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inputted. The altitude (semi-major axis) and inclination are defined for each test case. All of the test cases, with exception to those focused on Molniya orbits, have an eccentricity of 0, which also prevents the use of argument of perigee and true anomaly. Altitude, inclination, and eccentricity account for three of the inputs.

Since there is no argument of perigee or true anomaly, this necessitates the use of argument of latitude, which is the sum of the argument of perigee and true anomaly, if they were defined. Since these variables are not defined for a circular orbit, the argument of latitude is defined as the angle between the nodal vector and the satellite’s position vector. For most test cases, the argument of latitude is set to 0°, which is indicative of a deployment occurring over the equator, specifically at the ascending node. The test cases that do not have an argument of latitude set to 0° fall into three subsets. One subset is used to test how the location of the deployment (argument of latitude) effects the dispersal of the constellation of CubeSats, in which case the argument of latitude is set to a desired test setting. The second subset is used to construct the deployments that involve the delayed deployment of part of the constellation. More on how this is accomplished is presented later in this chapter.

The last subset that does not involve an argument of latitude equal to zero is for test cases involving Molniya orbits. Due to the eccentricity of a Molniya orbit, there are only two places on the orbit where the velocity vector and nadir vectors are perpendicular to one another, apogee and perigee. The way the deployment parameters are defined for this investigation is based on the velocity and nadir vectors being perpendicular so, for Molniya orbits, the argument of perigee is set to 270° and the true anomaly is set to 0°.

the algorithm used to create the deployment vectors for the individual satellites in this investigation. The final input is the right ascension of the ascending node (RAAN), which is also set to 0°.

The position and velocity vectors are then inputted into a MATLAB®²⁸ script that recreates the control satellite within a new scenario and creates the eight satellites of the specific constellation for that particular test case based off of the deployment geometry, separation velocity, and location in the orbit that is assigned to that test case.

STK’s High Precision Orbit Propagator (HPOP) is used in order to propagate each satellite forward in time. HPOP is more computationally intensive than STK’s analytical propagators and must integrate from an initial state to determine the satellite’s state at any other time.²⁹ This propagator is used because it allows the user greater control of which specific forces are used during the simulation.

During the creation of each satellite in the simulation, which is automated using MATLAB®, the force model that is used by HPOP is programmed. The first two programmed settings concern third body gravity. Neither the gravitational effects of the Moon or the Sun on the constellation are taken into account. Similar to third body gravity, solar radiation pressure experienced by the individual satellites is also not taken into account. These effects were not accounted for during the simulations because, for

more commonly used but, since only a 2x2 gravity model is utilized in the simulations, the differences between the two gravity models is very small (the largest difference in any coefficient used is 1*10^-9). A 2x2 gravity model is utilized using the HPOP in order to accurately model J₂ effects while also accounting for atmospheric drag.

The main reason for using HPOP within STK as opposed to the much faster J2 or J4 propagators is that it allows the user to also model atmospheric drag. A spherical Jacchia-Roberts atmospheric density model is applied for all of the test cases. A drag coefficient, CD, of 2.2 and an area to mass ratio of 0.005 m²/kg is used. The area to mass ratio used is the area to mass ratio of a 3U CubeSat with a mass of 6 kg. Note that by using a fixed area to mass ratio, two assumptions are being made. The first assumption is that the satellites’ mass does not change over time. The second assumption is that the body of the spacecraft is either spherical or has a fixed orientation with respect to the Earth. The second assumption is due to the cross-sectional area of the satellite not changing during the simulation. Since the body of a 3U CubeSat is not spherical, the assumption is that the orientation of the satellite’s body is fixed with respect to the Earth.

Figure 14 is a screenshot from STK of the propagator specific force model that is utilized for the simulations.

Figure 14: STK HPOP Force Model Settings – STK screenshot³

The duration of each test case is determined by the lifetime of the individual satellites contained within the test case. Each simulation is terminated once either all nine satellites deorbit or after three years, whichever duration is shorter.