# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## 4 Case Studies
### Case Study 1: The PQ8 Satellite
#### 4.1.4 Semi-Major Axis Decay and RAAN Drift  
Since the system relies on differential drag control, it inherently induces a difference in the semi-major axis between the satellites. This presents two key concerns. First, a difference in the semi-major axis leads to a variation in the satellites' mean motion, causing the in-plane phase angle to drift apart. Second, it causes differential nodal precession within the constellation.  
The first concern is mitigated by the differential drag control strategy, particularly through the analytical solutions provided in Equations (3) and (4). These equations are solved to ensure a final angular separation rate \(\dot{\theta}\) of zero, which directly correlates with mean motion. Essentially, the differential drag controller reduces the semi-major axis of the higher-altitude satellite at a faster rate than that of the lower-altitude satellites until they converge to approximately the same value. As a result, their mean motions become nearly identical. This effect is illustrated in Figure 12, which shows the semi-major axis profiles of the leading PQ1 satellite and the reference satellite. Here, the reference satellite, \(PQ1_{0}\), is in high-drag mode, until its mean motion converges with that of \(PQ1_{7}\), which is in low-drag mode.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Conditions** & **Phase Time** \\ \hline Ideal & No ADCS inaccuracy, \(f=0.9\) & 255.0 days \\ ADCS inaccuracy & \(\pm 5^{\circ}\) during dispersal only, \(f=0.9\) & 256.5 days \\ \(\pm 5^{\circ}\) during dispersal and disengagement, \(f=0.90\) & 256.5 days \\ ADCS inaccuracy & \(\pm 5^{\circ}\) during dispersal and disengagement, \(f=0.85\) & 254.0 days \\ + BC Modeling & \(\pm 5^{\circ}\) during dispersal and disengagement, \(f=1.00\) & 256.8 days \\ \hline \hline \end{tabular}
\end{table}
Table 5: Phase times uncertainties (\(\Delta v=0.35\) ms\({}^{-1}\)).  
The second concern remains valid, as some drift in RAAN between satellites is inevitable when using this method for dispersal. The formula for nodal precession is given by:  
\[\hat{\Omega}=-\frac{3}{2}\bigg{(}\frac{R_{E}}{(a(1-e^{2}))}\bigg{)}^{2}J_{2} \omega\text{cos}(i)\]  
where \(R_{E}\) is Earth's radius, a is the semi-major axis, \(J_{2}\) is Earth's second dynamic form factor, \(\omega\) is the angular velocity of the satellite, \(e\) is the eccentricity, and \(i\) is the inclination. The extent of this drift is strongly influenced by orbital inclination. At high inclinations, particularly near-polar orbits, RAAN drift is minimal, due to the term \(\cos(i)\), and negligible over typical phasing durations. However, at lower inclinations, the effect can become more pronounced, with drift reaching several degrees depending on the length of the deployment phase. In general, a shorter phasing period helps minimize this divergence. The degree of RAAN drift that can be tolerated is ultimately mission dependent. In this context, [PERSON] et al. proposed a differential drag control scheme that addresses cross-track motion using convex optimization [12]. If nodal precession poses a concern for constellation geometry or coverage, such schemes can be integrated to help mitigate its long-term impact.