(1)
William J. Cooke1, David B. Spencer2, B. Jeffrey Anderson3, and Robert M. Suggs4
1Computer Sciences Corporation, bill.cooke@msfc.nasa.gov
2Aerospace Engineering, Pennsylvania State University, dbs9@psu.edu
3Space Environments Team, Marshall Space Flight Center, jeff.anderson@msfc.nasa.gov
4 Space Environments Team, Marshall Space Flight Center,
rob.suggs@msfc.nasa.gov
Abstract A number of considerations on tether interactions with the orbital environment came to light at a recent Workshop on Orbital Debris, Large Space Structures and Tethers, held at the Marshall Space Flight Center in Alabama. Specialists in the areas of orbital debris, meteoroids, orbital dynamics, and tethers convened for the purpose of providing a forum for presentations and discussions of problems and issues regarding tethers and large space structures. In this paper, we present some of the results achieved at this meeting, focusing on:
Risk calculations: We present some views of workshop participants as to how the NASA Orbital Debris Engineering Model (ORDEM-96) and current meteoroid model can be applied to the question of tether survivability. An approach to calculating the effective debris and meteoroid cross-sectional area is discussed, and, as tethers essentially fly in a gravity gradient mode, show that a modification of the Earth shielding factor is needed when using the simple isotropic meteoroid flux approximation.
Collision avoidance issues regarding tethers: Current Space Surveillance Network (SSN) capabilities for tracking tethered systems are, at best, marginal for supporting applications such as collision avoidance for manned systems. Uncertainties in orbital position from SSN tether tracking rapidly exceed the collision avoidance “watch box” for ISS and Shuttle. This would lead to unacceptably high numbers of maneuvers for either vehicle. One difficulty is the inability of detection systems to separately identify the tether end masses, which leads to frequent misidentification of object element sets. Likewise, current systems are not set up to utilize GPS derived data in the tracking process, nor can they treat constantly thrusting vehicles.
Momentum for the use of tethers in space is clearly building, and current interest in space solar power, solar sails, large space-borne telescopes, and interferometers makes it clear that other large structures are also on the way. The LEO (Low Earth Orbit) population, including the orbital debris and meteoroid environments, proves to be a significant constraint to the development of these structures, and so the proper evaluation of the interaction of large structures with the environment has become very important. Unfortunately, there have been several cases where substantially different answers for a single problem were obtained by various workers, and discussions within the community indicate there are several subtle issues regarding the calculation of cross sections, spatial densities and directional effects which it would be useful to examine and clarify.
As Marshall Space Flight Center is actively engaged in the development of space tethers (a propulsive tether, ProSEDS, is scheduled for a mid-summer launch), the Environments Group of the Center's Engineering Directorate hosted a small workshop attended by several experts in the orbital debris, tether dynamics, and tether construction fields in an attempt to resolve the discrepancies in the calculation of tether lifetimes, and to examine in more detail the ramifications of placing such large structures into an environment becoming crowded by satellites, empty upper stages, explosive fragments, paint flakes, and other detritus, both natural and man-made. This paper shall outline the method of calculating the probability of a tether sever that many of the participants deemed sound, and will present results of recent work regarding tether interactions with active satellites, including manned vehicles such as the International Space Station (ISS).
For a conventional spacecraft, the probability of penetration by a meteoroid or orbital debris fragment is given by
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(1) |
where P is the probability of penetration, F is the cumulative flux of meteoroids and debris capable of penetrating the vehicle, A is the appropriate cross-sectional area, and T is the time the spacecraft remains in orbit (or the anticipated duration of the mission). This equation seems relatively straightforward, and is, for common types of spacecraft; however, the thinness and orientation (approximately gravity-gradient) of tethers introduce a few differences in the calculations of the flux and the effective cross-sectional area. To be more specific, the probability of severing a tether can be expressed as
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(2) |
where dF is the differential (with respect to size) flux of particles capable of severing the tether and A is the effective cross-sectional area of the tether-particle combination (also a function of particle size).
The Flux
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The flux of orbital debris on a long, thin object in a gravity gradient orientation may be properly calculated by realizing that orbital debris will intersect the object in the plane of motion (see Figure 1), necessitating the use of the “cross-sectional area” flux. Many debris models, such as NASA's ORDEM96, return the cumulative cross-sectional area flux, so their output may be used (the documentation should be consulted to determine if this is indeed the case), provided the results are then differentiated with regard to particle diameter to obtain the differential flux. Meteoroids will approach the tether from all directions, except from the bottom, where the Earth provides some degree of shielding. The cumulative surface area flux (which is one-fourth the cross-sectional area flux) returned by meteoroid models, such as that by Grün, is therefore also appropriate, and can be used along with the given gravitational focussing factors, provided that it too is differentiated with respect to particle size. However, the orientation of the tether requires a new Earth shielding factor, which is simply the integral of a sin2 term over the appropriate limits and normalized to 1. Kessler (2000) has given the tether shielding factor as
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(3) |
where re is the radius of the Earth and h is the tether altitude, both in km. Once this has been applied to the differential meteoroid flux, the total differential flux on the tether, dF, may be obtained by adding this result to the differential orbital debris flux.
The Area
Differences in the methods of calculating the effective tether cross-sectional area account for the bulk of the discrepancies in tether meteoroid and orbital debris (M/OD) lifetimes, as the final results are very sensitive to the areas used in the calculations. For example, many novices simply take the product of the tether length and diameter (or width) to be this area, or may equate it to the tether's surface area. Both are erroneous, as the sizes of the particles that can sever a tether are not small compared to its diameter, and must be taken into account in the calculation of the collision cross-section. Indeed, it is the fact that the tether diameter is not very large compared to the severing particle that necessitates the use of the differential fluxes, rather than the cumulative values used in the calculations for other spacecraft. The consensus of the participants at the Marshall workshop is that the cross-sectional area of the tether to be used in M/OD lifetime calculations is given by
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(4) |
where d is the particle diameter, t is the tether diameter or width, L is the tether length, and ds is the size of the smallest particle needed to sever. If we follow current geometrical arguments and assume that the size of the smallest severing particle is independent of speed, (4) may be rewritten as
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(5) |
where f is the ratio of ds to the tether diameter.
Example
Using the method outlined above, a 20 km long tether with a diameter of 2 mm at an altitude of 400 km has a Pcut of 0.88 over one year, provided a particle of one-third the tether diameter can cut the tether. The integrated product of df A yields the sever rate, which amounts to 2.15 cuts per year.
Note: Kessler (2000) has shown that the use of the cumulative flux in back-of-the-envelope tether lifetime calculations underestimates the sever rate by ~20%, and points out that this uncertainty is small compared to uncertainties in the environment models and relations estimating sizes of particles required to cut a tether.
The LEO Population
As a by-product of guarding the United States against incoming missiles, U.S. Space Command (SPACECOM) keeps track of the approximately 10,000 objects in Earth orbit that are greater than 10 cm or so in size. The object types are diverse - active and inactive satellites, inert upper stages, explosion debris, radioactive droplets leaked from space nuclear reactors, and so on. As might be expected, the distribution of these objects is anything but uniform, especially in LEO. Drag induced by the Earth's atmosphere keeps space relatively clean out to altitudes around 550 km; objects placed in orbits higher than that will tend to stay in orbit for decades, centuries, or longer. If the distribution of cataloged objects out to 1200 km altitude is plotted on an inclination-altitude plot, several concentrations or groups immediately become apparent (Figure 2). Intuition would indicate that placing a tether or other large space structure in an orbit that would cause it to pass through these “more populated” regions of LEO may result in an enhanced chance of a collision, as the number of close approaches to trackable objects (which are of such size that a tether would be severed, regardless of its type) would increase. As many of these regions (such as the sun-synchronous band) contain expensive, active satellites (in military parlance, “valuable assets”), recent work at Marshall has been focused on determining the probabilities of tether close approaches to objects in the catalog. We have defined a “close approach” to have an encounter distance of 500 m or less between the object and any part of a tether; it is felt that such encounter would be close enough to induce nervousness into the owners/operators of an expensive imager, a military satellite, or a manned spacecraft. The close approach probabilities were investigated in two different manners, the first being an analytic technique developed at Johnson Space Center and published by Mark Matney (2000). The second approach is obvious, as it involves propagating tether in an arbitrary orbit along with relevant objects in the catalog, calculating approach distances at each time step.
FIGURE 2. Distribution of cataloged objects in LEO (December, 2000)
The Method of Matney
Incorporating the now-standard spatial density relation developed by Kessler (1981), this method was used to estimate the probability of collision between a 4 km tether in a circular orbit and the International Space Station (~0.01 to 0.02 per year). The authors have modified the calculations slightly in that we evaluate the probability of any part of a 20 km tether passing within a sphere of 500 m radius centered on a target object, which was not too difficult, considering that the calculations scale linearly with the encounter distance. It is therefore no great surprise that the results presented below in Figure 3 are approximately an order of magnitude greater than those Matney found for ISS, as our “target sphere” has a diameter ten times larger than that of his.
One particularly illuminating aspect of Figure 3 is that it shows that tethers launched into orbits with altitudes around 800 km and inclinations greater than 80º have a high rate of encounter with sun-synchronous satellites. Indeed, if the tether orbit has an inclination that is the complement of the sun-synchronous inclination for that altitude, then it can encounter every sun-synchronous satellite at that altitude in a year!
FIGURE 3. Tether encounters per object per year for select tether orbits.
Combining the above (which gives the number of encounters per object per year) with a catalog (which gives the number of objects that can intersect the tether orbit) enables estimates for the number of encounters with cataloged objects per year for specific tether orbits. The results are displayed in Table 1; note that the number of encounters increases with both altitude and inclination.
TABLE 1. Encounters per year as a function of tether orbit.
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Tether Altitude (km) |
Tether Inclination |
Encounters per year |
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400 |
28º.5 |
9 |
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51º.6 |
10 |
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88º |
16 |
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800 |
28º.5 |
166 |
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51º.6 |
187 |
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88º |
296 |
The “Brute Force” Approach
As mentioned previously, the second method used to investigate the encounter rate involved placing a tether in an arbitrary orbit, and propagating it along with catalog objects capable of close approaches, calculating the approach distances at each time step throughout the period of propagation. For the initial phase of this study, 20 km tethers were placed in 400 km, 28º.5 and 800 km, 51º.6 circular orbits and were propagated for 10 day time spans against the public catalog of December 6, 2000. The results are quite interesting, and seem to at least qualitatively bear out the findings of the first technique. The tether in the 400 km orbit had a close encounter with an object at most once every 10 days (usually nothing), whereas the tether in the higher orbit had anywhere from 2 to 7 encounters during the same time. Assuming that these numbers are typical, and scaling them to one year results in ~12 encounters per year for the 400 km, 28º.5 orbit and ~180 encounters per year for the 800 km, 51º.6 orbit, in surprising agreement with Table 1.
One advantage of this approach is that the types of objects encountering the tether can be determined. The tether at 400 km altitude passed near rocket bodies and exhausted upper stages, whereas objects passing near the higher altitude tether were debris fragments, rocket bodies, and satellites, both active and inert. In one particular run, spanning December 1 through 10, the high altitude tether passed within 5 km of 5 active satellites (2 DMSP, 2 DoD, and ASUSat). It is worth mentioning that some of these encounters were well away from the tether center of mass (in this case the geometrical center), where the dynamical libration of the tether could greatly reduce the approach distances under the right circumstances.
Discussion
Matney's expressions and the numerical simulations indicate that a 20 km tether will have close encounters with cataloged objects on the order of once per month for a 400 km orbit and every 2 days or so for an 800 km orbit, a difference of over an order of magnitude. That this is a consequence of the greater population at higher LEO altitudes is borne out by the numerical simulations, which show that approximately 2100 objects (roughly 25% of the catalog) have orbits that cross 800 km, compared to the 450 which can intersect 400 km. Clearly, tether projects can minimize the risk to themselves and other spacecraft by avoiding the densely populated regions of LEO, though the effects of atmospheric drag at 400 to 500 km altitudes would preclude long-lived missions, unless the tether was propulsive, in which case it could maintain altitude.
The study also reveals another way of reducing the number of encounters - shorten the tether length to the minimum necessary to accomplish the mission. Simulations done using the orbital elements of the TiPS tether (1000 km altitude, 63º.4 inclination) show that it encounters 4 to 5 times fewer objects than a 20 km tether, no doubt due to the fact that its length is 4 km, one-fifth the size of the other tethers studied. This is intuitive - a shorter length means that the tether spans a smaller range in altitude, which reduces the number of objects that can intersect it.
Finally, estimates of the probability of collision between an active satellite and 20 km tethers at 400 and 800 km altitudes can now be made. If it is assumed that the characteristic size of a typical active satellite is of the order of 10 m, and that about 6% of objects in the catalog are active satellites, then the expected number of collisions is approximately 180 encounters per year x 0.01 (ratio of satellite size to 1 km sphere used in simulations) x 0.06, or ~0.11 collisions per year. This is for a tether at 800 km altitude and 51º.6 inclination. Down at 400 km, the collision rate drops by an order of magnitude, to approximately 0.01 per year. Recouching in terms of probabilities, it is found that a tether in an 800 km orbit has a collision chance of approximately 10% per year, compared to 1% for 400 km.
The next tether mission to be launched, ProSEDS, will be initially placed in an altitude close to that of ISS, and will use electrodynamic propulsion to create an artificial drag force, resulting in a rapid decay of the orbit (re-entry is expected to occur after a few days, at most 2 weeks). Its successor, AirSEDS, is a much more ambitious mission, using the same electrodynamic motive forces to change altitude between 300km and 1100 km over the course of a year. Given that both these tethers have the potential to cross the orbits of ISS or the Space Shuttle, and that these manned assets must have their safety maximized, it is important to consider the consequences of flying tethers at manned altitudes. If the tether can be adequately tracked, the shuttle and ISS have the capability to maneuver out of the way of a projected collision. Question 1 then becomes “Can SPACECOM track a tether with sufficient accuracy to make collision avoidance feasible?,” followed by “Will the presence of one or more tethers in this regime increase the maneuver rate of ISS, and impact microgravity experiments onboard?”
ISS Collision Avoidance and Tracking Tethers
The current procedure for determining whether or not the International Space Station (or the Space Shuttle, when it is on-orbit) must maneuver out of the way of an incoming piece of debris is based on the intersection of two position error ellipsoids. The first of these describes the uncertainty in the ISS position and it is typically small (not much bigger than the ISS itself). The second measures the uncertainty in the position of the debris object at the projected time of ISS encounter; it can be rather large, depending on how well the object's orbit is determined. The degree to which these ellipsoids intersect determines the probability of a collision; if it is above the “red” threshold, ISS or Shuttle must maneuver. If it is between the red threshold and a lower “yellow” threshold, then the maneuver occurs at the discretion of management. Critical to the success of this scheme is the ability of SPACECOM's Space Surveillance Network (SSN) to track the debris object (or tether) with sufficient accuracy to construct a meaningful error ellipsoid.
Unfortunately, the procedure outlined above will not work for tethers. There are three main reasons:
Efforts by SPACECOM in tracking the TiPS tether illustrate the difficulties involved. Because of the erroneous center of mass determinations, the orbital elements produced for TiPS produced positions that grew in error at the rate of ~100 km per day. It was only after considerable post-processing by Naval Research Labs (NRL) (essentially an effort to properly re-identify the end masses) that the position error was reduced to a more reasonable 4.4 km. However, even this is considered quite large in the ISS collision avoidance scheme.
Some have proposed that the old “warning box” procedure employed by the Space Shuttle in earlier days be used for tethers. The idea is quite simple - construct a warning “box” big enough in size to encompass the volume of space where the tether might be, and maneuver if ISS is projected to pass through this box. Such a box, however, would be huge, roughly equivalent to a 6 km diameter sphere, assuming a 20 km tether with a libration angle of 10º, and the ability to propagate the center of mass to better than 4.4 km. This would force ISS to maneuver at least 2 times per year to avoid the tether.
REFERENCES
Kessler, D.J., “Derivation of the Collision Probability Between Orbiting Objects: the Lifetimes of Jupiter's Outer Moons,” Icarus, 48, pp. 39-48, 1981.
Kessler, D. J., Presentation at NASA Marshall Space Flight Center (2000).
McBride, N. and Taylor, E.A., “The Risk to Satellite Tethers from Meteoroid and Debris Impacts,” in Second European Conference on Space Debris, edited by B. Kaldeich-Schurmann and B. Harris, European Space Agency, 1997, pp. 643-8.
Matney, M., Kessler, D., and Johnson, N., “Calculation of Collision Probabilities for Space Tethers,” presented at 51st International Astronautical Congress, Rio de Janeiro, Brazil, October, 2000.
