# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## 4 Case Studies  
To demonstrate the feasibility of the proposed dispersal method and the impact of disengagement velocity and phase angle selection on phasing times, two case studies are presented. The first case study presents a system of eight PocketQube-sized satellites in an initial configuration resembling a 1U CubeSat. The second presents the case of four 1U CubeSat satellites in a \(2\times 2\times 1\) initial configuration.  
The simulations are performed using the MATLAB R2024a Aerospace Blockset. In both case studies, the initial perigee altitude is set to 500 km, matching standard rideshare options, with an eccentricity of 0.01. Other orbital elements are set to zero and are summarized in Table 3.  
The motion models used for the case studies are governed by the high-precision numeric orbital propagator in MATLAB's Aerospace Toolbox. The initial orbital position and velocity are presented in the ICRF inertial frame. The block computes the acceleration acting on the satellite from the perturbation models such that:  
\[\overset{\rightarrow}{a}_{ICRF}=\overset{\rightarrow}{a}_{CentralBody}+ \overset{\rightarrow}{a}_{ThirdBody}+\overset{\rightarrow}{a}_{Drag}+ \overset{\rightarrow}{a}_{SRP}\]  
where \(\overset{\rightarrow}{a}_{CentralBody}\) is the acceleration due to the gravitational potential of Earth using the EGM2008 spherical harmonics model, \(\overset{\rightarrow}{a}_{ThirdBody}\) is the acceleration due to the Moon and the Sun modeled as point masses, and \(\overset{\rightarrow}{a}_{SRP}\) is the acceleration caused by Solar Radiation  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Orbital Element** & **Initial Value** \\ \hline Epoch & 1 December 2020 \\ Altitude (perigee) & 500 km \\ Eccentricity & 0.01 \\ Inclination & Case Study 1: 90\({}^{\circ}\) \\  & Case Study 2: 63.4\({}^{\circ}\) \\   RAAN & 0\({}^{\circ}\) \\   Arg. of perigee & 0\({}^{\circ}\) \\   True anomaly & 0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation initial orbit setup.  
Pressure (SRP) and equated in conjunction with a dual-cone shadow model for eclipse. Additionally, \(\overrightarrow{a}_{Drag}\) is the acceleration due to atmospheric drag obtained from the equation:  
\[\overrightarrow{a}_{Drag}=\frac{1}{2}\rho\bigg{(}\frac{c_{D}A}{m}\bigg{)}v_{ rel}^{2}\frac{\overrightarrow{v}_{rel}}{v_{rel}}\]  
where \(v_{rel}\) is the satellite velocity relative to the atmosphere. The atmospheric density is obtained from the NRLMSISE-00 atmospheric model in conjunction with CelesTrak space weather data. The epoch is set during the solar minima of Solar Cycle 25, leading to reduced atmospheric densities and lower differential accelerations, which present the worst-case weather conditions for a dispersal method dependent on drag. The EGM2008 and NRLMSISE-00 models were selected due to them being readily available in MATLAB. Additionally, NRLMSISE-00 has become the standard for satellite drag modeling over the last few years [17].  
These accelerations are summed and integrated using MATLAB's ode113 solver, as it is well suited for smooth, non-stiff orbital dynamics to propagate satellite state in the ICRF frame. The disengagement-induced change in velocity was modeled as an instantaneous impulse in the LVLH frame, transformed into the ICRF frame, and manually applied to the velocity vector at the moment of separation.