Copyright (c) 2001 by the Space Frontier Foundation. All rights reserved.
[Original word document also available online.]
Abstract
Over 5,000 tons of junk in orbit around the Earth cost billions and billions to launch. This debris should be salvaged and serve as ballast for a Low Earth Orbit (LEO) tether. Such a tether can double the payload delivered by an orbiter to LEO. Used as a junk man's ladder, this tether can forward payload to the Moon’s orbit, cutting the total mass to support a Moon mission in half. Design of a junk collector to retrieve 5 to 10 ton pieces of junk will make these savings plausible.
Tethers are an attractive option in space flight because they can accumulate energy gradually, and release it to toss a payload. An Electrodynamic Tether (EDT) can use electricity to push against the Earth's magnetic field to climb or to replace orbital energy from a payload catch or toss. Using such an EDT to gather some junk for ballast for the ladder would be cheaper than launching mass from Earth. A junk collector design based on a 20-km tether powered by 80 kW of solar cells is proposed. Using junk for ballast, the LEO tether could be built for a mass investment of 140 tons launched to orbit. This "Junk Man's ladder" can lift 10 tons every six days from LEO and send it to the Moon, with 1MW of solar cells.
Tethers
Tethers (two objects
in orbit connected by a wire) have exciting potential for frequent flyer
miles. In the simple case, the two
masses will hang so that the wire points to the center of the Earth. Thus we have one mass in a low “orbit”,
moving slower than local circular velocity, and the other mass in a high
“orbit” moving faster than the local circular velocity.
Copyright
© 2001 by the Space Frontier Foundation.
All rights reserved.
For simplicity, this review will assume circular
orbits. Tethers can be in an ellipse;
those are just more complicated to discuss.
The two connected objects will move at the orbital
velocity of their center of mass. If
they have unequal masses, the lighter end will be farther from that center of
mass than the heavy end. The distance
from the center of mass determines the energy that can be given to either
object by a toss. If the wire is cut, or
released, the two masses will separate and follow a pair of ellipses. The high
mass will be in a ellipse whose low point (perigee) is the location of the
release, and whose high point (apogee) will depend upon the length of the
tether and the ratio of this mass to the combined mass before the release.
If the heavy end has 10 times the mass of the light end, the light end will be 10 times farther from the center of mass. If this tether was 66 km long, the light end will be 60 km from the center of mass. Figure 1 shows this case. For short tethers, the tether toss adds about 6 times the distance from the center of mass to the ellipse that is the new orbit. If released, the light end would be in an ellipse that ranged from 60 km to 420 km from the original center of mass orbit. The heavy end would only be 6 km from the center of mass. When separated, the heavy end will range from 6 to 42 kilometers from the original center of mass orbit.
For lifting freight from the Earth to higher orbits, large mass will make most of the orbit change in the payload, with only minor changes to the orbit of the lifting station. If the heavy mass were 100 times the mass of the light end, in the example above, the light end would be 65.3 km from the center of mass. Then the heavy end will shift its orbit by 0.65 to 4.5 km.
For long tethers, calculating the new orbit is more
complicated (its not just 6 times). The
ratio of the masses remains an important consideration because it determines
the distance from the center of mass to the payload.
Guidebook
Many of the calculations in this paper are based on
the Guidebook for Analysis of Tether Applications; (Ref. #3) often referred to as the Guidebook
in this paper.
Most of the numbers for an EDT are subject to
variation. This is an elaborate back of
the envelope exercise. Precision is not
claimed.
Junk Man's Ladder
The ladder described here consists of a trapeze at the low end and a sling at the high end. There is also a large central mass, which will act as a flywheel to store kinetic energy for either end of the ladder. Payloads will be caught at the trapeze, and then carried up the ladder an appropriate distance and tossed outward. For a low catch or a high release, the flywheel will have to supply energy. For a low release or a high catch, the payload may put energy back into the flywheel.
The high end of the ladder is moving faster than the inertial mass
(ballast) and acts as a sling to throw payloads to higher orbits. By using a large inertial mass, and a long
tether, such a sling can slay Goliath sized problems. A payload tossed from the right point could
be in an ellipse that reaches the orbit of the Moon or Geosyncronous Earth
Orbit (GEO – a 24-hour or Clarke orbit).
The high end can catch a payload, but most of today's focus is on payload leaving Earth, so it will usually be called the sling end of the tether.
If the ladder is supporting traffic leaving Earth, it will have to replace the energy used to drive that traffic. This may be done if there is solar power that can drive an electric motor that uses the Earth's magnetic field as part of that motor.
Any mass can serve another use and also be valuable
as ballast. This mass can be anyplace on
the ladder, only the center of mass needs to be at the right altitude. Hotels or low gravity industry can be served
by the ladder and add to the ballast.
Variable apparent gravity is available.
Low gee environments exist near the center of mass. The trapeze end provides 20% of gravity. Industry that needs low gravity can pick any
level they want.
A hotel at
400 km altitude would have an apparent gravity (local gravity offset by
centripetal force) of about 11% of earth standard gravity. High enough to keep things in place and
provide orientation but low enough to be a novelty. Hotels that can tolerate occasional jerks,
(and most can) could do double duty as ballast for the tether.
Illustrating some Tosses
Figure 2 and table 1
display some information about a tether with center of mass at 700-km
altitude. This seemed to be high enough
to lower the trapeze velocity, but stay under the Van Allen belt. At this altitude, circular velocity is 7.504
km/s.
The first 5 rows in table 1 display data about the
use of this ladder for sending payload to the Moon. The trapeze has a length of 540-km (making it
160 km above the Earth). The (sub-)
orbital velocity is only 6.932 km/s.
Something standing or hanging on the trapeze would have a net
acceleration of about 20% of Earth gravity. It would require 2.44 kW hours to
raise a kilogram from the trapeze to the center of mass.
The sling or tossing end has a length of 1772 kilometers (the total length of the ladder is 2312 km). This sling must move at 9.383 km/s to keep up with the center of mass. Released from this radius, the ellipse touches the orbit of the Moon (384,400-km), with a slight excess. Ignoring the Moon, an additional velocity change (or delta V) of 0.8 km/s would be needed to circularize the orbit. The gravity of the Moon makes this interesting, but useless data. Note that prior to release, the payload would have a net centripetal force of almost half the gravity on Earth.
Raising a payload from 160-km altitude to 2472-km altitude can be done with an elevator or maglev motor. It would take 9.98-kilowatt (kW) hours of energy to move one kilogram from the trapeze to the top of this ladder. In California, when we can get kW-hours, we have been paying about 10 cents for one. That translates to about $1 per kg, or $0.45 per pound. Using launch cost assumptions later in this paper, LEO solar-cell kW-hours cost 25 cents, so the cost is closer to $2.5 per kg. The energy at the trapeze is less than half the energy at the sling. (The orbiter supplies 7.07 kW-hours per kg.)
The Orbit, Apogee, and Launch (rows 6-8 in table 1)
were an effort to estimate the velocity a conventional orbiter needs to get
into a 160-km altitude orbit. The launch
velocity (7.954 km/sec) produces an ellipse whose apogee is at an altitude of
160 km. Circular orbit velocity is 7.808
km/sec, so the orbiter needs an additional 0.05 km/sec (or 50 meters/sec) of
delta V. The trapeze end of the ladder
is moving at 6.932 km/sec. A more
eccentric ellipse that reaches 160-km altitude with a velocity of 6.932 km/sec
needs only 7.242 km/sec at the surface of the Earth.
These numbers ignore atmospheric losses and the
boost from the rotation of the Earth. If
we were in an equatorial orbit, the rotation of the Earth would add 0.464
km/sec – so an equatorial launched rocket only needs to supply 6.778 km/sec to
catch the trapeze. By contrast, a simple
equatorial launch would reduce the 7.808 km/sec to a relativistic 7.563 km/sec.
The trapeze reduces the rocket delta V from 7.56
km/sec to 6.78 km/sec, saving 0.78 km/s.
Putting these two numbers in the rocket equation gives an estimate of
the larger payload: between 2 and 2.5 times the original payload.
Non-equatorial launches have higher velocity
costs. A launch from Las Vegas would
only get a boost of 0.375 km/sec. Cape
Canaveral still has 0.408-km/sec rotational boost. An equatorial launch is the optimum, but we
have been living with lower assists for 40 years.
Rows 9-12 in table 1 look at a toss to GEO. The circular velocity at GEO is calculated
first. A release from 1317 km above the
center of mass follows an ellipse that reaches to GEO. An additional delta V of
1.3 km would be needed to convert to circular velocity of 3.072 km/s. At the 1317-km point, the ladder is moving at
8.9 km/s. The apparent gravity would be
38% of Earth normal (centripetal force up is greater than local gravity). Climbing the ladder from the trapeze to 1317
km above the center of mass costs 8.07-kilowatt hours. This ladder can support solar power
satellites or communications satellites as well as Moon traffic.
Another interesting “toss” is at a sling length of
1173 km, when the payload ellipse reaches 31,450 km. This is interesting only if there is another
ladder at GEO with a trapeze length of 10750 km. This GEO ladder, with a sling length of 8805
km, would forward payloads to the Moon. Some of the cleanup junk could be
tossed toward GEO, to serve as inertial mass for the second tether.
Two
cooperating tethers probably need to be in equatorial orbits. Orbits that cross the equator get a torque
that moves the plane of the orbit. The
rate of change is different for different altitudes above the Earth. It would be difficult to keep two tethers in
the same plane, unless they orbit over the equator.
Two such tethers would form a conveyer belt; the
payload will take weeks to get from LEO to the release from the GEO
tether. This combination reduces the
demands on the LEO tether. It only takes
7.48 kW hours to lift 1 Kg to 117k km.
The remaining 2.5 kW hours for the Moon can be supplied at GEO, where
the sun is always up. Unfortunately, it
takes building two tethers, so will not be discussed in detail.
Hazards
Unfortunately, the rosy picture just described has
some brutal facts that appear to put it out of reach.
The first brutal fact: at current and projected
launch costs, putting a large inertial mass (ballast) into orbit is
prohibitively expensive.
Second brutal fact: there is an erosion problem due
to interplanetary dust falling to the Earth.
These are mostly very small particles, which will pit and erode a
tether. A wire has an expected lifetime
that is a function of the diameter of the wire.
The assumptions are that there is some excess strength designed into the
wire that can survive erosion by dust, but that the wire will eventually be cut
by a larger particle. For individual
wires or strands of a complete tether, the micrometeorite hazard is important.
Third brutal fact: for tethers over 10-mm diameter,
the risk of impact with debris (junk) becomes larger than the risk due to
micrometeorites. There are about 700
pieces of junk (with mass greater than 5 tons each) in orbits between 900 and
4000 km altitude.
On page 27
of the Guidebook, there is a calculation that an Earth-based Space Elevator
would expect about 1.2 cuts per year due to the debris orbiting less than 4,000
km above the Earth. The ladder discussed
in this paper reaches between 160-km and 2500-km altitudes. It would face the same risk of cuts.