# Rapid Construction of Aerocapture Attainability Sets Using Sequential Convex Programming
## 5 Numerical Demonstrations
### Geometric Structure of the Aerocapture Attainability Sets  
Figure 7a shows the aerocapture attainability set for the initial state given in Table 1, plotted in the state space. The flight-path angle has a wide range for a low terminal velocity. As the terminal velocity increases, the boundary of the attainability set eventually shrinks to point \(A\). Without a lower boundary limit given by Equation (6), the attainability set extends infinitely close to \(\gamma_{f}=0\) and even to \(\gamma_{f}\leq 0\), where the vehicle falls into the atmosphere. The corresponding orbit elements are plotted in Figure 7b. The eccentricity of the terminal orbit is calculated using Keplerian orbital mechanics [28],  
\[e_{f}=\sqrt{\frac{a_{f}-V_{f}^{2}r_{atm}^{2}\cos^{2}\gamma_{f}}{a_{f}}} \tag{30}\]  
The nondimensional semimajor axis is \(a_{f}=1/\left(2/r_{atm}-V_{f}^{2}\right)\), and the periapsis of the terminal orbit has a nondimensional altitude \(h_{p,f}=a_{f}(1-e_{f})-1\). As shown in the figure, the larger the terminal flight-path angle, the larger the eccentricity of the terminal orbit. Besides, the left boundary of the attainability set, the curve \(A\)-\(B\) in Figure 7a, gives a stable upper boundary of the altitude of periapsis, the curve \(A1\)-\(B1\) in Figure 7b.  
The altitude, velocity, and flight-path angle profiles are shown in Figure 8. The state profiles plotted with black and grey colors have terminal states on the left and  
Figure 7: Aerocapture attainability set for the initial state given in Table 1.  
right boundaries of the attainability set in Figure 6(a), respectively, and the state profiles corresponding to the point \(A\) are highlighted with the red color. Figure 7(a) shows that the atmospheric flights with a terminal state on the right boundary of the attainability set have a lower flight altitude in the atmosphere than those on the left boundary. They require a larger change in flight-path angle, and the vehicle must fly at a lower altitude to obtain enough lift to increase \(d\gamma/dt\). Figure 7(b) shows that the time of atmospheric flight for a terminal state on the left boundary of the attainability set is longer than that on the right boundary. Meanwhile, Figure 7(c) indicates that the flight-path angle for a terminal state on the left boundary changes more smoothly during the atmospheric ascent phase than that on the right boundary.  
Note that, in the aforementioned computing environment, it took a total of 7.47 s to calculate the envelope of the attainability set in Figure 7. Specifically, a total of 57 extreme values of points were calculated, and the average calculation time per extreme value was about 0.131 s. That is, it took only about a hundred milliseconds. At the same time, in the process of solving all extreme values, the successive approximation was generally completed between 4-7 steps, and the calculation time of each step was about 0.02-0.03 s, which is equivalent to the time of solving a convex subproblem. The statistical results of the calculation time demonstrate the high efficiency of the proposed method, and it also verifies the feasibility of the method to determine the aerocapture attainability set onboard for an aerocapture mission.