# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## 4 Case Studies
### Case Study 1: The PQ8 Satellite
#### 4.1.3 Constellation Phase Times  
Figure 10 depicts the phase times for PQ1\({}_{7}\) for various \(\Delta v\) magnitudes with respect to \(\theta_{PQL,initial}\). The plot for the largest \(\Delta v\) magnitude, 0.35 m s\({}^{-1}\), shows the quickest phasing times. Notably, all cases start with a gradual gradient before rapidly increasing to match the gradients of the other plots. The gradual gradient occurs when the reference satellite is the first to enter the low-drag configuration post-disengagement, while the other PQ1 remains in high-drag mode. During this period, the phasing time is not sensitive to changes in \(\theta_{PQL,initial}\), as reflected by the slight gradient. However, after a certain threshold value, \(\theta_{PQL,thresh}\), the gradient increases sharply. Beyond this point, the reference satellite is in high-drag mode immediately following disengagement, with the other PQ1 in low drag, making phasing times highly sensitive to \(\theta_{PQL,initial}\). Additionally, disengagement performed at \(\theta_{PQL,thresh}\) introduces an edge condition in which the acceleration phase is entirely omitted, as the peak separation velocities are reached through the disengagement \(\Delta v\)'s alone and, thus, require only deceleration to achieve the desired relative angular separation. As \(\dot{\theta}_{i}\) is linearly proportional to the disengagement \(\Delta v\), an increase in \(\Delta v\) reduces \(\theta_{PQL,thresh}\). At \(\Delta v\) magnitudes of about 0.4 m s\({}^{-1}\) or greater, the slight gradient would no longer be present. This suggests that increasing the \(\Delta v\) magnitude beyond a certain threshold, \(\Delta v_{thresh}\) is not viable. In this case, \(\Delta v_{thresh}\) would be about 0.4 m s\({}^{-1}\). Comparing the plots indicates that increasing the \(\Delta v\) magnitude while remaining below \(\Delta v_{thresh}\) results in better phasing times in the small-gradient region. This implies that the optimal phase times occur when \(\Delta v=\Delta v_{thresh}\) and that there is a trade-off between \(\Delta v\), phase times, and the range of viable \(\theta_{PQL,initial}\). If the best-case \(\Delta v\) magnitude is used, any delay in the PQ2-to-PQ1 disengagement will cause a significant increase in phase times as the solution shifts into the large-gradient region. Conversely, when \(\Delta v\) is less than \(\Delta v_{thresh}\), a window exists in which delays have little impact on phase times. This window could be leveraged to accommodate necessary delays for disengagement maneuvers, such as collision avoidance or fault handling. Among the \(\Delta v\) magnitudes simulated for this scenario, the 0.35 m s\({}^{-1}\) value provided the best phasing time, approximately 255 days. However, this represents the phasing time  
Figure 9: PQ1 insertion point for \(\Delta v=0.15\) ms\({}^{-1}\). for PQ1\({}_{7}\), with the other satellites reaching their phase angles more quickly, as shown in Figure 11. In each simulation, the total phase time was determined by the phasing time of PQ1\({}_{7}\). This insight could be used to reduce simulation times in future analyses by focusing primarily on the phasing of the leading satellite, \(S_{kmax,n-1}\). To achieve this phase time, perfect attitude control is assumed. In practice, Alba Orbital has demonstrated that their Unicorn-2 PocketQube can attain a pointing accuracy of approximately 5\({}^{\circ}\) (\"Unicorn-2 Features and Specifications,\" Alba Orbital, Glasgow, Scotland, 2019. [[https://www.albaorbital.com/unicorn-2-features-and-specifications](https://www.albaorbital.com/unicorn-2-features-and-specifications)]([https://www.albaorbital.com/unicorn-2-features-and-specifications](https://www.albaorbital.com/unicorn-2-features-and-specifications)) (accessed 14 April 2025)). Considering a worst-case scenario where both the satellite and the reference exhibit this level of inaccuracy, the resulting range of ballistic coefficients spans from 14.5 to 29 kg m\({}^{-2}\) for the final phase.