
\tan\alpha = \cos\epsilon\,\tan(\bar{\lambda}+q),
\end{displaymath} (109)

where $\epsilon$ is the inclination of the ecliptic to the celestial equator, $q(M)$ the sun's equation of center, and $M$ its mean anomaly. Now, neglecting the small time variation of the longitude of the sun's perigee [i.e., setting $\varpi_1=0$ in Eq. (95)], we can write [see Eqs. (100), (101), and (103), as well as Table 30]
\begin{displaymath}
M = \bar{\lambda} + M_0-\bar{\lambda}_0 = \bar{\lambda} +77.213^\circ.
\end{displaymath} (110)

It follows that, to first order in the solar eccentricity, $e$, we have
\begin{displaymath}
\Delta\alpha = \bar{\alpha}-\alpha = \lambda - \tan^{-1}(\cos \epsilon\,\tan\lambda) - 2\,e\,\sin\,M,
\end{displaymath} (111)

where
\begin{displaymath}
M = \lambda + 77.213^\circ.
\end{displaymath} (112)

Now,
\begin{displaymath}
\Delta t = \Delta\alpha(^\circ)/15^\circ
\end{displaymath} (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 $15^\circ$ an hour), and is known as the equation of time. If $\Delta t$ 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 $t_0= 2\,451\,545.0$ 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 $3.8246\times 10^{-5}\,\,^\circ/{\rm day}$.
Object $a\,(AU)$ $e$ $n\,(^\circ/{\rm day})$ $\tilde{n}\,(^\circ/{\rm day})$ $\bar{\lambda}_0\,(^\circ)$ $M_0\,(^\circ)$
             
Mercury $0.387098$ $0.205636$ $4.09237703$ $4.09233439$ $252.087$ $174.693$
Venus $0.723334$ $0.006777$ $1.60216872$ $1.60213040$ $181.973$ $49.237$
Sun $1.000000$ $0.016711$ $0.98564735$ $0.98560025$ $280.458$ $357.588$
Mars $1.523706$ $0.093394$ $0.52407118$ $0.52402076$ $355.460$ $19.388$
Jupiter $5.202873$ $0.048386$ $0.08312507$ $0.08308100$ $34.365$ $19.348$
Saturn $9.536651$ $0.053862$ $0.03350830$ $0.03348152$ $50.059$ $317.857$



Table 31: Arc minute to decimal fraction conversion table.
$00.0'$ .000 $10.0'$ .167 $20.0'$ .333 $30.0'$ .500 $40.0'$ .667 $50.0'$ .833
$00.2'$ .003 $10.2'$ .170 $20.2'$ .337 $30.2'$ .503 $40.2'$ .670 $50.2'$ .837
$00.4'$ .007 $10.4'$ .173 $20.4'$ .340 $30.4'$ .507 $40.4'$ .673 $50.4'$ .840
$00.6'$ .010 $10.6'$ .177 $20.6'$ .343 $30.6'$ .510 $40.6'$ .677 $50.6'$ .843
$00.8'$ .013 $10.8'$ .180 $20.8'$ .347 $30.8'$ .513 $40.8'$ .680 $50.8'$ .847
$01.0'$ .017 $11.0'$ .183 $21.0'$ .350 $31.0'$ .517 $41.0'$ .683 $51.0'$ .850
$01.2'$ .020 $11.2'$ .187 $21.2'$ .353 $31.2'$ .520 $41.2'$ .687 $51.2'$ .853
$01.4'$ .023 $11.4'$ .190 $21.4'$ .357 $31.4'$ .523 $41.4'$ .690 $51.4'$ .857
$01.6'$ .027 $11.6'$ .193 $21.6'$ .360 $31.6'$ .527 $41.6'$ .693 $51.6'$ .860
$01.8'$ .030 $11.8'$ .197 $21.8'$ .363 $31.8'$ .530 $41.8'$ .697 $51.8'$ .863
$02.0'$ .033 $12.0'$ .200 $22.0'$ .367 $32.0'$ .533 $42.0'$ .700 $52.0'$ .867
$02.2'$ .037 $12.2'$ .203 $22.2'$ .370 $32.2'$ .537 $42.2'$ .703 $52.2'$ .870
$02.4'$ .040 $12.4'$ .207 $22.4'$ .373 $32.4'$ .540 $42.4'$ .707 $52.4'$ .873
$02.6'$ .043 $12.6'$ .210 $22.6'$ .377 $32.6'$ .543 $42.6'$ .710 $52.6'$ .877
$02.8'$ .047 $12.8'$ .213 $22.8'$ .380 $32.8'$ .547 $42.8'$ .713 $52.8'$ .880
$03.0'$ .050 $13.0'$ .217 $23.0'$ .383 $33.0'$ .550 $43.0'$ .717 $53.0'$ .883
$03.2'$ .053 $13.2'$ .220 $23.2'$ .387 $33.2'$ .553 $43.2'$ .720 $53.2'$ .887
$03.4'$ .057 $13.4'$ .223 $23.4'$ .390 $33.4'$ .557 $43.4'$ .723 $53.4'$ .890
$03.6'$ .060 $13.6'$ .227 $23.6'$ .393 $33.6'$ .560 $43.6'$ .727 $53.6'$ .893
$03.8'$ .063 $13.8'$ .230 $23.8'$ .397 $33.8'$ .563 $43.8'$ .730 $53.8'$ .897
$04.0'$ .067 $14.0'$ .233 $24.0'$ .400 $34.0'$ .567 $44.0'$ .733 $54.0'$ .900
$04.2'$ .070 $14.2'$ .237 $24.2'$ .403 $34.2'$ .570 $44.2'$ .737 $54.2'$ .903
$04.4'$ .073 $14.4'$ .240 $24.4'$ .407 $34.4'$ .573 $44.4'$ .740 $54.4'$ .907
$04.6'$ .077 $14.6'$ .243 $24.6'$ .410 $34.6'$ .577 $44.6'$ .743 $54.6'$ .910
$04.8'$ .080 $14.8'$ .247 $24.8'$ .413 $34.8'$ .580 $44.8'$ .747 $54.8'$ .913
$05.0'$ .083 $15.0'$ .250 $25.0'$ .417 $35.0'$ .583 $45.0'$ .750 $55.0'$ .917
$05.2'$ .087 $15.2'$ .253 $25.2'$ .420 $35.2'$ .587 $45.2'$ .753 $55.2'$ .920
$05.4'$ .090 $15.4'$ .257 $25.4'$ .423 $35.4'$ .590 $45.4'$ .757 $55.4'$ .923
$05.6'$ .093 $15.6'$ .260 $25.6'$ .427 $35.6'$ .593 $45.6'$ .760 $55.6'$ .927
$05.8'$ .097 $15.8'$ .263 $25.8'$ .430 $35.8'$ .597 $45.8'$ .763 $55.8'$ .930
$06.0'$ .100 $16.0'$ .267 $26.0'$ .433 $36.0'$ .600 $46.0'$ .767 $56.0'$ .933
$06.2'$ .103 $16.2'$ .270 $26.2'$ .437 $36.2'$ .603 $46.2'$ .770 $56.2'$ .937
$06.4'$ .107 $16.4'$ .273 $26.4'$ .440 $36.4'$ .607 $46.4'$ .773 $56.4'$ .940
$06.6'$ .110 $16.6'$ .277 $26.6'$ .443 $36.6'$ .610 $46.6'$ .777 $56.6'$ .943
$06.8'$ .113 $16.8'$ .280 $26.8'$ .447 $36.8'$ .613 $46.8'$ .780 $56.8'$ .947
$07.0'$ .117 $17.0'$ .283 $27.0'$ .450 $37.0'$ .617 $47.0'$ .783 $57.0'$ .950
$07.2'$ .120 $17.2'$ .287 $27.2'$ .453 $37.2'$ .620 $47.2'$ .787 $57.2'$ .953
$07.4'$ .123 $17.4'$ .290 $27.4'$ .457 $37.4'$ .623 $47.4'$ .790 $57.4'$ .957
$07.6'$ .127 $17.6'$ .293 $27.6'$ .460 $37.6'$ .627 $47.6'$ .793 $57.6'$ .960
$07.8'$ .130 $17.8'$ .297 $27.8'$ .463 $37.8'$ .630 $47.8'$ .797 $57.8'$ .963
$08.0'$ .133 $18.0'$ .300 $28.0'$ .467 $38.0'$ .633 $48.0'$ .800 $58.0'$ .967
$08.2'$ .137 $18.2'$ .303 $28.2'$ .470 $38.2'$ .637 $48.2'$ .803 $58.2'$ .970
$08.4'$ .140 $18.4'$ .307 $28.4'$ .473 $38.4'$ .640 $48.4'$ .807 $58.4'$ .973
$08.6'$ .143 $18.6'$ .310 $28.6'$ .477 $38.6'$ .643 $48.6'$ .810 $58.6'$ .977
$08.8'$ .147 $18.8'$ .313 $28.8'$ .480 $38.8'$ .647 $48.8'$ .813 $58.8'$ .980
$09.0'$ .150 $19.0'$ .317 $29.0'$ .483 $39.0'$ .650 $49.0'$ .817 $59.0'$ .983
$09.2'$ .153 $19.2'$ .320 $29.2'$ .487 $39.2'$ .653 $49.2'$ .820 $59.2'$ .987
$09.4'$ .157 $19.4'$ .323 $29.4'$ .490 $39.4'$ .657 $49.4'$ .823 $59.4'$ .990
$09.6'$ .160 $19.6'$ .327 $29.6'$ .493 $39.6'$ .660 $49.6'$ .827 $59.6'$ .993
$09.8'$ .163 $19.8'$ .330 $29.8'$ .497 $39.8'$ .663 $49.8'$ .830 $59.8'$ .997



Table 32: Mean motion of the sun. Here, $\Delta t = t-t_0$, $\Delta\bar{\lambda} = \bar{\lambda}-\bar{\lambda}_0$, and $\Delta M = M - M_0$. At epoch ( $t_0= 2\,451\,545.0$ JD), $\bar{\lambda}_0 = 280.458^\circ$, and $M_0 = 357.588^\circ$.
$\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$ $\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$ $\Delta t$(JD) $\Delta\bar{\lambda}(^\circ)$ $\Delta M(^\circ)$
                 
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.
$M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$ $M(^\circ)$ $q(^\circ)$ $100\,\zeta$
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 $m$ and $s$ denote minutes and seconds.
Aries Taurus Gemini Cancer Leo Virgo
$\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$
00$^\circ$ -07$^m$28$^s$ 00$^\circ$ +01$^m$02$^s$ 00$^\circ$ +03$^m$30$^s$ 00$^\circ$ -01$^m$42$^s$ 00$^\circ$ -06$^m$27$^s$ 00$^\circ$ -02$^m$44$^s$
02$^\circ$ -06$^m$51$^s$ 02$^\circ$ +01$^m$27$^s$ 02$^\circ$ +03$^m$21$^s$ 02$^\circ$ -02$^m$09$^s$ 02$^\circ$ -06$^m$30$^s$ 02$^\circ$ -02$^m$11$^s$
04$^\circ$ -06$^m$15$^s$ 04$^\circ$ +01$^m$50$^s$ 04$^\circ$ +03$^m$10$^s$ 04$^\circ$ -02$^m$36$^s$ 04$^\circ$ -06$^m$31$^s$ 04$^\circ$ -01$^m$36$^s$
06$^\circ$ -05$^m$38$^s$ 06$^\circ$ +02$^m$12$^s$ 06$^\circ$ +02$^m$56$^s$ 06$^\circ$ -03$^m$03$^s$ 06$^\circ$ -06$^m$29$^s$ 06$^\circ$ +00$^m$59$^s$
08$^\circ$ -05$^m$01$^s$ 08$^\circ$ +02$^m$31$^s$ 08$^\circ$ +02$^m$41$^s$ 08$^\circ$ -03$^m$29$^s$ 08$^\circ$ -06$^m$24$^s$ 08$^\circ$ +00$^m$21$^s$
10$^\circ$ -04$^m$24$^s$ 10$^\circ$ +02$^m$48$^s$ 10$^\circ$ +02$^m$23$^s$ 10$^\circ$ -03$^m$53$^s$ 10$^\circ$ -06$^m$16$^s$ 10$^\circ$ +00$^m$18$^s$
12$^\circ$ -03$^m$48$^s$ 12$^\circ$ +03$^m$03$^s$ 12$^\circ$ +02$^m$04$^s$ 12$^\circ$ -04$^m$17$^s$ 12$^\circ$ -06$^m$06$^s$ 12$^\circ$ +00$^m$58$^s$
14$^\circ$ -03$^m$12$^s$ 14$^\circ$ +03$^m$16$^s$ 14$^\circ$ +01$^m$43$^s$ 14$^\circ$ -04$^m$39$^s$ 14$^\circ$ -05$^m$54$^s$ 14$^\circ$ +01$^m$40$^s$
16$^\circ$ -02$^m$36$^s$ 16$^\circ$ +03$^m$26$^s$ 16$^\circ$ +01$^m$20$^s$ 16$^\circ$ -05$^m$00$^s$ 16$^\circ$ -05$^m$38$^s$ 16$^\circ$ +02$^m$22$^s$
18$^\circ$ -02$^m$01$^s$ 18$^\circ$ +03$^m$34$^s$ 18$^\circ$ +00$^m$57$^s$ 18$^\circ$ -05$^m$19$^s$ 18$^\circ$ -05$^m$20$^s$ 18$^\circ$ +03$^m$05$^s$
20$^\circ$ -01$^m$28$^s$ 20$^\circ$ +03$^m$40$^s$ 20$^\circ$ +00$^m$32$^s$ 20$^\circ$ -05$^m$35$^s$ 20$^\circ$ -05$^m$00$^s$ 20$^\circ$ +03$^m$49$^s$
22$^\circ$ +00$^m$55$^s$ 22$^\circ$ +03$^m$43$^s$ 22$^\circ$ +00$^m$06$^s$ 22$^\circ$ -05$^m$50$^s$ 22$^\circ$ -04$^m$37$^s$ 22$^\circ$ +04$^m$32$^s$
24$^\circ$ +00$^m$23$^s$ 24$^\circ$ +03$^m$43$^s$ 24$^\circ$ +00$^m$20$^s$ 24$^\circ$ -06$^m$03$^s$ 24$^\circ$ -04$^m$12$^s$ 24$^\circ$ +05$^m$16$^s$
26$^\circ$ +00$^m$06$^s$ 26$^\circ$ +03$^m$41$^s$ 26$^\circ$ +00$^m$47$^s$ 26$^\circ$ -06$^m$14$^s$ 26$^\circ$ -03$^m$45$^s$ 26$^\circ$ +06$^m$00$^s$
28$^\circ$ +00$^m$34$^s$ 28$^\circ$ +03$^m$37$^s$ 28$^\circ$ -01$^m$14$^s$ 28$^\circ$ -06$^m$22$^s$ 28$^\circ$ -03$^m$15$^s$ 28$^\circ$ +06$^m$44$^s$
30$^\circ$ +01$^m$02$^s$ 30$^\circ$ +03$^m$30$^s$ 30$^\circ$ -01$^m$42$^s$ 30$^\circ$ -06$^m$27$^s$ 30$^\circ$ -02$^m$44$^s$ 30$^\circ$ +07$^m$28$^s$
Libra Scorpio Sagittarius Capricorn Aquarius Pisces
$\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$ $\lambda$ $\Delta t~~~~~$
00$^\circ$ +07$^m$28$^s$ 00$^\circ$ +15$^m$40$^s$ 00$^\circ$ +13$^m$55$^s$ 00$^\circ$ +01$^m$42$^s$ 00$^\circ$ -10$^m$58$^s$ 00$^\circ$ -13$^m$58$^s$
02$^\circ$ +08$^m$10$^s$ 02$^\circ$ +15$^m$55$^s$ 02$^\circ$ +13$^m$22$^s$ 02$^\circ$ +00$^m$43$^s$ 02$^\circ$ -11$^m$32$^s$ 02$^\circ$ -13$^m$46$^s$
04$^\circ$ +08$^m$53$^s$ 04$^\circ$ +16$^m$08$^s$ 04$^\circ$ +12$^m$46$^s$ 04$^\circ$ +00$^m$15$^s$ 04$^\circ$ -12$^m$02$^s$ 04$^\circ$ -13$^m$31$^s$
06$^\circ$ +09$^m$34$^s$ 06$^\circ$ +16$^m$17$^s$ 06$^\circ$ +12$^m$08$^s$ 06$^\circ$ -01$^m$13$^s$ 06$^\circ$ -12$^m$30$^s$ 06$^\circ$ -13$^m$14$^s$
08$^\circ$ +10$^m$14$^s$ 08$^\circ$ +16$^m$23$^s$ 08$^\circ$ +11$^m$26$^s$ 08$^\circ$ -02$^m$11$^s$ 08$^\circ$ -12$^m$54$^s$ 08$^\circ$ -12$^m$55$^s$
10$^\circ$ +10$^m$53$^s$ 10$^\circ$ +16$^m$26$^s$ 10$^\circ$ +10$^m$42$^s$ 10$^\circ$ -03$^m$07$^s$ 10$^\circ$ -13$^m$16$^s$ 10$^\circ$ -12$^m$33$^s$
12$^\circ$ +11$^m$30$^s$ 12$^\circ$ +16$^m$26$^s$ 12$^\circ$ +09$^m$55$^s$ 12$^\circ$ -04$^m$03$^s$ 12$^\circ$ -13$^m$34$^s$ 12$^\circ$ -12$^m$10$^s$
14$^\circ$ +12$^m$06$^s$ 14$^\circ$ +16$^m$23$^s$ 14$^\circ$ +09$^m$07$^s$ 14$^\circ$ -04$^m$57$^s$ 14$^\circ$ -13$^m$49$^s$ 14$^\circ$ -11$^m$44$^s$
16$^\circ$ +12$^m$41$^s$ 16$^\circ$ +16$^m$16$^s$ 16$^\circ$ +08$^m$16$^s$ 16$^\circ$ -05$^m$50$^s$ 16$^\circ$ -14$^m$01$^s$ 16$^\circ$ -11$^m$17$^s$
18$^\circ$ +13$^m$13$^s$ 18$^\circ$ +16$^m$06$^s$ 18$^\circ$ +07$^m$23$^s$ 18$^\circ$ -06$^m$41$^s$ 18$^\circ$ -14$^m$09$^s$ 18$^\circ$ -10$^m$48$^s$
20$^\circ$ +13$^m$43$^s$ 20$^\circ$ +15$^m$52$^s$ 20$^\circ$ +06$^m$29$^s$ 20$^\circ$ -07$^m$30$^s$ 20$^\circ$ -14$^m$15$^s$ 20$^\circ$ -10$^m$17$^s$
22$^\circ$ +14$^m$12$^s$ 22$^\circ$ +15$^m$36$^s$ 22$^\circ$ +05$^m$33$^s$ 22$^\circ$ -08$^m$16$^s$ 22$^\circ$ -14$^m$17$^s$ 22$^\circ$ -09$^m$45$^s$
24$^\circ$ +14$^m$38$^s$ 24$^\circ$ +15$^m$15$^s$ 24$^\circ$ +04$^m$37$^s$ 24$^\circ$ -09$^m$01$^s$ 24$^\circ$ -14$^m$17$^s$ 24$^\circ$ -09$^m$12$^s$
26$^\circ$ +15$^m$01$^s$ 26$^\circ$ +14$^m$52$^s$ 26$^\circ$ +03$^m$39$^s$ 26$^\circ$ -09$^m$42$^s$ 26$^\circ$ -14$^m$13$^s$ 26$^\circ$ -08$^m$38$^s$
28$^\circ$ +15$^m$22$^s$ 28$^\circ$ +14$^m$25$^s$ 28$^\circ$ +02$^m$41$^s$ 28$^\circ$ -10$^m$22$^s$ 28$^\circ$ -14$^m$07$^s$ 28$^\circ$ -08$^m$03$^s$
30$^\circ$ +15$^m$40$^s$ 30$^\circ$ +13$^m$55$^s$ 30$^\circ$ +01$^m$42$^s$ 30$^\circ$ -10$^m$58$^s$ 30$^\circ$ -13$^m$58$^s$ 30$^\circ$ -07$^m$28$^s$



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Next: The Moon Up: The Sun Previous: Determination of Equinox and
Richard Fitzpatrick 2010-07-21