# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 2 Simplified Thrust Model  
This section illustrates the simplified MTC thrust model that has been used to optimize the CubeSat transfer trajectory in two deep space mission scenarios discussed later in this paper; see Sections 3.2 and 4. This specific thrust model can be considered a variation of the elegant surrogate model proposed by [PERSON] et al. [18] in their interesting work regarding the selection of the possible set of target asteroids during the preliminary design of the M-ARGO interplanetary mission [25]. In that paper [18], in fact, the authors consider the actual characteristics (at that design stage, at least) of the M-ARGO CubeSat in order to derive a simple mathematical model of the spacecraft thrust vector, which can be used to perform the trajectory analysis with a reduced computational effort. In particular, the mathematical model presented in ref. [18] considers a CubeSat whose total mass at the beginning of the heliocentric transfer towards the target asteroid is \(m_{0}=22.6\,\mathrm{kg}\), of which \(m_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{ \mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{ \mathrm{p}_{\mathrm{p}}}}}}}}}}}}}}=2.8\,\mathrm{kg}\) is the maximum propellant mass stored onboard. This value, which gives a propellant mass fraction of 12.4%, can be used to complete the interplanetary transfer with a flight time less than or equal to a maximum value of \(\Delta t_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{ \mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}_{\mathrm{p}}_{\mathrm{p}_{ \mathrm{p}}}}}}}}}}}}=3\,\mathrm{years}\). The continuous-thrust electric propulsion system modeled in ref. [18  
where \(\{a_{0},\,a_{1}\}\) are two best-fit coefficients defined as (see the second column in Table 3 of ref. [18])  
\[a_{0}=-0.7253\,\text{mN}\quad,\quad a_{1}=0.02481\,\frac{\text{mN}}{\text{W}} \tag{2}\]  
Note that \(a_{0}+a_{1}\,\text{H}_{\text{in}_{\text{max}}}\simeq 2.25\,\text{mN}\) and, according to Equation (1), one has \(T_{\text{max}}\in[0,\,2.25]\,\text{mN}\). A similar approach is used to describe the variation in the specific impulse \(I_{\text{sp}}\) with \(P_{\text{in}}\), which is modeled by the three-order polynomial function  
\[I_{\text{sp}}= \begin{cases}b_{0}+b_{1}\,P_{\text{in}}+b_{2}\,P_{\text{in}}^{2 }+b_{3}\,P_{\text{in}}^{3}&\text{if}\quad P_{\text{in}}\in[P_{\text{in}_{\text {min}}},\,P_{\text{in}_{\text{max}}}]\\ b_{0}+b_{1}\,P_{\text{in}_{\text{max}}}+b_{2}\,P_{\text{in}_{\text{max}}}^{2} +b_{3}\,P_{\text{in}_{\text{max}}}^{3}&\text{if}\quad P_{\text{in}}>P_{\text {in}_{\text{max}}}\end{cases}  \tag{3}\]  
where \(\{b_{0},\,b_{1},\,b_{2},\,b_{3}\}\) are best-fit (dimensional) coefficients defined as  
\[b_{0}=2652\,\text{s}\quad,\quad b_{1}=-18.123\,\frac{\text{s}}{\text{W}}\quad,\quad b_{2}=0.3887\,\frac{\text{s}}{\text{W}^{2}}\quad,\quad b_{3}=-0.00174\, \frac{\text{s}}{\text{W}^{3}} \tag{4}\]  
Note that the case of \(P_{\text{in}}<P_{\text{in}_{\text{min}}}\) is not considered in Equation (3) because that condition gives a zero value of the maximum thrust magnitude \(T_{\text{max}}\) as described by the first row of Equation (1). The condition \(T_{\text{max}}=0\), which is, for example, consistent with a shut-down electric thruster, gives a zero value of the propellant mass flow rate. Moreover, from Equation (3), observing that \(\partial I_{\text{sp}}/\partial P_{\text{in}}>0\) for \(P_{\text{in}}\in[P_{\text{in}_{\text{min}}},\,P_{\text{in}_{\text{max}}}]\) and bearing in mind that \(b_{0}+b_{1}\,P_{\text{in}_{\text{max}}}+b_{2}\,P_{\text{in}_{\text{max}}}^{2 }+b_{3}\,P_{\text{in}_{\text{max}}}^{3}\simeq 3068\,\text{s}\) during the entire interplanetary transfer. The thruster input power \(P_{\text{in}}\) depends on the spacecraft solar panels power output that, in its turn, can be approximately considered a function of the Sun-spacecraft distance \(r\) as analyzed in refs.