Astron. Astrophys. 332, 1142-1146 (1998)

## 5. Real syzygy calculations

Because the Moon's average motion is per day, this calendar gets 1 Xian 0.082 day, so one anomalistic month 1B' 336 Xian, within this period, the Moon moves:

per Xian.

To solve the real syzygy, we must first get the values of V(t'). The V(t') is defined as the velocity of the lunar motion, the unit is degree per Xian. Because this calendar had not detailed information about how to compute and obtain the V(t') value, we could not get the V(t') value from the Shoushi Calendar directly, but it could be read in the Datong Calendar (another Chinese ancient calendar in the Ming dynasty, which has almost the same fundamental constants and calculating methods as Shoushi, epoch of AD 1304, so it is thought that this calendar is roughly little different from Shoushi). There is only one value of V(t') per Xian. By analysing the Datong Calendar and its V(t') table, we give one method to compute V(t') values, so it is not necessary to read them from the table inconveniently (Li and Zhang 1996a).

The function Q(g) is given as

where g is another parameter, and its unit is Xian. The unit of Q(g) is the degree, the unit of V(t') is degree per Xian. So

The difference between the value computed by this method and the one read from the table is about 10-4 degree of Shoushi.

Eq. (7) really represents the velocity of the Moon's motion, which also has correspondence to contemporary lunar motion model. We compared approximate computing models of lunar longitude (mentioned above) with V(t'). The lunar mean longitude is

The parameter t' is TDB from J2000, and the unit is the Julian Century. We change the unit of t' to the day and (t') to degree of Shoushi. So the velocity (from the perigee just before Winter Solstice of 1280) is

where t' is the same as that in S(t'), the unit of V'(t') is degree of Shoushi per Xian.

Fig. 3 plots both functions of V(t') and V'(t'). By comparing them, we see that the two models are in good correspondence, almost the same amplitude and phase.

 Fig. 3. The velocity function V(t') (solid) of lunar motion of Shoushi Calendar and contemporary approximate lunar model for computing the velocity of the longitude V'(t') (dashed) within one period (B') from perigee just before Winter Solstice of 1280, the unit is degree of Shoushi per Xian, 1 Xian 0.082 day, when t' 0 (perigee), V(t') (maximum); when t' 0.5B' (apogee), V(t') (minimum).

According to the Shoushi Calendar, the Ganzhi numbers of the real new Moon, , is calculated by

the unit is the day.

In fact, since V(t') represents the velocity of lunar motion within one anomalistic month, and both models of T(t) and -S(t') just stand for the solar and lunar fluctuations, so the T(t) S(t') is really the difference between the motions of the Sun and Moon. This is the crux of the matter for the Shoushi Calendar to calculate the real syzygys. The models of solar (Fig. 1) and lunar (Fig. 2) motions in the Shoushi Calendar and contemporary astronomical ephemerides are almost in agreement. As regards the calculating method, the results of Shoushi are by approximate calculation, the time equals to distance (T(t) S(t')) divided by velocity (V(t')). The modern way is by iterative methods between longitude calculations of the Sun and Moon.

Now we have given all the formulas for calculating the real new Moon. However, how about the full Moon calculation? In these equations, if we take (n B 0.5B) in the place of (n B), so we could get the values of the real full Moon.

© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998