# Three-Dimensional Guidance Laws for Spacecraft Propelled by a SWIFT Propulsion System
## Appendix A SWIFT Thrust Vector Model and Spacecraft Dynamics for 2D Mission Scenarios  
In this appendix, we briefly summarize the thrust vector model proposed in Ref. [9] for the study of a 2D heliocentric mission scenario. That mathematical model has been used to design the optimal transfer trajectory between two coplanar, Keplerian orbits of assigned characteristics by using a classical indirect approach [46; 47] and the well-known Pontryagin's maximum principle [24; 29; 48]. In particular, the thrust model illustrated in Ref. [9] originates from the recent seminal work by [PERSON] et al. [1] that, in turn, refines the mathematical model of the SWIFT propulsion system concept, which was proposed in an interesting set of conference papers by the same authors [2; 49; 50; 51]. More precisely, the mathematical approach used in Ref. [9] to describe the SWIFT-induced propulsive acceleration vector is focused on the trajectory analysis and optimization. In this sense, the thrust model described in Ref. [9] was designed, according to the physical results illustrated in Ref. [1], with the aim of obtaining a set of analytical relations simple enough to be used in a trajectory optimization software [52; 53; 54], which can be used for a preliminary mission analysis and design. Bearing in mind that the spacecraft total mass is constant, according to [PERSON] et al. [1] and using the results of Ref. [9], in a 2D heliocentric mission scenario, the vector \(\mathbf{a}\) can be written as the (vector) sum of two contributions: (1) the part due to the radial component of the SWIFT-induced force (i.e., a sort of \"drag\" originated by focusing the solar wind stream [1]), which is directed along the Sun-spacecraft unit vector \(\mathbf{\hat{r}}\) and (2) the part due to the acceleration of the collected charged particles out of the vehicle (i.e., a sort of \"thrust\" term, thus borrowing the terminology used to describe the performance of a typical electric thruster), which can be steered within a wider field and is directed along the unit vector \(\mathbf{\hat{t}}\). From a mathematical point of view, and using the nomenclature introduced in Ref. [9], the vector \(\mathbf{a}\) can be written in a compact form as  
\[\mathbf{a}=a_{D_{\oplus}}\left(\frac{r_{\oplus}}{r}\right)^{2}\left(\mathbf{\hat{r}}+ K\mathbf{\hat{t}}\right) \tag{1}\]  
where \(r\) is the Sun-spacecraft distance, \(r_{\oplus}=1\,\mathrm{au}\) is a reference distance that roughly coincides with the semimajor axis of the Earth's orbit around the Sun, \(a_{D_{\oplus}}\) is the reference acceleration defined as the value of the magnitude of the \"drag\" part of the propulsive acceleration vector when \(r=r_{\oplus}\), and \(K>0\) is a dimensionless parameter, which coincides with the ratio of the exhaust velocity of the charged particles stream over the local velocity of the solar wind stream. In particular, the two terms \(\{a_{D_{\oplus}},\,K\}\) should be considered as propulsive system performance parameters [1] so that their values depend on both the characteristics of the SWIFT and the spacecraft mass breakdown. In this regard, Ref. [9] contains a mathematical model that is useful to estimate the values of the pair \(\{a_{D_{\oplus}},\,K\}\) as a function of the geometric, inertial, and propulsive characteristics of the SWIFT-based spacecraft. In the case of a 2D heliocentric mission scenario as that studied in Ref. [9], during the flight, the unit vector \(\mathbf{\hat{t}}\) is assumed to belong to the plane of the initial parking orbit so that the propelled trajectory of the spacecraft is 2D. In this context, the spacecraft dynamics can be studied by using a classical polar reference frame with the origin in the Sun's center of mass, in which \(\mathbf{\hat{\theta}}\) is the transverse unit vector, while the radial unit vector coincides with \(\mathbf{\hat{r}}\). Recall that the spacecraft inertial velocity vector has a positive component along the direction of \(\hat{\mathbf{\theta}}\).