Determination of Ecliptic Longitude

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 longitude model from Keplerian orbit theory alone. Instead, we shall employ a greatly simplified version of modern lunar theory. According to such theory, the time variation of the ecliptic longitude of the moon is fairly well represented by the following formulae (see http://jgiesen.de/moonmotion/index.html, or Astronomical Algorithms, J. Meeus, Willmann-Bell, 1998):

$$\bar{\lambda} = \bar{\lambda}_0+ n\,(t-t_0),$$ (114)

$$M = M_0 + \tilde{n}\,(t-t_0),$$ (115)

$$\bar{F} = \bar{F}_0 + \breve {n}\,(t-t_0),$$ (116)

$$\tilde{D} = \bar{\lambda} - \lambda_S,$$ (117)

$$q_1 = 2\,e\,\sin M + 1.430\,e^2\,\sin 2M,$$ (118)

$$q_2 = 0.422\,e\,\sin (2\tilde{D} - M),$$ (119)

$$q_3 = 0.211\,e\,(\sin 2\tilde{D} - 0.066\,\sin \tilde{D}),$$ (120)

$$q_4 = -0.051\,e\,\sin M_S,$$ (121)

$$q_5 = -0.038\,e\,\sin 2 \bar{F},$$ (122)

$$\lambda = \bar{\lambda} + q_1+q_2+q_3+q_4+q_5.$$ (123)

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 Cha. 4). 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. Note that Ptolemy's lunar theory only takes the first two lunar anomalies into account. 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 27-29. 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. 5.1.
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.