Analysis

By combining fundamental astrodynamic concepts and the tether drag equations presented earlier in this paper, a simulation can be constructed of the proposed mission. A simulation helps explore how all of the forces acting on the satellite interact. Furthermore, it provides a tool to develop an optimal solution in terms of minimizing mass and maintaining a stable controllable satellite.


Figure 10: Timeline

The above diagram outlines the mission sequence from start to finish. The launch vehicle deploys the satellite into a nearly circular 300 km altitude orbit. For the fist section of the mission the spacecraft remains in the stowed configuration with the tether stored in the ballast body. The main mission begins as the satellite extends the material tray and begins a two week exposure period for the first sample. Upon completion of day fourteen, the material sample tray retracts. Using a compressed spring or similar ejection method, the satellite provides the initial impetus to deploy the ballast mass downward, allowing the gradient force to take over and extend the tether to its full length. Engaging the current switch allows electrons to flow through the tether inducing drag that rapidly lowers the satellite to a radial position of 6600 km. At this altitude the satellite disengages the current switch, eliminating electrodynamic drag, and conducts the second material sample exposure. Upon completion of the three day test, the satellite again employs electrodynamic drag to move to an altitude of 125 km, for the third test, ten minutes in duration. As the satellite moves into the lower thermosphere atmospheric drag starts to dominate. With the final material exposure completed, this drag reduces the orbit into the mesosphere. The resulting friction incinerates the unprotected tether and ballast mass, as the ablative heat shielding protects the primary body. When the satellite enters the troposphere, a parachute deploys to further reduce the descent speed and provide a softer landing. Finally, a radio beacon guides the recovery operation.

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

Second, the low orbit and pass through the atmosphere on reentry entails consideration of atmospheric drag. This drag, by dissipating energy, has secular effects on the semi-major axis, inclination and eccentricity by lowering and circularizing the orbit. King-Hele developed expressions for the rate of change of orbital elements on a per revolution basis.


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

The Jj terms are first kind modified Bessel functions of order j, evaluated at c. The terms c, delta, and Q are defined as follows.


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

Calculating the density, rho, relies on an exponential model.


Equation 40: Atmospheric Density Model

Using the scale heights and reference densities given by Vallado yields an accurate density model from zero to 1000 km.[11]


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, J2 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

Constructing a realistic model necessitates estimating the value of a number of input parameters. Based on the small scientific payload and minimal equipment, a 0.3 m diameter sphere should accommodate the components in the main body. Using aluminum and a 5 mm wall thickness results in a primary mass of 3.69 kg. It can be further assumed that the contents inside the main body, including the parachute, battery, electronics, and material samples, have a combined mass of roughly 10 kg. The .4mm diameter tether represents a negligible amount of mass. A spherical body simplifies the computation of the drag due to its constant cross sectional area, and established drag coefficient of 2.0. A few trial runs identified the optimum values for the remaining parameters, the tether length and the ballast mass. Tether length selection balanced to need to provide adequate drag, which scales upward with increasing length, with the desire to minimize the higher risk associated with longer tether deployment. Moreover, as discussed previously, the tensile force and resulting stress increases linearly with increasing tether length. Considering these factors a 75 m long tether seems appropriate for the nature and scale of the mission. At this length the risk of failure, due to deployment malfunction or micrometeoroid impact, remains low. The relatively short length also poses minimal danger to other spacecraft, making mission control easier. Despite the short length, the tether still reduces the semi-major axis by roughly 0.5 km per revolution at a radial distance of 6650 km. Requiring only a matter of days to adjust the orbit between the various testing altitudes, the 75 m long tether keeps the mission relatively short, allowing for simple non-recharging battery power. Selecting the ballast mass focused on maintaining gravity gradient stability, while minimizing the mass. As the Earth rotates the relative orientation of the magnetic field varies resulting in a variable electrodynamic force acting on the tether. This causes a peridic fluctuation in the angle of the tether with respect to the radial direction. A small ballast mass, results in a low gradient force. Under these circumstances the cyclical electrodynamic force dominates the gradient force, inducing large swings in the tether orientation and therefore dramatic changes in the drag magnitude. This is unfavorable for stable drag operation. For example, the electrodynamic force causes a 2 kg ballast mass to fluctuate more than 60 in orientation, swinging through the horizontal (alpha=90), and putting the satellite in a bi-directional instability configuration where the ballast mass could swing up towards 180 or back down towards 0.


Figure 12: Unstable Tether

The ultimate design choice, a ballast mass of 3.25 kg, limits the oscillation to 25 and keeps the ballast below 45. Including this mass brings the total satellite mass to 17 kg. An additional 0.5 kg was added in to account for the anode and current switch masses. The final program parameters, the initial Keplerian elements of the satellite and launch time, were determined by considering the mission goals, the typical values for possible launch vehicles, and the effect on the reentry point. Foremost, the low eccentricity, 0.0002, keeps the radial distance from varying much, thereby maintaining a more constant environment to conduct the material exposures. Also, the initial inclination of 50 matches a typical inclination for the International Space Station. The time of deployment, as constrained by an initial qgst of 3.5 radians, was set to provide a reentry path over North America. Likewise, the tweaking of the initial semi-major axis helped orient the landing corridor. Other elements, primarily the right ascension of ascending node and argument of perigee, were picked fairly arbitrarily given the lack of mission impact from these parameters. Also due to the large secular changes in these values due to J2 perturbations, a typical value for the space station is difficult to define.

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

The different phases of the mission are highlighted, with the first sample exposure occupying the initial two weeks. During this period the semi-major axis decreases 14.1 km from 6674.3 to 6660.2 km due to atmospheric drag. With the deployment of the tether the orbit rapidly drops to 6600 km in just over seven days. The second exposure period, lasting three days, has a semi-major axis range of 20 km, decreasing to a final value of 6580 km. A second tether induced orbital change reduces the value to 6500 km. A close-up of the last two days of the mission better illustrates this second tether orbital change.


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

The following figure illustrates the time rate of change in the semi-major axis on a per revolution basis for the first twenty-five days of the mission.


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

Emphasizing the impact of the tether on the orbit, the graph also shows the cyclical dependency of electrodynamic drag on the Earth's rotation. The period of variation matches Earth's rotational period. Plotting the last minutes of the mission shows the final material exposure, lasting ten minutes and bringing the semi-major axis to 6486.6 km.


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

The rapidly thickening atmosphere brings the satellite out of orbit in an additional 112 seconds. Plotting the other orbital elements reveals further insights. The eccentricity reduces due to the circularizing effects of atmospheric drag.


Figure 17: Eccentricity Variation During Mission


Figure 18: Inclination Variation During Mission

Inclination remains nearly constant, with a small change, mainly on reentry, resulting from atmospheric friction. Both the right ascension of the ascending node and argument of perigee show linear procession due to the obliquity of the equator.


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


Figure 20: Argument of Perigee Variation During Mission

Reentry occurs over the northern United States, with a predicted landing point at 48.0562 N. latitude, and 106.5305 W. longitude. This occurs near the Fort Peck Dam in the flat sparsely populated region of northeast Montana. As such, the reentry should pose little danger to residents and would make for an easy recovery operation.


Figure 21: Reentry Path


Figure 22: Predicted Landing Zone [I-2]


Conclusions