Back in 2006, I helped a high-school student (Daniel Rodrigues) who was interested in momentum-exchange tethers to write a paper for a high-school class about a concept for a tether that would remove spent geosynchronous satellites from their orbits quickly, putting them into an elliptical orbit with a perigee that would intersect the atmosphere. He recently recovered the paper and sent it to me at my request, and I am publishing some of the more relevant sections of it here, with minor edits and occasional expansions for explanation.
Background on Momentum Exchange Tethers
The Momentum Exchange Tether is a concept originally pioneered by Hans Moravec in 1977. This orbital facility, essentially a rotating cable in orbit of the planet, had the ability to touch the surface of the planet every 20 minutes, and lift payloads into orbit. Carroll, in 1991, evolved this design into a totally in-space system able to lift payloads from sub-orbital trajectories and toss them into higher orbits . The Momentum Exchange Tether system was refined further in 1998 by Bangham, Lorenzini, and Vestal, who designed the system to transfer payloads from LEO to Geostationary Transfer Orbit (GTO), and who concluded that the system should be composed of two separate facilities: one at an altitude of 2019 kilometers, and another at 25048 kilometers . In 1999, Hoyt and Uphoff reestablished the one tether design due to simplicity concerns, but retained the LEO to GTO configuration. Since this study, the Momentum Exchange tether has been refined by Hoyt once again in 2000, by Sorensen et. al. in 2003, and finally Hoyt once more, also in 2003. As of now, the standard Momentum exchange tether is situated in a GTO, rotating so that its angular velocity and orbital velocity, when combined, equal the orbital velocity of it’s payload in LEO. At the tether’s perigee, it rendezvouses with the payload, rotates 180 degrees, and releases the payload. This adds momentum to the payload, but takes it away from the facility, consequently lowering its orbit. However, ballast at one end of the tether disallows for a significant drop in altitude. The station is then reboosted, and is then ready for another payload. The largest portion of the system, the cabling, is composed of a series of interlocking primary and secondary lines, a design known as the Hoytether .
The Problem with Orbital Debris
This investigation aims to apply the momentum exchange concept to the deorbiting of unwanted satellites (otherwise known as orbital debris, or simply debris). Orbital debris consists of inactive spacecraft, spent rocket stages, spacecraft fragments, and other miscellaneous objects . Objects in the .01 to 1 cm size category can cause significant system damage, and objects larger than a centimeter can conceivably be catastrophic . Additionally, spacecraft can only be shielded against debris up to 1 cm, due to mass practicalities . Therefore, it can be concluded that orbital debris is a considerable threat to spacecraft.
Orbital debris can be especially troublesome in Geosynchronous Earth Orbit (or GEO), a very valuable region in space. At GEO altitude (35,678 km above the Earth’s surface) satellites orbit the Earth at the same rate as the Earth rotates. This means that the satellites stay over a fixed point on the planet, which as obvious commercial value for communication satellites. However, this region in space is rapidly filling with debris. As more satellites are put into GEO, the amount of debris only increases. As of January, 2005 there were 153 tracked, uncontrolled objects in GEO, or about 14% of all known objects in GEO . These uncontrolled objects will only endanger the, on average, 21.1 satellites that will be launched annually into GEO from now into the near future, which is a number certain to increase in time .
A momentum exchange tether will be designed that is capable of capturing uncontrolled satellites and orbital debris of moderate dimensions in Geostationary Earth Orbit (GEO) with a maximum mass of 5400 kilograms (encompassing 83% of satellites expected to be launched, into 2013), and propelling the debris to a negative perigee altitude, ensuring the destruction of the debris. The momentum exchange tether will also be capable of capturing satellites in Geostationary Transfer Orbit (GTO), and slinging the payload into Geostationary Earth Orbit, losing momentum gained as a consequence of deorbiting debris.
The design of the tether system began with the analysis of the change in velocity required to remove a satellite from GEO into a deorbit ellipse intersecting with Earth’s atmosphere before perigee. The perigee of this ellipse was selected to be at an altitude of -100 km. This was done for several reasons: First, it ensured the destruction of the debris with a moderate margin for error, and second, it allowed for some flexibility in the coming design stages. The delta-v was set up between the initial orbit of the debris, GEO (circular equatorial orbit with an altitude of 35786 kilometers) and the target ellipse, which intersected with the Earth’s atmoshere. An infinitely massive ballast was also assumed in these initial, preliminary calculations, in order to simplify the center of mass variables. To calculate the speed of a satellite in GEO, the following equation was used:
This equation solves for the tangential velocity of an orbiting body (v), taking into account the Gravitational Parameter (mu) and the radius of the orbit (r). Afterward, the specific mechanical energy of the target ellipse was calculated. The specific mechanical energy (epsilon) is essentially a value that gives the energy per unit of mass of an orbiting body. The equation for this value is as follows:
where “a” is the semi-major axis. From the specific mechanical energy, tangential velocity of an orbiting body can be calculated using an alternate from of the equation for specific mechanical energy:
This equation was used to calculate the tangential velocity of the payload at the apogee of the target ellipse (after release from the tether). When the velocity of the payload in GEO is subtracted from this value, a new value of -1.5017 km/sec is calculated. This means that a vector with a magnitude of 1.5017 km/sec must be applied to the payload, in the opposite direction of the path of the payload, which accounts for the negative value.
Tip Velocity and Rendezvous
Once it was calculated how much the debris needed to be slowed down by (1.5017 km/sec), the actual tether facility could be designed. The first step in this process was to determine how the tether would perform its intended task. In this particular configuration, the tether would be situated below its payload. The tether’s center of mass (CM), about which it rotated, would be traveling in the same direction as the payload, albeit slower. The tip of the tether configured to capture the debris would also rotate in the direction the debris was traveling. Capture would occur with the payload directly above the tether’s CM, when the tether reached its apogee. The tether would then release the debris one-half of a rotation later, decelerating the debris into the above-discussed deorbit ellipse. Possibly the most critical value in this scenario would be the tether’s tip velocity, which is the velocity the tether would impose onto its payload. As mentioned previously, the debris needed to be slowed by 1.5017 km/sec. But because the tether imparts this velocity onto the payload in two ways (this will be elaborated upon in a moment), this value was divided by two, which gave the value of 0.75085 km/sec as a tip velocity.
Rendezvous between the payload and tether tip would not be possible if the velocities were not matched . Therefore to find the speed of the tether’s CM, the tip velocity was subtracted from the velocity of the debris (3.074 km/sec). Therefore, the combination of the tether’s CM vector and the tip vector would equal the vector of the debris. In this scenario, the tether’s CM was calculated to be traveling at 2.32315 km/sec, at apogee. From the debris’ point of view, because vectors along the x-axis would be matched, the tip of the tether would be approaching from below it, along the y-axis. When the payload is released, 180 degrees later, the tip vector is once again added to the tether’s CM vector. However the tip vector is not 0.75085 km/sec, but is now -0.75085 km/sec. Therefore, the payload is released at the desired velocity of 1.5017 km/sec, and enters the deorbit ellipse. Also, it is now clear why the tip velocity was equal to the desired velocity divided by two: the other half of the velocity to be imposed on the payload came from the CM vector. This allowed for a capture with a 0 km/sec relative velocity when the tip was rotating with the debris and the CM, and allowed for the desired velocity of 1.5017 to be attained once the CM and tip were traveling in opposite directions. It is important to note that because an infinitely massive ballast mass was assumed in these initial calculations, actual velocities before capture would be different. However, these variables would change commensurately, retaining the 0 km/sec relative velocity.
The next component that must be analyzed is the ballast mass. The ballast serves as a mass that stores momentum, allowing for smaller changes in altitude after release . The simplest and most efficient way to fix a ballast mass to a tether station is to utilize the rocket the station was launched in, which lowers initial launch cost by preventing the launch of additional mass to be used as ballast. In this study, the Ariane 5-ECA launch vehicle was chosen as the most suitable rocket. Its high capacity (10,500 kg) allows for massive satellites such as the tether station to be launched, and it has the capability to launch into a GTO . The upper stage on the vehicle, the ESC-A, has a dry mass of 4540 kg , which is therefore the mass of the ballast.
On the opposite side of the station is the capture device. The purpose of this device is to physically anchor the payload (in this case, orbital debris) to the tether itself, once position and velocity is matched. The design of the device must incorporate some margin of error. A mass of 200kg was estimated for this device, which will be capable of securing payloads of at least 1 meter long in any dimension. This is because debris in GEO that is tracked must be at least a meter wide to be traced, due to limitations in radar technology .
Concept of Operations
The facility will be launched in an Ariane 5-ECA rocket equipped with an ESC-A upper stage, into its orbit. The tether can then be fully deployed, and spun up to an angular velocity of 0.01882 radians/sec. This value will remain constant. The system is now ready for operation. The angular velocity of the system gives the tip a tangential velocity of 1.0921 km/sec. Added to the CM velocity of 1.9819 km/sec characteristic to the system’s current orbit will yield a value of 3.074 km/sec, identical to the orbital velocity of debris in GEO. After capture of debris, the CM of the tether shifts, toward the tip. This slows the tip velocity to 0.75085 km/sec, while accelerating the CM velocity to 2.32315 km/sec. This is because the tangential velocity of the point where the CM was shifting to is added to the old CM orbital velocity. This acceleration causes the altitude of the facility to increase, and is the first illustration of momentum exchange. After half a rotation, the debris is released. Once again, the CM shifts up, and accelerates, increasing its velocity. This also increases the station’s altitude, just as the previous maneuver. After release, the debris enters its deorbit ellipse. The velocity of the debris is now 1.5723 km/sec, at a radial distance (from the Earth’s center) of 42066.2 km. The debris under these circumstances would travel in an ellipse with a perigee altitude of -67.5 km. This means that the debris would enter the Earth’s atmosphere before its minimum altitude, allowing for a significant margin of error.
After release, the tether would be in a significantly higher state of energy, and therefore a higher orbit. The tether must be brought back to an altitude where it can perform its task of deorbiting unwanted space junk. There is a nearly “free,” and almost instantaneous way to bring down the facility. This would involve capturing a functional satellite on its way to GEO in a GTO intersecting with the fully boosted tether facility. The new satellite would interact with the tether in the exactly opposite way as the debris, as long as it has the same mass as the debris that was deorbited. If not, on board thrusters could complete the slowing of the facility to its original orbit. But inserting a functional satellite into GEO would save a significant amount of propellant, and would eliminate the need for an entire upper stage on the new satellite, saving more money. On a 5400 kg satellite, almost 2000 kg is used as on-board propellant and thrusters. This would mean a reduction in launch costs from Earth into GTO, which is typically about $10,000 per pound.
Of course, sending debris crashing through Earth’s atmosphere raises the question of safety. However, reentry of space debris is a very common occurrence. In the past 40 years, there have been over 16,000 known re-entries of cataloged space objects, without significant damage or injury. This is due to both the fact that most if not all of the debris disintegrates in the atmosphere, and the fact that any remnants of the doomed spacecraft have a very low probability of impacting populated areas. Regardless of the unlikeliness of a ground impact in a populated area, the decision to deorbit any particular non-functional spacecraft will have to be made on a case-by-case basis. Large objects have been known to survive reentry in past, and the reentry of any spacecraft containing radioactive substances is out of the question.
This investigation has outlined the basic configuration of a momentum exchange tether in GTO capable of both deorbiting debris, and putting new satellites in GEO. This tether will increase lifetimes of satellites in GEO by reducing the threat of debris, while reducing the cost of launching new satellites. Orbital debris is accumulating rapidly, and a solution such as the momentum exchange tether needs to be considered. GEO is a valuable natural resource that needs to be conserved, just as any other. The combination of a less hazardous environment in space with lower launch costs is certain to stimulate the development of a stronger space infrastructure, undeniably helping humanity expand its horizons.
1.Hans Moravec, A Non-Synchronous Orbital Skyhook, AI Lab, Computer Science Dept., Stanford University, Stanford, Ca. 94305
2.Carroll, Preliminary Design of a 1 km/sec Tether Transport Facility, March 1991, Tether Applications Final Report on NASA Contract NASW-4461 with NASA/HQ
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18. R. P. Patera, and W. H. Ailor. The Realities of Reentry Disposal, A98-43901 12-12, Spaceflight Mechanics 1998: Proceedings of the AAS/AIAA Space Flight Mechanics Meeting, Monterey, California, February 9–11, 1998
19.Foward, Hoyt, Failsafe Multiline Hoytether Lifetimes, 31st Joint Propulsion Conference and Exhibit, AIAA 95-2890, 1995
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