## 4 Analytical Method for the Approximation of Flight Time and Initial Costates  
In this section, an analytical approximation of the three unknowns \(\{\tilde{f}_{f},\delta,\lambda_{r}(\tilde{I}_{0})\}\), that is, the flight time and the initial costates, is detailed. In particular, the proposed procedure can be applied when \(\tilde{a}_{m}\ll 1\). This is the important case of a spacecraft equipped with a low-performance (or a very low-performance) propulsion system, which gives a propulsive acceleration whose maximum value is only a (small) fraction of the gravitational acceleration \(\mu/r_{0}^{2}\) along the circular parking orbit. In this scenario, the total flight time is usually very high (a few dozen, or even hundreds, of parking orbit periods), and the knowledge of an accurate guess of the triplet \(\{\tilde{f}_{f},\delta,\lambda_{r}(\tilde{I}_{0})\}\) is a crucial point in reducing the computation time required to solve the associated TPBVP. In this context, the last part of this section describes a method that can be used to rapidly estimate, as a function of the mission scenario characteristics in terms of \(\tilde{a}_{m}\) and \(\tilde{r}_{f}\), the maximum value of \(\tilde{a}_{m}\) compatible with the proposed, analytical approximate procedure. In fact, the proposed method is based on the observation that, when \(\tilde{a}_{m}\ll 1\), the optimal transfer trajectory resembles a tight spiral, i.e., the value of the spacecraft (dimensionless) radial acceleration \(u^{\prime}\) is nearly zero during the entire transfer between the two (coplanar) circular orbits. In this case, the numerical simulations indicate that the direction of the propulsive acceleration vector is substantially transverse (i.e., the radial component of \(\mathbf{a}_{p}\) is very low when compared with the transverse component) so that the thrust angle \(\alpha\) is substantially a constant of motion with  
\[\alpha\simeq \begin{cases}\dfrac{\pi}{2}\,\text{rad}&\text{if}\quad\tilde{r}_{ f}>1\\ -\dfrac{\pi}{2}\,\text{rad}&\text{if}\quad\tilde{r}_{f}<1\end{cases} \quad \quad\text{for}\quad t\in[t_{0},\,t_{f}] \tag{24}\]  
Note that the latter equation can be rewritten in a compact form as  
\[\alpha\simeq\text{sign}\Big{(}\tilde{r}_{f}-1\Big{)}\,\dfrac{\pi}{2}\,\text{ rad}\quad\quad\text{for}\quad t\in[t_{0},\,t_{f}] \tag{25}\]  
where \(\text{sign}(\Box)\in\{-1,\,1\}\) is the signum function. Accordingly, the value of the first of the three unknowns, i.e., the auxiliary angle \(\delta\) which coincides with the initial value of the thrust angle, is approximated through the simple equation  
\[\delta\simeq\text{sign}\Big{(}\tilde{r}_{f}-1\Big{)}\,\dfrac{\pi}{2}\,\text{ rad} \tag{26}\]  
Recall that the value of \(\delta\) allows the two initial costates \(\lambda_{\tilde{a}}(\tilde{I}_{0})\) and \(\lambda_{\tilde{\
u}}(\tilde{I}_{0})\) to be calculated using Equation (23) as a function of the maximum magnitude of the dimensionless propulsive acceleration \(\tilde{a}_{m}\). The second unknown, i.e., the value of the dimensionless flight time \(\tilde{I}_{f}\), can be approximated using a simplified form of the spacecraft dynamics. In fact, bearing in mind that \(\tilde{a}_{p}=\tilde{a}_{m}\) during the entire transfer (see first of Equation (12)), the last of Equation (6) gives  
\[\tilde{I}^{\prime}=\tilde{a}_{m}\,\text{sign}\Big{(}\tilde{r}_{f}-1\Big{)}\,\tilde {r} \tag{27}\]  
where \(\tilde{h}=\tilde{r}\,\tilde{\sigma}\) is the dimensionless magnitude of the spacecraft-specific angular momentum vector \(\mathbf{h}\), i.e., \(\tilde{h}=\|\mathbf{h}\|/\sqrt{\mu\,r_{0}}\).