Differences between revisions 1 and 10 (spanning 9 versions)
 ⇤ ← Revision 1 as of 2011-06-24 22:25:26 → Size: 3434 Editor: KeithLofstrom Comment: ← Revision 10 as of 2011-06-26 18:24:26 → ⇥ Size: 11279 Editor: KeithLofstrom Comment: Deletions are marked like this. Additions are marked like this. Line 1: Line 1: #format jsmath Line 2: Line 3: == "Landing" Launch Loop payloads without rockets == == "Landing" Launch Loop payloads in a destination without rockets == Line 4: Line 5: Imagine a long and relatively heavy orbiting tether, running vertically from 29900 km radius, through GEO (42164 km), to a counterweight above, perhaps at (50000 km). The tether has a thin, passive conductive rail on it, such that a magnet rail will be held off it by eddy current repulsion while travelling next to it, "held down" by coriolis acceleration. Imagine a long and heavy orbiting tether, running vertically from 29900 km radius, through GEO (42164 km), to a counterweight above. The tether has a thin, passive conductive rail around it. It will be used to magnetically capture vehicles and move them to GEO, using the sliding rail to push payloads transversely while they magnetically slide upwards. Line 6: Line 7: The tether is made with not-excessively-tapered Kevlar. We will need many tons of it. More mass is better for this system. THIS is where we put the hotels, the radiationshielding, and the heavy buffet tables for obese tourists. This is not the normal apogee capture system, where we launch into an orbit whose perigee transverse velocity matches a hanging tether at the right altitude. Instead, we launch the vehicle a little faster and a little later. That puts the vehicle in an orbit with an apogee trailing the tether. The vehicle will never get to that apogee, because we will slide in front of the tether below apogee, where we still have a high (1800 m/s) radial velocity. We will still match the tether's transverse velocity and altitude, though. Line 8: Line 9: We launch a vehicle off the launch loop at 10148.7 m/s, and with very good radar and some trim thrusters, we "land" on the bottom of the rail at 29979.7 km altitude. At that point, we have the same angular frequency and circular orbital velocity, and a large vertical velocity component, 1746.8 m/s . As we slide upwards, we magnetically levitate/push against the east side of the rail with Coriolis acceleration (0.25 m/s2 at 1746.8 m/s, 0.03 m/s2 at 200 m/s, nicely proportional to eddy currents!) getting some angular acceleration from the rail while we slow down vertically from the small (0.28 m/s^2) and rapidly decreasing centripedal acceleration (gravity minus rotational centrifugal acceleration). We slide upwards, we slow down vertically, we speed up horizontally. In the rotating tether frame of reference, the vehicle will approach from below at high speed, from the rear, and decelerate in the traverse ("horizontal") direction as it approaches the tether. The vehicle is rising radially, conserving angular momentum, and losing angular velocity as the radius increases. In the tether's rotating frame, this looks like Coriolis acceleration, equal to twice the velocity times the tether's angular velocity. The angular orbital velocity for a GEO tether is 7.2921E-5 radians per second, and the radial ("vertical") velocity will be around 1800 meters per second, so the Coriolis acceleration is 0.26 m/s^2^. If the vehicle misses the tether rendezvous, it will continue to accelerate retrograde in the rotating frame, with the backwards velocity turning into downwards Coriolis acceleration in the rotating frame, reaching apogee above the intended attach point but well below GEO. Line 10: Line 11: All those numbers result in us passing through GEO at about 200 m/s vertically, about 5.8 hours later. Hit a strip of passive braking magnets and stop, 2 kilometers at 1 gee. The tether is made with not-excessively-tapered Kevlar. We will need many tons of it, especially at GEO, as an angular momentum bank. More mass is better for this system. THIS is where we put the hotels, the radiation shielding, and the heavy buffet tables for obese tourists. Line 12: Line 13: The above assumes zero drag - given that there will be some eddy current drag, we launch from the loop and land on the tether rail a little faster. That speeds up transit time! We launch a vehicle off the launch loop at 10148.7 m/s, and with very good radar and some trim thrusters, we "land" on the bottom of the rail at 29979.7 km altitude. At that point, we have the same angular frequency and circular orbital velocity, and a large vertical velocity component, 1746.8 m/s . Line 14: Line 15: We can also launch a little faster (10154.1 m/s), hit the rail at 29988 km radius and 1804.8 m/s vertical velocity, and travel up the rail faster, cutting the vertical transit time significantly. 500 m/s gets us up the tether in 3.9 hours - we will need 13 kilometers to stop, though. As the vehicle slides up the tether, the weak gravity slows it down radially, as it accelerates forwards in orbit to remain on the "faster at higher altitude" tether. Line 16: Line 17: We can reverse the process, using a magnetic accelerator to launch a vehicle down a rail on the west side of the tether, falling off the end in a transfer orbit back to the upper atmosphere, restoring most of the momentum. We will probably need to make up some energy and momentum, certainly if we accumulate more upward vehicles than downward ones. But overall, the energy will be tiny compared to the the launch loop energies, and we can float a lot of solar cell around GEO. As the vehicle slides upwards, it magnetically levitates/pushes against the east side of the rail with Coriolis acceleration. That is 26 centimeters per second squared at 1800 m/s vertically, and 7 centimeters per second squared at 400 m/s, our speed as we approach GEO altitude. The Coriolis acceleration and the eddy currents both reduce with relative velocity - most convenient! The vertical deceleration (gravity minus rotational centrifugal acceleration) decreases to zero as we approach GEO, still with significant velocity, though with less than the velocity of a frictionless ballistic because of the eddy currents. Line 18: Line 19: If we deorbit our trash much faster than we receive payloads (aiming for the ocean, or the occasional email spammer) we can add more momentum by exploiting Coriolis force acceleration. We can also launch mass from the moon, or launch mass from the loop in slingshot orbits around it. However, the moon will not always be conveniently located for this, and vehicles may make many 20 day orbits before arriving at a suitable mean anomaly. As the vehicle approaches GEO, it is still moving fast, which is good, because it has 12000 kilometers to travel. The trip will take about 4 hours. About 10000 meters below GEO station, the rail surface changes to increase eddy current drag, slowing down the vehicle. The vehicle reaches the drag section at 400 meters per second, and deceleration increases from a few millimeters per second to 10 m/s^2^, a bit more than 1 gee (with passengers facing backwards, backs into chairs). The vehicle slows to 25 meters per second about 40 meters below the station, and with the aid of some linear motors, reduces deceleration and velocity to zero over the next 5 seconds. The payload or passenger compartment is plucked off the magnet-rail and wing, and pushed into the station through an airlock. Line 20: Line 21: The rocket thrust needed will be pure velocity correction, centimeters per second if we've done our radar and orbital mechanics calculations correctly. If we miss, we just reenter normally - a delay, but not a disaster. Besides a little correction exhaust, and whatever we need to add to for momentum restoration, the system is mass conservative and mostly energy-recycling. Since the tether rail is not magnetic or active (that would be far too heavy for a 12000 kilometer gossamer structure) the vehicle magnet-rail will need to wrap partly around it in some way. 26 cm/s^2^ Coriolis acceleration is far too weak to hold the vehicle against a rippling tether at high speed. For example, a ripple with a 20 kilometer wavelength and a 20 meter peak-to-peak amplitude will shake the vehicle back and forth at 0.3 gees ( 12x Coriolis ) and a period of 11 seconds - a mild roller coaster. Such a transverse ripple will not be a standing wave, but traveling up or down the tether at around 1 km per second. It will be a challenge to remove it, perhaps by periodic transverse shaking at GEO by the large momentum mass. It would be more difficult to remove by shaking the bottom counterweight vertically, inducing Coriolis accelerations, because a lot of power is required.We can reverse the process. Using a magnetic accelerator to launch a vehicle down a rail on the west side of the tether, we falling off the end in a transfer orbit back to the upper atmosphere. This restores most of the GEO rail momentum. The rocket thrust needed by GEO rail vehicles will be pure velocity correction, centimeters per second if we've done our radar and orbital mechanics calculations correctly. If we miss, we just reenter normally - a delay, but not a disaster. Besides a little correction exhaust, and whatever we need to add to for momentum restoration, the system is mass conservative and mostly energy-recycling. Line 23: Line 28: == Restoring GEO Rail orbital momentum ==If we de-orbit GEO station trash much faster than we receive payloads (aiming for empty and lifeless portions of the ocean) we can add more momentum by exploiting Coriolis force acceleration. We can also receive mass (and momentum) from the moon, or launch mass from the loop in slingshot orbits around the moon. However, the moon will not always be conveniently located for this, and vehicles may make many 20 day orbits before arriving with a suitable orbital position.A slightly stronger tether can swing like a pendulum. That allows vehicles to arrive from the launch loop with higher angular velocity, and leave with lower angular velocity. This variation needs further study.We will probably need to make up some energy and momentum losses with plasma rocket engines at GEO station, especially if we accumulate more upward vehicles than downward ones. But overall, the energy will be tiny compared to the the launch loop energies, and we can float a lot of solar cell around GEO as part of our "momentum anchor". As part of an overall "space anti-litter" ethic, we should make sure the plasma rocket engines are offset so they are firing exhaust away from the rest of the geosynchronous ring, and that the rocket exhaust is traveling fast enough to escape the solar system ( >16 km/sec at noon when the rockets are pushing retrograde to the Earth's orbit around the sun, >76 km/sec (!!) at midnight, when the rockets are pushing retrograde ). A [[ http://en.wikipedia.org/wiki/VASIMR | Variable Specific Impulse Magnetoplasma Rocket ]] (VASIMR) engine operates optimally at 50 km/sec, so operation at 80 km/sec is close to optimum. The reaction mass is argon, which can be frozen at -200C with a density of 1600 kg/m3 and a vapor pressure of 0.1 atmosphere for transit from earth. We will need about 1 kg of argon reaction mass shipped up for every 50 kg of uncompensated mass shipped up from Earth. Argon is 0.93% of the atmosphere, and costs about $1/kg (liquid) in high quantities. It is a byproduct of the production of liquid oxygen.We will expend about 40 kilowatt hours per "uncompensated vehicle kilogram" operating the plasma rockets, including coolers and radiator operation. Typical solar cell weights for existing satellites are 65 kg/kilowatt, so 1 kilogram of solar cells provides enough energy in a year to bring aboard 3.5 kilograms of mass - a power doubling time of 5 months. However, if we can use "server sky" style ultra-thin InP cells, closer to 1000 kg/kW, we can approach 50 kilograms of mass per kilogram of solar cell. In the very long term, solar cells made from lunar materials will bring both momentum and cheaper launch to the GEO rail, but solar cells are a high-tech undertaking, and it will be a long time before that level of manufacturing technology can operate practically in space.In the shorter term, we may be forced to get by with lower ISP engines than 80 km/sec VASIMR, perhaps operating only around noon with an exhaust velocity of 20km/sec. That increases our argon needs 4X, but reduces our power needs 16X. We will only be operating our engines with about a 30 percent duty cycle, so our solar panel needs only drop 5.3X. Still, our power doubling time drops to a month or so. Probably worth it during the building phase. We may even be rascals, and run the VASIMR engines 24 hours for a while, dumping a few tons of midnight argon into orbits in the inner solar system. == Operation of an M288 rail base ==A similar rail system can supply the M288 [[ http://server-sky.com | server sky orbits ]] MORE LATER== M288 rail bases for orbital debris capture ==Server sky thinsats are cheaper to launch if they are lighter, but below a minimum mass-to-area ratio, the orbits are unstable under light pressure. However, we can add mass ballast to a very light satellite, using gram-weight chunks of space debris. The processing is simple, just let the debris shatter as it is collision-captured inside a cornucopia-shaped funnel. Larger chunks can be cut apart at a simple processing plant at the M288 rail base, smaller chunks can be welded together.Capture vehicles can drop below the tether, in intercept orbits where the debris crosses the equatorial plane. Depending on their orbit, they can return to the same rail base, but more likely a different rail base will be more convenient.MORE LATER== The Math ==We will neglect oblateness, light pressure, and the gravity of other bodies for now. Simple Kepler two body orbits around the earth.The vehicle leaves the launch loop at 80km altitude, the apogee of an elliptical orbit with a perigee of$ r_p $= 6458km and velocity$ v_a $. Given those two values, we define the orbit:$ h ~ = ~ r_p v_p $  angular momentum$ a ~ = ~ { \Large 1 \over \LARGE { { {v_p}^2 } \over { \huge \mu } } - { { \huge 2 } \over { r_p } } } $  semi-major axis$ e ~ = ~ 1 - { \Large { { r_p } \over a } } ~ = ~ { \Large { { r_p {v_p}^2 } \over \mu } } - 1 $  eccentricity$ v_0 ~ = ~ { \Large { { v_p } \over { 1 + e } } } ~ = ~ { \Large { { v_p } \over { 2 ~ - ~ { \LARGE { r_p {v_p}^2 } \over { \huge \mu } } } } } $  nominal velocity$ r_a ~ = ~ ( 1 + e ) a ~ = ~ { \Large 1 \over { \LARGE { { \huge 2 \mu } \over { r_p {v_p}^2 } } ~ - ~1 } } $  apogeeThe angular velocity of the tether in geostationary orbit is$ 2 \pi / 86164.0905 $= 7.2921159E-5 radians per secondMORE LATER # GEO Rail ## "Landing" Launch Loop payloads in a destination without rockets Imagine a long and heavy orbiting tether, running vertically from 29900 km radius, through GEO (42164 km), to a counterweight above. The tether has a thin, passive conductive rail around it. It will be used to magnetically capture vehicles and move them to GEO, using the sliding rail to push payloads transversely while they magnetically slide upwards. This is not the normal apogee capture system, where we launch into an orbit whose perigee transverse velocity matches a hanging tether at the right altitude. Instead, we launch the vehicle a little faster and a little later. That puts the vehicle in an orbit with an apogee trailing the tether. The vehicle will never get to that apogee, because we will slide in front of the tether below apogee, where we still have a high (1800 m/s) radial velocity. We will still match the tether's transverse velocity and altitude, though. In the rotating tether frame of reference, the vehicle will approach from below at high speed, from the rear, and decelerate in the traverse ("horizontal") direction as it approaches the tether. The vehicle is rising radially, conserving angular momentum, and losing angular velocity as the radius increases. In the tether's rotating frame, this looks like Coriolis acceleration, equal to twice the velocity times the tether's angular velocity. The angular orbital velocity for a GEO tether is 7.2921E-5 radians per second, and the radial ("vertical") velocity will be around 1800 meters per second, so the Coriolis acceleration is 0.26 m/s2. If the vehicle misses the tether rendezvous, it will continue to accelerate retrograde in the rotating frame, with the backwards velocity turning into downwards Coriolis acceleration in the rotating frame, reaching apogee above the intended attach point but well below GEO. The tether is made with not-excessively-tapered Kevlar. We will need many tons of it, especially at GEO, as an angular momentum bank. More mass is better for this system. THIS is where we put the hotels, the radiation shielding, and the heavy buffet tables for obese tourists. We launch a vehicle off the launch loop at 10148.7 m/s, and with very good radar and some trim thrusters, we "land" on the bottom of the rail at 29979.7 km altitude. At that point, we have the same angular frequency and circular orbital velocity, and a large vertical velocity component, 1746.8 m/s . As the vehicle slides up the tether, the weak gravity slows it down radially, as it accelerates forwards in orbit to remain on the "faster at higher altitude" tether. As the vehicle slides upwards, it magnetically levitates/pushes against the east side of the rail with Coriolis acceleration. That is 26 centimeters per second squared at 1800 m/s vertically, and 7 centimeters per second squared at 400 m/s, our speed as we approach GEO altitude. The Coriolis acceleration and the eddy currents both reduce with relative velocity - most convenient! The vertical deceleration (gravity minus rotational centrifugal acceleration) decreases to zero as we approach GEO, still with significant velocity, though with less than the velocity of a frictionless ballistic because of the eddy currents. As the vehicle approaches GEO, it is still moving fast, which is good, because it has 12000 kilometers to travel. The trip will take about 4 hours. About 10000 meters below GEO station, the rail surface changes to increase eddy current drag, slowing down the vehicle. The vehicle reaches the drag section at 400 meters per second, and deceleration increases from a few millimeters per second to 10 m/s2, a bit more than 1 gee (with passengers facing backwards, backs into chairs). The vehicle slows to 25 meters per second about 40 meters below the station, and with the aid of some linear motors, reduces deceleration and velocity to zero over the next 5 seconds. The payload or passenger compartment is plucked off the magnet-rail and wing, and pushed into the station through an airlock. Since the tether rail is not magnetic or active (that would be far too heavy for a 12000 kilometer gossamer structure) the vehicle magnet-rail will need to wrap partly around it in some way. 26 cm/s2 Coriolis acceleration is far too weak to hold the vehicle against a rippling tether at high speed. For example, a ripple with a 20 kilometer wavelength and a 20 meter peak-to-peak amplitude will shake the vehicle back and forth at 0.3 gees ( 12x Coriolis ) and a period of 11 seconds - a mild roller coaster. Such a transverse ripple will not be a standing wave, but traveling up or down the tether at around 1 km per second. It will be a challenge to remove it, perhaps by periodic transverse shaking at GEO by the large momentum mass. It would be more difficult to remove by shaking the bottom counterweight vertically, inducing Coriolis accelerations, because a lot of power is required. We can reverse the process. Using a magnetic accelerator to launch a vehicle down a rail on the west side of the tether, we falling off the end in a transfer orbit back to the upper atmosphere. This restores most of the GEO rail momentum. The rocket thrust needed by GEO rail vehicles will be pure velocity correction, centimeters per second if we've done our radar and orbital mechanics calculations correctly. If we miss, we just reenter normally - a delay, but not a disaster. Besides a little correction exhaust, and whatever we need to add to for momentum restoration, the system is mass conservative and mostly energy-recycling. More rockets bite the dust ... ## Restoring GEO Rail orbital momentum If we de-orbit GEO station trash much faster than we receive payloads (aiming for empty and lifeless portions of the ocean) we can add more momentum by exploiting Coriolis force acceleration. We can also receive mass (and momentum) from the moon, or launch mass from the loop in slingshot orbits around the moon. However, the moon will not always be conveniently located for this, and vehicles may make many 20 day orbits before arriving with a suitable orbital position. A slightly stronger tether can swing like a pendulum. That allows vehicles to arrive from the launch loop with higher angular velocity, and leave with lower angular velocity. This variation needs further study. We will probably need to make up some energy and momentum losses with plasma rocket engines at GEO station, especially if we accumulate more upward vehicles than downward ones. But overall, the energy will be tiny compared to the the launch loop energies, and we can float a lot of solar cell around GEO as part of our "momentum anchor". As part of an overall "space anti-litter" ethic, we should make sure the plasma rocket engines are offset so they are firing exhaust away from the rest of the geosynchronous ring, and that the rocket exhaust is traveling fast enough to escape the solar system ( >16 km/sec at noon when the rockets are pushing retrograde to the Earth's orbit around the sun, >76 km/sec (!!) at midnight, when the rockets are pushing retrograde ). A Variable Specific Impulse Magnetoplasma Rocket (VASIMR) engine operates optimally at 50 km/sec, so operation at 80 km/sec is close to optimum. The reaction mass is argon, which can be frozen at -200C with a density of 1600 kg/m3 and a vapor pressure of 0.1 atmosphere for transit from earth. We will need about 1 kg of argon reaction mass shipped up for every 50 kg of uncompensated mass shipped up from Earth. Argon is 0.93% of the atmosphere, and costs about$1/kg (liquid) in high quantities. It is a byproduct of the production of liquid oxygen.

We will expend about 40 kilowatt hours per "uncompensated vehicle kilogram" operating the plasma rockets, including coolers and radiator operation. Typical solar cell weights for existing satellites are 65 kg/kilowatt, so 1 kilogram of solar cells provides enough energy in a year to bring aboard 3.5 kilograms of mass - a power doubling time of 5 months. However, if we can use "server sky" style ultra-thin InP cells, closer to 1000 kg/kW, we can approach 50 kilograms of mass per kilogram of solar cell. In the very long term, solar cells made from lunar materials will bring both momentum and cheaper launch to the GEO rail, but solar cells are a high-tech undertaking, and it will be a long time before that level of manufacturing technology can operate practically in space.

In the shorter term, we may be forced to get by with lower ISP engines than 80 km/sec VASIMR, perhaps operating only around noon with an exhaust velocity of 20km/sec. That increases our argon needs 4X, but reduces our power needs 16X. We will only be operating our engines with about a 30 percent duty cycle, so our solar panel needs only drop 5.3X. Still, our power doubling time drops to a month or so. Probably worth it during the building phase. We may even be rascals, and run the VASIMR engines 24 hours for a while, dumping a few tons of midnight argon into orbits in the inner solar system.

## Operation of an M288 rail base

A similar rail system can supply the M288 server sky orbits

MORE LATER

## M288 rail bases for orbital debris capture

Server sky thinsats are cheaper to launch if they are lighter, but below a minimum mass-to-area ratio, the orbits are unstable under light pressure. However, we can add mass ballast to a very light satellite, using gram-weight chunks of space debris. The processing is simple, just let the debris shatter as it is collision-captured inside a cornucopia-shaped funnel. Larger chunks can be cut apart at a simple processing plant at the M288 rail base, smaller chunks can be welded together.

Capture vehicles can drop below the tether, in intercept orbits where the debris crosses the equatorial plane. Depending on their orbit, they can return to the same rail base, but more likely a different rail base will be more convenient.

MORE LATER

## The Math

We will neglect oblateness, light pressure, and the gravity of other bodies for now. Simple Kepler two body orbits around the earth.

The vehicle leaves the launch loop at 80km altitude, the apogee of an elliptical orbit with a perigee of r_p = 6458km and velocity v_a . Given those two values, we define the orbit:

h ~ = ~ r_p v_p                      angular momentum

a ~ = ~ { \Large 1 \over \LARGE { { {v_p}^2 } \over { \huge \mu } } - { { \huge 2 } \over { r_p } } }                  semi-major axis

e ~ = ~ 1 - { \Large { { r_p } \over a } } ~ = ~ { \Large { { r_p {v_p}^2 } \over \mu } } - 1       eccentricity

v_0 ~ = ~ { \Large { { v_p } \over { 1 + e } } } ~ = ~ { \Large { { v_p } \over { 2 ~ - ~ { \LARGE { r_p {v_p}^2 } \over { \huge \mu } } } } }       nominal velocity

r_a ~ = ~ ( 1 + e ) a ~ = ~ { \Large 1 \over { \LARGE { { \huge 2 \mu } \over { r_p {v_p}^2 } } ~ - ~1 } }   apogee

The angular velocity of the tether in geostationary orbit is 2 \pi / 86164.0905 = 7.2921159E-5 radians per second

MORE LATER

CaptureRail (last edited 2021-06-20 03:27:51 by KeithLofstrom)