# Solar Sail Transfer Trajectory Design for Comet 29P/Schwassmann-Wachmann 1 Rendezvous
## 2 Mathematical Preliminaries and Models
### Spacecraft Dynamics
Bearing in mind that in an orbit-to-orbit transfer the (initial) spacecraft angular position is left free, one obtains five initial conditions by combining Equation (2) and the data of the first row of Table 1. In that case, and according to [25], the result is given by the following equations  
\[p(t_{0})=p_{0}\triangleq 1.0005\,\mathrm{au}\;,\;f(t_{0})=f_{0} \triangleq-3.5430\times 10^{-3}\;,\;g(t_{0})=g_{0}\triangleq 1.5542\times 10^{-2}\;,\\ h(t_{0})=h_{0}\triangleq-2.4765\times 10^{-5}\;,\;k(t_{0})=k_{0} \triangleq 9.0802\times 10^{-6} \tag{13}\]  
The three components of vector \(\boldsymbol{a}\) which appear in Equation (10) coincide with the components of the propulsive acceleration vector in a typical radial-tangential-normal (RTN) reference frame of unit vectors \(\{\boldsymbol{\hat{t}}_{\boldsymbol{x}},\;\boldsymbol{\hat{t}}_{\boldsymbol {r}},\;\boldsymbol{\hat{s}}_{\boldsymbol{x}}\}\). In particular, the RTN reference frame is sketched in Figure 2, in which \(\boldsymbol{r}\) is the Sun-spacecraft vector and \(\boldsymbol{\hat{r}}\triangleq\boldsymbol{r}/r\) is the radial unit vector, where \(r\) is the Sun-spacecraft distance. Note that \(\hat{\mathbf{t}}_{\rm{n}}\equiv\hat{\mathbf{r}}\), the plane (\(\hat{\mathbf{t}}_{\rm{n}}\), \(\hat{\mathbf{t}}_{\rm{r}}\)) coincides with the plane of the osculating orbit, and the normal unit vector \(\hat{\mathbf{t}}_{\rm{N}}\) is aligned with the spacecraft specific angular momentum vector. The analytical expression of the components of \(\mathbf{a}\) in the RTN reference frame depends on the solar sail thrust model which is adopted to describe the propulsive performance of such a propellantless thruster. In this context, assuming a flat sail, i.e., a solar sail with a rigid and flat reflecting membrane, a common choice in the preliminary trajectory design is to use the optical force model without degradation [31; 32]. The interested reader should note that the solar sail thrust vector can be described through a more refined (and complex) mathematical model. In this sense, the work of [PERSON] [33], which can be considered one of the founders of the photonic solar sail astrodynamics, discusses the effect of wrinkles on the thrust vector characteristics. The optical force model, which is detailed in the classical [PERSON] textbook [34], allows the propulsive acceleration vector to be written, in a compact analytical form, as a function of the radial unit vector \(\hat{\mathbf{r}}\) and the normal unit vector \(\hat{\mathbf{n}}\), which is defined as the unit vector normal to the sail (nominal) plane in the direction opposite to the Sun [18; 34]. Bearing in mind the approach described in [35], the compact expression of the propulsive acceleration vector is  
\[\mathbf{a}=a_{\rm{c}}\left(\frac{r_{\circ}}{r}\right)^{2}\left(\hat{\mathbf{r}}\cdot \hat{\mathbf{n}}\right)\left[b_{1}\,\hat{\mathbf{r}}+(b_{2}\,\hat{\mathbf{r}}\cdot\hat\mathbf{{n}}+b_{3})\hat{\mathbf{n}}\right] \tag{14}\]  
where \(r_{\circ}\triangleq 1\) au is a reference distance, \(a_{\rm{c}}\) is the characteristic acceleration defined as the propulsive acceleration magnitude experienced by a solar sail whose reference plane is oriented perpendicular to the solar rays when \(r=r_{\circ}\), and \(\{b_{1},\,b_{2},\,b_{3}\}\) are dimensionless (normalized) force coefficients where \(b_{1}+b_{2}+b_{3}=1\), whose value depends on the optical characteristics of the reflecting membrane. In particular, assuming a membrane with a highly reflective aluminum-coated front side and a highly emissive chromium-coated back side [36], the values of the normalized force coefficients are reassumed in Table 1, which also details the values of \(\{b_{1},\,b_{2},\,b_{3}\}\) for an ideal force model [25]. Figure 2: Typical RTN reference frame and characteristics of the spacecraft osculating orbit. Note that the term \(a_{c}\) is the common solar sail performance parameter, whose value depends on the characteristics of both the solar sail and the whole spacecraft (including the value of the payload mass and other subsystems).