# Continuous-Thrust Circular Orbit Phasing Optimization of Deep Space CubeSats
## 2 Simplified Thrust Model
The figure also shows that the derivative \(\partial I_{\rm sp}/\partial r\) is a continuous function of \(r\) because one has \(\partial I_{\rm sp}/\partial r\simeq 0\) when \(r\simeq 0.93\,\)AU. In the next section, the MTC optimal transfer trajectory is analyzed using an indirect approach, in which the spacecraft equations of motion and the Euler-Lagrange equations [31; 32] are numerically integrated with a suitable set of initial conditions. In particular, the derivative \(\partial T_{\rm max}/\partial r\) appears in one of the Euler-Lagrange equations. Therefore, in order to facilitate the numerical integration of the (coupled and non-linear) Euler-Lagrange equations, an approximation of the function \(T_{\rm max}=T_{\rm max}(r)\) is now proposed to avoid the presence of the discontinuity in its derivative with respect to \(r\). To this end, the Curve Fitting Toolbox of Matlab [33] is employed to obtain an approximation of \(T_{\rm max}=T_{\rm max}(r)\) by using a rational function in the form  
\[T_{\rm max}=T_{\rm max_{\odot}}\frac{d_{N_{1}}(r/r_{\odot})^{4}+d_{N_{2}}(r/r_ {\odot})^{3}+d_{N_{3}}(r/r_{\odot})^{2}+d_{N_{4}}(r/r_{\odot})+d_{N_{5}}}{(r/r _{\odot})^{3}+d_{D_{1}}(r/r_{\odot})^{2}+d_{D_{2}}(r/r_{\odot})+d_{D_{3}}} \tag{12}\]  
where \(r_{\odot}\triangleq 1\,\)AU is a reference distance, and \(T_{\rm max_{\odot}}\triangleq 1.8897\,\)mN is the actual value of \(T_{\rm max}\) when \(r=r_{\odot}\) [the latter is obtained by using Equation (1)], while \(\{d_{N_{i}},d_{D_{j}}\}\) with \(i\in\{1,2,\ldots,5\}\) and \(j\in\{1,2,3\}\) are dimensionless best-fit coefficients defined as  
\[d_{N_{1}}=-1.6239,\ d_{N_{2}}=6.6115,\ d_{N_{3}}=-9.7377,\ d_{N_{4 }}=6.1927,\ d_{N_{5}}=-1.4378,\\ d_{D_{1}}=-2.4888,\ d_{D_{2}}=2.0463,\ d_{D_{3}}=-0.5527 \tag{13}\]  
Note that Equation (12) gives a derivative \(\partial T_{\rm max}/\partial r\) which is continuous in the range \(r\in[0.75,\,1.25]\,\)AU. Although not strictly necessary, for reasons of consistency of the thrust model, we also use an analytical approximation of the function \(I_{\rm sp}=I_{\rm sp}(r)\) with a rational function similar to that described in Equation (12). In this regard, the approximated expression obtained by using the Curve Fitting Toolbox is  
\[I_{\text{sp}}=I_{\text{sp}_{\oplus}}\,\frac{e_{N_{1}}(r/r_{\oplus})^{4}+e_{N_{2}} (r/r_{\oplus})^{3}+e_{N_{3}}(r/r_{\oplus})^{2}+e_{N_{4}}(r/r_{\oplus})+e_{N_{5}} }{(r/r_{\oplus})^{3}+e_{D_{1}}(r/r_{\oplus})^{2}+e_{D_{2}}(r/r_{\oplus})+e_{D_{3 }}} \tag{14}\]  
where \(I_{\text{sp}_{\oplus}}\triangleq 3022.6\,\text{s}\) is the actual value of \(I_{\text{sp}}\) when \(r=r_{\oplus}\), which is obtained from Equation (3), and \(\{e_{N_{i}},e_{D_{j}}\}\) is a set of eight dimensionless best-fit coefficients given by  
\[e_{N_{1}}=-0.3556,\ e_{N_{2}}=2.2133,\ e_{N_{3}}=-4.0643,\ e_{N_{4 }}=2.9771,\ e_{N_{5}}=-0.7599,\\ e_{D_{1}}=-2.5148,\ e_{D_{2}}=2.0994,\ e_{D_{3}}=-0.5740 \tag{15}\]  
The accuracy of Equations (12) and (14) in approximating the expressions of the MTC surrogate thrust model given by Equations (1) and (3) is confirmed by Figure 4, which shows that the proposed model substantially gives a thrust (and specific impulse) variation with the solar distance, which is superimposed with the results illustrated in ref. [18]. Accordingly, the two Equations (12) and (14) can be used to analytically describe the electric thruster-induced propulsive acceleration vector \(\mathbf{a}\) of the MTC.