Example Syzygy Calculations

Example 1: Sixth new moon in 2004 CE:
 
From Table 42, the date of first new moon in 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 in 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 in 2004 CE is $2453174.10 +(360.00-357.22)/12.1907491= 2453174.33$ JD. This corresponds to 20:00 UT on June 17th.

 
Example 2: Third full moon in 1982 CE:
 
From Table 42, the fractional Julian day number of first new moon in 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 in 1982 CE took place before January 25th. Hence, a rough estimate for the date of the third full moon in 1982 CE is one and a half synodic months after that of the first new moon: 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 in 1982 CE is $2445039.0 +(180.00-187.90)/12.1907491= 2445038.35$ JD. This corresponds to 20:00 UT on March 9th.