Analysis of Solar and Lunar Motions in the Seonmyeong Calendar

Abstract

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 Referring to historical documents of China, Korea, and Japan, we analyze the calendrical methods of calculating the daily apparent movements of the Sun and Moon in the SM calendar, which were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.

1. INTRODUCTION

The Xuanming (Manifest Enlightenment) calendar was compiled by Xu, Ang (birth and death years are unknown) of the Tang dynasty (A.D. 618–907) in A.D. 822 and has been used for 71 years in China (Lee 1985). This calendar was also used in Korea and Japan, which was called the Seonmyeong and Senmyo calendars, respectively. According to the preface of the astronomical chapter of the Goryeosa (History of the Goryeo Dynasty), the Goryeo court (A.D. 918–1392) of Korea used the Seonmyeong calendar (hereinafter SM calendar) from the beginning of the dynasty by succeeding it from the Silla kingdom (B.C. 57–A.D. 935) to the end of the dynasty, particularly for calculating solar and lunar eclipses although the Shoushi (Granting the Seasons) calendar of the Yuan dynasty (A.D. 1271–1368) in China was enforced in 1281 (Seo 2012; Chang 2014; Jun 2014). Therefore, the SM calendar was utilized for 475 years in Korea, at the least. Meanwhile, this calendar was introduced to Japan in A.D. 862 through the Balhae kingdom (A.D. 698–926) of Korea and used for 823 years until 1684 (Kim 2012).

In this study, we analyzed the calendrical methods of solar and lunar motions in the SM calendar referring to the Xintangshu (New History of Tang Dynasty) of China, Goryeosa of Korea, and Senmyo (Senmyo Calendar) and Chokei–Senmyoreki–Sanpo (Calendrical Method for the Senmyo Calendar of the Chokei, shortly Sanpo) of Japan. With regard to Japanese literature, the former is preserved in the National Library of Korea and is known to be made of a Japanese woodblock in 1644 but the author is unknown. Meanwhile, the latter was compiled by Ando, Yueki in 1654 (Soma 2014) and the Chokei (Changqing in Chinese) is the Chinese reign-title used for four years from 821 to 824.

Together with the Shoushi calendar, the SM calendar is a well-studied Chinese calendar, particularly by Japanese scholars. Sato (1968) reviewed the calendar in the study on the calendar history of Japan. Uchida (1975) utilized the Sanpo for identifying Japanese lunisolar calendar dates with Julian calendar dates. However, Soma (2014) found some mistakes in the daily movement of the Sun, which is tabulated in the Sanpo. According to our examination, the same mistakes exist in the Senmyo preserved in Korea. ISOBDU (1998) compared the contents of the SM calendar recorded on various versions of the Goryeosa with the Sanpo. Qu (2008) studied Chinese calendars including the SM calendar in terms of mathematical techniques (see also Qian 1981; Martzloff 2016). In Korea, the SM calendar was first studied by the scholars of North Korea in 1964 when translating the Goryeosa written in Chinese characters into Korean (EDS 1991). Recently, SADU (2016) retranslated the astronomical chapter of the Goryeosa including detailed annotations and some calculation examples.

The SM calendar is known as a good Chinese calendar, particularly for the calculations of eclipses (Lee 1985). According to the work of Hahn et al. (2016), a quadratic equation of type x(x-a) was employed in the SM calendar to consider the horizontal parallax of the Moon and was utilized in subsequent Chinese calendars such as the Shoushi calendar. However, the accuracy of the SM calendar has not been analyzed through comparison with modern calculations. The purpose of this study is to investigate the accuracy of the solar and lunar motions by the SM calendar. The remainder of this paper is structured as follows. In Section 2, we introduce the units and astronomical constants in the SM calendar, which are used in this study. We analyze the numerical tables relating to solar and lunar motions listed in calendar books and compare with the results of modern calculations in Section 3. Finally, we summarize our findings in Section 4.

2. UNITS AND ASTRONOMICAL CONSTANTS

In the SM calendar, Il (Day), Bun (Minute) and Cho (Second) were used as the units of an hour (hereinafter, d_H, min_H, and sec_H, respectively) and Do (Degree), Bun (Arcminute), and Cho (Arcsecond) as those of an angle (hereinafter, d_A, min_A, and sec_A, respectively): 1 d_H = 84 or 8400 min_H = 672 or 67200 sec_H (i.e., 1 minH = 8 sec_H) and 1 d_A = 84 or 8400 min_A = 25200 or 2520000 sec_A (i.e., 1 min_A = 300 sec_A). Because two scales were used in units of min_H and min_A (i.e., 84 and 8400 called the Gakbeob and SeonmyeongTongbeob, respectively), we distinguished them using the superscripts G (i.e., Gakbeob) and T (i.e., SeonmyeongTongbeob) such as min^G_H or min^T_A.

As in other Chinese calendars, the beginning of the calendrical calculation for a given year, Y, was to determine the instant of the winter solstice in its previous year, Y-1, called the Cheonjeong-Dongji (hereinafter, CJDJ) of year Y. In addition, although the calendrical epoch of the SM calendar was the instant of the CJDJ of 822 (i.e., instant of the winter solstice in 821), that of the CJDJ of a given year was obtained by counting from the Super Epoch (i.e., an ancient epoch where the beginnings of the solar, lunar, and sexagenary cycles coincided). The length between the Super Epoch and calendrical epoch of the SM calendar, SangwonJeoknyeon, was 7070138 years. The abolition of the Super Epoch is a remarkable characteristic in the Shoushi calendar compared to its previous calendars (Sivin 2008)

Regarding the solar motion, the length of a tropical year called the Jangse was 3068055 min^T_H or 365 d_H 2055 min^T_H, and a year was divided into 24 sub-seasons (hereinafter, 24 solar terms) in Chinese calendars. A method of determining the instants of 24 solar terms is to equally divide the length of the tropical year into 24 intervals, as mentioned in Section 3.1. In this case, the length of a solar term is 15 d_H 1835 min^T_H 5 sec_H and this value was called the Jungjeol. The celestial sphere was divided into 28 lunar loges (refer to Mihn et al. 2017), and the length of the celestial perimeter called the Jucheon is 365 d_A 2153 min^T_A in the Goryeosa. However, the value of the Tang dynasty, i.e., 365 d_A 2153 min^T_A 299 sec_A, is also presented in cases of the Jucheon as well as some other constants. When converting the Tang value into units of sec_A, it becomes 920446199 that is recorded as the Sangsu in the Goryeosa. Therefore, it appears that the Goryeo astronomer used up to the unit of min_A for the value of the Jucheon presumably to simplify the calculations. From the lengths of the tropical year and celestial perimeter, we could deduce that the mean motion of the Sun in the ecliptic was nearly 1 d_A per day although this value was not explicitly defined in historical literature. In addition, the lengths of a synodic month called the Hapcheok and of an anomalistic month called the Yeokjuil was 29 d_H 4457 min^T_H and 27 d_H 4658 min^T_H 19 sec_H, respectively. Similarly, the 24 solar terms were utilized as the cardinal points in relation to the solar motion, and the periods of 7 d_H 7465 min^T_H and 14 d_H 6529 min^T_H derived from an anomalistic month were utilized as the reference points in relation to the lunar motion as mentioned below. The values of 7465 and 6529 were called the Chosu of the 7th and 14th day, respectively, and the values subtracted the Chosu from 8400, i.e., 935 and 1871, were called the Malsu of the 7th and 14th days, respectively. Finally, although the mean motion of the Moon among the stars was also not mentioned, it could be deduced as 13 d_A 31 min^G_A per day, as estimated in Section 3.2. In Table 1, we summarize the values of astronomical constants in the SM calendar that are used in this study.

Table 1. Summary of astronomical constants in the SM calendar used in this study

^† Value of the Tang dynasty, as indicated in the Goryeosa.

^‡ Super Epoch.

3. SOLAR AND LUNAR MOTIONS

3.1 Solar motion

The inequality of the solar motion has been known as early as A.D. 570 in China (Needham 1959). In the Goryeosa, the apparent movements of the Sun considering its unequal motion are tabulated with respect to the solar term, as in the case of the Chongxiu-Daming calendar (refer to Choi et al. 2018a). A year was divided into 24 solar terms, as mentioned above, and two methods were utilized for determining the instants of the solar terms: mean and corrected (literal translation) methods. The former method is to equally divide the length of the tropical year into 24 intervals while the latter method is to equally divide the ecliptic into 24 intervals conceptually. Currently, the corrected method is utilized for determining the dates of 24 solar terms (KASI 2018). In both methods, the starting point is the instant of a winter solstice of the preceding year. With regard to calculating the daily movement of the Sun from each solar term, only the calculation procedure is explained in the Goryeosa. In the Japanese literature, meanwhile, the amounts of daily Sun movement in each solar term are tabulated.

We present the ready reckoner relating to the solar motion in Table 2, which is tabulated in the Goryeosa. In column 1 of the table, i represents the solar term notated with an Arabic number. For reference, i = 1, 7, 13, and 19 represent the winter solstice, spring equinox, summer solstice, and autumnal equinox, respectively. In column 2, ∆U^G_S is the unequal amount of the solar motion in units of min^G_H during the length of each solar term by the mean solar term method (i.e., Jungjeol) compared to its mean motion. In column 3, JG is the length of the ith solar term by the corrected solar term method given in units of d_H, min^T_H, and sec_H. Because the Sun moves faster and slower around the winter and summer solstices, respectively, than its mean motion, the values of JG could be obtained by subtracting or adding the value of ∆U^G_S × 100 to the Jungjeol (the units of ∆U^G_S is min^G_H not T minH). As an example, the value of JG in the second solar term, JG₂, is 14 d_H 5235 min^T_H 5 sec_H by subtracting +50 × 100 min^T_H to the Jungjeol (i.e., 15 d_H 1835 min^T_H 5 sec_H). In this sense, the value of 2335 min^T_H for JG₁₈ in the Goryeosa is the typographical error of 2435 min^T_H, which is the same that for JG₇. In all tables, the values marked with the asterisk symbol are the corrected values in this study. In column 4, U^G_S is the accumulation of ∆U^G_S until the preceding solar term; hence, the value of U^G_S in the first term, U^G_S,1, is zero. In column 5, ∆U^M_S is the value dividing ∆U^G_S × 100 by the mean motion of the Moon per day, u^G_M . Although the value of u^G_M is not defined, it was 13 d_A 31 min^G_A (or 13.3689 d_A), according to our estimation in the next section. Adopting this value, we found that the values of ∆U^M_S were the results rounded half up. In addition, the value of in the 14th solar term, ∆U^M_S,14, should be +374 recorded as the value of the Tang dynasty in the Goryeosa. The last column, U^M_S, is the accumulation of the ∆U^M_S until the preceding solar term; hence, the value of U^M_S in the first term, U^M_S,1, is zero. The value of U^M_S,17 is the typographical error in the Goryeosa. Although the Goryeosa states that the value of the Tang dynasty is 1246, it is recorded as 1346 in the version of the Xintangsh that we refer to in this study.

Table 2. Ready reckoner in relation to the solar motion

* Corrected values in this study.

As shown in Table 2, the solar motion is symmetrical around two solstices and equinoxes. The typographical errors discussed above can be verified from this perspective. In reality, however, the Earth revolves around the Sun and its motion is fast around the perihelion and slow around the aphelion. To estimate the accuracy of the solar motion in the SM calendar, we calculated the daily apparent movement of the Sun following the calculation procedures described in the Goryeosa. In Appendix A, we present the procedures using mathematical symbols and formulas to facilitate the understanding of contemporary concepts. As shown in the appendix, a quadratic equation was used to consider the inequality of the Sun, as in the Shoush calendar (refer to Choi et al. 2018b).

3.2 Lunar motion

The length of the anomalistic month, 27 d_H 4658 min^T_H (19 sec_H), was divided into two ranges referring to the apogee (A) and perigee (P). Hereinafter, Sectors I and II stand for the AP and PA ranges, respectively. To calculate the position of the Moon for a given time, the SM calendar used the daily apparent movement of the Moon in each sector, which was given in a tabular form. In Table 3, we present the table listed in the Goryeosa in relation to the lunar motion. In the table, column 1, denoted as m_A or m_P (integer), represents the day number since the instants of the apogee and perigee passages, respectively (hereinafter, m is the common designation for m_A and m_P). Because the day number was counted from the first day (i.e., m = 1), the last day number in each sector was the 14th day. In column 2, u^G_M is the apparent movement of the Moon in units of min^G_A on its orbital during the day (i.e., 8400 T minH) on the mth day. In column 3, ∆u^G_M is the difference of two successive values in u^G_M for m ≤ 13, i.e., (∆u^G_M(m) = u^G_M(m+1)-u^G_M(m) , and 0 for m = 14. In column 4, u^G_M is the same quantity with u^G_M but converted in units of d_A and min^G_A, which is listed in a separate table in the Goryeosa. For instance, because the value of u^G_M on the third day in Sector II is 1203 min^G_A (= 14 × Gakbeob + 27 min^G_A), hence that of u^G_M on the day is 14 d_A 27 min^G_A . Suppose that the u^G_M is the mean motion of the Moon per day and δu^G_M is the difference between u^G_M and u^G_M . In columns 6 and 7, ∆u^M_M and U^M_M are the physical quantities dividing δu^G_M by u^G_M and accumulating ∆u^M_M until the preceding day, respectively; hence, U^M_M is zero for m = 1. Strictly speaking, the value of ∆u^M_M for the mth day, ∆u^M_M(m), is

\(\Delta u_{M}^{M}(m)=\alpha \times \frac{\delta u_{M}^{G}(m)}{\bar{u}_{M}^{G}(m) / 8400}\left[\min _{\mathrm{H}}^{\mathrm{T}}\right]\)      (1)

where α is 1 except for m = 7 and 14. As shown in Table 3, two values are listed in ∆u^M_M for m = 7. These values can be obtained from Eq. (1) using the values of the Chosu and Malsu on the 7th day. That is, two values in each sector can be obtained using α of 7465/8400 and 935/8400. Meanwhile,

Table 3. Ready reckoner in relation to the lunar motion

* Corrected values in this study

3.3 Accuracy estimation

We estimated the accuracy of the solar and lunar motions by the SM calendar when the calendar was used in Korea by the Goryeo dynasty, and not in China by the Tang dynasty. Hence, we selected the periods of 919, 1155, and 1392 (i.e., the beginning, middle, and ending periods of the Goryeo dynasty, respectively) and compared the amounts of daily Sun and Moon movements listed in the Goryeosa with the results of modern calculations for each period. As mentioned earlier, the first step of the calendrical calculation in the SM calendar is to determine the instant of the CJDJ of a given year. However, in this study, we will not discuss the procedures of calculating the instant of the CJDJ in detail because they are well explained in the work of SADU (2016) where examples are provided. In principle, the instants of the CJDJ and those of the apogee passage are determined by accumulating the lengths of a tropical year and anomalistic month since the Super Epoch. However, it is noteworthy that the day of the winter solstice (also CJDJ) should be placed in the lunar November in Chinese calendars. In other words, a lunar November is the month containing the winter solstice day. In addition, it is noteworthy that the calendar days are obtained as a sexagenary circle instead of Arabic numbers. Hence, we referred to the work of Han (2001) to convert lunisolar days obtained as a sexagenery circle into Julian calendar days. Because the hour system of the SM calendar was based on the apparent solar time (AST), we also converted the AST at the Gaeseong (i.e., a capital of the Goryeo dynasty) into the mean solar time in Korean standard time (KST, i.e., at the meridian of 135° E) by correcting the longitudinal difference, equation of time (ET), and ΔT (i.e., time difference between the terrestrial time and universal time) (see also Lee et al. 2011). We assumed the longitude of the Gaeseong as 126° 33′ E, i.e., the longitudinal difference is –33.8 min (Lee 2017), and referred to the work of Morris and Stephenson (2004) for the value of ΔT (e.g., 34.5 min in 918). Finally, we used the astronomical algorithms of Meeus (1998) and the ephemeris of Standish et al. (1997) in modern calculations.

To estimate the accuracy of the solar motion, we first reproduced the instants of the CJDJ of the years used as the examples in the work of SADU (2016) and obtained the same results. Subsequently, we determined the instants of CJDJ of 919, 1155, and 1392 and summarized the results in Table 4 including the perihelion passage times in those days from modern calculations. In the table, YY, MM, DD, and HH represent the year, month, day, and hour, respectively. In the lunisolar date, the day is expressed with the notation utilized by Lee et al. (2012) and the unit of hour is min^T_H as that in the AST. For instance, #55 November 1154 in the lunisolar calendar corresponds to 15 December 1154 in the Julian calendar, and 1905 min^T_H in AST at the Gaeseong to 4.55 h in KST by correcting the longitudinal difference of –33.8 min, equation of time of –0.72 min, and ΔT of 14.7 min. According to our study, the difference between the CJDJ instant of 1155 and the perihelion passage time at that time was the smallest, i.e., –0.14 days, among the three periods. After determining the CJDJ instant, we computed the dates of the 24 solar terms by accumulating the Jungjeol and calculated the daily Sun movement by the SM calendar since the instant of each solar term throughout the procedures described in Appendix A. Subsequently, we compared the movements for 15 days in each solar term (i.e., for a total of 360 days) with the results of modern calculations. Hence, the interval between the last day in a solar term and the first day in the next solar term (i.e., next solar term day itself) is 1 d_H 1835 min^T_H 5 sec_H, and not 8400 min^T_H (i.e., a day).

Table 4. Summary of the CJDJ (Cheonjeong-Dongji) instants of 919, 1155, and 1392 by the SM calendar and the perihelion passage times in those days by modern calculations​​​​​​​

Fig. 1 shows the differences in the daily apparent movement of the Sun between the SM calendar, S^SM, and modern calculations, S^MC, for the periods of 919, 1155, and 1392. The horizontal axis is the elapsed date since the CJDJ instant of each year and the vertical axis is S^SM_i (n) - S^MC_i (n) in units of (modern) arcmin, where i is the solar term from 1 to 24 and n is the day number since each solar term from 0 to 14. In the figure, the red solid, green dot, and blue shortdash lines represent the results for the periods of 919, 1155, and 1392, respectively; the vertical dotted lines represent the elapsed dates of the spring equinox, summer solstice, and autumn equinox, from left to right, since the CJDJ instant of each year. For a quantitative comparison, we also estimated the mean absolute difference (MAD) value defined as follows:

\(\mathrm{MAD}=\frac{1}{N} \sum_{i=1}^{i=N}\left|X^{S M}-X^{M C}\right|_{i}\),       (2)

where X^SM and X^MC are physical quantities from the SM calendar and modern calculations, respectively, and N is the number of data used in the calculations. We found that the MAD values are 0.33, 0.30, and 0.31 arcmin for the periods of 919, 1155, and 1392, respectively. In addition, the pattern in Fig. 1 is similar to that in the result of the ChongxiuDaming calendar of the Jin dynasty for the solar motion: the maximum differences are indicated around the winter and summer solstices, and the minimum around the spring and autumn equinoxes (refer to Choi et al. 2018a).

Fig. 1. Differences in daily solar motion between the SM calendar, S^SM, and modern calculations S^MC: red solid, green dot, and blue short-dash lines represent the results for the periods of 919, 1155, and 1392, respectively. The horizontal axis is the elapsed dates since the CJDJ (Cheonjeong-Dongji) instant of each year and the vertical axis is S^SM_i (n) - S^MC_i (n) in units of (modern) arcmin. The vertical dotted lines are the elapsed dates of the spring equinox, summer solstice, and autumn equinox, from left to right, since the CJDJ instant of each year.

In the lunar motion section of the Goryeosa, the methods of determining the elapsed dates of four phases of the Moon (i.e., new, first quarter, full, and last quarter moons) since the instant of an apogee passage are described. In addition, the instant of the apogee passage to calculate the elapsed date of the new moon in lunar November was computed using the length of the Yeonyeo (i.e., the difference in instants between the CJDJ and apogee passage around that time). For this study, however, we obtained the instant of the apogee passage immediately before the CJDJ instant (hereinafter, the CJAP instant) by accumulating the length of the anomalistic month from the Super Epoch. In Table 5, we present the instants of the CJAP referring to the instants of the CJDJ of 919, 1155, and 1392 together with the results of modern calculations.

Table 5. Summary of the instants of CJAP (apogee passage immediately before the CJDJ) of 919, 1155, and 1392 by the SM calendar and the differences with apogee passage times in those days by modern calculations​​​​​​​

To estimate the accuracy of the lunar motion in approximately one tropical year, we first determined 12 successive instants of the apogee passage in each year by accumulating the anomalistic month length to the CJAP instant. Subsequently, we calculated the daily Moon movement during each anomalistic month length using the DE405 ephemeris. As indicated in Section 3.2, however, the lunar movement listed in Table 3, u^G_M , is the daily amount (i.e., for 8400 min^T_H) for 28 days since the instant of an apogee passage although the length of the anomalistic month is 27 d_H 4658 min^T_H 19 sec_H. Therefore, in modern calculations for the daily lunar movement, we determined the beginning point of each day by accumulating 8400 min^T_H from the instant of an apogee passage but reset the point on the days of m_A = 8 and m_P = 1 and 8 using the Chosu of the 7th and 14th days. For example, the beginning point on m_A = 8 is 7 d_H 7465 min^T_H and not 8 d_H 0 min^T_H if the instant of each apogee passage is 1 d_H 0 min^T_H. From the comparison with the movements listed in the Goryeosa, we found that the MAD values are 0.217, 0.284, and 0.240 (modern) degrees for the periods of 919, 1155, and 1392, respectively. In Fig. 2, we present the differences in daily Moon movement between the SM calendar, M^SM, and modern calculations, M^MC. The horizontal axis is the day number since the instant of an apogee passage, and the vertical axis is M^SM( m ) - M^MC ( m ) in units of (modern) degrees, which are averaged over the 12 cycles of anomalistic month length for each day number. In the figure, the legend is the same as that in Fig. 1: red solid, green dot, and blue short-dash lines represent the results for the daily Moon movement for the periods of 919, 1155, and 1392, respectively.

Fig. 2. Differences in daily lunar motion between the SM calendar, M^SM, and modern calculations, M^MC : red solid, green dot, and blue short-dash lines represent the results for the periods of 919, 1155, and 1392, respectively. The horizontal axis is the day number since the instant of the apogee passage and the vertical axis is M^SM( m ) - M^MC ( m ) in units of (modern) degrees, which are averaged over the 12 cycles of anomalistic month length for each day number.

4. SUMMARY

We investigated the solar and lunar motions in the SM calendar of the Tang dynasty in China. According to the preface of the Goryeosa, this calendar was used in Korea during the Goryeo dynasty succeeding the Silla Kingdom. Although it was known that the Goryeo dynasty introduced the Shoushi calendar in 1281, the SM calendar had been utilized until the end of the dynasty for the prediction of solar and lunar eclipse times. Primarily referring to the Goryeosa, we analyzed the calendrical methods of the SM calendar to calculate the solar and lunar motions. Subsequently, we compared the motions with modern calculations using the DE405 ephemeris for three periods: 919, 1155, and 1392, i.e., the beginning, middle, and ending of the Goryeosa dynasty, respectively. Regarding the solar motion, we found that a quadratic equation was employed to calculate the daily Sun movement. Furthermore, we found that the MAD values were approximately 0.31 arcmin for the three periods, although the minimum was indicated in 1155, where the CJDJ instant of 1155 was the closest to the perihelion passage time among the three periods. Meanwhile, we found that relatively large MAD values for the lunar motion: 0.247 degrees on average. In addition, we found that the MAD value indicated the maximum in 1155, unlike the cases of the solar motion. This might be the reason for the largest difference in apogee passage time between the SM calendar and modern calculations in 1155 among the three periods analyzed in this study.

In conclusion, we believe that this study will facilitate in the understanding of calendrical methods in the SM calendar, such as the sunrise and sunset times, eclipse times, and so forth.