3 December 2004

Abstract:
This document describes a conceptual design for a tether-equipped transfer station in high Earth orbit, which serves as a home port and main propulsion system for cruise ships carrying passengers between the Earth orbit and the Moon. The system serves compact ships that shuttle passengers on the hours-long trip to and from a station in Low Earth Orbit (LEO), providing 2,800-4,000 m.p.h. of velocity change; and more spacious ships that make the three day long journey to the Moon, which receive 2,600 m.p.h. of velocity change. Its orbit is selected to allow daily launches to both locations (without such a system, launch opportunities between LEO and the Moon only occur once or twice per month). This system conserves momentum, so that when inbound traffic balances outbound traffic, no propellant is required for the orbit transfers. The system can be built with existing tether material.

Introduction

Launch Windows and Orbital Planes

The Tethers and Trajectories

Station-Keeping: Follow that Moon

The Fixed Plane Tether

Ballast Requirements

Inclination Change Tether

Use with a Geosynchronous Space Elevator

Lagrangia Access

Conclusion

References, Notes, & Website Links

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Advanced rocket engines such as nuclear-electric ion or solar-electric ion engines do not solve the whole problem either. They greatly reduce the propellant cost, but they have poor thrust-to-mass ratio so the one-way travel time to the Moon would be many weeks^note_1. This makes Lunar voyages less appealing for tourism and also results in high cost.

A partial solution could be borrowed from the idea for "cycling" trans-Mars ships. In the early 1980's orbits were discovered that could allow a ship to "cycle" repeatedly back and forth between the Earth and Mars, with no major propulsive maneuvers required after the initial departure ^note_2. The cycler's economy comes from the fact that it does not stop at either destination; instead, a small tending craft would be sent out to meet it as it passes by.

Using the cycling concept for a trans-Lunar ship solves the propellant problem, and the transit time problem; however, when a cycling ship returned from the Moon, it would not be in position for another round trip for one or two months ^note_3. This is better than the Mars cyclers, which spend years getting repositioned for departure, but it's still a cost problem for tourism.

The approach described herein combines the cycler's benefits with a quick turn around time. It uses a long rotating tether (100 to 200 miles long) and a heavy ballast (two orders of magnitude heavier than the ship) to capture a ship returning from the Moon, and hold it in a circular Earth orbit, with a period of less than one day. In this way, the ship will be in position for another departure the following day. The momentum and energy the ship has upon arrival are stored (the ship holds its speed as it spins around, tethered to the ballast); when the ship is released from the tether (at the appropriate time), it will coast all the way to the Moon again.

This system then, can allow each cruise ship to travel between Earth and the Moon about every seven days. It can therefore serve a fleet of seven ships to provide a departure and arrival every day.

Because of its high orbital altitude, this system has the very desirable feature that if the main tether is cut or breaks, the ship cannot crash down to Earth; and at most a small propulsive maneuver (under 500 m.p.h.) will prevent it from leaving Earth orbit.

Launch Windows and Orbital Planes
Another requirement for daily Lunar departures is orbital plane alignment.

As depicted in the diagram below, most low Earth orbits (including those that originate at fixed locations in the United States, Europe, Australia, and the equator) are inclined with respect to the orbital plane of the Moon. This presents a serious impediment to frequent travel between terrestrial space ports and the Moon due to the inconvenience of changing orbital planes.

Opportunities to fly directly from an inclined Earth orbit to the Moon do occur twice per lunar month (when the Moon crosses the plane of the Earth orbit). This results in an inclined Lunar orbit which is different for the first and second crossing each month, so that a given landing site can only be visited once per month.

When tethers are used for other parts of an Earth-Moon transportation system (specifically a Space Elevator in LEO and a Lunavator in Lunar orbit), it becomes more economical to have small daily flights rather than large monthly flights (because most tether costs scale with payload mass, not usage rate). To provide daily flights to the Moon then, the spacecraft inclination must be matched to that of the Moon, using a intermediate orbit with a period of less than one day.

It should be noted that a mobile ocean-based space port (such as Boeing's Sealaunch system) could be positioned for a Lunar launch at any day of the month, however a low Earth orbiting tether system can not support such a system. This is because the LEO tether is close enough to Earth that it suffers from rapid nodal regression which will shift its orbital plane so that it is no longer aligned with the Moon's orbital plane.

For the system described herein, spacecraft would depart from a tether station in an inclined low Earth orbit then dock at a Transfer Station in a circular high Earth orbit which is in the plane of the Moon's orbit. They would wait in the high orbit a few hours, then depart the Transfer Station on a direct flight to the Moon.

The Transfer Station would use a rotating tether to conserve as much momentum as possible when providing the velocity changes required by the spacecraft's arrival and departure.

If this type of system is used for one-way cargo transportation, then solar electric ion engines can be provided to replenish the momentum carried away by the cargo.

The Tethers and Trajectories
This system uses a transfer station with three tethers. The main tether attaches the ballast to the winch station and is about 100-200 miles long. The winch will extend a very short (perhaps under a mile) tether to the Lunar cruise ship, which will assist the ship with docking. The third tether is long and thin (a 50-100 miles long) and is used for docking and trajectory control of the LEO shuttle.

The transfer station will orbit at an altitude of 21,700 miles, and the orbital plane matches that of the Moon. The orbital period is therefore 23.17 hours, or one and 1/28th orbit per day. This means that every day, there will be an arrival and a departure window for trans-Lunar trajectories. As shown in the diagram below, the trans-Lunar departure window will always be about 5 hours after the trans-Lunar arrival window, so the station must accommodate two ships during the 5 hour overlap.

During each orbit, there will be two windows for transfer to and from LEO (180 degrees apart), which correspond to the intersection of transfer station's and the LEO station's orbital planes. However, only one of the windows will be used (the one that has the best separation from the trans-Lunar windows). The capture and release of the inbound and outbound LEO ships is done simultaneously. In general, the orbit of the LEO station need not be synchronized to that of the transfer station. Instead, the LEO ship's velocity will be adjusted so that the travel time (nominally 5.5 hours) will be varied by up to 2 hours in order for the LEO ship to rendezvous with the LEO station (by boosting the departure velocity from LEO by up to 65 mph, and that from this station by up to 320 mph).

At the station's altitude, the velocity required for a circular orbit is 6,960 m.p.h. and the escape velocity is 9,843 m.p.h. To reach the Moon, the cruise ship will have to leave with nearly escape velocity, about 2,600 m.p.h. faster than the circular orbit velocity. When rotating around the tether system center of rotation with this speed at a distance of 200 miles, the ship will therefore feel 0.40 times Earth-normal gravity due to the centripetal acceleration (accel=velocity^2/radius).

The main tether is under quite a large stress, both from the load of the ship and the winch facility, and it own mass. Using currently available tether materials (using a carbon fiber material called Thornel T-40, derated by a safety factor of 2.88), its mass must be 2.1 times that of its tip load (i.e. the ships, the shuttle tether, and winch facility) ^note_4. Placing the tether system in a higher altitude orbit would decrease the mass of the main tether, but each orbit would take longer, therefore trans-Lunar departures would be less frequent; also, the travel time down to LEO would be longer.

From the tether system's circular orbit (before allowing for inclination change or travel time adjustment), a velocity decrease of 2800 m.p.h can send a payload on a transfer orbit down to a Low Earth Orbit (LEO), of 1610 mile altitude, with a travel time of 5.51 hours (for a 200 mile LEO, the velocity change would be 3283 m.p.h, and the travel time is 5.14 hours) ^note_5.

Station-Keeping: Follow that Moon
To allow daily departures to the Moon, the Transfer Station must orbit in the plane of the Moon's orbit. Because of gravitational pertubations, it must use small rocket thrusters to periodically correct its course, in order to maintain this orbit. The two largest sources of pertubations are the Sun's effect on the Moon, and the Earth's equatorial bulge's effect on the Station. As described below, this station keeping is feasible at the high orbit proposed for this station, but is not feasible for a station in Low Earth Orbit.

The Moon orbits in a plane which is inclined by 5.13 degrees to the ecliptic (the plane in which the Earth orbits the Sun). As the Moon passes above and below the plane of the Sun, the Sun's gravity acts to pull it back into it's plane. This pull causes the orbit to precess like a gyroscope: the nodes (or places where the orbital planes cross) will rotate in the eclliptic, and complete a rotation in 18.6 years. The Transfer Station orbit will also precess for the same reason, however, it's much closer to the Earth, and so has a larger orbital velocity. The Sun's effect is proportionally smaller and the precession period is much longer: 550 years). This difference in precession rates must be corrected for by rocket thrusts ^note_6.

Because the Transfer Station and the Moon are not in the plane of the equator, the Earth's equatorial bulge will also cause their orbits to precess. In this case, the Transfer Station's closer distance will cause it to suffer the higher precession rate. If uncorrected, its plane would wrap around the equator once every 80 years (the effect on the Moon is negligible).

As shown in the following table, the annual cost of keeping the Transfer Station in the proper orbit is a few hundred mile per hour of velocity change (delta V). Note that even though the precession due to the equatorial bulge has a slower period than the Moon's precession, the higher inclination makes the correction delta V about the same.

The station would use solar-electric ion engines to provide this correction. If the engines had an effective specific impulse of 3,000 seconds, (or exhaust velocity=65.8k m.p.h.), then each year, the station would consume 0.70% of its mass in propellant. Probably, most of the mass of the station will be ballast, composed of dirt sent from the Moon. During initial construction, arriving Lunar ballast must be slowed by 2,600 m.p.h.; therefore it would use propellant amounting to 4.0% of the ballast mass using the same engines (100%*2,600 mph = 4%*65.8 kmph).

For comparison, a space station in low Earth orbit would suffer a precession rate 365 times higher (at 770 miles) due to its proximity to the equatorial bulge, and would require over 200,000 m.p.h. of course correction every year if it were to try to stay in plane with the Moon!

The Fixed Plane Tether
The Transfer Station would orbit in the plane of the Moon's orbit. Ships traveling from this station to a LEO station will generally need to change to a different orbital plane as they depart. The simplest form of transfer tether station will also rotate only in the plane of the Moon's orbit, and therefore not be able to help with the inclination change process, so a rocket engine must be used for that (so it would only make sense for small inclination changes, such as for an LEO station in an equatorial orbit). However, the inclination change in high orbit will use much less propellant than would be required for the same maneuver in low orbit (due to the lower orbital velocity); also since the ride from this station to LEO is only a few hours, the ships used can be smaller than those used on the Moon-bound leg, which also reduces the propellant requirement.

The following table shows the rocket burn required as a function of LEO inclination. The in-plane velocity change comes from the tether, and the cross-plane change is the required rocket burn. Note the inclination in the table really is the change from LEO to the Moon's orbital plane. (the cross-plane velocity component is Vxfer*sin(angle), and the inplane component is Vcirc-Vxfer*cos(angle), where Vcirc is the circular orbit velocity, Vxfer is the speed at the top of the transfer orbit, and angle is the inclination change).

The length of the required tether is inversely proportional to the desired G-load at the payload. For a 2,600 m.p.h. tip velocity, a radius of rotation of 200 miles provides 40% of Earth-normal gravity (and a rotation period of 29 minutes). Longer tethers would offer lower acceleration, and are not more massive (they would be correspondingly thinner) however, they would have a larger likelihood of a collision with space debris. Passengers returning from the Moon will have become acclimated to zero gravity during the journey would certainly find anything greater than one Gee objectionable, so 40% of a gee is probably a reasonable compromise. [For one of the orbital plane change systems discussed below, a shorter is preferable.]

Using modern tether materials, the tether's mass will be a few times the payload mass. However, the tether must be designed to accommodate two payloads docked on the tip at once if this Transfer Station must provide an inbound and outbound spacecraft transfer every day. That's because the transferring spacecraft will each typically stay for several hours, and the stay times will often overlap.

Ballast Requirements
The amount of ballast the Transfer Station must carry depends on how widely the Station's orbit is allowed to vary. With a total mass of 100 times that of the payload, then the 2,600 m.p.h. boost it gives to the payload upon docking or releasing will cause a 26 m.p.h. change in the station velocity, 15 min change in orbit period, and a resulting 380 mile altitude change.

The payload transfers to and from low Earth orbit will occur at about the same point in the Transfer Station's orbit, so matched pairs will not accumulate distortion to the orbit shape (in fact, the capture and release events are normally simultaneous). The trans-lunar launches and receptions however, will not occur at the same point in the orbit (since the Moon moves a lot during the 3 day journey), so orbital perturbation will accumulate for several days before being averaged out over the monthly period.

Computer simulations suggestion that the 100x mass ratio will give an orbital altitude variation of about +-1,700 miles ^note_7. Assuming daily arrivals and departures, it should be possible to have several transfer stations in the same altitude band, provided they share the same average orbit period.

Inclination Change Tether
It is possible to build a tether system that can store and re-use the orbital energy of inclination changes. The simplest system would extend the small LEO transfer ship from the winch station on tether, which swings (librates) in the cross-plane direction. Winch pumping would be used to build-up the libration energy (like a child pumping on a swing-set). After an initial rocket-burn to start the swinging, the ship is winched out as it swings away through center, and winched back in a little as it starts back to center. The libration would be built-up in this way as the tether was extended about 100 miles. This only works for small inclination changes, and the ship must have low mass compared to the winch station, otherwise the swinging gets chaotic.

The following system is more general, and could be used to send the larger trans-Lunar ships down to an LEO station. It would use a large momentum wheel connected to a grapple via two tethers (with one tether connected to each end of the wheel's axle) with winches to change the plane of rotation of a payload.  The change in plane of rotation is accomplished by alternately tensioning the upper and lower tethers as the payload rotates around the center of gravity.

Like the fixed plane tether, this system would start out rotating in the plane of the Moon's orbit (at an altitude of perhaps 20,915 miles). First it receives an incoming payload from the Moon, then waits a couple of hours and releases the outbound craft towards the moon. It then spends several hours precessing its plane of rotation and adjusting its spin rate so the payload has the correct velocity and angle. When it's positioned correctly, it will release the payload toward the destination in low Earth orbit (LEO) and simultaneously capture the incoming payload from LEO. It then precesses back to the Lunar plane to repeat the process.

The momentum wheel must have the same amount of angular momentum as the payload (including the tether with grapple tip), so it might have a rim diameter of 40 miles (about one fifth of the payload tether length), and a rim mass of ten times the payload mass, and rim velocity equal to tether tip velocity. The wheel's axle is the hardest part to build, as it must be a few miles long (the length is proportional to the required precession rate ^note_8).

The tip velocity (relative to the system center of gravity) will generally need to be different for the trans-LEO capture and release than it is for the trans-lunar capture and release. This velocity change could be done by winching in or out the payload (the tip velocity will change to satisfy conservation of angular momentum) or by spinning down the payload tether (this also spins down the momentum wheel). In order for a motor to spin down the tether in a few hours, the tether has to attach to the motor with a rigid piece (or hub) about a mile across.

Use with a Geosynchronous Space Elevator
With the development of carbon nanotubes, the geosynchronous space elevator may someday offer an alternative to rockets as a means to leave the surface of Earth. In this remarkable system, a tether is strung from a satellite in geosynchronous orbit (22,300 miles up, so that its orbital period is 24 hours, and it position in the sky appears fixed), all the way down to the surface of the Earth; another tether is strung upwards to a counterbalance. An elevator is provide to carry passengers and cargo up and down the tether to geosynchronous orbit, with no rockets required. The difficult part of building one is that the tether must be very strong (for its weight); recently a new type of carbon fiber called nanotubes has been discovered, and it appears to be strong enough for the space elevator.

NASA is currently funding a program to study the geosynchronous elevator ^note_9.

The geosynchronous elevator can be used to send ships away from the Earth at escape velocity, by simply releasing them from the elevator at the correct altitude (about 28,700 miles). Getting to the Moon is a little more complicated, again because the Moon's orbital plane does not coincide with that of the space elevator, which must be the Earth's equatorial plane. The Moon will only pass through the space elevator's plane twice per month, which then forms the limit of how often the space elevator can send ships to the Moon.

The solution is again to use a transfer station equipt with a rotating tether and momentum wheel for inclination change capability.

In this case, as before, the transfer station orbits in the plane with the Moon, and will have an orbit period of 24 hours to match the space elevator (and therefore its altitude is 22,300 miles). The transfer station's position within the orbit is selected so that it passes through the transfer nodes (the two spots where the space elevator goes through the plane of the Moon) about 5 minutes behind the space elevator (this spacing will be selected to be large enough to prevent the two tethers from colliding, yet small enough for easy access).

An outbound ship would then be released from the space elevator at geosynchronous altitude, it would spend a few hours adjusting its position so that it lags the space elevator by the same five minutes, then when it reached the transfer node, it would be captured by the transfer station tether,which would pull it into the transfer station's orbital plane (and simultaneously release an inbound ship into the space elevator's plane).

It would arrive at the transfer station with a relative velocity of 2,700 +-550 m.p.h. (= circular orbital velocity * sine of the inclination change), and will be rotating around the station out-of-plane. The transfer station would correct the out-of-plane rotation and adjust the rotation speed so that the ship can be sent to the Moon. As before, a ship will generally arrive from the Moon about 5 hours before the Moon-bound departure window opens.

This system will have a departure window for the Moon about every 25 hours (it must complete one Earth orbit, then go 1/28th orbit more due to the Moon's orbital motion).

Lagrangia Access
Since this facility orbits nominally in the plane of the Moon, it can also provide frequent repeated access to all five Earth-Moon Lagrange points. The one way travel time is about 6 days, and the ship will arrive with a relative velocity of 1300 mph (rough approximations calculated using Hohmann transfer orbits ignoring the Moon).

Conclusion
Because of inclination differences, direct flights between low Earth Orbit and the Moon (for example) can only depart once or twice per month.  The tether described here acts as a transfer station, which can be reached on a daily basis from tethers in LEO or from a Lunavator at the Moon. When a ship arrives from the Moon, it latches onto one end of the tether, spinning about the ballast, to preserve its momentum, so it can later depart for the Moon again with no additional propulsion.

The large amount of ballast required to hold the station in a stable orbit during tether operations will drive up the cost of station keeping against orbital perturbations, but the amount of propellant required will be small compared to that saved by the use of the tether for propulsion.

Comments and/or questions about this document may be addressed to Nathan2go@aol.com.

References, Notes, & Websites:
1. To estimate travel time with solar electric ion engines, we can start with an estimate based solely on the solar panels:

For a specific impulse (Isp) of 3,000 seconds (10x better than LOx-Kerosene): the exhaust velocity is =Isp*g=3,000*9.81 =30,000 m/s. So each kg of exhaust has energy of 1/2*M*V^2 = 4.5*10^8 Watt*seconds. So at 1kg of exhaust/sec, the thrust is 3,000 kg (force), and the power is 4.5*10^8 Watts (1.5*10^5 W per kg thrust). The "Cambridge Encyclopedia of Space" (1990) says flexible solar panels mass about 1kg/50 Watts. So if the electrical efficiency is 50%, and the solar panels are 50% of the ship's mass:

the thrust/mass = (3000 kgf)/((2*1kg/(50 Watts)) * 2*4.5*10^8 Watts) = 8.3*10^-5 gees *(21.8 mph/s/gee)
= 1.82*10^-3 mph/s = 157 mph/day.

A fast rocket burn in LEO can send a ship to the Moon with a delta-V of about 7,000 mph, but spiraling out slowly is less efficient. It takes about 15,000 mph from LEO or 5,000 mph from geosynchronous orbit (I calculated these delta-Vs with a spiral consisting of a large number of Hohmann transfers). So it takes about 32 days =5,000 mph/(157 mph/day), one-way, assuming the ship never comes within 22,300 miles of Earth.
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2. "Starbound", from Time-Life's "Voyage Through the Universe" series, 1991; seems to attribute the idea for cycling Earth-Mars ships to American engineer John Niehoff and astronaut Buzz Aldrin; see "Starbound" p. 40. Also, in "The Case for Mars" (1996), Robert Zubrin describes several "Free return trajectories between Earth and Mars", the preferred one takes 180 days to reach Mars, and 2 years to return to Earth.
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3. Cycling ("consecutive close approach") Earth-Moon orbits, as discussed in "Theory of Orbits, the restricted problem of three bodies", by Szebehely, Victor, 1967 (which references R.F.Arenstorf, "Existence of Periodic Solutions Passing Near Both Masses of the Restricted Three Body Problem", AIAA J. 1, 238, 1963).
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4. Main tether mass (est. 2.1* tip-load mass) is calculated assuming: the material is a carbon fiber call Thornel T-40, which has a tensile strength of 5650 n/mm^2 (5.65 GPa) and a density of g/cc (similar results would be obtained with Spectra-2000 (3250 n/mm^2 and 0.97 g/cc). We derate the strength by 2.88 (safety factor) to allow for a design with Hoytether type redundancy, protective coating, margin for aging, etc.

Hans Moravec derived a closed form equation, but simpler numeric integration was used here.
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5. Orbital transfers are calculated using the Velocity Equation, which gives the velocity (v) of an orbiting object, given its distance from the center of the Earth (r), and the orbit's semimajor axis (a = half the distance from the highest to the lowest point in the orbit):

v^2 = GM*(2/r - 1/a);

where GM is G= gravitational constant= 6.67*10^-11 N*(m/kg)^2, M= mass of the Earth = 5.98*10^24 kg.
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6. Orbital precession is calculated with equations from "Lunar-Solar Perturbations of an Earth Satellite", by Leon Blitzer, 1959, American Journal of Physics (1959 p. 634-645), and reprinted in "Kinematics and Dynamics of Satellite Orbits", American Institute of Physics.

The precession rate (in Rads/sec) causes by the Earth's oblateness is (for circular orbit) =
We = -0.00164*((R/a)^2)*((GM/(a^3))^0.5)*cos(i),

where R= radius of the Earth, a = R+(orbital alitude), G= gravitational constant= 6.67*10^-11 N*(m/kg)^2, M= mass of the Earth = 5.98*10^24 kg, and i is the orbital inclination (I used the average value of 23.5 deg here, the actual value could be higher or lower by 5 deg due to the Moon's orbital inclination).

The required course correction per year is = We*60*60*24*365*(orbit_velocity)*sin(i).

Following the Moon's precession, the required course correction per year is = 2*Pi/(18.6_years)*(orbit_velocity)*sin(5.1 deg). (based on the observed Lunar precession rate of one revolution per 18.6 years also report by Blitzer.)
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7. Trajectory simulations by the author.
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8. A good estimate of the precession rate for the Momentum wheel is (for small H/L):
(0.4 gees*21.8 m.p.h./sec/gee)*(H/2/L)*(20%)*(1/2600 m.p.h.*180 degrees/Pi *3600 sec/hour)
= H/L*69 degrees/hour, assuming tip acceleration is 0.4 gees, and the cyclic tether tensioning occurs for 20% (effectively) of the revolution period, H= height of the axel, L= length of main tethers.

So with L=200 miles, and a target of 30 degress in 3 hours, we need H= 29 miles. If we decrease L to 100 miles (which increases the tip accel. to 0.8 gees), then H= 7.2 miles.

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9. NASA Geosynchronous Elevator <add link>.
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