# Rapid Construction of Aerocapture Attainability Sets Using Sequential Convex Programming
## 2 Formulation of Aerocapture Attainability Set  
The main phase of an aero capture process is the planetary atmospheric flight. For an unpowered atmospheric flight, regardless of the transversal modulation and modeling in the nonrotating planet frame, the vehicle state at any time \(t\) during the aero capture is defined as \(\mathbf{x}(t)=[r,V,\gamma]^{\mathrm{T}}\), where \(r\), \(V\), and \(\gamma\) are the radial distance, velocity, and flight-path  
Figure 1: Potential missions achieved by aerocapture. angle, respectively. The motion of the vehicle is described by the nondimensional dynamic differential equations [28]  
\[\frac{dr}{dt}=V\sin\gamma \tag{1}\]  
\[\frac{dV}{dt}=-D-\frac{\sin\gamma}{r^{2}} \tag{2}\]  
\[\frac{d\gamma}{dt}=\frac{L\cos\sigma}{V}+(\frac{V^{2}}{r}-\frac{1}{r^{2}})\frac {\cos\gamma}{V} \tag{3}\]  
The radial distance \(r\) is normalized by the radius of planet \(R_{M}\), the relative velocity \(V\) is normalized by \(\sqrt{\mu_{M}/R_{M}}\), and the nondimensional time \(t\) is normalized by \(\sqrt{R_{M}^{3}/\mu_{M}}\), where \(\mu_{M}\) is the gravitational parameter of the planet. The bank angle \(\sigma\) is the only control variable. The lift and drag acceleration \(L\) and \(D\) are normalized by \(g_{M}=\mu_{M}/R_{M}^{2}\),  
\[\left\{ \begin{aligned} L&=0.5\rho V^{2}R_{M}SC_{L}/m\\ D&=0.5\rho V^{2}R_{M}SC_{D}/m\end{aligned} \right. \tag{4}\]  
where \(m\) represents the mass of the vehicle, \(S\) indicates the dimensional reference area, \(C_{L}\) and \(C_{D}\) represent the lift coefficient and drag coefficient, respectively, and \(\rho=\rho_{0}\exp(-R_{M}h/h_{s})\) is the dimensional atmospheric density, in which \(\rho_{0}\) is the dimensional atmospheric density at \(R_{M}\), \(h_{s}\) is the atmospheric density coefficient, and \(h\) is the nondimensional altitude. Given the initial entry state of the atmospheric flight \(\mathbf{x}(t_{0})=[r_{0},V_{0},\gamma_{0}]\), all the terminal states of the atmospheric flight form the attainability set of the aerocapture,  
\[\mathcal{A}=\Big{\{}\mathbf{x}^{i}(t_{f})|\mathbf{x}(t_{0})\,\Big{\}}, \tag{5}\]  
where \(\mathbf{x}_{i}(t_{f})=[r_{f}^{i},V_{f}^{i},\gamma_{f}^{i}]^{\mathrm{T}}\) is the terminal state of the \(i\)-th atmospheric flight trajectory. Note that as the height of atmosphere \(h_{atm}\) is usually a constant for a given planet, both \(r_{0}\) and \(r_{f}\) are constant and equal to the radius of atmosphere \(r_{atm}\), which is the height of atmosphere plus the radius of planet \(R_{M}\). The terminal states are then distinguished only by \(V_{f}\) and \(\gamma_{f}\). Additionally, note that as an infinite small change in bank angle \(\sigma\) leads to a sufficiently small difference in terminal state, the aerocapture attainability set \(\mathcal{A}\) is a simply connected region in the \(V_{f}-\gamma_{f}\) space. An illustration of an aerocapture attainability set is shown in Figure 2, where the boundary of the attainability set is denoted as \(\mathcal{O}\). In this paper, the attainability set consists of only the terminal states of the vehicle exiting the atmosphere after aerocapture. If the vehicle drops into the atmosphere and eventually collides with the planet, the terminal state is excluded  
Figure 2: Illustration of the aerocapture attainability set in state space. because the aerocapture failed. The demarcation of a successful and a failed aerocapture is \(\gamma_{f}=0\), which is delineated by the line \(D\)-\(E\) in Figure 3a. Given that the target orbit after aerocapture is far from the atmosphere, a minimum altitude of periapsis \(h_{a,\min}>h_{atm}\) is also constrained. Given the terminal state \(x(t_{f})\) of atmospheric flight, the constrain can be derived from the general energy equation of the elliptical orbit [28],  
\[\frac{V_{f}^{2}}{2}-\frac{\mu_{M}}{r_{atm}}+\frac{\mu_{M}}{2}\Bigg{(}\frac{2}{r _{a,\min}}-\frac{r_{atm}^{2}V_{f}^{2}\cos^{2}\gamma_{f}}{\mu_{M}r^{2}{}_{a,\min} }\Bigg{)}=0 \tag{6}\]  
where \(r_{a,\min}=h_{a,\min}+R_{M}\) is the radial distance of periapsis.