Analysis of Solar and Lunar Motions in the Seonmyeong Calendar

We investigate solar and lunar motions in the Seonmyeong (SM) calendar that was compiled by Xu, Ang of the Tang dynasty (A.D. 618-907) in China and used for 71 years from 822 to 892. This calendar was also used in Korea during the Goryeo dynasty (A.D. 918-1392) and in Japan for 823 years from 862 to 1684, the longest time among the three countries. Referring to historical documents of China, Korea, and Japan, we analyze the calendrical methods of calculating the daily apparent movements of the Sun and Moon in the SM calendar, which were considered their unequal motions, and compare the movements with the results of modern calculations for three periods in the Goryeo dynasty: 919, 1155, and 1392 years (i.e., the beginning, middle, and ending of the dynasty, respectively). We find that a quadratic equation was employed to obtain the daily movement of the Sun using physical quantities on the instant of each solar term, which was tabulated in its calendar book such as the Goryeosa (History of the Goryeo Dynasty). For quantitative analysis, we compute the mean absolute difference (MAD) of the daily apparent movement between the SM calendar and modern calculations and obtain 0.33, 0.30, and 0.31 arcmin for the periods of 919, 1155, and 1392 years, respectively. Meanwhile, we find relatively large MAD values in the daily movement of the Moon: 0.217, 0.284, and 0.240 degrees for each corresponding year. An interesting point is that the MAD value in the lunar motion shows the maximum in 1155 years, and is the minimum in the solar motion. In conclusion, we believe that this study will facilitate in the understanding of the SM calendar further, particularly in the calendrical methods of calculating sunrise, sunset, and eclipse times.