Abstract: According to the study on the eigenvalues of the K index, it is found that the larger K index indicated the greater synodic degree of the planetary system in a certain radial direction. The variation of the solar orbit radius calibrated by the extreme value of the K index has a significant nesting relationship of quasi20a, 179a and 2400a periods. With the increase of time scale, the maximum of K index shows a significant quasi2400a period, and the duration of the mass center of the solar system inside and outside the solar body also has a quasi2400a period. This implies the quasi2400a period of the solar orbital motion without bypassing the mass center of the solar system. Previous reports about variation in the distance between the Sun and the mass center of the solar system (including K-index images) have failed in revealing this rule. In this paper, taking the connection between Jupiter and the Sun as the reference direction, the planetary synodic vector index K is established. The K-vector index image clearly shows the quasi2400-year cycle of the Sun orbital motion without bypassing the mass center of the solar system, which is composed of 13~14 quasi179a envelope periods (the trough K values of 3~4 periods appear < 0). This discovery is a further step forward in the scientific identification of the complex characteristics of the Suns orbital motion.

Key words: Sun orbital motion, mass center of the solar system, planetary synodic index K, planetary synodic vector index K, quasi2400-year cycle