## 2 BIT-3 Thrust Model for Preliminary Mission Analysis
### Case of Two Engine Units  
In this part of the section, we will analyze the case where the CubeSat hosts two identical engine units which are both based on the BIT-3 RF ion thruster (with gimbal) [33]. The total propellant mass stored on board the spacecraft is therefore \(2\times 1.5\,\mathrm{kg}=3\,\mathrm{kg}\), the mass of the combined propulsion system is \(2\times 2.9\,\mathrm{kg}=5.8\,\mathrm{kg}\), and the total occupied volume is \(2\times 1.6\,\mathrm{U}=3.2\,\mathrm{U}\). The interesting case where there are three engine units can be easily modeled by extending the procedure outlined below, while a scenario with four units appears difficult to reconcile with the typical dimensions and weights of CubeSats designed today for scientific applications [50]. In this situation, let us indicate with the subscript 1 the first of these two identical BIT-3-based engine units, and with the subscript 2 the remaining unit, so that the symbol Id\({}_{\mathcal{D}}=[0,1,2,3,4,5]\) (or Id\({}_{\mathcal{R}}=[0,1,2,3,4,5]\)) indicates the thrust level of the first (or the second) engine unit. Let us also assume that each of the two engine units is independent and that therefore its operating level can be selected independently from that of the other unit. This means, among other things, that the electric power input to the PPU of the generic unit can be selected autonomously during the flight. From the point of view of the preliminary design of the spacecraft transfer trajectory, an important aspect to take into account regarding the effect linked to the presence of two engine units is the possibility of having a certain number of different combinations of total thrust magnitude \(T\) and total propellant mass flow rate \(\dot{m}_{p}\), which obviously correspond to different absorbed (total) electric power levels \(P\). More precisely, in the presence of two independent BIT-3-based engine units, from the point of view of the study of spacecraft dynamics, it is more convenient to consider a sort of virtual propulsion system whose characteristics, in terms of \(\{T,\dot{m}_{p},P\}\), depend on the values chosen for Id\({}_{\mathcal{D}}\) and Id\({}_{\mathcal{R}}\), as sketched in Figure 6.  
Figure 5: Thrust bubble when the operating level is Id \(=5\) and the engine unit gives the maximum thrust magnitude of \(1.1\,\mathrm{mN}\). The operating level \(\mathrm{Id}\) of such a virtual propulsion system is indicated, as a function of the values of \(\mathrm{Id}_{\mathcal{X}}\) and \(\mathrm{Id}_{\mathcal{Z}}\), in Table 4. For example, according to Table 4, the operating level \(\mathrm{Id}=0\) is obtained when \(\mathrm{Id}_{\mathcal{X}}=0\) and \(\mathrm{Id}_{\mathcal{Z}}=0\), so that in this case, bearing in mind the data in Table 3, the total thrust given by the virtual propulsion system is \(T=2\times 0.01\,\mathrm{mN}=0.02\,\mathrm{mN}\), the total propellant mass flow rate is \(\dot{m}_{p}=2\times 50.98\,\mathrm{\,\mu g/s}=101.96\,\mathrm{\,\mu g/s}\), and the total absorbed electric power is \(P=2\times 42\,\mathrm{W}=82\,\mathrm{W}\). On the other hand, the operating level \(\mathrm{Id}=8\) corresponds to a case in which \(\mathrm{Id}_{\mathcal{X}}=3\) and \(\mathrm{Id}_{\mathcal{Z}}=1\). Accordingly, when \(\mathrm{Id}=8\), the total thrust of the virtual engine is \(T=0.89\,\mathrm{nN}+0.66\,\mathrm{nN}=1.55\,\mathrm{mN}\), the total propellant mass flow rate is \(\dot{m}_{p}=52.15\,\mathrm{\,\mu g/s}+52.17\,\mathrm{\,\mu g/s}=104.32\,\mathrm{ \,\mu g/s}\), and the required electric power is \(P=65\,\mathrm{W}+55\,\mathrm{W}=120\,\mathrm{W}\).