H.M.Lewis^{1}, W.J. Cooke^{2}
^{1} NASA Marshall Space Flight Center (MSFC), Huntsville, AL 35812, USA
^{2} Computer Sciences Corporation, Huntsville, AL
Alternative concepts of transferring payloads to various Earth or interplanetary orbits often involve the construction (in low Earth orbit, or LEO) of very large space structures, such as tethers hundreds of kilometers in length or the “space elevator”, in which mass is moved directly from the ground (or near it) into a geostationary orbit via a huge tower. Proponents of these ideas have devoted some thought as to how these structures could be made survivable against the small debris population; for example, multi-strand, or “Hoyt”, tethers have been calculated to have lifetimes of many decades, if not centuries, when the debris flux of sub-centimeter particles is used in the computations. However, practically all advocates of such structures have ignored the cataloged population, consisting of debris greater than a few centimeters in size and satellites, both active and dormant, and it is this component, rather than the much more numerous small bodies, that ultimately determines the lifetimes of very large space constructs. We present the results from some numerical simulations that show that collisions between trackable space objects and such futuristic devices will occur on time scales as short as a few days, effectively eliminating LEO as a possible home for these structures.
Even though NASA considers low Earth orbit (LEO) to end at 2000 kilometers altitude, users of the Department of Defense SGP/SDP low precision propagators effectively put the dividing line at 12,250 kilometers, as this is the altitude below which the SGP propagator is used. While both models take as input a two-line element set (TLE), the SDP is an extension of the SGP model, used for “deep-space” satellites where lunar/solar perturbations are important. The element sets are periodically updated so as to maintain a reasonable prediction capability on all trackable space objects. It is important to note that these element sets are restricted to use in a prediction method that is compatible with the way the element sets were generated. The element sets are “mean” values constructed by removing the periodic variations and they must be reconstructed in the SGP/SDP model in exactly the same way in which they were removed. Use of the NORAD element sets in a different model, a numerical integrator, even if it is more accurate, will result in degraded predictions.
The collection of these element sets - all trackable objects - for nearly the same epoch constitutes a “catalog,” the unclassified version made publicly available from the United States Space Command. A glance at a current catalog shows that there are nearly 9,000 man-made objects, softball size and larger, orbiting Earth. About seven percent of these objects are operational satellites and 15 percent are rocket bodies, with the remainder being fragmentation debris and inactive satellites. Such a “quick look” is sufficient to show that near-Earth space is not the vast “emptiness” that many believe, and that the potential for impacts between catalog objects and any proposed large space structure should be carefully evaluated.
Figure 1. Distribution of trackable objects in LEO.
A glance at figure 1 clearly shows that the distribution of objects in LEO is not uniform with altitude and inclination. Drag keeps altitudes less than 550 kilometers fairly clean, whereas altitudes between 700 and 1100 kilometers constitute the “prime real estate” for many scientific, military, and commercial missions - a region densely populated for inclinations greater than 60 degrees. This higher density obviously translates into a greater collision risk; quantifying just how much greater was the motivation for this study.
In order to assess the collision risk between large LEO structures and catalog objects, three hypothetical space structures - a 400 kilometer tether (similar to the momentum transfer tethers proposed to move payloads from the upper atmosphere to geostationary (GEO) orbit and beyond), a 4000 kilometer tether, and a space elevator (proposed to move payloads from the ground to GEO) - were chosen. Each object was placed in a circular orbit at zero degree inclination. The 400 km tether was placed at 400 km altitude (geometrical center of mass) with the tether extending from 200 km to 600 km. The 4000 km structure's geometrical center of mass was placed at 2000 km altitude with the structure covering 0 to 4000 km altitude. The space elevator's geometrical center of mass was placed at GEO extending from the ground to GEO. Each object was then propagated for one and seven day spans of time against 12 catalogs with epochs from January to April 2001. The time step was one second with linear interpolation to determine close approach distance. Only approaches within 30 km were accepted. Encounter speeds relative to the target were computed for each approach and the point of closest approach was expressed in terms of distances parallel and perpendicular to the structure. The perpendicular distance is equivalent to the approach distance while the parallel distance defines where on the structure the approach occurs.
The results of each catalog run were scaled to 30 days (one calendar month) and are displayed below.
Approach Distance (km) |
Number of Encounters - 400 km tether |
Number of Encounters - 4000 km structure |
Number of encounters - GEO elevator |
0.0 - 0.10 |
0 |
0 |
0 |
0.0 - 0.50 |
0 |
12.5 |
12.5 |
0.0 - 1.00 |
2.5 |
65 |
52.5 |
0.0 - 5.00 |
82.5 |
1137.5 |
1077.5 |
0.0 - 10.0 |
227.5 |
2577.5 |
2410 |
0.0 - 20.0 |
462.5 |
5545 |
5082.5 |
0.0 - 30.0 |
765 |
8370 |
7812.5 |
Table 1. Number of encounters per 30 days from simulations.
The encounters do not occur uniformly along the length of the structures (see figure 2). There are two major peaks, the first at an altitude of 910 kilometers, followed by a second at 1525 kilometers. The encounters at the first peak are caused by objects such as sun-synchronous imaging satellites similar to the LandSats, former Soviet Union meteorological satellites, and other payloads. The second peak is due to mostly fragmentation debris and about 360 Strela satellites deployed by the Soviet military and intelligence forces for their communications, which were phased out of service in 1992. It is also important to note that very few encounters occur above 2500 km altitude, identifying LEO as the predominate area of risk to large space objects.
Figure 2. Number of encounters as a function of altitude.
Given that the model structures are long, thin cylinders, one can show that the cumulative number of encounters should decrease linearly with approach distance:
N_{e} = Kd, |
(1) |
where N_{e} is the number of encounters, K is a constant for a given tether or elevator length, and d is the close approach distance perpendicular to the structure. This equation can be used to calculate a theoretical curve of the number of encounters, which, when compared to the computed cumulative number of encounters from the study, match nicely until the approach distance gets close to 6 km (figure 3). The deviation from theory at approaches below 6 km is probably a result of the chosen time step (in one second, a cataloged object can move approximately eight kilometers) and the failure of the linear interpolation to completely compensate for this coarseness in time. Nonetheless, the overall match with basic theory increases the confidence in the simulations.
The numerical simulations can be used to determine the constant K in equation 1), resulting in
N_{e} = 6.7x10^{-5} Ld, |
(2) |
where L is the length of the tether or elevator in km (100 km < L < 2500 km) and d is the close approach distance in meters. Table 2 presents the results calculated according to relation 2). Note that a space elevator has a close approach within 100 meters ~ once per day, and that even a 400 km tether in LEO will have 2-3 such approaches per month.
Approach Distance (km) |
Number of Encounters - 400 km tether |
Number of Encounters - 4000 km structure |
Number of encounters - GEO elevator |
0.0 - 0.10 |
2.5 |
27.9 |
26 |
0.0 - 0.50 |
12.75 |
139.5 |
130.2 |
0.0 - 1.00 |
25.5 |
279 |
260.4 |
0.0 - 5.00 |
127.5 |
1395 |
1302.1 |
0.0 - 10.0 |
255 |
2790 |
2604.2 |
0.0 - 20.0 |
510 |
5580 |
5208.3 |
0.0 - 30.0 |
765 |
8370 |
7812.5 |
Table 2. Corrected number of encounters per 30 days for objects in LEO.
Figure 3. Comparison of simulation results with theory.
Figure
4. Encounters with 400 km tether.
A paradox appears to present itself in comparing the encounter data for the 4000 km tether with the space elevator - How can the tether have more encounters than the much longer elevator? This can be answered by the average encounter speed between the cataloged objects and the structures. The average encounter speed between objects and the 4000 km structure is roughly 9.3 km/sec, whereas for the GEO space elevator it is 7.3 km/sec. Recall that the number of encounters is proportional to the flux, which is given by:
Flux = ρv, |
(3) |
where ρ is the number density and v is the encounter speed. The number density of objects for the 4000 km structure and the GEO elevator are essentially the same, but the higher encounter speed results in a higher flux for the 4000 km structure, which in turn yields a greater number of encounters.
Another interesting point resulting from the simulations with the 400 km tether is that most of the encounters occur near the top of the tether (figure 4), which is not too surprising in that drag is proportional to atmospheric density, which decreases exponentially with height (more objects near the top). However, it does seem to indicate that placing any power stations with large solar arrays to facilitate reboost, etc. at this end would be unwise.
To recap:
1) The number of close approaches between the cataloged population and large structures in LEO peaks at about 910 and 1525 km altitude, and diminishes to near zero beyond 2500 km altitude.
2) A space elevator will be approached to within 100 meters by a cataloged object practically every day, resulting in a collision probability of
where d is the tether diameter in meters and T is the orbit lifetime in months. An elevator 1 meter in diameter would have an 87% chance of a collision in only a year's time.
3) A momentum transfer tether hundreds of kilometers long and extending down into the atmosphere is more likely to be struck by cataloged objects near the top of the tether, which is the location proposed for the “power station” needed to keep the tether's orbit from decaying. The probability of a collision between such a tether and a cataloged object is approximately
P_{hit} = 0.0268 d T,
where d and T are defined as above. As an example, a 400 km tether has a collision probability of about 3% over 1 year, assuming an “average” catalog object size of 10 cm (in the case of thin tethers, it is the size of the “impactor”, not the tether, that determines the frequency of collision). This neglects the dimensions of any “power station” or other unit of similar size attached to the tether, which will significantly increase the collision chances.
It is clear that LEO is far from “empty”; indeed, it is probably not stretching the truth to say that the Earth is now surrounded by a shell of man-made particles of all sizes, the number of which increases every year as space-faring nations continue to make use of near-Earth space. This aspect of the orbital environment, which is considered a “nuisance” to conventional spacecraft, poses real engineering challenges to those who wish to place kilometer-size or larger objects in low Earth orbit. It certainly seems possible to design a multi-strand tether to take an impact by a millimeter-size debris particle; however, designing a tether or space elevator to withstand a hypervelocity impact by a Delta upper stage is orders of magnitude more difficult. Collision avoidance schemes, similar to that used by the International Space Station to “get out of the way”, may offer hope, but more analysis is needed to demonstrate the viability of such an approach. At present, the population of LEO is now to the point where there is little, if any, room for “elephants,” and, unless strict debris mitigation procedures are adopted by all nations, the future looks even worse.
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