Example Longitude Calculations

Example 1: May 5, 2005 CE, 00:00 UT:
 
According to Tables 27-29, $t = 2\,453\,495.5$ 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. 2.6. 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 UT on May 5, 2005 CE was 14TA36.

 
Example 2: December 25, 1800 CE, 00:00 UT:
 
According to Tables 27-29, $t = 2\,378\,854.5$ 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 UT on December 25, 1800 CE was 3CP03.