## 2 BIT-3 Thrust Model for Preliminary Mission Analysis
It is therefore a discrete dimensionless control variable, which clearly involves a complication in the definition of the control law during a generic orbital transfer, since the thrust magnitude and the mass consumption of the small spacecraft do not have, in general, a continuous variation in time, as will be discussed later in the paper. Figure 2: Propulsive characteristics \(\{T,P,I_{\text{sp}},C\}\) of the BIT-3 engine unit as a function of the operating level \(\text{Id}=[0,1,2,3,4,5]\). Data retrieved from Ref. [44] and detailed in Table 1.  
From the trajectory design point of view, the performance parameters of the propulsion system to be considered are essentially the thrust magnitude \(T\), which defines the propulsive acceleration value, and the propellant mass flow rate \(\dot{m}_{p}\), on which the instantaneous mass of the spacecraft depends. In missions with a power generation system based on solar panels and a heliocentric scenario--where the available electric power depends on the distance of the spacecraft from the Sun (which is generally variable during the flight)--an additional constraint must be considered: the value of the absorbed electric power, \(P\). This is because the ability to activate a generic operating level depends on the available power. Consequently, a reduced (and slightly modified) version of the actual thrust table will be used in the remainder of the paper. More precisely, Table 3 summarizes the main thrust parameters \(\{T,\dot{m}_{p},P\}\) used in the design of the spacecraft trajectory and in the numerical simulations illustrated in Section 4, as a function of the control parameter \(\mathrm{Id}\). As regards the direction of the thrust vector, which is defined by the unit vector \(\hat{\mathbf{T}}\), the usual working hypothesis used in the design of space trajectories based on the use of solar electric propulsion systems is adopted. In particular, it is assumed that the unit vector \(\hat{\mathbf{T}}\) is not constrained and that, therefore, the direction of the BIT-3-induced thrust vector \(\mathbf{T}\) can be oriented in any direction during the flight of the spacecraft. Therefore, the thrust vector can be simply written as  
\[\mathbf{T}=T\,\hat{\mathbf{T}} \tag{3}\]  
where value of the thrust magnitude \(T=T(\mathrm{Id})\) is indicated in Table 3 as a function of the operating level \(\mathrm{Id}\). From the point of view of spacecraft dynamics, usually the thrust vector \(\mathbf{T}\) is more conveniently expressed in a typical radial-transverse-normal reference frame (RTN) whose unit vectors are defined, as a function of the spacecraft position \(\mathbf{r}\) and velocity \(\mathbf{v}\) vector, as \(\hat{\mathbf{t}}_{\mathrm{R}}\triangleq\mathbf{r}/\|\mathbf{r}\|\), \(\hat{\mathbf{t}}_{\mathrm{N}}\triangleq(\mathbf{r}\times\mathbf{v})/\|\mathbf{r}\times\mathbf{v}\|\), \(\hat{\mathbf{t}}_{\mathrm{T}}\triangleq\hat{\mathbf{t}}_{\mathrm{N}}\times\hat{t}_{ \mathrm{R}}\). In RTN, the thrust vector is rewritten as  
\[\mathbf{T}=T_{R}\,\hat{\mathbf{t}}_{\mathrm{R}}+T_{T}\,\hat{\mathbf{t}}_{\mathrm{T}}+T_{N }\,\hat{\mathbf{t}}_{\mathrm{N}} \tag{4}\]  
where \(T_{R}\), \(T_{T}\), and \(T_{N}\) are the radial, transverse, and normal components of the thrust vector, respectively, with \(\sqrt{T_{R}^{2}+T_{T}^{2}+T_{N}^{2}}\equiv T(\mathrm{Id})\). In the case where the vector \(\hat{\mathbf{T}}\) is not constrained, for a given operating level \(\mathrm{Id}\), the tip of the thrust vector describes a sphere in the RTN frame. Figure 5 shows such a kind of thrust bubble in the case where the BIT-3 engine unit provides the maximum value of \(T\), that is, in the case of \(\mathrm{Id}=5\). Finally, the thrust model needed to describe the dynamics of the spacecraft includes the differential equation for the total mass variation. Assuming for simplicity that during the flight, the propulsion system is always in TM (i.e., we do not consider the possibility of