## 3 Mathematical Modeling
### Hovering Within the Triple NEA System (153591) 2001 SN263
#### 4.2.2 Tracking Long-Lasting Orbits Within the System  
We now consider the case of observing one or more of the bodies in the triplet from trajectories that may not be connected to the relative equilibria engendered within the triplet. Admitting that the spacecraft will navigate well inside ALPHA's Hill radius \(R_{{}_{H}}\sim 180\) km, only the forces due to the gravitational interaction within the triplet members will be considered. Following [43], other external perturbations were not incorporated as their effects, four orders of magnitude smaller than the internal forces, will not impact the results for the adopted two-month time span. In this case, we will use the model called \"Processing Inclined Bi-Elliptical 4 BP with Radiation Pressure--PIBEPRP\" developed by [61], in which a Restricted Four-Body Problem, consisting of a spacecraft orbiting the triple system 2001 SN263, is separated in two interacting Two-Body Problems and also including the Solar Radiation Pressure and the precession engendered by the oblateness of the central body in the noncoplanar orbits of the two moonlets. Given the coordinates \((x,\,y,\,z)\) of the spacecraft with respect to the central body (ALPHA), \((x_{s},\,y_{s},\,z_{*})\) the coordinates of asteroid \"\(*\)\" related to an inertial reference and \(r_{*}\) the distance from the spacecraft to the asteroid \"\(*\)\", we define the respective distances  
\[r =\sqrt{x^{2}+y^{2}+z^{2}} \tag{35}\] \[r_{*} =\sqrt{\left(x-x_{\alpha}\right)^{2}+\left(y-y_{\alpha}\right)^{2 }+\left(z-z_{\alpha}\right)^{2}}\] \[r_{\beta} =\sqrt{\left(x-x_{\beta}\right)^{2}+\left(y-y_{\beta}\right)^{2 }+\left(z-z_{\beta}\right)^{2}}\] \[r_{\gamma} =\sqrt{\left(x-x_{\gamma}\right)^{2}+\left(y-y_{\gamma}\right)^ {2}+\left(z-z_{\gamma}\right)^{2}}\,\]  
and the equations governing the motion of the satellite are written as [61]  
\[\dot{x} =-\mu_{*}\,\frac{\left(x-x_{\alpha}\right)}{r_{\alpha}^{3}}\ -\ \mu_{\beta}\,\frac{\left(x-x_{\beta}\right)}{r_{\beta}^{3}}\ -\ \mu_{\gamma}\,\frac{\left(x-x_{\gamma}\right)}{r_{\gamma}^{3}}\ +\] \[-\frac{3\,\mu_{*}\,j_{2\alpha}\,r_{\alpha}^{2}}{2r^{5}}\,\left(1- \frac{5z^{2}}{r^{2}}\right)\cdot x\ +A_{x}\]  
\[\dot{y} =-\mu_{*}\,\frac{\left(y-y_{\alpha}\right)}{r_{\alpha}^{3}}\ -\ \mu_{ \beta}\,\frac{\left(y-y_{\beta}\right)}{r_{\beta}^{3}}\ -\ \mu_{\gamma}\,\frac{\left(y-y_{\gamma}\right)}{r_{\gamma}^{3}}\ + \tag{36}\] \[-\frac{3\,\mu_{*}\,j_{2\alpha}\,r_{\alpha}^{2}}{2r^{5}}\,\left(1- \frac{5z^{2}}{r^{2}}\right)\cdot y\ +A_{y}\]  
\[\dot{z} =-\mu_{*}\,\frac{\left(z-z_{\alpha}\right)}{r_{\alpha}^{3}}\ -\ \mu_{ \beta}\,\frac{\left(z-z_{\beta}\right)}{r_{\beta}^{3}}\ -\ \mu_{\gamma}\,\frac{\left(z-z_{\gamma}\right)}{r_{\gamma}^{3}}\ +\] \[-\frac{3\,\mu_{*}\,j_{2\alpha}\,r_{\alpha}^{2}}{2r^{5}}\,\left(3- \frac{5z^{2}}{r^{2}}\right)\cdot z\ +A_{z}\]  
in which \(\mu_{*}=GM_{*}\), \(M_{*}\) is the mass of asteroid \"\(*\)\", \(G=6.674287\times 10^{20}\,\mathrm{km}^{3}\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}\) is the universal gravitational constant and the vector \((A_{x},\,A_{y},\,A_{z})\) stands for the accelerations imparted to the spacecraft, comprising control accelerations and the disturbances that may considered in the model. Figures 11 and 12 depict representative characteristics of the system geometry and dynamics. Figure 11 shows a \"long-lasting\" orbit of a spacecraft (black) with the initial position in direct opposition to the secondary body Beta (the stars mark initial positions). The time span corresponds to 5 orbits of Beta (around 30 days). Figure 12 shows the disrupting case of a spacecraft's orbit with initial position \(\pi/5\) ahead of Beta escaping after a time span of 15 days.