## 2 The LEO Environment: Drag, Momentum Buildup (Aerospinning), and Meteoroids
Hence, it is possible to cancel aerospinning by slightly offsetting the re-boost engine thrust axis from the tether center of mass, although at the cost of added engineering complexity. As the numerical integration algorithm may suffer from high sensitivity to input parameter variations, the orbital lifetime model was verified against Table I-1 in [24]. No discrepancy was found for small tether lengths, indicating general stability of reentry (and therefore, air drag) numerical integration. For longer tethers, at altitudes below 200 km, a braking effect of the tether tip entering dense layers of atmosphere become noticeable, while, for higher altitudes, drag was only slightly dependent on tether length because of the longer atmospheric density scale in the thermosphere. Additional drag is rather small as, for normalized tethers, additional drag in inhomogeneous atmospheres is the second-order correction to the drag of compact surfaces. This additional drag is  
Figure 3: The normalized reentry time for tethers on equatorial circular orbits. visible in Figure 3 as a slight curvature of the reentry time contours below 3000 s of normalized reentry time. Figure 4 shows the steady-state speed of the tether tip for average solar activity. Solar activity greatly changes the density scale of the upper atmosphere. Therefore, the indicated speeds can vary as much as three times those at low solar activity (resulting in greater speeds at low orbits) and high solar activity (resulting in lower speeds at low orbits). However, as the molar weight of the upper atmosphere is increased during the solar maximum, an altitude range (550-750 km) exist such that the effects of increased temperature and increased molar weight roughly cancel each other, resulting in a more stable atmospheric density scale and, therefore, a relatively stable steady-state rotation speed. The important feature visible on Figure 5 is the range of orbit altitudes and tether lengths, whereby the steady-state rotation speed of the tether is nearly independent of the phase of the solar cycle. At this range, the tethers will rotate in a steady-state, with a period of roughly 170 s. Due to the large (compared to the Solar cycle period) spin-up time scale, it is expected that tethers on the rotation stability line will experience variability of their steady-state tip speed below 10%. Figure 4: Steady-state tip speed of the rotating tether on the equatorial circular orbit in average solar conditions (based on MSISE-90 atmospheric model). Figure 5: The characteristic spin-up time of tethers on equatorial circular orbits. Due to the gravity gradient stabilization, tether rotation cannot begin spontaneously as deployed. An initial tether angular velocity must be supplied by the momentum wheel or tip jet to start the rotation. To calculate the rotating tether tip minimal speed at the nadir (or zenith) point, it is possible to equate the rotational energy of the tether at this point to the tidal energy of the tether stretched vertically (Equation (1a)). Equation (1a) states that to sustain rotation, the tether must have enough rotational energy to librate from a stable vertical orientation to an unstable horizontal orientation, with inertia of the tether acting against the tidal forces. Solving Equation (1a) with the gravity potential of Earth yields Equation (1b). \[\frac{I_{\mathit{ether}}\omega_{\mathit{natr}}^{2}}{2}=E_{\mathit{gravity}} \left(\mathit{horizontal\_orientation}\right)-E_{\mathit{gravity}}\left( \mathit{vertical\_orientation}\right) \tag{1a}\] \[\upsilon_{\mathit{min}}=\sqrt{\left(\frac{L}{R_{\mathit{earth}}+h_{c}}+\ln \frac{R_{\mathit{earth}}+h_{c}-L/2}{R_{\mathit{earth}}+h_{c}+L/2}\right)} \cdot\frac{12 GM_{\mathit{earth}}}{L} \tag{1b}\]  
where \(L\) represents the length of the tether; \(h_{c}\), the tether orbit altitude; \(R_{\mathit{earth}}\), the radius of Earth; \(G\), the gravity constant; \(M_{\mathit{earth}}\), mass of Earth.