The Moon

The orbit of the moon around the earth is strongly perturbed by the gravitational influence of the sun. It follows that we cannot derive an accurate lunar model from Keplerian orbit theory alone. Instead, our approach will be purely empirical in nature. Careful observation of the moon over a period of many years reveals that the variation of its ecliptic longitude with time is well represented by the following formulae (see http://jgiesen.de/moonmotion/index.html):

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (98)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (99)

$$\bar{F} = \bar{F}_0 + \breve {n}\,(t-t_0),$$ (100)

$$\tilde{D} = \bar{\lambda} - \lambda_S,$$ (101)

$$q_1 = 2\,e\,\sin M + 1.430\,e^2\,\sin 2M,$$ (102)

$$q_2 = 0.422\,e\,\sin (2\tilde{D} - M),$$ (103)

$$q_3 = 0.211\,e\,(\sin 2\tilde{D} - 0.066\,\sin \tilde{D}),$$ (104)

$$q_4 = -0.051\,e\,\sin M_S,$$ (105)

$$q_5 = -0.038\,e\,\sin 2 \bar{F},$$ (106)

$$\lambda = \bar{\lambda} + q_1+q_2+q_3+q_4+q_5.$$ (107)

Here, $\lambda_S$ and $M_S$ are the longitude and mean anomaly of the sun, respectively. Moreover, $e$, $\lambda$, $\bar{\lambda}$, $\bar{F}$, and $q_i$ are the eccentricity, longitude, mean longitude, mean argument of latitude, and $i$th anomaly of the moon, respectively. The moon's first anomaly is due to the eccentricity of its orbit, and is very similar in form to that obtained from Keplerian orbit theory (see Sect. 3). The moon's second, third, and fourth anomalies are knows as evection, variation, and the annual inequality, respectively, and originate from the perturbing influence of the sun. Finally, the moon's fifth anomaly is called the reduction to the ecliptic, and is a consequence of the fact that the moon's orbit is slightly tilted with respect to the plane of the ecliptic. The moon's orbital elements--$e$, $n$, $\tilde{n}$, $\breve{n}$, $\bar{\lambda}_0$, $M_0$, and $F_0$--for the J2000 epoch are listed in Table 35. Note that the lunar perigee precesses in the direction of the moon's orbital motion at the rate of $n-\tilde{n} = 0.11140\, ^\circ$ per day, or $360^\circ$ in 8.85 years. This very large precession rate (more than 2000 times the corresponding precession rate for the sun's apparent orbit) is another consequence of the strong perturbing influence of the sun on the moon's orbit. The above formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean error of $5'$ and a maximum error of $14'$.

The ecliptic longitude of the moon can be calculated with the aid of Tables 36 and 37. Table 36 allows the lunar mean longitude, $\bar{\lambda}$, mean anomaly, $M$, and mean argument of latitude, $\bar{F}$, to be determined as functions of time. Table 37 specifies the lunar 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 moon'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. Calculate the ecliptic longitude, $\lambda_S$, and the mean anomaly, $M_S$, of the sun using the procedure set out in Sect. 5.
3. Enter Table 36 with the digit for each power of 10 in ${\Delta} t$ and take out the corresponding values of $\Delta\bar{\lambda}$, $\Delta M$, and $\Delta\bar{F}$. If $\Delta t$ is negative then the values are minus those shown in the table. 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. Finally, the value of the mean argument of latitude, $\bar{F}$, is the sum of all the $\Delta\bar{F}$ values plus the value of $\bar{F}$ at the epoch. Add as many multiples of $360^\circ$ to $\bar{\lambda}$, $M$, and $\bar{F}$ as is required to make them all fall in the range $0^\circ$ to $360^\circ$.
4. Form $\tilde{D}=\bar{\lambda}-\lambda_S$.
5. Form the five arguments $a_1=M$, $a_2=2\tilde{D} - M$, $a_3=\tilde{D}$, $a_4 = M_S$, $a_5=2\bar{F}$. 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.
6. Enter Table 37 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.
7. The moon's ecliptic longitude is given by $\lambda=\bar{\lambda} + q_1+q_2+q_3+q_4+q_5$. 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:
 
From Sect. 5, $t-t_0=1950.5$ JD, $\lambda_S = 44.602^\circ$, and $M_S= 120.001^\circ$. Making use of Table 36, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$	$\bar{F}(^\circ)$
+1000	$216.396$	$104.993$	$269.350$
+900	$338.757$	$238.494$	$26.415$
+50	$298.820$	$293.250$	$301.468$
+.5	$6.588$	$6.532$	$6.615$
Epoch	$218.322$	$134.916$	$93.284$
	$1078.883$	$778.185$	$697.132$
Modulus	$358.883$	$58.185$	$337.132$

It follows that

$$\tilde{D}=\bar{\lambda}-\lambda_S = 358.883-44.602 = 314.281^\circ,$$

Thus,

$$a_1=M\simeq 58^\circ,~~~a_2=2\tilde{D}-M = 2\times 314.281-58.185=570.377\simeq 210^\circ,$$

$$a_3=\tilde{D}\simeq 314^\circ,~~~a_4 = M_S\simeq 120^\circ,$$

$$a_5=2\bar{F} = 2\times 337.132=674.264\simeq 314^\circ.$$

Table 37 yields

$$q_1(a_1)=5.555^\circ,~~q_2(a_2)= -0.663^\circ,~~q_3(a_3) = -0.631^\circ,$$

$$q_4(a_4)=-0.139^\circ,~~q_5(a_5)= 0.086^\circ.$$

Hence,

$$\lambda = \bar{\lambda} + q_1+q_2+q_3+q_4+q_5=358.883+5.555-0.663-0.631-0.139+0.086= 363.091^\circ,$$

or

$$\lambda =3.091\simeq 3^\circ 05'.$$

Thus, the ecliptic longitude of the moon at 00:00 GMT on May 5, 2005 CE was 3AR05.

Example 2: December 25, 1800 CE, 00:00 GMT:
 
From Sect. 5, $t-t_0=-72\,690.5$ JD, $\lambda_S = 273.055^\circ$, and $M_S= 353.814^\circ$. Making use of Table 36, we find:

$t$(JD)	$\bar{\lambda}(^\circ)$	$M(^\circ)$	$\bar{F}(^\circ)$
-70,000	$-27.752$	$-149.506$	$-134.519$
-2,000	$-72.793$	$-209.986$	$-178.701$
-600	$-345.838$	$-278.996$	$-17.610$
-90	$-105.876$	$-95.849$	$-110.642$
-.5	$-6.588$	$-6.532$	$-6.615$
Epoch	$218.322$	$134.916$	$93.284$
	$-340.525$	$-605.953$	$-354.803$
Modulus	$19.475$	$114.047$	$5.197$

It follows that

$$\tilde{D}=\bar{\lambda}-\lambda_S = 19.475-273.055 = -253.580^\circ,$$

Thus,

$$a_1=M\simeq 114^\circ,~~~a_2=2\tilde{D}-M = -2\times 253.580-114.047=-621.207 \simeq 99^\circ,$$

$$a_3=\tilde{D}\simeq 106^\circ,~~~a_4 = M_S\simeq 354^\circ,$$

$$a_5=2\bar{F} = 2\times 5.197=10.394\simeq 10^\circ.$$

Table 37 yields

$$q_1(a_1)=5.562^\circ,~~q_2(a_2)= 1.311^\circ,~~q_3(a_3) = -0.394^\circ,$$

$$q_4(a_4)=0.017^\circ,~~q_5(a_5)= -0.021^\circ.$$

Hence,

$$\lambda = \bar{\lambda} + q_1+q_2+q_3+q_4+q_5=19.475+5.562+1.311-0.394+0.017-0.021= 25.950^\circ,$$

or

$$\lambda =25.950\simeq 25^\circ 57'.$$

Thus, the ecliptic longitude of the moon at 00:00 GMT on December 25, 1800 CE was 25AR57.

Figure 19: The orbit of the moon about the earth. Here, $G$, $L$, $N$, $N'$, $\Omega$, $F$, and $\Upsilon$ represent the earth, moon, ascending node, descending node, longitude of the ascending node, argument of latitude, and vernal equinox, respectively. View is from northern ecliptic pole. The moon orbits counterclockwise.

A model of the moon's ecliptic latitude is needed in order to predict the occurrence of solar and lunar eclipses. Figure 19 shows a top view of the moon's orbit about the earth. The plane of this orbit is fixed, but slightly tilted with respect to the plane of the ecliptic (i.e., the plane of the sun's apparent orbit about the earth). Let the two planes intersect along the line of nodes, $NGN'$. Here, $N$ is the point at which the orbit crosses the ecliptic plane from south to north (in the direction of the moon's orbital motion), and is termed the ascending node. Likewise, $N'$ is the point at which the orbit crosses the ecliptic plane from north to south, and is called the descending node. Incidentally, the line of nodes must pass through the earth, $G$, since it is common to both planes. The angle, $\Omega$, subtended between $GN$ and the radius vector, $G\Upsilon$, connecting the earth to the vernal equinox, is known as the longitude of the ascending node. Note, incidentally, that the ascending node precessess in the opposite direction to the moon's orbital motion at the rate $\breve{n}-n = 5.2954\times 10^{-2}\, ^\circ$ per day, or $360^\circ$ in 18.6 years. This unusually large precession rate is another consequence of the sun's strong perturbing influence on the moon's orbit. Let the line $MGM'$ lie in the plane of the moon's orbit such that it is perpendicular to $NGN'$. The inclination, $i$, of the moon's orbital plane is the angle that $GM$ subtends with its projection onto the ecliptic plane. Likewise, the moon's ecliptic longitude, $\beta$, is the angle that $GL$ subtends with its projection onto the ecliptic plane. Simple geometry yields $\sin\beta = \sin i\,\sin F$, where $F$ is the angle between $GL$ and $GN$. This angle is termed the argument of latitude. Now, it is easily seen that $F \simeq \lambda-\Omega$, where $\lambda$ is the moon's ecliptic longitude (i.e., the angle subtended between $GL$ and $G\Upsilon$). Here, we are assuming that $i$ is relatively small. The mean argument of latitude is defined $\bar{F} = \bar{\lambda}-\Omega$. Hence, our model for the moon's ecliptic latitude becomes

$$F = \bar{F}+q_1+q_2+q_3+q_4+q_5,$$ (108)

$$\sin\beta = \sin i\,\sin F.$$ (109)

The value of the lunar inclination, $i$, for the J2000 epoch is specified in Table 35. The above model is capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean error of $6'$, and a maximum error of $11'$.

The ecliptic latitude of the moon can be calculated with the aid of Table 38. The procedure for using this table is as follows:

1. Determine the fractional Julian day number, $t$, corresponding to the date and time at which the moon's ecliptic latitude 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. Calculate the lunar mean argument of latitude, $\bar{F}$, and the five lunar anomalies, $q_1$-$q_5$, using the procedure outlined earlier in this section.
3. Form the argument $F = \bar{F}+q_1+q_2+q_3+q_4+q_5$. Add as many multiples of $360^\circ$ to $F$ as is required to make it fall in the range $0^\circ$ to $360^\circ$. Round $F$ to the nearest degree.
4. Enter Table 38 with the value of $F$ and take out the lunar ecliptic latitude, $\beta$. It is necessary to interpolate if $F$ is odd.

For example, we have already seen that at 00:00 GMT on May 5, 2005 CE the lunar mean argument of latitude, and the lunar anomalies, were $\bar{F}=337.132^\circ$, and $q_1=5.555^\circ$, $q_2=-0.663^\circ$, $q_3=-0.631^\circ$, $q_4=-0.139^\circ$, and $q_5=0.086^\circ$, respectively. Hence, $F=\bar{F}+q_1+q_2+q_3+q_4+q_5 =337.132+5.555-0.663-0.631-0.139+0.086\simeq 341^\circ$. Thus, according to Table 38, the ecliptic latitude of the moon at 00:00 GMT on May 5, 2005 CE was $-1.680^\circ\simeq -1^\circ 41'$.

Figure 20: The moon, $L$, as viewed by a hypothetical observer, $C$, at the center of the earth, and a real observer, $X$, on the surface of the earth.

Now, it turns out that the moon is sufficiently close to the earth that its position in the sky is significantly modified by parallax. All of our previous analysis applies to a hypothetical observer situated at the center of the earth. Consider a real observer situated on the earth's surface. It can be seen from Fig. 20 that the altitude of the moon is $a'$ for the real observer, and $a$ for the hypothetical observer. Simple trigonometry reveals that $a' = a-\delta a$, which implies that the real observer sees the moon at a lower altitude than the hypothetical observer. Let $R$ be the radius of the earth, and $r$ the distance from the center of the earth to the moon. More simple trigonometry yields

$$\sin \delta a = \frac{R}{r}\,\cos a'.$$ (110)

Let us assume that the moon's orbit is elliptical to first order in its eccentricity. It follows, from Sect. 3, that

$$r \simeq a_M\,(1 - e\,\cos M),$$ (111)

where $a_M$, $e$, and $M$ are major radius, eccentricity, and mean anomaly of the lunar orbit. Assuming that $\delta a$ is small, we obtain

$$\delta a \simeq \delta a_0\,\cos a\,(1+e\,\cos M),$$ (112)

where $\delta a_0 = R/a_M = 0.0166$ (since $R=6371$ km and $a_M=384399$ km).

According to Eq. (112), lunar parallax can be written in the form

$$\delta a = \delta (a)\,[1+\zeta (M)],$$ (113)

where $a$, $a-\delta a$, and $M$ are the moon's geocentric altitude (i.e., the altitude seen from the center of the earth), true altitude, and mean anomaly, respectively. The functions $\delta(a)=\delta a_0\,\cos a$ and $\zeta (M)= e\,\cos M$ are tabulated in Table 39. It can be seen from the table that lunar parallax increases with decreasing lunar altitude, reaching a maximum value of about $57'$ when the moon is close to the horizon. For example, if $a=44^\circ$ and $M=100^\circ$ then Table 39 yields $\delta = 41.050'$ and $\zeta =- 0.00953$. Hence, $\delta a = 41.050\,(1-0.00953) \simeq 41'$, and the true altitude of the moon becomes $43^\circ 19'$.

Figure 21: Parallactic shifts in the moon's ecliptic longitude and latitude.

It now remains to investigate how parallax affects the moon's ecliptic longitude and latitude. Figure 21 shows a detail of Fig. 13. Point $Y$ is the moon's geocentric position on the celestial sphere. $DB$ is a line passing through this point which is parallel to the local ecliptic circle, whereas $ZC$ is a small section of an altitude circle passing through $Y$. The angle subtended between the ecliptic and the altitude circle is $\mu$. Let $F$ be the true position of the moon. It follows that $\delta a = YF$. The changes in the moon's ecliptic longitude and latitude are $\delta\lambda = YE$ and $-\delta\beta = EF$, respectively. Assuming that the arcs $\delta a$, $\delta\lambda$, and $\delta\beta$ are all fairly small, the triangle $YEF$ can be treated as a plane triangle. Hence, we obtain

$$\delta\lambda = - \delta a\,\cos\mu,$$ (114)

$$\delta\beta = - \delta a\,\sin\mu.$$ (115)

Actually, the above formulae only apply to the situation in which the ecliptic culminates south of the zenith. In the opposite case, in which the ecliptic culminates north of the zenith, we have

$$\delta\lambda = \delta a\,\cos\mu,$$ (116)

$$\delta\beta = - \delta a\,\sin\mu.$$ (117)

For example, consider a day on which the geocentric ecliptic longitude and mean anomaly of the moon are $\lambda=210^\circ$ (i.e., 00Sc00) and $M=90^\circ$, respectively. Suppose that the moon is viewed from an observation site located at terrestrial latitude $+10^\circ$. The ``Scorpio'' entry in Table 22 gives the moon's geocentric altitude, $a$, as a function of time, as well as the value of the ecliptic orientation angle $\mu$. Making use of this data, in combination with Table 39 and Eqs. (114)-(117), we can calculate the parallax-induced changes in the moon's ecliptic longitude and latitude as it transits the sky. Data from such a calculation is given in the table below. The first column specifies time since the moon's upper transit (thus, $t=1$ hrs. means one hour after the upper transit), the second column gives the moon's geocentric altitude, the third column the decrease in its real altitude due to parallax, and the fourth and fifth columns the parallax-induced changes in its ecliptic longitude and latitude.

$t$ (hrs.)	$a$	$\delta a$	$\delta\lambda$	$\delta\beta$
$-5.51$	$00^\circ 00'$	$57'$	$+56'$	$-10'$
$-5.00$	$12^\circ 26'$	$56'$	$+55'$	$-07'$
$-4.00$	$26^\circ 37'$	$51'$	$+51'$	$-03'$
$-3.00$	$40^\circ 23'$	$43'$	$+43'$	$-03'$
$-2.00$	$53^\circ 15'$	$34'$	$+33'$	$-08'$
$-1.00$	$63^\circ 52'$	$25'$	$+21'$	$-14'$
$+0.00$	$68^\circ 32'$	$21'$	$+07'$	$-20'$
$+1.00$	$63^\circ 52'$	$25'$	$-06'$	$-24'$
$+2.00$	$53^\circ 15'$	$34'$	$-20'$	$-28'$
$+3.00$	$40^\circ 23'$	$43'$	$-31'$	$-30'$
$+4.00$	$26^\circ 37'$	$51'$	$-40'$	$-32'$
$+5.00$	$12^\circ 26'$	$56'$	$- 46'$	$-31'$
$+5.51$	$00^\circ 00'$	$57'$	$-49'$	$-29'$

It can be seen that parallax causes the moon's apparent location to shift by almost $2^\circ$ relative to the fixed stars as it transits the sky. Note that the above calculation is somewhat inaccurate because it does not take into account the moon's motion along the ecliptic (which can easily amount to $6^\circ$ during the course of a night). However, the calculation does illustrate how the data contained in Tables 21-29, in combination with the data in Table 39, permits the parallax-induced shift in the moon's ecliptic position to be calculated for a wide range of different lunar phases, observation sites, and observation times.

Table 35: Orbital elements of the moon for the J2000 epoch (i.e., 12:00 GMT, January 1, 2000 CE, which corresponds to $t_0= 2\,451\,545.0$ JD).

$e$	$n(^\circ/{\rm day})$	$\tilde{n}(^\circ/{\rm day})$	$\breve{n}(^\circ/{\rm day})$	$\bar{\lambda}_0(^\circ)$	$M_0(^\circ)$	$F_0(^\circ)$	$i(^\circ)$
0.054881	13.17639646	13.06499295	$13.22935027$	218.322	134.916	93.284	5.161

Table 36: Mean motion of the moon. Here, $\Delta t = t-t_0$, $\Delta\bar{\lambda} = \bar{\lambda}-\bar{\lambda}_0$, $\Delta M = M - M_0$, and $\Delta\bar{F}= \bar{F}-\bar{F}_0$. At epoch ( $t_0= 2\,451\,545.0$ JD), $\bar{\lambda}_0 = 218.322^\circ$, $M_0 = 134.916^\circ$, and $\bar{F}_0 = 93.284^\circ$.

$\Delta t$(JD)	$\Delta\bar{\lambda}(^\circ)$	$\Delta M(^\circ)$	$\Delta \bar{F}(^\circ)$	$\Delta t$(JD)	$\Delta\bar{\lambda}(^\circ)$	$\Delta M(^\circ)$	$\Delta \bar{F}(^\circ)$
10,000	3.965	329.930	173.503	1,000	216.396	104.993	269.350
20,000	7.929	299.859	347.005	2,000	72.793	209.986	178.701
30,000	11.894	269.788	160.508	3,000	289.189	314.979	88.051
40,000	15.858	239.718	334.011	4,000	145.586	59.972	357.401
50,000	19.823	209.648	147.513	5,000	1.982	164.965	266.751
60,000	23.788	179.577	321.016	6,000	218.379	269.958	176.102
70,000	27.752	149.506	134.519	7,000	74.775	14.951	85.452
80,000	31.717	119.436	308.022	8,000	291.172	119.944	354.802
90,000	35.681	89.366	121.524	9,000	147.568	224.937	264.152
100	237.640	226.499	242.935	10	131.764	130.650	132.294
200	115.279	92.999	125.870	20	263.528	261.300	264.587
300	352.919	319.498	8.805	30	35.292	31.950	36.881
400	230.559	185.997	251.740	40	167.056	162.600	169.174
500	108.198	52.496	134.675	50	298.820	293.250	301.468
600	345.838	278.996	17.610	60	70.584	63.900	73.761
700	223.478	145.495	260.545	70	202.348	194.550	206.055
800	101.117	11.994	143.480	80	334.112	325.199	338.348
900	338.757	238.494	26.415	90	105.876	95.849	110.642
1	13.176	13.065	13.229	0.1	1.318	1.306	1.323
2	26.353	26.130	26.459	0.2	2.635	2.613	2.646
3	39.529	39.195	39.688	0.3	3.953	3.919	3.969
4	52.706	52.260	52.917	0.4	5.271	5.226	5.292
5	65.882	65.325	66.147	0.5	6.588	6.532	6.615
6	79.058	78.390	79.376	0.6	7.906	7.839	7.938
7	92.235	91.455	92.605	0.7	9.223	9.145	9.261
8	105.411	104.520	105.835	0.8	10.541	10.452	10.583
9	118.588	117.585	119.064	0.9	11.859	11.758	11.906

Table 37: Anomalies of the moon. The common argument corresponds to $M$, $2\tilde{D}-M$, $\tilde{D}$, $M_S$, and $2\bar{F}$ 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	-0.160	-0.119
002/(358)	0.237	0.046	0.045	-0.006	-0.004	092/(268)	6.268	1.326	-0.090	-0.160	-0.119
004/(356)	0.473	0.093	0.089	-0.011	-0.008	094/(266)	6.239	1.324	-0.136	-0.160	-0.119
006/(354)	0.709	0.139	0.133	-0.017	-0.012	096/(264)	6.203	1.320	-0.182	-0.159	-0.119
008/(352)	0.943	0.185	0.177	-0.022	-0.017	098/(262)	6.160	1.314	-0.226	-0.159	-0.118
010/(350)	1.176	0.230	0.219	-0.028	-0.021	100/(260)	6.109	1.307	-0.270	-0.158	-0.118
012/(348)	1.408	0.276	0.261	-0.033	-0.025	102/(258)	6.051	1.298	-0.313	-0.157	-0.117
014/(346)	1.637	0.321	0.301	-0.039	-0.029	104/(256)	5.986	1.288	-0.354	-0.156	-0.116
016/(344)	1.864	0.366	0.339	-0.044	-0.033	106/(254)	5.915	1.276	-0.394	-0.154	-0.115
018/(342)	2.088	0.410	0.376	-0.050	-0.037	108/(252)	5.836	1.262	-0.432	-0.153	-0.114
020/(340)	2.310	0.454	0.411	-0.055	-0.041	110/(250)	5.751	1.247	-0.468	-0.151	-0.112
022/(338)	2.527	0.497	0.444	-0.060	-0.045	112/(248)	5.660	1.230	-0.502	-0.149	-0.111
024/(336)	2.741	0.540	0.475	-0.065	-0.049	114/(246)	5.562	1.212	-0.533	-0.147	-0.109
026/(334)	2.951	0.582	0.504	-0.070	-0.052	116/(244)	5.458	1.193	-0.562	-0.144	-0.107
028/(332)	3.157	0.623	0.529	-0.075	-0.056	118/(242)	5.348	1.172	-0.589	-0.142	-0.106
030/(330)	3.358	0.663	0.553	-0.080	-0.060	120/(240)	5.233	1.149	-0.613	-0.139	-0.103
032/(328)	3.554	0.703	0.573	-0.085	-0.063	122/(238)	5.111	1.125	-0.634	-0.136	-0.101
034/(326)	3.746	0.742	0.591	-0.090	-0.067	124/(236)	4.985	1.100	-0.652	-0.133	-0.099
036/(324)	3.931	0.780	0.605	-0.094	-0.070	126/(234)	4.853	1.074	-0.667	-0.130	-0.097
038/(322)	4.111	0.817	0.617	-0.099	-0.074	128/(232)	4.716	1.046	-0.678	-0.126	-0.094
040/(320)	4.285	0.853	0.625	-0.103	-0.077	130/(230)	4.575	1.017	-0.687	-0.123	-0.092
042/(318)	4.454	0.888	0.630	-0.107	-0.080	132/(228)	4.428	0.986	-0.693	-0.119	-0.089
044/(316)	4.615	0.922	0.632	-0.111	-0.083	134/(226)	4.277	0.955	-0.695	-0.115	-0.086
046/(314)	4.770	0.955	0.631	-0.115	-0.086	136/(224)	4.122	0.922	-0.694	-0.111	-0.083
048/(312)	4.919	0.986	0.627	-0.119	-0.089	138/(222)	3.963	0.888	-0.689	-0.107	-0.080
050/(310)	5.061	1.017	0.620	-0.123	-0.092	140/(220)	3.799	0.853	-0.682	-0.103	-0.077
052/(308)	5.195	1.046	0.609	-0.126	-0.094	142/(218)	3.632	0.817	-0.671	-0.099	-0.074
054/(306)	5.323	1.074	0.595	-0.130	-0.097	144/(216)	3.462	0.780	-0.657	-0.094	-0.070
056/(304)	5.443	1.100	0.579	-0.133	-0.099	146/(214)	3.288	0.742	-0.640	-0.090	-0.067
058/(302)	5.555	1.125	0.559	-0.136	-0.101	148/(212)	3.111	0.703	-0.620	-0.085	-0.063
060/(300)	5.660	1.149	0.536	-0.139	-0.103	150/(210)	2.931	0.663	-0.597	-0.080	-0.060
062/(298)	5.757	1.172	0.511	-0.142	-0.106	152/(208)	2.748	0.623	-0.571	-0.075	-0.056
064/(296)	5.847	1.193	0.483	-0.144	-0.107	154/(206)	2.562	0.582	-0.542	-0.070	-0.052
066/(294)	5.929	1.212	0.453	-0.147	-0.109	156/(204)	2.375	0.540	-0.511	-0.065	-0.049
068/(292)	6.002	1.230	0.420	-0.149	-0.111	158/(202)	2.184	0.497	-0.477	-0.060	-0.045
070/(290)	6.068	1.247	0.385	-0.151	-0.112	160/(200)	1.992	0.454	-0.442	-0.055	-0.041
072/(288)	6.126	1.262	0.348	-0.153	-0.114	162/(198)	1.798	0.410	-0.404	-0.050	-0.037
074/(286)	6.176	1.276	0.309	-0.154	-0.115	164/(196)	1.603	0.366	-0.364	-0.044	-0.033
076/(284)	6.218	1.288	0.269	-0.156	-0.116	166/(194)	1.406	0.321	-0.322	-0.039	-0.029
078/(282)	6.252	1.298	0.227	-0.157	-0.117	168/(192)	1.207	0.276	-0.279	-0.033	-0.025
080/(280)	6.278	1.307	0.184	-0.158	-0.118	170/(190)	1.008	0.230	-0.235	-0.028	-0.021
082/(278)	6.296	1.314	0.139	-0.159	-0.118	172/(188)	0.807	0.185	-0.189	-0.022	-0.017
084/(276)	6.306	1.320	0.094	-0.159	-0.119	174/(186)	0.606	0.139	-0.143	-0.017	-0.012
086/(274)	6.308	1.324	0.048	-0.160	-0.119	176/(184)	0.404	0.093	-0.095	-0.011	-0.008
088/(272)	6.302	1.326	0.002	-0.160	-0.119	178/(182)	0.202	0.046	-0.048	-0.006	-0.004
090/(270)	6.289	1.327	-0.044	-0.160	-0.119	180/(180)	0.000	0.000	-0.000	-0.000	-0.000

Table 38: Ecliptic latitude of the moon. The latitude is minus the value shown in the table if the argument is in parenthesies.

$F (^\circ)$	$\beta(^\circ)$	$F (^\circ)$
000/180	0.000	(180)/(360)
002/178	0.180	(182)/(358)
004/176	0.360	(184)/(356)
006/174	0.539	(186)/(354)
008/172	0.718	(188)/(352)
010/170	0.896	(190)/(350)
012/168	1.073	(192)/(348)
014/166	1.248	(194)/(346)
016/164	1.422	(196)/(344)
018/162	1.595	(198)/(342)
020/160	1.765	(200)/(340)
022/158	1.933	(202)/(338)
024/156	2.099	(204)/(336)
026/154	2.263	(206)/(334)
028/152	2.423	(208)/(332)
030/150	2.581