Lunar-Solar Syzygies and Eclipses

Let $\lambda_S$ and $\lambda_M$ represent the ecliptic longitudes of the sun and the moon, respectively. The lunar-solar elongation is defined

$$D = \lambda_M - \lambda_S.$$ (118)

Since the moon is only visible because of light reflected from the sun, there is a fairly obvious relationship between lunar-solar elongation and lunar phase--see Fig. 22. For instance, a new moon corresponds to $D = 0^\circ$, a quarter moon to $D = 90^\circ$ or $270^\circ$, and a full moon to $D = 180^\circ$. New moons and full moons are collectively known as lunar-solar syzygies.

Figure 22: The phases of the moon.

We can predict the dates and times of lunar-solar syzygies by combining the solar and lunar models described in the previous two sections. The syzygy model is as follows:

$$\bar{D} = \bar{\lambda}_M - \bar{\lambda}_S,$$ (119)

$$q_1 = 2\,e_M\,\sin M_M + 1.430\,e^2\,\sin 2M_M,$$ (120)

$$q_2 = 0.422\,e_M\,\sin (2\bar{D} - M_M),$$ (121)

$$q_3 = 0.211\,e_M\,(\sin 2\bar{D} - 0.066\,\sin \bar{D}),$$ (122)

$$q_4 = -(0.051\,e_M+2\,e_S)\,\sin M_S - (5/4)\,e_S^{\,2}\,\sin 2 M_S,$$ (123)

$$q_5 = -0.038\,e_M\,\sin 2 \bar{F}_M,$$ (124)

$$D = \bar{D} + q_1+q_2+q_3+q_4+q_5.$$ (125)

Here, $e_S$, $M_S$, and $\bar{\lambda}_S$ are the eccentricity, mean anomaly, and mean longitude of the sun's apparent orbit about the earth, respectively. Moreover, $e_M$, $M_M$, $\bar{\lambda}_M$, and $\bar{F}_M$ are the eccentricity, mean anomaly, mean longitude, and mean argument of latitude of the moon's orbit, respectively.

The lunar-solar elongation can be calculated with the aid of Tables 40 and 41. Table 40 allows the mean lunar-solar elongation, $\bar{D}$, the mean lunar argument of latitude, $\bar{F}_M$, the mean anomaly of the sun, $M_S$, and the mean anomaly of the moon, $M_M$, to be determined as functions of time. Table 41 specifies the anomalies $q_1$-$q_5$ as functions of their various arguments.

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 lunar-solar elongation 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 40 with the digit for each power of 10 in ${\Delta} t$ and take out the corresponding values of $\Delta \bar{D}$, $\Delta \bar{F}_M$, $\Delta M_S$, and $\Delta M_M$. If $\Delta t$ is negative then the values are minus those shown in the table. The value of the mean lunar-solar elongation, $\bar{D}$, is the sum of all the $\Delta \bar{D}$ values plus the value of $\bar{D}$ at the epoch. Likewise, the value of the mean lunar argument of latitude, $\bar{F}_M$, is the sum of all the $\Delta \bar{F}_M$ values plus the value of $\bar{F}_M$ at the epoch. Moreover, the value of the solar mean anomaly, $M_S$, is the sum of all the $\Delta M_S$ values plus the value of $M_S$ at the epoch. Finally, the value of the lunar mean anomaly, $M_M$, is the sum of all the $\Delta M_M$ values plus the value of $M_M$ at the epoch. Add as many multiples of $360^\circ$ to $\bar{D}$, $\bar{F}_M$, $M_S$, and $M_M$ as is required to make them all fall in the range $0^\circ$ to $360^\circ$.
3. Form the five arguments $a_1=M_M$, $a_2=2\bar{D} - M_M$, $a_3=\bar{D}$, $a_4 = M_S$, $a_5=2\bar{F}_M$. Add as many multiples of $360^\circ$ to the arguments as is required to make them all fall in the range $0^\circ$ to $360^\circ$. Round each argument to the nearest degree.
4. Enter Table 41 with the value of each of the five arguments $a_1$-$a_5$ and take out the value of each of the five corresponding anomalies $q_1$-$q_5$. It is necessary to interpolate if the arguments are odd.
5. The lunar-solar elongation is given by $D=\bar{D} + q_1+q_2+q_3+q_4+q_5$. If necessary, convert $D$ into an angle in the range $0^\circ$ to $360^\circ$. The decimal fraction can be converted into arc minutes using Table 31.

In order to facilitate the calculation of syzygies, the above model has been used to contruct Table 42, which lists the dates and fractional Julian day numbers of the first new moons of the years 1900-2099 CE. Two examples of syzygy calculations are given below.

Example 1: Sixth new moon of 2004 CE:
 
From Table 42, the date of first new moon of 2004 CE is 2453026.4 JD. Now, the lunar-solar elongation increases at the mean rate $n_M - n_S = 13.17639646-0.98564735=12.1907491^\circ$ per day, or $360^\circ$ in $29.53$ days--the latter time period is known as a synodic month. Hence, a rough estimate for the date of the sixth new moon of 2004 CE is five synodic months after that of the first: i.e., $2453026.4 + 5\times 29.53\simeq 2453174.1$ JD. It follows that $\Delta t = 2453174.1-2451545.0=1629.1$ JD. Let us calculate the lunar-solar elongation at this date. From Table 40:

$t$(JD)	$\bar{D}(^\circ)$	$\bar{F}_M(^\circ)$	$M_S(^\circ)$	$M_M(^\circ)$
+1000	$310.749$	$269.350$	$265.600$	$104.993$
+600	$114.449$	$17.610$	$231.360$	$278.996$
+20	$243.815$	$264.587$	$19.712$	$261.300$
+9	$109.717$	$119.064$	$8.870$	$117.585$
+.1	$1.219$	$1.323$	$0.099$	$1.306$
Epoch	$297.864$	$93.284$	$357.588$	$134.916$
	$1077.813$	$765.218$	$883.229$	$899.096$
Modulus	$357.813$	$45.218$	$163.229$	$179.096$

Thus,

$$a_1=M_M\simeq 179^\circ,~~~a_2=2\bar{D}-M_M = 2\times 357.813-179.082\simeq 177^\circ,$$

$$a_3=\bar{D}\simeq 358^\circ,~~~a_4 = M_S\simeq 163^\circ,$$

$$a_5=2\bar{F}_M = 2\times 45.218\simeq 90^\circ.$$

Table 41 yields

$$q_1(a_1)=0.101^\circ,~~q_2(a_2)= 0.070^\circ,~~q_3(a_3) = -0.045^\circ,$$

$$q_4(a_4)=-0.596^\circ,~~q_5(a_5)= -0.119^\circ.$$

Hence,

$$D = \bar{D} + q_1+q_2+q_3+q_4+q_5=357.813+0.101+0.070-0.045-0.596-0.119\simeq 357.22^\circ.$$

Now, the actual new moon takes place when $D=360.00^\circ$. Thus, a far better estimate for the date of the sixth new moon of 2004 CE is $2453174.10 +(360.00-357.22)/12.1907491= 2453174.33$ JD. This corresponds to 20:00 hrs. GMT on June 17th.

Example 2: Third full moon of 1982 CE:
 
From Table 42, the fractional Julian day number of first new moon of 1982 CE is 2444994.7 JD, which corresponds to January 25th. Since there is more than half a synodic month between this event and the start of year, we conclude that the first full moon of 1982 CE took place before January 25th. Hence, a rough estimate for the date of the third new moon of 1982 CE is one and a half synodic months after that of the first: i.e., $2444994.7 + 1.5\times 29.53\simeq 2445039.0$ JD. It follows that $\Delta t = 2445039.0 -2451545.0=-6506.0$ JD. Let us calculate the lunar-solar elongation at this date. From Table 40:

$t$(JD)	$\bar{D}(^\circ)$	$\bar{F}_M(^\circ)$	$M_S(^\circ)$	$M_M(^\circ)$
-6000	$-64.495$	$-176.102$	$-153.601$	$-269.958$
-500	$-335.375$	$-134.675$	$-132.800$	$-52.496$
-6	$-73.144$	$-79.376$	$-5.914$	$-78.390$
Epoch	$297.864$	$93.284$	$357.588$	$134.916$
	$-175.150$	$-296.869$	$65.273$	$-265.928$
Modulus	$184.131$	$63.062$	$65.273$	$94.072$

Thus,

$$a_1=M_M\simeq 94^\circ,~~~a_2=2\bar{D}-M_M = 2\times 184.850-94.072\simeq 276^\circ,$$

$$a_3=\bar{D}\simeq 185^\circ,~~~a_4 = M_S\simeq 65^\circ,$$

$$a_5=2\bar{F}_M = 2\times 63.062\simeq 126^\circ.$$

Table 41 yields

$$q_1(a_1)=6.239^\circ,~~q_2(a_2)= -1.320^\circ,~~q_3(a_3) = 0.119^\circ,$$

$$q_4(a_4)=-1.896^\circ,~~q_5(a_5)= -0.097^\circ.$$

Hence,

$$D = \bar{D} + q_1+q_2+q_3+q_4+q_5=184.850+6.239-1.320+0.119-1.896-0.097\simeq 187.895^\circ.$$

Now, the actual full moon takes place when $D=180.00^\circ$. Thus, a far better estimate for the date of the third full moon of 1982 CE is $2445039.0 +(180.00-187.90)/12.1907491= 2445038.35$ JD. This corresponds to 20:00 hrs. GMT on March 9th.

A solar eclipse--or, more accurately, a lunar-solar occultation--occurs when the moon blocks the light of the sun. Clearly, this is only possible at a new moon--see Fig. 22. On the other hand, a lunar eclipse occurs when the moon falls into the shadow of the earth. Of course, this is only possible at a full moon. It follows that eclipses can only take place at lunar-solar syzygies.

In order to determine whether a particular lunar-solar syzygy conincides with an eclipse, we first need to calculate the angular radii of the sun, the moon, and the earth's shadow in the sky. Using the small angle approximation, the angular radius of the sun is given by $\rho_S = R_S/r_S$, where $R_S$ is the solar radius, and $r_S$ the earth-sun distance. However, $r_S\simeq a_S\,(1-e_S\,\cos M_S)$, where $a_S$, $e_S$, and $M_S$ are the major radius, eccentricity, and mean anomaly of the sun's apparent orbit around the earth, respectively. Hence,

$$\rho_S \simeq \rho_{S\,0}\,(1+e_S\,\cos M_S),$$ (126)

where $\rho_{S\,0}=R_S/a_S=6.960\times10^5\,{\rm km}/1.496\times 10^8\,{\rm km}\simeq 15.99'$. Likewise, the angular radius of the moon is

$$\rho_M\simeq \rho_{M\,0}\,(1+e_M\,\cos M_M),$$ (127)

where $\rho_{M\,0} = R_M/a_M =1743\,{\rm km}/384399\,{\rm km}\simeq 15.59'$. Here, $R_M$, $a_M$, $e_M$, and $M_M$ are the radius of the moon, and the major radius, eccentricity, and mean anomaly of the moon's orbit, respectively. As was shown in the previous section, lunar parallax causes the angular position of the moon in the sky to shift by up to

$$\delta_M = \frac{R_E}{r_M}= \delta_{M\,0}\,(1+e_M\,\cos M_M),$$ (128)

where $\delta_{M\,0} = R_E/a_M = 6371\,{\rm km}/384399\,{\rm km}=56.99'$. Here, $R_E$ is the radius of the earth. Finally, simple trigonometry reveals that the angular size of the earth's shadow (i.e., umbra) at the radius of the moon's orbit is

$$\rho_U = \delta_M-\rho_S.$$ (129)

This can be seen from Fig. 23. The radius of the umbra at the position of the moon is $R_U = R_E-x= R_E - r_M\,\rho_S$. Hence, the angular radius of the umbra is $\rho_U = R_U/r_M = \delta_M-\rho_S$. Incidentally, the identification of two of the angles in the figure with $\rho_S = R_S/r_S$ follows because $R_S\gg R_E$.

Figure 23: The earth's umbra.

A solar eclipse does not take place every new moon, nor a lunar eclipse every full moon, because of the inclination of the moon's orbit to the ecliptic plane, which causes the moon to pass either above or below the sun, or the earth's shadow, respectively, in the majority of cases. It follows that the critical parameter which determines the occurrence of eclipses is the ecliptic latitude of the moon at syzygy, $\beta_{syz}$. Of course, once the date and time of a syzygy has been established, $\beta_{syz}$ can be calculated from Table 38. However, the lunar argument of latitude, $F$, must first be determined using

$$F = \bar{F}_M + q_1+q_2+q_3+q_{4'}+q_5,$$ (130)

where $\bar{F}_M$ comes from Table 40, $q_1$, $q_2$, $q_3$, and $q_5$ are obtained from Table 41, and $q_{4'}$ is the $q_4$ from Table 37. For instance, we have seen that for the third new moon of 1982 CE, $\bar{F}_M=63.131$, $M_S\simeq 65^\circ$, $q_1=6.239^\circ$, $q_2=-1.320^\circ$, $q_3=0.119^\circ$, and $q_5=-0.097^\circ$. According to Table 37, $q_{4'}(M_S) = -0.145^\circ$. Hence, $F = \bar{F}_M + q_1+q_2+q_3+q_{4'}+q_5 =63.139+6.239-1.320+0.119-0.145-0.097=67.926\simeq 68^\circ$. It follows from Table 38 that $\beta_{syz} = 4.790^\circ$.

Figure 24: The limiting cases for a total lunar eclipse (left) and a partial lunar eclipse (right).

The criterion for a lunar eclipse is particularly simple, since it is not complicated by lunar parallax. A total lunar eclipse, in which the moon is completely immersed in the earth's shadow, must take place at a full moon if $\vert\beta_{syz}\vert < \rho_U-\rho_M$ (see Fig. 24), or

$$\vert\beta_{syz}\vert < \delta_M-\rho_M-\rho_S,$$ (131)

and either a total or a partial lunar eclipse, in which the moon is only partially immersed in the earth's shadow, must take place if $\vert\beta_{syz}\vert < \rho_U+\rho_M$ (see Fig. 24), or

$$\vert\beta_{syz}\vert < \delta_M+\rho_M-\rho_S.$$ (132)

Note that lunar eclipses are simultaneously visible at all observation sites on the earth at which the moon is above the horizon, since the earth's shadow is larger than the moon, and the relative position of the moon and the earth's shadow is not affected by parallax (since both the moon and the shadow are the same distance from the earth). The criterion for a solar eclipse is modified by lunar parallax, which causes the angular position of the moon relative to the sun to shift by up to $\delta_M$ from its geocentric position. The amount of the shift depends on the observation site. However, a site can always be found at which the shift takes its maximum value in any particular direction. Note that the sun has negligible parallax, since it is much further from the earth than the moon. Taking parallactic shifts into account, a total solar eclipse, in which the sun is totally obscured by the moon, must take place if $\rho_M>\rho_S$ and

$$\vert\beta_{syz}\vert < \delta_M + \rho_M - \rho_S,$$ (133)

an annular solar eclipse, in which all of the sun apart from a thin outer ring is obscured by the moon, must take place if $\rho_S>\rho_M$ and

$$\vert\beta_{syz}\vert < \delta_M + \rho_S-\rho_M,$$ (134)

and either a total, an annular, or a partial solar eclipse, in which the sun is only partially obscured by the moon, must take place if

$$\vert\beta_{syz}\vert< \delta_M + \rho_M + \rho_S.$$ (135)

As a consequence of lunar parallax, and the fact that the angular sizes of the sun and moon in the sky are very similar, solar eclipses are only visible in very localized regions of the earth. Note, finally, that the above criteria represent necessary, but not sufficient, conditions for the occurrence of the various eclipses with which they are associated. This is the case because the point of closest approach of the moon and the earth's shadow, in the case of a lunar eclipse, and the moon and sun, in the case of a solar eclipse, does not necessarily occur exactly at the syzygy, because of the inclination of the moon's orbit to the ecliptic. However, since the said inclination is fairly gentle, it turns out that the above criteria are very accurate predictors of eclipses.

The criterion for a total lunar eclipse can be written $\vert\beta_{syz}\vert< \beta_{Mt}$, where

$$\beta_{Mt} = 25.41' + \delta\beta_1(M_M)-\delta\beta_2 (M_M)-\delta\beta_3(M_S).$$ (136)

Here, the functions $\delta\beta_1=\delta_{M\,0}\,e_M\,\cos M_M$, $\delta\beta_2= \rho_{M\,0}\,e_M\,\cos M_M$, and $\delta\beta_3=\rho_{S\,0}\,e_S\,\cos M_S$ are tabulated in Table 43. The criterion for any type of lunar eclipse becomes $\vert\beta_{syz}\vert< \beta_{M}$, where

$$\beta_{M} = 56.59' + \delta\beta_1(M_M)+\delta\beta_2 (M_M)-\delta\beta_3(M_S).$$ (137)

The criterion for a total solar eclipse can be written $\vert\beta_{syz}\vert<\beta_{St}$ and $\beta_{St}>\beta_{Sa}$, where

$$\beta_{St} = 56.59' + \delta\beta_1(M_M)+\delta\beta_2 (M_M)-\delta\beta_3(M_S),$$ (138)

and

$$\beta_{Sa} = 57.39' + \delta\beta_1(M_M)-\delta\beta_2 (M_M)+\delta\beta_3(M_S),$$ (139)

The criterion for an annular solar eclipse is $\vert\beta_{syz}\vert<\beta_{Sa}$ and $\beta_{Sa}>\beta_{St}$. Finally, the criterion for any type of solar eclipse is $\vert\beta_{syz}\vert< \beta_{S}$, where

$$\beta_{S} = 88.57' + \delta\beta_1(M_M)+\delta\beta_2 (M_M)+\delta\beta_3(M_S).$$ (140)

Consider a very large collection of lunar-solar syzygies. For such a collection, we expect the lunar argument of latitude, $F$, the lunar mean anomaly, $M_M$, and the solar mean anomaly, $M_S$, to be statistically independent of one another, and randomly distributed in the interval $0^\circ$ to $360^\circ$. 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. (109) and the criteria (136)-(140). For a new moon we find:

Probability of total solar eclipse:	$4.2\%$
Probability of annular solar eclipse:	$7.7\%$
Probability of partial solar eclipse:	$6.6\%$
Probability of any solar eclipse:	$18.5\%$

For a full moon we get:

Probability of total lunar eclipse:	$5.2\%$
Probability of partial lunar eclipse:	$6.5\%$
Probability of any lunar eclipse:	$11.7\%$

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 $12.37\times 0.185\simeq 2.3$, whereas the mean number of lunar eclipses per year is about $12.37\times 0.117\simeq 1.4$. Clearly, solar eclipses are more common that lunar eclipses. However, 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.

Finally, let us use our model to examine the lunar-solar syzygies of the year 1992 CE, in order to see whether any of them were associated with solar or lunar eclipses. The table below shows the dates and times of the new moons of 1992 CE, calculated using the method described at the beginning of this section. Also shown is the magnitude of the moon's ecliptic latitude at each syzygy, $\vert\beta_{syz}\vert$, calculated from Eqs. (109) and (130), as well as the critical values of this parameter for a general, total, and annular solar eclipse. The latter are calculated from Eqs. (138)-(140). It can be seen that the criterion for a total solar eclipse (i.e., $\vert\beta_{syz}\vert<\beta_{St}$ and $\beta_{St}>\beta_{Sa}$) is satisfied for the syzygy marked with a T, the criterion for an annular solar eclipse (i.e., $\vert\beta_{syz}\vert<\beta_{Sa}$ and $\beta_{Sa}>\beta_{St}$) for the syzygy marked with an A, and the criterion for a partial solar eclipse (i.e., $\beta_{St},\beta_{Sa}<\vert\beta_{syz}\vert< \beta_{S}$) for the syzygy marked with a P. It is easily verified that a total solar eclipse, an annular solar eclipse, and a partial solar eclipse did indeed take place in 1992 CE at the dates and times indicated.

Date	Time (GMT)	$\beta_{S}(')$	$\beta_{St}(')$	$\beta_{Sa}(')$	$\vert\beta_{syz}\vert(')$
04/01/1992	23:00	85.0	52.5	55.5	22.8	A
03/02/1992	16:00	84.9	52.5	55.4	177.9
04/03/1992	09:00	85.7	53.4	55.8	282.7
03/04/1992	00:00	87.1	55.1	56.5	307.5
02/05/1992	14:00	88.7	57.0	57.4	245.8
01/06/1992	01:00	90.3	58.8	58.3	115.6
30/06/1992	12:00	91.5	60.1	59.0	46.3	T
29/07/1992	21:00	92.2	60.7	59.4	195.3
28/08/1992	05:00	92.3	60.7	59.5	290.2
26/09/1992	14:00	91.8	59.9	59.2	304.8
26/10/1992	00:00	90.7	58.5	58.7	234.8
24/11/1992	11:00	89.2	56.8	57.8	99.3
24/12/1992	01:00	87.4	54.9	56.9	64.3	P

The table below shows the dates and times of the full moons of 1992 CE. Also shown is the magnitude of the moon's ecliptic latitude at each syzygy, as well as the critical values of this parameter for a general and a total lunar eclipse. The latter are calculated from Eqs. (137) and (136), respectively. It can be seen that the criterion for a total lunar eclipse (i.e., $\vert\beta_{syz}\vert< \beta_{Mt}$) is satisfied for the syzygy marked with a T, whereas the criterion for a partial lunar eclipse (i.e., $\beta_{Mt}<\vert\beta_{syz}\vert< \beta_{M}$) is satisfied for the syzygy marked with a P. It is easily verified that a total lunar eclipse, and a partial lunar eclipse did indeed take place in 1992 CE at the dates and times indicated.

Date	Time (GMT)	$\beta_{M}(')$	$\beta_{Mt}(')$	$\vert\beta_{syz}\vert(')$
19/01/1992	20:00	60.3	27.4	104.2
18/02/1992	05:00	60.1	27.3	237.8
18/03/1992	14:00	59.3	26.9	305.6
17/04/1992	01:00	57.9	26.2	288.5
16/05/1992	13:00	56.3	25.3	190.8
15/06/1992	03:00	54.7	24.4	39.4	P
14/07/1992	20:00	53.4	23.7	123.4
13/08/1992	13:00	52.8	23.3	251.6
12/09/1992	05:00	53.1	23.5	308.6
11/10/1992	21:00	54.3	24.1	278.2
10/11/1992	12:00	55.8	24.9	169.6
10/12/1992	01:00	57.5	25.8	13.8	T
08/01/1993	12:00	59.0	26.7	145.4

Table 40: Mean motion of the lunar-solar elongation. Here, $\Delta t = t-t_0$, $\Delta \bar{D}= \bar{D}-\bar{D}_0$, $\Delta \bar{F}_M= \bar{F}_M-\bar{F}_{M\,0}$, $\Delta M_S = M_S - M_{S\,0}$, and $\Delta M_M= M_M-M_{M\,0}$. At epoch ( $t_0= 2\,451\,545.0$ JD), $\bar{D}_0 = 297.864^\circ$, $\bar{F}_{M\,0} = 93.284^\circ$, $M_{S\,0} = 357.588^\circ$, and $M_{M\,0} = 134.916^\circ$.

$\Delta t$(JD)	$\Delta \bar{D}(^\circ)$	$\Delta \bar{F}_M(^\circ)$	$\Delta M_S(^\circ)$	$\Delta M_M (^\circ)$	$\Delta t$(JD)	$\Delta \bar{D}(^\circ)$	$\Delta \bar{F}_M(^\circ)$	$\Delta M_S(^\circ)$	$\Delta M_M (^\circ)$
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 $M_M$, $2\bar{D}-M_M$, $\bar{D}$, $M_S$, and $2\bar{F}_M$ for the case of $q_1$, $q_2$, $q_3$, $q_4$, and $q_5$, respectively. If the argument is in parenthesies then the anomalies are minus the values shown in the table.

Arg. ($^\circ$)	$q_1(^\circ)$	$q_2(^\circ)$	$q_3(^\circ)$	$q_4(^\circ)$	$q_5(^\circ)$	Arg. ($^\circ$)	$q_1(^\circ)$	$q_2(^\circ)$	$q_3(^\circ)$	$q_4(^\circ)$	$q_5(^\circ)$
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