## 2 The \({}^{3}\)ColStar Satellite Mission
### Preliminary Design
#### 2.3.2 Thermal Subsystem  
The thermal design of a satellite is crucial for ensuring the health and longevity of a spacecraft in Earth's orbit. The objectives of the Thermal Subsystem are to analyze the thermal space environment specific to the CubeSat's orbit, identify potential thermal challenges, and establish survivability and operational temperature limits for every component of the CubeSat [68]. This approach ensures that the CubeSat can withstand the extreme temperature fluctuations experienced in space and maintain optimal operating conditions for its components. In order to conduct a preliminary analysis of the thermal subsystem, a thermal model was developed using the lumped parameter method. For the computation of the radiative heat loads, this model employed an in-house MATLAB algorithm that calculates the solar radiation \(\dot{Q}_{s,j}\), terrestrial infrared radiation \(\dot{Q}_{e,j}\), and albedo radiation \(\dot{Q}_{a,i}\). The method includes a set of in-house-developed MATLAB scripts that allow for the numerical integration of the following equations [69; 70]:  
\[\dot{Q}_{s,i}=\alpha_{i}(\mathbf{\hat{n}}_{i}\cdot\mathbf{\hat{r}}_{\oplus \rightarrow\odot})I_{\odot}A_{i} \tag{1}\]  
\[\dot{Q}_{e,i}=\int_{\mathcal{S}_{e}}\varepsilon_{i}\sigma T_{e}^{4}A_{i}\frac {(\mathbf{\hat{r}}_{\oplus\rightarrow dS}\cdot\mathbf{\hat{r}}_{dS\to sat })(-\mathbf{\hat{r}}_{dS\to sat}\cdot\mathbf{\hat{n}}_{i})}{\pi\| \mathbf{r}_{dS\to sat}\|^{2}}dS \tag{2}\]  
\[\dot{Q}_{a,i}=\int\int_{\mathcal{S}_{a}}\alpha_{i}\gamma_{\oplus}\frac{( \mathbf{\hat{r}}_{\oplus\rightarrow dS}\cdot\mathbf{\hat{r}}_{dS\to sat })(\mathbf{\hat{r}}_{dS\to sat}\cdot\mathbf{\hat{n}}_{i})}{\pi\|\mathbf{ r}_{dS\to sat}\|^{2}}d\dot{Q}_{\oplus}dA_{i} \tag{3}\]  
where  
\[d\dot{Q}_{\oplus}=I_{\odot}(\mathbf{\hat{r}}_{\oplus\rightarrow dS}\cdot \mathbf{\hat{r}}_{\oplus\rightarrow\odot})dS \tag{4}\] \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Mode**} & **Power (W)** & **Energy (Wh)** \\ \hline Standby (Sb) & 0 & 0 \\ \hline Released (R) & 9.00 & 1.58 \\ \hline Pre-detumbling (PD) & 6.11 & 3.04 \\ \hline Detumbling (D) & 6.66 & 4.17 \\ \hline Detumbled (Dd) & 6.66 & 4.17 \\ \hline Basic (Ba) & 7.19 & 4.53 \\ \hline Nominal IoT DL (Nid) & 7.19 & 4.08 \\ \hline Nominal IoT UL (Niu) & 3.43 & 2.77 \\ \hline Nominal SW (Ns) & 4.54 & 5.09 \\ \hline Nominal SW DL (Nsd) & 7.19 & 4.08 \\ \hline Sun Safe (SS) & 1.51 & 3.14 \\ \hline Ground Mode (GM) & 7.02 & 3.25 \\ \hline Survival (Sv) & 6.82 & 4.21 \\ \hline Satellite Off (OFF) & 1.87 & 3.48 \\ \hline Decommissioning (D) & 9.52 & 9.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of maximum power demand by subsystem and modes per time. where \(\alpha_{i}\) and \(\varepsilon_{i}\) are the effective absorptivity and emmisivity of the i-th surface, \(A_{i}\) (\(m^{2}\)) is the area of the i-th surface, \(\gamma_{\oplus}\) is the albedo factor, \(I_{\odot}\) (\(W/m^{2}\)) is the solar constant, \(T_{e}\) (\(K\)) is the mean Earth temperature, and \(dQ\oplus\) (\(W\)) is the power of solar radiation incident on a differential surface element of the Earth \(dS\). \(\mathbf{r}_{\oplus\to sat}\) describes the position of the satellite in space, \(\mathbf{\hat{n}}_{i}\) is a normal unit vector to the \(i\)-th surface of the satellite, \(\mathbf{r}_{\oplus\to dS}\) represents the position vector of a differential surface element of the Earth, and \(\mathbf{\hat{r}}_{\oplus\rightarrow\odot}\) describes the solar vector. The components of \(\mathbf{r}_{dS\to sat}\) can be calculated as \(\mathbf{r}_{dS\to sat}=\mathbf{r}_{\oplus\to sat}-\mathbf{r}_{\oplus\to dS}\). The script includes several routines to calculate the sun's position and transform the satellite's position and orientation from the body coordinate system to a geocentric reference frame. Based on these data, the integration of the power received by each surface is performed.