Figure 10: Timeline

Simulating the mission requires propagating the initial Keplerian elements forward in time, which in turn requires knowing the time rate of change in the elements due to the applied forces. Under the ideal two-body problem, only the true anomaly varies through the mean anomaly, which has a time rate of change equal to the mean motion of the satellite.

Equation 28: Rate of Change of the Mean Motion

Including other disturbing forces yields a more accurate model. First, the Earth's equatorial bulge exerts a torque on the orbit changing the orientation of the orbital plane through secular changes in the right ascension of the ascending node, the argument of perigee, and the mean anomaly.

Equations 29-31: Rate of Change of the Right Ascension of the Ascending Node,

Argument of Perigee, and Mean Anomaly Due to J2 Perturbation

Equations 32-36: Change in the Semi-major Axis, Eccentricity, Inclination, Right Ascension of

the Ascending Node, and Argument of Perigee Due to Atmospheric Drag

Equations 37-39: Definitions of Terms Used in Equations 27-31

Equation 40: Atmospheric Density Model

Figure 11: Density as a Function of Altitude

The final disturbing force, the electrodynamic drag, alters the semi-major axis, as discussed in the
tether concept section. Combining the two-body and disturbing force functions provides a propagation
scheme for all six elements. This set of equations form the basis for the Matlab simulation of the
mission, where the subscript rev-atmosphere refers to the rate of change due to the atmospheric drag,
J_{2} refers to the rate of change due to Earth's obliquity, and
tether refers to the rate of change due to the electrodynamic drag.

Equations 41-46: Propogation of the Semimajor Axis, Eccentricity, Inclination, Right Ascension of

the Ascending Node, Argument of Perigee, and Mean Anomaly

Figure 12: Unstable Tether

A complete mission simulation demonstrates the fulfillment of the goals outlined. A mission duration of 26.617337963 days results from a material exposure time of 17.006944444 days (14 days + 3 days + 10 minutes) and an orbital adjustment interval employing the balance. Thus, exposure time represents 63.9% of the total duration, highlighting the effectiveness of the tether in changing the altitude rapidly. By comparison, an atmospheric drag only mission lasts 47.589 days reducing the exposure time percentage to 35.74%. This comparison would be even more dramatic for missions operating at higher altitudes where atmospheric drag becomes negligible. The first plot displays the semi-major axis, the control variable, versus time.

Figure 13: Semi-Major Axis Variation During Mission

Figure 14: Semi-Major Axis Variation Enlargement of the Last 2 Days

Figure 15: Rate of Change in the Semi-Major Axis

Figure 16: Semi-Major Axis Variation Enlargement of the Last 12 Minutes

Figure 17: Eccentricity Variation During Mission

Figure 18: Inclination Variation During Mission

Figure 19: Right Ascension of the Ascending Node Variation During Mission

Figure 20: Argument of Perigee Variation During Mission

Figure 21: Reentry Path

Figure 22: Predicted Landing Zone [I-2]

Conclusions