Research on the expansion–contraction difference for the inner planets in ancient China

Abstract

In ancient China, astronomers attempted to correct mean geocentric longitude of the inner planets using planetary “Expansion–Contraction Difference” (yingsuo cha 盈縮差) to obtain the true geocentric longitude. They used the “Limit Degree” (xiandu 限度) as the independent variable for the “Expansion–Contraction Difference”. Although this idea was relatively ideal and operationally simple, the algorithm of the “Expansion–Contraction Difference” designed by ancient Chinese astronomers had significant flaws in terms of its actual computational effectiveness for calculating the position of the inner planets. The reason for this flaw is that, based on its intended purpose, the “Expansion–Contraction Difference” should be a three-variable function, including the planetary equation of center, the solar equation of center and the phase angle of the planet, and each variable has different independent variables. However, ancient Chinese astronomers attempted to simplify this complex three-variable function into a single-variable function, and such simplification was unsuccessful. Further research indicates that the starting point of the expansion phase in the “Table of Expansion–Contraction Difference” (Yingsuo Li 盈縮曆) for the inner planets in ancient Chinese astronomical systems did not accurately depict the position of the planetary perihelion. Adjusting the starting point based on the longitude can improve the accuracy to some extent. Although the special coefficients of “double it for Venus, triple it for Mercury” can enhance the accuracy of calculations on the position of the inner planets, the result is not as ideal as expected. This study highlights that within the framework of ancient Chinese planetary theory, the algorithm of the “Expansion–Contraction Difference” for the inner planets possessed inherent and irreparable flaws, resulting in significant errors in the calculation of the apparent position of the inner planets.