4. Corrections of lunar motion
To understand the details of the Moon's real motion in one anomalistic month (B', named Zhuanzhong, B' ), the function P(D) in the Shoushi Calendar was made up for this purpose. The correction of lunar motion is called Yueli (i.e. lunar equation). According to this calendar, the P(D) was defined as
where D is also named as Chuxian or Moxian, the unit of P(D) is the degree.
The parameter t', days from the mean syzygy to the start of the anomalistic month (perigee) just before it (t' B'), is obtained by
ZY, named Zhuanying, days from the epoch to the start of the anomalistic month which is just before it, . The earlier adopted value is 13.1904 in the Shoushi Calendar.
One anomalistic month (B') was divided into two parts, Jili (, expanding area in one period) and Chili (, contracting area in one period). In this paper, the lunar corrective function is S(t'), which was defined by the calendar, the unit is the degree, the coefficient ,
The function S(t') also resembles a trigonometric function. The amplitude is . The t' are the days from the beginning of the anomalistic month. When t' 0 (perigee), 0.5B' (apogee), B' (perigee), S(t') 0, the Moon is at its mean position, and there are no fluctuations. When t' is around 0.25B', S(t') = (minimum); when t' is about 0.75B', S(t') (maximum).
Eq. (6) has correspondence to contemporary lunar motion model. We compare the computing model of lunar longitude of Chapront-Touzé and Chapront (1983 & 1988) with S(t'). In this paper, we only discuss the main terms of this model (amplitude 1000". There are three terms. The largest is the equation of the center in the same way as for the Sun. Due to this consideration, the approximate fluctuation of lunar longitude of the epoch (from the perigee just before Winter Solstice of 1280) within one anomalistic month is:
where t' is the same as that in S(t'), the unit of S'(t') is the degree of Shoushi.
Fig. 2 plots both functions of S(t') and -S'(t'). By comparing them, the two models are of good correspondence, almost the same amplitude, but they have phase reversal. Due to Shoushi, the results calculated by S(t') are reversed to the real motion of the Moon, because the function S(t') given in the Shoushi Calendar is just for counting the real syzygys. This is why both phases are reversed. The function -S(t') used by ancient Chinese astronomers at that time approximately shows the real motions of the Moon.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998