## 3 A Few Notes on the Mathematical Model
### Simplified CubeSat Mass Model  
The study of the transfer trajectory in a heliocentric mission scenario requires a definition of the CubeSat's inertia characteristics, which must indicate both the mass \(m_{0}\) of the vehicle at the beginning of the mission (time \(t_{0}\triangleq 0\)) and the total mass \(m_{p}\) of the propellant stored on board the CubeSat. The value of the pair \(\{m_{0},m_{p}\}\) imposes a constraint on the minimum final mass of the spacecraft, i.e., the total mass of the vehicle at the end of the transfer, which must clearly be greater than the difference \(m_{0}-m_{p}\). From the point of view of the spacecraft mass distribution, we resort to the simplified analytical mathematical model recently proposed by the author in Ref. [41] to schematize the inertial characteristics of an interplanetary CubeSat equipped with an array of BIT-3-based engine units.  
In that simplified model, the spacecraft mass was divided into five components: the dry mass of the engine units \(m_{e}\); the total mass of the propellant \(m_{p}\); the mass of the power generation subsystem \(m_{\rm pow}\); the mass of the payload \(m_{\rm pay}\); and the mass of the remaining subsystems \(m_{\rm orth}\), including the structure and a suitable contingency term. As for the payload, a series of COTS components with a total mass of \(m_{\rm pay}=4\,\)kg are considered, while for the power subsystem, high-efficiency solar arrays such as those made available by MMA [42] are used, with a power-to-mass ratio of \(133\,\)W/kg. In particular, the power subsystem was sized to provide the maximum electric power required by the propulsion system and an additional \(25\,\)W (required for the operation of the other subsystems, including the scientific payload) up to a solar distance of \(1.1\,\)AU. Therefore, the electric power \(P_{\oplus}\) given by the subsystem at a solar distance of \(1\,\)AU is approximatively \(P_{\oplus}=(75\,\)W\(+25\,\)W\()\times 1.1^{2}\simeq 121\,\)W for the case of a single engine unit, while one has \(P_{\oplus}=(2\times 75\,\)W\(+25\,\)W\()\times 1.1^{2}\simeq 212\,\)W when there are two engine units; see also the power requirements of a BIT-3-based engine unit summarized in Table 1. According to the procedure described in Ref. [41], when the CubeSat hosts a single engine unit, one has \(m_{0}\simeq 13\,\)kg and \(m_{p}=1.5\,\)kg, so the final mass of the vehicle must be greater than \(11.5\,\)kg. On the other hand, when there are two engine units, the spacecraft initial mass is \(m_{0}\simeq 19\,\)kg and the propellant mass is \(m_{p}=3\,\)kg [41], with a CubeSat's final mass greater than roughly \(16\,\)kg. A donut chart of the CubeSat's mass distribution is reported in Figure 8 as a function of the number of engine units. Note that the mass \(m_{\rm orth}\) is, by assumption, a fixed percentage (i.e., 40%) of the total mass \(m_{0}\).  
Figure 8: Mass distribution of the CubeSat with a payload mass of \(4\,\)kg as a function of the number of engine units. **(a)** Case of a single engine unit with \(m_{0}\simeq 13\,\)kg; **(b)** case of two engine units with \(m_{0}\simeq 19\,\)kg.