# Robust Near-Optimal Aerocapture Guidance Method Based on Saturation Function
## 2 Aerocapture Modeling and Maneuver Description
### Impulsive Orbital Maneuvers  
At the entry interface (EI), the spacecraft enters the atmosphere. After atmospheric flight and energy dissipation, the spacecraft exits the atmosphere. Then in the exo-atmospheric flight phase, one or two impulse maneuvers are required to establish the desired target orbit. Figure 1 shows the complete single-impulse aerocapture maneuver. In this case, the apogee radius of post-atmospheric flight is equal to the radius of the apogee of the target orbit. The single velocity increment \(\Delta V\) required to establish the target orbit is  
\[\Delta V=\sqrt{2\mu}\left(\sqrt{\frac{1}{r_{atgt}}-\frac{1}{r_{atgt}+r_{ptgt}} }-\sqrt{\frac{1}{r_{a0}}-\frac{1}{r_{p0}+r_{a0}}}\right) \tag{13}\]  
where \(r_{atgt}\), \(r_{ptgt}\) are the apogee radius and perigee radius of the target orbit, respectively, and for a single-impulse maneuver, \(r_{a0}=r_{atgt}\); \(r_{a0}\), \(r_{p0}\)are the apogee and perigee radii of the orbit after aerocapture, which are determined by the following formulas:  
\[r_{a0}=a\left(1+\sqrt{1-\frac{V_{\mathrm{exit}}^{2}r_{\mathrm{EI}}^{2}\cos^{2 }\gamma_{\mathrm{exit}}}{\mu a}}\right) \tag{14}\]\[r_{p0}=a\left(1-\sqrt{1-\frac{V_{\text{exit}}^{2}r_{EI}^{2}\text{cos}^{2}\gamma_{ \text{exit}}}{\mu a}}\right) \tag{15}\]  
where \(a\) is the semi-major axis of the orbit after aerocapture, \(r_{\text{EI}}\), \(V_{\text{exit}}\), and \(\gamma_{\text{exit}}\) are the radius, inertial velocity, and flight path angle, respectively, at the atmospheric exit. In addition, \(a\) is determined by the kinetic energy and potential energy at the atmospheric exit:  
\[a=\frac{\mu}{-V_{\text{exit}}^{2}+2\mu/r_{\text{EI}}} \tag{16}\]  
On the other hand, when the post-aerocapture orbit's apogee radius does not match that of the target orbit, it is necessary to carry out a two-impulse maneuver to establish the target orbit. Figure 2 shows the situation that the apogee of the post-aerocapture orbit is lower and higher than that of the target orbit, respectively, where the first impulse (collinear with the velocity at the apogee) raises the apogee to the desired altitude, then after flying half an ellipse, the second impulse raises or lowers the new perigee to the target orbit. The required total velocity increment of the two impulses is  
\[\begin{split}\Delta V&=\sqrt{2\mu}\left(\sqrt{ \frac{1}{r_{a0}}-\frac{1}{r_{ptgt}+r_{a0}}}-\sqrt{\frac{1}{r_{a0}}-\frac{1}{r _{p0}+r_{a0}}}\right)\\ &+\sqrt{2\mu}\left|\sqrt{\frac{1}{r_{ptgt}}-\frac{1}{r_{ptgt}+ r_{atgt}}}-\sqrt{\frac{1}{r_{ptgt}}-\frac{1}{r_{ptgt}+r_{a0}}}\right| \end{split} \tag{17}\]  
If the target orbit is a circular orbit, that is, \(r_{atgt}=r_{ptgt}=r_{tgt}\), then the above formula can be simplified as  
\[\begin{split}\Delta V&=\sqrt{2\mu}\left(\sqrt{ \frac{1}{r_{a0}}-\frac{1}{r_{tgt}+r_{a0}}}-\sqrt{\frac{1}{r_{a0}}-\frac{1}{r _{p0}+r_{a0}}}\right)\\ &+\sqrt{2\mu}\left|\sqrt{\frac{1}{2r_{tgt}}}-\sqrt{\frac{1}{r_{ tgt}}-\frac{1}{r_{tgt}+r_{a0}}}\right|\end{split} \tag{18}\]  
Figure 1: Single-impulse aerocapture maneuver, \(r_{a0}=r_{atgt}\).