## 3 Mathematical Modeling
### Equations of Motion
#### 3.1.2 Zero-Velocity Curves and Equilibrium Points for Some Aspherical Bodies  
Changes in body characteristics, like density, rotation speed, geometry, etc., may affect the number, location, and stability of the equilibrium points [83; 84], thus affecting the structure/shape of the \(ZVS\)s (6), which are intrinsically related to the stability of the critical points of the effective potential (3). The stability analysis for these points will permit the identification of families of orbits, delimited by the \(ZVS\), that will last stable in the vicinity of the NEO or lead the spacecraft to impact on the central body or to escape from its vicinity. The Jacobi integral (energy) may be used to map families of \(ZVS\)s for the system in the body-fixed position space \(Oxyz\). In the present work, the near-equilibrium motion will be analyzed assuming a planar-equatorial orbit (\(z=0\)) around symmetric oblate bodies, thus reducing the analysis to the Zero-Velocity Curve (ZVC) corresponding to \(z=0\) at the given ZVS. Several studies have computed four equilibrium points around uniformly rotating small bodies, such as the asteroids (1620) Geographos, (4769) Castalia and (216) Kleopatra [80; 85]. Ref. [84] studied the influence of the rotation speed on the behavior of the equilibrium points in the gravitational field of the asteroid (216) Kleopatra. They observed that longer rotationperiods engender topological changes and cause some points to collide and annihilate each other until only one point remains inside the object. Ref. [33] presents the characteristics of the geophysical environment of (16) Psyche and studies how density changes affect the distribution of equilibrium points. The linear stability and topological classification of the equilibrium points in the gravity field of more than 20 small celestial bodies were investigated in [80], according to the classification established in [85]. The NEO's Hill radius \(R_{{}_{H}}\) will be assumed as the superior limit for the spacecraft's orbit radius so as to assure that the NEO's gravitational influence prevails in the dynamics, allowing the disregarding of external perturbations such as those arising from solar radiation pressure or third-body (moonlets) gravitational forces [86]. On the other hand, in order to guarantee the validity of the Harmonic Expansion of the gravity potential in the vicinity of an aspheric NEO surface, its Brillouin radius, \(r_{{}_{B}}\), will be used as an inferior limit to the spacecraft's orbit radius. So, it will be assumed the spacecraft's orbit radius will obey \(r_{{}_{B}}\leq r\leq R_{{}_{H}}\). Table 1 presents physical characteristics for some typical asteroids and Tables 2-6 show some of the planar (\(z=0\)) equilibrium points, along with their respective eigenvalues, eigenvectors and Jacobi constants. The shown \"Dimension\" corresponds to the \(x\times y\times z\) full extensions (\"diameters\") of the bodies. Some ZVCs for these bodies are shown in Figures 1-5, depicting the saddles (\(\lambda_{1}<0<\lambda_{2}\)) and centers (\(\lambda_{1},\lambda_{2}<0\)) corresponding to the, respectively, **(U)**nstable points, crossings of the dashed separatrices, and **(S)**table equilibrium points. The black (dotted) curves at the center of the figures, except for Icarus and Alpha, represent the equatorial contour of the portrayed body. From the cases shown here, one can see some intricacies endowed by mass and spin rate on the stability of motion around these small bodies. For instance, (253) Mathilde is \(\sim\)250 times less massive than (16) Psyche and \(\sim\)2500 times less massive than (4) Vesta but is rotating \(\sim\)100 times faster than the other two bodies. One can see that, while the distances of the equilibrium points from the larger bodies' surfaces correspond to their respective radii, the equilibria of (253) Mathilde are located at a distance corresponding \(\sim\)30 times the body's radius, not prone to pre-close-approach data acquisition.