# An Integrated Thermal-Electrical Model for Simulations of Battery Behavior in CubeSats
## 2 Methodology
### Irradiance Model  
The net radiation heat rate (\(Q_{r_{w}}\)) expresses the radiation heat exchange of the CubeSat's external surfaces, which depends on the orientation of the surface \(w\). For each surface \(w\), except \(w=7\), there is incoming radiation (\(Q_{r_{w_{n}}}\)) given by the direct absorbed radiation from the Sun (\(Q_{\text{Sun}}\)), radiation absorbed from the reflections off the Earth's surface called albedo (\(Q_{\text{alb}}\)), and absorbed infrared radiation emitted by the Earth (\(Q_{\text{IR}}\)) [27]. There is still energy emission (\(Q_{r_{w_{\text{out}}}}\)) from the satellite's surfaces to outer space (\(Q_{\text{sat}\rightarrow\text{spa}}\)), as shown in:  
\[Q_{r_{w}}(t)=Q_{r_{w_{n}}}(t)-Q_{r_{w_{\text{out}}}}(t) \tag{15}\]  
Or:  
\[Q_{r_{w_{n}}}(t)=Q_{\text{Sun}_{w}}(t)+Q_{\text{alb}_{w}}(t)+Q_{\text{IR}_{w}} (t) \tag{16}\]  
\[Q_{r_{w_{\text{out}}}}(t)=Q_{\text{sat}_{w}\rightarrow\text{spa}}(t) \tag{17}\]  
Figure 2: Attitude of the satellite. Equation (15) is a boundary condition of the thermal problem, which requires thermal parameters and also variables from the orbit and attitude models. The terms in Equations (16) and (17) are:  
**Solar radiation:** It is the primary heat source for satellites in Low Earth Orbit (LEO). In this work, the solar flux radiation (\(Q_{\text{Sun}}^{\prime\prime}\)) is constant, and the solar rays are parallel. Equation (18) estimates the total radiation absorbed from the Sun. \[Q_{\text{Sun}_{w}}^{\prime\prime}(t)=\alpha_{w}A_{\tau_{w}}Q_{s_{w}}^{\prime \prime}, \tag{18}\]  
where:  
\[Q_{s_{w}}^{\prime\prime}(t)=F_{w\to\text{Sun}}(t)\xi(t)Q_{\text{Sun}}^{\prime \prime} \tag{19}\]  
In these equations, \(\alpha_{w}\) is the surface's absorptivity of the satellite, \(A_{\tau_{w}}\) is the external area of each side \(w\), \(F_{w\to\text{Sun}}\) is the view factor between the surface \(w\) and the Sun, and \(\xi\) is a step function to account for the eclipse, so it assumes the value 0 when the satellite is under the shadow of the Earth and 1 when it is outside. Both \(F_{w\to\text{Sun}}(t)\) and \(\xi(t)\) are functions of the attitude and orbit models, respectively. The view factor between the surface \(w\) and the Sun is calculated by Equation (20), which is basically the projection of that surface towards the Sun, while \(\xi\) is estimated by Equation (21), based on Figure 3.  
\[F_{w\to\text{Sun}}(t)=|\min\left(\vec{\mu}_{w}(t).\vec{S},\,0\right)| \tag{20}\]  
\[\xi(t)= \begin{cases}0,&\cos^{-1}\left(\frac{\vec{R}_{\text{sat}}(t).\vec{S}}{ |\vec{R}_{\text{sat}}(t)|.|\vec{S}|}\right)\geq\tan^{-1}\left(\frac{R_{\text{ e}}}{R_{e}+h}\right)+\frac{\pi}{2}\\ 1,&\text{otherwise}\end{cases}  \tag{21}\]  
**Albedo radiation:** defined by Equation (22), it is the amount of radiation from the Sun that hits the Earth's surface and reflects back to outer space. \[Q_{\text{alb}_{w}}(t)=a\alpha_{w}A_{\tau_{w}}F_{w\to e}(t)U(t)Q_{\text{Sun}}^{ \prime\prime} \tag{22}\]  
The term \(a\) is the albedo coefficient, which is the solar radiation reflected by the Earth's surface, and \(F_{w\to e}(t)\) is the view factor between the satellite's surface \(w\) and the Earth's surface. The parameter \(U(t)\) projects the Earth's surface towards the solar radiation, therefore its peak coincides with the Equator's line and the Sun at the zenith, calculated through Equation (23). \[U(t)= \begin{cases}\left|\frac{\vec{R}_{\text{sat}}(t).\vec{S}}{|\vec{R}_{ \text{sat}}(t)|}\right|,&\vec{R}_{\text{sat}}(t).\vec{S}>0\\ 0,&\text{otherwise}\end{cases}  \tag{23}\]  
To estimate the view factor \(F_{w\to e}(t)\), the Earth is idealized as a perfectly spherical emitter, much bigger than the satellite's dimensions [28]. Therefore, it depends on the satellite and Earth's distance, and so do the angles between each satellite and the Earth's surface. Figure 4 illustrates these parameters. Figure 3: Parameters to calculate the eclipse.