## 4 Analytical Method for the Approximation of Flight Time and Initial Costates
Furthermore, according to the third of Equation (6), when \(\cos\alpha\simeq 0\) and \(u^{\prime}\simeq 0\), the dimensionless magnitude \(\tilde{h}\) can be approximated as  
\[\tilde{h}=\tilde{r}\,\tilde{\sigma}\simeq\sqrt{\tilde{r}} \tag{28}\]so that, with the aid of the previous equation, one has \(\bar{r}\simeq\bar{h}^{2}\), and Equation (27) can be easily solved to give an analytical approximation of the dimensionless (minimum) flight time  
\[\bar{t}_{f}=\frac{1}{\bar{a}_{m}\,\text{sign}\Big{(}\bar{r}_{f}-1\Big{)}}\,\left( 1-\frac{1}{\sqrt{\bar{r}_{f}}}\right) \tag{29}\]  
Note that the expression of the minimum flight time given by Equation (29) is consistent with the classical result obtained by [PERSON] et al. [8; 9; 10] using the elegant concept of the accumulated velocity change. The same concept has also been used by the author to evaluate the optimal performance of a solar-sail-based spacecraft in a typical heliocentric mission scenario [21; 22]. The last of the three unknowns required to initialize the associated TPBVP, i.e., the value of the initial costate \(\lambda_{\bar{r}}(\bar{t}_{0})\), can be approximated by firstly enforcing in Equation (15) the initial conditions given by Equation (7), viz. \[\lambda_{\bar{a}}^{\prime}(\bar{t}_{0})=\lambda_{\bar{\varrho}}(\bar{t}_{0})- \lambda_{\bar{r}}(\bar{t}_{0}) \tag{30}\]  
Now, according to Equations (23) and (26), the value of \(\lambda_{\bar{\varrho}}(\bar{t}_{0})\) is given by  
\[\lambda_{\bar{\varrho}}(\bar{t}_{0})=\frac{\text{sign}\Big{(}\bar{r}_{f}-1 \Big{)}}{\bar{a}_{m}} \tag{31}\]  
while \(\lambda_{\bar{a}}\simeq 0\) is substantially constant during the transfer (remember that the direction of the propulsive acceleration vector is essentially transverse during the transfer) so that  
\[\lambda_{\bar{a}}^{\prime}(\bar{t}_{0})=0 \tag{32}\]  
The value of the last unknown \(\lambda_{\bar{r}}(\bar{t}_{0})\) is then obtained by substituting Equations (31) and (32) in Equation (30), and the result is  
\[\lambda_{\bar{r}}(\bar{t}_{0})=\frac{\text{sign}\Big{(}\bar{r}_{f}-1\Big{)}}{ \bar{a}_{m}} \tag{33}\]  
or, equivalently, \(\lambda_{\bar{r}}(\bar{t}_{0})=\lambda_{\bar{\varrho}}(\bar{t}_{0})\). _Maximum Value of \(\bar{a}_{m}\) Compatible with the Proposed Method_  
The approximation of the triplet \(\{\bar{t}_{f},\delta,\lambda_{\bar{r}}(\bar{t}_{0})\}\) given by Equations (26), (29) and (33) is accurate when the optimal transfer trajectory resembles a tight spiral so that the osculating orbit is substantially circular and the radial component of the spacecraft acceleration \(\bar{u}^{\prime}\) is nearly zero. The latter is, in fact, a scenario analyzed in detail by [PERSON] et al. [8]. In particular, reference [8] discusses that the accumulated velocity change \(\Delta V\) defined as  
\[\Delta V=\int_{t_{0}}^{t_{f}}a_{p}\,\text{d}t \tag{34}\]  
in this specific case (i.e., when the osculating orbit is nearly circular) can be approximated through a very simple equation that depends on the radius of both the parking and the target orbit, viz. \[\Delta V=a_{m}\,t_{f}\simeq \begin{cases}\sqrt{\frac{\mu}{r_{0}}}-\sqrt{\frac{ \mu}{r_{f}}}&\text{if}\quad r_{f}>r_{0}\\ \sqrt{\frac{\mu}{r_{f}}}-\sqrt{\frac{\mu}{r_{0}}}&\text{if}\quad r_{f}<r_{0} \end{cases}  \tag{35}\]  
In particular, the latter equation indicates that the accumulated velocity change \(\Delta V\) coincides with the difference between the spacecraft velocity along the target and the parking orbit. Note also that Equation (35) is consistent with the model proposed in this paper; see, for instance, Equation (29). Using the results detailed in reference [8] in terms of plots of the accumulated velocity change \(\Delta V\) as a function of the characteristics of the propulsion system \(a_{m}\) and the radius of the target circular orbit, we obtain that Equation (35) holds (i.e., the approximation of an osculating nearly circular orbit is valid) when the spacecraft completes, during the transfer, a number \(n\) of revolutions around the primary body at least equal to 5.