# Design of CubeSat-Based Multi-Regional Positioning Navigation and Timing System in Low Earth Orbit
## 3 Results
### Space Debris Mitigation  
The mission plan does not involve any intentional break-up in orbit. The components susceptible to accidental breaking-up and releasing debris are the two GOMSpace NanoPower BP8 battery packs, the CubeSpace reaction wheels, and the Enpulsion NANO  
Figure 5: Walker Delta 61\({}^{\circ}\): 100/10/1 constellation at 550 km that tracks Germany.  
population system. Thus, their health shall be monitored through an array of sensors that capture their functional and environmental state throughout the mission lifetime. Apart from that, the space debris mitigation analysis is conducted via the DRAMA software [13].  
Each S/C's cross section is calculated for different orientations via the DRAMA-CROC as seen in Figure 6 and Table 3.  
Additionally, the probability of space debris or meteoroid impact was calculated via DRAMA-MIDAS as depicted in Figure 7.  
Since the total number of catastrophic impacts N is known, the probability of catastrophic impacts can be calculated via \(P_{cat}=1-e^{-N}\)[14]. The results are shown in Table 4. In all cases, it is lower than the critical value of \(10^{-4}\), as stated in ESA Space Debris Mitigation Requirements document, and is, therefore, considered acceptable [15].  
Furthermore, an orbital decay analysis was performed using DRAMA-OSCAR to verify compliance with the 5-year deorbit requirement.  
The S/Cs shall have station-keeping capabilities; thus, their original altitude of 550 km shall be maintained for the full 3-year nominal operations phase. Based on the \(\Delta\)V budget, in the worst case, 70 m/s will be allocated to lower the S/Cs's altitude before allowing them to passively decay. It is assumed that a single retrograde burn of 70 m/s is performed in a random point of the initial orbit.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Object Diameter** & **1 mm to 1 cm** & **Above 1 cm** \\ \hline Probability of catastrophic impact during 2029 & \(7.63\times 10^{-9}\) & \(5.09\times 10^{-6}\) \\ Probability of catastrophic impact during 2030 & \(1.06\times 10^{-8}\) & \(7.11\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Probability of catastrophic impact before the end of the mission.  
Figure 6: CROC 3D model.  
Then, via the Vis-viva equation, it is possible to calculate the new orbit after the maneuver.  
\[u^{2}=GM\bigg{(}\frac{2}{r}-\frac{1}{a}\bigg{)}\]  
where  
\(u\) is the orbital velocity of the satellite;  
\(r\) is the distance between the satellite and the Earth's center;  
\(a\) is the length of the semi-major axis;  
\(G\) is the gravitational constant (\(6.67\times 10^{-11}\) N\(\cdot\)m\({}^{2}\cdot\)kg\({}^{-2}\));  
\(M\) is the Earth's mass (\(5.972\times 10^{24}\) kg).  
The new semi major axis is calculated to be \(a^{\prime}=6811\) km, while the eccentricity is \(e=0.017\). By having these elements to be used as inputs in the DRAMA-OSCAR tool, the orbital decay analysis can be performed. As illustrated in Figure 8, it results that the S/Cs shall deorbit in 1.7 years.  
Figure 7: Number of impacts (**a**) and catastrophic impacts (**b**) vs. time for objects between 1 mm and 1 cm and number of impacts (**c**) and catastrophic impacts (**d**) vs. time for objects above 1 cm.  
Finally, the S/C's re-entry is modeled through DRAMA-SARA. The S/C was modeled as a solid 8U cuboid (0.8 \(\times\) 0.2 \(\times\) 0.1 m) made entirely of aluminum (DRAMA-AA7075 material from DRAMA's library). As depicted in Figure 9, the analysis concluded that the S/C is completely disintegrated in less than 1 h after atmospheric re-entry and at the altitude of 70 km.