tokens: of Fig. 5 , since the contact between the QS and HS surfaces does not occur yet, QS and HS surfaces coexist independently and we use ‘QSHS’ to denote this stage. When = −1.502164, the contact between the QS and HS surfaces begins to occur, and we define this critical value as the upper boundary of of the TS motion, . In the blue region of Fig. 5 , the compound isosurface is made up of QS, TS, and HS portions, and we use ‘QS + TS + HS’ to denote this stage. Figure 5. Evolution of isosurfaces with i m = 15°, where blue points denote values used in the following figures. Fig. 6 displays phase structures with (a) i m = 45° and (b) i m = 75° in the torus space, where the colours denote different values of . Compared with the result in Fig. 2(b) , we can observe that i m has a significant influence on the phase structure. The evolution of isosurfaces of i m = 45° and 75° with the decrease of are shown in Fig. 7 . Different the result in Fig. 5 , we can observe that when i m = 45°, is larger than . Numerical computation indicates that when i m = 29.65°, is equal to ; if i m > 29.65°, is larger than . QS and TP surfaces coexist independently in the yellow region of Fig. 7(a) , and when the contact between the QS and TP surfaces begins to emerge, the isosurface is made up of QS, TS, and TP portions (see the blue region in Fig. 7a ). If further deceases, the TP portion on the compound surface will be replaced by the HS portion. Numerical computation also shows that with the increase of i m , gradually decreases and approach , so the yellow region in Fig. 7(a) shrinks with the increase of i m . When i m ≥ 61°, is equal to and independent TP isosurfaces totally disappear. Hence, as we can see from the result of i m = 75° in Fig. 7(b) , the compound isosurface in the blue region is made up of QS and TS portions. Figure 6. isosurface with (a) i m = 45° and (b) i m = 75° in the torus space, where the colours denote different values of . Figure 7. Evolution of isosurfaces with (a) i m = 45° and (b) i m = 75°, where blue points denote values used in the following figures. Fig. 8 displays collision curves of i m = 15°, 45°, and 75° in (a) torus space and (b) ω − e projection. From Fig. 8(a) , we find that the φ range of collision curves starting from the point P = ( e 0 , 0, 0) expands with the increase of i m . In the ω − e projection, we can observe that with the increase of i m , the angle between the branches of collision curves increases. In the following section, we will demonstrate that collision curves play an important role in the co-orbital motion and the above findings can explain some co-orbital phenomena. Figure 8. Collision curves of i m = 15°, 45°, and 75° in (a) torus space and (b) ω − e projection. 4 LOCI IN THE TORUS SPACE In this section, we implement numerical integration in the Sun–Jupiter CRTBP to demonstrate the validity of the semi-analytical analysis in the last section and the detail of the co-orbital motion. Numerical integration is
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