## 4 Materials and Methods for Designing Tether-Deployed Constellations
### Orbit Design
#### 4.2.3 Orbit Design Question 3: Single or Multiple Semimajor Axes for Released Payloads
Figure 22: Basic payload orbit progression behavior with prograde release orbit distribution optimization for an example set of 8 prograde-released payloads and 8 retrograde-released payloads. 2.5 Orbit Design Question 5: Prograde or Retrograde Release Orbit Optimization Dependent on Period Ratio  
If choosing to optimize a prograde or retrograde release orbit payload group, orbit geometry and timing of the deployer's trajectory and release characteristics must be closely considered. This relationship between the deployer and optimized prograde/retrograde release orbit periods for proper release scheduling will here be referred to as the period ratio, \(T_{i}/T_{o}\), where \(T_{i}\) is the period of the innermost orbit for optimization, and \(T_{o}\) the period of the outermost orbit for optimization. For prograde release orbit optimization, the innermost orbit would be the deployer orbit with the outermost orbit being the prograde release orbit. For retrograde release orbit optimization, the innermost orbit would be the retrograde release orbit with the outermost orbit being the deployer orbit. This can be visualized in Figure 24.  
The period of a single orbit \(T\) can be defined by Equation (1), where and \(a\) is the orbit's semimajor axis, \(\mu\) is the gravitational parameter specific to the body being orbited, and \(R_{a}\) and \(R_{p}\) being the orbit's apoapsis and periapsis. \[T=2\pi\sqrt{\frac{a^{3}}{\mu}}=2\pi\sqrt{\frac{\left(R_{a}+R_{p}\right)^{3}}{ 8\mu}} \tag{1}\]  
The necessary period ratio \(T_{i}/T_{o}\) for a selected pair of orbits containing a deployer orbit and an evenly distributed collection of payloads in a prograde or retrograde orbit is defined in Equation (2). \[\frac{T_{i}}{T_{o}}=\frac{kn}{kn+1}=\left(\frac{a_{i}}{a_{o}}\right)^{\frac{3 }{2}}=\left(\frac{\left(R_{a}+R_{p}\right)_{i}}{\left(R_{a}+R_{p}\right)_{o} }\right)^{\frac{3}{2}} \tag{2}\]  
In this equation, \(n\) is defined as an integer number of payloads to be uniformly distributed in the optimized orbit, and \(k\) is defined as the integer number of instances where the deployer intersects the optimized prograde or retrograde orbit between releases. The lowest value for \(k\) in this case is 1, which would result in the deployer releasing a payload every orbit. A value of \(k\) equal to 2 would result in the deployer releasing a payload every two orbits, and so on. Figure 24: Basic inner and outer orbits. With this in mind, it can be seen that there is a balance between speed of deployment and the geometry of the orbits (apoapsis, periapsis, and eccentricity) of that must be decided on when given a set number of payloads to be dispersed. As \(T_{i}/T_{o}\) approaches 1, the deployer and optimized release orbits approach similarity, but this requires a high value for \(k\) orbit intersections between releases, which would require a longer time to execute. Conversely, as \(T_{i}/T_{o}\) approaches 0 (until \(k\) is minimized to 1), the fastest time for dispersal is achieved with one release per orbit, but this then involves more dissimilar geometries between the deployer and optimized release orbits as a consequence of the largely different orbital periods, potentially resulting in high eccentricity for one or both of them. These factors must also be considered with the resulting \(\Delta V\) requirements during deployment, where similar orbit geometries with slower dispersion times require much less \(\Delta V\) at deployment than dissimilar orbits with more rapid dispersion times. More discussion on the determination of \(\Delta V\) is included later in this article.