# Solar Sail-Based Mars-Synchronous Displaced Orbits for Remote Sensing Applications
## 2 Mathematical Preliminaries
### Sail Performance Required to Maintain a Circular DNKO  
Consider a heliocentric mission scenario in which a solar sail-based spacecraft traces a DNKO designed to obtain continuous observation of Mars high-latitude regions. With reference to the scheme in Figure 1, we introduce a perifocal reference system \(\mathcal{T}_{O}(O;\,p,\,q,\,w)\), where \(O\) is the center of mass of the Sun, axis \(p\) is aligned with the eccentricity vector of Mars heliocentric orbit (omitted in the figure for simplicity), and axis \(w\) is parallel to the angular momentum of Mars orbit. Therefore, the plane \((p,q)\) coincides with the orbital plane of Mars in its motion around the Sun. The heliocentric DNKO to be maintained is a circular trajectory of constant radius \(\rho\), which is centered on the point \(O^{\prime}\) belonging to axis \(w\); see Figure 1. The plane of the displaced orbit (i.e., the green disk in the previous figure) is parallel to the orbital plane of Mars (i.e., the gray disk in the figure), and the distance between the two planes is denoted by \(H\), which is positive (or negative) if \(O^{\prime}\) belongs to the positive (or negative) semiaxis \(w\). Furthermore, the solar sail-based spacecraft travels along the circular DNKO at a constant angular velocity \(\omega\), which corresponds to the mean motion of Mars around the Sun, that is, \(\omega\triangleq\sqrt{\mu_{\odot}/a^{3}}\simeq 0.5240\,\mathrm{deg}/\mathrm{ day}\), where \(a\simeq 1.5237\,\mathrm{au}\) is the semimajor axis of Mars heliocentric orbit. In this way, the period of revolution of the spacecraft matches that of Mars, thus allowing remote sensing applications of the high-latitude regions of the Red Planet to be carried out. For an assigned value of angular velocity \(\omega\), a circular DNKO may also be equivalently characterized by the elevation angle \(\gamma\), that is, the angle between the Sun-spacecraft line and the orbital plane of Mars, and the Sun-spacecraft distance \(r\), as it is sketched in Figure 1. Note that the conversion from the pair \(\{r,\,\gamma\}\) to the pair \(\{\rho,\,H\}\) is given by the two simple equations  
\[\rho=r\,\cos\gamma\quad,\quad H=r\,\sin\gamma \tag{3}\]  
while the reverse conversion is given by  
\[r=\sqrt{\rho^{2}+H^{2}}\quad,\quad\gamma=\arctan\biggl{(}\frac{H}{\rho}\biggr{)} \tag{4}\]  
The heliocentric motion of the solar sail-based spacecraft is analyzed in a spherical reference frame \(\mathcal{T}_{S}(O;\,r,\,\theta,\,\gamma)\), with the origin \(S\) in the spacecraft center of mass and unit vectors \(\{\hat{\mathbf{p}}_{r},\,\hat{\mathbf{p}}_{\theta},\,\hat{\mathbf{p}}_{\gamma}\}\) defined as  
\[\hat{\mathbf{p}}_{r}\triangleq\frac{\mathbf{r}}{r}\quad,\qquad\hat{\mathbf{p}}_{\theta} \triangleq\frac{\hat{\mathbf{k}}\times\hat{\mathbf{p}}}{\|\hat{\mathbf{k}}\times\hat\mathbf{{p}}\|}\quad,\qquad\hat{\mathbf{p}}_{\gamma}\triangleq\hat{\mathbf{p}}_{r}\times\hat{ \mathbf{p}}_{\theta} \tag{5}\]  
where \(\mathbf{r}\) is the Sun-spacecraft position vector and \(\hat{\mathbf{k}}\) is the unit vector of Mars orbital angular momentum, that is, the direction of \(\hat{\mathbf{k}}\) is perpendicular to the orbital plane of Mars. In particular, the position of the spacecraft is determined by three parameters, namely, the radial distance \(r\), the azimuthal angle \(\theta\in[0,\,2\,\pi)\) rad, and the elevation angle \(\gamma\in[-\pi/2,\,\pi/2]\) rad. With reference to Figure 2, \(\theta\) is the angle (measured counterclockwise) between \(p\) and the projection of vector \(\mathbf{r}\) onto the \((p,q)\) plane, while \(\gamma\) is the angle between the direction of \(\mathbf{r}\) and the \((p,q)\) plane. Note that the sign of \(\gamma\) corresponds to the (conventional) sign of the DNKO displacement \(H\); compare Figures 1 and 2. Figure 1: Conceptual scheme of a heliocentric, circular DNKO with a displacement equal to \(H\) with respect to the orbital plane of Mars (the gray disk in the sketch).