# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 2 Simplified Thrust Model
[26; 27] by using a simplified model for the performance of the power subsystem unit in a typical interplanetary transfer, which is derived from the more refined model proposed by [PERSON] [28; 29]. In this context, Equation (3) of ref. [18] provides a useful polynomial formula that directly expresses the power \(P_{\text{in}}\) absorbed by the electric thruster as a function of the solar distance \(r\in[0.75,\,1.25]\,\text{AU}\). That formula, in fact, bypasses the introduction of a mathematical model for the solar panels performance as a function of both the Sun-spacecraft distance and the flight time [30]. In this case, the simplified function \(P_{\text{in}}=P_{\text{in}}(r)\) proposed by ref. [18], which refines the classical relationship \(P_{\text{in}}\propto r^{-2}\), can be written in a compact form as  
\[P_{\text{in}}= \begin{cases}0&\text{if}\quad P_{\text{in}}^{\star}<P_{\text{ in}_{\text{min}}}\\ P_{\text{in}}^{\star}&\text{if}\quad P_{\text{in}}^{\star}\in[P_{\text{in}_{ \text{min}}},\,P_{\text{in}_{\text{max}}}]\\ P_{\text{in}_{\text{max}}}&\text{if}\quad P_{\text{in}}^{\star}>P_{\text{in}_{ \text{max}}}\end{cases}  \tag{5}\]  
where \(P_{\text{in}}  
regard, the design parameters of that specific CubeSat indicates a constraint in the solar distance in terms of a maximum (or minimum) value of \(r\) equal to 1.25 AU (or 0.75 AU). Accordingly, in order to be consistent with the results of the best-fit procedure used to obtain the analytical relationships illustrated in ref. [18], in the rest of this paper, we will consider the additional constraint regarding the distance of the spacecraft from the Sun in the form  
\[r\in[r_{\text{min}},\,r_{\text{max}}]\qquad\text{with}\quad r_{\text{min}} \triangleq 0.75\,\text{AU}\quad\text{and}\quad r_{\text{max}}\triangleq 1.25\, \text{AU} \tag{8}\]  
Therefore, observing that \(\partial P_{\text{in}}/\partial r<0\) and bearing in mind that  
\[c_{0}+c_{1}\,r_{\text{min}}+c_{2}\,r_{\text{min}}^{2}+c_{3}\,r_{\text{min}}^{ 3}+c_{4}\,r_{\text{min}}^{4}\simeq 168.86\,\text{W}>P_{\text{in}_{\text{max}}} \tag{9}\]  
and  
\[c_{0}+c_{1}\,r_{\text{max}}+c_{2}\,r_{\text{max}}^{2}+c_{3}\,r_{\text{max}}^{ 3}+c_{4}\,r_{\text{max}}^{4}\simeq 69.5\,\text{W}>P_{\text{in}_{\text{min}}} \tag{10}\]  
the expression of \(P_{\text{in}}\) given by Equation (5) is simplified as  
\[P_{\text{in}}= \begin{cases}P_{\text{in}}^{\star}&\text{if}\quad P_{\text{in}} ^{\star}\in[P_{\text{in}_{\text{min}}},\,P_{\text{in}_{\text{max}}}]\\ P_{\text{in}_{\text{max}}}&\text{if}\quad P_{\text{in}}^{\star}>P_{\text{in}_{ \text{max}}}\end{cases}  \tag{11}\]  
The expression of \(P_{\text{in}}=P_{\text{in}}(r)\) given by Equation (11) can be combined with the expressions of \(T_{\text{max}}=T_{\text{max}}(P_{\text{in}})\) and \(I_{\text{sp}}=I_{\text{sp}}(P_{\text{in}})\) given by Equations (1) and (3) to obtain the variation in the pair \(\{T_{\text{max}},\,I_{\text{sp}}\}\) with the Sun-spacecraft distance \(r\). In this regard, Figure 2 shows the functions \(P_{\text{in}}=P_{\text{in}}(r)\), \(T_{\text{max}}=T_{\text{max}}(r)\), and \(I_{\text{sp}}=I_{\text{sp}}(r)\) obtained with the mathematical model previously described. Note that the graphs in Figure 2 are consistent with the curves sketched in Figure 1 of ref. [18]. The analysis of the curves shown in Figure 2 indicates that the derivative of the function \(T_{\text{max}}=T_{\text{max}}(r)\) presents a discontinuity of the first kind when \(r\simeq 0.93\,\text{AU}\). This behavior is due to the value of \(P_{\text{in}}\), which reaches 120 W at that solar distance, according  
Figure 2: Variation in the MTC thrust model characteristics \(\{P_{\text{in}},\,T_{\text{max}},\,I_{\text{sp}}\}\) as a function of the solar distance \(r\), according to the mathematical model proposed in the recent literature by [PERSON] et al. [18]. to Equation (5). This aspect is more evident in Figure 3, which shows the variation with \(r\) in the two derivatives \(\partial T_{\rm max}/\partial r\) and \(\partial I_{\rm sp}/\partial r\).