## 6 Thermal System Design  
### Defining the Thermal Load  
The sun is an extremely powerful body of energy which will not only power the CubeSat's electrical system but will also input radiation into nanosatellites \"boundary conditions\" as shown in Figure 44 below. The following estimates the value of direct solar flux from solar radiation [10]:  
\[T_{sun}=5800\;\mathrm{K}\]  
\[P_{sun}=3.856*10^{26}\]  
The solar radiation intensity at any distance from the Sun can be found using the relationship:  
\[J_{\mathrm{s}}=\frac{P_{sun}}{4*PI*d^{2}}=\frac{3.85*10^{26}}{2.81*10^{23}}=1369 \;\mathrm{Wm}^{-2} \tag{27}\]  
This value of the solar constant ignores atmospheric attenuation. At an orbiting altitude of 600 km this value is assumed to be the same as at Earth's surface. Earth's surface also reflects the incoming solar radiation from the Sun back into the atmosphere, known as Albedo Radiation. The albedo radiation is calculated using the following formula:  
\[\mathrm{J_{a}}=\mathrm{J_{s}}*\mathrm{a}*\mathrm{F} \tag{28}\]  
where a is the albedo, assumed to be 0.34 average for Earth. For open ocean, this value drops to 0.05 and rises to 0.6 for high cloud and icecaps. F is the visibility factor found from the following Figure 45.  
Figure 44: Earths Radiation Loading [22]. Figure 45: Visibility Factor Alterations against Orbit Altitude [10,63]. With an altitude of 600 km and Sun-synchronous orbit the Beta angle of this mission will be 0 degrees. The beta angle is the minimum angle between the orbit plane and the solar vector, which is permitted to vary between \(-90\) to 90 degrees [20]. The visibility factor is 1.05 as this value on Figure 46 above corresponds to an altitude of 600 km and a Beta angle of 0 degrees. The beta angle can be calculated using the following formula (see Figure 46) [10]:  
\[\beta=\arcsin(\cos\delta_{s}*\sin(i)*\sin(\alpha-\alpha_{s})+\sin(\delta_{s})* \cos(i) \tag{29}\]  
where:  
\(\delta_{s}=\) declanation of the sun,  
\(i=\) orbit inclination,  
\(\alpha_{s}\) and \(\alpha\) are the right ascension of the sun and sun CubeSat respectively,  
\(\beta=\) Beta angle. \[J_{a\ average}=1369\;\mathrm{Wm}^{-2}*0.34*1.05=488.733\;\mathrm{Wm}^{-2} \tag{30}\]  
The maximum albedo radiation would occur over high clouds and icecaps:  
\[J_{a\ max}=1369\;\mathrm{Wm}^{-2}*0.6*1.05=862.47\;\mathrm{Wm}^{-2} \tag{31}\]  
The Earth also emits planetary radiation which must be considered during CubeSat thermal design. The Earth emits infra-red radiation. The observed values are as follows:  
\(Earth_{\ Characteristic\ temp}=288\;\mathrm{K}\)  
\(Earth_{average\ flux}=237\;\mathrm{Wm}^{-2}\)  
The value of planetary radiation at any altitude can be found at any altitude per the following formula:  
\[P_{sun}=3.856*10^{26}\]0  
\[P_{sun}=3.856*10^{26}\]1  
\[P_{sun}=3.856*10^{26}\]2 Polar Orbit Launch Configurations [20]. In reality, this value of planetary radiation varies significantly throughout a day as shown on Figure 47 below which presents experimental data of ] values over a 3 h period from a spacecraft in LEO. The intensity of incoming radiation from the Sun varies throughout Earth's elliptical orbit and depends on the weather season on Earth. For example, in the Northern Hemisphere's summer the intensity is 1372 W/m\({}^{2}\) and at the winter solstice the intensity reaches 1472 W/m\({}^{2}\). The intensity varies due to Earth's elliptical orbit around the Sun, meaning the distance between the two bodies constantly alters. To estimate the Intensity of the radiation (solar constant) for any given day of the year the following formula may be implemented:  
\[P_{sun}=3.856*10^{26}\]3  
where \(n\) is equal to the day number of the year, and \(I_{e}\) equals the current solar constant. At an altitude of 600 km the environment is very radiative causing extreme temperature variations to the CubeSat's components, particularly the outer surfaces.