Research on the Expansion-Contraction Difference and Limit Degree in ancient Chinese planetary theory: The case of outer planets

Planetary Expansion-Contraction Difference ( Ying suo cha 盈縮差) in ancient Chinese planetary theory was mainly used to correct non-uniform motion of the planet. The independent variable of planetary Expansion-Contraction Difference is the Argument Degree Entry ( Ru li du 入曆度) of the planet, which is related to the Limit Degree ( Xian du 限度) in the table of planetary Grades of Motion ( Wu xing dong tai biao 五星動態表). In this paper, we mainly discuss astronomical meanings of the Expansion-Contraction Difference and Limit Degree of the outer planets. First, we go through the historical development of the astronomical meaning of planetary Expansion-Contraction Difference and Limit Degree in ancient Chinese planetary theory. Then, we reconstruct a new mathematical model of Expansion-Contraction Difference for the outer planets. Finally, we discuss the errors in the longitude of the outer planets in the Era Epoch System ( Ji yuan li 紀元曆, A.D.1106) of Northern Song北宋Dynasty (A.D.960–1127) using different mathematical models of planetary Expansion-Contraction Difference and Limit Degree. The result shows that the Expansion-Contraction Difference and Limit Degree of the outer planets in the Era Epoch System and other astronomical systems should be the results of continuous refinement by astronomers on the basis of long-term actual observations and measurements, and their combination was a strong guarantee to ensure that the Era Epoch System is comparable to the Copernican system in the calculations of the planetary position.