In this section, we calculate the location, number, and stability of the external and internal equilibrium points of comet 67P, using Eq. (5). ∇V (r) = 0. (5) We identified five equilibrium points based on an adequate 3-D polyhedral shape model of comet 67P, which comprises 96,834 faces and 48,420 vertices [17]. Figure 7 shows the location of the five equilibrium points with different topological characteristics viewed in the xOy plane. The equilibrium points E1 and E3 are unstable and classified as Case II [33]. Equilibrium point E4 is also unstable and classified as Case V. Unlike Scheeres [16], which identified four equilibrium points around comet 67P, all of them unstable, 8 the equilibrium points E2 and E5 were linearly stable and classified as Case I, indicating that particles surrounding these equilibrium points may have timescales greater than the unstable equilibrium points E1, E3, and E4.

[Figure: Fig. 6: Surface slopes mapped across the surface of comet 67P viewed in six perspectives (±x, ±y, and ±z). The color bar indicates the angle θ (degrees) computed numerically from Eq. (4) over the surface of comet 67P.]

These equilibrium point regions may offer reduced fuel costs under idealized conditions. Additionally, equilibrium point E5 is located inside the comet 67P close to its center of mass. Furthermore, Tab. 2 shows the location of the equilibrium points of comet 67P. In addition, we indicate their latitude φ (degrees), longitude λ (degrees), radial distance r from the center of mass (km), and geopotential V (r) (km2s−2). The colored contour lines indicate the Jacobi constant (J) values associated with zero-velocity curves, in (km2s−2). The zero-velocity curve behavior confirms that 67P has a total of five equilibrium points. The equilibrium points within the body’s rotational Roche lobe limit (Fig. 7) may eventually trap particles [5, 34]. Our results show that the rotational Roche lobe, the teardrop-shaped region surrounding comet 67P, limited by the black line in Fig. 7, intersects itself at the E1 equilibrium point, while in [16], it intersects itself at the equilibrium point E3. This feature indicates that equilibrium point E1 has the minimum value for the geopotential, which differs from previous works [16], which found the minimum of the geopotential at equilibrium point E3. The equilibrium points pair E1-E5 has a large z-component (∼10−1) when compared with the others equilibrium points (∼10−2). Table 2 also illustrates that the radial distance to the center of mass highlights an axial symmetry that occurs particularly between the equilibrium point pairs E1–E3 and E2–E4, which are located on opposite sides of the comet 67P. Furthermore, the equilibrium points E1 and E3 lie above the xOy plane, while the equilibrium points E2, E4, and E5 remain below it.

[Figure: Fig. 7: Location of the five equilibrium points around comet 67P in the xOy projection plane. Equilibrium points denoted by crosses are topologically classified as Case II (hyperbolically unstable), equilibrium points with circles are referred to as Case I (linearly stable), and equilibrium points with a triangle are denominated as Case V (complexly unstable). The region delimited by the black line intersecting the equilibrium point E1 represents the rotational Roche lobe of comet 67P.]

[Table: Table 2: Location of equilibrium points around comet 67P and their geopotential values. Equilibria are computed through the polyhedron method with the Minor-Equilibria package [6] and an accuracy of 10−5.
Point X (km) Y (km) Z (km) φ (°) λ (°) r (km) V (r) (×10−7km2s−2)
E1]