# Modeling and Analysis of Staged Constellation Deployment from a Single-Unit System
## Appendix A  
In this work, the drag coefficient is computed following the methodology outlined by [PERSON] et al. [18]. Specifically, it is assumed that incident gas particles interacting with the satellite's surfaces undergo re-emission that includes both a diffuse and a quasi-specular component. The diffuse contribution to the drag coefficient, denoted as \(c_{D,\;Diffuse}\), is calculated using [PERSON]'s model [21]. The quasi-specular component, \(c_{D,\;Specular}\), is derived from [PERSON]'s quasi-specular accommodation model [22]. The total drag coefficient is then obtained as a weighted sum of these two components:  
\[c_{D}=fc_{D,\;Diffuse}+(1-f)c_{D,\;Specular}\]  
where \(f\) represents the fraction of interactions resulting in diffuse re-emission, assumed to be 0.9 in this paper for a corresponding altitude of approximately 500 km. This value was selected based on the observations in [23] that the emissions become increasingly specular as the altitude increases. Additionally, at an altitude of 225 km, Space Shuttle data indicate that the factor is approximately 0.98 [24].  
The satellite is modeled as a flat plate with a square cross-sectional area. For a gas particle incident at an angle \(\beta\) with respect to the normal surface, [PERSON]'s model provides the diffuse drag coefficient as:  
\[c_{d}A_{ref}=\frac{P}{\sqrt{\pi}}+\gamma QZ+\frac{\gamma v_{r}}{2v_{i}}\big{(} \gamma\sqrt{\pi}Z+P\;\big{)}\]  
where \(\gamma=\cos(\beta)\), and the intermediate variables are defined as \(P=\frac{1}{3}e^{-\gamma^{2}g^{2}}\), \(Q=1+\frac{1}{2g^{2}}\), and \(Z=1+erf(\gamma s)\), with \(erf(x)=\int_{0}^{x}\exp(-y^{2})dy\) and \(v_{i}\) is the incident velocity of the particle with respect to the satellite. Here, \(s\) is the speed ratio, representing the ratio of the satellite's speed to the most probable speed of the ambient atmospheric particles. The term \(v_{r}\) denotes the most probable velocity of the re-emitted particle and is given by:  
\[v_{r}=\sqrt{\frac{2}{3}}v_{i}\bigg{[}1+\alpha\bigg{(}\frac{E_{w}}{Ei}-1\bigg{)} \bigg{]}^{\frac{1}{2}}\]  
where \(E_{w}\) is the kinetic energy of the remitted molecule with a velocity corresponding to the satellite's surface temperature, and \(E_{i}\) is the average kinetic energy of the incident molecules. The energy accommodation coefficient, \(\alpha\), is calculated using [PERSON]'s formula, as recommended by [PERSON] et al. (2005):  
\[\alpha=\frac{3.6\;u\;sin(\beta)}{(1+u)^{2}}\]  
where \(u\) represents the ratio of the mass of incoming molecules to that of the surface molecules.  
To obtain the quasi-specular component, [PERSON]'s model is applied to the flat-plate geometry. The specular accommodation, as proposed by [PERSON] for a flat plate, is then given by ([PERSON], 2005):  
\[c_{D,\;Specular}=2(\;1-rcos(2\theta))\]  
where \(\theta\) is the incidence angle, and \(r\) is the ratio of the satellite's speed to the root-mean-square (rms) speed of ambient atmospheric particles.  
Values for the molecule types, average molecular weights of molecules, and atmospheric temperatures were obtained through the NRLMSISE-00 model for the ephemeris data of the satellite. Furthermore, the satellite surfaces are covered in solar cells, and thus, the surface molecular mass is approximated as 60 amu, corresponding to glass ([PERSON] et al., 2013).