# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## 2 Defining the Deployment Problem
### Satellite Dispersal and Phasing  
The model follows that presented by Planet Labs satellites in [5] for the Dove satellites' in-plane phasing. A discretized set of Cartesian position and velocity vectors of the constellation is converted into separation angles, \(\theta\), and their rate of change, \(\dot{\theta}\), with respect to a designated reference satellite over a 24 h period, k. The separation angles experience acceleration due to the application of differential drag, with the magnitude given by Equation (2). \[\ddot{\theta}_{k}=\frac{3q_{ref,k}}{\overline{a}_{ref}}\bigg{(}\frac{1}{\beta _{Min}}-\frac{1}{\beta_{Max}}\bigg{)} \tag{2}\]  
Here, \(\beta=\frac{m}{s_{C0}}\), \(\overline{a}_{ref}\) is the mean semi-major axis of the reference satellite, and \(q_{ref,k}\) is the mean dynamic pressure over the discretized period k given by \(q_{ref,k}=\frac{1}{2}\overline{\rho v^{2}}\). This angular acceleration serves as the control variable, providing either acceleration or deceleration between a satellite and reference pair. It is controlled through the attitude determination and control system (ADCS) commands using a binary control scheme, in which the satellites orient themselves to align with either \(\beta_{Max}\) and \(\beta_{Min}\), corresponding to low-drag and high-drag mode ballistic coefficients, respectively. Effectively, this splits the phasing time of the two satellites into an acceleration period and a deceleration period, denoted by \(\Delta t_{a}\) and \(\Delta t_{b}\). Minimizing the sum of these two periods results in the optimal phasing time. [PERSON] et al. also provide an analytical solution for the two-satellite phasing problem in [5], which is directly applicable to each satellite-reference pair in this deployment strategy. For a given pair with initial and final states defined by \(\left[\theta\,\dot{\theta}\right]_{0}^{T}\) and the desired final relative motion \(\left[\theta\,\dot{\theta}\right]_{f}^{T}\), the optimal switching times are given by Equations (3) and (4):  
\[\Delta t_{B}=\frac{\Delta\dot{\theta}-\ddot{\theta}_{A}\Delta t_{A}}{\ddot{ \theta}_{B}} \tag{3}\]  
and  
\[\left[\frac{1}{2}\ddot{\theta}_{A}\left(\frac{\ddot{\theta}_{A}}{\dot{\theta} _{B}}-1\right)\right]\Delta t_{A}^{2}+\left[\dot{\theta}_{0}\left(\frac{\ddot {\theta}_{A}}{\dot{\theta}_{B}}-1\right)\right]\Delta t_{A}+\left[\Delta\theta -\frac{\Delta\dot{\theta}}{\dot{\theta}_{B}}\bigg{(}\dot{\theta}_{0}+\frac{1} {2}\Delta\dot{\theta}\bigg{)}\right]=0 \tag{4}\]where \(\ddot{\theta}_{A}=\ddot{\theta}\), \(\ddot{\theta}_{B}=-\ddot{\theta}\), \(\Delta\theta=\theta_{f}-\theta_{0}\), and \(\Delta\dot{\theta}=\dot{\theta}_{f}-\dot{\theta}_{0}\). The solution is the \(\Delta t_{a}\) and \(\Delta t_{b}\) that provide positive, real roots for the given relative motion and angular acceleration. In a many-in-one system, the total phasing process begins immediately after separation from the launch vehicle at \(\mathrm{k}=0\) and continues through each deployment stage (\(\mathrm{k}=1,2,\ldots,\mathrm{k}_{\mathrm{max}}-1\)). Each stage consists of a disengagement maneuver followed by a phasing interval governed by Equations (3) and (4). The phasing duration of each stage is thus determined by the initial and final angular states, which evolve recursively based on the disengagement history and the differential drag control authority. Starting from Stage 1, the problem consists of two satellites, \(S_{1,0}\) and \(S_{1,1}\). The reference satellite will enter low-drag mode first, allowing \(S_{1,1}\) to accelerate by some value, \(\ddot{\theta}_{1,1}\), until \(\left[\theta_{1,1}\ \dot{\theta}_{1,1}\right]^{T}_{f}\) is reached. At the next perigee pass, both satellites perform a disengagement maneuver, which results in four satellites. This includes \(S_{2,2}\) and \(S_{2,3}\), which previously formed \(S_{1,1}\), as well as \(S_{2,0}\) and \(S_{2,1}\), which previously formed \(S_{1,0}\). This process repeats until the final configuration of \(n\) satellites is reached.