The Sun

Our solar model is sketched in Figure 17. From a geocentric point of view, the sun, $S$, appears to execute a (counterclockwise) Keplerian orbit of major radius $a$, and eccentricity $e$, about the earth, $G$. As has already been mentioned, the circle traced out by the sun on the celestial sphere is known as the ecliptic circle. This circle is inclined at $23^\circ 26'$ to the celestial equator, which is the projection of the earth's equator onto the celestial sphere. Suppose that the angle subtended at the earth between the vernal equinox (i.e., the point at which the sun passes the celestial equator from south to north) and the sun's perigee (i.e., the point of closest approach to the earth) is $\varpi$. This angle is termed the longitude of the perigee, and is assumed to vary linearly with time:

$$\varpi = \varpi_0 + \varpi_1\,(t-t_0).$$ (79)

Figure 17: The apparent orbit of the sun about the earth. Here, $S$, $G$, $\Pi$, $A$, $\varpi$, $T$, $\lambda$, and $\Upsilon$ represent the sun, earth, perigee, apogee, longitude of the perigee, true anomaly, ecliptic longitude, and vernal equinox, respectively. View is from northern ecliptic pole. The sun orbits counterclockwise.

The sun's ecliptic longitude is defined as the angle subtended at the earth between the sun and the vernal equinox. Hence, from Fig. 17,

$$\lambda = \varpi + T,$$ (80)

where $T$ is the true anomaly. By analogy, the mean longitude is written

$$\bar{\lambda} = \varpi + M,$$ (81)

where $M$ is the mean anomaly. It follows from Eq. (23) that

$$\lambda = \bar{\lambda} + q,$$ (82)

where

$$q = 2\,e\,\sin M + (5/4)\,e^2\,\sin\,2M,$$ (83)

is called the equation of center. Note that $\lambda$, $\bar{\lambda}$, $T$, and $M$ are usually written as angles in the range $0^\circ$ to $360^\circ$, whereas $q$ is generally written as an angle in the range $-180^\circ$ to $+180^\circ$.

The mean longitude increases uniformly with time (since both $\varpi$ and $M$ increase uniformly with time) as

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t -t_0),$$ (84)

where $\bar{\lambda}_0$ is termed the mean longitude at epoch, $n$ the rate of motion in mean longitude, and $t_0$ the epoch. We can also write

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (85)

where

$$M_0 = \bar{\lambda}_0 - \varpi_0$$ (86)

is called the mean anomaly at epoch, and

$$\tilde{n} = n - \varpi_1$$ (87)

the rate of motion in mean anomaly.

Our procedure for determining the ecliptic longitude of the sun is described below. The requisite orbital elements (i.e., $e$, $n$, $\tilde{n}$, $\bar{\lambda}_0$, and $M_0$) for the J2000 epoch (i.e., 12:00 GMT on January 1, 2000 CE, which corresponds to $t_0= 2\,451\,545.0$ JD) are listed in Table 30. These elements are calculated on the assumption that the vernal equinox precesses at the uniform rate of $-3.8246\times 10^{-5}\,\,^\circ/{\rm day}$. The ecliptic longitude of the sun is specified by the following formulae:

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (88)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (89)

$$q = 2\,e\,\sin M + (5/4)\,e^2\,\sin\,2M,$$ (90)

$$\lambda = \bar{\lambda} + q.$$ (91)

These formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE (see http://ssd.jpl.nasa.gov/) with a mean error of $0.2'$ and a maximum error of $0.7'$.

The ecliptic longitude of the sun can be calculated with the aid of Tables 32 and 33. Table 32 allows the mean longitude, $\bar{\lambda}$, and mean anomaly, $M$, of the sun to be determined as functions of time. Table 33 specifies the equation of center, $q$, as a function of the mean anomaly.

The procedure for using the tables is as follows:

1. Determine the fractional Julian day number, $t$, corresponding to the date and time at which the sun's ecliptic longitude is to be calculated with the aid of Tables 1-3. Form $\Delta t = t-t_0$, where $t_0= 2\,451\,545.0$ is the epoch.
2. Enter Table 32 with the digit for each power of 10 in ${\Delta} t$ and take out the corresponding values of $\Delta\bar{\lambda}$ and $\Delta M$. If $\Delta t$ is negative then the corresponding values are also negative. The value of the mean longitude, $\bar{\lambda}$, is the sum of all the $\Delta\bar{\lambda}$ values plus the value of $\bar{\lambda}$ at the epoch. Likewise, the value of the mean anomaly, $M$, is the sum of all the $\Delta M$ values plus the value of $M$ at the epoch. Add as many multiples of $360^\circ$ to $\bar{\lambda}$ and $M$ as is required to make them both fall in the range $0^\circ$ to $360^\circ$. Round $M$ to the nearest degree.
3. Enter Table 33 with the value of $M$ and take out the corresponding value of the equation of center, $q$, and the radial anomaly, $\zeta$. (The latter step is only necessary if the ecliptic longitude of the sun is to be used to determine that of a planet.) It is necessary to interpolate if $M$ is odd.
4. The ecliptic longitude, $\lambda$, is the sum of the mean longitude, $\bar{\lambda}$, and the equation of center, $q$. If necessary, convert $\lambda$ into an angle in the range $0^\circ$ to $360^\circ$. The decimal fraction can be converted into arc minutes using Table 31. Round to the nearest arc minute.

Two examples of the use of this procedure are given below.

Example 1: May 5, 2005 CE, 00:00 GMT:
 
According to Tables 1-3, $t = 2\,453\,495.5$ JD. Hence, $t-t_0 = 2\,453\,495.5-2\,451\,545.0=1\,950.5$ JD. Making use of Table 32, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$
+1000	$265.647$	$265.600$
+900	$167.083$	$167.040$
+50	$49.280$	$49.280$
+.5	$0.493$	$0.493$
Epoch	$280.458$	$357.588$
	$762.961$	$840.001$
Modulus	$42.961$	$120.001$

Rounding the mean anomaly to the nearest degree, we obtain $M\simeq 120^\circ$. It follows from Table 33 that

$$q(120^\circ)= 1.641^\circ,$$

so

$$\lambda =\bar{\lambda} + q =42.961+ 1.641=44.602\simeq 44^\circ36'.$$

Here, we have converted the decimal fraction into arc minutes using Table 31, and then rounded the final result to the nearest arc minute.

Following the practice of the Ancient Greeks (and modern-day astrologers), we shall express ecliptic longitudes in terms of the signs of the zodiac, which are listed in Sect. 4.6. The ecliptic longitude $44^\circ36'$ is conventionally written 14TA36: i.e., $14^\circ36'$ into the sign of Taurus. Thus, we conclude that the position of the sun at 00:00 GMT on May 5, 2005 CE was 14TA36.

Example 2: December 25, 1800 CE, 00:00 GMT:
 
According to Tables 1-3, $t = 2\,378\,854.5$ JD. Hence, $t-t_0 = 2\,378\,854.5-2\,451\,545.0=-72\,690.5$ JD. Making use of Table 32, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$
-70,000	$-235.315$	$-232.017$
-2,000	$-171.295$	$-171.200$
-600	$-231.388$	$-231.360$
-90	$-88.708$	$-88.704$
-.5	$-0.493$	$-0.493$
Epoch	$280.458$	$357.588$
	$-446.741$	$-366.186$
Modulus	$273.259$	$353.814$

We conclude that $M\simeq 354^\circ$. From Table 33,

$$q(354^\circ)= -0.204^\circ,$$

so

$$\lambda =\bar{\lambda} + q = 273.259 - 0.204=273.055\simeq 273^\circ03'.$$

Thus, the position of the sun at 00:00 GMT on December 25, 1800 CE was 3CP03.

We can also use Tables 32 and 33 to calculate the dates of the equinoxes and solstices, and, hence, the lengths of the seasons, in a given year. The vernal equinox (i.e., the point on the sun's apparent orbit at which it passes the celestial equator from south to north) corresponds to $\lambda = 0^\circ$, the summer solstice (i.e., the point at which the sun is furthest north of the celestial equator) to $\lambda = 90^\circ$, the autumnal equinox (i.e., the point at which the sun passes the celestial equator from north to south) to $\lambda = 180^\circ$, and the winter solstice (i.e., the point at which the sun is furthest south of the celestial equator) to $\lambda = 270^\circ$--see Fig. 18. Furthermore, spring is defined as the period between the spring equinox and the summer solstice, summer as the period between the summer solstice and the autumnal equinox, autumn as the period between the autumnal equinox and the winter solstice, and winter as the period between the winter solstice and the following vernal equinox. Consider the year 2000 CE. For the case of the vernal equinox, we can first estimate the time at which this event takes place by approximating the solar longitude as the mean solar longitude: i.e.,

$$\lambda\simeq \bar{\lambda} = \bar{\lambda}_0 + n\,(t-t_0) = 280.458 + 0.98564735\,(t-t_0),$$

We obtain

$$t \simeq t_0+(360-280.458)/0.98564735 \simeq t_0+81\,{\rm JD}.$$

Calculating the true solar longitude at this time, using Tables 32 and 33, we get $\lambda = 2.177^\circ.$ Now, the actual vernal equinox occurs when $\lambda = 0^\circ$. Thus, a much better estimate for the date of the vernal equinox is

$$t = t_0 + 81 -2.177/0.98564735 \simeq t_0 + 78.8\,{\rm JD},$$

which corresponds to 7:00 GMT on March 20. Similar calculations show that the summer solstice takes place at

$$t = t_0+ 171.6\, {\rm JD},$$

corresponding to 2:00 GMT on June 21, that the autumnal equinox takes place at

$$t = t_0+265.2\,{\rm JD},$$

corresponding to 17:00 GMT on September 22, and that the winter solstice takes place at

$$t =t_0+355.1\, {\rm JD},$$

corresponding to 14:00 GMT on December 21. Thus, the length of spring is $92.8$ days, the length of summer $93.6$ days, and the length of autumn $89.9$ days. Finally, the length of winter is the length of the tropical year (i.e., the time period between successive vernal equinoxes), which is $360/0.98564735 = 325.24$ days, minus the sum of the lengths of the other three seasons. This gives $88.9$ days.

Figure 18 illustrates the relationship between the equinox and solstice points, and the lengths of the seasons. The earth is displaced from the geometric center of the sun's apparent orbit in the direction of the perigee, which presently lies between the winter solstice and the vernal equinox. This displacement (which is greatly exaggerated in the figure) has two effects. Firstly, it causes the arc of the sun's apparent orbit between the summer solstice and autumnal equinox to be longer than that between the winter solstice and the vernal equinox. Secondly, it causes the sun to appear to move faster in winter than in summer, in accordance with Kepler's second law, since the sun is closer to the earth in the former season. Both of these effects tend to lengthen summer, and shorten winter. Hence, summer is presently the longest season, and winter the shortest.

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. 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. Greenwich mean time (GMT) is defined such that 12:00 hrs. GMT coincides with mean noon every day at an observation site of terrestrial longitude $0^\circ$. If we define local mean time (LMT) as $LMT = GMT- \phi(^\circ)/15^\circ$ hrs., where $\phi$ is the terrestrial longitude of the observation site, then 12:00 hrs. LMT 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

$$\bar{\alpha} = \bar{\lambda},$$ (92)

where $\bar{\lambda}$ is the sun's mean ecliptic longitude. Moreover, it follows from Eqs. (41) and (82) that the right ascension of the true sun is given by

$$\tan\alpha = \cos\epsilon\,\tan(\bar{\lambda}+q),$$ (93)

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. (79)], we can write [see Eqs. (84), (85), and (87), as well as Table. 30]

$$M = \bar{\lambda} + M_0-\bar{\lambda}_0 = \bar{\lambda} +77.213^\circ.$$ (94)

It follows that, to first order in the solar eccentricity, $e$, we have

$$\Delta\alpha = \bar{\alpha}-\alpha = \lambda - \tan^{-1}(\cos \epsilon\,\tan\lambda) - 2\,e\,\sin\,M,$$ (95)

where

$$M = \lambda + 77.213^\circ.$$ (96)

Now,

$$\Delta t({\rm hrs.}) = \Delta\alpha(^\circ)/15^\circ$$ (97)

represents the time difference 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.

Figure 18: The sun's apparent orbit around the earth, $G$, showing the vernal equinox (VE), summer solstice (SS), autumnal equinox (AE), and winter solstice (WS). Here, $\lambda$, $\Pi$, $A$, and $C$ are the ecliptic longitude, perigee, apogee, and geometric center of the orbit, respectively. The lengths of the seasons (in days) are indicated.

Table 30: Keplerian orbital elements for the sun and the five visible planets at the J2000 epoch (i.e., 12:00 GMT, 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$