## 4 Materials and Methods for Designing Tether-Deployed Constellations
### Conceptual Design Execution
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Release** & **Propose** & **True Anomaly** & **Date** **and Time of** \\
**Number** & **Bodies Considered for Scheduling** & **Difference** & **Release (UTC)** \\ \hline
1 & End of Deployer’s first orbit after insertion & N/A & 1 October 2023 01:34:31.657 \\
2 & Deployer and Prograde Payload 1 & \(\left|\Delta\theta_{Deployer}-\Delta\theta_{Prograde}\,1\right|\) & 2 October 2023 10:37:48.280 \\  & & \(\approx 60^{\circ}\) & \\
3 & Deployer and Retrograde Payload 1 & \(\left|\Delta\theta_{Deployer}-\Delta\theta_{Retgrade}\,1\right|\) & 3 October 2023 21:15:34.810 \\  & & \(\approx 120^{\circ}\) & \\
4 & Deployer and Prograde Payload 1 & \(\left|\Delta\theta_{Deployer}-\Delta\theta_{Prograde}\,1\right|\) & 5 October 2023 03:10:18.854 \\  & & \(\approx 180^{\circ}\) & \\
5 & Deployer and Retrograde Payload 1 & \(\left|\Delta\theta_{Deployer}-\Delta\theta_{Retgrade}\,1\right|\) & 6 October 2023 16:56:14.554 \\  & & \(\approx 240^{\circ}\) & \\
6 & Deployer and Prograde Payload 1 & \(\left|\Delta\theta_{Deployer}-\Delta\theta_{Prograde}\,1\right|\) & 7 October 2023 19:42:22.354 \\  & & \(\approx 300^{\circ}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Tethered payload release scheduling. Note that the first release is not preceded by any releases, hence not having an applicable true anomaly difference condition to trigger the release. and factors of spacecraft mass, atmospheric drag, solar radiation pressure, and radiation pressure were neglected, as those factors are highly dependent on spacecraft design, and are assumed to have little effect on the results of this basic conceptual demonstration. More detailed analysis incorporating these kinds of effects on spacecraft with masses up to 500 kg each, along with varying tether designs, will be published in future work. The deployment system was assumed to operate in idealized conditions where the payload release vectors are directly tangent with the deployer's trajectory, and releases are treated as impulsive velocity change maneuvers independent of mass and size. While allowing the deployer and onboard payloads to propagate forward in time, and releasing payloads based on the release scheduling criteria previously described, the dates and times of potential releases could be determined, and are also included in Table 1. The release maneuvers deployed payloads away at exactly 20 m/s directly prograde or retrograde. Visuals of the state of the constellation at each of the release times are given in Appendix A. The resulting constellation produces two very similar but separate orbital planes, each with six satellites roughly evenly distributed around them, and is shown in Figure 28. Payloads that were released into prograde directions are shown in red, retrograde payloads in blue, and the deployer in white. The prograde payloads were placed into orbits with an apoapsis altitude of \(\sim\)559 km, and a periapsis altitude of 500 km, and the retrograde payloads were placed into orbits with an apoapsis altitude of 500 km, and a periapsis altitude of \(\sim\)416 km. The altitude bounds of 416-559 km can be narrowed closer to the 500 km median altitude with a lower release velocity than the 20 m/s used in this analysis, but this comes with the consequence of longer time periods between releases, increasing the overall dispersion time. Conversely, more rapid dispersion periods can be achieved with higher release velocities beyond 20 m/s, but this comes with the consequence of a wider altitude range, where retrograde-released payloads can be sent to precariously low altitudes and experience increased effects of atmospheric drag. With each plane having different orbital periods because of their differing semimajor axes, there are occasional brief instances where satellites in two different planes can be in close proximity to each other in terms of true anomaly \(\theta\), but this is soon followed by their separation as one passes by another.