# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## 3 Methodology
The algorithms are employed as follows: First, the initial orbital parameters, satellite properties, and epoch of the single-unit satellite are input into Algorithm 1, along with the disengagement velocity vector and its direction in the LVLH reference frame. The algorithm begins by propagating the satellite to its first perigee, where the initial disengagement is applied (Algorithm 1, Lines 1-3). Following this, the two resulting satellites are each propagated for a fixed interval, \(t_{k}\), to allow for sufficient separation and prevent collisions prior to the next disengagement event. In this study, \(t_{k}\) is set to 7 days. After this interval has elapsed, both satellites are propagated to their respective next perigees, where the subsequent disengagement occurs. Figure 4: Algorithm 1 top-level flow chart. This process is iteratively repeated until the penultimate stage is reached (Algorithm 1, Lines 4-7). Figure 5: Algorithm 2 top-level flow chart. ```
1: Procedure: Generate Stage \(k_{max}\) Insertion Points
2: Propagate \(S_{0,0}\) until the first disengagement command. 3: Perform \(S_{0,0}\) disengagement maneuver to obtain \(S_{1,1}\) and \(S_{1,0}\)
4:for\(k=1;k<k_{max}-1;k++\),do
5: Propagate \(S_{k,i}\) by \(t_{k}\) and to next perigee
6: Perform \(S_{k,i}\) disengagement maneuvers to obtain \(S_{k+1,i}\)
7:endfor
8:for\(i=1;i<2^{k_{max}-1};i++\),do
9:for\(\theta_{(k_{max}-1),\ i}=0;\ \theta_{(k_{max}-1),\ i}<\theta_{(k_{max}-1),\ i\ max};\ \theta_{(k_{max}-1),\ i}+\Delta\theta\),do
10: Propagate \(S_{(k_{max}-1),i}\) until \(\theta_{(k_{max}-1),\ i}\), store \(t_{(k_{max}-1),\ i}\)
11: Perform \(S_{(k_{max}-1),i}\) to \(S_{(k_{max},j),\ i}\) disengagement maneuver
12: Calculate \(S_{(k_{max}),j}\) initial relative motion (insertion points)
13:endfor
14:endfor
15: Output: \(S_{k_{max},\ j}\) insertion points for every \(\theta_{(k_{max}-1),\ i}\) and associated \(t_{(k_{max}-1),\ i}\)
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**Algorithm 1**Generate Stage \(k_{max}\) Insertion Points  
At the penultimate stage, each satellite is propagated until it reaches a discretized set of angular separations relative to a designated reference satellite. Upon reaching the target angular separation, the satellite is propagated to its subsequent perigee, where the final disengagement takes place. The resulting relative angular velocity and separation angle at this point are recorded, forming an insertion point \(\left[\theta\ \dot{\theta}\right]_{i}^{k_{max}}\). This process is repeated for all satellites in the penultimate stage until an insertion point has been obtained for every value in the discretized angle set (Algorithm 1, Lines 8-14). Once the complete set of insertion points has been generated, it is passed to Algorithm 2, along with the desired final angular separation for the last-stage satellites. In Algorithm 2, each insertion point is evaluated using the target separation and Equations (3) and (4), which determine the required phasing times (Algorithm 2, Lines 3-6). The total phase time for a given final-stage satellite is computed by summing the phasing contributions from each stage with the time derived from Equations (3) and (4). Each total phase time is compared against those obtained from other insertion points, and the smallest value is retained as the best-case phase time for that satellite. This process is repeated for all insertion points associated with each final-stage satellite and then again across all final-stage satellites (Algorithm 2, Lines 2-13). The maximum among these best-case satellite values is taken as the best-case phase time for the whole constellation. In practice and in the following case studies, the phase time of each satellite at every insertion point is stored and plotted, enabling enumerative analysis of phasing performance across the complete set of candidate configurations. In summary, the deployment analysis focuses on minimizing the total phasing time \(\sum_{k=1}^{k_{max}}t_{k}\).