# Circular Orbit Flip Trajectories Generated by E-Sail
## 2 Problem Description and Mathematical Approach  
Consider a heliocentric, two-dimensional mission scenario in which a spacecraft initially (i.e., at time \(t=t_{0}\triangleq 0\)) follows a circular orbit of assigned radius \(r_{0}\) around the Sun. Note that the mathematical model for an elliptic parking orbit can be obtained by paralleling the procedure described in the recent Ref. [27]. The spacecraft primary propulsion system is an E-sail with an assigned value of characteristic acceleration \(a_{c}\), which is defined as the maximum magnitude of the E-sail propulsive acceleration \(\mathbf{a}\) when the distance from the Sun is \(r_{\oplus}\triangleq 1\) au. In particular, according to the mathematical model proposed by [PERSON] et al. [28], the analytical expression of \(\mathbf{a}\) is  
\[\mathbf{a}=\tau\,\frac{a_{c}}{2}\left(\frac{r_{\oplus}}{r}\right)\left[\mathbf{\hat{r} }+\left(\mathbf{\hat{n}}\cdot\mathbf{\hat{r}}\right)\mathbf{\hat{n}}\right] \tag{1}\]  
where \(\tau\in\{0,\,1\}\) is a dimensionless switching parameter that models the thruster operating mode (either on, when \(\tau=1\), or off, when \(\tau=0\)), \(\mathbf{\hat{r}}\) is the Sun-spacecraft (or radial) unit vector, \(r\) is the Sun-spacecraft distance, and \(\mathbf{\hat{n}}\) is the unit vector normal to the sail nominal plane (i.e., the plane that nominally contains the conducting tethers) such that \(\mathbf{\hat{r}}\cdot\mathbf{\hat{n}}>0\). In particular, if the E-sail attitude is fixed with respect to an orbital reference frame, then the magnitude of \(\mathbf{a}\) is inversely proportional to \(r\)[15; 29]. In the context of this mission scenario it is useful to introduce the reference propulsive acceleration \(a_{0}\), defined as the maximum magnitude of \(\mathbf{a}\) at distance \(r_{0}\) from the Sun. Equivalently, \(a_{0}\) is the maximum magnitude of the propulsive acceleration in the circular parking orbit. Recall that the maximum acceleration is obtained in a Sun-facing attitude condition [30] (that is, when \(\mathbf{\hat{n}}\equiv\mathbf{\hat{r}}\)) and \(\tau=1\). From the definition of \(a_{0}\), it follows that  
\[a_{0}=a_{c}\left(\frac{r_{\oplus}}{r_{0}}\right) \tag{2}\]from which the design parameter  
\[\beta\triangleq\frac{a_{0}}{\mu_{\odot}/r_{0}^{2}}=a_{c}\left(\frac{r_{0}\,r_{ \oplus}}{\mu_{\odot}}\right) \tag{3}\]  
can be used as a dimensionless value of reference propulsive acceleration, being defined as the ratio of \(a_{0}\) to the Sun's gravitational acceleration at a distance equal to \(r_{0}\). Taking Equations (2) and (3) into account, the propulsive acceleration of Equation (1) becomes  
\[\mathbf{a}=\tau\,\frac{\beta\,\mu_{\odot}}{2\,r_{0}\,r}\,[\mathbf{\hat{r}}+(\mathbf{\hat{n} }\cdot\mathbf{\hat{r}})\,\mathbf{\hat{n}}]. \tag{4}\]  
The last equation suggests that the propulsive acceleration may be written in dimensionless form by using the ratio \(\mu_{\odot}/r_{0}^{2}\) as a sort of acceleration unit, viz. \[\widetilde{\mathbf{a}}\triangleq\frac{\mathbf{a}}{\mu_{\odot}/r_{0}^{2}}=\tau\,\frac {\beta}{2\,\widetilde{r}}\,[\mathbf{\hat{r}}+(\mathbf{\hat{n}}\cdot\mathbf{\hat{r}})\,\mathbf{ \hat{n}}] \tag{5}\]  
where \(\widetilde{r}\triangleq r/r_{0}\) is the dimensionless radial distance. The thrust vector model of Equation (5) will be used to describe the spacecraft dynamics, as detailed in the next subsection. To that end, the components of \(\widetilde{\mathbf{a}}\) are calculated in a polar reference frame \(\mathcal{T}(O;r,\theta)\), see Figure 1, in which the origin \(O\) coincides with the Sun's center of mass and \(\theta\) represents the (time-variable) polar angle, that is, the angle measured counterclockwise from the Sun-spacecraft line at \(t=t_{0}\) to the current Sun-spacecraft line.